isomorphism conjectures in k and l-theory · 2019-05-03 · and ii are in reasonable but de nitely...

Wolfgang Luck

Isomorphism Conjectures in K-and L-TheorySPIN Springer’s internal project number, if known

Mathematics – Monograph (English)

May 3, 2019

Springer

Preface

This manuscript is not at all finished. Part Iand II are in reasonable but definitely not finalform, whereas part III is completely missing. Atthe moment the material is to be used at ownrisk. Comments are very welcome.

The Isomorphism Conjectures due to Farrell-Jones and Baum-Connes pre-dict the algebraic K-and L-theory of group rings and the topological K-theory of reduced group C∗-algebras. These theories are of major interest formany reasons. For instance, the algebraic L-groups are the recipients for var-ious surgery obstructions and hence highly relevant for the classification ofmanifolds. Other important obstructions such as Wall’s finiteness obstructionand Whitehead torsion take values in algebraic K-groups. The topological K-groups of C∗-algebras play a central role in index theory and the classificationof C∗-algebras.

In general these K- and L-groups are very hard to analyze for group ringsor group C∗-algebras. The Isomorphism Conjectures identify them with equi-variant homology groups of classifying spaces of families of subgroups. As anillustration, let us consider the special case that G is a torsionfree groupand R is a regular ring (with involution). Then the Isomorphism Conjecturespredict that the so called assembly maps

Hn(BG; K(R))∼=−→ Kn(RG);

Hn(BG; L〈−∞〉(R))∼=−→ L〈−∞〉n (RG);

Kn(BG)∼=−→ Kn(C∗r (G)),

are isomorphisms for all n ∈ Z. The target is the algebraic K-theory ofthe group ring RG, the algebraic L-theory of RG with decoration 〈−∞〉,or the topological K-theory of the reduced group C∗-algebra C∗r (G). Thesource is the evaluation of a specific homology theory on the classifying space

BG, where Hn(•; K(R)) ∼= Kn(R), Hn(•; L〈−∞〉(R)) ∼= L〈−∞〉n (R) and

Kn(•) ∼= Kn(C) for all n ∈ Z.Since the sources of these assembly maps are much more accessible than the

targets, the Isomorphism Conjectures are key ingredients in explicit compu-tations of the K-and L-groups of group rings and reduced group C∗-algebras.These often are motivated by and have applications to concrete problems thatarise, for instance, in the classification of manifolds or C∗-algebras.

The Farrell-Jones Conjecture and the Baum-Conjecture imply many otherprominent conjectures. In a lot of cases these conjectures were not known tobe true for certain groups until the Farrell-Jones or the Baum-Connes Conjec-ture were proved for them. Examples for such prominent conjectures are theBorel Conjecture about the topological rigidity of aspherical closed manifolds,

V

VI Preface

the (stable) Gromov-Lawson-Rosenberg Conjecture about the existence ofRiemannian metrics with positive scalar curvature on closed Spin-manifolds,the Kaplansky Conjecture and the Kadison Conjecture on the non-existenceof non-trivial idempotents in the group ring or the reduced group C∗-algebraof torsionfree groups, the Novikov Conjecture about the homotopy invarianceof higher signatures, and the conjectures about the vanishing of the reducedprojective class group of ZG and the Whitehead group of G for a torsionfreegroup G.

The Farrell-Jones Conjecture and the Baum-Connes Conjecture are stillopen (at the time of writing). However, tremendous progress has been madeon the class of groups for which they are known. The techniques of the sophis-ticated proofs range from functional analysis, algebra, geometry, topology todynamical systems.

The Farrell-Jones Conjecture and the Baum-Connes Conjecture seem tobe rather focused on the first glance. It is surprising and intriguing that theyhave strong impact on problems in and involve advanced techniques frommany different areas in mathematics. For instance, one would not expect thata solution to the purely algebraic question whether the group ring of a tor-sionfree group contains no non-trivial idempotents, requires input from spec-tral analysis or flow spaces. This extreme broad scope of the Baum-ConnesConjecture and the Farrell-Jones Conjecture is both the main challenge andmain motivation for writing this book. We hope that after reading parts ofthis monograph the reader will share the enthusiasm of the author for theIsomorphism Conjectures.

Organization

The monograph consists of three parts.In the first part “Introduction to K- and L-Theory”, which encompasses

Chapters 1 to 8, we introduce and motivate the relevant theories, namely,algebraic K-theory, algebraic L-theory and topological K-theory. In thesechapters we present some applications and special more accessible cases ofthe Farrell-Jones and the Baum-Connes Conjecture.

In the second part “The Isomorphism Conjectures”, which consists ofChapters 9 to Chapter 16, we introduce the Farrell-Jones Conjecture andthe Baum-Connes Conjecture in its most general form, namely, for arbitrarygroups and arbitrary twisted coefficients. We discuss further applications andin particular how they can be used for computations. We give a report aboutthe status of these conjectures and discuss open problems.

The third part “Methods of Proofs”, which ranges from Chapter 17 toChapter 21, we discuss the strategies and ingredients of the proofs.

We have inserted exercises which contain useful additional information andcan be though as additional lemmas, examples or remarks. Their solutionsare presented in Chapter 22 and hence one can refer to them in articles.

Preface VII

A User’s Guide

The monograph is a guide for and gives a panorama of Isomorphism Con-jectures and related topics. It presents or at least indicates the (at the timeof writing) most advanced results and developments. References for furtherreading and information have been inserted.

A reader who wants to get specific information or focus on a certain topicshould consult the detailed table of contents and the index in order to findthe right place in the monograph. We have written the text in a way such thatone can read small units independently from the rest, concentrate on certainaspects and extract specific information easily and quickly and can startreading in any of the chapters. We hopefully have found the right mixturebetween definitions, theorems, examples and remarks so that reading thebook is entertaining and illuminating. One can use selected chapters to runseminars on specific topics.

We recommend the reader to try to solve the exercises. This is gives theopportunity to recover from reading and to work on its own, thus improv-ing intuition and familiarity with the definitions, concepts, results and theirproofs.

We require that the reader is familiar with basic notions in topology(CW -complexes, chain complexes, homology, homotopy groups, manifolds,coverings, cofibrations, fibrations . . . ), functional analysis (Hilbert spaces,bounded operators, differential operators, . . . ), algebra (groups, modules,group rings, elementary homological algebra, . . . ), group theory (presenta-tions, Cayley graphs, hyperbolic groups, . . . ) and category theory (functors,transformations, additive categories, . . . ).

Other survey articles on the Farrell-Jones Conjecture and the Baum-Connes Conjecture are [61, 74, 91, 295, 385, 536, 546, 606, 692, 797].

Acknowledgments

We want to thank the following present and former members of the topol-ogy groups in Bonn and Munster who read through the monograph and madea lot of useful comments, corrections and suggestions.

We want to thank the Deutsche Forschungsgemeinschaft and the Euro-pean Research Council which have been and are financing the collaborativeresearch centers “Geometrische Strukturen in der Mathematik” and “Groups,Geometry and Actions”, the research training groups “Analytische Topolo-gie und Metageometrie” and “Homotopy and Kohomologie”, the cluster ofexcellence “Hausdorff center for Mathematics” and the Leibniz award andAdvanced ERC-Grant of the author. These made it possible to invite guestsand run workshops on the topics of the monograph.

The author thanks Paul Baum, Alain Connes, Tom Farrell and Low-ell Jones for their beautiful Isomorphism Conjectures. They were originallystated in [91, Conjecture 3.15 on page 254] and [295, 1.6 on page 257].

VIII Preface

Finally, the author wants to express his deep gratitude to his wife Sibylleand our children and grandchildren Christian, Isabel, Julian, Nora, Tobiasand Severina. Comment 1: Add names.

Bonn, November 2018, Wolfgang Luck

last edited on 19.04.2019last compiled on May 3, 2019

name of texfile: ic

Contents

0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Motivation for the Baum-Connes Conjecture . . . . . . . . . . . . . . . 1

0.1.1 Topological K-Theory of Reduced Group C∗-Algebras 10.1.2 Homological Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1.3 The Baum-Connes Conjecture for Torsionfree Groups . 40.1.4 The Baum-Connes Conjecture . . . . . . . . . . . . . . . . . . . . . 40.1.5 Reduced versus Maximal Group C∗-Algebras . . . . . . . . 70.1.6 Applications of the Baum-Connes Conjecture . . . . . . . . 7

0.2 Motivation for the Farrell-Jones Conjecture for K-Theory . . . 70.2.1 Algebraic K-Theory of Group Rings . . . . . . . . . . . . . . . . 70.2.2 Appearance of Nil-Terms . . . . . . . . . . . . . . . . . . . . . . . . . . 80.2.3 The Farrell-Jones Conjecture for K∗(RG) for Regular

Rings and Torsionfree Groups . . . . . . . . . . . . . . . . . . . . . . 90.2.4 The Farrell-Jones Conjecture for K∗(RG) for Regular

Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90.2.5 The Farrell-Jones Conjecture for K∗(RG) . . . . . . . . . . . 100.2.6 Applications of the Farrell-Jones Conjecture for

K∗(RG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110.3 Motivation for the Farrell-Jones Conjecture for L-Theory . . . . 11

0.3.1 Algebraic L-Theory of Group Rings . . . . . . . . . . . . . . . . . 110.3.2 The Farrell-Jones Conjecture for L∗(RG)[1/2] . . . . . . . . 120.3.3 The Farrell-Jones Conjecture for L∗(RG) . . . . . . . . . . . . 130.3.4 Applications of the Farrell-Jones Conjecture for L∗(RG) 14

0.4 Status of the Baum-Connes and the Farrell-Jones Conjecture 150.5 Structural Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

0.5.1 The Meta-Isomorphism Conjecture . . . . . . . . . . . . . . . . . 160.5.2 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

0.6 Computational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170.7 Methods of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

0.7.1 Controlled Topology Methods . . . . . . . . . . . . . . . . . . . . . . 180.7.2 Coverings, Flow Spaces and Transfer . . . . . . . . . . . . . . . . 19

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0.7.3 Analytic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190.7.4 Cyclic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

0.8 Are the Baum-Connes Conjecture and the Farrell-JonesConjecture True in General? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

0.9 Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Part I: Introduction to K- and L-theory

1 The Projective Class Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.2 Definition and Basic Properties of the Projective Class Group 241.3 The Projective Class Group of a Dedekind domain . . . . . . . . . . 281.4 Swan’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.5 Wall’s Finiteness Obstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.6 Geometric Interpretation of Projective Class Group and

Finiteness Obstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7 Universal Functorial Additive Invariants . . . . . . . . . . . . . . . . . . . 361.8 Variants of the Farrell-Jones Conjecture for K0(RG) . . . . . . . . 381.9 The Kaplansky Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.10 The Bass Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.10.1 The Bass Conjecture for fields of characteristic zeroas coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.10.2 The Bass Conjecture for integral domains as coefficients 471.11 Survey on Computations of K0(RG) for Finite Groups . . . . . . 481.12 Survey on Computations of K0(C∗r (G)) and K0(N (G)) . . . . . . 521.13 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2 The Whitehead Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.2 Definition and Basic Properties of K1(R) . . . . . . . . . . . . . . . . . . 572.3 Whitehead Group and Whitehead Torsion . . . . . . . . . . . . . . . . . 642.4 Geometric Interpretation of Whitehead Group and

Whitehead Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.5 The s-Cobordism Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.6 Reidemeister Torsion and Lens Spaces . . . . . . . . . . . . . . . . . . . . 762.7 The Bass-Heller-Swan Theorem for K1 . . . . . . . . . . . . . . . . . . . . 81

2.7.1 The Bass-Heller-Swan Decomposition for K1 . . . . . . . . . 812.7.2 The Grothendieck Decomposition for G0 and G1 . . . . . 852.7.3 Regular Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

2.8 The Mayer-Vietoris K-Theory Sequence of a Pullback of Rings 882.9 The K-Theory Sequence of a two-sided Ideal . . . . . . . . . . . . . . . 902.10 Swan Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

2.10.1 The Classical Swan Homomorphism . . . . . . . . . . . . . . . . 932.10.2 The Classical Swan Homomorphism and Free

Homotopy Representations . . . . . . . . . . . . . . . . . . . . . . . . 95

Contents XI

2.10.3 The Generalized Swan Homomorphism . . . . . . . . . . . . . . 972.10.4 The Generalized Swan Homomorphism and Free

Homotopy Representations . . . . . . . . . . . . . . . . . . . . . . . . 982.11 Variants of the Farrell-Jones Conjecture for K1(RG) . . . . . . . . 992.12 Survey on Computations of K1(ZG) for Finite Groups . . . . . . 1002.13 Survey on Computations of Algebraic K1(C∗r (G)) and

K1(N (G)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.14 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3 Negative Algebraic K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053.2 Definition and Basic Properties of Negative K-Groups . . . . . . 1053.3 Geometric Interpretation of Negative K-Groups . . . . . . . . . . . . 1123.4 Variants of the Farrell-Jones Conjecture for Negative

K-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.5 Survey on Computations of Negative K-Groups for Finite

Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1143.6 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4 The Second Algebraic K-Group . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.2 Definition and Basic Properties of K2(R) . . . . . . . . . . . . . . . . . . 1174.3 The Steinberg Group as Universal Extension . . . . . . . . . . . . . . . 1184.4 Extending Exact Sequences of Pullbacks and Ideals . . . . . . . . . 1194.5 Steinberg Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.6 The Second Whitehead Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.7 A Variant of the Farrell-Jones Conjecture for the Second

Whitehead group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.8 The Second Whitehead Group of Some Finite Groups . . . . . . . 1234.9 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5 Higher Algebraic K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.2 The Plus-Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.3 Survey on Main Properties of Algebraic K-Theory of Rings . . 128

5.3.1 Compatibility with Finite Products . . . . . . . . . . . . . . . . . 1285.3.2 Morita Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.3.3 Compatibility with Directed Colimits . . . . . . . . . . . . . . . 1295.3.4 The Bass-Heller-Swan Decomposition . . . . . . . . . . . . . . . 1295.3.5 Algebraic K-Theory of Finite Fields . . . . . . . . . . . . . . . . 1305.3.6 Algebraic K-Theory of the Ring of Integers in a

Number Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.4 Algebraic K-Theory with Coefficients . . . . . . . . . . . . . . . . . . . . . 1325.5 Other Constructions of Connective Algebraic K-Theory . . . . . 1335.6 Non-Connective Algebraic K-Theory of Additive Categories . 135

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5.7 Survey on Main Properties of Algebraic K-Theory ofCategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.7.1 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.7.2 Resolution Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.7.3 Devissage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.7.4 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.8 Torsionfree Groups and Regular Rings . . . . . . . . . . . . . . . . . . . . 1385.9 Mayer-Vietoris Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.10 Homotopy Algebraic K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.11 Algebraic K-Theory and Cyclic Homology . . . . . . . . . . . . . . . . . 1455.12 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6 Algebraic K-Theory of Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.2 Pseudo-isotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.3 Whitehead Spaces and A-Theory . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.3.1 Categories with Cofibrations and Weak Equivalences . . 1516.3.2 The wS•-Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.3.3 A-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546.3.4 Whitehead Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7 Algebraic L-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.2 Symmetric and Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.2.1 Symmetric Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607.2.2 The Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627.2.3 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.3 Even-Dimensional L-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1677.4 Intersection and Selfintersection Pairings . . . . . . . . . . . . . . . . . . 170

7.4.1 Intersections of Immersions . . . . . . . . . . . . . . . . . . . . . . . . 1717.4.2 Selfintersections of Immersions . . . . . . . . . . . . . . . . . . . . . 173

7.5 The Surgery Obstruction in Even Dimensions . . . . . . . . . . . . . . 1767.5.1 Poincare Duality Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1767.5.2 Normal Maps and the Surgery Step . . . . . . . . . . . . . . . . . 1787.5.3 The Surgery Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1797.5.4 The Even-Dimensional Surgery Obstruction . . . . . . . . . 181

7.6 Formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1897.7 Odd-Dimensional L-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1917.8 The Surgery Obstruction in Odd Dimensions . . . . . . . . . . . . . . . 1927.9 Surgery Obstructions for Manifolds with Boundary . . . . . . . . . 1957.10 Decorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

7.10.1 L-groups with K1-Decorations . . . . . . . . . . . . . . . . . . . . . 1977.10.2 The Simple Surgery Obstruction . . . . . . . . . . . . . . . . . . . 2007.10.3 Decorated L-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Contents XIII

7.10.4 The Rothenberg Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 2027.10.5 The Shaneson Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

7.11 The Farrell-Jones Conjecture for Algebraic L-Theory forTorsionfree Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

7.12 The Surgery Exact Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2047.12.1 The Structure Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2057.12.2 Realizability of Surgery Obstructions . . . . . . . . . . . . . . . 2067.12.3 The Surgery Exact Sequence . . . . . . . . . . . . . . . . . . . . . . . 208

7.13 Surgery Theory in the PL and in the Topological Category . . 2107.14 The Novikov Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

7.14.1 The Original Formulation of the Novikov Conjecture . . 2137.14.2 Invariance Properties of the L-Class . . . . . . . . . . . . . . . . 2157.14.3 The Novikov Conjecture and surgery theory . . . . . . . . . 216

7.15 Topologically Rigidity and the Borel Conjecture . . . . . . . . . . . . 2187.15.1 Aspherical Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2187.15.2 Formulation and Relevance of the Borel Conjecture . . . 2207.15.3 The Farrell-Jones and the Borel Conjecture . . . . . . . . . . 222

7.16 Homotopy Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2247.17 Poincare Duality Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2247.18 Boundaries of Hyperbolic Groups . . . . . . . . . . . . . . . . . . . . . . . . . 2277.19 Product Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2287.20 Automorphisms of Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2307.21 Survey on Computations of L-Theory of Group Rings of

Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2317.22 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

8 Topological K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2338.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2338.2 Topological K-Theory of Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 233

8.2.1 Complex Topological K-Theory of Spaces . . . . . . . . . . . 2338.2.2 Real Topological K-Theory of Spaces . . . . . . . . . . . . . . . 2368.2.3 Equivariant Topological K-Theory of Spaces . . . . . . . . . 237

8.3 Topological K-Theory of C∗-Algebras . . . . . . . . . . . . . . . . . . . . . 2428.3.1 Basics about C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 2428.3.2 Basic Properties of the Topological K-Theory of

C∗-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2468.4 The Baum-Connes Conjecture for Torsionfree Groups . . . . . . . 251

8.4.1 The Trace Conjecture in the Torsionfree Case . . . . . . . . 2538.4.2 The Kadison Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 2548.4.3 The Zero-in-the-Spectrum Conjecture . . . . . . . . . . . . . . . 255

8.5 Kasparov’s KK -Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2568.5.1 Basic Properties of KK -theory for C∗-Algebras . . . . . . 2568.5.2 The Kasparov’s Intersection Product . . . . . . . . . . . . . . . . 258

8.6 Equivariant Topological K-Theory and KK -Theory . . . . . . . . . 2598.7 Comparing Algebraic and Topological K-theory of C∗-Algebras263

XIV Contents

8.8 Comparing Algebraic L-Theory and Topological K-theoryof C∗-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

8.9 Topological K-Theory for Finite Groups . . . . . . . . . . . . . . . . . . . 2658.10 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Part II: The Isomorphism Conjectures

9 Classifying Spaces for Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2679.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2679.2 Definition and Basic Properties of G-CW -Complexes . . . . . . . . 2679.3 Proper G-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2719.4 Maps between G-CW -Complexes . . . . . . . . . . . . . . . . . . . . . . . . . 2729.5 Definition and Basic Properties of Classifying Spaces for

Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2749.6 Special Models for the Classifying Space of Proper Actions . . 275

9.6.1 Simplicial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2769.6.2 Operator Theoretic Model . . . . . . . . . . . . . . . . . . . . . . . . . 2769.6.3 Discrete Subgroups of Almost Connected Lie Groups . 2769.6.4 Actions on Simply Connected Non-Positively Curved

Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2779.6.5 Actions on Trees and Graphs of Groups . . . . . . . . . . . . . 2779.6.6 Actions on CAT(0)-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 2779.6.7 The Rips Complex of a Hyperbolic Group . . . . . . . . . . . 2789.6.8 Arithmetic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2799.6.9 Outer Automorphism Groups of Finitely Generated

Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2799.6.10 Mapping Class Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2809.6.11 Special Linear Groups of (2,2)-Matrices . . . . . . . . . . . . . 2809.6.12 Groups with Appropriate Maximal Finite Subgroups . . 2819.6.13 One-Relator Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

9.7 Special Models for the Classifying Space for the Family ofVirtually Cyclic Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2849.7.1 Groups with Appropriate Maximal Virtually Cyclic

Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2849.8 Finiteness Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

9.8.1 Review of Finiteness Conditions on BG . . . . . . . . . . . . . 2859.8.2 Cohomological Criteria for Finiteness Properties in

Terms of Bredon Cohomology . . . . . . . . . . . . . . . . . . . . . . 2879.8.3 Finite Models for the Classifying Space for Proper

Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2879.8.4 Models of Finite Type for the Classifying Space for

Proper Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2889.8.5 Finite Dimensional Models for the Classifying Space

for Proper Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

Contents XV

9.8.6 Brown’s Problem about Virtual CohomologicalDimension and the Dimension of the ClassifyingSpace for Proper Actions . . . . . . . . . . . . . . . . . . . . . . . . . . 290

9.8.7 Finite Dimensional Models for the Classifying Spacefor the Family of Virtually Cyclic Subgroups . . . . . . . . . 291

9.8.8 Low Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2939.8.9 Finite Models for the Classifying Space for the Family

of Virtually Cyclic Subgroups . . . . . . . . . . . . . . . . . . . . . . 2939.9 On the Homotopy Type of the Quotient Space of the

Classifying Space for Proper Actions . . . . . . . . . . . . . . . . . . . . . . 2949.10 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

10 Equivariant Homology Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29710.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29710.2 Basics about G-Homology Theories . . . . . . . . . . . . . . . . . . . . . . . 29710.3 Basics about Equivariant Homology Theories . . . . . . . . . . . . . . 30210.4 Constructing Equivariant Homology Theories Using Spectra . 30510.5 Equivariant Homology Theories Associated to K- and

L-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31010.6 Two Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

10.6.1 The Equivariant Atiyah-Hirzebruch Spectral Sequence 31210.6.2 The p-Chain Spectral Sequence . . . . . . . . . . . . . . . . . . . . 313

10.7 Equivariant Chern Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31510.7.1 Mackey Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31610.7.2 The Equivariant Chern Character . . . . . . . . . . . . . . . . . . 317

10.8 Some Rational Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31910.8.1 Green Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31910.8.2 Induction Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32210.8.3 Rational Computation of the source of the assembly

maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32510.9 Some Integral Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32610.10Equivariant Homology Theory over a Group and Twisting

with Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32910.11Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

11 The Farrell-Jones Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33311.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33311.2 The Farrell-Jones Conjecture with Coefficients in Rings . . . . . 334

11.2.1 The K-Theoretic Farrell-Jones Conjecture withCoefficients in Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

11.2.2 The L-Theoretic Farrell-Jones Conjecture withCoefficients in Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

11.3 The Farrell-Jones Conjecture with Coefficients in AdditiveCategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

XVI Contents

11.3.1 The K-theoretic Farrell-Jones Conjecture withCoefficients in Additive G-Categories . . . . . . . . . . . . . . . 337

11.3.2 The L-theoretic Farrell-Jones Conjecture withCoefficients in Additive G-Categories with Involution . 338

11.4 Finite Wreath Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34011.5 The Full Farrell-Jones Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 34111.6 Inheritance Properties of the Farrell-Jones Conjecture . . . . . . 34111.7 Splitting the Assembly Map from FIN to VCY . . . . . . . . . . . . 34411.8 Splitting Rationally the Assembly Map from T R to FIN . . . 34511.9 Reducing the Family of Subgroups for the Farrell-Jones

Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34611.9.1 Reducing the Family of Subgroups for the

Farrell-Jones Conjecture for K-Theory . . . . . . . . . . . . . . 34711.9.2 Reducing the Family of Subgroups for the

Farrell-Jones Conjecture for L-Theory . . . . . . . . . . . . . . 35311.10 The Farrell-Jones Conjecture Implies All Its Variants . . . . . . . 354

11.10.1 List of Variants of the Farrell-Jones Conjecture . . . . . . 35411.10.2 Proof of the Variants of the Farrell-Jones Conjecture . 357

11.11 Summary of the Applications of the Farrell-Jones Conjecture 36011.12 G-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36411.13 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

12 The Baum-Connes Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36712.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36712.2 The Analytic Version of the Baum-Connes Assembly Map . . . 36812.3 The Version of the Baum-Connes Assembly Map in Terms

of Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36912.4 The Baum-Connes Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37012.5 Variants of the Baum-Connes Conjecture . . . . . . . . . . . . . . . . . . 372

12.5.1 The Baum-Connes Conjecture for Maximal GroupC∗-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

12.5.2 The Bost Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37312.5.3 The Baum-Connes Conjecture for Frechet Algebras . . . 37412.5.4 The Strong and the Integral Novikov Conjecture . . . . . 37412.5.5 The Coarse Baum Connes Conjecture . . . . . . . . . . . . . . . 375

12.6 Inheritance Properties of the Baum-Connes Conjecture . . . . . . 37612.7 Reducing the Family of Subgroups for the Baum-Connes

Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37912.8 Applications of the Baum-Connes Conjecture . . . . . . . . . . . . . . 380

12.8.1 The Modified Trace Conjecture . . . . . . . . . . . . . . . . . . . . 38012.8.2 The Stable Gromov-Lawson-Rosenberg Conjecture . . . . 38112.8.3 L2-Rho-Invariants and L2-Signatures . . . . . . . . . . . . . . . 384

12.9 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

Contents XVII

13 The (Fibered) Meta-Isomorphism Conjecture and OtherIsomorphism Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38713.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38713.2 The Meta-Isomorphism Conjecture . . . . . . . . . . . . . . . . . . . . . . . 38813.3 The Fibered Meta-Isomorphism Conjecture . . . . . . . . . . . . . . . . 38913.4 The Farrell-Jones Conjecture with Coefficients in Additive

G-Categories is Fibered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39013.5 Transitivity Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39113.6 Inheritance Properties of the Fibered Meta-Isomorphism

Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39513.7 Actions on Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39813.8 The Meta-Isomorphism Conjecture for Functors from Spaces

to Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40113.9 Proof of the Inheritance Properties . . . . . . . . . . . . . . . . . . . . . . . 40613.10 The Farrell-Jones Conjecture for A-Theory and Pseudoisotopy41213.11 The Farrell-Jones Conjecture for Topological Hochschild

and Cyclic Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41313.11.1 Topological Hochschild Homology . . . . . . . . . . . . . . . . . . 41413.11.2 Topological Cyclic Homology . . . . . . . . . . . . . . . . . . . . . . 414

13.12 The Farrell-Jones Conjecture for Homotopy K-Theory . . . . . . 41513.12.1 The Farrell-Jones Conjecture for Totally

Disconnected Groups and Hecke Algebras . . . . . . . . . . . 41813.13 Relations among the Isomorphisms Conjectures . . . . . . . . . . . . 419

13.13.1 The Farrell-Jones Conjecture for K-Theory and forA-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

13.13.2 The Farrell-Jones Conjecture for A-Theory and forPseudoisotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

13.13.3 The Farrell-Jones Conjecture for K-Theory and fortopological cyclic homology . . . . . . . . . . . . . . . . . . . . . . . . 420

13.13.4 The L-Theoretic Farrell-Jones Conjecture and theBaum-Connes Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 421

13.13.5 Mapping Surgery to Analysis . . . . . . . . . . . . . . . . . . . . . . 42313.13.6 The Baum-Connes Conjecture and the Bost Conjecture42713.13.7 The Farrell-Jones Conjecture for K-theory and for

homotopy K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42713.14 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

14 Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43114.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43114.2 Status of the Full Farrell-Jones Conjecture . . . . . . . . . . . . . . . . . 43114.3 Status of the Baum-Conjecture (with coefficients) . . . . . . . . . . 43414.4 Injectivity Results in the Baum-Connes Setting . . . . . . . . . . . . 43814.5 Injectivity Results in the Farrell-Jones Setting . . . . . . . . . . . . . 44014.6 Status of the Novikov Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 44214.7 Review of and Status report for some classes of groups . . . . . . 444

XVIII Contents

14.7.1 Hyperbolic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44414.7.2 Lacunary Hyperbolic Groups . . . . . . . . . . . . . . . . . . . . . . 44414.7.3 Relative Hyperbolic Groups . . . . . . . . . . . . . . . . . . . . . . . . 44514.7.4 Systolic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44514.7.5 Finite dimensional CAT(0) Groups . . . . . . . . . . . . . . . . . 44514.7.6 Fundamental Groups of Complete Riemannian

Manifolds with Non-Positive Sectional Curvature . . . . . 44614.7.7 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44714.7.8 S-Arithmetic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44714.7.9 Linear Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44814.7.10Subgroups of Almost Connected Lie Groups . . . . . . . . . 44814.7.11Virtually Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . 44814.7.12A-T-menable, Amenable and Elementary Amenable

Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44814.7.13 Three-Manifold Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 45014.7.14 One-Relator Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45014.7.15Selfsimilar Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45114.7.16(Strongly) Poly-Surface Groups . . . . . . . . . . . . . . . . . . . . 45114.7.17 Poly-Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45214.7.18 Braid Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45314.7.19 Coxeter Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45314.7.20 Mapping Class Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 45314.7.21 Out(Fn) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45414.7.22 Thompson’s Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45414.7.23 Groups Satisfying Homological Finiteness Conditions . 454

14.8 Open Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45614.9 How Can We Find Counterexamples? . . . . . . . . . . . . . . . . . . . . 457

14.9.1 Exotic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45714.9.2 Infinite Direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 45814.9.3 Exotic Closed Aspherical Manifolds . . . . . . . . . . . . . . . . . 45914.9.4 Some Results Which Hold for All Groups . . . . . . . . . . . . 460

14.10 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

15 Guide for Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46315.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46315.2 K- and L- Groups for Finite Groups . . . . . . . . . . . . . . . . . . . . . . 46315.3 The passage from FIN to VCY . . . . . . . . . . . . . . . . . . . . . . . . . . 46415.4 Rational Computations for Infinite Groups . . . . . . . . . . . . . . . . 464

15.4.1 Rationalized Algebraic K-Theory . . . . . . . . . . . . . . . . . . . 46415.4.2 Rationalized Algebraic L-Theory . . . . . . . . . . . . . . . . . . . 46515.4.3 Rationalized Topological K-Theory . . . . . . . . . . . . . . . . . 46615.4.4 The Complexified Comparison Map from Algebraic

to Topological K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 46615.5 Integral Computations for Infinite Groups . . . . . . . . . . . . . . . . . 467

15.5.1 Groups satisfying conditions (M) and (NM) . . . . . . . . . . 467

Contents XIX

15.5.2 Fuchsian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46915.5.3 Torsionfree Hyperbolic Groups . . . . . . . . . . . . . . . . . . . . . 47115.5.4 L-theory of Torsionfree Groups . . . . . . . . . . . . . . . . . . . . . 47215.5.5 Cocompact NEC-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 47315.5.6 Crystallographic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 47315.5.7 Virtually finitely generated free abelian groups . . . . . . . 47515.5.8 SL3(Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

15.6 Applications of Some Computations . . . . . . . . . . . . . . . . . . . . . . 47615.6.1 Classification of some C∗-algebras . . . . . . . . . . . . . . . . . . 47615.6.2 Unstable Gromov-Lawson Rosenberg Conjecture . . . . . 47615.6.3 Classification of certain manifolds with infinite not

torsionfree fundamental groups . . . . . . . . . . . . . . . . . . . . . 47715.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

16 Assembly Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47916.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47916.2 Homological Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48016.3 Extension from Homogenous Spaces to G-CW -Complexes . . . 48016.4 Homotopy Colimit Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48116.5 Universal Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48116.6 Identifying assembly maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48716.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

Part III: Methods of Proofs

17 Controlled Topology Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48917.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48917.2 Controlled Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

17.2.1 Geometric Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48917.2.2 Imposing Control Conditions . . . . . . . . . . . . . . . . . . . . . . 49117.2.3 Germs at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49317.2.4 Some control condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49417.2.5 The assembly map as a forget control map. . . . . . . . . . . 499

17.3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

18 Coverings and Flow Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50718.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50718.2 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

19 Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50919.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50919.2 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

XX Contents

20 Analytic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51120.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51120.2 The Dirac-Dual Dirac Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 51120.3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

21 Cyclic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51321.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51321.2 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

22 Solutions of the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

23 Comments (temporary chapter) . . . . . . . . . . . . . . . . . . . . . . . . . . 61523.1 Mathematical Comments and Problems . . . . . . . . . . . . . . . . . . . 61523.2 Mathematical Items or Explanations that have to be Added . 61623.3 Layout and presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61623.4 Papers which we have cited but could not downloaded and

checked . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61623.5 Comments about Latex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61623.6 Comments about the Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61723.7 Comments about Spelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61823.8 Reminders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61823.9 Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

Chapter 0

Introduction

This version of the introduction is very old and far away fromits final form. It will be changed completely. It needs still somestrategic thoughts and more input when more chapters of thebook have been written.

The Isomorphism Conjectures due to Baum-Connes and Farrell-Jones areimportant conjectures which have many interesting applications and conse-quences. However, they are not easy to formulate and it is a priori not clearwhy the actual versions are the most promising ones. They are the finalupshot of a longer process which lead step by step to the final versions. Ithas been influenced and steered by the various new results which have beenproved during the last decades and given new insight into the objects, prob-lems and constructions these conjectures aim at. In this introduction we wantto explain how one can be lead to these conjectures by general considerationsand certain facts, what implications these conjectures have, what their statusis and what methods of proof are known.

0.1 Motivation for the Baum-Connes Conjecture

We will start with the easiest and most convenient to handle IsomorphismConjecture, the Baum-Connes Conjecture for the topological K-theory ofreduced group C∗-algebras, and then pass to the more complicated Farrell-Jones Conjecture for the algebraic K- and L-theory of group rings.

0.1.1 Topological K-Theory of Reduced Group C∗-Algebras

The target of the Baum-Connes Conjecture is the topological (complex) K-theory of the reduced C∗-algebra C∗r (G) of a group G. We will considerdiscrete groups G only. One defines the topological K-groups Kn(A) forany Banach algebra A to be the abelian group Kn(A) = πn−1(GL(A)) forn ≥ 1. The famous Bott Periodicity Theorem gives a natural isomorphism

Kn(A)∼=−→ Kn+2(A) for n ≥ 1. Finally one defines Kn(A) for all n ∈ Z so

that the Bott isomorphism theorem is true for all n ∈ Z. It turns out thatK0(A) is the same as the projective class group of the ring A which is the

1

2 0 Introduction

Grothendieck group of the abelian monoid of isomorphism classes of finitelygenerated projective A-modules with the direct sum as addition. The topolo-gical K-theory of C is trivial in odd dimensions and is isomorphic to Z in evendimensions. More generally, for a finite group G the topological K-theory ofC∗r (G) is the complex representation ring RC(G) in even dimensions and istrivial in odd dimensions.

Let P be an appropriate elliptic differential operator (or more generally anelliptic complex) on a closed n-dimensional Riemannian manifold M , for in-stance the Dirac operator or the signature operator. Then one can consider itsindex in Kn(C) which is zero for odd n and dimC(ker(P ))−dimC(coker(P )) ∈Z for even n. If M comes with an isometric G-action of a finite group Gand P is compatible with the G-action, then ker(P ) and coker(P ) are com-plex (finite dimensional) G-representations and one obtains an element inKn(C∗r (G)) = RC(G) by [ker(P )] − [coker(P )] for even n. Suppose that Gis an arbitrary discrete group and that M is a (not necessarily compact)n-dimensional smooth manifold without boundary with a proper cocompactG-action, a G-invariant Riemannian metric and an appropriate elliptic differ-ential operator P compatible with the G-action. An example is the universalcovering M = N of a closed n-dimensional Riemannian manifold N withG = π1(N) and the lift P to N of an appropriate elliptic differential operatorP on N . Then one can define an equivariant index of P which takes values inKn(C∗r (G)). Therefore the interest of K∗(C

∗r (G)) comes from the fact that it

is the natural recipient for indices of certain equivariant differential operators.

0.1.2 Homological Aspects

A first basic problem is to compute K∗(C∗r (G)) or to identify it with more

familiar terms. The key idea comes from the observation that K∗(C∗r (G))

has some homological properties. More precisely, if G is the amalgamatedproduct G = G1 ∗G0 G2 for subgroups Gi ⊆ G, then there is a long exactsequence

(0.1)

· · · ∂n+1−−−→ Kn(C∗r (G0))Kn(C∗r (i1))⊕Kn(C∗r (i2))−−−−−−−−−−−−−−−−→ Kn(C∗r (G1))⊕Kn(C∗r (G2))

Kn(C∗r (j1))−Kn(C∗r (j2))−−−−−−−−−−−−−−−−→ Kn(C∗r (G))∂n−→ Kn−1(C∗r (G0))

Kn−1(C∗r (i1))⊕Kn−1(C∗r (i2))−−−−−−−−−−−−−−−−−−−→ Kn−1(C∗r (G2))⊕Kn−1(C∗r (G1))

Kn−1(C∗r (j1))−Kn−1(C∗r (j2))−−−−−−−−−−−−−−−−−−−→ Kn−1(C∗r (G))∂n−1−−−→ · · ·

where i1,i2, j1 and j2 are the obvious inclusions (see [663, Theorem 18 onpage 632]). If φ : G→ G is a group automorphism and Goφ Z the associatedsemidirect product, then there is a long exact sequence

0.1 Motivation for the Baum-Connes Conjecture 3

(0.2)

· · · ∂n+1−−−→ Kn(C∗r (G))Kn(C∗r (φ))−id−−−−−−−−−→ Kn(C∗r (G))

Kn(C∗r (k))−−−−−−−→ Kn(C∗r (GoφZ))

∂n−→ Kn−1(C∗r (G))Kn−1(C∗r (φ))−id−−−−−−−−−−−→ Kn−1(C∗r (G))

Kn−1(C∗r (k))−−−−−−−−→ · · ·

where k is the obvious inclusion (see [662, Theorem 3.1 on page 151] or moregenerally [663, Theorem 18 on page 632]).

We compare this with group homology in order to explain the analogywith homology. Recall that the classifying space BG of a group G is an as-pherical CW -complex whose fundamental group is isomorphic to G and thataspherical means that all higher homotopy groups are trivial, or, equivalently,that its universal covering is contractible. It is unique up to homotopy. If onehas an amalgamated product G = G1 ∗G0

G2, then one can find modelsfor the classifying spaces such that BGi is a CW -subcomplex of BG andBG = BG1 ∪ BG2 and BG0 = BG1 ∩ BG2. Thus we obtain a pushout ofinclusions of CW -complexes

BG0Bi1 //

Bi2

BG1

Bj1

BG2

Bj2

// BG

It yields a long Mayer-Vietoris sequence for the cellular or singular homology

(0.3) · · · ∂n+1−−−→ Hn(BG0)Hn(Bi1)⊕Hn(Bi2)−−−−−−−−−−−−→ Hn(BG1)⊕Hn(BG2)

Hn(Bj1)−Hn(Bj2)−−−−−−−−−−−−→ Hn(BG)∂n−→ Hn−1(BG0)

Hn−1(Bi1)⊕Hn−1(Bi2)−−−−−−−−−−−−−−−→ Hn−1(BG2)⊕Hn−1(BG1)

Hn−1(Bj1)−Hn−1(Bj2)−−−−−−−−−−−−−−−→ Hn−1(BG)∂n−1−−−→ · · ·

If φ : G → G is a group automorphism, then a model for B(G oφ Z) is themapping torus of Bφ : BG→ BG which is obtained from the cylinder BG×[0, 1] by identifying the bottom and the top with the map Bφ. Associated toa mapping torus there is the long exact sequence

(0.4) · · · ∂n+1−−−→ Hn(BG)Hn(Bφ)−id−−−−−−−→ Hn(BG)

Hn(Bk)−−−−−→ Hn(B(Goφ Z))

∂n−→ Hn−1(BG)Hn−1(Bφ)−id−−−−−−−−−→ Hn−1(BG)

Hn(Bk)−−−−−→ · · ·

where k is the obvious inclusion of BG into the mapping torus.

4 0 Introduction

0.1.3 The Baum-Connes Conjecture for Torsionfree Groups

There is an obvious analogy between the sequences (0.1) and (0.3) and thesequences (0.2) and (0.4). On the other hand we get for the trivial groupG = 1 that Hn(B1) = Hn(•) is Z for n = 0 and trivial for n 6= 0 so thatthe group homology of BG cannot be the same as the topological K-theoryof C∗r (1). But there is a better candidate, namely take the topological K-homology of BG instead of the singular homology. Topological K-homologyis a homology theory defined for CW -complexes. At least we mention thatfor a topologist its definition is a routine, namely, it is the homology theoryassociated to the K-theory spectrum which defines topological K-theory ofCW -complexes, i.e., the cohomology theory which comes from consideringvector bundles over CW -complexes. In contrast to singular homology, thetopological K-homology of a point Kn(•) is Z for even n and is trivial forn odd. So we still get exact sequences (0.3) and (0.4) if we replace H∗ by K∗everywhere and we have Kn(B1) ∼= Kn(C∗r (1) for all n ∈ Z. This leadsto the following conjecture

Conjecture 0.5 (Baum-Connes Conjecture for torsionfree groups).Let G be a torsionfree group. Then there is for n ∈ Z an isomorphism calledassembly map

Kn(BG)∼=−→ Kn(C∗r (G)).

This is indeed a formulation which will turn out to be equivalent to theBaum-Connes Conjecture, provided that G is torsionfree. Conjecture 0.5cannot hold in general as already the example of a finite group G shows.Namely, if G is finite, then the obvious inclusion induces an isomorphism

Kn(B1) ⊗Z Q∼=−→ Kn(BG) ⊗Z Q for all n ∈ Z, whereas K0(C∗r (1) →

K0(C∗r (G)) agrees with the map RC(1) → RC(G) which is rationally bi-jective if and only if G itself is trivial. Hence Conjecture 0.5 is not true fornon-trivial finite groups.

0.1.4 The Baum-Connes Conjecture

What is going wrong? The sequences (0.1) and (0.2) do exist regardlesswhether the groups are torsionfree or not. More generally, if G acts on atree, then they can be combined to compute the K-theory K∗(C

∗r (G)) of a

group G by a certain Mayer-Vietoris sequence from the stabilizers of the ver-tices and edges (see Pimsner [663, Theorem 18 on page 632]). In the specialcase, where all stabilizers are finite, one sees that K∗(C

∗r (G)) is built by the

topological K-theory of the finite subgroups of G in a homological fashion.This leads to the idea that K∗(C

∗r (G)) can be computed in a homological

way, but the building blocks do not only consist of K∗(C∗r (1)) alone but

0.1 Motivation for the Baum-Connes Conjecture 5

of K∗(C∗r (H)) for all finite subgroups H ⊆ G. This suggest to study equi-

variant topological K-theory. It assigns to every proper G-CW -complex X asequence of abelian groups KG

n (X) for n ∈ Z such that G-homotopy invari-ance holds and Mayer-Vietoris sequences exist. A proper G-CW -complex isa CW -complex with G-action such that for g ∈ G and every open cell e withe ∩ g · e 6= ∅ we gave gx = x for all x ∈ e and all isotropy groups are finite.Two interesting features are that KG

n (G/H) agrees with Kn(C∗r (H)) for ev-ery finite subgroup H ⊆ G and that for a free G-CW -complex X and n ∈ Zwe have a natural isomorphism KG

n (X)∼=−→ Kn(G\X). Recall that EG is a

free G-CW -complex which is contractible and that EG → G\EG = BG isthe universal covering of BG. We can reformulate Conjecture 0.5 by statingan isomorphism

KGn (EG)

∼=−→ Kn(C∗r (G)).

Now suppose that G acts on a tree T with finite stabilizers. Then the com-putation of Pimsner [663, Theorem 18 on page 632]) mentioned above can berephrased to the statement that there is an isomorphism

KGn (T )

∼=−→ Kn(C∗r (G)).

In particular the left hand side is independent of the tree T on which Gacts by finite stabilizers. This can be explained as follows. It is known thatfor every finite subgroup H ⊆ G the H-fixed point set T is again a non-empty tree and hence contractible. This implies that two trees T1 and T2

on which G acts with finite stabilizers are G-homotopy equivalent and hencehave the same equivariant topological K-theory. The same remark applies toKn(BG) and Kn(EG), namely, two models for BG are homotopy equivalentand two models for EG are G-homotopy equivalent and therefore Kn(BG)and KG

n (EG) are independent of the choice of a model. This leads to the ideato look for an appropriate proper G-CW -complex EG which is characterizedby a certain universal property and is unique up to G-homotopy such thatfor a torsionfree group G we have EG = EG and for a tree on which G actswith finite stabilizers we have EG = T and that there is an isomorphism

KGn (EG)

∼=−→ Kn(C∗r (G)).

In particular for a finite group we would like to have EG = G/G = •and then the desired isomorphism above is true for trivial reasons. Recallthat EG is characterized up to G-homotopy by the property that it is a G-CW -complex such that EGH is empty for H 6= 1 and is contractible forH = 1. Having the case of a tree on which G acts with finite stabilizersin mind, we define the classifying space for proper G-actions EG to be aG-CW -complex such that EGH is empty for |H| =∞ and is contractible for|H| < ∞. Indeed two models for EG are G-homotopy equivalent, a tree onwhich G acts with finite stabilizers is a model for EG, we have EG = EG

6 0 Introduction

if (and only if) G is torsionfree and EG = G/G = • if (and only if) G isfinite. This leads to the following

Conjecture 0.6 (Baum-Connes Conjecture). Let G be a group. Thenthere is for all n ∈ Z an isomorphism called assembly map

KGn (EG)

∼=−→ Kn(C∗r (G)).

The conjecture above makes sense for all groups, and no counterexamplesare known at the time of writing. This conjecture reduces in the torsionfreecase to Conjecture 0.5 and is consistent with the results by Pimsner [663,Theorem 18 on page 632]) for G-acting on a tree with finite stabilizers. It isalso true for finite groups G. Pimsner’s result does hold more generally forgroups acting on trees with not necessarily finite stabilizers. So one should getthe analogous result for the left hand side of the isomorphism appearing inthe Baum-Connes Conjecture 0.6. Essentially this boils down to the question,whether the analogues of the long exact sequences (0.1) and (0.2) holds for theleft side of the isomorphism appearing in the Baum-Connes Conjecture 0.6.This follows for (0.1) from the fact that for G = G1 ∗G0

G2 one can findappropriate models for the classifying spaces for proper G-actions such thatthere is a G-pushout of inclusions of proper G-CW -complexes

G×G0EG0

//

G×G1EG1

G×G2

EG2// EG

and for a subgroup H ⊆ G and a proper H-CW -complex X there is a naturalisomorphism

KHn (X)

∼=−→ KGn (G×H X).

Thus the associated long exact Mayer-Vietoris sequence yields the long exactsequence

· · · ∂n+1−−−→ KG0n (EG0)→ KG1

n (EG1)⊕KG2n (EG2)→ KG

n (EG)∂n−→

KG0n−1(EG0)→ KG1

n−1(EG1)⊕KG2n−1(EG2)→ KG0

n−1(EG)→ · · ·

which corresponds to (0.1). For (0.2) one uses fact that for a group automor-

phism φ : G∼=−→ G the GoφZ-CW -complex given by the to both sides infinite

mapping telescope of the φ-equivariant map Eφ : EG → EG is a model forE(Goφ Z).

In general KGn (EG) is much bigger than KG

n (EG) ∼= Kn(BG) and thecanonical map KG

n (EG)→ KGn (EG) is rationally injective but not necessar-

ily integrally injective.

0.2 Motivation for the Farrell-Jones Conjecture for K-Theory 7

0.1.5 Reduced versus Maximal Group C∗-Algebras

All the arguments above do also apply to the maximal group C∗-algebrawhich does even have better functorial properties than the reduced groupC∗-algebra. So a priori one may think that one should use the maximal groupC∗-algebra instead of the reduced one. However, the version for the maximalgroup C∗-algebra is not true in general and the version for the reduced groupC∗-algebra seems to be the right one. This will be discussed in more detailin subsection 12.5.1.

If one considers instead of the reduced group C∗-algebra the Banach groupalgebra l1(G), one obtains the Bost Conjecture 12.22.

0.1.6 Applications of the Baum-Connes Conjecture

The assembly map appearing in the Baum-Connes Conjecture 0.6 has anindex theoretic interpretation. An element in KG

0 (EG) can be representedby a pair (M,P ∗) consisting of a cocompact proper smooth n-dimensionalG-manifold M with a G-invariant Riemannian metric together with an el-liptic G-complex P ∗ of differential operators of order 1 on M and its imageunder the assembly map is a certain equivariant index indC∗r (G)(M,P ∗) inKn(C∗r (G)). There are many important consequences of the Baum-ConnesConjecture such as the Kadison Conjecture, see Subsection 8.4.2, the stableGromov-Lawson-Rosenberg Conjecture, see Subsection 12.8.2. Novikov Con-jecture, see Section 7.14, and the (Modified) Trace Conjecture, see Subsec-tions 8.4.1 and 12.8.1

0.2 Motivation for the Farrell-Jones Conjecture forK-Theory

Next we want to deal with the algebraic K-groups Kn(RG) of the group ringRG of a group G with coefficients in an associative ring R with unit.

0.2.1 Algebraic K-Theory of Group Rings

For an associative ring with unit R one defines K0(R) to be the projec-tive class group of R and K1(R) to be the abelianization of GL(R) =colimn→∞GLn(R). The higher algebraic K-groups Kn(R) for n ≥ 1 are thehomotopy group groups of a certainK-theory space associated to the categoryof finitely generated projective R-modules. One can define negative K-groups

8 0 Introduction

Kn(R) for n ≤ −1 by a certain contracting procedure applied to K0(R). Fi-nally there exists a K-theory spectrum K(R) such that πn(K(R)) = Kn(R)for all n ∈ Z. If Z → R is the obvious ring map sending n to n · 1R, thenone defines the reduced K-groups to be the cokernel of the induced mapKn(Z)→ Kn(R). The Whitehead group Wh(G) of a group G is the quotientof K1(ZG) by elements given by (1, 1)-matrices of the shape (±g) for g ∈ G.

The reduced projective class group K0(ZG) is the recipient for the finite-ness obstruction of a finitely dominated CW -complex X with fundamentalgroup G = π1(X). Finitely dominated means that there is a finite CW -complex Y and maps i : X → Y and r : Y → X such that r i is homotopicto the identity on X. The Whitehead group Wh(G) is the recipient of theWhitehead torsion of a homotopy equivalence of finite CW -complexes andof a compact h-cobordism over a closed manifold with fundamental groupG. An h-cobordism W over M consists of a manifold W whose boundary isthe disjoint union ∂W = ∂0W

∐∂1W such that both inclusions ∂iW → W

are homotopy equivalences together with a diffeomorphism M∼=−→ ∂0W . The

finiteness obstruction and the Whitehead torsion are very important topo-logical obstructions whose vanishing has interesting geometric consequences.The vanishing of the finiteness obstruction says that the finitely dominatedCW -complex under consideration is homotopy equivalent to a finite CW -complex. The vanishing of the Whitehead torsion of a compact h-cobordismW over M of dimension ≥ 6 implies that the W is trivial, i.e., is diffeomorphicto a cylinder M × [0, 1] relative M = M ×0. This explains why topologistsare interested in Kn(ZG) for groups G.

0.2.2 Appearance of Nil-Terms

The situation for algebraic K-theory of RG is more complicated than the onefor the topological K-theory of C∗r (G). As a special case of the sequence (0.2)we obtain an isomorphism

Kn(C∗r (G× Z)) = Kn(C∗r (G))⊕Kn−1(C∗r (G)).

For algebraic K-theory the analogue is the Bass-Heller-Swan decomposition

Kn(R[Z]) ∼= Kn(R)⊕Kn−1(R)⊕NKn(R)⊕NKn(R),

where certain additional terms, the Nil-terms NKn(R) appear. If one replacesR by RG, one gets

Kn(R[G× Z]) ∼= Kn(RG)⊕Kn−1(RG)⊕NKn(RG)⊕NKn(RG).

0.2 Motivation for the Farrell-Jones Conjecture for K-Theory 9

Such correction terms in form of Nil-terms appear also, when one wants toget analogues of the sequences (0.1) and (0.2) for algebraic K-theory (seeSection 5.9).

0.2.3 The Farrell-Jones Conjecture for K∗(RG) for Regular Ringsand Torsionfree Groups

Let R be a regular ring, i.e., it is Noetherian and every R-module possessesa finite dimensional projective resolution. For instance, any principal idealdomain is a regular ring. Then one can prove in many cases for torsionfreegroups that the analogues of the sequences (0.1) and (0.2) do hold for alge-braic K-theory (see Waldhausen [804] and [807]). The same reasoning as inthe Baum-Connes Conjecture for torsionfree groups leads to

Conjecture 0.7. (Farrell-Jones Conjecture for K∗(RG) for torsion-free groups and regular rings). Let G be a torsionfree group and let Rbe a regular ring. Then there is for n ∈ Z an isomorphism

Hn(BG; K(R))∼=−→ Kn(RG).

HereH∗(−; K(R)) is the homology theory associated to theK-theory spec-trum of R. It is a homology theory with the property that Hn(•; K(R)) =πn(K(R)) = Kn(R) for n ∈ Z.

0.2.4 The Farrell-Jones Conjecture for K∗(RG) for Regular Rings

If one drops the condition that G is torsionfree but requires that the orderof every finite subgroup of G is invertible in R, then one still can prove inmany cases that the analogues of the sequences (0.1) and (0.2) do hold foralgebraic K-theory. The same reasoning as in the Baum-Connes Conjectureleads to

Conjecture 0.8. (Farrell-Jones Conjecture for K∗(RG) for regularrings). Let G be a group. Let R be a regular ring such that |H| is invertiblein R for every finite subgroup H ⊆ G. Then there is an isomorphism

HGn (EG; KR)

∼=−→ Kn(RG).

Here HGn (−; KR) is an appropriate G-homology theory with the property

that HGn (G/H; KR) ∼= HH

n (•; KR) ∼= Kn(RH) for every subgroup H ⊆ Gand the isomorphism above is induced by the G-map EG → •. Conjec-ture 0.8 reduces to Conjecture 0.7 if G is torsionfree.

10 0 Introduction

0.2.5 The Farrell-Jones Conjecture for K∗(RG)

Conjecture 0.7 can be applied in the case R = Z what is not true for Con-jecture 0.8. So what is the right formulation for arbitrary rings R? The ideais that one does not only need to take all finite subgroups into account butalso all virtually cyclic subgroups. A group is called virtually cyclic if it isfinite or contains Z as subgroup of finite index. Namely, let EG = EVCY(G)be the classifying space for the family of virtually cyclic subgroups, i.e., aG-CW -complex EG such that EGH is contractible for every virtually cyclicsubgroup H ⊆ G and is empty for every subgroup H ⊆ G which is notvirtually cyclic. The G-space EG is unique up to G-homotopy.

Conjecture 0.9. (Farrell-Jones Conjecture for K∗(RG)). Let G be agroup. Let R be an associative ring with unit. Then there is for all n ∈ Z anisomorphism called assembly map which is induced by the G-map EG→ •

HGn (EG; KR)

∼=−→ Kn(RG).

The conjecture above makes sense for all groups and rings, and no coun-terexamples are known at the time of writing. We have absorbed all the Nil-phenomena into the source by replacing EG by EG. There is a certain prizewe have to pay, since often there are nice small geometric models for EG,whereas the spaces EG are much harder to analyze and are in general huge.There are up to G-homotopy unique G-maps EG → EG and EG → EGwhich yield maps

Hn(BG; K(R)) = HGn (EG; KR)→ HG

n (EG; KR)→ HGn (EG; KR).

We will later see that there is a splitting, see Theorem 11.26,

HGn (EG; KR) ∼= HG

n (EG; KR)⊕HGn (EG,EG; KR),(0.10)

where HGn (EG; KR) is the comparatively easy homological part and all Nil-

type information is contained in HGn (EG,EG; KR). If R is regular and the

order of any finite subgroup of G is invertible in R, then HGn (EG,EG; KR) is

trivial and hence the natural mapHGn (EG; KR)

∼=−→ HGn (EG; KR) is bijective.

Therefore Conjecture 0.9 reduces to Conjecture 0.7 and Conjecture 0.8 whenthey apply.

In the Baum-Connes setting the natural map KGn (EG)

∼=−→ KGn (EG) is

always bijective.

0.3 Motivation for the Farrell-Jones Conjecture for L-Theory 11

0.2.6 Applications of the Farrell-Jones Conjecture for K∗(RG)

Since Kn(Z) = 0 for n ≤ −1 and the maps Z∼=−→ K0(Z), which sends n

to the class of Zn, and ±1 → K1(Z), which sends ±1 to the class of the(1, 1)-matrix (±1), are bijective, an easy spectral sequence argument showsthat Conjecture 0.7 implies

Conjecture 0.11. (Farrell-Jones Conjecture Kn(ZG) in dimensions

n ≤ 1). Let G be a torsionfree group. Then Kn(ZG) = 0 for n ∈ Z, n ≤ 0and Wh(G) = 0.

In particular the finiteness obstruction and the Whitehead torsion arealways zero for torsionfree fundamental groups. This implies in particular thatevery h-cobordism over a simply connected d-dimensional closed manifoldfor d ≥ 5 is trivial and thus the Poincare Conjecture in dimensions ≥ 6(and with some extra effort also in dimension d = 5). This will be explainedin Section 2.5. The Farrell-Jones Conjecture for K-theory 0.9 implies theBass Conjecture (see Section 1.10). The Kaplansky Conjecture follows fromthe Farrell-Jones Conjecture for K-theory 0.7 as explained in Section 1.9.Further applications of the Farrell-Jones Conjecture for K-theory 0.9 , e.g.,to pseudo-isotopy and to automorphisms of manifolds, will be discussed inSection 7.20.

0.3 Motivation for the Farrell-Jones Conjecture forL-Theory

Next we want to deal with the algebraic L-groups Lεn(RG) of the group ringRG of a group G with coefficients in an associative ring R with unit andinvolution.

0.3.1 Algebraic L-Theory of Group Rings

Let R be an associative ring with unit. An involution of rings R→ R, r 7→ ron R is a map satisfying r + s = r + s, rs = s r, 0 = 0, 1 = 1 and r = rfor all r, s ∈ R. Given a ring with involution, the group ring RG inher-its an involution by

∑g∈G rg · g =

∑g∈G r · g−1. If the coefficient ring R

is commutative, we usually use the trivial involution r = r. Given a ringwith involution, one can associate to it quadratic L-groups Lhn(R) for n ∈ Z.The abelian group Lh0 (R) can be identified with the Witt group of quadraticforms on finitely generated free R-modules, where every hyperbolic quadraticforms represent the zero element and the addition is given by the orthogo-nal sum of quadratic forms. The abelian group Lh2 (R) is essentially given

12 0 Introduction

by the skew-symmetric versions. One defines Lh1 (R) and Lh3 (R) in terms ofautomorphism of quadratic forms. The L-groups are four-periodic, i.e., there

is a natural isomorphism Lhn(R)∼=−→ Lhn+4(R) for n ∈ Z. If one uses finitely

generated projective R-modules instead of finitely generated free R-modules,one obtains the quadratic L-groups Lpn(RG) for n ∈ Z. If one uses finitelygenerated based free RG-modules and takes the Whitehead torsion into ac-count, then one obtains the quadratic L-groups Lsn(RG) for n ∈ Z. For every

j ∈ −∞ q j ∈ Z | j ≤ 2 there are versions L〈j〉n (RG), where 〈j〉 is called

decoration. The decorations j = 0, 1 correspond to the decorations p, h andj = 2 is related to the decoration s.

The relevance of the L-groups comes from the fact that they are the recip-ients for various surgery obstructions. The fundamental surgery problem isthe following. Consider a map f : M → X from a closed oriented manifold Mto a finite Poincare complex X. We want to know whether we can change itby a process called surgery to a map g : N → X with a closed manifold N assource and the same target such that g is a homotopy equivalence. This cananswer the question whether a finite Poincare complex X is homotopy equiv-alent to a closed manifold. Notice that a space which is homotopy equivalentto a closed oriented manifold must be a finite Poincare complex but not everyfinite Poincare complex is homotopy equivalent to a closed oriented manifold.If f comes with additional bundle data and has degree 1, we can find g if andonly if the so called surgery obstruction of f vanishes which takes values inLhn(ZG) for n = dim(X) and G = π1(X). If we want g to be a simple homo-topy equivalence, the obstruction lives in Lsn(ZG). We see that analogous to

the finiteness obstruction in K0(ZG) and the Whitehead torsion in Wh(G)the algebraic L-groups are the recipients for important obstructions whosevanishing has interesting geometric consequences. Also the question whethertwo closed manifolds are diffeomorphic or homeomorphic can be decided viasurgery theory of which the L-groups are a part.

0.3.2 The Farrell-Jones Conjecture for L∗(RG)[1/2]

If we invert 2, i.e., if we consider the localization L〈−j〉n (RG)[1/2], then there

is no difference between the various decorations and the analogues of thesequences (0.1) and (0.2) are true for L-theory (see Cappell [153]). The samereasoning as for the Baum-Connes Conjecture leads to

Conjecture 0.12. (Farrell-Jones Conjecture for L∗(RG)[1/2]). Let Gbe a group. Let R be an associative ring with unit and involution. Thenthere is for all n ∈ Z and all decorations j an isomorphism

HGn (EG; L

〈j〉R )[1/2]

∼=−→ L〈j〉n (RG)[1/2].

0.3 Motivation for the Farrell-Jones Conjecture for L-Theory 13

Here HGn (−; L

〈j〉R ) is an appropriate G-homology theory with the property

that HGn (G/H; L

〈j〉R ) ∼= HH

n (•; L〈j−∞〉R ) ∼= L〈j〉n (RH) for every subgroup

H ⊆ G and the isomorphism is induced by the G-map EG→ •.

0.3.3 The Farrell-Jones Conjecture for L∗(RG)

In general the L-groups L〈j〉n (RG) do depend on the decoration and often the

2-torsion carries sophisticated information and is hard to handle. Recall thatas a special case of the sequence (0.2) we obtain an isomorphism

Kn(C∗r (G× Z)) = Kn(C∗r (G))⊕Kn−1(C∗r (G)).

The L-theory analogues is given by the Shaneson splitting [752]

L〈j〉n (R[Z]) ∼= L〈j−1〉n−1 (R)⊕ L〈j〉n (R).

Here for the decoration j = −∞ one has to interpret j − 1 as −∞. Since S1

is a model for BZ, we get an isomorphisms

Hn(BZ; L〈j〉(R)) ∼= L〈j〉n−1(R)⊕ L〈j〉n (R).

Therefore the decoration −∞ shows the right homological behavior and isthe right candidate for the formulation of an isomorphism conjecture.

The analogues of the sequences (0.1) and (0.2) do hold not hold for

L〈j〉∗ (RG), certain correction terms, the UNil-terms come in, which are inde-

pendent of the decoration and are always 2-torsion (see Cappell [152], [153]).As in the algebraic K-theory case this leads to the following

Conjecture 0.13. (Farrell-Jones Conjecture for L∗(RG)). Let G be agroup. Let R be an associative ring with unit and involution. Then there isfor all n ∈ Z an isomorphism called assembly map

HGn (EG; L

〈−∞〉R )

∼=−→ L〈−∞〉n (RG).

Here HGn (−; L

〈−∞〉R ) is an appropriate G-homology theory with the prop-

erty that HGn (G/H; L

〈−∞〉R ) ∼= HH

n (•; L〈−∞〉R ) ∼= L〈−∞〉n (RH) for every sub-

group H ⊆ G, and the assembly map is induced by the map EG→ •. Theconjecture above makes sense for all groups and rings with involution, andno counterexamples are known at the time of writing.

After inverting 2 Conjecture 0.13 is equivalent to Conjecture 0.12.There is an L-theory version of the splitting (0.10)

HGn (EG; L

〈−∞〉R ) ∼= HG

n (EG; L〈−∞〉R )⊕HG

n (EG,EG; L〈−∞〉R ),(0.14)

14 0 Introduction

provided that there exists an integer i0 such that Ki(RV ) = 0 holds for allvirtually cyclic subgroups V ⊆ G and i ≤ i0.

0.3.4 Applications of the Farrell-Jones Conjecture for L∗(RG)

For applications in geometry the groups Lsn(ZG) are the interesting ones.The difference between the various decorations is measured by the so calledRothenberg sequences and given in terms of the Tate cohomology of Z/2 with

coefficients in Kn(ZG) for n ≤ 0 and Wh(G) with respect to the involutioncoming from the involution on the group ring ZG. Hence the decorations donot matter if Kn(ZG) for n ≤ 0 and Wh(G) vanish. This leads in view ofConjecture 0.11 to the following version of Conjecture 0.13 for torsionfreegroups

Conjecture 0.15. (Farrell-Jones Conjecture for L∗(ZG) for torsion-free groups). Let G be a torsionfree group. Then there is for n ∈ Z and alldecorations j an isomorphism

Hn(BG; L〈j〉(Z))∼=−→ L〈j〉n (RG)

and the source, target and the map itself are independent of the decorationj.

Here Hn(−; L〈j〉(Z)) is the homology theory associated to the L-theory

spectrum L〈−j〉(Z) and satisfies Hn(•; L〈j〉(Z)) ∼= πn(L〈j〉(Z)

) ∼= L〈j〉n (Z).

The L-theoretic assembly map appearing in Conjecture 0.15 has a geo-metric meaning. It appears in the so called long exact surgery sequence. LetLs(Z)〈1〉 be the 1-connected cover Ls(Z)〈1〉 of Ls(Z). There is a canonicalmap ι : Hn(BG; Ls(Z)〈1〉)→ Hn(BG; Ls(Z)). Let N be an aspherical closedoriented manifold with fundamental group G, i.e., a closed oriented manifoldhomotopy equivalent to BG. Then G is torsionfree, the source of the com-posite Hn(BG; Ls(Z)〈1〉)→ Lsn(RG) of the assembly map Conjecture 0.15 iwith ι consists of bordism classes of normal maps M → N with N as targetand the composite sends such a normal map to its surgery obstruction. Thisis analogous to the Baum-Connes setting, where the assembly map can bedescribed by assigning to an equivariant index problem its index.

The third term in the surgery sequence is given by the so called struc-ture set of N . It is the set of equivalence classes of homotopy equivalencesf0 : M0 → N with a closed topological manifold as source and N as target,where f0 : M0 → N and f1 : M1 → N are equivalent if there is a homeomor-phism g : M0 → M1 such that f1 g and f0 are homotopic. Conjecture 0.15implies that this structure set is trivial provided that the dimension of N isgreater or equal to five. Hence Conjecture 0.15 implies in dimensions ≥ 5 thefamous

0.4 Status of the Baum-Connes and the Farrell-Jones Conjecture 15

Conjecture 0.16 (Borel Conjecture). Let M and N be two asphericalclosed topological manifolds whose fundamental groups are isomorphic. Thenthey are homeomorphic and every homotopy equivalence from M to N ishomotopic to a homeomorphism.

The Borel Conjecture is a topological rigidity theorem for aspherical man-ifolds and analogous to the Mostow Rigidity Theorem which says that twohyperbolic closed Riemannian manifolds with isomorphic fundamental groupsare isometrically diffeomorphic. The Borel Conjecture is false if one replacestopological manifold by smooth manifold and homeomorphism by diffeomor-phism. The connection to the Borel Conjecture is one of the main features ofthe Farrell-Jones Conjecture. More details will be given in Subsections 7.15.2and 7.15.3

The Farrell-Jones Conjecture for L-theory 0.13 implies the Novikov Con-jecture (see Section 7.14). It also has applications to the problem whetherPoincare duality groups and torsionfree hyperbolic groups with spheres asboundary are fundamental groups of closed aspherical manifolds (see Sec-tions 7.17 and 7.18). Product decompositions of closed aspherical manifoldsare treated in Section 7.19.

0.4 Status of the Baum-Connes and the Farrell-JonesConjecture

A detailed report on the groups for which these conjectures are been provedwill be given in Chapter 14. For example, the Baum-Connes Conjecture 0.6is known for a class of groups which includes amenable groups, hyperbolicgroups, knot groups and one-relator groups, but is open for SL(n,Z) for n ≥3. The class of groups for which the Farrell-Jones Conjectures 0.9 and 0.13have been proved contains hyperbolic groups, CAT(0)-groups, fundamentalgroups of 3-manifolds, solvable groups, lattices in virtually connected Liegroups and arithmetic groups, but they are open for amenable groups ingeneral. If one allows twisted coefficients, one can prove for the Baum-ConnesConjecture and the Farrell-Jones Conjecture inheritance properties, e.g., theclass of groups for which they are true is closed under taking subgroups,finite direct products, free products, directed colimits over directed system(with not necessarily injective structure maps). This will be explained inSections 11.6 and 12.6.

The Farrell-Jones Conjecture (with coefficients) is known to be true forsome groups with unusual properties, e.g., groups with expanders, Tarskymonsters, lacunary groups, subgroups of finite products of hyperbolic groups,selfsimilar groups, see Theorem 14.1. At the time of writing we have no spe-cific candidate of a group or of a general property of groups such that theFarrell-Jones Conjecture (with twisted coefficients) or one of its consequences,

16 0 Introduction

e.g., the Novikov Conjecture and the Borel Conjecture, might be false. Sowe have no good starting point for a search for counterexamples (see Sec-tion 14.9). Finding a counterexample will probably require some new ideas,maybe from logic or random groups.

At the time of writing no counterexamples to the Baum-Connes Conjectureis known to the author. There exists a counterexample to the Baum-ConnesConjecture with coefficients, as explained in Section 14.9.

0.5 Structural Aspects

0.5.1 The Meta-Isomorphism Conjecture

The formulations of the Baum-Connes Conjecture 0.6 and of the Farrell-Jones Conjecture 0.9 and 0.13 are very similar in the homological picture. Itallows a formulation of the following Meta-Isomorphism Conjecture of whichboth conjectures are special cases and which has also other very interestingspecializations, e.g., for pseudoisotopy, A-theory, topological Hochschild andtopological cyclic homology (see Section 13.2).

Meta-Isomorphism Conjecture 0.17. Given a group G, a G-homologytheory HG∗ and a family F of subgroups of G, we say that the Meta-Isomorphism Conjecture is satisfied, if the G-map EF (G) → • inducesfor all n ∈ Z an isomorphism

AF : HGn (EF (G))→ HGn (•).

This general formulation is an excellent framework to construct trans-formations from one conjecture to the other. For instance, the cyclotomictrace relates the K-theoretic Farrell-Jones Conjecture with coefficients in Zto the Isomorphism Conjecture for topological cyclic homology, see Subsec-tion 13.13.3, and via symmetric signatures one can link the Farrell-JonesConjecture for algebraic L-theory with coefficient in Z to the Baum-ConnesConjecture, see Subsection 13.13.4. Moreover, basic computational tools andtechniques for equivariant homology theories apply both to the Baum-ConnesConjecture 0.6 and the Farrell-Jones Conjectures 0.9 and 0.13.

0.5.2 Assembly

One important idea is the assembly principle which leads to assembly mapsin a canonical and universal way by asking for the best approximation ofa homotopy invariant functor from G-spaces to spectra by an equivarianthomology theory. It is an important ingredient for the identification of the

0.6 Computational Aspects 17

various descriptions of assembly maps appearing in the Baum-Connes Con-jecture and the Farrell-Jones Conjecture. For instance, the assembly mapappearing in Baum-Connes Conjecture 0.6 can be interpreted as assigningto an appropriate equivariant elliptic complex its equivariant index, and theassembly map appearing in the L-theoretic Farrell-Jones Conjecture 0.13 isrelated to the map appearing the surgery sequence which assigns to a surgeryproblem its surgery obstruction. We have already explained above that theseidentification are the basis for some of applications of the Isomorphisms Con-jectures and we will see that there are also important for proofs. There is ahomotopy theoretic approach to the assembly map based on homotopy col-imits over the orbit category which motivates the name assembly. All thiswill be explained in Chapter 16.

This parallel treatment of the Baum-Connes Conjecture and the Farrell-Jones Conjecture and of other variants is one of the topics of this book.However, the geometric interpretations of the assembly maps in terms ofindices, surgery obstructions or forget control are quite different. Thereforethe methods of proof for the Farrell-Jones Conjecture and the Baum-ConnesConjecture use different input. Although there are some similarities in theproofs, its is not clear how to export methods of proof from one conjectureto the other.

0.6 Computational Aspects

In general the target Kn(C∗r (G)) of the assembly map appearing in the Baum-Connes Conjecture 0.6 is very hard to compute, whereas the source KG

n (EG)is much more accessible, because one can apply standard techniques fromalgebraic topology such as spectral sequences and equivariant Chern char-acters and there are often nice small geometric models for EG. For theFarrell-Jones Conjecture 0.9 and 0.13 this holds for the part HG

n (EG; KR)

or HGn (EG; L

〈−∞〉R ) respectively appearing in the splittings (0.10) and (0.14).

The other part HGn (EG,EG; KR) or HG

n (EG,EG; L〈−∞〉R ) is harder to han-

dle since it involves Nil- or UNil-terms respectively and the G-CW -complexEG is not proper and in general huge. Most of the known computations of

Kn(C∗r (G)), Kn(RG) and L〈−j〉n (RG) are based on the Baum-Connes Con-

jecture 0.6 and the Farrell-Jones Conjecture 0.9 and 0.13.Classifications of manifolds and of C∗-algebras rely on and thus motivate

explicit calculations of K- and L-groups. In this context it is often impor-tant, not only to determine the K- and L-groups abstractly, but to developdetection techniques so that one can identify or distinguish specific elementsassociated to the original classification problem or give geometric or index-theoretic interpretations to elements in the K- and L-groups.

18 0 Introduction

A general guide for computations and a list of known cases including ap-plications to classification problems will be given in Chapter 15.

0.7 Methods of Proof

Here is a brief discussion of the methods of proofs.

0.7.1 Controlled Topology Methods

For the Farrell-Jones Conjecture 0.9 and 0.13 controlled topology and con-trolled algebra is one of the main important tools. Here the basic idea is thatgeometric objects or algebraic objects come with a reference map to a met-ric space so that one can measure sizes. For instance, for an h-cobordismone wants to measure the size of handles. In algebra one considers geometricmodules which assign to each point in a metric space a finitely generatedZ-module with a basis such that the non-trivial modules are distributed ina locally finite way. A typical transition from geometry to algebra wouldbe to assign to an h-cobordism the cellular chain complex coming from ahandlebody decomposition but taking into account, where the handle sits.

One of the basic features of a homology theory is excision. It often comesfrom the fact that a representing cycle can be found with arbitrarily goodcontrol. An example is the technique of subdivision. It allows to make therepresenting cycles for simplicial or singular homology arbitrarily small con-trolled, i.e., the diameter of any simplex appearing with non-zero coefficientis very small. One may say that requiring control conditions amounts to im-plementing homological properties. In agreement with the assembly principleabove, the assembly map can be viewed as a forget control map.

With this interpretation it is clear what the main task in the proof ofsurjectivity of the assembly map is: achieve control, i.e., manipulate cycleswithout changing their homology class so that they become sufficiently con-trolled. There is a general principle that a proof of surjectivity also gives in-jectivity. Namely, proving injectivity means that one must construct a cyclewhose boundary is a given cycle, i.e., one has to solve a surjectivity problemin a relative situation,

Comment 2: We should say something about the axiomatic approach.

0.7 Methods of Proof 19

0.7.2 Coverings, Flow Spaces and Transfer

In order to get control one needs some geometric input from the groups, forinstance, to be the fundamental group of a closed negative curved Riemannianmanifold. Often flows and transfer maps are used to improve control. In acase of a negatively closed manifold the transfer is used to pass from themanifold to its sphere tangent bundle and then use the geodesic flow which isavailable on the sphere tangent bundle but not on the manifold itself. Whenone wants to deal with more general groups, one has to extend these methodstoo much more singular spaces such as CAT(0)-spaces. These techniques willbe explained in Comment 3: Add reference.

0.7.3 Analytic Methods

The main methods of proof for the Baum-Connes Conjecture 0.6 are of an-alytic nature. In particular the Dirac-Dual Dirac method is very important.The main input will be Kasparov’s equivariant KK-theory and the Kasparovproduct. They allow to define the assembly map appearing in the Baum-Connes Conjecture 0.6 and its inverse by specifying two elements, the Diracelement and the dual Dirac element, in the equivariant KK-groups and show-ing that the Kasparov products of these elements is the identity. The con-struction of these elements requires some input from the group and its geom-etry or functional analytic properties. For instance, some proofs need actionsof the group on Hilbert spaces with certain properties or an appropriate em-bedding of the group into a Hilbert space. This will be explained in moredetail in Comment 4: Add reference. .

0.7.4 Cyclic Methods

There are also homotopy theoretic approaches to a proof of the the Farrell-Jones Conjecture, at least to injectivity results about the assembly map. Theyaim at constructing detection maps from the target of the conjectures to amore familiar object. A prototype is the Dennis-trace map which allows todetect parts of the algebraic K-theory in Hochschild homology. The Dennistrace map is of linear nature. A much more advanced tool to detect thealgebraic K-theory is the cyclotomic trace which takes values in topologicalcyclic homology. All these constructions are on the level of spectra and cannotbe carried out using chain complexes as in the case of the Dennis trace mapand Hochschild homology or related theories such as cyclic homology. Thesemethods can be used to get injectivity results but not surjectivity resultabout the Farrell-Jones assembly map. In order to apply these methods, one

20 0 Introduction

does not need geometric input but homotopy theoretic input from the groupG such as certain finiteness conditions about the classifying space EG forproper G-actions. For the Baum Connes Conjecture Connes’ cyclic homologyand Chern characters are important tools for the proof of injectivity results.All these methods will be discussed in Comment 5: Add reference. .

0.8 Are the Baum-Connes Conjecture and theFarrell-Jones Conjecture True in General?

The title of this section is the central and at the time of writing unsolvedquestion. One motivation for writing this monograph is to stimulate some veryclever mathematician to work on this problem and finally find an answer. Letus speculate about the possible answer.

We are skeptical about the Baum-Connes Conjecture for two reasons: thereare counterexamples for the version with coefficients, and the left side of theBaum-Connes assembly map is functorial under arbitrary group homomor-phisms, whereas the right side is not. The Bost Conjecture which predicts anisomorphism

KGn (EG)→ Kn(l1(G))

has a much better chance to be true in general. The possible failure of theBaum-Connes Conjecture may come from the possible failure of the canonicalmap Kn(l1(G))→ Kn(Cr∗(G) to be bijective.

In spite of the Baum-Connes Conjecture, we do not see an obvious flawwith the Bost Conjecture or the Farrell-Jones Conjecture. As explained above,we have no starting point for a construction of a counterexample, and allabstract properties we know for the right side do hold for the left side ofthe assembly map and vice versa. In particular for the Bass Conjecture andfor the Novikov Conjecture the class of groups for which they are knownto be true is impressive. There are some conclusions from the Farrell-JonesConjecture which are not trivial and true for all groups. These are argumentsare in favor of a positive answer

The following arguments are in favor of a negative answer. The universeof groups is overwhelming large. We have Gromov’s saying on our neck thata statement which is true for all groups is either trivial or false. We have nophilosophical reason why the Bost Conjecture or the Farrell-Jones Conjectureshould be true.

The upshot of this discussion is that the author does not believe in theBaum-Connes Conjecture, but sees a chance for the other conjectures, inparticular for the Novikov Conjecture, to be true for all groups.

We will elaborate on this discussion in Section 14.9.

0.9 Notations and Conventions 21

0.9 Notations and Conventions

Comment 6: This section is in a very preliminary form.Here is a briefing on our main conventions and notations. Details are of

course discussed in the text.

• Ring will means associative ring with unit unless not explicitly stateddifferently;

• Module means always left unless not explicitly stated differently;• Groups means discrete group unless not explicitly stated differently;• We will always work in the category of compactly generated spaces, com-

pare [764] and [832, I.4]. In particular every space is automatically Haus-dorff.:

• For our conventions concerning spectra see Section 10.4. Spectra are de-noted with boldface letters such as E.Comment 7: Shall we add a table of spectra and associated homologytheories or shall we completely rely on the index of symbols and evendiscard the next table?

• We use the standards symbols Z, Q, R and C for the integers, the rationalnumbers, the real numbers and the complex numbers. Moreover, we denote

symbol name

Z/n finite cyclic group of order n

Sn symmetric group of permutations of the set 1, 2, . . . nAn alternating group of even permutations of the set 1, 2, . . . nD∞ infinite dihedral group

Dn dihedral group of order nSDn semidihedral(=quasidihedral) group of order n

Q2n (generalized) quaternion group of order 2n


name of texfile: ic

Chapter 1

The Projective Class Group

1.1 Introduction

This chapter is devoted to the projective class group K0(R) of a ring R.We give three equivalent definitions, namely, by the universal additive

invariant for finitely generated projective modules, by the Grothendieck con-struction applied to the abelian monoid of isomorphism classes of finitelygenerated projective modules, and by idempotent matrices. We explain somecalculations for principal ideal domains and Dedekind rings and the signifi-cance of K0(R) for the category of finitely generated projective modules.

We explain connections to geometry. We prove Swan’s Theorem whichrelates K0(C0(X)) for the ring C0(X) of continuous functions on a compactspace X to the Grothendieck group of the abelian monoid of isomorphismclasses of vector bundles over X. The relevance of K0(ZG) for topologists isillustrated by Wall’s finiteness obstruction which also leads to a geometricdescription of K0(ZG) in terms of finitely dominated spaces.

We introduce variants of the K-theoretic Farrell-Jones Conjecture. A pro-totype asserts that for a torsionfree group G and a regular ring R, e.g., R = Zor R a field, the change of rings map

K0(R)∼=−→ K0(RG)

is bijective. It implies the conjecture that for a torsionfree group G the re-duced projective class group K0(ZG) vanishes, which is for finitely presentedG equivalent to the conjecture that every finitely dominated CW -complexwith π1(X) ∼= G is homotopy equivalent to a finite CW -complex. We alsointroduce a version, where the group is not necessarily torsionfree, but R isa regular ring with Q ⊆ R or a field of prime characteristic.

We discuss applications. The Kaplansky Conjecture asserts for a torsionfreegroup G and a field F of characteristic zero that FG contains no non-trivialidempotents. It is a consequence of the Farrell-Jones Conjecture. We alsodiscuss various Bass Conjectures which are also implied by the Farrell-JonesConjecture.

Finally, we give a survey of K0(ZG) for finite groups G and of K0(C∗r (G))and K0(N (G)), where C∗r (G) is the reduced group C∗-algebra and N (G) thegroup von Neumann algebra.

23

24 1 The Projective Class Group

1.2 Definition and Basic Properties of the ProjectiveClass Group

Definition 1.1 (Projective class group K0(R)). Let R be an (associative)ring (with unit). Define its projective class group K0(R) to be the abeliangroup whose generators are isomorphism classes [P ] of finitely generated pro-jective R-modules P and whose relations are [P0] + [P2] = [P1] for any exactsequence 0→ P0 → P1 → P2 → 0 of finitely generated projective R-modules.

Define G0(R) analogously but replacing finitely generated projective byfinitely generated.

Given a ring homomorphism f : R→ S, we can assign to an R-module Man S-module f∗M by S⊗RM , where we consider S as a right R-module usingf . We say that f∗M is obtained by induction with f from M . If M is finitelygenerated or free or projective, the same is true for f∗M . This construction isnatural, compatible with direct sums and sends an exact sequence 0→ P0 →P1 → P2 → 0 of finitely generated projective R-modules to an exact sequence0 → f∗P0 → f∗P1 → f∗P2 → 0 of finitely generated projective S-modules.Hence we get a homomorphism of abelian groups

f∗ : K0(R)→ K0(S), [P ] 7→ [f∗P ],(1.2)

which is also called change of rings homomorphism. Thus K0 becomes acovariant functor from the category of rings to the category of abelian groups.

Remark 1.3 (The universal property of the projective class group).One should view K0(R) together with the assignment sending a finitely gener-ated projective R-module P to its class [P ] in K0(R) as the universal additiveinvariant or the universal dimension function for finitely generated projec-tive R-modules. Namely, suppose that we are given an abelian group and anassignment d which associates to a finitely generated projective R-modulean element d(P ) ∈ A such that d(P0) + d(P2) = d(P1) holds for any exactsequence 0→ P0 → P1 → P2 → 0 of finitely generated projective R-modules.Then there is precisely one homomorphism of abelian groups φ : K0(R)→ Asuch that φ([P ]) = d(P ) holds for every finitely generated projective R-module P . The analogous statement holds for G0(R) if we consider finitelygenerated R-modules instead of finitely generated projective R-modules.

A ring is an integral domain if every zero-divisor is trivial, i.e., if r, s ∈ Rsatisfy rs = 0, then r = 0 or s = 0. A principal ideal domain is a commutativeintegral domain for which every ideal is a principal ideal, i.e., of the form(r) = r′r | r′ ∈ R for some r ∈ R.

Example 1.4 (K0(R) and G0(R) of a principal ideal domain). Let Rbe a principal ideal domain. Then we get isomorphisms of abelian groups

1.2 Definition and Basic Properties of the Projective Class Group 25

Z∼=−→ K0(R), n 7→ [Rn];

K0(R)∼=−→ G0(R), [P ] 7→ [P ].

This follows from the structure theorem of finitely generated R-modules overprincipal ideal domains. It implies for any finitely generated R-module Mthat it can be written as a direct sum Rn ⊕ T for some torsion R-moduleT for which there exists an exact sequence of R-modules of the shape 0 →Rs → Rs → T → 0. Moreover, M is projective if and only if T is trivial andRm = Rn ⇔ m = n.

Definition 1.5 (Reduced projective class group K0(R)). Define the re-

duced projective class group K0(R) to be the quotient of K0(R) by the abeliansubgroup [Rm]− [Rn] | n,m ∈ Z,m, n ≥ 0 which is the same as the abeliansubgroup generated by the class [R].

We conclude from Example 1.4 that the reduced projective class groupK0(R) is isomorphic to the cokernel of the homomorphism

f∗ : K0(Z)→ K0(R)

where f is the unique ring homomorphism Z→ R, n 7→ n · 1R.

Remark 1.6 (The projective class group as a Grothendieck group).Let Proj(R) be the abelian semigroup of isomorphisms classes of finitelygenerated projective R-modules with the addition coming from the directsum. Let K ′0(R) be the associated abelian group given by the Grothendieckconstruction applied to Proj(R). There is a natural homomorphism

φ : K ′0(R)∼=−→ K0(R)

sending the class of a finitely generated projective R-module P in K ′0(R) toits class in K0(R). This is a well-defined isomorphism of abelian groups.

The analogous definition of G′0(R) and the construction of a homomor-phism G′0(R) → G0(R) makes sense but the latter map is not bijective ingeneral. It works for K0(R) because every exact sequence of projective R-modules 0 → P0 → P1 → P2 → 0 splits and thus yields an isomorphismP1∼= P0 ⊕ P2. In general K-theory deals with exact sequences, not with di-

rect sums. Therefore Definition 1.1 of K0(R) reflects better the underlyingidea of K-theory than its definition in terms of the Grothendieck construc-tion.

Exercise 1.7. Prove that the homomorphism φ : K ′0(R)→ K0(R) appearingin Remark 1.6 is a well-defined isomorphism of abelian groups.

Remark 1.8 (What does the reduced projective class group mea-sure?). Let P be a finitely generated projective R-module. Then we conclude


from Remark 1.6 that its class [P ] ∈ K0(R) is trivial if and only if P is stablyfinitely generated free, i.e., P ⊕ Rr ∼= Rs for appropriate integers r, s ≥ 0.So the reduced projective class group K0(R) measures the deviation of afinitely generated projective R-module to be stably finitely generated free.Notice that stably finitely generated free does in general not imply finitelygenerated free as Examples 1.9 and 1.29 will show.

Example 1.9 (Dunwoody’s example). An interesting ZG-module P whichis stably finitely generated free, but not finitely generated free is constructedby Dunwoody [256] for G the torsionfree one-relator group 〈a, b | a2 = b3〉which is the fundamental group of the trefoil knot. Notice that K0(ZG) isknown to be trivial, in other words, every finitely generated projective RG-module is stably finitely generated free. It is also worth while mentioningthat ZG contains no idempotent besides 0 and 1. Hence any direct summandin ZG is free.

More examples of this kind are given in Berridge-Dunwoody [108].

One basic feature of algebraic K-theory is Morita equivalence.

Theorem 1.10 (Morita equivalence for K0(R)). For every ring R andinteger n ≥ 1 there is a natural isomorphism

µ : K0(R)∼=−→ K0(Mn(R)).

Proof. We can consider Rn as Mn(R)-R-bimodule, denoted by Mn(R)RnR.

Then µ sends [P ] to [Mn(R)RnR ⊗R P ]. We can also consider Rn as an R-

Mn(R)-bimodule denoted by RRnMn(R). Define ν : K0(Mn(R)) → K0(R)

by sending [Q] to [RRnMn(R) ⊗Mn(R) Q]. Then µ and ν are inverse to one

another. ut

Exercise 1.11. Check that µ and ν are inverse to one another.

We omit the easy proof of

Lemma 1.12. Let R0 and R1 be rings. Denote by pri : R0 × R1 → Ri fori = 0, 1 the projection. Then we obtain an isomorphism

(pr0)∗ × (pr1)∗ : K0(R0 ×R1)∼=−→ K0(R0)×K0(R1).

Example 1.13 (Rings with non-trivial K0(R)). We conclude from Ex-ample 1.4 and Lemma 1.12 that for a principal ideal domain R we have

K0(R×R) ∼= Z⊕ Z;

K0(R×R) ∼= Z.

The R × R-module R × 0 is finitely generated projective but not stably

finitely generated free. It is a generator of the infinite cyclic group K0(R×R).

1.2 Definition and Basic Properties of the Projective Class Group 27

Notation 1.14 (M(R), GL(R) and Idem(R)). Let Mm,n(R) be the set of(m,n)-matrices over R. For A ∈ Mm,n(R) let rA : Rm → Rn, x → xA bethe R-homomorphism of (left) R-modules given by right multiplication withA. Let Mn(R) be the ring of (n, n)-matrices over R. Denote by GLn(R) thegroup of invertible (n, n)-matrices over R. Let Idemn(R) be the subset ofMn(R) of idempotent matrices A, i.e., (n, n)-matrices satisfying A2 = A.

There are embeddings it,n : Mn(R) → Mn+1(R), A 7→(A 00 t

)for t = 0, 1

and n ≥ 1. The embedding i1,n induces an embedding GLn(R)→ GLn+1(R)of groups. Let GL(R) be the union of the GLn(R)-s which is a group. De-note by M(R) the union of the Mn(R)-s with respect to the embeddings i0.This is a ring without unit. Let Idem(R) be the set of idempotent elementsin M(R). This is the same as the union of the Idemn(R)-s with respectto the embeddings Idemn(R) → Idemn+1(R) coming from the embeddingsi0,n : Mn(R)→Mn+1(R).

Remark 1.15 (The projective class groups in terms of idempotentmatrices). The projective class groups K0(R) can also be defined in terms ofidempotent matrices. Namely, the conjugation action of GLn(R) on Mn(R)induces an action ofGL(R) onM(R) which leaves Idem(R) fixed. One obtainsa bijection of sets

φ : GL(R)\ Idem(R)→ Proj(R), [A] 7→ im (rA : Rn → Rn) .

This becomes a bijection of abelian semigroups if we equip the source with the

addition coming from (A,B) 7→(A 00 B

)and the target with the one coming

from the direct sum. So we can identify K0(R) with the Grothendieck groupassociated to the abelian semigroup GL(R)\ Idem(R) by Remark 1.6.

Exercise 1.16. Show that the map φ appearing in Remark 1.15 is a well-defined isomorphism of abelian semigroups.

Example 1.17 (A ring R with trivial K0(R)). Let F be a field and letV be an F -vector space with an infinite countable basis. Consider the ringR = endF (V ). Next we prove that K0(R) is trivial.

By Remark 1.15 it suffices to show for every integer n ≥ 0 and two idem-potent matrices A,B ∈ Idemn(R) that the matrices A⊕ 0⊕ 1 and B ⊕ 1⊕ 0in Mn+2(R) are conjugated by an element in GLn+2(R). This follows fromthe observation that the both the kernel and the image of the F -linear en-domorphisms rA⊕0⊕1 and rB⊕0⊕1 of V n+2 have infinite countable dimensionand hence are isomorphic as F -vector spaces.

Lemma 1.18. Let G be a group. Let R be a commutative integral domainwith quotient field F . Then we obtain an isomorphism


K0(RG)∼=−→ K0(RG)⊕ Z, [P ] 7→ ([P ],dimF (F ⊗RG P )) ,

where F is considered as an RG-module with respect to the trivial G-actionand the inclusion of rings j : R→ F .

Proof. Since F⊗RGP is a finite dimensional F -vector space for finitely gener-ated P and F⊗RG (P⊕Q) ∼=G (F ⊗RG P )⊕(F ⊗RG Q), this is a well-definedhomomorphism. Bijectivity follows from dimF (F ⊗RG RGn) = n. ut

1.3 The Projective Class Group of a Dedekind domain

Let R be a commutative integral domain with quotient field F . A non-zeroR-submodule I ⊂ F is called a fractional ideal if for some r ∈ R we haverI ⊆ R. A fractional ideal I is called principal if I is of the form

rab | r ∈ R

for some a, b ∈ R with a, b 6= 0.

Definition 1.19 (Dedekind domain). A commutative integral domain Ris called a Dedekind ring if for any fractional ideal I there exists anotherfractional ideal J with IJ = R.

Notice that in Definition 1.19 the fractional ideal J must be given byx ∈ F | x · I ⊆ R.

The fractional ideals in a Dedekind ring form by definition a group undermultiplication of ideals with R as unit. The principal fractional ideals forma subgroup. The class group C(R) is the quotient of these abelian groups.

A proof of the next theorem can be found for instance in [592, Corollary 11on page 14] and [704, Theorem 1.4.12 on page 20].

Theorem 1.20 (The reduced projective class group and the classgroup of Dedekind domains). Let R be a Dedekind domain. Then everyfractional ideal is a finitely generated projective R-module and we obtain anisomorphism of abelian groups

Z⊕ C(R)∼=−→ K0(R), (n, [I]) 7→ n · [R] + [I]− [R].

In particular we get an isomorphism

C(R)∼=−→ K0(R), [I] 7→ [I].

A ring is called hereditary, if every ideal is projective, or, equivalently, ifevery submodule of a projectiveR-module is projective (see [165, Theorem 5.4in Chapter I.5 on page 14]).

Theorem 1.21 (Characterization of Dedekind domains). The follow-ing assertions are equivalent for a commutative integral domain with quotientfield F :

1.3 The Projective Class Group of a Dedekind domain 29

(i) R is a Dedekind domain;(ii) For every pair of ideals I ⊆ J of R there exists an ideal K ⊆ R with

I = JK;(iii) R is hereditary;(iv) Every finitely generated torsionfree R-module is projective;(v) R is Noetherian and integrally closed in its quotient field F and every

non-zero prime ideal is maximal.

Proof. This follows from [214, Proposition 4.3 on page 76 and Proposition 4.6on page 77] and the fact that a finitely generated torsionfree module over anintegral domain R can be embedded into Rn for some integer n ≥ 0. Seealso [53, Chapter 13]. ut

Remark 1.22 (The class group in terms of ideals of R). If one callstwo ideals I and J in R equivalent if there exists non-zero elements r and sin R with rI = sJ , then C(R) is the same as the equivalence classes of idealsunder multiplication of ideals and the class given by the principal ideals asunit. Two ideals I and J of R define the same element in C(R) if and only ifthey are isomorphic as R-modules, see [704, Proposition 1.4.4 on page 17].

Recall that an algebraic number field is a finite algebraic extension of Qand the ring of integers in F is the integral closure of Z in F .

Theorem 1.23 (The class group of a ring of integers is finite). LetR be the ring of integers in an algebraic number field. Then R is a Dedekinddomain and its class group C(R) and hence its reduced projective class group

K0(R) are finite.

Proof. See [704, Theorem 1.4.18 on page 22 and Theorem 1.4.19 on page 23].ut

Exercise 1.24. Let d be a squarefree integer. Let R be the ring of integers

in Q[√d]. Show that R is Z[

√d] if d = 2, 3 mod 4 and is Z

[1+√d

2

]if d = 1

mod 4.

Remark 1.25 (Class group of Z[exp(2πi/p)]). Let p be a prime num-ber. The ring of integers in the algebraic number field Q[exp(2πi/p)] isZ[exp(2πi/p)]. Its class group C(Z[exp(2πi/p)]) is finite by Theorem 1.23.But its structure as a finite abelian group is only known for finitely manysmall primes, see [592, Remark 3.4 on page 30] or [818, Tables §3 on page352ff].

Example 1.26 (K0(Z[√−5])). The reduced projective class group K0(Z[

√−5])

of the Dedekind domain Z[√−5] is cyclic of order two. A generator is given

by the maximal ideal (3, 2 +√−5) in Z[

√5]. (For more details see [704, Ex-

ercise 1.4.20 on page 25]).


1.4 Swan’s Theorem

Let F be the field R or C. Let X be a compact space. Denote by C(X,F )or briefly by C(X) the ring of continuous functions from X to F . Let ξ andη be (finite dimensional locally trivial) F -vector bundles over X. Denote byC(ξ) the F -vector space of continuous sections of ξ. This becomes a C(X)-module by the pointwise multiplication. If F denotes the trivial 1-dimensionalvector bundle X × F → X, then C(F ) and C(X) are isomorphic as C(X)-modules. If ξ and η are isomorphic as F -vector bundles, then C(ξ) and C(η)are isomorphic as C(X)-modules. There is an obvious isomorphism of C(X)-modules

C(ξ)⊕ C(η)∼=−→ C(ξ ⊕ η).(1.27)

Since X is compact, every F -vector bundle has a finite bundle atlas andadmits a Riemannian metric. This implies the existence of a F -vector bundleξ′ such that ξ⊕ξ′ is isomorphic as F -vector bundle to a trivial F -vector bundleFn. Hence C(ξ) is a finitely generated projective C(X)-module. Denote byhom(ξ, η) the C(X)-module of morphisms of F -vector bundles from ξ toη, i.e., of continuous maps between the total spaces which commutes withthe bundle projections to X and induce linear (not necessarily injective orbijective) maps between the fibers over x for all x ∈ X. This becomes aC(X)-module by the pointwise multiplication. Such a morphism f : ξ → ηinduces a C(X)-homomorphism C(f) : C(ξ) → C(η) by composition. Thenext result is due to Swan [776].

Theorem 1.28 (Swan’s Theorem). Let X be a compact space and F =R,C. Then:

(i) Let ξ and η be F -vector bundles. Then we obtain an isomorphism of C(X)-modules

Γ (ξ, η) : hom(ξ, η) −→ homC(X)(C(ξ), C(η)), f 7→ C(f);

(ii) We have ξ ∼= η ⇔ C(ξ) ∼=C(X) C(η);(iii) If P is a finitely generated projective C(X)-module, then there exists an

F -vector bundle ξ satisfying C(ξ) ∼=C(X) P .

Proof. (i) Obviously Γ (ξ⊕ ξ′, η) can be identified with Γ (ξ, η)⊕Γ (ξ′, η) andΓ (ξ, η ⊕ η′) can be identified with Γ (ξ, η) ⊕ Γ (ξ, η′′) under the identifica-tion (1.27). Since a direct sum of two maps is a bijection if and only if each ofthe maps is a bijection and for every ξ there is ξ′ such that ξ⊕ ξ′ is trivial, itsuffices to treat the case, where ξ = Fm and η = Fn for appropriate integersm,n ≥ 0. There is an obvious commutative diagram

1.4 Swan’s Theorem 31

hom(Fm, Fn)Γ (Fm,Fn) //

∼=

homC(X)(C(Fm), C(Fn))

∼=

Mm,n(hom(F , F ))Mm,n(Γ (F,F ))

// Mm,n(C(F ))

Hence it suffices to treat the claim for m = n = 1 which is obvious.

(ii) This follows from assertion (i).

(iii) Given a finitely generated projective C(X)-module P , choose a C(X)-map p : C(X)n → C(X)n satisfying p2 = p and im(p) ∼=C(X) P . Because ofassertion (ii) we can choose a morphism of F -vector bundles q : Fn → Fn

with Γ (Fn, Fn)(q) = p. We conclude q2 = q from p2 = p and the injectivityof Γ (Fn, Fn). Elementary bundle theory shows that the image of q and theimage of 1−q are F -subvector bundles in Fn satisfying im(q)⊕im(1−q) = Fn.One easily checks C(im(q)) ∼=C(X) P . ut

One may summarize Theorem 1.28 by saying that we obtain an equivalenceof C(X)-additive categories from the category of F -vector bundles over X tothe category of finitely generated projective C(X)-modules by sending ξ toC(ξ).

Example 1.29 (C(TSn)). Consider the n-dimensional sphere Sn. Let TSn

be its tangent bundle. Then C(TSn) is a finitely generated projective C(Sn)-module. It is free if and only if TSn is trivial. This is equivalent to thecondition that n = 1, 3, 7 (see [121]). On the other hand C(TSn) is alwaysstably finitely generated free as a C(Sn)-module since TSn is stably finitelygenerated free as an F -vector bundle because the direct sum of TSn andthe normal bundle ν(Sn,Rn+1) of the standard embedding Sn ⊆ Rn+1 isTRn+1|Sn and both F -vector bundles ν(Sn,Rn+1) and TRn+1|Sn are trivial.

Exercise 1.30. Consider an integer n ≥ 1. Show that there exists a C(Sn)-module M with C(TSn) ∼=C(Sn) C(Sn) ⊕ M if and only if Sn admits anowhere vanishing vector field. (This is equivalent to requiring that χ(Sn) =0, or, equivalently, that n is odd.)

Remark 1.31 (Topological K-theory in dimension 0). Let X be a com-pact space. Let VectF (X) be the abelian semigroup of isomorphism classes ofF -vector bundles over X, where the addition comes from the Whitney sum.Let K0(X) be the abelian group obtained from the Grothendieck construc-tion to it. It is called the 0-th topological K-group of X. If f : X → Y is amap of compact spaces, the pullback construction yields a homomorphismK0(f) : K0(Y ) → K0(X). Thus we obtain a contravariant functor K0 fromthe category of compact spaces to the category of abelian groups. Since thepullback of a vector bundle with two homotopic maps yields isomorphic vec-tor bundles, K0(f) depends only on the homotopy class of f . Actually there is


a sequence of such homotopy invariants covariant functor Kn for n ∈ Z whichconstitutes a generalized cohomology theory K∗ called topological K-theory.It is 2-periodic if F = C, i.e., there are natural so called Bott isomorphism

Kn(X)∼=−→ Kn+2(X) for n ∈ Z. If F = R, it is 8-periodic. We will give further

explanations and generalization of topological K-theory later in Section 8.2Swan’s Theorem 1.28 yields an identification

K0(X) ∼= K0(C(X)) [ξ] 7→ [C0(ξ)].(1.32)

Exercise 1.33. Let f : X → Y be a map of compact spaces. Composi-tion with f yields a ring homomorphism C(f) : C(Y ) → C(X). Show thatunder the identification (1.32) the maps K0(f) : K0(Y ) → K0(X) andC(f)∗ : K0(C(Y ))→ K0(C(X)) coincide.

Exercise 1.34. Compute K0(C(Dn)) for the n-dimensional disk Dn for n ≥0.

1.5 Wall’s Finiteness Obstruction

We now discuss the geometric relevance of K0(ZG).Let X be a CW -complex. It is called finite if it consists of finitely many

cells. This is equivalent to the condition that X is compact. We call Xfinitely dominated if there exists a finite domination (Y, i, r), i.e., a finiteCW -complex Y together with maps i : X → Y and r : Y → X such thatr i is homotopic to the identity on X. If X is finitely dominated, its set ofpath components π0(X) is finite and the fundamental group π1(C) of eachcomponent C of X is finitely presented.

While studying existence problems for compact manifolds with prescribedproperties (like for example the existence of certain group actions), it happensoccasionally that it is relatively easy to construct a finitely dominated CW -complex within a given homotopy type, whereas it is not at all clear whetherone can also find a homotopy equivalent finite CW -complex. If the goal is toconstruct a compact manifold, this is a necessary step in the construction.Wall’s finiteness obstruction which we will explain below decides the question.

An example of such a geometric problem is the spherical space form prob-lem, i.e., the classification of closed manifolds M whose universal coveringsare diffeomorphic or homeomorphic to the standard sphere. Such examplesarise as unit sphere in unitary representations of finite groups but there arealso examples which do not occur in this way. This problem initiated not onlythe theory of the finiteness obstruction but also surgery theory for closed man-ifolds with non-trivial fundamental group. We refer to the survey articles [226]and [565] for more information about the spherical space form problem.

1.5 Wall’s Finiteness Obstruction 33

The finiteness obstruction also appears in the Ph.D.-thesis [754] of Sieben-mann who dealt the problem whether a given smooth or topological manifoldcan be realized as the interior of a compact manifold with boundary.

At least we give a brief description of its definition illustrating that it is akind of Euler characteristic but now counting elements in the projective classgroup instead of counting ranks of finitely generated free modules.

Definition 1.35 (Types of chain complexes). We call an R-chain com-plex finitely generated, free, or projective respectively, if each chain module isfinitely generated, free or projective respectively. It is called positive if Cn = 0for n ≤ −1. It is called finite dimensional, if there exists a natural numberN such that Cn = 0 for |n| ≤ N . It is called finite if it is finite dimensionaland finitely generated.

In the sequel we ignore base point questions. This is not a real problemsince an inner automorphism of a group G induces the identity on K0(RG).

Given a finitely dominated connected CW -complex X with fundamentalgroup π, we consider its universal covering X and the associated cellularZπ-chain complex C∗(X). Given a finite domination (Y, i, r), we regard theπ-covering Y over Y associated to the epimorphism r∗ : π1(Y )→ π1(X). Thepullback construction yields a π-covering i∗Y over X. Then F∗ = C∗(i

∗Y )is a finite free Zπ-chain complex. The maps i and r yield Zπ-chain mapsr∗ : F∗ → C∗(X) and i∗ : C∗(X)→ F∗ such that r∗i∗ is Zπ-chain homotopic

to the identity on C∗(X). Thus (F∗, i∗, r∗) is the chain complex version of afinite domination. This implies that there exists a finite projective Zπ-chaincomplex P∗ which is Zπ-chain homotopy equivalent to C∗(X). Now definethe unreduced finiteness obstruction

o(X) :=∑n∈Z

(−1)n · [Pn] ∈ K0(Zπ).(1.36)

This is indeed independent of the choice of P∗. Define the finiteness ob-struction o(X) to be the image of o(X) under the canonical projection

K0(Zπ) → K0(Zπ). Obviously o(X) = 0 if X is homotopy equivalent to

a finite CW -complex Z since in this case we can take P∗ = C∗(Z) and C∗(Z)is a finite free Zπ-chain complex. The next result is due to Wall (see [812]and [813]).

Theorem 1.37 (Properties of the Finiteness Obstruction). Let Xbe a finitely dominated connected CW -complex with fundamental group π =π1(X).

(i) The space X is homotopy equivalent to a finite CW -complex if and only

if o(X) = 0 ∈ K0(Zπ1(X));

(ii) Every element in K0(ZG) can be realized as the finiteness obstruction o(X)of a finitely dominated connected CW -complex X with G = π1(X), pro-vided that G is finitely presented.


Theorem 1.37 illustrates why it is important to study the algebraic objectK0(Zπ) when one is dealing with geometric or topological questions. The

favorite case is of course, where K0(Zπ) vanishes because then the finite-ness obstruction is obviously zero and one does not have to go to a specificcomputation.

Exercise 1.38. Let X be a finitely dominated connected CW -complex withfundamental group π. Define a homomorphism of abelian groups

ψ : K0(Zπ)→ Z, [P ] 7→ dimQ(Q⊗Zπ P ).

Show that ψ sends o(X) to the Euler characteristic χ(X).

Exercise 1.39. Let P∗ be a finite projective R-chain complex. Define itsunreduced finiteness obstruction by o(P∗) :=

∑n∈Z(−1)n · [Pn] ∈ K0(R).

Define its finiteness obstruction o(P∗) ∈ K0(R) to be the image of o(P∗)

under the canonical projection K0(R)→ K0(R). Prove

(i) o(P∗) depends only on the chain homotopy type of P∗;(ii) P∗ is R-chain homotopy equivalent to a finite free R-chain complex if and

only if o(P∗) = 0;(iii) If 0 → P∗ → P ′∗ → P ′′∗ → 0 is an exact sequence of finite projective

R-chain complexes, then

o(P∗)− o(P ′∗) + o(P ′′∗ ) = 0.

For more information about the finiteness obstruction we refer for instanceto [308, 310, 520, 543, 604, 607, 619, 684, 798, 812, 813].

1.6 Geometric Interpretation of Projective Class Groupand Finiteness Obstruction

Next we give a geometric construction of K0(Zπ) which is in the spirit of thewell-known interpretation of the Whitehead group in terms of deformationretractions which we will present later in Section 2.4. The material of thissection is taken from [520], where more information and details of the proofscan be found.

Given a space Y , we want to define an abelian group Wa(Y ). The underly-ing set is the set of equivalence classes of an equivalence relation ∼ defined onthe set of maps f : X → Y with finitely dominated CW -complexes as sourceand the given space Y as target. We call f0 : X0 → Y and f4 : X4 → Yequivalent if there exists a commutative diagram

1.6 Geometric Interpretation of Projective Class Group and Finiteness Obstruction 35

X0i0 //

f0

""

X1j1 //

f1

X2

f2

X3j3oo

f3

X4i4oo

f4

||Y

such that j1 and j3 are homotopy equivalences and i0 and i4 are inclusionsof CW -complexes such that the larger one is obtained from the smaller oneby attaching finitely many cells. Obviously this relation is symmetric andreflexive. It needs some work to show transitivity and hence that it is anequivalence relation. The addition in Wa(Y ) is given by the disjoint sum,i.e., define the sum of the class of f0 : X0 → Y and f1 : X1 → Y to be theclass of f0

∐f1 : X0

∐X1 → Y . It is easy to check that this is compatible

with the equivalence relation. The neutral element is represented by ∅ → Y .The inverse of the class [f ] of f : X → Y is constructed as follows. Choosea finite domination (Z, i, r) of X. Construct a map F : cyl(i)→ X from themapping cylinder of i to Y such that F |X = idX and F |Z = r. Then aninverse of [f ] is given by the class [f ′] of the composite

f ′ : cyl(i) ∪X cyl(i)F∪idX

F−−−−−→ X

f−→ Y.

This finishes the definition of the abelian group Wa(Y ). A map f : Y0 → Y1

induces a homomorphism of abelian groups Wa(f) : Wa(Y0) → Wa(Y1) bycomposition. Thus Wa defines a functor from the category of spaces to thecategory of abelian groups.

Exercise 1.40. Show that [f ] + [f ′] = 0 holds for the composite f ′ above.

Given a finitely dominated CW -complex X, define its geometric finitenessobstruction ogeo(X) ∈Wa(X) by the class of idX .

Theorem 1.41 (The geometric finiteness obstruction). Let X be afinitely dominated CW -complex. Then X is homotopy equivalent to a finiteCW -complex if and only if ogeo(X) = 0 in Wa(X).

Proof. Obviously ogeo(X) = 0 if X is homotopy equivalent to a finite CW -complex. Suppose ogeo(X) = 0. Hence there are a CW -complex Y , a mapr : Y → X and a homotopy equivalence h : Y → Z to a finite CW -complex Zsuch that Y is obtained from X by attaching finitely many cells and ri = idXholds for the inclusion i : X → Y . The mapping cylinder cyl(r) is built fromthe mapping cylinder cyl(i) by attaching a finite number of cells. Choose ahomotopy equivalence g : cyl(i)→ Z. Consider the push-out


cyl(i)i //

g

cyl(r)

g′

Z

i′// Z ′

where i is the inclusion. Since g is a homotopy equivalence, the same is truefor g′. Hence X is homotopy equivalent to the finite CW -complex Z ′. ut

Theorem 1.42 (Identifying the finiteness obstruction with its geo-metric counterpart). Let Y be a space. Then there is a natural isomor-phism of abelian groups

Φ : Wa(Y )∼=−→

⊕C∈π0(Y )

K0(Zπ1(C)).

Proof. We only explain the definition of Φ. Consider an element [f ] ∈Wa(Y )represented by a map f : X → Y from a finitely dominated CW -complexX to Y . Given a path component C of X, let Cf be the path componentof Y containing f(C). The map f induces a map f |C : C → Cf and hence

a map (f |C)∗ : K0(Zπ1(C)) → K0(Zπ1(Cf )). Since X is finitely dominated,every path component C of X is finitely dominated and we can consider itsfiniteness obstruction o(C) ∈ K0(Zπ1(C)). Let φ([f ])C be the image of o(C)under the composite

K0(Zπ1(C))(f |C)∗−−−−→ K0(Zπ1(Cf ))→

⊕C∈π0(Y )

K0(Zπ1(C)).

Since π0(X) is finite, we can define

φ([f ]) :=∑

C∈π0(X)

φ([f ])C .

We omit the proof that this is compatible with the equivalence relation ap-pearing in the definition of Wa(Y ), that φ is a homomorphism of abeliangroups and that Theorem 1.37 implies that Φ is bijective. ut

1.7 Universal Functorial Additive Invariants

In this section we describe the pair (K0(Zπ1(X)), o(X)) by an abstract prop-erty.

Definition 1.43 (Functorial additive invariant for finitely dominatedCW -complexes). A functorial additive invariant for finitely dominated

1.7 Universal Functorial Additive Invariants 37

CW -complexes consists of a covariant functor A from the category of finitelydominated CW -complexes to the category of abelian groups together withan assignment a which associates to every finitely dominated CW -complexX an element a(X) ∈ A(X) such that the following axioms are satisfied:

• Homotopy invariance of AIf f, g : X → Y are homotopic maps between finitely dominated CW -complexes, then A(f) = A(g);

• Homotopy invariance of a(X)If f : X → Y is a homotopy equivalence of finitely dominated CW -complexes, then A(f)(a(X)) = a(Y );

• AdditivityLet

X0i1 //

i2

j0 !!

X1

j1

X2

j2// X

be a cellular pushout, i.e., the diagram is a pushout, the map i1 is an in-clusion of CW -complexes, the map i2 is cellular and X carries the inducedCW -structure. Suppose that X0, X1, X2 are finitely dominated.Then X is finitely dominated and

a(X) = A(j1)(a(X1)) +A(j2)(a(X2))−A(j0)(a(X0));

• Normalizationa(∅) = 0.

Example 1.44 (Componentwise Euler characteristic). Let A be thecovariant functor sending a finitely dominated CW -complexX toH0(X;Z) =⊕

C∈π0(X) Z. Let a(X) ∈ A(X) be the componentwise Euler characteristic,

i.e., the collection of integers χ(C) | C ∈ π0(X). Then (A, a) is a functorialadditive invariant for finitely dominated CW -complexes.

Definition 1.45 (Universal functorial additive invariant for finitelydominated CW -complexes). A universal functorial additive invariant forfinitely dominated CW -complexes (U, u) is a functorial additive invariantwith the property that for any functorial additive invariant (A, a) thereis precisely one natural transformation T : U → A with the property thatT (X)(u(X)) = a(X) holds for every finitely dominated CW -complex X.

Exercise 1.46. Show that the functorial additive invariant defined in Exam-ple 1.44 is the universal one if we restrict to finite CW -complexes.

Obviously the universal additive functorial invariant is unique (up tounique natural equivalence) if it exists. It is also easy to construct it. How-


ever, it turns out that there exists a concrete model, namely, the followingtheorem is proved in [520, Theorem 4.1].

Theorem 1.47 (The finiteness obstruction is the universal functorialadditive invariant). The covariant functor X 7→

⊕C∈π0(X)K0(Zπ1(C))

together with the componentwise finiteness obstruction o(C) | C ∈ π0(X)is the universal functorial additive invariant for finitely dominated CW -complexes.

Exercise 1.48. (i) Construct for finitely dominated CW -complexes X andY a natural pairing

P (X,Y ) : U(X)⊗Z U(Y )→ U(X × Y )

sending u(X)⊗u(Y ) to u(X ×Y ), where (U, u) is the universal functorialadditive invariant for finitely dominated CW -complexes;

(ii) LetX be a finitely dominated CW -complex. Let Y be a finite CW -complexsuch that χ(C) = 0 for every component C of Y . Show that X × Y ishomotopy equivalent to a finite CW -complex.

1.8 Variants of the Farrell-Jones Conjecture for K0(RG)

In this section we state variants of the Farrell-Jones Conjecture for K0(RG).The Farrell-Jones Conjecture itself will give a complete answer for arbitrarygroups and rings but to formulate the full version some additional effort willbe needed. If one assumes that R is regular and G torsionfree or that R isregular and Q ⊆ R, then the conjecture reduces to easy to formulate state-ments which we will present next. Moreover, these special cases are alreadyvery interesting.

Definition 1.49 (Projective resolution). Let M be an R-module. Aprojective resolution (P∗, φ∗) of M is a positive projective R-chain com-plex P∗ with Hn(P∗) = 0 for n ≥ 1 together with an R-isomorphism

φ : H0(P∗)∼=−→M . It is called finite, finitely generated, free, finite dimensional,

or d-dimensional if the R-chain complex P∗ has this property.

A ring R is Noetherian if any submodule of a finitely generated R-module isagain finitely generated. A ring R is called regular if it is Noetherian and anyfinitely generated R-module has a finite dimensional projective resolution.Any principal ideal domain such as Z or a field is regular.

Conjecture 1.50 (Farrell-Jones Conjecture for K0(R) for torsionfreeG and regular R). Let G be a torsionfree group and let R be a regularring. Then the map induced by the inclusion of the trivial group into G

1.8 Variants of the Farrell-Jones Conjecture for K0(RG) 39

K0(R)∼=−→ K0(RG)

is bijective.In particular we get for any principal ideal domain R and torsionfree G

K0(RG) = 0.

Remark 1.51 (Relevance of Conjecture 1.50). In view of Remark 1.8Conjecture 1.50 is equivalent to the statement that for a torsionfree groupG and a regular ring R every finitely generated projective RG-module isstably finitely generated free. This is the algebraic relevance of this conjec-ture. Its geometric meaning comes from the following conclusion of Theo-rem 1.37. Namely, if R = Z and G is a finitely presented torsionfree group,it is equivalent to the statement that every finitely dominated CW -complexwith π1(X) ∼= G is homotopy equivalent to a finite CW -complex.

Definition 1.52 (Family of subgroups). A family F of subgroups of agroup G is a set of subgroups which is closed under conjugation with elementsof G and under passing to closed subgroups.

Our main examples of families are listed below

Notation 1.53.notation subgroupsT R trivial group

FCY finite cyclic subgroups

FIN finite subgroups

CYC cyclic subgroups

VCY virtually cyclic subgroups

ALL all subgroups

Definition 1.54 (Orbit category). The orbit category Or(G) has as ob-jects homogeneous spaces G/H and as morphisms G-maps. Given a family Fof subgroups of G, let the F-restricted orbit category OrF (G) be the full sub-category of Or(G) whose objects are homogeneous spaces G/H with H ∈ F .

Definition 1.55 (Subgroup category). The subgroup category Sub(G)has as objects subgroups H of G. For H,K ⊆ G let conhomG(H,K) be theset of all group homomorphisms f : H → K, for which there exists a groupelement g ∈ G such that f is given by conjugation with g. The group of innerautomorphisms inn(K) consists of those automorphisms K → K, which aregiven by conjugation with an element k ∈ K. It acts on conhom(H,K) fromthe left by composition. Define the set of morphisms in Sub(G) from H to Kto be inn(K)\ conhom(H,K). Composition of group homomorphisms definesthe composition of morphisms in Sub(G).

Given a family F , define the F-restricted category of subgroups SubF (G)to be the full subcategory of Sub(G) which is given by objects H belongingto F .


Exercise 1.56. Show that SubF (G) is a quotient category of OrF (G).

Note that there is a morphism from H to K only if H is conjugatedto a subgroup of K. Clearly K0(R(−)) yields a functor from SubF (G) toabelian groups since inner automorphisms on a group K induce the identityon K0(RK). Using the inclusions into G, one obtains a map

colimH∈SubF (G)K0(RH)→ K0(RG).

We briefly recall the notion of a colimit of a covariant functor F : C →ABEL from a small category C into the category of abelian groups, wheresmall means that the objects of C form a set. Given an abelian group A,let CA be the constant functor C → ABEL which sends every object in Cto A and every morphism in C to idA. Given a homomorphism f : A → Bof abelian groups, let Cf : CA → CB be the obvious transformation. Thecolimit, or sometimes also called direct limit, of F consists of an abelian groupcolimC F together with a transformation TF : F → CcolimC F such that for anyabelian group B and transformation T : F → CB there exists precisely onehomomorphism of abelian groups φ : colimC F → B satisfying Cφ TF = T .The colimit is unique (up to unique isomorphism) and always exists. If wereplace abelian group by ring or by R-module respectively, we get the notionof a colimit, or sometimes also called direct limit of functors from a smallcategory to rings or R-modules respectively.

Conjecture 1.57 (Farrell-Jones Conjecture for K0(RG) for regular Rwith Q ⊆ R). Let R be a regular ring and G be a group such that for everyfinite subgroup H ⊆ G the element |H| · 1R of R is invertible in R.

Then the homomorphism

IFIN (G,F ) : colimH∈SubFIN (G)K0(RH)→ K0(RG).(1.58)

coming from the various inclusions of finite subgroups of G into G is a bijec-tion.

We mention that the surjectivity of the map IFIN (G,F ) is equivalent tothe surjectivity of the map induced by the various inclusions of subgroupsH ∈ FIN into G ⊕

H∈FINK0(RH) → K0(RG).

because this map factorizes as⊕H∈FIN

K0(RH)ψ−→ colimH∈SubFIN (G)K0(RH)

IFIN (G,F )−−−−−−−→ K0(RG),

where the first map ψ is surjective.

Remark 1.59 (Module-theoretic relevance of Conjecture 1.57). Con-jecture 1.57 implies that for a regular ring R with Q ⊆ R every finitely


generated projective R-module is up to adding finitely generated free RG-modules a direct sum of modules of the shape RG ⊗RH P for a finite sub-group H ⊆ G and a finitely generated projective RH-module P . So it pre-dicts the (stable) structure of finitely generated projective RG-modules inthe most elementary way. We mention, however, that the situation is muchmore complicated in the case, where we drop the assumption that R is reg-ular and Q ⊆ R. In particular for R = Z new phenomena will occur asexplained later which are related to so called negative K-groups and Nil-groups. For instance, the obvious inclusion Z/6 → Z × Z/6 does not in-

duce a surjection K0(Z[Z/6]) → K0(Z[Z × Z/6]), since K0(Z[Z/6]) = 0 and

K0(Z[Z×Z/6]) ∼= Z, whereas by K0(Q[Z/6])→ K0(Q[Z×Z/6]) is known tobe bijective as predicted by Conjecture 1.57.

Remark 1.60 (Conjecture 1.57 and the Atiyah Conjecture). Conjec-ture 1.57 plays a role in a program aiming at a proof of the Atiyah Conjectureabout L2-Betti numbers as explained in [527, Section 10.2]. Atiyah defined

the n-th L2-Betti number of the universal covering M of a closed Riemannianmanifold M to be the non-negative real number

b(2)n (M) := lim

t→∞

∫F

tr(e−t∆n(x,x)

)dx,

where F is a fundamental domain for the π1(M)-action and e−t∆n(x,x) de-

notes the heat kernel on M . The version of the Atiyah Conjecture, which we

are interested in and which is at the time of writing open, says that d·b(2)n (M)

is an integer if d is an integer such that the order of any finite subgroup of

π1(M) divides d. In particular b(2)n (M) is expected to be an integer if π1(M)

is torsionfree. This gives an interesting connection between the analysis ofheat kernels and the projective class group of complex group rings CG.

If one drops the condition that there exists a bound on the order of finitesubgroups of π1(M), then also transcendental real numbers can occur as L2-

Betti number of the universal covering M of a closed Riemannian manifoldM , see [54, 346, 661].

AR-moduleM is called Artinian if for any descending series of submodulesM1 ⊇M2 ⊇ . . . there exists an integer k such that Mk = Mk+1 = Mk+2 = . . .holds. An R-module M is called simple or irreducible if M 6= 0 and Mcontains only 0 and M as submodules. A ring R is called Artinian if bothR considered as left R-module is Artinian and R considered as right R-moduleis Artinian, or, equivalently, every finitely generated left R-module and everyfinitely generated right R-module is Artinian. Skewfields and finite rings areArtinian, whereas Z is not Artinian.

Conjecture 1.61 (Farrell-Jones Conjecture for K0(RG) for an Ar-tinian ring R). Let G be a group and R be an Artinian ring.

Then the canonical map


I(G,R) : colimH∈SubFIN (G)K0(RH)→ K0(RG)

is an isomorphism

1.9 The Kaplansky Conjecture

In this section we discuss:

Conjecture 1.62 (Kaplansky Conjecture). Let R be an integral domainand let G be a torsionfree group. Then all idempotents of RG are trivial, i.e.,equal to 0 or 1.

Remark 1.63 (The Kaplansky Conjecture for prime characteristic).If p is a prime and we additionally assume that p is not a unit in R, then

a reasonable version of the Conjecture 1.62 is obtained by replacing the con-dition torsionfree by the weaker condition that all finite subgroups of G arep-groups.

The Kaplansky Conjecture 1.62 says roughly that idempotents in RG ei-ther come from idempotents in R or by the following construction.

Example 1.64 (Construction of idempotents). Let G be a group andg ∈ G be an element of finite order. Suppose that the order |g| is invertible

in R. Define an element x := |g|−1 ·∑|g|i=1 g

i. Then x2 = x, i.e., x is anidempotent in RG.

Remark 1.65 (Sofic groups). In the next theorem we will use the notionof a sofic group that was introduced by Gromov and originally called suba-menable group. Every residually amenable group is sofic but the converse isnot true. The class of sofic groups is closed under taking subgroups, directproducts, free amalgamated products, colimits and inverse limits, and, if His a sofic normal subgroup of G with amenable quotient G/H, then G is sofic.To the authors’ knowledge there is no example of a group which is not sofic.It is unknown but likely to be true that all hyperbolic groups are sofic. Formore information about the notion of a sofic group we refer to [270].

Definition 1.66 (Directly finite). An R-module M is called directly finiteif every R-module N satisfying M ∼=R M ⊕ N is trivial. A ring R is calleddirectly finite (or von Neumann finite) if it is directly finite as a module overitself, or, equivalently, if r, s ∈ R satisfy rs = 1, then sr = 1. A ring is calledstably finite if the matrix algebra M(n, n,R) is directly finite for all n ≥ 1.

Remark 1.67 (Stable finiteness). Stable finiteness for a ring R is equiv-alent to the following statement. Every finitely generated projective R-module P whose class in K0(R) is zero is already the trivial module, i.e.,0 = [P ] ∈ K0(R) implies P ∼= 0.

1.9 The Kaplansky Conjecture 43

If F is a field of characteristic zero, then FG is stably finite for every groupG. This is proved by Kaplansky [440] (see also Passman [644, Corollary 1.9on page 38]). If R is a skew-field and G is a sofic group, then RG is stablyfinite. This is proved for free-by-amenable groups by Ara-Meara-Perera [32]and extended to sofic groups by Elek-Szabo [269, Corollary 4.7]. These resultshave been extended to extensions with a finitely generated residually finitegroup as kernel and a sofic finitely generated group as quotient by Berlai [103].

The next theorem is taken from [74, Theorem 1.12].

Theorem 1.68 (The Farrell-Jones Conjecture and the KaplanskyConjecture). Let G be a group. Let R be a ring whose idempotents areall trivial. Suppose that

K0(R)⊗Z Q −→ K0(RG)⊗Z Q

is an isomorphism.Then 0 and 1 are the only idempotents in RG if one of the following

conditions is satisfied:

(i) RG is stably finite;(ii) R is a field of characteristic zero;

(iii) R is a skew-field and G is sofic.

Remark 1.69 (The Farrell-Jones Conjecture and the KaplanskyConjecture). Theorem 1.68 implies that for a skew-field D of character-istic zero and a torsionfree group G the Kaplansky Conjecture 1.62 is truefor DG, provided that Conjecture 1.50 holds and that D is commutative orG is sofic.

Remark 1.70 (The Farrell-Jones Conjecture and the KaplanskyConjecture for prime characteristic). Suppose that D is a skew-fieldof prime characteristic p, that Conjecture 1.61 holds for G and D, and that

all finite subgroups of G are p-groups. Then K0(D)∼=−→ K0(DG) is an isomor-

phism since for a finite p-group H the group ring DH is a local ring (see [214,

Theorem 5.24 on page 114]) and hence K0(DH) = 0 by Lemma 1.106. If wefurthermore assume that G is sofic, then Theorem 1.68 implies that all idem-potents in DG are trivial.

Remark 1.71 (Reducing the case of a field of characteristic zero toC). Let F be a field of characteristic zero and let u =

∑g∈G xg · g ∈ FG

be an element. Let K be the finitely generated field extension of Q given byK = Q(xg | g ∈ G) ⊂ F . Then u is already an element in KG. The field Kembeds into C: since K is finitely generated, it is a finite algebraic extensionof a transcendental extension K ′ of Q (see [501, Theorem 1.1 on p. 356]), andK ′ has finite transcendence degree over Q. Since the transcendence degreeof C over Q is infinite, there exists an embedding K ′ → C induced by an


injection of a transcendence basis of K over Q into a transcendence basis ofC over Q. It extends to an embedding K → C because C is algebraicallyclosed. Hence u can be viewed as an element in CG. This reduces the case offields F of characteristic zero to the case F = C.

Next we mention some further results.Formanek [325, Theorem 9] (see also [143, Proposition 4.2]) has shown

that all idempotents of FG are trivial, provided that F is a field of charac-teristic zero and there are infinitely many primes p for which there do not

exist an element g ∈ G, g 6= 1 and an integer k ≥ 1 such that g and gpk

areconjugate. Torsionfree hyperbolic groups satisfy these conditions. Hence For-manek’s results imply that all idempotents in FG are trivial if G is torsionfreehyperbolic and F is a field of characteristic zero.

Delzant [241] has proved the Kaplansky Conjecture 1.62 for all integraldomains R for a torsionfree hyperbolic group G provided that G admits anappropriate action with large enough injectivity radius. Delzant actually dealswith zero-divisors and units as well.

1.10 The Bass Conjectures

1.10.1 The Bass Conjecture for fields of characteristic zero ascoefficients

Let G be a group. Let con(G) be the set of conjugacy classes (g) of elementsg ∈ G. Denote by con(G)f the subset of con(G) consisting of those conju-gacy classes (g) for which each representative g has finite order. Let R bea commutative ring. Let class(G,R) and class(G,R)f be the free R-modulewith the set con(G) and con(G)f as basis. This is the same as the R-moduleof R-valued functions on con(G) and con(G)f with finite support. Define theuniversal R-trace

truRG : RG→ class(G,R),∑g∈G

rg · g 7→∑g∈G

rg · (g).(1.72)

It extends to a function truRG : Mn(RG) → class(G,R) on (n, n)-matricesover RG by taking the sum of the traces of the diagonal entries. Let P be afinitely generated projective RG-module. Choose a matrix A ∈Mn(RG) suchthat A2 = A and the image of the RG-map rA : RGn → RGn given by rightmultiplication with A is RG-isomorphic to P . Define the Hattori-Stallingsrank of P to be

HSRG(P ) = truRG(A) ∈ class(G,R).(1.73)

1.10 The Bass Conjectures 45

The Hattori-Stallings rank depends only on the isomorphism class of theRG-module P . It induces an R-homomorphism, the Hattori-Stallings homo-morphism,

HSRG : K0(RG)⊗Z R → class(G,R), [P ]⊗ r 7→ r ·HSRG(P ).(1.74)

Let F be a field of characteristic zero. Fix an integerm ≥ 1. Let F (ζm) ⊃ Fbe the Galois extension given by adjoining the primitive m-th root of unityζm to F . Denote by Γ (m,F ) the Galois group of this extension of fields, i.e.,the group of automorphisms σ : F (ζm)→ F (ζm) which induce the identity onF . It can be identified with a subgroup of Z/m∗ by sending σ to the unique

element u(σ) ∈ Z/m∗ for which σ(ζm) = ζu(σ)m holds. Let g1 and g2 be two

elements of G of finite order. We call them F -conjugated if for some (andhence all) positive integers m with gm1 = gm2 = 1 there exists an element σ in

the Galois group Γ (m,F ) with the property that gu(σ)1 and g2 are conjugated.

Two elements g1 and g2 are F -conjugated for F = Q, R or C respectively ifthe cyclic subgroups 〈g1〉 and 〈g2〉 are conjugated, if g1 and g2 or g1 and g−1

2

are conjugated, or if g1 and g2 are conjugated respectively.Denote by conF (G)f the set of F -conjugacy classes (g)F of elements g ∈ G

of finite order. Let classF (G)f be the F -vector space with the set conF (G)fas basis, or, equivalently, the F -vector space of functions conF (G)f → F withfinite support. There are obvious inclusions of F -modules

classF (G)f ⊆ class(G,F )f ⊆ class(G,F ).

Lemma 1.75. Suppose that F is a field of characteristic zero and H is afinite group. Then the Hattori-Stallings homomorphism, see (1.74), inducesan isomorphism

HSFH : K0(FH)⊗Z F∼=−→ classF (H) = classF (H)f .

Proof. Since H is finite, an FH-module is a finitely generated projective FH-module if and only if it is a (finite dimensional) H-representation with coeffi-cients in F and K0(FH) is the same as the representation ring RF (H). TheHattori-Stallings rank HSFH(V ) and the character χV of a G-representationV with coefficients in F are related by the formula

χV (h−1) = |CG〈h〉| ·HSFH(V )(h)(1.76)

for h ∈ H, where CG〈h〉 is the centralizer of h in G. Hence Lemma 1.75 followsfrom representation theory, see for instance [748, Corollary 1 in Chapter 12on page 96]. ut

Exercise 1.77. Prove formula (1.76).


The following conjecture is the obvious generalization of Lemma 1.75 toinfinite groups.

Conjecture 1.78 (Bass Conjecture for fields of characteristic zero ascoefficients). Let F be a field of characteristic zero and let G be a group.The Hattori-Stallings homomorphism (see (1.74)) induces an isomorphism

HSFG : K0(FG)⊗Z F → classF (G)f .

Lemma 1.79. Suppose that F is a field of characteristic zero and G is agroup. Then the composite

(1.80) colimH∈SubFIN (G)K0(FH)⊗Z FIFIN (G,F )⊗ZidF−−−−−−−−−−−→ K0(FG)⊗Z F

HSFG−−−−→ class(G,F )

is injective and has as image classF (G)f , where IFIN (G,F ) is the map de-fined in (1.58).

Proof. This follows from the following commutative diagram, compare [523,Lemma 2.15 on page 220].

colimH∈SubFIN (G)K0(FH)⊗Z F

colimH∈SubFIN (G) HSFH ∼=

IFIN (G,F )⊗ZidF // K0(FG)⊗Z F

HSFG

colimH∈SubFIN (G) classF (H)f

j

∼=// classF (G)f

i // class(G,F )

Here the isomorphism j is the direct limit over the obvious maps classF (H)f →classF (G)f given by extending a class function in the trivial way and the mapi is the natural inclusion and in particular injective. ut

Exercise 1.81. Let F be a field of characteristic zero. Show that the groupG must be torsionfree if K0(FG) is a torsion group.

Theorem 1.82 (The Farrell-Jones Conjecture and the Bass Conjec-ture for fields of characteristic zero). The Farrell-Jones Conjecture 1.57for K0(RG) for regular R and Q ⊆ R implies the Bass Conjecture for fieldsof characteristic zero as coefficients (see Conjecture 1.78).

Proof. This follows from Lemma 1.79. ut

The Bost Conjecture 12.22 implies the Bass Conjecture 1.78 for fields ofcharacteristic zero as coefficients, provided that F = C, see [105, Theorem 1.4and Lemma 1.5].

1.10 The Bass Conjectures 47

Exercise 1.83. Let F be field of characteristic zero and let G be a group.Suppose that the Farrell-Jones Conjecture 1.57 for K0(RG) for regular R andQ ⊆ R holds for FG. Consider any finitely generated projective FG-moduleP . Then the Hattori-Stallings rank HS(P ) evaluated at the unit e ∈ G belongsto Q ⊆ F .

Remark 1.84 (Zalesskii’s Theorem). Zalesskii [854] (see also [143, The-orem 3.1]) has shown for every field F , every group G, and every idempotentx ∈ FG that HS(P ) evaluated at the unit e ∈ G belongs to the prime fieldof F .

1.10.2 The Bass Conjecture for integral domains as coefficients

Conjecture 1.85 (Bass Conjecture for integral domains as coeffi-cients). Let R be a commutative integral domain and let G be a group. Letg ∈ G be an element in G. Suppose that either the order |g| is infinite or thatthe order |g| is finite and not invertible in R.

Then for every finitely generated projective RG-module the value of itsHattori-Stallings rank HSRG(P ) at (g) is trivial.

Sometimes the Bass Conjecture 1.85 for integral domains as coefficients iscalled the Strong Bass Conjecture (see [86, 4.5]). The Weak Bass Conjecture(see [86, 4.4]) states for a finitely generated projective ZG-module P thatthe evaluation of its Hattori-Stallings ring at the unit HS(P )(1) agrees withdimZ(Z⊗ZG P ).

Exercise 1.86. Show that the Weak Bass Conjecture follows from the BassConjecture 1.85 for integral domains as coefficients.

The Bass Conjecture 1.85 can be interpreted topologically. Namely, theBass Conjecture 1.85 is true for a finitely presented group G in the case R = Zif and only if every homotopy idempotent selfmap of an oriented smoothclosed manifold whose dimension is greater than 2 and whose fundamentalgroup is isomorphic to G is homotopic to one that has precisely one fixedpoint (see [106]). The Bass Conjecture 1.85 for G in the case R = Z (orR = C) also implies for a finitely dominated CW -complex with fundamentalgroup G that its Euler characteristic agrees with the L2-Euler characteristicof its universal covering (see [265, 0.3]).

The next results follows from the argument in [301, Section 5].

Theorem 1.87 (The Farrell-Jones Conjecture and the Bass Conjec-ture for integral domains). Let G be a group. Suppose that

I(G,F )⊗Z Q : colimOrFIN (G)K0(FH)⊗Z Q→ K0(FG)⊗Z Q


is surjective for all fields F of prime characteristic.Then the Bass Conjecture 1.85 is satisfied for G and every commutative

integral domain R.In particular the Bass Conjecture 1.85 follows from the Farrell-Jones Con-

jecture 1.61.

For finiteG andR an integral domain such that no prime dividing the orderof |G| is a unit in R, Conjecture 1.85 was proved by Swan [774, Theorem 8.1](see also [86, Corollary 4.2]). The Bass Conjecture 1.85 has been proved byBass [86, Proposition 6.2 and Theorem 6.3] for R = C and G a torsionfreelinear group and by Eckmann [263, Theorem 3.3] for R = Q provided that Ghas at most cohomological dimension 2 over Q.

The following result is due to Linnell [514, Lemma 4.1].

Theorem 1.88 (The Bass Conjecture for integral domains and ele-ments of finite order). Let G be a group.

(i) Let p be a prime, and let P be a finitely generated projective Z(p)G-module.Suppose for g ∈ G that HS(P )(g) 6= 0. Then there exists an integer n ≥ 1such that g and gp

n

are conjugated in G and we get for the Hattori-Stallingsrank HS(P )(g) = HS(P )(gp

n

);(ii) Let P be a finitely generated projective ZG-module. Suppose for g ∈ G that

g 6= 1 and HS(P )(g) 6= 0. Then there exist subgroups C,H of G such thatg ∈ C, C ⊆ H, C is isomorphic to the additive group Q, H is finitelygenerated, and the elements of C lie in finitely many H-conjugacy classes.In particular the order of g is infinite.

More information about the Bass Conjectures can be found in [85, 105,107, 143, 181, 273, 274, 275, 442, 527, 641, 735, 736].

1.11 Survey on Computations of K0(RG) for FiniteGroups

In this section we give a brief survey about computations of K0(RG) forfinite groups G and certain rings R. The upshot will be that the reducedprojective class group K0(ZG) is a finite abelian group, but in most casesit is non-trivial and unknown, and that for F a field of characteristic zeroK0(FG) is a well-known finitely generated free abelian group.

The following result is due to Swan [774, Theorem 8.1 and Proposition 9.1].

Theorem 1.89 (K0(ZG) is finite for finite G). Let G be a finite group.

Let R be a ring of algebraic integers, e.g., R = Z. Then K0(RG) is finite.

A proof of the next theorem will be given in Section 2.8. It was originallyproved by Rim [696].

1.11 Survey on Computations of K0(RG) for Finite Groups 49

Theorem 1.90 (Rim’s Theorem). Let p be a prime number. The ho-momorphism induced by the ring homomorphism Z[Z/p] → Z[exp(2πi/p)]sending the generator of Z/p to the primitive p-th root of unity exp(2πi/p)

K0(Z[Z/p])∼=−→ K0(Z[exp(2πi/p)])

is a bijection.

Example 1.91 (K0(Z[Z/p])). Let p be a prime. We have already mentionedin Remark 1.23 that Z[exp(2πi/p)] is the ring of integers in the algebraic num-ber field Q[exp(2πi/p)] and hence a Dedekind domain and that the structureof its ideal class group C(Z[exp(2πi/p)]) is only known for a few primes. Thusthe message of Rim’s Theorem 1.90 is that we know the structure of the fi-nite abelian group K0(Z[Z/p]) only for a few primes. Here is a table takenfrom [592, page 30] or [818, Tables §3 on page 352ff].

p K0(Z[Z/p])≤ 19 023 Z/329 Z/2⊕ Z/2⊕ Z/231 Z/937 Z/3741 Z/11⊕ Z/1143 Z/21147 Z/5⊕ Z/139

Remark 1.92 (Strategy to study K0(ZG) for finite G). A Z-order Λ isa Z-algebra which is finitely generated projective over Z. Its locally free classgroup is defined as the subgroup of K0(Λ)

Cl(Λ) :=

[P ]− [Q] | P(p)∼=Λ(p)

Q(p) for all primes p

(1.93)

where (p) denotes localization at the prime p. This is the part of K0(Λ) whichcan be described by localization sequences. Its significance for Λ = ZG liesin the result of Swan [774] (see also Curtis-Reiner [214, Theorem 32.11 on

page 676] and [215, (49.12 on page 221]) that K0(ZG) ∼= Cl(ZG) for everyfinite group G. Now fix a maximal Z-order ZG ⊆M ⊆ QG. Such a maximalorder has better ring properties than ZG, namely, it is a hereditary ring.The map i∗ : Cl(ZG) → Cl(M) induced by the inclusion i : ZG → M issurjective. Define

D(ZG) = ker (i∗ : Cl(ZG)→ Cl(M)) .(1.94)

The definition of D(ZG) is known to be independent of the choice of the

maximal orderM. Thus the study of K0(ZG) splits into the study of D(ZG)and Cl(M). The analysis of Cl(M) can be intractable and involves study-


ing cyclotomic fields, whereas the analysis of D(ZG) essentially uses p-adiclogarithms.

Remark 1.95 (Finiteness obstructions and D(ZG)). Often calculations

concerning finiteness obstructions are done first in Cl(M) = K0(ZG)/D(ZG),and then in D(ZG), where often some geometric input is needed for the firstpart. For instance Mislin [603] proved that the finiteness obstruction for ev-ery finitely dominated homologically nilpotent space with the finite group Gas fundamental group lies in D(ZG) but that not every element in D(ZG)occurs this way. Questions concerning the spherical space form problem in-volve direct computations in D(ZG) (see for instance Bentzen [98], Bentzen-Madsen [99] and Milgram [584]). The group D(ZG) enters also in the workof Oliver on actions of finite groups on disks (see [626, 627].)

For computations of D(ZG) for finite p-groups we refer to Oliver [628, 629]and Oliver-Taylor [632].

A survey on D(ZG) and the methods of its computations can be found inOliver [630].

Theorem 1.96 (Vanishing results for D(ZG)).

(i) Let G be a finite abelian group G. Then D(ZG) = 0 holds precisely for thefollowing groups:

(a) G has prime order;(b) G is cyclic of order 4, 6, 8, 9, 10, 14;(c) G is Z/2× Z/2;

(ii) If G is a finite group which is not abelian and satisfies D(ZG) = 0, thenit is Dn for n ≥ 6, or A4, A5 or S4;

(iii) One has D(ZG) = 0 if G is A4, A5 or S4;(iv) D(ZDn) = 0 for n < 120 and D(ZD120) = Z/2;(v) D(ZDn) = 0 if n satisfies one of the following conditions:

(a) n/2 is an odd prime;(b) n/2 is a power of a regular odd prime;(c) n/2 is a power of 2.

Proof. (i) This is proved by Cassou-Nogues [168] (see also [215, Theo-rem 50.16 on page 253]).

(ii) This is proved in Endo-Hironaka [276] (see also [215, Theorem 50.29 onpage 266]).

(iii) This follows from Reiner-Ulom [695] (see also [215, Theorem 50.29 onpage 266]).

(iv) This is proved in Endo-Miyata [277] (also [215, Theorem 50.30 onpage 266]).

(v) See [215, Theorem 50.29 on page 266]. ut

1.11 Survey on Computations of K0(RG) for Finite Groups 51

Theorem 1.97 (Finite groups with vanishing K0(ZG)).

(i) Let G be a finite abelian group G. Then K0(ZG) = 0 holds precisely forthe following groups:

(a) G is cyclic of order n for 1 ≤ n ≤ 11;(b) G is cyclic of order 13, 14, 17, 19;(c) G is Z/2× Z/2;

(ii) If G is a non-abelian finite group with K0(ZG) = 0, then G is Dn forn ≥ 6 or A4, A5 or S4;

(iii) We have K0(ZG) = 0 for G = A4, S4, D6, D8, D12.

Proof. (i) This is proved by Cassou-Nogues [168] (see also [215, Corol-lary 50.17 on page 253]).

(ii) This follows from Theorem 1.96 (ii).

(iii) The cases G = A4, S4, D6, D8 are already treated in [694, Theorem 6.4and Theorem 8.2]. Because of Theorem 1.96 (iii) it suffices to show for themaximal order M for the groups G = A4, S4, D6, D8, D12 that Cl(M) = 0.This follows from the fact that QG is a products of matrix algebras over Qand hence the maximal Z-order M is a products of matrix rings over Z. ut

Exercise 1.98. Determine all finite groups G of order ≤ 9 for which K0(ZG)is non-trivial.

Theorem 1.99 (K0(RG) for finite G and an Artinian ring R). LetR be an Artinian ring. Let G be a finite group. Then RG is also an Ar-tinian ring. There are only finitely many isomorphism classes [P1], [P2], . . .,[Pn] of irreducible finitely generated projective RG-modules and we obtain anisomorphism of abelian groups

Zn∼=−→ K0(RG), (k1, k2, . . . kn) 7→

n∑i=1

ki · [Pi].

Proof. This follows from [214, Proposition 16.7 on page 406 and the paragraphafter Corollary 6.22 on page 132]. ut

Let F be a field of characteristic zero or a prime number p not dividing|G|. Then K0(FG) is the same as the representation ring RF (G) of G withcoefficients in the field F since the ring FG is semisimple i.e., every submod-ule of a module is a direct summand. If F is a field of characteristic zero,then representations are detected by their characters (see Lemma 1.75). Formore information about modules over FG for a finite group G and a field Fwe refer for instance to Curtis-Reiner [214, Chapter 1 and Chapter 2] andSerre [748].


Exercise 1.100. Compute K0(FD8) for F = Q, R and C.

1.12 Survey on Computations of K0(C∗r (G)) and

K0(N (G))

Let G be a group. Let B(L2(G)) denote the bounded linear operators onthe Hilbert space L2(G) whose orthonormal basis is G. The reduced groupC∗-algebra C∗r (G) is the closure in the norm topology of the image of theregular representation CG → B(L2(G)), which sends an element u ∈ CG tothe (left) G-equivariant bounded operator L2(G) → L2(G) given by rightmultiplication with u. The group von Neumann algebra N (G) is the closurein the weak topology. There is an identification N (G) = B(L2(G))G. One hasnatural inclusions

CG ⊆ C∗r (G) ⊆ N (G) ⊆ B(L2(G)).

We have CG = C∗r (G) = N (G) if and only if G is finite. If G = Z, then theFourier transform gives identifications C∗r (Z) = C(S1) and N (Z) = L∞(S1).

Remark 1.101 (K0(C∗r (G)) versus K0(CG)). We will later see that thestudy of K0(C∗r (G)) is not done by its algebraic nature. Instead we will in-troduce and analyze the topological K-theory of C∗r (G) and explain thatin dimension 0 the algebraic and the topological K-theory of C∗r (G) agree.In order to explain the different flavour of K0(C∗r (G)) in comparison withK0(CG), we mention the conclusion of the Baum-Connes Conjecture for tor-sionfree groups 8.44 that for torsionfree G there exists an isomorphism⊕

n≥0

H2n(BG;Q)∼=−→ K0(C∗r (G))⊗Z Q.

The space BG is the classifying space of the group G, which is up to homotopycharacterized by the property that it is a CW -complex with π1(BG) ∼= Gwhose universal covering is contractible. We denote by H∗(X,R) the singu-lar or cellular homology of a space or CW -complex X with coefficient in acommutative ring R. We can identify H∗(BG;R) with the group homologyof G with coefficients in R.

We see that K0(C∗r (G)) can be huge also for torsionfree groups, whereasK0(CG) ∼= Z for torsionfree G is a conclusion of the Farrell-Jones Conjec-ture 1.50 for K0(R) for torsionfree G and regular R. We see already here ahomological behavior of K0(C∗r (G)) which is not yet evident in the case ofgroup rings so far and will become clear later.

Remark 1.102 (K0(N (G))). The projective class group K0(A) can be com-puted for any von Neumann algebra A using the center-valued universal trace

1.13 Miscellaneous 53

(see for instance [527, Section 9.2]). In particular one gets for a finitely gen-erated group G which does not contain Zn as subgroup of finite index anisomorphism

K0(N (G)) ∼= Z(N (G))Z/2.

Here Z(N (G)) is the center of the group von Neumann algebra and the Z/2-action comes from taking the adjoint of an operator in B(L2(G)) (see [527,Example 9.34 on page 353]). If G is a finitely generated group which does notcontain Zn as subgroup of finite index and for which the conjugacy class (g)of an element g different from the unit is always infinite, then Z(N (G)) = Cand one obtains an isomorphism

K0(N (G)) ∼= R.

A pleasant feature of N (G) is that there is no difference between stablyisomorphic and isomorphic in the sense that for three finitely generated pro-jective N (G)-modules P0, P1 and Q we have P0 ⊕ Q ∼=N (G) P1 ⊕ Q if andonly if P0

∼=N (G) P1.We see that in the case of the group von Neumann algebra we can com-

pute K0(N (G)) completely, but the answer does not show any homologicalbehavior in G. In fact, the Farrell-Jones Conjecture and the Baum-ConnesConjecture have no analogues for group von Neumann algebras.

Exercise 1.103. Let G be an torsionfree hyperbolic group which is notcyclic. Prove K0(N (G)) ∼= R.

Remark 1.104 (Change of rings homomorphisms for K0 for ZG →CG→ C∗r (G)→ N (G)). We summarize what is conjectured or known aboutthe string of change of rings homomorphism

K0(ZG)i1−→ K0(CG)

i2−→ K0(C∗r (G))i3−→ K0(N (G))

coming from the various inclusion of rings. The first map i1 is conjecturedto be rationally trivial (see [546, Conjecture 85 on page 754]) and may alsobe integrally trivial (see [546, Remark 89 on page 756]). The second map i2is conjectured to be rationally injective (compare [526, Theorem 0.5]) but isnot surjective in general. The map i3 is in general not injective, not surjectiveand not trivial. It is known that the composite i3 i2 i1 is trivial (see forinstance [527, Theorem 9.62 on page 362]).

1.13 Miscellaneous

Algebraic K-theory is compatible with direct limits as explained for the pro-jective class group next. A directed set I is a non-empty set with a partial


ordering ≤ such that for two elements i0 and i1 there exists an element i withi0 ≤ i and i1 ≤ i. A directed system of rings is a set of rings Ri | i ∈ Iindexed by a directed set I together with a choice of a ring homomorphismφi,j : Ri → Rj for i, j ∈ I with i ≤ j such that φi,k = φj,k φi,j holds fori, j, k ∈ I with i ≤ j ≤ k and φi,i = id holds for i ∈ I. The colimit, sometimesalso called direct limit, of Ri | i ∈ I is a ring denoted by colimi∈I Ri to-gether with ring homomorphisms ψj : Ri → colimi∈I Ri for every j ∈ I suchthat ψj φi,j = ψi holds for i, j ∈ I with i ≤ j and the following universalproperty is satisfied: For every ring S and every system of ring homomor-phisms µi : Ri → S | i ∈ I such that µj φi,j = µi holds for i, j ∈ Iwith i ≤ j there is precisely one ring homomorphism µ : colimi∈I Ri → Ssatisfying µ ψi = µi for every i ∈ I. If we replace ring by group or modulerespectively everywhere, we get the notion of directed system and direct limitof groups or modules respectively. This is a special case of the direct limitof a functor, namely, consider I as category with the set I as objects andprecisely one morphism from i to j if i ≤ j, and no other morphisms.

Let Ri | i ∈ I be a direct system of rings. For every i ∈ I we obtaina change of rings homomorphism (ψi)∗ : K0(Ri) → K0(R). The universalproperty of the direct limit yields a homomorphism

colimi∈I(ψi)∗ : colimi∈I K0(Ri)∼=−→ K0(R),(1.105)

which turns out to be an isomorphism (see [704, Theorem 1.2.5]).We denote by R× the group of units in R. A ring R is called local if the

set I := R −R× forms a (left) ideal. If I is a left ideal, it is automatically atwo-sided ideal and it is maximal both as a left ideal and as a right ideal. Aring R is local if and only if it has a unique maximal left ideal and a uniquemaximal right ideal and these two coincide. An example of a local ring isthe ring of formal power series F [[t]] with coefficients in a field F . If R is acommutative ring and I is a prime ideal, then the localization RI of R at Iis a local ring.

Theorem 1.106 (K0(R) of local rings). Let R be a local ring. Then everyfinitely generated projective R-module is free and K0(R) is infinite cyclic with[R] as generator.

Proof. See for instance [592, Lemma 1.2 on page 5] or [704, Theorem 1.3.11on page 14]. ut

Its proof is based on Nakayama’s Lemma which says for a ring R and afinitely generated R-module M that rad(R)M = M ⇔ M = 0 holds. Hererad(R) is the radical, or Jacobson radical, i.e., the two sided ideal which isgiven by the intersection of all maximal left ideals, or, equivalently, of allmaximal right ideals of R. The radical is the same as the set of elementr ∈ R for which there exists s ∈ S such that 1− rs has a left inverse in R.

If R is a commutative ring and spec(R) is its spectrum consisting of itsprime ideals and equipped with the Zariski topology, then we obtain for


every finitely generated projective R-module P a continuous rank functionSpec(R)→ Z by sending a prime ideal I to the rank of the finitely generatedfree RI -module PI = P ⊗R RI . This makes sense because of Theorem 1.106since RI is local. If R is a commutative integral domain, this rank functionis constant. For more details we refer for instance to [704, Proposition 1.3.12on page 15].

Exercise 1.107. Prove for an integer n ≥ 1 that K0(Z/n) is the free abeliangroup whose rank is the number of prime numbers dividing n.

A ring is called semilocal ifR/ rad(R) is Artinian, or, equivalently,R/ rad(R)is semisimple. If R is commutative, then R is semilocal if and only if it hasonly finitely many maximal ideas (see [755, page 69].) For a semilocal ringR the projective class group K0(R) is a finitely generated free abelian group(see [755, Proposition 14 on page 28]. More information about semilocal ringscan be found for instance in[496, § 20].

Lemma 1.108. For any ring R and nilpotent two-sided ideal I ⊆ R, the mapK0(R)→ K0(R/I) induced by the projection R→ R/I is bijective.

Proof. See [826, Lemma 2.2 in Section II.2 on page 69]. ut

Given two groups G1 and G2, let G1∗G2 by the free amalgamated product.Then the natural maps Gk → G0 ∗ G1 for k = 1, 2 induce an isomorphism(see [341, Theorem 1.1])

K0(Z[G1])⊕ K0(Z[G1]) ∼= K0(Z[G1 ∗G2]).(1.109)

This is a first glimpse of a homological behavior of K0 if one compares thiswith the corresponding isomorphism of group homology

Hn(G1)⊕ Hn(G1) ∼= Hn(G1 ∗G2).

Exercise 1.110. Show that the projections prk : G1 ×G2 → Gk for k = 1, 2do not in general induce isomorphisms

K0(Z[G1 ×G2])→ K0(Z[G1])× K0(Z[G2]).

There are also equivariant versions of the finiteness obstructions, see forinstance [30], [520],[521, Chapter 3 and 11]. Finiteness obstructions for cate-gories are investigated in [319, 318].


name of texfile: ic

Chapter 2

The Whitehead Group

2.1 Introduction

This chapter is devoted to the first K-group K1(R) of a ring R and theWhitehead group Wh(G) of a group G.

We give two equivalent definitions of K1(R), namely, as the universal de-terminant and in terms of invertible matrices. We explain some elementarycalculations of K1(R) for rings with Euclidian algorithm, local rings, andrings of integers in algebraic number fields.

We introduce the Whitehead group of a group and the Whitehead torsionof a homotopy equivalence of finite CW -complexes algebraically and geo-metrically. The relevance of these notions are illustrated by the s-CobordismTheorem and its applications to the classification of manifolds, and by theclassification of lens spaces by Reidemeister torsion.

The next topic is the Bass-Heller-Swan decomposition and the long exactsequence associated to a pullback of rings and to a two-sided ideal. These areimportant tools for computations and relate K0(R) and K1(R).

We discuss Swan homomorphisms and free homotopy representations.Thus we will provide a link between torsion invariants and finite obstruc-tions.

We explain the variant of the Farrell-Jones Conjecture that for a torsion-free group G the reduced projective class group K0(ZG) and the Whiteheadgroup Wh(G) vanish. It implies that any h-cobordism with torsionfree fun-damental group and dimension ≥ 6 is trivial.

Finally, we give a survey of computations of K1(ZG) for finite groups Gand of the algebraic K1-group of commutative Banach algebras, commutativeC∗-algebras and of some group von Neumann algebras.

2.2 Definition and Basic Properties of K1(R)

Definition 2.1 (K1-group K1(R)). Let R be a ring. Define the K1-group ofa ring K1(R) to be the abelian group whose generators are conjugacy classes[f ] of automorphisms f : P → P of finitely generated projective R-moduleswith the following relations:

• AdditivityGiven a commutative diagram of finitely generated projective R-modules

57

58 2 The Whitehead Group

0 // P1i //

f1

P2p //

f2

P3//

f3

0

0 // P1i // P2

p // P3// 0

with exact rows and automorphisms as vertical arrows, we get

[f1] + [f3] = [f2];

• Composition formulaGiven automorphisms f, g : P → P of a finitely generated projective R-module P , we get

[g f ] = [f ] + [g].

Define G1(R) analogously but replacing finitely generated projective byfinitely generated everywhere.

Given a ring homomorphism f : R → S, we obtain a change of rings ho-momorphism

f∗ : K1(R)→ K1(S), [g : P → P ] 7→ [f∗g : f∗P → f∗P ](2.2)

analogously as we have defined it for the projective class group in (1.2). ThusK1 becomes a covariant functor from the category of rings to the category ofabelian groups.

Exercise 2.3. Show that K1(R) = 0 holds for the ring R appearing in Ex-ample 1.17.

Remark 2.4 (The universal property of K1(R)). One should viewK1(R) together with the assignment sending an automorphism f : P → P ofa finitely generated projective R-module P to its class [f ] ∈ K1(R) as theuniversal determinant. Namely, for any abelian group A and assignment awhich sends the automorphism f of a finitely generated projective R-moduleto a(f) ∈ A such that (A, a) satisfies additivity and the composition for-mula (see Definition 2.1), there exists precisely one homomorphism of abeliangroups φ : K1(R)→ A such that φ([f ]) = a(f) holds for every automorphismf of a finitely generated projective R-module.

We always have the following map of abelian groups

i : R×/[R×, R×] → K1(R), [x] 7→ [rx : R→ R].(2.5)

It is neither injective nor surjective in general. However

Theorem 2.6 (K1(F ) of skewfields). The map i defined in (2.5) is anisomorphism if R is a skewfield or, more generally, a local ring. It is surjective(with an explicitly described kernel) if R is a semilocal ring.

2.2 Definition and Basic Properties of K1(R) 59

Proof. See for instance [755, Corollary 43 on page 133], [704, Corollary 2.2.6on page 69] and [755, Proposition 53 on page 140]. ut

Exercise 2.7. Let H be the skewfield of quaternions a + bi + cj + dk |a, b, c, d ∈ R. Since H is a 4-dimensional vector space, there is an embeddingGLn(H) → GL4n(R). Its composition with the determinant over R yieldsa homomorphism µn : GLn(H) → R>0 to the multiplicative group of posi-tive real numbers. Show that the system of homomorphisms µn induces anisomorphism

µ : K1(H)∼=−→ R>0.

The proof of the next two results is analogous to the one of Theorem 1.10and Lemma 1.12.

Theorem 2.8 (Morita equivalence for K1(R)). For every ring R andinteger n ≥ 1 there is natural isomorphism

µ : K1(R)∼=−→ K1(Mn(R)).

Lemma 2.9. Let R0 and R1 be rings. Denote by pri : R0 × R1 → Ri fori = 0, 1 the projection. Then we obtain an isomorphism

(pr0)∗ × (pr1)∗ : K1(R0 ×R1)∼=−→ K1(R0)×K1(R1).

Lemma 2.10. Define the abelian group Kf1 (R) analogous to K1(R) but with

finitely generated projective replaced by finitely generated free everywhere.Then the canonical homomorphism

α : Kf1 (R)

∼=−→ K1(R), [f ] 7→ [f ]

is an isomorphism.

Proof. Given an automorphism f : P → P of a finitely generated projectiveR-module P , we can choose a finitely generated projective R-module Q, a

finitely generated free R-module F and an R-isomorphism φ : P ⊕ Q∼=−→ F .

We obtain an automorphism φ (f ⊕ idQ)φ−1 : F → F and thus an element

[φ (f ⊕ idQ) φ−1] ∈ Kf1 (R). One easily checks that it is independent of the

choice of Q and φ and then that it depends only on [f ] ∈ K1(R). Thus we

obtain a homomorphism of abelian groups β : K1(R) → Kf1 (R). One easily

checks that α and β are inverse to one another. ut

Next we give a matrix description of K1(R). Denote by En(i, j) for n ≥ 1and 1 ≤ i, j ≤ n the (n, n)-matrix whose entry at (i, j) is one and is zeroelsewhere. Denote by In the identity matrix of size n. An elementary (n, n)-matrix is a matrix of the form In + r · En(i, j) for n ≥ 1, 1 ≤ i, j ≤ n, i 6= jand r ∈ R. Let A be an (n, n)-matrix. The matrix B = A · (In+r ·En(i, j)) is


obtained from A by adding the i-th column multiplied with r from the rightto the j-th column. The matrix C = (In + r · En(i, j)) · A is obtained fromA by adding the j-th row multiplied with r from the left to the i-th row. LetE(R) ⊂ GL(R) be the subgroup generated by all elements in GL(R) whichare represented by elementary matrices.

Lemma 2.11. The subgroup E(R) of GL(R) coincides with the commutatorsubgroup [GL(R), GL(R)].

Proof. For n ≥ 3, pairwise distinct numbers 1 ≤ i, j, k ≤ n and r ∈ R we canwrite In + r · En(i, k) as a commutator in GL(n,R), namely,

In + r · En(i, k)

= (In+r ·En(i, j)) · (In+En(j, k)) · (In+r ·En(i, j))−1 · (In+En(j, k))−1.

This implies E(R) ⊂ [GL(R), GL(R)].Let A and B be two elements in GL(n,R). Let [A] and [B] be the elements

in GL(R) represented by A and B. Given two elements x and y in GL(R),we write x ∼ y if there are elements e1 and e2 in E(R) with x = e1ye2, inother words, if the classes of x and y in E(R)\GL(R)/E(R) agree. One easilychecks

[AB] ∼[(

AB 00 In

)]∼[(

AB A0 In

)]∼[(

0 A−B In

)]∼[(

0 A−B 0

)],

since each step is given by multiplication from the right or left with a block

matrix of the form

(In 0C In

)or

(In C0 In

)and such a block matrix is obviously

obtained from I2n by a sequence of column and row operations and hence itsclass in GL(R) belongs to E(R). Analogously we get

[BA] ∼[(

0 B−A 0

)].

Since the element in GL(R) represented by

(0 −InIn 0

)belongs to E(R), we

conclude [(0 A−B 0

)]∼[(

A 00 B

)]∼[(

0 B−A 0

)].

and hence[AB] ∼ [BA].

This implies for any element x ∈ GL(R) and e ∈ E(R) that xex−1 ∼ ex−1x =e and hence xex−1 ∈ E(R). Therefore E(R) is normal. Given a commutatorxyx−1y−1 for x, y ∈ GL(R), we conclude for appropriate elements e1, e2, e3

in E(R)


xyx−1y−1 = e1yxe2x−1y−1 = e1(yx)e2(yx)−1 = e1e3 ∈ E(R).

ut

Theorem 2.12 (K1(R) equals GL(R)/[GL(R), GL(R)]). There is a nat-ural isomorphism

GL(R)/[GL(R), GL(R)]∼=−→ K1(R).

Proof. Because of Lemma 2.10 it suffices to construct to one another inversehomomorphisms of abelian groups α : GL(R)/[GL(R), GL(R)]→ Kf

1 (R) and

β : Kf1 (R) → GL(R)/[GL(R), GL(R)]. The map α sends the class [A] of

A ∈ GLn(R) to the class [rA] of rA : Rn → Rn. This is a well-defined homo-morphism of abelian groups since [rAB ] = [rA]+[rB ], [rA⊕I1 ] = [rA] holds forall n ∈ Z, n ≥ 1 and A,B ∈ GLn(R), and K1(R) is abelian. The map β sendsthe class [f ] of an automorphism f of a finitely generated free R-module Fto the class [A(f,B)] of the invertible (n, n)-matrix A(f,B) associated tof after a choice of some R-basis B for F . This class is independent of thechoice of B, since for another choice of a bases B′ there exists U ∈ GLn(R)with UA(f,B)U−1 = A(f,B′) which implies [A(f,B′)] = [UA(f,B)U−1] =[U ][A(f,B)][U ]−1 = [U ][U ]−1[A(f,B)] = [A(f,B)]. Thus we have defined βon generators. It remains to check the relations. Obviously the compositionformula is satisfied. Additivity is satisfied because of the following calculationin GL(R)/[GL(R), GL(R)] for A ∈ GLm(R), B ∈ GLn(R) and C ∈Mm,n(R)based on Lemma 2.11[(

A 0B C

)]=

[(A 00 In

)·(Im 00 C

)·(

Im 0C−1B In

)]=

[(A 00 In

)]·[(

Im 00 C

)]·[(

Im 0C−1B In

)]= [A] · [C] · [Im+n] = [A] · [C].

One easily checks that α and β are inverse to one another. ut

Remark 2.13 (What K1(R) measures). We conclude from Lemma 2.11and Theorem 2.12 that two matricesA ∈ GLm(R) andB ∈ GLn(R) representthe same class in K1(R) if and only if B can be obtained from A by a sequenceof the following operations:

(i) Elementary row operationB is obtained from A by adding the k-th row multiplied with x from theleft to the l-th row for x ∈ R and k 6= l;

(ii) Elementary column operationB is obtained from A by adding the k-th column multiplied with x fromthe right to the l-th row for x ∈ R and k 6= l;


(iii) StabilizationB is obtained by taking the direct sum of A and I1, i.e., B looks like the

block matrix

(A 00 1

);

(iv) DestabilizationA is the direct sum of B and I1. (This is the inverse operation to (iii).)

Since multiplication from the left or right with an elementary matrix cor-responds to the operation (i) or the operation (ii), the abelian group K1(R)is trivial if and only if any invertible matrix A ∈ GLn(R) can be reduced bya sequence of the operations above to the empty matrix.

One could delete the operation (i) or the operation (ii) from the list abovewithout changing the conclusion. This follows from the fact that E(R) is anormal subgroup of GL(R).

One easily checks

Lemma 2.14. Let R be a commutative ring. Then the determinant definesa homomorphism of abelian groups

det : K1(R)→ R×, [f ] 7→ det(f).

It satisfies det i = idR× for the map i defined in (2.5).

Hence i is injective for commutative rings R.

Definition 2.15 (SK1(R) of a commutative ring R). Let R be a com-mutative ring. Define

SK1(R) := ker(det : K1(R)→ R×

).

We will see in Section 2.12 that there are commutative group rings ZG forwhich the surjective map det : K1(ZG) → ZG× is not injective, or, equiva-lently, with non-trivial SK1(ZG). Here is another example.

Example 2.16. The following example is taken from [88, Example 4.4], seealso [704, Exercise 2.3.11 on page 82]. Let Λ be obtained from the polynomialring R[x, y] by dividing out the ideal generated by x2 + y2 − 1. This is aDedekind domain. The matrix

M :=

(x y−y x

)∈ SL2(Λ)

represents a non-trivial element in SK1(Λ). The proof uses Mennicke symbolsand is based on the observation that the function

S1 → SLn(R), (x, y) 7→

x y 0−y x 00 0 In−2


represents a non-trivial element in π1(SLn(R)) ∼= π1(SO(n)) ∼= Z/2 for n ≥ 3.

Theorem 2.17 (K1(R) = R× for commutative rings with Euclideanalgorithm). Let R be a commutative ring with Euclidean algorithm (see [704,2.3.1 on page 74]), for instance a field or Z.

Then the determinant induces an isomorphism

det : K1(R)∼=−→ R×.

Proof. Because of Lemma 2.14 it suffices to show for A ∈ GLn(R) withdet(A) = 1 that it can be reduced to the empty matrix by a sequence ofoperations appearing in Remark 2.13. But this is a well-known result of ele-mentary algebra (see for instance [704, Theorem 2.3.2 on page 74]). ut

Exercise 2.18. Prove K1(Z[i]) ∼= 1,−1, i,−i ∼= Z/4.

Remark 2.19 (K1(R) of principal ideal domains). There exists principalideal domains R such that det : K1(R) → R× is not bijective. For instanceGrayson [348] gives such an example, namely, take Z[x] and invert x and allpolynomials of the shape xm − 1 for m ≥ 1. Other examples can be found inIschebeck [416].

Theorem 2.20 (Vanishing of SK1 of ring of integers in an algebraicnumber field). Let R be the ring of integers in an algebraic number field.Then the determinant induces an isomorphism

det : K1(R)∼=−→ R×.

Proof. See [88, page 77] or [592, Corollary 16.3 on page 159]. ut

The proof of the next classical result can be found for instance in [703,Theorem 2.3.8 on page 79].

Theorem 2.21 (Dirichlet Unit Theorem). Let R be the ring of integersin an algebraic number field F . Let r1 be the number of distinct embeddingsof F into R and let r2 be the number of distinct conjugate pairs of embeddingsof F into C with image not contained in R. Then

(i) r1 + 2r2 is the degree [F : Q] of the extension Q ⊆ F ;(ii) The abelian group R× is finitely generated:

(iii) The torsion subgroup of R× is the finite cyclic group of roots of unity inF ;

(iv) The rank of R× is r1 + r2 − 1.

Exercise 2.22. Let R be the ring of integers in an algebraic number field F .Then K1(R) is finite if and only if F is Q or an imaginary quadratic field.


2.3 Whitehead Group and Whitehead Torsion

In this section we will assign to a homotopy equivalence f : X → Y of fi-nite CW -complexes its Whitehead torsion τ(f) in the Whitehead groupWh(π(Y )) associated to Y . A basic feature is that the Whitehead torsioncan distinguish manifolds or spaces which are homotopy equivalent. Thenotion of Whitehead torsion goes back to the papers by J.H.C. White-head [833, 834, 835].

The reduced K1-group K1(R) is defined to be the cokernel of the mapK1(Z) → K1(R) induced by the unique ring homomorphism Z → R. Thehomomorphism det : K1(Z) → ±1 is a bijection, because Z is a ring with

Euclidean algorithm (see Theorem 2.17). Hence K1(R) is the same as thequotient of K1(R) by the cyclic subgroup of at most order two generated bythe class of the (1, 1)-matrix (−1).

Definition 2.23 (Whitehead group). Define the Whitehead group Wh(G)of a group G to be the cokernel of the map G×±1 → K1(ZG) which sends(g,±1) to the class of the invertible (1, 1)-matrix (±g).

Exercise 2.24. Using the ring homomorphism f : Z[Z/5] → C which sendsthe generator of Z/5 to exp(2πi/5) and the norm of a complex number, definea homomorphism of abelian groups

φ : Wh(Z/5)→ R>0.

Show that 1−t−t−1 is a unit in Z[Z/5] whose class in Wh(Z/5) is an elementof infinite order. (Actually Wh(Z/5) is an infinite cyclic group with this classas generator).

For a ring R and a group G we denote by

A0 = K0(i) : K0(R)→ K0(RG)(2.25)

the map induced by the inclusion i : R→ RG. Sending (g, [P ]) ∈ G×K0(R) tothe class of the RG-automorphism R[G]⊗RP → R[G]⊗RP, u⊗x 7→ ug−1⊗xdefines a map Φ : G/[G,G]⊗Z K0(R)→ K1(RG). Define a homomorphism

A1 := Φ⊕K1(i) : (G/[G,G]⊗Z K0(R))⊕K1(R)→ K1(RG).(2.26)

Definition 2.27 (Generalized Whitehead group). For a regular ring Rand a group G we define the generalized Whitehead group WhR1 (G) as thecokernel of the map A1. Denote by WhR0 (G) the cokernel of the map A0.

Notice that the abelian group WhZ1 (G) of Definition 2.27 agrees with the

abelian group Wh(G) of Definition 2.23.Next we will define torsion invariants on the level of chain complexes.

2.3 Whitehead Group and Whitehead Torsion 65

We begin with some input about chain complexes. Let f∗ : C∗ → D∗ be achain map of R-chain complexes for some ring R. Define cyl∗(f∗) to be thechain complex with n-th differential

Cn−1 ⊕ Cn ⊕Dn

−cn−1 0 0− id cn 0fn−1 0 dn

−−−−−−−−−−−−−→ Cn−2 ⊕ Cn−1 ⊕Dn−1.

Define cone∗(f∗) to be the quotient of cyl∗(f∗) by the obvious copy of C∗.Hence the n-th differential of cone∗(f∗) is

Cn−1 ⊕Dn

−cn−1 0fn−1 dn

−−−−−−−−−−→ Cn−2 ⊕Dn−1.

Given a chain complex C∗, define ΣC∗ to be the quotient of cone∗(idC∗) bythe obvious copy of C∗, i.e., the chain complex with n-th differential

Cn−1−cn−1−−−−→ Cn−2.

Definition 2.28 (Mapping cylinder and mapping cone). We call cyl∗(f∗)the mapping cylinder, cone∗(f∗) the mapping cone of the chain map f∗, andΣC∗ the suspension of the chain complex C∗.

These algebraic notions of mapping cylinder, mapping cone and suspen-sion are modelled on their geometric counterparts. Namely, the cellular chaincomplex of a mapping cylinder of a cellular map f of CW -complexes is themapping cylinder of the chain map induced by f . As suggested alreadyfrom the geometric picture, there exists obvious exact sequences such as0→ C∗ → cyl∗(f∗)→ cone∗(f∗)→ 0 and 0→ D∗ → cone∗(f∗)→ ΣC∗ → 0.

A chain contraction γ∗ for an R-chain complex C∗ is a collection of R-homomorphisms γn : Cn → Cn+1 for n ∈ Z such that cn+1 γn + γn−1 cn =idCn holds for all n ∈ Z. We call a finite free R-chain complex based free if eachR-chain module Cn comes with a preferred (finite ordered) basis. Supposethat C∗ is a finite based free R-chain complex which is contractible, i.e., whichpossesses a chain contraction. Put Codd = ⊕n∈ZC2n+1 and Cev = ⊕n∈ZC2n.Let γ∗ and δ∗ be two chain contractions. Define R-homomorphisms

(c∗ + γ∗)odd : Codd → Cev;

(c∗ + δ∗)ev : Cev → Codd.

Let A be the matrix of (c∗+γ∗)odd with respect to the given bases. Let B bethe matrix of (c∗ + δ∗)ev with respect to the given bases. Put µn := (γn+1 −δn+1) δn and νn := (δn+1 − γn+1) γn. One easily checks that (id +µ∗)odd,(id +ν∗)ev and both compositions (c∗ + γ∗)odd (id +µ∗)odd (c∗ + δ∗)ev and(c∗ + δ∗)ev (id +ν∗)ev (c∗ + γ∗)odd are given by upper triangular matrices


whose diagonal entries are identity maps. Hence A and B are invertible andtheir classes [A], [B] ∈ K1(R) satisfy [A] = −[B]. Since [B] is independent ofthe choice of γ∗, the same is true for [A]. Thus we can associate to a finitebased free contractible R-chain complex C∗ an element

τ(C∗) := [A] ∈ K1(R).(2.29)

Let f∗ : C∗ → D∗ be a homotopy equivalence of finite based free R-chaincomplexes. Its mapping cone cone(f∗) is a contractible finite based free R-chain complex. Define the Whitehead torsion of f∗ by

τ(f∗) := τ(cone∗(f∗)) ∈ K1(R).(2.30)

Now we can pass to CW -complexes. Let f : X → Y be a cellular homo-topy equivalence of connected finite CW -complexes. Let pX : X → X andpY : Y → Y be the universal coverings. Identify π1(Y ) with π1(X) using theisomorphism induced by f . (We ignore base point questions here and in thesequel. This can be done since an inner automorphisms of a group G inducesthe identity on K1(ZG) and hence also on Wh(G).) There is a lift f : X → Ywhich is π1(Y )-equivariant. It induces a Zπ1(Y )-chain homotopy equivalence

C∗(f) : C∗(X)→ C∗(Y ). The CW -structure defines a basis for each Zπ1(Y )-

chain module Cn(X) and Cn(Y ) which is unique up to multiplying each basiselement with a unit of the form ±g ∈ Zπ1(Y ) and permuting the elementsof the basis. Pick such a cellular basis for each chain module. We can ap-ply (2.30) to it and thus obtain an element in K1(Zπ1(Y )) which we canproject down to Wh(π1(Y )) and thus get

τ(f) ∈Wh(π1(Y )).(2.31)

Since we consider τ(f) in Wh(π1(Y )), the choice of the cellular basis doesnot matter anymore.

If f : X → Y is any homotopy equivalence of connected finite CW -complexes, we can choose by the Cellular Approximation Theorem a cellularmap f ′ : X → Y which is homotopic to f , and define the Whitehead torsionτ(f) by τ(f ′). By the homotopy invariance of the Whitehead torsion, this isindependent of the choice of f ′.

If f : X → Y is a homotopy equivalence of finite CW -complexes, thendefine Wh(π1(Y )) :=

⊕C∈π0(Y ) Wh(π1(C)) and τ(f) ∈ Wh(π1(Y )) by the

collection of the Whitehead torsions of the homotopy equivalences inducebetween path components. Obviously a map g : Y1 → Y2 induces a homo-morphism of abelian groups g∗ : Wh(π1(Y1)) → Wh(π1(Y2)) by the homo-morphisms between the various fundamental groups of the path componentsinduced by g.

Definition 2.32 (Whitehead torsion). We call τ(f) the (algebraic) White-head torsion of the homotopy equivalence f : X → Y of finite CW -complexes.

2.3 Whitehead Group and Whitehead Torsion 67

Exercise 2.33. Let 0 → C∗i∗−→ D∗

p∗−→ E∗ → 0 be an exact sequence ofprojective R-chain complexes. Suppose that E∗ is contractible. Construct anR-chain map s∗ : E∗ → D∗ such that p∗ s∗ = idE∗ . Show that i∗⊕ s∗ : C∗⊕E∗ → D∗ is an isomorphism of R-chain complexes. Give a counterexampleto the conclusion if one drops the condition that E∗ is contractible.

The basic properties of this invariant are summarized in the followingtheorem whose proof can be found for instance in [190, (22.1), (22.4), (23.1),and (23.2)], [206, Chapter 2] or [525, Chapter 2].

Theorem 2.34 (Basic properties of Whitehead torsion).

(i) Sum formula

Let the following two diagrams be pushouts of finite CW -complexes

X0i1 //

i2

X1

j1

X2

j2// X

Y0k1 //

k2

Y1

l1

Y2

l2

// Y

where the left vertical arrows are inclusions of CW -complexes, the upperhorizontal maps are cellular, and X and Y are equipped with the inducedCW -structure. Let fi : Xi → Yi be homotopy equivalences for i = 0, 1, 2satisfying f1 i1 = k1 f0 and f2 i2 = k2 f0. Put l0 = l1 k1 = l2 k2.Denote by f : X → Y the map induced by f0, f1 and f2 and the pushoutproperty.Then f is a homotopy equivalence and

τ(f) = (l1)∗τ(f1) + (l2)∗τ(f2)− (l0)∗τ(f0);

(ii) Homotopy invariance

Let f ' g : X → Y be homotopic maps of finite CW -complexes. Then thehomomorphisms f∗, g∗ : Wh(π1(X)) → Wh(π1(Y )) agree. If additionallyf and g are homotopy equivalences, then

τ(g) = τ(f);

(iii) Composition formula

Let f : X → Y and g : Y → Z be homotopy equivalences of finite CW -complexes. Then

τ(g f) = g∗τ(f) + τ(g);

(iv) Product formula

Let f : X ′ → X and g : Y ′ → Y be homotopy equivalences of connectedfinite CW -complexes. Then


τ(f × g) = χ(X) · j∗τ(g) + χ(Y ) · i∗τ(f),

where χ(X), χ(Y ) ∈ Z denote the Euler characteristics, j∗ : Wh(π1(Y ))→Wh(π1(X × Y )) is the homomorphism induced by j : Y → X × Y, y 7→(y, x0) for some base point x0 ∈ X and i∗ is defined analogously.

Let X be a finite simplicial complex. Let X ′ be its barycentric subdivision.Then one can show τ(f) = 0 for the map f : X → X ′ whose underlyingmap of spaces is the identity. However, if X1 and X2 are two finite CW -complexes with the same underlying space, it is not at all clear that τ(f) =0 holds for the map f : X1 → X2 whose underlying map of spaces is theidentity. This problem is solved by the following (in comparison with theother statements above much deeper) result due to Chapman [175], [176](see also [190, Appendix]).

Theorem 2.35 (Topological invariance of Whitehead torsion). TheWhitehead torsion of a homeomorphism f : X → Y of finite CW -complexesvanishes.

2.4 Geometric Interpretation of Whitehead Group andWhitehead Torsion

In this section we introduce the concept of a simple homotopy equivalencef : X → Y of finite CW -complexes geometrically. We will show that theobstruction for a homotopy equivalence f : X → Y of finite CW -complexesto be simple is the Whitehead torsion.

We have the inclusion of spaces Sn−2 ⊂ Sn−1+ ⊂ Sn−1 ⊂ Dn, where

Sn−1+ ⊂ Sn−1 is the upper hemisphere. The pair (Dn, Sn−1

+ ) carries an ob-

vious relative CW -structure. Namely, attach an (n − 1)-cell to Sn−1+ by the

attaching map id: Sn−2 → Sn−2 to obtain Sn−1. Then we attach to Sn−1

an n-cell by the attaching map id: Sn−1 → Sn−1 to obtain Dn. Let X be aCW -complex. Let q : Sn−1

+ → X be a map satisfying q(Sn−2) ⊂ Xn−2 and

q(Sn−1+ ) ⊂ Xn−1. Let Y be the space Dn ∪q X, i.e., the pushout

Sn−1+

q //

i

X

j

Dn

g// Y

where i is the inclusion. Then Y inherits a CW -structure by putting Yk =j(Xk) for k ≤ n − 2, Yn−1 = j(Xn−1) ∪ g(Sn−1) and Yk = j(Xk) ∪ g(Dn)for k ≥ n. Notice that Y is obtained from X by attaching one (n − 1)-celland one n-cell. Since the map i : Sn−1

+ → Dn is a homotopy equivalence and

2.4 Geometric Interpretation of Whitehead Group and Whitehead Torsion 69

cofibration, the map j : X → Y is a homotopy equivalence and a cofibration.We call j an elementary expansion and say that Y is obtained from X by anelementary expansion. There is a map r : Y → X with r j = idX . This mapis unique up to homotopy relative j(X). We call any such map an elementarycollapse and say that X is obtained from Y by an elementary collapse.

Definition 2.36 (Simple homotopy equivalence). Let f : X → Y be amap of finite CW -complexes. We call it a simple homotopy equivalence ifthere is a sequence of maps

X = X[0]f0−→ X[1]

f1−→ X[2]f2−→ · · · fn−1−−−→ X[n] = Y

such that each fi is an elementary expansion or elementary collapse and f ishomotopic to the composition of the maps fi.

Remark 2.37 (Combinatorial meaning of simple homotopy equiva-lence). The idea of the definition of a simple homotopy equivalence is thatsuch a map can be written as a composition of elementary maps, namely,elementary expansions and collapses, which are obviously homotopy equiva-lences and in some sense the smallest and most elementary steps to pass fromone finite CW -complex to another without changing the homotopy type. Ifone works with simplicial complexes, an elementary map has a purely com-binatorial description. An elementary collapse means to delete a simplex andone of its faces which is not shared by another simplex. So one can describethe passage from one finite simplicial complex to another coming from a sim-ple homotopy equivalence by finitely many combinatorial data. This does notwork for two finite simplicial complexes which are homotopy equivalent butnot simple homotopy equivalent.

This approach is similar to the idea in knot theory that two knots areequivalent if one can pass from one knot to the other by a sequence of el-ementary moves, the so called Reidemeister moves. A Reidemeister moveobviously does not change the equivalence class of a knot and, indeed, itturns out that one can pass from one knot to a second knot by a sequenceof Reidemeister moves if and only if the two knots are equivalent (see forinstance [142, Chapter 1] or [817]). The analogous statement is not true forhomotopy equivalences f : X → Y of finite CW -complexes because there isan obstruction for f to be simple, namely, its Whitehead torsion.

Exercise 2.38. Consider the simplicial complex X with four vertices v0, v1,v2 and v3, the edges v0, v1, v1, v2, v0, v2 and v2, v3 and one 2-simplexv0, v1, v2. Describe a sequence of elementary collapses and expansions trans-forming it to the one-point-space •.

Recall that the mapping cylinder cyl(f) of a map f : X → Y is defined bythe pushout


X × 0f //

Y

X × [0, 1] // cyl(f)

There are natural inclusions iX : X = X ×1 → cyl(f) and iY : Y → cyl(f)and a natural projection p : cyl(f)→ Y . Notice that iX is a cofibration andp iX = f and pY iY = idY . Define the mapping cone cone(f) by thequotient cyl(f)/iX(X).

Lemma 2.39. Let f : X → Y be a cellular map of finite CW -complexes andA ⊂ X be a CW -subcomplex. Then the inclusion cyl(f |A) → cyl(f) and (inthe case A = ∅) iY : Y → cyl(f) is a composition of elementary expansionsand hence a simple homotopy equivalence.

Proof. It suffices to treat the case, whereX is obtained from A by attaching ann-cell by an attaching map q : Sn−1 → X. Then there is an obvious pushout

Sn−1 × [0, 1] ∪Sn−1×0 Dn × 0 //

cyl(f |A)

Dn × [0, 1] // cyl(f)

and an obvious homeomorphism (Dn × [0, 1], Sn−1 × [0, 1] ∪Sn−1×0 Dn ×

0)→ (Dn+1, Sn+). ut

Lemma 2.40. A map f : X → Y of finite CW -complexes is a simple ho-motopy equivalence if and only if iX : X → cyl(f) is a simple homotopyequivalence.

Proof. This follows from Lemma 2.39 since a composition of simple homotopyequivalence and a homotopy inverse of a simple homotopy equivalence is againa simple homotopy equivalence. ut

We only sketch the proof of the next result. More details can be found forinstance in [190, (22.2)] or [525, Chapter 2]. However, we try to give enoughinformation about its proof to illustrate, why the geometric problem to decidewhether a homotopy equivalence is simple, is equivalent to a question aboutan invertible matrix A, which has a positive answer precisely if the class ofA vanishes in the Whitehead group. The key will be Remark 2.13.

Theorem 2.41 (Whitehead torsion and simple homotopy equiva-lences).

(i) Let X be a finite CW -complex. Then for any element x ∈ Wh(π1(X))there is an inclusion i : X → Y of finite CW -complexes such that i is ahomotopy equivalence and i−1

∗ (τ(i)) = x;

2.4 Geometric Interpretation of Whitehead Group and Whitehead Torsion 71

(ii) Let f : X → Y be a homotopy equivalence of finite CW -complexes. Thenits Whitehead torsion τ(f) ∈ Wh(π1(Y )) vanishes if and only if f is asimple homotopy equivalence.

Proof. (i) We can assume without loss of generality that X is connected. Putπ = π1(X). Choose an element A ∈ GL(n,Zπ) representing x ∈ Wh(π).Choose n ≥ 2. In the sequel we fix a zero-cell in X as base point. PutX ′ = X ∨ ∨nj=1S

n. Let bj ∈ πn(X ′) be the element represented by theinclusion of the j-th copy of Sn into X ′ for j = 1, 2 . . . , n. Recall that πn(X ′)is a Zπ-module. Choose for i = 1, 2 . . . , n a map fi : S

n → X ′n such that[fi] =

∑nj=1 ai,j ·bj holds in πn(X ′). Attach to X ′ for each i ∈ 1, 2 . . . , n an

(n+1)-cell by fi : Sn → X ′n. Let Y be the resulting CW -complex and i : X →

Y be the inclusion. Then i is an inclusion of finite CW -complexes and inducesan isomorphism on the fundamental groups. In the sequel we identify π andπ1(Y ) by π1(i). The cellular Zπ-chain complex C∗(Y , X) is concentrated indimensions n and (n + 1) and its (n + 1)-differential is given by the matrix

A with respect to the cellular basis. Hence C∗(Y , X) is a contractible finite

based free Zπ-chain complex with τ(C∗(Y , X)) = [A] in Wh(π). This impliesthat i : X → Y is a homotopy equivalence with i−1

∗ (τ(i)) = x.

(ii) Suppose that f is a simple homotopy equivalence. We want to showτ(f) = 0. Because of Theorem 2.34 (iii) it suffices to prove for an elementaryexpansion j : X → Y that its Whitehead torsion τ(j) ∈Wh(Y ) vanishes. Wecan assume without loss of generality that Y is connected. In the sequel wewrite π = π1(Y ) and identify π = π1(X) by π1(f). The following diagram ofbased free finite Zπ-chain complexes

0 // C∗(X)C∗(j) // C∗(Y )

pr∗ // C∗(Y , X) // 0

0 // C∗(X)id∗ //

id∗

OO

C∗(X)pr∗ //

C∗(j)

OO

0 //

0∗

OO

0

has based exact rows and Zπ-chain homotopy equivalences as vertical arrows.Elementary facts about chain complexes, in particular the conclusion fromExercise 2.33, imply

τ(C∗(j)

)= τ

(id∗ : C∗(X)→ C∗(X)

)+ τ(0∗ : 0→ C∗(Y , X)

)= 0 + τ

(C∗(Y , X)

)= τ

(C∗(Y , X)

).

The Zπ-chain complex C∗(Y , X) is concentrated in two consecutive dimen-sions and its only non-trivial differential is id : Zπ → Zπ if we identify the twonon-trivial Zπ-chain modules with Zπ using the cellular basis. This impliesτ(C∗(Y , X)) = 0 and hence τ(j) := τ(C∗(j)) = 0.


Now suppose that τ(f) = 0. We want to show that f is simple. We canassume without loss of generality that X is connected, otherwise treat eachpath component separately. Because of Lemma 2.40 we can assume thatf is an inclusion i : X → Y of connected finite CW -complexes which is ahomotopy equivalence. We have to show that we can achieve by a sequenceof elementary collapses and expansions that Y = X, i.e., we must get rid ofall the cells in Y −X.

Since χ(X) = χ(Y ), it is clear that one cannot remove a single cell, thisalways has to be done in pairs. In the first step one shows for an n-dimensionalcell en that one can attach one new (n+ 1)-cell en+1 and a new (n+ 2)-cellen+2 by an elementary expansion and then get rid of en and en+1 by anelementary collapse. The outcome is that one can replace an n-cell by a(n + 2)-cell. Analogously one can show that one can replace an (n + 2)-cellby a n-cell. Thus one can arrange for some integer n ≥ 2 that Y is obtainedfrom X by attaching k cells of dimension n trivially and then attaching kcells of dimension (n+ 1). Hence the cellular Zπ-chain complex C∗(Y , X) isconcentrated in dimension n and (n+1). After we have picked a cellular basis,its (n+ 1)-differential is given by an invertible (k, k)-matrix A. By definitionτ(f) is the class of this matrix in Wh(π). In Remark 2.13 we have describedwhat τ(f) = [A] = 0 means, namely, there is a sequence of operations whichtransform A to the empty matrix. Notice that X = Y precisely if A is theempty matrix. Now the main idea is to show that each of this operations canbe realized by elementary expansions and collapses. ut

Next we describe the Whitehead group geometrically. Fix a finite CW -complex X. Consider two pairs of finite CW -complexes (Y,X) and (Z,X)such that the inclusions of X into Y and Z are homotopy equivalences. Wecall them equivalent, if there is a chain of pairs of finite CW -complexes

(Y,X) = (Y [0], X), (Y [1], X), (Y [2], X), . . . , (Y [n], X) = (Z,X),

such that for each k ∈ 1, 2, . . . , n either Y [k] is obtained from Y [k − 1] byan elementary expansion or Y [k− 1] is obtained from Y [k] by an elementaryexpansion. Denote by Whgeo(X) the equivalence classes [Y,X] of such pairs(Y,X). This becomes an abelian group under the addition [Y,X] + [Z,X] :=[Y ∪X Z,X]. The zero element is given by [X,X]. The inverse of [Y,X] isconstructed as follows. Choose a map r : Y → X with rX = id. Let p : X ×[0, 1] → X be the projection. Then [(cyl(r) ∪p X) ∪r X,X] + [Y,X] = 0. Amap g : X → X ′ induces a homomorphism g∗ : Whgeo(X) → Whgeo(X ′) bysending [Y,X] to [Y ∪g X ′, X ′]. We obviously have id∗ = id and (g h)∗ =g∗ h∗. In other words, we obtain a covariant functor on the category of finiteCW -complexes with values in abelian groups.

Given a homotopy equivalence of finite CW -complexes f : X → Y , de-fine its geometric Whitehead torsion τgeo(f) ∈ Whgeo(X) to be the class of(cyl(f), X). Because of Lemma 2.40 we have τgeo(f) = 0 if and only f is asimple homotopy equivalence

2.5 The s-Cobordism Theorem 73

The next result is essentially a consequence of Theorem 2.41. Details ofits proof can be found in [190, §21].

Theorem 2.42 (Geometric and algebraic Whitehead groups).

(i) Let X be a finite CW -complex. The map

τ : Whgeo(X)→Wh(π1(X))

sending [Y,X] to i−1∗ τ(i) for the inclusion i : X → Y is a natural isomor-

phism of abelian groups.It sends τgeo(f) to f−1

∗ τ(f) for a homotopy equivalence f : X → Y of finiteCW -complexes.

(ii) A homotopy equivalence f : X → Y is a simple homotopy equivalence ifand only if τ(f) ∈Wh(Y ) vanishes.

Exercise 2.43. Let Y be a simply connected finitely dominated CW -complex.Show that there exists a finite CW -complex X and a homotopy equivalencef : X → Y . Prove that for any two finite CW -complexes X0 and X1 and ho-motopy equivalences fi : Xi → Y for i = 0, 1 there exists a simple homotopyequivalence g : X0 → X1 with f1 g ' f0.

2.5 The s-Cobordism Theorem

One of the main applications of Whitehead torsion is the theorem below.

Theorem 2.44 (s-Cobordism Theorem). Let M0 be a closed connectedmanifold of dimension n ≥ 5 with fundamental group π = π1(M0). Then

(i) Let (W ;M0, f0,M1, f1) be an h-cobordism over M0. Then W is trivial overM0 if and only if its Whitehead torsion τ(W,M0) ∈Wh(π) vanishes;

(ii) For any x ∈ Wh(π) there is an h-cobordism (W ;M0, f0,M1, f1) over M0

with τ(W,M0) = x ∈Wh(π);(iii) The function assigning to an h-cobordism (W ;M0, f0,M1, f1) over M0 its

Whitehead torsion yields a bijection from the diffeomorphism classes rela-tive M0 of h-cobordisms over M0 to the Whitehead group Wh(π).

Here are some explanations. An n-dimensional cobordism (sometimes alsocalled just bordism) (W ;M0, f0,M1, f1) consists of a compact n-dimensionalmanifold W , closed (n − 1)-dimensional manifolds M0 and M1, a disjointdecomposition ∂W = ∂0W

∐∂1W of the boundary ∂W of W and diffeomor-

phisms f0 : M0 → ∂W0 and f1 : M1 → ∂W1. If we want to specify M0, wesay that W is a cobordism over M0. If ∂0W = M0, ∂1W = M1 and f0 andf1 are given by the identity or if f0 and f1 are obvious from the context,we briefly write (W ; ∂0W,∂1W ). We call a cobordism (W ;M0, f0,M1, f1) an


h-cobordism, if the inclusions ∂iW → W for i = 0, 1 are homotopy equiva-lences. Two cobordisms (W ;M0, f0,M1, f1) and (W ′;M0, f

′0,M

′1, f′1) over M0

are diffeomorphic relative M0 if there is a diffeomorphism F : W →W ′ withF f0 = f ′0. We call an h-cobordism over M0 trivial , if it is diffeomorphicrelative M0 to the trivial h-cobordism (M0 × [0, 1];M0 × 0, (M0 × 1)).Notice that the choice of the diffeomorphisms fi do play a role although theyare often suppressed in the notation.

The Whitehead torsion of an h-cobordism (W ;M0, f0,M1, f1) over M0

τ(W,M0) ∈Wh(π1(M0))(2.45)

is defined to be the preimage of the Whitehead torsion (see Definition 2.32)

τ(M0

f0−→ ∂0Wi0−→W

)∈Wh(π1(W ))

under the isomorphism (i0 f0)∗ : Wh(π1(M0))∼=−→ Wh(π1(W )), where the

map i0 : ∂0W → W is the inclusion. Here we use the fact that each smoothclosed manifold has a CW -structure, which comes for instance from a smoothtriangulation, or that each closed topological manifold of dimension differentfrom 4 has a CW -structure, which comes from a handlebody decomposition,and that the choice of CW -structure does not matter by the topologicalinvariance of the Whitehead torsion (see Theorem 2.35).

The idea of the proof of Theorem 2.44 is analogous to the one of Theo-rem 2.41 but now one uses a handlebody decomposition instead of a CW -structure and carries out the manipulation for handlebodies instead of forcells. Here a handlebody of index k corresponds to a k-dimensional cell. Moredetails can be found for instance in [206, Chapter 1].

The s-Cobordism Theorem 2.44 is due to Barden, Mazur, Stallings. Itstopological version was proved by Kirby and Siebenmann [473, Essay II].More information about the s-cobordism theorem can be found for instancein [468], [525, Chapter 1], [590],[591], [721, page 87–90]. The s-cobordism the-orem is known to be false (smoothly) for n = dim(M0) = 4 in general, bythe work of Donaldson [252], but it is true for n = dim(M0) = 4 for so called“good” fundamental groups in the topological category by results of Freed-man [328], [329]. The trivial group is an example of a “good” fundamentalgroup. Counterexamples in the case n = dim(M0) = 3 are constructed byCappell and Shaneson [156].

Exercise 2.46. Show for n ≥ 6 that there exists an n-dimensional h-cobordism (W ;M0,M1) which is not trivial such that the h-cobordism(W × S3;M0 × S3,M1 × S3) is trivial.

Since the Whitehead group of the trivial group vanishes (see Theo-rem 2.17), the s-Cobordism Theorem 2.44 implies (see also [590])

2.5 The s-Cobordism Theorem 75

Theorem 2.47 (h-Cobordism Theorem). Let M0 be a simply connectedclosed n-dimensional manifold with dim(M0) ≥ 5. Then every h-cobordism(W ;M0, f0,M1, f1) over M0 is trivial.

Theorem 2.48 (Poincare Conjecture). The Poincare Conjecture is truefor a closed n-dimensional manifold M with dim(M) ≥ 5, namely, if M issimply connected and its homology Hp(M) is isomorphic to Hp(S

n) for allp ∈ Z, then M is homeomorphic to Sn.

Proof. We only give the proof for dim(M) ≥ 6. Since M is simply connectedand H∗(M) ∼= H∗(S

n), one can conclude from the Hurewicz Theorem andWhitehead Theorem [832, Theorem IV.7.13 on page 181 and Theorem IV.7.17on page 182] that there is a homotopy equivalence f : M → Sn. Let Dn

i ⊂Mfor i = 0, 1 be two embedded disjoint disks. Let W be obtained from M byremoving the interior of the two disks Dn

0 and Dn1 . Then W turns out to be a

simply connected h-cobordism. Hence we can find because of Theorem 2.47 ahomeomorphism F : (∂Dn

0 × [0, 1], ∂Dn0 × 0, ∂Dn

0 × 1)→ (W,∂Dn0 , ∂D

n1 )

which is the identity on ∂Dn0 = ∂Dn

0 × 0 and induces some (unknown)homeomorphism f1 : ∂Dn

0 × 1 → ∂Dn1 . By the Alexander trick one can

extend f1 : ∂Dn0 = ∂Dn

0 × 1 → ∂Dn1 to a homeomorphism f1 : Dn

0 → Dn1 .

Namely, any homeomorphism f : Sn−1 → Sn−1 extends to a homeomorphismf : Dn → Dn by sending t · x for t ∈ [0, 1] and x ∈ Sn−1 to t · f(x). Nowdefine a homeomorphism h : Dn

0 × 0 ∪i0 ∂Dn0 × [0, 1] ∪i1 Dn

0 × 1 → Mfor the canonical inclusions ik : ∂Dn

0 × k → ∂Dn0 × [0, 1] for k = 0, 1 by

h|Dn0×0 = id, h|∂Dn0×[0,1] = F and h|Dn0×1 = f1. Since the source of h isobviously homeomorphic to Sn, Theorem 2.48 follows.

In the case dim(M) = 5 one uses the fact that M is the boundary of acontractible 6-dimensional manifold W and applies the s-cobordism theoremto W with an embedded disc removed. ut

The Poincare Conjecture (see Theorem 2.48) is nowadays also known indimension 3 by work of Perelman (see [474, 612, 613, 655, 656, 657]) and indimension 4 by work of Freedman (see [328], [329]). It is obviously true indimensions 1 and 2.

Remark 2.49 (Exotic Spheres). Notice that the proof of the PoincareConjecture in Theorem 2.48 works only in the topological category but notin the smooth category. In other words, we cannot conclude the existence of adiffeomorphism h : Sn →M . The proof in the smooth case breaks down whenwe apply the Alexander trick. The construction of f given by coning f yieldsonly a homeomorphism f and not a diffeomorphism, even if we start witha diffeomorphism f . The map f is smooth outside the origin of Dn but notnecessarily at the origin. Indeed, not every diffeomorphism f : Sn−1 → Sn−1

can be extended to a diffeomorphism Dn → Dn and there exist so calledexotic spheres, i.e., closed manifolds which are homeomorphic to Sn but notdiffeomorphic to Sn. The classification of these exotic spheres is one of the


early very important achievements of surgery theory and one motivation forits further development. For more information about exotic spheres we referfor instance to [206, Chapter 11], [469], [497], [509] and [525, Chapter 6].

Remark 2.50 (The surgery program). In some sense the s-CobordismTheorem 2.44 is one of the first theorems, where diffeomorphism classesof certain manifolds are determined by an algebraic invariant, namely, theWhitehead torsion. Moreover, the Whitehead group Wh(π) depends only onthe fundamental group π = π1(M0), whereas the diffeomorphism classes ofh-cobordisms over M0 a priori depend on M0 itself. The s-Cobordism The-orem 2.44 is one step in a program to decide whether two closed manifoldsM and N are diffeomorphic what is in general a very hard question. Theidea is to construct an h-cobordism (W ;M,f,N, g) with vanishing White-head torsion. Then W is diffeomorphic to the trivial h-cobordism over Mwhat implies that M and N are diffeomorphic. So the surgery program is:

(i) Construct a simple homotopy equivalence f : M → N ;(ii) Construct a cobordism (W ;M,N) and a map (F, f, id) : (W ;M,N) →

(N × [0, 1], N × 0, N × 1);(iii) Modify W and F relative boundary by so called surgery such that F be-

comes a simple homotopy equivalence and thusW becomes an h-cobordismwhose Whitehead torsion is trivial.

The advantage of this approach will be that it can be reduced to prob-lems in homotopy theory and algebra which can sometimes be handled bywell-known techniques. In particular one will get sometimes computable ob-structions for two homotopy equivalent manifolds to be diffeomorphic. Oftensurgery theory has proved to be very useful when one wants to distinguishtwo closed manifolds which have very similar properties. The classificationof homotopy spheres is one example. Moreover, surgery techniques can beapplied to problems which are of different nature than of diffeomorphismor homeomorphism classifications, for instance for the construction of groupactions.

More information about surgery theory will be given in Chapter 7.

2.6 Reidemeister Torsion and Lens Spaces

In this section we briefly deal with Reidemeister torsion which was definedearlier than Whitehead torsion and motivated the definition of Whiteheadtorsion. Reidemeister torsion was the first invariant in algebraic topologywhich could distinguish between spaces which are homotopy equivalent butnot homeomorphic. Namely, it can be used to classify lens spaces up to home-omorphism (see Reidemeister [693]). We will give no proofs. More informationand complete proofs can be found in [190, Chapter V] and [525, Section 2.4].

2.6 Reidemeister Torsion and Lens Spaces 77

Let X be a finite CW -complex with fundamental group π. Let U be anorthogonal finite dimensional π-representation. Denote by Hp(X;U) the ho-mology of X with coefficients in U , i.e., the homology of the R-chain complexU ⊗Zπ C∗(X). Suppose that X is U -acyclic, i.e., Hn(X;U) = 0 for all n ≥ 0.

If we fix a cellular basis for C∗(X) and some orthogonal R-basis for U , then

U ⊗Zπ C∗(X) is a contractible based free finite R-chain complex and yields

an element τ(U ⊗Zπ C∗(X)) ∈ K1(R), see (2.29). Define the Reidemeistertorsion

ρ(X;U) ∈ R>0(2.51)

to be the image of τ(U ⊗Zπ C∗(X)) ∈ K1(R) under the homomorphism

K1(R) → R>0 sending the class [A] of A ∈ GL(n,R) to |det(A)|. Noticethat for any trivial unit ±γ the automorphism of U given by multiplicationwith ±γ is orthogonal and that the absolute value of the determinant of anyorthogonal automorphism of U is 1. Therefore ρ(X;U) ∈ R>0 is independent

of the choice of cellular basis for C∗(X) and the orthogonal basis for U , andhence is an invariant of the CW -complexX and the orthogonal representationU .

We state without proof the next result, which essential says that White-head torsion of a homotopy equivalence is related to the difference of Reide-meister torsion of the target and the source when defined.

Lemma 2.52. Let f : X → Y be a homotopy equivalence of connected finiteCW -complexes and let U be an orthogonal finite dimensional π = π1(Y )-representation. Suppose that Y is U -acyclic. Let f∗U be the orthogonalπ1(X)-representation obtained from U by restriction with the isomorphismπ1(f). Let dU : Wh(π(Y )) → R>0 be the map sending the class [A] ofA ∈ GL(n,Zπ1(Y )) to |det(idU ⊗ZπrA : U ⊗Zπ Zπn → U ⊗Zπ Zπn)|.

Thenρ(Y ;U)

ρ(X; f∗U)= dU (τ(f)).

Next we introduce lens spaces. Let G be a cyclic group of finite order |G|.Let V be a unitary finite dimensional G-representation. Define its unit sphereSV and its unit disk DV to be the G-subspaces SV = v ∈ V | ||v|| = 1 andDV = v ∈ V | ||u|| ≤ 1 of V . Notice that a complex finite dimensional vec-tor space has a preferred orientation as real vector space, namely the one givenby the R-basis b1, ib1, b2, ib2, . . . , bn, ibn for any C-basis b1, b2, . . . , bn.Any C-linear automorphism of a complex finite dimensional vector space pre-serves this orientation. Thus SV and DV are compact oriented Riemannianmanifolds with isometric orientation preserving G-action. We call a unitaryG-representation V free if the induced G-action on its unit sphere SV is free.Then SV → G\SV is a covering and the quotient space L(V ) := G\SVinherits from SV the structure of an oriented closed Riemannian manifold.


Definition 2.53 (Lens space). We call the closed oriented Riemannian ma-nifold L(V ) the lens space associated to the free finite dimensional unitaryrepresentation V of the finite cyclic group G.

Exercise 2.54. Show that the 3-dimensional real projective space RP3 is alens space. Let R− be the non-trivial orthogonal Z/2-representation. Showthat RP3 is R−-acyclic and compute the Reidemeister torsion ρ(RP3;R−).

One can specify these lens spaces also by numbers as follows.

Notation 2.55. Let Z/t be the cyclic group of order t ≥ 2. The 1-dimensionalunitary representation Vk for k ∈ Z/t has as underlying vector space C andl ∈ Z/t acts on it by multiplication with exp(2πikl/t). Notice that Vk is freeif and only if k ∈ (Z/t)×, and is trivial if and only if k = 0 in Z/t. Definethe lens space L(t; k1, . . . , kc) for an integer c ≥ 1 and elements k1, . . . , kcin (Z/t)× by L(⊕ci=1Vki).

Lens spaces form a very interesting family of manifolds which can be com-pletely classified as we will see. Two lens spaces L(V ) and L(W ) of the samedimension n ≥ 3 have the same homotopy groups, namely, their fundamentalgroup is G and their p-th homotopy group is isomorphic to πp(S

n). They alsohave the same homology with integral coefficients, namely Hp(L(V )) ∼= Z forp = 0, 2n− 1, Hp(L(V )) ∼= G for p odd and 1 ≤ p < n and Hp(L(V )) = 0 forall other values of p. Also their cohomology groups agree. Nevertheless not allof them are homotopy equivalent. Moreover, there are homotopy equivalentlens spaces which are not diffeomorphic (see Example 2.59).

We state without proof the following result.

Theorem 2.56 (Homotopy Classification of Lens Spaces). The lensspaces L(t; k1, . . . , kc) and L(t; l1, . . . , lc) are homotopy equivalent if and onlyif there is e ∈ (Z/t)× such that we get

∏ci=1 ki = ±ec ·

∏ci=1 li in (Z/t)×.

The lens spaces L(t; k1, . . . , kc) and L(t; l1, . . . , lc) are oriented homotopyequivalent if and only if there is e ∈ (Z/t)× such that we get

∏ci=1 ki =

ec ·∏ci=1 li in (Z/t)×.

Theorem 2.57 (Diffeomorphism Classification of Lens Spaces).

(i) Let G be a finite cyclic group. Let L(V ) and L(W ) be two lens spaces ofthe same dimension n ≥ 3. Then the following statements are equivalent:

(a) There is an automorphism α : G → G such that V and α∗W are iso-morphic as orthogonal G-representations;

(b) There is an isometric diffeomorphism L(V )→ L(W );(c) There is a diffeomorphism L(V )→ L(W );(d) There is a homeomorphism L(V )→ L(W );(e) There is a simple homotopy equivalence L(V )→ L(W );

2.6 Reidemeister Torsion and Lens Spaces 79

(f) There is an automorphism α : G→ G such that for any orthogonal finitedimensional representation U with UG = 0

ρ(L(W );U) = ρ(L(V );α∗U)

holds;(g) There is an automorphism α : G → G such that for any non-trivial

1-dimensional unitary G-representation U

ρ(L(W ); resU) = ρ(L(V );α∗ resU)

holds, where the orthogonal representation resU is obtained from U byrestricting the scalar multiplication from C to R;

(ii) Two lens spaces L(t; k1, . . . , kc) and L(t; l1, . . . , lc) are homeomorphic ifand only if there are e ∈ (Z/t)×, signs εi ∈ ±1 and a permutationσ ∈ Σc such that ki = εi · e · lσ(i) holds in (Z/t)× for i = 1, 2, . . . , c.

Proof. We give only a sketch of the proof of assertion (i). Assertion (ii) is adirect consequence of assertion (i).

The implications (ia)⇒ (ib)⇒ (ic)⇒ (id) and (if)⇒ (ig) are obvious. Theimplication (id)⇒ (ie) follows from Theorem 2.35. The implication (ie)⇒ (if)follows from Lemma 2.52. The hard part of the proof is the implication (ig)⇒ (ia). It involves proving the formula

ρ(L(V ⊕W ); resU) = ρ(L(V ); resU) · ρ(L(W ); resU)

for two free unitary G-representations V and W and then directly computingρ(L(V ); resU) for every free 1-dimensional unitary representation V . Finallyone has to show that the values of the Reidemeister torsion do distinguish theunitary representations V and W up to automorphisms of G. This proof isbased on the following number theoretic result mentioned below whose proofcan be found for instance in Franz [326] or in [235]. ut

Lemma 2.58 (Franz’ independence Lemma). Let t ≥ 2 be an integerand S = j ∈ Z | 0 < j < t, (j, t) = 1. Let (aj)j∈S be a sequence ofintegers indexed by S such that

∑j∈S aj = 0, aj = at−j for j ∈ S and∏

j∈S(ζj − 1)aj = 1 holds for every t-th root of unity ζ 6= 1. Then aj = 0 forj ∈ S.

Example 2.59. We conclude from Theorem 2.56 and Theorem 2.57 (ii) thefollowing facts:

(i) Any homotopy equivalence L(7; k1, k2) → L(7; k1, k2) has degree 1. ThusL(7; k1, k2) possesses no orientation reversing selfdiffeomorphism;

(ii) L(5; 1, 1) and L(5; 2, 1) have the same homotopy groups, homology groupsand cohomology groups, but they are not homotopy equivalent;

(iii) L(7; 1, 1) and L(7; 2, 1) are homotopy equivalent, but not homeomorphic.


Example 2.60 (h-cobordisms between lens spaces). The rigidity of lensspaces is illustrated by the following fact. Let (W,L,L′) be an h-cobordism oflens spaces which is compatible with the orientations and the identificationsof π1(L) and π1(L′) with G. Then W is diffeomorphic relative L to L× [0, 1]and L and L′ are diffeomorphic [591, Corollary 12.13 on page 410].

Remark 2.61 (Differential geometric characterization of lens spaces).Lens spaces with their preferred Riemannian metric have constant positivesectional curvature. A closed Riemannian manifold with constant positivesectional curvature and cyclic fundamental group is isometrically diffeomor-phic to a lens space after possibly rescaling the Riemannian metric with aconstant [840].

Remark 2.62 (de Rham’s Theorem). The results above when interpretedas statements about unit spheres in free representations are generalized byDe Rham’s Theorem [234] (see also [519, Proposition 3.2 on page 478], [522,page 317], [720, section 4]) as follows. It says for a finite group G and two or-thogonal G-representations V and W whose unit spheres SV and SW are G-diffeomorphic that V and W are isomorphic as orthogonal G-representations.This remains true if one replaces G-diffeomorphic by G-homeomorphic pro-vided that G has odd order (see [405], [570]), but not for any finite group G(see [155, 157, 366, 368, 369]).

We refer to [190],[525, Chapter 2] and [591] for more information aboutReidemeister torsion and lens spaces.

Remark 2.63 (Further appearance of Reidemeister torsion). TheAlexander polynomial of a knot can be interpreted as a kind of Reidemeis-ter torsion of the canonical infinite cyclic covering of the knot complement(see [589], [794]). Reidemeister torsion appears naturally in surgery the-ory [566]. Counterexamples to the (polyhedral) Hauptvermutung that twohomeomorphic simplicial complexes are already PL-homeomorphic are givenby Milnor [588] (see also [688]) and detected by Reidemeister torsion. Seiberg-Witten invariants for 3-manifolds are closely related to torsion invariants (seeTuraev [793]).

Remark 2.64 (Analytic Reidemeister torsion). Ray-Singer [691] de-fined the analytic counterpart of topological Reidemeister torsion using aregularization of the zeta-function. Ray and Singer conjectured that the ana-lytic and topological Reidemeister torsion agree. This conjecture was provedindependently by Cheeger [183] and Muller [614]. Manifolds with boundaryand manifolds with symmetries, sum (= glueing) formulas and fibration for-mulas are treated in [139, 216, 519, 522, 556, 799]. For a survey on analytic andtopological torsion we refer for instance to [538]. There are also L2-versionsof these notions, see for instance [144, 160, 517], [527, Chapter 3], [553, 574].

2.7 The Bass-Heller-Swan Theorem for K1 81

2.7 The Bass-Heller-Swan Theorem for K1

In the section we want to compute K1(R[Z]) for a ring R. This computation,the so called Bass-Heller-Swan decomposition, marks the beginning of the(long) way towards the final formulation of the Farrell-Jones Conjecture foralgebraic K-theory.

2.7.1 The Bass-Heller-Swan Decomposition for K1

We need some preparation to formulate it. In the sequel we write R[Z] as thering R[t, t−1] of finite Laurent polynomials in t with coefficients in R. Definethe ring homomorphisms

ε : R[t] → R,∑n∈Z rnt

n 7→∑n∈Z rn;

i′ : R → R[t], r 7→ r · t0;i : R → R[t, t−1], r 7→ r · t0.

The ring homomorphism ε is the one induced by the group homomorphismZ→ 1, whereas i is the one induced by the inclusion of groups 1 → Z.

Definition 2.65 (NKn(R)). Define for n = 0, 1

NKn(R) := ker (ε∗ : Kn(R[t])→ Kn(R)) .

Example 2.66. Let F be a field. Put R = F [t]/(t2). Every element in R canbe uniquely written as a+bt for a, b ∈ F . We have (1+bt)·(1−bt) = 1−b2t2 = 1in R. Hence the element a+bt ∈ R is a unit if and only if a 6= 0. We concludethat R is a local ring with (t) = bt | b ∈ F as the unique maximal ideal.Since R is commutative, the homomorphism

iR : R×∼=−→ K1(R), [x] 7→ [rx : R→ R]

is bijective by Theorem 2.6. Let ε : R→ F be the ring homomorphism sendinga+bt to a. Its kernel is (t). It induces a group homomorphism R[x]× → F [x]×.Since F [x]× is the multiplicative group of non-trivial polynomials over F ofdegree 0 and (1 + tvx) · (1 − tvx) = 1 − v2t2x2 = 1 holds in R[x] for allv ∈ F [x], we obtain an isomorphism of abelian groups

φ : R× ⊕ F [x]∼=−→ R[x]×, (u, v) 7→ u · (1 + tvx).

Since R[x] is commutative, the map iR[x] : R[x]×∼=−→ K1(R[x]) is injective,

a retraction is given by the determinant. We conclude that the followingcomposition is an injection of abelian groups


F [x]φ|F [x]−−−−→ coker

(R× → R[x]×

) i−→ coker (K1(R)→ K1(R[x])) ∼= NK1(R),

where i is the map induced by iR and iR[x]. This implies that NK1(R) is anabelian group which is not finitely generated.

Example 2.66 illustrates the following fact. If R is any ring, then NK1(R)is either trivial or infinitely generated as abelian group (see Theorem 5.20). Soin general NK1(R) is hard to compute. At least we have the following usefulresults. IfR is a ring of finite characteristicN , then we getNKn(R)[1/N ] = 0for n = 0, 1 (see Theorem 5.17). If NKn(R) = 0 and G is finite, thenNKn(RG)[1/|G|] = 0 for n = 0, 1 (see Theorem 5.18).

Recall that an endomorphism f : P → P of an R-module P is callednilpotent if there exists a positive integer n with fn = 0.

Definition 2.67 (Nil-group Nil0(R)). Define the 0-th Nil-group Nil0(R) tobe the abelian group, whose generators are conjugacy classes [f ] of nilpotentendomorphisms f : P → P of finitely generated projective R-modules withthe following relation. Given a commutative diagram of finitely generatedprojective R-modules

0 // P1i //

f1

P2p //

f2

P3//

f3

0

0 // P1i // P2

p // P3// 0

with exact rows and nilpotent endomorphisms as vertical arrows, we get

[f1] + [f3] = [f2].

Let ι : K0(R) → Nil0(R) be the homomorphism sending the class [P ] ofa finitely generated projective R-module P to the class [0 : P → P ] of thetrivial endomorphism of P .

Definition 2.68 (Reduced Nil-group Nil0(R)). Define the reduced 0-th

Nil-groups Nil0(R) by the cokernel of the map ι.

The homomorphism Nil0(R)→ K0(R), [f : P → P ] 7→ [P ] is a retractionof the map ι. So we get a natural splitting

Nil0(R)∼=−→ Nil0(R)⊕K0(R).

Denote byj : NK1(R)→ K1(R[t])

the inclusion. Letτ± : R[t]→ R[t, t−1]

be the inclusion of rings sending t to t±1. Define


j± := (τ±)∗ j : NK1(R)→ K1(R[t, t−1]).

The homomorphismB : K0(R)→ K1(R[t, t−1])

sends the class [P ] of a finitely generated projective R-module P to theclass [rt ⊗R idP ] of the R[t, t−1]-automorphism rt ⊗R idP : R[t, t−1]⊗R P →R[t, t−1]⊗R P which maps u⊗ p to ut⊗ p. The homomorphism

N ′ : Nil0(R)→ K1(R[t])

sends the class [f ] of the nilpotent endomorphism f : P → P of the finitelygenerated projective R-module P to the class [id−rt−1⊗R f ] of the R[t, t−1]-automorphism

id−rt−1 ⊗R f : R[t, t−1]⊗R P → R[t, t−1]⊗R P,u⊗ p 7→ u⊗ p− u(t− 1)⊗ f(p).

This is indeed an automorphism. Namely, if fn+1 = 0, then an inverse is givenby∑nk=0 (rt−1 ⊗R f)

k. The composition of N ′ with both ε∗ : K1(R[t]) →

K1(R) and ι : K0(R)→ Nil0(R) is trivial. Hence N ′ induces a homomorphism

N : Nil0(R)→ NK1(R).

The proof of the following theorem can be found for instance in [87] (forregular rings), [84, Chapter XII], [704, Theorem 3.2.22 on page 149] and [826,3.6 on page 225].

Theorem 2.69 (Bass-Heller-Swan decomposition for K1). The fol-lowing maps are isomorphisms of abelian groups, natural in R,

N : Nil0(R)∼=−→ NK1(R);

j ⊕ i′∗ : NK1(R)⊕K1(R)∼=−→ K1(R[t]);

B ⊕ i∗ ⊕ j+ ⊕ j− : K0(R)⊕K1(R)⊕NK1(R)⊕NK1(R)∼=−→ K1(R[t, t−1]).

One easily checks that Theorem 2.69 applied to R = ZG implies the fol-lowing reduced version

Theorem 2.70 (Bass-Heller-Swan decomposition for Wh(G×Z)). LetG be a group. Then there is an isomorphism of abelian groups, natural in G

B⊕ i∗⊕j+⊕j− : K0(ZG)⊕Wh(G)⊕NK1(ZG)⊕NK1(ZG))∼=−→Wh(G×Z).

Example 2.71 (K0(ZG) affects Wh(G)). The Whitehead group Wh(Sn)of the symmetric group Sn is trivial (see Theorem 2.113 (iii)), whereas

K0(Z[Sn])) is a finite non-trivial group for n ≥ 5 (see Theorem 1.97 (ii).)


In the sequel we let n ≥ 5. We conclude from Theorem 2.70 that Wh(Sn×Z)is non-trivial for n ≥ 5, whereas the obvious map

colimH∈SubFIN (Sn×Z) Wh(H)→Wh(Sn × Z)

is the zero map and hence not surjective. Also the map

colimH∈SubFIN (G)K1(ZH)→ K1(Z[Sn × Z])

cannot be surjective. Hence there is no hope that a formula, where one com-putes a specific K-group in terms of its values on all finite subgroups (suchas appearing in Conjecture 1.57) is true in general. The general picture willbe that a computation of a K or L-group in dimension n does involve K-and L-groups in all dimensions ≤ n.

Denote byk± : R→ R[t±1]

the ring homomorphism sending r to r · t0. Obviously τ± k± = i. Define amap

C : K1(R[t, t−1])→ K0(R)

by sending the class [f ] of anR[t, t−1]-automorphism f : R[t, t−1]n → R[t, t−1]n

to the element [P (f, k)]−k ·[R], where k is a large enough positive integer andP (f, k) is the finitely generated projective R-module f(tk−1 · R[t−1]) ∩ R[t].We omit the proof that P (f, k) is a finitely generated projective R-modulefor large enough k, that the class [P (f, k)] − k · [R] is independent of k anddepends only on [f ], and that the map C is a well-defined homomorphism ofabelian groups.

Theorem 2.72 (Fundamental Theorem of K-theory in dimension 1).The following sequence is natural in R and exact

0→ K1(R)(k+)∗⊕−(k−)∗−−−−−−−−−→ K1(R[t])⊕K1(R[t−1])

(τ+)∗⊕(τ−)∗−−−−−−−−→ K1(R[t, t−1])

C−→ K0(R)→ 0.

If we regard it as an acyclic Z-chain complex, there exists a chain contraction,natural in R.

Proof. One checks C B = idK0(R) and C i∗ = C j− = C j+ = 0. Nowapply Theorem 2.69. ut


2.7.2 The Grothendieck Decomposition for G0 and G1

There is also a G-theory version of the Bass-Heller-Swan decomposition whichis due to Grothendieck. Its proof can be found in [87] or [704, Theorem 3.2.12on page 141, Theorem 3.2.16 on page 143 and Theorem 3.2.19 on page 147].

Theorem 2.73 (Grothendieck decomposition for G0 and G1). Let Rbe a Noetherian ring. Then

(i) The inclusions R→ R[t] and R→ R[t, t−1] induce isomorphisms of abeliangroups

G0(R)∼=−→ G0(R[t]);

G0(R)∼=−→ G0(R[t, t−1]);

(ii) There are natural isomorphism

i′∗ : G1(R)∼=−→ G1(R[t]);

B ⊕ i∗ : G0(R)⊕G1(R)∼=−→ G1(R[t, t−1]),

where i′∗, B and i∗ are defined analogously to the maps appearing in The-orem 2.69.

Exercise 2.74. Show that the map Z∼=−→ G0(R[Zn]) sending n to n · [R[Zn]]

is an isomorphism for a principal ideal domain R and n ≥ 0.

2.7.3 Regular Rings

Theorem 2.75 (Hilbert Basis Theorem). If R is Noetherian, then R[t]and R[t, t−1] are Noetherian.

Proof. See for instance [704, Theorem 3.2.1 on page 133 and Corollary 3.2.2on page 134]. ut

Let (P ) be a property of groups, e.g., being finite or being cyclic. A groupG is called virtually (P) if G contains a subgroup H ⊂ G of finite index suchthat H has property (P). A group G is poly-(P) if there is a finite sequenceof subgroups 1 = G0 ⊂ G1 ⊂ G2 ⊂ . . . Gr = G such that Gi is normalin Gi+1 and the quotient Gi+1/Gi has property (P) for i = 0, 1, 2, . . . , r − 1.Thus the notions of virtually finitely generated abelian, virtually free, virtuallynilpotent, poly-cyclic, poly-Z, and virtually poly-cyclic are defined.

Theorem 2.76 (Noetherian group rings). If R is a Noetherian ring andG is a virtually poly-cyclic group, then RG is Noetherian.


Proof. See for instance [527, Lemma 10.55 on page 397]. ut

No counterexample is known to the conjecture that CG is Noetherian ifand only if G is virtually poly-cyclic.

Theorem 2.77 (Regular group rings).

(i) The rings R[t] and R[t, t−1] are regular, if R is regular;(ii) The ring RG is regular, if R is regular and G is poly-Z;

(iii) The ring RG is regular, if R is regular, Q ⊆ R and G is virtually poly-cyclic;

Proof. (i) This is proved for instance in [704, Theorem 3.2.3 on page 134 andCorollary 3.2.4 on page 136].

(ii) This is follows from [725, Theorem 8.2.2 on page 533 and Theorem 8.2.18on page 537] in the case, where R is a field.

(iii) This is follows from [725, Theorem 8.2.2 on page 533 and Theorem8.2.20on page 538] in the case, where R is a field. ut

A ring is called semihereditary, if every finitely generated ideal is projec-tive, or, equivalently, if every finitely generated submodule of a projectiveR-module is projective (see [165, Proposition 6.2 in Chapter I.6 on page 15]).

Theorem 2.78 (Bass-Heller-Swan decomposition for K1 for regularrings). Suppose that R is semihereditary or regular. Then

Nil0(R) = NK1(R) = 0;

and the Bass-Heller-Swan decomposition (see Theorem 2.69) reduces to theisomorphism

B ⊕ i∗ : K0(R)⊕K1(R)∼=−→ K1(R[t, t−1]).

Proof. The proof for regular R can be found for instance in [704, Exer-cise 3.2.25 on page 152] or [777, Corollary 16.5 on page 226].

Suppose that R is semihereditary. Consider a nilpotent endomorphismf : P → P of the finitely generated projective R-module P . Define I1(f) =im(f) and K1(f) = ker(f). Let f |I1(f) : I1(f)→ I1(f) be the endomorphisminduced by f . Since R is semihereditary, I1(f) is a finitely generated projec-tive R-module. We obtain a commutative diagram

0 // K1(f)i //

0

Pf //

f

I1(f) //

f |I1(f)

0

0 // K1(f)i // P

f // I1(f) // 0


with exact rows and nilpotent endomorphisms of finitely generated projectiveR-modules as vertical arrows. Hence we get [f : P → P ] = [I1(f) : I1(f)→I1(f)] in Nil0(R). Define inductively In+1(f) = I1

(f |In(f)

). Hence we get for

all n ≥ 1[f : P → P ] = [f |In(f) : In(f)→ In(f)].

Since f is nilpotent, there exists some n with In(f) = 0. This implies [f ] = 0

in Nil0(R). Now apply Theorem 2.69 ut

Exercise 2.79. Prove that K0(Z[Zn]) = Wh(Zn) = 0 for all n ≥ 0.

Remark 2.80 (Glimpse of a homological behavior of K-theory). Inthe case that R is regular, Theorem 2.78 shades some homological flavour onK-theory. Just observe the analogy between the two formulas

K1(R[Z]) ∼= K0(R[1])⊕K0(R[1]);H1(Z;A) ∼= H0(1;A)⊕H1(1;A),

where in the second line we consider group homology with coefficients in someabelian group A which corresponds to the role of R in the first line.

Remark 2.81 (Von Neumann algebras are semihereditary but notNoetherian). Notice that any von Neumann algebra is semihereditary. Thisfollows from the facts that any von Neumann algebra is a Baer ∗-ring andhence in particular a Rickart C∗-algebra [100, Definition 1, Definition 2 andProposition 9 in Chapter 1.4] and that a C∗-algebra is semihereditary if andonly if it is Rickart [31, Corollary 3.7 on page 270]. The group von Neumannalgebra N (G) is Noetherian if and only if G is finite (see [527, Exercise 9.11on page 367]).

Lemma 2.82. If R is regular, then the canonical homomorphism

f : K0(R)∼=−→ G0(R), [P ] 7→ [P ]

is a bijection.

Proof. We have to define an inverse homomorphism

r : G0(R)→ K0(R).

Given a finitely generated R-module M , we can choose a finite projectiveresolution P∗ = (P∗, φ) since R is by assumption regular. We want to define

r([M ]) :=∑n≥0

(−1)n · [Pn].

The Fundamental Lemma of Homological Algebra implies for two projectiveresolutions P∗ and Q∗ of M the existence of an R-chain homotopy equivalence


f∗ : P∗ → Q∗ (see for instance [825, Comparison Theorem 2.2.6 on page 35]).We conclude from Exercise 1.39∑

n≥0

(−1)n · [Pn] = o(P∗) = o(Q∗) =∑n≥0

(−1)n · [Qn].

Hence the choice of projective resolution does not matter in the definitionof r([M ]). It remains to show for an exact sequence of finitely generatedR-modules 0 → M → M ′ → M ′′ → 0 that r(M) − r(M ′) + r(M ′′) = 0holds. This follows again from Exercise 1.39 since we can construct fromfinite projective R-resolutions P∗ of M and P ′′∗ of M ′′ a finite projective R-resolution P ′∗ of M ′ such that there exists a short exact sequence of R-chaincomplexes 0 → P∗ → P ′∗ → P ′′∗ → 0. Hence r is well-defined. One easilychecks that r and f are inverse to one another. ut

2.8 The Mayer-Vietoris K-Theory Sequence of aPullback of Rings

In this section we want to explain

Theorem 2.83 (Mayer-Vietoris sequence for middle K-theory of apullback of rings). Consider a pullback of rings

Ri1 //

i2

R1

j1

R2

j2// R0

such that j1 or j2 is surjective. Then there exists a natural exact sequence ofsix terms

K1(R)(i1)∗⊕(i2)∗−−−−−−−→ K1(R1)⊕K1(R2)

(j1)∗−(j2)∗−−−−−−−→ K1(R0)∂1−→

K0(R)(i1)∗⊕(i2)∗−−−−−−−→ K0(R1)⊕K0(R2)

(j1)∗−(j2)∗−−−−−−−→ K0(R0).

Its construction and its proof requires some preparation. In particularwe need the following basic construction due to Milnor [592, page 20]. Letjk : Pk → (jk)∗Pk be the map sending x ∈ Pk to 1⊗x ∈ R0⊗jkPk for k = 1, 2.Define a ring homomorphism i0 = j1i1 = j2i2 : R→ R0. Given Rk-modules

Pk for k = 0, 1, 2 and isomorphisms of R0-modules fk : (jk)∗Pk∼=−→ P0 for

k = 1, 2, define an R-moduleM = M(P1, P2, f1, f2) by the pullback of abeliangroups

2.8 The Mayer-Vietoris K-Theory Sequence of a Pullback of Rings 89

M //

P1

f1j1

P2f2j2

// P0

together with the R-multiplication on M induced by the R-actions on Pkwhich comes from the ring homomorphisms ik : R→ Rk for k = 0, 1, 2.

Lemma 2.84.(i) The R-module M is projective if P0 and P1 are projective.The R-module M is finitely generated projective if P0 and P1 are finitelygenerated projective;

(ii) Every projective R-module P can be realized up to isomorphism as M forappropriate projective Rk-modules Pk for k = 0, 1, 2 and isomorphisms of

R0-modules fk : (jk)∗Pk∼=−→ P0 for k = 1, 2;

(iii) The Rk-modules (ik)∗M and Pk are isomorphic for k = 1, 2.

Proof. This is proved in Milnor [592, Theorems 2.1, 2.2 and 2.3 on page 20]or in[755, Proposition 59 on page 155, Proposition 60 on page 157, Proposi-tion 61 on page 158]. ut

Now we can give the proof of Theorem 2.83

Proof. The main step is to construct the boundary homomorphism ∂1. Given

an element x ∈ K1(R0), we can find an automorphism f : Rn0∼=−→ Rn0 of a

finitely generated free R-module with x = [f ] (see Lemma 2.10). The R0-module M(Rn1 , R

n2 , idRn0 , f) is a finitely generated projective R0-module by

Lemma 2.84 (i). Define

∂1(x) := [M(Rn1 , Rn2 , idRn0 , f)]− [Rn0 ].

This is a well-defined homomorphism of abelian groups (see [755, page 164]).The elementary proof of the exactness of the sequence of six terms can befound in [755, Proposition 63 on page 164]. ut

Now we are ready to give the promised proof of Rim’s Theorem 1.90

Proof. Consider the pullback of rings

Z[Z/p] i1 //

i2

Z[exp(2πi/p)]

j1

Z

j2// Fp

where Fpm denotes the field with pm elements, i1 sends the generator of Z/pto exp(2πi/p), the map i2 sends the generator of Z/p to 1 ∈ Z, the map j2 is


the projection and the homomorphism j1 sends exp(2πi/p) to 1. Obviouslyj1 and j2 are surjective. Hence we get from Theorem 2.83 an exact sequence

K1(Z[Z/p]) (i1)∗⊕(i2)∗−−−−−−−→ K1(Z[exp(2πi/p)])⊕K1(Z)(j1)∗−(j2)∗−−−−−−−→ K1(Fp)

∂1−→

K0(Z[Z/p]) (i1)∗⊕(i2)∗−−−−−−−→ K0(Z[exp(2πi/p)])⊕K0(Z)(j1)∗−(j2)∗−−−−−−−→ K0(Fp).

The map (j2)∗ : K0(Z) → K0(Fp) is bijective by Example 1.4. Hence it re-mains to prove that (j1)∗ : K1(Z[exp(2πi/p)]) → K1(Fp) is surjective. Be-cause of Theorem 2.17 we have to find for each integer k with 1 ≤ k ≤ p− 1a unit u ∈ Z[exp(2πi/p)]× satisfying j1(u) = k. Put ξ = exp(2πi/p). Choosean integer l such that kl = 1 mod p. Define

u := 1 + ξ + ξ2 + · · ·+ ξk−1;

v := 1 + ξk + ξ2k + · · ·+ ξ(l−1)k.

Since (ξ − 1)u = ξk − 1 and (ξk − 1) · v = ξ − 1 and Z[exp(2πi/p)] is anintegral domain, we get uv = 1 and hence u ∈ Z[exp(2πi/p)]×. Obviouslyj1(u) = k. ut

2.9 The K-Theory Sequence of a two-sided Ideal

Let I ⊆ R be a two-sided ideal in the ring R. The double of the ring Ralong the ideal I is the subring D(R, I) of R × R consisting of pairs (r1, r2)satisfying r1 − r2 ∈ I. Let pk : D(R, I)→ R send (r1, r2) to rk for k = 1, 2.

Definition 2.85 (Kn(R, I)). Define for n = 0, 1 the abelian group Kn(R, I)to be the kernel of the homomorphism

(p1)∗ : Kn(D(R, I))→ Kn(R).

Theorem 2.86 (Exact sequence of a two-sided ideal for middle K-theory). We obtain an exact sequence, natural in I ⊆ R

K1(R, I)j1−→ K1(R)

pr1−−→ K1(R/I)∂1−→ K0(R, I)

j1−→ K0(R)pr0−−→ K0(R/I).

Proof. We obtain a pullback of rings

D(R, I)p1 //

p2

R

pr

R

pr// R/I

2.9 The K-Theory Sequence of a two-sided Ideal 91

such that pr is surjective. We get from Theorem 2.83 the exact sequence

K1(D(R, I))(p1)∗⊕(p2)∗−−−−−−−−→ K1(R)⊕K1(R)

− pr∗ + pr∗−−−−−−−→ K1(R/I)∂−→

K0(D(R, I))(p1)∗⊕(p2)∗−−−−−−−−→ K0(R)⊕K0(R)

− pr∗ + pr∗−−−−−−−→ K0(R/I).

This yields the desired exact sequence if we define jn : Kn(R, I)→ Kn(R) tobe the restriction of (p2)∗ : Kn(D(R, I)) → Kn(R) to Kn(R, I) for n = 0, 1and let ∂1 be the map induced by ∂. ut

Next we give alternative descriptions of K0(R, I).Let S be a ring, but now for some time we do not require that it has a unit.

If we want to emphasize that we do not require this, we say that S is a ringwithout unit although it may have one. The point is that a homomorphismof rings without units f : S → S′ is a map compatible with the abelian groupstructure and the multiplication but no requirement about the unit is made.The ring obtained from S by adjoining a unit

S+(2.87)

has a underlying group S ⊕ Z. The multiplication is given by

(s1, n1) · (s2, n2) := (s1s2 + n1s2 + n2s1, n1n2).

The unit in S+ is given by (0, 1). We obtain a natural embedding iS : S → S+

by sending s to (s, 0). Let pS : S+ → Z be the homomorphism of rings withunit sending (s, n) to n. We obtain an exact sequence of rings without unit

0→ SiS−→ S+

pS−→ Z→ 0. If f : S → S′ is a homomorphism of rings withoutunit, we obtain a homomorphism f+ : S+ → S′+ of rings with unit by sending(s, n) to (f(s), n). If S does has a unit 1S , then we obtain an isomorphism of

rings with unit uS : S+

∼=−→ S × Z by sending (s, n) to (s+ n · 1S , n).

Definition 2.88 (Kn(S) for rings without unit). Let S be a ring withoutunit. Define for n = 0, 1

Kn(S) := ker ((pS)∗ : Kn(S+)→ Kn(Z)) .

Given a homomorphism f : S → S′ of rings without unit, the homomor-phism (f+)∗ : Kn(S+)→ Kn(S′+) induces a homomorphism of abelian groupsf∗ : Kn(S)→ Kn(S′). Thus we obtain a covariant functor from the categoryof rings without unit to the category of abelian groups by sending S to Kn(S).

If S happens to have already a unit, we get back the old definition (up

to natural isomorphism). Namely, the isomorphism K0(uS) : Kn(S+)∼=−→

Kn(S × Z) sends ker((pS)∗) to the kernel of the map (prZ)∗ : Kn(S × Z) →Kn(Z) given by the projection prZ : S × Z → Z and the inclusion j : S →


S × Z, s 7→ (s, 0) induces an isomorphism of Kn(S) to the kernel of themap prZ by Theorem 1.12 and Theorem 2.9.

Lemma 2.89. Let I be a two-sided ideal in the ring R. Let K0(I) be theprojective class group of the ring I without unit (see Definition 2.88). Thenthere is a natural isomorphism

K0(I)∼=−→ K0(R, I).

In particular K0(R, I) depends only on the ring without unit I but not on R.

Proof. The isomorphism is induced by the homomorphism of rings with unitI+ → D(R, I) sending (s, n) to (n ·1R, n ·1R+s). The proof that it is bijectivecan be found for instance in [704, Theorem 1.5.9 on page 30]. ut

Exercise 2.90. Let n be a positive integer. Compute

K0((n)) ∼=

0 if n = 2;

(Z/n)×/±1 if n ≥ 3,

for the ideal (n) = mn | m ∈ Z ⊆ Z. Prove for the ideal (NZ/2) ⊆ Z[Z/2]generated by the norm element that (NZ/2) and (2Z) are isomorphic as ringswithout unit. Conclude

K0(Z[Z/2]) = 0.

Next we give an alternative description of K1(R, I). Define GL(R, I) tobe the kernel of the map GL(R)→ GL(R/I) induced by the projection R→R/I. Let E(R, I) be the smallest normal subgroup of E(R) which containsall matrices of the shape In + r · Eni,j for n ∈ Z, n ≥ 1, i, j ∈ 1, 2, . . . , n,i 6= j, r ∈ I. Notice that E(R, I) ⊆ GL(R, I). The proof of the next resultcan be found for instance in [704, Theorem 2..5.3 on page 93].

Theorem 2.91 (Relative Whitehead Lemma). Let I ⊆ R be a two-sidedideal. Then

(i) The subgroup E(R, I) of GL(R) is normal;(ii) There is an isomorphism, natural in (R, I)

GL(R, I)/E(R, I)∼=−→ K1(R, I);

(iii) The center of GL(R)/E(R, I) is GL(R, I)/E(R, I);(iv) We have E(R, I) = [E(R), E(R, I)] = [GL(R), E(R, I)].

Example 2.92 (K1(R, I) depends on R). In contrast to K0(R, I) it is nottrue that K1(R, I) is independent of R as shown by Swan [779, Section 1].Let S be a ring and put

2.10 Swan Homomorphisms 93

R =

(a b0 d

)| a, b, d ∈ S

;

R′ =

(n b0 n

)| n ∈ Z, b ∈ S

;

I =

(0 b0 0

)| b ∈ S

.

Then K1(R, I) = 0 and K1(R′, I) ∼= S.

Remark 2.93 (Congruence Subgroup Problem). Given a commutativering R, the Congruence Subgroup Problem asks if every normal subgroupof GL(R) is of the form SL(R, I) := A ∈ GL(R, I) | det(A) = 1 forsome two-sided ideal I ⊆ R. Bass has shown that for any normal subgroupH ⊆ GL(R) there exists an ideal I ⊆ R satisfying E(R, I) ⊆ H ⊆ GL(R, I).(see [84, Theorem 2.1 (a) on page 229] or [703, Exercise 2.5.21 on page 106]).Hence the Congruence Subgroup Problem has a positive answer if and onlyfor every two-sided ideal I ⊆ R we have E(R, I) = SL(R, I) (see [703,Exercise 2.5.21 on page 106]). More information about this problem can befound for instance in [88].

Exercise 2.94. Show that the Congruence Subgroup Problem has a positiveanswer for every field F .

2.10 Swan Homomorphisms

2.10.1 The Classical Swan Homomorphism

The definitions and results of this subsection are taken from Swan [775]. Thispaper marked the beginning of a development which finally leads to a solu-tion of the spherical space form problem mentioned before in Section 1.5. Itpresents a nice and illuminating interaction between geometry, group theoryand algebraic K-theory.

Let G be a finite group. Let NG ∈ ZG be the norm element, i.e., NG :=∑g∈G g. Consider the following pullback of rings

ZG i1 //

i2

ZG/(NG)

j1

Z

j2// Z/|G|


where (NG) ⊆ ZG is the ideal generated by NG, i1 and j2 are the obvi-ous projections, i2 is induced by the group homomorphism G → 1 andj1 is the unique ring homomorphism for which the diagram above com-mutes. One easily checks that it is a pullback and that the maps i1 andj1 are surjective. Hence we can apply Theorem 2.83 and obtain a bound-ary homomorphism ∂ : K1(Z/|G|) → K0(ZG). The obvious homomorphismi : Z/|G|× → K1(Z/|G|) is an isomorphism by Theorem 2.6 since the com-mutative finite ring Z/|G| is a commutative semilocal ring and hence thedeterminant det : K1(Z/|G|)→ Z/|G|× is an inverse of i.

Definition 2.95 (Swan homomorphism). The (classical) Swan homo-morphism is the composition

swG : Z/|G|× i−→ K1(Z/|G|) ∂−→ K0(ZG).

Lemma 2.96. Let n ∈ Z/|G|× be an element represented by n ∈ Z. Then theideal (n,NG) ⊆ ZG generated by n and NG is a finitely generated projectiveZG-module and

sw(n) = [(n,NG)]− [ZG].

Proof. Let P1 be the Z-module Z, P2 be the ZG/(NG)-module ZG/(NG) andP0 be the Z/|G|-module Z/|G|. Consider the automorphism rn : Z/|G| →Z/|G| given by multiplication with n. Define a ZG-module P by the pullback

P //

i2

ZG/(NG)

rnj1

Zj2// Z/|G|.

One easily checks that the ZG maps (n,NG) → Z, which sends n to n andNG to |G|, and (n,NG)→ ZG/(NG), which sends n to the class of 1 and NG

to 0, induce an isomorphism of ZG-modules (n,NG)∼=−→ P . We conclude from

Lemma 2.84 (i) that (r,NG) is a finitely generated projective ZG-module andthat sw(n) = [(n,NG)]− [ZG]. utRemark 2.97 (Another description of the Swan homomorphism).For any n ∈ Z with (n, |G|) = 1 the abelian group Z/n with the triv-ial G-action is a ZG-module which possesses a finite projective resolu-tion P∗ (see [134, Theorem VI.8.12 on page 152]). Since two finite projec-tive resolutions of Z/n are ZG-chain homotopic, their finiteness obstruc-tions agree (see Exercise 1.39). Thus we can define [Z/n] ∈ K0(ZG) byo(P∗) =

∑n≥0(−1)n · [Pn] for any finite projective resolution P∗. We get

sw(n) = −[Z/n]

for any integer n ∈ Z with (n, |G|) = 1. This follows essentially from [775,Lemma 6.2] and Lemma 2.96.


Exercise 2.98. Show that swG is trivial for finite cyclic groups G.

2.10.2 The Classical Swan Homomorphism and Free HomotopyRepresentations

Let G be a finite group. A free d-dimensional G-homotopy representation Xis a d-dimensional CW -complex X together with a G-action such that forany open cell e we have ge ∩ e 6= ∅ ⇒ g = 1, the space X is homotopyequivalent to Sn and G\X is a finitely dominated CW -complex. Let f : X →Y be a G-map of free d-dimensional G-homotopy representations for d ≥ 2.Let n be a positive integer such that the homomorphism of infinite cyclicgroups Hd(f) : Hd(X) → Hd(Y ) sends a generator of Hd(X) to ±n-timesthe generator of Hd(Y ). Let o(G\X), o(G\Y ) ∈ K0(ZG) be the finitenessobstructions of X and Y with respect to the obvious identification G =π1(X) = π1(Y ).

Lemma 2.99. Let G be a finite group of order ≥ 3.

(i) The G-action on Hn(X) is trivial for n ≥ 0 and d is odd;(ii) We have (n, |G|) = 1 and

swG(n) = o(G\Y )− o(G\X),

Proof. (i) Let C∗(X) be the cellular ZG-chain complex. The conditions aboutthe G-actions imply that C∗(X) is a free ZG-chain complex and is the same

as C∗(G\X). Since G\X is finitely dominated, we can find a finite projectiveZG-chain complex P∗ which is ZG-chain homotopy equivalent to C∗(X).Since CG is semisimple, every submodule of a finitely generated CG-moduleis finitely generated projective again. This implies the following equality inK0(CG) = RC(G):∑

n≥0

(−1)n · [Pn ⊗ZG CG] = [H0(X;C)] + (−1)d · [Hd(X;C)]

The Bass Conjecture for integral domains (see 1.85) has been proved for finitegroups andR = Z by Swan [774, Theorem 8.1]. This implies that Pn⊗ZGCG isa finitely generated free CG-module for every n. Since P∗⊗ZGZ ' C∗(G\X),we conclude

∑n≥0(−1)n · [Pn ⊗ZG CG] = χ(G\X) · [CG]. Hence we get the

following equality in RC(G)

χ(G\X) · [CG] = [H0(X;C)] + (−1)d · [Hd(X;C)].

ObviouslyH0(X;C) is CG-isomorphic to the trivial 1-dimensionalG-represen-tation [C]. Since Hd(X) ∼= Z, there is a group homomorphism w : G→ ±1


such that Hd(X;C) is the 1-dimensional G-representation Cw for which g ∈ Gacts by multiplication with w(g). Thus we get in RC(G)

χ(G\X) · [CG] = [C] + (−1)d · [Cw].

Computing the characters on both sides yields the following equalities forg ∈ G

χ(G\X) · |G| = 1 + (−1)d;

0 = 1 + (−1)d · w(g) for g 6= 1.

Since we assume |G| ≥ 3 and χ(G\X) is an integer, the first equality impliesthat d is odd. The second inequality implies that w(g) = 1 for all g ∈ G.Hence G acts trivially on Hn(X) for all n ≥ 0.

(ii) Let C∗(X) and C∗(Y ) be the free cellular ZG-chain complexes. Choosefinite projective ZG-chain complexes P∗ and Q∗ together with ZG-chainhomotopy equivalences u∗ : P∗ → C∗(X) and v∗ : Q∗ → C∗(Y ). The mapf : X → Y induces a ZG-chain map C∗(f) : C∗(X)→ C∗(Y ). Choose a ZG-chain map h∗ : P∗ → Q∗ such that v∗h∗ ' C∗(f)u∗. Let cone∗ = cone∗(h∗)be the mapping cone of h∗. It is a (d+1)-dimensional free ZG-chain complexsuch that Hn(cone∗) = 0 for n 6= d and Hd(cone(C∗(f)) is ZG-isomorphicto Z/n with the trivial G-action. This follows from the long exact homol-ogy sequence associated to the short exact sequence of ZG-chain complexes0 → Q∗ → cone(h∗) → ΣP∗ → 0 and assertion (i). Let D∗ be the ZG-chainsubchain complex of cone∗ such that Dd+1 = coned+1, Dd is the kernel of thed-th differential of cone∗ and Dk = 0 for k 6= d, d+1. Then D∗ is a projectiveZG-chain complex and the inclusion D∗ → cone∗ induces an isomorphismon homology and hence is a ZG-chain homotopy equivalence. In particularwe get a short exact sequence 0 → Dd+1 → Dd → Z/n → 0. Suppose that(n, |G|) = 1 is not true. Then we can find a prime number p such that Z/p isa subgroup of G and Z/pl is a direct summand in Z/n for some l ≥ 1. Thisimplies that the cohomological dimension of the trivial Z[Z/p]-module Z/plis bounded by 1. An easy computation shows that ExtnZ[Z/p](Z,Z/pl) doesnot vanish for all n ≥ 2, a contradiction. Hence (n, |G|) = 1.

We conclude from Exercise 1.39

(−1)d · [Z/n] = (−1)d+1 · [Dd+1] + (−1)d · [Dd] = o(D∗) = o(cone∗)

= [Q∗]− [P∗] = o(G\Y )− o(G\X).

Since d is odd by assertion (i), we conclude sw(n) = o(G\Y )− o(G\X) fromRemark 2.97. ut

Exercise 2.100. Let X be a free d-dimensional G-homotopy representationof the finite cyclic group G. Then G\X is homotopy equivalent a finite CW -complex.


2.10.3 The Generalized Swan Homomorphism

In this subsection we briefly introduce the generalized Swan homomorphism.For proofs and more information we refer to [521, Chapter 19].

Fix a finite group G. Let m be its order |G|. We obtain a pullback of rings

ZG //

Z[1/m]G

Z(m)G // QG.

Despite the fact that neither the right horizontal arrow nor the lower verticalarrow are surjective, one obtains a long exact sequence which is an exampleof a localization sequence

(2.101) K1(ZG)→ K1(Z[1/m]G)⊕K1(Z(m)G)→ K1(QG)∂−→ K0(ZG)

→ K0(Z[1/m]G)⊕K0(Z(m)G)→ K0(QG).

We denote in the sequel by K1(QG)/K1(Z(m)G) the cokernel of the changeof rings homomorphism K1(Z(m)G)→ K1(QG).

Definition 2.102 (Generalized Swan homomorphism). The generalizedSwan homomorphism

swG : Z/m× → K1(QG)/K1(Z(m)G)

sends r to element in K1(QG)/K1(Z(m)G) which is given by the element inK1(QG) represented by the QG-automorphism r · id : Q → Q of the trivialQG-module Q.

This is well-defined by the argument in [521, page 381]. The followingresult is taken from [521, Theorem 19.4 on page 381]

Theorem 2.103 (The generalized Swan homomorphism). Let G be afinite group of order m.

(i) The composite of the generalized Swan homomorphism swG : Z/m× →K1(QG)/K1(Z(m)G) introduced in Definition 2.102 with the homomor-

phism ∂ : K1(QG)/K1(Z(m)G)→ K0(ZG) induced by the boundary homo-morphism of the localization sequence (2.101) is the classical Swan homo-morphism swG : Z/m× → K0(ZG) of Definition 2.95;

(ii) The generalized Swan homomorphism swG : Z/m× → K1(QG)/K1(Z(m)G)is injective.


2.10.4 The Generalized Swan Homomorphism and FreeHomotopy Representations

In this subsection we briefly discuss Reidemeister torsion for free homotopyrepresentations. For proofs and more information we refer to [521, Chap-ter 20].

Let G be a finite group of order m = |G|. Let X be a free d-dimensionalG-homotopy representation. Suppose that we have fixed an orientation, i.e.,a generator of Hd(X;Z). Then we can define a kind of Reidemeister torsionof X

ρG(X) ∈ K1(QG)/K1(Z(m)G)(2.104)

as follows. The change of rings map K0(ZG)→ K0(Z(m)G) is trivial (see [774,Theorem 7.1 and Theorem 8.1]). Hence there is a finite free Z(m)G-chaincomplex F∗ together with a Z(m)G-chain homotopy equivalence f∗ : F∗ →C∗(X)⊗ZGZ(m)G. Choose a Z(m)G-basis for F∗. Then F∗⊗Z(m)GQG is a finitebased free QG-chain complex. Notice that we have preferred isomorphisms ofabelian group H0(X) ∼= Z and Hd(X) ∼= Z and G acts trivially on H0(X) andHd(X). This induces preferred QG-isomorphisms Hi(F∗⊗Z(m)GQG) ∼= Q fori = 0, d, where we equip Q with the trivial G-action. This enables us to definea torsion invariant τ(F∗ ⊗Z(m)G QG) ∈ K1(QG) although F∗ ⊗Z(m)G QG is

not acyclic. Define ρG(X) to be its image under the projection K1(QG) →K1(QG)/K1(Z(m)G). One easily checks that ρG(X) is independent of thechoice of F∗, f∗ and the choice of the Z(m)G-basis for F∗. The proof of thefollowing result is a special case of the results in [521, Theorem 20.37 onpage 403 and Corollary 20.39 on page 404].

Theorem 2.105 (Torsion and free homotopy representations). LetG be a finite group of order m = |G| ≥ 3. Let X and Y be free orientedG-homotopy representations.

(i) The homomorphism ∂ : K1(QG)/K1(Z(m)G)→ K0(ZG) sends the torsion

ρG(X) to the finiteness obstruction o(G\X);(ii) Let f : X → Y be a G-map which always exists. Then its degree deg(f) is

prime to m andswG(deg(f)) = ρG(Y )− ρG(X);

(iii) The free G-homotopy representations X and Y are oriented G-homotopyequivalent if and only if ρG(X) = ρG(Y ).

Theorem 2.105 gives an interesting relation between torsion invariants andfinite obstructions and generalizes the homotopy classification of lens spacesto free G-homotopy representations.

All this can be extended to not necessarily free G-homotopy represen-tations (see [521, Section 20]. The theory of homotopy representations wasinitiated by tom Dieck-Petrie [790].


2.11 Variants of the Farrell-Jones Conjecture forK1(RG)

In this section we state variants of the Farrell-Jones Conjecture for K1(RG).The Farrell-Jones Conjecture itself will give a complete answer for arbitraryrings but to formulate the full version some additional effort will be needed.If one assumes that R is regular and G torsionfree, the conjecture reduces toan easy to formulate statement which we will present next. Moreover, thisspecial case is already very interesting.

Conjecture 2.106 (Farrell-Jones Conjecture for K0(RG) and K1(RG)for regular R and torsionfree G). Let G be a torsionfree group and letR be a regular ring. Then the maps defined in (2.25) and (2.26)

A0 : K0(R)∼=−→ K0(RG);

A1 : G/[G,G]⊗Z K0(R)⊕K1(R)∼=−→ K1(RG),

are both isomorphisms. In particular the groups WhR0 (G) and WhR1 (G) (seeDefinition 2.27) vanish.

We mention the following important special case of Conjecture 2.106.

Conjecture 2.107 (Farrell-Jones Conjecture for K0(ZG) and Wh(G)

for torsionfree G). Let G be a torsionfree group. Then K0(ZG) and Wh(G)vanish.

We have already discussed the K0-part of the two conjectures above inSection 1.8. The following exercise shows that we cannot expect to have ananalogue for K1(RG) of the Conjecture 1.57.

Exercise 2.108. Let G be a group and let R be a ring. Suppose that themap

colimH∈SubFIN (G×Z)K1(RH)→ K1(R[G× Z]).

is surjective. Show that then K0(RG) = 0 and hence K0(R) = 0. In particu-lar, R cannot be a commutative integral domain.

Remark 2.109 (Relevance of Conjecture 2.107). In view of Remark 2.13Conjecture 2.107 predicts for a torsionfree group G that any matrix A ∈GLn(ZG) can be transformed by a sequence of the operations mentioned inRemark 2.13 to a (1, 1)-matrix of the form (±g) for some g ∈ G. This isthe algebraic relevance of this conjecture. Its geometric meaning comes fromthe following conclusion of the s-Cobordism Theorem 1.37. Namely, if G isa finitely presented torsionfree group, and n an integer with n ≥ 6, then itimplies that every compact n-dimensional h-cobordism is trivial.


2.12 Survey on Computations of K1(ZG) for FiniteGroups

In contrast to K0(ZG) for finite groups G, the Whitehead group Wh(G) of afinite group is very well understood. The key source for the computation ofWh(G) for finite groups G is the book written by Oliver [631].

Definition 2.110 (SK1(ZG) and Wh′(G)). Let G be a finite group. Define

SK1(ZG) := ker((K1(ZG)→ K1(QG)) ;

Wh′(G) = Wh(G)/ tors(Wh(G)).

Remark 2.111 (SK1(ZG) and reduced norms). Let G be a finite group.The reduced norm on CG is defined as the composite of isomorphisms ofabelian groups

nrCG : K1(CG)φ∗−→ K1

(k∏i=1

Mri(C)

)∼=−→

k∏i=1

K1(Mri(C))

∼=−→k∏i=1

K1(C)∏ki=1 det

−−−−−−→k∏i=1

C×,

where the isomorphism of rings φ : CG∼=−→∏ki=1Mri(C) comes from Wed-

derburn’s Theorem applied to the semisimple ring CG and the remainingthree isomorphisms come from Theorem 2.6, Lemma 2.8 and Lemma 2.9.The reduced norm on RG for R = Z,Q is defined as the composite

nrRG : K1(RG)iR−→ K1(CG)

nrCG−−−→k∏i=1

C×,

where iR is the obvious change of rings homomorphism. The map iQ is injec-tive (see [631, Theorem 2.5 on page 43]). Thus we can identify

SK1(ZG) = ker

(nrZG : K1(ZG)→

k∏i=1

C×).

This identification is useful for investigating SK1(ZG) and Wh′(G). We con-clude that for abelian groups the two definitions of SK1(ZG) appearing inDefinition 2.15 and Definition 2.110 agree.

We denote by rF (G), the number of isomorphism classes of irreduciblerepresentations of the finite G over the field F . Recall that rF = | conF (G)|by Lemma 1.75. The proof of the next result can be found for instance in [631,Theorem 2.5 on page 48] and is based on the Dirichlet Unit Theorem.

2.12 Survey on Computations of K1(ZG) for Finite Groups 101

Theorem 2.112 (SK1(ZG) = tors(Wh(G))). Let G be a finite group. Thenthe abelian group SK1(ZG) is finite and agrees with the torsion subgrouptors(Wh(G)) of Wh(G). The group Wh′(G) = Wh(G)/ tors(Wh(G)) is afinitely generated free abelian group of rank rR(G)− rQ(G).

Hence the next step is to compute SK1(ZG). This is done using localizationsequences (see [631, Theorem 1.17 on page 36 and Section 3c] which do alsoinvolve the second algebraic K-group (see Chapter 4) and are consequencesof the general result of Quillen stated in Theorem 5.40. Define

SK1(ZpG) := ker (K1(ZpG)→ K1(QpG)) .

Put

Cl1(ZG) := ker

SK1(ZG)→∏p| |G|

SK1(ZpG)

,

where p runs over all prime numbers dividing |G|. Then one obtains an exactsequence

0→ Cl1(ZG)→ SK1(ZG)→∏p| |G|

SK1(ZpG)→ 0.

The analysis of Cl1(ZG) and SK1(ZpG) is carried out independently andwith different methods. Besides localization sequences p-adic logarithms playa key role. Details can be found in Oliver [631].

Given a groups G and Q the wreath product G o Q is defined to be thesemidirect product

∏QG o Q, where Q acts on

∏QG o Q permuting the

factors.

Theorem 2.113 (Finite groups with vanishing Wh(G) or SK1(ZG)).Let G be a finite group.

(i) Let p be a prime number. If the p-Sylow subgroup SpG of G is isomorphicto Z/pn or Z/pn × Z/p for some n ≥ 0, then SK1(ZG)(p) = 0, i.e., thefinite abelian group SK1(ZG) contains no p-torsion;

(ii) Let G be a finite abelian group. Then SK1(ZG) = 0 if and only if for everyprime p the p-Sylow subgroup SpG is isomorphic to Z/pn or Z/pn × Z/pfor some n ≥ 0 or if G = (Z/2)n for some n ≥ 1;

(iii) Let CWh be the smallest class of groups which is closed under finite productsand wreath products with Sn for every n ≥ 2 and contains the trivialgroup. Let CSK1 be the smallest class of groups which is closed under finiteproducts and wreath products with Sn for every n ≥ 2 and contains thedihedral groups Dn for n ≥ 2.Then Wh(G) = 0 for G ∈ CWh and SK1(ZG) = 0 if G ∈ CSK1

;(iv) We have SK1(ZG) = 0 if G is one of the following groups

(a) G is finite cyclic;


(b) Z/pn × Z/p for some prime p and n ≥ 1;(c) (Z/2)n for n ≥ 1;(d) G is any symmetric group;(e) G is any dihedral group;(f) G is any semidihedral 2-group.

Proof. (i) See Oliver [631, Theorem 14.2 (i) on page 330].

(ii) See Oliver [631, Theorem 14.2 (iii) on page 330].

(iii) See Oliver [631, Theorem 14.1 on page 328].

(iv) This follows essentially from the other assertions. See Oliver [631, Ex-amples 1 and 2 on page 14]. ut

The group SK1(ZG) can be computed for many examples. We mentionthe following example taken from [631, Theorem 14.6 on page 336].

Example 2.114 (SK1(Z[An])). We have SK1(Z[An]) ∼= Z/3 if we canwrite n =

∑ri=1 3mi such that m1 > m2 > · · · > mr > 0 and

∑ri=1mi is odd.

Otherwise we get SK1(Z[An]) = 0.

Exercise 2.115. Show that the Whitehead group Wh(Z/m) of the finitecyclic group Z/m of order m is a free abelian group of rank bm/2c+1−δ(M),where bm/2c is the greatest integer less or equal to m/2 and δ(m) is thenumber of divisors of m.

Let p be a prime. Show that Wh(Z/p) is isomorphic to Z(p−1)/2−1 if p isodd and is trivial if p = 2.

2.13 Survey on Computations of Algebraic K1(C∗r (G))

and K1(N (G))

Define SL(R) := A ∈ GL(R) | det(A) = 1. Let B be a commutativeBanach algebra. Then GLn(B) inherits a topology, namely the subspace

topology for the obvious embedding GLn(B) ⊆ Mn(B) =∏n2

i=1B. EquipGL(B) =

⋃n≥1GLn(B) with the weak topology, i.e., a subset A ⊂ GL(B) is

closed if and only if A ∩GLn(R) is a closed subset of GLn(B) for all n ≥ 1.Equip SL(B) ⊆ GL(R) with the subspace topology.

The following results are due to Milnor [592, Corollary 7.2 on page 57 andCorollary 7.3 on page 58].

Theorem 2.116 (K1(B) of a commutative Banach algebra). Let B bea commutative Banach algebra. Then there is a natural isomorphism

K1(B)∼=−→ B× × π0(SL(B)).


Define the infinite special orthogonal group SO =⋃n≥1 SO(n) and infinite

special unitary group SU =⋃n≥1 SU(n) where SO(n) = A ∈ GLn(R) |

AAt = I, det(A) = 1 is the special n-th orthogonal group and SU(n) =A ∈ GLn(C) | AA∗ = I, det(A) = 1 is the special n-th unitary group.Denote by [X,SO] and [X,SU ] respectively the set of homotopy classes ofmaps from X to SO and SU respectively.

Theorem 2.117 (K1(C(X)) of a commutative C∗-algebra C(X)). LetX be compact space. Then there are natural isomorphisms

K1(C(X,R))∼=−→ C(X,R)× × [X,SO];

K1(C(X,C))∼=−→ C(X,C)× × [X,SU ];

The sets [X,SO] and [X,SU ] are closely related to the topological K-groups KO−1(X) and K−1(X).

If B is a group C∗-algebra C∗r (G), then not much is known about thealgebraic K-group K1(B) in general. However, the algebraic K1-group of avon Neumann algebra is fully understood (see [527, Section 9.3],[551]). Wemention the special case (see [527, Example 9.34 on page 353]) that for afinitely generated group G which is not virtually finitely generated abelianthe Fuglede-Kadison determinant induces an isomorphism

K1(N (G))∼=−→ Z(N (G))+,inv,(2.118)

where Z(N (G))+,inv consists of the elements of the center of N (G) which areboth positive and invertible.

The connection between the algebraic and the topological K-theory of aC∗-algebra will be discussed in Section 8.7.

2.14 Miscellaneous

A universal property describing the Whitehead group and the Whiteheadtorsion similar to the description of the finiteness obstruction in Section 1.7is presented in [521, Theorem 6.11].

Geometric versions or analogues of maps related to the Bass-Heller-Swandecomposition are described in [285], [308], [521, (7.34) on page 130], and [686,§ 10].

Given two groups G1 and G2, let G1∗G2 by the free amalgamated product.Then the natural maps Gk → G0 ∗ G1 for k = 1, 2 induce an isomorphism(see [761])

Wh(G1)⊕Wh(G2) ∼= Wh(G1 ∗G2).(2.119)

Compare this with the analog for the reduced projective class group (see (1.109)).


Exercise 2.120. Show that the projections prk : G1 ×G2 → Gk for k = 1, 2do not in general induce an isomorphism

Wh(G1 ×G2)∼=−→ Wh(G1)×Wh(G2).

There are also equivariant versions of the Whitehead torsion, see for in-stance [521, Chapter 4 and Chapter 12], where more references can be found.


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Chapter 3

Negative Algebraic K-Theory

3.1 Introduction

In this chapter we introduce negative K-groups.They are designed such that the Bass-Heller-Swan decomposition and the

long exact sequence of a pullback of rings and of a two-sided ideal extendbeyond K0. We give a geometric interpretation of negative K-groups of grouprings in terms of bounded h-cobordisms. We state variants of the Farrell-Jones Conjecture for negative K-groups and give a survey of computationsfor group rings of finite groups.

3.2 Definition and Basic Properties of NegativeK-Groups

Recall that we get from Theorem 2.72 an isomorphism

K0(R) = coker(K1(R[t])⊕K1(R[t−1])→ K1(R[t, t−1])

).

This motivates the following definition of negative K-groups due to Bass.

Definition 3.1. Given a ring R, define inductively for n = −1,−2, . . .

Kn(R) := coker(Kn+1(R[t])⊕Kn+1(R[t−1])→ Kn+1(R[t, t−1])

).

Define for n = −1,−2, . . .

NKn(R) := coker (Kn(R)→ Kn(R[t])) .

The Bass-Heller-Swan decomposition 2.69 for K1(R[t, t−1]) extends to neg-ative K-theory.

Theorem 3.2 (Bass-Heller-Swan decomposition for middle and lowerK-theory). There are isomorphisms of abelian groups, natural in R, forn = 1, 0,−1,−2, . . .

NKn(R)⊕Kn(R)∼=−→ Kn(R[t]);

Kn(R)⊕Kn−1(R)⊕NKn(R)⊕NKn(R)∼=−→ Kn(R[t, t−1]).

105

106 3 Negative Algebraic K-Theory

The following sequence is natural in R and exact for n = 1, 0,−1, . . .

0→ Kn(R)(k+)∗⊕−(k−)∗−−−−−−−−−→ Kn(R[t])⊕Kn(R[t−1])

(τ+)∗⊕(τ−)∗−−−−−−−−→ Kn(R[t, t−1])

Cn−−→ Kn−1(R)→ 0.

If we regard it as an acyclic Z-chain complex, there exists a chain contraction,natural in R.

Proof. We give the proof only for n = 0, then an iteration of the argumentproves the claim for all n ≤ 0. Take S = R[Z] = R[x, x−1]. We obtain acommutative diagram

0

0

K0(R)

f1

//

(k+)∗⊕−(k−)∗

K1(S)

(k+)∗⊕−(k−)∗

K0(R[t])⊕K0(R[t−1])

f2

//

(τ+)∗⊕(τ−)∗

K1(S[t])⊕K1(S[t−1])

(τ+)∗⊕(τ−)∗

K0(R[t, t−1])

f3

//

C′

K1(S[t, t−1])

C

K−1(R)

f4

//

K0(S)

0 0

where the right column is the exact sequence appearing in Theorem 2.72,the map C ′ is the canonical projection, the maps f1, f2 and f3 come fromthe Bass-Heller-Swan decompositions for S = R[x, x−1], S[t] = R[t][x, x−1],S[t−1] = R[t−1][x, x−1] and S[t, t−1] = R[t, t−1][x, x−1] and the map f4 is theunique map which makes the diagram commutative. There are natural re-tractions rk of fk for k = 1, 2, 3 for which the diagram remains commutative,and a natural chain contraction of the right column (see Theorem 2.69). Letr4 : K0(S)→ K−1(R) be the unique map which satisfies r4 C = C ′ r3. Aneasy diagram shows that r4 is well-defined since C ′ r3 sends the kernel of Cto zero. One easily checks r4 f4 = id. Since the right column is exact and allvertical maps have retractions for which the diagram remains commutative,the left column is exact as well. This establishes the exact sequence. Since thechain contraction of the right column is also compatible with the left verticalarrows and the retractions, we obtain a natural chain contraction for the leftcolumn as well. ut

3.2 Definition and Basic Properties of Negative K-Groups 107

Remark 3.3 (Extending exact sequences to negative K-theory). TheMayer-Vietoris sequence of a pullback of rings (see Theorem 2.83) can be ex-tended to negative K-theory and also to K2 as we will explain in Theorem 4.8.Similarly, the long exact sequence of a two-sided ideal (see Theorem 2.86) canbe extended to negative K-theory and also to K2, as we will explain in The-orem 4.11.

Exercise 3.4. Let R and S be rings. Show for n ≤ 1 that the projectionsinduce an isomorphism

Kn(R× S)∼=−→ Kn(R)×Kn(S).

Definition 3.5. Define for n ≤ 1 inductively for p = 0, 1, 2, . . .

N0Kn(R) := Kn(R);

Np+1Kn(R) := coker (NpKn(R)→ NpKn(R[t])) .

Obviously N1Kn(R) agrees with NKn(R).

Theorem 3.6 (Bass-Heller-Swan decomposition for lower and mid-dle K-theory for regular rings). Suppose that R is regular. Then

Kn(R) = 0 for n ≤ −1;

NpKn(R) = 0 for n ≤ 1 and p ≥ 1,

and the Bass-Heller-Swan decomposition (see Theorem 3.2) reduces for n ≤ 1to the natural isomorphism

Kn−1(R)⊕Kn(R)∼=−→ Kn(R[t, t−1]).

Proof. The Bass-Heller-Swan decomposition (see Theorem 3.2) applied to Rand R[t] together with the obvious maps i : R → R[t] and ε : R[t] → Rsatisfying ε i = idR yield a natural Bass-Heller-Swan decomposition

(3.7) NKn(R)⊕NKn−1(R)⊕N2Kn(R)⊕N2Kn(R)∼=−→ NKn(R[Z]).

Hence NKn−1(R) = 0 if NKn(R[Z]) = 0. If R is regular, then R[Z] is regularby Theorem 2.77 (i). Hence NKn−1(R) vanishes for all regular rings R ifNKn(R) vanishes for all regular rings. We have shown in Theorem 2.78 thatNK1(R) vanishes for all regular rings R. We conclude by induction over nthat NKn(R) vanishes for all regular rings R and n ≤ 1. Obviously NpKn(R)is a direct summand in NKn(R[t]) and R[t] is regular by Theorem 2.77 (i).Hence NpKn(R) vanishes for p ≥ 1 and n ≤ 1 if R is a regular ring.

Next we show that K−1(R) = 0 for every regular ring. It suffices to showthat the obvious map K0(R[t])→ K0(R[t, t−1]) is surjective. The homomor-phism


α : G0(R[t])→ G0(R[t, t−1]), [M ]→ [M ⊗R[t] R[t, t−1]]

is well-defined since R[t, t−1] is a localization of R[t] and hence flat asR[t]-module. Since R by assumption and hence R[t] and R[t, t−1] by The-orem 2.77 (i) are regular, we conclude from Lemma 2.82 that it remains toprove surjectivity of α. Let M be a finitely generated R[t, t−1]-module. SinceR[t, t−1] is Noetherian, we can find a matrix A ∈ Mm,n(R[t, t−1]) such that

there exists an exact sequence of R[t, t−1]-modules R[t, t−1]mA−→ R[t, t−1]n →

M → 0. Since t is invertible in R[t, t−1], the sequence remains exact if wereplace A by tkA for some k ≥ 1. Hence we can assume without loss of gen-erality that A ∈ Mm,n(R[t]). Define the R[t]-module N to be the cokernel

of R[t]mA−→ R[t]n. Then N ⊗R[t] R[t, t−1] is R[t, t−1]-isomorphic to M and

hence α([N ]) = [M ].Now Kn(R) = 0 follows inductively for n ≤ −1 for every regular ring from

Theorem 2.77 (i) and the Bass-Heller-Swan decomposition 3.2.Finally apply Theorem 3.2. ut

Exercise 3.8. Let R be a regular ring. Prove

K1(R[Zk]) = K1(R)⊕k⊕i=1

K0(R);

K0(R[Zk]) ∼= K0(R);

Kn(R[Zk]) ∼= 0 for n ≤ −1.

Example 3.9 (Negative K-theory of Z[Z/p × Zk]). Let p be a primenumber. We want to show

Kn(Z[Z/p× Zk]) = 0 for n ≤ −1 and k ≥ 0,

and thatK0(Z[Z/p×Zk]) is finitely generated for k ≥ 0. Consider the pullbackof rings appearing in the proof of Rim’s Theorem in Section 2.8.

Z[Z/p] i1 //

i2

Z[exp(2πi/p)]

j1

Z

j2// Fp

If we apply −⊗Z Z[Zk], we obtain the pullback of rings


Z[Z/p× Zk]i1 //

i2

Z[exp(2πi/p)][Zk]

j1

Z[Zk]

j2// Fp[Zk]

The ring Z[exp(2πi/p)] is a Dedekind domain (see Theorem 1.23) and inparticular regular. The rings Z and Fp are regular as well. Hence the ringsZ[exp(2πi/p)][Zk], Z[Zk] and Fp[Zk] are regular by Theorem 2.77 (i). Thenegative K-groups of Z[exp(2πi/p)][Zk], Z[Zk] and Fp[Zk] vanish by Theo-rem 3.6. The obvious maps

K0(Z)∼=−→ K0(Z[Zk]);

K0(Z[exp(2πi/p)])∼=−→ K0(Z[exp(2πi/p)][Zk]);

K0(Fp)∼=−→ K0(Fp[Zk]),

are bijective because of Theorem 3.6. Hence the associated long exact Mayer-Vietoris sequence (see Remark 3.3) implies that Kn(Z[Z/p× Zk]) = 0 holdsfor n ≤ −2 and that we get the exact sequence

K1(Fp[Zk])→ K0(Z[Z/p× Zk])

→ K0(Z)⊕K0(Z[exp(2πi/p)])→ K0(Fp)→ K−1(Z[Z/p× Zk])→ 0.

Since Fp is a field and hence K0(Fp) is generated by [Fp] (see Example 1.4), weconclude K−1(Z[Z/p×Zk]) = 0. Example 1.4, Theorem 2.17 and Theorem 3.6imply K1(Fp[Zk]) ∼= K1(Fp) ⊕ K0(Fp)k ∼= (Fp)× ⊕ Zk. The abelian groupK0(Z) ⊕ K0(Z[exp(2πi/p)]) is finitely generated by Theorem 1.23. HenceK0(Z[Z/p× Zk]) is finitely generated.

Exercise 3.10. Show for k ≥ 0 and n ≤ 0 that Kn(Z[Z/3× Zk]) = 0. Provethat NpKn(Z[Z/3× Zk]) = 0 holds for n ≤ −1 and p ≥ 0 and for n = 0 andp ≥ 1.

Example 3.11 (Negative K-theory of Z[Z/6]). We want to show

Kn(Z[Z/6]) ∼=

Z n = −1;

0 n ≤ −2.

Consider the pullback of rings


Z[Z/2]i1 //

i2

Z

j1

Z

j2// Z/2

where i1 sends a+bt to a−b and i2 sends a+bt to a+b for t ∈ Z/2 the generatorand the two maps from Z to Z/2 are the canonical projections. Since Z[Z/3]is free as abelian group, this remains to be a pullback of rings if we apply−⊗Z Z[Z/3]. We have isomorphisms of rings Z[Z/2]⊗Z Z[Z/3] = Z[Z/6] andZ⊗ZZ[Z/3] = Z[Z/3]. From the pullback for p = 3 appearing in Example 3.9we obtain an isomorphism of rings

F2 ⊗Z Z[Z/3] ∼= F2 × (F2 ⊗Z Z[exp(2πi/3)]).

The ring Z[exp(2πi/3)] is as abelian group free with two generators 1 and ω =exp(2πi/3) and the multiplication is uniquely determined by ω2 = −1 − ω.Hence F2 ⊗Z Z[exp(2πi/3)] contains four elements, namely 0, 1, 1 ⊗ ω andthe sum 1 + 1⊗ω. Since (1⊗ω) · (1 + 1⊗ω) = 1, it is the field F4 consistingof four elements. Hence we obtain a pullback of rings

Z[Z/6]i1 //

i2

Z[Z/3]

j1

Z[Z/3]

j2// F2 × F4

Since Kn(F2 × F4) ∼= Kn(F2)×Kn(F4) vanishes for n ≤ −1 and Kn(Z[Z/3])vanishes for n ≤ −1 by Example 3.9, the associated long exact Mayer-Vietorissequence (see Remark 3.3) implies that Kn(Z[Z/6]) = 0 holds for n ≤ −2and there is an exact sequence

K0(Z[Z/3])⊕K0(Z[Z/3])→ K0(F2 × F4)→ K−1(Z[Z/6])→ 0.

Since K0(Z[Z/3]) is trivial (see Example 1.91) and the projections induce

an isomorphism K0(F2 × F4)∼=−→ K0(F2) × K0(F4) ∼= Z ⊕ Z, we conclude

K−1(Z[Z/6]) ∼= Z.

Exercise 3.12. Consider k ∈ 0, 1, 2, . . .. Compute

Kn(Z[Zk × Z/6]) ∼=

Zk for n = 0;

Z for n = −1;

0 for n ≤ −2,

and prove NpKn(Z[Z/6× Zk]) = 0 for p ≥ 1 and n ≤ 0.


The Bass-Heller-Swan decomposition can be used to show that certainresults about the K-groups in a fixed degree m have implications to all theK-groups in degree n ≤ m, as illustrated by the next result.

Lemma 3.13. Consider a ring R and m ∈ Z with m ≤ 1. Suppose thatfor every k ≥ 1 the map Km(R) → Km(R[Zk]) induced by the inclusionR→ R[Zk] is bijective.

Then Kn(R[Zl]) = 0 for n ≤ m − 1 and NKn(R[Zl]) = 0 for n ≤ m holdfor all l ≥ 0.

Proof. Since the bijectivity of Km(R)→ Km(R[Zk]) for all k ≥ 1 implies thebijectivity of Km(R[Zl]) → Km((R[Zl])[Zk]) for all k, l ≥ 0 because of theidentification (R[Zl])[Zk] = R[Zk+l], it suffices to treat the case l = 0.

Consider any integer k ≥ 1. The assumptions in Lemma 3.13 imply thatthe map Km(R[Zk−1]) → Km(R[Zk]) induced by the inclusion R[Zk−1] →R[Zk] is bijective. Theorem 3.2 applied to the ring R[Zk−1] together withthe identity R[Zk] = (R[Zk−1])[Z] shows that Km−1(R[Zk−1]) = 0 andNKm(R[Zk−1]) = 0. Using Theorem 3.2 and Bass-Heller-Swan-decompositionfor NK (see (3.7)), one shows inductively for i = 0, 1, . . . , (k − 1) thatKm−1−j(R[Zk−i−1]) = 0 and NKm−j(R[Zk−i−1]) = 0 holds for j = 0, 1 . . . , i.Then the case i = k − 1 shows that Kn(R) = 0 for m− k ≤ n ≤ m− 1 andNKn(R) = 0 for m− k+ 1 ≤ n ≤ m. Since k ≥ 1 was arbitrary, Lemma 3.13follows. ut

Exercise 3.14. Consider a ring R and m ∈ Z with m ≤ 1. Suppose thatKm(R[Zk]) = 0 for every k ≥ 1. Then Ki(R[Zl]) = NKi(R[Zl]) = 0 holds fori ≤ m and l ≥ 0.

Theorem 3.15 (The middle and lower K-theory of RG for finite Gand Artinian R). Let G be a finite group and let R be an Artinian ring.Then:

(i) For every k ≥ 0 the map

K0(RG)∼=−→ K0(RG[Zk])

induced by the inclusion is bijective;(ii) Given any k ≥ 0, we have Kn(RG[Zk]) = 0 for n ≤ −1 and NKn(RG[Zk]) =

0 for n ≤ 0.

Proof. (i) Denote by J = rad(RH) ⊆ RH the Jacobson radical of RH. SinceR and hence RH are Artinian, there exists a natural number l with JJ l = J l.By Nakayama’s Lemma (see [755, Proposition 8 in Chapter 2 on page 20]) J l

is 0, in other words, J is nilpotent. The ring RH/J is a semisimple Artinianring (see [496, Definition 20.3 on page 311 and (20.3) on page 312]) and inparticular regular. Theorem 2.77 (ii) implies that (RH/J)[Zk] is regular for


all k ≥ 1. We derive from Theorem 3.6 that Kn((RH/J)[Zk]) = 0 for n ≤ −1and NKn((RH/J)[Zk]) for n ≤ 0 hold for all k ≥ 0. We conclude fromTheorem 3.6 by induction over k = 0, 1, 2 . . . that the inclusion RH/J →(RH/J)[Zk] induces an isomorphism

K0(RH/J)∼=−→ K0((RH/J)[Zk])

for all n ≥ 0.The following diagram

K0(RH) //

K0(RH[Zk])

K0(RH/J) // K0((RH/J)[Zk])

commutes. Since J is a nilpotent two-sided ideal of RH, J [Zk] is a nilpo-tent two-sided ideal of RH[Zk]. Obviously (RH/J)[Zk] can be identified with(RH[Zk])/(J [Zk]). Hence the vertical arrows in the diagram above are bijec-tive by Lemma 1.108. Since the lower horizontal arrow is bijective for everyk ≥ 1, the upper horizontal arrow is bijective for every k ≥ 1.

(ii) This follows from assertion (i) and Lemma 3.13 applied in the case m = 0to the ring RG. ut

3.3 Geometric Interpretation of Negative K-Groups

One possible geometric interpretation of negative K-groups is in terms ofbounded h-cobordisms.

Comment 8: Maybe later this will have to be adjusted when we introducecontrolled topology later.

We consider manifolds W parametrized over Rk, i.e., manifolds which areequipped with a surjective proper map p : W → Rk. Recall that proper meansthat preimages of compact subsets are compact again. We will always assumethat the fundamental group(oid) is bounded, compare [650, Definition 1.3].A map f : W →W ′ between two manifolds parametrized over Rk is boundedif p′ f(x)− p(x) | x ∈W is a bounded subset of Rk.

A bounded cobordism (W ;M0, f0,M1, f1) is defined just as in Section 2.5but compact manifolds are replaced by manifolds parametrized over Rk andthe parametrization for Ml is given by pW fl. If we assume that the inclusionsil : ∂kW → W are homotopy equivalences, then there exist deformationsrl : W × I → W such that rl|W×0 = idW and rl(W × 1) ⊂ ∂lW . Abounded cobordism is called a bounded h-cobordism if the inclusions il arehomotopy equivalences and additionally the deformations can be chosen such

3.4 Variants of the Farrell-Jones Conjecture for Negative K-Groups 113

that the two sets

Sl = pW (rl(x, t))− pW (rl(x, 1)) | x ∈W, t ∈ [0, 1]

are bounded subsets of Rk.The following theorem (compare [650] and [827, Appendix]) contains the

s-Cobordism Theorem 2.44 as a special case, gives another interpretation ofelements in K0(Zπ) and explains one aspect of the geometric relevance ofnegative K-groups.

Theorem 3.16 (Bounded h-Cobordism Theorem). Suppose that M0

is parametrized over Rk and satisfies dimM0 ≥ 5. Let π be its fundamen-tal group(oid). Equivalence classes of bounded h-cobordisms over M0 mod-ulo bounded diffeomorphism relative M0 correspond bijectively to elements inκ1−k(π), where

κ1−k(π) =

Wh(π) if k = 0;

K0(Zπ) if k = 1;

K1−k(Zπ) if k ≥ 2.

3.4 Variants of the Farrell-Jones Conjecture forNegative K-Groups

In this section we state variants of the Farrell-Jones Conjecture for negativeK-theory. The Farrell-Jones Conjecture itself will give a complete answer forarbitrary rings but to formulate the full version some additional effort will beneeded. If one assumes that R is regular and G torsionfree or that R = Z, theconjecture reduces to an easy to formulate statement which we will presentnext.

Conjecture 3.17 (The Farrell-Jones Conjecture for negative K-theoryand regular coefficient rings). Let R be a regular ring and G be a groupsuch that for every finite subgroup H ⊆ G the element |H| · 1R of R isinvertible in R. Then

Kn(RG) = 0 for n ≤ −1.

Exercise 3.18. Prove that Conjecture 3.17 is true if G is finite.

Conjecture 3.19 (The Farrell-Jones Conjecture for negative K-theoryof the ring of integers in an algebraic number field). Let R be ofintegers in an algebraic number field. Then, for every group G, we have

K−n(ZG) = 0 for n ≥ 2,


and the map

colimH∈SubFIN (G)K−1(ZH)∼= // K−1(ZG)

is an isomorphism.

Conjecture 3.20 (The Farrell-Jones Conjecture for negative K-theoryand Artinian rings as coefficient rings). Let G be a group and let Rbe a Artinian ring. Then

Kn(RG) = 0 for n ≤ −1.

3.5 Survey on Computations of Negative K-Groups forFinite Groups

The following result is due to Carter [167]. (See also [84, Theorem 10.6 onpage 695].)

Theorem 3.21 (Negative K-theory of ZG for finite groups G). LetR be a Dedekind domain of characteristic zero. Let k be its fraction field. Forany maximal ideal P of R, let kP be the P -adic completion. Let G be a finitegroup of order n.

For a field F we denote by rF (G) the number of isomorphism classes ofirreducible representations of G over the field F . Then

(i) Km(RG) = 0 for m ≤ −2;(ii) K−1(RG) is a finitely generated group;

(iii) Suppose that no prime divisor of n is invertible in R. Then the rank r ofthe finitely generated abelian group K−1(RG) is given by

r = 1− rk(G) +∑p|nR

rkP (G)− rR/P (G),

where the sum runs over all maximal (= non-zero prime) ideals P dividingnR;

(iv) If R is the ring of integers in an algebraic number field k, then

K−1(RG) = Zr ⊕ Z/2s

There is an explicit description of the integer s in terms of global and localSchur indices.If G contains a normal abelian subgroup of odd index, then s = 0;

(v) Let A be a finite abelian group. Then K−1(ZA) vanishes if and only if |A|is a prime power.


If R = Z, then r = 1−rQ(G)+∑p|n rQp(G)−rFp(G), where p runs through

the prime numbers dividing n.

3.6 Miscellaneous

The following result is taken from [364, Corollary 1.1].

Theorem 3.22 (Vanishing of NKn(ZG) for n ≤ 1 and G finite ofsquarefree order). Let G be a finite group and let p be a prime. Supposethat p2 does not divide the order of G. Then

NKn(ZG)(p) = 0

for n ≤ 1.

As an application, we get a new proof of the fact that NKn(ZG) = 0, forn ≤ 1, if the order of G is square-free (see Harmon[373]).

Exercise 3.23. Let G be a finite group of square-free order. Show for allk ≥ 1

Kn(Z[G× Zk]) =

K1(ZG)⊕K0(ZG)k ⊕K−1(ZG)k(k−1)/2 if n = 1;

K0(ZG)⊕K−1(ZG)k if n = 0;

K−1(ZG) if n = −1;

0 if n ≤ −2.

More information about NKn(RG) for all n ∈ Z will be given in Theo-rem 5.17, Theorem 5.18 and Theorem 5.19.

More information about negative K-groups can be found for instance in[25, 84, 166, 167, 297, 426, 560, 559, 571, 649, 650, 673, 686, 704].


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Chapter 4

The Second Algebraic K-Group

4.1 Introduction

This chapter is devoted to the second algebraic K-group.We give two equivalent definitions, namely, in terms of the Steinberg group

and in terms of the universal central extension of E(R). We extend the longexact sequence associated to a pullback of rings and to a two-sided idealbeyond K1 to K2. The long exact sequence associated to a pullback of ringscannot be extended to the left to higher algebraic K-groups, whereas thiswill be done for the long exact sequence associated to a two-sided ideal later.

We will introduce the second Whitehead group and state a variant of theFarrell-Jones Conjecture for it, namely, that it vanishes for torsionfree groups.Finally we give some information about computations of the second algebraicK-group.

4.2 Definition and Basic Properties of K2(R)

Definition 4.1 (n-th Steinberg group). For n ≥ 3 and a ring R define itsn-th Steinberg group Stn(R) to be the group given by generators and relationsas follows. The set of generators is

xri,j | i, j ∈ 1, 2, . . . , n and r ∈ R.

The relations are

(i) xri,j · xsi,j = xr+si,j for i, j ∈ 1, 2, . . . , n and r, s ∈ R;(ii) [xri,j , x

sj,k] = xrsi,k for i, j, k ∈ 1, 2, . . . , n with i 6= k and r, s ∈ R;

(iii) [xri,j , xsk,l] = 1 for i, j, k, l ∈ 1, 2, . . . , n with i 6= k, j 6= l and r, s ∈ R,

where [a, b] denotes the commutator aba−1b−1.

The idea behind the Steinberg group is that for every ring R the corre-sponding relations hold in GLn(R) if we replace xri,j by the matrix In + r ·En(i, j) appearing in Section 2.2. Hence we get a canonical group homomor-phism

φRn : Stn(R)→ GLn(R), xri,j 7→ In + r · En(i, j).

117

118 4 The Second Algebraic K-Group

The image of φRn is by definition the subgroup of GLn(R) generated by allelements of the form In + r ·En(i, j) for i, j ∈ 1, 2, . . . , n and r ∈ R. Thereis an obvious inclusion Stn(R)→ Stn+1(R) sending a generator xri,j to xri,j .

Definition 4.2 (Steinberg group). Define the Steinberg group St(R) to bethe union of the groups Stn(R).

The set of maps φRn | n ≥ 3 defines a homomorphism of groups

φR : St(R)→ GL(R)(4.3)

The image of φR is just the group E(R) which agrees with [GL(R), GL(R)](see Lemma 2.11).

Definition 4.4 (K2(R)). Define the algebraic K2-group K2(R) of a ring Rto be the kernel of the group homomorphism φR : St(R)→ GL(R) of (4.3).

Exercise 4.5. Show that there is a natural exact sequence

0→ K2(R)→ St(R)→ GL(R)→ K1(R)→ 0.

4.3 The Steinberg Group as Universal Extension

A central extension of a group Q is a surjective group homomorphism φ : G→Q with Q as target such that the kernel of φ is contained in the centerg ∈ G | g′g = gg′ for all g′ ∈ G of G. A central extension φ : U → Q of agroup Q is called universal if for every central extension ψ : G→ Q there isprecisely one group homomorphism f : U → G with ψ f = φ. If a group Qadmits a universal central extension, it is unique up to unique isomorphism.A group Q possesses a universal central extension if and only if it is perfect,i.e., it is equal to its commutator subgroup (see [592, Theorem 5.7 on page 44]or [704, Theorem 4.1.3 on page 163]). In this case the kernel of the universalcentral extension φ : U → Q is isomorphic to the second homology H2(Q;Z)of Q (see [592, Corollary 5.8 on page 46] or [704, Theorem 4.1.3 on page 163]).A central extension φ : G → Q of a group Q is universal if and only if G isperfect and every central extension ψ : H → G of G splits, i.e., there is ahomomorphism s : G → H with ψ s = idG (see [592, Theorem 5.3 onpage 43] or [704, Theorem 4.1.3 on page 163]). A central extension φ : G→ Qof a perfect group Q is universal if and only if H1(G;Z) = H2(G;Z) = 0(see [704, Corollary 4.1.18 on page 177]). The proof of the next result can befound in [592, Theorem 5.10 on page 47] or [704, Theorem 4.2.7 on page 190]

Theorem 4.6 (K2(R) and universal central extensions of E(R)). Thecanonical epimorphism φR : St(R)→ E(R) coming from the map (4.3) is theuniversal central extension of E(R).

4.4 Extending Exact Sequences of Pullbacks and Ideals 119

Exercise 4.7. Prove K2(R) ∼= H2(E(R);Z).

4.4 Extending Exact Sequences of Pullbacks and Ideals

Theorem 4.8 (Mayer-Vietoris sequence for K-theory in degree ≤ 2of a pullback of rings). Consider a pullback of rings

Ri1 //

i2

R1

j1

R2

j2// R0

such that both j1 and j2 are surjective. Then there exists a natural exact se-quence, infinite to the right,

K2(R)(i1)∗⊕(i2)∗−−−−−−−→ K2(R1)⊕K2(R2)

(j1)∗−(j2)∗−−−−−−−→ K2(R0)∂2−→

K1(R)(i1)∗⊕(i2)∗−−−−−−−→ K1(R1)⊕K1(R2)

(j1)∗−(j2)∗−−−−−−−→ K1(R0)∂1−→

K0(R)(i1)∗⊕(i2)∗−−−−−−−→ K0(R1)⊕K0(R2)

(j1)∗−(j2)∗−−−−−−−→ K0(R0)∂0−→

K−1(R)(i1)∗⊕(i2)∗−−−−−−−→ K−1(R1) ⊕ K−1(R2)

(j1)∗−(j2)∗−−−−−−−→ K−1(R0)∂−1−−→

· · · .

Proof. See [592, Theorem 6.4 on page 55] for the extension to K2. The exten-sion for negative K-theory follows for example from the fact that the passagegoing from R to R[Z] sends a pullback of rings to a pullback of rings. ut

Remark 4.9 (Surjectivity assumption is necessary). Swan [779, Corol-lary 1.2] has shown that the assumption that both j1 and j2 are surjectivein Theorem 4.8 is necessary. It is not enough that j1 or j2 is surjective, incontrast to the weaker Theorem 2.83.

Remark 4.10 (No exact sequence for pullbacks in higher degrees).Swan [779, Corollary 6.9] has shown that it is not possible to define a functorK3 so that the exact sequence appearing in Theorem 4.8 can be extended toK3.

Theorem 4.11 (Exact sequence of a two-sided ideal K-theory in de-gree ≤ 2). Given a two-sided ideal I ⊂ R, we obtain an exact sequence,natural in I ⊆ R and infinite to the right,


K2(R)pr∗−−→ K2(R/I)

∂2−→ K1(R, I)j1−→ K1(R)

pr∗−−→ K1(R/I)

∂1−→ K0(R, I)j0−→ K0(R)

pr∗−−→ K0(R/I)∂0−→ K−1(R, I)

j−1−−→ K−1(R)pr∗−−→ · · · .

where pr: R→ R/I the projection.

Proof. See [592, Theorem 6.2 on page 54], [704, Theorem 3.3.4. on page 155and Theorem 4.3.1 on page 200] or [826, Theorem 5.7.1 on page 243] ut

Remark 4.12 (Dependence of Kn(R, I) on R). The group Kn(R, I) canbe identified for n ≤ 0 with Kn(I) (see Definition 2.88) and hence dependsonly on the structure of I as a ring without unit but not on the embeddingI ⊆ R. But for n ≥ 1 the group Kn(R, I) does depend on the embeddingI ⊆ R (see Example 2.92).

Often one wants to get information about K2 in order to compute K1-groups using for instance Theorem 4.11. This is illustrated by the followingexample.

Example 4.13. Let R be the ring of integers in an algebraic number fieldand let P be a non-zero prime ideal. Then the exact sequence appearing inTheorem 4.11 induces an exact sequence

K2(R/P )→ SK1(R,P )→ SK1(R)→ SK1(R/P ),

where SK1(R) has been defined in Definition 2.15 and we put:

SK1(R,P ) := (SL(R) ∩GL(R,P ))/E(R,P )∼= ker

(det : GL(R,P )→ r ∈ R | r ≡ 1 mod P

).

Since R/P is a finite field, SK1(R/P ) and K2(R/P ) vanish by Theorem 2.17and Theorem 4.16 (v). Hence we obtain an isomorphism

SK1(R,P )∼=−→ SK1(R).

The group SK1(R) vanishes by [592, Corollary 16.3]. Hence also SK1(R,P )vanishes.

4.5 Steinberg Symbols

Let R be a commutative ring and u, v ∈ R×. Consider the elementsd1,2(u), d1,3(v) ∈ E(R) given by the invertible (3, 3)-matricesu 0 0

0 u−1 00 0 1

and

v 0 00 1 00 0 v−1

4.5 Steinberg Symbols 121

Let d1,2(u) and d1,3(v) be any preimages of d1,2(u) and d1,3(v) under the

canonical map φR : St(R)→ E(R). Then the commutator [d1,2(u), d1,3(v)] ∈St(R) defines an element in the kernel of φR : St(R) → E(R) and hence inK2(R). It depends only on u and v. The proof of the facts above can be foundfor instance in [704, page 192].

Definition 4.14 (Steinberg symbol). Let R be a commutative ring andu, v ∈ R×. The element in K2(R) given by the construction above is calledthe Steinberg symbol of u and v and is denoted by u, v.

Exercise 4.15. Prove that the Steinberg symbol of Definition 4.14 is well-defined.

Theorem 4.16 (Properties of the Steinberg symbol). Let R be a com-mutative ring. Then:

(i) The Steinberg symbol defines a bilinear skew-symmetric pairing

R× ×R× → K2(R), (u, v) 7→ u, v,

i.e., u1·u2, v = u1, v+u2, v and u, v = −v, u for all u1, u2, u, v ∈R×;

(ii) For u ∈ R× we have u,−u = 0;(iii) If for u ∈ R× also 1− u ∈ R×, then u, 1− u = 0;(iv) (Matsumoto’s Theorem) If F is a field, then K2(F ) is isomorphic to the

abelian group given by the generators u, v for u, v ∈ F× and the rela-tions:

(a) u, 1− u = 0 for u ∈ F with u 6= 0, 1;(b) u1 · u2, v = u1, v+ u2, v for u1, u2, v ∈ F×;(c) u, v1 · v2 = u, v1+ u, v2 for u, v1, v2 ∈ F×;

(v) If F is a finite field, then K2(F ) = 0;(vi) We have K2(Z) = Z/2. A generator is given by the Steinberg symbol−1,−1;

(vii) Let m ≥ 2 be an integer. If m 6= 0 mod 4, then K2(Z/m) = 0. If m = 0mod 4, then K2(Z/m) = Z/2 and a generator is given by the Steinbergsymbol 1, 1;

(viii) (Tate) We have K2(Q) = Z/2 ×∏p F×p , where p runs through the odd

prime numbers;(ix) (Bass, Tate) Let R be a Dedekind domain with quotient field F . Then there

is an exact sequence

K2(F )→⊕P

K1(R/P )→ K1(R)→ K1(F )

→⊕P

K0(R/P )→ K0(R)→ K0(F )→ 0,


where P runs through the maximal ideals of R.

Proof. (i) See [592, Theorem 8.2 on page 64] or [704, Lemma 4.2.14 onpage 194].

(ii) and (iii) See [592, Theorem 9.8 on page 74] or [704, Theorem 4.2.17 onpage 197].

(iv) See [592, Theorem 11.1 on page 93] or [704, Theorem 4.3.15 on page 214].

(v) See [592, Theorem 9.13 on page 78] or [704, Theorem 4.3.13 and Re-mark 4.3.14 on page 213].

(vi) See [592, Corollary 10.2 on page 81].

(vii) See [592, Corollary 10.8 on page 92], [247, Theorem 5.1], and [704, Ex-ercise 4.3.19 on page 217].

(viii) See [592, Theorem 11.6 on page 101].

(ix) See [592, Corollary 13.1 on page 123] and [84, pages 702, 323]. ut

4.6 The Second Whitehead Group

Let R be a ring. Consider u ∈ R× and integers i, j ≥ 1. If xui,j is the canonicalgenerator of St(R) (see Definition 4.1), then define

wui,j := xui,jx−u−1

j,i xuij ∈ St(R).

Let G be a group. Let WG be the subgroup of St(ZG) generated by allelements of the shape wgi,j for g ∈ G and integers i, j ≥ 1. Recall that we canthink of K2(ZG) as a subgroup of St(ZG).

Definition 4.17 (The second Whitehead group). Let G be a group.Define the second Whitehead group of G by

Wh2(G) := K2(ZG)/(K2(ZG) ∩WG).

Exercise 4.18. Show that the second Whitehead group of the trivial groupvanishes using the fact (see [704, Example 4.2.19 on page 198]) that w1,2(1)4 =−1,−1 holds in St(Z).

Let I denote the unit interval [0, 1]. Let M be a closed smooth manifold.A smooth pseudoisotopy of M is a diffeomorphism h : M ×I →M ×I, whichrestricted to M ×0 ⊆M × I is the obvious inclusion. The group PDiff(M)of smooth pseudoisotopies is the group of all such diffeomorphisms undercomposition. Pseudoisotopies play an important role if one tries to understandthe homotopy type of the group Diff(M) of self-diffeomorphisms of M . Twoselfdiffeomorphisms f0, f1 : M → M are called isotopic if there is a smoothmap h : M× [0, 1]→M called isotopy such that ht : M →M,x 7→ h(x, t) is a

4.8 The Second Whitehead Group of Some Finite Groups 123

selfdiffeomorphism for each t ∈ [0, 1] and hk = fk for k = 0, 1. They are calledpseudoisotopic if there exists a diffeomorphism H : M × [0, 1] → M × [0, 1]such that H(x, k) = (fk(x), k) for all x ∈ M and k = 0, 1. If h is an isotopy,then we obtain a pseudoisotopy by H(x, k) = (h(x, k), k). Hence isotopicselfdiffeomorphisms are pseudoisotopic. The converse is not true in general,there is no reason why a pseudoisotopy is level preserving, i.e., sends M×tto M × t for every t ∈ [0, 1].

In order to decide whether two selfdiffeomorphisms are isotopic, it is oftenvery useful to firstly decide whether they are pseudoisotopic what is in generaleasier.

The set of path components π0(Diff(M)) of the space Diff(M) agrees withthe set of isotopy classes of selfdiffeomorphisms of M . The group PDiff(M)acts on Diff(M) by h · f := h1 f . If PDiff(M) is path-connected, thentwo pseudoisotopic diffeomorphisms M → M are isotopic since the orbitof the identity idM : M → M under the PDiff(M)-action consists of thediffeomorphisms M → M that are pseudoisotopic to the identity. If M issimply connected, PDiff(M) is known to be path connected by a result ofCerf [169, 170] if dim(M) ≥ 5.

The relevance of the second Whitehead group comes from the followingresult of Hatcher-Wagoner [374].

Theorem 4.19 (Pseudoisotopy and the second Whitehead group).Let M be a smooth closed manifold of dimension ≥ 5. Then there is anepimorphism

π0(PDiff(M))→Wh2(π1(M)).

More information about pseudoisotopy and its relation to algebraic K-theory will be given in Chapter 6. The Farrell-Jones Conjecture for pseu-doisotopy will be stated in 13.57.

4.7 A Variant of the Farrell-Jones Conjecture for theSecond Whitehead group

Conjecture 4.20 (Farrell-Jones Conjecture for Wh2(G) for torsion-free G). Let G be a torsionfree group. Then Wh2(G) vanishes.

4.8 The Second Whitehead Group of Some FiniteGroups

We give some information about K2(ZG) and Wh2(G) for some finite groups.We have


Wh2(G) = 0, for G = 1,Z/2,Z/3,Z/4;

|Wh2(Z/6)| ≤ 2;

Wh2(D6) ∼= Z/2,

where D6 is the dihedral group of order six. The claim for the finite cyclicgroups follow from [257, page 482] and [765, pages 218 and 221]. We getK2(ZD6) ∼= (Z/2)3 from [765, Theorem 3.1]. This implies Wh2(D6) ∼= Z/2as explained in [557, Theorem 3.2.d.iii].

The 2-rank of the finite abelian group Wh2

((Z/2)n

)is at least (n − 1) ·

2n − (n+2)(n−1)2 by [242, Corollary 7]. If p is an odd prime, then the p-rank

of Wh2

((Z/p)n

)is at least (n − 1) · (pn − 1) −

(p+n−1

p

)− n(n−1)

2 by [242,

Corollary 8]. In particular Wh2

((Z/p)n

)is non-trivial for a prime p and

n ≥ 2.

Exercise 4.21. Determine all integers n ≥ 1 for which Ki(Z[Z/n]) for alli ≤ 0, Wh(Z/n) and Wh2(Z/n) vanish.

4.9 Miscellaneous

We have already mentioned that often computations involving K1 use in-formation about K2 since there are various long exact sequences relatingK-groups of different rings. Examples of such sequences have been given inTheorem 4.8, Theorem 4.11 and Theorem 4.16 (ix). Another important classof such exact sequences are given by localization sequences (see [631, Chap-ter 3]).

The second algebraic K-group of fields plays also a role in number theory,as for instance explained in [592, Chapters 11, 15, 16], [760, Chapter 8]and [703, Chapter 4, Section 4]. Keywords are Hilbert symbols, Gauss’ lawsof quadratic reciprocity, Brauer groups and the Mercurjev-Suslin Theorem.Relations to operator theory are discussed in [592, Chapter 7], and [703,Chapter 4, Section 4].

Further references to K2 and the second Whitehead group are [20, 243,244, 245, 246, 247, 375, 766].


name of texfile: ic

Chapter 5

Higher Algebraic K-Theory

5.1 Introduction

In this chapter we extend the definition of the algebraic K-groups Kn(R) toall integers n ∈ Z.

We first present the plus-construction to define higher algebraic K-theoryand record the basic properties. We introduce algebraic K-theory with coef-ficients in Z/k. We discuss other constructions of K-theory which apply tomore general situations such as to exact categories or Waldhausen categories.The previous constructions lead only to spaces and one can find deloopingswhich result in spectra whose homotopy groups are the algebraic K-groupsalso in negative degrees. We present the K-theoretic Farrell-Jones Conjec-ture for torsionfree groups and regular rings. We introduce Mayer-Vietorissequences for amalgamated products and Wang sequence for HNN exten-sions for the algebraic K-theory of group rings. The appearance of Nil-termsin these exact sequences is responsible for some complications concerningalgebraic K-theory and the Farrell-Jones Conjecture. We discuss homotopyK-theory, a theory which is on the one hand close to algebraic K-theoryand on the other hand is free of Nil-phenomena. We briefly explain relationsbetween algebraic K-theory and cyclic homology.

5.2 The Plus-Construction

Let R be a ring. So far the algebraic K-groups Kn(R) for n ≤ 2 have beendescribed in a purely algebraic fashion by generators and relations. The defi-nition of the higher algebraic K-groups Kn(R) for n ≥ 3 has been achievedtopologically, namely, one assigns to a ring R a spaceK(R) and definesKn(R)by the n-th homotopy group πn(K(R)) for n ≥ 0. This will coincide with theprevious definitions. There are various definitions of the space K(R) whichextend to more general settings as explained below and are appropriate indifferent situations. We briefly recall the technically less demanding one, theplus-construction.

A space Z is called acyclic if it has the homology of a point, i.e., thesingular homology with integer coefficients Hn(Z) vanishes for n ≥ 1 and isisomorphic to Z for n = 0.

125

126 5 Higher Algebraic K-Theory

Exercise 5.1. Prove that an acyclic space is path connected and that itsfundamental group π is perfect and satisfies H2(π;Z) = 0.

In the following we will suppress choices of and questions about basepoints. The homotopy fiber hofib(f) of a map f : X → Y of path connectedspaces has the property that it is the fiber of a fibration pf : Ef → Y whichcomes with a homotopy equivalence h : Ef → X satisfying pf = fh (see [832,Theorem 7.30 in Chapter I.7 on page 42]). The long exact homotopy sequenceassociated to f (see [832, Corollary 8.6 in Chapter IV.8 on page 187]) lookslike

(5.2)

· · · ∂3−→ π2(hofib(f))π2(i)−−−→ π2(X)

π2(f)−−−→ π2(Y )∂2−→ π1(hofib(f))

π1(i)−−−→ π1(X)

π1(f)−−−→ π1(Y )∂1−→ π0(hofib(f))

π0(i)−−−→ π0(X)π0(f)−−−→ π0(Y )→ 0.

Definition 5.3 (Acyclic map). Let X and Y be path connected CW -complexes. A map f : X → Y is called acyclic if its homotopy fiber hofib(f)is acyclic.

We conclude for an acyclic map f : X → Y from the long exact homotopysequence (5.2) that f1 : π1(X)→ π1(Y ) is surjective and its kernel is a perfectsubgroup P of π1(X) since P is a quotient of the perfect group π1(hofib(f))and π0(hofib(f)) consists of one element. Obviously a space Z is acyclic ifand only if the map Z → • is acyclic.

Definition 5.4 (Plus-construction). Let X be a connected CW -complexand P ⊆ π1(X) be a perfect subgroup. A map f : X → X+ to a CW -complexis called a plus-construction of X relative to P if f is acyclic and the kernelof f1 : π1(X)→ π1(X+) is P .

The next result is due to Quillen. A proof can be found for instance in [704,Theorem 5.2.2 on page 266 and Proposition 5.2.4 on page 268].

Theorem 5.5 (Properties of the plus-construction). Let X be a con-nected CW -complex and let P ⊆ π1(X) be a perfect subgroup. Then:

(i) There exists a plus-construction f : X → X+ relative to P . (One canconstruct X+ by attaching 2- and 3-cells to X);

(ii) Let f : X → X+ be a plus-construction relative to P and let g : X → Ybe a map such that the kernel of π1(g) : π1(X)→ π1(Y ) contains P . Thenthere is a map g : X+ → Y which is up to homotopy uniquely determinedby the property that g f is homotopic to g;

(iii) If f1 : X → X+1 and f2 : X → X+

2 are two plus-constructions for X relativeto P , then there exists a homotopy equivalence g : X+

1 → X+2 which is up

to homotopy uniquely determined by the property g f1 ' f2;

5.2 The Plus-Construction 127

(iv) If f : X → X+ is a plus-construction relative to P , then π1(f) : π1(X)→π1(X+) can be identified with the canonical projection π1(X)→ π1(X)/P ;

(v) If f : X → X+ is a plus-construction, then Hn(f ;M) : Hn(X; f∗M) →Hn(X+;M) is bijective for all n ≥ 0 and all local coefficient systems Mon X+.

Remark 5.6 (Perfect radical). Every group G has a unique largest perfectsubgroup P ⊆ G, called the perfect radical of G. In the following we willalways use the perfect radical of G for P unless explicitly stated differently.

Exercise 5.7. Show that every group has a unique largest perfect subgroup.

Exercise 5.8. Show that E(R) = [GL(R), GL(R)] is the perfect radical ofGL(R).

Definition 5.9 (Higher algebraic K-groups of a ring). Let BGL(R)→BGL(R)+ be a plus-construction in the sense of Definition 5.4 for the clas-sifying space BGL(R) of GL(R) (relative to the perfect radical of GL(R)which is E(R)). Define the K-theory space associated to R

K(R) := K0(R)×BGL(R)+,

where we view K0(R) with the discrete topology. Define the n-th algebraicK-group

Kn(R) := πn(K(R)) for n ≥ 0.

This definition makes sense because of Theorem 5.5 (i) and (iii). Noticethat for n ≥ 1 we have Kn(R) = πn(BGL(R)+).

Exercise 5.10. Show that the Definition 5.9 of Kn(R) for n = 0, 1 is com-patible with the one of Definitions 1.1 and 2.1.

For n = 0, 1, 2 Definition 5.9 is compatible with the previous Defini-tions 1.1, 2.1 and 4.4, and we have K3(R) ∼= H3(St(R)) and Kn(R) =πn(BSt(R)+) for n ≥ 3 (see [703, Corollary 5.2.8 on page 273], [343].)

A ring homomorphism f : R → S induces maps GL(R) → GL(S) andhence maps BGL(R) → BGL(S) and BGL(R)+ → BGL(S)+. We have amap K0(R)→ K0(S). Therefore f induces a map K(f) : K(R)→ K(S).

Definition 5.11 (Relative K-groups). Define for a two-sided ideal I ⊆ Rand n ≥ 0

Kn(R, I) := πn(hofib(K(pr)) : K(R)→ K(R/I))

)for pr : R→ R/I the projection.


The long exact homotopy sequence (5.2) associated to K(pr) : K(R) →K(R/I) together with Theorem 4.11 implies

Theorem 5.12 (Long exact sequence of a two-sided ideal for higheralgebraic K-theory). Let I ⊆ R be a two sided ideal. Then there is a longexact sequence, infinite to both sides

· · · ∂3−→ K2(R, I)j2−→ K2(R)

pr∗−−→ K2(R/I)∂2−→ K1(R, I)

j1−→ K1(R)

pr∗−−→ K1(R/I)∂1−→ K0(R, I)

j0−→ K0(R)pr∗−−→ K0(R/I)

∂0−→ K−1(R, I)j−1−−→ K−1(R)

pr∗−−→ K−1(R/I)∂−1−−→ · · · .

The existence of the long exact sequence of a two-sided ideal of Theo-rem 5.12 has been one important requirement of an extension of middle andlower algebraic K-theory to higher degrees.

For more information about the plus-construction we refer for instanceto [104], [703, Chapter 5], [760, Chapter 2].

5.3 Survey on Main Properties of Algebraic K-Theoryof Rings

5.3.1 Compatibility with Finite Products

The basic idea of the proof of the following result can be found in [670, (8)in §2 on page 20].

Theorem 5.13 (Algebraic K-theory and finite products). Let R0 andR1 be rings. Denote by pri : R0 × R1 → Ri for i = 0, 1 the projection. Thenwe obtain for n ∈ Z isomorphisms

(pr0)n × (pr1)n : Kn(R0 ×R1)∼=−→ Kn(R0)×Kn(R1).

5.3.2 Morita Equivalence

The idea of the proof of the next result is essentially the same as of Theo-rem 1.10

Theorem 5.14 (Morita equivalence for algebraic K-theory). For ev-ery ring R and integer k ≥ 1 there are for all n ∈ Z natural isomorphisms

µn : Kn(R)∼=−→ Kn(Mk(R)).

5.3 Survey on Main Properties of Algebraic K-Theory of Rings 129

5.3.3 Compatibility with Directed Colimits

We conclude from [670, (12) in §2 on page 20], [742].

Theorem 5.15 (Algebraic K-theory and directed colimits). Let Ri |i ∈ I be a directed system of rings. Then the canonical map

colimi∈I Kn(Ri)∼=−→ Kn

(colimi∈I Ri

)is bijective for n ∈ Z.

5.3.4 The Bass-Heller-Swan Decomposition

We have already explained the following result for n ≤ 1 in Theorem 2.69and Theorem 3.2. Definition 2.65 of NKn(R) makes sense for every n ∈ Z.The proof for higher algebraic K-theory can be found in [760, Theorem 9.8on page 207] (see also [703, Theorem 5.3.30 on page 295]). More generalversions, where twistings are allowed and additive categories are consideredare presented in [347, 349, 364, 427, 431, 495, 560].

Theorem 5.16 (Bass-Heller-Swan decomposition for algebraic K-theory).

(i) There are isomorphisms of abelian groups, natural in R, for n ∈ Z

NKn(R)⊕Kn(R)∼=−→ Kn(R[t]);

Kn(R)⊕Kn−1(R)⊕NKn(R)⊕NKn(R)∼=−→ Kn(R[t, t−1]).

The following sequence is exact and natural in R for all n ∈ Z

0→ Kn(R)(k+)∗⊕−(k−)∗−−−−−−−−−→ Kn(R[t])⊕Kn(R[t−1])

(j+)∗⊕(j−)∗−−−−−−−−→ Kn(R[t, t−1])

Cn−−→ Kn−1(R)→ 0,

where k+, k−, j+, and j− are the obvious inclusions. If we regard it as anacyclic Z-chain complex, there exists a chain contraction, natural in R;

(ii) If R is regular, then

NKn(R) = 0 for n ∈ Z;

Kn(R) = 0 for n ≤ −1.

The proof of the next result can be found in Weibel [823, Corollary 3.2].

Theorem 5.17 (NKn(R)[1/N ] vanishes for characteristic N). Let R bea ring of finite characteristic N . Then we get for n ∈ Z


NKn(R)[1/N ] = 0.

Theorem 5.18 (Vanishing criterion of NKn(RG) for finite groups).Let R be a ring and let G be a finite group. Fix n ∈ Z. Suppose NKn(R) = 0.Then

NKn(RG)[1/|G|] = 0.

Proof. This follows from Hambleton-Luck [364, Theorem A]. ut

The following result is taken from Hambleton-Luck [364, Corollary B].

Theorem 5.19 (p-elementary induction for NKn(RG)). Let R be a ringand let G be a finite group. For all n ∈ Z, the sum of the induction maps⊕

E

NKn(RE)(p) → NKn(RG)(p),

is surjective, where E runs through all p-elementary subgroups.

The following theorem due to Prasolov [665] is an extension of a result dueto Farrell [286] for n = 1 to n ≥ 1.

Theorem 5.20 (NKn(R) is trivial or infinitely generated for n ≥ 1).Let R be a ring. Then NKn(R) is either trivial or infinitely generated as

abelian group for n ≥ 1.

5.3.5 Algebraic K-Theory of Finite Fields

The following result has been proved by Quillen [669].

Theorem 5.21 (Algebraic K-theory of finite fields). Let Fq be a finitefield of order q. Then Kn(Fq) vanishes if n = 2k for some integer k ≥ 1, andis a finite cyclic group of order qk − 1 if n = 2k − 1 for some integer k ≥ 1.

Recall that K0(F ) ∼= Z and Kn(F ) = 0 for n ≤ −1 if F is a field (seeExample 1.4 and Theorem 3.6).

5.3.6 Algebraic K-Theory of the Ring of Integers in a NumberField

The computation of the higher algebraic K-groups of Z or, more generally, ofthe ring of integers R in an algebraic number field F is very hard. Quillen [669]showed that these are finitely generated as abelian group. Their ranks asabelian groups have been determined by Borel [119].

5.3 Survey on Main Properties of Algebraic K-Theory of Rings 131

Theorem 5.22 (Rational Algebraic K-theory of ring of integers ofnumber fields). Let R be a ring of integers in an algebraic number field.Let r1 be the number of distinct embeddings of F into R and let r2 be thenumber of distinct conjugate pairs of embeddings of F into C with image notcontained in R. Then

Kn(R)⊗Z Q ∼=

0 n ≤ −1;

Q n = 0;

Qr1+r2−1 n = 1;

Qr1+r2 n ≥ 2 and n = 1 mod 4;

Qr2 n ≥ 2 and n = 3 mod 4;

0 n ≥ 2 and n = 0 mod 2.

We have Kn(Z) = 0 for n ≤ −1 and the first values of Kn(Z) forn = 0, 1, 2, 3, 4, 5, 6, 7 are given by Z, Z/2, Z/2, Z/48, 0, Z, 0, Z/240.

The Lichtenbaum-Quillen Conjecture makes a prediction about the torsion(see [512, 513]) relating the algebraic K-groups to number theory via the zeta-function. We refer to the survey article of Weibel [822], where a completepicture about the algebraic K-theory of ring of integers in algebraic numberfields and in particular of K∗(Z) is given, and a list of relevant references canbe found. See also Weibel [826, VI.10 on page 579ff].

An outline how the next corollary follows from Theorem 5.40 can be foundin [670, page 29] and [703, page 294]. It is a basic tool for computations.

Corollary 5.23. Let R be a Dedekind domain with quotient field F . Thenthere is an exact sequence

· · · → Kn+1(F )→⊕P

Kn(R/P )→ Kn(R)→ Kn(F )→⊕P

Kn−1(R/P )

→ · · · →⊕P

K0(R/P )→ K0(R)→ K0(F )→ 0,

where P runs through the maximal ideals of R.

Exercise 5.24. Consider the part of the sequence

K1(Z)→ K1(Q)∂1−→⊕p

K0(Z/p)→ K0(Z)→ K0(Q)→ 0

of Corollary 5.23 for R = Z. Compute the five terms appearing in it. Guesswhat the map ∂1 is and determine the others.


5.4 Algebraic K-Theory with Coefficients

By invoking the Moore space associated to Z/k, one can introduce K-theorymod k Kn(R;Z/k) for any integer k ≥ 2. Its main feature is that there existsa long exact sequence

(5.25) · · · → Kn+1(R;Z/k)→ Kn(R)k·id−−→ Kn(R)→ Kn(R;Z/k)

→ Kn−1(R)k·id−−→ Kn−1(R)→ Kn−1(R;Z/k)→ · · · .

The next theorem is due to Suslin [771].

Theorem 5.26 (Algebraic K-theory mod k of algebraically closedfields). The inclusion of algebraically closed fields induces isomorphisms onK∗(−;Z/k).

Let p be a prime number. Quillen [669] has computed the algebraic K-groups for any algebraic extension of the field Fp of p-elements for everyprime p. One can determine Kn(Fp;Z/k) for the algebraic closure Fp of Fpfrom (5.25). Hence one obtains Kn(F ;Z/k) for any algebraically closed fieldof prime characteristic p by Suslin’s Theorem 5.26.

The next theorem is due to Suslin [772]. We will explain the topological K-groups Ktop

n (R) and Ktopn (C) of the C∗-algebras R and C in Subsection 8.3.2.

There are mod k versions Ktopn (R;Z/k) and Ktop

n (C;Z/k) for which a longexact sequence analogous to the one of (5.25) exists.

Theorem 5.27 (Algebraic and topological K-theory mod k for R andC). The comparison map from algebraic to topological K-theory induces forall integers k ≥ 2 and all n ≥ 0 isomorphisms

Kn(R;Z/k)∼=−→ Ktop

n (R;Z/k);

Kn(C;Z/k)∼=−→ Ktop

n (C;Z/k).

Generalizations of Theorem 5.27 to C∗-algebras will be discussed in Sec-tion 8.7.

Since Ktopn (C) is Z for n even and vanishes for n odd and for every alge-

braically closed field F of characteristic 0 we have an injection Q→ F for thealgebraic closure Q of Q, Theorem 5.26 and Theorem 5.27 imply for everyalgebraically closed field F of characteristic zero.

Kn(F ;Z/k) ∼=

Z/k n ≥ 0, n even;

0 n ≥ 1, n odd.

0 n ≤ −1.

5.5 Other Constructions of Connective Algebraic K-Theory 133

Exercise 5.28. Using the fact that Ktopn (R) is 8-periodic and its values for

n = 0, 1, 2, 3, 4, 5, 6, 7 are given by Z, Z/2, Z/2, 0, Z, 0 ,0, 0, computeKn(R;Z/k) and Ktop

n (R;Z/k) for n ∈ Z and k ≥ 3 an odd natural number.

5.5 Other Constructions of Connective AlgebraicK-Theory

The plus-construction works for rings and finitely generated free or projec-tive modules. However, it turns out that it is important to consider moregeneral situations, where one can feed in categories with certain extra struc-tures. The main examples are Quillen’s Q-construction (see [670, §2], [703,Chapter 5], [760, Chapter 4]), designed for exact categories, the group com-pletion construction (see [347, 746]) designed for symmetric monoidal cate-gories, and Waldhausen’s wS•-construction (see [809] and Subsection 6.3.2)designed for categories with cofibrations and weak equivalences. Given a ringR, the category of finitely generated projective R-modules yields examples ofthe type of categories above and the appropriate construction yields alwaysthe same, namely the K-groups as defined by the plus-construction above.The Q-construction and exact categories can be used to define K-theory forthe category of finitely generated R-modules (dropping projective) or thecategory of locally free OX -modules of finite rank over a scheme X. One im-portant feature is that the notion of exact sequences can be different from theone given by split exact sequences, or, equivalently, by direct sums. Whereasin Quillen’s setting one needs exact structures in an algebraic sense, Wald-hausen’s wS•-construction is also suitable for categories, where the input arespaces and one can replace isomorphisms by weak equivalences.

We briefly recall the setup of exact categories beginning with additive cat-egories. A category C is called small if its objects form a set. An additivecategory A is a small category A such that for two objects A and B themorphism set morA(A,B) has the structure of an abelian group, there ex-ists a zero-object, the direct sum A⊕ B of two objects A and B exists, andthe obvious compatibility conditions hold, e.g., composition of morphismsis bilinear. A functor of additive categories F : A0 → A1 is a functor re-specting the zero-objects such that for two objects A and B in A0 the mapmorA0

(A,B) → morA1(F (A), F (B)) sending f to F (f) respects the abelian

group structures and F (A⊕B) is a model for F (A)⊕ F (B).A skeleton D of a category C is a full subcategory such that D is small

and the inclusion D → C is an equivalence of categories, or, equivalently, forevery object C ∈ C there is an object D in D together with an isomorphism

C∼=−→ D in C.


Definition 5.29 (Exact category). An exact category P is a full additivesubcategory of some abelian category A with the following properties:

• P is closed under extensions in A, i.e., for any exact sequence 0 → P0 →P1 → P2 → 0 in A with P0, P2 in P we have P1 ∈ P;

• P has a small skeleton.

An exact functor F : P0 → P1 is a functor of additive categories that sendsexact sequences to exact sequences.

Examples of exact categories are abelian categories possessing a smallskeleton, the category of finitely generated projective R-modules, the cat-egory of finitely generated R-modules, the category of vector bundles overa compact space, the category of algebraic vector bundles over a projectivealgebraic variety, and the category of locally free sheaves of finite rank on ascheme.

An additive category becomes an exact category in the sense of Quillenwith respect to split exact sequences. On the other hand there are interestingexact categories where the exact sequences are not necessarily split exactsequences.

The Q-construction, see [670, §2], [703, Chapter 5], [760, Chapter 4], as-signs to any exact category P its K-theory space K(P) and one definesKn(P) := πn(K(P)) for n ≥ 0. If P is the category of finitely generated pro-jective R-modules, this definition coincides with the Definition 5.9 of Kn(R)coming from the plus-construction.

The Q-construction allows to define algebraic K-theory for objects natu-rally appearing in algebraic geometry, arithmetic geometry and number the-ory since these give exact categories as described above.

Example 5.30 (The category of nilpotent endomorphism). Let NIL(R)be the exact category whose objects are pairs (P, f) of finitely generated pro-jective R-modules together with nilpotent endomorphisms f : P → P . Its

K-theory Niln(R) := Kn(NIL(R)) splits as Kn(R) ⊕ Niln(R) for n ≥ 0,

where Niln(R) is the cokernel of the homomorphism Kn(R)→ Kn(NIL(R))induced by the obvious functor sending a finitely generated projective R-module P to 0: P → P . We get for n ≥ 1

NKn(R) = Niln−1(R).

This has been considered for n = 1 already in Theorem 2.69.

5.6 Non-Connective Algebraic K-Theory of Additive Categories 135

5.6 Non-Connective Algebraic K-Theory of AdditiveCategories

The approaches mentioned in Section 5.5 will always yield spaces K(R) suchthat the algebraic K-groups are defined to be its homotopy groups. Since aspace has no negative homotopy groups, this definition will not encompassthe negative algebraic K-groups. In order to take these into account, one hasto find appropriate deloopings.

So the task is to replace the space K(R) by a (non-connective) spectrumK(R) such that one can define Kn(R) by πn(K(R)) for n ∈ Z and thisdefinition coincides with the other definitions for all n ∈ Z. For rings this hasbeen achieved by Gersten [342] and Wagoner [803].

We would like to feed in additive categories. An additive category A iscalled flasque if (after possibly replacing it by an equivalent additive categoryfor which there exists a model for the direct sum) there exists a functor ofadditive categories S : A → A together with a natural equivalence idA⊕S →S.

The category of spectra SPECTRA will be introduced in Section 10.4. De-note by ADD-CAT the category of additive categories. There is an obviousnotion of the direct sum of two additive categories. We will use a constructionof Pedersen-Weibel [653] (see also [159]) or of Luck-Steimle [558] of a functor

K : ADD-CAT → SPECTRA, A 7→ K(A).(5.31)

Definition 5.32 (Algebraic K-groups of additive categories). We callK(A) the non-connective K-theory spectrum associated to an additive cate-gory. Define for n ∈ Z the n-th algebraic K-group of an additive category Aby

Kn(A) := πn(K(A)).

We conclude from Pedersen-Weibel [653] (see also [159]) or from of Luck-Steimle [558]

Theorem 5.33 (Properties of K(A)).

(i) If R is a ring and A is the additive category of finitely generated projectiveR-modules, then Kn(A) coincides with Kn(R) for n ∈ Z;

(ii) Let F1 and F2 be functors of additive categories. If there exists a naturalequivalence of such functors from F1 to F2, then the maps of spectra K(F1)and K(F2) are homotopic;In particular a functor F : A → A′ of additive categories which is anequivalence of categories induces a homotopy equivalence K(F ) : K(A)→K(A′);

(iii) If A is flasque, then K(A) is contractible.

Exercise 5.34. Give a definition of K0(A) and K1(A) as abelian groups interms of generators and relations such that in the case, where R is a ring and


A is the category of finitely generated projective R-modules this definitioncoincides with the ones appearing in Definitions 1.1 and 2.1. Show that K0(A)and K1(A) are trivial if A is flasque.

Exercise 5.35. Let A be the category of countably generated R-modules.Show that Kn(A) = 0 for all n ∈ Z.

Remark 5.36 (Non-connective K-theory spectra for exact categories).Schlichting [742] has constructed for an exact category P a delooping of thespace K(P). Thus he can assign to an exact category P a (non-connective)spectrum K(P) and define Kn(P) := πn(K(P)) for n ∈ Z. If P is the cate-gory of finitely generated projective R-modules, this definition coincides withour previous definition of Kn(R). If the exact sequences in P are given bysplit exact sequences, this definition agrees with the one of Definition 5.32when we consider P as an additive category.

5.7 Survey on Main Properties of Algebraic K-Theoryof Categories

Although we have for simplicity mainly dealt with the algebraic K-theory ofrings only, one should in general work with categories such as exact categoriesor categories with cofibrations and weak equivalences instead of the categoryof finitely generated projective R-modules only. There are some basic resultsabout algebraic K-theory which become in particular useful if one considerscategories. We mention some of them without detailed explanations.

5.7.1 Additivity

For a proof of the next result we refer for instance to [670, Corollary 1 in §3 onpage 22], [760, Corollary 4.3 on page 41], (at least in the connective setting),and [742].

Theorem 5.37 (Additivity Theorem for exact categories). Let 0 →F0

i−→ F1p−→ F2 → 0 be an exact sequence of functors Fk : P1 → P2 of exact

categories P1 and P2, i.e., i and p are natural transformations such that for

each object P the sequence 0→ F0(P )i(P )−−−→ F1(P )

p(P )−−−→ F2(P )→ 0 is exact.Then we get for the induced morphisms (Fk)n : Kn(P1)→ Kn(P2) for everyn ∈ Z

(F1)n = (F0)n + (F2)n.

5.7 Survey on Main Properties of Algebraic K-Theory of Categories 137

5.7.2 Resolution Theorem

Let M and P be exact categories which are contained in the same abeliancategory A. Suppose that P is a full subcategory ofM. A finite resolution ofan object M of M by objects in P is an exact sequence 0→ Pn → Pn−1 →· · · → P1 → P0 →M → 0 for some natural number n. We say that P is closedunder extensions in M if for any exact sequence 0→ M0 → M1 → M2 → 0inM with M0,M2 in P we have M1 ∈ P. For a proof of the next Theorem werefer for instance to [670, Corollary 1 in §4 on page 25] or [760, Theorem 4.6on page 41], (at least in the connective setting), and [742].

Theorem 5.38 (Resolution Theorem). LetM and P be exact categorieswhich are contained in the same abelian category A. Suppose that P is a fullsubcategory of M and is closed under extensions in M. Suppose that everyobject in M has a finite resolution by objects in P.

Then the inclusion P →M induces for every n ∈ Z an isomorphism

Kn(P)∼=−→ Kn(M).

5.7.3 Devissage

An object N in an abelian category is called simple if N 6= 0 and anymonomorphism M → N is the zero-homomorphism or an isomorphism. For asimple object M its ring of automorphisms endA(M) is a skew-field (Schur’sLemma). An object N in an abelian category is called semisimple if it isisomorphic to a finite direct sum of simple objects. A zero object is calledan object of length 0. Call the simple objects of an abelian category objectsof type 1. We define inductively for n ≥ 2 an object M to be of length n ifthere exists an exact sequence 0→M1 →M →M2 → 0 for an object M1 oflength 1 and an object M2 of length (n − 1). An object is of finite length ifit has length n for some natural number n. For a proof of the next result werefer for instance to [670, Corollary 1 in §5 on page 28],[760, Theorem 4.8 onpage 42].

Theorem 5.39 (Devissage). Let A be an abelian category. Suppose thatthere is a subset S of the set of objects of A with the property that anysimple object in A is isomorphic to precisely one object in S. Let Ass be thefull subcategory of A consisting of semisimple objects and let Afl be the fullsubcategory consisting of objects of finite length. Then we obtain for everyn ∈ Z, n ≥ 0 an isomorphism⊕

M∈SKn(endA(M))

∼=−→ Kn(Ass)∼=−→ Kn(Afl).


If A is a Noetherian abelian category, the its negative K-groups vanish andTheorem 5.39 holds also for negative K-groups of trivial reasons, see [742].

In particular we get in the situation of Theorem 5.39 from Example 1.4and Theorem 2.6.

K0(Afl) ∼=⊕S

Z;

K1(Afl) ∼=∏S

endA(S)×/[endA(S)×, endA(S)×].

5.7.4 Localization

Theorem 5.40 (Localization). Let A be a small abelian category and let Bbe an additive subcategory such that for any exact sequence 0→M0 →M1 →M2 → 0 in A the object M1 belongs to B if and only if both M0 and M2 belongto B. Then there exists a well-defined quotient abelian category A/B. It hasthe same objects as A, and its morphisms are obtained from those in A byformally inverting morphisms whose kernel and cokernel belong to B.

Then there are obvious functors B → A and A → A/B which induce along exact sequence

· · · → Kn+1(A/B)→ Kn(B)→ Kn(A)→ Kn(A/B)→ · · · .

A description ofA/B can be found in [760, Appendix B]. A proof of the lasttheorem can be found in [670, Theorem 5 in §5 on page 29], [760, Theorem 4.9on page 42], (at least in the connective setting), and [742].

5.8 The K-Theoretic Farrell-Jones Conjecture forTorsionfree Groups and Regular Rings

The Farrell-Jones Conjecture for algebraic K-theory, which we will formulatein full generality in Conjecture 11.1 reduces for a torsionfree group and aregular ring to the following conjecture. Under the additional assumptionthat there is a finite model for BG it appears already in [404]. Comment9: Take a look at this reference and make it more precise.

Conjecture 5.41 (Farrell-Jones Conjecture for torsionfree groupsand regular rings for K-theory). Let G be a torsionfree group. LetR be a regular ring. Then the assembly map

Hn(BG; K(R))→ Kn(RG)

5.8 Torsionfree Groups and Regular Rings 139

is an isomorphism for n ∈ Z.

Here H∗(−; K(R)) denotes the homology theory which is associated to the(non-connective) K-spectrum K(R) of (5.31). Recall that Hn(•; K(R)) isKn(R) for n ∈ Z, where here and elsewhere • denotes the space consistingof one point. The space BG is the classifying space of the group G, which upto homotopy is characterized by the property that it is a CW -complex withπ1(BG) ∼= G whose universal covering is contractible. The technical detailsof the construction of Hn(−; K(R)) and the assembly map will be explainedin a more general setting in Sections 10.4 and 10.5.

The point of Conjecture 5.41 is that on the right-hand side of the assemblymap we have the group Kn(RG) we are interested in, whereas the left-handside is a homology theory and hence much easier to compute. A basic toolfor the computation of a homology theory is the Atiyah-Hirzebruch spectralsequence, which in our case has as E2-term E2

p,q = Hp(BG;Kq(R)) andconverges to Hp+q(BG; K(R)).

Remark 5.42 (The conditions appearing in Conjecture 5.41 are nec-essary). The condition that G is torsionfree and that R is regular are nec-essary in Conjecture 5.41. If one drops one of these conditions, one obtainscounterexamples as follows.

If G is a finite group, then we obtain an isomorphism

Kn(R)⊗Z Q ∼= Hn(•; K(R))⊗Z Q∼=−→ Hn(BG; K(R))⊗Z Q.

Hence Conjecture 5.41 would predict for a finite group that the change of rings

homomorphism Kn(R)⊗Z Q∼=−→ Kn(RG)⊗Z Q is bijective. This contradicts

for instance Lemma 1.75.In view of the Bass-Heller-Swan decomposition 5.16, Conjecture 5.41 is

true for G = Z in degree n only if NKn(R) vanishes.

Exercise 5.43. Let R be a regular ring. Let G = G1 ∗G0 G2 be an amalga-mated free product of torsionfree groups, where G0 is a common subgroupof G1 and G2. Suppose that Conjecture 5.41 is true for G0, G1, G2, andG with coefficients in the ring R. Show that then there exists a long exactMayer-Vietoris sequence

· · · → Kn(RG0)→ Kn(RG1)⊕Kn(RG2)→ Kn(RG)

→ Kn−1(RG0)→ Kn−1(RG1)⊕Kn−1(RG2)→ · · ·

Exercise 5.44. Let R be a regular ring. Let φ : G→ G be an automorphismof the torsionfree group G. Suppose that Conjecture 5.41 is true for G andthe semidirect product GoφZ with coefficients in the ring R. Show that thenthere exists a long exact Wang sequence


· · · → Kn(RG)id−φ∗−−−−→ Kn(RG)→ Kn(R[Goφ Z])

→ Kn−1(RG)id−φ∗−−−−→ Kn−1(RG)→ · · ·

Remark 5.45 (K∗(ZG) ⊗Z Q for torsionfree G). Rationally the Atiyah-Hirzebruch spectral sequence always collapses and the homological Cherncharacter gives an isomorphism

ch:⊕p+q=n

Hp(BG;Q)⊗Q (Kq(R)⊗Z Q)∼=−→ Hn(BG; K(R))⊗Z Q.

The Atiyah-Hirzebruch spectral sequence and the Chern character willbe discussed in a much more general setting in Subsection 10.6.1 and Sec-tion 10.7.

Because of Theorem 5.22 the left hand side of the isomorphism described inRemark 5.45 specializes for R = Z to Hn(BG;Q)⊕

⊕∞k=1Hn−(4k+1)(BG;Q).

Hence Conjecture 5.41 predicts for a torsionfree group G

Kn(ZG)⊗Z Q ∼= Hn(BG;Q)⊕∞⊕k=1

Hn−(4k+1)(BG;Q).(5.46)

Conjecture 5.47 (Nil-groups for regular rings and torsionfree groups).Let G be a torsionfree group and let R be a regular ring. Then

NKn(RG) = 0 for all n ∈ Z.

Exercise 5.48. Show that a torsionfree group G satisfies Conjecture 5.47 forall regular rings R if it satisfies Conjecture 5.41 for all regular rings R.

5.9 Mayer-Vietoris Sequences for AmalgamatedProducts and Wang Sequences for HNN-Extensions

We have seen in the introduction that for the topological K-theory of reducedgroup C∗-algebras there exist Mayer-Vietoris sequences associated to amal-gamated products (see (0.1)) and long exact Wang sequences for semi-directproducts of the shape G = HoφZ (see (0.2)). These lead to the final formula-tion of the Baum-Connes Conjecture 0.6. Because of Exercises 5.43 and 5.44one can expect similar long exact sequences to exists for algebraic K-theoryof group rings for torsionfree groups and regular rings, but not in general, asone can derive for instance from the Bass-Heller-Swan decomposition 5.16.

5.9 Mayer-Vietoris Sequences 141

We want to explain the more complicated general answer for algebraicK-theory of group rings which is given by Waldhausen [805] and [806].

A ring R is called regular coherent if every finitely presented R-modulepossesses a finite projective resolution. A ring R is regular if and only if itis regular coherent and Noetherian. A group G is called regular or regularcoherent respectively, if for any regular ring R the group ring RG is regularor regular coherent respectively. If G = G1 ∗G0

G2 for regular coherent groupsG1 and G2 and a regular group G0, or if G = H oφ Z for a regular group H,then G is regular coherent. In particular Zn is regular and regular coherent,whereas a non-abelian finitely generated free group is regular coherent butnot regular. For proofs of the claims above and for more information aboutregular coherent groups we refer to [806, Theorem 19.1].

The maps of spectra appearing in the theorem below are all induced byobvious functors between categories.

Theorem 5.49 (Waldhausen’s cartesian squares for non-connectivealgebraic K-theory). Let G = G1 ∗G0

G2 be a free amalgamated productand let R be a ring.

(i) The exists a homotopy cartesian square of spectra

Nil(RG0;RG1, RG2)j //

i

K(RG1) ∨K(RG2)

k1

K(RG0)

k0

// K(RG)

where Nil(RG0;RG1, RG2) is a certain non-connective Nil-spectrum as-sociated to G = G1 ∗G0G2 and R, and K is the (non-connective) K-theoryspectrum;

(ii) There is a map f : K(RG0) ∨ K(RG0) → Nil(RG0;RG1, RG2) and fork = 1, 2 a map gk : Nil(RG0;RG1, RG2) → K(RG0) with the followingproperties. The composite gk f : K(RG0) ∨ K(RG0) → K(RG0) is theprojection to the k-th summand, the composite

K(RG0) ∨K(RG0)f−→ Nil(RG0;RG1, RG2)

j−→ K(RG1) ∨K(RG2)

is homotopic to K(j1) ∨K(j2) for jk : G0 → Gk the canonical inclusion,and i f is homotopic to id∨ id : K(RG0) ∨K(RG0)→ K(RG0);

(iii) If R is regular and G0 is regular coherent, then f : K(RG0) ∨K(RG0)→Nil(RG0;RG1, RG2) is a weak homotopy equivalence;

(iv) The composite of the map ΩK(RG)→ Nil(RG0;RG1, RG2) associated tothe homotopy cartesian square of assertion (i) with the canonical map tothe homotopy cofiber of the map f induces a split surjection on homotopygroups.


Proof. All these claims are proved for connectiveK-theory in Waldhausen [806,11.2, 11.3, 11.6]. In [69, Section 9 and 10] the definitions and assertions are ex-tended to the non-connective version except for assertion (iv). Assertion (iv)can be derived from the connective version by using the Bass-Heller-Swandecomposition 5.16. ut

Theorem 5.50 (Mayer-Vietoris sequence of an amalgamated prod-uct for algebraic K-theory). Let G = G1 ∗G0

G2 be a free amalgamatedproduct and let R be a ring. Denote by ik : G0 → Gk and jk : Gk → G theobvious inclusions. Define NKn(RG0;RG1, RG2) to be the (n−1)-homotopygroup of the homotopy cofiber of the map f appearing in Theorem 5.49 (ii). Letpn : Kn(RG) → NKn(RG0;RG1, RG2) be the split surjection coming fromTheorem 5.49 (iv). Then

(i) We obtain a splitting

Kn(RG) ∼= ker(pn)⊕NKn(RG0;RG1, RG2);

(ii) There exists a long exact Mayer-Vietoris sequence sequence

· · · ∂n+1−−−→ Kn(RG0)(i1)∗⊕(i2)∗−−−−−−−→ Kn(RG1)⊕Kn(RG2)

(j1)∗−(j2)∗−−−−−−−→ ker(pn)

∂n−→ Kn−1(RG0)(i1)∗⊕(i2)∗−−−−−−−→ Kn−1(RG1)⊕Kn−1(RG2)

(j1)∗−(j2)∗−−−−−−−→ · · · ;

(iii) If G0 is regular coherent and R is regular, then

NKn(RG0;RG1, RG2) = 0 for n ∈ Z

and the sequence of assertion (ii) reduces to the long exact sequence

· · · ∂n+1−−−→ Kn(RG0)(i1)∗⊕(i2)∗−−−−−−−→ Kn(RG1)⊕Kn(RG2)

(j1)∗−(j2)∗−−−−−−−→ Kn(RG)

∂n−→ Kn−1(RG0)(i1)∗⊕(i2)∗−−−−−−−→ Kn−1(RG1)⊕Kn−1(RG2)

(j1)∗−(j2)∗−−−−−−−→ · · · .

Exercise 5.51. Show that Theorem 5.49 implies Theorem 5.50.

Analogously one gets from Waldhausen [805] and [806] using [69, Section 9and 10]

Theorem 5.52 (Wang sequence associated to an HNN-extension foralgebraic K-theory). Let α, β : H → K be two injective group homomor-phisms. Let G be the associated HNN-extension and let j : K → G be thecanonical inclusion. Then there are certain Nil-groups NKn(RH,RK,α, β)and homomorphisms pn : Kn(RG) → NKn(RH,RK,α, β) such that the fol-lowing holds:

(i) There is a long exact Wang sequence

5.10 Homotopy Algebraic K-Theory 143

· · · ∂n+1−−−→ Kn(RH)α∗−β∗−−−−→ Kn(RK)

j∗−→ ker(pn)

∂n−→ Kn−1(RH)α∗−β∗−−−−→ Kn−1(RK)

j∗−→ · · · ;

(ii) The map pn : Kn(RG)→ NKn(RH,RK,α, β) is split surjective;(iii) If R is regular and H is regular coherent, then NKn(RH,RK,α, β) van-

ishes for all n ∈ Z. In this case the Wang sequence reduces to

· · · ∂n+1−−−→ Kn(RH)α∗−β∗−−−−→ Kn(RK)

j∗−→ Kn(RG)

∂n−→ Kn−1(RH)α∗−β∗−−−−→ Kn−1(RK)

j∗−→ · · · .

Remark 5.53 (Wang sequence of a semi-direct product G = K oφ Zfor algebraic K-theory). A semi-direct product G = K oφ Z for a groupautomorphism φ : K → K is a special case of an HNN-extensions, namelytake H = K, α = id and β = φ. In this case the Wang sequence appearingin Theorem 5.52 (i) takes the form

· · · ∂n+1−−−→ Kn(RK)id−φ∗−−−−→ Kn(RK)

j∗−→ ker(pn)

∂n−→ Kn−1(RK)id−φ∗−−−−→ Kn−1(RK)

j∗−→ · · ·

and we get an isomorphism

N+Kn(RK,φ)⊕N−Kn(RK,φ)∼=−→ NKn(RK,RK, id, φ),

whereN±Kn(RK,φ) is the image of the injectionKn(RKφ[t])→ Kn(RKφ[t, t−1])induced by the ring homomorphism RKφ[t] → RKφ[t, t−1] of twisted poly-nomial and Laurent rings sending t to t±1.

We mention the following computation from [559, Theorem 9.4].

Theorem 5.54 (Vanishing of NKn(RK,φ)). Let R be a regular ring. Let

K be a finite group of order r and let φ : K∼=−→ K be an automorphism of

order s.Then N±Kn(RK,φ)[1/rs] = 0 holds for all n ∈ Z. In particular we get

N±Kn(RK,φ)⊗Z Q = 0 for all n ∈ Z.

The non-connective K-theory of additive categories with automorphismsis treated in [560].

5.10 Homotopy Algebraic K-Theory

Homotopy algebraic K-theory has been introduced by Weibel [824]. He con-structs for a ring R a spectrum KH(R) and defines


KHn(R) := πn(KH(R)) for n ∈ Z.(5.55)

The main feature of homotopy K-theory is that it is homotopy invariant,i.e., for every ring R and every n ∈ Z the canonical inclusion induces anisomorphism [824, Theorem 1.2 (i)]

KHn(R)∼=−→ KHn(R[t]).(5.56)

Notice that homotopy invariance does not hold for algebraic K-theory unlessR is regular, see Theorem 5.16.

A consequence of homotopy invariance is that we get for every ring R andn ∈ Z isomorphisms (see [824, Theorem 1.2 (iii)])

KHn(R)⊕KHn−1(R)∼=−→ KHn(RZ).(5.57)

Hence the are no Nil-terms appearing for the trivial HNN-extension G×Z.It turns out that there are no Nil-phenomena concerning amalgamated prod-ucts and HNN-extensions in general. Namely, we conclude from [69, Theo-rem 11.3]

Theorem 5.58 (Mayer-Vietoris sequence of an amalgamated prod-uct for homotopy K-theory). Let G = G1 ∗G0 G2 be a free amalgamatedproduct and let R be a ring. Denote by ik : G0 → Gk and jk : Gk → G theobvious inclusions.

Then there exists a Mayer-Vietoris sequence

· · · ∂n+1−−−→ KHn(RG0)(i1)∗⊕(i2)∗−−−−−−−→ KHn(RG1)⊕KHn(RG2)

(j1)∗−(j2)∗−−−−−−−→ KHn(RG)∂n−→ KHn−1(RG0)

(i1)∗⊕(i2)∗−−−−−−−→ KHn−1(RG1)⊕KHn−1(RG2)(j1)∗−(j2)∗−−−−−−−→ · · · ;

Theorem 5.59 (Wang sequence associated to an HNN-extension forhomotopy K-theory). Let α, β : H → K be two injective group homomor-phisms. Let G be the associated HNN-extension and let j : K → G be thecanonical inclusion. Then there is a long exact Wang sequence

· · · ∂n+1−−−→ KHn(RH)α∗−β∗−−−−→ KHn(RK)

j∗−→ KHn(RG)

∂n−→ KHn−1(RH)α∗−β∗−−−−→ KHn−1(RK)

j∗−→ · · · .

There is a natural map of (non-connective) spectra K(R)→ KH(R) andhence one obtains natural homomorphisms

Kn(R) → KHn(R) for n ∈ Z.(5.60)

5.11 Algebraic K-Theory and Cyclic Homology 145

This map is in general neither injective nor surjective. It is bijective if R isregular. In some sense homotopy algebraic K-theory is the best approxima-tion of algebraic K-theory by a homotopy invariant functor.

Exercise 5.61. Let R = R0⊕R1⊕R2⊕ . . . be a graded ring. Show that the

inclusion i : R0 → R induces isomorphisms KHn(R0)∼=−→ KHn(R) for n ∈ Z.

The same argument as for the Baum Conjecture in Subsection 0.1.3 leadsto

Conjecture 5.62 (Farrell-Jones Conjecture for torsionfree groupsfor homotopy K-theory). Let G be a torsionfree group. Then the as-sembly map

Hn(BG; KH(R))→ KHn(RG)

is an isomorphism for every n ∈ Z and every ring R.

The next result is taken from [74, Lemma 2.11].

Lemma 5.63.(i) Let R be a ring of finite characteristic N . Then the canon-ical map from algebraic K-theory to homotopy K-theory induces an iso-morphism

Kn(R)[1/N ]∼=−→ KHn(R)[1/N ]

for all n ∈ Z;(ii) Let H be a finite group. Then the canonical map from algebraic K-theory

to homotopy K-theory induces an isomorphism

Kn(Z[H])⊗Z Q∼=−→ KHn(Z[H])⊗Z Q

for all n ∈ Z.

Conjecture 5.64 (Comparison of algebraic K-theory and homotopyK-theory for torsionfree groups). Let R be a regular ring and let G bea torsionfree group. Then the canonical map

Kn(RG)→ KHn(RG)

is bijective for all n ∈ Z.

Notice that Conjecture 5.64 follows from Conjecture 5.41 and Conjec-ture 5.62.

5.11 Algebraic K-Theory and Cyclic Homology

Fix a commutative ring k, referred to as the ground ring. Let R be a k-algebra.We denote by HH⊗k∗ (R) the Hochschild homology of R relative to the ground


ring k, and similarly by HC⊗k∗ (R), HP⊗k∗ (R), and HN⊗k∗ (R) the cyclic, theperiodic cyclic, and the negative cyclic homology of R relative to k. Hochschildhomology receives a map from the algebraic K-theory, which is known as theDennis trace map. There are variants of the Dennis trace taking values incyclic, periodic cyclic and negative cyclic homology (sometimes called Cherncharacters), as displayed in the following commutative diagram.

HN⊗k∗ (R) //

h

HP⊗k∗ (R)

K∗(R)

ntr

88

dtr // HH⊗k∗ (R) // HC⊗k∗ (R).

(5.65)

For the definition of these maps, see [516, Chapters 8 and 11] and [547,Section 5]. In [547] the question is investigated which parts of Kn(RG)⊗Z Qcan be detected by using the linear traces above.

Remark 5.66. In [118], Bokstedt, Hsiang and Madsen define the cyclotomictrace, a map out of K-theory which takes values in topological cyclic homol-ogy. The cyclotomic trace map can be thought of as an even more elaboraterefinement of the Dennis trace map. In contrast to the Dennis trace, thecyclotomic trace has the potential to detect almost all of the rationalizedK-theory of an integral group ring. This question is investigated in detail byLuck-Rognes-Reich-Varisco [548, 549].

5.12 Miscellaneous

A good source of survey articles about algebraic K-theory is the handbookof K-theory, edited by Friedlander and Grayson [332]. There the relevanceof higher algebraic K-theory for algebra, topology, arithmetic geometry, andnumber theory is explained. Other good sources are the books by Rosen-berg [704] and Weibel [826].

The relation of the exact sequences for amalgamated products and HNN-extensions appearing in Sections 5.9 and 5.10 to the Farrell-Jones Conjectureis explained in Section 13.7.

The exact sequences for amalgamated products and HNN-extensions ap-pearing in Sections 5.9 and 5.10 are the main ingredients in the proof thatConjecture 5.41 holds for a certain class of groups CL, see [806, Theorem 19.4on page 249] in the connective case and [69, Corollary 0.12] in general. Theclass CL is described and analyzed in [806, Definition 19.2 on page 248 andTheorem 17.5 on page 250] and [69, Definition 0.10]. It is closed under takingsubgroups and contains for instance all torsionfree 1-relator groups.

The compatibility of algebraic K-theory with infinite products of cate-gories is examined in [161] and [466].



name of texfile: ic

Chapter 6

Algebraic K-Theory of Spaces

6.1 Introduction

We give a brief introduction to the K-theory of spaces called A-theory. Thistheory was initialized by Waldhausen. Its benefit is that it allows to study in-teresting spaces of geometric structures such as groups of diffeomorphisms orhomeomorphism of manifolds, pseudoisotopy spaces, spaces of h-cobordismsand Whitehead spaces. It is the instance of a very successful strategy in topol-ogy to extend algebraic notions to spaces. Other examples of this type aretopological Hochschild homology and topological cyclic homology.

6.2 Pseudo-isotopy

Let I denote the unit interval [0, 1]. A topological pseudoisotopy of a compactmanifold M is a homeomorphism h : M × I → M × I, which restricted toM×0∪∂M×I is the obvious inclusion. The space P (M) of pseudoisotopiesis the group of all such homeomorphisms, where the group structure comesfrom composition. If we allow M to be non-compact, we will demand thath has compact support, i.e., there is a compact subset C ⊆ M such thath(x, t) = (x, t) for all x ∈M − C and t ∈ [0, 1].

Pseudoisotopies play an important role if one tries to understand the ho-motopy type of the space Top(M) of self-homeomorphisms of a closed ma-nifold M . We will see in Section 7.20 how the results about pseudoisotopiesdiscussed in this section combined with surgery theory lead to quite explicitresults about the homotopy groups of Top(M) for a closed aspherical mani-fold M .

There is a stabilization map P (M)→ P (M × I) given by crossing a pseu-doisotopy with the identity on the interval I and the stable pseudoisotopyspace is defined as P(M) = colimj P (M × Ij). There exist also smooth ver-sions PDiff(M) and PDiff(M) = colimj P

Diff(M × Ij). The PL-version agreesfor closed manifolds of dimension ≥ 6 with the topological version (see [145]).

The natural inclusions P (M)→ P(M) and PDiff(M)→ PDiff(M) induceisomorphisms on the i-th homotopy group if the dimension n of M is largecompared to i, roughly for i ≤ n/3 see [146], [374], [376] and [411].

Next we want to define a delooping of P (M). Let p : M × Rk × I → Rkdenote the natural projection. For a manifold M the space Pb(M ;Rk) of

149

150 6 Algebraic K-Theory of Spaces

bounded pseudoisotopies is the space of all self-homeomorphism h : M×Rk×I →M×Rk×I satisfying: i.) The restriction of h to M×Rk×0∪∂M×R×[0, 1] is the inclusion, ii.) the map h is bounded in the Ri-direction, i.e., the setph(y)−p(y) | y ∈M×Rk×I is a bounded subset of Rk, and iii.) the maph has compact support in the M -direction, i.e., there is a compact subset C ⊆M such that h(x, y, t) = (x, y, t) for all x ∈M−C, y ∈ Ri and t ∈ [0, 1]. Thereis an obvious stabilization map Pb(M ;Rk) → Pb(M × I;Rk) and a stablebounded pseudoisotopy space Pb(M ;Rk) = colimj Pb(M×Ij ;Rk). There is ahomotopy equivalence Pb(M ;Rk) → ΩPb(M ;Rk+1), see [377, Appendix II].Comment 10: Take a look at this reference. Hence the sequences of spacesPb(M ;Rk) for k = 0, 1, 2, . . . and Ω−iPb(M) for i = 0,−1,−2, . . . definean Ω-spectrum P(M). Analogously one defines the differentiable boundedpseudoisotopies PDiff

b (M ;Rk) and an Ω-spectrum PDiff(M).There is a simplicial construction which allows to extend these definitions

for manifolds to all topological spaces. We actually obtain covariant functorsfrom the category of topological spaces to the category of Ω-spectra. Webriefly indicate the definition of Pi(X) for a space X as a simplicial complex.The general idea is to consider pseudoisotopies parametrized over the stan-dard k-simplex. ∆k. The following description is taken from [295, 1.1], wherefor more details the authors refer to [377] and [673].

A k-simplex of Pi(X) is given by a natural number n, a compact smoothcodimension zero submanifold Y ⊆ Rn, a map h : ∆k × Y × Ri × [0, 1] →∆k×Y ×Ri×[0, 1], and a map g : ∆k×Y → X satisfying (i.) for each x ∈∆k

the map h induces a map hx : x × Y ×Ri × [0, 1]∼=−→ x × Y ×Ri × [0, 1],

(ii.) the map hx is a topological pseudoisotopy bounded in the Ri factorwith a bound independent of x, i.e., for each x ∈ ∆ the map hx is a self-homeomorphisms of x×Y ×Ri× [0, 1] such that its restriction to x×Y ×Rk×0∪x×∂Y ×Ri×[0, 1] is the inclusion and there exists a real numberR with the property that for the projection px : x×Y ×Ri× [0, 1]→ Ri thesubset px hx(z)−px(z) | z ∈ x×Y ×Ri× [0, 1] of Ri is contained in theball around the origin of radius R for all x ∈ ∆k. Two such triples Y ⊆ Rn,h : ∆k×Y ×Ri× [0, 1]→∆k×Y ×Ri× [0, 1], g : ∆k×Y → X and Y ′ ⊆ Rn′ ,h′ : ∆k ×Y ′×Ri× [0, 1]→∆k ×Y ′×Ri× [0, 1], g : ∆k ×Y ′ → X representthe same simplex if there is a codimension zero submanifold A ⊆ Y such thatthe support of h is contained in ∆k×A×Ri× [0, 1], there is an identificationY ′ = A× Iq, the map h′ agrees with the obvious map induced by h, and idIq

and g|∆k×A pr = g′ holds for the projection pr : ∆k × A × Iq → ∆k × A.Functoriality in X is given by composing the reference map g to X with amap X → X ′.

Comment 11: Add references to the papers by Enkelmann and Pieper.

Definition 6.1 ((Non-connective) pseudo-isotopy spectrum). We callthe Ω-spectra P(X) and PDiff(X) associated to a topological space X the(non-connective) pseudoisotopy spectrum and the smooth (non-connective)pseudoisotopy spectrum of X.

6.3 Whitehead Spaces and A-Theory 151

We conclude from [377, Proposition 1.3].

Theorem 6.2 (Pseudo-isotopy is a homotopy-invariant functor). Letf : X → Y be a weak homotopy equivalence. Then the induced maps

P(f) : P(X)→ P(Y );

PDiff(f) : PDiff(X)→ PDiff(Y ),

are weak homotopy equivalences.

Comment 12: Mention the contributions of Enkelmann and Pieper andmention that there are some flaws concerning functoriality.

6.3 Whitehead Spaces and A-Theory

6.3.1 Categories with Cofibrations and Weak Equivalences

The following definition is a generalization of the notion of an exact categoryof Definition 5.29 in the sense of Quillen. It allows to deal with spaces insteadof algebraic objects such as modules. It is due to Waldhausen.

A category C is called pointed if it comes with a distinguished zero-object,i.e., an object which is both initial and terminal.

Definition 6.3 (Category with cofibrations and weak equivalences).A category with cofibrations and weak equivalences is a small pointed categorywith a subcategory coC, called category of cofibrations in C and a subcategorywC, called category of weak equivalences in C such that the following axiomsare satisfied:

(i) The isomorphisms in C are cofibrations, i.e., belong to coC;(ii) For every object C the map ∗ → C is a cofibration, where ∗ is the distin-

guished zero-object;

(iii) If in the diagram A Booioo f // C the left arrow is a cofibration, the

pushout

A //i //

f

B

f

C //i // D

exists and i is a cofibration;(iv) The isomorphisms in C are contained in wC;(v) If in the commutative diagram


B

'

Aoooo //

'

C

'

B′ A′oooo // C ′

the horizontals arrow on the left are cofibrations, and all vertical arrowsare weak equivalences, then the induced map on the pushout of the upperrow to the pushout of the lower row is a weak isomorphism.

Example 6.4 (Exact categories are categories with cofibrations andweak equivalences). Let P ⊆ A be an exact category in the sense ofDefinition 5.29. The zero-object is just a zero-object in the abelian categoryA. An cofibration in P is a morphism i : A → B which occurs in an exactsequence 0 → A → B → C → 0 of P. The weak equivalences are given bythe isomorphisms.

Exercise 6.5. Let C be the category of finite projective R-chain complexes.Define cofibrations to be chain maps i∗ : C∗ → D∗ such that in : Cn → Dn issplit injective for all n ≥ 0. Define weak equivalences to be homology equiva-lences. Show that C is a category with cofibrations and weak equivalences inthe sense of Definition 6.3 ignoring the fact that C is not small.

Example 6.6 (The category R(X) of retractive spaces). Let X be aspace. A retractive space over X is a triple (Y, r, s) consisting of a spaceY and maps s : X → Y and r : Y → X such that s is a cofibration andr s = idX . A morphism from (Y, r, s) to (Y ′, r′, s′) is a map f : X → X ′

satisfying r′ f = r and f s = s′. The zero-object is (X, idX , idX). Amorphism f : (Y, r, s) → (Y ′, r′, s′) is declared to be a cofibration if the un-derlying map of spaces f : Y → Y ′ is a cofibration. Now there are severalpossibilities to define weak equivalences. One may require that f : Y → Y ′ isa homeomorphism, a homotopy equivalence, weak homotopy equivalence ora homology equivalence with respect to some fixed homology theory. Thenone obtains a category R(X) with cofibrations and weak equivalences in thesense of Definition 6.3 except that R(X) is not small.

To achieve that R(X) is small and later to get interesting K-theory, onemay for instance require that (Y,X) is a relative CW -complex which is rel-atively finite, and s : X → Y is the inclusion and morphisms to be cellu-lar maps. Denote this category with cofibrations and weak equivalences byRf (X), where we choose all weak homotopy equivalences as weak equiva-lences and inclusion of relative CW -complexes as cofibrations.


6.3.2 The wS•-Construction

Let C be a category with cofibrations and weak equivalences. Next we brieflyrecall Waldhausen’s wS•-construction (see [809, Section 1.3]).

For an integer n ≥ 0 let [n] be the ordered set 0, 1, 2, . . . , n. Let ∆be the category whose set of objects is [n] | n = 0, 1, 2 . . . and whoseset of morphisms from [m] to [n] consists of the order preserving maps. Asimplicial category is a contravariant functor from ∆ to the category CATof categories. Analogously, a simplicial category with cofibrations and weakequivalences is a contravariant functor from ∆ to the category CATCOFWEQof categories with cofibrations and weak equivalences Now we assign to C asimplicial category with cofibrations and weak equivalences S•C as follows.Define SnC to be the category for which an object is a sequence of cofibrations

A0,1k0,1−−→ A0,2

k0,2−−→ · · · k0,n−1−−−−→ A0,n together with explicit choices of quotientobjects pri,j : A0,j → Ai,j = A0,j/A0,i for i, j ∈ 1, 2, . . . , n, i < j, i.e., wefix pushouts

A0,i

k0,j−1···k0,i//

A0,,j

pri,j

0 // Ai,j

Morphisms are given by collection of morphisms fi,j which make the obvi-ous diagram commute.

With these explicit choices of quotient objects it is easy to define therelevant face and degeneracy maps. For instance the face map di : SnC →Sn−1C is given for i ≥ 1 by dropping A0,i and for i = 0 by passing toA0,2/A0,1 → A0,3/A0,1 → · · · → A0,n/A0,1. An arrow in SnC is declared tobe a cofibration if each arrow Ai,j → Ai′,j′ is a cofibration and analogouslyfor weak equivalences.

We obtain a simplicial category wS•C by considering the category of weakequivalences of S•C. Let |wS•C| be the geometric realization of the simpli-cial category wS•C which is the geometric realization of the bisimplicial setobtained by the composite of the functor nerve of a category with wS•C.

Definition 6.7 (Algebraic K-theory space of a category with cofi-brations and weak equivalences). Let C be a category with cofibrationsand weak equivalences. Its algebraic K-theory space K(C) is defined by

K(C) := Ω|wS•C|.

The 1-skeleton in the S• direction of |wS•C| is obtained from |wSC| ×[0, 1] = |wS1C| ×∆1 by collapsing ∗ × [0, 1] ∪ |wSC| × 0 to a point since|wS0| = •. Hence there is a canonical map |wC| → Ω|wS•C| which is theadjoint of the obvious identification of the 1-skeleton in the S•-direction of|wS•C| with the reduced suspension |wC| ∧ S1. If we apply the construction


to SnC, we obtain a map of spaces |wSnC| → Ω|wS•SnC|. The collectionof these maps for n ≥ 0 yields a map of simplicial spaces and hence bygeometric realization a map of spaces |wS•C| → Ω|wS•S•C| By iterating thisconstruction, we obtain a sequence of maps

|wC| → Ω|wS•C| → ΩΩ|wS•S•C| → ΩΩΩ|wS•S•S•C| → · · ·

such that all maps except the first one are weak homotopy equivalences. SoK(C) is an infinite loop space beyond the first term.

6.3.3 A-Theory

Next we recall Waldhausen’s definition of A-theory of a topological space(see [809, Chapter 2]).

Definition 6.8 (Connective A-theory). Let X be a topological space.Let Rf (X) be the category with cofibrations and weak equivalences definedin Example 6.6. Define the A-theory space A(X) associated to X to be thealgebraic K-theory space K(Rf (X)) in the sense of Definition 6.7.

Remark 6.9 (The wS•-construction encompasses the Q-construction).Waldhausen’s construction encompasses theQ-construction of Quillen (see [809,Section 1.9]).

As in the case of algebraic K-theory of rings or pseudoisotopy it will becrucial for us to consider a non-connective version. Vogell [800] has defined adelooping of A(X) yielding a non-connective Ω-spectrum A(X) for a topo-logical space. The idea is similar to the construction of the (non-connective)pseudo-isotopy spectrum in Section 6.2, where one considers parametrizationsover Rn and imposes control conditions. This construction actually yields acovariant functor from the category of topological spaces to the category ofΩ-spectra

A : TOP → Ω-SPECTRA(6.10)

Definition 6.11 (Non-connective A-theory). We call A(X) the (non-connective) A-theory spectrum associated to the topological space X. Wewrite for n ∈ Z

An(X) := πn(A(X))

Notice that An(X) agrees with πn(A(X)) for n ≥ 1 if A(X) is the spaceappearing in Definition 6.8. Actually there is a map of spectra, natural in X,

i(X) : A(X)→ A(X)(6.12)

which induces isomorphisms πn(i(X)) : πn(A(X))∼=−→ πn(A(X)) for n ≥ 1.


Remark 6.13 (π0(A(X))). If X is path connected, then A0(X) ∼= Z. Theisomorphism comes from taking the Euler characteristic of a relative relativelyfinite CW -complex (Y,X).

One may replace in the definition of A(X) the category Rf (X) by thefull subcategory of R(X) of those triples (Y, r, s) such that (Y,X) is a rel-ative CW -complex consisting of countably many cells, s : X → Y is theinclusion and the object (Y, r, s) is up to homotopy the retract of an object(Y ′, r′, s′) such that (Y ′, X) is a relatively finite relative CW -complex. Thenπn(A(X)) is unchanged for n ≥ 1, whereas π0(A(X) can now be identifiedwith K0(Z[π1(X)]) if X is path connected. The identification comes fromtaking an appropriate finiteness obstruction. With this new definition themap π0(i) : π0(A(X))→ π0(A(X)) is bijective.

For the proof of the next result see [809, Proposition 2.1.7]).

Theorem 6.14 (A-theory is a homotopy-invariant functor). Let f : X →Y be a weak homotopy equivalence. Then the induced maps

A(f) : A(X)→ A(Y );

A(f) : A(X)→ A(Y ),


Let X be a connected space with fundamental group π = π1(X) which

admits a universal covering pX : X → X. Consider an object in Rf (X). Re-call that it is given by a relatively finite relative CW -complex (Y,X) together

with a map r : Y → X satisfying r|X = idX . Let Y → Y be the π-covering ob-

tained from pX : X → X by the pullback construction applied to r : Y → X.The cellular Zπ-chain complex C∗(Y , X) of the relative free π-CW -complex

(Y , X) is a finite free Zπ-chain complex. This yields a functor of categorieswith cofibrations and weak equivalences from Rf (X) to the category of finitefree Zπ-chain complexes. The algebraic K-theory of the category of finitefree Zπ-chain complexes agrees with the one of the finitely generated freeZπ-modules. Hence we get a natural map of spectra called linearization map

L(X) : A(X)→ K(Zπ1(X))(6.15)

The next result follows by combining [801, Section 4] and [808, Proposi-tion 2.2 and Proposition 2.3].

Theorem 6.16 (Connectivity of the linearization map). Let X be aconnected CW -complex. Then:

(i) The linearization map L(X) of (6.15) is 2-connected, i.e., the map

Ln := πn(L(X)) : An(X)→ Kn(Zπ1(X))

is bijective for n ≤ 1 and surjective for n = 2.


(ii) Rationally the map Ln is bijective for all n ∈ Z provided that X is aspher-ical.

Exercise 6.17. Show that the canonical map of spectra A(•) → A(•)is a weak homotopy equivalence.

6.3.4 Whitehead Spaces

Waldhausen (see [808], [809]) defines the functor WhPL(X) from spaces toinfinite loop spaces which can be viewed as connective Ω-spectra, and afibration sequence

X+ ∧A(•)→ A(X)→WhPL(X).(6.18)

Here X+ ∧ A(•) → A(X) is an assembly map. After taking homotopygroups, it can be compared with the algebraic K-theory assembly map thatappears in Conjecture 5.41 via a commutative diagram

πn(X+ ∧A(•))

∼=

// πn(A(X))

πn(X+ ∧A(•)) = Hn(X; A(•))

Hn(X;L)

// πn(A(X))

πn(L)

Hn(Bπ1(X); K(Z)) // Kn(Zπ1(X)).

(6.19)

Here the vertical arrows from the first row to the second row come from themap i of (6.12). The left one of these is bijective for n ∈ Z by Exercise 6.17and the right one of these is bijective for n ≥ 1. The vertical arrows fromthe second row to the third row come from the linearization map L of (6.15).Because of Theorem 6.16 the left one of these is bijective for n ≤ 1 andrationally bijective for n ∈ Z.

In the case where X is aspherical, the lower right vertical map πn(L) isbijective for n ≤ 1 and rationally bijective for all n ∈ Z because of Theo-rem 6.16. Because of (6.18) and the fact that

Ω2 WhPL(X) ' P(X),(6.20)

see [260, Section 9] and [811], Conjecture 5.41 implies rational vanishingresults for the groups πn(P(M)) if M is an aspherical closed manifold.

Lemma 6.21. Suppose that M is a closed aspherical manifold and Conjec-ture 5.41 holds for R = Z and G = π1(M). Then we get for all n ≥ 0


πn(WhPL(M))⊗Z Q = 0;

πn(P(M))⊗Z Q = 0.

Exercise 6.22. Show that π1(WhPL(BG)) is Wh(G).

There is also a smooth version of the Whitehead space WhDiff(X) and wehave

Ω2 WhDiff(X) ' PDiff(X).(6.23)

Again there is a close relation to A-theory via the natural splitting of con-nective spectra due to Waldhausen [808], [810], [811]

A(X) ' Σ∞(X+) ∨WhDiff(X).(6.24)

Here Σ∞(X+) denotes the suspension spectrum associated to X+. Since forevery space πn(Σ∞(X+))⊗Z Q ∼= Hn(X;Q), Conjecture 5.41 combined withRemark 5.45 and Theorem 6.16 yields the following result.

Lemma 6.25. Suppose that M is a closed aspherical manifold and Conjec-ture 5.41 holds for R = Z and G = π1(M). Then we get for all n ≥ 0

πn(WhDiff(M))⊗Z Q ∼=∞⊕k=1

Hn−4k−1(M ;Q);

πn(PDiff(M))⊗Z Q ∼=∞⊕k=1

Hn−4k+1(M ;Q).

Exercise 6.26. Show that there is no connected closed manifold M such thatthe homomorphism induced by the forgetful map πn(WhDiff(M)) ⊗Z Q ∼=πn(WhPL(M))⊗ZQ is bijective for all n ≥ 0. Use the fact that the compositeof the obvious inclusion of WhDiff(X) into Σ∞(X+) ∨WhDiff(X) with theinverse of the splitting (6.24) and the map A(X)→WhPL(X) of (6.18) is upto homotopy the obvious forgetful map WhDiff(M)→WhPL(M).

6.4 Miscellaneous

One of the basic tools to investigate algebraic K-theory of spaces is theAdditivity Theorem, see [577] and [809, Theorem 1.4.2]. If C is a categorywith cofibrations and weak equivalences, we can assign to it a category withcofibrations and weak equivalences E(C) whose objects are exact sequences

Ai−→ B

p−→ C, where exact means that the map i is a cofibration and thefollowing diagram is a pushout


Ai //

B

p

∗ // C

Theorem 6.27 (Additivity Theorem for categories with cofibrationsand weak equivalences). Let F1 and F3 respectively be the functorsE(C) → C of categories with cofibrations and weak equivalences sending an

object Ai−→ B

p−→ C to A and C respectively. Then we obtain a weak homotopyequivalence

K(F1)×K(F3) : K(E(C)) '−→ K(C)×K(C).

Another useful tool is the Approximation Theorem (see [809, Theo-rem 1.6.7]) which gives a criterion to decide when a functor of categorieswith cofibrations and weak equivalences induces a weak homotopy equiva-lence on the K-theory spaces.

There is also a space of parametrized h-cobordisms H(M) for a closed to-pological manifold M . Roughly speaking, the space is designed such thata map N → H(M) is the same as a bundle over N whose fibers are h-cobordisms over M . The set of path component π0(H(M)) agrees with theisomorphism classes of h-cobordisms over M . In particular the s-CobordismTheorem 2.44 is equivalent to the statement that for dim(M) ≥ 5 we obtain a

bijection π0(H(M))∼=−→Wh(π1(M)) coming from taking the Whitehead tor-

sion, or, equivalently that we obtain a bijection π0(H(M))∼=−→ π0(ΩWh(M)).

There is also a stable version, the space of stable parametrized h-cobordismsK(M) = colimj H(M × Ij).

Theorem 6.28 (The stable parametrized h-cobordism Theorem). IfM is a closed topological manifold, then there is a homotopy equivalence

K(M)'−→ ΩWh(M).

There is also a smooth version of the result above. For the proof and moreinformation about the stable parametrized h-cobordism Theorem we referto [811].


name of texfile: ic

Chapter 7

Algebraic L-Theory

7.1 Introduction

Comment 13: Probably this chapter is too long and we should add morereferences to the book by Crowley-Luck-Macko [206], where more details aregiven anyway. When this book is finished, we should rewrite this chapteraccordingly.

In Remark 2.50 we have briefly discussed the surgery program. An impor-tant step is to start with a map of degree one of connected closed orientedmanifolds f : M → N and modify it by surgery steps so that it becomes ahomotopy equivalence. This will change the source but not the target and canonly be carried out if the map f is covered by bundle data. With the bundledata, one is able to make the map highly connected, but in the last steptowards a homotopy equivalence an obstruction, the surgery obstruction, ap-pears whose appearance is due to Poincare duality. This surgery obstructiontakes values in the algebraic L-groups Ln(ZG) for G = π1(N). An introduc-tion to the surgery obstruction and the algebraic L-groups will be given inthis chapter. These are the key tools for the classification of manifolds besidesthe s-Cobordism Theorem 2.44.

We will present the L-theoretic Farrell-Jones Conjecture for torsionfreegroups 7.109 which relates the algebraic L-groups Ln(ZG) to the homologyof BG with coefficient in the L-theory spectrum, analogous to Farrell-JonesConjecture for torsionfree groups and regular rings for K-theory 5.41. Thistogether with the Surgery Exact Sequence of Section 7.12 opens the doorto many applications. We will discuss the Novikov Conjecture 7.131 predict-ing the homotopy invariance of higher signatures, the Borel Conjecture 7.152about the topologically rigidity of aspherical closed manifolds and the prob-lems whether a given Poincare duality group occurs as the fundamental groupof an aspherical closed manifold, which hyperbolic groups have spheres as theirboundary and when does a product decomposition of the fundamental groupof an aspherical closed manifold already implies a product decomposition ofthe manifold itself.

159

160 7 Algebraic L-Theory

7.2 Symmetric and Quadratic Forms

7.2.1 Symmetric Forms

Definition 7.1 (Ring with involution). A ring with involution R is anassociative ring R with unit together with an involution of rings

: R→ R, r 7→ r,

i.e., a map satisfying r = r, r + s = r + s, r · s = s · r and 1 = 1 for r, s ∈ R.

If R is commutative, we can equip it with the trivial involution r = r.Below we fix a ring R with involution. Module is to be understood as left

module unless explicitly stated differently.

Example 7.2 (Involutions on groups rings). Let w : G → ±1 be agroup homomorphism. Then the group ring RG inherits an involution, theso called w-twisted involution that sends

∑g∈G rg · g to

∑g∈G w(g) · rg · g−1.

Remark 7.3 (Dual modules). The main purpose of the involution is toensure that the dual of a left R-module can be viewed as a left R-moduleagain. Namely, let M be a left R-module. Then M∗ := homR(M,R) carriesa canonical right R-module structure given by (fr)(m) = f(m) · r for ahomomorphism of left R-modules f : M → R and m ∈ M . The involutionallows us to view M∗ = homR(M,R) as a left R-module, namely, define rffor r ∈ R and f ∈M∗ by (rf)(m) := f(m) · r for m ∈M .

Notation 7.4. Given a finitely generated projective R-module P , denote by

e(P ) : P∼=−→ (P ∗)∗ the canonical isomorphism of (left) R-modules that sends

p ∈ P to the element in (P ∗)∗ given by P ∗ → R, f 7→ f(p).

We will often use the following elementary fact. Let f : P → Q be a ho-momorphism of finitely generated projective R-modules. Then the followingdiagram commutes

Pf //

e(P )

Q

e(P )

(P ∗)∗

(f∗)∗ // (Q∗)∗

(7.5)

Exercise 7.6. Show that the map e(P ) : P → (P ∗)∗ of Notation (7.4) is awell-defined isomorphism of finitely generated projective R-modules, is com-patible with direct sums and is natural, i.e., the diagram (7.5) commutes.

Definition 7.7 (Non-degenerate ε-symmetric form). Let ε ∈ ±1. Anε-symmetric form (P, φ) over an associative ring R with unit and involution

7.2 Symmetric and Quadratic Forms 161

is a finitely generated projective (left) R-module P together with an R-map

φ : P → P ∗ such that the composition Pe(P )−−−→ (P ∗)∗

φ∗−→ P ∗ agrees with ε ·φ.A morphism f : (P, φ)→ (P ′, φ) of ε-symmetric forms is anR-homomorphism

f : P → P ′ satisfying f∗ φ′ f = φ.We call an ε-symmetric form (P, φ) non-degenerate if φ is an isomorphism.

If ε is 1 or −1 respectively, we often replace ε-symmetric by symmetricor skew-symmetric respectively. The direct sum of two ε-symmetric forms isdefined in the obvious way. The direct sum of two non-degenerate ε-symmetricforms is again a non-degenerate ε-symmetric form.

Remark 7.8 (ε-symmetric forms as pairings). We can rewrite an ε-symmetric form (P, φ) as a pairing

λ : P × P → R, (p, q) 7→ φ(p)(q).

The conditions that φ is R-linear and that φ(p) is R-linear for all p ∈ Ptranslates to

λ(p, r1 · q1 + r2 · q2, ) = r1 · λ(p, q1) + r2 · λ(p, q2);

λ(r1 · p1 + r2 · p2, q) = λ(p1, q) · r1 + λ(p2, q) · r2.

The condition φ = ε · φ∗ e(P ) translates to λ(q, p) = ε · λ(p, q).

If we consider the real numbers R as a ring with involution by the trivialinvolution, then a non-degenerate 1-symmetric form φ on a finite dimensionalR-vector space V such that φ(x)(x) ≥ 0 holds for all x ∈ Rn is the same asa scalar product on V . If we consider the complex numbers C as a ring withinvolution by taking complex conjugation, then the corresponding statementholds for a finite dimensional complex vector space.

Definition 7.9 (The standard hyperbolic ε-symmetric form). Let Pbe a finitely generated projective R-module. The standard hyperbolic ε-symmetric form Hε(P ) is given by the R-module P ⊕ P ∗ and the R-isomorphism

φ : (P ⊕ P ∗)

0 1ε 0

−−−−−→ P ∗ ⊕ P id⊕e(P )−−−−−→ P ∗ ⊕ (P ∗)∗

ρ−→ (P ⊕ P ∗)∗,

where ρ is the obvious R-isomorphism.

If we write the standard hyperbolic ε-symmetric form Hε(P ) as a pairing(see Remark 7.8), we obtain

(P ⊕ P ∗)× (P ⊕ P ∗)→ R, ((p, α), (p′, α′)) 7→ α′(p) + ε · α(p′).


7.2.2 The Signature

Consider a non-degenerate symmetric bilinear pairing s : V × V → R fora finite dimensional real vector space V , or, equivalently, a non-degeneratesymmetric form of finitely generated free R-modules. Choose a basis for Vand let A be the square matrix describing s with respect to this basis. Since sis symmetric and non-degenerate, A is symmetric and invertible. Hence A canbe diagonalized by an orthogonal matrix U to a diagonal matrix whose entrieson the diagonal are non-zero real numbers. Let n+ be the number of positiveentries and n− be the number of negative entries on the diagonal. Thesetwo numbers are independent of the choice of the basis and the orthogonalmatrix U . Namely n+ is the maximum of the dimensions of subvector spacesW ⊂ V on which s is positive-definite, and analogous for n−. Obviouslyn+ + n− = dimR(V ).

Definition 7.10 (Signature). Define the signature of the non-degeneratesymmetric bilinear pairing s : V × V → R for a finite dimensional real vectorspace V to be the integer

sign(s) := n+ − n−.

Define the signature of a non-degenerate symmetric form over Z to be thesignature of the associated non-degenerate symmetric form over R.

Lemma 7.11. Let s : V × V → R be a non-degenerate symmetric bilinearpairing for a finite dimensional real vector space V . Then sign(s) = 0 if andonly if there exists a subvector space L ⊂ V such that dimR(V ) = 2 ·dimR(L)and s(a, b) = 0 for a, b ∈ L.

Proof. Suppose that sign(s) = 0. Then one can find an orthogonal (withrespect to s) basis b1, b2, . . . , bn+

, c1, c2, . . . , cn− such that s(bi, bi) > 0 ands(cj , cj) < 0 holds. Since 0 = sign(s) = n+ − n−, we can define L to be thesubvector space generated by bi + ci | i = 1, 2, . . . , n+. One easily checksthat L has the desired properties.

Suppose such an L ⊂ V exists. Choose subvector spaces V+ and V− of Vsuch that s is positive-definite on V+ and negative-definite on V− and thatV+ and V− are maximal with respect to this property. Then V+ ∩ V− = 0and V = V+ ⊕ V−. Obviously V+ ∩ L = V− ∩ L = 0. From

dimR(V±) + dimR(L)− dimR(V± ∩ L) ≤ dimR(V ),

we conclude dimR(V±) ≤ dimR(V )−dimR(L). Since 2·dimR(L) = dimR(V ) =dimR(V+) + dimR(V−) holds, we get dimR(V±) = dimR(L). This implies

sign(s) = dimR(V+)− dimR(V−) = dimR(L)− dimR(L) = 0.

ut


If M is an orientable connected closed manifold of dimension d, thenHd(M) is infinite cyclic. An orientation on M is equivalent to a choice ofgenerator [M ] ∈ Hd(M) called fundamental class. This definition extends toa (not necessarily connected) orientable closed manifold M of dimension dby defining [M ] ∈ Hd(M) to be the image of [C] | C ∈ π0(M) under the

canonical isomorphism⊕

C∈π0(M)Hdim(M)(C)∼=−→ Hdim(M)(M).

Example 7.12 (Intersection pairing). Let M be a closed oriented mani-fold of even dimension 2n. Then we obtain a (−1)n-symmetric form on thefinitely generated free Z-module Hn(M)/ tors(Hn(M))

i : Hn(M)/ tors(Hn(M))×Hn(M)/ tors(Hn(M))→ Z

by sending [x], [y] for x, y ∈ Hn(M) to 〈x ∪ y, [M ]〉, where 〈u, v〉 denotes theKronecker product. It is non-degenerate by Poincare duality.

Next we define a fundamental invariant of a closed oriented manifold. Thisis the first kind of surgery obstruction we will encounter.

Definition 7.13 (Signature of a closed oriented manifold). Let M bea closed oriented manifold of dimension n. If n is divisible by four, thenthe signature sign(M) of M is defined to be the signature of its intersectionpairing. If n is not divisible by four, define sign(M) = 0.

One easily checks sign(M) =∑C∈π0(M) sign(C).

Exercise 7.14. Let M be a closed oriented 4k-dimensional manifold. Letχ(M) be its Euler characteristic. Show sign(M) ≡ χ(M) mod 2.

The signature can also be defined for compact oriented manifolds with pos-sibly non-empty boundary Comment 14: Add reference or the definition.and has the following properties.

Theorem 7.15 (Properties of the signature of closed oriented man-ifolds).

(i) The signature is an oriented bordism invariant, i.e., if M is a (4k + 1)-dimensional oriented compact manifold with boundary ∂M , then

sign(∂M) = 0;

(ii) Let M and N be compact oriented manifolds and f : ∂M → ∂N be an ori-entation reversing diffeomorphism. Then M ∪f N inherits an orientationfrom M and N and

sign(M ∪f N) = sign(M) + sign(N);


(iii) Let p : M →M be a finite covering with d sheets of closed oriented mani-folds. Then

sign(M) = d · sign(M);

(iv) If the closed oriented connected manifolds M and N are oriented homotopyequivalent, then

sign(M) = sign(N);

(v) If M is an oriented closed manifold and M− is obtained from M by re-versing the orientation, then

sign(M−) = − sign(M).

Proof. (i) Let i : ∂M → M be the inclusion. Then the following diagramcommutes

H2k(M ;R)H2k(i) //

?∩[M,∂M ] ∼=

H2k(∂M ;R)δ2k //

?∩∂4k+1([M,∂M ]) ∼=

H2k+1(M,∂M ;R)

?∩[M,∂M ] ∼=

H2k+1(M,∂M ;R)∂2k+1 // H2k(∂M ;R)

H2k(i) // H2k(M ;R)

This implies dimR(ker(H2k(i))) = dimR(im(H2k(i))). Since R is a field, weget from the Kronecker pairing an isomorphism H2k(M ;R) ∼= (H2k(M ;R))∗

and analogously for ∂M . Under these identificationsH2k(i) becomes (H2k(i))∗.Hence dimR(im(H2k(i))) = dimR(im(H2k(i))). From

dimR(H2k(∂M ;R)) = dimR(ker(H2k(i))) + dimR(im(H2k(i)))

we conclude

dimR(H2k(∂M ;R)) = 2 · dimR(im(H2k(i))).

We have for x, y ∈ H2k(M ;R)

〈H2k(i)(x) ∪H2k(i)(y), ∂4k+1([M,∂M ])〉 = 〈H4k(i)(x ∪ y), ∂4k+1([M,∂M ])〉= 〈x ∪ y,H4k(i) ∂4k+1([M,∂M ])〉= 〈x ∪ y, 0〉 = 0.

If we apply Lemma 7.11 to the non-degenerate symmetric bilinear pairing

H2k(∂M ;R)⊗R H2k(∂M ;R)

∪−→ H4k(∂M ;R)〈?,∂4k+1([M,∂M ])〉−−−−−−−−−−−−→ R

with L the image of H2k(i) : H2k(M ;R) → H2k(∂M ;R), we see that thesignature of this pairing is zero.

(ii) This is due to Novikov. For a proof see for instance [49, Proposition 7.1


on page 588].

(iii) For a smooth manifold M this follows from Atiyah’s L2-index theo-rem [42, (1.1)]. Topological closed manifolds are treated in [734, Theorem 8].

(iv) The two intersection pairings are isomorphic and hence have the samesignatures.

(v) This follows from [M−] = −[M ]. ut

Exercise 7.16. Compute for n ≥ 1 the signature of:

(i) the complex projective space CPn;(ii) the total space STM of the sphere tangent bundle of a closed oriented

n-dimensional manifold M ;(iii) a closed oriented n-dimensional manifold M admitting an orientation re-

versing selfdiffeomorphism.

7.2.3 Quadratic Forms

Next we introduce quadratic forms which are refinements of symmetric forms.

Notation 7.17. For a finitely generated projective R-module P define aninvolution of R-modules

T = T (P ) : homR(P, P ∗)→ homR(P, P ∗), f 7→ f∗ e(P ).

Notation 7.18. Let P be a finitely generated projective R-module. Defineabelian groups

Qε(P ) := ker ((1− ε · T ) : homR(P, P ∗)→ homR(P, P ∗)) ;

Qε(P ) := coker ((1− ε · T ) : homR(P, P ∗)→ homR(P, P ∗)) .

An R-homomorphism f : P → Q induces a homomorphism of abelian groups

Qε(f) : Qε(Q)→ Qε(P ), g 7→ f∗ g f ;

Qε(f) : Qε(Q)→ Qε(P ), [g] 7→ [f∗ g f ].

Let(1 + ε · T ) : Qε(P )→ Qε(P )

be the homomorphism which sends the class represented by ψ : P → P ∗ to theelement ψ + ε · T (ψ).

Definition 7.19 (Non-degenerate ε-quadratic form). Let ε ∈ ±1. Anε-quadratic form (P,ψ) is a finitely generated projective R-module P together


with an element ψ ∈ Qε(P ). It is called non-degenerate if the associated ε-symmetric form (P, (1+ε·T )(ψ)) is non-degenerate, i.e. (1+ε·T )(ψ) : P → P ∗

is bijective.A morphism f : (P,ψ) → (P ′, ψ′) of two ε-quadratic forms is an R-

homomorphism f : P∼=−→ P ′ such that the induced map Qε(f) : Qε(P

′) →Qε(P ) sends ψ′ to ψ.

Given a non-degenerate ε-symmetric form (P, φ), a quadratic refinementis a non-degenerate ε-quadratic form (P,ψ) with φ = (1 + ε · T )(ψ).

There is an obvious notion of a direct sum of two ε-quadratic forms. Thedirect sum of two non-degenerate ε-quadratic forms is a non-degenerate ε-quadratic form.

Remark 7.20 (ε-quadratic forms as pairings). An ε-quadratic form(P, φ) is the same as a triple (P, λ, µ) consisting of a pairing

λ : P × P → R

satisfying

λ(p, r1 · q1 + r2 · q2, ) = r1 · λ(p, q1) + r2 · λ(p, q2);

λ(r1 · p1 + r2 · p2, q) = λ(p1, q) · r1 + λ(p2, q) · r2;

λ(q, p) = ε · λ(p, q).

and a mapµ : P → Qε(R) = R/r − ε · r | r ∈ R

satisfying

µ(rp) = rµ(p)r;

µ(p+ q)− µ(p)− µ(q) = pr(λ(p, q));

λ(p, p) = (1 + ε · T )(µ(p)),

where pr : R→ Qε(R) is the projection and (1 + ε · T ) : Qε(R)→ R the mapsending the class of r to r + ε · r. Namely, put

λ(p, q) = ((1 + ε · T )(ψ)(p)) (q);

µ(p) = ψ(p)(p).

Definition 7.21 (The standard hyperbolic ε-quadratic form). Let Pbe a finitely generated projective R-module. The standard hyperbolic ε-quadratic form Hε(P ) is given by the R-module P ⊕ P ∗ and the class inQε(P ⊕ P ∗) of the R-homomorphism

φ : (P ⊕ P ∗)

0 10 0

−−−−−→ P ∗ ⊕ P id⊕e(P )−−−−−→ P ∗ ⊕ (P ∗)∗

ρ−→ (P ⊕ P ∗)∗,

7.3 Even-Dimensional L-groups 167

where ρ is the obvious R-isomorphism.

If we write the standard hyperbolic ε-quadratic form Hε(P ) as a pairing(see Remark 7.20), we obtain

λ : (P ⊕ P ∗)× (P ⊕ P ∗)→ R, ((p, α), (p′, α′)) 7→ α′(p) + ε · α(p′).µ : P ⊕ P ∗ → Qε(R), (p, α) 7→ pr(α(p)).

In particular the ε-symmetric form associated to the standard ε-quadraticform Hε(P ) is just the standard ε-symmetric form Hε(P ).

Exercise 7.22. Let λ : P×P → Z be a non-degenerate symmetric Z-bilinearpairing on the finitely generated free Z-module P . Show that it has, whenconsidered as a non-degenerate symmetric form, a quadratic refinement ifand only if λ(x, x) is even for all x ∈ P .

Remark 7.23. Suppose that 1/2 ∈ R. Then the homomorphism

(1 + ε · T ) : Qε(P )→ Qε(P ), [ψ] 7→ [ψ + ε · T (ψ)]

is bijective. The inverse sends [φ] to [φ/2]. Hence any ε-symmetric form carriesa unique ε-quadratic structure. Therefore there is no difference between thesymmetric and the quadratic setting if 2 is invertible in R.

7.3 Even-Dimensional L-groups

Next we define even-dimensional L-groups. Below R is an associative ringwith unit and involution.

Definition 7.24 (L-groups in even dimensions). For an even integer n =2k define the abelian group Ln(R), called the n-th quadratic L-group, of R tobe the abelian group of equivalence classes [(F,ψ)] of non-degenerate (−1)k-quadratic forms (F,ψ) whose underlying R-module F is a finitely generatedfree R-module with respect to the following equivalence relation: We call(F,ψ) and (F ′, ψ′) equivalent if and only if there exists integers u, u′ ≥ 0 andan isomorphism of non-degenerate (−1)k-quadratic forms

(F,ψ)⊕Hε(R)u ∼= (F ′, ψ′)⊕Hε(R)u′.

Addition is given by the sum of two (−1)k-quadratic forms. The zero elementis represented by [H(−1)k(R)u] for any integer u ≥ 0. The inverse of [F,ψ] isgiven by [F,−ψ].

There is an analogous definition of the the n-th symmetric L-group Ln(R).


A morphism u : R → S of rings with involution induces homomorphismsu∗ : Ln(R) → Ln(S) and u∗ : Ln(R) → Ln(S) for even n ∈ Z by induction.One easily checks (u v)∗ = u∗ v∗ and (idR)∗ = idLk(R) for k = 0, 2.

Next we will present a criterion for an ε-quadratic form (P,ψ) to representzero in L1−ε(R). Let (P,ψ) be an ε-quadratic form. A subLagrangian L ⊂ Pis an R-submodule such that the inclusion i : L → P is split injective, theimage of ψ under the map Qε(i) : Qε(P )→ Qε(L) is zero and L is containedin its annihilator L⊥ which is by definition the kernel of

P(1+ε·T )(ψ)−−−−−−−→ P ∗

i∗−→ L∗.

A subLagrangian L ⊂ P is called Lagrangian if L = L⊥. Equivalently, aLagrangian L ⊂ P is an R-submodule L with inclusion i : L → P such thatthe sequence

0→ Li−→ P

i∗(1+ε·T )(ψ)−−−−−−−−−→ L∗ → 0.

is exact.

Lemma 7.25. Let (P,ψ) be an ε-quadratic form. Let L ⊂ P be a subLa-grangian. Then L is a direct summand in L⊥ and ψ induces the structure ofa non-degenerate ε-quadratic form (L⊥/L, ψ⊥/ψ). Moreover, the inclusioni : L→ P extends to an isomorphism of ε-quadratic forms

Hε(L)⊕ (L⊥/L, ψ⊥/ψ)∼=−→ (P,ψ).

In particular a non-degenerate ε-quadratic form (P,ψ) is isomorphic toHε(Q) if and only if it contains a Lagrangian L ⊂ P which is isomorphic asR-module to Q. The analogous statement holds for ε-symmetric forms.

Proof. Choose an R-homomorphism s : L∗ → P such that i∗ (1+ ε ·T )(ψ)sis the identity on L∗. Our first attempt is the obvious split injection i ⊕s : L⊕L∗ → P . The problem is that it is not necessarily compatible with theε-quadratic structure. To be compatible with the ε-quadratic structure it isnecessary to be compatible with the ε-symmetric structure, i.e., the followingdiagram must commute

L⊕ L∗ i⊕s // 0 1ε 0

P

ψ+ε·T (ψ)

L∗ ⊕ L P ∗

(i∗⊕s∗)∆P∗oo

where ∆P∗ : P ∗ → P ∗ ⊕ P ∗ is the diagonal map and we write s∗ for the

composition P ∗s∗−→ (L∗)∗

e(L)−1

−−−−→ L. The diagram above commutes if andonly if s∗ (ψ + ε · T (ψ)) s = 0. Notice that s is not unique, we can replaces by s′ = s + i v for any R-map v : L∗ → L. For this new section s′ the

7.3 Even-Dimensional L-groups 169

diagram above commutes if and only if s∗ (ψ+ ε · T (ψ)) s+ v∗ + ε · v = 0.This suggests to take v = −εs∗ ψ s. Now one easily checks that

g := i⊕ (s′ − ε · i (s′)∗ ψ s′) : L⊕ L∗ → P

is split injective and compatible with the ε-quadratic structures and henceinduces a morphism g : Hε(L)→ (P,ψ) of ε-quadratic forms.

Let im(g)⊥ be the annihilator of im(g). Denote by j : im(g) → P theinclusion. We obtain an isomorphism of ε-quadratic forms

g ⊕ j : Hε(L)⊕ (im(g)⊥, j∗ ψ j)→ (P,ψ).

The inclusion im(g)⊥ → L⊥ induces an isomorphism h : im(g)⊥∼=−→ L⊥/L.

Let ψ⊥/ψ be the ε-quadratic structure on L⊥/L for which h becomes anisomorphism of ε-quadratic forms. This finishes the proof in the quadraticcase. The proof in the symmetric case is analogous. ut

Exercise 7.26. Show for a non-degenerate ε-quadratic form (M,φ) that(M,φ) ⊕ (M,−φ) is isomorphic to Hε(M) and hence an inverse of [(M,φ)]in L1−ε(R) is given by [(M,−φ)].

Exercise 7.27. Show that the signature defines an isomorphism L0(R)∼=−→ Z.

Finally we state the computation of the even-dimensional L-groups of theintegers. Consider an element (P, φ) in L0(Z). By tensoring over Z with R andonly taking the symmetric structure into account, we obtain a non-degeneratesymmetric R-bilinear pairing λ : R⊗Z P ×R⊗Z P → R. It turns out that itssignature is always divisible by eight.

Theorem 7.28 (L-groups of the integers in dimension 4n). The sig-nature defines for n ∈ Z an isomorphism

1

8· sign: L4n(Z)

∼=−→ Z, [P,ψ] 7→ 1

8· sign(R⊗Z P, λ).

Comment 15: Shall we add a reference, for instance to [206, Chapter 7]?Shall we say something about the classification of quadratic forms over Z ormention at least the form E8? Maybe we should add a reference to [593].

Consider a non-degenerate quadratic form (P,ψ) over the field F2 of two el-ements. Write (P,ψ) as a triple (P, λ, µ) as explained in Remark 7.20. Chooseany symplectic basis b1, b2, . . . , b2m for P , where symplectic means thatλ(bi, bj) is 1 if i − j = m and 0 otherwise. Such a symplectic basis alwaysexists. Define the Arf invariant of (P,ψ) by

Arf(P,ψ) :=

m∑i=1

µ(bi) · µ(bi+m) ∈ Z/2.(7.29)


It turns out that the Arf invariant of (P,ψ) is 1 if and only if µ sends a(strict) majority of the elements of P to 1, see [132, Corollary III.1.9 onpage 55]. (Because of this property sometimes the Arf invariant is called thedemocratic invariant) This description shows that (7.29) is independent ofthe choice of symplectic basis.

Exercise 7.30. Let V be a two-dimensional F2-vector space. Classify all non-degenerate quadratic forms on V up to isomorphism and compute their Arfinvariants.

The Arf invariant defines an isomorphism

Arf : L2n(F2)∼=−→ Z/2,

essentially, since two non-degenerate quadratic forms over F2 on the samefinite dimensional F2-vector space are isomorphic if and only if they have thesame Arf invariant (see [132, Theorem III.1.12 on page 55]). The change ofrings homomorphism Z→ F2 induces an isomorphism Comment 16: Addreference, for instance [206, Chapter 7].

L4n+2(Z)∼=−→ L4n+2(F2).

This implies

Theorem 7.31 (L-groups of the integers in dimension 4n + 2). TheArf invariant defines for n ∈ Z an isomorphism

Arf : L4n+2(Z)∼=−→ Z/2, [(P,ψ)] 7→ Arf(F2 ⊗Z (P,ψ)).

For more information about forms over the integers and the Arf invariantwe refer for instance to [132, 593]. Implicitly the computation of Ln(Z) isalready in [469].

7.4 Intersection and Selfintersection Pairings

The notions of an ε-symmetric form as presented in Remark 7.8 and of an ε-quadratic form as presented in Remark 7.20 are best motivated by consideringintersections and selfintersection pairings. When trying to solve a surgeryproblem in even dimensions, one faces in the final step, namely, when dealingwith the middle dimension, the problem to decide whether we can change animmersion f : Sk → M within its regular homotopy class to an embedding,where M is a compact manifold of dimension n = 2k. This problem leadsin a natural way to selfintersection pairings and to the notion of ε-quadraticform as explained next.

7.4 Intersection and Selfintersection Pairings 171

7.4.1 Intersections of Immersions

Let k ≥ 2 be a natural number and let M be a compact connected smoothmanifold of dimension n = 2k. We fix base points s ∈ Sk and b ∈M . We willconsider pointed immersions (f, w), i.e., an immersion f : Sk → M togetherwith a path w from b to f(s). A pointed regular homotopy from (f0, w0) to(f1, w1) is a smooth homotopy h : Sk × [0, 1] → M from h0 = f0 to h1 = f1

such that ht : Sk → M, x 7→ H(x, t) is an immersion for each t ∈ [0, 1] and

w0 ∗ h(s, ?) and w1 are homotopic paths relative end points. Here h(s, ?) isthe path from f0(s) to f1(s) given by restricting h to s × [0, 1]. Denoteby Ik(M) the set of pointed regular homotopy classes of pointed immersionsfrom Sk to M . We need the paths to define the structure of an abelian groupon Ik(M). The sum of [(f0, w0)] and [(f1, w1)] is given by the connected sumalong the path w−0 ∗w1 from f0(s) to f1(s). The zero element is given by thecomposition of the standard embedding Sk → Rk+1 ⊂ Rk+1 × Rk−1 = Rnwith some embedding Rn ⊂ M and any path w from b to the image of s.The inverse of the class of (f, w) is the class of (f a,w) for any base pointpreserving diffeomorphism a : Sk → Sk of degree −1.

The fundamental group π = π1(M, b) operates on Ik(M) by composing thepath w with a loop at b. Thus Ik(M) inherits the structure of a Zπ-module.

Next we want to define the intersection pairing

λ : Ik(M)× Ik(M)→ Zπ.(7.32)

For this purpose we will have to fix an orientation of TbM at b. Considerα0 = [(f0, w0)] and α1 = [(f1, w1)] in Ik(M). Choose representatives (f0, w0)and (f1, w1). We can arrange without changing the pointed regular homotopyclass that D = im(f0) ∩ im(f1) is finite, for any y ∈ D both the preimagef−1

0 (y) and the preimage f−11 (y) consists of precisely one point and for any

two points x0 and x1 in Sk with f0(x0) = f1(x1) we have Tx0f0(Tx0

Sk) +Tx1

f1(Tx1Sk) = Tf0(x0)M . Consider d ∈ D. Let x0 and x1 in Sk be the

points uniquely determined by f0(x0) = f1(x1) = d. Let ui be a path in Sk

from s to xi. Then we obtain an element g(d) ∈ π by the loop at b givenby the composition w1 ∗ f1(u1) ∗ f0(u0)− ∗ w−0 . Recall that we have fixed anorientation of TbM . The fiber transport along the path w0 ∗ f(u0) yields anisomorphism TbM ∼= TdM which is unique up to isotopy. Hence TdM inheritsan orientation from the given orientation of TbM . The standard orientation ofSk determines an orientation on Tx0

Sk and Tx1Sk. We have the isomorphism

of oriented vector spaces

Tx0f0 ⊕ Tx1

f1 : Tx0Sk ⊕ Tx1

Sk∼=−→ TdM.

Define ε(d) = 1 if this isomorphism respects the orientations and ε(d) = −1otherwise. The elements g(d) ∈ π and ε(d) ∈ ±1 are independent of thechoices of u0 and u1 since Sk is simply connected for k ≥ 2. Define


λ(α0, α1) :=∑d∈D

ε(d) · g(d).

Lift b ∈M to a base point b ∈ M . Let fi : Sk → M be the unique lift of fi

determined by wi and b for i = 0, 1. Let λZ(f0, f1) be the Z-valued intersection

number of f0 and f1. This is the same as the algebraic intersection numberof the classes in the k-th homology with compact support defined by f0 andf1 which obviously depends only on the homotopy classes of f0 and f1. Then

λ(α0, α1) =∑g∈π

λZ(f0, lg−1 f1) · g,(7.33)

where lg−1 denotes left multiplication with g−1. This shows that λ(α0, α1)depends only on the pointed regular homotopy classes of (f0, w0) and (f1, w1).

Below we use the w1(M)-twisted involution on Zπ which sends∑g∈π ag ·g

to∑g∈π w1(M)(g) · ag · g−1, where w1(M) : π → ±1 is the first Stiefel-

Whitney class of M . One easily checks

Lemma 7.34. For α, β, β1, β2 ∈ Ik(M) and u1, u2 ∈ Zπ we have

λ(α, β) = (−1)k · λ(β, α);

λ(α, u1 · β1 + u2 · β2) = u1 · λ(α, β1) + u2 · λ(α, β2).

Remark 7.35 (Intersection pairing and (−1)k-symmetric forms).Lemma 7.34 says that the pair (Ik(M), λ) satisfies all the requirements ap-pearing in Remark 7.8 except that Ik(M) may not be finitely generated freeover Zπ.

Remark 7.36 (The intersection pairing as necessary obstruction forfinding an embedding). Suppose that the normal bundle of the immersionf : Sk →M has a nowhere vanishing section. (In the typical situation whichappears in surgery theory it actually will be trivial.) Suppose that f is regularhomotopic to an embedding g. Then the normal bundle of g has a nowherevanishing section σ. Let g′ be the embedding obtained by moving g a little bitin the direction of this normal vector field σ. Choose a path wf from f(s) tob. Then for appropriate paths wg and wg′ we get pointed embeddings (g, wg)and (g′, wg′) such that the pointed regular homotopy classes of (f, w), (g, wg)and (g′, wg′) agree. Since g and g′ have disjoint images, we conclude

λ([f, w], [f, w]) = 0.

Hence the vanishing of λ([f, w], [f, w]) is a necessary condition for findingan embedding in the regular homotopy class of f , provided that the normalbundle of f has a nowhere vanishing section. It is not a sufficient condition.To get a sufficient condition we have to consider self-intersections what wewill do next.


7.4.2 Selfintersections of Immersions

Let α ∈ Ik(M) be an element. Let (f, w) be a pointed immersion representingα. Recall that we have fixed base points s ∈ Sk, b ∈ M and an orientationof TbM . Since we can find arbitrarily close to f an immersion which is ingeneral position, we can assume without loss of generality that f itself is ingeneral position. This means that there is a finite subset D of im(f) suchthat f−1(y) consists of precisely two points for y ∈ D and of precisely onepoint for y ∈ im(f)−D and for two points x0 and x1 in Sk with x0 6= x1 andf(x0) = f(x1) we have Tx0f(Tx0S

k) + Tx1f(Tx1Sk) = Tf0(x0)M . Now fix for

any d ∈ D an ordering x0(d), x1(d) of f−1(d). Analogously to the constructionabove one defines ε(x0(d), x1(d)) ∈ ±1 and g(x0(d), x1(d)) ∈ π = π1(M, b).Consider the element

∑d∈D ε(x0(d), x1(d)) ·g(x0(d), x1(d)) of Zπ. It does not

only depend on f but also on the choice of the ordering of f−1(d) for d ∈ D.One easily checks that the change of ordering of f−1(d) has the followingeffect for w = w1(M) : π → ±1

g(x1(d), x0(d)) = g(x0(d), x1(d))−1;

w(g(x1(d), x0(d))) = w(g(x0(d), x1(d)));

ε(x1(d), x0(d)) = (−1)k · w(g(x0(d), x1(d))) · ε(x0(d), x1(d));

ε(x1(d), x0(d)) · g(x1(d), x0(d)) = (−1)k · ε(x0(d), x1(d)) · g(x0(d), x1(d)).

We have defined the abelian group Q(−1)k(Zπ,w) in Notation 7.18 Define theselfintersection element for α ∈ Ik(M)

µ(α) :=

[∑d∈D

ε(x0(d), x1(d)) · g(x0(d), x1(d))

]∈ Q(−1)k(Zπ,w).(7.37)

The passage from Zπ to Q(−1)k(Zπ,w) ensures that the definition is inde-pendent of the choice of the order on f−1(d) for d ∈ D. It remains to showthat it depends only on the pointed regular homotopy class of (f, w). Let hbe a pointed regular homotopy from (f, w) to (g, v). We can arrange that his in general position. In particular each immersion ht is in general positionand comes with a set Dt. The collection of the Dt-s yields a bordism W fromthe finite set D0 to the finite set D1. Since W is a compact one-dimensionalmanifold, it consists of circles and arcs joining points in D0 ∪D1 to points inD0 ∪D1. Suppose that the point e and the point e′ in D0 ∪D1 are joint byan arc. Then one easily checks that their contributions to

µ(f, w)− µ(g, w) :=

[ ∑d0∈D0

ε(x0(d0), x1(d0)) · g(x0(d0), x1(d0))

−∑d1∈D1

ε(x0(d1), x1(d1)) · g(x0(d1), x1(d1))

]


cancel out. This implies µ(f, w) = µ(g, w).

Lemma 7.38. Let µ : Ik(M) → Q(−1)k(Zπ,w) be the map given by the self-intersection element (see (7.37)) and let λ : Ik(M) × Ik(M) → Zπ be theintersection pairing (see (7.32)). Then

(i) Let (1 + (−1)k · T ) : Q(−1)k(Zπ,w)→ Zπ be the homomorphism of abelian

groups which sends [u] to u+(−1)k ·u. Denote for α ∈ Ik(M) by χ(α) ∈ Zthe Euler number of the normal bundle ν(f) for any representative (f, w) ofα with respect to the orientation of ν(f) given by the standard orientationon Sk and the orientation on f∗TM given by the fixed orientation on TbMand w. Then

λ(α, α) = (1 + (−1)k · T )(µ(α)) + χ(α) · 1;

(ii) We get for pr: Zπ → Q(−1)k(Zπ,w) the canonical projection and α, β ∈Ik(M)

µ(α+ β)− µ(α)− µ(β) = pr(λ(α, β));

(iii) For α ∈ Ik(M) and u ∈ Zπ we get with respect to the obvious Zπ-bimodulestructure on Q(−1)k(Zπ,w)

µ(u · α) = uµ(α)u.

Proof. (i) Represent α ∈ Ik(M) by a pointed immersion (f, w) which is ingeneral position. Choose a section σ of ν(f) which meets the zero sectiontransversally. Notice that then the Euler number satisfies

χ(f) =∑

y∈N(σ)

ε(y),

where N(σ) is the (finite) set of zero points of σ and ε(y) is a sign whichdepends on the local orientations. We can arrange that no zero of σ is thepreimage of an element in the set of double points Df of f . Now move f alittle bit in the direction of this normal field σ. We obtain a new immersiong : Sk → M with a path v from b to g(s) such that (f, w) and (g, v) arepointed regularly homotopic.

We want to compute λ(α, α) using the representatives (f, w) and (g, v).Notice that any point in d ∈ Df corresponds to two distinct points x0(d) andx1(d) in the set D = im(f) ∩ im(g) and any element n ∈ N(σ) correspondsto one point x(n) in D. Moreover any point in D occurs as xi(d) or x(n) ina unique way. Now the contribution of d to λ([(f, w)], [(g, v)]) is ε(d) · g(d) +(−1)k · ε(d) · g(d) and the contribution of n ∈ N(σ) is ε(n) · 1. Now assertion(i) follows.

(ii) and (iii) The proof of these assertions are left to the reader. ut

Remark 7.39 (Selfintersection pairing and (−1)k-quadratic forms).Lemma 7.38 says that the triple (Ik(M), λ, µ) satisfies all the requirements


appearing in Remark 7.20 except that Ik(M) may not be finitely generatedfree over Zπ and we have to require χ(α) = 0 which will be satisfied in thecases of interest.

Theorem 7.40 (Selfintersections and embeddings). Let M be a com-pact connected manifold of even dimension n = 2k. Fix base points s ∈ Skand b ∈ M and an orientation of TbM . Let (f, w) be a pointed immersionof Sk in M . Suppose that k ≥ 3. Then (f, w) is pointed homotopic to apointed immersion (g, v) for which g : Sk → M is an embedding, if and onlyµ(f, w) = 0.

Proof. If f is represented by an embedding, then µ(f, w) = 0 by definition.Suppose that µ(f, w) = 0. We can assume without loss of generality that f isin general position. Since µ(f, w) = 0, we can find d and e in the set of doublepoints Df of f and a numeration x0(d), x1(d) of f−1(d) and x0(e), x1(e) off−1(e) such that

ε(x0(d), x1(d)) = −ε(x0(e), x1(e));

g(x0(d), x1(d)) = g(x0(e), x1(e)).

Therefore we can find arcs u0 and u1 in Sk such that u0(0) = x0(d), u0(1) =x0(e), u1(0) = x1(d) and u1(1) = x1(e), the path u0 and u1 are disjoint fromone another, f(u0((0, 1))) and f(u1((0, 1))) do not meet Df and f(u0) andf(u1) are homotopic relative endpoints. We can change u0 and u1 withoutdestroying the properties above and find a smooth map U : D2 → M whoserestriction to S1 is an embedding (ignoring corners on the boundary) and isgiven on the upper hemisphere S1

+ by u0 and on the lower hemisphere S1− by

u1 and which meets im(f) transversally. There is a compact neighborhood Kof S1 such that U |K is an embedding. Since k ≥ 3 we can find arbitrarily closeto U an embedding which agrees with U on K. Hence we can assume withoutloss of generality that U itself is an embedding. The Whitney trick (see [590,Theorem 6.6 on page 71], [837]) allows to change f within its pointed regularhomotopy class to a new pointed immersion (g, v) such that Dg = Df−d, eand µ(g, v) = 0. By iterating this process we achieve Df = ∅. ut

Remark 7.41 (The dimension assumption dim(M) ≥ 5). The conditiondim(M) ≥ 5 which arises in high-dimensional manifold theory ensures in theproof of Theorem 7.40 that k ≥ 3 and hence we can arrange U to be an em-bedding. If k = 2, one can achieve that U is an immersion but not necessarilyan embedding. This is the technical reason why surgery in dimension 4 ismuch more complicated than in dimensions ≥ 5.

Exercise 7.42. Let f : Sk → M be an immersion into a compact 2k-dimensional manifold. Suppose that it is in general position and the set ofdouble points consists of precisely one element. Show that f is not regularhomotopic to an embedding.


Exercise 7.43. Construct an immersion f : M → N of closed connectedmanifolds which is homotopic but not regularly homotopic to an embedding.

7.5 The Surgery Obstruction in Even Dimensions

We give a brief introduction to the surgery obstruction in even dimension tomotivate the relevance of the L-groups for topology.

7.5.1 Poincare Duality Spaces

Consider a connected finite CW -complex X with fundamental group π anda group homomorphism w : π → ±1. Below we use the w-twisted involu-

tion on Zπ. Denote by C∗(X) the cellular Zπ-chain complex of the universal

covering. It is a finite free Zπ-chain complex. The product X × X equippedwith the diagonal π-action is again a π-CW -complex. The diagonal mapD : X → X × X sending x to (x, x) is π-equivariant but not cellular. By theequivariant cellular Approximation Theorem (see for instance [521, Theorem2.1 on page 32]) there is up to cellular π-homotopy precisely one cellular π-

map D : X → X × X which is π-homotopic to D. It induces a Zπ-chain mapunique up to Zπ-chain homotopy

C∗(D) : C∗(X)→ C∗(X × X).(7.44)

There is a natural isomorphism of Zπ-chain complexes

i∗ : C∗(X)⊗Z C∗(X)∼=−→ C∗(X × X).(7.45)

Definition 7.46 (Dual chain complex). Given an R-chain complex of leftR-modules C∗ and n ∈ Z, we define its dual chain complex Cn−∗ to be thechain complex of left R-modules whose p-th chain module is homR(Cn−p, R)and whose p-th differential is given by

homR(cn−p+1, id) : (Cn−∗)p = homR(Cn−p, R)

→ (Cn−∗)p−1 = homR(Cn−p+1, R).

Denote by Zw the Zπ-module whose underlying abelian group is Z and onwhich g ∈ π acts by w(g) · id. Given two projective Zπ-chain complexes C∗and D∗ we obtain a natural Z-chain map unique up to Z-chain homotopy

s : Zw ⊗Zπ (C∗ ⊗Z D∗)→ homZπ(C−∗, D∗)(7.47)

7.5 The Surgery Obstruction in Even Dimensions 177

by sending 1⊗ x⊗ y ∈ Z⊗ Cp ⊗Dq to

s(1⊗ x⊗ y) : homZπ(Cp,Zπ)→ Dq, (φ : Cp → Zπ) 7→ φ(x) · y.

The composite of the chain map (7.47) for C∗ = D∗ = C∗(X), the inverseof the chain map (7.45) tensored with Zw ⊗Zπ − and the chain map (7.44)tensored with Zw ⊗Zπ − yields a Z-chain map

Zw ⊗Zπ C∗(X)→ homZπ(C−∗(X), C∗(X)).

Notice that the n-th homology of homZπ(C−∗(X), C∗(X)) is the set of Zπ-

chain homotopy classes [Cn−∗(X), C∗(X)]Zπ of Zπ-chain maps from Cn−∗(X)

to C∗(X). Define Hn(X;Zw) := Hn(Zw⊗Zπ C∗(X)). Taking the n-th homol-ogy group yields a well-defined Z-homomorphism

∩ : Hn(X;Zw)→ [Cn−∗(X), C∗(X)]Zπ(7.48)

which sends a class x ∈ Hn(X;Zw) = Hn(Zw ⊗Zπ C∗(X)) to the Zπ-chain

homotopy class of a Zπ-chain map denoted by ? ∩ x : Cn−∗(X)→ C∗(X).

Definition 7.49 (Poincare complex). A connected finite n-dimensionalPoincare complex is a connected finite CW -complex of dimension n togetherwith a group homomorphism w = w1(X) : π1(X)→ ±1 called orientationhomomorphism and an element [X] ∈ Hn(X;Zw) called fundamental class

such that the Zπ-chain map ?∩ [X] : Cn−∗(X)→ C∗(X) is a Zπ-chain homo-topy equivalence. We will call it the Poincare Zπ-chain homotopy equivalence.

Exercise 7.50. Show that the orientation homomorphism w : π1(X) →±1 is uniquely determined by the homotopy type of the finite n-dimensionalPoincare complex X.

Obviously there are two possible choices for [X], since it has to be a gen-erator of the infinite cyclic group Hn(X,Zw) ∼= H0(X;Z) ∼= Z. A choice of[X] will be part of the Poincare structure.

A map f : Y1 → Y2 of connected Poincare complexes has degree one ifw1(Y2) π1(f) = w1(Y2) and the map Hn(Y1,Zw1(Y1)) → Hn(Y2,Zw1(Y2))induced by f sends [Y1] to [Y2].

Theorem 7.51. Let M be a connected closed manifold of dimension n. ThenM carries the structure of a connected finite n-dimensional Poincare complex.

For a proof we refer for instance to [815, Theorem 2.1 on page 23].Below an orientation of a compact connected manifold M of dimension n

is a choice of a generator [M ] of the infinite cyclic group Hn(M,∂M ;Zw1(M)).


Remark 7.52 (Poincare duality as obstruction for being homotopyequivalent to a closed manifold). Theorem 7.51 gives us the first ob-struction for a topological space X to be homotopy equivalent to a connectedclosed n-dimensional manifold. Namely, X must be homotopy equivalent toa connected finite n-dimensional Poincare complex.

7.5.2 Normal Maps and the Surgery Step

Definition 7.53 (Normal map of degree one). Let X be a connectedfinite n-dimensional Poincare complex together with a m-dimensional vectorbundle ξ : E → X. A normal m-map (M, i, f, f) with (X, ξ) as target consistsof an oriented connected closed manifold M of dimension n together with anembedding i : M → Rn+m and a bundle map (f, f) : ν(M)→ ξ, where ν(M)denotes the normal bundle ν(i) of the embedding i : M → Rm+n. A normalmap of degree one is a normal map such that the degree of f : M → X isone.

Given a normal map (M, i, f, f) with (X, ξ) as target, we obtain for

k ≥ 1 a normal map (M, i, f, f′) with (X, ξ ⊕ Rk) as target as follows. Let

i′ : M → Rn+m+k be the composite of the embedding i : M → Rn+m withthe standard inclusion Rn+m → Rn+m+k. Then ν(i′) is the Whitney sum

ν(i)⊕Rk, where Rk is the trivial k-dimensional bundle. Let f′: ν(i′)→ ξ⊕Rk

be the stabilization of f . We call (M, i′, f, f′) a stabilization of (M, i, f, f).

The next result is due to Whitney [836, 837].

Theorem 7.54 (Whitney’s Approximation Theorem). Let M and Nbe closed manifolds of dimensions m and n. Then any map f : M → N isarbitrarily close and in particular homotopic to an immersion, provided that2m ≤ n, and arbitrarily close and in particular homotopic to an embedding,provided that 2m < n.

Remark 7.55 (Existence of a normal map of degree one as obstruc-tion for being homotopy equivalent to a closed manifold). Given afinite n-dimensional Poincare complex X, the existence of a normal map ofdegree one with (X, ξ) as target for some vector bundle ξ over X is necessaryfor X to be homotopy equivalent to a closed manifold. Namely, if f : M → Xis such a homotopy equivalence, choose a homotopy inverse g : X → M andput ξ = g∗ν(i) for some embedding i : M ⊆ Rn+m. Such an embedding existsalways for n < m by Theorem 7.54. Obviously f can be covered by a bundlemap f : ν(M)→ ξ and f has degree one.

Definition 7.56 (Normal bordism). Consider two normal maps of degreeone (Mk, ik, fk, fk) with the same target (X, ξ) for k = 0, 1. A normal bordismfrom (f0, f0) to (f1, f1) consists of a compact connected oriented manifold W


with boundary ∂W , an embedding j : (W,∂W ) → (Rn+m × [0, 1],Rn+m ×0, 1), a map (F, ∂F ) : (W,∂W )→ (X× [0, 1], X×0, 1) of degree one cov-ered by a bundle map F : ν(W )→ ξ and an orientation preserving diffeomor-

phism u : ∂W∼=−→M0 qM1 satisfying the obvious compatibility conditions.

We call (M0, i0, f0, f0) and (M1, i1, f1, f1) normally bordant if after sta-bilization and reversing the orientation of M1 there exists a normal bordismbetween them.

Exercise 7.57. Let (M, i0, f, f0) be normal map of degree one with target(X, ξ). Let i1 : M → Rn+k be an embedding. Show that there exists a normalmap of degree one (M, i1, f, f1) with target (X, ξ) which is normally bordantto (M, i0, f, f0).

Below we will often suppress the embedding i : M → Rn+m in the notation.

7.5.3 The Surgery Step

So the question is whether we can modify a normal map of degree one with(X, ξ) as target (without changing the target) so that the underlying map f isa homotopy equivalence. There is a procedure in the world of CW -complexesto turn a map into a weak homotopy equivalence, namely by attaching cells.If f : Y1 → Y2 is already k-connected, we can attach (k + 1) cells to Y1 toobtain an extension f ′ : Y ′1 → Y2 of f which is (k+ 1)-connected. In principlewe want to do the same for a normal map of degree one with target (X, ξ).However, there are two fundamental difficulties. First of all we have to keepthe manifold structure on the source and cannot therefore just attach cells.Moreover, by Poincare duality any modification in dimension k will cause adual modification in dimension n− k if n is the dimension of X so that oneencounters at any rate problems when n happens to be 2k.

Consider a normal map (f, f) : ν(M) → ξ such that f : M → X is a k-connected map. Consider an element ω ∈ πk+1(f) represented by a diagram

Skq //

j

M

f

Dk+1

Q// X

We cannot attach a single cell to M without destroying the manifold struc-ture. But one can glue two manifolds together along a common boundary suchthat the result is a manifold. Suppose that the map q : Sk → M extends toan embedding q : Sk ×Dn−k →M . (This assumption will be justified later.)Let int(im(q)) be the interior of the image of q. Then M − int(im(q)) is amanifold with boundary im(q|Sk×Sn−k−1). We can get rid of the boundary by


attaching Dk+1 × Sn−k−1 along im(q|Sk×Sn−k−1). Call the result

M ′ := Dk+1 × Sn−k−1 ∪im(q|Sk×Sn−k−1 ) (M − int(im(q))) .

Here and elsewhere we apply without further mentioning the techniqueof straightening the angle in order to get a well-defined smooth structure(see [131, Definition 13.11 on page 145 and (13.12) on page 148] and [398,Chapter 8, Section 2]). Choose a map Q : Dk+1×Dn−k → X which extends Qand f q. The restriction of f to M−int(im(q)) extends to a map f ′ : M ′ → Xusing Q|Dk+1×Sn−k . Notice that the inclusion M − int(im(q)) → M is(n − k − 1)-connected since Sk × Sn−k−1 → Sk × Dn−k is (n − k − 1)-connected. So the passage from M to M − int(im(q)) will not affect πj(f) forj < n− k− 1. All in all we see that πl(f) = πl(f

′) for l ≤ k and that there isan epimorphism πk+1(f)→ πk+1(f ′) whose kernel contains ω, provided that2(k + 1) ≤ n. The condition 2(k + 1) ≤ n can be viewed as a consequence ofPoincare duality. Roughly speaking, if we change something in a manifold indimension l, Poincare duality forces also a change in dimension (n− l). Thisphenomenon will cause surgery obstructions to appear.

It is important to notice that f : M → X and f ′ : M ′ → X are bordant.The relevant bordism is given by W = Dk+1 ×Dn−k ∪qM × [0, 1], where wethink of q as an embedding Sk × Dn−k → M × 1. In other words, W isobtained from M × [0, 1] by attaching a handle Dk+1 × Dn−k to M × 1.Then M appears in W as M × 0 and M ′ as other part of the boundary ofW . Define F : W → X by f × id[0,1] and Q. Then F restricted to M and M ′

is f and f ′.Why can we assume that the map q : Sk → M extends to an embedding

q : Sk × Dn−k → M? This will be ensured by the bundle data in the case2k + 1 < n by the following argument.

Because of Theorem 7.54 we can arrange that q is an embedding since amap which is sufficiently close to h of compact manifolds is automaticallyhomotopic to h.

The extension q exists if and only if the normal bundle ν(q) of the embed-ding q : Sk → M is trivial. Since Dk+1 is contractible, every vector bundleover Dk+1 is trivial. Hence Q∗ξ is a trivial vector bundle over Dk+1. Re-call that i : M → Rm+n is a fixed embedding and ν(M) is define to be thenormal bundle ν(i) of i. Pullbacks of trivial vector bundles are trivial again.This implies that q∗ν(M) ∼= q∗f∗ξ ∼= j∗Q∗ξ is a trivial vector bundle overSk. Since ν(q)⊕q∗ν(M) ∼= ν(i : Sk → Rn+m) is trivial, ν(q) is a stably trivial(n − k)-dimensional vector bundle over Sk. Since 2k + 1 ≤ n, this impliesthat ν(q) itself is trivial.

So we see that the bundle data are needed to carry out the desired surgerystep. It is important to notice that the construction yields a map f ′ : M ′ → Xof degree one and a bundle map f ′ : ν(M ′) → ξ covering f ′ so that we endup with a normal map of degree one with target X again. Hence we are ableto repeat this surgery step over and over again in dimensions 2k − 1 ≤ n.


Actually, also the bordism W together with the map F : W → X come by abundle map F : ν(W ) → ξ covering F and is therefore a normal bordism inthe sense of Definition 7.56. In particular surgery does not change the normalbordism class.

Lemma 7.58. Consider a normal map of degree one (f, f) : ν(M)→ ξ cov-ering f : M → X, where M is an oriented connected closed manifold of di-mension n and X is a connected finite Poincare complex of dimension n. Letk be the natural number given by n = 2k or n = 2k + 1.

Then we can always arrange by finitely many surgery steps that for theresulting normal map of degree one (f ′, f ′) : ν(M ′) → ξ its underlying mapf ′ : M ′ → X is k-connected.

Now assume that n is even, let us say n = 2k. As mentioned above, we canarrange that f is k-connected. If we can achieve that f is (k + 1)-connected,then by Poincare duality the map f is a homotopy equivalence.

But in this last step we encounter a problem which actually leads to thesurgery obstruction in the even-dimensional case. Namely, in the argumentabove we used at one point that the map q : Sk →M can be arranged to be anembedding by general position if 2k+ 1 ≤ n and that certain normal bundleare trivial. In the situation n = 2k we can arrange q to be an immersionby Theorem 7.54 and simultaneously ensure that the bundle data carry overto the desired normal bordism essentially because of a systematic use ofTheorem 7.60 below. However, the latter fixes the regular homotopy class ofthe immersion q. Hence one open problem is to ensure that we can change q toan embedding within its regular homotopy class. We have already introducedthe main obstruction for that, the selfintersection element in (7.37). We alsoencounter the problem that by Poincare duality any change in the homologyof the middle dimension comes with a dual change and one has to ensure thatthere two have the desired effect and do not disturb one another. Next weexplain how this leads to the so called surgery obstruction in L2k(Zπ1(X))with respect to the w1(X)-twisted involution on Zπ.

7.5.4 The Even-Dimensional Surgery Obstruction

For the rest of this subsection we fix a normal map of degree one (f, f) : ν(M)→ξ covering f : M → X, where M is an oriented connected closed manifoldof dimension n and X is a connected finite Poincare complex of dimensionn. Suppose that f induces an isomorphism on the fundamental groups. Fix

a base point b ∈ M together with lifts b ∈ M of b and f(b) ∈ X of f(b).

We identify π = π1(M, b) = π(X, f(b)) by π1(f, b). The choices of b and

f(b) determine π-operations on M and on X and a lift f : M → X which isπ-equivariant.


Definition 7.59 (Surgery kernels). Let Kk(M) be the kernel of the

Zπ-map Hk(f) : Hk(M) → Hk(X). Denote by Kk(M) the cokernel of

the Zπ-map Hk(f) : Hk(X) → Hk(M) which is the Zπ-map induced by

C∗(f) : C∗(X)→ C∗(M). We call Kk(M) the surgery kernel.

Given two vector bundles ξ : E → M and η : F → N , we have so far onlyconsidered bundle maps (f, f) : ξ → η which are fiberwise isomorphisms.We need to consider, at least for the next theorem, more generally bundlemonomorphisms, i.e., we only will require that the map is fiberwise injective.Consider two bundle monomorphism (f0, f0), (f1, f1) : ξ → η. Let ξ × [0, 1]be the vector bundle ξ × id : E × [0, 1] → M × [0, 1]. A homotopy of bundlemonomorphisms (h, h) from (f0, f0) to (f1, f1) is a bundle monomorphism(h, h) : ξ×[0, 1]→ η whose restriction toX×j is (f j , fj) for j = 0, 1. Denoteby π0(Mono(ξ, η)) the set of homotopy classes of bundle monomorphisms.

For a proof of the following result we refer to Haefliger-Poenaru [360],Hirsch [397] and Smale [758]. Denote by π0(Imm(M,N)) the set of regularhomotopy classes of immersions from M to N .

Theorem 7.60 (Immersions and Bundle Monomorphisms). Let Mbe a m-dimensional and N be a n-dimensional closed manifold.

(i) Suppose that 1 ≤ m < n. Then taking the differential of an immersionyields a bijection

T : π0(Imm(M,N))∼=−→ π0(Mono(TM, TN));

(ii) Suppose that 1 ≤ m ≤ n and that M has a handlebody decompositionconsisting of q-handles for q ≤ n − 2. Then taking the differential of animmersion yields a bijection

T : π0(Imm(M,N))∼=−→ colima→∞ π0(Mono(TM ⊕ Ra, TN ⊕ Ra)),

where the colimit is given by stabilization.

Lemma 7.61.(i) The cap product with [M ] induces isomorphisms

? ∩ [M ] : Kn−k(M)∼=−→ Kk(M);

(ii) Suppose that f is k-connected. Then there is the composition of naturalZπ-isomorphisms

hk : πk+1(f)∼=−→ πk+1(f)

∼=−→ Hk+1(f)∼=−→ Kk(M);

(iii) Suppose that f is k-connected and n = 2k. Then there is a natural Zπ-homomorphism

tk : πk+1(f)→ Ik(M).


Proof. (i) The following diagram commutes and has isomorphisms as verticalarrows

Hn−k(M)

∼=?∩[M ]

Hn−k(X)Hn−k(f)oo

Hk(M)Hk(f)

// Hk(X)

?∩[X]∼=

OO(7.62)

Hence the composition Kk(M)→ Hk(M)(?∩[M ])−1

−−−−−−−→ Hn−k(M)→ Kn−k(M)is bijective.

(ii) The commutative square (7.62) above implies that Hl(f) : Hl(M) →Hl(X) is split surjective for all l. We conclude from the long exact sequence

of C∗(f) that the boundary map

∂ : Hk+1(f) := Hk+1(cone(C∗(f)))→ Hk(M)

induces an isomorphism

∂k+1 : Hk+1(f)∼=−→ Kk(M).

Since f and hence f is k-connected, the Hurewicz homomorphism

πk+1(f)∼=−→ Hk+1(f)

is bijective [832, Corollary IV.7.10 on on page 181]. The canonical map

πk+1(f)→ πk+1(f)

is bijective. The composition of the maps above or their inverses yields anatural isomorphism hk : πk+1(f)→ Kk(M).

(iii) Notice that an element in πk+1(f, b) is given by a commutative diagram

Skq //

M

Dk+1

Q// X

together with a path w from b to f(s) for a fixed base point s ∈ Sk. We leavethe details of the rest of the proof which is based on Theorem 7.60 (ii) to thereader. The necessary bundle monomorphisms come from the bundle data of(f, f), the stable triviality of TSk and the fact that any vector bundle overDk+1 is trivial. ut


Suppose that n = 2k. The Kronecker product 〈 , 〉 : Hk(M) ×Hk(M) →Zπ is induced by the evaluation pairing homZπ(Cp(M),Zπ)× Cp(M)→ Zπwhich sends (β, x) to β(x). It induces a pairing

〈 , 〉 : Kk(M)×Kk(M)→ Zπ.

Together with the isomorphism

? ∩ [M ] : Kn−k(M)∼=−→ Kk(M);

of Theorem 7.61 (i) it yields the intersection pairing

s : Kk(M)×Kk(M)→ Zπ.(7.63)

We get from Lemma 7.61 (ii) and (iii) a Zπ-homomorphism

α : Kk(M)→ Ik(M).(7.64)

We leave it to the reader to check

Lemma 7.65. The following diagram commutes, where the upper pairing isdefined in (7.63), the lower pairing in (7.32) and the left vertical arrows in(7.64)

Kk(M)×Kk(M)s //

α×α

Zπ

id

Ik(M)× Ik(M)

λ// Zπ

Exercise 7.66. Let f : X → Y be a map of connected finite Poincare com-plexes of dimension n ≥ 4. Suppose that f has degree one and that f is(k+ 1)-connected, where k is given by n = 2k if n is even and by n = 2k+ 1if n is odd. Show that then f is a homotopy equivalence.

Lemma 7.67. Let D∗ be a finite projective R-chain complex. Suppose for afixed integer r that Hi(D∗) = 0 for i < r. Suppose that Hr+1(homR(D∗, V )) =0 for any R-module V . Then im(dr+1) is a direct summand in Dr and thereare canonical exact sequences

0→ ker(dr)→ Dr → Dr−1 → · · · → D0 → 0

and0→ im(dr+1)→ ker(dr)→ Hr(D∗)→ 0.

In particular Hr(D∗) is finitely generated projective. If Hi(D∗) = 0 for i > r,we obtain an exact sequence


. . .→ Dr+3dr+3−−−→ Dr+2

dr+2−−−→ Dr+1dr+1−−−→ im(dr+1)→ 0.

Proof. If we apply the assumption Hr+1(homR(D∗, V )) = 0 in the case V =im(dr+1), we obtain an exact sequence

homZπ(Dr, im(dr+1))homZπ(dr+1,id)−−−−−−−−−−→ homZπ(Dr+1, im(dr+1))

homZπ(dr+2,id)−−−−−−−−−−→ homZπ(Dr+2, im(dr+1)).

Since dr+1 ∈ homZπ(Dr+1, im(dr+1)) is mapped to zero under homZπ(dr+2, id),we can find an R-homomorphism ρ : Dr → im(dr+1) with ρ dr+1 = dr+1.Hence im(dr+1) is a direct summand in Dr. The other claims are obvious. ut

Recall that R-module V is called stably finitely generated free if for somenon-negative integer l the R-module V ⊕ Rl is a finitely generated free R-module.

Lemma 7.68. If f : X → Y is k-connected for n = 2k or n = 2k + 1, thenKk(M) is stably finitely generated free.

Proof. We only give the proof for n = 2k. The proof for n = 2k + 1 is alongthe same lines using Poincare duality for the kernels (see Lemma 7.61 (i)).

Consider the finitely generated free Zπ-chain complex D∗ := cone(C∗(f)).

Its homology Hp(D∗) is by definition Hp(f). Since f is k-connected and D∗is projective, there is an R-subchain complex E∗ ⊂ D∗ such that E∗ is finiteprojective, Ei = 0 for i ≤ k and the inclusion E∗ → D∗ is a homologyequivalence and hence an R-chain homotopy equivalence. Namely, take Ei =Di for i ≥ k + 2, Ek+1 = ker(dk+1) and Ei = 0 for i ≤ k. We get fromthe commutative square (7.62) a Zπ-chain homotopy equivalence Dn+1−∗ →D∗. This implies for any R-module V since homR(D∗, V ) is chain homotopyequivalent to homR(E∗, V ).

Hn+1−i(homR(D∗, V )) = 0 for i ≤ k;

Hi(D) = 0 for i ≥ n+ 1− k;

We conclude from Lemma 7.61.

Hp+1(D∗) ∼=Kk(M) , if p = k;0 , if p 6= k;

Hk+2(homR(D∗, R)) = 0.

Now apply Lemma 7.67 to D∗. ut

Example 7.69 (Effect of trivial surgery). Consider the normal map(f, f) : ν(M)→ ξ covering the k-connected map of degree one f : M → X foran oriented connected closed n-dimensional manifolds M for n = 2k. If we


do surgery on the zero element in πk+1(f), then the effect on M is that M is

replaced by the connected sum M ′ = M](Sk × Sk). The effect on Kk(M) is

that it is replaced by Kk(M ′) = Kk(M)⊕(Zπ⊕Zπ). The intersection pairing

on this new kernel is the sum of the given intersection pairing on Kk(M) to-

gether with the standard hyperbolic symmetric form H(−1)k(Zπ). Moreover,taking the selfintersections into account, the non-degenerate (−1)k-quadraticform on the new kernel is direct sum of the the one of the old kernel and thestandard hyperbolic (−1)k-quadratic form H(−1)k(Zπ). In particular we canarrange by finitely many surgery steps on the trivial element in πk+1(f) that

Kk(M) is a finitely generated free Zπ-module.

Remark 7.70. Let (f, f) : ν(M)→ ξ be a normal map of degree one coveringf : M → X, where M is an oriented connected closed manifold of dimensionn and X is a connected finite Poincare complex of dimension n. Suppose thatn = 2k and f is k-connected.

By Lemma 7.68 and Example 7.69 we can do finitely many trivial surgerysteps to achieve that the kernel Kk(M) is a finitely generated free Zπ-module.By the intersection pairing s of (7.63) we obtain a non-degenerate (−1)k-

symmetric form (Kk(M), s) (see Remark 7.8).So far we have not used the bundle data. They come now into play when we

want to refine (Kk(M), s) to a non-degenerate (−1)k-quadratic form. Because

of Remark 7.20 we have to specify a map t : Kk(M) → Q(−1)k(Zπ). We willtake the composite

Kk(M)h−1k−−→ πk+1(f)

tk−→ Ik(M)µ−→ Q(−1)k(Zπ),

where µ has been defined (7.37), and the isomorphism hk and the map tkhave been introduced in Lemma 7.61. This is indeed a quadratic refinementby Lemma 7.38 and Lemma 7.65.

Definition 7.71 (Even-dimensional surgery obstruction). Consider anormal map of degree one (f, f) : ν(M) → ξ covering f : M → X, where Mis an oriented connected closed manifold of even dimension n = k and X isa connected finite Poincare complex of dimension n with fundamental groupπ. Perform surgery below the middle dimension and trivial surgery in themiddle dimension so that we obtain a k-connected normal map of degree one(f ′, f ′) : ν(M) → ξ such that Kk(M ′) is finitely generated free Zπ-module.Define the surgery obstruction of (f, f) : ν(M)→ ξ

σ(f, f) ∈ L2k(Zπ,w1(X))

by the class of the non-degenerate (−1)k-quadratic form (Kk(M ′, s, t) of Re-mark 7.70.

We omit the proof that this element is well-defined, e.g., independent ofthe previous surgery steps.


Theorem 7.72 (Surgery obstruction in even dimensions). Considera normal map of degree one (f, f) : ν(M) → ξ covering f : M → X, whereM is an oriented connected closed manifold of even dimension n = 2k andX is a connected finite Poincare complex of dimension n with fundamentalgroup π. Then:

(i) Suppose k ≥ 3. Then σ(f, f) = 0 in Ln(Zπ,w1(X)) if and only if we can doa finite number of surgery steps to obtain a normal map (f ′, f ′) : ν(M ′)→ξ which covers a homotopy equivalence f ′ : M ′ → X;

(ii) The surgery obstruction σ(f, f) depends only on the normal bordism classof (f, f).

Proof. We only give the proof of assertion (i). Because of Lemma 7.58, Ex-ample 7.69, Lemma 7.67 and the definition of L2k(Zπ,w) we can arrange byfinitely many surgery steps that the non-degenerate (−1)k-quadratic form

(Kk(M), s, t) is isomorphic to H(−1)k(Zπv). Thus we can choose for some

natural number v a Zπ-basis b1, b2, . . . , bv, c1, c2, . . . , cv for Kk(M) suchthat

s(bi, ci) = 1 i ∈ 1, 2, . . . , v;s(bi, cj) = 0 i, j ∈ 1, 2, . . . , v, i 6= j;s(bi, bj) = 0 i, j ∈ 1, 2, . . . , v;s(ci, cj) = 0 i, j ∈ 1, 2, . . . , v;t(bi) = 0 i ∈ 1, 2, . . . , v.

Notice that f is a homotopy equivalence if and only if the number vis zero. Hence it suffices to explain how we can lower the number v to(v − 1) by a surgery step on an element in πk+1(f). Of course our candi-date is the element ω in πk+1(f) which corresponds under the isomorphism

h : πk+1(f)→ Kk(M) (see Lemma 7.61 (ii)) to the element bv. By construc-tion the composition

πk+1(f)tk−→ Ik(M)

µ−→ Q(−1)k(Zπ,w)

of the maps defined in (7.37) and Lemma 7.61 (iii) sends ω to zero. NowTheorem 7.40 ensures that we can perform surgery on ω. Notice that theassumption k ≥ 3 and the quadratic structure on the kernel become relevantexactly at this point. Finally it remains to check whether the effect on Kk(M)is the desired one, namely, that we get rid of one of the hyperbolic summandsHε(Zπ), or equivalently, v is lowered to v − 1.

We have explained earlier that doing surgery yields not only a new ma-nifold M ′ but also a bordism from M to M ′. Namely, take W = M ×[0, 1] ∪Sk×Dn−k Dk+1 × Dn−k, where we attach Dk+1 × Dn−k by an em-bedding Sk × Dn−k → M × 1, and M ′ := ∂W −M , where we identifyM = M ×0. The manifold W comes with a map F : W → X× [0, 1] whoserestriction to M is the given map f : M = M×0 → X = X×0 and whoserestriction to M ′ is a map f ′ : M ′ → X × 1. The definition of the kernels


makes also sense for pair of maps. We obtain an exact braid combining thevarious long exact sequences of pairs

0

$$

Kk+1(W , M)

$$

Kk(M)

$$

Kk(W , M ′)!!

0

0

>>

Kk+1(W , ∂W )

>>

Kk(W )

>>

0

>>

0

>>

;;Kk(M ′)

>>

== 0

>>

The (k+ 1)-handle Dk+1×Dn−k defines an element φk+1 in Kk+1(W , M)

and the associated dual k-handle defines an element ψk ∈ Kk(W , M ′). These

elements constitute a Zπ-basis for Kk+1(W , M) ∼= Zπ and Kk(W , M ′) ∼= Zπ.

The Zπ-homomorphism Kk+1(W , M) → Kk(M) maps φ to bv. The Zπ-

homomorphism Kk(M)→ Kk(W , M ′) sends x to s(bv, x) ·ψk. Hence we can

find elements b′1, b′2 , . . ., b′v and c′1, c′2 , . . ., c′v−1 in Kk+1(W , ∂W ) uniquelydetermined by the property that b′i is mapped to bi and c′i to ci under the

Zπ-homomorphism Kk+1(W , ∂W )→ Kk(M). Moreover, these elements form

a Zπ-basis for Kk+1(W , ∂W ) and the element φk+1 is mapped to b′v under

the Zπ-homomorphism Kk+1(W , M)→ Kk+1(W , ∂W ). Define b′′i and c′′i fori = 1, 2, . . . , (v−1) to be the image of b′i and c′i under the Zπ-homomorphism

Kk+1(W , ∂W ) → Kk(M ′). Then b′′i | i = 1, 2 . . . , (v − 1)∐c′′i | i =

1, 2 . . . , (v−1) is a Zπ-basis for Kk(M ′). One easily checks for the quadratic

structure (s′, t′) on Kk(M ′)

s′(b′′i , c′′i ) = s(bi, ci) = 1 i ∈ 1, 2, . . . , (v − 1);

s′(b′′i , c′′j ) = s(bi, cj) = 0 i, j ∈ 1, 2, . . . , (v − 1), i 6= j;

s′(b′′i , b′′j ) = s(bi, bj) = 0 i, j ∈ 1, 2, . . . , (v − 1);

s′(c′′i , c′′j ) = s(ci, cj) = 0 i, j ∈ 1, 2, . . . , (v − 1);

t′(b′′i ) = t(bi) = 0 i ∈ 1, 2, . . . , (v − 1).

This finishes the proof of assertion (i) of Theorem 7.72. ut

Exercise 7.73. Let M be a stably framed manifold of dimension (4k + 2),i.e., a closed (4k+ 2)-dimensional manifold together with a choice of a stabletrivialization of its tangent bundle. Assign to it an element α(M) ∈ Z/2 suchthat α(M) = α(N) depends only on the stably framed bordism class of M .(The easy solution α(M) := 0 is not allowed).

7.6 Formations 189

7.6 Formations

In this subsection we explain the algebraic objects, so called formations,which describe the surgery obstruction and will be the typical elements inthe surgery obstruction group in odd dimensions. Throughout this section Rwill be an associative ring with unit and involution and ε ∈ ±1.

Definition 7.74 (Formation). An ε-quadratic formation (P,ψ;F,G) con-sists of a non-degenerate ε-quadratic form (P,ψ) together with two La-grangians F and G.

An isomorphism f : (P,ψ;F,G) → (P ′, ψ′;F,′ , G′) of ε-quadratic forma-tions is an isomorphism f : (P,ψ) → (P ′, ψ′) of non-degenerate ε-quadraticforms such that f(F ) = F ′ and f(G) = G′ holds.

Definition 7.75 (Trivial formation). The trivial ε-quadratic formationassociated to a finitely generated projective R-module P is the formation(Hε(P );P, P ∗). A formation (P,ψ;F,G) is called trivial if it isomorphic to thetrivial ε-quadratic formation associated to some finitely generated projectiveR-module. Two formations are stably isomorphic if they become isomorphicafter taking the direct with trivial formations.

Remark 7.76 (Formations and automorphisms). We conclude fromLemma 7.25 that any formation is isomorphic to a formation of the type(Hε(P );P, F ) for some Lagrangian F ⊂ P ⊕ P ∗. Given an automorphism

f : Hε(P )∼=−→ Hε(P ) of the standard hyperbolic ε-quadratic form Hε(P )

for some finitely generated projective R-module P , we get a formation by(Hε(P );P, f(P )).

Consider a formation (P,ψ;F,G) such that P , F and G are finitely gen-erated free and suppose that R has the property that Rn and Rm are R-isomorphic if and only if n = m. Then (P,ψ;F,G) is stably isomorphic to(Hε(Q);Q, f(Q)) for some finitely generated free R-module Q by the follow-ing argument. Because of Lemma 7.25 we can choose isomorphisms of non-

degenerate ε-quadratic forms f : Hε(F )∼=−→ (P,ψ) and g : Hε(G)

∼=−→ (P,ψ)such that f(F ) = F and g(G) = G. Since F ∼= Ra and G ∼= Rb by assump-tion and R2a ∼= F ⊕ F ∗ ∼= P ∼= G⊕G∗ ∼= R2b, we conclude a = b. Hence wecan choose an R-isomorphism u : F → G. Then we obtain an isomorphism ofnon-degenerate ε-quadratic forms by the composition

v : Hε(F )Hε(u)−−−−→ Hε(G)

g−→ (P,ψ)f−1

−−→ Hε(F )

and an isomorphism of ε-quadratic formations

f : (Hε(F );F, v(F ))∼=−→ (P,ψ;F,G).

Recall that K1(R) is defined in terms of automorphisms of finitely gener-ated free R-modules. Hence it is plausible that the odd-dimensional L-groups


will be defined in terms of formations which is essentially the same as in termsof automorphisms of the standard hyperbolic form over a finitely generatedfree R-module.

Definition 7.77 (Boundary formation). Let (P,ψ) be a (not necessarilynon-degenerate) (−ε)-quadratic form. Define its boundary ∂(P,ψ) to be theε-quadratic formation (Hε(P );P, Γψ), where Γψ is the Lagrangian given bythe image of the R-homomorphism

P → P ⊕ P ∗, x 7→ (x, (1− ε · T )(ψ)(x)) .

One easily checks that Γψ appearing in Definition 7.77 is indeed a La-grangian. Two Lagrangians F,G of a non-degenerate ε-quadratic form (P,ψ)are called complementary if F ∩G = 0 and F +G = P .

Lemma 7.78. Let (P,ψ;F,G) be an ε-quadratic formation. Then:

(i) (P,ψ;F,G) is trivial if and only F and G are complementary to one an-other;

(ii) (P,ψ;F,G) is isomorphic to a boundary if and only if there is a LagrangianL ⊂ P such that L is a complement of both F and G;

(iii) There is an ε-quadratic formation (P ′, ψ′;F ′, G′) such that (P,ψ;F,G)⊕(P ′, ψ′;F ′, G′) is a boundary;

(iv) An (−ε)-quadratic form (Q,µ) is non-degenerate if and only if its boundaryis trivial.

Proof. (i) The inclusions of F and G in P induce an R-isomorphism f : F ⊕G→ P . Let (

a bc d

): F ⊕G→ (F ⊕G)∗ = F ∗ ⊕G∗

be f∗ ψ f for some representative ψ : P → P ∗ of ψ ∈ Qε(P ) and letψ′ ∈ Qε(F ⊕ G) be the associated class. Then f is an isomorphism of non-degenerate ε-quadratic forms (F ⊕ G,ψ′) → (P,ψ) and F and G are La-grangians in (F ⊕G,ψ′). This implies that the isomorphism ψ′ + ε · T (ψ′) isgiven by (

a bc d

)+ ε · T

(a bc d

)=

(a+ ε · a∗ b+ εc∗

c+ εb∗ d+ εd∗

)=

(0 ef 0

)Hence (b+ ε · c∗) : G→ F ∗ is an isomorphism. Define an R-isomorphism

u :

(1 00 (b+ ε · c∗)−1

): F ⊕ F ∗

∼=−→ P.

One easily checks that u defines an isomorphism of formations

u : (Hε(F );F, F ∗)∼=−→ (F ⊕G,ψ′;F,G).

7.7 Odd-Dimensional L-groups 191

(ii) One easily checks that for an (−ε)-quadratic form (P,ψ) the LagrangianP ∗ in its boundary ∂(P,ψ) := (Hε(P );P, Γψ) is complementary to both Pand Γψ. Conversely, suppose that (P,ψ;F,G) is an ε-quadratic formationsuch that there exists a Lagrangian L ⊂ P which is complementary to bothF and G. By the argument appearing in the proof of assertion (i) we find anisomorphism of ε-quadratic formations

f : (Hε(F );F, F ∗)∼=−→ (P,ψ;F,L)

which is the identity on F . The preimage G′ := f−1(G) is a Lagrangian inHε(F ) which is complementary to F ∗. Write the inclusion of G′ into F ⊕F ∗as (a, b) : G′ → F ⊕ F ∗. Consider the (−ε)-quadratic form (F,ψ′), whereψ′ ∈ Q−ε(F ) is represented by b a∗ : F → F ∗. One easily checks thatits boundary is precisely (Hε(F );F,G′) and f induces an isomorphism ofε-quadratic formations

∂(F,ψ′) = (Hε(F );F,G′)∼=−→ (P,ψ;F,G).

(iii) Because of Lemma 7.25 we can find Lagrangians F ′ and G′ suchthat F and F ′ are complementary and G and G′ are complementary. Put(P ′, ψ′;F ′, G′) = (P,−ψ, F ′, G′). Then M = (p, p) | p ∈ P ⊂ P ⊕ P is aLagrangian in the direct sum

(P,ψ;F,G)⊕ (P ′, ψ′;F ′, G′) = (P ⊕ P,ψ ⊕ (−ψ), F ⊕ F ′, G⊕G′)

which is complementary to both F ⊕F ′ and G⊕G′. Hence the direct sum isisomorphic to a boundary by assertion (ii).

(iv) The Lagrangian Γψ in the boundary ∂(Q,µ) := (Hε(P );P, Γψ) is com-plementary to P if and only if (1 − ε · T )(µ) : P → P ∗ is an isomorphism.This finishes the proof of Lemma 7.78. ut

7.7 Odd-Dimensional L-groups

Now we can define the odd-dimensional L-groups

Definition 7.79 (Odd-dimensional L-groups). Let R be an associativering with unit and involution. For an odd integer n = 2k+1 define the abeliangroup Ln(R) called the n-th quadratic L-group of R to be the abelian groupof equivalence classes [(P,ψ;F,G)] of (−1)k-quadratic formations (P,ψ;F,G)such that P , F and G are finitely generated free R-modules with respect tothe following equivalence relation. We call (P,ψ;F,G) and (P ′, ψ′;F ′, G′)equivalent if and only if there exist (−(−1)k)-quadratic forms (Q,µ) and(Q′, µ′) for finitely generated free R-modules Q and Q′ and finitely generatedfree R-modules S and S′ together with an isomorphism of (−1)k-quadratic


formations

(P,ψ;F,G)⊕ ∂(Q,µ)⊕ (Hε(S);S, S∗)∼= (P ′, ψ′;F ′, G′)⊕ ∂(Q′, µ′)⊕ (Hε(S

′);S′, (S′)∗).

Addition is given by the sum of two (−1)k-quadratic formations. The zeroelement is represented by ∂(Q,µ) ⊕ (H(−1)k(S);S, S∗) for any (−(−1)k)-quadratic form (Q,µ) for any finitely generated free R-module Q and anyfinitely generated free R-module S. The inverse of [(P,ψ;F,G)] is repre-sented by (P,−ψ;F ′, G′) for any choice of Lagrangians F ′ and G′ in Hε(P )such that F and F ′ are complementary and G and G′ are complementary.

A morphism u : R → S of rings with involution induces homomorphismsu∗ : Lk(R) → Lk(S) for k = 1, 3 by induction. One easily checks (u v)∗ =u∗ v∗ and (idR)∗ = idLk(R) for k = 1, 3.

The odd-dimensional L-groups of the ring of integers vanish. Comment17: Add reference.

Theorem 7.80 (Vanishing of the odd-dimensional L-groups of Z).We have L2k+1(Z) = 0 for all k ∈ Z.

Remark 7.81 (Four-periodicity of the L-groups). Obviously the L-groups are four-periodic, i.e., Ln(R) = Ln+4k(R) holds for all k, n ∈ Z.

7.8 The Surgery Obstruction in Odd Dimensions

Next we define the odd-dimensional surgery obstruction.Consider a normal map of degree one (f, f) : ν(M)→ ξ covering f : M →

X, where M is a closed oriented manifold of dimension n and X is a connectedfinite Poincare complex of dimension n for odd n = 2k+ 1. To these data wewant to assign an element σ(f, f) ∈ Ln(Zπ,w) such that the following holds

Theorem 7.82 (Surgery obstruction in odd dimensions). We get un-der the conditions above:

(i) Suppose k ≥ 2. Then σ(f, f) = 0 in Ln(Zπ,w) if and only if we can do afinite number of surgery steps to obtain a normal map (f ′, f ′) : ν(M ′)→ ξcovering a homotopy equivalence f ′ : M ′ → X;

(ii) The surgery obstruction σ(f, f) depends only on the normal bordism classof (f, f).

At least we outline the poof of assertion (i) of Theorem 7.82. We can ar-range by finitely many surgery steps that f is k-connected (see Lemma 7.58).Consider a normal bordism (F , F ) : ν(W ) → ξ covering a map F : W →X × [0, 1] of degree one from the given normal map (f, f) : ν(M) → pr∗ ξ

7.8 The Surgery Obstruction in Odd Dimensions 193

covering f : M → X to a new k-connected normal map (f ′, f ′) : ν(M ′) → ξcovering f ′ : M ′ → X, where pr : X × [0, 1]→ X is the projection. We finallywant to arrange that f ′ is a homotopy equivalence. By applying Lemma 7.58to the interior of W without changing ∂W we can arrange that W is (k+ 1)-connected. The kernels fit into an exact sequence

0→ Kk+1(M ′)Kk+1(j)−−−−−→ Kk+1(W )

Kk+1(i′)−−−−−−→ Kk+1(W , M ′)∂k+1−−−→ Kk(M ′)→ 0,

where j : M ′ →W and i′ : W → (W,M ′) are the inclusions. Hence f ′ is a ho-

motopy equivalence if and only if Kk+1(i′) : Kk+1(W )→ Kk+1(W , M ′) is bi-

jective. Therefore we must arrange thatKk+1(i′) : Kk+1(W )→ Kk+1(W , M ′)is bijective.

We can associate to the normal bordism (F , F ) : ν(W ) → pr∗ ξ from thegiven k-connected normal map of degree one (f, f) to another k-connectednormal map of degree one (f ′, f ′) a (−1)k-quadratic formation (H(−1)k(F );F,G)

as follows. The underlying non-degenerate (−1)k-quadratic form is H(−1)k(F )

for F := Kk+1(W , M ′). The first Lagrangian is F . The second Lagrangian Gis given by the image of the map

(Kk+1(i′), u (? ∩ [W,∂W ])−1 Kk+1(i)) : Kk+1(W )

→ F ⊕ F ∗ = Kk+1(W , M ′)⊕Kk+1(W , M ′)∗,

where (?∩ [W,∂W ]) : Kk+1(W , M ′)∼=−→ Kk+1(W , M) is the Poincare isomor-

phism, i : W → (W,M) is the inclusion and u is the canonical map

u : Kk+1(W , M ′)∼=−→ Kk+1(W , M ′)∗, α 7→ 〈α, ?〉

which is bijective by an elementary chain complex argument or by the uni-versal coefficient spectral sequence.

What is the relation between this formation and the problem whether f ′

is a homotopy equivalence? Suppose that f ′ is a homotopy equivalence. ThenKk+1(i′) : Kk+1(W ) → Kk+1(W , M ′) is an isomorphism. Define a (−1)k+1-

quadratic form (H,ψ) by H = Kk+1(W ) and

ψ : H := Kk+1(W )Kk+1 (i)−−−−−→ Kk+1(W , M)

(?∩[W,∂W ])−1

−−−−−−−−−→ Kk+1(W , M ′)

u−→ Kk+1(W , M ′)∗Kk+1(i′)∗−−−−−−→ Kk+1(W )∗ = H∗.

One easily checks that the isomorphismKk+1(i′) : Kk+1(W )→ Kk+1(W , M ′)induces an isomorphism of (−1)k-quadratic formations

∂(H,ψ)∼=−→ (H(−1)k(F );F,G).


Hence we see that f ′ is a homotopy equivalence only if (H(−1)k(F );F,G) isisomorphic to a boundary.

One decisive step is to prove that the class of this (−1)k-quadratic forma-tion (H(−1)k(F );F,G) in L2k+1(Zπ,w) is independent of the choice of the(k + 1)-connected nullbordism. We will not give the proof of this fact. Thisfact enables us to define the surgery obstruction of (f, f)

σ(f, f) := [(H(−1)k(F );F,G)] ∈ L2k+1(Zπ,w),(7.83)

where (H(−1)k(F );F,G) is the (−1)k-quadratic formation associated to any

normal bordism of degree one (F , F ) from a normal map (f0, f0) to some nor-mal map (f ′, f ′) such that (f0, f0) and (f ′, f ′) are k-connected, F is (k+ 1)-connected and (f0, f0) is obtained from the original normal map of degreeone (f, f) by surgery below the middle dimension. From the discussion aboveit is clear that the vanishing of σ(f, f) is a necessary condition for the ex-istence of a normal map (f ′, f ′) which is normally bordant to f and whoseunderlying map is a homotopy equivalence. We omit the proof that for k ≥ 2this condition is also sufficient. This finishes the outline of the proof of The-orem 7.82 (i)

More details can be found in [815, chapter 6]. Comment 18: Maybe wecan add a reference to the book by Crowley-Macko-Luck [206].

Example 7.84 (The surgery obstruction in the simply-connectedcase). Consider a normal map of degree one (f, f) : ν(M) → ξ coveringf : M → X, where M is an oriented connected closed manifold of dimensionn and X is a connected finite Poincare complex of dimension n. Suppose thatX is simply connected.

If n is odd, Ln(Z) is trivial and hence σ(f, f) = 0. In particular we canarrange by finitely many surgery steps that the underlying map is a homotopyequivalence provided n ≥ 5.

If n is divisible by four, we obtain an isomorphism Ln(Z)∼=−→ Z by sending

a quadratic form to its signature divided by eight (see Theorem 7.28). Itturns out that under this isomorphism we get

σ(f, f) =sign(X)− sign(M)

8.

Notice that in this case the surgery obstruction depends only on M and Xbut not on f and f . This is not true in general.

If n is even but not divisible by four, then the Arf invariant yields an

isomorphism Ln(Z)∼=−→ Z/2. It turns out that σ(f, f) depends not only on f

but also on the bundle data f . For instance, for different framings of T 2 oneobtains different invariants α(T 2) in Exercise 7.73

7.9 Surgery Obstructions for Manifolds with Boundary 195

7.9 Surgery Obstructions for Manifolds with Boundary

Next we deal with manifolds with boundary.

Definition 7.85 (Poincare pairs). The notion of a Poincare complex canbe extended to pairs as follows. Let X be a connected finite n-dimensionalCW -complex with fundamental group π together with a subcomplex A ⊂X of dimension (n − 1). Denote by A ⊂ X the preimage of A under the

universal covering X → X. We call (X,A) a finite n-dimensional Poincarepair with respect to the orientation homomorphism w : π1(X) → ±1 ifthere is a fundamental class [X,A] ∈ Hn(X,A;Zw) such that the Zπ-chain

maps ?∩ [X,A] : Cn−∗(X, A)→ C∗(X) and ?∩ [X,A] : Cn−∗(X)→ C∗(X, A)are Zπ-chain equivalences.

We call (X,A) simple if the Whitehead torsion of these Zπ-chain mapsvanish.

If M is a compact connected manifold of dimension n with boundary ∂M ,then (M,∂M) is a simple finite n-dimensional Poincare pair.

We want to extend the notion of a normal map from closed manifoldsto manifolds with boundary. The underlying map f is a map of pairs(f, ∂f) : (M,∂M)→ (X, ∂X), where M is a compact oriented manifold withboundary ∂M and (X, ∂X) is a finite Poincare pair, the degree of f is one and∂f : ∂M → ∂X is required to be a homotopy equivalence. The bundle dataare unchanged, they consist of a vector bundle ξ over X and a bundle mapf : ν(M)→ ξ. Next we explain what a nullbordism in this setting means.

A manifold triad (W ; ∂0W,∂1W ) of dimension n consists of a compactmanifoldW of dimension n whose boundary decomposes as ∂W = ∂0W∪∂1Wfor two compact submanifolds ∂0W and ∂1W with boundary such that ∂0W∩∂1W = ∂(∂0W ) = ∂(∂1W ). A Poincare triad (X; ∂0X, ∂1X) of dimension nconsists of a finite Poincare pair (X, ∂X) of dimension n together with (n−1)-dimensional finite subcomplexes ∂0X and ∂1X such that ∂X = ∂0X ∪ ∂1X,the intersection ∂0X ∩ ∂1X is (n − 2)-dimensional, ∂X is equipped withthe structure of a Poincare complex induced by Poincare pair structure on(X, ∂X) and both (∂0X, ∂0X ∩ ∂1X) and (∂1X, ∂0X ∩ ∂1X) are equippedwith the structure of a Poincare pair induced by the given structure of aPoincare complex on ∂X. We use the convention that the fundamental class of

[X, ∂X] is sent to [∂X] under the boundary homomorphism Hn(X, ∂X)∂n−→

Hn−1(∂X) and [∂X] is sent to ([∂0X, ∂0X ∩∂1X],−[∂1X, ∂0X ∩∂1X]) underthe composite

Hn−1(∂X)→ Hn−1(∂0X ∪ ∂1X, ∂0X ∩ ∂1X)∼= Hn−1(∂0X, ∂0X ∩ ∂1X)⊕Hn−1(∂1X, ∂0X ∩ ∂1X).

A manifold triad (W ; ∂0W,∂1W ) together with an orientation for W is aPoincare triad. We allow that ∂0X or ∂1X or both are empty.


A normal nullbordism (F , F ) : ν(W ) → η for a normal map of degreeone (f, f) with underlying map (f, ∂f) : (M,∂M) → (X, ∂X) consists ofthe following data. Put n = dim(M). We have a compact oriented (n + 1)-dimensional manifold triad (W ; ∂0W,∂1W ), a finite (n + 1)-dimensionalPoincare triad (Y ; ∂0Y, ∂1Y ), a map (F ; ∂0F, ∂1F ) : (W ; ∂0W,∂1W )→ (Y ; ∂0Y, ∂1Y ),an orientation preserving diffeomorphism (u, ∂u) : (M,∂M)→ (∂0W,∂(∂0W ))and an orientation preserving homeomorphism (v, ∂v) : (X, ∂X)→ (∂0Y, ∂0Y ∩∂1Y ). We require that F has degree one, ∂1F is a homotopy equivalence and∂0F u = v f . Moreover, we have a bundle map F : ν(W ) → η coveringF and a bundle map v : ξ ⊕ Rc → η covering v such that F , Tu and v fittogether.

We call two such normal maps (f, f) and (f ′, f ′) normally bordant if afterpossible stabilization and after changing the orientation for M appearing in(f, f) their disjoint union possesses a normal nullbordism.

The definition and the main properties of the surgery obstruction carryover from normal maps for closed manifolds to normal maps for compactmanifolds with boundary. The main reason is that we require ∂f : ∂M → ∂Xto be a homotopy equivalence so that the surgery kernels “do not feel theboundary”. All arguments such as making a map highly connected by surgerysteps and intersection pairings and selfintersection can be carried out in theinterior of M without affecting the boundary. Thus we get

Theorem 7.86. (Surgery Obstruction for Manifolds with Bound-ary).Let (f, f) be a normal map of degree one with underlying map (f, ∂f) : (M,∂M)→(X, ∂X) such that ∂f is a homotopy equivalence. Put n = dim(M). Then

(i) We can associate to it its surgery obstruction

σ(f, f) ∈ Ln(Zπ,w).(7.87)

(ii) The surgery obstruction depends only on the normal bordism class of (f, f);(iii) Suppose n ≥ 5. Then σ(f, f) = 0 in Ln(Zπ,w) if and only if we can do a

finite number of surgery steps on the interior of M leaving the boundaryfixed to obtain a normal map (f ′, f ′) which covers a homotopy equivalenceof pairs (f ′, ∂f ′) : (M ′, ∂M ′)→ (X, ∂X) with ∂M ′ = ∂M and ∂f ′ = ∂f .

7.10 Decorations

Next we want to modify the L-groups and the surgery obstruction so that thesurgery obstruction is the obstruction to achieve a simple homotopy equiva-lence. This will force us to study L-groups with decorations.

7.10 Decorations 197

7.10.1 L-groups with K1-Decorations

We begin with the L-groups. It is clear that this requires to take equivalenceclasses of bases into account. Suppose that we have specified a subgroup U ⊂K1(R) such that U is closed under the involution on K1(R) coming from theinvolution of R and contains the image of the change of ring homomorphismK1(Z)→ K1(R).

Two bases B and B′ for the same finitely generated free R-module Vare called U -equivalent if the change of basis matrix defines an element inK1(R) which belongs to U . Notice that the U -equivalence class of a basis Bis unchanged if we permute the order of elements of B. We call an R-moduleV U -based if V is finitely generated free and we have chosen a U -equivalenceclass of bases.

Let V be a stably finitely generated free R-module. A stable basis for Vis a basis B for V ⊕Ru for some integer u ≥ 0. Denote for any integer v thedirect sum of the basis B and the standard basis Sa for Ra by B

∐Sa which

is a basis for V ⊕Ru+a. Let C be a basis for V ⊕Rv. We call the stable basisB and C stably U -equivalent if and only if there is an integer w ≥ u, v suchthat B

∐Sw−u and C

∐Sw−v are U -equivalent basis. We call an R-module

V stably U -based if V is stably finitely generated free and we have specifieda stable U -equivalence class of stable basis for V .

Let V and W be stably U -based R-modules. Let f : V ⊕Ra∼=−→W ⊕Rb be

an R-isomorphism. Choose a non-negative integer c together with basis forV ⊕Ra+c and W⊕Rb+c which represent the given stable U -equivalence classes

of basis for V and W . Let A be the matrix of f⊕idRc : V ⊕Ra+c∼=−→W⊕Rb+c

with respect to these bases. It defines an element [A] in K1(R). Define theU -torsion

τU (f) ∈ K1(R)/U(7.88)

by the class represented by [A]. One easily checks that τ(f) is independent ofthe choices of c and the basis and depends only on f and the stable U -basisfor V and W . Moreover, one easily checks

τU (g f) = τU (g) + τU (f);

τU(f 0u v

)= τU (f) + τU (v);

τU (idV ) = 0

for R-isomorphisms f : V0

∼=−→ V1, g : V1

∼=−→ V2 and v : V3

∼=−→ V4 and an R-homomorphism u : V0 → V4 of stably U -based R-modules Vi. Let C∗ be acontractible stably U -based finite R-chain complex, i.e., a contractible R-chain complex C∗ of stably U -based R-modules which satisfies Ci = 0 for|i| > N for some integer N . The definition of Whitehead torsion in (2.29)


carries over to the definition of the U -torsion

τU (C∗) = [A] ∈ K1(R)/U.(7.89)

Analogously we can associate to an R-chain homotopy equivalence f : C∗ →D∗ of stably U -based finite R-chain complexes its U -torsion (cf. (2.30))

τU (f∗) := τ(cone∗(f∗)) ∈ K1(R)/U.(7.90)

We will consider U -based ε-quadratic forms (P,ψ), i.e. ε-quadratic formswhose underlying R-module P is a U -based finitely generated free R-module

such that U -torsion of the isomorphism (1 + ε · T )(ψ) : P∼=−→ P ∗ is zero in

K1(R)/U . An isomorphism f : (P,ψ)→ (P ′, ψ′) of U -based ε-quadratic formsis U -simple if the U -torsion of f : P → P ′ vanishes in K1(R)/U . Notice thatthe ε-quadratic form Hε(R) inherits a basis from the standard basis of R.The sum of two stably U -based ε-quadratic forms is again a stably U -basedε-quadratic form. It is worthwhile to mention the following U -simple versionof Lemma 7.25.

Lemma 7.91. Let (P,ψ) be a U -based ε-quadratic form. Let L ⊂ P be aLagrangian such that L is U -based and the U -torsion of the following 2-dimensional U -based finite R-chain complex

0→ Li−→ P

i∗(1+ε·T )(ψ)−−−−−−−−−→ L∗ → 0

vanishes in K1(R)/U . Then the inclusion i : L → P extends to a U -simpleisomorphism of ε-quadratic forms

Hε(L)∼=−→ (P,ψ).

Next we give the simple version of the even-dimensional L-groups.

Definition 7.92. Let R be an associative ring with unit and involution. Forε ∈ ±1 define LU1−ε(R) to be the abelian group of equivalence classes [(F,ψ)]of U -based non-degenerate ε-quadratic forms (F,ψ) with respect to the fol-lowing equivalence relation. We call (F,ψ) and (F ′, ψ′) equivalent if andonly if there exists integers u, u′ ≥ 0 and a U -simple isomorphism of non-degenerate ε-quadratic forms

(F,ψ)⊕Hε(R)u ∼= (F ′, ψ′)⊕Hε(R)u′.

Addition is given by the sum of two ε-quadratic forms. The zero element isrepresented by [Hε(R)u] for any integer u ≥ 0. The inverse of [F,ψ] is givenby [F,−ψ].

For an even integer n define the abelian group LUn (R) called the n-th U -decorated quadratic L-group of R by


LUn (R) :=

LU0 (R) if n ≡ 0 mod 4;LU2 (R) if n ≡ 2 mod 4.

A U -based ε-quadratic formation (P,ψ;F,G) consists of an ε-quadratic for-mation (P,ψ;F,G) such that (P,ψ) is a U -based non-degenerate ε-quadraticform, the Lagrangians F and G are U -based R-modules and the U -torsion ofthe following two contractible U -based finite R-chain complexes

0→ Fi−→ P

i∗(1+ε·T )(ψ)−−−−−−−−−→ F ∗ → 0

and

0→ Gj−→ P

j∗(1+ε·T )(ψ)−−−−−−−−−→ G∗ → 0

vanish in K1(R)/U , where i : F → P and j : G→ P denote the inclusions. Anisomorphism f : (P,ψ;F,G)→ (P ′, ψ′;F ′, G′) of U -based ε-quadratic forma-tions is U -simple if underlying the U -torsion of the induced R-isomorphisms

P∼=−→ P ′, F

∼=−→ F ′ and G∼=−→ G′ vanishes in K1(R)/U . Notice that the trivial

ε-quadratic formation (Hε(Ru);Ru, (Ru)∗) inherits a U -basis from the stan-

dard basis on Ru. Given a U -based (−ε)-quadratic form (Q,ψ), its boundary∂(Q,ψ) is a U -based ε-quadratic formation. Obviously the sum of two U -based ε-quadratic formations is again a U -based ε-quadratic formation. Nextwe give the simple version of the odd-dimensional L-groups.

Definition 7.93. Let R be an associative ring with unit and involution. Forε ∈ ±1 define LU2−ε(R) to be the abelian group of equivalence classes[(P,ψ;F,G)] of U -based ε-quadratic formations (P,ψ;F,G) with respect tothe following equivalence relation. We call two U -based ε-quadratic forma-tions (P,ψ;F,G) and (P ′, ψ′;F ′, G′) equivalent if and only if there existsU -based (−ε)-quadratic forms (Q,µ) and (Q′, µ′) and non-negative integersu and u′ together with a U -simple isomorphism of ε-quadratic formations

(P,ψ;F,G)⊕ ∂(Q,µ)⊕ (Hε(Ru);Ru, (Ru)∗)

∼= (P ′, ψ′;F ′, G′)⊕ ∂(Q′, µ′)⊕ (Hε(Ru′);Ru

′, (Ru

′)∗).

Addition is given by the sum of two ε-quadratic forms. The zero element isrepresented by ∂(Q,µ)⊕(Hε(R

u);Ru, (Ru)∗) for any U -based (−ε)-quadraticform (Q,µ) and non-negative integer u. The inverse of [(P,ψ;F,G)] is rep-resented by (P,−ψ;F ′, G′) for any choice of stably U -based Lagrangians F ′

and G′ in Hε(P ) such that F and F ′ are complementary and G and G′ are

complementary and the U -torsion of the obvious isomorphism F ⊕ F ′∼=−→ P

and G⊕G′∼=−→ P vanishes in K1(R)/U .

For an odd integer n define the abelian group LUn (R) called the n-th U -decorated quadratic L-group of R

LUn (R) :=

LU1 (R) if n ≡ 1 mod 4;LU3 (R) if n ≡ 3 mod 4.


Notation 7.94. If R = Zπ with the w-twisted involution and U ⊂ K1(Zπ)is the abelian group of elements of the shape (±g) for g ∈ π, then we write

Lsn(Zπ,w) := LUn (Zπ,w);

Lhn(Zπ,w) := Ln(Zπ,w);

where Ln(R) is the L-group introduced in Definitions 7.24 and 7.79 andLUn (R) is the L-group introduced in Definitions 7.92 and 7.93. The L-groupsLsn(Zπ,w) are called simple quadratic L-groups.

7.10.2 The Simple Surgery Obstruction

Let (f, f) be a normal map of degree one with (f, ∂f) : (M,∂M)→ (X, ∂X)as underlying map such that (X, ∂X) is a simple finite Poincare complex and∂f is a simple homotopy equivalence. Then the definition of the surgery ob-struction of (7.87) can be modified to the simple setting. Notice that the dif-ference between the L-groups Lhn(Zπ,w) and the simple L-groups Lsn(Zπ,w)is the additional structure of a U -basis. The definition of the simple surgeryobstruction

σ(f, f) ∈ Lsn(Zπ,w).(7.95)

is the same as the one in (7.87) except that we must explain how the varioussurgery kernels inherit a U -basis.

The elementary proof of the following lemma is left to the reader. Noticethat for any stably U -based R-module V and element x ∈ K1(R)/U we canfind another stable U -basis C for V such that the U -torsion τU (id : (V,B)→(V,C) is x. This is not true in the unstable setting. For instance, there existsa ring R with an element x ∈ K1(R)/U for U the image of K1(Z)→ K1(R)such that x cannot be represented by a unit in R, in other words x is not theU -torsion of any R-automorphism of R.

Lemma 7.96. Let C∗ be a contractible finite stably free R-chain complex andr be an integer. Suppose that each chain module Ci with i 6= r comes with astable U -basis. Then Cr inherits a preferred stable U -basis which is uniquelydefined by the property that the U -torsion of C∗ vanishes in K1(R)/U .

Now Lemma 7.67 has the following version in the simple homotopy setting.

Lemma 7.97. Let D∗ be a stably U -based finite R-chain complex. Supposefor a fixed integer r that Hi(D∗) = 0 for i 6= r. Suppose that Hr+1(homR(D∗, V )) =0 for any R-module V . Then Hr(D∗) is stably finitely generated free and in-herits a preferred stable U -basis.

Proof. Combine Lemma 7.96 and Lemma 7.67. ut


Now we can prove the following version of Lemma 7.68

Lemma 7.98. If f : X → Y is k-connected for n = 2k or n = 2k + 1, thenKk(M) is stably finitely generated free and inherits a preferred stable U -basis.

Proof. The proof is the same as the one of the proof of Lemma 7.68 exceptthat we apply Lemma 7.97 instead of Lemma 7.67. ut

Next we can give the simple version of the surgery obstruction theorem.Notice that simple normal bordism class means that in the definition of nor-mal nullbordisms the pairs (Y, ∂Y ), (∂0Y, ∂0Y ∩ ∂1Y ) and (∂1Y, ∂0Y ∩ ∂1Y )are required to be simple finite Poincare pairs and the map ∂1F : ∂1M → ∂1Yis required to be a simple homotopy equivalence.

Theorem 7.99. (Simple surgery obstruction for manifolds with bound-ary) Let (f, f) be a normal map of degree one with underlying map(f, ∂f) : (M,∂M) → (X, ∂X) such that (X, ∂X) is a simple finite Poincarecomplex and ∂f is a simple homotopy equivalence. Put n = dim(M). Then:

(i) The simple surgery obstruction depends only on the simple normal bordismclass of (f, f);

(ii) Suppose n ≥ 5. Then σ(f, f) = 0 in Lsn(Zπ,w) if and only if we can doa finite number of surgery steps on the interior of M leaving the bound-ary fixed to obtain a normal map (f ′, f ′) : νM ′ → ξ which covers a sim-ple homotopy equivalence of pairs (f ′, ∂f ′) : (M ′, ∂M ′) → (X, ∂X) with∂M ′ = ∂M and ∂f ′ = ∂f .

Exercise 7.100. Let W be a compact manifold of dimension n whose bound-ary is the disjoint union M q N . Let (f, f) be a normal map such that theunderlying maps of pairs is of the shape f : (W,∂W )→ (X×[0, 1], X×0, 1)for some closed manifold X and induces a simple homotopy equivalence∂W → X × 0, 1. Show that M and N are diffeomorphic provided thatthe simple surgery obstruction σ(f, f) of (7.95) vanishes and n ≥ 6.

7.10.3 Decorated L-Groups

L-groups are designed as obstruction groups for surgery problems. The dec-oration reflects what kind of surgery problem one is interested in.

The L-groups Ln(R) of Subsection 7.10.1 for U the image of the map

K1(Z) → K1(R) are also denoted L〈2〉n (R). The group Ln(R) of Defini-

tions 7.24 and 7.79 are also denoted by denoted L〈1〉n (R) or by Lhn(R). If one

works with finitely generated projective modules instead of finitely generatedfree R-modules in Definitions 7.24 and 7.79 , one obtains projective quadratic

L-groups Lpn(R) which are also denoted by L〈0〉n (R). The negative decorations


L〈j〉n (R) for j ∈ Z, j ≤ −1 can be obtained using suitable categories of mod-

ules parametrized over Rk. There are forgetful maps L〈j+1〉n (R) → L

〈j〉n (R)

for j ∈ Z, j ≤ 1. The group L〈−∞〉n (R) is defined as the colimit over these

maps. For details the reader should consult [679], [686].

For group rings we also define Lsn(RG) similar to L〈2〉n (RG) but we re-

quire the torsion to lie in imA1 ⊂ K1(RG), where A1 is the map definedin (2.26). Observe that Lsn(RG) really depends on the pair (R,G) and differs

from L〈2〉n (RG) in general. If R = Z, this definition agrees with the one of

Notation 7.94.

7.10.4 The Rothenberg Sequence

Next we explain how decorated L-groups can be computed from one another.

For j ≤ 1 there are forgetful maps L〈j+1〉n (R) → L

〈j〉n (R) which sit inside

the following sequence, which is known as the Rothenberg sequence [683,Proposition 1.10.1 on page 104], [686, 17.2].

(7.101) . . .→ L〈j+1〉n (R)→ L〈j〉n (R)→ Hn(Z/2; Kj(R))

→ L〈j+1〉n−1 (R)→ L

〈j〉n−1(R)→ . . . .

Here Hn(Z/2; Kj(R)) is the Tate-cohomology of the group Z/2 with coef-

ficients in the Z[Z/2]-module Kj(R). The involution on Kj(R) comes fromthe involution on R. There is a similar sequence relating LsnZG) and Lhn(ZG),where the third term is the Z/2-Tate-cohomology of Wh(G). There is a similarsequence relating LhnZG) and Lpn(ZG), where the third term is the Z/2-Tate-

cohomology of K0(ZG).

Theorem 7.102 (Independence of decorations). Let G be a group such

that Wh(G), K0(ZG) and Kn(ZG) for all n ∈ Z, n ≤ −1 vanish. Then forevery j ∈ Z, j ≤ −1 and every n ∈ Z the forgetful maps induce isomorphisms

Ls(ZG)∼=−→ Lh(ZG)

∼=−→ Lp(ZG)∼=−→ L〈j〉(ZG)

∼=−→ L〈−∞〉(ZG).

Proof. This follows from the various Rothenberg sequences. ut

Exercise 7.103. Show that for every group G, every j ∈ Z, j ≤ −1 andevery n ∈ Z the forgetful maps induces isomorphisms after inverting 2

Ls(ZG)[1/2]∼=−→ Lh(ZG)[1/2]

∼=−→ Lp(ZG)[1/2]∼=−→ L〈j〉(ZG)[1/2]

∼=−→ L〈−∞〉(ZG)[1/2].


7.10.5 The Shaneson Splitting

The Bass-Heller-Swan decomposition in K-theory (see Theorem 5.16) has thefollowing analogue for the algebraic L-groups.

Theorem 7.104 (Shaneson splitting). For every group G, every ring withinvolution R, every j ∈ Z, j ≤ 2 and n ∈ Z there is a natural isomorphism

L〈j〉n (RG)⊕ L〈j−1〉n−1 (RG)

∼=−→ L〈j〉n (R[G× Z]).

The map appearing in the theorem above comes from the map L〈j〉n (RG)→

L〈j〉n (R[G×Z]) coming from the inclusionG→ G×Z and a map L

〈j−1〉n−1 (RG)→

L〈j〉n (R[G × Z]) which is essentially given by the cartesian product with S1.

The latter explains the raise from (n− 1) to n. But why does the decorationraises from j − 1 to j? The reason is the product formula for Whiteheadtorsion (see Theorem 2.34 iv). It predicts for any (not necessarily simple) ho-motopy equivalence f : X → Y of finite CW -complexes that the homotopyequivalence f × idS1 : X × S1 → Y × S1 is a simple homotopy equivalence.There is also a product formula for the finiteness obstruction which predictsfor a finitely dominated (not necessarily up to homotopy finite) CW -complexX that X ×S1 is homotopy equivalent to a finite CW -complex. The originalproof of the Shaneson splitting for the case j = 2 and R = Z i.e., for theisomorphism

Lsn(ZG)⊕ Lhn−1(ZG)∼=−→ Lsn(Z[G× Z])

can be found in [752]. The proof for arbitrary j and R is given [686, 17.2].Notice that for j = 1 we obtain an isomorphism

Lhn(RG)⊕ Lpn−1(RG)∼=−→ Lhn(R[G× Z])

and for j = −∞ we obtain an isomorphism

L〈−∞〉n (RG)⊕ L〈−∞〉n−1 (RG)∼=−→ L〈−∞〉n (R[G× Z]).(7.105)

Remark 7.106 (Nil-groups and L-theory). Even though in the Shanesonsplitting (7.105) above there are no terms analogous to the Nil-terms in theBass-Heller-Swan decomposition in K-theory (see Theorem 5.16) such Nil-phenomena do also occur in L-theory, as soon as one considers amalgamatedfree products. The corresponding groups are the UNil-groups. They vanishif one inverts 2, see [153]. For more information about the UNil-groups werefer for instance to [58, 150, 151, 196, 197, 200, 287, 687]. How the Farrell-Jones Conjecture predicts such exact sequences after inverting 2 is explainedin Section 13.7.

Exercise 7.107. Compute L〈j〉n (Z[Z]).


7.11 The Farrell-Jones Conjecture for AlgebraicL-Theory for Torsionfree Groups

The Farrell-Jones Conjecture for algebraic L-theory, which will later be for-mulated in full generality in Chapter 11, reduces for a torsionfree group anda regular ring to the following conjecture. Given a ring with involution R,there exists a L-spectrum associated to R with decoration 〈−∞〉

L〈−∞〉(R)(7.108)

with the property that πn(L〈−∞〉(R)

)= L

〈−∞〉n (R) holds for n ∈ Z.

Conjecture 7.109 (Farrell-Jones Conjecture for torsionfree groupsfor L-theory). Let G be a torsionfree group. Let R be any ring with invo-lution.

Then the assembly map

Hn(BG; L〈−∞〉(R))→ L〈−∞〉n (RG)


In the special case G = Z, the left hand side is

Hn(BG; L〈−∞〉(R)) ∼= Hn(•; L〈−∞〉(R))⊕Hn−1(•; L〈−∞〉(R))

∼= L〈−∞〉n (R)⊕ L〈−∞〉n−1 (R).

The Farrell-Jones Conjecture for torsionfree groups for L-theory 7.109 pre-

dicts that this is isomorphic to L〈−∞〉n (RZ). In view of the Shaneson splitting

of Subsection 7.10.5 it is now obvious why we have passed to the decoration〈−∞〉.Exercise 7.110. Let Fg be the closed orientable surface of genus g. Compute

L〈j〉n (Z[π1(Fg)]) for all j ∈ Z, j ≤ 2 and n ∈ Z.

7.12 The Surgery Exact Sequence

In this section we introduce the surgery exact sequence. It is the realizationof the surgery program which we have explained in Remark 2.50. The surgeryexact sequence is the main theoretical tool in solving the classification prob-lem of manifolds of dimensions greater than or equal to five.

7.12 The Surgery Exact Sequence 205

7.12.1 The Structure Set

Definition 7.111 (Simple structure set). Let X be a closed oriented ma-nifold of dimension n. We call two orientation preserving simple homotopyequivalences fi : Mi → X from closed oriented manifolds Mi of dimension nto X for i = 0, 1 equivalent if there exists an orientation preserving diffeomor-phism g : M0 →M1 such that f1 g is homotopic to f0. The simple structureset Ssn(X) of X is the set of equivalence classes of orientation preservingsimple homotopy equivalences M → X from closed oriented manifolds of di-mension n to X. This set has a preferred base point, namely the class of theidentity id : X → X.

The simple structure set Ssn(X) is the basic object in the study of manifoldswhich are diffeomorphic to X. Notice that a simple homotopy equivalencef : M → X is homotopic to a diffeomorphism if and only if it represents thebase point in Ssn(X). A manifold M is oriented diffeomorphic to N if and onlyif for some orientation preserving simple homotopy equivalence f : M → Nthe class of [f ] agrees with the preferred base point. Some care is necessarysince it may be possible that a given orientation preserving simple homotopyequivalence f : M → N is not homotopic to a diffeomorphism although Mand N are diffeomorphic. Hence it does not suffice to compute Ssn(N), onealso has to understand the operation of the group of homotopy classes ofsimple orientation preserving selfequivalences of N on Ssn(N). This can berather complicated in general. But it will be no problem in the case N = Sn,because any orientation preserving selfhomotopy equivalence Sn → Sn ishomotopic to the identity.

There is also a version of the structure set which does not take Whiteheadtorsion into account.

Definition 7.112 (Structure set). Let X be a closed oriented manifoldof dimension n. We call two orientation preserving homotopy equivalencesfi : Mi → X from closed oriented manifolds Mi of dimension n to Xfor i = 0, 1 equivalent if there is a manifold triad (W ; ∂0W,∂1W ) with∂0W ∩∂1W = ∅ and an orientation preserving homotopy equivalence of triads(F ; ∂0F, ∂1F ) : (W ; ∂0W,∂1W )→ (X× [0, 1];X×0, X×1) together withorientation preserving diffeomorphisms g0 : M0 → ∂0W and g1 : M−1 → ∂1Wsatisfying ∂iF gi = fi for i = 0, 1. (Here M−1 is obtained from M1 byreversing the orientation.) The structure set Shn(X) of X is the set of equiv-alence classes of orientation preserving homotopy equivalences M → X froma closed oriented manifold M of dimension n to X. This set has a preferredbase point, namely the class of the identity id : X → X.

Remark 7.113 (The simple structure set and s-cobordisms). If werequire in Definition 7.112 the homotopy equivalences F , f0 and f1 to besimple homotopy equivalences, we get the simple structure set Ssn(X) of Def-inition 7.111, provided that n ≥ 5. We have to show that the two equivalence


relations are the same. This follows from the s-Cobordism Theorem 2.44.Namely, W appearing in Definition 7.112 is an h-cobordism and is even ans-cobordism if we require F , f0 and f1 to be simple homotopy equivalences(see Theorem 2.34). Hence there is a diffeomorphism Φ : ∂0W × [0, 1] → Winducing the obvious identification ∂0W × 0 → ∂0W and some orienta-tion preserving diffeomorphism φ1 : (∂0W )− = (∂0W × 1)− → ∂1W . Thenφ : M0 →M1 given by g−1

1 φ1g0 is an orientation preserving diffeomorphismsuch that f1 φ is homotopic to f0. The other implication is obvious.

7.12.2 Realizability of Surgery Obstructions

In this section we explain that any element in the L-groups can be realized asthe surgery obstruction of a normal map (f, f) covering a map f : (M,∂M)→(Y, ∂Y ) if we allow M to have non-empty boundary ∂M .

Theorem 7.114 (Realizability of the surgery obstruction). Supposen ≥ 5. Consider a connected compact oriented manifold X possibly withboundary ∂X. Let π be its fundamental group and let w : π → ±1 be itsorientation homomorphism. Consider an element x ∈ Ln(Zπ,w).

Then we can find a normal map of degree one (f, f)covering a map oftriads

f = (f ; ∂0f, ∂1f) : (M ; ∂0M,∂1M)→ (X×[0, 1], X×0∪∂X×[0, 1], X×1)

with the following properties:

(i) ∂0f is a diffeomorphism and f |∂0M is a stabilization of T (∂f0);(ii) ∂1f is a homotopy equivalence;

(iii) The surgery obstruction σ(f, f) in Ln(Zπ,w) (see Definition 7.71 and (7.83))is the given element x.

The analogous statement holds for x ∈ Lsn(Zπ,w) if we require ∂1f to be asimple homotopy equivalence and we consider the simple surgery obstruction(see (7.95)).

Sketch of proof. We give at least the idea of the proof in the case n = 2k,more details can be found in[815, Theorem 5.8 on page 54, Theorem 6.5 onpage 68].

Recall that the element x ∈ Ln(Zπ,w) is represented by a non-degenerate(−1)k-quadratic form (P,ψ) with P a finitely generated free Zπ-module. Wefix such a representative and write it as a triple (P, µ, λ) as explained inRemark 7.20. Let b1, br, . . . , br be a Zπ-basis for P .

Choose r disjoint embeddings ji : D2k−1 → X − ∂X into the interior of X

for i = 1, 2, . . . , r. Let F 0 : Sk−1 ×Dk → D2k−1 be the standard embedding.Define embeddings F 0

i : Sk−1 ×Dk → X by the composition of the standard


embedding F 0 and ji for i = 1, 2, . . . , r. Let f0i : Sk−1 → X be the restriction

of Fi to Sk−1 × 0 ⊂ Sk−1 × Dk. Fix base points b ∈ X − ∂X and s ∈Sk−1 and paths wi from b to f0

i (s) for i = 1, 2, . . . , r. Thus each f0i is a

pointed immersion. Next we construct regular homotopies ηi : Sk−1 → X

from f0i to a new embedding f1

i which are modelled upon (P, µ, λ). Theseregular homotopies define associated immersions η′i : S

k−1 × [0, 1] → X ×[0, 1], (x, t) 7→ (ηi(x), t). Since these immersions η′i are embeddings on theboundary, we can define their intersection and selfintersection number asin (7.32) and (7.37).

We want to achieve that the intersection number of η′i and η′j for i < jis λ(bi, bj) and the selfintersection number µ(η′i) is µ(bi) for i = 1, 2, . . . , r.Notice that these numbers are additive under stacking regular homotopies to-gether. Hence it suffices to explain how to introduce a single intersection withvalue ±g between η′i and η′j for i < j or a selfintersection for η′i with value±g for i ∈ 1, 2, . . . , r without introducing other intersections or selfinter-sections. We explain the construction for introducing an intersection betweenη′i and η′j for i < j, the construction for a selfintersection is analogous. Thisconstruction will only change ηi, the other regular homotopies ηj for j 6= iwill be unchanged.

Join f0i (s) and f0

j (s) by a path v such that v is an embedding, does not

meet any of the f0i -s and the composition wj ∗ v ∗ w−i represents the given

element g ∈ π. Now there is an obvious regular homotopy from f0i to another

embedding along the path v inside a small tubular neighborhood N(v) ofv which leaves f0

i fixed outside a small neighborhood U(s) of s and movesf0i |U(s) within this small neighborhood N(v) along v very close to f1

i (s). Now

use a disc Dk which meets f0j transversally at the origin to move f0

i further,thus introducing the desired intersection with value ±g. Inside the disc themove looks like pushing the upper hemisphere of the boundary down to thelower hemisphere of the boundary, thus having no intersection with the originat any point of time with one exception. One has to check that there is enoughroom to realize both possible signs.

Since f1i is regularly homotopic to f0

i which is obtained from the trivialembeddings F 0

i , we can extend f1i to an embedding F 1

i : Sk−1 ×Dk → X −∂X. Now attach for each i ∈ 1, 2 . . . , r a handle (F 1

i ) = Dk × Dk toX × [0, 1] by F 1

i × 1 : Sk−1 × Dk → X × 1. Let M be the resultingmanifold. We obtain a manifold triad (M ; ∂0M,∂1M) if we put ∂0M = X ×0 ∪ ∂X × [0, 1] and ∂1M = ∂M − int(∂0X). There is an extension of theidentity id : X× [0, 1]→ X× [0, 1] to a map f : M → X× [0, 1] which inducesa map of triads (f ; ∂0f, ∂1f) such that ∂0f is a diffeomorphism. This mapis covered by a bundle map f which is given on ∂0M by the differential of∂0f . These claims follow from the fact that f1

i is regularly homotopic to f0i

and the f0i were trivial embeddings so that we can regard this construction as

surgery on the identity map X → X. The kernel Kk(M) = Kk(M, ˜X × [0, 1])has a preferred basis corresponding to the cores Dk × 0 ⊂ Dk × Dk of


the new handles (F 1i ). These cores can be completed to immersed spheres

Si by adjoining the images of the η′i in X × [0, 1] and finally the obviousdiscs Dk in X × 0 whose boundaries are given by f0

i (Sk−1). Then theintersection of Si with Sj is the same as the intersection of η′i with η′j andthe selfintersection of Si is the selfintersection of η′i with itself. Hence byconstruction λ(Si, Sj) = λ(bi, bj) for i < j and the selfintersection numberµ(S′i) is µ(bi) for i = 1, 2, . . . , r. This implies λ(Si, Sj) = λ(bi, bj) for i, j ∈1, 2, . . . , r. Comment 19: Add reference? The isomorphism P → P ∗

associated to the given non-degenerate (−1)k-quadratic form can be identified

with Kk(M) → Kk(M, ∂1M) if we use for Kk(M, ∂1M) the basis given by

the cores of the handles (F 1i ). Hence Kk(∂1M) = 0 and we conclude that

∂1f is a homotopy equivalence. By definition and construction σ(f, f) inLn(Zπ,w) is the class of the non-degenerated (−1)k-quadratic form (P, µ, λ).This finishes the outline of the proof of Theorem 7.114. ut

Remark 7.115 (Surgery obstructions of closed manifolds). It is nottrue that for any closed oriented manifold N of dimension n with fundamentalgroup π and orientation homomorphism w : π → ±1 and any element x ∈Ln(Zπ,w) there is a normal map (f, f) covering a map of closed orientedmanifolds f : M → N of degree one such that σ(f, f) = x. Notice that inTheorem 7.114 the target manifold X × [0, 1] is not closed. The same remarkholds for Lsn(Zπ,w). These questions are discussed in in [361, 365, 585, 586].

7.12.3 The Surgery Exact Sequence

Now we can establish one of the main tools in the classification of man-ifolds, the surgery exact sequence. We have already extended the notionof a normal map for closed manifolds to manifolds with boundary in Sec-tion 7.9. Below we will always assume that the maps on the boundaries arediffeomorphisms. We have already explained the notion of a normal bordism.We will call two normal maps with the same target (X, ∂X) normally bor-dant if there is a normal bordism between them whose target is the triad(X × [0, 1];X × 0, 1, ∂X × [0, 1]) and whose underlying map induces a dif-feomorphism ∂1W → ∂X × [0, 1]. This is more restrictive than we neededfor Theorem 7.86 (ii), where the target manifold was not necessarily of theshape (X × [0, 1];X × 0, 1, ∂X × [0, 1]) and could be some triad and werequired on the boundaries only homotopy equivalences and not necessarilydiffeomorphisms.

Definition 7.116. Let (X, ∂X) be a compact oriented manifold of dimensionn with boundary ∂X. Define the set of normal maps to (X, ∂X)

Nn(X, ∂X)


to be the set of normal bordism classes of normal maps of degree one (f, f)with underlying map (f, ∂f) : (M,∂M)→ (X, ∂X) for which ∂f : ∂M → ∂Xis a diffeomorphism.

Let X be a closed oriented connected manifold of dimension n ≥ 5. Denoteby π its fundamental group and by w : π → ±1 its orientation homomor-phism. Let Nn+1(X× [0, 1], X×0, 1) and Nn(X) be the set of normal mapsof degree one as introduced in Definition 7.116. Let Ssn(X) be the structureset of Definition 7.111. Denote by Lsn(Zπ) the simple surgery obstructiongroup (see Notation 7.94.) Denote by

σsn+1 : Nn+1(X × [0, 1], X × 0, 1)→ Lsn+1(Zπ,w);(7.117)

σsn : Nn(X)→ Lsn(Zπ,w)(7.118)

the maps which assign to the normal bordism class of a normal map of de-gree one its simple surgery obstruction (see (7.95)). This is well-defined byTheorem ii (7.86). Let

ηsn : Ssn(X)→ Nn(X)(7.119)

be the map which sends the class [f ] ∈ Ssn(X) represented by a simple ho-motopy equivalence f : M → X to the normal bordism class of the followingnormal map of degree one. Choose a homotopy inverse f−1 : X → M and ahomotopy h : idM ' f−1 f . Put ξ = (f−1)∗TM . Up to isotopy of bundlemaps there is precisely one bundle map (h, h) : TM × [0, 1] → TM cover-ing h : M × [0, 1] → M whose restriction to TM × 0 is the identity mapTM × 0 → TM . The restriction of h to X × 1 induces a bundle mapf : TM → ξ covering f : M → X. Put η([f ]) := [(f, f)]. One easily checksthat the normal bordism class of (f, f) depends only on [f ] ∈ Ssn(X) andhence that η is well-defined.

Next we define an action of the abelian group Lsn+1(Zπ,w) on the structureset Ssn(X)

ρsn : Lsn+1(Zπ,w)× Ssn(X)→ Ssn(X).(7.120)

Fix x ∈ Lsn+1(Zπ,w) and [f ] ∈ Ssn(X) represented by a simple homotopy

equivalence f : M → X. By Theorem 7.114 we can find a normal map (F , F )covering a map of triads (F ; ∂0F, ∂1F ) : (W ; ∂0W,∂1W ) → (M × [0, 1],M ×0,M × 1) such that ∂0F is a diffeomorphism and ∂1F is a simple ho-motopy equivalence and σ(F , F ) = x. Then define ρsn(x, [f ]) by the class[f ∂1F : ∂1W → X]. We have to show that this is independent of the choiceof (F , F ). Let (F ′, F ′) be a second choice. We can glue W ′ and W− togetheralong the diffeomorphism (∂0F )−1∂0F

′ : ∂0W′ → ∂0W and obtain a normal

bordism from (F |∂1W , ∂1F ) to (F ′|∂1W ′ , ∂1F′). The simple obstruction of this

normal bordism is


σ(F ′, F ′)− σ(F , F ) = x− x = 0.

Because of Theorem 7.99 (ii) we can perform surgery relative boundary onthis normal bordism to arrange that the reference map from it to X × [0, 1]is a simple homotopy equivalence. In view of Remark 7.113 this shows thatf ∂1F and f ∂1F

′ define the same element in Ssn(X). One easily checksthat this defines a group action since the surgery obstruction is additive understacking normal bordisms together. The next result is the main result of thischapter and follows from the definitions and Theorem 7.99 (ii).

Theorem 7.121 (The surgery Exact Sequence). Under the conditionsand in the notation above the so called surgery sequence

Nn+1(X × [0, 1], X × 0, 1)σsn+1−−−→ Lsn+1(Zπ,w)

∂sn+1−−−→ Ssn(X)

ηsn−→ Nn(X)σsn−−→ Lsn(Zπ,w)

is exact for n ≥ 5 in the following sense. An element z ∈ Nn(X) lies inthe image of ηsn if and only if σsn(z) = 0. Two elements y1, y2 ∈ Ssn(X)have the same image under ηsn if and only if there exists an element x ∈Lsn+1(Zπ,w) with ρsn(x, y1) = y2. For two elements x1, x2 in Lsn+1(Zπ) wehave ρsn(x1, [id : X → X]) = ρsn(x2, [id : X → X]) if and only if there isu ∈ Nn+1(X × [0, 1], X × 0, 1) with σsn+1(u) = x1 − x2.

There is an analogous surgery exact sequence

N hn+1(X × [0, 1], X × 0, 1)

σhn+1−−−→ Lhn+1(Zπ,w)∂hn+1−−−→ Sh(X)

ηhn−→ Nn(X)σhn−−→ Lhn(Zπ,w)

where Sh(X) is the structure set of Definition 7.112 and Lhn(Zπ,w) :=Ln(Zπ,w) have been introduced in Definitions 7.24 and 7.79.

Remark 7.122 (Extending the surgery exact sequence to the left).The surgery sequence of Theorem 7.121 can be extended to infinity to theleft. In the range far enough to the left it is a sequence of abelian groups.

7.13 Surgery Theory in the PL and in the TopologicalCategory

One can also develop surgery theory in the PL-category or in the topologi-cal category [473]. This requires to carry over the notions of vector bundlesand tangent bundles to these categories. There are analogs of the sets ofnormal invariants NPL

n (X) and N TOPn (X) and the structure sets SPL,hn (X),

SPL,sn (X) STOP,hn (X) and STOP,sn (X). There are analogs PL and TOP of

7.13 Surgery Theory in the PL and in the Topological Category 211

the group O = colimn→∞O(n). Let G = colimn→∞G(n), where G(n) is themonoid of self homotopy equivalences of Sn.

Theorem 7.123 (The set of normal maps and G/O, G/PL and G/TOP ).Let X be a connected finite oriented n-dimensional Poincare complex. Sup-

pose that Nn(X), NPLn (X), or N TOP

n (X) respectively, is non-empty.Then there is a canonical group structure on the set [X,G/O] [X,G/PL],

or [X,G/TOP ] respectively, of homotopy classes of maps from X to G/PLand a transitive free operation Nn(X), NPL

n (X), or N TOPn (X) respectively.

In particular we get bijections

[X,G/O]∼=−→ Nn(X);

[X,G/PL]∼=−→ NPL

n (X);

[X,G/TOP ]∼=−→ N TOP

n (X);

respectively.

There are analogs of the surgery exact sequence, see Theorem 7.121, forthe PL-category and the topological category.

Theorem 7.124 (The Surgery Exact Sequence). There is a surgerysequence

NPLn+1(X × [0, 1], X × 0, 1)

σsn+1−−−→ Lsn+1(Zπ,w)∂sn+1−−−→ SPL,sn (X)

ηsn−→ NPLn (X)

σsn−−→ Lsn(Zπ,w)

which is exact for n ≥ 5 in the sense of Theorem 7.121. There is an analogoussurgery exact sequence

NPLn+1(X × [0, 1], X × 0, 1)

σhn+1−−−→ Lhn+1(Zπ,w)∂hn+1−−−→ SPL,hn (X)

ηhn−→ NPLn (X)

σhn−−→ Lhn(Zπ,w)

The analogous sequences exists in the topological category, namely there is asurgery sequence

N TOPn+1 (X × [0, 1], X × 0, 1)

σsn+1−−−→ Lsn+1(Zπ,w)∂sn+1−−−→ STOP,sn (X)

ηsn−→ N TOPn (X)

σsn−−→ Lsn(Zπ,w)

which is exact for n ≥ 5 in the sense of Theorem 7.121, and an analogoussurgery exact sequence


N TOPn+1 (X × [0, 1], X × 0, 1)

σhn+1−−−→ Lhn+1(Zπ,w)∂hn+1−−−→ STOP,hn (X)

ηhn−→ N TOPn (X)

σhn−−→ Lhn(Zπ,w)

Notice that the surgery obstruction groups are the same in the smoothcategory, PL-category and in the topological category. Only the set of normalinvariants and the structure sets are different. The set of normal invariants inthe smooth category, PL-category or topological category do not depend onthe decoration h and s, whereas the structure sets and the surgery obstructiongroups depend on the decoration h and s. In particular the structure setdepends on both the choice of category and choice of decoration.

As in the smooth setting the surgery sequence above can be extended toinfinity to the left.

Some interesting constructions can be carried out in the topological cate-gory which do not have smooth counterparts.

Remark 7.125 (The total surgery obstruction). Given a finite Poincarecomplex X of dimension ≥ 5, a single obstruction, the so called total surgeryobstruction, is constructed in [685, §17]. It vanishes if and only if X is homo-topy equivalent to a closed topological manifold. It combines the two stagesof the classical obstructions, namely, the problem whether the Spivak normalfibration has a reduction to a TOP -bundle (which is equivalent to the con-dition that N TOP (X) is non-empty) and whether the surgery obstruction ofthe associated normal map is trivial.

Remark 7.126 (Group structures on the surgery sequence). An alge-braic surgery sequence is constructed in [685, §14, §18] and identified with thegeometric surgery sequence above in the topological category. Moreover, inthe topological situation one can find abelian group structures on STOP,sn (X),STOP,hn (X) and N TOP

n (X) such that the surgery sequence becomes a se-quence of abelian groups. The main point is to find the right addition onG/TOP .

There cannot be a group structure known for Shn(X) and Nn(X) such

that Shn(X)η−→ Nn(X)

σ−→ Lhn(Zπ,w) is a sequence of groups (and analogousfor the simple version), see Crowley [205]. Notice that the composition, seeTheorem 7.123

[X;G/O] ∼= Nn(X)σsn−−→ Lsn(Zπ,w)

is a map whose source and target come with canonical group structures butit is not a homomorphism of abelian groups in general. These difficulties areconnected to the existence of homotopy spheres. The same problem ariseswith the decoration h. Comment 20: Add reference. For instance [815,page 114].

Remark 7.127 (The homotopy type of G/O and G/PL). The computa-tion of the homotopy type of the space G/O and G/PL due to Sullivan [770]is explained in detail in [568, Chapter 4]. One obtains homotopy equivalences

7.14 The Novikov Conjecture 213

G/TOP

[1

2

]' BO

[1

2

];

G/TOP(2) '∏j≥1

K(Z(2), 4j)×∏j≥1

K(Z/2, 4j − 2),

where K(A; i) denotes the Eilenberg-MacLane space of type (A, i), i.e., aCW -complex such that πn(K(A, i)) is trivial for n 6= i and is isomorphic toA if n = i, (2) stands for localizing at (2), i.e., all primes except 2 are inverted,

and[

12

]stands for localization of 2, i.e. 2 is inverted. In particular we get for

a space X isomorphisms

[X,G/TOP ]

[1

2

]∼= KO

0(X)

[1

2

];

[X,G/TOP ](2)∼=∏j≥1

H4j(X;Z(2))×∏j≥1

H4j−2(M ;Z/2),

where KO∗ is K-theory of real vector bundles, see Subsection 8.2.2.

The various groups G, TOP , PL and their (homotopy theoretic) quotientsG/PL, PL/O and G/PL fit into a braid by inspecting long exact sequences offibrations. This braid can be interpreted geometrically in terms of L-groups,bordism groups, and homotopy groups of exotic spheres in dimensions ≥ 5,see for instance Crowley-Macko-Luck [206, Chapter 11]. Comment 21:Make reference more precise when the book is finished.

Kirby and Siebenmann [473, Theorem 5.5 in Essay V on page 251] (seealso [728]) have proved

Theorem 7.128. The space TOP/PL is an Eilenberg MacLane space of type(Z/2, 3).

7.14 The Novikov Conjecture

In this section we introduce the Novikov Conjecture in its original form interms of higher signatures and make a first link to surgery theory. It followsfrom both the Baum-Connes Conjecture and the Farrell-Jones Conjectureand has been an important interface between topology and non-commutativegeometry.

7.14.1 The Original Formulation of the Novikov Conjecture

Let G be a (discrete) group. Let u : M → BG be a map from a closed orientedsmooth manifold M to BG. Let


L(M) ∈⊕

k∈Z,k≥0

H4k(M ;Q)(7.129)

be the L-class of M . Its k-th entry L(M)k ∈ H4k(M ;Q) is a certain homo-geneous polynomial of degree k in the rational Pontrjagin classes pi(M ;Q) ∈H4i(M ;Q) for i = 1, 2, . . . , k such that the coefficient sk of the monomialpk(M ;Q) is different from zero. It is defined in terms of multiplicative se-quences, see for instance [594, § 19]. We mention at least the first values

L(M)1 =1

3· p1(M ;Q);

L(M)2 =1

45·(7 · p2(M ;Q)− p1(M ;Q)2

);

L(M)3 =1

945·(62 · p3(M ;Q)− 13 · p1(M ;Q) ∪ p2(M ;Q) + 2 · p1(M ;Q)3

).

The L-class L(M) is determined by all the rational Pontrjagin classes and viceversa. Recall that the k-th rational Pontrjagin class pk(M,Q) ∈ H4k(M ;Q)is defined as the image of k-th Pontrjagin class pk(M) under the obviouschange of coefficients map H4k(M ;Z) → H4k(M ;Q). The L-class dependson the tangent bundle and thus on the differentiable structure of M . Forx ∈

∏k≥0H

k(BG;Q) define the higher signature of M associated to x andu to be

signx(M,u) := 〈L(M) ∪ u∗x, [M ]〉 ∈ Q.(7.130)

where [M ] denotes the fundamental class of a closed oriented d-dimensionalmanifold M in Hd(M ;Z) or its image under the change of coefficients mapHd(M ;Z)→ Hd(M ;Q). Recall that for dim(M) = 4n the signature sign(M)of M is the signature of the non-degenerate bilinear symmetric pairing onthe middle cohomology H2n(M ;R) given by the intersection pairing (a, b) 7→〈a∪ b, [M ]〉. Obviously sign(M) depends only on the oriented homotopy typeof M . We say that signx for x ∈ H∗(BG;Q) is homotopy invariant if for twoclosed oriented smooth manifolds M and N with reference maps u : M → BGand v : N → BG we have

signx(M,u) = signx(N, v),

whenever there is an orientation preserving homotopy equivalence f : M → Nsuch that v f and u are homotopic.

Conjecture 7.131 (Novikov Conjecture). The group G satisfies theNovikov Conjecture if signx is homotopy invariant for all x ∈

∏k∈Z,k≥0H

k(BG;Q).

This conjecture appears for the first time in the paper by Novikov [623,§ 11]. A survey about its history can be found in [311].


7.14.2 Invariance Properties of the L-Class

One motivation for the Novikov Conjecture comes from the Signature Theo-rem due to Hirzebruch [399, 400].

Theorem 7.132 (Signature Theorem). Let M be an oriented closed ma-nifold of dimension n. Then the higher signature sign1(M,u) = 〈L(M), [M ]〉associated to 1 ∈ H0(M) and some map u : M → BG coincides with the sig-nature sign(M) of M , if dim(M) = 4n, and is zero, if dim(M) is not divisibleby four.

The Signature Theorem 7.132 leads to the question, whether the Pontrja-gin classes or the L-classes are homotopy invariants or homeomorphism in-variants. They are obviously invariants of the diffeomorphism type. However,the Pontrjagin classes pk(M) ∈ H4k(M ;Z) for k ≥ 2 are not homeomorphisminvariants, see for instance [477, Theorem 4.8 on page 31]. On the other hand,there is the following deep result due to Novikov [620, 621, 622].

Theorem 7.133 (Topological invariance of rational Pontrjagin classes).The rational Pontrjagin classes pk(M,Q) ∈ H4k(M ;Q) are topological in-

variants, i.e., for a homeomorphism f : M → N of closed smooth manifoldswe have

H4k(f ;Q)(pk(N ;Q)) = pk(M ;Q)

for all k ≥ 0 and in particular H∗(f ;Q)(L(N)) = L(M).

Example 7.134 (The L-class is not a homotopy invariant). The ra-tional Pontrjagin classes and the L-class are not homotopy invariants as thefollowing example shows. There exists for k ≥ 1 and large enough j ≥ 0 a(j+1)-dimensional vector bundle ξ : E → S4k with Riemannian metric whosek-th Pontrjagin class pk(ξ) is not zero and which is trivial as a fibration. Thetotal space SE of the associated sphere bundle is a closed (4k+j)-dimensionalmanifold which is homotopy equivalent to S4k × Sj and satisfies

pk(SE) = −pk(ξ) 6= 0;

L(SE)k = sk · pk(SE) 6= 0,

where sk 6= 0 is the coefficient of pk in the polynomial defining the L-class. Butpk(S4k×Sj) and L(S4k×Sj)k vanish since the tangent bundle of S4k×Sj isstably trivial. In particular SE and S4k×Sj are simply-connected homotopyequivalent closed manifolds, which are not homeomorphic. This example istaken from [687, Proposition 2.9] and attributed to Dold and Milnor there.See also [687, Proposition2.10] or [594, Section 20].

Remark 7.135 (The homological version of the Novikov Conjec-ture). One may understand the Novikov Conjecture as an attempt to figureout how much of the L-class is a homotopy invariant of M . If one considers the


oriented homotopy type and the simply-connected case, it is just the expres-sion 〈L(M), [M ]〉 or, equivalently, the top component of L(M). In the NovikovConjecture one asks the same question but now taking the fundamental groupinto account by remembering the classifying map uM : M → Bπ1(M), or,more generally, a reference map u : M → BG. The Novikov Conjecture canalso be rephrased by saying that for a given group G each pair (M,u) consist-ing of an oriented closed manifold M of dimension n together with a referencemap u : M → BG the term

u∗(L(M) ∩ [M ]) ∈⊕k∈Z

Hn+4k(BG;Q)

depends only on the oriented homotopy type of the pair (M,u). This followsfrom the elementary computation for x ∈ H∗(BG;Q)

〈L(M) ∪ u∗x, [M ]〉 = 〈u∗x,L(M) ∩ [M ]〉 = 〈x, u∗(L(M) ∩ [M ])〉.

and the fact that the Kronecker product 〈−,−〉 for rational coefficients isnon-degenerate. Notice that − ∩ [M ] : Hn−i(M ;Q) → Hi(M ;Q) is an iso-morphism for all i ≥ 0 by Poincare duality. Hence L(M) ∩ [M ] carries thesame information as L(M).

Exercise 7.136. Let f : M → N be an orientation preserving homotopyequivalence of oriented closed manifolds which are aspherical. Assume thatthe Novikov Conjecture 7.131 holds for G = π1(M). Show that then L(M) =f∗L(N) must be true.

7.14.3 The Novikov Conjecture and surgery theory

Remark 7.137 (The Novikov Conjecture and assembly map). Theexists an assembly map

asmbGn :⊕k∈Z

Hn+4k(BG;Q)→ Lhn(ZG)⊗Z Q,

which fits into the following commutative diagram


STOP,hn (M)ηhn //

s

""

N TOPn (M)

σsn //

b

Lhn(Zπ1(M))

u∗

[M,G/TOP ]

c

Lhn(ZG)

i

⊕k∈ZHn+4k(BG;Q)

asmbGn // Lhn(ZG)⊗Z Q

The map i is the obvious inclusion and u∗ is the homomorphism coming fromπ1(u) : π1(M)→ π1(BG) = G. The bijection b is taken from Theorem 7.123and c comes from the rational version of the homotopy equivalences describingG/TOP appearing in Remark 7.127. The composite c b sends the class ofa normal map (f, f) with underlying map f : N → M of degree one to(u f)∗(L(N)∩ [N ])− u∗(L(M)∩ [M ]). The map s is defined analogously, itsends the class [f ] of an orientation preserving homotopy equivalence f : N →M to the difference (u f)u∗(L(N) ∩ [N ]) − u∗(L(M) ∩ [M ]). We concludefrom Remark 7.135 that the Novikov Conjecture 7.131 is equivalent to thestatement that s is trivial. The upper row is part of the surgery exact sequenceof Theorem 7.124. This implies that the composite

STOP,hn (M)s−→⊕k∈Z

Hn+4k(BG;Q)asmbGn−−−−→ Lhn(ZG)⊗Z Q

is trivial.Thus we can conclude that the group G satisfies the Novikov Conjec-

ture 7.131 if the map asmbGn :⊕

k∈ZHn+4k(BG;Q) → Lhn(ZG) ⊗Z Q is in-jective. See also Kaminker-Miller [437] for a proof. Notice that the last mapinvolves only G. This conclusion will be a key ingredient in the proof thatthe L-theoretic Farrell-Jones Conjecture for G implies the Novikov Conjec-ture 7.131, see Theorem 11.52 (x).

Remark 7.138 (The converse of the Novikov Conjecture). A kindof converse to the Novikov Conjecture 7.131 is the following result. Let Nbe a closed connected oriented smooth manifold of dimension n ≥ 5. Letu : N → BG be a map inducing an isomorphism on the fundamental groups.Consider any element l ∈

∏i≥0H

4i(N ;Q) such that u∗(l ∩ [N ]) = 0 holdsin H∗(BG;Q). Then there exists a non-negative integer K such that for anymultiple k of K there is a homotopy equivalence f : M → N of closed orientedsmooth manifolds satisfying

f∗(L(N) + k · l) = L(M).

A proof can be found for instance in [219, Theorem 6.5]. This shows thatthe top dimension part of the L-class L(M) is essentially the only homo-


topy invariant rational characteristic class for simply-connected closed 4k-dimensional manifolds.

More information about the Novikov Conjecture can be found for instancein [312, 313, 477, 709]. An algebraic geometric and an equivariant version ofthe Novikov Conjecture is introduced in [707] and [714].

7.15 Topologically Rigidity and the Borel Conjecture

In this section we deal with the Borel Conjecture and how it follows from theFarrell-Jones Conjecture in dimensions ≥ 5.

7.15.1 Aspherical Spaces

Definition 7.139 (Aspherical). A space X is called aspherical if it is pathconnected and all its higher homotopy groups vanish, i.e., πn(X) is trivial forn ≥ 2.

Remark 7.140 (Homotopy classification of aspherical CW -complexes).A CW complex is aspherical if and only if it is connected and its universalcovering is contractible. Given two aspherical CW -complexes X and Y , themap from the set of homotopy classes of maps X → Y to the set of group ho-momorphisms π1(X) → π1(Y ) modulo inner automorphisms of π1(Y ) givenby the map induced on the fundamental groups is a bijection. In particular,two aspherical CW -complexes are homotopy equivalent if and only if theyhave isomorphic fundamental groups and every isomorphism between theirfundamental groups comes from a homotopy equivalence.

Remark 7.141 (Classifying space of a group). An aspherical CW -complex X with fundamental group π is the same as an Eilenberg Mac-Lanespace K(π, 1) of type (π, 1) and the same as the classifying space Bπ for thegroup π.

Exercise 7.142. Let F → E → B be a fibration. Suppose that F and B areaspherical. Show that then E is aspherical.

Exercise 7.143. Let X be an aspherical CW -complex of finite dimension.Show that π1(X) is torsionfree.

Example 7.144 (Examples of aspherical manifolds).

7.15 Topologically Rigidity and the Borel Conjecture 219

(i) A connected closed 1-dimensional manifold is homeomorphic to S1 andhence aspherical;

(ii) Let M be a connected closed 2-dimensional manifold. Then M is eitheraspherical or homeomorphic to S2 or RP2;

(iii) A connected closed 3-manifold M is called prime if for any decompositionas a connected sum M ∼= M0]M1 one of the summands M0 or M1 is home-omorphic to S3. It is called irreducible if any embedded sphere S2 boundsa disk D3. Every irreducible closed 3-manifold is prime. A prime closed3-manifold is either irreducible or an S2-bundle over S1. The followingstatements are equivalent for a closed 3-manifold M :

• M is aspherical;• M is irreducible and its fundamental group is infinite and contains no

element of order 2;• The fundamental group π1(M) cannot be written in a non-trivial way

as a free amalgamated product of two groups, is infinite, different fromZ, and contains no element of order 2.

• The universal covering of M is homeomorphic to R3.

(iv) Let L be a Lie group with finitely many path components. Let K ⊆ L be amaximal compact subgroup. Let G ⊆ L be a discrete torsionfree subgroup.Then M = G\L/K is a closed aspherical manifold with fundamental groupG since its universal covering L/K is diffeomorphic to Rn for appropriaten;

(v) Every closed Riemannian manifold with non-positive sectional curvaturehas a universal covering which is diffeomorphic to Rn and is in particularaspherical.

Exercise 7.145. Classify all simply connected aspherical closed manifolds.

Exercise 7.146. Suppose that M is a connected sum M1]M2 of two closedmanifolds M1 and M2 of dimension n ≥ 3 which are not homotopy equivalentto a sphere. Show that M is not aspherical.

There exists exotic aspherical manifolds as the following results illustrate.The following theorem is due to Davis-Januszkiewicz [232, Theorem 5a.4].

Theorem 7.147 (Non-PL-example). For every n ≥ 4 there exists a closedaspherical n-manifold which is not homotopy equivalent to a PL-manifold

The following result is proved by Davis-Fowler-Lafont [231] using the workof Manolescu [573, 572].

Theorem 7.148 (Non-triangulable aspherical closed manifolds). Thereexists for each n ≥ 6 an n-dimensional aspherical closed manifold which can-not be triangulated.


The proof of the following theorem can be found in [229], [232, Theo-rem 5b.1].

Theorem 7.149 (Exotic universal covering of aspherical closed man-ifolds). For each n ≥ 4 there exists a closed aspherical n-dimensional ma-nifold such that its universal covering is not homeomorphic to Rn.

By the Hadamard-Cartan Theorem (see [336, 3.87 on page 134]) the mani-fold appearing in Theorem 7.149 above cannot be homeomorphic to a smoothmanifold with Riemannian metric with non-positive sectional curvature.

More information about fundamental groups of closed aspherical manifoldswith unusual properties we refer for instance to [730].

The question when the isometry group of the universal covering of anaspherical closed manifold is non-discrete is studied by Farb-Weinberger [282].

Remark 7.150 (S1-actions on aspherical closed manifolds). If S1 actson a closed aspherical manifold, then the orbit circle is a non-trivial ele-ment in the center by a result of Borel, see for instance [191, Lemma 5.1 onpage 242]. Comment 22: Take a look at the reference. There is the con-jecture of Conner-Raymond [191, page 229] Comment 23: Take a look atthe reference. stating that the converse is true, namely, if the fundamentalgroup of an aspherical manifold has nontrivial center, then the manifold hasa circle action, such that the orbit circle is a nontrivial central element of thefundamental group. A counterexample in dimensions ≥ 6 was constructed byCappell-Weinberger-Yan [158].

For more information about aspherical spaces we refer for instance to [537].

7.15.2 Formulation and Relevance of the Borel Conjecture

Definition 7.151 (Topologically rigid). We call a closed (topological) ma-nifoldN topologically rigid if any homotopy equivalenceM → N with a closedmanifold M as source is homotopic to a homeomorphism.

Conjecture 7.152 (Borel Conjecture (for a group G in dimensionn)). The Borel Conjecture for a group G in dimension n predicts for twoaspherical closed topological manifolds M and N of dimensions n withπ1(M) ∼= π1(N) ∼= G, that M and N are homeomorphic and any homo-topy equivalence M → N is homotopic to a homeomorphism.

The Borel Conjecture says that every aspherical closed topological mani-fold is topologically rigid.

Remark 7.153 (The Borel Conjecture in low dimensions). The BorelConjecture is true in dimension ≤ 2. It is true in dimension 3 if Thurston’sGeometrization Conjecture is true. This follows from results of Waldhausen


(see Hempel [380, Lemma 10.1 and Corollary 13.7]) and Turaev (see [792]) asexplained for instance in [478, Section 5]. A proof of Thurston’s Geometriza-tion Conjecture is given in [474, 613] following ideas of Perelman. Some in-formation in dimension 4 can be found in Davis [220].

Remark 7.154 (Topological rigidity for non-aspherical manifolds).Topological rigidity phenomenons do hold also for some non-aspherical closedmanifolds. For instance the sphere Sn is topologically rigid by the PoincareConjecture. The Poincare Conjecture is known to be true in all dimensions.This follows in high dimensions from the h-cobordism theorem, in dimensionfour from the work of Freedman [328], in dimension three from the workof Perelman as explained in [474, 612] and and in dimension two from theclassification of surfaces.

Many more examples of classes of manifolds which are topologically rigidare given and analyzed in Kreck-Luck [478]. For instance, the connected sumof closed manifolds of dimension ≥ 5 which are topologically rigid and whosefundamental groups do not contain elements of order two, is again topologi-cally rigid. The product Sk×Sn is topologically rigid if and only if k and n areodd. An integral homology sphere of dimension n ≥ 5 is topologically rigidif and only if the inclusion Z → Z[π1(M)] induces an isomorphism of sim-ple L-groups Lsn+1(Z)→ Lsn+1

(Z[π1(M)]

). Every 3-manifold with torsionfree

fundamental group is topologically rigid.

Exercise 7.155. Give an example of a closed orientable 3-manifold with fi-nite fundamental group which is not topologically rigid.

Exercise 7.156. Give an example of two topologically rigid orientable smoothclosed manifolds whose cartesian product is not topologically rigid.

Remark 7.157 (The Borel Conjecture does not hold in the smoothcategory). The Borel Conjecture 7.152 is false in the smooth category, i.e.,if one replaces topological manifold by smooth manifold and homeomorphismby diffeomorphism. The torus Tn for n ≥ 5 is an example (see [815, 15A]).Other counterexample involving negatively curved manifolds are constructedby Farrell-Jones [292, Theorem 0.1].

Remark 7.158 (The Borel Conjecture versus Mostow rigidity). Theexamples of Farrell-Jones [292, Theorem 0.1] give actually more. Namely, ityields for given ε > 0 a closed Riemannian manifold M0 whose sectionalcurvature lies in the interval [1− ε,−1 + ε] and a closed hyperbolic manifoldM1 such that M0 and M1 are homeomorphic but no diffeomorphic. The ideaof the construction is essentially to take the connected sum of M1 with exoticspheres. Notice that by definition M0 were hyperbolic if we would take ε = 0.Hence this example is remarkable in view of Mostow rigidity, which predictsfor two closed hyperbolic manifolds N0 and N1 that they are isometrically


diffeomorphic if and only if π1(N0) ∼= π1(N1) and any homotopy equivalenceN0 → N1 is homotopic to an isometric diffeomorphism.

One may view the Borel Conjecture as the topological version of Mostowrigidity. The conclusion in the Borel Conjecture is weaker, one gets only home-omorphisms and not isometric diffeomorphisms, but the assumption is alsoweaker, since there are many more aspherical closed topological manifoldsthan hyperbolic closed manifolds.

Remark 7.159 (The work of Farrell-Jones). Farrell-Jones have madedeep contributions to the Borel Conjecture. They have proved it in dimen-sion ≥ 5 for non-positively curved closed Riemannian manifolds, for com-pact complete affine flat manifolds and for closed aspherical manifolds whosefundamental group is isomorphic to the fundamental group of a completenon-positively curved Riemannian manifold which is A-regular. Relevant ref-erences are [293, 294, 296, 298, 299].

The Borel Conjecture for higher dimensional graph manifolds is studiedby Frigerio-Lafont-Sisto [333].

7.15.3 The Farrell-Jones and the Borel Conjecture

Theorem 7.160 (The Farrell-Jones and the Borel Conjecture). LetG be a finitely presented group. Suppose that it satisfies the versions of the K-theoretic Farrell-Jones Conjecture stated in 2.107 and 3.19 and the version ofthe L-theoretic Farrell-Jones Conjecture stated in 7.109 for the ring R = Z.

Then every closed aspherical manifold of dimension ≥ 5 with G as funda-mental group is topologically rigid, in other words, the Borel Conjecture 7.152holds for G in dimensions ≥ 5.

For its proof we need the following lemma.

Lemma 7.161. Let M be a closed topological manifold. Suppose Wh(π1(M)) =0. Then M is topologically rigid if and only if the simple topological structureset STOP,s(M) consists of precisely on element, namely the class of idM .

Proof. Suppose that M is topologically rigid. Consider any element in η ∈STOP,s(M). Choose a simple homotopy equivalence f : N →M representingη. SinceM is topologically rigid, f is homotopic to a homeomorphism h : N →M . Hence idM h ' f . This implies that η is represented by idM .

Suppose that STOP,s(M) consists only of one class, the one represented byidM . Consider any homotopy equivalence f : N →M . Since Wh(π1(M)) = 0holds by assumption, f is a simple homotopy equivalence and thus representsan element in STOP,s(M). Since it represents the same class as idM by as-sumption, there exists a homeomorphism h : N → M such that h = idM his homotopic to f . ut


Lemma 7.162. Let M be a closed topological manifold of dimension n ≥ 5.Let w : π := π1(M) → ±1 be given by its first Stiefel-Whitney class. Sup-pose Wh(π1(M)) = 0. Assume that the homomorphism of abelian groupsσsn+1 : N TOP

n+1 (M × [0, 1],M × 0, 1) → Lsn+1(Zπ,w) of (7.117) is surjec-tive and that the preimage of 0 under the map σsn : NPL

n (X) → Lsn(Zπ,w)of (7.118) consists of one point, where these maps are taken from the simpletopological exact surgery sequence.

Then M is topologically rigid.

Proof. This follows from the simple topological surgery exact sequence ofTheorem 7.123 and Lemma 7.161. ut

Now we can give a sketch of the proof of Theorem 7.160.

Sketch of the proof of Theorem 7.160. We deal for simplicity with the ori-entable case only. Let L〈−∞〉(Z) the L-theory spectrum appearing in theversion of the L-theoretic Farrell-Jones Conjecture 7.109. Since it holds byassumption, the so called assembly map

asmb〈−∞〉k : Hk(Bπ; L〈−∞〉(Z))→ L

〈−∞〉k (Zπ)

is bijective for all k. Let L〈−∞〉(Z)〈1〉 be the 1-connected cover of L〈−∞〉(Z).This spectrum comes with a map of spectra i : L〈−∞〉(Z)〈1〉 → L〈−∞〉(Z)such that πk(i) is bijective for k ≥ 1 and πk(L〈−∞〉(Z)〈1〉) = 0 for k ≤ 0. Fork ≥ 1 there is a connective version of the assembly map asmbk above

asmb〈−∞〉k 〈1〉 : Hk

(Bπ; L〈−∞〉(Z)〈1〉

)→ L

〈−∞〉k (Zπ)

such that πk(i) asmbk〈1〉 = asmbk holds. A comparison argument of theAtiyah-Hirzebruch spectral sequence shows that the bijectivity of asmbk 〈−∞〉for k = n, n + 1 implies that asmb

〈−∞〉n+1 〈1〉 is bijective and in particular sur-

jective and asmb〈−∞〉n 〈1〉 is injective, if n is the dimension of the aspher-ical closed manifold under consideration. Because by assumption Conjec-tures 2.107 and 3.19 hold for π, one concludes from the various Rothenbergsequences of Subsection 7.10.4 that the simple versions of the 1-connectiveassembly maps

asmbsk〈1〉 : Hk

(Bπ; Ls(Z)〈1〉

)→ Lsk(Zπ)

agree with the maps asmb〈−∞〉k 〈1〉. One can identify the map asmbsn+1〈1〉 with

the map σsn+1 : N TOPn+1 (M× [0, 1],M×0, 1)→ Lsn+1(Zπ) of (7.117) and the

map asmbsn〈1〉 with the map σsn : NPLn (X) → Lsn(Zπ) of (7.118), see [685,

Theorem 18.5 on page 198], [680], [485] using Remark 16.14, Remark 16.15,and Example 16.19.

Now Theorem 7.160 follows from Lemma 7.162. ut


Remark 7.163 (Dimension 4). The conclusion of Theorem 7.160 hold alsoin dimension 4 provided that the fundamental group is good in the senseof Freedman (see [328, 329]). Groups of subexponential growth are good,see [330, Theorem 0.1].

7.16 Homotopy Spheres

A smooth closed oriented manifold is called a homotopy sphere if it is ho-motopy equivalent to the standard sphere. By the Poincare Conjecture ahomotopy sphere is always homeomorphic to a standard sphere and actuallytopologically rigid. However, it does need to be diffeomorphic to a standardsphere and in this case it is called an exotic homotopy sphere.

The classification of homotopy spheres due to Kervaire-Milnor[469] marksthe beginning of surgery theory. In order to understand the surgery machineryand in particular the long exact surgery sequence, we recommend to thereader to study the case homotopy spheres which boils down to computeSsn(Sn). Moreover, there are some beautiful constructions of exotic spheresand results about the curvature properties of Riemannian metric on an exoticsphere. We refer for instance to the following survey articles [423], [497], [509],and [525, Chapter 6].

7.17 Poincare Duality Groups

The following definition is due to Johnson-Wall [424].

Definition 7.164 (Poincare duality group).A group G is called a Poincare duality group of dimension n if the following

conditions holds:

(i) The group G is of type FP, i.e., the trivial ZG-module Z possesses a finitedimensional projective ZG-resolution by finitely generated projective ZG-modules;

(ii) We get an isomorphism of abelian groups

Hi(G;ZG) ∼=0 for i 6= n;Z for i = n.

The next definition is due to Wall [814]. Recall that a CW -complex Xis called finitely dominated if there exists a finite CW -complex Y and mapsi : X → Y and r : Y → X with r i ' idX .

A topological space X is called an absolute neighborhood retract or brieflyANR if for every normal space Z, every closed subset Y ⊆ Z and every

7.17 Poincare Duality Groups 225

(continuous) map f : Y → X there exists an open neighborhood U of Y in Ztogether with an extension F : U → X of f to U . A compact n-dimensionalhomology ANR-manifold X is a compact absolute neighborhood retract suchthat it has a countable basis for its topology, has finite topological dimensionand for every x ∈ X the abelian group Hi(X,X − x) is trivial for i 6= nand infinite cyclic for i = n. A closed n-dimensional topological manifold isan example of a compact n-dimensional homology ANR-manifold (see [217,Corollary 1A in V.26 page 191]).

Exercise 7.165. Show that the product of two Poincare duality groups isagain a Poincare duality group.

Theorem 7.166 (Homology ANR-manifolds and finite Poincare com-plexes). Let M be a closed topological manifold, or more generally, a compacthomology ANR-manifold of dimension n. Then M is homotopy equivalent toa finite n-dimensional Poincare complex.

Proof. A closed topological manifold, and more generally a compact ANR,has the homotopy type of a finite CW -complex (see [473, Theorem 2.2].[830]). The usual proof of Poincare duality for closed manifolds carries overto homology manifolds. ut

Theorem 7.167 (Poincare duality groups). Let G be a group and n ≥ 1be an integer. Then:

(i) The following assertions are equivalent:

(a) G is finitely presented and a Poincare duality group of dimension n;(b) There exists a finitely dominated n-dimensional aspherical Poincare

complex with G as fundamental group;

(ii) Suppose that K0(ZG) = 0. Then the following assertions are equivalent:

(a) G is finitely presented and a Poincare duality group of dimension n;(b) There exists a finite n-dimensional aspherical Poincare complex with G

as fundamental group;

(iii) A group G is a Poincare duality group of dimension 1 if and only if G ∼= Z;(iv) A group G is a Poincare duality group of dimension 2 if and only if G is

isomorphic to the fundamental group of a closed aspherical surface;

Proof. (i) Every finitely dominated CW -complex has a finitely presented fun-damental group since every finite CW -complex has a finitely presented fun-damental group and a group which is a retract of a finitely presented groupis again finitely presented, see [812, Lemma 1.3]. If there exists a CW -modelfor BG of dimension n, then the cohomological dimension of G satisfiescd(G) ≤ n and the converse is true provided that n ≥ 3 (see [134, Theo-rem 7.1 in Chapter VIII.7 on page 205], [268], [812], [813]). This implies that


the implication (ib) =⇒ (ia) holds for all n ≥ 1 and that the implication(ia) =⇒ (ib) holds for n ≥ 3. For more details we refer to [424, Theorem 1].The remaining part to show the implication (ia) =⇒ (ib) for n = 1, 2 followsfrom assertions (iii) and (iv).

(ii) This follows in dimension n ≥ 3 from assertion i and Wall’s results aboutthe finiteness obstruction which decides whether a finitely dominated CW -complex is homotopy equivalent to a finite CW -complex and takes values inK0(Zπ) (see [310, 604, 812, 813]). The implication (iib) =⇒ (iia) holds forall n ≥ 1. The remaining part to show the implication (iia) =⇒ (iib) holdsfollows from assertions (iii) and (iv).

(iii) Since S1 = BZ is a 1-dimensional closed manifold, Z is a finite Poincareduality group of dimension 1 by Theorem 7.166. We conclude from the (easy)implication (ib) =⇒ (ia) appearing in assertion i that Z is a Poincare du-ality group of dimension 1. Suppose that G is a Poincare duality group ofdimension 1. Since the cohomological dimension of G is 1, it has to be a freegroup (see [762, 778]). Since the homology group of a group of type FP isfinitely generated, G is isomorphic to a finitely generated free group Fr ofrank r. Since H1(BFr) ∼= Zr and H0(BFr) ∼= Z, Poincare duality can onlyhold for r = 1, i.e., G is Z.

(iv) This is proved in [266, Theorem 2]. See also [110, 111, 264, 267]. ut

Conjecture 7.168 (Manifold structures on aspherical Poincare com-plexes). Every finitely dominated aspherical Poincare complex is homotopyequivalent to a closed topological manifold.

Remark 7.169 (Existence and uniqueness part of the Borel Conjec-ture). Conjecture 7.168 can be viewed as the existence part of the BorelConjecture 7.152, namely, the question whether an aspherical finite Poincarecomplex carries up to homotopy the structure of a closed topological mani-fold, and the Borel 7.152 as stated above is the uniqueness part.

Conjecture 7.170 (Poincare duality groups). A finitely presented groupis an n-dimensional Poincare duality group if and only if it is the fundamentalgroup of a closed aspherical n-dimensional topological manifold.

The disjoint disk property says that for any ε > 0 and maps f, g : D2 →Mthere are maps f ′, g′ : D2 → M so that the distance between f and f ′ andthe distance between g and g′ are bounded by ε and f ′(D2) ∩ g′(D2) = ∅.

Theorem 7.171 (Poincare duality groups and aspherical compacthomology ANR-manifolds). Suppose that the torsionfree group G is afinitely presented Poincare duality group of dimension n ≥ 6 and satisfies theversions of the K-theoretic Farrell-Jones Conjecture stated in 2.107 and 3.19and the version of the L-theoretic Farrell-Jones Conjecture stated in 7.109for the ring R = Z.

Then BG is homotopy equivalent to an aspherical compact homology ANR-manifold satisfying the disjoint disk property.

7.18 Boundaries of Hyperbolic Groups 227

Proof. See [685, Remark 25.13 on page 297], [136, Main Theorem on page 439and Section 8] and [137, Theorem A and Theorem B]. ut

Remark 7.172 (Compact homology ANR-manifolds versus closedtopological manifolds). In the following all manifolds have dimension ≥ 6.One would prefer if in the conclusion of Theorem 7.171 one could replace“compact homology ANR-manifold” by “closed topological manifold”. Theproblem is that in the geometric exact surgery sequence one has to workwith the 1-connective cover L〈1〉 of the L-theory spectrum L, whereas in theassembly map appearing in the Farrell-Jones setting one uses the L-theoryspectrum L. The L-theory spectrum L is 4-periodic, i.e., πn(L) ∼= πn+4(L) forn ∈ Z. The 1-connective cover L〈1〉 comes with a map of spectra f : L〈1〉 → Lsuch that πn(f) is an isomorphism for n ≥ 1 and πn(L〈1〉) = 0 for n ≤ 0. Sinceπ0(L) ∼= Z, one misses a part involving L0(Z) of the so called total surgeryobstruction due to Ranicki, i.e., the obstruction for a finite Poincare complexto be homotopy equivalent to a closed topological manifold, if one deals withthe periodic L-theory spectrum L and picks up only the obstruction for afinite Poincare complex to be homotopy equivalent to a compact homologyANR-manifold, the so called four-periodic total surgery obstruction. The dif-ference of these two obstructions is related to the resolution obstruction ofQuinn which takes values in L0(Z). Any element of L0(Z) can be realized byan appropriate compact homology ANR-manifold as its resolution obstruc-tion. There are compact homology ANR-manifolds that are not homotopyequivalent to closed manifolds. But no example of an aspherical compacthomology ANR-manifold that is not homotopy equivalent to a closed topolo-gical manifold is known. For an aspherical compact homology ANR-manifoldM , the total surgery obstruction and the resolution obstruction carry thesame information. So we could replace in the conclusion of Theorem 7.171“compact homology ANR-manifold” by “closed topological manifold” if andonly if every aspherical compact homology ANR-manifold with the disjointdisk property admits a resolution.

We refer for instance to [136, 309, 674, 675, 685] for more informationabout this topic.

Question 7.173 (Vanishing of the resolution obstruction in the as-pherical case). Is every aspherical compact homology ANR-manifold ho-motopy equivalent to a closed manifold?

7.18 Boundaries of Hyperbolic Groups

If G is the fundamental group of an n-dimensional closed Riemannian mani-fold with negative sectional curvature, then G is a hyperbolic group in thesense of Gromov (see for instance [124], [129], [344], [352]). Moreover sucha group is torsion-free and its boundary ∂G is homeomorphic to a sphere.


This leads to the natural question whether a torsion-free hyperbolic groupwith a sphere as boundary occurs as fundamental group of a closed aspher-ical manifold (see Gromov [353, page 192]). In high dimensions this ques-tion is answered by the following two theorems taken from Bartels-Luck-Weinberger [76] For the notion of and information about the boundary of ahyperbolic group and its main properties we refer for instance to [441].

Theorem 7.174 (Hyperbolic groups with spheres as boundary). LetG be a torsion-free hyperbolic group and let n be an integer ≥ 6. Then:

(i) The following statements are equivalent:

(a) The boundary ∂G is homeomorphic to Sn−1;(b) There is a closed aspherical topological manifold M such that G ∼=

π1(M), its universal covering M is homeomorphic to Rn and the com-

pactification of M by ∂G is homeomorphic to Dn;

(ii) The aspherical manifold M appearing in the assertion above is unique upto homeomorphism.

Theorem 7.175 (Hyperbolic groups with Cech-homology spheres asboundary). Let G be a torsion-free hyperbolic group and let n be an integer≥ 6. Then

(i) The following statements are equivalent:

(a) The boundary ∂G has the integral Cech cohomology of Sn−1;(b) G is a Poincare duality group of dimension n;(c) There exists a compact homology ANR-manifold M homotopy equiva-

lent to BG. In particular, M is aspherical and π1(M) ∼= G;

(ii) If the statements in assertion i hold, then the compact homology ANR-manifold M appearing there is unique up to s-cobordism of compact ANR-homology manifolds.

One of the main ingredients in the proof of the two theorems above is thefact that both the K-theoretic and the L-theoretic the Farrell-Jones Con-jecture hold for hyperbolic groups (see [72] and[73]). This is also true forFerry-Lueck-Weinberger [306], where a stable version of the Cannon Conjec-ture is proved.

7.19 Product Decompositions

In this section we show that, roughly speaking, a closed aspherical manifoldM is a product M1 ×M2 if and only if its fundamental group is a productπ1(M) = G1×G2 and that such a decomposition is unique up to homeomor-phism. The following theorem is taken from [537, Theorem 6.1].

7.19 Product Decompositions 229

Theorem 7.176 (Product decompositions of aspherical closed mani-folds). Let M be a closed aspherical topological manifold of dimension n withfundamental group G = π1(M). Suppose we have a product decomposition

p1 × p2 : G∼=−→ G1 ×G2.

Suppose that G, G1 and G2 satisfy the versions of the K-theoretic Farrell-Jones Conjecture stated in 2.107 and 3.19 and the version of the L-theoreticFarrell-Jones Conjecture stated in 7.109 for the ring R = Z.

Then G, G1 and G2 are Poincare duality groups whose cohomological di-mensions satisfy

n = cd(G) = cd(G1) + cd(G2).

Suppose in the following:

• the cohomological dimension cd(Gi) is different from 3, 4 and 5 for i = 1, 2.• n ≥ 5 or n ≤ 2 or (n = 4 and G is good in the sense of Freedman);

Then:

(i) There are topological closed aspherical manifolds M1 and M2 together withisomorphisms

vi : π1(Mi)∼=−→ Gi

and mapsfi : M →Mi

for i = 1, 2 such that

f = f1 × f2 : M →M1 ×M2

is a homeomorphism and vi π1(fi) = pi (up to inner automorphisms) fori = 1, 2;

(ii) Suppose we have another such choice of topological closed aspherical man-ifolds M ′1 and M ′2 together with isomorphisms

v′i : π1(M ′i)∼=−→ Gi

and mapsf ′i : M →M ′i

for i = 1, 2 such that the map f ′ = f ′1 × f ′2 is a homotopy equivalence andv′i π1(f ′i) = pi (up to inner automorphisms) for i = 1, 2. Then there arefor i = 1, 2 homeomorphisms hi : Mi → M ′i such that hi fi ' f ′i andvi π1(hi) = v′i holds for i = 1, 2.


7.20 Automorphisms of Manifolds

We record two the following to results which deduce information about thehomotopy groups of the automorphism group of a closed aspherical manifoldfrom the Farrell-Jones Conjecture and the material from Chapter 6 aboutpseudoisotopy spaces.

Theorem 7.177 (Homotopy Groups of Top(M)). Let M be an ori-entable closed aspherical manifold of dimension > 10 with fundamental groupG. Suppose the L-theory assembly map

Hn(BG; L〈−∞〉(Z))→ L〈−∞〉n (ZG)

is an isomorphism for all n and suppose the K-theory assembly map

Hn(BG; K(Z))→ Kn(ZG)

is an isomorphism for n ≤ 1 and a rational isomorphism for n ≥ 2. Then for1 ≤ i ≤ (dimM − 7)/3 one has

πi(Top(M))⊗Z Q =

center(G)⊗Z Q if i = 1,0 if i > 1

In the differentiable case one additionally needs to study involutions onthe higher K-theory groups. The corresponding result reads:

Theorem 7.178 (Homotopy Groups of Diff(M)). Let M be an ori-entable closed aspherical differentiable manifold of dimension > 10 with fun-damental group G. Then under the same assumptions as in Theorem 7.177we have for 1 ≤ i ≤ (dimM − 7)/3

πi(Diff(M))⊗Z Q =

center(G)⊗Z Q if i = 1;⊕∞

j=1H(i+1)−4j(M ;Q) if i > 1 and dimM odd ;

0 if i > 1 and dimM even .

For a proof see for instance [289], [294, Section 2] and [288, Lecture 5]. Fora survey on automorphisms of manifolds we refer to [829].

Comment 24: Add discussion about the implications about the homologygroups of Top(M) as described on the slides for the conference in Oxford inSeptember 2015.


7.21 Survey on Computations of L-Theory of GroupRings of Finite Groups

Recall that we denote by rF (G) the number of isomorphism classes of ir-reducible representations of the finite G over the field F = R,C, andrF = | conF (G)| holds by Lemma 1.75.

Theorem 7.179 (Algebraic L-theory of ZG for finite groups). Let Gbe a finite group. Then

(i) The groups L〈j〉n (Z) are independent of the decoration j and given by Z,

0, Z/2, 0 for n ≡ 0, 1, 2, 3 mod (4);

(ii) For each j ≤ 1 the groups L〈j〉n (ZG) are finitely generated as abelian groups

and contain no p-torsion for odd primes p. Moreover, they are finite forodd n;

(iii) Let r(G) be the number of isomorphisms classes of irreducible real G-representations. Let rC(G) be the number of isomorphisms classes of ir-reducible real π-representations V which are of complex type. For everydecoration 〈j〉 we have

L〈j〉n (ZG)[1/2] ∼= L〈j〉n (QG)[1/2] ∼= L〈j〉n (RG)[1/2]

∼=

Z[1/2]r(G) n ≡ 0 (4);Z[1/2]rC(G) n ≡ 2 (4);0 n ≡ 1, 3 (4);

(iv) If G has odd order and n is odd, then Lεn(ZG) = 0 for ε = p, h, s.

Proof. (i) Comment 25: Add a reference to the book by Crowley-Macko-Luck [206].

(ii) See for instance [371].

(iii) See [685, Proposition 22.34 on page 253].

(iv) See [55, 371]. ut

7.22 Miscellaneous

We will relate the algebraic L-theory of C∗-algebras to their topological K-theory in Theorem 8.78. In particular we get for all n ∈ Z natural isomor-phisms

Ln(C∗r (G,R))[1/2] ∼= Ktopn (C∗r (G;R))[1/2];

Ln(C∗r (G,C))[1/2] ∼= Ktopn (C∗r (G;C))[1/2].


Conjecture 13.80 deals with the passage for L-theory from QG to RG toC∗r (G;R) and its connection to the Baum-Connes Conjecture and the Farrell-Jones Conjecture is analyzed in Lemma 13.81.

Our definition of the L-groups follows the original approach due to Wall.A much more satisfactory and elegant approach via chain complexes is dueto Mishchenko and Ranicki and is of fundamental importance for many ap-plications and generalizations, see for instance [206, 598, 599, 600, 681, 682,683, 685].

We mention that a different approach to surgery has been developed byKreck. A survey about his approach is given in [476]. Its advantage is thatone does not have to get a complete homotopy classification first. The price topay is that the L-groups are much more complicated, they are not necessarilyabelian groups any more. This approach is in particular successful when themanifolds under consideration are already highly connected. See for instance[479, 480, 767].

More information about surgery theory can be found for instance in [132,148, 149, 206, 525, 689, 815].

There is also a version of the Borel Conjecture for manifolds with bundarywhich is implied by the Farrell-Jones Conjecture, see for instance [311, page 17and page 31].

There is an obvious equivariant version of the Borel Conjecture, where onereplaces EG with the classifying space for proper G-actions EG, see Defini-iton 9.18. This version is not true in general and investigated for instancein [193, 194, 195, 199].

Another survey article about topological rigidity is [677].


name of texfile: ic

Chapter 8

Topological K-Theory

8.1 Introduction

In this chapter we deal with topological K-theory of reduced group C∗-algebras which is the target of the Baum-Connes Conjecture, in contrast to al-gebraic K- and L-theory of group rings which is the target of the Farrell-JonesConjecture. We begin with reviewing the topological K-theory of spaces andits equivariant version for proper actions of possibly infinite discrete groups.Then we pass to its generalization to C∗-algebras. We discuss the Baum-Connes Conjecture for torsionfree groups 8.44 and present two applications,namely, to the Trace Conjecture about the integrality of the trace map andto the Kadison Conjecture about idempotents in reduced group C∗-algebrasof torsionfree groups. Then we briefly state the main properties of Kasparov’sKK-theory and its equivariant version (without explaining its construction).This will later be needed in Chapter 12 to explain the analytic Baum-Connesassembly map and state the Baum-Connes Conjecture for arbitrary groupsand with coefficients in a G-C∗-algebra.

8.2 Topological K-Theory of Spaces

8.2.1 Complex Topological K-Theory of Spaces

Complex topological K-theory of spaces, sometimes also called complex topo-logical K-cohomology of spaces, is a generalized cohomology theory, i.e., it as-signs to a pair of CW -complexes (X,A) a Z-graded abelian group K∗(X;A)and a homomorphism of degree one δ∗ : K∗(A) → K∗+1(X,A) and to amap f : (X,A)→ (Y,B) of such pairs a homomorphism K∗(f) : K∗(Y,B)→K∗(X,A) of Z-graded abelian groups such that the Eilenberg-Zilber axiomsof a cohomology theory are satisfied, i.e., one has naturality, homotopy invari-ance, the long exact sequence of a pair and excision. Moreover, the disjointunion axiom holds, see Definition 10.1. In contrast to singular cohomologythe dimension axiom is not satisfied, actually Kn(•) is Z if n is even andis trivial if n is odd. A very important feature is that topological complex K-theory satisfies Bott periodicity, i.e., there is a natural isomorphism of degreetwo compatible with the boundary map in the long exact sequence of pairs

233

234 8 Topological K-Theory

β∗(X,A) : K∗(X,A)∼=−→ K∗+2(X,A).

Topological complex K-theory comes with a multiplicative structure. Com-ment 26: Add reference to the place in this book, where multiplicative struc-tures are described in detail.

It can be constructed by the so called complex topological K-theory spec-trum Ktop

C which is the following Ω-spectrum. (Spectra will be defined inSection 10.4.) The n-th space is Z × BU for even n and Ω(Z × BU) forodd n. The n-th structure map is given by the identity id : Ω(Z × BU) →Ω(Z× BU) for odd n and by an explicit homotopy equivalence due to Bott

Z × BU '−→ Ω2(Z × BU) for even n. As usual, associated to this spectrumis also a generalized homology theory K∗(X,A) called topological complexK-homology of spaces such that Kn(•) is Z if n is even and is trivial if n isodd. A proof of a universal coefficient theorem for complex K-theory can befound in [24] and [845, (3.1)], the homological version then follows from [12,Note 9 and 15].

Rational one can compute complex topological K-theory by Chern char-acters (Equivariant versions will be explained in Section 10.7.) Namely, weget for any pair of CW -complexes (X,A) a natural Q-isomorphism⊕

p∈Z,p≡n(2)

Hp(X,A;Q)∼=−→ Kn(X,A)⊗Z Q,(8.1)

and for any pair of finite CW -complexes (X,A) a natural Q-isomorphism

Kn(X,A)⊗Z Q∼=−→

∏p∈Z,p≡n(2)

Hp(X,A;Q).(8.2)

The condition that (X,A) is finite is needed in (8.2). The cohomologicalChern character (8.2) is compatible with the multiplicative structures.

For integral computations one has to use the Atiyah-Hirzebruch spectralsequence which does not collapse in general.

Exercise 8.3. Let X be a finite CW -complex. Show for its Euler character-istic

χ(X) = rkZ(K0(X))− rkZ(K1(X)) = rkZ(K0(X))− rkZ(K1(X)).

The groups K∗(BG) can be computed explicitly for all finite groups Gusing the Completion Theorem due to Atiyah-Segal [39, 47], see for in-stance [533, Theorem 0.3]. Namely, if for a prime p we denote by r(p) thenumber of conjugacy classes (g) of elements g ∈ G whose order |g| is pd forsome integer d ≥ 1, and by Zp the p-adic integers, then there are isomor-phisms of abelian groups

8.2 Topological K-Theory of Spaces 235

K0(BG) ∼= Z×∏

p prime

(Zp)r(p);(8.4)

K1(BG) ∼= 0.(8.5)

One can also figure out the multiplicative structure on K0(BG) in (8.4). Thisshows how accessible topological K-theory is, for instance, one does not knowthe group cohomology H∗(BG) of all finite groups G.

If X is a finite CW -complex, K∗(X) can be described in terms of vectorbundles. For instance, K0(X) is the Grothendieck group associated to theabelian monoid of isomorphism classes of (finite dimensional complex) vectorbundles over X under the Whitney sum. Naturality comes from the pullbackconstruction, the multiplicative structure from the tensor product of vectorbundles.

There are a Thom isomorphism and a Kunneth Theorem for finite CW -complexes for topological complex K-cohomology see [44, Corollary 2.7.12 onpage 111 and Corollary 2.7.15 on page 113].

Using exterior powers one can construct the so called Adams operationson topological complex K-cohomology. They were a key ingredient in thework of Adams on the Hopf invariant one problem, see [3, 14], and on linearindependent vector fields on spheres, see [4, 5, 6]. Atiyah [40] introducedthe groups J(X), where vector bundles are considered up to fiber homotopyequivalence. They were studied by Adams [8, 9, 10, 11].

Complex topological K-theory was the first generalized cohomology the-ory. Meanwhile there are other generalized cohomology theories such as com-plex bordism, see for instance [690], Morava K-theory, see for instance [843],elliptic cohomology, see for instance [563, 783] and topological modular formstmf, see for instance [402, 403], which have been in the focus of algebraictopology over the last decades.

The connection between topological K-theory and spaces of Fredholm op-erators was explained by Janich [418]. Namely, there exists a natural bijectionof abelian groups for finite CW -complexes X

[X,Fred] ∼= K0(X),(8.6)

where Fred is the space of Fredholm operators, i.e., bounded operators withfinite dimensional kernel and cokernel. This shows that there is a relation be-tween topological K-theory and index theory. For instance, we get from (8.6)applied to X = • an isomorphism π0(Fred) = K0(•) ∼= Z that sendsa Fredholm operator to its classical index which is the difference of thedimension of its kernel and the dimension of its cokernel. The bijectionof (8.6) assigns to a map X → Fred, which can be interpreted as a familyof Fredholm operators parametrized by X, its family index which is essen-tially the difference of the class of the vector bundle over X whose fiberover x is the kernel of the Fredholm operator associated to x ∈ X andthe vector bundle over X whose fiber over x is the cokernel of the Fred-


holm operator associated to x ∈ X. Good introductions to index theory arethe seminal papers [46, 48, 49, 51, 52]. Other references about index theoryare [114, 697, 844]

8.2.2 Real Topological K-Theory of Spaces

There is also real topological K-theory of spaces, sometimes also called realtopological KO-cohomology of spaces, KO∗(X,A) and real topological K-homology KO∗(X,A), where one considers real vector bundles instead ofcomplex vector bundles and BO instead of BU . One uses a specific homo-

topy equivalence Z × BO'−→ Ω8(Z × BO) to construct so the called real

K-theory spectrum KtopR . A much more sophisticated and structured sym-

metric spectrum representing real K-theory in terms of Fredholm operatorswas constructed by Joachim [420, 421] and Mitchener [608] based on ideas ofAtiyah-Singer [50].

The main difference between the real and the complex version is that KO∗is 8-periodic and KOn(•) = KO−n(•) is given by Z,Z/2,Z/2, 0,Z, 0, 0, 0for n = 0, 1, 2, 3, 4, 5, 6, 7. There are natural transformations i∗ : KO∗(X) →K∗(X) and r∗ : K∗(X) → KO∗(X) which corresponds to assigning to areal vector bundle its complexification and to a complex vector bundle itsrestriction to a real vector bundle. They satisfy r i = 2 · id. They also existon K-homology. It is sometimes useful to consider the real topological K-theory instead of the complex version, since one does loose information whenpassing to the complex topological version. On the other hand computationsfor the real topological K-theory are harder than for the complex topologicalK-theory, since the the real version is 8-periodic and its value at • contains2-torsion, whereas the complex version is 2-periodic and its evaluation at •is much simpler than for the real version.

Rationally we get again a Chern character, namely, for any pair of CW -complexes (X,A) a natural Q-isomorphism⊕

p∈Z,p≡n(4)

Hp(X,A;Q)∼=−→ KOn(X,A)⊗Z Q,(8.7)

and for any pair of finite CW -complexes (X,A) a natural Q-isomorphism

KOn(X,A)⊗Z Q∼=−→

∏p∈Z,p≡n(4)

Hp(X,A;Q).(8.8)

There is a natural transformation of homology theories called KO-orien-tation of Spin bordism due to Atiyah-Bott-Shapiro [45], which can be inter-preted by sending a Spin manifold to the index class of the associated Diracoperator


D : ΩSpinn (X)→ KOn(X).(8.9)

It plays an important role for the question when a closed spin manifold admitsa Riemannian metric of positive sectional curvature, see Subsection 12.8.2.

A relation of KO-theory to surgery theory has already been explained inRemark 7.127.

Another variant of topological K-theory denoted by KR∗(X,A) was de-fined by Atiyah [41]. Twisted topological K-theory has recently been in thefocus of interest, see for instance [37, 38, 327, 447].

More information about topological K-theory of spaces can be found forinstance in [7, 36, 43, 44, 407, 408, 446, 504].

8.2.3 Equivariant Topological K-Theory of Spaces

Equivariant topological K-theory has been considered for compact topolo-gical groups acting on compact spaces, see for instance [44, 745]. For ourpurpose it will be important to treat the more general case of a proper actionof a not necessarily compact group. It suffices for our purposes to consider dis-crete groupsG and properG-CW -complexes, or, equivalently, CW -complexeswith a G-action such that all isotropy groups are finite and for every opencell e of X with g · e ∩ e 6= ∅ we have gx = x for all x ∈ e. This is difficultenough, but not as hard as the much less understood case of a topologicalgroup acting properly on locally compact Hausdorff space.

If G is a discrete group, G-cohomology theories K∗G and KO∗G are con-structed by Luck-Oliver [544] for pairs of proper G-CW -complexes (X,A)using classifying spaces for G-vector bundles. More precisely, for every pairof proper G-CW -complexes (X,A) one obtains Z-graded abelian groupsK∗G(X,A) and KO∗G(X,A) such that one has naturality, G-homotopy in-variance, long exact sequence of pairs, excision, and the disjoint union axiomholds, see Definition 10.1. The complex version K∗G is 2-periodic, the realversion is 8-periodic.

Let H ⊆ G be a finite subgroup, Then

KnG(G/H) =

RepC(H) if n is even;

0 if n is odd.(8.10)

There is a decomposition of the real group ring RH as a direct product∏ri=0Mni,ni(Di) of matrix algebras over skew-fields Di, where Di is R, C or

H. Then one obtains a decomposition for each n ∈ Z

KO−nG (G/H) =

r∏i=1

KO−nG (G/H)i,(8.11)


where

KO−nG (G/H)i =

KOn(•) if Di = R;

Kn(•) if Di = C;

KOn+4(•) if Di = H.

There is a natural external multiplicative structure, i.e., there is a naturalpairing

KmG (X,A)⊗Z K

nH(X,B)→ Km+n

G×H((X,A)× (Y,B))(8.12)

for discrete groups G and H and a pair (X,A) of proper G-CW -complexesand a pair (Y,B) of proper H-CW -complexes. There exists a natural restric-tion homomorphism for any inclusion i : H → G of discrete groups

i∗ : K∗G(X,A)→ K∗H(i∗(X,A)),(8.13)

where (X,A) is a pair of proper G-CW -complexes and i∗(X,A) is its restric-tion to H. Applying this to the diagonal map G → G × G and the externalproduct and using the diagonal embedding X → X×X, one obtains a naturalinternal multiplicative structure, i.e., natural pairings

KmG (X,A)⊗Z K

nG(X,B)→ Km+n

G (X,A ∪B)(8.14)

for a discrete group G and a proper G-CW -complex X with G-CW -subcom-plexes A and B. In particular K∗G(X) becomes a Z-graded algebra for anyproper G-CW -complex X. Given a group homomorphism α : H → G, thereis an induction homomorphism

indα : K∗H(X,A)→ K∗G(indα(X,A)),(8.15)

where (X,A) is a proper H-CW -complex and indα(X,A) is the proper G-CW -complex G ×α (X,A). If ker(α) acts freely on (X,A), the map indα isbijective.

All the constructions and results above are carried out in [544] and thecorresponding statements do hold also for the real version KO∗G. If G is finite,they all reduce to the classical constructions and results.

One can give a description for pairs (X,A) of finite proper G-CW -complexes for a discrete groupG in terms ofG-vector bundles such that for in-stance K0

G(X) and KO0G(X) respectively agree with the Grothendieck groups

of isomorphism classes of G-equivariant complex and real respectively vectorbundles over the finite proper G-CW -complex X. This follows from [545,Theorem 3.2 and Theorem 3.15] and [544, Proposition 1.5]. (A C∗-theoreticanalogue of this result is discussed in [95, Section 6].) However, the inter-pretation of K0

G(X) in terms of vector bundles does not hold if G is a Liegroup, as explained in [545, Section 5]. A description in terms of infinite di-


mensional G-vector bundles is discussed by Phillips [658]. The question howthe Grothendieck group of isomorphism classes of G-vector bundles over aclassifying space BG of a compact Lie group G looks like and is related toK0(BG) is treated in [417]. (Notice that this is a non-trivial question alreadyfor finite groups since BG does not have a finite dimensional CW -model fornon-trivial finite groups.)

Let G be a discrete group. For any cyclic group C ⊆ G of order n < ∞we denote by Z[ζC ] ⊆ Q(ζC) the cyclotomic ring and field generated bythe n-th roots of unity, but regarded as quotient rings of the group ringsZ[hom(C,C∗)] ⊆ Q[hom(C,C∗)]. In other words, we fix an identification ofthe n-th roots of unity in Q(ζC) with the irreducible characters of C. Let C(G)be a set of conjugacy class representatives for the cyclic subgroups C ⊆ Gof finite order. Denote by CGC the centralizer and by NGC the normalizerof C in G. Then for any pair of finite proper G-complexes (X,A), there isthe following version of an equivariant Chern character, namely, a naturalisomorphism of rings

K∗G(X;A)⊗Z Q∼=−→

∏C∈C(G)

(H∗((X,A)C/CGC;Q(ζC))

)NGC/CGC,(8.16)

where NGC/CGC acts via the conjugation action on Q(ζC) and on XC/CGCin terms of the given G-action on X.

Equivariant Chern characters can be used to compute K∗(BG) ⊗Z Q forinfinite groups possessing a finite G-CW -model for its classifying space ofproperG-actions, i.e., for instance for hyperbolic groupsG or compact latticesG in connected Lie groups, see [533] and also [15, 16]. More information aboutK∗(BG) for infinite groups can be found in [422, Theorem 0.1], and aboutcohomological Chern characters in [530].

Exercise 8.17. Let G be an abelian group. Let X be a finite proper G-CW -complex. Show that there is a Q-isomorphism

K∗G(X)⊗Z Q ∼=∏

C∈C(G)

H∗(XC/G;Q)ϕ(|C|)

for the Euler Phi-function ϕ.

We will construct in Section 8.6 the equivariant K-homology KG∗ which

is a G-homology theory defined for pairs of G-CW -complexes for discretegroups G and satisfies the disjoint union axiom.

An equivariant Universal Coefficient Theorem for equivariant complex K-theory for discrete groups G and finite proper G-CW -complexes X is givenin [422, Theorem 0.3], namely, there are short exact sequences, natural in X,

0→ ExtZ(KG∗−1(X),Z)→ K∗G(X)→ homZ(KG

∗ (X),Z)→ 0;(8.18)

0→ ExtZ(K∗+1G (X),Z)→ KG

∗ (X)→ homZ(K∗G(X),Z)→ 0.(8.19)


It reduces for a finite group G to the one of Bokstedt [115] as explainedin [422, Remark 5.21],

An external Kunneth Theorem for complex K-theory relating K∗G×H(X ×Y ) to K∗G(X) and K∗H(Y ) is given in [595] for compact Lie groups G and Hand finite G-CW -complexes X and Y , namely, there is a short exact sequence

(8.20) 0→⊕i+j=n

KiG(X)⊗Z K

jH(Y )→ Kn

G×H(X × Y )

→⊕

i+j=n+1

TorZ(KiG(X),Kj

H(Y ))→ 0.

The situation is much more complicated and much less understood if onewants to relate K∗G(X × Y ) to K∗G(X) and K∗G(Y ) for a finite group G andfinite G-CW -complexes X and Y , see [401, 708]. This complication is notsurprising since it is related to the difficult question to compute K∗G(X × Y )for the diagonal G-action on X ×Y from KG×G(X ×Y ) for a finite group Gand finite G-CW -complexes X and Y .

Exercise 8.21. Let G and H be discrete groups. Let (X,A) be a pair offinite proper G-CW -complexes and let (Y,B) be a pair of finite proper H-CW -complexes. Suppose that either Ki

G(X) is torsionfree for i ∈ Z or that

KjH(Y ) is torsionfree for all j ∈ Z.Then the external multiplicative structure induces for every n ∈ Z an

isomorphism⊕i+j=n

KiG(X,A)⊗Z K

jH(Y,B)

∼=−→ KnG×H((X,A)× (Y,B)).

Consider a discrete group G and a complex G-vector bundle p : E → Xwith Hermitian metric over a finite proper G-CW -complex. Let pDE : DE →X be the disk bundle and pSE : SE → X be the sphere bundle associated top whose fiber over x ∈ X is the disk and sphere in p−1(x). Then there existsa Thom class λE ∈ K0

G(DE,SE) and the composite

TE : K∗G(X)K∗G(pDE)−−−−−−→ K∗G(DE)

−∪λE−−−−→ K∗G(DE,SE)(8.22)

is an isomorphism of Z-graded abelian groups called Thom isomorphism,see [545, Theorem 3.14].

Exercise 8.23. For a discrete group G and a complex G-vector bundlep : E → X over a finite proper G-CW -complex define its Euler classe(p) ∈ KG

0 (X) to be the image of the Thom class under the composite

K0G(DE,SE)

K0G(j)−−−−→ K0

G(DE)K0G(pDE)−1

−−−−−−−−→ K0G(X) for j : DE → (DE,SE)

the inclusion. Show that there exists a long exact Gysin sequence


(8.24) · · · δn−1

−−−→ KnG(X)

−∪e(p)−−−−→ KnG(X)

KnG(pSE)−−−−−−→ Kn

G(SE)

δn−→ Kn+1G (X)

−∪e(p)−−−−→ Kn+1G (X)

Kn+1G (pSE)−−−−−−−→ · · · .

A Completion Theorem for complex and real topological K-theory allowingfamilies of subgroups is proved in [544, Theorem 6.5] for a discrete groupG and a finite proper G-CW -complex X in terms of isomorphisms of pro-systems, see also [545, Theorem 4.3]. Let p : EG → BG be the universalcovering of BG, or, equivalently, the universal principal G-bundle. Up toG-homotopy EG is uniquely characterized by the property that it is a freG-CW -complex which is (after forgetting the group action) contractible. Aconsequence of the Completion Theorem is that the inverse system

K∗((EG×GX)(n)

)n≥0

satisfies the Mittag-Leffler condition and we obtain isomorphisms

K∗G(X)I ∼= K∗(EG×GX) ∼= invlimn→∞K∗((EG×GX)(n)

).(8.25)

Here K∗G(X)I is the completion of K∗G(X) with respect to the so calledaugmentation ideal I which is the kernel of the dimension map K0

G(EG)→ Zfor EG the classifying space for proper G-actions, and we have to assume thatthere is a finite dimensional model for EG. If G is finite and we take X = •,this reduces to the classical Atiyah-Segal Completion Theorem predicting anisomorphism

Kn(BG) =

RepC(G)I n even;

0 n odd,

where I is the augmentation ideal, i.e., kernel of the map given by takingcomplex dimension RepC(G)→ Z.

There is also a version for the real topological K-theory. A CocompletionTheorem for the topological complex K-homology for discrete groups and finiteproper G-CW -complexes is proved in [422, Theorem 0.2]. It assigns to a finiteproper G-CW -complex X a short exact sequence

(8.26) 0→ colimn≥1 Ext1Z(K∗+1

G (X)/In ·K∗+1G (X),Z)→ K∗(EG×G X)

→ colimn≥1 homZ(K∗G(X)/In ·K∗G(X),Z)→ 0.

The Completion and Cocompletion Theorems are not only interesting in itsown right, they are needed in the computation of the topological K-theoryof certain group C∗-algebras, see for instance [224, 225, 503].

Another important tool for equivariant K-theory over compact Lie groupsis the Localization Theorem for equivariant topological complex K-theory ofSegal [745, Proposition 4.1]. Given a prime ideal P of RepC(G) = K0

G(•),


there is a topologically cyclic group S associated to P, its so called support.If X is a finite G-CW -complex, let X(S) be the G-CW -subcomplex G ·XS .Then after localization at P the inclusion X(S) → X induces an isomorphism

K∗G(X)(P)

∼=−→ K∗G(X(S))(P).(8.27)

Localization for equivariant cohomology theories for compact Lie groups istreated in general in [787, Chapter 7] and [788, III.3 and III.4].

Equivariant topological K-theory was designed for and is a key ingredientwhen one considers indices of equivariant operators. See for instance [46,48, 49], where also applications such as Lefschetz Theorems, Riemann-RochTheorems and G-Signature Theorems are treated for compact Lie groups.

The KG-degree of G-maps between spheres of unitary G-representationsfor a compact Lie group G is an important tool, see [788, II.5].

A discussion about equivariant K-theory and orbifold K-theory can befound in [17, Chapter 3].

A geometric description of equivariant K-homology for proper actionsin term cycles built by proper cocompact G-Spinc-manifolds and smoothcomplex G-vector bundles over them is given in [95], extending the non-equivariant version of Baum-Douglas [92, 94].

8.3 Topological K-Theory of C∗-Algebras

8.3.1 Basics about C∗-algebras

For this section let F be R or C. For λ ∈ F , denote by λ the complexconjugate of λ.

A Banach algebra over F is an associative F -algebra A = (A,+, ·) togetherwith a norm || || for the underlying F -vector space such that the underlyingF -vector space is complete with respect to the given norm and we have theinequality ||a · b|| ≤ ||a|| · ||b|| for all elements a, b ∈ A.

A Banach ∗-algebra is a Banach algebra together with an involution∗ : A → A, a 7→ a∗ satisfying (a∗)∗ = a, (a · b)∗ = b∗ · a∗, (λ · a + µ · b)∗ =λ · a∗ + µ · b∗, and ||a∗|| = ||a|| for a, b ∈ A and λ, µ ∈ F . If G is a discretegroup, L1(G,F ) carries the structure of a Banach ∗-algebra coming from theconvolution product, the L1-norm and the involution sending

∑g∈G λg · g to∑

g∈G λg · g−1.A C∗-algebra is a Banach ∗-algebra A which satisfies additionally the C∗-

identity ||a∗a|| = ||a||2 for all a ∈ A. A homomorphism of C∗-algebras f : A→B is a homomorphism of F -algebras in the algebraic sense which respects theinvolutions. A consequence of the C∗-identity is that a homomorphism ofC∗-algebras f : A → B automatically satisfies ||f(a)|| ≤ ||a|| for all a ∈ Aand is in particular continuous. Moreover, any injective homomorphism of

8.3 Topological K-Theory of C∗-Algebras 243

C∗-algebras f : A→ B is automatically isometric, i.e., satisfies ||f(a)|| = ||a||for all a ∈ A, and two C∗-algebras, which are isomorphic as F -algebras withinvolutions in the purely algebraic sense, are automatically isomorphic asC∗-algebras. Two homomorphisms f, g : A → B are homotopic if there is apath γt | t ∈ [0, 1] of homomorphisms of C∗-algebras γt : A→ B such thatγ0 = f and γ1 = g and for every a the evaluation map [0, 1] → B, t 7→γt(a) is continuous with respect to the C∗-norm on B. Equivalently, thereis a homomorphism of C∗-algebras γ : A → C([0, 1], B) to the C∗-algebra ofcontinuous functions from [0, 1] to B under the supremum norm such that itscomposition with the evaluation maps at t = 0 and t = 1 from C([0, 1], B) toB are f and g.

If H is a Hilbert F -space, then the algebra of bounded operators B(H)with the involution given by taking adjoint operators and the operator normis a C∗-algebra. In particular any subalgebra A ⊆ B(H) which is closed inthe norm topology and closed under taking adjoints, inherits the structure ofa C∗-algebra and any C∗-algebra is isomorphic as C∗-algebra to such A.

We are not requiring a unit for the multiplication. If the Banach algebraor C∗-algebra A has a unit for the multiplication, we call A a unital Banachalgebra or unital C∗-algebra.

Given a C∗-algebra A, an ideal in A is a two-sided ideal of the underlyingF -algebra which is closed in the norm topology. It is automatically closedunder the involution and hence inherits the structure of a C∗-algebra. Thequotient A/I inherits the structure of a C∗-algebra by the obvious F -algebrastructure and the norm ||a + I||A/I := inf||a + i||A | i ∈ I. Kernels ofC∗-homomorphisms f : A → B are ideals A and each ideal in A is the ker-nel of some homomorphism of C∗-algebras with A as source, namely, of theprojection A→ A/I.

Fix an infinite dimensional separable F Hilbert space H. Let B be theunital C∗-algebra of bounded operators H → H. An element T ∈ B(H) iscompact if for any bounded subset B ⊆ H the closure of T (B) is a compactsubset of H. The compact operators form an ideal K in B. The Calkin algebrais the unital C∗-algebra B/K

Let X be a locally compact Hausdorff space. Denote by C0(X,F ) theC∗-algebra of continuous functions f : X → F which vanish at infinity, i.e.,for every ε > 0 there exists a compact subset C ⊆ X such that |f(x)| ≤ εholds for all x ∈ X \ C. If F is clear from the context, we often abbreviateC0(X) = C0(X,F ). Define an involution ∗ : C0(X,F )→ C0(X,F ) by sendingf to the function mapping x ∈ X to f(x). Equip C0(X,F ) with the supremumnorm. Then C0(X;F ) is a C∗-algebra. If X is compact, the constant functionon X with value 1 is a unit. Moreover, C0(X,F ) is unital if and only if X iscompact.

Example 8.28 (One-point and Stone-Cech compactification). If X isa locally compact Hausdorff space, then we can assign to it two compactifi-cations, the one-point compactification X+ and the Stone-Cech compactifi-cation βX, see [615, page 183 and Section 5.3]. Then C0(X+, F ) agrees with


C0(X,F )+ and C(βX,F ) agrees with Cb(X,F ), the C∗-algebra of boundedcontinuous functions X → F . (Actually Cb(X,F ) is the so called multiplieralgebra of C0(X;F ).) See for instance [819, Example 2.1.2 on page 28 andExample .2.2.4 on page 32].

Let L2(G,F ) be the Hilbert F -space whose orthonormal basis is G. IfF is clear from the context, we often abbreviate L2(G) = L2(G,F ). LetB(L2(G,F )) denote the bounded linear operators on the Hilbert F -spaceL2(G,F ).The reduced group C∗-algebra C∗r (G,F ) is the closure in the normtopology of the image of the regular representation FG → B(L2(G,F )),which sends an element u ∈ FG to the (left) G-equivariant bounded operatorL2(G,F ) → L2(G,F ) given by right multiplication with u. Let L1(G,F ) bethe Banach ∗-algebra of formal sums

∑g∈G λg · g with coefficients in F such

that∑g∈G |λg| < ∞. If F is clear from the context, we often abbreviate

L1(G) = L1(G,F ). There are natural inclusions

FG ⊆ L1(G,F ) ⊆ C∗r (G,F ) ⊆ B(L2(G,F ))G ⊆ B(L2(G,F )).

Exercise 8.29. Show for a discrete group G that L1(G;F ) is a C∗-algebraif and only if G is trivial or (G has order 2 and F = R).

For a group G let C∗m(G,F ) be its maximal group C∗-algebra, which is thenorm closure of the image of the so called universal representation FG →B(Hu), compare [654, 7.1.5 on page 229]. The maximal group C∗-algebrahas the advantage that every homomorphism of groups φ : G → H inducesa homomorphism C∗m(G,F ) → C∗m(H,F ) of C∗-algebras. This is not truefor the reduced group C∗-algebra C∗r (G,F ). Here is a counterexample: sinceC∗r (G,F ) is a simple algebra if G is a non-abelian free group [664], there isno unital algebra homomorphism C∗r (G,F ) → C∗r (1, F ) = F . There is acanonical homomorphism of C∗-algebras C∗m(G,F )→ C∗r (G,F ) which is anisomorphism of C∗-algebra, if and only if G is amenable, see [654, Theorem7.3.9 on page 243].

If F is clear from the context, we often abbreviate C∗r (G) = C∗r (G,F ) andC∗m(G) = C∗m(G,F ).

Given a discrete group G, a G-C∗-algebra A is a C∗-algebra together witha G-action ρ : G→ aut(A) by C∗-automorphisms. One can associate to a G-C∗-algebra A two new C∗-algebras, its reduced crossed product A or G andits maximal crossed product A om G, see [654, 7.6.5 on page 257 and 7.7.4on page 262]. There is a canonical homomorphism from the maximal crossedproduct to the reduced crossed product which is an isomorphism if G isamenable, see [654, Theorem 7.7.7. on page 263]. If we take A = F with thetrivial G-action, then F or G and F om G are just C∗r (G,F ) and C∗m(G,F ).

Let Ai | i ∈ I be a directed system of C∗-algebras. Then its col-imit, often also called inductive limit, or direct limit, is a C∗-algebra de-noted by colimi∈I Ai, together with homomorphisms of C∗-algebras ψj : Ai →colimi∈I Ai for every j ∈ I such that ψj φi,j = ψi holds for i, j ∈ I with i ≤ j


and the following universal property is satisfied: For every C∗-algebra B andevery system of homomorphisms of C∗-algebras µi : Ai → B | i ∈ I suchthat µj φi,j = µi holds for i, j ∈ I with i ≤ j, there is precisely one homo-morphism of C∗-algebras µ : colimi∈I Ai → B satisfying µψi = µi for everyi ∈ I. The colimit exists and is unique up to isomorphism of C∗-algebras.

An extensive discussions about tensor products A⊗B of C∗-algebras canbe found in [819, Appendix T]. There are various ways for two C∗-algebras Aand B to complete their algebraic tensor product A⊗FB to a new C∗-algebraA⊗B. One is the spatial norm which turns out to be the minimal normand leads to the spatial tensor product, sometimes also called the minimaltensor product. A second is the maximal norm which leads to the maximaltensor product. Any C∗-norm on the algebraic tensor product lies betweenthe minimal and the maximal norm. The favorite situation is the case, whereA is a so called nuclear C∗-algebra, i.e., the minimal and the maximal normon the algebraic tensor product A ⊗F B agree for any C∗-algebra B. Thenfor any C∗-algebra B there exists only one C∗-norm on the algebraic tensorproduct A⊗F B and hence there is a unique tensor product C∗-algebra A⊗B.Commutative C∗-algebras and finite dimensional C∗-algebras are nuclear.The class of nuclear C∗-algebras is closed under taking colimits limits overdirected systems and extensions. In particular the C∗-algebra of compactoperators K is nuclear. Ideals in and quotients of nuclear C∗-algebras areagain nuclear. The reduced group C∗-algebra of G is nuclear if and only if Gis amenable.

Given a C∗-algebra A, define Mn(A) = A⊗Mn(F ) which is well definedsince Mn(F ) = B(Fn) is nuclear. Actually, the underlying F -algebra ofMn(A) is the algebraic tensor product A⊗F Mn(F ) itself, one does not haveto complete.

The C∗-algebra K of compact operators on an infinite dimensional separa-ble Hilbert F -space is the colimit of the directed system M1(F )→M2(F )→M3(F ) → · · · where the structure maps are given by taking the block sumwith the (1, 1)-identity matrix (1). Given a C∗-algebra A, the tensor productA⊗K is the colimit of the directed system M1(A)→M2(A)→M3(A)→ · · · .

A C∗-algebra is called separable if its underlying topological space is sep-arable, i.e., contains a dense countable subset.

A C∗-algebra SA is called stable if A is isomorphic as C∗-algebra to A⊗K.Since K⊗K is isomorphic to K, the tensor product A⊗K is a stable C∗-algebrafor every C∗-algebra A.

More information about C∗-algebras can be found for instance in [34, 113,192, 218, 250, 316, 435, 436, 654].


8.3.2 Basic Properties of the Topological K-Theory ofC∗-Algebras

Topological K-theory assigns to any (not necessarily unital) C∗-algebra A aZ-graded abelian group K∗(A) such that the following properties hold:

(i) FunctorialityA homomorphism f : A → B of C∗-algebras induces a map of Z-gradedabelian groups K∗(f) : K∗(A) → K∗(B). If g : B → C is another homo-morphism of C∗-algebras, we have K∗(g f) = K∗(g) K∗(f). MoreoverK∗(idA) = idK∗(A);

(ii) Homotopy invarianceHomotopic homomorphisms of C∗-algebras induce the same map on thetopological K-theory;

(iii) Finite direct productsIf A and B are C∗-algebras, their direct product A×B inherits the struc-ture of a C∗-algebra by ||(a, b)|| = max||a||, ||b||. The projections to thefactors are homomorphisms of C∗-algebras and induce a natural isomor-phism of Z-graded abelian groups

K∗(A×B)∼=−→ K∗(A)×K∗(B);

(iv) Compatibility with colimits over directed systemsLet Ai | i ∈ I be a directed system of C∗-algebras. Then the canonicalmap of Z-graded abelian groups is an isomorphism

colimi∈I K∗(Ai)∼=−→ K∗

(colimi→I Ai

);

(v) Morita equivalenceThe inclusion of C∗-algebras A → Mn(A) sending a ∈ A to the diagonalmatrix whose diagonal entries are all equal to a induces an isomorphismK∗(A)→ K∗(Mn(A));

(vi) StabilizationThe canonical inclusion F = M1(F ) → K yields an inclusion iA : A →A⊗K. The induced map of Z-graded abelian groups K∗(iA) : K∗(A) →K∗(A⊗K) is an isomorphism;

(vii) Long exact sequence of an idealLet I be a (two-sided closed) ideal in the C∗-algebra A. Denote by i : I → Athe inclusion and by p : A→ A/I the projection. Then there exists a longexact sequence, natural in (A, I) and infinite to both sides,

· · · ∂n+1−−−→ Kn(I)Kn(i)−−−−→ Kn(A)

Kn(p)−−−−→ Kn(A/I)∂n−→ Kn−1(I)

Kn−1(i)−−−−−→ Kn−1(A)Kn−1(p)−−−−−→ Kn−1(A/I)

∂n−1−−−→ · · · ;


(viii) Bott periodicityFor any C∗-algebra A over F there exists an isomorphism of degree b(F )

β∗(A) : K∗(A)∼=−→ K∗+b(F )(A),

which is natural in A, compatible with the boundary operator ∂∗ of thelong exact sequence of an ideal, where b(F ) = 2 if F = R and b(F ) = 8 ifF = R;

(ix) Commutative C∗-algebrasLet X be a finite CW -complex (or more generally, compact Hausdorffspace). Then there are isomorphisms of Z-graded abelian groups, naturalin X,

K∗(X)∼=−→ K∗(C(X,C));

KO∗(X)∼=−→ K∗(C(X,R)),

from the topological complex or real K-theory of X to the topological K-theory of the unital C∗-algebra C(X,F ) of continuous functions X → F .

Of course the last property about commutative C∗-algebras is closely re-lated to the material in Section 1.4 about Swan’s Theorem 1.28.

Notation 8.30 (K and KO). If one considers a real C∗-algebra, one of-ten writes KO∗(A) instead of K∗(A) to indicate that the C∗-algebra underconsideration lives over R.

The 0-th topological K-group K0(A) of a C∗-algebra A agrees with theprojective class group K0(A) of the underlying ring (possibly without unit)in the sense of Definition 2.88. In contrast to K0(A) the topology of A entersin the definition of K1(A) as explained next.

If A is a C∗-algebra (with or without unit), then we define the unitalC∗-algebra A+ as follows. The underling unital F -algebra is A⊕ F with theaddition (a, λ)+(b, µ) = (a+b, λ+µ), multiplication (a, λ)·(b, µ) = (a·b+λ·b+µ·a, λ·µ) and unit (0, 1). The involutions sends (a, λ) to (a∗, λ). The C∗-normis explained for instance in [654, 1.1.3 on page 1] or [819, Proposition 2.1.7on page 30]. Let p : A+ → F be the canonical projection sending (a, λ) toλ. It induces maps Mn(A+) → Mn(F ) and GLn(A+) → GLn(F ), denotedagain by p. Define

GL+n (A) := B ∈ GLn(A+) | p(B) = 1.(8.31)

This becomes a topological group by the subspace topology with respect tothe inclusion GL+

n (A) ⊆ Mn(A+). There is an obvious directed system oftopological groups

GL+1 (A) ⊆ GL+

2 (A) ⊆ GL+3 (A) ⊆ · · ·


coming from embedding Mn(A+) into Mn+1(A+) by taking the block sumwith the (1, 1)-identity matrix (1). Its colimit is a topological group denotedby GL+(A). Let GL+(A)0 be the path component of the unit element inGL+(A). Then we get

K1(A) = GL+(A)/GL+(A)0 = π0(GL+(A)).(8.32)

More generally, we have

Kn(A) = πn−1(GL+(A)) for n ≥ 1.(8.33)

IfA is unital, then one defines the topological groupGL(A) = colimn→∞GLn(A)and obtains a canonical isomorphism

Kn(A) ∼= πn−1(GL(A)) for n ≥ 1.(8.34)

Exercise 8.35. Compute for F = C the topological K-theory of B, K andB/K.

Remark 8.36 (Six term sequence of an ideal). Let F = C in this Re-mark 8.36. Since K∗ is two-periodic, one thinks often about it as a Z/2-graded

theory. The long exact sequence of an extension 0 → Ii−→ A

p−→ A/I → 0becomes the six-term exact sequence of an ideal

K1(I)K1(i) // K1(A)

K1(p)// K1(A/I)

∂1

K0(A/I)

∂0

OO

K0(A)K0(p)oo K0(I)

K0(i)oo

Remark 8.37 (Topological K-theory in terms of unitary groups).Let F = C in this Remark 8.37. Let Un(A) be the group of unitary (n, n)-matrices over A, i.e., (n, n)-matrices U which are invertible and satisfyU−1 = U∗, where U∗ is defined by transposing and applying to each en-try the involution on A. Define U+

n (A) := U ∈ Un(A+) | p(U) = 1. PutU(A) = colimn→∞ Un(A) and U+(A) := colimn→∞ U+

n (A). Then then wehave isomorphisms of groups, see [819, Proposition 4.2.4 on page 77] Com-ment 27: Check that we obtain homotopy equivalences U+(A) → GL+(A)by polar decomposition. Shall we mention this?

K1(A) = GL+(A)/GL+(A)0∼= GL(A+)/GL(A+)0

∼= U+(A)/U+(A)0∼= U(A+)/U(A+)0.

Example 8.38 (On the boundary map and indices). Let F = C inthis Example 8.38. Let A be a unital C∗-algebra, I ⊆ A be an ideal and


p : A→ A/I be the projection. Let u be a unitary element in A/I. Let a ∈ Abe any element in A with p(a) = u and ||a|| = 1. Consider the (2, 2)-matricesover A

P :=

(aa∗ a(1a∗a)1/2

a∗(1− aa∗)1/2 1− a∗a

);

Q :=

(1 00 0

),

where (1−aa∗)1/2 is uniquely determined by the properties that it is positive,i.e., of the form b∗b for some b ∈ A, and satisfies (1− aa∗)1/2 · (1− aa∗)1/2 =1−aa∗, and analogously for (1−a∗a)1/2. Then P is a projection, i.e., P 2 = Pand P ∗ = P , and Q is a projection. Moreover, P − Q lies in M2(I). Definematrices in M2(I+) by

P+ :=

((aa∗ − 1, 1) (a(1a∗a)1/2, 0)

(a∗(1− aa∗)1/2, 0) (1− a∗a, 0)

)Q :=

((0, 1) (0, 0)(0, 0) (0, 0)

)One easily checks P 2

+ = P+ and Q2+ = Q+ and P+ − Q+ ∈ I. Hence P+

and Q+ determine elements [P+], [Q+] ∈ K0(I+) such that the difference[P+] − [Q+] is mapped under the canonical projection K0(I+) → K0(C) tozero. Hence [P+] − [Q+] defines an element in K0(I). It turns out that theimage ∂1([u]) of the class [u] ∈ K1(A) under the boundary homomorphism∂1 : K1(A/I)→ K0(I) is the class [P+]−[Q+], see [392, Proposition 4.8.10 (a)on page 109].

If we can additionally arrange that a is a partial isometry, i.e., a∗a is aprojection, then 1 − a∗a and 1 − aa∗ lie in I and are projections, and weobtain an element [1 − a∗a] − [1 − aa∗] in K0(I) which agrees with ∂1([u]),see [392, Proposition 4.8.10 (b) on page 109].

Now we apply this to A = B = B(H) and I = K = K(H) for an infinitedimensional separable Hilbert space H. Let a ∈ B be a Fredholm operatorsuch that a is a partial isometry. Then 1 − a∗a is the orthogonal projectiononto the kernel of a and 1−aa∗ is the orthogonal projection onto the cokernelof a. Hence the element [1 − a∗a] − [1 − aa∗] ∈ K0(K) becomes under thestandard identification K0(K) ∼= Z the difference of the dimension of thekernel of a and the dimension of the cokernel of a which is by definition theclassical index of the Fredholm operator a. This shows that ∂1 : K1(B/K)→K0(K) ∼= Z sends the class of [a] to the classical index of a.

It will often occur in many more general and important situations that ∂1

can be viewed as an index map.

Example 8.39 (Suspensions and cones). The suspension of a C∗-algebraA is the C∗-algebra ΣA of continuous functions f : [0, 1] → A with f(0) =f(1) = 1 equipped with the obvious algebra structure and involution and the


supremums norm inherited from A. Denote by Σn(A) the n-fold suspension.It can be identified with the tensor product of C∗-algebras A⊗C0(Rn). Thecone is defined analogously as the C∗-algebra cone(A) of continuous functionsf : [0, 1] → A with f(0) = 0. It can be identified with the tensor product ofC∗-algebras A⊗C0((0, 1]). There is an obvious exact sequence of C∗-algebras0 → ΣA → cone(A) → A → 0. Moreover, the C∗-algebra cone(A) is con-tractible, i.e., the zero and the identity endomorphism are homotopic. Thedesired homotopy is given by the formula ft(s) := f(ts). Hence K∗(cone(A))is trivial and the boundary operator in the associated long exact sequenceinduces isomorphisms

∂n : Kn(A)∼=−→ Kn−1(ΣA).

For complex C∗-algebras A and B for which A lies in the so calledbootstrap category N a Kunneth Theorem ,i.e., an exact sequence 0 →K∗(A) ⊗ K(B) → K∗(A⊗B) → TorZ(K∗(A),K∗(B)) → 0 is establishedin [743]. The case of real C∗-algebras is treated in [116].

Remark 8.40 (Topological K-theory and the classification of C∗-algebras). One prominent feature is that for certain classes of C∗-algebrastheir isomorphism type is determined by their topological K-theory, some-times taking the order structure on K0(A) coming from the positive cone ofthose elements which are represented by finitely generated projective mod-ules into account. If one considers the topological K-theory of spaces suchnice classification results are not available.

One example is the class of AF-algebras, i.e., C∗- algebras which occuras a colimit of a sequence of finite dimensional C∗-algebras, due to Elliot,see [271], [700, Chapter 7], [819, 12.1]. The index n of the Cuntz C∗-algebraOn is determined by the topological K-theory since K0(On) ∼= Z/n andK1(On) = 0, see [209], [819, 12.2]. A very important result about the clas-sification of so called Kirchberg C∗-algebras in terms of their topological K-theory is due to Kirchberg, see for instance [699, Chapter 8].

Remark 8.41 (Topological K-theory and generalized index theory).One important motivation to study the topological K-theory of C∗-algebrasis index theory and its generalizations. A first introduction how one canassign to a Fredholm operator over a C∗-algebra A an element in K0(A) isgiven in [819, Chapter 17] following Mingo [597]. There are many other indextheorems taking values in the topological K-theory of C∗-algebras. Oftenthey are generalizations of the classical family index theorem for familiesof operators parametrized over a closed manifold M which take values inK∗(M) = K∗(C(M)).

One can attach to geometric or topological situations new C∗-algebras andconsiders their topological K-theory and indices of appropriate operators,where it is not possible anymore to work with topological spaces. Examplesare foliations and coarse geometry. There are also plenty other generalizations

8.4 The Baum-Connes Conjecture for Torsionfree Groups 251

of the classical index theorems using topological K-theory of C∗-algebras. Forinformation about these topics we refer for instance to [192, 213, 392, 602].

More information about the topological K-theory of C∗-algebras can befound for instance in [112, 192, 213, 392, 700, 819].

8.4 The Baum-Connes Conjecture for TorsionfreeGroups

Let G be a group. Then there exists for all n ∈ Z assembly maps

asmbG,C(BG)n : Kn(BG)→ Kn(C∗r (G;C));(8.42)

asmbG,R(BG)n : KOn(BG)→ KOn(C∗r (G;R)).(8.43)

Conjecture 8.44 (Baum-Connes Conjecture for torsionfree groups).The assembly maps appearing in (8.42) and (8.43) are isomorphisms for alln ∈ Z, provided that G is torsionfree.

It is crucial for the Baum-Connes Conjecture to work with the reducedgroup C∗-algebra, it is definitely not true for the maximal group C∗ algebrain general. Moreover, Conjecture 8.44 in general fails for groups with torsion.The general version which makes sense for all groups will be discussed inChapter 12.

Exercise 8.45. Show for a finite group G that the following statements areequivalent:

(i) K0(BG) and K0(C∗r (G)) are rationally isomorphic;(ii) KO0(BG) and KO0(C∗r (G)) are rationally isomorphic;

(iii) G is trivial.

One benefit of Conjecture 8.44 is that the right side is of great interest be-cause of index theory but hard to compute, whereas the left side is accessibleto standard methods from algebraic topology.

Example 8.46 (Three-dimensional Heisenberg group). Let Hei(R) bethe three-dimensional Heisenberg group which is the subgroup of GL3(R)consisting of upper triangular matrices whose diagonal entries are all equalto 1. The three-dimensional discrete Heisenberg group Hei is the intersectionof Hei(R) with GL3(Z). Obviously Hei is a torsionfree discrete subgroup ofthe contractible Lie group Hei(R). Hence Hei \Hei(R) is a model for BHeiwhich is an orientable closed 3-manifold.

Define elements in Hei


u :=

1 1 00 1 00 0 1

; v :=

1 0 10 1 00 0 1

; w :=

1 0 00 1 10 0 1

.

Then we get the presentation

Hei = 〈u, v, w | [u,w] = v, [u, v] = 1, [w, v] = 1〉.

Therefore we have a central extension 1→ Z i−→ Heip−→ Z2 → 1, where i sends

the generator of Z to v and p sends v to (0, 0), u to (1, 0) and w to (0, 1).Hence the map H1(BHei) → H1(BZ2) is an isomorphism. Using Poincareduality we conclude

Hn(BHei) =

Z if n = 0, 3;

Z2 if n = 1, 2.

We conclude from the Chern character (8.1) for every n ∈ Z.

Kn(BHei)⊗Z Q ∼= Q3.

Next we consider the Atiyah-Hirzebruch spectral sequence converging toKp+q(BHei) whose E2-term is E2

p,q = Hp(BHei;Kq(•)). Its E2-page looksas follows

......

......

......

Z Z2 Z2 Z 0 0 · · ·

0 0 0 0 0 · · ·

Z Z2 Z2 Z 0 0 · · ·

0 0 0 0 0 · · ·

......

......

......

Each entry is a finitely generated free Z-module and we have for every n ∈ Z∑p+q=n

dimQ(E2p,q)⊗Z Q = 3 = Kn(BHei)⊗Z Q.


This implies that all differentials must vanish and we get for every n ∈ Z

Kn(BHei) ∼= Z3.

Conjecture 8.44 is known to be true for Hei and hence we conclude for everyn ∈ Z

Kn(C∗r (Hei)) ∼= Z3.

Exercise 8.47. Let G be the semi-direct product ZoZ, where the generatorof Z acts on Z by − id. Compute K∗(C

∗r (G)) using the fact that Conjec-

ture 8.44 is known to be true for G.

Next we discuss some consequences of the Baum-Connes Conjecture fortorsionfree groups 8.44.

8.4.1 The Trace Conjecture in the Torsionfree Case

The assembly map appearing in the Baum-Connes Conjecture has an inter-pretation in terms of index theory. Namely, an element η ∈ K0(BG) can berepresented by a pair (M,P ∗) consisting of a cocompact free proper smoothG-manifold M with a G-invariant Riemannian metric together with an el-liptic G-complex P ∗ of differential operators of order 1 on M , see [92]. Tosuch a pair one can assign an index indC∗r (G)(M,P ∗) in K0(C∗r (G)), see [602]which is the image of η under the assembly map K0(BG)→ K0(C∗r (G)) ap-pearing in Conjecture 8.44. With this interpretation the surjectivity of theassembly map for a torsionfree group says that any element in K0(C∗r (G))can be realized as an index. This allows to apply index theorems to get in-teresting information. It is of the same significance as the interpretation ofthe L-theoretic assembly map as the map σ appearing in the exact surgerysequence discussed in the proof of Theorem 7.160.

Here is a prototype of such an argument. The standard trace

trC∗r (G) : C∗r (G)→ C(8.48)

sends an element f ∈ C∗r (G) ⊆ B(l2(G)) to 〈f(1), 1〉l2(G). Applying the traceto idempotent matrices yields a homomorphism

trC∗r (G) : K0(C∗r (G))→ R.

Let pr : BG → • be the projection. For a group G the following diagramcommutes


K0(BG)

K0(pr)

asmbG,C(BG)∗ // K0(C∗r (G))trC∗r (G) // R

K0(•)∼= // K0(C)

trC

∼= // Z.

i

OO(8.49)

where i : Z → R is the inclusion. This non-trivial statement follows fromAtiyah’s L2-index theorem [42]. Atiyah’s theorem says that the L2-indextrC∗r (G) asmb∗(η) of an element η ∈ K0(BG), which is represented by a pair(M,P ∗), agrees with the ordinary index of (G\M ;G\P ∗), which is given bytrC K0(pr)(η) ∈ Z.

The following conjecture is taken from [90, page 21].

Conjecture 8.50 (Trace Conjecture for torsionfree groups). For atorsionfree group G the image of

trC∗r (G) : K0(C∗r (G))→ R

consists of the integers.

The commutativity of diagram (8.49) shows

Lemma 8.51. If the Baum-Connes assembly map K0(BG) → K0(C∗r (G))of (8.42) is surjective, then the Trace Conjecture for Torsionfree Groups 8.50holds for G.

A Modified Trace Conjecture for not necessarily torsionfree groups is dis-cussed in Subsection 12.8.1.

8.4.2 The Kadison Conjecture

Conjecture 8.52 (Kadison Conjecture). If G is a torsionfree group, thenthe only idempotent elements in C∗r (G) are 0 and 1.

Lemma 8.53. The Trace Conjecture for Torsionfree Groups 8.50 implies theKadison Conjecture 8.52.

Proof. Assume that p ∈ C∗r (G) is an idempotent different from 0 or 1. Fromp one can construct a non-trivial projection q ∈ C∗r (G), i.e. q2 = q, q∗ = q,with im(p) = im(q) and hence with 0 < q < 1. Since the standard tracetrC∗r (G) is faithful, we conclude trC∗r (G)(q) ∈ R with 0 < trC∗r (G)(q) < 1. SincetrC∗r (G)(q) is by definition the image of the element [im(q)] ∈ K0(C∗r (G))under trC∗r (G) : K0(C∗r (G)) → R, we get a contradiction to the assumptionim(trC∗r (G)) ⊆ Z. ut


Remark 8.54 (The Kadison Conjecture 8.52 and the KaplanskyConjecture 1.62). Obviously the Kadison Conjecture 8.52 implies the Ka-plansky Conjecture 1.62 in the case that R can be embedded in C. Becauseof Remark 1.71 the Kadison Conjecture 8.52 implies the Kaplansky Conjec-ture 1.62 if R is any field of characteristic zero. The Bost Conjecture 12.22implies that there are no non-trivial idempotents in L1(G) and hence theKaplansky Conjecture 1.62 for fields of characteristic zero, see [105, Corol-lary 1.6].

8.4.3 The Zero-in-the-Spectrum Conjecture

The following Conjecture is due to Gromov [351, page 120].

Conjecture 8.55 (Zero-in-the-spectrum Conjecture). Suppose that

M is the universal covering of an aspherical closed Riemannian manifold M(equipped with the lifted Riemannian metric). Then zero is in the spectrumof the minimal closure

(∆p)min : L2Ωp(M) ⊃ dom(∆p)min → L2Ωp(M),

for some p ∈ 0, 1, . . . ,dimM, where ∆p denotes the Laplacian acting on

smooth p-forms on M .

Theorem 8.56 (The strong Novikov Conjecture implies the Zero-in-the-spectrum Conjecture). Suppose that M is an aspherical closedRiemannian manifold with fundamental group G, then the injectivity of theassembly map

K∗(BG)⊗Z Q→ K∗(C∗r (G))⊗Z Q

implies the Zero-in-the-spectrum Conjecture 8.55 for M .

Proof. We give a sketch of the proof. More details can be found in [518,Corollary4]. We only explain that the assumption that in every dimension

zero is not in the spectrum of the Laplacian on M , yields a contradiction inthe case that n = dim(M) is even. Namely, this assumption implies that theC∗r (G)-valued index of the signature operator twisted with the flat bundle

M ×G C∗r (G) → M in K0(C∗r (G)) is zero, where G = π1(M). This index isthe image of the class [S] defined by the signature operator in K0(BG) underthe assembly map K0(BG)→ K0(C∗r (G)). Since by assumption the assemblymap is rationally injective, this implies [S] = 0 in K0(BG)⊗Z Q. Notice thatM is aspherical by assumption and hence M = BG. The homological Cherncharacter defines an isomorphism

K0(BG)⊗Z Q = K0(M)⊗Z Q∼=−→⊕p≥0

H2p(M ;Q)


which sends [S] to the Poincare dual L(M) ∩ [M ] of the Hirzebruch L-class L(M) ∈

⊕p≥0H

2p(M ;Q). This implies that L(M) ∩ [M ] = 0 andhence L(M) = 0. This contradicts the fact that the component of L(M) inH0(M ;Q) is 1. ut

More information about the Zero-in-the-spectrum Conjecture 8.55 can befound for instance in [518] and [527, Section 12].

8.5 Kasparov’s KK -Theory

Kasparov introduced the bivariant KK-theory which assigns to two separableC∗-algebras A and B a Z-graded abelian group KK∗(A,B). We give a verybrief summary of it. In the sequel all C∗-algebras are assumed to be separable.

8.5.1 Basic Properties of KK -theory for C∗-Algebras

(i) Bi-functorialityA homomorphism f : A→ B of C∗-algebras induces a homomorphisms ofZ-graded abelian groups

KK∗(f, idD) : KK∗(B,D)→ KK∗(A,D);

KK∗(idD, f) : KK∗(D,A)→ KK∗(D,B).

If g : B → C is another homomorphism of C∗-algebras, we have KK∗(g f, id) = KK∗(f, idD) K∗(g, idD) and KK∗(idD, g f) = KK∗(idD, g) K∗(idD, f). MoreoverK∗(idA, idB) = idKK∗(A,B). In particularKK∗(−, D)is a contravariant and KK∗(D,−) is a covariant functor from the categoryof C∗-algebras to the category of Z-graded abelian groups;

(ii) Homotopy invarianceIf f, g : A → B are homotopic homomorphisms of C∗-algebras, thenKK∗(f, idC) = KK∗(g, idC) and KK∗(idC , f) = KK∗(idC , g);

(iii) Finite direct productsIf A and B are C∗-algebras, there are a natural isomorphism of Z-gradedabelian groups

KK∗(A×B,C)∼=−→ KK∗(A,C)×KK∗(B,C);

KK∗(C,A×B)∼=−→ KK∗(C,A)×KK∗(C,B);

(iv) Countable direct sums in the first variableIf A =

⊕∞i=0Ai is a countable direct sum of C∗-algebras, then there is a

natural isomorphism

8.5 Kasparov’s KK -Theory 257

KKn

( ∞⊕i=0

Ai, B) ∼=−→ ∞∏

i=0

Kn(Ai, B);

(v) Morita equivalenceFor any integers m,n ≥ 1 there are natural isomorphisms of Z-graded

abelian groups KK∗(A,B)∼=−→ KK∗(Mm(A),Mn(B));

(vi) StabilizationThere are natural isomorphisms of Z-graded abelian groups

KK∗(A,B)∼=−→ KK∗(A⊗K, B)

∼=−→ KK∗(A,B⊗K)∼=−→ KK∗(A⊗K, B⊗K);

(vii) Long exact sequence of an ideal

Let 0→ Ii−→ A

p−→ A/I → 0 be an extensions of (separable) C∗-algebras.If it is semisplit in the sense of [112, Definition 19.5.1. on page 195] (what isautomatically true if A is nuclear,) then there exists a long exact sequence,natural in (A, I) and infinite to both sides,

· · · δn−1−−−→ KKn(A/I,B)KKn(p,idB)−−−−−−−−→ KKn(A,B)

KKn(i,idB)−−−−−−−→ KKn(I,B)

δn−→ KKn+1(A/I,B)KKn+1(p,idB)−−−−−−−−−→ KKn+1(A,B)

KKn+1(i,idB)−−−−−−−−−→ KKn+1(I,B, )δn+1−−−→ · · · ,

If the extension is semisplit or if B is nuclear, then there exists a longexact sequence, natural in (A, I) and infinite to both sides,

· · · ∂n+1−−−→ KKn(B, I)KKn(idB ,i)−−−−−−−→ KKn(B,A)

KKn(idB ,p)−−−−−−−−→ KKn(B,A/I)

∂n−→ KKn−1(B, I)KKn−1(idB ,i)−−−−−−−−−→ KKn−1(B,A)

KKn−1(idB ,p)−−−−−−−−−→ KKn−1(B,A/I)∂n−1−−−→ · · · ;

(viii) Bott periodicityThere exists an isomorphism of degree b(F )

β∗(A) : KK∗(A,B)∼=−→ KK∗+b(F )(A,B),

which is natural in A and B, and compatible with the boundary operators∂∗ of the long exact sequence of an ideal, where b(F ) = 2 if F = R andb(F ) = 8 if F = R;

(ix) Connection to topological K-theoryThere is a natural isomorphism of Z-graded abelian groups

K∗(A)∼=−→ KK∗(F,A);


(x) Homomorphisms of C∗-algebrasA homomorphism f : A → B of C∗-algebras defines an element [f ] inKK∗(A,B).

Remark 8.57 (Some failures). The second variable is in general not com-patible with countable direct sums and in particular not with colimits overdirected sets. However, in the special case A = C, this is the case, since thenKK∗(C, B) is just the topological K-theory of B.

The conditions about the existence of long exact sequence of an ideal suchas semi-split or B being nuclear are needed.

8.5.2 The Kasparov’s Intersection Product

One of the basic features of KK-theory is Kasparov’s intersection productwhich is a bilinear pairing of Z-graded abelian groups

⊗B : KK∗(A,B)⊗KK∗(B,C)→ KK∗(A,C).(8.58)

It has the following properties

(i) NaturalityIt is natural in A, B and C;

(ii) AssociativityIt is associative;

(iii) Composition of homomorphismsIf f : A → B and g : B → C are homomorphisms of C∗-algebras, then weget for the associated elements [f ] ∈ KK0(A,B), [g] ∈ KK0(B,C) and[g f ] ∈ KK0(A,C)

[g f ] = [f ]⊗B [g];

(iv) UnitsThere is a unit 1A := [idA] in KK0(A,A) for the intersection product.

Remark 8.59 (KK-equivalence). One of the basic features of the productis that an element x inKK0(A,B) induces a homomorphism−⊗Bx : Kn(A) =KKn(F,A)→ Kn(B) = KKn(F,B). Of course −⊗B [f ] agrees with Kn(f) iff : A→ B is a homomorphism of C∗-algebras. An element x ∈ KK0(A,B) iscalled a KK-equivalence if there exists an element y ∈ KK0(B,A) satisfyingx⊗By = 1A and y⊗Ax = 1B . The basic feature of a KK-equivalence is that−⊗Bx : Kn(A) = KKn(F,A) → Kn(B) = KKn(F,B) is automatically anisomorphism, the inverse is −⊗By.

Remark 8.60 (K-homology of C∗-algebras). One can define topologicalK-homology of a C∗-algebra K∗(A) by Kn(A) := KK−n(A,F ). It is in somesense dual to the topological K-theory K∗(A). Moreover, the intersectionproduct yields the index pairing

8.6 Equivariant Topological K-Theory and KK -Theory 259

Kn(A)⊗Z Kn(A)→ KK0(F, F ) = Z, (x, y) 7→ 〈x, y〉 := x⊗Ay.

If we take n = 0 and A = C(M) for a smooth closed Riemannian manifoldM , then an appropriate elliptic operator P over M defines an element in [P ]in K0(C(M)) = K0(M) and a vector bundle ξ over M defines an element inK0(C(M)) = K0(M) and the pairing 〈[ξ], [P ]〉 is the classical index of theelliptic operator obtained from P by twisting with ξ.

There are Universal Coefficient Theorems and Kunneth Theorems forKK-theory, see for instance [116, 117, 712, 743]. The Pimsner-Voiculescusequences associated to an automorphisms of a C∗-algebra are explained forKK-theory in [112, Theorem 19.6.1 on page 198].

More information about KK-theory, for instance about its constructionin terms of Kasparov modules or quasi-homomorphisms, other bivariant the-ories such as Ext for extensions of C∗-algebras, kk-theory, E-theory, andtheir relation to KK-theory, generalizations of these theories to more gen-eral operator algebras than C∗-algebras, universal properties of these theo-ries, applications to index theory and the relevant literature can be foundfor instance in [112, 213, 382, 384, 392, 419], or in the papers of Kas-parov [453, 454, 455, 456].

8.6 Equivariant Topological K-Theory and KK -Theory

Recall that groups are always assumed to be discrete. Given a group G, thereexists an equivariant version of KK-theory. It assigns to two G-C∗-algebras Aand B an Z-graded abelian group KKG

∗ (A,B) and has essentially the samebasic properties as non-equivariant KK-theory. Namely, it is a bi-functor,contravariant in the first and covariant in the second variable, is homotopyinvariant, satisfies Morita equivalence and stabilization, is split exact, i.e.,has long exact sequences for appropriate ideals, satisfies Bott periodicity, iscompatible with finite direct products in both variables and countable directsums in the first variable, and a homomorphism of G-C∗-algebras f : A→ Bdefines an element [f ] ∈ KKG

0 (A,B). There is also an equivariant version ofKasparov’s intersection product

⊗B : KKGj (A,B)⊗KKG

i (B,C)→ KKi+j(A,C)

which has all the expected properties as in the non-equivariant case.In particular we get on KKG

0 (F, F ) an interesting structure of a commu-tative ring with unit and it is sometimes called representation ring of G. IfG is finite, KKG

0 (F, F ) is indeed isomorphic as ring to RepF (G).There exists certain additional structures in the equivariant setting. Given

a homomorphism α : H → G, there are natural restriction homomorphisms


α∗ : KKG∗ (A,B)→ KKH(α∗A,α∗B)(8.61)

where α∗A and α∗B are the H-C∗-algebras obtained from the G-C∗-algebrasA and B by restring the H-action to a G-action using α. It is compatiblewith the equivariant Kasparov product.

Let i : H → G be the inclusion of groups. Given an H-C∗-algebra A,we define its induction i∗A, to be the G-C∗-algebra of bounded functionsf : G → A which satisfy f(gh) = h−1 · f(g) and vanish at infinity, i.e., forevery ε > 0 there exists a finite subset S ⊆ G/H such that for every g ∈ Gwith gH /∈ S we have ||f(g)|| ≤ ε. The norm is the supremums norm. Giveng ∈ G and such a function f , define g · f to be the function sending g′ ∈ Gto f(g−1g′).

Notice that the left FG-module FG⊗FH A, which is the algebraic induc-tion of A viewed as FH-module, embeds as a dense FG-submodule into i∗Aby sending g ⊗ a to the function which maps gh to h−1a for h ∈ H andg′ ∈ G with g′H 6= gH to zero. In other words, we can think of FG ⊗FH Aas the set of elements f ∈ i∗A such that gH ∈ G/H | f(g) 6= 0 is finite.In contrast to modules over group rings induction i∗ and restriction i∗ donot form an adjoint pair (i∗, i

∗) for equivariant C∗-algebras as the followingexercise illustrates.

Exercise 8.62. Let i : 1 → G be the inclusion of the trivial group into aninfinite discrete group G. Show that homG(i∗F, F ) and hom1(F, i

∗F ) arenot isomorphic, where F = R,C denotes both the obvious 1-C∗-algebraand the obvious G-C∗-algebra with trivial G-action.

If X is a proper H-CW -complex, then G×HX is a proper G-CW -complex

and we obtain an isomorphism of G-C∗-algebras i∗C0(X)∼=−→ C0(G ×H X)

which sends f ∈ i∗C0(X) to the function G ×H X → F (g, x) 7→ f(g)(x).Given a H-C∗-algebra A and a H-C∗-algebra B, there is a natural inductionhomomorphism

i∗ : KKH∗ (A,B)→ KKG(i∗A, i∗B).(8.63)

It is compatible with the equivariant Kasparov’s intersection product respect-ing the units. If j : G→ K is an inclusion, we get (j i)∗ = j∗ i∗.

There are descent homomorphisms

jGr : KKG∗ (A,B)→ KK∗(Aor G,B or G);(8.64)

jGr : KKG∗ (A,C)→ KK∗(Aor G,C);(8.65)

jGm : KKG∗ (A,B)→ KK∗(Aom G,B om G).(8.66)

The dual of the Green-Julg Theorem says that (8.65) is an isomorphism.The descent homomorphisms are natural and compatible with Kasparov’sintersection products respecting the units.

8.6 Equivariant Topological K-Theory and KK -Theory 261

Define the equivariant complex K-homology of a pair of proper G-CW -complexes (X,A) with coefficients in the complex G-C∗-algebra B by

KGn (X,A;B) := colimC⊆X KK

Gn (C0(C,C ∩A), B),(8.67)

where the colimit is taken over the directed system of cocompact proper G-CW -subcomplexes C ⊆ X, directed by inclusion, and C0(C,C ∩ A) is theG-C∗-algebra of continuous functions C → C which vanish on C ∩ A and atinfinity. This group is often denoted by RKn(X,A;B) in the literature andcalled equivariant K-homology with compact support but from a topologistspoint of view it is better to call it equivariant K-homology in view of itsdescription in terms of spectra, see Section 10.4. If B is C with the trivialG-action, we just write KG

∗ (X,A) for KG∗ (X,A;C), and this is precisely the

Z-graded abelian group which we have mentioned already in Subsection 8.2.3and will be constructed in terms of spectra in Section 10.4.

Next we explain the equivariant Chern character for equivariant complexK-homology. Denote for a proper G-CW -complex X by F(X) the set of allsubgroups H ⊂ G, for which XH 6= ∅, and by

ΛG(X) := Z[

1

F(X)

](8.68)

the ring Z ⊂ ΛG(X) ⊂ ΛG obtained from Z by inverting the orders of allsubgroups H ∈ F(X). Denote by JG(X) the set of conjugacy classes (C)of finite cyclic subgroups C ⊂ G for which XC is non-empty. Let C ⊂ Gbe a finite cyclic subgroup. Let CGC be the centralizer and NGC be thenormalizer of C ⊂ G. Let WGC be the quotient NGC/CGC. For a specificidempotent θC ∈ ΛC ⊗Z RepQ(C) defined in [528, Section 3] the cokernel of

⊕D⊂C,D 6=C

indCD :⊕

D⊂C,D 6=C

Z[

1

|C|

]⊗Z RepC(D)→ Z

[1

|C|

]⊗Z RepC(C)

is isomorphic to the image of the idempotent endomorphism

θC : Z[

1

|C|

]⊗Z RepC(C)→ Z

[1

|C|

]⊗Z RepC(C).

The element θC ∈ ΛC ⊗Z RepQ(C) is uniquely determined by the propertythat its character sends a generator of C to 1 and all other elements in C to0.

The next theorem is taken from [528, Theorem 0.7].

Theorem 8.69 (Equivariant Chern character for equivariant K-ho-mology). Let X be a proper G-CW -complex. Put Λ = ΛG(X) and J =JG(X). Let im(θC) ⊆ Λ ⊗Z RepC(C) be the image of θC : Λ ⊗Z RepC(C) →Λ⊗Z RepC(C).


Then there is for n ∈ Z a natural isomorphism

chGp (X) :⊕

(C)∈J

Λ⊗Z Kn(CGC\XC)⊗Λ[WGC] im(θC)∼=−→ Λ⊗Z K

Gn (X).

Notice that the isomorphism appearing in Theorem 8.69 exists alreadyover Λ, one does not have to pass to Q or C. This will be important whenwe will deal with the Modified Trace Conjecture in Subsection 12.8.1.

Example 8.70. In the special case, where G is finite, X is the one-point-space ∗ and n = 0, the equivariant Chern character appearing in Theo-rem 8.69 reduces to an isomorphism⊕

(C)∈JGZ[

1

|G|

]⊗Z[ 1

|G| ][WGC] im(θC)∼=−→ Z

[1

|G|

]⊗Z RepC(G).

where JG is the set of conjugacy classes (C) of cyclic subgroups C ⊂ G. Thisis a strong version of the well-known theorem of Artin that the map inducedby induction ⊕

(C)∈JGQ⊗Z RepC(C)→ Q⊗Z RepC(G)

is surjective for any finite group G.

Exercise 8.71. Let p be an odd prime and let V be an orthogonal Z/p-representation of dimension d such that V Z/p 6= 0. Denote by SV theZ/p-CW -complex consisting of elements v ∈ V of norm 1. Show

Z[1/p]⊗Z Kn(SV ) ∼=Z[1/p]

Z[1/p]p if d is even;

Z[1/p]2p if d is odd and n is even;

0 if d is odd and n is even.

Analogously to the complex case one defines equivariant real K-homologyKOG∗ (X,A;B) of a pair of proper G-CW -complexes (X,A) with coeffi-cients in the real G-C∗-algebra B. We will abbreviate KOG∗ (X,A) :=KOGn (X,A;R), where R carries the trivial G-action, This is precisely theZ-graded abelian group which we will be constructed in terms of spectra inSection 10.4.

For discussions of universal coefficient theorems for equivariant K-theorysee [569, 711, 712].

Further information about equivariant KK-theory can be found for in-stance in [112, Section 20], [457], and [785].

8.7 Comparing Algebraic and Topological K-theory of C∗-Algebras 263

8.7 Comparing Algebraic and Topological K-theory ofC∗-Algebras

Let A be a C∗-algebra. Then Kn(A) denotes in most cases topological K-theory, but it can also mean the algebraic K-theory of A considered just as aring. To avoid this ambiguity, we will use in this Section 8.7 the superscriptstop and alg to make clear what we mean.

There is for any C∗-algebra over R or C a canonical map of spectra

t(A) : Kalg(A)→ Ktop(A)(8.72)

from the non-connective algebraic K-theory spectrum of A just consideredas ring to the topological K-theory spectrum associated to the C∗-algebraA, see [706, Theorem 4 on page 851]. It induces homomorphisms of abeliangroups for all n ∈ Z

tn(A) = Kn(t(A)) : Kalgn (A)→ Ktop

n (A).(8.73)

It is always an isomorphism for n = 0, but in general far from being a bijectionas illustrated by the following

Exercise 8.74. Let X be a finite non-empty CW -complex. Prove that thecomparison map K1(C(X))→ Ktop

1 (C(X)) is never injective.

The situation is different if A is stable or if one uses coefficients in Z/k.Namely, the following result is proved in [773, Theorem 10.9] over C andn ≥ 1, but holds in the more general form below by [706, Theorem 19 onpage 863], see also Higson [383].

Theorem 8.75 (Karoubi’s Conjecture). Karoubi’s Conjecture is true,i.e., for any stable C∗-algebra A over R or C the canonical map t of (8.72)is weak homotopy equivalence i.e., the maps tn of (8.73) are bijective forn ∈ Z.

Given an integer k ≥ 2, we have introduced Kalgn (A;Z/k) in Section 5.4.

The analogous construction works for topological K-theory and there is theanalogue of (8.73), a natural homomorphisms

tn(A;Z/k) : Kalgn (A;Z/k)→ Ktop

n (A;Z/k).(8.76)

We mention the following conjecture of Rosenberg [702, Conjecture 4.1]or [706, Conjecture 26 on page 869].

Conjecture 8.77 (Comparing algebraic and topological K-theorywith coefficients for C∗-algebras). If A is a C∗-algebra and k ≥ 2 aninteger, then the comparison map


Kn(A;Z/k)→ Ktopn (A;Z/k)

is bijective for n ≥ 1.

The map Kn(A;Z/k) → Ktopn (A;Z/k) appearing in Conjecture 8.77 is

known to be bijective for n = 1 and to be surjective for n ≥ 1 by [445,Theorem 2.5]. Conjecture 8.77 is true if A is stable by Theorem 8.75, or ifA is commutative, see [320],[666], Rosenberg [702, Theorem 4.2] and [706,Theorem 27 on page 870].

A discussion about Ki-regularity and the homotopy invariance of Kn(A)for n ≤ −1 is discussed for C∗-algebras in [706, Section 3.3.4 on page 865ff].

More information about the relation between algebraic and topologicalK-theory can be found in [202].

8.8 Comparing Algebraic L-Theory and TopologicalK-theory of C∗-Algebras

Whereas the algebraic and the topological K-theory of a C∗-algebra are verydifferent in general, the topological K-theory of a C∗ algebra is closely relatedto the L-theory of the C∗-algebra just considered as ring with involution. Thisis illustrated by the following result.

Theorem 8.78 (L-theory and topological K-theory of C∗-algebras).

(i) A generalized signature defines for any unital C∗-algebra over R or C anatural isomorphism

Lp0(A)∼=−→ K0(A);

(ii) Let A be a unital C∗-algebra over C. Then there is for all n ∈ Z a naturalisomorphism

Kn(A)∼=−→ Lpn(A);

(iii) Let A be a unital C∗-algebra over R. Then there is a natural homomor-phism

K1(A)∼=−→ Lh1 (A)

which is surjective and whose kernel has at most order two;(iv) For any unital C∗-algebra over R or C there are natural isomorphisms

Kn(A)[1/2] ∼= Lpn(A)[1/2].

Proof. (i) See [705, Theorem 1.6].

(ii) See [587, Theorem 0.2], [601], [705, Theorem 1.8].

(iii) See [705, Theorem 1.9].


(iv) See [705, Theorem 1.11], where this result described as a consequence ofKaroubi [443, 444]. ut

8.9 Topological K-Theory for Finite Groups

Notice that CG = l1(G) = C∗r (G) = C∗max(G) holds for a finite group, andanalogous for the real versions.

Theorem 8.79 (Topological K-theory of the C∗-algebra of finitegroups). Let G be a finite group.

(i) We have

Kn(C∗r (G)) ∼=R(G) ∼= ZrC(G) for n even;0 for n odd.

(ii) There is an isomorphism of topological K-groups

Kn(C∗r (G;R)) ∼= Kn(R)rR(G;R) ×Kn(C)rR(G;C) ×Kn(H)rR(G;H).

Moreover Kn(C) is 2-periodic with values Z, 0 for n = 0, 1, Kn(R) is8-periodic with values Z, Z/2, Z/2, 0, Z, 0, 0, 0 for n = 0, 1, . . . , 7 andKn(H) = Kn+4(R) for n ∈ Z.

Proof. One gets isomorphisms of C∗-algebras

C∗r (G) ∼=rC(G)∏j=1

Mni(C)

and

C∗r (G;R) ∼=rR(G;R)∏i=1

Mmi(R)×rR(G;C)∏i=1

Mni(C)×rR(G;H)∏i=1

Mpi(H)

from [748, Theorem 7 on page 19, Corollary 2 on page 96, page 102, page106].Now the claim follows from Morita invariance and the well-known values forKn(R), Kn(C) and Kn(H) (see for instance [780, page 216]). ut

8.10 Miscellaneous

Bivariant algebraic K-theory is investigated in [204, 339].



name of texfile: ic

Chapter 9

Classifying Spaces for Families

9.1 Introduction

This chapter is devoted to classifying spaces for families of subgroups. Theyare key input in the general formulations of the Baum-Connes Conjectureand the Farrell-Jones Conjecture.

If one only wants to understand these conjectures, one only needs to knowthat a family of subgroups F is a set of subgroups of G, closed under con-jugation and passing to subgroups, a G-CW -model for the classifying spaceEF (G) is a G-CW -complex whose isotropy groups belong to F and whose H-fixed point set is contractible for every H ∈ F , such a G-CW -model alwaysexists, and two such G-CW -models are G-homotopy equivalent. Only if oneis interested in concrete computations, it is very useful to know situationswhere one can find small G-CW -models for specific G and F .

Nevertheless, we give much more information about classifying spaces forfamilies since they are interesting in their own right and are important toolsfor investigating groups. After we have explained the basic G-homotopy the-oretic aspects, we pass to the classifying space EG = ECOM(G) for properaction which is the same as the classifying space for the family COM of com-pact subgroups. There are many prominent groups for which the are nicegeometric models for EG. The G-CW -complex EG is relevant for the Baum-Connes Conjecture. For the Farrell-Jones Conjecture we also have to dealwith EG = EVCY(G) for the family VCY of virtually cyclic subgroups whichis much harder to analyze. We systematically address the question whetherthere are finite or finite dimensional G-CW -models and what the minimaldimension of such G-CW -models for EF (G) are for F = FIN ,VCY.

9.2 Definition and Basic Properties ofG-CW -Complexes

Remark 9.1 (Compactly generated spaces). In the sequel we will workin the category of compactly generated spaces. This convenient category isexplained in detail in [561, Appendix A], [764] and [832, I.4]. A reader mayignore this technical point without harm, but we nevertheless give a shortexplanation.

267

268 9 Classifying Spaces for Families

A Hausdorff space X is called compactly generated if a subset A ⊆ Xis closed if and only if A ∩ K is closed for every compact subset K ⊆ X.Given a topological space X, let k(X) be the compactly generated topolo-gical space with the same underlying set as X and the topology for whicha subset A ⊆ X is closed if and only if for every compact subset K ⊆ Xthe intersection A ∩ K is closed in the given topology on X. The identityinduces a continuous map k(X)→ X which is a homeomorphism if and onlyif X is compactly generated. The spaces X and k(X) have the same compactsubsets. Locally compact Hausdorff spaces and every Hausdorff space whichsatisfies the first axiom of countability are compactly generated. In particularmetrizable spaces are compactly generated.

Working in the category of compactly generated spaces means that oneonly considers compactly generated spaces and whenever a topological con-struction such as the cartesian product or the mapping space leads outof this category, one retopologizes the result as described above to get acompactly generated space. The advantage is for example that in the cate-gory of compactly generated spaces the exponential map map(X × Y, Z) →map(X,map(Y,Z)) is always a homeomorphism, for an identification p : X →Y the map p × idZ : X × Z → Y × Z is always an identification and for afiltration by closed subspaces X1 ⊂ X2 ⊆ . . . ⊆ X such that X is the colimitcolimn→∞Xn, we always get X × Y = colimn→∞(Xn × Y ).

One may also work in the category of weak Hausdorff spaces, see for in-stance Strickland [769].

In the sequel G is a topologically group (which is compactly generated).Subgroups are understood to be closed.

Definition 9.2 (G-CW -complex). A G-CW -complex X is a G-space to-gether with a G-invariant filtration

∅ = X−1 ⊆ X0 ⊂ X1 ⊆ . . . ⊆ Xn ⊆ . . . ⊆⋃n≥0

Xn = X

such that X carries the colimit topology with respect to this filtration (i.e., aset C ⊆ X is closed if and only if C∩Xn is closed in Xn for all n ≥ 0) and Xn

is obtained from Xn−1 for each n ≥ 0 by attaching equivariant n-dimensionalcells, i.e., there exists a G-pushout

∐i∈In G/Hi × Sn−1

∐i∈In q

ni //

Xn−1

∐i∈In G/Hi ×Dn

∐i∈In Q

ni // Xn

The space Xn is called the n-skeleton of X. Notice that only the filtra-tion by skeletons belongs to the G-CW -structure but not the G-pushouts,

9.2 Definition and Basic Properties of G-CW -Complexes 269

only their existence is required. An equivariant open n-dimensional cell isa G-component of Xn − Xn−1, i.e., the preimage of a path component ofG\(Xn − Xn−1). The closure of an equivariant open n-dimensional cell iscalled an equivariant closed n-dimensional cell . If one has chosen the G-pushouts in Definition 9.2, then the equivariant open n-dimensional cellsare the G-subspaces Qi(G/Hi × (Dn − Sn−1)) and the equivariant closedn-dimensional cells are the G-subspaces Qi(G/Hi ×Dn).

The attaching maps in the G-pushout explaining how Xn−1 is obtainedfrom Xn are not part of the structure of a G-CW -complex, only their exis-tence is required.

It is obvious what a pair of G-CW -complexes is.

Remark 9.3 (G-CW -complexes versus CW -complexes with G-action).Suppose that G is discrete. A G-CW -complex X is the same as a CW -complex X with G-action such that for each open cell e ⊆ X and each g ∈ Gwith ge ∩ e 6= ∅ left multiplication with g induces the identity on e.

The definition of a G-CW -complex appearing in Definition 9.2 has theadvantage that it makes also sense for topological groups.

Example 9.4 (Simplicial actions). Let X be a simplicial complex onwhich the group G acts by simplicial automorphisms. Then G acts also onthe barycentric subdivision X ′ by simplicial automorphisms. The filtration ofthe barycentric subdivision X ′ by the simplicial n-skeleton yields the struc-ture of a G-CW -complex what is not necessarily true for X. This becomesclear if one considers the standard 2-simplex with the obvious actions of thesymmetric group S3 given by permuting the three vertices.

A map f : X → Y between G-CW -complexes is called cellular if f(Xn) ⊆Yn holds for all n ≥ 0.

For a subgroup H ⊆ G denote by NGH = g ∈ G | gHg−1 = H itsnormalizer and by WGH = NGH/H its Weyl group.

Lemma 9.5.

(i) Let X be a G-CW -complex and let Y be an H-CW -complex. Then X ×Ywith the product G×H-action is a G×H-CW -complex;

(ii) Let X be a G-CW -complex and let H ⊆ G be a subgroup. Suppose thatG is discrete or that H is open and closed in G. Then X viewed as anH-space by restriction inherits the structure of an H-CW -complex;

(iii) Consider a G-pushout

X0i1 //

i2

X1

j1

x2

j2// X

Suppose that Xi for i = 0, 1, 2 is a G-CW -complex and that i1 is cellularand i2 is the inclusion of a pair of G-CW -complexes. Then X inherits thestructure of a G-CW -complex;


(iv) Let X be a G-CW -complex and let H ⊆ G be a subgroup. Then XH viewedas an WGH-space inherits the structure of a WGH-CW -complex providedthat G is discrete, or that K ⊆ G is open and closed, or that G is a Liegroup and H ⊆ G is compact;

(v) Let X be a G-CW -complex and let H ⊆ G be a normal subgroup. ThenX/H viewed as an G/H-space inherits the structure of a G/H-CW -complex.

Proof. (9.5) The skeletal filtration on X × Y is given by

(X × Y )n =⋃

p+q=n

Xp × Yq.

To ensure that X × Y is the colimit colimn→∞(X × Y )n one needs to workin the category of compactly generated spaces.

(ii) Use the same filtration on X viewed as an H-space as for the G-CW -complex X.

(iii) Define the filtration on XH given by

Xn = j1((X1)n

)∪ j2

((X2)n

).

(iv) The G-action on X induces a NGH-action on XH which in turn passes toa WGH-action on XH . Take the n-skeleton of XH to be (Xn)H . Use the factthat for every K ⊆ G the space (G/K)H is a disjoint union of WGH-orbitswhat is obvious if G is discrete, or if K ⊆ G is open and closed, and followsfor a Lie group G and compact K ⊆ G for instance from [521, Theorem 1.33on page 23].

(v) The n-skeleton of X/H is the image of Xn under the canonical projectionX → X/H. ut

Exercise 9.6. Let p : X → X be the universal covering of the connectedCW -complex X with fundamental group π. Show that the π-space X inheritsthe structure of a π-CW -complex.

Exercise 9.7. Let p be a prime number and let X be a compact Z/p-CW -complex. Show that X and XZ/p are compact CW -complexes and their Eulercharacteristics satisfy

χ(X) ≡ χ(XZ/p) mod p.

Definition 9.8 (Type of a G-CW -complex). A G-CW -complex is calledfinite if it is built by finitely many equivariant cells.

A G-CW -complex is called of finite type if each n-skeleton is a finite G-CW -complex.

9.3 Proper G-Spaces 271

A G-CW -complex is called of dimension ≤ n if X = Xn. It is called n-dimensional or of dimension n if X = Xn and X 6= Xn−1 holds. It is calledfinite dimensional if it is of dimension ≤ n for some integer n.

Remark 9.9 (Slice Theorem). A Slice Theorem for G-CW -complexes isproved in [561, Theorem 7.1]. It says roughly, that for a G-CW -complex X wecan find for any x ∈ X an arbitrary small Gx-subspace Sx and an arbitrarysmall open G-invariant neighborhood U of x such that the closure of Sx iscontained in U , the inclusion x → Sx is a Gx-homotopy equivalence andthe canonical G-map

G×Gx Sx → U, (g, s) 7→ g · s

is a G-homeomorphism.

9.3 Proper G-Spaces

Definition 9.10 (Proper G-space). A G-space X is called proper if foreach pair of points x and y in X there are open neighborhoods Vx of x andWy of y in X such that the closure of the subset g ∈ G | gVx ∩Wy 6= ∅ ofG is compact.

Lemma 9.11. A G-CW -complex X is proper if and only if all its isotropygroups are compact.

Proof. This is shown in [521, Theorem 1.23]. ut

In particular a free G-CW -complex is always proper. However, not everyfree G-space is proper.

Exercise 9.12. Find a free Z-space which is not proper.

Remark 9.13 (Lie Groups acting properly and smoothly on mani-folds). Let G be a Lie group. If M is a (smooth) proper G-manifold, then anequivariant smooth triangulation induces a G-CW -structure on M . For theproof and for equivariant smooth triangulations we refer to [412, Theorem Iand II].

Exercise 9.14. Let p be an odd prime. Show that there is no smooth freeZ/p-action on an even-dimensional sphere.


9.4 Maps between G-CW -Complexes

Theorem 9.15 (Equivariant cellular approximation Theorem). Let(X,A) be a pair of G-CW -complexes and let Y be a G-CW -complex. Letf : X → Y be a G-map such that f |A : A→ Z is cellular.

Then there exists a cellular G-map f ′ : X → Y such that f |A = f ′|A andf and f ′ are G-homotopic relative A.

Proof. Since X = colimn→∞Xn by definition, it suffices to construct induc-tively for n = −1, 0, 1, 2, . . . G-maps

hn : Xn × [0, 1] ∪X × 0 → Y

such that hn(x, 0) = f(x) for every x ∈ Xn and hn(x, t) = hn−1(x, t) forevery x ∈ Xn−1 and t ∈ [0, 1] hold and the map f ′n : X → Y sending x ∈ Xn

to hn(x, 1) is cellular. The induction beginning n = −1 is trivial, defineh−1 : A× [0, 1] ∪X × 0 → Y by sending (x, t) to f(x). The induction stepfrom (n− 1) to n is done as follows. Choose a G-pushout


∐i∈In qi //

Xn−1 ∪A

∐i∈In G/Hi ×Dn

∐i∈In Qi // Xn ∪A

It yields the G-pushout

∐i∈In G/Hi ×

(Sn−1 × [0, 1] ∪Dn × 0

) ∐i∈In q

′i //

Xn−1 × [0, 1] ∪X × 0

∐i∈In G/Hi ×Dn × [0, 1]

∐i∈In Q

′i // Xn × [0, 1] ∪X × 0

Because of the G-pushout property it suffices to explain for every i ∈ Inhow to extend the composite

φi : G/Hi ×(Sn−1 × [0, 1] ∪Dn × 0

) q′i−→ Xn−1 × [0, 1] ∪X × 0 hn−1−−−→ Y

to a G-mapΦi : G/Hi ×Dn × [0, 1]→ Y

such that Φi(G/Hi×Dn×1

)⊆ Yn. This is the same as the non-equivariant

problem to extend the map

φ′i : Sn−1 × [0, 1] ∪Dn × 0 → Y H

9.4 Maps between G-CW -Complexes 273

obtained from φi by restriction to eHi ×(Sn−1 × [0, 1] ∪Dn × 0

)to a

mapΦ′i : D

n × [0, 1]→ Y H

such that Φ′i(Dn × 1

)⊆ Yn, since we can then define Φi(gH, x, t) := g ·

Φ′i(x, t). It is not hard to check that this non-equivariant problem can besolved if the inclusion Y Hm−1 → Y Hm is m-connected for every m ≥ 0. Sincewe have the pushout of spaces

∐i∈Im G/H

Hi × Sm−1

∐i∈Im qmi //

Ym−1

∐i∈In G/H

Hi ×Dm

∐i∈In Q

mi // Ym

the inclusion G/HHi × Sm−1 → G/HH

i ×Dm is m-connected and G/HHi ×

Sm−1 is a deformation retract of an open neighborhood in G/HHi ×Dm, this

follows from Blakers-Massey excision theorem, see [789, Proposition 6.4.2 onpage 133]. ut

A map f : X → Y of spaces is called a weak homotopy equivalence if finduces a bijection π0(f) : π0(X) → π0(Y ) and for every x ∈ X and n ≥ 1an isomorphism πn(f, x) : πn(X,x) → πn(Y, f(x)). A G-map f : X → Y ofG-spaces is called a weak G-homotopy equivalence if fH : XH → Y H is aweak equivalence of spaces for all subgroups H ⊆ G.

Theorem 9.16 (Equivariant Whitehead Theorem).

(i) Let f : Y → Z be a G-map between G-spaces. Then f is a weak G-homotopyequivalence if for every G-CW -complex X the map induced by f betweenthe G-homotopy classes of G-maps

f∗ : [X,Y ]G → [X,Z]G, [h] 7→ [f h]

is bijective;(ii) Let f : Y → Z be a G-map between G-CW -complexes. Then the following

assertions are equivalent:

(a) f is a G-homotopy equivalence;(b) f is a weak G-homotopy equivalence;(c) For every H ⊆ G which occurs as isotropy group of some point in X or

Y , the map fH : XH → Y H is a weak homotopy equivalence of spaces.

Proof. See [788, II.2.6], [521, Theorem 2.4 on page 36]. ut

Exercise 9.17. Let Y be a G-space. A G-CW -approximation of Y is a G-CW -complex X together with a weak G-homotopy equivalence f : X → Y .Show that two G-CW -approximations of Y are G-homotopy equivalent.


9.5 Definition and Basic Properties of ClassifyingSpaces for Families

Recall that we have defined the notion of a family of subgroups of a groupG in Definition 1.52, namely, to be a set of subgroups of G which is closedunder conjugation with elements of G and under passing to subgroups, andlisted some examples in Notation 1.53, for instance the family T R consistingof the trivial subgroup, the family FIN of finite subgroups, the family VCYof virtually cyclic subgroups, and the family ALL of all subgroups. (Actuallyone could replace the condition that F is closed under taking subgroups bythe weaker condition that the intersection of finitely many elements of Fbelongs to F . Then the set of compact open subgroups is a family.)

Definition 9.18 (Classifying G-CW -complex for a family of sub-groups). Let F be a family of subgroups of G. A model EF (G) for theclassifying G-CW -complex for the family F of subgroups of G or classifyingspaces for the family F of subgroups of G is a G-CW -complex EF (G) whichhas the following properties:

(i) All isotropy groups of EF (G) belong to F ;(ii) For any G-CW -complex Y , whose isotropy groups belong to F , there is

up to G-homotopy precisely one G-map Y → X.

We abbreviate EG := ECOM(G) and call it the universal G-CW -complexfor proper G-actions.

If G is discrete, we have EG := EFIN (G).In other words, EF (G) is a terminal object in the G-homotopy category

of G-CW -complexes, whose isotropy groups belong to F . In particular twomodels for EF (G) are G-homotopy equivalent and for two families F0 ⊆ F1

there is up to G-homotopy precisely one G-map EF0(G)→ EF1(G).

Theorem 9.19 (Homotopy characterization of EF (G)). Let F be afamily of subgroups.

(i) There exists a model for EF (G) for any family F ;(ii) A G-CW -complex X is a model for EF (G) if and only if all its isotropy

groups belong to F and for each H ∈ F the H-fixed point set XH isweakly contractible, i.e., XH is path connected and πn(XH , y) vanishesfor all n ≥ 1 and one (and hence all) basepoints y ∈ XH ;

Proof. (i) A model can be obtained by attaching equivariant cells G/H ×Dn

for all H ∈ F to make the H-fixed point sets weakly contractible. See forinstance [521, Proposition 2.3 on page 35]. There are also functorial construc-tions for discrete G generalizing the bar construction (see [222, Section 3 andSection 7]).

(ii) Suppose that the G-CW -complex X is a model for EF (G). Let Y be any

9.6 Special Models for the Classifying Space of Proper Actions 275

CW -complex and let H ∈ F . Then there is up to G-homotopy precisely oneG-map G/H × Y → X. Hence there is up to homotopy precisely one mapY → XH . This is equivalent to the condition that XH is weakly contractible.

Suppose that XH is weakly contractible for every H ∈ F . Let (Y,B) be aG-CW -pair and f−1 : B → X be any G-map. We next show the existence of aG-map f : Y → X extending f−1. Obviously this implies that X is a model forEF (G). Since Y is the colimit over the skeletons Yn for n ≥ −1 and Y−1 = B,it suffices to prove for n ≥ 0 that for a given G-map fn−1 : Yn−1 → X thereexists a G-map fn : Yn → X with fn|Yn−1

= fn−1. Recall that by definitionthere exists a G-pushout


∐i∈In q

ni //

Yn−1

∐i∈In G/Hi ×Dn

∐i∈In Q

ni // Yn

such that each Hi belong to F . Because of the universal property of a G-pushout it remains to show for every H ∈ F that every G-map u : G/H ×Sn−1 → X can be extended to a G-map v : G/H ×Dn → X. This is equiv-alent to showing that every map u′ : Sn−1 → XH can be extended to amap v′ : Dn → XH . This follows from the assumption that XH is weaklycontractible. ut

A model for EALL(G) is G/G. A model for ET R(G) is the same as a modelfor EG i.e, the universal covering of BG, or, equivalently, the total space ofthe universal G-principal bundle. In Section 9.6 we will give many interestinggeometric models for classifying spaces EG = EFIN (G).

Exercise 9.20. Show for a discrete groupG that there exists a zero-dimensionalmodel for EF (G) if and only if G ∈ F . Is there a non-trivial connected Liegroup L with a 0-dimensional model for EL?

9.6 Special Models for the Classifying Space of ProperActions

In this section we present some interesting geometric models for the clas-sifying space of proper actions EG for some discrete groups. These modelswill often be small in the sense that they are finite, of finite type or finitedimensional. We will restrict ourselves to discrete groups G in this section.More information, also for non-discrete groups, can be found for instancein [91, 532].


9.6.1 Simplicial Model

Let P∞(G) be the geometric realization of the simplicial set whose k-simplicesconsist of (k+1)-tupels (g0, g1, . . . , gk) of elements gi in G. There is an obvioussimplicial G-action of G on P∞(G) coming from the group structure. We getfor instance from [1, Example 2.6].

Theorem 9.21 (Simplicial model). P∞(G) is a model for EG.

9.6.2 Operator Theoretic Model

Let C0(G) be the Banach space of complex valued functions of G with finitesupport, where the Banach structure comes from the supremum norm. Thegroup G acts isometrically on C0(G) by (g · f)(x) := f(g−1x) for f ∈ C0(G)and g, x ∈ G. Let PC0(G) be the subspace of C0(G) consisting of functionsf such that f is not identically zero and has non-negative real numbers asvalues.

Let XG be the space

XG =

f : G→ [0, 1]

∣∣∣∣ f has finite support,∑g∈G

f(g) = 1

with the topology coming from the supremum norm.

Theorem 9.22 (Operator theoretic model). Both PC0(G) and XG areG-homotopy equivalent to a G-CW -model for EG.

Proof. See[1, Theorem 2.4] and [91, page 248]. ut

Remark 9.23 (Comparing P∞(G) and XG). The simplicial G-complexP∞(G) of Theorem 9.21 and the G-space XG of Theorem 9.22 have the sameunderlying sets but in general they have different topologies. The identitymap induces a (continuous) G-map P∞(G) → XG which is a G-homotopyequivalence, but in general not a G-homeomorphism (see also [797, A.2]).

9.6.3 Discrete Subgroups of Almost Connected Lie Groups

The next result is a special case of a much more general result due to Abels [1,Corollary 4.14].

Theorem 9.24 (Discrete subgroups of almost connected Lie groups).Let L be a Lie group. Suppose that L is almost connected, i.e., it consists of

finitely many path components. Let G ⊆ L be a discrete subgroup.


Then L contains a maximal compact subgroup K which is unique up toconjugation, and the G-space L/K is a model for EG.

9.6.4 Actions on Simply Connected Non-Positively CurvedManifolds

Theorem 9.25 (Actions on simply connected non-positively curvedmanifolds). Suppose that G acts properly and isometrically on the simply-connected complete Riemannian manifold M with non-positive sectional cur-vature. Then M is a model for EG.

Proof. See [1, Theorem 4.15]. ut

9.6.5 Actions on Trees and Graphs of Groups

A tree is a 1-dimensional CW -complex which is contractible.

Theorem 9.26 (Actions on trees). Suppose that G acts on a tree T suchthat for each element g ∈ G and each open cell e with g · e ∩ e 6= ∅ we havegx = x for any x ∈ e. Assume that the isotropy group of each x ∈ T is finite.

Then T is a model for EG.

Proof. Obviously T is a G-CW -complex (see Remark 9.3). Let H ⊆ G befinite. If e0 is a zero-cell in T , then H · e0 is finite. Let T ′ be the union ofall geodesics with extremities in H · e. This is an H-invariant subtree of Tof finite diameter. One shows now inductively over the diameter of T ′ thatT ′ has a vertex which is fixed under the H-action (see [750, page 20] or [248,Proposition 4.7 on page 17]). Hence TH is non-empty. If e and f are verticesin TH , the geodesic in T from e to f must be H-invariant. Hence TH is aconnected CW -subcomplex of the tree T and hence is itself a tree. This showsthat TH is contractible. Now apply Theorem 9.19 (ii). ut

9.6.6 Actions on CAT(0)-Spaces

For the notion of a CAT(0)-space we refer for instance to [129, Definition 1.1in II.1 on page 158].

Theorem 9.27 (Actions on CAT(0)-spaces). Let X be a proper G-CW -complex. Suppose that X has the structure of a complete CAT(0)-space onwhich G acts by isometries. Then X is a model for EG.


Proof. By [129, Corollary 2.8 in II.2 on page 179] the K-fixed point set of Xis a non-empty convex subset of X and hence contractible for any compactsubgroup K ⊂ G. ut

This result contains as special case Theorem 9.25 and Theorem 9.26 sincesimply-connected complete Riemannian manifolds with non-positive sectionalcurvature and trees are complete CAT(0)-spaces.

9.6.7 The Rips Complex of a Hyperbolic Group

A metric space X = (X, d) is called δ-hyperbolic for a given real numberδ ≥ 0 if for any four points x, y, z, t the following inequality holds

d(x, y) + d(z, t) ≤ maxd(x, z) + d(y, t), d(x, t) + d(y, z)+ 2δ.(9.28)

A group G with a finite set S of generators is called δ-hyperbolic if themetric space (G, dS) given by G and the word-metric dS with respect to thefinite set of generators S is δ-hyperbolic.

The Rips complex Pd(G,S) of a group G with a finite set S of generatorsfor a natural number d is the geometric realization of the simplicial set whoseset of k-simplices consists of (k+ 1)-tuples (g0, g1, . . . gk) of pairwise distinctelements gi ∈ G satisfying dS(gi, gj) ≤ d for all i, j ∈ 0, 1, . . . , k. TheobviousG-action by simplicial automorphisms on Pd(G,S) induces aG-actionby simplicial automorphisms on the barycentric subdivision Pd(G,S)′ (seeExample 9.4).

Theorem 9.29 (Rips complex). Let G be a group with a finite set S ofgenerators. Suppose that (G,S) is δ-hyperbolic for the real number δ ≥ 0. Letd be a natural number with d ≥ 16δ + 8. Then the barycentric subdivision ofthe Rips complex Pd(G,S)′ is a finite G-CW -model for EG.

Proof. See [578], [579]. ut

A metric space is called hyperbolic if it is δ-hyperbolic for some real numberδ ≥ 0. A finitely generated group G is called hyperbolic if for one (and henceall) finite set S of generators the metric space (G, dS) is a hyperbolic metricspace. Since for metric spaces the property hyperbolic is invariant underquasiisometry and for two finite sets S1 and S2 of generators of G the metricspaces (G, dS1

) and (G, dS2) are quasiisometric, the choice of S does not

matter. Theorem 9.29 implies that for a hyperbolic group there is a finiteG-CW -model for EG.

The notion of a hyperbolic group is due to Gromov and has intensively beenstudied (see for example [129, 344, 352]). The prototype is the fundamentalgroup of a closed hyperbolic manifold.


9.6.8 Arithmetic Groups

An arithmetic group A in a semisimple connected linear Q-algebraic grouppossesses a finite A-CW -model for EA. Namely, let G(R) be the R-pointsof a semisimple Q-group G(Q) and let K ⊆ G(R) be a maximal compactsubgroup. If A ⊆ G(Q) is an arithmetic group, then G(R)/K with the leftA-action is a model for EA as already explained in Theorem 9.24. The A-space G(R)/K is not necessarily cocompact. The Borel-Serre completion ofG(R)/K (see [120], [749]) is a finite A-CW -model for EG as pointed outin [18, Remark 5.8], where a private communication with Borel and Prasadis mentioned.

9.6.9 Outer Automorphism Groups of Finitely Generated FreeGroups

Let Fn be the free group of rank n. Denote by Out(Fn) the group of outerautomorphisms of Fn, i.e., the quotient of the group of all automorphismsof Fn by the normal subgroup of inner automorphisms. Culler and Vogt-mann [208, 802] have constructed a space Xn, called outer space, on whichOut(Fn) acts with finite isotropy groups. It is analogous to the Teichmullerspace of a surface with the action of the mapping class group of the sur-face. Fix a graph Rn with one vertex v and n-edges and identify Fn withπ1(Rn, v). A marked metric graph (g, Γ ) consists of a graph Γ with all ver-tices of valence at least three, a homotopy equivalence g : Rn → Γ calledmarking, and to every edge of Γ there is assigned a positive length whichmakes Γ into a metric space by the path metric. We call two marked metricgraphs (g, Γ ) and (g′, Γ ′) equivalent of there is a homothety h : Γ → Γ ′ suchthat g h and h′ are homotopic. Homothety means that there is a constantλ > 0 with d(h(x), h(y)) = λ · d(x, y) for all x, y. Elements in outer space Xn

are equivalence classes of marked graphs. The main result in [208] is that Xis contractible. Actually, for each finite subgroup H ⊆ Out(Fn) the H-fixedpoint set XH

n is contractible [483, Proposition 3.3 and Theorem 8.1], [831,Theorem 5.1].

The space Xn contains a spine Kn which is an Out(Fn)-equivariant de-formation retraction. This space Kn is a simplicial complex of dimension(2n−3) on which the Out(Fn)-action is by simplicial automorphisms and co-compact. Actually the group of simplicial automorphisms of Kn is Out(Fn)[130]. Hence the barycentric subdivision K ′n is a finite (2n − 3)-dimensionalmodel of EOut(Fn).


9.6.10 Mapping Class Groups

Let Γ sg,r be the mapping class group of an orientable compact surface F sg,rof genus g with s punctures and r boundary components. This is the groupof isotopy classes of orientation preserving selfdiffeomorphisms F sg,r → F sg,r,which preserve the punctures individually and restrict to the identity on theboundary. We require that the isotopies leave the boundary pointwise fixed.We will always assume that 2g + s + r > 2, or, equivalently, that the Eulercharacteristic of the punctured surface F sg,r is negative. It is well-known thatthe associated Teichmuller space T sg,r is a contractible space on which Γ sg,racts properly.

Theorem 9.30 (Mapping class group). The Teichmuler space T sg,r is amodel for EΓ sg,r

Proof. This follows from [467]. ut

Remark 9.31 (Finite model for EΓ sg,r). There exist a finite Γ sg,r-CW -model for EΓ sg,r, see [605].

9.6.11 Special Linear Groups of (2,2)-Matrices

In order to illustrate some of the general statements above we consider thespecial example SL2(Z).

Let H2 be the 2-dimensional hyperbolic space. We will use either the up-per half-plane model or the Poincare disk model. The group SL2(R) acts byisometric diffeomorphisms on the upper half-plane by Moebius transforma-

tions, i.e., a matrix

(a bc d

)acts by sending a complex number z with positive

imaginary part to az+bcz+d . This action is proper and transitive. The isotropy

group of z = i is SO(2). Since H2 is a simply-connected Riemannian ma-nifold, whose sectional curvature is constant −1, the SL2(Z)-space H2 is amodel for ESL2(Z) by Theorem 9.25.

One easily checks that SL2(R) is a connected Lie group and SO(2) ⊆SL2(R) is a maximal compact subgroup. Since the SL2(R)-action on H2 istransitive and SO(2) is the isotropy group at i ∈ H2, we see that the SL2(R)-manifolds SL2(R)/SO(2) and H2 are SL2(R)-diffeomorphic.

As SL2(Z) is a discrete subgroup of SL2(R), the space H2 = SL2(R)/SO(2)with the obvious SL2(Z)-action is a model for ESL2(Z) by Theorem 9.24.

The group SL2(Z) is isomorphic to the amalgamated product Z/4∗Z/2Z/6.This implies that SL2(Z) acts cell preserving with finite stabilizers on a treeT , which has alternately two and three edges emanating from each vertex(see [750, Theorem 7 in I.4.1 on page 32 and Example 4.2 (c) in I.4.2 onpage 35]). This tree is a model for ESL2(Z) by Theorem 9.26.


The other model H2 is a manifold. These two models must be SL2(Z)-homotopy equivalent. They can explicitly be related by the following con-struction.

Divide the Poincare disk or the half plane model H2 into fundamental do-mains for the SL2(Z)-action. Each fundamental domain is a geodesic trianglewith one vertex at infinity, i.e., a vertex on the boundary sphere, and twovertices in the interior. Then the union of the edges, whose end points lie inthe interior of the Poincare disk, is a tree T with SL2(Z)-action. This is thetree model above. The tree is a SL2(Z)-equivariant deformation retraction ofH2. A retraction is given by moving a point p in H2 along a geodesic startingat the vertex at infinity, which belongs to the triangle containing p, throughp to the first intersection point of this geodesic with T . (See for instance [750,Example 4.2 (c) in I.4.2 on page 35]).

Exercise 9.32. Let G be a one-relator group. Let M ⊆ G be a maximalcyclic subgroup. Show that the inclusion induces for n ≥ 3 an isomorphism

Hn(BM)∼=−→ Hn(BG).

Exercise 9.33. Let G be a finitely generated group. Suppose that for everyinteger K there is k ≥ K with Hk(BG;Q) 6= 0. Show that G cannot be ahyperbolic group, an arithmetic group, Out(Fn), a mapping class group or aone-relator group.

9.6.12 Groups with Appropriate Maximal Finite Subgroups

Let G be a discrete group. Let MFIN be the subset of FIN consistingof elements in FIN which are maximal with respect to inclusion in FIN .Consider the following assertions concerning G:

(M) Every non-trivial finite subgroup of G is contained in a unique maximalfinite subgroup;

(NM) M ∈MFIN ,M 6= 1 =⇒ NGM = M ;

For such a group there is a nice model for EG with as few non-free cellsas possible. Let Mi | i ∈ I be a complete set of representatives for theconjugacy classes of maximal finite subgroups of G, i.e, each Mi is a maximalfinite subgroup of G and any maximal finite subgroup of G is conjugatedto Mi for precisely one element i ∈ I. By attaching free G-cells we get aninclusion of G-CW -complexes j1 :

∐i∈I G×Mi EMi → EG, where EG is the

same as ET R(G), i.e., a contractible free G-CW -complex.

Theorem 9.34 (Passage from EG to EG). Suppose that G satisfies (M)and (NM). Let X be the G-CW -complex define by the G-pushout


∐i∈I G×Mi

EMij1 //

u1

EG

f1

∐i∈I G/Mi

k1

// X

where u1 is the obvious G-map obtained by collapsing each EMi to a point.Then X is a model for EG.

Proof. We have to explain why EG is a model for the classifying space forproper actions of G. Obviously it is a G-CW -complex. Its isotropy groupsare all finite. We have to show for H ⊆ G finite that XH contractible. Webegin with the case H 6= 1. Because of conditions (M) and (NM) there isprecisely one index i0 ∈ I such that H is subconjugated to Mi0 and is notsubconjugated to Mi for i 6= i0 and we get(∐

i∈IG/Mi

)H= (G/Mi0)

H= •.

Hence XH = •. It remains to treat H = 1. Since u1 is a non-equivarianthomotopy equivalence and j1 is a cofibration, f1 is a non-equivariant homo-topy equivalence and hence EG is contractible (after forgetting the groupaction). ut

Here are some examples of groups Q which satisfy conditions (M) and(NM):

• Extensions 1→ Zn → G→ F → 1 for finite F such that the conjugationaction of F on Zn is free outside 0 ∈ Zn.

The conditions (M) and (NM) are satisfied by [557, Lemma 6.3].• Fuchsian groups F

The conditions (M) and (NM) are satisfied by [557, Lemma 4.5]. Com-ment 28: This does not seem to be what this reference says. We shoulddiscard this example here. does not seem to be true and we should discardthis. In [557] the larger class of cocompact planar groups (sometimes alsocalled cocompact NEC-groups) is treated.

• One-relator groups G

Let G be a one-relator group. Let G = 〈(qi)i∈I | r〉 be a presentation withone relation. We only have to consider the case, where G contains torsion.Let F be the free group with basis qi | i ∈ I. Then r is an element inF . There exists an element s ∈ F and an integer m ≥ 2 such that r = sm,the cyclic subgroup C generated by the class s ∈ G represented by s hasorder m, any finite subgroup of G is subconjugated to C and for any g ∈ Gthe implication g−1Cg ∩C 6= 1⇒ g ∈ C holds. These claims follows from


[564, Propositions 5.17, 5.18 and 5.19 in II.5 on pages 107 and 108]. HenceG satisfies conditions (M) and (NM).

Remark 9.35 (Passing to larger families). Theorem 9.34 is a specialcase of a general recipe to construct for two families F ⊆ G an efficient modelfor EG(G) from EF (G) in [562, Section 2]. These models are important forconcrete calculations of the left hand side appearing in the Baum-Conjectureor the Farrell-Jones Conjecture, see Chapter 15.

9.6.13 One-Relator Groups

Let G be a one-relator group. Let G = 〈(qi)i∈I | r〉 be a presentation with onerelation. There is up to conjugacy one maximal finite subgroup C which iscyclic. Let p : ∗i∈I Z→ G be the epimorphism from the free group generatedby the set I to G, which sends the generator i ∈ I to qi. Let Y →

∨i∈I S

1

be the G-covering associated to the epimorphism p. There is a 1-dimensionalunitary C-representation V and a C-map f : SV → resCG Y such that theinduced action on the unit sphere SV is free and the following is true: Ifwe equip SV with the C-CW -structure with precisely one equivariant 0-celland precisely one equivariant 1-cell DV with the C-CW -complex structurescoming from the fact that DV is the cone over SV , then the C-map f can bechosen to be cellular and we obtain a G-CW -model for EG by the G-pushout

G×C SVf //

Y

G×C DV // EG

where f sends (g, x) to gf(x). Thus we get a 2-dimensional G-CW -model forEG such that EG is obtained from G/C for a maximal finite cyclic subgroupC ⊆ G by attaching free cells of dimensions ≤ 2. The CW -CW -complexstructure on EG has precisely one 0-cell G/C×D0, one 0-cell G×D0, (2 · |I|many 1-cells G × D1 and |I| many 2-cells G × D2. All these claims followfrom [134, Exercise 2 (c) II. 5 on page 44].

If G is torsionfree, the 2-dimensional complex associated to a presentationwith one relation is a model for BG (see also [564, Chapter III §§9 -11]).


9.7 Special Models for the Classifying Space for theFamily of Virtually Cyclic Subgroups

In general the G-CW -models for EG are not as nice and small than theones for EG. We illustrate this in the case G = Zn for n ≥ 2. Then a Zn-CW -model for EZn = EZn is Rn with the standard translation action ofZn.

An explicite Zn-CW -model for EZn can be constructed as follows. Choosean enumeration Ci | i ∈ Z of the infinite cyclic subgroups of Zn. Considerthe space Rn ×R. For each i ∈ Z we identify in Rn × i the subspace givenby the R-span of Ci ⊆ Zn ⊆ Rn to a point. Then we obtain a Zn-CW -complex X. Since the Ci-fixed point set of X consists of precisely one point,the underlying topological space X is contractible, and all isotropy groupsof the Zn-action are infinite cyclic or trivial, X is a Zn-CW -model for EZn.Notice that the dimension of X is (n + 1). One can actually show that anyZn-CW -model for EZn has dimension greater or equal to (n + 1), see [562,Example 5.21].

9.7.1 Groups with Appropriate Maximal Virtually CyclicSubgroups

Let G be a discrete group. Let MVCY be the subset of VCY consistingof elements in VCY which are maximal with respect to inclusion in VCY.Consider the following assertions concerning G:

(M) Every infinite virtually cyclic subgroup of G is contained in a uniquemaximal virtually cyclic subgroup;

(NM) V ∈MVCY, |V | =∞ =⇒ NGV = V .

For such a group there is a nice model for EG with as few cells of type G/Vwith infinite virtually cyclic V as possible. Let Vi | i ∈ I be a complete setof representatives for the conjugacy classes of maximal infinite virtually cyclicsubgroups of G. By attaching G-cells of the type G/H for finite subgroupsH ⊆ G we get an inclusion of G-CW -complexes j1 :

∐i∈I G ×NGVi EVi →

EG.The next result is proved in [562, Corollary 2.11].

Theorem 9.36 (Passage from EG to EG). Suppose that G satisfies (M)and (NM). Let X be the G-CW -complex define by the G-pushout

9.8 Finiteness Conditions 285

∐i∈I G×NGVi EVi

j1 //

u1

EG

f1

∐i∈I G/Vi k1

// X

where u1 is the obvious G-map obtained by collapsing each EVi to a point.Then X is a model for EG.

A useful criterion for a group G to satisfy both (M) and (NM) can befound in [562, Theorem 3.1]. It implies that any hyperbolic group satisfiesboth (M) and (NM), see [562, Example 3.6]. On the other hand the Kleinbottle group Zo Z does not satisfy (M), see [562, Example .3.7]. This is theone of the few instances, where EG behaves nicer than EG since the class ofgroups for which both (M) and (NM) hold is much richer than the class forwhich both (M) and (NM) hold.

Theorem 9.36 will be very helpful for computating the left hand side ap-pearing in the Farrell-Jones Conjecture, see Section 15.5.

9.8 Finiteness Conditions

It has been very fruitful in group theory to investigate the question whetherone can find small models for BG, for instance a finite CW -model, a CW -model of finite type or a finite dimensional CW -model, or equivalently, smallG-CW -models for EG. The same question can be asked for EG and EG. Fortorsionfree groups there is no difference between EG and EG, but for groupswith torsion the space EG seems to carry much more information than EG.In this section we collect some information about finite conditions on EG, EGand EG. Having small models is also important for computation of the lefthand sides appearing in the Baum-Connes Conjecture and the Farrell-JonesConjecture, see Subsection 15.5.1.

Throughout this section G will be a discrete group.

9.8.1 Review of Finiteness Conditions on BG

As an illustration we review what is known about finiteness properties ofG-CW -models for EG for a discrete group G. This is equivalent to the samequestion about BG.

We introduce the following notation. Let R be a commutative associativering with unit. The trivial RG-module is R viewed as RG-module by thetrivial G-action. The cohomological dimension cdR(M) of a RG-module Mis∞ if there is no finite dimensional projective RG-resolution and is equal to


the integer n if there exists a projective resolution of dimension ≤ n for M butnot of dimension ≤ n−1. Notice that M possesses a projective RG-resolutionof dimension n if and only if for any RG-module N we have ExtiRG(M,N) = 0for i ≥ n + 1. The cohomological dimension over R of a group G, which isdenoted by cdR(G), is the cohomological dimension of trivial RG-module R.If R = Z, we abbreviate cd(G) := cdZ(G).

An RG-module M is of type FPn, if it admits a projective RG-resolutionP∗ such that Pi is finitely generated for i ≤ n, and of type FP∞ if it admitsa projective RG-resolution P∗ such that Pi is finitely generated for all i. Agroup G is of type FPn or FP∞ respectively if the trivial ZG-module Z is oftype FPn or FP∞ respectively.

Here is a summary of well-known statements about finiteness conditionson BG.

Theorem 9.37 (Finiteness conditions for BG). Let G be a discretegroup.

(i) If there exists a finite dimensional model for BG, then G is torsionfree;(ii)(a) There exists a CW -model for BG with finite 1-skeleton if and only if

G is finitely generated;(b) There exists a CW -model for BG with finite 2-skeleton if and only if

G is finitely presented;(c) For n ≥ 3 there exists a CW -model for BG with finite n-skeleton if and

only if G is finitely presented and of type FPn;(d) There exists a CW -model for BG of finite type, i.e., all skeleta are

finite, if and only if G is finitely presented and of type FP∞;(e) There exists groups G which are of type FP2 and which are not finitely

presented;(iii) There is a finite CW -model for BG if and only if G is finitely presented

and there is a finite free ZG-resolution F∗ for the trivial ZG-module Z;(iv) The following assertions are equivalent:

(a) The cohomological dimension over Z of G is ≤ 1;(b) There is a model for BG of dimension ≤ 1;(c) G is free;

(v) The following assertions are equivalent for d ≥ 3:

(a) There exists a CW -model for BG of dimension d;(b) The cohomological dimension over Z of G is d;

(vi) For Thompson’s group F there is a CW -model of finite type for BG butno finite dimensional model for BG.

Proof. (i) Suppose we can choose a finite dimensional model for BG. Let C ⊆G be a finite cyclic subgroup. Then C\BG = C\EG is a finite dimensionalmodel for BC. Hence there is an integer d such that we have Hi(BC) = 0


for i ≥ d. This implies that C is trivial [134, (2.1) in II.3 on page 35]. HenceG is torsionfree.

(ii) See [109] and [134, Theorem 7.1 in VIII.7 on page 205].

(iii) See [134, Theorem 7.1 in VIII.7 on page 205].

(iv) See [762] and [778].

(v) See [134, Theorem 7.1 in VIII.7 on page 205].

(vi) See [135]. ut

9.8.2 Cohomological Criteria for Finiteness Properties in Termsof Bredon Cohomology

We have seen that we can read off finiteness properties of BG or EG fromthe group cohomology of G. If one wants to investigate the same questionfor EF (G) analogous statements are true if one considers modules over theF-restricted orbit category OrF (G) in the sense of Definition 1.54. This isexplained in [532, Subsection 5.2]. For instance, if d ≥ 3 is a natural number,then there is a G-CW -model of dimension ≤ d for EF (G) if and only if thetrivial ZOrF (G)-module Z has projective ZOrF (G)-resolution of dimension≤ d, see [532, Theorem 5.2 (i)]. The role of the cohomology of a group isnow played by the Bredon cohomology of EF (G). We will deal with Bredoncohomology in Example 10.2.

Other papers related to the topic of connecting geometric dimension orother finiteness properties for classifying spaces for families to algebraic ana-logues are [126, 322, 324, 624, 625].

9.8.3 Finite Models for the Classifying Space for Proper Actions

The specific constructions of Sections 9.6 show that there is a finite G-CW -model for the classifying space of proper actions EG if G is a cocompactdiscrete subgroups of an almost connected Lie group, a hyperbolic group, anarithmetic group, the outer automorphism group of a finitely generated freegroups, a mapping class group, or a finitely generated one-relator group. Thisis also the case for an elementary amenable group of type FP∞, see [481].

If 1 → K → G → Q → 1 is an extension of groups and there are finitemodels for EK and EQ, one may ask whether there is a finite model for EG.Some sufficient conditions for this question are given in [524, Theorem 3.2and Theorem 3.3], for instance that K is hyperbolic or virtually poly-cyclic.However, even in the case that Q is finite and K is torsionfree with a finitemodel for BK, it can happen that there is no finite model for EG, see [507,Example 3 on page 149 in Section 7].


9.8.4 Models of Finite Type for the Classifying Space for ProperActions

The following result is proved in [524, Theorem 4.2].

Theorem 9.38 (Models for EG of finite type).The following statements are equivalent for the group G.

(i) There is a G-CW -model for EG of finite type;(ii) There are only finitely many conjugacy classes of finite subgroups of G and

for any finite subgroup H ⊂ G there is a CW -model for BWGH of finitetype, where WGH := NGH/H;

(iii) There are only finitely many conjugacy classes of finite subgroups of G andfor any finite subgroup H ⊂ G the Weyl group WGH is finitely presentedand is of type FP∞.

The comments about extensions in Subsection 9.8.3 for finite models carryover to models of finite type.

9.8.5 Finite Dimensional Models for the Classifying Space forProper Actions

The following result follows from Dunwoody [258, Theorem 1.1].

Theorem 9.39 (A criterion for 1-dimensional models for EG). Let Gbe a discrete group. Then there exists a 1-dimensional model for EG if andonly the cohomological dimension of G over Q is less or equal to one.

If G is finitely generated, then there is a 1-dimensional model for EG ifand only if G contains a finitely generated free subgroup of finite index [448,Theorem 1]. If G is torsionfree, we rediscover the results due to Swan andStallings stated in Theorem 9.37 (iv) from Theorem 9.39.

If G is virtually torsionfree, one defines its virtual cohomological dimensionvcd(G) by the cohomological dimension cd(H) of any torsionfree subgroupH ⊆ G of finite index. Since for any other torsionfree subgroup K ⊆ G offinite index we have cd(H) = cd(K), this notion is well-defined.

Definition 9.40 (Homotopy dimension). Given a G-space X, the homo-topy dimension hdimG(X) ∈ 0, 1, . . . q ∞ of X is defined to be the infi-mum over the dimensions of all G-CW -complexes Y which are G-homotopyequivalent to X.

Notation 9.41. Put for a group G

gd(G) := hdimG(EG);

gd(G) := hdimG(EG).


Lemma 9.42. Suppose that G is virtually torsionfree. Then

vcd(G) ≤ gd(G).

Proof. Choose a torsionfree subgroup H ⊆ G of finite index. Then the re-striction of EG to H is a model for EH. This implies cd(H) ≤ dim(EG) andhence vcd(G) ≤ gd(G). ut

The next result is taken from [532, Theorem 5.24]

Theorem 9.43 (Dimension of EG for a discrete subgroup G of analmost connected Lie group). Let L be a Lie group with finitely manypath components. Then L contains a maximal compact subgroup K which isunique up to conjugation. Let G ⊆ L be a discrete subgroup of L. Then L/Kwith the left G-action is a model for EG.

Suppose additionally that G is virtually torsionfree. Then we have

vcd(G) ≤ dim(L/K)

and equality holds if and only if G\L is compact.

The next result follows from [321, Theorem 1 and inequalities (1) and (2)on page 7], where also the notion of the Hirsch length for elementary amenablegroups due to Hillman [396] is recalled. In the special case that there is afinite sequence G = G0 ⊇ G1 ⊇ G2 ⊇ · · · ⊇ Gn−1 ⊇ Gn = 1 of subgroupssuch that Gi+1 is normal in Gi and Gi/Gi+1 is finitely generated abelian for

i = 0, 1, . . . , (n− 1), the Hirsch length h(G) is∑n−1i=0 rkZ(Gi/Gi+1).

Theorem 9.44 (Dimension of EG for countable elementary amenablegroups of finite Hirsch length). If G is an elementary amenable group,then its Hirsch length satisfies

h(G) ≤ gd(G).

If G is a countable elementary amenable group, then

gd(G) ≤ max3, h(G) + 1.

If F is a virtually poly-cyclic group G, then G is virtually torsionfree,vcd(G) is finite and satisfies vcd(G) = h(G) = gd(G), see [532, Example 5.26].

If H ⊆ G is a subgroup of finite index [G : H]. If there is a H-CW -modelfor EH of dimension ≤ d, then there is a G-CW -model for EG of dimension≤ d · [G : H], see [524, Theorem 2.4]. In particular gd(G)) ≤ [G : H] · gd(H).

Theorem 9.45 (Dimension of EG and extension). Let 1→ K → G→Q → 1 be an exact sequence of groups. Suppose that there exists a positiveinteger d which is an upper bound on the orders of finite subgroups of Q.Then

gd(G) ≤ d · gd(K) + gd(Q).


Remark 9.46 (gd(G) for locally finite groups). For a locally finite groupof cardinality ℵn the inequality gd(G) ≤ n+1 is proved in [249] and [562, The-orem 5.31]. The equality gd(G) = n+ 1 is explained in [562, Example 5.32].

Exercise 9.47. Let F be a finite group. Put H =⊕

Z F . Let H o Z bethe semidirect product with respect to the shift automorphism of H. Showgd(H) = 1 and gd(H o Z) = 2.

9.8.6 Brown’s Problem about Virtual Cohomological Dimensionand the Dimension of the Classifying Space for ProperActions

The following problem, whether the converse of Lemma 9.42 is true, is statedby Brown [133, page 32].

Problem 9.48 (Brown’s problem about vcd(G) = dim(EG)). For whichvirtually torsionfree groups G does the equality

vcd(G) = gd(G)

hold?

The length l(H) ∈ 0, 1, . . . of a finite group H is the supremum over alll for which there is a nested sequence H0 ⊂ H1 ⊂ . . . ⊂ Hl of subgroups Hi

of H with Hi 6= Hi+1. The following result is proved in [524, Theorem 6.4]and was motivated by Brown’s Problem 9.48.

Theorem 9.49 (Estimate on dim(EG) in terms of vcd(G)). Let G bea group with virtual cohomological dimension vcd(G) ≤ d. Let l ≥ 0 be aninteger such that the length l(H) of any finite subgroup H ⊂ G is bounded byl.

Then there is a G-CW -model for EG such that for any finite subgroupH ⊂ G

dim(EGH) = max3, d+ l − l(H)

holds. In particular gd(G) ≤ max3, d+ l.

However, we obtain from Leary-Petroysan [508, Corollary 1.2], see alsoLeary-Nucinkis [507, Example 12 on page 153 in Section 7].

Theorem 9.50 (Brown’s Problem 9.48 has a negative answer in gen-eral). Given a natural number m, there exists a group G such that there isa finite model for EG and we have vcd(G) = 2m and gd(G) ≥ 3m.

Moreover, Leary-Petroysan [508, page 732] show that the estimate in The-orem 9.49 cannot be improved, even if one considers only finite models forEG.


9.8.7 Finite Dimensional Models for the Classifying Space for theFamily of Virtually Cyclic Subgroups

The following problem has triggered a lot of activities

Problem 9.51 (Relating the dimension of EG and EG). For whichcountable groups G do the inequalities

gd(G)− 1 ≤ gd(G) ≤ gd(G) + 1

hold?

The inequality appearing in Problem 9.51 holds for countable elementaryamenable groups, see [240, Corollary 4.4]. There are groups of type FP forwhich the difference gd(G)−gd(G) is arbitrary large, see [240, Example 6.5].

All possible cases of the inequality appearing in Problem 9.51 can occur,in particular there are examples of finitely presented groups G with gd(G) <

gd(G), see Remark 9.55.The next result is proved in [240, Theorem A].

Theorem 9.52 (Dimension of EG for elementary amenable groups offinite Hirsch length). If G is an elementary amenable group of cardinalityℵn such that the Hirsch length h(G) of G is finite, then

gd(G) ≤ h(G) + n+ 2.

Theorem 9.53 (The dimension of EG).

(i) We have for any group G

gd(G) ≤ 1 + gd(G);

(ii) We havegd(G×H) ≤ gd(G) + gd(H),

andgd(G×H) ≤ gd(G) + gd(H) + 3,

and these inequalities cannot be improved in general;(iii) If G satisfies condition (M) and (NM), then

gd(G)

= gd(G) if gd(G) ≥ 2;

≤ 2 if gd(G) ≤ 1;

(iv) If H ⊆ G is a subgroup of finite index [G : H] then

gd(G) ≤ [G : H] · gd(H).


Proof. (i) See [562, Corollary 5.4 (1)].

(ii) This is obvious for gd(G×H) and proved for gd(G×H) in [562, Corol-

lary 5.6 and Remark 5.7].

(iii) See [562, Theorem 5.8 (2)].

(iv) This is proved in [524, Theorem 2.4]. ut

Exercise 9.54. If G is the fundamental group of a hyperbolic closed Rie-mannian manifold M , then

cd(G) = dim(N) = gd(G) = gd(G).

Remark 9.55 (Virtually-poly-cyclic-groups). In [562, Theorem 5.13] acomplete computation of gd(G) is presented for virtually poly-Z groups. The

answer is much more complicated than the one for gd(G) which is equalto both vcd(G) and the Hirsch length h(G), see [532, Example 5.26]. Thisleads to some interesting examples in [562, Subsection 5.4]. For instance,one can construct, for k = −1, 0, 1, automorphisms fk : Hei → Hei of thethree-dimensional Heisenberg group Hei such that

gd(HeiofkZ) = 4 + k.

Notice that gd(HeiofZ) = cd(HeiofZ) = 4 holds for any automorphismf : Hei→ Hei.

The following result is taken from [535, Theorem 1.1].

Theorem 9.56 (Dimensions of EG and EG for groups acting onCAT(0)-spaces). Let G be a discrete group which acts properly and iso-metrically on a complete proper CAT(0)-space X. Let top-dim(X) be thetopological dimension of X.

(i) We havegd(G) ≤ top-dim(X);

(ii) Suppose that G acts by semisimple isometries. (This is the case if weadditionally assume that the G-action is cocompact.)Then

gd(G) ≤ top-dim(X) + 1.

Remark 9.57 (gd(G) for locally virtually cyclic groups). For a locally

virtually cyclic group of cardinality ℵn the inequality gd(G) ≤ n + 1 is a

special case of [562, Theorem 5.31].

The next result is taken from [237, Theorem A].


Theorem 9.58 (Finite dimensional models for EG for discrete sub-groups of GLn(R)). Every discrete subgroup G of GLn(R) admits a fi-nite dimensional model for EG. More precisely, if m is the dimension of theZariski closure of G in GLn(R), then

gd(G) ≤ m+ 1.

9.8.8 Low Dimensions

Besides Theorem 9.39 we have the following result proved in [562, Theo-rem 5.33].

Theorem 9.59 (Low-dimensional models for EG and EG).

(i) Let G be a countable group which is locally virtually cyclic. Then

gd(G) =

0 if G is finite;

1 if G is infinite and either locally finite

or virtually cyclic;

2 otherwise,

and

gd(G) =

0 if G is virtually cyclic;

1 otherwise;

(ii) Let G be a countable group satisfying gd(G) ≤ 1. Then

gd(G) =

0 if G is virtually cyclic;

1 if G is locally virtually cyclic but

not virtually cyclic;

2 otherwise.

Exercise 9.60. Let G be a countable group. Show that G is infinite locallyfinite if and only if gd(G) = gd(G) = 1 holds.

9.8.9 Finite Models for the Classifying Space for the Family ofVirtually Cyclic Subgroups

If G is virtually cyclic, a model for EG is • = G/G which is in particularfinite. There is no group known such that EG has a finite G-CW -model and


G is not virtually cyclic. This leads to the following conjecture of Juan-Pinedaand Leary [428, Conjecture 1],

Conjecture 9.61 (Finite Models for EG). If a group G has a finiteG-CW -model for EG, then G is virtually cyclic.

Conjecture 9.61 is known to be true in many cases, since the existenceof a finite G-CW -model for EG implies that there is a finite G-CW -modelfor EG, see [562, Corollary 5.4 (2)], and that there are only finitely manyconjugacy classes of infinite virtually cyclic groups ofG. Conjecture 9.61 holdsfor instance for hyperbolic groups, see [428, Corollary 12], and elementaryamenable groups, see [475, Corollary 5.8].

9.9 On the Homotopy Type of the Quotient Space ofthe Classifying Space for Proper Actions

One may think that there are more homotopy classes of CW -complexes thanisomorphisms classes of groups. Namely, we can assign to any group G itsclassifying space BG and for two groups G and H the spaces BH and BGare homotopy equivalent if and only if G and H are isomorphic, and thereare CW -complexes which are not homotopy equivalent to BG for any groupG. However, here is a result due to Leary-Nucinkis [506, Theorem 1], whichis in some sense the converse.

Theorem 9.62 (Every CW -complex occurs up to homotopy as quo-tient of a classifying space for proper group actions). Let X be aCW -complex. Then there exists a group G such that G\EG is homotopyequivalent to X. Moreover one can arrange that G contains a torsionfreesubgroup of index two.

Exercise 9.63. Let X be a CW -complex. Show that there exists a Z/2-CW -complex Y such that Y is aspherical and X is homotopy equivalent to theZ/2-quotient space of Y .

Remark 9.64 (Metric Kan-Thurston Theorem). Leary proves a meticKan-Thurston Theorem in [505, Theorem A]. It yields the following vari-ant of Theorem 9.62, see [505, Theorem 8.3]. Given a group G and proper

simplicial G-complex X with connected G\X, there exists a group G, a cu-bical CAT(0)-complex E with simplicial G-action, an epimorphism of groups

p : G → G and a map f : E → X such that E is a model for EG, the mapf is p : G → G-equivariant and for any equivariant homology theory in thesense of Definition 10.9 the pair (p, f) induces for all n ∈ Z isomorphisms

HGn (E) → HGn (X). An application to Isomorphism Conjectures is discussedin [505, Section 10].


The understanding of G\EG and G\EG will be important for the compu-tation of the left hand side appearing in the Baum-Conjecture or the Farrell-Jones Conjecture, see Chapter 15.

In contrast to the trivial family T R, where EG and BG = G\EG carrythe same information, this is not true for EG and G\EG. For instance, G\EGis contractible, if G is the infinite dihedral group D∞ ∼= ZoZ/2 ∼= Z/2∗Z/2,what can be seen by direct inspection, or if G = SL3(Z), see [759, Corollaryon page 8].

9.10 Miscellaneous

The notion of a classifying space for a family was introduced by tomDieck [786].

Classifying spaces for families play a role in computations of equivarianthomology and cohomology for compact Lie groups such as equivariant bor-dism as explained in [787, Chapter 7] and [788, Chapter III].

Classifying spaces for topological groups and appropriate families of sub-groups play a key role in the construction of classifying equivariant principalbundles in [561] or the construction of the topological K-cohomology forarbitrary proper equivariant CW -complexes in [544].

More information about classifying spaces for families can be found forinstance in [1, 91, 60, 198, 238, 239, 240, 323, 482, 532, 542, 562, 668, 788].


name of texfile: ic

Chapter 10

Equivariant Homology Theory

10.1 Introduction

This section is devoted to equivariant homology theories. They are key inputin the general formulations of the Baum-Connes Conjecture and the Farrell-Jones Conjecture. If one316 only wants to understand these conjectures, oneonly needs to browse through the Definition 10.1 of a G-homology theory.Since G-homology theories are of general importance, we have added morematerial to this section which will also be useful for concrete computations ofK- and L-groups of group rings and group C∗-algebras based on the Baum-Connes Conjecture and the Farrell-Jones Conjecture.

The notion of a G-homology theoryHG∗ is the obvious generalization of thenotion of a (generalized) homology theory in the non-equivariant sense. Animportant step is to pass to an equivariant homology theory H?

∗ which assignsto every group G a G-homology theory HG∗ and links for any group homomor-phisms α : H → G the theoriesHH∗ andHG∗ by a so called induction structure.This global point of view is the key for many applications and computa-tions and most interesting theories arise as equivariant homology theories.Whenever one has a covariant functor from the category of small connectedgroupoids GROUPOIDS to the category of spectra SPECTRA, one obtainsan associated equivariant homology theory whose coefficients are given bytaking homotopy groups of the spectra. Thus one can construct our mainexamples for equivariant homology theories which are based on K- and L-groups of group rings and group C∗-algebras. We will also provide tools forcomputations, namely the equivariant Atiyah-Hirzebruch spectral sequence,the p-chain spectral sequence and the equivariant Chern character. We willpresent some concrete examples of such computations.

10.2 Basics about G-Homology Theories

In this section we describe the axioms of a (proper) G-homology theory andgive some basic examples. The main examples for us will be the sources of theassembly maps appearing in the Farrell-Jones Conjecture and Baum-ConnesConjecture.

Fix a discrete group G and an associative commutative ring Λ with unit.

297

298 10 Equivariant Homology Theory

Definition 10.1 (G-homology theory). A G-homology theory HG∗ withvalues in Λ-modules is a collection of covariant functorsHGn from the categoryof G-CW -pairs to the category of Λ-modules indexed by n ∈ Z together withnatural transformations

∂Gn (X,A) : HGn (X,A)→ HGn−1(A) := HGn−1(A, ∅)

for n ∈ Z such that the following axioms are satisfied:

• G-homotopy invariance

If f0 and f1 are G-homotopic G-maps of G-CW -pairs (X,A) → (Y,B),then HGn (f0) = HGn (f1) for n ∈ Z;

• Long exact sequence of a pair

Given a pair (X,A) of G-CW -complexes, there is a long exact sequence

. . .HGn+1(j)−−−−−→ HGn+1(X,A)

∂Gn+1−−−→ HGn (A)HGn (i)−−−−→ HGn (X)

HGn (j)−−−−→ HGn (X,A)∂Gn−−→ . . . ,

where i : A→ X and j : X → (X,A) are the inclusions;• Excision

Let (X,A) be a G-CW -pair and let f : A → B be a cellular G-map ofG-CW -complexes. Equip (X ∪f B,B) with the induced structure of a G-CW -pair. Then the canonical map (F, f) : (X,A)→ (X ∪f B,B) inducesan isomorphism

HGn (F, f) : HGn (X,A)∼=−→ HGn (X ∪f B,B);

• Disjoint union axiom

Let Xi | i ∈ I be a family of G-CW -complexes. Denote by ji : Xi →∐i∈I Xi the canonical inclusion. Then the map

⊕i∈IHGn (ji) :

⊕i∈IHGn (Xi)

∼=−→ HGn

(∐i∈I

Xi

)

is bijective.

If HG∗ is defined or considered only for proper G-CW -pairs (X,A), we callit a proper G-homology theory HG∗ with values in Λ-modules.

Example 10.2 (Bredon Homology). The most basic G-homology theoryis Bredon homology, which was originally introduced in [127]. Recall thatOr(G) denotes the orbit category of G. Let X be a G-CW -complex. It definesa contravariant functor from the orbit category Or(G) to the category of CW -complexes by sending G/H to mapG(G/H,X) = XH . Composing it with thefunctor cellular chain complex yields a contravariant functor

10.2 Basics about G-Homology Theories 299

COr(G)∗ (X) : Or(G)→ Z-CHCOM

to the category of Z-chain complexes. Let Λ be a commutative ring and let

M : Or(G)→ Λ-MODULES

be a covariant functor. If N : Or(G) → Z-MODULES is a contravariantfunctor, one can form the tensor product over the orbit category N ⊗ΛOr(G)

M , see for instance [521, 9.12 on page 166]. It is the quotient of the Λ-module⊕G/H∈ob(Or(G))

N(G/H)⊗Z M(G/H)

by the Λ-submodule generated by

xf ⊗ y − x⊗ fy | f : G/H → G/K, x ∈ N(G/K), y ∈M(G/H) ,

where xf stands for N(f)(x) and fy for M(f)(y). Since this is natural,

we obtain a Λ-chain complex COr(G)∗ (X) ⊗ZOr(G) M . The homology of

COr(G)∗ (X)⊗ZOr(G) M is the Bredon homology of X with coefficients in M

HGn (X;M) := Hn(C

Or(G)∗ (X)⊗ZOr(G) M).(10.3)

This extends in the obvious way to G-CW -pairs. Thus we get a G-homologytheory HG

∗ with values in Λ-modules.

The description of COr(G)∗ (X) ⊗ZOr(G) M in terms of the orbit category

is conceptually the right one since it is intrinsically defined and the basicproperties are easily checked following closely the non-equivariant case. Forcomputation, the following explicit description is useful.

Fix G-pushouts


∐i∈In q

ni //

Xn−1

∐i∈In G/Hi ×Dn

∐i∈In Q

ni // Xn

as they appear in Definition 9.2. Then the n-th Λ-chain module of COr(G)∗ (X)⊗ZOr(G)

M is given by

COr(G)n (X)⊗ZOr(G) M =

⊕i∈In

M(G/Hi).

In order to specify the n-th differential


cn :⊕i∈In

M(G/Hi)→⊕

j∈In−1

M(G/Hj)

we specify for each pair (i, j) ∈ In×In−1 a Λ-homomorphism αi,j : M(G/Hi)→M(G/Hj) such that for fixed i ∈ In there are only finitely many j ∈ In−1

such that αi,j 6= 0.We begin with the case n = 1. Consider the composite of G-maps for i ∈ I1

and j ∈ I0G/Hi × S0 q1i−→ X0 =

∐j∈I0

G/Hj

prj−−→ G/Hj

where prj is the obvious projection. Thus we obtain two G-maps f± : G/Hi →G/Hj if we restrict this composite to G/Hi × ±1. Define

αi,j = M(f+)−M(f−).

Next we deal with the case n ≥ 2. Let Xn−1,j be the quotient of Xn−1,where we collapse all the equivariant (n−1)-cells except the one for the indexj to a point. The pushout above, but now for (n − 1) instead of n, yields aG-homeomorphism

Qn−1j :

∨G/Hi

Sn−1 =(G/Hj ×Dn−1

)/(G/Hj × Sn−2

) ∼=−→ Xn−1,j ,

where∨G/Hi

Sn−1 is the one-point union or wedge of as many copies of Sn−1

as there are elements inG/Hj . If pgHj :∨G/Hi

Sn−1 → Sn−1 is the projection

onto the summand belonging to gHj ∈ G/Hj , k : Sn−1 → G/Hi×Sn−1 is theobvious inclusion to the summand belonging to eHi and prj : Xn−1 → Xn−1,j

the obvious projection, then we obtain a selfmap of Sn−1 by the followingcomposite

Sn−1 k−→ G/Hi × Sn−1 qni−→ Xn−1

prj−−→ Xn−1,j

Qn−1j

−1

−−−−−→∨G/Hj

Sn−1pgHj−−−→ Sn−1.

Define di,j,gHj ∈ Z to be the mapping degree of the map above. For gHj ∈G/HHi

j we obtain a G-map

rgHj : G/Hi → G/Hj g′Hi 7→ g′gHj .

Defineαi,j : M(G/Hi)→M(G/Hj)

10.2 Basics about G-Homology Theories 301

to be the some of the maps∑gHj∈G/H

Hij

di,j,gHj ·M(rgHj ). Since because

of the compactness of Sn−1 there are for fixed i ∈ In−1 only finitely manypairs (j, gHj) for j ∈ In−1 and gHj ∈ G/Hj with di,j,gHj 6= 0, the definitionof αi,j makes sense and we can indeed define cn by sending xi | i ∈ In to∑i∈In αi,j(xi) | j ∈ In−1.

Obviously Bredon homology reduces for G = 1 to the cellular homologyof a CW -complex with coefficients in the abelian group M . It is the obviousgeneralization of this concept to the equivariant setting if one keeps in mindthat in the equivariant situation the building blocks are equivariant cells givenby G-spaces G/Hi ×Dn.

Exercise 10.4. Let Z/2 act on S2 := (x0, x1, x2) | xi ∈ R, x20 + x2

1 + x22 =

1 by the involution which sends (x0, x1, x2) to (x0, x1,−x2). Consider thecovariant functor

RC : Or(Z/2)→ Z-MODULES

which sends (Z/2)/H to the complex representation ring RC(H), any endo-morphism in Or(Z/2) to the identity and the morphism pr: (Z/2)/1 →(Z/2)/(Z/2) to the homomorphism RC(1) → RC(Z/2) given by inductionwith the inclusion 1 → Z/2.

Show that S2 becomes a Z/2-CW -complex if we take (1, 0, 0) as 0-skeleton, (x0, x1, 0) | x2

0 + x21 = 1 as 1-skeleton and S2 itself as 2-skeleton

and compute the abelian groups HZ/2∗ (Sn;RC).

Lemma 10.5. Let HG∗ be a G-homology theory. Let X be a G-CW -complexand Xi | i ∈ I be a directed system of G-CW -subcomplexes directed byinclusion such that X =

⋃i∈I Xi. Then for all n ∈ Z the natural map

colimi∈I HGn (Xi)∼=−→ HGn (X)

is bijective.

Proof. Compare for example with [780, Proposition 7.53 on page 121], wherethe non-equivariant case for I = N is treated. The general case is an obviousvariation. The basic idea is the following. Let Mi | i ∈ I be a directedsystem of Λ-modules. Then there is an exact an exact sequence of Λ-modules⊕

i,j∈I,i≤j

Mi →⊕j∈I

Mj → colimi∈IMi → 0,

which allows to play back questions of a colimit over some directed indexingset to the direct sum of modules. This can be used to reduce the claim forcolimit of a directed system of CW -complexes to disjoint unions. ut

LetHG∗ andKG∗ beG-homology theories. A natural transformation T∗ : HG∗ →KG∗ of G-homology theories is a sequence of natural transformations Tn : HGn →


KGn of functors from the category ofG-CW -pairs to the category of Λ-moduleswhich are compatible with the boundary homomorphisms.

Lemma 10.6. Let T∗ : HG∗ → KG∗ be a natural transformation of G-homologytheories. Suppose that Tn(G/H) is bijective for every homogeneous spaceG/H and n ∈ Z.

Then Tn(X,A) is bijective for every G-CW -pair (X,A) and n ∈ Z.

Notice that one needs in Lemma 10.6 the existence of the natural trans-formation T . Namely, there exists two (non-equivariant) homology theoriesH∗ and K∗ such that H(•) ∼= Kn(•) holds for n ∈ Z but there exists aCW -complex X and m ∈ Z such that Hm(X) and Km(X) are not isomor-phic. An example is topological K-homology theory K∗ and the homologytheory H∗ =

⊕n∈ZH∗+2n for H∗ singular homology.

Exercise 10.7. Give the proof of Lemma 10.6.

10.3 Basics about Equivariant Homology Theories

In this section we describe the axioms of a (proper) equivariant homologytheory and give some basic examples. The point is that an equivariant ho-mology theory assign to every group a G-homology theory and links them byan induction structure. It will play a key role in computations, various proofsand the construction of the equivariant Chern character.

Let α : H → G be a group homomorphism. Given an H-space X, definethe induction of X with α to be the G-space

indαX = G×α X(10.8)

which is the quotient of G×X by the right H-action (g, x)·h := (gα(h), h−1x)for h ∈ H and (g, x) ∈ G × X. If α : H → G is an inclusion, we also writeindGH instead of indα.

Definition 10.9 (Equivariant homology theory). A (proper) equivarianthomology theory with values in Λ-modules H?

n assigns to each group G a(proper) G-homology theory HG∗ with values in Λ-modules (in the sense ofDefinition 10.1) together with the following so called induction structure:

Given a group homomorphism α : H → G and a (proper) H-CW -pair(X,A), there are for every n ∈ Z natural homomorphisms

indα : HHn (X,A)→ HGn (indα(X,A))(10.10)

satisfying:

10.3 Basics about Equivariant Homology Theories 303

• Compatibility with the boundary homomorphisms

∂Gn indα = indα ∂Hn ;• Functoriality

Let β : G→ K be another group homomorphism. Then we have for n ∈ Z

indβα = HKn (f1) indβ indα : HHn (X,A)→ HKn (indβα(X,A)),

where f1 : indβ indα(X,A)∼=−→ indβα(X,A), (k, g, x) 7→ (kβ(g), x) is the

natural K-homeomorphism;• Compatibility with conjugation

For n ∈ Z, g ∈ G and a (proper) G-CW -pair (X,A) the homomorphismindc(g) : G→G : HGn (X,A) → HGn (indc(g) : G→G(X,A)) agrees with HGn (f2)for the G-homeomorphism f2 : (X,A) → indc(g) : G→G(X,A) which sendsx to (1, g−1x) in G×c(g) (X,A);

• Bijectivity

If ker(α) acts freely on X \ A, then indα : HHn (X,A) → HGn (indα(X,A))is bijective for all n ∈ Z.

Exercise 10.11. Let H?∗ be an equivariant homology theory. Show for any

group G, any element g ∈ G and n ∈ Z that induction with c(g) : G → Ginduces the identity on HGn (•).

We will later need

Lemma 10.12. Consider finite subgroups H,K ⊂ G and an element g ∈ Gwith gHg−1 ⊂ K. Let Rg−1 : G/H → G/K be the G-map sending g′H tog′g−1K and c(g) : H → K be the homomorphism sending h to ghg−1. Letpr: (indc(g) : H→K•) → • be the projection. Then the following diagramcommutes

HHn (•)HKn (pr)indc(g) //

indGH∼=

HKn (•)

indGK∼=

HGn (G/H)HGn (Rg−1 )

// HGn (G/K)

Proof. Let f1 : indc(g) : G→G indGH• → indGK indc(g) : H→K• be the bijec-tive G-map sending (g1, g2, •) in G×c(g) G×H • to (g1gg2g

−1, 1, •) inG×K K ×c(g) •. The condition that induction is compatible with compo-sition of group homomorphisms means precisely that the composite

HHn (•) indGH−−−→ HGn (indGH•)indc(g) : G→G−−−−−−−−→ HGn (indc(g) : G→G indGH•)

HGn (f1)−−−−−→ HGn (indGK indc(g) : H→K•)

agrees with the composite


HHn (•)indc(g) : H→K−−−−−−−−→ HKn (indc(g) : H→K•)

indGK−−−→ HGn (indGK indc(g) : H→K•).

Naturality of induction implies HGn (indGK pr) indGK = indGK HKn (pr). Hencethe following diagram commutes

HHn (•)HKn (pr)indc(g)H→K //

indGH∼=

HKn (•)

indGK∼=

HGn (G/H)HGn (indGK pr)HGn (f1)indc(g) : G→G // HGn (G/K)

By the axioms indc(g) : G→G : HGn (G/H) → HGn (indc(g) : G→GG/H) agreeswith HGn (f2) for the map f2 : G/H → indc(g) : G→GG/H which sends g′H to

(g′g−1, 1H) in G ×c(g) G/H. Since the composite (indGK pr) f1 f2 is justRg−1 , Lemma 10.12 follows. ut

Example 10.13 (Borel homology). Let K∗ be a homology theory for (non-equivariant) CW -pairs with values in Λ-modules. Examples are singular ho-mology, oriented bordism theory or topological K-homology. Then we obtaintwo equivariant homology theories with values in Λ-modules in the sense ofDefinition 10.9 by the following constructions

HGn (X,A) = Kn(G\X,G\A);

HGn (X,A) = Kn(EG×G (X,A)).

The second one is called the equivariant Borel homology associated to K.In both cases HG∗ inherits the structure of a G-homology theory from the

homology structure on K∗. Induction for a group homomorphism α : H → G

is induced by the following two maps a and b. Let a : H\X∼=−→ G\(G ×α

X) be the homeomorphism sending Hx to G(1, x). Define b : EH ×H X →EG ×G G ×α X by sending (e, x) to (Eα(e), 1, x) for e ∈ EH, x ∈ X andEα : EH → EG the α-equivariant map induced by α. Induction for a grouphomomorphism α : H → G is induced by these maps a and b. If the kernelker(α) acts freely on X, then the map b is a homotopy equivalence and hencein both cases indα is bijective.

Example 10.14 (Equivariant bordism). For a proper G-CW -pair (X,A),one can define the G-bordism group NG

n (X,A) as the abelian group of G-bordism classes of maps f : (M,∂M) → (X,A) whose sources are smoothmanifolds with cocompact proper smooth G-actions. Cocompact means thatthe quotient space G\M is compact. The definition is analogous to the onein the non-equivariant case. This is also true for the proof that this defines aproper G-homology theory. There is an obvious induction structure comingfrom induction of equivariant spaces which is, however, only defined if thekernel of α acts freely on X\A. It is well-defined because of the following fact.

10.4 Constructing Equivariant Homology Theories Using Spectra 305

If α : H → G is a group homomorphism, M is an smooth H-manifold withcocompact proper smooth H-action, and ker(α) acts freely, then indαM is asmooth G-manifold with cocompact proper smooth G-action. The boundaryof indαM is indα ∂M .

Example 10.15 (Equivariant topological K-theory). We have explainedthe notion of equivariant topological K-theory K?

∗ in (8.67). If RC(H) denotesthe complex representation ring of the finite subgroup H ⊆ G, then

KGn (G/H) ∼= KH

n (•) ∼=

RC(H) n even;

0 n odd.

Exercise 10.16. Compute KD∞∗ (ED∞).

10.4 Constructing Equivariant Homology TheoriesUsing Spectra

We briefly fix some conventions concerning spectra. Let SPACES+ be thecategory of pointed compactly generated spaces. (One may also work withweakly Hausdorff spaces.) Here the objects are (compactly generated) spacesX with base points for which the inclusion of the base point is a cofibration.Morphisms are pointed maps. If X is a space, denote by X+ the pointedspace obtained from X by adding a disjoint base point. For two pointedspaces X = (X,x) and Y = (Y, y) define their smash product to be thepointed space

X ∧ Y = X × Y/(x × Y ∪X × y),(10.17)

and the reduced cone to be the pointed space

cone(X) := X × [0, 1]/(X × 1 ∪ x × [0, 1]).(10.18)

A spectrum E = (E(n), σ(n)) | n ∈ Z is a sequence of pointedspaces E(n) | n ∈ Z together with pointed maps called structure mapsσ(n) : E(n) ∧ S1 −→ E(n + 1). A map of spectra f : E → E′ is a sequenceof maps f(n) : E(n)→ E′(n) which are compatible with the structure mapsσ(n), i.e., we have f(n+ 1) σ(n) = σ′(n) (f(n) ∧ idS1) for all n ∈ Z. Mapsof spectra are sometimes called functions in the literature, they should not beconfused with the notion of a map of spectra in the stable category (see [13,III.2.]). The category of spectra and maps will be denoted SPECTRA. Recallthat the homotopy groups of a spectrum are defined by

πi(E) := colimk→∞ πi+k(E(k)),(10.19)


where the ith structure map of the system πi+k(E(k)) is given by the com-posite

πi+k(E(k))S−→ πi+k+1(E(k) ∧ S1)

σ(k)∗−−−→ πi+k+1(E(k + 1))

of the suspension homomorphism S and the homomorphism induced by thestructure map. A weak equivalence of spectra is a map f : E → F of spectrainducing an isomorphism on all homotopy groups. A spectrum E is calledan Ω-spectrum if the adjoint En → ΩEn+1 of each structure map is a weakhomotopy equivalence.

Given a spectrum E and a pointed space X, we can define their smashproduct X ∧E by (X ∧E)(n) := X ∧E(n) with the obvious structure maps.It is a classical result that a spectrum E defines a homology theory by setting

Hn(X,A; E) = πn((X+ ∪A+

cone(A+)) ∧E).

We want to extend this to G-homology theories. This requires the consid-eration of spaces and spectra over the orbit category. Our presentation fol-lows [222], where more details can be found.

In the sequel C is a small category. Our main example will be the orbitcategory Or(G).

Definition 10.20. A covariant (contravariant) C-space X is a covariant(contravariant) functor

X : C → SPACES.

A map between C-spaces is a natural transformation of such functors. Anal-ogously a pointed C-space is a functor from C to SPACES+ and a C-spectruma functor to SPECTRA.

Example 10.21. Let Y be a left G-space. Define the associated contravari-ant Or(G)-space mapG(−, Y ) by

mapG(−, Y ) : Or(G)→ SPACES, G/H 7→ mapG(G/H, Y ) = Y H .

If Y has a G-invariant base point, then mapG(−, Y ) takes values in pointedspaces.

Let X be a contravariant and Y be a covariant C-space. Define their bal-anced product to be the space

X ×C Y :=∐

c∈ob(C)

X(c)× Y (c)/ ∼(10.22)

where ∼ is the equivalence relation generated by (xφ, y) ∼ (x, φy) for allmorphisms φ : c→ d in C and points x ∈ X(d) and y ∈ Y (c). Here xφ standsfor X(φ)(x) and φy for Y (φ)(y). If X and Y are pointed, then one definesanalogously their balanced smash product to be the pointed space


X ∧C Y :=∧

c∈ob(C)

X(c) ∧ Y (c)/ ∼ .(10.23)

In [222] the notation X ⊗C Y was used for this space. Performing the sameconstruction level-wise, one defines the balanced smash product X ∧C E of acontravariant pointed C-space and a covariant C-spectrum E.

The proof of the next result is analogous to the non-equivariant case.Details can be found in [222, Lemma 4.4], where also cohomology theoriesare treated.

Theorem 10.24 (Constructing G-Homology Theories). Let E be acovariant Or(G)-spectrum. It defines a G-homology theory HG

∗ (−; E) by

HGn (X,A; E) := πn

(mapG

(−, (X+ ∪A+ cone(A+))

)∧Or(G) E

).

In particular we have

HGn (G/H; E) = πn(E(G/H)).

A version of the Brown representability Theorem is proved for G-homologytheories and Or(G)-spectra in [59]. Comment 29: Shall we also referto [236]?

Example 10.25 (Bredon homology in terms of spectra). Consider acovariant functorM : Or(G)→ Z-MODULES. Composing it with the functorsending a Z-module N to its Eilenberg-MacLane spectrum HN , which is aspectrum such that π0(HN ) ∼= N and πn(HN ) = 0 for n 6= 0, yields acovariant functor

HM : Or(G)→ SPECTRA.

Then theG-homology theoryHG∗ (−; HM ) associated to HM in Theorem 10.24

agrees with Bredon homology HG∗ (−;M) defined in Example 10.2.

Recall that we seek an equivariant homology theory and not only a G-homology theory. If the Or(G)-spectrum in Theorem 10.24 is obtained froma GROUPOIDS-spectrum in a way we will now describe, then automaticallywe obtain the desired induction structure.

Let GROUPOIDS be the category of small connected groupoids withcovariant functors as morphisms. Recall that a groupoid is a category inwhich all morphisms are isomorphisms. A covariant functor f : G0 → G1 ofgroupoids is called injective, if for any two objects x, y in G0 the induced mapmorG0(x, y)→ morG1(f(x), f(y)) is injective. (We are not requiring that theinduced map on the set of objects is injective.) Let GROUPOIDSinj be thesubcategory of GROUPOIDS with the same objects and injective functorsas morphisms. For a G-set S we denote by GG(S) its associated transportgroupoid. Its objects are the elements of S. The set of morphisms from s0 tos1 consists of those elements g ∈ G which satisfy gs0 = s1. Composition in


GG(S) comes from the multiplication in G. Thus we obtain for a group G acovariant functor

GG : Or(G)→ GROUPOIDSinj, G/H 7→ GG(G/H).(10.26)

A functor of small categories F : C → D is called an equivalence if thereexists a functor G : D → C such that both F G and G F are naturallyequivalent to the identity functor. This is equivalent to the condition that Finduces a bijection on the set of isomorphisms classes of objects and for anyobjects x, y ∈ C the map morC(x, y) → morD(F (x), F (y)) induced by F isbijective.

Theorem 10.27 (Constructing equivariant homology theories usingspectra). Consider a covariant GROUPOIDSinj-spectrum

E : GROUPOIDSinj → SPECTRA.

Comment 30: Attention!! If we use inj, we get an induction structure onlyfor injective group homomorphisms!. This has to be corrected. Suppose thatE respects equivalences, i.e., it sends an equivalence of groupoids to a weakequivalence of spectra. Then E defines an equivariant homology theory

H?∗(−; E),

whose underlying G-homology theory for a group G is the G-homology theoryassociated to the covariant Or(G)-spectrum E GG : Or(G) → SPECTRA inthe previous Theorem 10.24, i.e.,

HG∗ (X,A; E) = HG

∗ (X,A; E GG).

In particular we have

HGn (G/H; E) ∼= HH

n (•; E) ∼= πn(E(I(H))),

where I(H) denotes H considered as a groupoid with one object. The wholeconstruction is natural in E.

Proof. We have to specify the induction structure for a homomorphismα : H → G. We only sketch the construction in the special case A = ∅.

The functor induced by α on the orbit categories is denoted in the sameway

α : Or(H)→ Or(G), H/L 7→ indα(H/L) = G/α(L).

There is an obvious natural transformation of functors Or(H)→ GROUPOIDSinj

T : GH → GG α.

Its evaluation at H/L is the functor GH(H/L) → GG(G/α(L)) which sendsan object hL to the object α(h)α(L) and a morphism given by h ∈ H to the


morphism α(h) ∈ G. The desired homomorphism

indα : HHn (X; E GH)→ HG

n (indαX; E GG)

is induced by the following map of spectra

mapH(−, X+) ∧Or(H) E GH id∧E(T )−−−−−→ mapH(−, X+) ∧Or(H) E GG α∼=←− (α∗mapH(−, X+))∧Or(G) EGG

∼=←− mapG(−, indαX+)∧Or(G) EGG.

Here α∗mapH(−, X+) is the pointed Or(G)-space which is obtained from thepointed Or(H)-space mapH(−, X+) by induction, i.e., by taking the balancedproduct over Or(H) with the (discrete) Or(H)-Or(G) biset morOr(G)(??, α(?))(see [222, Definition 1.8]). Notice that E GG α is the same as the re-striction of the Or(G)-spectrum E GG along α which is often denoted byα∗(E GG) in the literature (see [222, Definition 1.8]). The second mapis given by the adjunction homeomorphism of induction α∗ and restric-tion α∗ (see [222, Lemma 1.9]). The third map is the homeomorphism ofOr(G)-spaces which is the adjoint of the obvious map of Or(H)-spacesmapH(−, X+) → α∗mapG(−, indαX+) whose evaluation at H/L is givenby indα.

It remains to show indα is a weak equivalence, provided that ker(α) actsfreely on X. Because the second and third maps appearing in the definitionabove are homeomorphisms, this boils down to prove that id∧E(T ) is a weakequivalence, provided that ker(α) acts freely on X. This follows from the factthat T (H/L) is an equivalence of groupoids and hence E(T )(G/L) is a weakequivalence of spectra for all subgroups L ⊆ G appearing as isotropy group inX since for such L the restriction of α to L induces a bijection L→ α(L). ut

Example 10.28 (Bredon Homology). Let M be a a covariant functorfrom GROUPOIDSinj to Z-MODULES. Then Bredon homology yields anequivariant homology theory if we define its value at G as the Bredon homol-ogy with coefficients in the covariant functor MG : Or(G) → Z-MODULESsending to G/H to M(GG(G/H)). This is the same as the equivarianthomology theory we obtain from applying Theorem 10.27 to the functorGROUPOIDSinj → SPECTRA which sends a groupoid G to the Eilenberg-MacLane spectrum associated with M(G).

Example 10.29 (Borel homology in terms of spectra). Let E be aspectrum. Let H(−; E) be the (non-equivariant) homology theory associatedto E. Given a groupoid G, denote by EG its classifying space. If G has onlyone object and the automorphism group of this object is G, then EG is amodel for EG. We obtain two covariant functors

cE : GROUPOIDS→ SPECTRA, G 7→ E;

bE : GROUPOIDS→ SPECTRA, G 7→ EG+ ∧E.


Thus we obtain two equivariant homology theories H∗∗ (−; cE) and H∗∗ (−; bE)from Theorem 10.27. These coincide with the ones associated to K∗ =H(−; E) in Example 10.13. Namely, we get for any group G and any G-CW -complex X natural isomorphisms

HGn (X; cE) ∼= Hn(G\X; E);(10.30)

HGn (X; bE) ∼= Hn(EG×G X; E).(10.31)

Exercise 10.32. Let E and F be covariant functors GROUPOIDSinj →SPECTRA. Let t : E → F be a natural transformation such that for everyG ∈ ob(GROUPOIDSinj) the map t(G) : E(G)→ F(G) is a weak equivalenceof spectra.

Show that the induced transformation of equivariant homology theoriesH?∗(−; t) : H?

∗(−; E)→ H?∗(−; F) is a natural equivalence.

10.5 Equivariant Homology Theories Associated to K-and L-Theory

In this section we explain our main examples for covariant functors fromGROUPOIDS or GROUPOIDSinj to SPECTRA.

Let RINGS be the category of associative rings with unit Let RINGSinv

be the category of rings with involution. Let C∗-ALGEBRAS be the categoryof C∗-algebras. There are classical functors for j ∈ −∞q j ∈ Z | j ≤ 2

K : RINGS → SPECTRA;(10.33)

L〈j〉 : RINGSinv → SPECTRA;(10.34)

Ktop : C∗-ALGEBRAS → SPECTRA.(10.35)

The construction of such a non-connective algebraic K-theory functor goesback to Gersten [342] and Wagoner [803]. The spectrum for quadratic al-gebraic L-theory is constructed by Ranicki in [685]. In a more geometricformulation it goes back to Quinn [671]. In the topological K-theory case aconstruction using Bott periodicity for C∗-algebras can easily be derived fromthe Kuiper-Mingo Theorem (see [744, Section 2.2]). The homotopy groups ofthese spectra give the algebraic K-groups of Quillen (in high dimensions) andof Bass (in negative dimensions), the decorated quadratic L-theory groups,and the topological K-groups of C∗-algebras.

We emphasize that in all three cases we need the non-connective versionsof the spectra, i.e., the homotopy groups in negative dimensions are non-trivial in general, in order to ensure later that the formulations of the variousIsomorphisms Conjectures do have a chance to be true.

10.5 Equivariant Homology Theories Associated to K- and L-Theory 311

Now let us fix a coefficient ring R (with involution). Then sending a groupG to the group ring RG yields functors R(−) : GROUPS → RINGS, respec-tively R(−) : GROUPS → RINGSinv, where GROUPS denotes the categoryof groups. Let GROUPSinj be the category of groups with injective group ho-momorphisms as morphisms. Taking the reduced group C∗-algebra defines afunctor C∗r : GROUPSinj → C∗-ALGEBRAS. The composition of these func-tors with the functors (10.33), (10.34) and (10.35) above yields functors

KR(−) : GROUPS → SPECTRA;(10.36)

L〈j〉R(−) : GROUPS → SPECTRA;(10.37)

KtopC∗r (−;F ) : GROUPSinj → SPECTRA,(10.38)

where F = R or C. They satisfy

πn(KR(G)) = Kn(RG);

πn(L〈j〉R(G)) = L〈j〉n (RG);

πn(KtopC∗r (G;F )) = Kn(C∗r (G;F )),

for all groups G and n ∈ Z. The next result essentially says that these functorscan be extended to groupoids.

Theorem 10.39 (K- and L-Theory Spectra over Groupoids). Let Rbe a ring (with involution). There exist covariant functors

KR : GROUPOIDS → SPECTRA;(10.40)

L〈j〉R : GROUPOIDS → SPECTRA;(10.41)

KtopF : GROUPOIDSinj → SPECTRA,(10.42)

with the following properties:

(i) If F : G0 → G1 is an equivalence of (small) groupoids, then the induced

maps KR(F ), L〈j〉R (F ) and Ktop(F ) are weak equivalences of spectra;

(ii) Let I : GROUPS → GROUPOIDS be the functor sending G to G consid-ered as a groupoid, i.e. to GG(G/G). This functor restricts to a functorGROUPSinj → GROUPOIDSinj.There are natural transformations from KR(−) to KR I, from L〈j〉R(−)

to L〈j〉R I and from KC∗r (−) to Ktop I such that the evaluation of each of

these natural transformations at a given group is an equivalence of spectra;(iii) For every group G and all n ∈ Z we have

πn(KR I(G)) ∼= Kn(RG);

πn(L〈j〉R I

inv(G)) ∼= L〈j〉n (RG);

πn(KtopF I(G)) ∼= Kn(C∗r (G;F )).


Proof. We only sketch the strategy of the proof. More details can be foundin [222, Section 2].

Let G be a groupoid. Similar to the group ring RG one can define an R-linear category RG by taking the free R-modules over the morphism sets of G.Composition of morphisms is extended R-linearly. By formally adding finitedirect sums one obtains an additive category RG⊕. Pedersen-Weibel [653],see also [159] and [558], define a non-connective algebraic K-theory functorwhich digests additive categories and can hence be applied to RG⊕. For thecomparison result one uses that for every ring R (in particular for RG) thePedersen-Weibel functor applied to R⊕ (a small model for the category offinitely generated free R-modules) yields the non-connective K-theory of thering R and that it sends equivalences of additive categories to equivalencesof spectra. In the L-theory case RG⊕ inherits an involution and one appliesthe construction of Ranicki [685, Example 13.6 on page 139] to obtain theL〈1〉 = Lh-version. The versions for j ≤ 1 can be obtained by a construc-tion which is analogous to the Pedersen-Weibel construction for K-theory,compare Carlsson-Pedersen [163, Section 4], or by iterating the Shanesonsplitting and then finally passing to a homotopy colimit, compare on thegroup level with [686, Section 17]. In the C∗-case one obtains from G a C∗-category C∗r (G) and assigns to it its topological K-theory spectrum. Thereis a construction of the topological K-theory spectrum of a C∗-category inDavis-Luck [222, Section 2]. However, the construction given there dependson two statements, which appeared in [315, Proposition 1 and Proposition3], and those statements are incorrect, as already pointed out by Thoma-son in [784]. The construction in [222, Section 2] can easily be fixed butinstead we recommend the reader to look at the more recent construction ofJoachim [421]. ut

Exercise 10.43. Compute HD∞n (ED∞; KR) for n ≤ 0 and R = Z,C.

10.6 Two Spectral Sequences

In this section we state two spectral sequences which are useful for compu-tations of equivariant homology theories.

10.6.1 The Equivariant Atiyah-Hirzebruch Spectral Sequence

Theorem 10.44 (The equivariant Atiyah-Hirzebruch spectral se-quence). Let G be a group and HG∗ be a G-homology theory with valuesin Λ-modules in the sense of Definition 10.1. Let X be a G-CW -complex.

Then there is a spectral (homology) sequence of Λ-modules

10.6 Two Spectral Sequences 313

(Erp,q, drp,q : Erp,q → Erp−r,q+r−1)

whose E2-term is given by the Bredon homology of Example 10.2

E2p,q = HG

p+q(X;HGq (−))

for the coefficient system given by the covariant functor

Or(G)→ Λ -MODULES, G/H 7→ HGq (G/H).

The E∞-term is given by

E∞p,q = colimr→∞Erp,q.

This spectral sequence converges to HGp+q(X), i.e., there is an ascending fil-

tration Fp,m−pHGp+q(X) of HGp+q(X) such that

Fp,qHGp+q(X)/Fp−1,q+1HGp+q(X) ∼= E∞p,q.

The construction of the equivariant Atiyah-Hirzebruch spectral sequenceis based on the filtration of X by its skeletons. More details, actually in themore general context of spaces over a category, and a version for cohomologycan be found in [222, Theorem 4.7].

Exercise 10.45. Let X be a proper G-CW -complex such that X/G with theinduced CW -structure has no odd-dimensional cells. Show that KG

n (X) = 0for odd n ∈ Z, where KG

∗ denotes the equivariant topological complex K-homology. Show that KG

n (X) for even n ∈ Z is a finitely generated freeabelian group, if we additionally assume that X/G is finite.

10.6.2 The p-Chain Spectral Sequence

Let G be a group. Recall that for a subgroup H ⊆ G we denote by NGHits normalizer and define the Weyl group WGH := NGH/H. We obtain abijection

WGH∼=−→ autG(G/H), gH 7→

(Rg−1 : G/H → G/H

),

where Rg−1 maps g′H to g′g−1H. Hence for any two subgroups H,K ⊆ Gthe set mapG(G/H,G/K) inherits the structure of a WGK-WGH-biset.

A p-chain is sequence of conjugacy classes of finite subgroups

(H0) < · · · < (Hp)


where (Hi−1) < (Hi) means that Hi−1 is subconjugated, but not conjugatedto (Hi). For p ≥ 1 define a WGHp-WGH0-set associated to such a p-chain by

S((H0) < · · · < (Hp))

:= map(G/Hp−1, G/Hp)G ×WGHp−1

· · · ×WGH1map(G/H0, G/H1)G.

For p = 0 put S(H0) = WGH0.Let X be a G-CW -complex. Then XH = mapG(G/H,X) inherits a right

WGH-action. In particular we get for a p-chain (H0) < · · · < (Hp) a rightWGH0-space X(G/Hp)×WGHp S((H0) < · · · < (Hp)).

Theorem 10.46 (The p-chain spectral sequence). Let G be a group andE be a covariant Or(G)-spectrum. Let X be a proper G-CW -complex.

Then there is a spectral sequence of Λ-modules, called p-chain spectral se-quence, which converges to HG

p+q(X; E) and whose E1-term is

E1p,q =

⊕(H0)<···<(Hp)

πq

((EWGH0×(XHp×WGHp S((H0) < · · · < (Hp)))

)+

∧WGH0E(G/H0)

),

where (H0) < · · · < (Hp) runs through all p-chains consisting of finite sub-groups Hi ⊆ G with XHp 6= ∅.

The p-chain spectral sequence is established in [223, Theorem 2.5 (a) andExample 2.14], actually more generally for spaces over a category. There isalso a more complicated version, where one drops the condition that X isproper. Since then the book-keeping gets more involved and in most applica-tions X is proper, we only deal with the proper case here.

Notice that the complexity of the equivariant Atiyah-Hirzebruch spectralsequence grows with the natural number n for which one wants to computeHGn (X). The complexity of the p-chain spectral sequence growth with themaximum over all natural numbers p for which there is a p-chain (H0) <· · · < (Hp) of finite subgroups such that XHp is non-empty.

Example 10.47 (Free G-CW -complex). Consider the situation of Theo-rem 10.46 and assume additionally that X is a free G-CW -complex. ThenE1p,q = 0 for p ≥ 1 and hence the p-chain spectral sequence predicts

HGq (X; E) = πq

((EG×X)+ ∧G E(G)

).

But this is obviously true since the right hand side of the last equation is bydefinition HG

q (EG×X; E) and the projection EG×X → X is a G-homotopy

equivalence and induces an isomorphism HGq (EG×X; E)

∼=−→ HGq (X; E).

10.7 Equivariant Chern Characters 315

Example 10.48 (G is finite cyclic of prime order). Let G be a finitecyclic group of prime order. Then G has only two subgroups, namely, G and1. Let E be a covariant Or(G)-spectrum and X be a G-CW -complex. Thep-chain spectral sequence of Theorem 10.46 satisfies E1

p,q = 0 for p ≥ 2 andhence reduces to a long exact sequence

. . .→ E11,n

d11,n−−−→ E10,n → HG

n (X; E)→ E11,n−1

d11,n−1−−−−→ E10,n−1 → . . .

We get

E10,n = πn

((EG×X)+ ∧G E(G)

)⊕ πn

(XG

+ ∧E(G/G));

E11,n = πn

((EG×XG)+ ∧G E(G)

)and the differential d1

1,n is given by the homomorphism

πn((EG×XG)+ ∧G E(G)

)→ πn

((EG×X)+ ∧G E(G)

)which is induced by the inclusion XG → X, and the homomorphism (up toa sign)

πn((EG×XG)+ ∧G E(G)

)→ πn

(XG

+ ∧E(G/G))

which is induced by the projection EG×XG → XG.Now suppose additionally that E is the constant functor Or(G) →

SPECTRA with value the spectrum F. Let H∗ be the (non-equivariant) ho-mology theory associated to F. Then HG

n (X; E) = Hn(X/G) and the longexact sequence above reduces to the long exact sequence

(10.49)

. . .→ Hn(EG×G XG)d11,n−−−→ Hn(EG×G X)⊕Hn(XG)

en−→ Hn(X/G)

→ Hn−1(EG×G XG)d11,n−1−−−−→ Hn−1(EG×G X)⊕Hn−1(XG)

en−1−−−→ . . . .

where the maps d1n,1 and en are up to sign induced by the obvious map on

space level.

Exercise 10.50. Give a direct construction of the long exact sequence (10.49).

10.7 Equivariant Chern Characters

If we rationalize and have a Mackey structure on the coefficient system ofan equivariant homology theory, then we can give a more direct and con-crete computation via equivariant Chern characters which does avoid all thedifficulties concerning spectral sequences.


10.7.1 Mackey Functors

Let Λ be an associative commutative ring with unit. Let FGINJ be the cat-egory of finite groups with injective group homomorphisms as morphisms.Let

M : FGINJ→ Λ−MODULES

be a bifunctor, i.e., a pair (M∗,M∗) consisting of a covariant functor M∗ and

a contravariant functor M∗ from FGINJ to Λ−MODULES, which agree onobjects. We will often denote for an injective group homomorphism f : H →G the map M∗(f) : M(H) → M(G) by indf and the map M∗(f) : M(G) →M(H) by resf and write indGH = indf and resHG = resf if f is an inclusionof groups. We call such a bifunctor M a Mackey functor with values in Λ-modules if

(i) For an inner automorphism c(g) : G→ G we have M∗(c(g)) = id: M(G)→M(G);

(ii) For an isomorphism of groups f : G∼=−→ H the composites resf indf and

indf resf are the identity;(iii) Double coset formula

We have for two subgroups H,K ⊂ G

resKG indGH =∑

KgH∈K\G/H

indc(g) : H∩g−1Kg→K resH∩g−1Kg

H ,

where c(g) is conjugation with g, i.e. c(g)(h) = ghg−1.

Important examples of Mackey functors will be RepF (H),Kq(RH), Lq(RH)and Ktop

q (Cr∗(H,F )), where R is an associative ring with unit and andF = R,C.

LetH?∗ be a proper equivariant homology theory with values in Λ-modules.

It defines a covariant functor

H?q(•) : FGINJ→ Λ-MODULES, H 7→ HGq (•).

It sends an injective homomorphism i : H → G to the compositeHH(•) indi−−→

HG(G×H•)HGn (pr)−−−−−→ HGn (•), where pr : G×H• → • is the projection.

We say that the coefficients of H?∗ extend to a Mackey functor, if there exists

a Mackey functor (M∗,M∗) such that M∗ is the functor H?

q(•) above.

Example 10.51. The functors defined in (10.36), (10.37), and (10.38) send-ing a group to the algebraic K- or L-theory of RG or to the topologicalK-theory of C∗r (G;F ) define Mackey functors with the obvious definition ofinduction and restriction.

10.7 Equivariant Chern Characters 317

10.7.2 The Equivariant Chern Character

We can associate to a proper equivariant homology theory with values inΛ-modules H?

∗ another Bredon type equivariant homology theory with valuesin Λ-modules BH?

∗. as follows. For a group G we define

BHGn (X) :=⊕p+q=n

HGp (X;HGq (−))

where HGp (X;HGq (−)) is the Bredon homology of X with coefficients in the

covariant functor Or(G)→ Λ -MODULES sending G/H to HGq (G/H). Next

we show that the collection of the G-homology theories BHG∗ (X,A) inheritsthe structure of a proper equivariant homology theory. We have to specifythe induction structure.

Let α : H → G be a group homomorphism and (X,A) be a proper H-CW -pair. Induction with α yields a functor denoted in the same way

α : OrFIN (H)→ OrFIN (G), H/K 7→ indα(H/K) = G/α(K).

There is a natural isomorphism of OrFIN (G)-chain complexes

indα COrFIN (H)∗ (X,A)

∼=−→ COrFIN (G)∗ (indα(X,A))

and a natural adjunction isomorphism, see [526, (2.5)](indα C

OrFIN (H)∗ (X,A)

)⊗ROrFIN (G) HGq (−)

∼=−→ COrFIN (H)∗ (X,A)⊗ROrFIN (H)

(resαHGq (−)

).

The induction structure on H?∗ yields a natural equivalence of ROrFIN (H)-

modulesHHq (H/?)

∼=−→ resαHGq (−).

The last three maps can be composed to a chain isomorphism

COrFIN (H)∗ (X,A)⊗ROrFIN (H)HHq (H/?)

∼=−→ C∗(indα(X,A))⊗ROr(G,FIN )HGq (−),

Since X is proper and hence the Bredon homology can be defined overOrFIN (H) instead of Or(G), it induces a natural map

indα : Hp(X,A;HHq (−))∼=−→ HG

p (indα(X,A);HGq (−)).

Thus we obtain the required induction structure.Define for a finite group H


(10.52) SH(HHq (•

):= coker

⊕K⊂HK 6=H

indHK :⊕K⊂HK 6=H

HKq (•)→ HHq (•)

.

Notice that SH(HHq (•

)carries a natural left Λ[NGH/H ·CGH]-module

structure, where NGH/H ·CGH is the quotient of NGH by the normal sub-group H · CGH := h · g | h ∈ H, g ∈ CGH. The obvious left-actionof WGH = NGH/H-action on XH yields a left NGH/H · CGH-action onCGH\XH and hence a right NGH/H · CGH-action by y · k := k−1 · y fory ∈ XH and k ∈ NGH/H · CGH.

The proof of the following result can be found in [526, Theorem 0.2 and0.3].

Theorem 10.53 (The equivariant Chern character). Let Λ be a com-mutative ring with Q ⊂ Λ. Let H?

∗ be a proper equivariant homology theorywith values in Λ-modules in the sense of Definition 10.9. Suppose that itscoefficients extend to a Mackey functor.

(i) There is an isomorphism of proper equivariant homology theories

ch?∗ : BH?

∗∼=−→ H?

∗;

(ii) Let I be the set of conjugacy classes (H) of finite subgroups H of G. Thenthere is for any group G and any proper G-CW -pair (X,A) a naturalisomorphism⊕

p+q=n

⊕(H)∈I

Hp(CGH\(XH , AH);Λ)⊗Λ[NGH/H·CGH] SH(HHq (•)

)∼=−→ BHGn (X,A).

Theorem 10.53 reduces the computation of HGn (X,A) to the computationof the singular or cellular homology Λ-modules Hp(CGH\(XH , AH);Λ) ofthe CW -pairs CGH\(XH , AH) including the obvious right WGH-operationand of the left Λ[WGH]-modules SH

(HHq (•)

)which only involve the values

HGq (G/H) = HHq (•).

Exercise 10.54. Let Λ be a commutative ring with Q ⊂ Λ. Let H?∗ be

a proper equivariant homology theory with values in Λ-modules. Supposethat its coefficients extend to a Mackey functor. Consider a group G anda proper G-CW -complex X. Show that all differentials of the equivariantAtiyah-Hirzebruch spectral sequence converging to HG

p+q(X) vanish.

Exercise 10.55. Let H?∗ be a proper equivariant homology theory with val-

ues in Q-modules in the sense of Definition 10.9. Suppose that its coefficients

10.8 Some Rational Computations 319

extend to a Mackey functor. Let G be a group. Consider two families of sub-groups F and G with F ⊆ G ⊆ FIN . Let ιF⊆G : EF (G) → EG(G) be theup to G-homotopy unique G-map. Show that for every n the induced mapHGn (ιF⊆G) : HGn (EF (G))→ HGn (EG(G)) is injective.

Remark 10.56 (Rationalizing an equivariant homology theory). LetH?∗ be an equivariant homology theory with values in Z-modules. Suppose

that its coefficients extend to a Mackey functor. Then we obtain an equiva-riant homology theory Q⊗Z H?

∗ with values in Q-modules whose coefficientsextend to a Mackey functor since Q ⊗Z − is a flat functor and commuteswith direct sums over arbitrary index sets. We can apply Theorem 10.53 toQ⊗Z H?

∗ and thus obtain a rational computation of H?∗.

10.8 Some Rational Computations

10.8.1 Green Functors

Let φ : Λ → Λ′ be a homomorphism of associative commutative rings withunit. Let M be a Mackey functor with values in Λ-modules and let N and Pbe Mackey functors with values in Λ′-modules. A pairing with respect to φis a family of maps

m(G) : M(G)×N(G)→ P (G), (x, y) 7→ m(G)(x, y) =: x · y,

where G runs through the finite groups and we require the following proper-ties for all injective group homomorphisms f : H → G of finite groups:

(x1 + x2) · y = x1 · y + x2 · y for x1, x2 ∈M(H), y ∈ N(H);x · (y1 + y2) = x · y1 + x · y2 for x ∈M(H), y1, y2 ∈ N(H);

(λx) · y = φ(λ)(x · y) for λ ∈ Λ, x ∈M(H), y ∈ N(H);x · λ′y = λ′(x · y) for λ′ ∈ Λ′, x ∈M(H), y ∈ N(H);

resf (x · y) = resf (x) · resf (y) for x ∈M(G), y ∈ N(G);indf (x) · y = indf (x · resf (y)) for x ∈M(H), y ∈ N(G);x · indf (y) = indf (resf (x) · y) for x ∈M(G), y ∈ N(H).

A Green functor with values in Λ-modules is a Mackey functor U withvalues in Λ-modules together with a pairing with respect to id : Λ → Λ andelements 1G ∈ U(G) for each finite group G such that for each finite groupG the pairing U(G) × U(G) → U(G) induces the structure of an Λ-algebraon U(G) with unit 1G and for any morphism f : H → G in FGINJ the mapU∗(f) : U(G) → U(H) is a homomorphism of Λ-algebras with unit. Let Ube a Green functor with values in Λ-modules and M be a Mackey functorwith values in Λ′-modules. A (left) U -module structure on M with respect


to the ring homomorphism φ : Λ → Λ′ is a pairing such that any of themaps U(G) ×M(G) → M(G) induces the structure of a (left) module overthe Λ-algebra U(G) on the Λ-module φ∗M(G) which is obtained from theΛ′-module M(G) by λx := φ(λ)x for r ∈ Λ and x ∈M(G).

The importance of the notion of a Green functor is due to the followingelementary lemma which allows to deduce induction theorems for all Mackeyfunctors which are modules over a given Green functor from the correspond-ing statement for the given Green functor.

Lemma 10.57. Let φ : Λ → Λ′ be a homomorphism of associative commu-tative rings with unit. Let U be a Green functor with values in Λ-modulesand let M be a Mackey functor with values in Λ′-modules such that M comeswith a U -module structure with respect to φ. Let S be a set of subgroups ofthe finite group G. Suppose that the map⊕

H∈SindGH :

⊕H∈S

U(H)→ U(G)

is surjective. Then the map⊕H∈S

indGH :⊕H∈S

M(H)→M(G)

is surjective.

Proof. By hypothesis there are elements uH ∈ U(H) for H ∈ S satisfying1G =

∑H∈S indGH uH in U(G). This implies for x ∈M(G).

x = 1G · x =

(∑H∈S

indGH uH

)· x =

∑H∈S

indGH(uH · resHG x

).

ut

Example 10.58 (Burnside ring). The Burnside ring A(G) of a (not nec-essarily finite) group G is the commutative associative ring with unit A(G)which is obtained by the additive Grothendieck construction applied to thecommutative associative semi-ring with unit given by the G-isomorphismclasses [S] of G-sets S of finite cardinality, i.e., |S| <∞, under disjoint unionand cartesian product and the unit element given by [G/G]. For more in-formation about the Burnside ring for not necessarily finite groups we referto [529].

The Burnside ring defines a Mackey functor A(?) by induction and re-striction. The ring structure and the Mackey structure fit together to thestructure of a Green functor A(?) with values in Z-modules.

Exercise 10.59. Let M be a Mackey functor with values in Λ-modules foran associative commutative ring Λ with unit. Let φ : Z → Λ be the unique


ring homomorphism. Show that M inherits the structure of a module overthe Green functor given by the Burnside ring with respect to φ.

Example 10.60 (Swan group). Let G be a (not necessarily) finite group.Let Λ be an associative commutative ring with unit. Denote by Swp(G;Λ) bethe abelian group whose generators are the isomorphism classes [M ] of ΛG-modules M , whose underlying Λ-module is finitely generated projective. Forevery short exact sequence 0 → M0 → M1 → M2 → 0 of such ΛG-modules,we require the relation [M0]− [M1] + [M2] in Swp(G;Λ). The tensor productover Λ with the diagonal G-action induces the structure of an associativecommutative ring with unit [Λ], where [Λ] is the class of Λ equipped with thetrivial G-action. We call Swp(G;Λ) the Swan ring.

Let R be an associative ring with unit. Let M be a ZG-module whoseunderlying Z-module is finitely generated free. If P is a finitely generatedprojective RG-module, then M ⊗Z P is a finitely generated projective RG-module if r ∈ R acts by r · (m⊗ p) = m⊗ rp and g ∈ G acts by g · (m⊗ p) :=gm⊗ gp. This yields a pairing

Swp(G)⊗Kn(RG)→ Kn(RG)

Using induction and restriction Swp(?) defines a Green functor with val-ues in Z-modules. There is a natural homomorphism of Green functors withvalues in Z-modules

A(G)→ Swp(G;Λ)

sending the class of a finite G-set S to the Λ-module with S as basis equippedwith the G-action coming from the G-action on S. If Λ = Z, we abbreviateSwp(G) := Swp(G;Z). Thanks to the pairing above the Mackey functor givenby Kn(RH) becomes a module over the Green functor given by Swp(G).

Example 10.61 (Rational representation ring). An important exampleof a Green functor with values in Q-modules is the rationalized representationring of rational representations Q⊗ZRQ(?). It assigns to a finite group G theQ-module Q⊗ZRepQ(G), where RepQ(G) denotes the rational representationring of G. Notice that RepQ(G) is the same as the projective class groupK0(QG) and also the same as Swp(G;Q). The Mackey structure comes frominduction and restriction of representations. The pairing Q ⊗Z RepQ(G) ×Q ⊗Z RepQ(G) → Q ⊗Z RepQ(G) comes from the tensor product P ⊗Q Q oftwo QG-modules P and Q equipped with the diagonal G-action. The unitelement is the class of Q equipped with the trivial G-action.

Recall that classQ(G) denotes the Q-vector space of functions G → Qwhich are invariant under Q-conjugation, i.e., we have f(h1) = f(h2) for twoelements g1, g2 ∈ G if the cyclic subgroups 〈g1〉 and 〈g2〉 generated by g1 andg2 are conjugate in G. Elementwise multiplication defines the structure of aQ-algebra on classQ(G) with the function which is constant 1 as unit element.


Taking the character of a rational representation yields an isomorphism ofQ-algebras [751, Theorem 29 on page 102]

χG : Q⊗Z RepQ(G)∼=−→ classQ(G).(10.62)

We define a Mackey structure on classQ(?) as follows. Let f : H → G bean injective group homomorphism. For a character χ ∈ classQ(H) define itsinduction with f to be the character indf (χ) ∈ classQ(G) given by

indf (χ)(g) =1

|H|·

∑l∈G,h∈Hf(h)=l−1gl

χ(h).

For a character χ ∈ classQ(G) define its restriction with f to be the characterresf (χ) ∈ classQ(H) given by

resf (χ)(h) := χ(f(h)).

One easily checks that this yields the structure of a Green functor on classQ(?)and that the family of isomorphisms χG defined in (10.62) yields an isomor-phism of Green functors from Q⊗Z RepQ(?) to classQ(?).

10.8.2 Induction Lemmas

As already explained by Lemma 10.57, Green functors play a prominentrole for induction theorems. In order to formulate two versions, we have tointroduce the following idempotents.

Let G be a finite group. There is a ring homomorphism

(10.63) card: A(G)→∏H

Z, [S] 7→ (|SH |)(H),

where the product is indexed over the conjugacy classes of subgroups of G and|SH | is the cardinality of the H-fixed point set. The ring homomorphism cardis injective and has a finite cokernel. In particular it induces an isomorphismof Q-algebras

cardQ : Q⊗Z A(G)∼=−→∏(H)

Q.

Now let eG ∈∏

(H) Q be the idempotent whose value at (G) is 1 and whose

value at (H) for H 6= G is 0. We then define the idempotent

ΘG := card−1Q (eG) ∈ Q⊗Z A(G).(10.64)

For a finite cyclic group C, define the idempotent


θC ∈ Q⊗Z RepQ(C)(10.65)

to be the element whose image under the isomorphism of (10.62) is the classfunction which sends an elements of C to 1 if it is a generator, and to 0otherwise. The image of ΘC under the map Q ⊗Z A(G) → Q ⊗Z RepQ(C),which sends a finite C-set S to the associated permutation module Q[S], isθC .

Lemma 10.66. Let φ : Q → Λ be a homomorphism of associative commu-tative rings with unit. Let M be a Mackey functor with values in Λ-moduleswhich is a module over the Green functor Q ⊗Z RepQ(H) with respect to φ.Then

(i) For a finite group H the map⊕C⊂HC cyclic

indHC :⊕C⊂HC cyclic

M(C)→M(H)

is surjective;(ii) Let C be a finite cyclic group. Let

θC : M(C)→M(C)

be the map induced by the Q ⊗Z RepQ(C)-module structure and multipli-cation with idempotent θC of (10.65). Then the inclusion of the imageof θC : M(C) → M(C) into M(C) composed with the projection onto thecokernel of ⊕

D⊂CD 6=C

indCD :⊕D⊂CD 6=C

M(D)→M(C)

is an isomorphism.

Proof. Let C ⊂ H be a cyclic subgroup of the finite group H. Then we getfor h ∈ H

1

[H : C]· indHC θC(h) =

1

[H : C]· 1

|C|·∑l∈H

l−1hl∈C

θC(l−1hl) =1

|H|·

∑l∈H

〈l−1hl〉=C

1.

This implies in Q⊗Z RepQ(H) ∼= classQ(H)

1H =∑C⊂HC cyclic

1

[H : C]· indHC θC(10.67)

since for any l ∈ H and h ∈ H there is precisely one cyclic subgroup C ⊂ Hwith C = 〈l−1hl〉. Now assertion (i) follows from the following calculation forx ∈M(H)


x = 1H ·x =

∑C⊂HC cyclic

1

[H : C]· indHC θC

·x =∑C⊂HC cyclic

1

[H : C]·indHC (θC ·resCH x).

It remains to prove assertion (ii). Obviously θC is an idempotent for anycyclic group C. We get for x ∈M(C) from (10.67)

(1C−θC) ·x =

∑D⊂CD 6=C

1

[C : D]· indCD θD

·x =∑D⊂CD 6=C

1

[C : D]· indCD(θD ·resDC x)

and for D ⊂ C,D 6= C and y ∈M(D)

θC · indCD y = indCD(resDC θC · y) = indCD(0 · y) = 0.

This finishes the proof of Lemma 10.66. ut

The proof of the next result is similar to the one of Lemma 10.66. De-tails can be found in [547, Lemma 7.2 and Lemma 7.4]. Key ingredients areLemma 10.57, Example 10.60, and the result of Swan [774, Corollary 4.2 onpage 560] which implies together with [652, page 890] that for every finitegroup H the cokernel of the map⊕

C⊆H,C cyclic

indHC :⊕C⊆HC cyclic

Swp(C)→ Swp(H)

is annihilated by |H|2.

Lemma 10.68. Let R be an associative ring with unit. Then

(i) For a finite group H and n ∈ Z the map⊕C⊂HC cyclic

indHC :⊕C⊂HC cyclic

Q⊗Z Kn(RC)→ Q⊗Z Kn(RH)

is surjective;(ii) Let C be a finite cyclic group. Let

ΘC : Q⊗Z Kn(RC)→ Q⊗Z Kn(RC)

be the map induced by the Q⊗Z A(C)-module structure and multiplicationwith the idempotent θC of (10.64). Then the inclusion of the image of themap θC : Q ⊗Z Kn(RC) → Q ⊗Z Kn(RC) into Q ⊗Z Kn(RC) with theprojection onto the cokernel of

10.8 Some Rational Computations 325⊕D⊂CD 6=C

indCD :⊕D⊂CD 6=C

M(D)→M(C)

is an isomorphism.

Remark 10.69 (L-theory analogue of Lemma 10.68). The L-theoryanalogue of Lemma 10.68 is also true, one has to use instead of Swan [774,Corollary 4.2 on page 560] the corresponding L-theory analogue of Dress [254,Theorem 2(a)].

For more information about Mackey and Green functors and inductiontheorems we refer for instance to [787, Section 6] and [254].

10.8.3 Rational Computation of the source of the assembly maps

We get from Theorem 10.53, Lemma 10.68 and Remark 10.69.

Theorem 10.70 (Rational computation of the source of the assem-bly maps appearing in the Farrell-Jones and Baum-Connes Con-jecture). Let R be an associative ring with unit and let F be R or C. LetG be a group. Denote by J be the set of conjugacy classes (C) of finite cyclicsubgroups C of G.

Then the rational Chern character of Theorem 10.53 induces isomorphisms⊕p+q=n

⊕(C)∈J

Hp(CGC;Q)⊗Q[WGC] ΘC ·(Q⊗Z Kq(RC)

)∼=−→ Q⊗Z H

Gn (EG; KR)

and⊕p+q=n

⊕(C)∈J

Hp(CGC;Q)⊗Q[WGC] ΘC ·(Q⊗Z L

〈−∞〉q (RC)

)∼=−→ Q⊗Z H

Gn (EG; L

〈−∞〉R ).

and⊕p+q=n

⊕(C)∈J

Hp(CGC;Q)⊗Q[WGC] ΘC ·(Q⊗Z Kq(C

∗r (C,F ))

)∼=−→ Q⊗Z H

Gn (EG; Ktop

F ).

Computations of Kq(RC) as Z[aut(C)]-module for finite cyclic groups Cand R = Z or R a field of characteristic zero can be found in [646].


The computations simplifies even more of we consider the case R = C, asthe following example shows which is taken from [526, Example 8.11].

Example 10.71 (Complex coefficients). Let T be the set of conjugacyclasses (g) of elements g ∈ G. If we tensor with C instead of Q and takeR = F = C, then the isomorphism appearing in Theorem 10.70 reduce tothe isomorphisms⊕

p+q=n

⊕(g)∈T

Hp(CG〈g〉;C)⊗Z Kq(C)∼=−→ C⊗Z H

Gn (EG; KC);

⊕p+q=n

⊕(g)∈T

Hp(CG〈g〉;C)⊗Z Lq(C)∼=−→ C⊗Z H

Gn (EG; L

〈−∞〉R );

⊕p+q=n

⊕(g)∈T

Hp(CG〈g〉;C)⊗Z Ktopq (C)

∼=−→ C⊗Z HGn (EG; Ktop

C ),

where we use in the definition of Lq(C) and Ln(CG) the involutions comingfrom complex conjugation. The targets of the maps above are isomorphic to

C⊗Z Kn(CG), C⊗Z L〈−∞〉n (CG) and C⊗Z Kn(C∗r (G;C) if the Farrell-Jones

Conjecture and the Baum-Connes Conjecture hold for G.

10.9 Some Integral Computations

Integral computations are of course harder than rational computations. Wehave already provided basic tools such as the equivariant Atiyah-Hirzebruchspectral sequence and the p-chain spectral sequence in Section 10.6.

Often we are considering an equivariant homology theory and want tocompute HGn (EG) or HGn (EG). Sometimes one gets easy and useful compu-tations by having good models for EG and EG. We illustrate this in thefollowing favorite case.

Let G be a discrete group. Let MFIN be the subset of FIN consist-ing of elements in FIN which are maximal with respect to inclusion inFIN . Throughout this subsection we suppose that G satisfies the conditions(M) and (NM) introduced in Subsection 9.6.12, where also examples of suchgroups G are given. Let Mi | i ∈ I be a complete set of representatives forthe conjugacy classes of maximal finite subgroups of G. Consider an equiva-riant homology theory H?

∗. In the sequel we put

BG := G\EG.(10.72)

Then we obtain from Theorem 9.34 long exact sequences

10.9 Some Integral Computations 327

(10.73) · · · →⊕i∈IH1n (BMi)→ H1n (BG)⊕

⊕i∈IHMin (•)→ HGn (EG)⊕

i∈IH1n−1(BMi)→ H1n−1(BG)⊕

⊕i∈IHMin−1(•)→ · · · .

(10.74) · · · →⊕i∈IH1n (BMi)→ H1n (BG)⊕

⊕i∈IH1n (•)→ H1n (BG)⊕

i∈IH1n−1(BMi)→ H1n−1(BG)⊕

⊕i∈IH1n−1(•)→ · · · .

We have the maps H1n (•) → HMin (•) induced by the inclusion 1 →

Mi and HMin (•) → H1n (•) induced by the projection Mi → 1. The

composite is the identity. Define

(10.75) HMin (•) := ker

(HMin (•)→ H1n (•)

).

Obviously we have an isomorphism

HMin (•) ∼= H1n (•)⊕ HMi

n (•).

One can splice the two long exact sequences (10.73) and (10.74) togetherto the long exact sequence

(10.76) · · · → H1n+1(BG)→⊕i∈IHMin (•)→ HGn (EG)→

→ H1n (BG)→⊕i∈IHMin−1(•)→ · · · .

Example 10.77 (Equivariant topological K-theory of EG for G =Z2 o Z/4). Consider the automorphism φ : Z2 → Z2, (x, y) 7→ (−y, x). Ithas order four. We want to show for the semi-direct product G = Z2 oα Z/4

KGn (EG) ∼=

Z9 if n is even;

0 if n is odd

In this case we have the presentation

Z2 o Z4 = 〈u, v, t : t4 = 1, uv = vu, tut−1 = v, tvt−1 = u−1〉.

The maximal finite subgroups are up to conjugacy given by

M0 = 〈t〉;M1 = 〈ut〉;M2 = 〈ut2〉.


We have M0∼= M1

∼= Z4 and M2∼= Z2. We have

KZ/mn (•) ∼=

Zm−1 if n is even;

0 if n is odd.

Obviously BG is the same as Z/4\T 2 for the obvious Z/4-action on thetwo-dimensional torus T 2 = Z2\EG = Z2\EZ2. This implies because we arein dimension two, that BG has a model which is a compact 2-dimensionalmanifold. The rational cohomology H∗(BG) agrees with H∗(T 2;Q)Z/4. SinceZ/4 is a subgroup of SL(2,Z), its action on T 2 is orientation preserving.This implies that Z/4 acts trivial on Hp(T 2;Q) for p = 0, 2. Since Z/4acts freely on Z2 = H1(T 2;Z) outside 0, we conclude H1(T 2;Q)Z/4 ∼=homZ(H1(T 2;Z)Z/4,Q) ∼= 0. We conclude that BG = Z/4\T 2 has the ra-tional cohomology of S2 and hence is homeomorphic to S2. This implies thatK0(BG) ∼= Z2 and K1(BG) = 0.

The group G satisfies conditions conditions (M) and (NM) by a directcheck or because of Subsection 9.6.12 since the Z/4 action on Z2 given by αis free outside 0. Now the claim follows from the long exact sequence (10.76)applied in the case H?

∗ = K?∗ .

Since G satisfies the Baum-Connes Conjecture, we have Kn(C∗r (G)) ∼=KGn (EG).

Exercise 10.78. Determine all finite subgroups F ⊆ SL2(Z) and computefor any of these KG

n (EG) for n ∈ Z and G = Z2 o F .

The long exact sequence (10.76) will be a key ingredient in computations of

Kn(RG), L〈−∞〉n (RG) and Kn(C∗r (G)), provided that G satisfies the Farrell-

Jones Conjecture and the Baum-Connes Conjecture, see Theorem 15.9.Already for group homology the long exact sequence (10.76) contains valu-

able information as we explain next.

Example 10.79 (Group homology). Suppose that G satisfies (M) and(NM). Let H∗ be given by the Borel homology, i.e., HG(X) := Hn(EG ×X) for Hn singular homology with coefficients in Z, see Example 10.13.Then (10.76) reduces to the long exact sequence, where Hn(G) := Hn(BG)

is the group homology and Hn(G) := ker(Hn(G)→ Hn(1)

· · · → Hn+1(BG)→⊕i∈I

Hn(Mi)→ Hn(G)

→ Hn(BG)→⊕i∈I

Hn−1(Mi)→ · · · .

In particular we get for n ≥ dim(BG) + 2 an isomorphism⊕i∈I

Hn(Mi)∼=−→ Hn(G).

10.10 Equivariant Homology Theory over a Group and Twisting with Coefficients 329

Example 10.80 (The group homology of certain extensions 1 →Zn → G → F → 1 for finite F ). Consider an extension 1 → Zn → G →F → 1 for finite F such that the conjugation action of F on Zn is free outside0 ∈ Zn. Then the conditions (M) and (NM) are satisfied by [557, Lemma 6.3]and there is an n-dimensional model for EG whose underlying space is R2.

Even in the case, where F is a finite cyclic group, the computation ofthe homology of G is not at all easy. It is carried in [224, Theorem 2.1]provided that |F | is a prime. More information in the case, where there areno restriction |F |, can be found in [502].

Example 10.81 (Group homology of one relator groups). Let G =〈(qi)i∈I | r〉 be a presentation with one relation. Let F be the free group withbasis qi | i ∈ I. Then r is an element in F . We have seen in Subsection 9.6.12that G satisfies conditions (M) and (NM) and there is up to conjugationprecisely one finite cyclic subgroup C which is maximal among the finitesubgroups and in Subsection 9.6.13 that there is a 2-dimensional model forEG. Hence we get

Hn(G) = Hn(C) for n ≥ 3;

and a short exact sequence

0→ H2(G)→ H2(BG)→ H1(C)→ H1(G)→ H1(BG)→ 0.

We conclude from the specific model for EG of Subsection 9.6.13 the fol-lowing facts. We have H2(BG) = 0, H2(G) = 0 and an exact sequence0 → H1(C) → H1(G) → H1(BG) → 0, if the relation r ∈ F does not be-long to the commutator [F, F ]. We have H2(BG) ∼= Z , H2(G) ∼= Z, andH1(G) ∼= H1(BG) is a free abelian group of rank |I| − 1, provided thatr ∈ [F, F ].

10.10 Equivariant Homology Theory over a Group andTwisting with Coefficients

Next we present a slight variation of the notion of an equivariant homologytheory introduced in Section 10.3. We have to treat this variation since welater want to study coefficients over a fixed group Γ which we will thenpullback via group homomorphisms with Γ as target. For instance, we maybe interested in the algebraic K-theory of a twisted groups ring RαG forsome homomorphism α : G→ aut(R). More generally, we will later consideradditive G-categories as coefficients.

Fix a group Γ . A group (G, ξ) over Γ is a group G together with a grouphomomorphism ξ : G → Γ . A map α : (G1, ξ1) → (G2, ξ2) of groups over Γ


is a group homomorphisms α : G1 → G2 satisfying ξ2 α = ξ1. Let Λ be anassociative commutative ring with unit.

Definition 10.82 (Equivariant homology theory over a group Γ ).An equivariant homology theory H?

∗ with values in Λ-modules over a group Γ

assigns to every group (G, ξ) over Γ a G-homology theory HG,ξ∗ with values inΛ-modules and comes with the following so called induction structure: givena homomorphism α : (H,µ) → (G, ξ) of groups over Γ and an H-CW -pair(X,A), there are for each n ∈ Z natural homomorphisms

indα : HH,µn (X,A)→ HG,ξn (α∗(X,A))(10.83)

satisfying

• Compatibility with the boundary homomorphisms

∂G,ξn indα = indα ∂H,µn ;• Functoriality

Let β : (G, ξ) → (K, ν) be another morphism of groups over Γ . Then wehave for n ∈ Z

indβα = HK,νn (f1) indβ indα : HH,µHn(X,A)→ HK,νn ((β α)∗(X,A)),

where f1 : β∗α∗(X,A)∼=−→ (β α)∗(X,A), (k, g, x) 7→ (kβ(g), x) is the

natural K-homeomorphism;• Compatibility with conjugation

Let (G, ξ) be a group over Γ . Fix g ∈ G such that ξ c(g) = ξ.Then the conjugation homomorphisms c(g) : G → G defines a morphismc(g) : (G, ξ) → (G, ξ) of groups over Γ . Let f2 : (X,A) → c(g)∗(X,A) bethe G-homeomorphism which sends x to (1, g−1x) in G×c(g) (X,A).Then for every n ∈ Z and every G-CW -pair (X,A) the homomorphismindc(g) : HG,ξn (X,A)→ HG,ξn (c(g)∗(X,A)) agrees with HGn (f2).

• Bijectivity

If ker(α) acts freely on X \A, then indα : HH,µn (X,A)→ HG,ξn (indα(X,A))is bijective for all n ∈ Z.

Definition 10.82 reduces to Definition 10.9, if one puts Γ = 1.The analog of Lemma 10.12 in this setting is obvious and easily checked.The proof of Theorem 10.27 to this setting as explained in [65, Lemma 7.1].

Theorem 10.84 (Constructing equivariant homology over a grouptheories using spectra). Let Γ be a group. Denote by GROUPOIDS ↓ Γthe category of small connected groupoids over Γ considered as a groupoidwith one object. Consider a covariant functor

E : GROUPOIDS ↓ Γ → SPECTRA

which sends equivalences of groupoids to weak equivalences of spectra.


Then we can associate to it an equivariant homology theory H?∗(,−; E)

with values in Z-modules over Γ such that for every group (G,µ) over Γ andsubgroup H ⊆ G we have a natural identification

HH,ξ|Hn (•; E) = HG,ξn (G/H,E) = πn(E(H, ξ|H)).

There are twisted analogues of the functors mentioned in Section 10.5,see (11.8) and (11.13).

More information about equivariant homology theories over a group canbe found in [65].

10.11 Miscellaneous

Comment 31: Shall we say something about RO(G)-gradings and Mackey-structures. The sentence at the very bottom on page 301 in [350] is interesting:It turns out that the Bredon cohomology theory H∗G(−;M) can be extendedto an RO(G)-graded theory if and only if the coefficient system M extends toa Mackey functor, where [510] is cited. We can extend the notion of Mackeystructure to infinite groups, what is not so clear for RO(G)-gradings. Maybefor RO(G)-gradings one has to replace representations by G-vector bundles.See also [236].

Equivariant stable cohomotopy has been introduced in [529] for arbitrarygroups G and proper finite G-CW -complexes and extended to proper G-CW -complexes in [236]. Comment 32: update reference when it has been postedon the arXive. A sytsematic study of the equivariant homotop categoryfor proper G-CW -complex can be found in [236]. Comment 33: updatereference when it has been posted on the arXive. A version of the SegalConjecture in this setting is proved in [541].

If one is dealing with equivariant topological K-theory, then there existsa Chern character, where one does not have to fully rationalize, it sufficesto invert the orders of all the isotropy groups of the proper G-CW -complexunder consideration, see [528].

There are also equivariant cohomology theories and a cohomological ver-sion of the equivariant Chern character, see [530]. It can be used to extendthe Atiyah-Segal Completion Theorem for finite groups to infinite groups andproper G-CW -complexes, see [544, 545]. It leads to rational computations ofK∗(BG) also for not necessarily finite groups, see [422, 533].

An equivariant Chern character for equivariant topological K-theory aftercomplexification has been introduced in [89].

Comment 34: We shall mention the paper on the Segal Conjecture forinfinite groups when we have put it on the arXive.



name of texfile: ic

Chapter 11

The Farrell-Jones Conjecture

11.1 Introduction

In this chapter we discuss the Farrell-Jones Conjecture for K- and L-theoryfor arbitrary groups and rings. It predicts that certain assembly maps

HGn (pr) : HG

n (EVCY(G); KR)→ Kn(RG);

HGn (pr) : HG

n (EVCY(G); L〈−∞〉R ) → L〈−∞〉n (RG),

are bijective for all n ∈ Z. The target is the algebraic K-or L-group of thegroup ring RG which one wants to understand. The source is an expressionwhich depends only on the values of these K-and L-groups on virtually cyclicsubgroups of G and is therefore much more accessible. The version above isoften the one which is relevant in concrete applications, but nevertheless wewill consider generalizations, for instance to twisted group rings and twistedinvolutions. The most general version will be the Full Farrell-Jones Conjec-ture 11.20. It implies all other variants of the Farrell-Jones Conjecture, whichappear in this book, see Section 11.10, and has very nice inheritance proper-ties, see Section 11.6.

A status report of the Full Farrell-Jones Conjecture 11.20 wil be given inTheorem 14.1, it is known for a large class of groups.

The main point about the Full Farrell-Jones Conjecture 11.20 is that itimplies a great variety of other prominent conjectures such as the ones dueto Bass, Borel, Kaplanski, and Novikov, and leads to very deep and inter-esting results about manifolds and groups, as we will record and explainin Section 11.11. Often these applications are much more appealing andeasier to comprehend than the rather technical Full Farrell-Jones Conjec-ture 11.20. The author’s favorite is the Borel Conjecture which predicts thattwo closed aspherical manifolds are homeomorphic if and only if their funda-mental groups are isomorphic, and in this case any homotopy equivalence ishomotopic to a homeomorphism.

The other sections are mainly concerned with the question whether onecan reduce the family of virtually cyclic subgroups to a smaller family of sub-groups, for instance to all finite subgroups or just to the family consisting ofthe trivial subgroup, except Section 11.12 which consists of a short discussionabout G-theory.

We have tried to keep this chapter as much as possible independent of theother chapters, so that one may start reading here directly.

333

334 11 The Farrell-Jones Conjecture

11.2 The Farrell-Jones Conjecture with Coefficients inRings

Let G be a (discrete) group. Recall that a G-homology theory HG∗ with valuesin Λ-modules for some commutative (associative) ring (with unit) Λ assigns toevery G-CW -pair (X,A) and integer n ∈ Z a Λ-module HGn (X,A) such thatthe obvious generalization to G-CW -pairs of the axioms of a (non-equivariantgeneralized) homology theory for CW -complexes holds, i.e., G-homotopy in-variance, the long exact sequence of a G-CW -pair, excision, and the disjointunion axiom are satisfied. The precise definition of a G-homology theory canbe found in Definition 10.1 and of a G-CW -complex in Definition 9.2, seealso Remark 9.3.

Recall that we have defined the notion of a family of subgroups of a groupGin Definition 1.52, namely, to be a set of subgroups of G which is closed underconjugation with elements of G and passing to subgroups. Denote by EF (G)a model for the classifying space for the family F of subgroups of G, i.e., aG-CW -complex EF (G) whose isotropy groups belong to F and for which foreach H ∈ F the H-fixed point set EF (G)H is weakly contractible. Such amodel always exists and is unique up to G-homotopy, see Definition 9.18 andTheorem 9.19. Recall that EG and EG are abbreviations for EFIN (G) andEVCY(G).

11.2.1 The K-Theoretic Farrell-Jones Conjecture withCoefficients in Rings

Given a ring R, there is a specific G-homology theory HGn (−; KR) with values

in Z-modules which has the property that HGn (G/H; KR) ∼= Kn(RH) holds

for all n ∈ Z and subgroups H ⊆ G, where Kn(RH) is the nth algebraicK-group of the group ring RH. Its construction can be used in the sequelas a black box. We have already given some details, namely, it is given bythe equivariant homology theory H?

∗(−; KR) evaluated at G, which is asso-ciated to the covariant functor KR : GROUPOIDS → SPECTRA of (10.40)in Theorem 10.27. Let VCY be the family of virtually cyclic subgroups, i.e.,subgroups which are either finite or contain Z as subgroup of finite index.

Conjecture 11.1 (K-theoretic Farrell-Jones Conjecture with coeffi-cients in the ring R). We say that G satisfies the K-theoretic Farrell-JonesConjecture with coefficients in the (associative) ring (with unit) R if the as-sembly map induced by the projection pr : EVCY(G)→ G/G

HGn (pr) : HG

n (EVCY(G); KR)→ HGn (G/G; KR) = Kn(RG)


11.2 The Farrell-Jones Conjecture with Coefficients in Rings 335

In many of the proofs the coefficients rings do not play a role and thereforeit is reasonable to consider the following stronger variant which is now astatement about the group G itself.

Conjecture 11.2 (K-theoretic Farrell-Jones Conjecture with coeffi-cients in rings). We say that G satisfies the K-theoretic Farrell-JonesConjecture with coefficients in rings if the K-theoretic Farrell-Jones Conjec-ture 11.1 with coefficients in R holds for every ring R.

Exercise 11.3. Show that Conjecture 11.2 does not hold for G = Z if onereplaces VCY by FIN in Conjecture 11.1.

11.2.2 The L-Theoretic Farrell-Jones Conjecture with Coefficientsin Rings

The situation for L-theory is similar. Namely, given a ring with involution

R, there is a specific G-homology theory HGn (−; L

〈−∞〉R ) with values in Z-

modules which has the property that HGn (G/H; L

〈−∞〉R ) ∼= L

〈−∞〉n (RH) holds

for all n ∈ Z and subgroups H ⊆ G, where L〈−∞〉n (RH) is the nth algebraic

L-groups of the group ring with involution RH with decoration 〈−∞〉. Itsconstruction can be used in the sequel as a black box. We have alreadygiven some details, namely, it is given by the equivariant homology theory

H?∗(−; L

〈−∞〉R ) evaluated at G, which is associated to the covariant functor

L〈−∞〉R : GROUPOIDS→ SPECTRA of (10.41) in Theorem 10.27.

Conjecture 11.4 (L-theoretic Farrell-Jones Conjecture with coeffi-cients in the ring with involution R). We say that G satisfies the L-theoretic Farrell-Jones Conjecture with coefficients in the (associative unital)ring with involution R if the assembly map given by the projection

HGn (pr) : HG

n (EVCY(G); L〈−∞〉R )→ HG

n (G/G; L〈−∞〉R ) = L〈−∞〉n (RG)


Exercise 11.5. Show that Conjecture 11.4 holds for G = Z if one replacesVCY by FIN in Conjecture 11.6.

Conjecture 11.6 (L-theoretic Farrell-Jones Conjecture with coeffi-cients in rings with involution). We say that G satisfies the L-theoreticFarrell-Jones Conjecture with coefficients in rings with involution if the L-theoretic Farrell-Jones Conjecture 11.4 with coefficients in R holds for everyring with involution R.


Remark 11.7 (The decoration 〈−∞〉 is necessary). One can define forany decoration j ∈ n ∈ Z | n ≤ 1 q −∞ an assembly map

HGn (pr) : HG

n (EVCY(G); L〈j〉R )→ HG

n (G/G; L〈j〉R ) = L〈j〉n (RG).

But in general one can only hope that it is bijective if one chooses j = −∞.Counterexamples for G = Z2×F for a finite group F , R = Z and j = −1, 0, 1,which is also sometimes denoted by j = p, h, s, are constructed in [300].

11.3 The Farrell-Jones Conjecture with Coefficients inAdditive Categories

There are situations, where one wants to consider twisted groups rings RαGfor some group homomorphism α : G → aut(R) into the group of ring auto-morphisms of R. Elements in RαG are given by formal finite sums

∑g∈G rg ·g,

and addition and multiplication is given by(∑g∈G

rg · g)

+

(∑g∈G

sg · g

):=∑g∈G

(rg + sg) · g;

(∑g∈G

rg · g

)·

(∑g∈G

sg · g

):=∑g∈G

( ∑h,k∈G,g=hk

rh · α(h)(sk)

)· g.

So the decisive relation for the multiplication is (r ·h)·(s·k) = (r ·α(h)(s))·hk.Or even, more generally, one may want to consider crossed product rings, seefor instance [71, Section 4].

When considering L-theory, one considers a ring with involution R andwants to allow to twist the involution on RG by an orientation homomor-phism w : G → center(R) satisfying w(g) = w(g), namely the w-twisted in-volution on RG is given by∑

g∈Grg · g :=

∑g∈G

w(g) · rg · g−1.

The situation becomes even more involved if one wants to consider crossedproduct rings with involution. Details are explained in [71, Section 4].

It turns out that one can nicely treat these generalization of group ringsand involutions by looking at additive G-categories (with involution).

There is another crucial reason why it is useful to look at coefficients inadditive G-categories (with involution). These versions of the Farrell-JonesConjecture with coefficients in additive G-categories (with involution) havemuch better inheritance properties than the one with coefficients in rings

11.3 The Farrell-Jones Conjecture with Coefficients in Additive Categories 337

(with involution) as we will explain below in Section 11.6, for instance theypass to subgroups.

The details are given for additive G-categories and K-theory in [77], thecase of additive G-categories with involution is treated for the K-theory tak-ing the involution into account and for L-theory in [71]. Since we can use thisgeneral approach essentially as a black box, we give only a brief summaryhere, following the notation of [71].

11.3.1 The K-theoretic Farrell-Jones Conjecture with Coefficientsin Additive G-Categories

Let A be an additive G-category in the sense of [71, Definition 2.1], i.e., anadditive categories with right G-action by functors of additive categories. LetGROUPOIDS ↓ G be the category of connected groupoids over G. Recall thatfor a group G we denote by I(G) the groupoid with one object and G as itsautomorphism group. We obtain from [71, Section 5] a contravariant functorto the category ADD-CAT of small additive categories

GROUPOIDS ↓ G→ ADD-CAT , pr: G → I(G) 7→∫GA pr .

Composing it with the functor sending an additive category to its non-connective K-theory spectrum, see for instance [159, 558, 653], yields a func-tor

KA : GROUPOIDS ↓ G→ SPECTRA.(11.8)

By Theorem 10.84 we obtain an equivariant homology theory over G in thesense of Definition 10.82. In particular its evaluation atG yields aG-homologytheory HG

∗ (−; KA).

Conjecture 11.9 (K-theoretic Farrell-Jones Conjecture with coeffi-cients in additive G-categories). We say that G satisfies the K-theoreticFarrell-Jones Conjecture with coefficients in additive G-categories if for anyadditive G-category A and any n ∈ Z the assembly map given by the projec-tion

Hn(pr) : HGn (EVCY(G); KA)→ HG

n (G/G; KA) = πn(KA(I(G))

)is bijective.

Remark 11.10 (The setting of additive G-categories encompassesthe setting with rings as coefficients). Let α : G → aut(R) be a grouphomomorphism. Then we have already introduced the twisted group ringRα(G) above. For a suitable choice of an additive G-category A the assembly


map



)appearing in Conjecture 11.9 reduces to the assembly map

Hn(pr) : HGn (EVCY(G); KR,α)→ HG

n (G/G; KR,α) = Kn(RαG)

where for any subgroup H ⊆ G and integer n ∈ Z we have πn(KR,α(I(H))

)=

Kn(Rα|HH). If α is trivial, this is precisely the assembly map appearing inConjecture 11.1. More details, even for crossed product rings, can be foundin [71, Section 4 and 6].

In particular we get that the K-theoretic Farrell-Jones Conjecture 11.2with coefficients in rings holds for G if the K-theoretic Farrell-Jones Conjec-ture 11.9 with coefficients in additive G-categories holds for G.

Exercise 11.11. Let R be a ring. Define a category A(R) as follows. Foreach integer m ∈ Z with m ≥ 0 we have one object [m]. For m,n ≥ 1 theset of morphisms from [m] to [n] is the set Mm,n(R) of (m,n)-matrices withentries in R. The set of morphisms from [0] to [m] and from [m] to [0] consistof precisely one element. Composition is given by matrix multiplication.

Show that A(R) can be equipped with the structure of a small additivecategory and that it is equivalent as an additive category to the category offinitely generated free R-modules.

Remark 11.12 (Involutions and K-theory). If A is an additive G-category with involution in the sense of [71, Definition 4.22], then the involu-tion induces involutions on the source and target of the K-theoretic assemblymap



)of Conjecture 11.9 and the assembly map is compatible with them.

11.3.2 The L-theoretic Farrell-Jones Conjecture with Coefficientsin Additive G-Categories with Involution

Let A be an additive G-category with involution in the sense of [71, Defi-nition 4.22]. We obtain from [71, Section 7] a contravariant functor to thecategory ADD-CAT inv of small additive categories with involution

GROUPOIDS ↓ G→ ADD-CAT inv, pr: G → I(G) 7→∫GA pr .

11.3 The Farrell-Jones Conjecture with Coefficients in Additive Categories 339

Composing it with the functor sending an additive category with involutionA to its L-theory spectrum L〈−∞〉(A), yields a functor

L〈−∞〉A : GROUPOIDS ↓ G→ SPECTRA,(11.13)

where for the construction of the spectrum L-theory L〈−∞〉A associated to an

additive category with involution A we refer to Ranicki [685, Chapter 13].By Theorem 10.84 we obtain an equivariant homology theory over G in thesense of Definition 10.82. In particular its evaluation atG yields aG-homology

theory HGn (−; L

〈−∞〉A ).

Conjecture 11.14 (L-theoretic Farrell-Jones Conjecture with coef-ficients in additive G-categories with involution). We say that Gsatisfies the L-theoretic Farrell-Jones Conjecture with coefficients in additiveG-categories with involution if for any additive G-category with involution Athe assembly map given by the projection

Hn(pr) : HGn (EVCY(G); L

〈−∞〉A )→ HG

n (G/G; L〈−∞〉A ) = πn

(L〈−∞〉A (I(G))

)is bijective.

Remark 11.15 (The setting of additive G-categories with involu-tion encompasses the setting with rings with involution as coeffi-cients). Let R be a ring with involution. Consider a group homomorphismα : G → aut(R) satisfying α(g)(r) = α(g)(r), and a group homomorphismw : G→ center(R) satisfying w(g) = w(g). Then we have already introducedthe twisted group ring Rα(G) above. It inherits an involution by∑

g∈Grg · g :=

∑g∈G

w(g) · α(g−1)(rg) · g−1

and we denote this ring with involution by Rα,wG. For a suitable choice ofan additive G-category with involution A the assembly map


〈−∞〉A )→ HG

n (G/G; L〈−∞〉A ) = L〈−∞〉n

(LA(I(G))

)appearing in Conjecture 11.14 reduces to the assembly map


〈−∞〉R,α,w)→ HG

n (G/G; L〈−∞〉R,α,w) = L〈−∞〉n (Rα,wG),

where for any subgroup H ⊆ G and integer n ∈ Z we have L〈−∞〉R,α,w(I(H)) =

L〈−∞〉n (Rα|HH,w|H). If α and w are trivial, this is precisely the assembly

map appearing in Conjecture 11.4. More details, even for crossed productrings, can be found in [71, Theorem 0.4, Section 4 and 8].

In particular we get that the L-theoretic Farrell-Jones Conjecture 11.6 withcoefficients in rings with involution holds for G if the L-theoretic Farrell-Jones


Conjecture 11.14 with coefficients in additive G-categories with involutionholds for G.

Exercise 11.16. Let F : A → B be a functor of additive categories. Showthat it is an equivalence of additive categories if and only if for every twoobjects A and B inA the induced map morA(A0, A1)→ morB(F (A0), F (A1))sending f to F (f) is bijective and for each object B in B there exists an objectA in A such that F (A) and B are isomorphic in B.

11.4 Finite Wreath Products

The versions of the Farrell-Jones Conjecture discussed above do not carryover to overgroups of finite index. To handle this difficulty, we consider finitewreath products.

Let G and F be groups. Their wreath product G oF is defined as the semi-direct product (

∏F G)oF , where F acts on

∏F G by permuting the factors.

For our purpose the following elementary lemma is crucial.

Lemma 11.17.

(i) There is an embedding (H o F1) o F2 → H o (F1 o F2);(ii) If F1 and F2 are finite, then F1 o F2 is finite;

(iii) Let G be an overgroup of H of finite index. Then there is subgroup N ⊆ Hof H, which satisfies [G : H] <∞ and is normal in G, and a finite groupF such that G embeds into N o F .

Proof. (i) See [484, Lemma 1.21].

(ii) This is obvious.

(iii) Let S denote a system of representatives of the cosets G/H. Since G/His by assumption finite, N :=

⋂s∈S sHs

−1 is a finite index normal subgroupof G and is contained in H. Now G can be embedded in N oG/N , see [251,Section 2.6]. ut

Conjecture 11.18 (K-theoretic Farrell-Jones Conjecture with coef-ficients in additive G-categories with finite wreath products). Wesay that G satisfies the K-theoretic Farrell-Jones Conjecture with coefficientsin additive G-categories with finite wreath products if for any finite group Fthe group G o F satisfies the K-theoretic Farrell-Jones Conjecture 11.9 withcoefficients in additive G o F -categories.

Conjecture 11.19 (L-theoretic Farrell-Jones Conjecture with coef-ficients in additive G-categories with involution with finite wreathproducts). We say that G satisfies the L-theoretic Farrell-Jones Conjecturewith coefficients in additive G-categories with involution with finite wreath

11.6 Inheritance Properties of the Farrell-Jones Conjecture 341

products if for any finite group F the group G o F satisfies the L-theoreticFarrell-Jones Conjecture 11.14 with coefficients in additive G o F -categorieswith involution.

11.5 The Full Farrell-Jones Conjecture

Next we can formulate the version of the Farrell-Jones Conjecture which isthe most general one, implies all other ones and has the best inheritanceproperties.

Conjecture 11.20 (Full Farrell-Jones Conjecture). We say that agroup satisfies the Full Farrell-Jones Conjecture if G satisfies both theK-theoretic Farrell-Jones Conjecture 11.18 with coefficients in additive G-categories with finite wreath products and the L-theoretic Farrell-Jones Con-jecture 11.19 with coefficients in additive G-categories with involution withfinite wreath products.

11.6 Inheritance Properties of the Farrell-JonesConjecture

In this section we discuss the inheritance properties of the various versionsof the Farrell-Jones Conjectures above. Both the K-theoretic Farrell-JonesConjecture 11.2 with coefficients in rings and the L-theoretic Farrell-JonesConjecture 11.6 with coefficients in rings with involution do not have goodinheritance properties. The reason why we have introduced the other variantsis that they do have some remarkable inheritance properties.

Comment 35: Introduce the notion of a Farrell-Jones group and the classFJ .

Theorem 11.21 (Inheritance properties of the Farrell-Jones Conjec-ture).

(i) Passing to subgroupsLet H ⊆ G be an inclusion of groups. If G satisfies the Full Farrell-Jones

Conjecture 11.20, then H satisfies the Full Farrell-Jones Conjecture 11.20;(ii) Passing to finite direct products

If the groups G0 and G1 satisfy the Full Farrell-Jones Conjecture 11.20,then G0 ×G1 satisfies the Full Farrell-Jones Conjecture 11.20;

(iii) Group extensionsLet 1 → K → G → Q → 1 be an extension of groups. Suppose that

for any virtually cyclic subgroup V ⊆ Q the group p−1(V ) satisfies theFull Farrell-Jones Conjecture 11.20 and that the group Q satisfies the Full


Farrell-Jones Conjecture 11.20. Comment 36: Maybe we can assumethat K satisfies the conjecture and we only need infinite cyclic groups.Then G satisfies the Full Farrell-Jones Conjecture 11.20;

(iv) Directed colimitsLet Gi | i ∈ I be a direct system of groups indexed by the directed set I

(with arbitrary structure maps). Suppose that for each i ∈ I the group Gisatisfies the Full Farrell-Jones Conjecture 11.20.Then the colimit colimi∈I Gi satisfies the Full Farrell-Jones Conjecture 11.20;

(v) Passing to finite free productsIf the groups G0 and G1 satisfy the Full Farrell-Jones Conjecture 11.20,

then G0 ∗G1 satisfies the Full Farrell-Jones Conjecture 11.20;(vi) Other versions

The obvious versions of assertions (i), (ii), (iii), (iv), and (v) are alsotrue for the versions 11.9, 11.14, 11.18, and 11.19 of the Farrell-JonesConjecture.

(vii) Passing to overgroups of finite indexLet G be an overgroup of H with finite index [G : H]. If H satisfies the

Full Farrell-Jones Conjecture 11.20, then G satisfies the Full Farrell-JonesConjecture 11.20;

(viii) Passing to finite wreath productsIf G satisfies the Full Farrell-Jones Conjecture 11.20, then G oF satisfies

the Full Farrell-Jones Conjecture 11.20 for any finite group F .

Proof. (i) This is proved in [77, Theorem 4.5] for Conjecture 11.9 and in [71,Theorem 0.10] for Conjecture 11.14. Now the claim follows for the Full Farrell-Jones Conjecture 11.20, since H oF is a subgroup of G oF for every subgroupH ⊆ G.

(ii) and(iii) The obvious version of assertion (iii) is proved in [71, Theorem 0.9]for Conjecture 11.14, the proof for Conjecture 11.9 is analogous.

The versions of the Farrell-Jones Conjecture 11.9 and 11.14 are true forvirtually finitely generated abelian groups by [66, Theorem 3.1]. Hence theyhold in particular for the product of two virtually cyclic subgroups. By in-specting the proof of [484, Lemma 3.15], we see that the obvious version ofassertion (ii) holds for versions of the Farrell-Jones Conjecture 11.9 and 11.14.Thus we have shown that the obvious versions of both assertion (ii) and asser-tion (iii) hold for the versions of the Farrell-Jones Conjecture 11.9 and 11.14.

Next we prove assertion (ii) for the Full Farrell-Jones Conjecture 11.20.Suppose the Full Farrell-Jones Conjecture 11.20 holds for G1 and G2. Let Fbe any finite subgroup. We have to show that versions of the Farrell-JonesConjecture 11.9 and 11.14 holds for (G1 ×G2) oF . By assumption they bothhold for G1 oF and G2 oF . Since (G1×G2)oF is a subgroup of (G1 oF )×(G2 oF )by [484, Lemma 1.197] and Conjecture 11.9 and 11.14 pass to subgroups bythe argument given in assertion (i), assertion (ii) holds for the Full Farrell-Jones Conjecture 11.20.

11.6 Inheritance Properties of the Farrell-Jones Conjecture 343

Finally we conclude from [484, Lemma 3.16] that the Full Farrell-JonesConjecture 11.20 satisfies assertion (iii).(iv) This is proved in [71, Theorem 0.8] for Conjecture 11.14, the proof forConjecture 11.9 is completely analogous. Now the claim follows for the FullFarrell-Jones Conjecture 11.20 since there is an obvious isomorphism for afinite group F , see [484, Lemma 1.20].

colimi→∞(Gi o F )∼=−→(colimi→∞Gi

)o F ).

(v) Let G1 and G2 be groups. Let pr : G1 ∗ G2 → G1 × G2 be the canoni-cal projection. Let V ⊆ G1 ×G2 be a virtually cyclic subgroup. Then thereexists a free group F and a finite group H such that pr−1(V ) is a subgroupof F o H, see [484, Lemma 3.21]. (In the statement of [484, Lemma 3.21]the assumption countable appears but the proof goes through in the gen-eral case without modifications.) A finitely generated free group satisfies theFull Farrell-Jones Conjecture 11.20 by [75, Remark 6.4], since it is a hy-perbolic group. Hence F satisfies the Full Farrell-Jones Conjecture 11.20 byassertion (iv). We conclude from assertion (viii) that F oH satisfies the FullFarrell-Jones Conjecture 11.20. Hence pr−1(V ) satisfies the Full Farrell-JonesConjecture 11.20 for every virtually cyclic subgroup V ⊆ G1 ×G2 by asser-tion (i). The product G1×G2 satisfies the Full Farrell-Jones Conjecture 11.20by assertion (ii). Now assertion (iii) implies that G1 ∗ G2 satisfies the FullFarrell-Jones Conjecture 11.20.

(vi) The proofs given above do cover these cases as well.

(vii) This follows from Lemma 11.17 (iii) and assertion (i).

(viii) This follows from using Lemma 11.17 (i) and (ii) and assertion (i). ut

Exercise 11.22. Consider an epimorphism of groups G → Q whose kernelis finite. Suppose that Q satisfies the Full Farrell-Jones Conjecture 11.20.

Show that G satisfies the Full Farrell-Jones Conjecture 11.20.

Exercise 11.23. Suppose that the Full Farrell-Jones Conjecture 11.20 holdsfor all groups which occur as fundamental groups of a connected closed ori-entable 4-manifold.

Show that then the Full Farrell-Jones Conjecture 11.20 holds for all groups.

Exercise 11.24. Let 1→ K → G→ Q→ 1 be an extension of groups. Sup-pose that for any infinite cyclic subgroup C ⊆ Q the group p−1(C) satisfiesthe Full Farrell-Jones Conjecture 11.20 and that the groups K and Q satisfythe Full Farrell-Jones Conjecture 11.20.

Show that then G satisfies the Full Farrell-Jones Conjecture 11.20.


11.7 Splitting the Assembly Map from FIN to VCY

In the sequel we denote for two families F ⊆ G by

(11.25) ιF⊆G : EF (G)→ EG(G)

the up to G-homotopy unique G-map. Notice that ιF⊆ALL : EF (G) →EALL(G) = G/G is the projection.

Theorem 11.26 (Splitting the K-theoretic assembly map from FINto VCY). Let G be a group and let A be an additive G-category. Let n beany integer.

Then

HGn

(ιFIN⊆VCY ; KA

): HG

n

(EFIN (G); KA

)→ HG

n

(EVCY(G); KA

)is split injective. In particular we obtain a natural splitting

HGn

(EVCY(G); KA

) ∼=−→ HGn

(EFIN (G); KA

)⊕HG

n

(ιFIN⊆VCY ; KA

).

Moreover, there exists an Or(G)-spectrum NKA and a natural isomorphism

HGn

(ιFIN⊆VCYI ; NKA

) ∼=−→ HGn

(ιFIN⊆VCY ; KA

)where VCYI is the family of virtually cyclic subgroups of type I.


Whereas in [79, Theorem 1.3] just a splitting is established, in [559, The-orem 0.1], an explicit Or(G)-spectra NKG

A is constructed and the relativeterms including the involution on the K-groups are further analyzed, in par-ticular they are identified with K-groups of Nil-categories. For rings see alsoLafont-Ortiz [493].

For L-theory one has at least the following version, which is mentionedafter Theorem 1.3 in [79] for rings. The argument carries over to additiveG-categories with involution.

Theorem 11.27 (Splitting the L-theoretic assembly map from FINto VCY). Let A be an additive G-category with involution such that thereexists an integer Nwith the property that πn

(KA(I(V ))

)= 0 for all virtually

cyclic subgroups V of G and all n ≤ N .Then

HGn

(ιFIN⊆VCY ; L

〈−∞〉A

): HG

n

(EFIN (G); L

〈−∞〉A

)→ HG

n

(EVCY(G); L

〈−∞〉A

)is split injective.

11.8 Splitting Rationally the Assembly Map from T R to FIN 345

It is not clear whether the condition about πn(KA(I(V ))

)appearing in

Theorem 11.27, which is needed for the proposed proof, is necessary. If weconsider group rings RG, this condition is automatically satisfied if R isregular, e.g., if R = Z or R is a field.

11.8 Splitting Rationally the Assembly Map from T Rto FIN

Lemma 11.28. Let G be a group and let R be a ring (with involution).Then the relative assembly maps

Hn(ιT R⊆FIN ; KR) : Hn(ET R(G); KR)→ Hn(EFIN (G); KR);

Hn(ιT R⊆FIN ; L〈−∞R ) : Hn(ET R(G); L

〈−∞R )→ Hn(EFIN (G); L

〈−∞R );

KGn (ιT R⊆FIN ) : KG

n (ET R(G)) → KGn (EFIN (G));

KOGn (ιT R⊆FIN ) : KOGn (ET R(G)) → KOGn (EFIN (G)),

are split injective after applying −⊗Z Q for n ∈ Z.

Proof. This follows from Theorem 10.53, see Exercise 10.55. ut

Example 11.29 (The L-theory assembly map for the trivial familyis not injective in general). Consider the group Z/3. Then

H1(BZ/3; L(Z))→ L1(Z[Z/3])

is not injective. Namely, the target is known to be trivial, but the source isnon-trivial. This can be seen by inspecting the Atiyah-Hirzebruch spectralsequence converging to Hp+q(BZ/3; L(Z)) with E2-term

Hp(BZ/3, Lq(Z)) =

Z/3 p ≥ 1, p odd, q ≡ 0 mod 4;

Lq(Z) p = 0;

0 otherwise.

Notice that Wh(Z/3), K0(Z[Z/3]) and Kn(Z[Z/3]) for n ≤ −1 vanish by The-orem 2.112, Theorem 2.113 (iva), and Theorem 3.21 (i) and (v), Example 1.91so that the decorations for the L-groups do not play a role by Theorem 7.102.

Comment 37: Can one find explicite examples also for algebraic K-theory?

Comment 38: If one wants to extend Lemma 11.28 to additive G-categories, one needs an equivariant Chern character for equivariant homol-ogy theories over a group.


11.9 Reducing the Family of Subgroups for theFarrell-Jones Conjecture

Next we explain that one sometimes can reduce the family of virtually cyclicsubgroups VCY to a smaller family.

A virtually cyclic group V is called of type I if it admits an epimorphismto the infinite cyclic group, and of type II if it admits an epimorphism ontothe infinite dihedral group. The elementary proof of the following result canbe found in [559, Lemma 1.1].

Lemma 11.30. Let V be an infinite virtually cyclic group.

(i) V is either of type I or of type II;(ii) The following assertions are equivalent:

(a) V is of type I;(b) H1(V ) is infinite;(c) H1(V )/ tors(V ) is infinite cyclic;(d) The center of V is infinite;

(iii) There exists a unique maximal normal finite subgroup KV ⊆ V , i.e., KV isa finite normal subgroup and every normal finite subgroup of V is containedin KV ;

(iv) Let QV := V/KV . Then we obtain a canonical exact sequence

1→ KViV−→ V

pV−−→ QV → 1.

Moreover, QV is infinite cyclic if and only if V is of type I and QV isisomorphic to the infinite dihedral group if and only if V is of type II;

(v) Let f : V → Q be any epimorphism onto the infinite cyclic group or ontothe infinite dihedral group. Then the kernel of f agrees with KV ;

Exercise 11.31. Let φ : V → W be a homomorphism of infinite virtuallycyclic groups with infinite image. Then φ maps KV to KW and we obtainthe following canonical commutative diagram with exact rows

1 // KViV //

φK

VpV //

φ

QV //

φQ

1

1 // KWiW // W

pW // QW // 1

with injective φQ.

Exercise 11.32. Show that a group G is infinite virtually cyclic if and onlyif it admits a proper cocompact isometric action on R.

11.9 Reducing the Family of Subgroups for the Farrell-Jones Conjecture 347

In the sequel we denote by VCYI the family of subgroups which are eitherfinite or infinite virtually cyclic of type I.

Definition 11.33 (Hyperelementary group). Let p be a prime. A (pos-sibly infinite) group G is called p-hyperelementary if it can be written as anextension 1 → C → G → P → 1 for a cyclic group C and a finite group Pwhose order is a power of p.

We call G hyperelementary if G is p-hyperelementary for some prime p.

If G is finite, this reduces to the usual definition. Notice that for a finitep-hyperelementary group G one can arrange that the order of the finite cyclicgroup C appearing in the extension 1 → C → G → P → 1 for a finite p-group P is prime to p. Subgroups and quotient groups of p-hyperelementarygroups are p-hyperelementary again. For a group G and a prime p let HEpand HE respectively be the class of (possibly infinite) subgroups which arep-hyperelementary or hyperelementary respectively.

The following result is taken from [66, Theorem 8.2].

Theorem 11.34 (Hyperelementary induction). Let G be a group andlet A be an additive G-category (with involution). Then both relative assemblymaps

Hn(ιHE,VCY ; KA) : HGn

(EHE(G); KA

)→ HG

n

(EVCY(G); KA

)and

Hn(ιHE,VCY ; L〈−∞〉A ) : HG

n

(EHE(G); L

〈−∞〉A

)→ HG

n

(EVCY(G); L

〈−∞〉A

)induced by the up to G-homotopy unique G-map ιHE,VCY ;EH(G)→ EVCY(G)are bijective for all n ∈ Z.

11.9.1 Reducing the Family of Subgroups for the Farrell-JonesConjecture for K-Theory

The following result taken from [227, Remark 1.6].

Theorem 11.35 (Passage from VCYI to VCY for K-theory). Let G bea group and A be an additive G-category. Then the relative assembly map

HGn

(ιVCYI⊆VCY ; KA

): HG

n

(EVCYI (G); KA

) ∼=−→ Hn

(EVCY(G); KA

)is bijective for all n ∈ Z.

We can combine Theorem 11.34 and Theorem 11.35 to get


Theorem 11.36 (Passage from HEI to VCY for K-theory). Let G bea group and A be an additive G-category. Let HEI be the family of subgroupsof G given by the intersection VCYI ∩HE.

Then the relative assembly map

HGn

(ιHEI⊆VCY ; KA

): HG

n

(EHEI (G); KA

) ∼=−→ Hn

(EVCY(G); KA

)is bijective for all n ∈ Z.

Proof. This follows from Theorem 11.34, Theorem 11.35, Theorem 13.9 (ii),and Lemma 13.14. ut

Theorem 11.36 implies that we get equivalent conjectures if we replace inConjectures 11.1, 11.2, 11.9 and 11.18 the family VCY by the smaller familyHEI .

Exercise 11.37. Fix a prime p. Show that an infinite subgroup H ⊂ Gbelongs to HEp ∩ VCYI if and only if there exists a natural number m ≥ 1,a finite p-subgroup P ⊆ G and an element t ∈ G such that t ∈ NGP andtpm

xt−pm

= x holds for all x ∈ P . In particular H ∈ HEp∩VCYI is isomorphicto P oφ Z for some automorphism φ : P → P whose order is a power of p.

Exercise 11.38. Let p be a prime. Let G be an infinite virtually cyclic groupof type I which is p-hyperelementary. Let R be a regular ring.

Show that the map induced by the projection pr : EFIN (G)→ G/G

HGn (EFIN (G); KR)→ HG

n (G/G; KR) = Kn(RG)

is bijective for all n ∈ Z after applying −⊗Z Z[1/p].

One can reduce the families by extending the classical induction theoremsfor finite groups due to Dress to our setting. This is carried out in detailin [70]. There only rings as coefficients are treated but the proofs carry overto the setting of additive G-categories. For instance for K-theory one has toextend the relevant pairing of the Swan group for group rings to additivecategories. We leave the details to the reader and just record some results.Recall that FCY is the family of finite cyclic subgroups.

Theorem 11.39 (Reductions to families contained in FIN for alge-braic K-theory). Let G be a group and R be a ring.

(i) Suppose that R is regular and Q ⊆ R. Then the relative assembly map

HGn

(ι(HE∩FIN )⊆VCY ; KR

): HG

n

(EHE∩FIN (G); KR

) ∼=−→ Hn

(EVCY(G); KR

)is bijective for all n ∈ Z;


(ii) Suppose that R is regular and Q ⊆ R. Let p be a prime. Then the relativeassembly map

HGn

(ι(HEp∩FIN )⊆VCY ; KR

): HG

n

(EHEp∩FIN (G); KR

) ∼=−→ Hn

(EVCY(G); KR

)is bijective for all n ∈ Z after applying Z(p) ⊗Z −;

(iii) Suppose that R is regular and Q ⊆ R. Then the relative assembly map

HGn

(ιFCY⊆VCY ; KR

): HG

n

(EFCY(G); KR

) ∼=−→ Hn

(EVCY(G); KR

)is bijective for all n ∈ Z after applying Q⊗Z −;

(iv) Suppose that R is regular and Q ⊆ R. Then the relative assembly map

HGn

(ιFIN⊆VCY ; KR

): HG

n

(EFIN (G); KR

) ∼=−→ Hn

(EVCY(G); KR

)is bijective for all n ∈ Z;

(v) Let G be a group and let R be a regular ring. Then

HGn (ιFIN⊆VCY ; KR

): HG

n (EFIN (G); KR)∼=−→ HG

n (EVCY(G); KR)

is rationally bijective for all n ∈ Z.

Proof. The proof of assertions (i), (ii), and (iii) is a modification of the proofof [70, Theorem 0.1], where for assertion (iii) one has to take into account [70,Lemma 4.1 (e)].

Assertion (iv) is proved in [546, Proposition 70 on page 744] or can be de-duced from assertion (i) using the Transitivity Principle, see Theorem 13.13.

Assertion (v) is proved in [559, Theorem 0.3]. ut

Exercise 11.40. Let G be a group satisfying the K-theoretic Farrell-JonesConjecture 11.1 with coefficients in the field F of characteristic zero. Showthat the various inclusions induce an isomorphism

colimH∈SubHE∩FIN (G)K0(FH)∼=−→ K0(FG).

For later purposes we state and prove the following results.

Lemma 11.41. Consider a ring R, a group G, and m ∈ Z with m ≤ 1.Suppose that, for every finite subgroup H of G, every group automorphism

φ : H∼=−→ H, and every n ≥ 1, the assembly map

HHoφZm (EZ; KR[Zn])→ H

HoφZm (•; KR[Zn]) = Km((R[Zn])[H oφ Z])

is an isomorphism, where we consider the Z-CW -complex EZ as a H oφ Z-CW -complex by restriction with the projection H oφ Z→ Z.



HGi (EFIN (G),KR)

∼=−→ HGi (EVCY(G),KR)

is bijective for i ≤ m.

Proof. Theorem 11.35 implies that for any group G and i ∈ Z the map

HGi (EVCYI (H oφ Z),KR)→ HG

i (EVCY(H oφ Z),KR)

is bijective. Hence it suffices to show that, for every group G and i ∈ Z withi ≤ m, the canonical map

HGi (EFIN (G),KR)

∼=−→ HGi (EVCYI (G),KR)

is bijective. Thanks to the Transitivity Principle appearing in Theorem 13.12,this has only to be done in the special case, where G is a virtually cyclic groupof type I.

Consider any finite group H and any group automorphism φ : H∼=−→ H.

Since EZ with the HoφZ action coming from restriction with the projectionHoφZ→ Z is a model for EFIN (H×φZ) and • is a model for EVCYI (H×φZ), it remains to show that for i ≤ m the assembly map

HHoφZi (EZ; KR)→ H

HoφZi (•; KR) = Ki(R[H oφ Z])

is bijective for i ≤ m. This will be achieved by proving inductively forn = 0, 1, 2, . . . that this map is bijective for m − n ≤ i ≤ m provided that

HHoφZm (EZ; KR[Zn])→ H

HoφZm (•; KR[Zn]) is bijective.

The induction beginning n = 0 is trivial. The induction step from (n− 1)to n is done as follows. The Bass-Heller-Swan decomposition for the ringR[Zn−1] can be implemented on the spectrum level (see for instance [558,Theorem 4.2]) and yields because of the identity (R[Zn−1])[Z] = R[Zn] forevery H oφ Z-CW -complex X and i ∈ Z an isomorphism, natural in X,

HHoφZm (X; KR[Zn−1])⊕H

HoφZm−1 (X; KR[Zn−1])⊕H

HoφZm (X; NKR[Zn−1])

⊕HHoφZm (X; NKR[Zn−1])

∼=−→ HHoφZm (X; KR[Zn]).

Since a direct sum of an isomorphism is again an isomorphism and we canapply the latter isomorphism to X = EZ and X = •, the map

HHoφZk (EZ; KR[Zn−1])→ H

HoφZk (E•; KR[Zn−1])

is bijective for k = m− 1,m. Now the induction shows that

HHoφZi (EZ; KR)→ H

HoφZi (•; KR)

is bijective for m− n ≤ i ≤ m. This finishes the proof of Lemma 3.13. ut


Consider a ring R together with a ring automorphism Ψ : R∼=−→ R. Let

K(Rψ|L [L]) be the non-connective algebraic K-theory spectrum of the Ψ |L-twisted group ring of Z with coefficient in R for the ring automorphismΨ |L := Ψ [Z:L] of R if L 6= 0, and the non-connective algebraic K-theoryspectrum of R, if L = 0. We obtain a covariant Or(Z)-spectrum KR,Ψ bysending Z/L to K(RΨ |L [L]). Notice that for two subgroups L,L′ ⊆ Z theset morOr(Z)(Z/L,Z/L′) is empty if L 6⊆ L′, and consists of precisely oneelement, the canonical projection Z/L→ Z/L′, if L ⊆ L′. In the case L ⊆ L′the functor KR,Ψ sends this morphism to the map of spectra induced by theinclusion of rings RΨ |L [L]→ RΨ |L′ [L

′].

Lemma 11.42. Let R be a regular and Ψ : R → R be a ring automorphism.Then the map

HZn(EZ; KR,Ψ )→ HZ

n(•; KR,Ψ ) = Kn(RΨ [Z])

is an isomorphisms for all n ∈ Z.

Proof. There is a twisted Bass-Heller-Swan decomposition for non-negativeK-theory (see [560, Theorem 0.1]) which reduces to the desired isomorphismif the twisted Nil terms NKn(R,Ψ) vanish. By inspecting the definition ofthe non-connective K-theory spectrum of [558] one sees that it suffices toshow the bijectivity

HZn(EZ; KR[Zm],Ψ [Zm])→ HZ

n(•; KR[Zm],Ψ [Zn]) = Kn((R[Zn])Ψ [Zn][Z])

for all n ≥ 1 and m ≥ 0. Since R is regular, Theorem 2.77 (ii) shows thatR[Zn] is regular for every n ≥ 0. Hence it suffices to prove Lemma 11.42only for n ≥ 1. This has already be done by Waldhausen [804, Theorem 4 onpage 138 and the Remark on page 216]. One may also refer to [349, Remarkon page 362]. Comment 39: Add reference to Bartels-Lueck(2018HeckeXI)when it has been uploaded ut

Consider a group H together with an automorphism φ : H → H. Letp : H oZ Z → Z be the projection. The we get from the adjunction betweenp∗ and p∗ (see [222, Lemma 1.9]) for any Z-CW -complex an isomorphism,natural in X

(11.43) HHoφZ0 (p∗X; KR[Zn])

∼=−→ HZ0 (X; p∗KR[Zn])

and from the definition we get p∗KR[Zn](Z/L) = KR[Zn](H oφ Z/p−1(L))for any object Z/L in Or(Z). Let Φ[Zn] : RH[Zn] → RH[Zn] be the ringautomorphism induced by φ : H → H. We have defined a covariant Or(Z)-spectrum KRH[Zn],φ[Zn] before Lemma 11.42, just take Ψ = Φ[Zn]. There

is an obvious weak equivalence of covariant Or(Z)-spectra KRH[Zn],Φ[Zn]

∼=−→p∗KR[Zn]. This implies using [222, Theorem 3.11]


Lemma 11.44. We get for every Z-CW -complex X and n ∈ Z an isomor-phism, natural in X

HZn(X; KRH[Zn],Φ[Zn])

∼=−→ HZn(X; p∗KR[Zn]).

Lemma 11.45. Let H be a finite group and let φ : H∼=−→ H be an automor-

phism. Let R be a Artinian ring. Then the map

HHoφZ0 (EZ; KR[Zn])→ H

HoφZ0 (•; KR[Zn]) = K0((R[Zn])[H oφ Z])

is an isomorphisms for all integers n ≥ 0.

Proof. We conclude from Lemma 11.44 that it remains to show that the map

HZ0 (EZ; KRH[Zn],Φ[Zn])→ HZ

0 (•; KRH[Zn],Φ[Zn]) = K0(RH[Zn]Φ[Zn][Z])

is bijective for all n ≥ 1.Denote by J ⊆ RH the Jacobson radical of RH. Since RH is Artinian,

there exists a natural number m with JJm = Jm. By Nakayama’s Lemma(see [755, Proposition 8 in Chapter 2 on page 20]) Jm is 0, in other words,J is nilpotent. The ring RH/J is a semisimple Artinian ring (see [496, Defi-nition 20.3 on page 311 and (20.3) on page 312]) and in particular regular.

The ring automorphism Φ : RH → RH induced by φ obviously satisfiesΦ(J) = J and hence induces a ring automorphism Φ : RH/J → RH/J . Hencewe get a commutative diagram induced by the projection RH → RH/J .

(11.46) HZ0 (EZ; KRH[Zn],Φ[Zn]) //

K0(RH[Zn] oΦ[Zn] [Z])

HZ

0 (EZ; K(RH/J)[Zn],Φ[Zn])// K0((RH/J)[Zn] oΦ[Zn] [Z]).

We have the short exact sequence of abelian groups 0 → J → RH →RH/J → 0. It induces a short exact sequence of abelian groups

0→ J [Zn] oΦ[Zn]|J[Zn][Z]→ RH[Zn] oΦ[Zn] [Z]

→ (RH/J)[Zn] oΦ[Zn] [Z]→ 0.

Hence we can identify the ring (RH/J)[Zn]oΦ[Zn] [Z] with the quotient of the

ring RH[Zn]oΦ[Zn] [Z] by the ideal J [Zn]oΦ[Zn]|J [Z]. Recall that an ideal I ina ring is nilpotent if and only if there is a natural number l such that for anycollection of l elements i1, i2, . . . il in I the product i1i2 · · · il vanishes. Since Jis nilpotent, we conclude that the ideal J [Zn]oΦ[Zn]|J [Z] is nilpotent. Hencethe right vertical arrow in the diagram (11.46) is bijective by Lemma 1.108.

Next we show that the left vertical arrow in the diagram (11.46) is bijective.Since EZ is a free Z-CW -complex, we conclude from the equivariant Atiyah-


Hirzebruch spectral sequence described in Theorem 10.44 that it suffices toshow for every i that the map Ki(RH[Zn]) → Ki((RH/J)[Zn]) is bijectivefor all i ≤ 0.

Since J is a nilpotent two-sided ideal of RH, J [Zn] is a nilpotent two-sidedideal ofRH[Zn]. Obviously (RH/J)[Zn] can be identified with (RH[Zn])/(J [Zn]).Hence K0(RH[Zn]) → K0((RH/J)[Zn]) is bijective by Lemma 1.108. Weconclude Ki(RH[Zn]) = 0 for i ≤ −1 from Theorem 3.15 (ii). Since RH/Jis regular and hence RH/J [Zn] is regular by Theorem 2.77 (ii), we concludefrom Theorem 3.6 that Ki((RH/J)[Zn]) = 0 for i ≤ −1. Hence the leftvertical arrow in the diagram (11.46) is bijective. The lower vertical arrowin the diagram (11.46) is bijective because of Lemma 11.42 applied to theautomorphism Φ[Zn]. We conclude that the upper vertical arrow in the dia-gram (11.46) is bijective. This finishes the proof of Lemma 11.45 ut

11.9.2 Reducing the Family of Subgroups for the Farrell-JonesConjecture for L-Theory

Theorem 11.47 (Passage from FIN to VCYI for L-theory). Let G bea group and let A be an additive G-category with involution. Let n be anyinteger. Then

HGn

(ιFIN⊆VCYI ; L

〈−∞〉A

): HG

n

(EFIN (G); L

〈−∞〉A

)→ HG

n

(EVCYI (G); L

〈−∞〉A

)is bijective.

Proof. The argument given in [531, Lemma 4.2] goes through since it is basedon the Wang sequence for a semi-direct product G o Z which can be gener-alized for additive G-categories with involutions as coefficients. ut

The last result is very useful when G does not contain virtually cyclicsubgroups of type II since then one can replace in Conjectures 11.4, 11.6and 11.14 the family VCY by the family FIN . (This is not true for Con-jecture 11.19, since G o F for a finite group F may contain a virtually cyclicsubgroup of type II even in the case that G does not contain a virtually cyclicsubgroups of type II.)

Exercise 11.48. Consider the group extension 1 → F → Gf−→ Zd → 1

for a finite group F . Show that there exists a spectral sequence converging

to L〈−∞〉p+q (ZG) whose E2-term is given by Hp(C∗(EZd) ⊗Z[Zd] L

〈−∞〉q (ZF )),

where the Zd-action on L〈−∞〉q (ZF ) is induced by the conjugation action of

G on F .

If we invert 2, we can refine ourselves in L-theory to the family FIN .The next result is taken from [546, Proposition 74 on page 747] and [70,


Theorem 0.3]. It is conceivable that it holds for additive G-categories withinvolution. Recall that after inverting 2 the decoration does not anymore playa role.

Let p be a prime. A finite group G is called p-elementary if it is isomorphicto C × P for a cyclic group C and a p-group P such that the order |C| isprime to p. Let Ep be the class of of finite subgroups which are p-elementary.

Theorem 11.49 (Bijectivity of the L-theoretic assembly map fromFIN to VCY after inverting 2). Let G be a group and let R be a ringwith involution.

(i) The relative assembly map

HGn

(ιFIN⊆VCY ; L

〈−∞〉R

): HG

n

(EFIN (G); L

〈−∞〉R

)→ HG

n

(EVCY(G); L

〈−∞〉R

)is bijective for all n ∈ Z after applying Z[1/2]⊗Z −;

(ii) Put

F =⋃

p prime,p6=2

Ep


HGn

(ιF⊆VCY ; L

〈−∞〉R

): HG

n

(EF (G); L

〈−∞〉R

) ∼=−→ Hn

(EVCY(G); L

〈−∞〉R

)is bijective for all n ∈ Z after applying Z[1/2]⊗Z −.

11.10 The Farrell-Jones Conjecture Implies All ItsVariants

In this section we give the proofs that the K-theoretic Farrell-Jones Conjec-ture 11.2 with coefficients in rings and the L-theoretic Farrell-Jones Conjec-ture 11.6 with coefficients in rings imply all the variants we have stated beforeat various places. Recall that the Full Farrell-Jones Conjecture 11.20 impliesConjectures 11.2 and 11.6, see Remarks 11.10 and 11.15. For the reader’sconvenience we recall all these variants below before we give their proofs.

11.10.1 List of Variants of the Farrell-Jones Conjecture

Conjecture 1.50 (Farrell-Jones Conjecture for K0(R) for torsionfreeG and regular R). Let G be a torsionfree group and let R be a regular ring.Then the map induced by the inclusion of the trivial group into G

K0(R)∼=−→ K0(RG)

11.10 The Farrell-Jones Conjecture Implies All Its Variants 355

is bijective.In particular we get for any principal ideal domain R and torsionfree G

K0(RG) = 0.

Conjecture 1.57 (Farrell-Jones Conjecture for K0(RG) for regularR with Q ⊆ R). Let R be a regular ring and G be a group such that forevery finite subgroup H ⊆ G the element |H| · 1R of R is invertible in R.

Then the homomorphism

IFIN (G,F ) : colimH∈SubFIN (G)K0(RH)→ K0(RG).

coming from the various inclusions of finite subgroups of G into G is a bijec-tion.

Conjecture 1.61 (Farrell-Jones Conjecture for K0(RG) for an Ar-tinian ring R) Let G be a group and R be an Artinian ring. Then thecanonical map

I(G,R) : colimH∈SubFIN (G)K0(RH)→ K0(RG)

is an isomorphism.

Conjecture 2.106 (Farrell-Jones Conjecture for K0(RG) and K1(RG)for regular R and torsionfree G). Let G be a torsionfree group and let Rbe a regular ring. Then the maps defined in (2.25) and (2.26)

A0 : K0(R)∼=−→ K0(RG);

A1 : G/[G,G]⊗Z K0(R)⊕K1(R)∼=−→ K1(RG),

are both isomorphisms. In particular the groups WhR0 (G) and WhR1 (G) (seeDefinition 2.27) vanish.

Conjecture 2.107 (Farrell-Jones Conjecture for K0(ZG) and Wh(G)

for torsionfree G). Let G be a torsionfree group. Then K0(ZG) and Wh(G)vanish.

Conjecture 3.17 (The Farrell-Jones Conjecture for negative K-theory and regular coefficient rings). Let R be a regular ring and Gbe a group such that for every finite subgroup H ⊆ G the element |H| · 1Rof R is invertible in R. Then

Kn(RG) = 0 for n ≤ −1.

Conjecture 3.19 (The Farrell-Jones Conjecture for negative K-theory of the ring of integers in an algebraic number field). Let


R be of integers in an algebraic number field. Then, for every group G, wehave

K−n(ZG) = 0 for n ≥ 2,

and the map

colimH∈SubFIN (G)K−1(ZH)∼= // K−1(ZG)

is an isomorphism.

Conjecture 3.20 (The Farrell-Jones Conjecture for negative K-theory and Artinian rings as coefficient rings) Let G be a group andlet R be a Artinian ring. Then

Kn(RG) = 0 for n ≤ −1.

Conjecture 4.20 (Farrell-Jones Conjecture for Wh2(G) for torsion-free G). Let G be a torsionfree group. Then Wh2(G) vanishes.

Conjecture 5.41 (Farrell-Jones Conjecture for torsionfree groupsand regular rings for K-theory). Let G be a torsionfree group. Let R bea regular ring. Then the assembly map



Conjecture 5.47 (Nil-groups for regular rings and torsionfree groups).Let G be a torsionfree group and let R be a regular ring. Then

NKn(RG) = 0 for all n ∈ Z.

Conjecture 5.62 (Farrell-Jones Conjecture for torsionfree groupsfor homotopy K-theory). Let G be a torsionfree group. Then the assemblymap

Hn(BG; KH(R))→ KHn(RG)

is an isomorphism for every n ∈ Z and every ring R.

Conjecture 5.64 (Comparison of algebraic K-theory and homotopyK-theory for torsionfree groups). Let R be a regular ring and let G bea torsionfree group. Then the canonical map

Kn(RG)→ KHn(RG)



Conjecture 7.109 (Farrell-Jones Conjecture for torsionfree groupsfor L-theory) Let G be a torsionfree group. Let R be any ring with involu-tion.

Then the assembly map

Hn(BG; L〈−∞〉(R))→ L〈−∞〉n (RG)


Finally we mention the following Novikov type conjectures

Conjecture 11.50 (K-theoretic Novikov Conjecture). A group G sat-isfies the K-theoretic Novikov Conjecture if the assembly map

Hn(BG; K(Z)) = HGn (EG; KZ))→ HG

n (G/G; K(Z)) = Kn(ZG)

is rationally injective for all n ∈ Z.

Conjecture 11.51 (L-theoretic Novikov Conjecture). A group G sat-isfies the L-theoretic Novikov Conjecture if the assembly map

Hn(BG; L〈−∞〉(Z)) = HGn (EG; L〈−∞〉(Z))

→ HGn (G/G; L〈−∞〉(Z)) = L〈−∞〉n (ZG)


11.10.2 Proof of the Variants of the Farrell-Jones Conjecture

Theorem 11.52 (The Full Farrell-Jones Conjecture implies all othervariants).

(i) The K-theoretic Farrell-Jones Conjecture 11.2 with coefficients in ringsimplies Conjecture 5.41 and the L-theoretic Farrell-Jones Conjecture 11.6with coefficients in rings implies Conjecture 7.109;

(ii) Conjecture 5.41 implies Conjectures 1.50, 2.106, and 2.107;(iii) The K-theoretic Farrell-Jones Conjecture 11.2 with coefficients in rings

implies Conjectures 1.57 and 3.17;(iv) The K-theoretic Farrell-Jones Conjecture 11.2 with coefficients in rings

implies Conjectures 1.61 and 3.20.(v) The K-theoretic Farrell-Jones Conjecture 11.1 implies Conjecture 3.19

(vi) Conjecture 5.41 implies Conjecture 4.20;(vii) Conjecture 5.41 implies Conjecture 5.47;

(viii) The K-theoretic Farrell-Jones Conjecture 11.2 with coefficients in ringsimplies Conjectures 5.62 and 5.64;

(ix) The K-theoretic Farrell-Jones Conjecture 11.1 with coefficients in the ringZ implies the K-theoretic Novikov Conjecture 11.50;


(x) The L-theoretic Farrell-Jones Conjecture 11.4 with coefficients in thering Z implies the L-theoretic Novikov Conjecture 11.51. The L-theoreticNovikov Conjecture 11.51 implies the Novikov Conjecture 7.131;

(xi) The Full Farrell-Jones Conjecture 11.20 implies all other variants of theFarrell-Jones Conjecture;

Proof. (i) Every torsionfree virtually cyclic group is isomorphic to Z byLemma 11.30 By the Transitivity Principle 13.13 applied to T R ⊆ VCYit suffices to show that the assembly maps

HZn(EZ; KR)→ Kn(RZ);

HZn(EZ; L

〈−∞〉R )→ L〈−∞〉n (RZ),

are bijective for n ∈ Z. This follows for K-theory from the Bass-Heller-Swandecomposition, see Theorem 5.16, and for L-theory from the Shaneson split-ting, see (7.105).

(ii) Since R is regular, the negative K-groups of R vanish by Theorem 3.6.Hence the Atiyah-Hirzebruch spectral sequence, which has E2-term E2

p,q =Hp(BG;Kq(R)) and converges to Hp+q(BG; K(R)), is a first quadrant spec-

tral sequence. The edge homomorphism H0(BG;K0(R))∼=−→ H0(BG; K(R))

at (0, 0) is bijective. There is an obvious identification H0(BG;K0(R)) ∼=K0(R). Under this identification the edge homomorphism composed with theassembly map appearing in Conjecture 5.41 turns out to be the change ofrings map K0(R)→ K0(RG). Hence we conclude from Conjecture 5.41 thatK0(R)→ K0(RG) is bijective as predicted by Conjecture 1.50. Inspecting theAtiyah-Hirzebruch spectral yields an exact sequence 0→ H0(BG;K1(R))→H1(BG; K(R)) → H1(BG;K0(R)) → 0. Under the obvious identificationH0(BG;K1(R)) = K1(R) the composite ofH0(BG;K1(R))→ H1(BG; K(R))with the assembly map appearing in Conjecture 5.41 turns out to be thechange of rings map K1(R)→ K1(RG). Since H1(BG;K0(R)) = G/[G,G]⊗K0(R), we obtain an exact sequence

0→ K1(R)→ K1(RG)→ G/[G,G]⊗K0(R)→ 0.

Next one checks that the composite of the map K1(RG)→ G/[G,G]⊗K0(R)appearing in the sequence above with the map A1 appearing in Conjec-ture 2.106 is the obvious projection. This implies Conjecture 2.106, and hencealso Conjecture 2.107.

(iii) See [74, Theorem 1.5] and [546, paragraph before Conjecture 79 on page750].

(iv) We conclude from Lemma 11.41 and Lemma 11.45 that the assemblymap HG

n (EFIN (G),KR) → Kn(RG) is an isomorphism for n ≥ 0. Wehave Ki(RH) = 0 for every finite group Hand every i ≤ −1 by Theo-rem 3.15 (ii). We conclude from the equivariant Atiyah-Hirzebruch spec-tral sequence described in Theorem 10.44 that HG

n (EFIN (G),KR) = 0


holds for n ≤ −1, and that HHoφZ0 (EFIN (H ×φ Z),KR) is the 0-th Bre-

don homology of EFIN (H ×φ Z) with coefficients in the covariant functorOr(G)→ Z-MODULES sending G/K to Kn(RK). This 0-th Bredon homol-ogy can be identified with colimG/H∈OrFIN (G)K0(RH). Under this identifi-cation the bijective assembly map HG

n (EFIN (G),KR) → Kn(RG) becomesthe canonical map colimG/H∈OrFIN (G)K0(RH)→ K0(RG).

(v) See [546, page 749]. The proof goes through if we replace Z by the ring Rof integers in an algebraic number field, since the results appearing in [297]for Z have been extended to R by Juan-Pineda [426].

(vi) We conclude from [515] that the second Whitehead group can be identi-fied with the cokernel of the assembly map

H2(pr; KR) : HG2 (EG; KR) = H2(BG; K(Z))→ HG

2 (EG; KR) = K2(ZG).

(vii) This follows from the obvious commutative diagram

Hn(BG; K(R[t]))∼= //

∼=

Kn(R[t]G)

Hn(BG; K(R)) ∼=

// Kn(RG)

whose horizontal arrows are bijective by the assumption that Conjecture 11.2holds and whose left vertical arrow is bijective since Kn(R[t]) → Kn(R) isbijective for all n ∈ Z by Theorem 5.16 (ii).

(viii) This follows from [69, Theorem 8.4 and Remark 8.6].

(ix) This follows from Theorem 11.26 and Lemma 11.28.

(x) The L-theoretic Novikov Conjecture 11.51 follows from the L-theoreticFarrell-Jones Conjecture 11.4 because of Theorem 11.26 and Lemma 11.28.

For the proof that the L-theoretic Novikov Conjecture 11.51 for G impliesthe Novikov Conjecture 7.131 forG. we refer to [477, Lemma 23.2 on page 192]and [687, Proposition 6 on page 300]. Or just take a look at Remark 7.137and use the fact that under the Chern character the assembly map

asmbGn :⊕k∈Z

Hn+4k(BG;Q)→ Lhn(ZG)⊗Z Q,

appearing in Remark 7.137 can be identified with the assembly map appear-ing in L-theoretic Novikov Conjecture 11.51 for G.

(xi) The Full Farrell-Jones Conjecture 11.20 implies Conjectures 11.2 and 11.6,see Remarks 11.10 and 11.15. Now the claim follows from all the other asser-tions which we have already proved. ut


11.11 Summary of the Applications of the Farrell-JonesConjecture

We have discussed at various places applications and consequences of thevarious versions of the Farrell-Jones Conjecture. In Theorem 11.52 we haveexplained that the Full Farrell-Jones Conjecture 11.20 implies all of thesevariants of the Farrell-Jones Conjecture and hence all these applications andconsequences. For the reader’s convenience we list now all these applicationsand where they are treated in this book or in the literature.

• Wall’s Finiteness ObstructionWall’s finiteness obstruction of a connected finitely dominated CW -complexX takes values in K0(Z[π1(X)]) and vanishes if and only if X is homo-topy equivalent to a finite CW -complex, see Section 1.5. For torsionfreeπ1(X) Conjecture 1.50 predicts that K0(Z[π1(X)]) vanishes and hence Xis always homotopy equivalent to a finite CW -complex, see Remark 1.51.

• The Kaplansky ConjectureThe Kaplansky Conjecture 1.62 predicts for an integral domain R and atorsionfree group G that all idempotents of RG are trivial. See Section 1.9.

• The Bass ConjecturesThe Bass Conjecture for fields of characteristic zero as coefficients 1.78 saysfor a field F of characteristic zero and a group G that the Hattori-Stallingshomomorphism of (1.74) induces an isomorphism

HSFG : K0(FG)⊗Z F → classF (G)f .

This essentially generalizes character theory for finite representations forfinite groups to infinite groups.The Bass Conjecture for integral domains as coefficients 1.85 predicts fora commutative integral domain R, a group G and g ∈ G such that eitherthe order |g| is infinite or that the order |g| is finite and not invertible inR that for every finitely generated projective RG-module the value of itsHattori-Stallings rank HSRG(P ) at (g) is trivial.For more information about the Bass Conjectures we refer to Section 1.10.

• Whitehead torsionOne can assign to a homotopy equivalence f : X → Y its Whitehead tor-sion τ(f) which takes values in the Whitehead group Wh(π1(Y )), see Sec-tions 2.3. It vanishes if and only f is a simple homotopy equivalence, seeSection 2.4An h-cobordism of dimension ≥ 5 is trivial if and only if its Whiteheadtorsion vanishes, see Theorem 2.5.If π1(Y ) is torsionfree, then Conjecture 2.107 predicts that Wh(π1(Y ))vanishes and hence f is a simple homotopy equivalence, see Remark 2.109.

• Bounded h-cobordismsThere are so called bounded h-cobordisms, controlled over Rk, for k ≥

11.11 Summary of the Applications of the Farrell-Jones Conjecture 361

1. They are trivial (for dimension ≥ 5) if and only if certain elements

in negative K-groups K1−k(ZG) vanish, see Section 3.3. Conjecture 3.17

predicts for a torsionfree group G the vanishing of Kn(ZG) for n ≤ 0.

• Pseudo-isotopy and the second Whitehead groupThere are certain obstruction for pseudoisotopies to be trivial which takevalues in Wh2(G), see Section 4.6. Conjecture 4.20 predicts for a torsionfreegroup G the vanishing of Wh2(G).

• Whitehead spaces and pseudoisotopy spacesOne can assign to a compact manifold M its pseudo-isotopy spaces P(M)and PDiff(M), Whitehead spaces WhPL(X) and WhDiff(X), and its A-theory A(X) in the sense of Waldhausen, see Section 6.2 and 6.3. Therealso exist non-connective versions. There are various relations betweenthese spaces. The homotopy groups of A(M) are related to the K-groupsKn(Z[π1(M)]).Conjecture 5.41 predicts for a torsionfree group G and a regular ring Rthat the assembly map


is an isomorphism for n ∈ Z. It implies for a closed aspherical manifoldM , see Lemma 6.21 and Lemma 6.25 for all n ≥ 0

πn(WhPL(M))⊗Z Q ∼= 0;

πn(P(M))⊗Z Q ∼= 0;

πn(WhDiff(M))⊗Z Q ∼=∞⊕k=1

Hn−4k−1(M ;Q);

πn(PDiff(M))⊗Z Q ∼=∞⊕k=1

Hn−4k+1(M ;Q).

• Automorphisms of manifoldsIf K-theoretic Farrell-Jones Conjecture 5.41 and the L-theoretic Farrell-Jones Conjecture 7.109 hold for the torsionfree group G and the ringR = Z, then some rational computations of the homotopy groups of theautomorphism group of an aspherical closed manifold M with G = π1(M)can be found in Theorems 7.177 and 7.178.

• Novikov ConjectureThe Novikov Conjecture 7.131 for a group G predicts the homotopy in-variants of the higher signatures

signx(M,u) := 〈L(M) ∪ u∗x, [M ]〉 ∈ Q(11.53)

of a closed oriented manifold M coming with a reference map f : M → BGfor an element x ∈

∏k≥0H

k(BG;Q), see Subsection 7.14.1.


We conclude from Theorem 11.52 (x) that the L-theoretic Farrell JonesConjecture 11.4 for the group G and the ring Z implies the Novikov Con-jecture 7.131 for G.

• Borel ConjectureThe Borel Conjecture 7.152 predicts that any closed aspherical manifold Mis topologically rigid, i.e, if N is another closed aspherical manifold withπ1(M) ∼= π1(N), then M and N are homeomorphic and any homotopyequivalence M → N is homotopic to a homeomorphism.Let G be a finitely presented group. Suppose that it satisfies the versionsof the K-theoretic Farrell-Jones Conjecture stated in 2.107 and 3.19 andthe version of the L-theoretic Farrell-Jones Conjecture stated in 7.109 forthe ring R = Z.Then Theorem 7.160 shows that every closed aspherical manifold of di-mension ≥ 5 with G as fundamental group is topologically rigid.

• Poincare duality groupsConjecture 7.170 predicts that a finitely presented group is an n-dimensionalPoincare duality group if and only if it is the fundamental group of a closedaspherical n-dimensional topological manifold.Suppose that the torsionfree group G is a finitely presented Poincardualitygroup of dimension n ≥ 6 and satisfies the versions of the K-theoreticFarrell-Jones Conjecture stated in 2.107 and 3.19 and the version of theL-theoretic Farrell-Jones Conjecture stated in 7.109 for the ring R = Z.Let X be a Poincare complex of dimension ≥ 6 with π1(X) ∼= G.Then X is homotopy equivalent to a compact homology ANR-manifoldsatisfying the disjoint disk property, see Theorem 7.171.

• Boundaries of hyperbolic groupsAs a consequence of the Farrell-Jones Conjecture we get Theorem 7.174which says for a torsion-free hyperbolic group G and let n ≥ 6 that thefollowing statements are equivalent:

– The boundary ∂G is homeomorphic to Sn−1;– There is a closed aspherical topological manifold M such that G ∼=π1(M), its universal covering M is homeomorphic to Rn and the com-

pactification of M by ∂G is homeomorphic to Dn;

Moreover the aspherical manifold M appearing above is unique up tohomeomorphism.

• Stable Cannon ConjectureA stable version of the Cannon Conjecture is proved in [306].

• Product decompositions of closed aspherical manifoldsTheorem 7.176 deals with the question when for a closed aspherical mani-fold M a given algebraic decomposition π1(M) = G1×G2 comes from thetopological decomposition M = M1×M2. Theorem 7.176 is a consequenceof the K-theoretic Farrell-Jones Conjecture stated in 2.107 and 3.19 and

11.11 Summary of the Applications of the Farrell-Jones Conjecture 363

the version of the L-theoretic Farrell-Jones Conjecture stated in 7.109 forthe ring R = Z.

• Classification of manifolds homotopy equivalent to certain torus bundlesover lens spaces.The K-theoretic Farrell-Jones Conjecture 11.1 and the L-theoretic Farrell-Jones Conjecture 11.4 play a key role in the paper [225], where a classifi-cation of manifolds homotopy equivalent to certain torus bundles over lensspaces is presented.

• Fibering manifoldsThe K-theoretic Farrell-Jones Conjecture 11.1 and the L-theoretic Farrell-Jones Conjecture 11.4 play a key role in the paper [304], where the questionis treated when for a closed aspherical manifold B and a map p : M →B from some closed connected manifold M the map p is homotopic toManifold Approximate Fibration.

• The Atiyah ConjectureConjecture 1.57 is related to the Atiyah Conjecture which makes predic-tions about the possibly values of the L2-Betti numbers of coverings ofclosed Riemannian manifolds, see Remark 1.60.

• Homotopy invariance of τ (2)(M) and of the L2-Rho-invariant ρ(2)(M)Suppose that the L-theoretic Farrell-Jones Conjecture11.4 with coefficientsin the ring R with involution is rationally true for R = Z, i.e., the ratio-nalized assembly map

Hn(BG; L〈−∞〉(Z))⊗Z Q→ L〈−∞〉n (ZG)⊗Z Q

is an isomorphism for n ∈ Z.Then Theorem 12.56 (ii) says that the invariant τ (2)(M) defined in Sub-section 12.8.3 is a homotopy invariant, and, if furthermore G is residuallyfinite, then the L2-Rho-invariant ρ(2)(M) is a homotopy invariant.

• Homotopy invariance of the (twisted) L2-torsionThe K-theoretic Farrell-Jones Conjecture 11.1 with coefficients in the ringR = Z implies the homotopy invariance of L2-torsion and of the L2-torsionfunction, see [540, Theorem 7.5 (4)]. The twisted L2-torsion function isrelated to the Thurston norm for appropriate 3-manifolds in [331].

• Vanishing of κ-classes for aspherical closed manifoldsThe vanishing of κ-classes for aspherical closed manifolds is analyzedin [379].

Comment 40: Is the list complete?


11.12 G-Theory

Instead of considering finitely generated projective modules, one may applythe standard K-theory machinery to the category of finitely generated mod-ules. This leads to the definition of the groups Gn(R) for n ≥ 0. We havedescribed G0(R) and G1(R) already in Definitions 1.1 and 2.1. One mayask whether versions of the Farrell-Jones Conjectures for G-theory instead ofK-theory might be true. The answer is negative as the following discussionexplains.

For a finite group H the ring CH is semisimple. Hence any finitely gener-ated CH-module is automatically projective and K0(CH) = G0(CH). Recallthat a group G is called virtually poly-cyclic if there exists a subgroup offinite index H ⊆ G together with a filtration 1 = H0 ⊆ H1 ⊆ H2 ⊆ . . . ⊆Hr = H such that Hi−1 is normal in Hi and the quotient Hi/Hi−1 is cyclic.More generally for all n ∈ Z the forgetful map

f : Kn(CG)→ Gn(CG)

is an isomorphism if G is virtually poly-cyclic, since then CG is regular [725,Theorem 8.2.2 and Theorem 8.2.20] and the forgetful map f is an isomor-phism for regular rings, compare [704, Corollary 53.26 on page 293]. In par-ticular this applies to virtually cyclic groups and so the left hand side ofthe Farrell-Jones assembly map does not see the difference between K- andG-theory if we work with complex coefficients. We obtain a commutativediagram

colimH∈SubFIN (G)K0(CH)

∼=

// K0(CG)

f

colimH∈SubFIN (G)G0(CH) // G0(CG)

where, as indicated, the left hand vertical map is an isomorphism. Conjec-ture 1.57 which follows from the K-theoretic Farrell-Jones Conjecture 11.1with coefficients in the ring C predicts that the upper horizontal arrow is anisomorphism. A G-theoretic analogue of Conjecture 11.1 would say that thelower horizontal map is an isomorphism. There are however cases where theupper horizontal arrow is known to be an isomorphism, but the forgetful mapf on the right is not injective or not surjective:

If G contains a non-abelian free subgroup, then the class [CG] ∈ G0(CG)vanishes [527, Theorem 9.66 on page 364] and hence the map f : K0(CG)→G0(CG) has an infinite kernel since [CG] generates an infinite cyclic subgroupin K0(CG). Notice that Conjecture 11.1 is known for non-abelian free groups.

Conjecture 11.1 is also known for A =⊕

n∈Z Z/2 and hence K0(CA)is countable, whereas G0(CA) is not countable [527, Example 10.13 onpage 375]. Hence the map f cannot be surjective.


At the time of writing we do not know the answer to the following ques-tions:

Question 11.54. If G is an amenable group, for which there is an up-per bound on the orders of its finite subgroups, is then the forgetful mapf : K0(CG)→ G0(CG) an isomorphism?

Question 11.55. If the group G is not amenable, is then G0(CG) = 0?

To our knowledge the answer to Question 11.55 is not even known in thespecial case G = Z ∗ Z.

For more information about G0(CG) we refer for instance to [527, Subsec-tion 9.5.3].

Exercise 11.56. Let H ⊆ G be a subgroup of G possessing an epimorphismf : H → Z. Show that the class of C[G/H] in G0(CG) is trivial.

11.13 Miscellaneous

The original formulation of the Farrell-Jones Conjecture appears in[295, 1.6on page 257]. Our formulations differ from the original one, but are equivalent.

Proofs of some of the inheritance properties above are also given in [370,722].

The inheritance properties of the Farrell-Jones Conjecture under actionsof trees is discussed in [69], see also Section 5.9 and Section 13.7. The situ-ation is much more complicated than for the Baum-Connes Conjecture withcoefficients 12.11, where the optimal result holds, see Theorem 12.30 (v) andRemark 12.34.

Comment 41: Mention the version for Hecke algebras when the manuscriptis written.

In the sequel we consider classes C of groups which are closed under tak-ing subgroups and passing to isomorphic groups. Examples are the classes ofvirtually cyclic or of finite groups. Given a group G, let C(G) be the family ofsubgroups of G which belong to C. The relevant family of subgroups appear-ing in Conjectures 11.1, 11.2, 11.4, 11.6, 11.9, 11.14, 11.18, 11.19, and 11.20is always given by C(G), where C is the class of virtually cyclic subgroups.We have proved various theorems, where C could be chosen to be smaller,for instance to be the class of virtually cyclic groups of type I or of hyperele-mentary groups, see Theorems 11.34, 11.35, and 11.36. One may ask whetherthere is always a class Cmin for which such a conjecture holds for all groups Gand which is minimal. Of course for the class of all groups such a conjecturewill hold for trivial reasons. In the worst case Cmin may be just the class ofall groups. A candidate for Cmin may be the intersection of all the classes


C of groups for which the conjecture is true for all groups, but we do notknow whether this intersection does satisfies the conjecture for all groups. Atleast we know that the intersection of two classes of groups C0 and C1 forwhich one of the Conjectures 11.9, 11.14, 11.18, 11.19 and 11.20 holds forall groups, satisfies this conjecture for all groups as well. We also know fortwo classes of subgroups C ⊆ D such that D satisfies one one of the Conjec-tures 11.9, 11.14, 11.18, 11.19 and 11.20 for all groups, if C does. These claimsfollow from Theorem 13.9 (ii) and (iv), Theorem 13.13 (ii) and Lemma 13.14.

Further application of the Farrell-Jones Conjecture to manifolds withgroup actions are given for instance in [193, 194, 199]. Comment 42: Isthe list complete?

Comment 43: Shall we add [61, Remark 1.13]? Shall we add an exercisemodelled upon [61, Remark 1.8]?


name of texfile: ic

Chapter 12

The Baum-Connes Conjecture

12.1 Introduction

In this chapter we discuss the Baum-Connes Conjecture 12.9 for the topolo-gical K-theory of the reduced group C∗-algebra C∗r (G;F ) for F = R,C. Itpredicts that certain assembly maps

KGn (EFIN (G)) → Kn(C∗r (G;C));

KOGn (EFIN (G)) → KOn(C∗r (G;R)),

are bijective for all n ∈ Z. The target is the topological K-theory of C∗r (G;F )which one wants to understand. The source is an expression which dependsonly on the values of these topological K-groups on finite subgroups of G andis therefore much more accessible. The version above is often the one whichis relevant in concrete applications, but there is also a more general version,the Baum-Connes Conjecture with coefficients 12.11, where one allows coef-ficients in a G-C∗-algebra. Notice that in contrast to the Full Farrell-JonesConjecture 11.20 it suffices to consider finite subgroups instead of virtuallycyclic subgroups.

A status report of the Baum-Connes Conjecture 12.9 and its version withcoefficients 12.11 will be given in Section 14.3.

The main point about the Baum-Connes Conjecture 12.9 is that it impliesa great variety of other prominent conjectures such as the ones due to Kadisonand Novikov, and leads to very deep and interesting results about manifoldsand C∗-algebras, as we will record and explain in Section 12.8.

Variants of the Baum-Connes Conjecture 12.9 and its version with coeffi-cients 12.11 are presented in Section 12.5.

We will discuss the inheritance properties of the Baum-Connes Conjecturewith coefficients 12.11 in Section 12.6.

We have tried to keep this chapter as much as possible independent of theother chapters, so that one may start reading here directly.

367

368 12 The Baum-Connes Conjecture

12.2 The Analytic Version of the Baum-ConnesAssembly Map

Let A be a G-C∗-algebra over F = R,C. Denote by A or G the C∗-algebraover F given by the reduced crossed product, see [654, 7.7.4 on page 262].If A is R or C with the trivial G-action, this is the reduced real or complexreduced group C∗-algebra C∗r (G;R) or C∗r (G;C), see Subsection 8.3.1. Denoteby Kn(A or G) and KO(A or G) their topological K-theory, as introducedin Subsection 8.3.2.

Let X be a proper G-CW -complex. Denote by KG∗ (X;A) and KOG∗ (X;A)

the complex and real topological K-theory of X with coefficients in A seeSection 8.6. Notice that KG

∗ (−;A) and KOG∗ (−;A) are G-homology theoriesin the sense of Definition 10.1 such that KG

n (G/H;A) = Kn(A o H) andKOGn (G/H;A) = KOn(A or H) hold for any finite subgroup H ⊆ G andn ∈ Z, provided that we consider proper G-CW -complexes only.

We want to explain the analytic Baum-Connes assembly map

asmbG,CA (X)n : KGn (X;A)→ Kn(Aor G);(12.1)

asmbG,RA (X)n : KOGn (X;A)→ KOn(Aor G).(12.2)

We will only treat the case F = C, the case F = R is analogous.We first consider the special case, where X is proper and cocompact and

then explain how the map extends by a colimit argument to arbitrary properG-CW -complexes. Notice that for a proper and cocompactG-CW -complexXwe can identify KG

n (X;A) with the equivariant KK-groups KKGn (C0(X), A),

see Section 8.6.One description is in terms of indices with values in C∗-algebras. Namely,

one assigns to a Kasparov cycle representing an element in KKGn (C0(X), A)

its C∗-valued index in Kn(AoG) in the sense of Mishchenko-Fomenko [602],thus defining a map KKG

n (C0(X), A) → Kn(A o G), provided that X isproper and cocompact. This is the approach appearing in [91].

The other equivalent approach is based on the Kasparov product. Given aproper cocompact G-CW -complex X, one can assign to it an element [pX ] ∈KKG

0 (C, C0(X) or G). Now define the map (12.1) by the composition of adescent map and a map coming from the Kasparov product

KKGn (C0(X), A)

jGr−−→ KKn(C0(X) or G,Aor G)

[pX ]⊗C0(X)orG−−−−−−−−−−−−→ KKn(C, Aor G) = Kn(AoG).

For some information about these two approaches and their identification, atleast for torsionfree G, we refer to [498].

This extends to arbitrary proper G-CW -complexes X by the following ar-gument. If f : X → Y is a G-map of proper cocompact G-CW -complexes,

12.3 The Version of the Baum-Connes Assembly Map in Terms of Spectra 369

then f is a proper map (after forgetting the group action). Hence compositionwith f defines a homomorphism of G-C∗-algebras C0(f) : C0(Y ) → C0(X).Denote by KKG

n (C0(f), idA) : KKGn (C0(X), A) → KKG

n (C0(Y ), A) the in-duced map on equivariant KK-groups. One easily checks asmbG,C(Y )n KKG

n (C0(f), idA) = asmbG,C(X)n. Recall from the definition (8.67) that forany proper G-CW -complex Y the canonical map

colimC⊆X KGn (C)

∼=−→ KGn (X)

is an isomorphism, where C runs through the finite G-CW -subcomplexes of Ydirected by inclusion. Hence by a colimit argument over the directed systemsof proper cocompact G-CW -subcomplexes the definition above for propercompact G-CW -complexes extends to the desired assembly maps (12.1) forany proper G-CW -complex X. Moreover, for any G-map of proper G-CW -complexes f : X → Y we obtain again by passing to the colimit a homomor-phism KG

n (f) : KGn (X;A)→ KG

n (Y ;A) satisfying

asmbG,C(Y )n KGn (f ;A) = asmbG,C(X)n;(12.3)

asmbG,R(Y )n KOGn (f ;A) = asmbG,R(X)n.(12.4)

12.3 The Version of the Baum-Connes Assembly Map inTerms of Spectra

There is also a version of the Baum-Connes assembly map which is very closeto the construction of the one for the Farrell-Jones Conjecture. Namely, if weapply Theorem 10.24 to the functor

KtopF : GROUPOIDSinj → SPECTRA,

of (10.42) for F = R,C, then we obtain an equivariant homology theoryH?∗(−; Ktop

F ) in the sense of Definition 10.9 such that we get for every inclusionH ⊆ G of groups natural identifications

HGn (G/H; Ktop

F ) ∼= HHn (H/H; Ktop

F ) ∼= πn(KtopF I(H)) = Kn(C∗r (H;F )).

Notice that H?n(X; Ktop

F ) is defined for any G-CW -complex X, whereas thedefinition of KG

n (X) and KOn(X) in terms of KK-theory only makes sensefor proper G-CW -complexes. For a cocompact proper G-CW complex onecan identify

HGn (X; Ktop

F ) =

KGn (X) if F = C;

KOGn (X) if F = R,

Consider a proper G-CW -complex X. One sometimes finds in the literaturthe definition of


RKGn (X) := colimC⊆X KK

Gn (C0(X),C),(12.5)

where C runs through the finite G-CW -subcomplexes of Y directed by in-clusion. Lemma 10.5 implies that we get identifications

(12.6) RKGn (X) = KG

n (X) = HGn (X; Ktop

C )

and analogous in the real case.The assembly maps induced by the projection

HGn (pr; Ktop

C ) : HGn (X; Ktop

C )→ HGn (G/G; Ktop

C ) = Kn(C∗r (G;C));(12.7)

HGn (pr; Ktop

R ) : HGn (X; Ktop

R )→ HGn (G/G; Ktop

R ) = Kn(R∗r(G;R)),(12.8)

can be identified with the assembly maps (12.1) and (12.2), see in [222,page 247-248].

The assembly maps (12.1) and (12.7) are identified in [222, Section 6].Unfortunately, the proof is based on an unpublished preprint by Carlsson-Pedersen-Roe. Another proof of the identification is given in [609] and [367,Corollary 8.4].

The identifications above carry over to the case, where one allows coeffi-cients in a G-C∗-algebra A. (Unfortunately, we do not know a satisfactoryplace in the literature, where all details of the identifications of the assem-bly maps (12.1), (12.3), and (12.7), and of the assembly maps (12.2), (12.4),and (12.8), are given allowing arbitrary coefficients and not necessarily tor-sionfree groups.)

12.4 The Baum-Connes Conjecture

Recall that a model for the classifying space for proper G-actions is a G-CW -complex EG = EFIN (G) such that EGH is non-empty and contractible foreach finite subgroup H ⊆ G and empty for each infinite subgroup H ⊆G. Two such models are G-homotopy equivalent. See Definition 9.18 andTheorem 9.19.

Conjecture 12.9 (Baum-Connes Conjecture). A group G satisfies theBaum-Connes Conjecture if the assembly maps

asmbG,C(EG)n : KGn (EG)→ Kn(C∗r (G));

asmbG,R(EG)n : KOGn (EG)→ KOn(C∗r (G)),

defined in (12.1) and (12.2) are bijective for all n ∈ Z in the special case thatA is C or R respectively with the trivial G-action.

Exercise 12.10. Show KGn (EG) ∼= Zk for k, n ∈ Z, k ≥ 1 and G = Z×Z/k.

12.4 The Baum-Connes Conjecture 371

Conjecture 12.11 (Baum-Connes Conjecture with coefficients). Agroup G satisfies the Baum-Connes Conjecture with coefficients if the assem-bly maps

asmbG,CA (EG)n : KGn (EG;A)→ Kn(Aor G);

asmbG,RA (EG)n : KOGn (EG;A)→ KOn(Aor G),

defined in (12.1) and (12.2) are bijective for all n ∈ Z and all G-C∗-algebrasA over F = R,C.

Remark 12.12 (Counterexample to the Baum-Connes Conjecturewith coefficients 12.11 and a modified version). We will discuss thestatus and further applications of the Baum-Connes Conjecture with coeffi-cients 12.11 in Section 12.8 and 14.3, but immediately want to point out thatthere exists counterexamples to the version 12.11 with coefficients, see [388],but no counterexample to the Baum-Connes Conjecture 12.9 is known.

In [93] a new formulation of the Baum-Connes Conjecture with coefficientsis given by considering a different crossed product for which the counterexam-ples mentioned above are not counterexamples anymore and no counterex-ample is known to the author’s knowledge. The new version takes care ofthe problem that there exists groups G together with short exact sequencesof G-C∗-algebras 0 → I → A → B → 0 for which the induced sequence0→ I oG→ AoG→ B oG→ 0 is not exact anymore and it is hence notclear that there exists a long exact sequence

. . .→ Kn(I oG)→ Kn(AoG)→ Kn(B oG)

→ Kn−1(I oG)→ Kn−1(AoG)→ Kn−1(B oG)→ . . .

whose existence is a consequence of the Baum-Connes Conjecture with coef-ficients 12.11. The new version still has the flaw that the left hand side of theassembly map is functorial under arbitrary group homomorphism, whereasthis is unknown for the right hand side, compare Remark 12.20.

The original source of the Baum-Connes Conjecture (with coefficients)is [91, Conjecture 3.15 on page 254].

Remark 12.13 (The complex case implies the real case). The complexversion of the Baum-Connes Conjecture 12.9 and 12.11 implies automaticallythe real version, see [96, 739].

Remark 12.14 (Reduction to the torsionfree case). There are always a

canonical isomorphismsKG∗ (EG)

∼=−→ K∗(BG) andKOG∗ (EG)∼=−→ KO∗(BG).

Suppose that G is torsionfree. Then EG is a model for EG and under theidentification above the assembly map appearing in the Baum-Connes Con-jecture 12.9 agrees with the one appearing in the Baum-Connes Conjecturefor torsionfree groups 8.44. Hence the Baum-Connes Conjecture for torsion-free groups 8.44 is a special case of the Baum-Connes Conjecture 12.9.


Exercise 12.15. Let f : H → G be a group homomorphism of torsionfreegroups. Suppose that H and G satisfy the Baum-Connes Conjecture 12.9and the induced map on group homology Hn(f) : Hn(H) → Hn(G) is bi-jective for n ∈ Z. Show that then Kn(C∗r (G;C)) ∼= Kn(C∗r (H;C)) andKOn(C∗r (G;R)) ∼= KOn(C∗r (H;R)) holds for all n ∈ Z.

12.5 Variants of the Baum-Connes Conjecture

In this section we discuss some variants of the Baum-Connes Conjecture.

12.5.1 The Baum-Connes Conjecture for Maximal GroupC∗-Algebras

There are also versions of the Baum-Connes assembly map using the maximalcrossed product A om G, see [654, 7.6.5 on page 257] for a G-C∗-algebra Aover F or the maximal group C∗-algebra C∗m(G;F ) for F = R,C. Namely,there are assembly maps

asmbG,C,mA (X)∗ : KG∗ (X;A)→ K∗(Aom G);(12.16)

asmbG,R,mA (X)∗ : KOG∗ (X;A)→ KO∗(Aom G),(12.17)

which reduce for A = R,C equipped with the trivial G-action to assemblymaps

asmbG,C,m(EG)n : KGn (EG) → Kn(C∗m(G;C));(12.18)

asmbG,R,m(EG)n : KOGn (EG) → KOn(C∗m(G;R)).(12.19)

In the sequel we only consider the complex case, the corresponding state-ments are true over R as well.

There is always a C∗-homomorphism p : AomG→ AorG, and we obtainthe following factorization of the Baum-Connes assembly map of (12.1)

asmbG,CA (X)∗ : KGn (X;A)

asmbG,C,mA (X)∗−−−−−−−−−−→ K∗(Aom G)K∗(p)−−−−→ K∗(Aor G).

The Baum-Connes Conjecture 12.11 implies that the map asmbG,C,mA (EG)∗is always injective, and that it is surjective if and only if the map K∗(p) isinjective.

Remark 12.20 (Functoriality of the Baum-Connes assembly map).Notice that the source of the assembly maps asmbG,C(EG)n : KG

n (EG) →

12.5 Variants of the Baum-Connes Conjecture 373

Kn(C∗r (G)) and asmbG,C.m(EG)n : KGn (EG) → Kn(C∗m(G)) are functorial

in G. The target Kn(C∗m(G)) is also functorial in G since C∗m(G) is functorialin G, and the assembly map asmbG,C.m(EG)n : KG

n (EG) → Kn(C∗m(G)) isnatural in G.

However, it is not known whether the target Kn(C∗r (G)) is functorialin G and we have already explained in Subsection 8.3.1 that not everygroup homomorphism α : G → H induces a homomorphism of C∗-algebrasC∗r (G) → C∗r (H). This is irritating since the Baum-Connes Conjecture 12.9implies that Kn(C∗r (G)) is also functorial in G.

The same problem is still present in the new formulation of the Baum-Connes Conjecture with coefficients in [93].

Remark 12.21 (The Baum-Connes Conjecture does not hold in gen-eral for the maximal group C∗-algebra). It is known that the assembly

map asmbG,C,mA (EG)∗ of (12.16) is in general not surjective. A countablegroup G is called K-amenable if the map p : C∗max(G) → C∗r (G) induces aKK-equivalence (compare [210]). This implies in particular that the mapKn(p) above is an isomorphism for all n ∈ Z. Note that for K-amenablegroups the Baum-Connes Conjecture 12.9 holds if and only if the “maximal”version of the assembly map Amax

FIN is an isomorphism for all n ∈ Z. A-T-menable groups are K-amenable, see [387]. But K0(p) is not injective if Gis any infinite group which has property (T), compare for instance the dis-cussion in [432]. There are infinite groups with property (T) for which theBaum-Connes Conjecture is known, see [486] and also [756] Hence there arecounterexamples to the conjecture that asmbG,C,m(EG)n is an isomorphism.

12.5.2 The Bost Conjecture

Some of the strongest results about the Baum-Connes Conjecture are provenusing the so called Bost Conjecture, see [488, page 798]. The Bost Conjectureis the version of the Baum-Connes Conjecture, where one replaces the reducedgroup C∗-algebra C∗r (G;F ) by the Banach algebra L1(G;F ). One still candefine the topological K-theory of L1(G;F ) and the assembly map in thiscontext.

Conjecture 12.22 (Bost Conjecture). The assembly maps

asmbG,C,L1

(EG)n : KGn (EG)→ Kn(L1(G;C))

asmbG,R,L1

(EG)n : KOGn (EG)→ KOn(L1(G;R))

are isomorphism for all n ∈ Z.

In the sequel we only consider the complex case, the corresponding state-ments are true over R as well.


Again the left hand side coincides with the left hand side of the Baum-Connes assembly map. There is a canonical map of Banach ∗-algebrasq : L1(G)→ C∗r (G). We obtain a factorization of the Baum-Connes assemblymap appearing in the Baum-Connes Conjecture 12.9

(12.23) asmbG,CA (EG)∗ : KGn (EG)

asmbG,C,L1(X)∗−−−−−−−−−−→ K∗(L

1(G))

K∗(q)−−−−→ K∗(C∗r (G))

Every group homomorphism G → H induces a homomorphism of Ba-nach algebras L1(G) → L1(H) and the assembly map appearing in Conjec-ture 12.22 is natural in G.

The disadvantage of L1(G) is however that indices of operators tend totake values in the topological K-theory of the group C∗-algebras, not inKn(L1(G)). Moreover the representation theory of G is closely related to thegroup C∗-algebra, whereas the relation to L1(G) is not well understood.

There is also a version of the Bost Conjecture with coefficients in a C∗-algebra:

(12.24) asmbG,C,L1

A (EG)∗ : KG∗ (EG;A)→ K∗(AoL1 G)

For more information about the Bost Conjecture 12.22 we refer for instanceto [65, 488, 489, 642, 643, 756].

12.5.3 The Baum-Connes Conjecture for Frechet Algebras

Comment 44: This section has to be written, when the manuscript in prepa-ration about this setting has been written.

12.5.4 The Strong and the Integral Novikov Conjecture

We mention the following conjectures which actually follow from the Baum-Connes Conjecture 12.9.

Conjecture 12.25 (Strong Novikov Conjecture). A group G satisfiesthe Strong Novikov Conjecture if the assembly maps appearing in (8.42)or (8.43)

asmbG,C(BG)∗ : Kn(BG)→ Kn(C∗r (G;C));

asmbG,R(BG)∗ : KOn(BG)→ KOn(C∗r (G;R)),

are rationally injective for all n ∈ Z.

12.5 Variants of the Baum-Connes Conjecture 375

Conjecture 12.26 (Integral Novikov Conjecture).A torsionfree group G satisfies the Integral Novikov Conjecture if the as-

sembly map appearing in (8.42) or (8.43) are injective for all n ∈ Z.

The assembly map appearing in the Integral Novikov Conjecture 12.25can be identified with the assembly maps appearing in the Baum-ConnesConjecture for torsionfree groups.

The Integral Novikov Conjecture makes only sense for torsionfree groups.

Exercise 12.27. Find a finite group G for which there cannot be an injectivemap from K1(BG) to K1(C∗r (G)).

Theorem 12.28 (The Baum-Connes Conjecture implies the NovikovConjecture). Given a group G, the Baum-Connes Conjecture 12.9 for Gimplies the Strong Novikov Conjecture 12.25 for G and the Strong NovikovConjecture 12.25 for G implies the Novikov Conjecture 7.131 for G.

Proof. The implication that the Baum-Connes Conjecture 12.9 implies theStrong Novikov Conjecture 12.25 follows from Lemma 11.28. For proofs thatthe Strong Novikov Conjecture 12.25 implies the Novikov Conjecture 7.131we refer to Kasparov [457, § 9], [449] or Kaminker-Miller [438]. ut

12.5.5 The Coarse Baum Connes Conjecture

We briefly explain the Coarse Baum-Connes Conjecture, a variant of theBaum-Connes Conjecture, which applies to metric spaces and not onlyto groups. Its importance lies in the fact that isomorphism results aboutthe Coarse Baum-Connes Conjecture can be used to prove injectivity re-sults about the classical assembly map for topological K-theory, see Theo-rem 14.14.

Let X be a metric space which is proper, i.e., closed balls are compact. LetHX a separable Hilbert space with a faithful nondegenerate ∗-representationof C0(X). Let T : HX → HX be a bounded linear operator. Its supportsuppT ⊂ X×X is defined as the complement of the set of all pairs (x, x′), forwhich there exist functions φ and φ′ ∈ C0(X) such that φ(x) 6= 0, φ′(x′) 6= 0and φ′Tφ = 0. The operator T is said to be a finite propagation operator, ifthere exists a constant α such that d(x, x′) ≤ α for all pairs in the support ofT . The operator is said to be locally compact if φT and Tφ are compact forevery φ ∈ C0(X). An operator is called pseudolocal if φTψ is a compact oper-ator for all pairs of continuous functions φ and ψ with compact and disjointsupports.

The Roe-algebra C∗(X) is the operator-norm closure of the ∗-algebraof all locally compact finite propagation operators on HX . The algebraD∗(X) is the operator-norm closure of the pseudolocal finite propagation


operators. One can show that the topological K-theory of the quotientK∗(D

∗(X)/C∗(X)) agrees with K-homology K∗−1(X). A metric space iscalled uniformly contractible if for every R > 0 there exists S > R suchthat for every xı8nX the inclusion of balls BR(x) → BS(x) is nullhomo-topic. For a uniformly contractible proper metric space the coarse assemblymap Kn(X)→ Kn(C∗(X)) is the boundary map in the long exact sequenceassociated to the short exact sequence of C∗-algebras

0→ C∗(X)→ D∗(X)→ D∗X)/C∗(X)→ 0.

For general metric spaces one first approximates the metric space by spaceswith nice local behavior, compare [698]. For simplicity we only explain thecase, where X is a discrete metric space. Let Pd(X) be the Rips complex fora fixed distance d, i.e., the simplicial complex with vertex set X, where asimplex is spanned by every collection of points in which every two pointsare a distance less than d apart. Equip Pd(X) with the spherical metric,compare [847].

A discrete metric space has bounded geometry if for each r > 0 there existsa N(r) such that for all x the ball of radius r centered at x ∈ X contains atmost N(r) elements.

Conjecture 12.29 (Coarse Baum-Connes Conjecture). Let X be aproper discrete metric space of bounded geometry. Then for n ∈ Z the coarseassembly map

colimd→∞Kn(Pd(X))→ colimd→∞Kn(C∗(Pd(X))) ∼= Kn(C∗(X))

is an isomorphism.

The conjecture is false if one drops the bounded geometry hypothesis. Acounterexample can be found in [848, Section 8].

The Coarse Baum-Connes Conjecture for a finitely generated discretegroup G (considered as a metric space) can be interpreted as a case of theBaum-Connes Conjecture with Coefficients 12.11 for the group G with a cer-tain specific choice of coefficients, compare [852].

Further information about the coarse Baum-Connes Conjecture can befound for instance in [186, 317, 334, 335, 389, 390, 392, 698, 838, 839, 846,847, 849, 850, 853].

12.6 Inheritance Properties of the Baum-ConnesConjecture

Similar to the Farrell-Jones Conjecture, the Baum-Connes Conjecture withcoefficients 12.11 has much better inheritance properties than the Baum-Connes Conjecture 12.9. Namely, we have

12.6 Inheritance Properties of the Baum-Connes Conjecture 377

Theorem 12.30 (Inheritance properties of the Baum-Connes Con-jecture with coefficients).

(i) Passing to subgroupsLet H ⊆ G be an inclusion of groups. If G satisfies the Baum-Connes

Conjecture with coefficients 12.11, then H satisfies the Baum-Connes Con-jecture with coefficients 12.11;

(ii) Group extensions

Let 1 → K → Gp−→ Q → 1 be an extension of groups. Suppose that for

any finite subgroup H ⊆ Q the group p−1(H) satisfies the Baum-ConnesConjecture with coefficients 12.11 and that the group Q satisfies the Baum-Connes Conjecture with coefficients 12.11.Then G satisfies the Baum-Connes Conjecture with coefficients 12.11;

(iii) Passing to finite direct productsIf the groups G0 and G1 satisfy the Baum-Connes Conjecture with coef-

ficients 12.11, then G0 × G1 satisfies the Baum-Connes Conjecture withcoefficients 12.11;

(iv) Directed unionsLet G be a union of the directed system of subgroups Gi | i ∈ I.

If each group Gi satisfies the Baum-Connes Conjecture with coefficients 12.11,then G satisfies the Baum-Connes Conjecture with coefficients 12.11;

(v) Actions on treesSuppose that G acts on a tree without inversion. Assume that the Baum-

Connes Conjecture with coefficients 12.11 holds for the stabilizers of anyof the vertices.Then the Baum-Connes Conjecture with coefficients 12.11 holds for G;

(vi) Amalgamated productsLet G0 be a subgroup of G1 and G2 and G be the amalgamated productG = G1 ∗G0

G2. Suppose Gi satisfies the Baum-Connes Conjecture withcoefficients 12.11 for i = 0, 1, 2.Then G satisfies Baum-Connes Conjecture with coefficients 12.11;

(vii) HNN extensionLet G be an HN extension of the group H. Suppose that G satisfies the

Baum-Connes Conjecture with coefficients 12.11.Then G satisfies the Baum-Connes Conjecture with coefficients 12.11;

Proof. (i) This has been stated in [91], a proof can be found for instancein [171, Theorem 2.5].

(ii) See [639, Theorem 3.1].

(iii) This follows from assertion (ii).

(iv) See [65, Theorem 1.8 (ii)].

(v) This is proved by Oyono-Oyono [640, Theorem 1.1].

(vi) and (vii) These are special case of assertion (v). ut


Exercise 12.31. Show that the Baum-Connes Conjecture with coefficients 12.11holds for any abelian group and any free group.

Exercise 12.32. Let G be the fundamental group of the closed orientablesurface of genus g ≥ 1. Show

Kn(C∗r (G;C)) =

Z2 n is even;

Zg n is odd.

Remark 12.33 (The Baum-Connes Conjecture with coefficients isis not compatible with colimits in general). The Baum-Connes Con-jecture with coefficients is not compatible with colimits in general. This is incontrast to the Full Farrell-Jones Conjecture 11.20, see Theorem 11.21 (iv)and to the Bost Conjecture with coefficients (12.24) see [65, Theorem 1.8 (i)].The Baum-Connes Conjecture with coefficients 12.11 is known for hyperbolicgroups, see [486, 756]. Now let G be a colimit of a directed system of hyper-bolic groups Gi | i ∈ I. Suppose that the Baum-Connes Conjecture withcoefficients 12.11 passes to colimits of directed systems of groups. Then theBaum-Connes Conjecture with coefficients 12.11 holds for G as well. However,there exists a group G which is a colimit of hyperbolic groups and containsappropriate expanders so that that [388] applies and hence the Baum-ConnesConjecture with coefficients 12.11 does not hold for G. The construction ofsuch a group is described in [35].

Remark 12.34 (The Farrell-Jones Conjecture and actions on trees).The inheritance properties of the Baum-Connes Conjecture with coeffi-cients 12.11 for actions on trees, see Theorem 12.30 (v), is very useful. Itdoes not hold for the Full Farrell-Jones Conjecture 11.20. The main reasonis that in the Baum-Connes setting the family FIN suffices, whereas in theFarrell-Jones setting we have to use the family VCY, since in the Farrell-Jones setting Nil-phenomenons occur which are not present in the Baum-Connes setting. Nevertheless, some partial results about this question in theFarrell-Jones setting can be found in [69]. Alternatively, one uses actions ontrees to compute HG

n (EFIN (G); KR), see Section 13.7, and treats the rela-tive group HG

n (EFIN (G)→ EVCY(G); KR) separately, for which the resultsof Section 11.9 are very useful. Thanks to the splitting results of Section 11.7one can put these two computations together to get a full description ofHGn (EVCY(G); KR). The analogous remark applies to L-theory.

Remark 12.35 (Passing to overgroups of finite index). It is not knownwhether the Baum-Connes Conjecture with coefficients 12.11 passes to over-groups of finite index. The same is true for the K- and L-theoretic Farrell-Jones Conjecture with coefficients in additive G-categories (with involution),see Conjecture 11.9 and Conjecture 11.14. This was the reason why we have

12.7 Reducing the Family of Subgroups for the Baum-Connes Conjecture 379

introduced in Section 11.4 the versions “with finite wreath products”. Onecan do the same in the Baum-Connes setting.

12.7 Reducing the Family of Subgroups for theBaum-Connes Conjecture

The following result is proved in [70, Theorem 0.5] based on a Comple-tion Theorem, see [544, Theorem 6.5] and a Universal Coefficient Theorem,see [115, 422]. An argument for the complex case using equivariant Eulerclasses is given by Mislin and Matthey [576] for the complex case. It is notclear to us whether it is possible to extend the methods of [576] to the realcase.

Theorem 12.36 (Reducing the family of subgroups for the Baum-Connes Conjecture). For any group G the relative assembly maps

KGn (EFCY(G))→ KG

n (EFIN (G));

KOGn (EFCY(G)) → KOGn (EFIN (G)),

are bijective for all n ∈ Z, where FCY is the family of finite cyclic subgroups.

Remark 12.37 (FCY is the smallest family for the Baum-ConnesConjecture). Let C be a finite cyclic group and F be a family of subgroupsof C. Then the assembly map

KC0 (EF (C)) → KC

0 (C/C) = RC(C)

is surjective if and only if F consists of all subgroups. This follows from [528,Theorem 0.7 and Lemma 3.4] since they predict that the homomorphisminduced by the various inclusions⊕

D∈FRC(D)→ RC(C)

is rationally surjective and hence C must be contained in F .Let C be a class of groups which is closed under taking subgroups and

passing to isomorphic groups. Examples are the classes of finite cyclic groupsor of finite groups. Given a group G, let C(G) be the family of subgroups ofG which belong to G. Suppose that for any group G the assembly map

KGn (EC(G)(G))→ KG

n (G/G)

is bijective. The considerations above imply that C has to contain all finitecyclic subgroups. So, roughly speaking, FCY is the smallest family for whichone can hope that the Baum-Connes Conjecture 12.9 is true for all groups.


12.8 Applications of the Baum-Connes Conjecture

The Baum-Connes Conjecture for torsionfree groups 8.44 follows from theBaum-Connes Conjecture 12.9, see Remark 12.14, and implies, see Subsec-tions 8.4.1 and 8.4.2,

• Trace Conjecture 8.50 for torsionfree groupsFor a torsionfree group G the image of

trC∗r (G) : K0(C∗r (G))→ R

consists of the integers.

• Kadison Conjecture 8.52If G is a torsionfree group, then the only idempotent elements in C∗r (G)are 0 and 1.

The Baum-Connes Conjecture 12.9 implies by Theorem 12.28

• Strong Novikov Conjecture 12.25The assembly maps

asmbG,C(BG)∗ : Kn(BG)→ Kn(C∗r (G;C));

asmbG,R(BG)∗ : KOn(BG)→ KOn(C∗r (G;R)).

of (8.42) and (8.43) are rationally injective for all n ∈ Z.

The strong Novikov Conjecture 12.25 (and hence also the Baum-Connes Con-jecture 12.9) implies, see Subsection 8.4.3,

• Zero-in-the-spectrum Conjecture 8.55If M is the universal covering of an aspherical closed Riemannian manifoldM , then zero is in the spectrum of the minimal closure of the pth Laplacianon M for some p ∈ 0, 1, . . . ,dimM.

Moreover, we have already shown in Theorem 12.28 that the Baum-ConnesConjecture 12.9 implies

• Novikov Conjecture 7.131Higher signatures are homotopy invariant.

Next we deal with some other conjectures which follows from the Baum-Connes Conjecture.

12.8.1 The Modified Trace Conjecture

Denote by ΛG the subring of Q which is obtained from Z by inverting allorders |H| of finite subgroups H of G, i.e.,

12.8 Applications of the Baum-Connes Conjecture 381

ΛG = Z[|H|−1 | H ⊂ G, |H| <∞

].(12.38)

The following conjecture generalizes Conjecture 8.50 to the case where thegroup need no longer be torsionfree. For the standard trace see (8.48).

Conjecture 12.39 (Trace Conjecture, modified). Let G be a group.Then the image of the homomorphism induced by the standard trace

trC∗r (G) : K0(C∗r (G))→ R(12.40)

is contained in ΛG.

The following result is proved in [528, Theorem 0.3].

Theorem 12.41. Let G be a group. Then the image of the composition

KG0 (EFIN (G))⊗Z Λ

G asmbG,C(EG)n⊗Zid−−−−−−−−−−−−→ K0(C∗r (G))⊗Z ΛG

trC∗r (G)−−−−−→ R

is ΛG. Here asmbG,C(EG)n is the map appearing in the Baum-Connes Con-jecture 12.9. In particular the Baum-Connes Conjecture 12.9 implies theModified Trace Conjecture 12.39.

The original version of the Trace Conjecture due to Baum and Connes [90,page 21] makes the stronger statement that the image of trC∗r (G) : K0(C∗r (G))→R is the additive subgroup of Q generated by all numbers 1

|H| , where H ⊂ Gruns though all finite subgroups of G. Roy has constructed a counterexam-ple to this version in [726] based on her article [727]. The examples of Roydo not contradict the Modified Trace Conjecture 12.39 or the Baum-ConnesConjecture 12.9.

Exercise 12.42. The G be a finite group. Show that the image of the tracemap trC∗r (G) : K0(C∗r (G))→ R is n · |G|−1 | n ∈ Z.

12.8.2 The Stable Gromov-Lawson-Rosenberg Conjecture

The Stable Gromov-Lawson-Rosenberg Conjecture is a typical conjecture re-lating Riemannian geometry to topology. It is concerned with the questionwhen a given manifold admits a metric of positive scalar curvature. It isrelated to the real version of the Baum-Connes Conjecture 12.9.

Let ΩSpinn (BG) be the bordism group of closed Spin-manifolds M of di-

mension n with a reference map to BG. Given an element [u : M → BG] ∈ΩSpinn (BG), we can take the C∗r (G;R)-valued index of the equivariant Dirac

operator associated to the G-covering M → M determined by u. Thus weget a homomorphism


indC∗r (G;R) : ΩSpinn (BG)→ KOn(C∗r (G;R)).(12.43)

A Bott manifold is any simply connected closed Spin-manifold B of dimension8 whose A-genus A(B) is 1. We fix such a choice, the particular choice doesnot matter for the sequel. Notice that indC∗r (1;R)(B) ∈ KO8(R) ∼= Z is agenerator and the product with this element induces the Bott periodicity

isomorphisms KOn(C∗r (G;R))∼=−→ KOn+8(C∗r (G;R)). In particular

indC∗r (G;R)(M) = indC∗r (G;R)(M ×B),(12.44)

if we identify KOn(C∗r (G;R)) = KOn+8(C∗r (G;R)) via Bott periodicity.

Conjecture 12.45 (Stable Gromov-Lawson-Rosenberg Conjecture).Let M be a closed connected Spin-manifold of dimension n ≥ 5. LetuM : M → Bπ1(M) be the classifying map of its universal covering. ThenM × Bk carries for some integer k ≥ 0 a Riemannian metric with positivescalar curvature if and only if

indC∗r (π1(M);R)([M,uM ]) = 0 ∈ KOn(C∗r (π1(M);R)).

If M carries a Riemannian metric with positive scalar curvature, thenthe index of the Dirac operator must vanish by the Bochner-Lichnerowiczformula [701]. The converse statement that the vanishing of the index impliesthe existence of a Riemannian metric with positive scalar curvature is thehard part of the conjecture. The following result is due to Stolz. A sketch ofthe proof can be found in [768, Section 3], details are announced to appearin a different paper.

Theorem 12.46 (The Baum-Connes Conjecture implies the StableGromov-Lawson-Rosenberg Conjecture). If the assembly map for thereal version of the Baum-Connes Conjecture 12.9 is injective for the groupG, then the Stable Gromov-Lawson-Rosenberg Conjecture 12.45 is true forall closed Spin-manifolds of dimension ≥ 5 with π1(M) ∼= G.

The requirement dim(M) ≥ 5 is essential in the Stable Gromov-Lawson-Rosenberg Conjecture, since in dimension four new obstructions, the Seiberg-Witten invariants, occur. The unstable version of this conjecture says thatM carries a Riemannian metric with positive scalar curvature if and onlyif indC∗r (π1(M);R)([M,uM ]) = 0. Schick [737] constructs counterexamples tothe unstable version using minimal hypersurface methods due to Schoen andYau (see also [259]). There are counterexamples with π ∼= Z4×Z/3. Howeverfor appropriate ρ : Z/3 → aut(Z4) the unstable version does hold for π ∼=Z4 oρ Z/3 and dim(M) ≥ 5, see [224, Theorem 0.7 and Remark 0.9].

Since the Baum-Connes Conjecture 12.9 is true for finite groups (for thetrivial reason that EFIN (G) = • for finite groups G), Theorem 12.46implies that the Stable Gromov-Lawson Conjecture 12.45 holds for finite


fundamental groups, see also [713]. It is not known at the time of writingwhether the unstable version is true for finite fundamental groups.

The index map appearing in (12.43) can be factorized as a composition(12.47)

indC∗r (G;R) : ΩSpinn (BG)

D−→ KOn(BG)asmbG,C(BG)n−−−−−−−−−→ KOn(C∗r (G;R)),

where D sends [M,u] to the class of the G-equivariant Dirac operator of theG-manifold M given by u and asmbG,C(BG)n is the real version of the classi-cal assembly map. The homological Chern character defines an isomorphism

KOn(BG)⊗Z Q∼=−→⊕p∈Z

Hn+4p(BG;Q).

Recall that associated to M there is the A-class

A(M) ∈∏p≥0

Hp(M ;Q)(12.48)

which is a certain polynomial in the Pontrjagin classes. The map D appear-ing in (12.47) sends the class of u : M → BG to u∗(A(M) ∩ [M ]), i.e., the

image of the Poincare dual of A(M) under the map induced by u in rational

homology. Hence D([M,u]) = 0 if and only if u∗(A(M) ∩ [M ]) vanishes. For

x ∈∏k≥0H

k(BG;Q) define the higher A-genus of (M,u) associated to x tobe

Ax(M,u) = 〈A(M) ∪ u∗x, [M ]〉 = 〈x, u∗(A(M) ∩ [M ])〉 ∈ Q.(12.49)

The vanishing of A(M) is equivalent to the vanishing of all higher A-genera

Ax(M,u). The following conjecture is a weak version of the Stable Gromov-Lawson-Rosenberg Conjecture.

Conjecture 12.50 (Homological Gromov-Lawson-Rosenberg Conjec-ture). Let G be a group. Then for any closed Spin-manifold M , which admits

a Riemannian metric with positive scalar curvature, the A-genus Ax(M,u)vanishes for all maps u : M → BG and elements x ∈

∏k≥0H

k(BG;Q).

From the discussion above we obtain the following result.

Lemma 12.51. If the assembly map

KOn(BG)⊗Z Q→ KOn(C∗r (G;R))⊗Z Q

is injective for all n ∈ Z, then the Homological Gromov-Lawson-RosenbergConjecture 12.50 holds for G.

The following conjecture is due to Gromov-Lawson [354, page 313].


Conjecture 12.52 (Aspherical closed manifolds carry no Rieman-nian metric with positive scalar curvature). A closed aspherical ma-nifold carries no Riemannian metric with positive scalar curvature.

Lemma 12.53. Let M be an aspherical closed Spin-manifold whose funda-mental group satisfies the Homological Gromov-Lawson-Rosenberg Conjec-ture 12.50.

Then M satisfies Conjecture 12.52, i.e., M carries no Riemannian metricwith positive scalar curvature.

Proof. Suppose M carries a Riemannian metric of positive scalar curvature.Since M is aspherical, we can take M = BG for G = π1(M) and f = idG in

Conjecture 12.50. Since A(M)0 = 1, we get for all x ∈ Hdim(M)(M ;Q) that〈x, [M ]〉 = 0 holds, a contradiction. ut

Exercise 12.54. Let F → M → S1 be a fiber bundle such that F is anorientable closed surface and M is a closed spin-manifold. Show that Mcarries a Riemannian metric with positive scalar curvature if and only if Fis S2.

The (moduli) space of metrics of positive scalar curvature of closed spinmanifolds is studied in [122, 123, 207, 372].

12.8.3 L2-Rho-Invariants and L2-Signatures

Let M be a closed connected orientable Riemannian manifold. Denote byη(M) ∈ R the eta-invariant of M and by η(2)(M) ∈ R the L2-eta-invariant of

the π1(M)-covering given by the universal covering M →M . Let ρ(2)(M) ∈ Rbe the L2-rho-invariant which is defined to be the difference η(2)(M)−η(M).These invariants were studied by Cheeger and Gromov [184, 185]. They showthat ρ(2)(M) depends only on the diffeomorphism type ofM and is in contrast

to η(M) and η(2)(M) independent of the choice of Riemannian metric on M .The following conjecture is taken from Mathai [575].

Conjecture 12.55 (Homotopy Invariance of the L2-Rho-Invariantfor Torsionfree Groups). If π1(M) is torsionfree, then ρ(2)(M) is a ho-motopy invariant.

Chang-Weinberger [174] assign to a closed connected oriented (4k − 1)-dimensional manifold M a Hirzebruch-type invariant τ (2)(M) ∈ R as fol-lows. By a result of Hausmann [378] there is a closed connected oriented4k-dimensional manifold W with M = ∂W such that the inclusion ∂W →Winduces an injection on the fundamental groups. Define τ (2)(M) as the differ-

ence sign(2)(W )−sign(W ) of the L2-signature of the π1(W )-covering given by


the universal covering W →W and the signature of W . This is indeed inde-pendent of the choice of W . It is reasonable to believe that ρ(2)(M) = τ (2)(M)is always true. Chang-Weinberger [174] use τ (2) to prove that if π1(M) is nottorsionfree there are infinitely many diffeomorphically distinct manifolds ofdimension 4k+ 3 with k ≥ 1, which are tangentially simple homotopy equiv-alent to M .

Theorem 12.56 (Homotopy Invariance of τ (2)(M) and ρ(2)(M)). LetMbe a closed connected oriented (4k− 1)-dimensional manifold M such thatG = π1(M) is torsionfree.

(i) If the assembly map K0(BG)→ K0(C∗max(G)) for the maximal group C∗-algebra, see Subsection 12.5.1, is surjective, then ρ(2)(M) is a homotopyinvariant;

(ii) Suppose that the L-theoretic Farrell-Jones Conjecture with coefficients inthe ring with involution 11.4 is rationally true for R = Z, i.e., the ratio-nalized assembly map

Hn(BG; L〈−∞〉(Z))⊗Z Q→ L〈−∞〉n (ZG)⊗Z Q

is an isomorphism for n ∈ Z. Then τ (2)(M) is a homotopy invariant. Iffurthermore G is residually finite, then ρ(2)(M) is a homotopy invariant.

Proof. (i) This is proved by Keswani [470], [471].

(ii) This is proved by Chang-Weinberger [174] using [555]. ut

Remark 12.57 (L2-signature Theorem). Let X be a 4n-dimensionalPoincare space over Q. Let X → X be a normal covering with torsionfree cov-ering group G. Suppose that the assembly map K0(BG)→ K0(C∗max(G)) forthe maximal group C∗-algebra is surjective see Subsection 12.5.1, or supposethat the rationalized assembly map for L-theory

H4n(BG; L〈−∞〉(Z))⊗Z Q→ L〈−∞〉4n (ZG)⊗Z Q

is an isomorphism. Then the following L2-signature theorem is proved inLuck-Schick [554, Theorem 0.3]

(12.58) sign(2)(X) = sign(X).

If one drops the condition that G is torsionfree this equality becomes false.Namely, Wall has constructed a finite Poincare space X with a finite G cover-ing X → X for which sign(X) 6= |G|·sign(X) holds (see [685, Example 22.28],[814, Corollary 5.4.1]). If X is a closed topological manifold, then (12.58) istrue for all groups G, see [554, Theorem 0.2].

Remark 12.59 (Obstructions for knots to be slice). Cochran-Orr-Teichner give in [189] new obstructions for a knot to be slice which are


sharper than the Casson-Gordon invariants. They use L2-signatures and theBaum-Connes Conjecture 12.9. We also refer to the survey article [188] aboutnon-commutative geometry and knot theory.

12.9 Miscellaneous

In contrast to the Farrell-Jones Conjecture the Baum-Connes Conjecture hasalso been formulated for and has important applications to (not necessarilydiscrete) topological groups, see for instance [91]. Comment 45: If theversion of the Farrell-Jones Conjecture for Hecke algebras appears in thebook, this has to be adjusted.

Comment 46: Mention the connection of the Baum-Connes Conjectureto representation theory.

The Baum-Connes assembly maps in terms of localizations of triangulatedcategories are considered in [413, 414, 415, 581, 582, 583].

Certain so called Cuntz-Lie C∗-algebras, see [211, 212], were classifiedin [511, Corollary 1.3]. The main difficulty is to compute the topological K-theory of these C∗-algebras which comes down to the computation of thetopological C∗-algebra of certain crystallographic groups. This in turn leadsvia the Baum-Connes Conjecture to an open conjecture about group homol-ogy which was solved in the case needed for this application, see [502, 503].

Other classification results, whose proof uses the Baum-Connes Conjec-ture 12.9, can be found in [262, Theorem 0.1].

For more information about the Baum-Connes Conjecture and its appli-cations we refer for instance to [91, 385, 393, 394, 395, 546, 606, 659, 710,738, 797].


name of texfile: ic

Chapter 13

The (Fibered) Meta-IsomorphismConjecture and Other IsomorphismConjectures

13.1 Introduction

In this section we deal with Isomorphism Conjectures in their most generalform. Namely, given a G-homology theory HG∗ the Meta-Isomorphism Con-jecture 13.2 predicts for a group G and a family F of subgroups of G thatthe map induced by the projection EF (G)→ G/G

HGn (EF (G))→ HGn (G/G)

is bijective for all n ∈ Z.If we take special examples forHG∗ and F , then we obtain the Farrell-Jones

Conjecture for a ring R (with involution), see Conjectures 11.1 and 11.4, andthe Baum-Connes Conjecture 12.9. We will also introduced a Fibered Meta-Isomorphism Conjecture 13.8 which is more general and has much better in-heritance properties, see Section 13.6. The versions of the Farrell-Jones Con-jecture with coefficients in additive categories, see Conjectures 11.9 and 11.14,and the Baum-Connes Conjecture 12.11 with coefficients are automaticallyfibered, see Theorem 13.9, and hence have good inheritance properties.

The main tool to reduce the family of subgroups is the Transitivity Prin-ciple which we discuss in Section 13.5.

Section 13.7 is devoted to actions on trees and their implications such asthe existence of Mayer-Vietoris sequences associated to amalgamated prod-ucts and Wang sequences associated to semidirect products with Z.

In Section 13.8 we pass to the special case, where the homology theorycomes from a functor from spaces to spectra which respects weak homotopyequivalences and disjoint unions, and discuss inheritance properties in thisframework.

By specifying the functor from spaces to spectra, we obtain the Farrell-Jones Conjecture for Waldhausen’s A-theory and for pseudoisotopy in Sec-tion 13.10. We also deal with topological Hochschild homology and cyclichomology in Section 13.11. We explain the Farrell-Jones Conjecture for ho-motopy K-theory in Section 13.12. The only instance, where we will considernot necessarily discrete groups is the Farrell-Jones Conjecture 13.72 for thealgebraic K-theory of the Hecke algebra of a totally disconnected locallycompact second countable Hausdorff group.

In Section 13.13 interesting relations between these conjectures are dis-cussed, namely, between the Farrell-Jones Conjecture for the K-theory ofgroups rings, for A-theory and for pseudoisotopy, between the L-theoretic

387

38813 The (Fibered) Meta-Isomorphism Conjecture and Other Isomorphism Conjectures

Farrell-Jones Conjecture and the Baum-Connes Conjecture, and betweenthe Farrell-Jones Conjecture for K-theory and homotopy K-theory. We willbriefly also relate the geometric surgery sequence in the topological cate-gory to an analytic surgery exact sequence. Comment 47: Depending onwhether we include something in this chapter at all, we have to mentionalso the Baum-Connes Conjecture and its versions for Frechet algebras andL1(G).

13.2 The Meta-Isomorphism Conjecture

Let G be a (discrete) group. Let HG∗ be a G-homology theory with valuesin Λ-modules for some commutative (associative) ring (with unit) Λ. Recallthat it assigns to every G-CW -pair (X,A) and integer n ∈ Z a Λ-moduleHGn (X,A) such that the obvious generalization to G-CW -pairs of the axiomsof a (non-equivariant generalized) homology theory for CW -complexes holds,i.e., G-homotopy invariance, the long exact sequence of a G-CW -pair, exci-sion, and the disjoint union axiom are satisfied. The precise definition of aG-homology theory can be found in Definition 10.1 and of a G-CW -complexin Definition 9.2, see also Remark 9.3.

Recall that we have defined the notion of a family of subgroups of a groupG in Definition 1.52, namely, to be a set of subgroups of G which is closedunder conjugation with elements of G and passing to subgroups. Let F bea family of subgroups of G. Denote by EF (G) a model for the classifyingG-CW -complex for the family F of subgroups of G, i.e., a G-CW -complexEF (G) whose isotropy groups belong to F and for which for each H ∈ F theH-fixed point set EF (G)H is weakly contractible. Such a model always existsand is unique up to G-homotopy, see Definition 9.18 and Theorem 9.19.

The projection pr : EF (G)→ G/G induces for all integers n ∈ Z a homo-morphism of Λ-modules

(13.1) HGn (pr) : HGn (EF (G))→ HGn (G/G)

which is sometimes called assembly map.

Conjecture 13.2 (Meta-Isomorphism Conjecture). The group G satis-fies the Meta-Isomorphism Conjecture with respect to the G-homology theoryHG∗ and the family F of subgroups of G if the assembly map

Hn(pr) : HGn (EF (G))→ HGn (G/G)

of (13.1) is bijective for all n ∈ Z.

If we choose F to be the family ALL of all subgroups, then G/G is amodel for EALL(G) and the Meta-Isomorphism Conjecture 13.2 is obviouslytrue. The point is to find an as small as possible family F . The idea of the

13.3 The Fibered Meta-Isomorphism Conjecture 389

Meta-Isomorphism Conjecture 13.2 is that one wants to compute HGn (G/G)which is the unknown and the interesting object, by assembling it from thevalues HGn (G/H) for H ∈ F , which are usually much more accessibly sincethe structure of the groups H is easy. For instance F could be the family FINof all finite subgroups or the family VCY of all virtually cyclic subgroups.

The various Isomorphism Conjectures are now obtained by specifying theG-homology theory HG∗ and the family F . For instance, the K-theoreticFarrell-Jones Conjecture 11.1 with coefficients in the ring R and the L-theoretic Farrell-Jones Conjecture 11.4 with coefficients in the ring with in-volution R are equivalent to the Meta-Isomorphism Conjecture 13.2 if we

choose F to be VCY and HGn to be HGn (−; KR) and HG

n (−; L〈−∞〉R ). The

Baum-Connes Conjecture 12.9 is equivalent the Meta-Isomorphism Conjec-ture 13.2 if we choose F to be FIN and HGn to be KG

n (−) = HGn (−; Ktop

C )

or KOGn (−) = HGn (−; Ktop

R ). The analogous statement holds for the ver-sions with coefficients in additive G-categories (with involutions), Conjec-tures 11.9, 11.14, and 12.11.

Exercise 13.3. Let H?∗ be an equivariant homology theory with values in Λ-

modules in the sense of Definition 10.9. Fix a class of groups C which is closedunder isomorphisms, taking subgroups and taking quotients, e.g., the classof finite groups or the class of virtually cyclic subgroups. For a group G letC(G) be the family of subgroups of G which belong to C. Then we obtain foreach group G an assembly map induced by the projection EC(G)(G)→ G/G.

HGn (EC(G)(G))→ HGn (G/G)

Explain that using the induction structure of H?∗ we can turn the source

and target to be functors from the category of groups to the category of Λ-modules such that the assembly maps yield a natural transformation of suchfunctors.

13.3 The Fibered Meta-Isomorphism Conjecture

Given a group homomorphism φ : K → G and a family F of subgroups of G,define the family of subgroups of K by

φ∗F := H ⊆ K | φ(H) ∈ F.(13.4)

If φ is an inclusion of subgroups, we also write

F|K := φ∗F = H ⊆ K | H ∈ F.(13.5)

If ψ : H → K is another group homomorphism, then


ψ∗(φ∗F) = (φ ψ)∗F .(13.6)

Exercise 13.7. Let φ : K → G be a group homomorphism. Consider a familyF of subgroups of G and a G-CW -model EF (G). Show that its restrictionto K by φ : K → G is a K-CW -complex which is a model for Eφ∗F (K).

Consider an equivariant homology theory H?∗ over the group Γ with values

in Λ-modules for a commutative ring in the sense of Definition 10.82.

Conjecture 13.8 (Fibered Meta-Isomorphism Conjecture). A group(G, ξ) over Γ satisfies the Fibered Meta-Isomorphism Conjecture with respectto H?

∗ and the family F of subgroups of G if for each group homomorphismφ : K → G the group K satisfies the Meta-Isomorphism Conjecture 13.2 withrespect to the K-homology theory HK,ξφ∗ and the family φ∗F of subgroupsof K.

13.4 The Farrell-Jones Conjecture with Coefficients inAdditive G-Categories is Fibered

We will see that it is important for inheritance properties to pass to thefibered version. It turns out that the fibered version is automatically builtinto the versions of the Farrell-Jones Conjecture with coefficients in additiveG-categories (with involution).

Theorem 13.9 (The Farrell-Jones Conjecture with coefficients inadditive G-categories (with involutions) is automatically fibered).

(i) Let φ : K → G be a group homomorphism. Let F be a family of subgroupsof G. Suppose that the assembly map

HGn (pr) : HG

n (EF (G); KA)→ HGn (G/G; KA) = πn

(KA(I(G))

)is bijective for every n ∈ Z and every additive G-category A.Then the assembly map

HKn (pr) : HK

n (Eφ∗F (K); KB)→ HKn (K/K; KB) = πn

(KB(I(K))

)is bijective for every n ∈ Z and every additive K-category B;

(ii) Suppose that G satisfies the K-theoretic Farrell-Jones Conjecture 11.9 withcoefficients in additive G-categories.Then the Fibered Meta-Isomorphism Conjecture 13.8 holds for the group(G, idG) over G, the family VCY and the equivariant homology theoryH?∗(−; KA) over G for every additive G-category A;

(iii) Let φ : K → G be a group homomorphism. Let F be a family of subgroupsof G. Suppose that the assembly map

13.5 Transitivity Principles 391

HGn (pr) : HG

n (EF (G); L〈−∞〉A )→ HG

n (G/G; L〈−∞〉A ) = πn

(L〈−∞〉A (I(G))

)is bijective for every n ∈ Z and every additive G-category with involutionA.Then the assembly map

HKn (pr) : HK

n (Eφ∗F (G); L〈−∞〉B )→ HK

n (K/K; L〈−∞〉B ) = πn

(KB(I(K))

)is bijective for every n ∈ Z and every additive K-category with involutionB;

(iv) Suppose that G satisfies the L-theoretic Farrell-Jones Conjecture 11.14with coefficients in additive G-categories with involution.Then the Fibered Meta-Isomorphism Conjecture 13.8 holds for the group(G, idG) over G, the family VCY and the equivariant homology theory

H?∗(−; L

〈−∞〉A ) over G for every additive G-category with involution A;

Proof. (i) See [77, Corollary 4.3].

(ii) This follows from assertion (i) by taking B = φ∗A since the K-homologytheory obtained by taking in H?

∗(−; KA) the variable ? to be φ is the same asthe K-homology theory HK

∗ (−; Kφ∗A) associated to the additive K-categoryφ∗A.

(iii) See [71, Theorem 11.3].

(iv) This follows from assertion (iii) by taking B = φ∗A. ut

It is useful to have the Fibered Meta Conjecture 13.8 available, since thereare other situations, where it is not known to formulate it with adequatecoefficients, as it is possible in the Farrell-Jones setting.

13.5 Transitivity Principles

In this subsection we treat only equivariant homology theories H?∗ to keep

the notation and exposition simple. The generalizations to an equivarianthomology theory over a group Γ are obvious, just equip each group occurringbelow with the appropriate reference map to Γ .

Lemma 13.10. Let G be a group and let F be a family of subgroups of G.Let m be an integer. Let Z be a G-CW -complex. For H ⊆ G let F|H bethe family of subgroups of H given by L ⊆ H | L ∈ F. Suppose for eachH ⊂ G, which occurs as isotropy group in Z, that the maps induced by theprojection prH : EF|H (H)→ H/H

HHn (prH) : HHn (EF|H (H)→ HHn (H/H)

satisfy one of the following conditions


(i) They are bijective n ∈ Z with n ≤ m;(ii) They are bijective for n ∈ Z with n ≤ m− 1 and surjective for n = m.

Then the maps induced by the projection pr2 : EF (G)× Z → Z

HGn (pr2) : HGn (EF (G)× Z)→ HGn (Z)

satisfies the same condition.

Proof. We first prove the claim for finite dimensional G-CW -complexes byinduction over d = dim(Z). The induction beginning dim(Z) = −1, i.e.Z = ∅, is trivial. In the induction step from (d − 1) to d we choose a G-pushout ∐

i∈Id G/Hi × Sd−1 //

Zd−1

∐i∈Id G/Hi ×Dd // Zd

If we cross it with EF (G), we obtain another G-pushout of G-CW -complexes.The various projections induce a map from the Mayer-Vietoris sequence ofthe latter G-pushout to the Mayer-Vietoris sequence of the first G-pushout.By the Five-Lemma (or its obvious variant, if we consider assumption (ii)) itsuffices to prove that the following maps

HGn (pr2) : HGn

(EF (G)×

∐i∈Id

G/Hi × Sd−1

)→ HGn

(∐i∈Id

G/Hi × Sd−1

);

HGn (pr2) : HGn (EF (G)× Zd−1)→ HGn (Zd−1);

HGn (pr2) : HGn

(EF (G)×

∐i∈Id

G/Hi ×Dd

)→ HGn

(∐i∈Id

G/Hi ×Dd

),

satisfy condition (i) or(ii). This follows from the induction hypothesis forthe first two maps. Because of the disjoint union axiom and G-homotopyinvariance of H?

∗ the claim follows for the third map if we can show for anyH ⊆ G which occurs as isotropy group in Z that the maps

HGn (pr2) : HGn (EF (G)×G/H)→ HG(G/H)(13.11)

satisfy condition (i) or(ii). The G-map

G×H resHG EF (G)→ G/H × EF (G) (g, x) 7→ (gH, gx)

is a G-homeomorphism where resHG denotes the restriction of the G-actionto an H-action. Since F|H = K ∩ H | K ∈ F, the H-space resHG EF (G)is a model for EF|H (H). We conclude from the induction structure that themap (13.11) can be identified with the map

13.5 Transitivity Principles 393

HHn (prH) : HHn (EF|H (H)) → HH(H/H)

which is satisfies condition (i) or(ii) by assumption. This finishes the proofin the case that Z is finite dimensional. The general case follows by a colimitargument using Lemma 10.5. ut

Theorem 13.12 (Transitivity Principle for equivariant homology).Suppose F ⊂ G are two families of subgroups of the group G. Suppose forevery H ∈ G that the maps induced by the projection

HHn (EF|H (H))→ HHn (H/H)

satsify one of the following conditions

(i) They are bijective n ∈ Z with n ≤ m;(ii) They are bijective for n ∈ Z with n ≤ m− 1 and surjective for n = m.

Then the maps induced by the up to G-homotopy unique G-map ιF⊆G : EF (G)→EG(G)

HGn (ιF⊆G) : HGn (EF (G))→ HGn (EG(G))

satisfy the same condition.

Proof. If we equip EF (G)×EG(G) with the diagonal G-action, it is a modelfor EF (G). Now apply Lemma 13.10 in the special case Z = EG(G). ut

This implies the following transitivity principle for the Fibered Isomor-phism Conjecture. At the level of spectra this transitivity principle is due toFarrell and Jones [295, Theorem A.10].

Theorem 13.13 (Transitivity Principle). Suppose F ⊆ G are two fami-lies of subgroups of G.

(i) Assume that for every element H ∈ G the group H satisfies the Meta-Isomorphism Conjecture 13.2 or the Fibered Meta-Isomorphism Conjec-ture 13.8 respectively for F|H .Then the group G satisfies the Meta-Isomorphism Conjecture 13.2 or theFibered Meta-Isomorphism Conjecture 13.8 respectively with respect to Gif and only if G satisfies the Meta-Isomorphism Conjecture 13.2 or theFibered Meta-Isomorphism Conjecture 13.8 respectively with respect to F ;

(ii) The group G satisfies the Fibered Meta-Isomorphism Conjecture 13.8 withrespect to G if G satisfies the Fibered Meta-Isomorphism Conjecture 13.8respectively with respect to F ;

Proof. (i) We first treat the (slightly harder) case of the Fibered Meta-Isomorphism Conjecture 13.8

Consider a group homomorphism φ : K → G. Then for every subgroup Hof K we conclude

(φ|H)∗(F|φ(H)) = (φ∗F)|H


from (13.6), where φ|H : H → φ(H) is the group homomorphism induced byφ. For every element H ∈ φ∗G the map

HHn (E(φ|H)∗(F|φ(H))(H)) = HHn (Eφ∗F|H (H))→ HHn (H/H)

is bijective for all n ∈ Z by the assumption that the element φ(H) ∈ G satisfiesthe Fibered Isomorphism Conjecture for F|φ(H). Hence by Theorem 13.12applied to the inclusion φ∗F ⊆ φ∗G of families of subgroups of K we get anisomorphism

HKn (ιφ∗F⊆φ∗G) : HKn (Eφ∗F (K))∼=−→ HKn (Eφ∗G(K)).

Therefore the map HKn (Eφ∗F (K)) → HKn (K/K) is bijective for all n ∈ Z ifand only if the map HKn (Eφ∗G(K))→ HKn (K/K) is bijective for all n ∈ Z.

The argument for the Meta-Isomorphism Conjecture 13.8 is analogous,just specialize the argument above to the case φ = idG.

(ii) We want to apply assertion (i). We have to show that for every elementH ∈ G the group H satisfies the Fibered Meta-Isomorphism Conjecture 13.8for F|H , provided that G satisfies the Fibered Meta-Isomorphism Conjec-ture 13.8 with respect to F . This follows from Lemma 13.16 since F|H = i∗Ffor the inclusion i : H → G. ut

Notice that assertion (ii) of Theorem 13.13 is only formulated for thefibered version.

The Fibered Isomorphism Conjecture is also well behaved with respect tofinite intersections of families of subgroups.

Lemma 13.14. Let G be a group and F and G be families of subgroups.Suppose that G satisfies the Fibered Meta-Isomorphism Conjecture 13.8 forboth F and G.

Then G satisfies the Fibered Meta-Isomorphism Conjecture 13.8 for thefamily F ∩ G := H ⊆ G | H ∈ F and H ∈ G.

Proof. Obviously F ∪ G := H ⊆ G | H ∈ F or H ∈ G is a family ofsubgroups of G.

Consider a group homomorphism φ : K → G. We have to show that theMeta-Isomorphism Conjecture 13.2 holds for G with respect to φ∗(F ∩ G).

Choose G-CW -models EF∩G(G), EF (G) and EG(G) such that EF∩G(G)is a G-CW -subcomplex of both EF (G) and EG(G). This can be arranged bya mapping cylinder construction. Define a G-CW -complex

X = EF (G) ∪EG(F∩G) EG(G).

For any subgroup H ⊆ G we get

XH = EF (G)H ∪EG(F∩G)H EG(G)H .

13.6 Inheritance Properties of the Fibered Meta-Isomorphism Conjecture 395

If EF (G)H and EG(G)H are empty, the same is true for XH . If EF (G)H

is empty, then EG(G)H = XH . If EG(G)H is empty, then EF (G)H = XH .If EF (G)H , EG(G)H and EF∩G(G)H are weakly contractible, the same istrue for XH . Hence X is a model for EF∪G(G). If we apply restriction withφ, we get a decomposition of Eφ∗(F∪G)(K) = φ∗EF∪G(G) as the union ofEφ∗F (K) = φ∗EF (G) and Eφ∗G(K) = φ∗EG(G) such that the intersectionof Eφ∗F (K) and Eφ∗G(K) is Eφ∗(F∩G)(K) = φ∗EF∩G(G). By assumptionand by Theorem 13.13 (ii) the Fibered Meta-Isomorphism Conjecture 13.8holds for G with respect to F , G and F ∪ G. Hence the Meta-IsomorphismConjecture 13.2 holds for G with respect to φ∗(F ∪ G), φ∗F and φ∗G. Usingthe Mayer-Vietoris sequence for the decomposition of Eφ∗F∪φ∗G(K) aboveand the Five-Lemma we conclude that Meta-Isomorphism Conjecture 13.2holds for G with respect to φ∗(F ∩ G). Since φ : K → G is an arbitrarygroup homomorphism with target G, the group G satisfies the Fibered Meta-Isomorphism Conjecture 13.8 for the family F ∩ G. ut

Exercise 13.15. Assume that the Fibered Meta-Isomorphism Conjecture 13.8holds for G = Z, the family F = FIN , and the equivariant homology theoryH?∗(−; KR) for a given ring R.Show that then we have NKn(RG) = 0 for every group G and n ∈ Z.

13.6 Inheritance Properties of the FiberedMeta-Isomorphism Conjecture

The Fibered Meta-Isomorphism Conjecture 13.8 has better inheritance prop-erties than the Meta-Isomorphism Conjecture 13.2.

In this subsection we treat only equivariant homology theories H?∗ for sim-

plicity. The generalizations to an equivariant homology theory over a groupΓ are obvious.

Lemma 13.16. Let φ : K → G be a group homomorphism and F be afamily of subgroups. If (G,F) satisfies the Fibered Meta-Isomorphism Con-jecture 13.8 then (K,φ∗F) satisfies the Fibered Meta-Isomorphism Conjec-ture 13.8.

Proof. If ψ : L → K is a group homomorphism, then ψ∗(φ∗F) = (φ ψ)∗Fby (13.6). ut

Exercise 13.17. Fix a class of groups C which is closed under isomorphismsand taking subgroups, e.g., the class of finite groups or the class of virtu-ally cyclic subgroups. For a group G let C(G) be the family of subgroups ofG which belong to C. Suppose that the Fibered Meta-Isomorphism Conjec-ture 13.8 holds for (G, C(G)). Let H ⊆ G be a subgroup.


Show that (H, C(H)) satisfies the Fibered Meta-Isomorphism Conjec-ture 13.8.

Lemma 13.18. Fix a class of groups C which is closed under isomorphisms,taking subgroups and taking quotients, e.g., the class of finite groups or theclass of virtually cyclic subgroups. For a group G let C(G) be the family of

subgroups of G which belong to C. Let 1→ K → Gp−→ Q→ 1 be an extension

of groups. Suppose that (Q, C(Q)) and (p−1(H), C(p−1(H)) for every H ∈C(Q) satisfy the Fibered Meta-Isomorphism Conjecture 13.8.

Then (G, C(G)) satisfies the Fibered Meta-Isomorphism Conjecture 13.8.

Proof. By Lemma 13.16 the pair (G, p∗C(Q)) satisfies the Fibered Meta-Isomorphism Conjecture 13.8. Obviously C(G) ⊆ p∗C(Q). Because of theTransitivity Principle 13.13 (i) it remains to show for each L ∈ p∗C(Q) thatthe pair (L, C(L)) satisfies the Fibered Meta-Isomorphism Conjecture 13.8.Since L ⊆ p−1(p(L)) and p(L) ∈ C(Q) holds, we conclude from Exercise 13.17that this follows from the assumption that (p−1(H), C(p−1(H)) satisfy theFibered Meta-Isomorphism Conjecture 13.8 for every H ∈ C(Q). ut

Fix an equivariant homology theory H?∗ with values in Λ-modules. Let X

be a G-CW -complex. Let α : H → G be a group homomorphism. Denote byα∗X the H-CW -complex obtained from X by restriction with α. Recall thatα∗Y denotes the induction of an H-CW -complex Y and is a G-CW -complex.The functors α∗ and α∗ are adjoint to one another. In particular the adjointof the identity on α∗X is a natural G-map

(13.19) f(X,α) : α∗α∗X → X.

It sends an element in G×α α∗X given by (g, x) to gx. Define the Λ-map

an = an(X,α) : HHn (α∗X)indα−−−→ HGn (α∗α

∗X)HGn (f(X,α))−−−−−−−−→ HGn (X).

If β : G→ K is another group homomorphism, then by the axioms of an in-

duction structure the composite HHn (α∗β∗X)an(β∗X,α)−−−−−−−→ HGn (β∗X)

an(X,β)−−−−−→HKn (X) agrees with an(X,β α) : HHn (α∗β∗X) = HHn ((β α)∗X)→ HGn (X)for a K-CW -complex X.

Consider a directed system of groups Gi | i ∈ I with G = colimi∈I Giand structure maps ψi : Gi → G for i ∈ I and φi,j : Gi → Gj for i, j ∈ I, i ≤ j.We obtain for every G-CW -complex X a system of Λ-modules HGin (ψ∗iX) |i ∈ I with structure maps an(ψ∗jX,φi,j) : HGin (ψ∗iX)→ HGjn (ψ∗jX). We geta map of Λ-modules(13.20)tGn (X,A) := colimi∈I an(X,ψi) : colimi∈I HGin (ψ∗i (X,A)) → HGn (X,A).

13.6 Inheritance Properties of the Fibered Meta-Isomorphism Conjecture 397

Definition 13.21 ((Strongly) continuous equivariant homology the-ory). An equivariant homology theory H?

∗ is called continuous if for everygroup G and every directed system of subgroups Gi | i ∈ I of G withG =

⋃i∈I Gi the Λ-map (see (13.20))

tGn (•) : colimi∈I HGin (•)→ HGn (•)

is an isomorphism for every n ∈ Z.An equivariant homology theory H?

∗ over Γ is called strongly continuousif for every group G and every directed system of groups Gi | i ∈ I withG = colimi∈I Gi and structure maps ψi : Gi → G for i ∈ I the Λ-map

tGn (•) : colimi∈I HGin (•)→ HGn (•)

is an isomorphism for every n ∈ Z.


Lemma 13.22. Consider a directed system of groups Gi | i ∈ I with G =colimi∈I Gi and structure maps ψi : Gi → G for i ∈ I. Let (X,A) be a G-CW -pair. Suppose that H?

∗ is strongly continuous.Then the Λ-homomorphism, see (13.20)

tGn (X,A) : colimi∈I HGin (ψ∗i (X,A))∼=−→ HGn (X,A)

is bijective for every n ∈ Z.

The proof of the next result is based on Lemma 13.22.

Lemma 13.23. Fix a class of groups C which is closed under isomorphisms,taking subgroups and taking quotients, e.g., the class of finite groups or theclass of virtually cyclic subgroups. For a group G let C(G) be the family ofsubgroups of G which belong to C. Let G be a group.

(i) Let G be the directed union of subgroups Gi | i ∈ I. Suppose that H?∗

is continuous and for every i ∈ I the Meta-Isomorphism Conjecture 13.2holds for Gi and C(Gi).Then the Meta-Isomorphism Conjecture 13.2 holds for G and C(G);

(ii) Let G be the directed union of subgroups Gi | i ∈ I. Suppose that H?∗ is

continuous and for every i ∈ I the assembly map appearing in the Meta-Isomorphism Conjecture 13.2 for Gi and C(Gi) is injective for all n ∈ Z.Then the assembly map appearing in the Meta-Isomorphism Conjecture 13.2for G and C(G) is injective for all n ∈ Z;

(iii) Let Gi | i ∈ I be a directed system of groups with G = colimi∈I Gi andstructure maps ψi : Gi → G. Suppose that H?

∗ is strongly continuous andfor every i ∈ I the Fibered Meta-Isomorphism Conjecture 13.8 holds forGi and C(Gi).Then the Fibered Meta-Isomorphism Conjecture 13.8 holds for G andC(G).


Proof. (i) is proved in [69, Proposition 3.4].

(ii) The proof of [69, Proposition 3.4] for isomorphism yields also a proof forthe injectivity version, since the colimit over a directed system is an exactfunctor and hence preserves injectivity.

(iii) See [65, Theorem 5.6]. ut

Remark 13.24 (Injectivity and the Transitivity principle). For col-imits over a directed system of subgroups we did get a statement aboutinjectivity in Lemma 13.23 (ii), essentially since the colimit over a directedsystem is an exact functor. We cannot prove such injectivity statement for as-sertion (iii) since its proof uses the Transitivity Principle 13.13 for which theinjectivity version is not true in general, essentially, because the Five-Lemmadoes not has a version for injectivity.

13.7 Actions on Trees

In this subsection we treat only equivariant homology theories H?∗ for sim-

plicity. The generalizations to an equivariant homology theory over a groupΓ are obvious.

Lemma 13.25. Suppose that G acts on a tree T by automorphisms of treeswithout inversion. Suppose that the Meta-Isomorphism Conjecture 13.2 holdsfor G with respect to FIN . Assume that for any isotropy group H of the G-action on T the Meta-Isomorphism Conjecture 13.2 holds for H with respectto FIN .

Then the projection T → • induces for all n ∈ Z an isomorphism

HGn (T )∼=−→ HGn (•).

Proof. Our assumptions on G imply that we can consider T as a G-CW -complex. The H-fixed point set TH is a non-empty subtree and hence con-tractible for every finite subgroup H ⊆ G (see [750, Theorem 15 in 6.1on page 58 and 6.3.1 on page 60]). Hence the projection EFIN (G) ×T → EFIN (G) is a G-homotopy equivalence. By assumption the projec-tion EFIN (G) → • induces for all n ∈ Z isomorphisms HGn (EFIN (G)) →HGn (•). Hence the projection EFIN (G) × T → • induces for all n ∈ Zisomorphisms HGn (EFIN (G)× T )→ HGn (•). By Lemma 13.10 and the as-sumptions on T the projection EFIN (G) × T → T induces for all n ∈ Zisomorphisms HGn (EFIN (G)× T )→ HGn (T ). Hence the projection T → •induces for all n ∈ Z isomorphisms HGn (T )→ HGn (•). ut

Example 13.26 (Amalgamated products). Let H?∗ be an equivariant ho-

mology theory with values in Λ-modules. Let G be the amalgamated productG1 ∗G0 G2 for a common subgroup G0 of the groups G1 and G2. Suppose

13.7 Actions on Trees 399

that Gi for i = 0, 1, 2 and G satisfy the Meta-Isomorphism Conjecture 13.2.with respect to the family FIN . Then there is a long exact sequence

· · · → HG0n (•)→ HG1

n (•)⊕HG1n (•)→ HGn (•)

→ HG0n−1(•)→ HG1

n−1(•)⊕HG1n−1(•)→ · · ·

Namely, there is a 1-dimensional G-CW -complex T whose underlying spaceis a tree such that the 1-skeleton is obtained from the 0-skeleton by theG-pushout

G/G0 × S0 q //

G/G1

∐G/G2

G/G0 ×D1 // T

where q is the disjoint union of the canonical projection G/G0 → G/G1 andG/G0 → G/G2, see [750, Theorem 7 in §4.1 on page 32]. Now the desiredlong exact sequence comes from the Mayer-Vietoris sequence associated withthe G-pushout above and Lemma 13.25.

Suppose that G0, G1, G2 and G satisfy the Baum-Connes Conjecture 12.9which is equivalent the Meta-Isomorphism Conjecture 13.2 if we choose F tobe FIN and HGn to be HG

n (−; KtopC ). Then we obtain a long exact sequence

(13.27)

· · · ∂n+1−−−→ Kn(C∗r (G0))Kn(C∗r (i1))⊕Kn(C∗r (i2))−−−−−−−−−−−−−−−−→ Kn(C∗r (G1))⊕Kn(C∗r (G2))

Kn(C∗r (j1))−Kn(C∗r (j2))−−−−−−−−−−−−−−−−→ Kn(C∗r (G))∂n−→ Kn−1(C∗r (G0))

Kn−1(C∗r (i1))⊕Kn−1(C∗r (i2))−−−−−−−−−−−−−−−−−−−→ Kn−1(C∗r (G2))⊕Kn−1(C∗r (G1))

Kn−1(C∗r (j1))−Kn−1(C∗r (j2))−−−−−−−−−−−−−−−−−−−→ Kn−1(C∗r (G))∂n−1−−−→ · · ·

where i1,i2, j1 and j2 are the obvious inclusions. Actually, such long exactMayer-Vietoris sequence exists always for a free amalgamated product G =G1 ∗G0

G2 (see Pimsner [663, Theorem 18 on page 632]).Suppose that G0, G1, G2 and G satisfy the K-theoretic Farrell Conjecture

Conjecture 11.1 with coefficients in the regular ring R with Q ⊆ R. Then weobtain using Theorem 11.39 (iv) a long exact sequence

(13.28) · · · ∂n+1−−−→ Kn(RG0)Kn(Ri1)⊕Kn(Ri2)−−−−−−−−−−−−→ Kn(RG1)⊕Kn(RG2)

Kn(Rj1)−Kn(Rj2)−−−−−−−−−−−−→ Kn(RG)∂n−→ Kn−1(RG0)

Kn−1(Ri1)⊕Kn−1(Ri2)−−−−−−−−−−−−−−−→ Kn−1(RG2)⊕Kn−1(RG1)

Kn−1(Rj1)−Kn−1(Rj2)−−−−−−−−−−−−−−−→ Kn−1(RG)∂n−1−−−→ · · ·


Without extra assumptions on R the long exact sequence above does notexist, certain Nil-terms enter, see Theorem 5.50.

Suppose that G0, G1, G2 and G satisfy the L-theoretic Farrell-Jones Con-jecture 11.1 with coefficients in the ring R with involution. Then we obtainusing Theorem 11.49 (i) a long exact sequence

(13.29) · · · ∂n+1−−−→ Ln(RG0)[1/2]

Ln(Ri1)[1/2]⊕Ln(Ri2)[1/2]−−−−−−−−−−−−−−−−−−→ Ln(RG1)[1/2]⊕ Ln(RG2)[1/2]

Ln(Rj1)[1/2]−Ln(Rj2)[1/2]−−−−−−−−−−−−−−−−−−→ Ln(RG)[1/2]∂n−→ Ln−1(RG0)[1/2]

Ln−1(Ri1)[1/2]⊕Ln−1(Ri2)[1/2]−−−−−−−−−−−−−−−−−−−−−→ Ln−1(RG2)[1/2]⊕ Ln−1(RG1)[1/2]

Ln−1(Rj1)[1/2]−Ln−1(Rj2)[1/2]−−−−−−−−−−−−−−−−−−−−−→ Ln−1(RG)[1/2]∂n−1−−−→ · · ·

Notice that the decoration of the L-groups does not play a role since we invert2. Actually, such long exact Mayer-Vietoris sequence exists always for a freeamalgamated product G = G1 ∗G0

G2 (see Cappell [153]). Without inverting2 the long exact sequence above does not exist, certain UNil-terms enter.

Exercise 13.30. Let H?∗ be an equivariant homology theory with values in

Λ-modules. Let φ : K → K be a group automorphism. Let G ×φ Z be theassociated semidirect product. Denote by i : G→ GoφZ the obvious inclusionSuppose that G and G ×φ Z satisfy the Meta-Isomorphism Conjecture 13.2with respect to the family FIN .

Prove the existence of a long exact sequence

· · · → HGn (•) φ∗−id−−−−→ HGn (•) k∗−→ HGoφZn (•)

→ HGn−1(•) φ∗−id−−−−→ HGn−1(•) k∗−→ · · ·

where φ∗ : HGn (•) → HGn (•) and k∗ come from the induction structureand the identification indφ• = • and the projection indi• → •.

Explain that this reduces in the case of the Baum-Connes Conjecture tothe long exact sequence

· · · → Kn(C∗r (G))Kn(C∗r (φ))−id−−−−−−−−−→ Kn(C∗r (G))

Kn(C∗r (k))−−−−−−−→ Kn(C∗r (Goφ Z))

→ Kn−1(C∗r (G))Kn−1(C∗r (φ))−id−−−−−−−−−−−→ Kn−1(C∗r (G))

Kn−1(C∗r (k))−−−−−−−−→ · · · .

and similar for the K-theoretic Farrell-Jones Conjecture for a regular ring Rwith Q ⊆ R and the L-theoretic Farrell-Jones Conjecture after inverting 2.

13.8 The Meta-Isomorphism Conjecture for Functors from Spaces to Spectra 401

13.8 The Meta-Isomorphism Conjecture for Functorsfrom Spaces to Spectra

Let S : SPACES→ SPECTRA be a covariant functor. Throughout this sectionwe will assume that it respects weak equivalences and disjoint unions, i.e., aweak homotopy equivalence of spaces f : X → Y is sent to a weak homotopyequivalence of spectra S(f) : S(X) → S(Y ) and for a collection of spacesXi | i ∈ I for an arbitrary index set I the canonical map∨

i∈IS(Xi)→ S

(∐i∈I

Xi

)is weak homotopy equivalence of spectra. We obtain a covariant functor

(13.31) SB : GROUPOIDS→ SPECTRA, G 7→ S(BG)

where BG is the classifying space of the category G which is the geometric re-alization of the simplicial set given by its nerve and denoted by BbarG in [222,page 227]. Denote by H?

∗(−; SB) the equivariant homology theory associatedto SB in Theorem 10.27. Let H?

n(−; SB) be the equivariant homology theoryin the sense of [526, Section 1] which is associated to SB by the constructionin [546, Proposition 175 on page 796]. It has the property that for any groupG and subgroup H ⊆ G we have canonical identifications

HGn (G/H; SB)

∼=−→ HHn (H/H; SB) ∼= S(BG).

Conjecture 13.32 (Meta-Isomorphism Conjecture for functors fromspaces to spectra). Let S : SPACES → SPECTRA be a covariant functorwhich respects weak equivalences and disjoint unions. The group G satisfiesthe Meta-Isomorphism Conjecture for S with respect to the family F ofsubgroups of G if it satisfies the Meta-Isomorphism Conjecture 13.2 for theG-homology theory HG∗ (−; SB), i.e., the assembly map

HGn (pr) : HGn (EF (G); SB)→ HGn (G/G; SB)


Example 13.33 (The Farrell-Jones Conjecture in the setting of func-tors from spaces to spectra). In the sequel Π(X) denotes the fundamentalgroupoid of a space X. If we take the covariant functor to be the one, which

sends a space X to KR(Π(X)), L〈−∞〉R (Π(X)) or Ktop

F (Π(X)) respectively,see Theorem 10.39, then the Meta-Isomorphism Conjecture 13.32 for S for agroup G and the family VCY, VCY or FIN respectively is equivalent to theK-theoretic Farrell-Jones Conjecture 11.1 with coefficients in the ring R, theL-theoretic Farrell-Jones Conjecture 11.4 with coefficients in the ring with


involution R, or the Baum-Connes Conjecture 12.9 respectively. This follows

from the obvious natural weak equivalence of groupoids G '−→ Π(BG).

Let G be a group and Z be a G-CW -complex. Define a covariant Or(G)-spectrum

(13.34) SGZ : Or(G)→ SPECTRA, G/H 7→ S(G/H ×G Z),

where G/H×GZ is the orbit space of the diagonal left G-action on G/H×S.

Notice that there is an obvious homeomorphism G/H ×G Z∼=−→ H\Z.

Conjecture 13.35 (Meta-Isomorphism Conjecture for functors fromspaces to spectra with coefficients). Let S : SPACES → SPECTRA bea covariant functor which respects weak equivalences and disjoint unions.The group G satisfies the Meta-Isomorphism Conjecture for S with coeffi-cients with respect to the family F of subgroups of G if for any free G-CW -complex Z the pair (G,F) satisfies the Meta-Isomorphism Conjecture 13.2for HG

∗ (−; SGZ ), i.e., the assembly map

HGn (EF (G); SGZ )→ HG

n (G/G; SGZ )


Exercise 13.36. Let S : SPACES→ SPECTRA be a covariant functor whichrespects weak equivalences and disjoint unions. Suppose that it satisfies theMeta-Isomorphism Conjecture 13.35 for every group G and the trivial familyT R consisting of one element, the trivial subgroup. Show that then we obtainfor every connected CW -complex X a weak homotopy equivalence

Eπ1(X)+ ∧π1(X) S(X)→ S(X).

Show that πn(Eπ1(X)+ ∧π1(X) S(X)

)and πn

(Bπ1(X)+ ∧ S(•)

)are not

isomorphic in general, but that they are isomorphic if X is contractible or S isof the shape Y 7→ T(Π(Y )) for some covariant functor T : GROUPOIDS→SPECTRA.

Example 13.37 (Z = EG). If we take Z = EG in Conjecture 13.35, thenConjecture 13.35 reduces to Conjecture 13.32 since there is a natural ho-

motopy equivalence G/H ×G EG'−→ BGG(G/H) and hence we get a weak

homotopy equivalence of Or(G)-spectra SGEG'−→ SB(GG(G/?)).

Remark 13.38 (Relation to the original formulation). In [295, Sec-tion 1.7 on page 262] Farrell and Jones formulate a fibered version of theirconjectures for a covariant functor S : SPACES→ SPECTRA for every (Serre)fibration ξ : Y → X over a connected CW-complex X. In our set-up this cor-responds to choosing Z to be the total space of the fibration obtained from


Y → X by pulling back along the universal covering X → X. This space Z isa free G-CW for G = π1(B). Note that an arbitrary free G-CW -complex Zcan always be obtained in this fashion from the fiber bundle EG×GZ → BGup to G-homotopy, compare [295, Corollary 2.2.1 on page 263].

We sketch the proof of this identification. Let A be a G-CW -complex.Let f : E(X) → X be the map obtained by taking the quotient of the G =π1(B)-action on the G-map A × X → X given by the projection. Denoteby p : E(ξ) → E(X) the pullback of ξ with f . Let q : E(ξ) → A/G be thecomposite of p with the map E(X) → A/G induced by the projection A ×X → A. This is a stratified fibration and one can consider the spectrumH(A/G;S(q)) in the sense of Quinn [672, Section 8]. Put

HGn (A; ξ) := πn(H(A/G;S(q)).

The projection pr : A→ G/G induces maps

(13.39) an(A) : HGn (A; ξ)→ HGn (G/G; ξ) = πn(S(Y )),

which is the assembly map in [295, Section 1.7 on page 262] if we take A =EVCY(G). The construction of HGn (A; ξ) := H(A/G;S(q) is very complicated,but, fortunately, for us only two facts are relevant. We obtain by HG∗ (−; ξ)a G-homology theory in the sense of Definition 10.1 and for every H ⊆ Gwe get a natural identification HGn (G/H; ξ) = SGZ (G/H). Hence the functorG-CW-COMPLEXES → SPECTRA sending A → H(A/G;S(q)) is weaklyexcisive and its restriction to Or(G) is the functor SGZ . Corollary 16.13 impliesthat the map (13.39) can be identified with the map induced by the projectionA→ G/G

HGn (A; SGZ )→ HG

n (G/G; SGZ ) = πn(S(Z/G)) = πn(S(Y )).

which appears in Meta-Isomorphism Conjecture 13.35 for functors fromspaces to spectra with coefficients.

Remark 13.40 (The condition free is necessary in Conjecture 13.35).In general Conjecture 13.35 is not true if we drop the condition that Z is free.Take for instance Z = G/G. Then Conjecture 13.35 predicts that the projec-tion EF (G)/G→ G/G induces for all n ∈ Z an isomorphism

Hn(pr; S(•)) : Hn(EF (G)/G; S(•))→ Hn(•,S(•))

where H∗(−; S(•)) is the (non-equivariant) homology theory associated tothe spectrum S(•). This statement is in general wrong, except in extremecases such as F = ALL.

The proof of the next theorem will be given at the end of Section 13.9.

Theorem 13.41 (Inheritance properties of the Meta-IsomorphismConjecture 13.35 for functors from spaces to spectra with coeffi-


cients). Let S : SPACES→ SPECTRA be a covariant functor which respectsweak equivalences and disjoint unions. Fix a class of groups C which is closedunder isomorphisms, taking subgroups and taking quotients.

(i) Suppose that the Meta-Isomorphism Conjecture 13.35 for functors fromspaces to spectra with coefficients holds for the group G and the family ofsubgroups C(G) := K ⊆ G,K ∈ C of G. Let H ⊆ G be a subgroup.Then Conjecture 13.35 holds for (H, C(H));

(ii) Let 1 → K → Gp−→ Q → 1 be an extension of groups. Suppose that

(Q, C(Q)) and (p−1(H), C(p−1(H)) for every H ∈ C(Q) satisfy Conjec-ture 13.35.Then (G, C(G)) satisfies Conjecture 13.35;

(iii) Suppose that Conjecture 13.35 is true for (H1×H2, C(H1×H2)) for everyH1, H2 ∈ C.Then for two groups G1 and G2 Conjecture 13.35 is true for the directproduct G1 × G2 and the family C(G1 × G2), if and only it is true for(Gk, C(Gk)) for k = 1, 2;

(iv) Suppose that for any directed systems of spaces Xi | i ∈ I indexed overan arbitrary directed set I the canonical map

hocolimi∈I S(Xi)→ S(hocolimi∈I Xi

)is a weak homotopy equivalence. Let Gi | i ∈ I be a directed systemof groups over a directed set I (with arbitrary structure maps). Put G =colimi∈I Gi. Suppose that Conjecture 13.35 holds for (Gi, C(Gi)) for everyi ∈ I;Then Conjecture 13.35 holds for (G, C(G)).

Exercise 13.42. Let C be the class of finite groups or let C be the class ofvirtually cyclic subgroups. Suppose that Conjecture 13.35 holds for (H, C(H))if H contains a subgroup K of finite index such that K is a finite product offinitely generated free groups.

Show that for a collection of groups Gi | i ∈ I Conjecture 13.35 is truefor the free product ∗i∈IGi and the family C(∗i∈IGi), if and only it is truefor (Gi, C(Gi)) for every i ∈ I.

Lemma 13.43. Suppose that there is a spectrum E such that S : SPACES→SPECTRA is given by Y 7→ Y+ ∧E.

(i) Then for any group G, any G-CW -complex X, which is contractible (afterforgetting the G-action), and any free G-CW -complex Z the projectionX → G/G induces for all n ∈ Z an isomorphism

HGn (X; SGZ )

∼=−→ HGn (G/G; SGZ );

(ii) Both Conjecture 13.32 and Conjecture 13.35 for S hold for every group Gand every family F of subgroups of G.


Proof. (i) There are natural isomorphisms of spectra

mapG(G/?, X)+ ∧Or(G)

((G/?×G Z)+ ∧E

)∼=−→((mapG(G/?), X)×Or(G) G/?)×G Z

)+∧E

∼=−→ (X ×G Z)+ ∧E,

where the second isomorphism comes from the G-homeomorphism

mapG(G/?), X)×Or(G) G/?∼=−→ X

of [222, Theorem 7.4 (1)]. Since Z is a free G-CW -complex and X is con-tractible (after forgetting the group action), the projection X ×G Z →G/G ×G Z is a homotopy equivalence and hence induces a weak homotopyequivalence

(X ×G Z)+ ∧E'−→ (G/G×G Z)+ ∧E,

Thus we get a weak homotopy equivalence

mapG(G/?), X)+ ∧Or(G)

((G/?×G Z)+ ∧E

)→ (G/G×G Z)+ ∧E.

Under the identifications coming from the definitions

HGn (X; SGZ ) := πn

(mapG(G/?), X)+ ∧Or(G)

((G/?×G Z)+ ∧E

)),

HGn (G/G; SGZ ) = πn ((G/G×G Z)+ ∧E) ,

this weak homotopy equivalence induces on homotopy groups the isomor-phism HG

n (X; SGZ )→ HGn (G/G; SGZ ).

(ii) This follows from assertion (i). ut

Lemma 13.44. Let S,T,U : SPACES → SPECTRA be covariant functorswhich respects weak equivalences and disjoint unions. Let i : S → T andp : T → U be natural transformations such that for any space Y the map

of spectra S(Y )i(Y )−−−→ T(Y )

p(Y )−−−→ U(Y ) is up to weak homotopy equivalencea cofibration of spectra.

(i) Then we obtain for every group G and all G-CW -complexes X and Z anatural long exact sequence

· · · → HGn (X; SGZ )→ HG

n (X; TGZ )→ HG

n (X; UGZ )

→ HGn−1(X; SGZ )→ HG

n−1(X; TGZ )→ HG

n−1(X; UGZ )→ · · · ;

(ii) Let G be a group and F be a family of subgroups of G. Then Conjec-ture 13.32 or Conjecture 13.35 respectively holds for all three functors S,T and U for (G,F) if Conjecture 13.32 or Conjecture 13.35 respectivelyholds for two of the functors S, T and U for (G,F).


Proof. (i) This is a consequence of the fact following from the version for spec-tra of [222, Theorem 3.11] that we obtain an up to weak homotopy equivalencecofiber sequence of spectra

mapG(G/?, X)+∧Or(G)S(G/?×GZ)→ mapG(G/?, X)+∧Or(G)T(G/?×GZ)

→ mapG(G/?, X)+ ∧Or(G) U(G/?×G Z).

(ii) This follows from assertion (i) and the Five-Lemma. ut

13.9 Proof of the Inheritance Properties

This section is entirely devoted to the proof of Theorem 13.41.Let S : SPACES → SPECTRA be a covariant functor. Throughout this

section we will assume that it respects weak equivalences and disjoint unions.

Lemma 13.45. Let ψ : K1 → K2 be a group homomorphism.

(i) If Z is a K1-CW -complex and X is a K2-CW -complex, then there is anatural isomorphism

HK1n (ψ∗X; SK1

Z )∼=−→ HK2

n (X; SK2

ψ∗Z);

(ii) If Z is a K2-CW -complex and X is a K1-CW -complex, then there is anatural isomorphism

HK1n (X; SK1

ψ∗Z)∼=−→ HK2

n (ψ∗X; SK2

Z );

Proof. (i) The fourth isomorphisms appearing in [222, Lemma 1.9] impliesthat it suffices to construct a natural weak homotopy equivalence of Or(K2)-spectra

u(ψ,Z) : ψ∗SK1

Z

∼=−→ SK2

ψ∗Z,

where ψ∗SK1

Z is the Or(K2)-spectrum obtained by induction in the senseof [222, Definition 1.8] with the functor Or(ψ) : Or(K1)→ Or(K2), K1/H 7→ψ∗(K1/H) applied to the Or(K1)-spectrum SK1

Z . For a homogeneous spaceK2/H we define u(ψ,Z)(K2/H) to be the composite

ψ∗SK1

Z (K2/H) := mapK2(ψ∗(K1/?),K2/H)×Or(K1) S (K1/?×K1

Z)∼=−→ mapK1

(K1/?), ψ∗(K2/H))×Or(K1) S(K1/?×K1Z)

∼=−→ S(ψ∗(K2/H)×K1Z)

∼=−→ S(K2/H ×K2ψ∗Z)

=: SK2

ψ∗Z(K2/H).

13.9 Proof of the Inheritance Properties 407

Here the first map comes from the adjunction isomorphism

mapK2(ψ∗(K1/?),K2/H)

∼=−→ mapK1(K1/?), ψ∗(K2/H)),

and the third map comes from the canonical homeomorphism

ψ∗(K2/H)×K1Z∼=−→ K2/H ×K2

ψ∗Z.

The second map is the special case T = ψ∗K2/? of the natural weak homotopyequivalence defined for any K1-set T

κ(T ) : mapK1(K1/?), T )×Or(K1) S (K1/?×K1 Z)

∼=−→ S(T ×K1 Z),

which is given by (u : K1/? → T ) × s 7→ S(u ×K1 idZ)(s). If T is a homoge-neous K1-set, then κ(T ) is an isomorphism by the Yoneda Lemma. Since ψis compatible with disjoint unions, S is compatible with disjoint unions up toweak homotopy equivalence by assumption, and every K1-set is the disjointunion of homogeneous K1-set, κ(T ) is a weak homotopy equivalence for allK1-sets T .

(ii) The third isomorphisms appearing in [222, Lemma 1.9] implies that itsuffices to construct a natural weak homotopy equivalence of Or(K1)-spectra

v(ψ,Z) : ψ∗SK2

Z

∼=−→ SK1

ψ∗Z ,

where ψ∗SK2

Z is the Or(K1)-spectrum obtained by restriction in the senseof [222, Definition 1.8] with the functor Or(ψ) : Or(K1)→ Or(K2), K1/H 7→ψ∗(K1/H) applied to the Or(K2)-spectrum SK2

Z . Actually, we obtain even anisomorphism v(ψ,Z) using the adjunction

ψ∗(K1/H)×K2Z ∼= K1/H ×K1

ψ∗Z

for any subgroup H ⊆ K1. ut

Notice that for a homomorphism φ : H → G the restriction φ∗Z of a freeG-CW -complex Z is free again if and only if φ is injective. We have alreadyexplained in Remark 13.40 that the assumption that Z is free is needed inConjecture 13.35. In the Fibered Meta-Isomorphism Conjecture 13.8 it iscrucial not to require that φ : H → G is injective since we want to havegood inheritance properties such as the one appearing in assertion (iii) ofLemma 13.23 which will be crucial for the proof of assertion (iv) of Theo-rem 13.41. Therefore we are forced to introduce the following construction.

Consider a group G and a G-CW -complex Z. We want to define an equi-variant homology theory H?

∗(−; S↓GZ ) over G in the sense of Definition 10.82.Given a homomorphism φ : K → G define the associated K-homology theory

HK,φ∗ (−; S↓GZ ) := HK

∗ (−; SKEK×φ∗Z).


Given group homomorphisms ψ : K1 → K2, φ1 : K1 → G and φ2 : K2 → Gwith φ2 ψ = φ1, a K1-CW -complex X, and n ∈ Z, we have to define anatural map

HK1n (X; SK1

EK1×φ∗1Z)→ HK2

n (ψ∗X; SK2

EK2×φ∗2Z).

We get the isomorphism HK2n (ψ∗X; SK2

EK2×φ∗2Z) = HK1

n (X; SK2

ψ∗(EK2×φ∗2Z))

from Lemma 13.45 (ii). Hence it suffices to specify a K1-map

EK1 × φ∗1Z → ψ∗(EK2 × φ∗2Z) = ψ∗(EK2)× φ∗1Z.

The homomorphism ψ : K1 → K2 induces a K1-map EK1 → ψ∗(EK2) andwe can take its product with idφ∗1Z .

The proof of the next Lemma is left to the reader.

Lemma 13.46. Given a group G and a G-CW -complex Z, all the axioms ofan equivariant homology theory over G, see Definition 10.82, are satisfied byH?∗(−; S↓GZ ).

Exercise 13.47. Let G be a group and Z be a G-CW -complex. Considerthe functor

E : GROUPOIDS ↓ G→ SPECTRA, p : G → I(G) 7→ S(EG ×G p∗Z

).

Here EG is the classifying G-CW -complex associated to G, see [222, Defini-tion 3.8], for which we use the functorial model EbarG of [222, page 230], weconsider Z as a I(G)-CW -complex and hence get a G-CW -complex p∗Z byrestriction with p : G → I(G), and the space EG ×G p∗Z is defined in [222,Definition 1.4].

Show that the equivariant homology theory H?∗(−; E) over G associated

to E in Theorem 10.84 is isomorphic to H?∗(−; S↓GZ ).

Lemma 13.48. Let φ : H → K and ψ : K → G be a group homomorphisms.

(i) Let X be a G-CW -complex and let Z be a K-CW -complex. Then we obtaina natural isomorphism

HH,φn (φ∗ψ∗X; S↓KZ )

∼=−→ HGn (X; SG(ψφ)∗(EH×φ∗Z));

(ii) Let X be a H-CW -complex and let Z be a G-CW -complex. Then we obtaina natural isomorphism

HH,φn (X; S↓Kψ∗Z)

∼=−→ HH,ψφn

(X; S↓GZ

).

Proof. (i) We have by definition

HH,φn (φ∗ψ∗X; S↓KZ ) := HH

n (φ∗ψ∗X; SHEH×φ∗Z).


Now apply Lemma 13.45 (i).

(ii) We get by definition

HH,φn (X; S↓Kψ∗Z) := HH

n (X; SHEH×φ∗ψ∗Z)

= HHn (X; SHEH×(ψφ)∗Z) =: HH,ψφ

n

(X; S↓GZ

).

ut

Conjecture 13.49 (Fibered Meta-Isomorphism Conjecture for a func-tor from spaces to spectra with coefficients). We say that S satisfiesthe Fibered Meta-Isomorphism Conjecture for a functor from spaces to spec-tra with coefficients for the group G and the family of subgroups F of Gif for any G-CW -complex Z the equivariant homology theory H?

∗(−; S↓G)over G satisfies the Fibered Meta-Isomorphism Conjecture 13.8 for the group(G, idG) over G and the family F .

Notice that Conjecture 13.35 is a statement about SG, whereas Conjec-ture 13.49 is a statement about S↓G.

Lemma 13.50. Let ψ : K → G be a group homomorphism.

(i) Suppose that the Meta Conjecture 13.35 with coefficients holds for thegroup G and the family F . Then the Fibered Meta Conjecture 13.49 withcoefficients holds for the group K and the family ψ∗F ;

(ii) If the Fibered Meta Conjecture 13.49 with coefficients holds for the groupG and the family F , then the Meta Conjecture 13.35 with coefficients holdsfor the group G and the family F ;

(iii) Suppose that the Fibered Meta Conjecture 13.49 with coefficients holds Kand the family F . Then for every G-CW -complex Z the Fibered Meta-Isomorphism Conjecture 13.8 holds for the equivariant homology theoryHn(−; S↓GZ ) over G for the group (K,ψ) over G and the family F of sub-groups of K.

Proof. (i) This follows from Lemma 13.48 (i) since in the notation usedthere we have φ∗ψ∗EF (G) = φ∗Eψ∗F (K) and φ∗ψ∗G/G = H/H, and (ψ φ)∗(EH × φ∗Z) is a free G-CW -complex.

(ii) This follows from the fact that for a free G-CW -complex Z the projectionEG × Z → Z is a G-homotopy equivalence and hence we get a naturalisomorphism

HG,idGn (X; S↓GZ ) := HG

n (X; SGEG×Z)∼=−→ HG

n (X; SGZ )

for every G-CW -complex X and n ∈ Z.

(iii) This follows from Lemma 13.48 (ii). ut


Lemma 13.51. Suppose that for any directed systems of spaces Xi | i ∈ Iindexed over an arbitrary directed set I the canonical map

hocolimi∈I S(Xi)→ S(hocolimi∈I Xi

)is a weak homotopy equivalence.

Then for every group G and G-CW -complex Z the equivariant homologytheory over G given by H?

∗(−S↓GZ ) is strongly continuous.

Proof. We only treat the case idG : G→ G, the case of a group ψ : K → G overG is completely analogous. Consider a directed system of groups Gi | i ∈ Iwith G = colimi∈I Gi. Let ψi : Gi → G be the structure map for i ∈ I.

The canonical map

(13.52) hocolimi∈I S(EGi ×Gi ψ∗i Z)→ S(hocolimi∈I(EGi ×Gi ψ∗i Z)

)is by assumption a weak homotopy equivalence. We have the homeomor-phisms

EGi ×Gi ψ∗i Z∼=−→ (ψi)∗EGi ×G Z;(

hocolimi∈I(ψi)∗EGi)×G Z

∼=−→ hocolimi∈I((ψi)∗EGi ×G Z

).

They induce a homeomorphism

(13.53) S(hocolimi∈I(EGi ×Gi ψ∗i Z)) ∼=−→ S

((hocolimi∈I(ψi)∗EGi)×G Z

)The canonical map

hocolimi∈I(ψi)∗EGi → EG

is a G-homotopy equivalence. The proof of this fact is a special case of theargument appearing in the proof of [562, Theorem 4.3 on page 516]. It inducesa weak homotopy equivalence

(13.54) S((hocolimi∈I(ψi)∗EGi)×G Z)→ S(EG×G Z).

Hence we get by taking the composite of the maps (13.52), (13.53) and (13.54)a weak homotopy equivalence

hocolimi∈I S(EGi ×Gi ψ∗i Z)→ S(EG×G Z).

It induces after taking homotopy groups for every n ∈ Z an isomorphism

colimi∈I πn(S(EGi ×Gi ψ∗i Z)

)→ πn

(S(EG×G Z)

)which is by definition the same as the canoncial map

colimi∈I HGi,ψin (Gi/Gi; S

↓GZ )→ HG,idG

n (G/G; S↓GZ ).


This finishes the proof of Lemma 13.51. ut

Proof of Theorem 13.41. (i) Consider a free H-CW -complex Z. Let i : H →G be the inclusion. Then i∗Z is a free G-CW -complex, i∗EC(G)(G) is a modelfor EC(H)(H) and i∗G/G = K/K. From Lemma 13.45 (i), we obtain a com-mutative diagram with isomorphisms as vertical maps

HHn (EC(H)(H); SHZ ) //

∼=

HHn (H/H; SGZ )

∼=

HGn (EC(G)(G); SGi∗Z) // HG

n (G/G; SGi∗Z)

where the horizontal maps are induced by the projections. The lower map isbijective by assumption. Hence the upper map is bijective as well.

(ii) Since (Q, C(Q)) and (p−1(H), C(p−1(H))) for every H ∈ C(Q) satisfy theMeta-Isomorphism Conjecture Conjecture 13.35 with coefficients by assump-tion, we conclude from Lemma 13.50 (i) that the Fibered Meta-IsomorphismConjecture 13.49 with coefficients holds for the group G and the familyp∗C(Q) and that for every H ∈ C(Q) the Fibered Meta-Isomorphism Con-jecture 13.49 with coefficients holds for p−1(H) and the family C(p−1(H)) =C(G)|p−1(H). Lemma 13.50 (iii) implies that for every H ∈ C(Q) and G-CW -complex Z the Fibered Meta-Isomorphism Conjecture 13.8 holds for the equi-variant homology theory H?

n(−; S↓GZ ) over G for the group (p−1(H) ⊆ G)over G and the family C(G)|p−1(H). Since for every L ∈ p∗C(Q) we havep(L) ∈ C(Q) and hence L ⊆ p−1(p(L)), we conclude from Lemma 13.16 thatfor every L ∈ p∗C(Q) and G-CW -complex Z the Fibered Meta-Isomorphism

Conjecture 13.8 holds for the equivariant homology theory H?n(−; S↓GZ ) over

G for the group (L ⊆ G) over G and the family C(G)|L. The Transitiv-ity Principle 13.13 (i) implies that for every G-CW -complex Z the FiberedMeta-Isomorphism Conjecture 13.8 holds for the equivariant homology the-ory H?

n(−; S↓GZ ) over G for the group (G, idG) over G and the family C(G),in other words, the Fibered Meta-Isomorphism Conjecture 13.49 with coeffi-cients holds for G and the family C(G). We conclude from Lemma 13.50 (ii)that the Meta-Isomorphism Conjecture 13.35 holds for the group G and thefamily C(G).

(iii) If the Meta-Isomorphism Conjecture 13.35 with coefficients holds for(G1×G1, C(G1×G2)), it holds for Gk and the family C(Gk) = C(G1×G2)|Gkfor k = 1, 2 by assertion (i).

Suppose that the Meta-Isomorphism Conjecture 13.35 with coefficientsholds for (Gk, C(Gk)) for k = 1, 2. By assertion (ii) applied to the obviousexact sequence 1 → H2 → G1 × H2 → G1 → 1, Conjecture 13.35 holds for(G1×H2, C(G1×H2)) for every H2 ∈ C(G2). By assertion (ii) applied to theobvious exact sequence 1→ G1 → G1×G2 → G2 → 1 Conjecture 13.35 withcoefficients holds for (G1 ×G2, C(G1 ×G2)).


(iv) Since the Meta-Isomorphisms Conjecture Conjecture 13.35 holds for Giand C(Gi) for every i ∈ I by assumption, we conclude from Lemma 13.50 (i)that the Fibered Meta-Isomorphism Conjecture 13.49 with coefficients holdsfor the group Gi and the family C(Gi) for every i ∈ I. Lemma 13.50 (iii)implies that for every i ∈ I and G-CW -complex Z the Fibered Meta-Isomorphism Conjecture 13.8 holds for the equivariant homology theoryHn(−; S↓GZ ) over G for the group ψi : Gi → G over G and the family C(Gi).We conclude from Lemma 13.23 (iii) and Lemma 13.51 that for every G-CW -complex Z the Fibered Meta-Isomorphism Conjecture 13.8 holds forthe equivariant homology theoryH?

∗(−; S↓GZ ) over G for the group (G, idG)over G and the family C(G), in other words, the Fibered Meta-IsomorphismConjecture 13.49 with coefficients holds for the group G and the family C(G).We conclude from Lemma 13.50 (ii) that the Meta-Isomorphism ConjectureConjecture 13.35 with coefficients holds for the group G and the family C(G).This finishes the proof of Theorem 13.41. ut

13.10 The Farrell-Jones Conjecture for A-Theory andPseudoisotopy

Conjecture 13.55 (Farrell-Jones Conjecture for A-theory (with co-efficients)). A group G satisfies the Farrell-Jones Conjecture for A-theoryif the Meta-Isomorphism Conjecture 13.32 for functors from spaces to spec-tra applied to the case S = A for the functor non-connective A-theory Aintroduced in (6.10) holds for (G,VCY).

A group G satisfies the Farrell-Jones Conjecture for A-theory with coef-ficients if the Meta-Isomorphism Conjecture 13.35 for functors from spacesto spectra with coefficients applied to the case S = A for the functor non-connective A-theory A introduced in (6.10) holds for (G,VCY).

Notice that A respects weak equivalences and disjoint unions, see Theo-rem 6.14.

Exercise 13.56. Suppose that G is torsionfree and satisfies the Farrell-JonesConjecture 13.55 for A-theory. Show that πn(A(BG)) = 0 for n ≤ −1 andπ0(A(BG)) ∼= Z.

Conjecture 13.57 (Farrell-Jones Conjecture for (smooth) pseudoiso-topy (with coefficients)). A group G satisfies the Farrell-Jones Conjec-ture for (smooth) pseudoisotopy if the Meta-Isomorphism Conjecture 13.32for functors from spaces to spectra applied to the case S = P or S = PDiff

for the functor non-connective (smooth) pseudoisotopy P and PDiff of Defi-nition 6.1 holds for (G,VCY).

13.11 The Farrell-Jones Conjecture for Topological Hochschild and Cyclic Homology413

A group G satisfies the Farrell-Jones Conjecture for (smooth) pseudoiso-topy with coefficients if the Meta-Isomorphism Conjecture 13.35 for func-tors from spaces to spectra with coefficients applied to the case S = P orS = PDiff(X) for the functor non-connective (smooth) pseudoisotopy P andPDiff of Definition 6.1 holds for (G,VCY).

Notice that P and PDiff respect weak equivalences and disjoint unions, seeTheorem 6.2.

Question 13.58 (Reducing VCY to VCYI for A-theory). Does it makesense to replace in Conjecture 13.55 VCY by VCYI?

Notice that for algebraic K-theory this replacement does make sense byTheorem 11.35. Question 13.58 is also interesting for pseudoisotopy.

Question 13.59 (Splitting of the relative assembly maps for FIN ⊆VCY for A-theory). Is there an analogue for A-theory of the splittingof the relative assembly maps for EFIN (G) and EVCY(G) in K-theory, seeTheorem 11.26?

The construction of such a splitting in [559, Theorem 0.1] for K-theory isbased on a twisted Bass-Heller-Swan Theorem for the algebraic K-theory ofadditive categories. For connective A-theory a Bass-Heller-Swan decomposi-tion is established and analyzed in [409, 410]. Question 13.59 makes sense forpseudoisotopy. Bass-Heller-Swan decompositions for connective pseudoiso-topy are treated in [406, 472]. Comment 48: Check reference and makeit more precise.

13.11 The Farrell-Jones Conjecture for TopologicalHochschild and Cyclic Homology

There are the notions of Hochschild homology and cyclic homology of algebraswhich are defined in the algebraic setting, see for instance Connes [192] orLoday [516]. One of the important insights of Waldhausen was that one candefine an analogue of algebraic K-theory for rings, where one “spacifies” theconstructions. This led to A-theory which we have described in Chapter 6.These circle of ideas motivated also the definition of topological Hochschildhomology by Bokstedt and then of topological cyclic homology by Bokstedt-Hsiang-Madsen [118] which are better approximations of the algebraic K-theory than their original algebraic counterparts. A systematic study howmuch algebraic cyclic homology detects from algebraic K-theory of grouprings is presented in [547] showing that the topological versions are muchmore effective. Roughly speaking, in the topological versions one replacesrings by ring spectra and tensor products by (highly structured and strictlycommutative) smash products. The role of the ring Z of integers, which is


initial in the category of rings, is now played by the sphere spectrum S, whichis initial in the category of ring spectra. We refer for further information tothe book by Dundas-Goodwillie-McCarthy [255] and the survey article byMadsen [567].

Given a symmetric ring spectrum A and a prime p, one can define functorssee [549, (14.1) and Example 14.3]

THHA : GROUPOIDS → SPECTRA;(13.60)

TCA;p : GROUPOIDS → SPECTRA,(13.61)

such that for a group G considered as groupoid I(G) the value of these func-tors is the topological Hochschild homology and the topological cyclic homol-ogy with respect to the prime p of the group ring spectrum A[G] := A∧G+.From Theorem 10.27 we obtain equivariant homology theoriesH?

∗(−; THHA)and H?

∗(−; TCA;p) satisfying for any group G and subgroup H ⊆ G

HGn (G/H; THHA) = HHn (H/H; THHA) = πn(THH(A[H])

);

HGn (G/H; TCA;p) = HHn (H/H; TCA;p) = πn(TC(A[H]; p)

).

13.11.1 Topological Hochschild Homology

The following theorem is taken from [549, Theorem 1.19].

Theorem 13.62 (The Farrell-Jones Conjecture holds for topologicalHochschild homology). Let G be a group and F be a family of subgroups.Then the map induced by the projection pr: EF (G)→ G/G

HGn (EF (G); THHA)→ HG

n (G/G; THHA) = πn(THH(A[G])

)is split injective for all n ∈ Z. If F contains all cyclic subgroups, then it isbijective for all n ∈ Z.

Topological Hochschild homology is one of the rare instances where anIsomorphism Conjecture is known for all groups and an interesting family ofsubgroups, namely the family of all cyclic subgroups, but the reasons are notcompletely elementary.

13.11.2 Topological Cyclic Homology

The following result is a special case of [548, Theorem 1.5], where othersymmetric spectra than the sphere spectrum are allowed.

Theorem 13.63. Let G be a group and p be a prime. Assume that there isa G-CW -model for EFIN (G) of finite type.

13.12 The Farrell-Jones Conjecture for Homotopy K-Theory 415

Then the map induced by the projection pr: EFIN (G)→ G/G

HGn (EFIN (G); TCS.p)→ HG

n (G/G; TCS;p) = πn(TC(S[G]; p)

)is split injective for all n ∈ Z.

Proof. See ut

One of the reasons why topological cyclic homology is much harder thantopological Hochschild homology is that in the construction of topologicalcyclic homology a homotopy inverse limits occurs and taking smash productdoes not commute with homotopy inverse limits in general, see [550]. This isalso one reason why we are hesitant to formulate an Isomorphism Conjecturefor topological cyclic homology.

Question 13.64 (Isomorphism Conjecture for topological cyclic ho-mology). For which groups and which symmetric ring spectra A is the mapinduced by the projection pr : EFIN (G)→ G/G

HGn (EFIN (G); TCA.p)→ HG

n (G/G; TCA;p) = πn(TC(A[G]; p)

)(rationally) split injective for all n ∈ Z?

For which groups and which symmetric ring spectra A is the map inducedby the projection pr : ECYC(G)→ G/G

HGn (ECYC(G); TCS.p)→ HG

n (G/G; TCS;p) = πn(TC(S[G]; p)

)(rationally) bijective for all n ∈ Z?

13.12 The Farrell-Jones Conjecture for HomotopyK-Theory

Let E : ADD-CAT → SPECTRA be a (covariant) functor from the categoryADD-CAT of small additive categories. In [558, Definition 8.1] its homotopystabilization is constructed which consists of a covariant functor

HE : ADD-CAT → SPECTRA

together with a natural transformation

h : E→ HE

We call E homotopy stable if h(A) is an equivalence for any object A inADD-CAT .

This construction has the following basic properties. Given an automor-phism Φ : A → A, let AΦ[t] be the additive category of twisted polynomials


with coefficients in A, see [560, Definition 1.2]. Let ev+0 : AΦ[t] → A be the

functor of additive categories given by taking t = 0 and let i+ : A → AΦ[t]be the obvious inclusion see [560, (1.10) and (1.12)]. The following result isproved in [558, Lemma 8.2].

Lemma 13.65. Let E : ADD-CAT → SPECTRA be a covariant functor.

(i) HE is homotopy stable;(ii) Suppose that E is homotopy stable. Let A be any additive category with an

automorphism Φ : A∼=−→ A. Then the maps

E(ev+0 ) : E(AΦ[t])

'−→ E(A);

E(i+) : E(A)'−→ E(AΦ[t]),


Remark 13.66 (Universal property of H). Notice that Lemma 13.65 (i)says that up to weak homotopy equivalence the transformation h : E→ HEis universal (from the left) among transformations f : E → F to homotopystable functors F : ADD-CAT → SPECTRA since we obtain a commutativesquare whose lower vertical arrow is a weak homotopy equivalence

Eh //

f

HE

Hf

Fh

' // HF

Lemma 13.65 (ii) essentially says that homotopy stable automatically im-plies homotopy stable in the twisted case.

Definition 13.67 (Homotopy K-theory). Let K : ADD-CAT → SPECTRAbe the covariant functor which sends an additive category to its non-connective K-theory spectrum, see for instance [159, 558, 653]. Define thehomotopy K-theory functors

HK : ADD-CAT → SPECTRA

to be the homotopy stabilization of K.


Theorem 13.68 (Bass-Heller-Swan decomposition for homotopy K-

theory). Let A be an additive category with an automorphism Φ : A∼=−→ A.

Then we get for all n ∈ Z a weak homotopy equivalence

a : TK(Φ−1)'−→ HK(AΦ[t, t−1]),

where TK(Φ−1) is the mapping torus of the self-map K(Φ−1) : K(A)→ K(A).

13.12 The Farrell-Jones Conjecture for Homotopy K-Theory 417

Remark 13.69 (Identification with Weibel’s definition). Weibel hasdefined a version of homotopy K-theory for a ring R by a simplicial con-struction in [824]. It is not hard to check using Remark 13.66, which appliesalso to the constructions of [824] instead of H, that πi(HK(R)) can be iden-tified with the one in [824], if R is a skeleton of the category of finitelygenerated free R-modules.

Conjecture 13.70 (Farrell-Jones Conjecture for homotopy K-theorywith coefficients in additive G-categories). We say that G satisfies theFarrell-Jones Conjecture with coefficients for homotopy K-theory in additiveG-categories if for every additive G-category A and every n ∈ Z the assemblymap given by the projection pr : EFIN (G)→ G/G

HGn (EFIN (G); HKA)→ HG

n (G/G; HKA) = πn(HKA(I(G))

)is bijective, where HKA : GROUPOIDS ↓ G → SPECTRA is analogouslydefined as the functor appearing in (11.8) but with K replaced by HK.

The version of Conjecture 13.70 has been treated for rings in [69].Notice that the decisive difference between the K-theoretic Farrell-Jones

Conjecture 11.9 with coefficients in additive G-categories and the Farrell-Jones Conjecture 13.70 for homotopy K-theory with coefficients in additiveG-categories is that in the latter we can use the smaller family FIN . Thisis essentially a consequence of and reflected by Theorem 13.68. It has theconsequence that one gets better inheritance properties as the following resulttaken from [558, Theorem 9.1 and Remark 9.3] illustrates.

Theorem 13.71 (Inheritance properties of the Farrell-Jones Conjec-ture for homotopy K-theory).

(i) Let 1 → K → G → Q → 1 be an extensions of groups. If Q and p−1(H)for every finite subgroup H ⊆ Q satisfy Conjecture 13.70, then also Gsatisfies Conjecture 13.70;

(ii) If G acts on a tree T such that every stabilizer group Gx of any vertex xin T satisfies Conjecture 13.70, then also G satisfies Conjecture 13.70;

(iii) Passing to subgroupsLet H ⊆ G be an inclusion of groups. If G satisfies Conjecture 13.70,

then also H satisfies Conjecture 13.70;(iv) Passing to finite direct products

If the groups G0 and G1 satisfy Conjecture 13.70, then G0 ×G1 satisfiesConjecture 13.70;

(v) Passing to finite free productsIf the groups G0 and G1 satisfy Conjecture 13.70, then G0 ∗G1 satisfies

Conjecture 13.70;(vi) Directed colimits

Let Gi | i ∈ I be a direct system of subgroups indexed by the directed set


I (with arbitrary structure maps). Suppose that for each i ∈ I the groupGi satisfies Conjecture 13.70.Then the colimit colimi∈I Gi satisfies Conjecture 13.70.

13.12.1 The Farrell-Jones Conjecture for Totally DisconnectedGroups and Hecke Algebras

There is one instance, where one can formulate the Farrell-Jones Conjecturefor non-discrete groups, namely, for the algebraic K-theory of a Hecke algebraH(G) of a totally disconnected locally compact second countable Hausdorffgroup G.

Denote by H(G) the Hecke algebra of G which consists of locally constantfunctions G→ C with compact support and inherits its multiplicative struc-ture from the convolution product. The Hecke algebra H(G) plays the samerole for G as the complex group ring CG for a discrete group G and reducesto this notion if G happens to be discrete. There is a G-homology theoryHG∗ with the property that for any open and closed subgroup H ⊆ G and alln ∈ Z we have HGn (G/H) = Kn(H(H)), where Kn(H(H)) is the algebraicK-group of the Hecke algebra H(H). There is also the notion of a classifyingspace EKO(G) for the family of compact-open subgroups of G. Notice thatKO is not closed under passing to subgroups but at least under finite inter-sections which suffices to our purposes. The space EKO(G) is characterizedby the property that for any G-CW -complex X whose isotropy groups arecompact-open, there is up to G-homotopy precisely one G-map from X toEKO(G). More information about this space and the comparison with theclassifying space for numerable G-spaces JKO(G) can be found in [532].

Conjecture 13.72 (The Farrell-Jones Conjecture for the algebraicK-theory of Hecke-Algebras). For a totally disconnected locally compactsecond countable Hausdorff group G the assembly map

HGn (EKO(G))→ HG(•) = Kn(H(G))(13.73)

induced by the projection pr : EKO(T )→ • is an isomorphism for all n ∈ Z.

In the case n = 0 this reduces to the statement that

colimG/H∈OrKO(G)K0(H(H)) → K0(H(G))(13.74)

is an isomorphism. For n ≤ −1 one obtains the statement that Kn(H(G)) =0. The group K0(H(G)) is closely related to the theory of the smooth re-presentations of G. The G-homology theory can be constructed using anappropriate functor KH : OrKO(G)→ SPECTRA and the recipe explained inTheorem 10.24. The desired functor KH is given in [732]. Comment 49:

13.13 Relations among the Isomorphisms Conjectures 419

Discuss the notions and results of the paper by Bartels and Luck on this topicwhen it is written.

13.13 Relations among the Isomorphisms Conjectures

13.13.1 The Farrell-Jones Conjecture for K-Theory and forA-Theory

Let G be a group and let X be a G-CW -complex. We get from the lineariza-tion map of (6.15) a natural map

(13.75) LGn (X) : HGn (X; AB)→ HG

n (X; KZ).

if we take Example 13.33 into account. We conclude from Theorem 6.16 andthe equivariant Atiyah Hirzebruch spectral sequence, see Theorem 10.44, thatLGn (X) is bijective for n ≤ 1, surjective for n = 2 and rationally bijective forall n ∈ Z. If we take X = EVCY(G) and X = G/G we obtain a commutativediagram, where the horizontal maps are assembly maps and the vertical mapsare given by the maps (13.75).

HGn (EVCY(G); AB) //

HGn (G/G; AB) = πn(A(BG))

HGn (EVCY(G); KZ) // HG

n (G/G; KZ) = Kn(ZG)

We conclude that for n ∈ Z with n ≤ 1 that the upper arrow is bijective ifand only if the lower arrow is bijective We also conclude for every for n ∈ Zand that the lower arrow is rationally bijective if and only if the lower arrowis rationally bijective for n ∈ Z. This gives some interesting relations betweenthe K-theoretic Farrell-Jones Conjecture 11.1 with coefficients in the ring Zand the Farrell-Jones Conjecture 13.55 for A-theory (without coefficients).For instance, they are equivalent in degrees n ≤ 1, and they are rationallyequivalent.

The case, where we allow in the Farrell-Jones Conjecture 13.55 for A-theory coefficients is more complicated since in Theorem 6.16 (ii) the as-sumption occurs that the space under consideration has to be aspherical.Consider a free G-CW -complex Z which is simply connected (but not neces-sarily contractible). Then π1(G/H ×G Z) ∼= H. We still get a commutativediagram


HGn (EVCY(G); AG

Z ) //

HGn (G/G; AG

Z ) = πn(A(G\Z))

HGn (EVCY(G); KZ) // HG

n (G/G; KZ) = Kn(ZG)

but we only know that the vertical arrows are bijective for n ≤ 1 and sur-jective for n = 2, but not anymore that they are rationally bijective for alln ∈ Z.

13.13.2 The Farrell-Jones Conjecture for A-Theory and forPseudoisotopy

The Farrell Jones Conjecture 13.55 for A-theory (with coefficients) andthe Farrell-Jones Conjecture 13.57 for (smooth) pseudoisotopy (with co-efficients) are equivalent. This follows from the non-connective analoguesof (6.18) and (6.20) for pseudo-isotopy, and from the non-connective ana-logues of (6.23) and (6.24) for smooth pseudo-isotopy using Lemma 13.43 (ii)and Lemma 13.44 (ii). Comment 50: Are there good references for the non-connective versions?

13.13.3 The Farrell-Jones Conjecture for K-Theory and fortopological cyclic homology

The basic reason why topological cyclic homology is a powerful approxima-tion of algebraic K-theory is the cyclotomic trace due to Bokstedt-Hsiang-Madsen [118]. It can extended to the equivariant setting and thus be usedtogether with the linearization map (6.15) to construct the following com-mutative diagram which is closely related to the main diagram in [550, (3.1)]for n ≥ 0


(13.76)

HGn (EVCY(G); K≥0

Z ) // HGn (G/G; K≥0

Z ) = Kn(ZG)

HGn (EFCY(G); K≥0

Z ) //

Hn(ιFCY⊆VCY ;K≥0Z ) ∼=Q

OO

HGn (G/G; K≥0

Z ) = Kn(ZG)

id∼=Q

OO

HGn (EFCY(G); A≥0) //

Hn(EFCY(G);L≥0) ∼=Q

OO

Hn(EFCY(G);ct≥0)

HGn (G/G; A≥0) = An(BG)

Ln∼=Q

OO

ctn

HGn (EFCY(G); TCS) // HG

n (G/G; TCS,p) = TCn(BG, p),

where FCY is the family of finite cyclic subgroups of G, the superscript ≥ 0indicates that we consider the 0-connective covers, the vertical arrows fromthe third row to the second row come from the linearization map and the ver-tical arrows from the third row to the fourth row come from the cyclotomictrace. All arrows marked with ∼=Q are known to be bijective. This followsfrom the maps induced by the linearization from Theorem 6.16. For the mapHn(ιFCY⊆VCY ; K≥0

Z ) this follows from Theorem 11.39 (v) and further com-putations based equivariant Chern characters using Theorem 10.70 and [550,

Example 12.12]. The natural map HGn (EVCY(G); K≥0

Z )→ HGn (EVCY(G); KZ)

is split injective and its cokernel is given by an expression involving the groupsK−1(ZC) for finite cyclic subgroups C ⊆ G. Thus the diagram (13.76) im-plies that the K-theoretic Farrell-Jones assembly map is rationally injective,ignoring certain contributions from the collection of the groups K−1(ZC) forfinite cyclic subgroups C ⊆ G, provided that the lowermost horizontal arrowis rationally injective. This is the basic idea in the proof of rational injectiv-ity results for the K-theoretic Farrell-Jones assembly map presented in [550],where the actual argument is more involved and uses the C-functors as well.

A rational computation of Kn(ZG) is given in Theorem 15.1, provided thatif G satisfies the satisfies the K-theoretic Farrell-Jones Conjecture 11.1 withcoefficients in the ring Z. With the methods mentioned above one can detectunder certain conditions the source of the map appearing in Theorem 15.1,if one ignores the summand for q = −1

13.13.4 The L-Theoretic Farrell-Jones Conjecture and theBaum-Connes Conjecture

In the sequel [1/2] stands for inverting 2 at the level of spectra or abeliangroups. Notice that for a spectrum E we have a natural isomorphism

πn(E)[1/2]∼=−→ πn(E[1/2]).


One can construct the following commutative diagram

(13.77) HGn (EG; L

〈−∞〉Z )[1/2] //

l ∼=

L〈−∞〉n (ZG)[1/2]

id ∼=

HGn (EG; L

〈−∞〉Z [1/2]) // L〈−∞〉n (ZG)[1/2]

HGn (EG; LpZ[1/2])

i1 ∼=

i0 ∼=

OO

// Lpn(ZG)[1/2]

j1∼=

j0∼=

OO

HGn (EG; LpQ[1/2])

i2 ∼=

// Lpn(QG)[1/2]

j2

HGn (EG; LpR[1/2])

i3 ∼=

// Lpn(RG)[1/2]

j3

HGn (EG; LpC∗r (?;R)[1/2]) // Lpn(C∗r (G;R))[1/2]

HGn (EG; Ktop

R [1/2]) //

i4 ∼=

OO

Ktopn (C∗r (G;R))[1/2]

j4∼=

OO

HGn (EG; Ktop

R )[1/2]

i5

l∼=

OO

Ktopn (C∗r (G;R))[1/2]

j5

∼=id

OO

HGn (EG; Ktop

C )[1/2] // Kn(C∗r (G))[1/2]

where all horizontal maps are assembly maps and the vertical arrows areinduced by transformations of functors GROUPOIDS → SPECTRA. Thesetransformations are induced by change of rings maps except the one fromKtop

R [1/2] to LpC∗r (?;R)[1/2] which is much more complicated and carried out

in [499]. Actually, it does not exists without inverting two on the spectrumlevel. Since it is a weak equivalence, the maps i4 and j4 are bijections.

On the level of homotopy groups the comparison between the algebra L-theory and the topological K-theory of a real and of a complex C∗-algebrahave already been explained in Theorem 8.78, namely we obtain isomor-phisms


Lpn(A)[1/2]∼=−→ KOn(A)[1/2], if A is a real C∗-algebra;(13.78)

Lpn(A)∼=−→ Kn(A), if A is a complex C∗-algebra.(13.79)

Notice that in the complex case one does not have to invert 2.Since for any finite group H each of the following maps is known to be a

bijection because of [685, Proposition 22.34 on page 252] and RH = C∗r (H;R)

Lpn(ZH)[1/2]∼=−→ Lpn(QH)[1/2]

∼=−→ Lpn(RH)[1/2]∼=−→ Lpn(C∗r (H;R)),

we conclude from the equivariant Atiyah Hirzebruch spectral sequence, seeTheorem 10.44, that the vertical arrows i1, i2, and i3 are isomorphisms. Thearrow j1 is bijective by [683, page 376]. The maps l are isomorphisms forgeneral results about localizations.

The lowermost vertical arrows i5 and j5 are known to be split injec-tive because the inclusion C∗r (G;R) → C∗r (G;C) induces an isomorphismC∗r (G;R)→ C∗r (G;C)Z/2 for the Z/2-operation coming from complex conju-gation C → C. Comment 51: Check this statement and add a reference.The following conjecture is already raised as question in [477, Remark 23.14on page 197], see also [499, Conjecture 1 in Subsection 5.2].

Conjecture 13.80 (Passage for L-theory from QG to RG to C∗r (G;R)).The maps j2 and j3 appearing in diagram (13.77) are bijective.

One easily checks

Lemma 13.81. Let G be a group.

(i) Suppose that G satisfies the L-theoretic Farrell-Jones Conjecture 11.4 withcoefficients in the ring R for R = Q and R = R and the complex versionof the Baum-Connes Conjecture 12.9. Then G satisfies Conjecture 13.80;

(ii) Suppose that G satisfies Conjecture 13.80. Then G satisfies the L-theoreticFarrell-Jones Conjecture 11.4 for the ring Z after inverting 2, if and onlyif G satisfies the real version of the Baum-Connes Conjecture 12.9 afterinverting 2;

(iii) Suppose that the assembly map appearing in the complex version of theBaum-Connes Conjecture 12.9 is (split) injective after inverting 2. Thenthe assembly map appearing in L-theoretic Farrell-Jones Conjecture 11.4with coefficients in the ring for R = Z is (split) injective after inverting 2.

Proof. This follows from Theorem 11.49 (i), Remark 12.13 and the dia-gram (13.77). ut

13.13.5 Mapping Surgery to Analysis

Let X be a connected CW -complex with fundamental group π. Let X →X be its universal covering. Denote by ε one of the decorations s, h or p.


We have constructed functors LεZ and KtopR : GROUPOIDS → SPECTRA in

Theorem 10.39. We obtain maps of spectra

X+ ∧ LεZ(∗) X+ ∧π LεZ(Gπ(π)) //'oo LεZ(Gπ(π/π))

X+ ∧KtopR (∗) X ∧π Ktop

R (Gπ(π)) //'oo KtopR (Gπ(π/π))

Here ∗ denotes the trivial groupoid with one object, the horizontal arrowspointing to the left are defined in the obvious way and are weak homotopyequivalences, since X is a free π-CW -complex with π\X = X and Gπ(π)→∗ is an equivalence of groupoids, and the horizontal arrows to the right are

assembly maps composed with maps induced by a fixed π-map X → Eπ. (If

one wants to get rid of the dependency of a choice of π-map X → Eπ, onecan consider Π(π/H ×π X) instead of Gπ(π/H) for objects π/H in Or(π).)

Denote by Sε(X) and D(X) respectively the homotopy fiber of the arrowpointing to the right in the first and second row above.

After taking homotopy groups we obtain long exact sequences

(13.82) · · · → Hn+1(X; LεZ(∗))→ Lεn+1(Zπ)→ πn(Sε(X))

→ Hn(X; LεZ(∗))→ Lεn(Zπ)→ · · · ,

and

(13.83) · · · → KOn+1(X)→ KOn+1(C∗r (π;R))→ πn(D(X))

→ KOn(X)→ KOn(C∗r (π;R))→ · · · .

After inverting 2 there is a zigzag of natural transformation from KtopR [1/2]

LεZ[1/2] as explained in Subsection 13.13.4. It yields a map between longexact sequence


(13.84)...

...

KOn+1(X)[1/2] //

Hn+1(X; LεZ(∗))[1/2]

KOn+1(C∗r (π;R)) //

Lεn+1(Zπ)[1/2][1/2]

πn(D(X))[1/2] //

πn(Sε(X))[1/2]

KOn(X)[1/2] //

Hn(X; LεZ(∗))[1/2]

KOn(C∗r (π;R))[1/2] //

Lεn(Zπ)[1/2]

...

...

Lemma 13.85. Suppose that π satisfies the L-theoretic Farrell-Jones Con-jecture 11.4 with coefficients in the ring with involution Z and the Baum-Connes Conjecture 12.9 for the real group C∗-algebra.

Then the map

πn(D(M))[1/2]∼=−→ πn(Sε(M))[1/2]

is bijective for n ∈ Z.

Proof. The first and fourth horizontal arrow in the diagram 13.84 are bijec-tive since there are given by transformation of homology theories and theirevaluation at • is known to be bijective. The Rothenberg sequence (7.101),Theorem 11.49 (i) and the diagram (13.77) together with the assumptionthat π satisfies the L-theoretic Farrell-Jones Conjecture 11.4 with coeffi-cients in the ring with involution Z and the Baum-Connes Conjecture 12.9for the real group C∗-algebra imply that the second and fifth horizontal ar-row in the diagram 13.84 are bijective. Now apply the Five-Lemma to thediagram (13.84). ut

Now consider the case X = M for a closed orientable topological manifoldM of dimension d. Then the part of the sequence (13.82) for n ≥ d can identi-fied with the long exact surgery sequence Comment 52: Add a reference tothe surgery exact sequence when the chapter about surgery theory is finished,


for instance to Theorem 7.124, and add a reference to [485]. Some extra careis necessary at the end in degree d since one has to pass to the 1-connectivecover of the L-theory spectrum. In particular we get an identification ofπd(S

s(M)) with the structure set Ssd(M), see Subsection 7.12.1, which is thecentral object of study in the classification of manifolds. Notice that in viewof Lemma 13.85 one can hope for an identification of Ssd(M) after invert-ing 2 with πd(D(M)) which is an object related to topological K-theory ofspaces and C∗-algebras. An analytic surgery exact sequence in terms of thetopological K-theory of C∗-algebra associated to M is constructed in [395,Section 1]

Problem 13.86 (Identification of analytic surgery exact sequences).Identify the real version of the analytic surgery sequence appearing in [395,

Section 1] with the exact sequence (13.83) for a closed orientable manifold ofdimension d.

Notice that Higson-Roe have to work with smooth manifolds since theywant to apply index theory. So they have to consider the surgery sequencein the smooth category. They construct a diagram relating the surgery exactsequence in the smooth category to their analytic surgery exact sequence.

A more direct approach to the map comparing the surgery sequence in thesmooth category to the analytic surgery exact sequence is given in Piazza-Schick [660].

A comparison map starting with the surgery exact sequence in the topolo-gical category is constructed in Zenobi [855] using the approach of [660] andLipschitz structures.

Recall that the surgery exact sequence in the topological category is anexact sequence of abelian groups, what is not true for the smooth category.It is not clear whether the construction in Zenobi [855] is compatible withthe structures of an abelian groups on the topological and analytic structuresets.

Notice that that the comparison maps appearing in [395, 660, 855] go inthe opposite direction, namely from L-theory to KO-theory. Comment 53:Do we have to elaborate on this?

So one can state the following problem after Problem 13.86 has beensolved:

Problem 13.87 (Identification of transformations from the surgeryexact sequence to its analytic counterpart). Identify the comparisonmap (13.84) from the surgery exact sequence in the topological category tothe analytic surgery sequence appearing in [395, Section 5] with the compar-ison map appearing in Zenobi [855].


13.13.6 The Baum-Connes Conjecture and the Bost Conjecture

Comment 54: Maybe we discard this section and say more in the relevantsection about the Baum-Connes Conjecture for Frechet algebras. The onlything we should discuss is the factorization of the Baum-Connes assemblymap though the K-theory of the group Frechet algebra and L1(G).

13.13.7 The Farrell-Jones Conjecture for K-theory and forhomotopy K-theory

Theorem 13.88 (The K-theoretic Farrell-Jones Conjecture impliesthe Farrell-Jones Conjecture for homotopy K-theory). If G satisfiesthe K-theoretic Farrell-Jones Conjecture 11.9 with coefficients in additiveG-categories, then G satisfies also the Farrell-Jones Conjecture 13.70 forhomotopy K-theory with coefficients in additive G-categories.

Proof. See [558, Theorem 9.1 (iii)]. ut

Remark 13.89 (Implications of the homotopy K-theory version tothe K-theory version). Next we discuss some cases, where the Farrell-Jones Conjecture 13.70 for homotopy K-theory with coefficients in additiveG-categories gives implications for the injectivity part of the K-theoreticFarrell-Jones Conjecture 11.1 with coefficients in the ring R. These all followby inspecting for a ring R the following commutative diagram

HGn (EVCY(G); KR) // HG

n (•; KR) = Kn(RG)

KH(h)

HGn (EFIN (G); KR)

ιFIN⊆VCY

OO

h

HGn (EFIN (G); HKR) // HG

n (•; HKR) = KHn(RG)

where the two vertical arrows pointing downwords are induced by the trans-formation h : K → HK, the map ιFIN⊆VCY is induced by the inclusion offamilies FIN ⊆ VCY and the two horizontal arrows are the assembly mapsfor K-theory and homotopy K-theory.

Suppose that R is regular and the order of any finite subgroup of G isinvertible in R. Then the two left vertical arrows are known to be bijections.This follows for ιFIN⊆VCY from [546, Proposition 70 on page 744] and for hfrom [222, Lemma 4.6] and the fact that RH is regular for all finite subgroupsH of G and hence Kn(RH)→ KHn(RH) is bijective for all n ∈ Z. Hence the


(split) injectivity of the lower horizontal arrow implies the (split) injectivityof the upper horizontal arrow.

Suppose that R is regular. Then the two left vertical arrows are rationalbijections. This follows for ιFIN⊆VCY from [559, Theorem 0.3]. To show it forh it suffices because of [222, Lemma 4.6] to show that Kn(RH)→ KHn(RH)is rationally bijective for each finite group H and n ∈ Z. By the version ofthe spectral sequence appearing in [824, 1.3] for non-connective K-theory, itremains to show that NpKn(RH) vanishes rationally for all n ∈ Z. Since R[t]is regular if R is, this boils down to show that NKp(RH) is rationally trivialfor any regular ring R and any finite group H. The proof that NKp(RH)is rationally trivial for any regular ring R and any finite group H can befound for instance in [559, Theorem 9.4]. Hence the upper horizontal arrowis rationally injective if the lower horizontal arrow is rationally injective.

The next conjecture generalizes Conjecture 5.64 from torsionfree groupsto arbitrary groups.

Conjecture 13.90 (K-theory versus homotopy K-theory for regularrings). Let G be a group. Suppose that R is regular and the order of anyfinite subgroup of G is invertible in R.

Then the natural map

Kn(RG)→ KHn(RG)

is an isomorphism for all n ∈ Z.

Exercise 13.91. Suppose that G satisfies the K-theoretic Farrell-Jones Con-jecture 11.9 with coefficients in additive G-categories. Then G satisfies Con-jecture 13.90.

Exercise 13.92. Let G be a group. Suppose that R is regular and the orderof any finite subgroup of G is invertible in R. Suppose that Conjecture 13.90is true for G. Show that then NKn(RG) = 0 holds for all n ∈ Z.

13.14 Miscellaneous

One can also define a version of the Meta-Isomorphism Conjecture 13.2 or ofthe Fibered Meta-Isomorphism Conjecture 13.8 with finite wreath products,compare Section 11.4. Let C be a class of groups closed under isomorphismsand taking subgroups and quotients. Let H?

∗ be an equivariant homologytheory.

Definition 13.93 (Fibered Meta-Isomorphism Conjecture with fi-nite wreath products).


A group G satisfies the Fibered Isomorphism Conjecture with finite wreathproducts with respect toH?

∗ and C if for any finite group F the wreath productG o F satisfies the Fibered Meta-Isomorphism Conjecture 13.8 with respectto H?

∗ and the family C(G o F ) consisting of subgroups of G o F which belongto C.

The inheritance properties for the Fibered Meta-Isomorphism Conjec-ture 13.8 plus the passage to overgroups of finite index do also hold for theFibered Meta-Isomorphism Conjecture with finite wreath products 13.93,see [484, Section 3].

Proofs of some of the inheritance properties above are also given in [370,722].

One may ask whether one can find abstractly for the Fibered Meta-Isomorphism Conjecture 13.8 a smallest family for which it is true. For in-stance what happens if one takes the intersection of all families for which theFibered Meta-Isomorphism Conjecture 13.8 is true. This questions turns outto be equivalent to the difficult and unsolved question whether Fibered Meta-Isomorphism Conjecture 13.8 holds for an infinite product of groups providedthat for each of these groups the Fibered Meta-Isomorphism Conjecture 13.8is true, see [667, Section 7].


name of texfile: ic

Chapter 14

Status

14.1 Introduction

In this chapter we give a status report about the class of groups forwhich the Full Farrell-Jones Conjecture 11.20, see Theorem 14.1, the Baum-Connes 12.11 with coefficients, see Theorem 14.6, the Baum-Connes Con-jecture 12.9, see Theorem 14.11, and the Novikov Conjecture 7.131, see Sec-tion 14.6 have been proved. We discuss also injectivity results in Sections 14.4and 14.5. In order to restrict the length of the exposition we do not presentthe history of these results and concentrate only on the current state of art,although this unfortunately means that certain papers, which were spectac-ular break throughs at the time of their writing and had a big impact on thefollowing papers, do not appear here.

A review of and status report for some classes of groups is given Sec-tion 14.7. This may be helpful for a reader who is interested in a certain classof groups, although this means that there are some repetitions of statementsof results.

At the time of writing no counterexamples to the Full Farrell-Jones Con-jecture 11.20, the Baum-Connes Conjecture 12.9, and the Novikov Conjec-ture 7.131 are known, but they are open in general. In Section 14.9 we explainthat the search for counterexamples is not easy at all. In Subsection 14.9.4we mention a few results which are consequences of the K-theoretic Farrell-Jones Conjecture 11.1 with coefficients in the ring R and for which there existindependent proofs for all groups.

14.2 Status of the Full Farrell-Jones Conjecture

The most general form of the Farrell-Jones Conjecture is the Full Farrell-Jones Conjecture 11.20. It has the best inheritance properties and all variantsof the Farrell-Jones Conjecture presented in this book are special cases of it.

Theorem 14.1 (Status of the Full Farrell-Jones Conjecture 11.20).Let FJ be the class of groups for which the Full Farrell-Jones Conjec-

ture 11.20 is true. Then

(i) The following classes of discrete groups belong to FJ :

431

432 14 Status

(a) Hyperbolic groups;(b) Finite dimensional CAT(0)-groups;(c) Virtually solvable groups;(d) (Not necessarily cocompact) lattices in second countable locally compact

Hausdorff groups with finitely many path components;(e) Fundamental groups of (not necessarily compact) connected manifolds

(possibly with boundary) of dimension ≤ 3;(f) The groups GLn(Q) and GLn(F (t)) for F (t) the function field over a

finite field F ;(g) S-arithmetic groups;(h) mapping class group of any oriented surface of finite type; Comment

55: Does this mean each Γ sg,r?(i) Fundamental groups of graphs of abelian groups;(j) Fundamental groups of graphs of virtually cyclic groups;(k) Artin’s full braid groups Bn;(l) Coxeter groups;

(ii) The class FJ has the following inheritance properties:

(a) Passing to subgroupsLet H ⊆ G be an inclusion of groups. If G belongs to FJ , then H

belongs to FJ ;(b) Passing to finite direct products

If the groups G0 and G1 belong to FJ , then also G0 × G1 belongs toFJ ;

(c) Group extensionsLet 1→ K → G→ Q→ 1 be an extension of groups. Suppose that for

any cyclic subgroup C ⊆ Q the group p−1(C) belongs to FJ and thatthe group Q belongs to FJ .Then G belongs to FJ ;

(d) Directed colimitsLet Gi | i ∈ I be a direct system of groups indexed by the directed

set I (with arbitrary structure maps). Suppose that for each i ∈ I thegroup Gi belongs to FJ .Then the colimit colimi∈I Gi belongs to FJ ;

(e) Passing to finite free productsIf the groups G0 and G1 belong to FJ , then G0 ∗G1 belongs to FJ ;

(f) Passing to overgroups of finite indexLet G be an overgroup of H with finite index [G : H]. If H belongs toFJ , then G belongs to FJ ;

(g) Graph productsA graph product of groups, each of which belongs to FJ , belongs to FJ

again.

Proof. (ia) This is proved for K-theory in [73, Main Theorem] and for L-theory in [72, Theorem B], but not including the “with finite wreath product”

14.2 Status of the Full Farrell-Jones Conjecture 433

property. How this can be included, is explained in [75, Remark 6.4].

(ib) This is proved for K theory in degree ≤ 1 and for L-theory in all degreesin [72, Theorem B]. The argument why the K-theory case holds in all degreescan be found in [820, Theorem 1.1 and Theorem 3.4]. Notice that for a finitedimensional CAT(0)-group G and a finite group F the wreath product G oFis a finite dimensional CAT(0)-group again so that the version with finitewreath products is automatically satisfied;

(ic) See [821, Theorem 1.1]. (The special case of nearly crystallographicgroups is treated by Farrell-Wu [302].)

(id) See [439, Theorem 11] whose proof is based on the case of a cocompactlattices in an almost connected Lie groups handled in [66, Theorem 1.2 andRemark 1.4].

(ie) In dimension 3 this is proved in [66, Corollary 1.3 and Remark 1.4], whereRoushon [722, 723] is used. The dimensions 1 and 2 can be handled directlyor reduced to dimension 3 by crossing with D1.

(if) See [729, Theorem 8.13].

(ig) This follows from assertion (if) and the inheritance property passing tosubgroups, see assertion (iia) since any S-arithmetic group is a subgroup ofGLn(Q) or of GLn(F (t)) for F (t) the function field over a finite field F .

(ih) See Bartels-Bestvina [64, Remark 9.4].

(ii) See Gandini-Meinert-Ruping [337].

(ij) See Wu [842].

(ik) The pure Artin braid group Pn is a strongly poly-surface group in thesense of Definition 14.22 by [33, Theorem 2.1]. Hence it satisfies the FullFarrell-Jones Conjecture 11.20 by Theorem 14.23. Since the full braid groupBn contains Pn as a subgroup of finite index, Bn satisfies the Full Farrell-Jones Conjecture 11.20 by assertion (iif).

(il) The argument in Bartels-Lueck [72, page 636] for the version without“finite wreath products” extends directly to the case with “finite wreathproducts”.

(ii) See Theorem 11.21 except for assertion (iic) and (iig) To prove or asser-tion (iic) we have to consider in Theorem 11.21 virtually cyclic subgroupsof Q instead of cyclic subgroups. But the more general case stated in asser-tion (ii) is a direct consequence of assertion (iif). Assertion (iig) is proved byGandini-Ruping [338]. ut

Exercise 14.2. Let G be a cocompact torsionfree lattices in an almost con-nected Lie group L with dim(L) ≥ 5. Let M be a closed aspherical manifoldwith fundamental group G. Let K ⊆ L be a maximal compact subgroup.Show that then M is homeomorphic to G\L/K.

434 14 Status

Exercise 14.3. Let Fi−→M

p−→ B be a locally trivial fiber bundle of compactconnected manifolds. Suppose that dim(M) ≤ 2 and that π1(B) satisfies theFull Farrell-Jones Conjecture 11.20. Show that then also π1(M) satisfies theFull-Farrell-Jones Conjecture 11.20.

Exercise 14.4. Let U be a group which is universal finitely presented, i.e.,any finitely presented group is isomorphic to a subgroup of G. (Such a groupexists by Higman [381, page 456], and there is even a universal finitely pre-sented groups which is the complement of an embedded S2 in S4, see [345,Corollary 3.4].) Show that the Full Farrell-Jones Conjecture 11.20 holds forall groups if and only if it holds for U .

Exercise 14.5. Let 1 → K → G → Q → 1 be an extension of groups.Suppose that K is finite and G is finitely generated residually finite.

Show that G belongs to FJ if and only if Q belongs to FJ .

14.3 Status of the Baum-Conjecture (with coefficients)

We have introduced the Baum-Connes Conjecture 12.11 with coefficients inSection 12.4.

Theorem 14.6 (Status of the Baum-Connes 12.11 with coefficients).Let BC be the class of groups for which the Baum-Connes Conjecture 12.11

with coefficients holds.

(i) The following classes of groups belong to BC.

(a) A-T-menable groups;(b) G is a countable subgroup of GL2(F ) for a field F ;(c) Hyperbolic groups;(d) One-relator groups;(e) Fundamental groups of compact 3-manifolds (possibly with boundary);(f) Artin’s full braid groups Bn;(g) Thompson’s groups F , T and V ;(h) Coxeter groups;

(ii) The class BC has the following inheritance properties:

(a) Passing to subgroupsLet H ⊆ G be an inclusion of groups. If G belongs to BC, then H

belongs to BC;(b) Passing to finite direct products

If the groups G0 and G1 belong to BC, the also G0×G1 belongs to BC;

14.3 Status of the Baum-Conjecture (with coefficients) 435

(c) Group extensionsLet 1→ K → G→ Q→ 1 be an extension of groups. Suppose that for

any finite subgroup F ⊆ Q the group p−1(F ) belongs to BC and that thegroup Q belongs to BC.Then G belongs to BC;

(d) Directed unionsLet Gi | i ∈ I be a direct system of subgroups of G indexed by the

directed set I such that G =⋃i∈I Gi. Suppose that Gi belongs to BC for

every i ∈ I.Then G belongs to BC;

(e) Actions on treesLet G be a countable discrete group acting without inversion on a treeT . Then G belongs to BC if and only if the stabilizers of each of thevertices of T belong to BC.In particular BC is closed under amalgamated products and HNN-extensions.

Proof. (ia). This is proved by Higson-Kasparov [387, Theorem 1.1]

(ib) Such groups are a-T-menable by [356, Theorem 4]. Now apply asser-tion (ia).

(ic) See Lafforgue [490, Theoreme 0.4]. The proof without coefficients can befound in Mineyev-Yu [596].

(id) See Oyono-Oyono [640, Corollary 1.3].

(ie) Let M be a closed Seifert manifold. Then there is an extension 1→ Z→π1(M) → Q → 1 such that Q contains a subgroup H of finite index whichis isomorphic to the fundamental group of a closed surface S, see [380, The-orem 12.2 on page 118]. If F carries the structure of a hyperbolic manifold,π1(S) and hence Q are hyperbolic and belongs to BC by assertion (ic). If Fdoes not carry the structure of a hyperbolic manifold, its fundamental groupand hence Q are virtually finitely generated abelian and hence belong to BCby assertion (ia). Now assertions (ia) and (iic) imply that π1(M) belongs toBC.

Let M be a closed hyperbolic 3-manifold. Then its fundamental group ishyperbolic and hence belongs to BC by assertion (ic).

Let M be a compact irreducible manifold with infinite fundamental groupsuch that its boundary is non-trivial or is Haken. Then π1(M) can be ob-tained from the trivial group by a finite number of HNN extensions and freeamalgamated products. See [806, Proof of Proposition 19.5 (6) on page 253],where the condition orientable is only assumed for simplicity, or see [380, The-orem 13.3 on page 141]. Hence π1(M) belongs to BC by assertion (iie). Let Mbe an irreducible closed 3-manifold. If it does not contain an incompressibletorus, it is either Seifert or hyperbolic by the proof of Thurston’s Geometriza-tion Conjecture due to Perelman, see for instance Morgan-Tian [613] andhence belongs to BC. If it contains an incompressible torus, it is Haken andhence belongs to BC by the argument above. We conclude that π1(M) belongs

436 14 Status

to BC for any compact irreducible 3-manifold. Since any prime 3-manifoldwhich is not irreducible is an S1-bundle over S2, see [380, Lemma 3.13 onpage 28], and hence belongs to BC by assertion (ia), any compact prime 3-manifold M belongs to BC. Since any compact 3-manifold is a connectedsum of prime compact 3-manifolds, see [380, Theorem 3.15 on page 31], as-sertion (ie) follows from assertion (iie).

(ig) These groups are a-T-menable by Farley [283] and hence we can applyassertion (ia).

(if) See Schick [740, Theorem 20].

(ih) Since a finitely generated Coxeter group is a-T-menable, it satisfies theBaum-Connes Conjecture 12.11 with coefficients by Theorem 14.6 (ia). By acolimit argument based on Theorem 14.6 (iid) every Coxeter group satisfiesthe Baum-Connes Conjecture 12.11 with coefficients.

(iia) See Chabert-Echterhoff [171, Theorem 2.5].

(iib) See Chabert-Echterhoff [171, Theorem 3.17], or Oyono-Oyono [639,Corollary 7.12].

(iic) See Oyono-Oyono [639, Theorem 3.1].

(iid) This follows from Bartels-Echterhoff-Luck [65, Theorem 5.6 (i) andLemma 6.2].

(iie) This is proved by Oyono-Oyono [640, Theorem 1.1]. ut

Exercise 14.7. Let 1 → K → G → Q → 1 be an extension of groups suchthat K and Q satisfy the Baum-Connes Conjecture 12.11 with coefficientsand Q is torsionfree. Show that then G satisfies the Baum-Connes Conjec-ture 12.11 with coefficients.

Exercise 14.8. Let G be a torsionfree group. Suppose that CG has an idem-potent different from 0, 1. Show that then G cannot be hyperbolic, a finitedimensional CAT(0)-group, a lattice in an almost connected Lie group, thefundamental group of a manifold of dimension ≤ 3, amenable, or a one-relatorgroup.

Remark 14.9 (Passing to overgroups of finite index). It is not knownin general whether a group G belongs to BC, i.e., G satisfies the Baum-ConnesConjecture 12.11 with coefficients, if a subgroup of finite index does. Partialanswers to this question are given by Schick [740, Theorem 20].

This suggests to systematically implement the with “finite wreath productversion” in the Baum-Connes setting, as we did in the Farrell-Jones setting,see Section 11.4.

Remark 14.10 (The Status of the Baum-Connes Conjecture for to-pological groups). We have only dealt with the Baum-Connes Conjecturefor discrete groups. The Baum Connes Conjecture (with coefficients) makes

14.3 Status of the Baum-Conjecture (with coefficients) 437

also sense for second countable locally compact Hausdorff groups. Here someresults in this setting.

Higson-Kasparov [387] treat the Baum-Connes Conjecture with coefficientsfor second countable locally compact Hausdorff groups.

Julg-Kasparov [434, Theorem 5.4 (i)] prove the Baum-Connes Conjecturewith coefficients for connected Lie groups L whose Levi-Malcev decomposi-tion L = RS into the radical R and semisimple part S is such that S is locallyof the form

S = K × SO(n1, 1)× · · · × SO(nk, 1)× SU(m1, 1)× · · · × SU(ml, 1)

for a compact group K. The Baum-Connes Conjecture with coefficients forSp(n, 1) is proved by Julg [433].

The Baum-Connes Conjecture (without coefficients) for second countablealmost connected Hausdorff groups has been proven by Chabert-Echterhoff-Nest [172] based on the work of Higson-Kasparov [387] and Lafforgue [489].

Next we deal with the Baum-Connes Conjecture 12.9 itself. Recall that allgroups which satisfy the Baum-Connes 12.11 with coefficients do in particularsatisfy the Baum-Connes Conjecture 12.9. Below are some case, which arenot covered by this implication.

A length function on G is a function L : G → R≥0 such that L(1) = 0,L(g) = L(g−1) for g ∈ G and L(g1g2) ≤ L(g1) + L(g2) for g1, g2 ∈ G holds.The word length metric LS associated to a finite set S of generators is anexample. A length function L on G has property (RD) (“rapid decay”) ifthere exist C, s > 0 such that for any u =

∑g∈G λg · g ∈ CG we have

||ρG(u)||∞ ≤ C ·

∑g∈G|λg|2 · (1 + L(g))2s

1/2

,

where ||ρG(u)||∞ is the operator norm of the bounded G-equivariant opera-tor l2(G) → l2(G) coming from right multiplication with u. A group G hasproperty (RD) if there is a length function which has property (RD). Thisnotion is due to Jolissaint [425]. More information about property (RD) canbe found for instance in [182], [487] and [797, Chapter 8]. Bolicity general-izes Gromov’s notion of hyperbolicity for metric spaces. A simply connectedcomplete Riemannian manifold with non-positive sectional curvature is bolic.We refer to [450, Section 2] for a precise definition.

Theorem 14.11 (Status of the Baum-Connes Conjecture). A group Gsatisfies the Baum-Connes Conjecture 12.9 if it satisfies one of the followingconditions.

(i) It is a discrete subgroup of a connected Lie groups L, whose Levi-Malcevdecomposition L = RS into the radical R and semisimple part S is suchthat S is locally of the form

438 14 Status

S = K × SO(n1, 1)× · · · × SO(nk, 1)× SU(m1, 1)× · · · × SU(ml, 1)

for a compact group K;(ii) The group G has property (RD) and admits a proper isometric action on

a strongly bolic weakly geodesic uniformly locally finite metric space;(iii) The group G is a discrete finite covolume subgroup of the isometry groups

of a simply connected complete Riemannian manifold with pinched negativesectional curvature;

(iv) G is a discrete subgroup of Sp(n, 1).

Proof. (i) See Julg-Kasparov [434].

(ii) See Lafforgue [486] or [756].(iii) See [182, Corollary 0.3].

(iv) See Julg [433]. ut

14.4 Injectivity Results in the Baum-Connes Setting

There are cases where one show that the assembly maps appearing in theFarrell-Jones setting or Baum-Connes setting are injective without knowingthat they are bijective. There is no case where one can prove surjectivity butdoes not know bijectivity as well. This is a common phenomenon in algebraictopology, where surjectivity arguments often contain an injectivity argument,essentially one applies the surjectivity argument to a cycle whose boundaryis the image of a cycle representing an element in the kernel of the assemblymap. Moreover, this shows that in general surjectivity results are harder thaninjectivity results.

The main value of surjectivity statements is that they allow to interpreteelements in the K- or L-groups homologically and thus to obtain valuableinformation. The injectivity statements are interesting since they imply theNovikov Conjecture or give some idea how large the K- and L-groups are.

Theorem 14.12 (Split injectivity of the assembly map appearing inthe Baum-Connes Conjecture 12.9 for fundamental groups of com-plete Riemannian manifolds with non-positive sectional curvature).The assembly map appearing in the Baum-Connes Conjecture 12.9 is split

injective if G is the fundamental group of complete Riemannian manifold withnon-positive sectional curvature.

Proof. See Kasparov [456, Theorem 6.7]. ut

More general results for bolic spaces are proved in Kasparov-Skandalis [451].A metric space (X, d) admits a uniform embedding into Hilbert space if

there exist a separable Hilbert space H, a map f : X → H and non-decreasingfunctions ρ1 and ρ2 from [0,∞)→ R such that ρ1(d(x, y)) ≤ ||f(x)−f(y)|| ≤

14.4 Injectivity Results in the Baum-Connes Setting 439

ρ2(d(x, y)) for x, y ∈ X and limr→∞ ρi(r) = ∞ for i = 1, 2. A metric isproper if for each r > 0 and x ∈ X the closed ball of radius r centeredat x is compact. The question whether a discrete group G equipped with aproper left G-invariant metric d admits a uniform embedding into Hilbertspace is independent of the choice of d, since the induced coarse structuredoes not depend on d, see [757, page 808]. For more information about groupsadmitting a uniform embedding into Hilbert space we refer to [253, 356].

The next result is due to Yu [850, Theorem 2.2 and Proposition 2.6].

Theorem 14.13 (Status of the Coarse Baum-Connes Conjecture).The Coarse Baum-Connes Conjecture 12.29 is true for a discrete metric

space X of bounded geometry if X admits a uniform embedding into Hilbertspace. In particular a countable group G satisfies the Coarse Baum-ConnesConjecture 12.29 if G equipped with a proper left G-invariant metric admitsa uniform embedding into Hilbert space.

Theorem 14.14 (Split injectivity of the assembly map appearing inthe Baum-Connes Conjecture 12.11 with coefficients). Let G be acountable group. Then for any C∗-algebra A the assembly map appearing inthe Baum-Connes Conjecture 12.11

KGn (EG;A)→ Kn(Aor G);

is split injective, if the group G has one of the following properties:

(i) The group G admits a G-invariant metric for which G admits a uniformembedding into Hilbert space;

(ii) The group G is a subgroup of GLn(F ) for some field F and natural numbern;

(iii) The group G is a subgroup of an almost connected Lie group.

Proof. (i) This is proved by Skandalis-Tu-Yu [757, Theorem 6.1] using ideasof Higson [386] and Theorem 14.13.

(ii) Assertion (i) applies to G by [356, Theorem 2 and 3].

(iii) Assertion (i) applies to G by [356, Theorem 7]. ut

Exercise 14.15. Show that one can drop the condition countable in Theo-rem 14.14, if one wants to prove injectivity only.

The split injectivity of the Baum-Connes assembly map (with coefficients)for groups embedding in Banach spaces is investigated by Kasparov-Yu [452,Theorem 1.3].

Split Injectivity of the Baum-Connes assembly map (for trivial coefficients)is proved under certain conditions about the compactifications of the modelfor the space of proper G-actions by Rosenthal [719] based on techniquesdeveloped by Carlsson-Pedersen [163].

440 14 Status

Remark 14.16 (Groups Acting Amenably on a Compact Space). Acontinuous action of a discrete group G on a compact space X is calledamenable if there exists a sequence

pn : X →M1(G) = f : G→ [0, 1] |∑g∈G

f(g) = 1

of weak-∗-continuous maps such that for each g ∈ G one has

limn→∞

supx∈X||g ∗ (pn(x)− pn(g · x))||1 = 0.

Note that a group G is amenable if and only if its action on the one-point-space is amenable. More information about this notion can be found forinstance in [22, 23].

Higson-Roe [391, Theorem 1.1 and Proposition 2.3] show that a finitelygenerated group equipped with its word length metric admits an amenable ac-tion on a compact metric space, if and only if it belongs to the class A definedin [850, Definition 2.1], and hence admits a uniform embedding into Hilbertspace. Hence Theorem 14.14 (i) implies the result of Higson [386, Theo-rem 1.1] that the assembly map the assembly map KG

n (EG;A)→ Kn(AorG)appearing in the Baum-Connes Conjecture 12.11 is split injective if G admitsan amenable action on some compact space.

14.5 Injectivity Results in the Farrell-Jones Setting

Theorem 14.17 (Split injectivity of the assembly map appearing inthe L-theoretic Farrell Jones Conjecture with coefficients in thering Z for fundamental groups of complete Riemannian manifoldswith non-positive sectional curvature). The assembly map appearingin the L-theoretic Farrell Jones Conjecture 11.4 with coefficients in the ringZ is split injective, if G is the fundamental group of complete Riemannianmanifold with non-positive sectional curvature.


The asymptotic dimension of a proper metric space X is the infimum overall integers n such that for any R > 0 there exists a cover U of X with theproperty that the diameter of the members of U is uniformly bounded andevery ball of radius R intersects at most (n + 1) elements of U (see [353,page 29]). The asymptotic dimension of a finitely generated group is theasymptotic dimension of its Cayley graph (and is independent of the choiceof set of finite generators.)

For a torsionfree group G with finite asymptotic dimension and a finitemodel for BG and any ring R the split injectivity of Hn(BG; K(R)) →

14.5 Injectivity Results in the Farrell-Jones Setting 441

Kn(RG) is proved by Bartels [80, Theorem 1.1] and by Carlsson-Goldfarb [162,Main Theorem on page 406]. The L-theory version is proved in Bartels [80,Section 7] as well, provided that there exists a natural number N withK−i(R) = 0 for i ≥ N .

The notion of finite decomposition complexity was introduced and stud-ied by Guentner-Tessera-Yu [357, 358]. It is a weaker notion than finiteasymptotic dimension. For a torsionfree group G with finite asymptotic di-mension and any ring R the split injectivity of Hn(BG; K(R)) → Kn(RG)

and Hn(BG; L〈−∞〉(R))→ L〈−∞〉n (RG) for a torsionfree group G with finite

model for BG and finite decomposition complexity is proved by Ramras-Tessera-Yu [678, Theorem 1.1] and Guentner-Tessera-Yu [357, page 334],provided that in the L-theory case there exists a natural number N withK−i(R) = 0 for i ≥ N .

Kasprowski [459, Theorem 8.1], proved for a group G with finite dimensio-nal model for EFIN (G) and finite quotient finite decomposition complexity, astrengthening of the notion of finite decomposition complexity, and a globalupper bound on the orders of the finite subgroups that the assembly mapHGn (EFIN (G); KR) → Kn(RG) is split injective for all n ∈ Z. An L-theory

version is proved in [459, Theorem 9.1].The paper [459] uses ideas of [78]. Kasprowski [459] points out a gap in the

proof of [78] which has the consequence that the results in [78] are only provedunder the additional assumption that there is a finite model for EFIN (G).

The papers by Kasprowski [461, 460] are based on [459] and lead to thefollowing two results.

Theorem 14.18 (Injectivity of the Farrell-Jones assembly map forFIN for subgroups of almost connected Lie groups). Let G be asubgroup of an almost connected Lie group. Suppose that G admits a finitedimensional model for the classifying space EFIN (G).

(i) Let A be an additive G-category. Then the assembly map

HGn (EFIN (G); KA)→ HG


)is split injective for all n ∈ Z;

(ii) Let A be an additive G-category with involution. Suppose that there existsN ≥ 0 such that π−i(KA(I(A))

)= 0 holds for all i ≥ N and all virtually

abelian subgroups A ⊆ G.Then the assembly map

HGn (EFIN (G); L

〈−∞〉A )→ HG

n (G/G; L〈−∞〉A ) = πn

(L〈−∞〉A (I(G))


(iii) A subgroup G of a virtually connected Lie group admits a finite dimensionalmodel for EFIN (G) if and only if there exists N ∈ N such that everyfinitely generated abelian subgroup of G has rank at most N .

442 14 Status

Proof. (i) and (ii) IfG is finitely generated, this is proved in [460, Theorem 1.1and Theorem 6.1]. Since every group is the union of its finitely generatedsubgroups, the general case for injectivity follows from Lemma 13.23 (ii).One obtains even split injectivity since also the retraction is natural, see [460,Section 7].

(iii) See [460, Proposition 1.3]. ut

Theorem 14.19 (Injectivity of the Farrell-Jones assembly map forFIN for linear groups). Let R be a commutative ring with unit and letG ⊆ GLn(R) be a subgroup. Suppose that G admits a finite dimensional modelfor the classifying space EFIN (G).

(i) Let A be any additive G-category. Then the assembly map

HGn (EFIN (G); KA)→ HG



(ii) Let A be any additive G-category with involution. Suppose that there existsN ≥ 0 such that π−i(KA(I(H)) = 0 holds for all i ≥ N and all virtuallynilpotent subgroups H ⊆ G.Then the assembly map

HGn (EFIN (G); L

〈−∞〉A )→ HG

n (G/G; L〈−∞〉A ) = πn

(L〈−∞〉A (I(G))

)is split injective for all n ∈ Z.

Proof. If G is finitely generated, this is proved in [461, Theorem 1.1]. Since ev-ery group is the union of its finitely generated subgroups, the general case forinjectivity follows from Lemma 13.23 (ii). One obtains even split injectivitysince also the retraction is natural, as explained in [460, Section 7]. ut

Split Injectivity of the K- and L-theoretic Farrell-Jones assembly map (fortrivial coefficients) is proved under certain conditions about the compactifi-cations of the model for the space of proper G-actions by Rosenthal [715,716, 717], based on techniques developed by Carlsson-Pedersen [163].

14.6 Status of the Novikov Conjecture

Recall that the Novikov Conjecture 7.131 holds for a group G if one of thefollowing conditions is satisfied:

• The assembly map

Hn(BG; L〈−∞〉(Z)) = HGn (EG; L〈−∞〉(Z))

→ HGn (G/G; L〈−∞〉(Z)) = L〈−∞〉n (ZG)

14.6 Status of the Novikov Conjecture 443

is rationally injective for all n ∈ Z, see Theorem 11.52 x• The assembly map

HGn (EFIN (G); L

〈−∞〉A )→ HG

n (G/G; L〈−∞〉A ) = πn

(L〈−∞〉A (I(G))

)is rationally injective, see Lemma 11.28 and Theorem 11.52 x;

• The L-theoretic Farrell-Jones Conjecture 11.4 with coefficients in the ringZ holds, see Theorem 11.52 (x);

• The assembly mapKn(BG)→ Kn(C∗r (G))

is split injective for all n ∈ Z, see Theorem 12.28;• The assembly map

KGn (EFIN (G)))→ Kn(C∗r (G))

is split injective for all n ∈ Z, see Lemma 11.28 and Theorem 12.28;• The Baum-Connes Conjecture 12.9 for G holds for G, see Theorem 12.28.

Hence all groups appearing in Theorems 14.1, 14.6, 14.11, 14.12, 14.14, 14.17,or 14.18 satisfy the Novikov Conjecture 7.131. In particular a group G sat-isfies Novikov Conjecture 7.131 if G is a (discrete) subgroup of one of thefollowing type of groups: Comment 56: Add the results by Engel [278] whoproves it for groups having polynomially bounded higher-order combinatorialfunctions. This includes all automatic groups.

• Hyperbolic groups (or more generally directed colimits of hyperbolicgroups)

• Finite dimensional CAT(0)-groups;• Almost connected Lie groups;• (Not necessarily cocompact) lattices in second countable locally compact

Hausdorff groups with finitely many path components;• Subgroups of GLn(F ) for a field F and a natural number n;• S-arithmetic groups;• mapping class groups;• Fundamental groups of complete Riemannian manifold with non-positive

sectional curvature;• Fundamental groups of (not necessarily compact) connected manifolds

(possibly with boundary) of dimension ≤ 3;• A-T-menable groups and hence also amenable and elementary amenable

groups;• One-relator groups;• Coxeter groups;• Thompson’s groups F , T and V ;• Artin’s full braid groups Bn;

444 14 Status

14.7 Review of and Status report for some classes ofgroups

14.7.1 Hyperbolic Groups

The definition and the basic properties of the notion of a hyperbolic group canbe found for instance in [129, 233, 344, 534]. Examples are free groups andfundamental groups of closed Riemannian manifolds with negative sectionalcurvature.

Almost all conjectures in this book about groups are satisfied for hyper-bolic groups, since they satisfy both the Full Farrell-Jones Conjecture 11.20,see Theorem 14.1 (ia), and the Baum-Connes Conjecture 12.11 with coeffi-cients, see Theorem 14.6 (ic).

14.7.2 Lacunary Hyperbolic Groups

A finitely generated group is a lacunary hyperbolic group if one of its asymp-totic cones is an R-tree, see Olshanskii-Osin-Sapir [635]. Since they are di-rected colimits of hyperbolic groups, they satisfy the Full Farrell-Jones Con-jecture 11.20, see Theorem 14.1 (ia) and (iid). It is not known whether thesatisfy the Baum-Connes Conjecture 12.9.

A lacunary hyperbolic group is finitely presented if and only if it is hyper-bolic. This is due to Kapovich-Kleiner, see [635, Theorem 8.1].

There are rather exotic examples of lacunary hyperbolic group. For in-stance a finitely generated torsionfree non-cyclic group, all whose propersubgroups are cyclic, is constructed by Ol’shanskii [633] which is a lacunaryhyperbolic group by [635, Theorem 1.1].

Other examples of lacunary hyperbolic groups are constructed in [35].These finitely generated groups do contain (in a weak sense) an infinite ex-pander. Hence they admit no coarse embedding into a Hilbert space (or intoany lp with 1 ≤ p < ∞) and any infinite dimensional linear representationof these groups has infinite image. Notice that for these group a counterex-ample to the Baum-Connes Conjecture with coefficients 12.11 is constructedby Higson-Lafforgue-Skandalis [388]. (This lead Baum-Guentner-Willet [93]to reformulate the Baum-Connes Conjecture with coefficients 12.11 by intro-ducing a new crossed product for which no counterexamples are known sofar.)

More examples of exotic lacunary hyperbolic groups are discussed in [635]and [731, Section 4].

14.7 Review of and Status report for some classes of groups 445

14.7.3 Relative Hyperbolic Groups

For the definition and basic information about relative hyperbolic groups werefer for instance to [125, 138, 281, 352, 638, 781, 782]

The following result is taken from [62, Remark 4.7], where the notion of arelative hyperbolic groups following Bowditch [125] is used.

Theorem 14.20 (The Full Farrell-Jones Conjecture and relativelyhyperbolic groups). Let G be a countable group which is relatively hy-perbolic to the subgroups P1, P2, . . . , Pn. If P1, P2, . . . , Pn satisfy the FullFarrell-Jones Conjecture 11.20, then G satisfies the Full Farrell-Jones Con-jecture 11.20.

14.7.4 Systolic Groups

Let G be a group which acts cocompactly and properly on a systolic complexwith the Isolated Flats Property by simplicial automorphisms. Then G isrelatively hyperbolic to the family of virtually abelian groups by Elsner [272,Theorem B]. Hence Theorem 14.20 implies that G satisfies the Full Farrell-Jones Conjecture 11.20. Comment 57: Does one need the Isolated FlatsProperty for a systolic group to satisfy Full Farrell-Jones Conjecture 11.20?It is needed to argue as above, see [272, Theorem 5.17].

14.7.5 Finite dimensional CAT(0) Groups

A CAT(0)-group is a group admitting a cocompact proper isometric actionon a CAT(0)-space X. We call it a finite dimensional CAT(0)-group if we canadditionally arrange that X has finite topological dimension. Basic propertiesof this notion of a can be found for instance in [129, 534]. Examples for finitedimensional CAT(0)-groups are fundamental groups of closed Riemannianmanifolds with non-positive sectional curvature.

A finite dimensional CAT(0)-group satisfies the Full Farrell-Jones Conjec-ture 11.20, see Theorem 14.1 (ib).

It is not known whether every finite dimensional CAT(0)-group satisfy theBaum-Connes Conjecture 12.11 with coefficients or the Baum-Connes Con-jecture 12.9. If G admits a cocompact proper isometric action on a CAT(0)-space with the isolated flat property in the sense of [182, Definition 3.1], thenthe Baum-Connes Conjecture 12.9 holds for G, see [182, Corollary 0.3 b].

Limit groups as they appear for instance in [747] have been in the focusof geometric group theory for the last years. Expositions about limit groupsare for instance [173, 647]. Alibegovic-Bestvina have shown that limit groups

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are CAT(0)-groups [19]. It is nord hard to check that their proof shows thata limit group is even a finite dimensional CAT(0)-group.

Right-angled Artin groups G are finite dimensional CAT(0)-groups, theuniversal covering of the Salvetti complex associated to G is a finite dimen-sional complete CAT(0)-space on which G acts cocompactly, properly andisometrically, see Charney-Davis [179]. Comment 58: Check the referenceand make it more precise or refer to [178, Theorem 3.6]. For more informationabout right-angled Artin groups for refer for instance to [178].

14.7.6 Fundamental Groups of Complete Riemannian Manifoldswith Non-Positive Sectional Curvature

Let π be the fundamental group of a complete Riemannian manifold M . Letsec denote its sectional curvature.

• M closed and sec(M) < 0If M is closed and has negative sectional curvature, then π is hyperbolicand hence satisfies both the Full Farrell-Jones Conjecture 11.20, see The-orem 14.1 (ia), and the Baum-Connes Conjecture 12.11 with coefficients,see Theorem 14.6 (ic).

• M closed and sec(M) ≤ 0If M is closed and has non-positive sectional curvature, then π is a fi-nite dimensional CAT(0)-group and satisfies the Full Farrell-Jones Con-jecture 11.20, see Theorem 14.1 (ib). It is not known whether all such πsatisfy the Baum-Connes Conjecture 12.11 with coefficients or the Baum-Connes Conjecture 12.9.

• C1 ≤ sec(M) ≤ C2 < 0 and finite volumeLet M be a complete Riemannian manifold, which is pinched negativelycurved and has finite volume. Then π satisfies the Full Farrell-Jones Con-jecture 11.20 since π is relatively hyperbolic with respect to the familyof virtually finitely generated nilpotent groups, see [125] Comment 59:Make the last reference more precise. or [281, Theorem 4.11]. and we canapply Theorem 14.1 (ic), and Theorem 14.20.If we additionally assume that the curvature tensor has bounded deriva-tives, then also the Baum-Connes Conjecture 12.9 holds for G by [182,Corollary 0.3 a]. Lattices G in rank one Lie groups are examples.

• M A-regular and sec(M) ≤ 0A complete Riemannian manifold M is called A-regular if there exists asequence of positive real numbers A0, A1, A2, . . . such that ||∇nK|| ≤ An,where ||∇nK|| is the supremum-norm of the n-th covariant derivative ofthe curvature tensor K. Every locally symmetric space is A-regular since∇K is identically zero.Let M be a complete Riemannian manifold with non-positive sectionalcurvature which is A-regular. Then π = π1(M) satisfies the K-theoretic


Farrell-Jones Conjecture 11.1 with coefficients in the ring Z in degree n ≤ 1and L-theoretic Farrell-Jones Conjecture 11.4 with coefficients in the ringwith involution Z, see [299, Proposition 0.10 and Lemma 0.12]. Since π is

torsionfree, this implies that Wh(π), K0(Zπ) and Kn(Zπ) for n ≤ −1 allvanish and Conjecture 7.109 holds for R = Z.

• C1 ≤ sec(M) ≤ C2 < 0Let M be a complete Riemannian manifold with pinched negative curva-ture. Then there is another Riemannian metric for which M is negativelycurved complete and A-regular. This fact is mentioned in [299, page 216]and attributed there to Abresch [2] and Shi [753]. Hence the conclusionsabove for complete Riemannian manifold M with non-positive sectionalcurvature which is A-regular do also hold for pinched negatively curvedcomplete Riemannian manifolds.

• sec(M) ≤ 0If M is a complete Riemannian manifold with non-positive sectional curva-ture, we have already stated some injectivity results for π in Theorem 14.12and Theorem 14.17.In particular π satisfies the Novikov Conjecture 7.131 by Theorem 11.52 (x)or Theorem 12.28.

14.7.7 Lattices

A discrete subgroup G of a locally compact second countable Hausdorff groupΓ is called a lattice if the quotient space Γ/G has finite covolume with respectto the Haar measure of Γ .

Every lattice satisfies the Full Farrell-Jones Conjecture 11.20, see Theo-rem 14.1 (id).

It is a big open problem to decided whether lattices satisfy the Baum-Connes Conjecture 12.11 with coefficients or the Baum-Connes Conjec-ture 12.9. This is not even known for cocompact lattices in Lie groups withfinitely many path components. The case SLn(Z) is still open for n ≥ 3.By [182, Corollary 0.3 a] lattices G in rank one Lie groups satisfy the Baum-Connes Conjecture 12.9. Some other lattices satisfying the Baum-ConnesConjecture 12.9 come from Theorem 14.11.

14.7.8 S-Arithmetic Groups

Every S-arithmetic group satisfies the Full Farrell-Jones Conjecture 11.20, seeTheorem 14.1 (ig). This is not known for the Baum-Connes Conjecture 12.11with coefficients or the Baum-Connes Conjecture 12.9, the group SLn(Z) forn ≥ 3 is still an open problem.

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14.7.9 Linear Groups

The Full Farrell-Jones Conjecture 11.20, and actually even the K-theoreticFarrell-Jones Conjecture 11.2 with coefficients in rings and the L-theoreticFarrell-Jones Conjecture 11.6 with coefficients in rings with involution, areopen for linear groups, i.e., GLn(F ) for some field F , in general The samestatement holds for the Baum-Connes Conjecture.

The Novikov-Conjecture holds by Theorem 12.28 and Theorem 14.14 (ii)and Exercise 14.15 for any subgroup of GLn(F ) for a field F .

14.7.10 Subgroups of Almost Connected Lie Groups

The Full Farrell-Jones Conjecture 11.20, and actually even the K-theoreticFarrell-Jones Conjecture 11.2 with coefficients in rings and the L-theoreticFarrell-Jones Conjecture 11.6 with coefficients in rings with involution, areopen for subgroups of almost connected Lie groups in general. The samestatement holds for the Baum-Connes Conjecture 12.9.

The Novikov-Conjecture holds by Theorem 12.28 and Theorem 14.14 (iii)and Exercise 14.15 for any subgroup of an almost connected Lie group.

14.7.11 Virtually Solvable Groups

Virtually solvable groups satisfy both the Full Farrell-Jones Conjecture 11.20,see Theorem 14.1 (ic) and the Baum-Connes Conjecture 12.11 with coeffi-cients, see Theorem 14.6 (ia).

14.7.12 A-T-menable, Amenable and Elementary AmenableGroups

A group G is called amenable if there is a (left) G-invariant linear operatorµ : L∞(G,R)→ R with µ(1) = 1 which satisfies for all f ∈ l∞(G,R)

inff(g) | g ∈ G ≤ µ(f) ≤ supf(g) | g ∈ G.

The latter condition is equivalent to the condition that µ is bounded andµ(f) ≥ 0 if f(g) ≥ 0 for all g ∈ G.

The class of elementary amenable groups is defined as the smallest classof groups which has the following properties:

(i) It contains all finite and all abelian groups;


(ii) It is closed under taking subgroups;(iii) It is closed under taking quotient groups;(iv) It is closed under extensions, i.e., if 1 → H → G → K → 1 is an exact

sequence of groups and H and K belong to the class, then also G;(v) It is closed under directed unions, i.e., if Gi | i ∈ I is a directed system

of subgroups such that G =⋃i∈I Gi and each Gi belongs to the class, then

G belongs to the class.

Since the class of amenable groups has all the properties mentioned above,every elementary amenable group is amenable. The converse is not true. Formore information about amenable and elementary amenable groups we referfor instance to [527, Section 6.4.1] or [645].

A group G is a-T-menable, or, equivalently, has the Haagerup property ifG admits a metrically proper isometric action on some affine Hilbert space.Metrically proper means that for any bounded subset B the set g ∈ G |gB ∩B 6= ∅ is finite.

An extensive treatment of such groups is presented in [187]. Any a-T-menable group is countable. The class of a-T-menable groups is closed un-der taking subgroups, under extensions with finite quotients and under fi-nite products. It is not closed under semi-direct products. Examples of a-T-menable groups are countable amenable groups, countable free groups, dis-crete subgroups of SO(n, 1) and SU(n, 1), Coxeter groups, countable groupsacting properly on trees, products of trees, or simply connected CAT(0) cu-bical complexes. A group G has Kazhdan’s property (T) if, whenever it actsisometrically on some affine Hilbert space, it has a fixed point. An infinitea-T-menable group does not have property (T). Since SL(n,Z) for n ≥ 3 hasproperty (T), it cannot be a-T-menable.

Every a-T-menable, every amenable, and every elementary-amenable groupsatisfies the Baum-Connes Conjecture 12.11 with coefficients. This followsfrom Theorem 14.6 (ia) in the a-T-menable case. Since every group is thedirected union of its finitely generated subgroups, every finitely generatedgroup is countable, and every countable amenable group is a-T-menable, theclaim follows for amenable groups and hence also for elementary amenablegroups from Theorem 14.6 (iid).

The Full Farrell-Jones Conjecture 11.20, and actually even the K-theoreticFarrell-Jones Conjecture 11.2 with coefficients in rings and the L-theoreticFarrell-Jones Conjecture 11.6 with coefficients in rings with involution, areopen for elementary amenable groups. The main problem in the Farrell-Jonessetting is that one has to deal with virtually cyclic subgroups in its formula-tion and for the inheritance property under extensions, see Theorem 14.6 (iic),whereas in the Baum-Connes setting finite subgroups suffice. The Full Farrell-Jones Conjecture 11.20 follows by an easy argument for elementary amenablegroups with finite Hirsch length directly from the inheritance properties andthe virtually solvable case, as explained in Wu [841].

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The L-theoretic Farrell-Jones Conjecture 11.4 with coefficients in any ringwith involution R holds after inverting two for elementary amenable groupsby [379, Proposition 5.2.1].

14.7.13 Three-Manifold Groups

Let M be a (not necessarily compact) manifold (possibly with boundary) ofdimension ≤ 3. Then π1(M) satisfies the Full Farrell-Jones Conjecture 11.20,see Theorem 14.1 (ie).

If we additionally assume that M is compact, π1(M) satisfies the Baum-Connes Conjecture 12.11 with coefficients, see Theorem 14.6 (ie).

The reason why in the Farrell-Jones etting we do not need compact, is theinheritance property under directed colimits of directed systems of subgroups,see Theorem 14.1 (iid), which is not available in the Baum-Connes setting,where we need that all structure maps are injective, see Theorem 14.6 (iid).

Exercise 14.21. Let G be the fundamental group of a knot complementand let R be a regular ring. Show that the projection pr : G→ G/[G,G] ∼= Zinduces for every ring R an isomorphism Kn(RG) → Kn(R[G/[G,G]]) andwe get an isomorphism Kn(RG) ∼= Kn(R)⊕Kn−1(R).

14.7.14 One-Relator Groups

The Baum-Connes Conjecture 12.11 with coefficients holds for one-relatorgroups by Theorem 14.6 (id).

The Full Farrell-Jones Conjecture 11.20, and actually even the K-theoreticFarrell-Jones Conjecture 11.2 with coefficients in rings and the L-theoreticFarrell-Jones Conjecture 11.6 with coefficients in rings with involution, areopen for torsionfree one-relator groups. Notice that not all one-relator groupsare solvable, hyperbolic or finite dimensional CAT(0)-groups, so that we can-not apply Theorem 14.1 in general. Nevertheless theK-theoretic Farrell-JonesConjecture 11.1 with coefficients in the ring R is known if R is regular andG is a one-relator group by [806, Theorem 19.4 on page 249] in the con-nective case and by [69, Corollary 0.12] for the non-connective version. TheL-theoretic Farrell-Jones Conjecture 11.4 with coefficients in any ring with in-volution R holds after inverting two for torsionfree one-relator groups by [153,Corollary 8].

A consequence of Newman’s spelling theorem (see [617]) is that a one-relator groups, which is not torsionfree, is hyperbolic and hence satisfies theFull Farrell-Jones Conjecture 11.20 by Theorem 14.1 (ia).


All Baumslag-Solitar groups satisfy the Full Farrell-Jones Conjecture 11.20,see Farrell-Wu [305] for the version without “finite wreath products” andGandini-Meinert-Ruping [337, Corollary 1.1].

14.7.15 Selfsimilar Groups

We use the notion of selfsimilar group as presented in [81, Section 3] which isslightly general than the classical notion defined for instance in [82, 616]. Self-similar groups are groups acting in a recursive manner on a regular rootedtree RTd. If the recursion of every element involves only a linearly growingsubtree of Td, the group is said to be bounded.

The Full Farrell-Jones Conjecture 11.20 is proved by Bartholdi [81, The-orem A] for bounded selfsimilar groups since these are subgroups of finitedimensional CAT(0)-groups and hence Theorem 14.1 (ib) and (iia) applies.Using Theorem 14.1 (ib) and (iib) Bartholdi [81, Theorem C] proves theFull Farrell-Jones Conjecture 11.20 for the Aleshin-Grigorchuk group, Gupta-Sidki groups, and generalized Grigorchuk groups, whose definition is reviewedin [81, Section 4]

14.7.16 (Strongly) Poly-Surface Groups

Definition 14.22 ((Strongly) poly-surface group). Let G be a groupwith a finite filtration 1 = G0 ⊆ G1 ⊆ . . . ⊆ Gd = G.

We call G strongly poly-surface if the filtration satisfies the following con-ditions:

(i) Gi is normal in G for i = 0, 1, 2, . . . , d;(ii) For every i ∈ 1, 2, . . . , d and g ∈ G, there is a (not necessarily compact)

surface S (possibly with boundary) with torsionfree π1(S), a diffeomor-

phism f : S → S, an isomorphism, and an isomorphism α : Gi/Gi−1

∼=−→π1(S) such that the following diagram commutes

Gi/Gi−1

cg //

α

Gi/Gi−1

α

π1(S)

π1(f) // π1(S)

where cg is induced by conjugation with g ∈ G.

Theorem 14.23 (The Full Farrell-Jones Conjecture for StronglyPoly-surface groups). A poly surface group G satisfies the Full Farrell-Jones Conjecture 11.20.

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Proof. Fix a filtration 1 = G0 ⊆ G1 ⊆ . . . ⊆ Gd = G as it occurs in Def-inition 14.22. We show by induction over i = 0, 1, 2, . . . , d that G/Gd−i istorsionfree and satisfies the Full Farrell-Jones Conjecture 11.20. The induc-tion beginning i = 0 is trivial, the induction step from (i − 1) to i done asfollows.

Consider the exact sequence 1→ Gd−i+1/Gd−i → G/Gd−ip−→ G/Gd−i+1 →

1. By induction hypothesis G/Gd−i+1 is torsionfree and satisfies the the FullFarrell-Jones Conjecture 11.20. Since Gd−i+1/Gd−i ∼= π1(S) is torsionfree,G/Gd−i is torsionfree. Consider any virtually cyclic subgroup V ⊆ G/Gd−i+1.Since G/Gd−i+1 is torsionfree, V is infinite cyclic and we can choose g ∈ Gsuch that the image of g under p : G/Gd−i → G/Gd−i+1 sends g to a gen-erator of C. Hence p−1(V ) is isomorphic to Gd−i+1/Gd−i ocg Z. From theassumptions about G, we get a diffeomorphism f : S → S of a surface S suchthat p−1(V ) is isomorphic to π1(Tf ). Since Tf is a 3-manifold, π1(Tf ) satisfiesthe Full Farrell-Jones Conjecture 11.20 by Theorem 14.1 (ie). We concludefrom Theorem 14.1 (iic) that G/Gd−i satisfies the Full Farrell-Jones Conjec-ture 11.20. ut

Exercise 14.24. Let G be a group with a filtration 1 = G0 ⊆ G1 ⊆ . . . ⊆Gd = G such that Gi−1 is normal in Gi and Gi/Gi−1 is torsionfree andisomorphic to the fundamental group of a compact manifold of dimension≤ 3 (possibly with boundary). Show that the Baum-Connes Conjecture 12.11with coefficients holds for G.

14.7.17 Poly-Free Groups

A group G is called poly-free if there is a finite filtration 1 = G0 ⊆ G1 ⊆. . . ⊆ Gd = G such that Gi−1 ⊆ Gi is normal and Gi/Gi−1 is free fori = 1, 2, . . . , d. The Baum-Connes Conjecture 12.11 with coefficients holdsfor poly-free groups G by Theorem 14.6 (iic) (iid) and (iie).

For the Full Farrell-Jones Conjecture 11.20 the knowledge about poly-free

groups is rather poor. For instance, if α : Fn∼=−→ Fn is an automorphism of the

free group Fn of rank n, then the Full Farrell-Jones Conjecture 11.20 is notknown for the semi-direct product Fnoα Z for all α, only for those for whichone knowns that Fn oα Z is hyperbolic, a CAT(0)-group or implemented byan automorphism of a compact surface with boundary. The problem is that incontrast to the Baum-Connes setting the inheritance under extensions doesnot apply in this case, since we have to deal in the Farrell-Jones setting withvirtually cyclic groups and not only with finite groups as in the Baum-Connessetting.


14.7.18 Braid Groups

Artin’s full braid groups Pn satisfies satisfy both the Full Farrell-Jones Con-jecture 11.20, see Theorem 14.1 (ik) and the Baum-Connes Conjecture 12.11with coefficients, see Theorem 14.6 (if).

14.7.19 Coxeter Groups

For the definition of and information about Coxeter groups we refer to [230].Every Coxeter group satisfies the Full Farrell-Jones Conjecture 11.20 by The-orem 14.1 (il) and the Baum-Connes Conjecture 12.11 with coefficients byTheorem 14.6 (ih).

14.7.20 Mapping Class Groups

The Full Farrell-Jones Conjecture 11.20 is know for the mapping class groupof any oriented surface of finite type, Comment 60: Does this include allΓ sg,r? Depending on the answer, we may adjust the text below. whereas theBaum-Connes Conjecture 12.9 is open for mapping class groups.

Let Ssg,r be the orientable compact surface of genus g with r boundarycomponents and s punctures, where s puncture means the choice of s pair-wise distinct points. Let Diff(Ssg,r, rel) be the group of orientation preserv-ing diffeomorphisms Ssg,r → Ssg,r which leave the boundary and the punc-tures pointwise fixed. Then the mapping class group Γ sg,r is defined to beπ0(Diff(Ssg,r, rel)), the group of isotopy classes of such diffeomorphisms. Weabbreviate Γg := Γ 0

g,0. For 2g + r + s > 2 there are exact sequence

(14.25) 1→ π1(Ssg,r)i−→ Γ s+1

g,rp−→ Γ sg,r → 1

such that for f ∈ Diff(Ss+1g,r , rel) conjugation with f induces an automorphism

cf : Γ s+1g,r → Γ s+1

g,r such that i π1(f) = cf i holds, and for 2g + r + s > 2and s ≥ 1 there are exact sequences

(14.26) 1→ Z→ Γ s−1g,r+1 → 1→ Γ sg,r → 1,

see for example Mislin [605, page 266].

Exercise 14.27. Consider g ≥ 2. Show that the Full Farrell-Jones Conjec-ture 11.20 for Γg implies the Full Farrell-Jones Conjecture 11.20 for Γ sg,r forall r, s ≥ 0.

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14.7.21 Out(Fn)

The Full Farrell-Jones Conjecture 11.20, and actually even the K-theoreticFarrell-Jones Conjecture 11.2 with coefficients in rings and the L-theoreticFarrell-Jones Conjecture 11.6 with coefficients in rings with involution, areopen for Out(Fn) for n ≥ 3. The same statement holds for the Baum-ConnesConjecture. Not even the Novikov Conjecture 7.131 has been proved forOut(Fn) for n ≥ 3.

At least the rational injectivity of the K-theoretic Farrell-Jones assemblymap with coefficients in Z (disregarding some K−1-term contribution) followsfrom [549] for Out(Fn).

14.7.22 Thompson’s Groups

Thompson defined the groups F , T , and V in some handwritten notess from1965. Thompsons group V is the group of right-continuous automorphisms fof [0, 1] that map dyadic rational numbers to dyadic rational numbers, thatare differentiable except at finitely many dyadic rational numbers, and suchthat, on each interval on which f is differentiable, f is affine with derivative apower of 2. The group F is the subgroup of V consisting of homeomorphisms.The group T is the subgroup of V consisting of those elements which inducehomeomorphisms of the circle, where the circle is regarded as [0, 1] with0 and 1. identified. These groups have some unusual properties. It is anopen question, whether F is amenable. It is known that F is not elementaryamenable.

Farley [283] has shown that F , T , and V are a-T-menable and hence satisfythe Baum-Connes Conjecture 12.11 with coefficients, see Theorem 14.6 (ia).

The Full Farrell-Jones Conjecture 11.20, and actually even the K-theoreticFarrell-Jones Conjecture 11.2 with coefficients in rings and the L-theoreticFarrell-Jones Conjecture 11.6 with coefficients in rings with involution, areopen for F , T , and V .

At least the rational injectivity of the K-theoretic Farrell-Jones assemblymap with coefficients in Z (disregarding some K−1-term contribution) followsfrom [549] for T using [340].

14.7.23 Groups Satisfying Homological Finiteness Conditions

So far the groups for which we were able to prove the Farrell-Jones Conjectureor the Baum-Connes Conjecture satisfy some geometric condition, often areminiscence of non-positive sectional curvature. At least for the K-theoreticFarrell-Jones Conjecture there are results, where no geometric conditions but


some finiteness conditions are required. The celebrated prototype of such aresult is the following theorem due to Boekstedt-Hsiang-Madsen [118].

Theorem 14.28 (Bokstedt-Hsiang-Madsen Theorem). Let G be agroup such that Hi(G;Z) is finitely generated for all i ≥ 0. Then G satis-fies the K-theoretic Novikov Conjecture 11.50, i.e., the assembly map

Hn(BG; K(Z))→ Kn(ZG)


This raises the question under which finiteness conditions one can showthat the assembly map appearing in the K-theoretic Farrell-Jones Conjec-ture 11.1 with coefficients in the ring Z is rationally injective. The followingresult taken from Luck-Steimle [559, Theorem 0.3] is interesting in this con-text since Z is a regular ring.

Theorem 14.29 (Rational bijectivity of the relative assembly mapfrom FIN to VCY for regular rings R as coefficients). Let G be agroup and let R be a regular ring.


HGn (EFIN (G); KG

R)∼=−→ HG

n (EVCY(G); KGR)

is rationally bijective for all n ∈ Z.

The source of the assembly map appearing in Theorem 14.29 has al-ready been computed rationally using equivariant Chern characters in The-orem 10.70

(14.30)⊕p+q=n

⊕(C)∈J

Hp(CGC;Q)⊗Q[WGC] ΘC ·(Q⊗Z Kq(ZC)

)∼=−→ Q⊗Z H

Gn (EFIN (G); KZ);

Using Theorem 14.29 and the isomorphism (14.30) the assembly map ap-pearing in the Farrell-Jones Conjecture 11.1 with coefficients in the ring Zbecomes rationally a map(14.31)⊕p+q=n

⊕(C)∈J

Hp(CGC;Q)⊗Q[WGC] ΘC ·(Q⊗Z Kq(ZC)

)→ Q⊗Z Kn(ZG)

So the question above is equivalent to the question whether the mapof (14.31) is rationally injective.

Its restriction to the summand corresponding to C = 1 is rationally thesame as the map appearing in in Theorem 14.28. Hence a positive answer tothe question above implies Theorem 14.28.

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The main result of [549] says that under certain finiteness assumptions,which are for instance satisfied if there is a model for EFIN (G) of finite type,and certain number theoretic conditions which are implied by the Leopoldt-Schneider Conjecture, the assembly map (14.31) is rational injective if weignore the summands for q = −1. This summand cannot be detected since to-pological cyclic homology does not see K−1. Notice that in the Theorem 14.30just detects the summand for C = 1 and does not see the ones for nnn-trivial C. The methods and proofs of [549] are based on the ideas of [118].

As an illustration we mention two easy to formulate consequences of theresults of [549, Main Theorem 1.13], where the necessary input from numbertheory is known to be true and therefore does not appear in the assumptions,similar to the situation in Theorem 14.28.

Theorem 14.32 (Rationally injectivity of the colimit map for finitesubgroups for the Whitehead group). Let G be a group. Assume thatfor every finite cyclic subgroup C of G the abelian groups H1(BZGC;Z) andH2(BZGC;Z) are finitely generated.


colimH∈SubG(FIN ) Q⊗Z Wh(H)→ Q⊗Z Wh(G)

is injective.

Theorem 14.33 (Eventual injectivity of the rational K-theoretic as-sembly map for R = Z). Let G be a group. Assume that there is a finiteG-CW -model for EFIN (G).

Then there exists an integer L > 0 such that the rationalized Farrell-Jonesassembly map (14.31) is injective for all n ≥ L. The bound L only dependson the dimension of EFIN (G) and on the orders of the finite cyclic subgroupsof G.

14.8 Open Cases

Here is a list of interesting groups for which the Full Farrell-Jones Conjec-ture 11.20 is open in general:

• elementary amenable, amenable, a-T-menable groups;• Out(Fn);• Fn oφ Z for automorphisms of the free groups Fn of rank n; Comment

61: Probably meanwhile proven by Bestvina-Fujiwara-Wigglesworth. Wuhas extended this to F o Z for any free group F .

• Thompson’s groups F , V , and T ;• One-relator groups;• linear groups;

14.9 How Can We Find Counterexamples? 457

• subgroups of almost connected Lie groups;• residual finite groups;

Here is a list of interesting groups for which the Baum-Connes Conjec-ture 12.11 with coefficients is open in general:

• Finite dimensional CAT(0)-groups.• fundamental groups of closed Riemannian manifolds with non-positive sec-

tional curvature;• lattices in almost connected Lie groups, for instance SLn(Z) for n ≥ 3;• S-arithmetic groups;• Out(Fn) for n ≥ 3;• mapping class groups (of higher genus);• linear groups;• subgroups of almost connected Lie groups;• residual finite groups;

14.9 How Can We Find Counterexamples?

There is no group known for which the Full Farrell-Jones Conjecture 11.20 isknow to be false, but it is not known for all groups. The same statement holdsfor the Baum-Connes Conjecture 12.9 and the Novikov Conjecture 7.131.

We have already mentioned that the groups which come from the construc-tion of Arzhantseva-Delzant [35] and yield counterexamples to the Baum-Connes Conjecture with coefficients 12.11 by Higson-Lafforgue-Skandalis [388],are colimits of hyperbolic groups and hence satisfy the Full Farrell-Jones Con-jecture 11.20 by Theorem 14.1 (ia) and (iid). Baum-Guentner-Willet [93] givea reformulation of the Baum-Connes Conjecture with coefficients 12.11 by in-troducing a new crossed product for which no counterexamples are known sofar. We have already discussed the problem about the Baum-Connes Conjec-ture 12.9, which does not occur for the Full Farrell-Jones Conjecture 11.20,that the left hand side of the Baum-Connes Conjecture is functorial undergroup homomorphism and there is no reason why the right hand side shouldhave this property, see Remark 12.12. The new version of Baum-Guentner-Willet [93] still faces this problem.

One may wonder whether all the conjectures above do or do not hold for allgroups. Here are some arguments why it is not easy to find counterexamples.

14.9.1 Exotic Groups

We have mentioned above some groups with exotic properties for which theFull Farrell-Jones Conjecture 11.20 is true, such as finitely generated tor-

458 14 Status

sionfree infinite groups all of whose proper subgroups are cyclic, groups withexpanders, and selfsimilar groups. Comment 62: List more typical examplesof such properties and give references to groups which do have this propertyand do satisfy the Farrell-Jones Conjecture, e.g., growth conditions, prop-erty (T), torsionfree simple, no finite-dimensional models models for BG orBG. Given a large enough prime p, there exists an infinite finitely gener-ated group all of whose proper subgroups are finite cyclic group of order p,see [634]. These groups are not necessarily lacunary hyperbolic but it seemsto be known to experts such as Denis Osin that they are directed colimits ofhyperbolic groups and hence satisfy the Full Farrell-Jones Conjecture 11.20.

The Full Farrell-Jones Conjecture 11.20 is known for any subgroup G ofa finite direct product of right-angled Artin groups by Theorem 14.1 (ia)(iia) since right-angled Artin group is a finite dimensional CAT(0)-group,see Charney-Davis [179]. Comment 63: Check the reference and make itmore precise, or refer to [178, Theorem 3.6]. Such groups G can have exoticproperties as explained in Bridson [128]. There are examples of such groupsG which are finitely presented and have unsolvable conjugacy problem.

One does not know of a property of a group for which one may expect thatgroups with this property may be counterexamples to the Full Farrell-JonesConjecture 11.20 or to the Baum-Connes Conjecture 12.9.

Also the results about groups with some homological finiteness conditionsof Subsection 14.7.23 indicate that the search for counterexamples for theFarrell-Jones Conjecture is not easy.

In order to find counterexamples one seems to need completely new ideas,maybe from random groups or logic.

14.9.2 Infinite Direct Products

Nothing is known about infinite products. It would be very interesting ifone can show that for family of groups Gi | i ∈ I (with infinite I) theFull Farrell-Jones Conjecture 11.20 is true for the direct product

∏i∈I Gi if

it holds for each Gi. (Notice that the corresponding statement is true forthe direct sum

⊕i∈I Gi by Theorem 14.1 (iib) and (iid).) In view of Theo-

rem 14.1 (iia) this would imply that the Full Farrell-Jones Conjecture 11.20 isstable under inverse limits over directed systems of groups. This would havethe immediate consequence that the Full Farrell-Jones Conjecture 11.20 istrue for all residually finite groups. On the other hand it may be worthwhileto look at infinite direct products in order to find a counterexample.

14.9 How Can We Find Counterexamples? 459

14.9.3 Exotic Closed Aspherical Manifolds

One may look also for counterexamples to one of the conjecture which fol-low from the Full Farrell-Jones Conjecture 11.20, for instance to the BorelConjecture 7.152. There are indeed closed aspherical manifolds with unusualproperties.

Davis constructed for every n ≥ 4 closed aspherical manifolds M whoseuniversal cover is not homeomorphic to Euclidean space [229, Corollary 15.8].In particular, these manifolds do not support metrics of non-positive sectionalcurvature. The fundamental groups of these examples are finite index sub-groups of Coxeter groups W . Thus they satisfy the Full Farrell-Jones Con-jecture 11.20 and the manifolds M are indeed topologically rigid, providedthat n ≥ 5.

Davis and Januszkiewicz [232, Theorem 5b.1] used Gromov’s hyperboliza-tion technique to construct for every n ≥ 5 a closed aspherical n-dimensionalmanifold M such that the universal covering M is a finite dimensionalCAT(0)-space whose fundamental group at infinite is non-trivial. In particu-lar, these universal covers are not homeomorphic to Euclidean space. Becausethese examples are in addition non-positively curved polyhedron, their funda-mental groups are finite dimensional CAT(0)-groups. There is a variation ofthis construction that uses the strict hyperbolization of Charney-Davis [180]and produce a closed aspherical manifolds M whose universal cover is nothomeomorphic to Euclidean space and whose fundamental group is hyper-bolic. The fundamental groups of these manifolds M satisfy the Full Farrell-Jones Conjecture 11.20 and these manifolds M are topologically rigid.

Davis-Januszkiewicz [232, Theorem 5a.1 and Corollary 5a.4] construct a 4-manifold N such that π1(N) is a finite dimensional CAT(0)-group and N×T kfor k ≥ 1 is not homotopy equivalent to a PL-manifold Since π1(N × T k) isa finite dimensional CAT(0)-groups and dim(N × T k) ≥ 5 for k ≥ 1, themanifolds N × T k for k ≥ 1 are topologically rigid.

Belegradek [97, Corollary 5.1], and Weinberger (see [228, Section 13]) haveshown that for every n ≥ 4 there is a closed aspherical manifold of dimensionn whose fundamental group has an unsolvable word problem. Notice that afinitely presented group with unsolvable word problem is not a CAT(0)-group,not hyperbolic, not automatic, not asynchronously automatic, not residuallyfinite and not linear over any commutative ring (see [97, Remark 5.2]). Sowe do not know whether it satisfies the Full Farrell-Jones Conjecture 11.20or the Borel Conjecture 7.152.

The proof of the results above is based on the reflection group trick as itappears for instance in [228, Sections 8,10 and 13]. It can be summarized asfollows.

Theorem 14.34 (Reflection group trick). Let G be a group which pos-sesses a finite model for BG. Then there is a closed aspherical manifold Mand a map i : BG→M and r : M → BG such that r i = idBG.

460 14 Status

An interesting immediate consequence of the reflection group trick is thatmany well-known conjectures about groups hold for every group which pos-sesses a finite model for BG if and only if it holds for the fundamental groupof every closed aspherical manifold, see also [228, Sections 11].

Exercise 14.35. Let R be a regular ring. Suppose that the K-theoreticFarrell-Jones Conjecture 11.1 with coefficients in the ring R holds for thefundamental group of any aspherical closed orientable manifold. Show thatit then holds for all groups G with a finite model for BG. If R is a ring withinvolution, the analogous statement is true for the L-theoretic Farrell-JonesConjecture 11.4 with coefficients in the ring with involution R.

14.9.4 Some Results Which Hold for All Groups

Here is a result which holds for all (discrete) groups, is non-trivial and re-lated to the Farrell-Jones Conjecture. Let i : H → G be the inclusion of anormal subgroup H ⊂ G. It induces a homomorphism i0 : Wh(H)→Wh(G).The conjugation action of G on H and on G induces a G-action on Wh(H)and on Wh(G) which turns out to be trivial on Wh(G). Hence i0 induceshomomorphisms

i1 : Z⊗ZG Wh(H)→Wh(G);(14.36)

i2 : Wh(H)G →Wh(G).(14.37)

Theorem 14.38 (Rational injectivity of Z ⊗ZG Wh(H) → Wh(G) fornormal finite H ⊆ G). Let i : H → G be the inclusion of a normal finitesubgroup H into an arbitrary group G. Then the maps i1 and i2 definedin (14.36) and (14.37) have finite kernel.

Proof. See [527, Theorem 9.38 on page 354]. ut

We omit the details of the proof that the result of Theorem 14.38 canbe also deduced from the K-theoretic Farrell-Jones Conjecture 11.1 withcoefficients in the ring Z.

Exercise 14.39. Let G be a group with vanishing Whitehead group. Showthat each element in the center has order order 1, 2, 3, 4 or 6.

Another non-trivial consequence of the Farrell-Jones Conjecture whichholds for all groups has been discussed in Remark 1.84.

Furthermore, Yu [851], see also Cortinas-Tartaglia [203], proves the injec-tivity of the K-theoretic assembly map Hn(BG; KS) → Kn(SG) for everygroup G, where S is the ring of Schatten class operators of an infinite dimen-sional separable Hilbert space.


14.10 Miscellaneous

The current status of the Farrell-Jones Conjecture for A-theory with coeffi-cients (see Conjecture 13.55), where one can additionally allow finite wreathproducts, is the same as the one as described in Theorem 14.1. This is followsfrom [280] and [465].

Injectivity results about theA-theoretic assembly mapHn(BG; A(•))→A(BG) for certain groups possessing nice compactifications of EG can befound in [164, Theorem A].

We have explained in Theorem 13.88 that a group, which satisfies theK-theoretic Farrell-Jones Conjecture 11.9 with coefficients in additive G-categories, also satisfies the Farrell-Jones Conjecture 13.70 for homotopy K-theory with coefficients in additive G-categories. Moreover, we have statedin Theorem 13.71 the inheritance properties of the Farrell-Jones Conjec-ture 13.70 for homotopy K-theory with coefficients in additive G-categorieswhich are better than the one for the K-theoretic Farrell-Jones Conjec-ture 11.9 with coefficients in additive G-categories.

There are groups for which the Full Farrell-Jones Conjecture 11.20 is notknown to be true but weaker versions of it have been proved. For example,the K-theoretic Farrell-Jones Conjecture 11.1 with coefficients in the ring Ris known if R is regular and G belongs to the class CL′ described in [69, Defini-tion 0.10]. The class CL′ contains for instance all torsionfree 1-relator groups.The class of groups, for which the L-theoretic Farrell-Jones Conjecture 11.4with coefficients in any ring with involution R holds after inverting two, isanalyzed in [379, Proposition 5.2.1]. It contains for instance all elementaryamenable groups.

A proof of the Full Farrell-Jones Conjecture 11.20 for CAT(0)-groups hasbeen extended to a larger class of group which also contains all hyperbolicgroups by Kasprowski-Ruping [463].

The bijectivity of the algebraic K-theoretic assembly map for certain coef-ficients coming from C∗-algebras is proved by Cortinas-Tartaglia [201, Corol-lary 1.5] for a-T-menable groups G by reducing it to the Baum-Connes Con-jecture.


name of texfile: ic

Chapter 15

Guide for Computations

15.1 Introduction

One major goal is to compute K- and L-groups such as Kn(RG), L〈−∞〉n (RG),

andKn(C∗r (G)). Assuming that theK-theoretic Farrell-Jones Conjecture 11.1with coefficients in the ring R, the L-theoretic Farrell-Jones Conjecture 11.4with coefficients in the ring with involution R, or the Baum-Connes Con-jecture 12.9 holds for G, this reduces to the computation of the left handside of the corresponding assembly maps, namely, of HG

n (EVCY(G); KR),

HGn (EVCY(G); L

〈−∞〉R ), or HG

n (EFIN (G); Ktop) = KGn (EFIN (G)). This is

much easier since here we can use standard methods from algebraic topology.The main general tools are the equivariant Atiyah-Hirzebruch spectral se-quence, see Theorem 10.44, the p-chain spectral sequence, see Theorem 10.46,and equivariant Chern characters, see Theorem 10.53. Nevertheless such com-putations can be pretty hard. Roughly speaking, one can obtain a generalreasonable answer after rationalization, but integral computations have onlybeen done case by case and there seems to be no general pattern. Often thekey is a good understanding of how one can built EFIN (G) from EG andhow one can built EVCY(G) from EFIN (G). These passages have alreadybeen studied in Theorems 9.34 and 9.36.

15.2 K- and L- Groups for Finite Groups

For the computations of HGn (EFIN (G); KR), HG

n (EFIN (G); L〈−∞〉R ), and

HGn (EFIN (G); Ktop) = KG

n (EFIN (G)) one needs to understand Kn(RH),

L〈−∞〉n (RH), and Kn(C∗r (H)) for finite groups H, since these are the values

of HGn (G/H; KR), HG

n (G/H; L〈−∞〉R ), and HG

n (G/H; Ktop) = KGn (G/H) for

homogeneous spaces G/H for finite subgroups H ⊆ G.For a finite group G we have given information about K0(ZG) in Sec-

tion 1.11, about K1(ZG) and Wh(G) in Section 2.12, about Kn(ZG) forn ≤ −1 in Examples 3.9 and 3.11, and in Section 3.5, about K2(ZG) and

Wh2(G) in Section 4.8, about L〈j〉n (ZG) in Section 7.21 and about Kn(C∗rG)

and KOn(C∗r (G;R)) in Section 8.9.

Let us summarize. A complete computation of K0(Z[Z/p]) for arbitraryprimes p is out of reach. There is a complete formula for K−1(ZG) andKn(ZG) = 0 for n ≤ −2 and one has a good understanding of Wh(G). A

463

464 15 Guide for Computations

complete computation of Kn(Z) is not known. We have already mentionedBorel’s formula for Kn(Z)⊗ZQ for all n ∈ Z in Theorem 5.22. The L-groups ofZG are pretty well understood for finite groups G. The values of Kn(C∗r (G))and Kn(C∗r (G;R)) are explicitly known for finite groups G and are in thecomplex case in contrast to the real case always torsionfree.

15.3 The passage from FIN to VCY

In the Baum-Connes setting it is enough to consider the family FIN . In theFarrell-Jones Conjecture we have to pass from FIN to VCY. This passagehas been discussed on detail already in Section 11.7. We get splittings

HGn

(EVCY(G); KR

)∼= HG

n

(EFIN (G); KR

)⊕HG

n

(EFIN (G)→ EVCY(G); KA

),

and under mild K-theoretic assumptions

HGn

(EVCY(G); L

〈−∞〉R

)∼= HG

n

(EFIN (G); L

〈−∞〉R

)⊕HG

n

(EFIN (G)→ EVCY(G); L

〈−∞〉R

).

We have also explained in Theorem 11.35 that in K-theory it suffices toreplace VCY by VCYI and in Theorem 11.47 that in L-theory there is nodifference between FIN and VCYI .

If we are only interested in rational information there is no differencebetween FIN and VCY, when we are dealing with the algebraic K-theoryof groups rings RG for regular rings R, see Theorem 11.39 (v) and when weare dealing with L-theory, see Theorem 11.49 (i).

15.4 Rational Computations for Infinite Groups

Next we state what is known rationally about the K- and L-groups of aninfinite (discrete) group, provided the Farrell-Jones Conjectures 11.1 or 11.4or the Baum-Connes Conjecture 12.9 holds.

15.4.1 Rationalized Algebraic K-Theory

The next result follows from Theorem 10.70 and Theorem 11.39 (v). ForR = Z see also Grunewald [355, Corollary on page 165].

15.4 Rational Computations for Infinite Groups 465

Theorem 15.1 (Rational computations of Kn(RG) for regular R).Let R be regular ring, e.g., R is Z. Suppose that the group G satisfies theK-theoretic Farrell-Jones Conjecture 11.1 with coefficients in the ring R.

Then we have for all n ∈ Z a natural isomorphism⊕p+q=n

⊕(C)∈J

Hp(CGC;Q)⊗Q[WGC] ΘC ·(Q⊗Z Kq(RC)

) ∼=−→ Q⊗Z Kn(RG)

where we use the notation from Theorem 10.70.

Computations of Kq(RC) as Z[aut(C)]-module for finite cyclic groups Cand R = Z or R a field of characteristic zero can be found in [646].

Exercise 15.2. If in Theorem 15.1 we drop the condition that R is regular,show that then we still know that the map appearing there is split injective.

Example 15.3 (A Formula for K0(ZG) ⊗Z Q). Suppose that the K-theoretic Farrell-Jones Conjecture 11.1 with coefficients in the ring Z is truefor G. Then Conjecture 3.19 is true by Theorem 11.52. Hence we obtain fromTheorem 15.1 an isomorphism

K0(ZG)⊗Z Q ∼=⊕

(C)∈(FCY)

H1(BCGC;Q)⊗Q[WGC] θC ·K−1(RC)⊗Z Q.

Notice that K0(ZG) ⊗Z Q contains only contributions from K−1(ZC) ⊗Z Qfor finite cyclic subgroups C ⊆ G.

Exercise 15.4. Suppose that the K-theoretic Farrell-Jones Conjecture 11.1with coefficients in the ring Z is true for G and that any element of finiteorder has prime power order. Show that then K0(ZG)⊗Z Q vanishes.

15.4.2 Rationalized Algebraic L-Theory

The next result follows from Theorem 7.102, Theorem 10.70, and Theo-rem 11.49 (i).

Theorem 15.5 (Rational computation of algebraic L-theory). Sup-pose that the group G satisfies L-theoretic Farrell-Jones Conjecture 11.4 withcoefficients in the ring with involution R. Then we get for all j ∈ Z, j ≤ 1and n ∈ Z an isomorphism⊕p+q=n

⊕(C)∈J

Hp(CGC;Q)⊗Q[WGC] ΘC ·(Q⊗Z L

〈j〉q (RC)

) ∼=−→ Q⊗Z L〈j〉n (RG).

where we use the L-theoretic version of the notation of Theorem 10.70.


Exercise 15.6. Let F be a finite group of odd order. Put G = F o Z. Show

that Q⊗Z L〈j〉n (ZG) vanishes for odd n and all j.

15.4.3 Rationalized Topological K-Theory

The next result is taken from [528, Theorem 0.7]. Let ΛG is the ring Z ⊆ ΛG ⊆Q which is obtained from Z by inverting the orders of the finite subgroups ofG.

Theorem 15.7 (Rational computation of topological K-theory).Suppose that the group G satisfies the Baum-Connes Conjecture 12.9.

Then there is an isomorphism⊕p+q=n

⊕(C)∈(FCY)

Kp(BCGC)⊗Z[WGC] ΘC ·Kq(C∗r (C))⊗Z Λ

G

∼=−→ Kn(C∗r (G))⊗Z ΛG,

where we use the notation of Theorem 10.70.If we tensor with Q, we get an isomorphism⊕p+q=n

⊕(C)∈(FCY)

Hp(BCGC;Q)⊗Q[WGC] θC ·Kq(C∗r (C))⊗Z Q.

∼=−→ Kn(C∗r (G))⊗Z Q.

15.4.4 The Complexified Comparison Map from Algebraic toTopological K-theory

If we consider R = C as coefficient ring and apply −⊗ZC instead of −⊗ZQ,the formulas simplify. Suppose that G satisfies the Baum-Connes Conjec-ture 12.9 and K-theoretic Farrell-Jones Conjecture 11.1 with coefficients inthe ring C. Recall that con(G)f is the set of conjugacy classes (g) of ele-ments g ∈ G of finite order. We denote for g ∈ G by 〈g〉 the cyclic subgroupgenerated by g.

Then we get the following commutative square, whose horizontal maps areisomorphisms and whose vertical maps are induced by the obvious change oftheory homomorphisms (see [526, Theorem 0.5])

15.5 Integral Computations for Infinite Groups 467⊕p+q=n

⊕(g)∈con(G)f

Hp(CG〈g〉;C)⊗Z Kq(C)∼= //

Kn(CG)⊗Z C

⊕p+q=n

⊕(g)∈con(G)f

Hp(CG〈g〉;C)⊗Z Ktopq (C)

∼= // Kn(C∗r (G))⊗Z C

Suslin [772, Theorem 4.9] has proved that the algebraic K-theory of Cin dimensions 2n for n ≥ 1 a unique divisible group and hence admits nonon-trivial map to Z. This implies that the canonical map from the algebraicK-theory of C to the topological K-theory of C is trivial in all dimensionsexcept dimension zero where it is a bijection. Thus rationally we understandby the diagram above the comparison map from algebraic K-theory of thecomplex group ring to the topological K-theory of the group C∗-algebraprovided that G satisfies the Baum-Connes Conjecture 12.9 and K-theoreticFarrell-Jones Conjecture 11.1 with coefficients in the ring C.

Remark 15.8 (Separation of Variables). In Theorems 15.1 15.5, 15.7 andin Subsection 15.4.4 we see a principle which we call separation of variables:There is a group homology part which is independent of the coefficient ringR and the K- or L-theory under consideration and a part depending only onthe values of the theory under consideration on RC or C∗r (C) for all finitecyclic subgroups C ⊆ G.

15.5 Integral Computations for Infinite Groups

As mentioned above, no general pattern for integral calculations is knownor expected. We give some examples, where computations are possible andwhich shall illustrate the techniques

15.5.1 Groups satisfying conditions (M) and (NM)

We mention at least one situation where a certain class of groups can betreated systematically. Let MFIN be the subset of FIN consisting of ele-ments in FIN which are maximal in FIN .

Consider the following assertions concerning G:

(M) Every non-trivial finite subgroup of G is contained in a unique maximalfinite subgroup;

(NM) M ∈MFIN ,M 6= 1 =⇒ NGM = M ;

Let Kn(C∗r (H)) be the cokernel of the map Kn(C∗r (1)) → Kn(C∗r (H))

and L〈j〉n (RG) be the cokernel of the map L

〈j〉n (R)→ L

〈j〉n (RG). This coincides


with L〈j〉n (R), which is the cokernel of the map L

〈j〉n (Z) → L

〈j〉n (R) if R = Z

but not in general. Denote by WhRn (G) the n-th Whitehead group of RGwhich is the (n−1)-th homotopy group of the homotopy fiber of the assemblymapBG+∧K(R)→ K(RG). The next result is taken from [223, Theorem 5.1]and is based on the existence of a nice model for EFIN (G), see Theorem 9.34.

Theorem 15.9. Let Z ⊆ Λ ⊆ Q be a ring such that the order of any finitesubgroup of G is invertible in Λ. Let (MFIN ) be the set of conjugacy classes(H) of subgroups of G such that H belongs toMFIN . Suppose that the groupG satisfies conditions (M) and (NM). Then:

(i) If G satisfies the Baum-Connes Conjecture 12.9, then for n ∈ Z there isan exact sequence of topological K-groups

0→⊕

(H)∈(MFIN )

Kn(C∗r (H))→ Kn(C∗r (G))→ Kn(G\EFIN (G))→ 0,

where the maps Kn(C∗r (H)) → Kn(C∗r (G)) are induced by the inclusionsH → G.It splits after applying −⊗Z Λ

(ii) Suppose that every infinite virtually cyclic subgroup of G is of type I, andG satisfies the L-theoretic Farrell-Jones Conjecture 11.4 with coefficientsin the ring with involution R.Then for all n ∈ Z there is an exact sequence

. . .→ Hn+1(G\EFIN (G); L〈−∞〉(R))→⊕

(H)∈(MFIN )

L〈−∞〉n (RH)

→ L〈−∞〉n (RG)→ Hn(G\EFIN (G); L〈−∞〉(R))→ . . .

where the maps L〈−∞〉n (RH)→ L

〈−∞〉n (RG) are induced by the inclusions

H → G.It splits after applying −⊗Z Λ, more precisely

L〈−∞〉n (RG)⊗Z Λ→ Hn(G\EFIN (G); L〈−∞〉(R))⊗Z Λ

is a split-surjective map of Λ-modules;(iii) If G satisfies the K-theoretic Farrell-Jones Conjecture 11.1 with coeffi-

cients in the ring R, then there is for n ∈ Z an isomorphism

Hn(EFIN (G)→ EVCY(G); KR)⊕⊕

(H)∈(MFIN )

WhRn (H)∼=−→WhRn (G),

where the maps WhRn (H) →WhRn (G) are induced by the inclusions H →G.

15.5 Integral Computations for Infinite Groups 469

Remark 15.10 (Role of G\EFIN (G)). Theorem 15.9 illustrates that forsuch computations a good understanding of the geometry of the orbit spaceG\EFIN (G) is necessary. This can be hard to figure out, even for on the firstglance nice groups with pleasant geometric properties such as crystallographicgroups. In general G\EFIN (G) can be very complicated, see Theorem 9.62.

Remark 15.11. In [223] it is explained that the following classes of groups dosatisfy the assumption appearing in Theorem 15.9 and what the conclusionsare in the case R = Z. Some of these cases have been treated earlier in [102,557].

• Extensions 1→ Zn → G→ F → 1 for finite F such that the conjugationaction of F on Zn is free outside 0 ∈ Zn;

• Fuchsian groups F ; Comment 64: This does not seem to be correct.Adjust the text.

• One-relator groups G.

Theorem 15.9 is generalized in [531], in order to treat for instance thesemi-direct product of the discrete three-dimensional Heisenberg group byZ/4. For this group G\EFIN (G) is S3.

Exercise 15.12. Let G be a group satisfying conditions (M) and (NM) ap-pearing in Subsection 9.7.1. Show that then we obtain for any ring R anisomorphism⊕

(V )

HGn (EFIN (V )→ •; KR)

∼=−→ HGn (EFIN (G)→ EVCY(G); KR),

where (V ) runs through the conjugacy classes of maximal infinite virtualcyclic subgroups.

More information about HGn (EFIN (V )→ •; KR) can be found in The-

orem 11.26 or in [559], where also identifications with twisted Nil-categoriesare discussed.

Many of the following results are based on Theorem 15.9.

15.5.2 Fuchsian groups

Here is an example of a group with torsion where one can carry out a completecalculation. Namely, let F be a cocompact Fuchsian group with presentation

F = 〈a1, b1, . . . , ag, bg, c1, . . . , ct |cγ11 = · · · = cγtt = c−1

1 · · · c−1t [a1, b1] · · · [ag, bg] = 1〉


for integers g, t ≥ 0 and γi > 1. Then F\EFIN (F ) is a closed orientablesurface of genus g. The following is a consequence of Theorem 15.9 and Re-mark 15.11, see [557] for details. Comment 65: It does not seem to be truethat (M) and (NM) are satisfied. But the reference to Luck-Steimle is okay.

Theorem 15.13 (K-and L-groups of Fuchsian groups).

(i) There are isomorphisms

Kn(C∗r (F )) ∼=

(2 +

∑ti=1(γi − 1)

)· Z n = 0;

(2g) · Z n = 1.

(ii) The inclusions of the maximal subgroups Z/γi = 〈ci〉 induce an isomor-phism

t⊕i=1

Whn(Z/γi)∼=−→Whn(F )

for n ≤ 1.(iii) There are isomorphisms

Ln(ZF )[1/2] ∼=

(1 +

∑ti=1

[γi2

])· Z[1/2] n ≡ 0 (4);

(2g) · Z[1/2] n ≡ 1 (4);(1 +

∑ti=1

[γi−1

2

])· Z[1/2] n ≡ 2 (4);

0 n ≡ 3 (4),

where [r] for r ∈ R denotes the largest integer less than or equal to r.From now on suppose that each γi is odd. Then the number m above isodd and we get for for ε = p and s

Lεn(ZF ) ∼=

Z/2

⊕(1 +

∑ti=1

γi−12

)· Z n ≡ 0 (4);

(2g) · Z n ≡ 1 (4);

Z/2⊕(

1 +∑ti=1

γi−12

)· Z q ≡ 2 (4);

(2g) · Z/2 n ≡ 3 (4).

For ε = h we do not know an explicit formula for Lεn(ZF ). The problem is

that no general formula is known for the 2-torsion contained in Lh2q(Z[Z/m]),

for m odd, since it is given by the term H2(Z/2; K0(Z[Z/m])), see [56, The-orem 2].

Exercise 15.14. Let F be a Fuchsian group as above. Show that its White-head group Wh(F ) is a free abelian group of rank

⊕ti=1bγi/2c + 1 − δ(γi),

where bγi/2c is the greatest integer less or equal to γi/2, and δ(γi) is thenumber of divisors of Γi.


15.5.3 Torsionfree Hyperbolic Groups

Theorem 15.15 (Farrell-Jones Conjecture for torsionfree hyperbolicgroups for K-theory). Let G be a non-trivial torsionfree hyperbolic group.

(i) We obtain for all n ∈ Z an isomorphism

Hn(BG; K(R))⊕⊕C

(NKn(R)⊕NKn(R)

) ∼=−→ Kn(RG),

where C runs through a complete system of representatives of the conjugacyclasses of maximal infinite cyclic subgroups;

(ii) The abelian groups Kn(ZG) for n ≤ −1, K0(ZG), and Wh(G) vanish;(iii) We get for every ring R with involution and n ∈ Z an isomorphism

Hn(BG; L〈−∞〉(R))∼=−→ L〈−∞〉n (RG).

For every j ∈ Z, j ≤ 2, and n ∈ Z, the natural map

L〈j〉n (ZG)∼=−→ L〈−∞〉n (ZG)

is bijective;(iv) We get for any n ∈ Z isomorphisms

Kn(BG)∼=−→ Kn(C∗r (G));

KOn(BG)∼=−→ KOn(C∗r (G;R)).

Proof. (i) By [562, Corollary 2.11, Theorem 3.1 and Example 3.6], see alsoTheorem 9.36, there is a G-pushout∐

C G×C EC

p

i // EFIN (G)

∐C G/C

// EVCY(G)

where i is an inclusion of G-CW -complexes, p is the obvious projection. Hencewe obtain using Theorem 5.16 an isomorphism

HGn

(EFIN (G)→ EVCY(G); KR

) ∼= ⊕C

HGn

(G×C EC → G/C; KR

)∼=⊕C

HCn

(EC → •; KR

)∼=⊕C

(NKn(R)⊕NKn(R)

).


We obtain from Theorem 11.26 an isomorphism

HGn

(EVCY(G); KR

) ∼= HGn

(EG; KA

)⊕HG

n

(EFIN (G)→ EVCY(G); KR

)∼= Hn(BG; K(R))⊕

⊕C

(NKn(R)⊕NKn(R)

).

SinceG satisfies the Full Farrell-Jones Conjecture 11.20, see Theorem 14.1 (ia)Theorem 11.52 implies that G satisfies the K-theoretic Farrell-Jones Conjec-ture 11.1 with coefficients in the ring R.

(ii) Since G satisfies the Full Farrell-Jones Conjecture 11.20, see Theo-rem 14.1 (ia) and hence by Theorem 11.52 Conjectures 2.107 and 3.17, as-sertion (ii) follows.

(iii) Since G satisfies the Full Farrell-Jones Conjecture 11.20, see Theo-rem 14.1 (ia), Theorem 11.52 implies that G satisfies Conjecture 7.109. Nowassertion (iii) follows from assertion (ii) and Theorem 7.102.

(iv) This follows from the fact that G satisfies the Baum-Connes Conjec-ture 12.11 with coefficients by Theorem 14.6 (ic) and from Remark 12.14. ut

Not necessarily torsionfree hyperbolic groups are treated in [552, Theo-rem 1.1].

15.5.4 L-theory of Torsionfree Groups

Throughout this subsection, let G be a torsionfree group satisfying the L-theoretic Farrell-Jones Conjecture 11.4 with coefficients in the ring with in-volution Z. Then Theorem 11.52 implies that G satisfies Conjecture 7.109,i.e., we obtain an isomorphism

Hn(BG; L〈−∞〉(Z))∼=−→ L〈−∞〉n (ZG).

Thus we obtain from Subsection 13.13.4 an isomorphism

(15.16) KO(BG)[1/2]∼=−→ L〈−∞〉n (ZG)[1/2].

Example 15.17 (p-torsion in L-groups). Let n ≥ 3 be an odd naturalnumber. Consider the group automorphism

α : Z2 → Z2, (a, b) 7→ (a+ nb, b).

Let G be the semi-direct product Z2oZ. Obviously there is a closed asphericaloriented 3-manifold M which is a fiber bundle T 2 → M → S1 whose fun-damental group is G, namely the mapping torus of the selfhomeomorphismT 2 → T 2 inducing α on π1(T 2) ∼= Z2. One easily computes


Hk(M ;Z) ∼=

Z k = 0, 1, 3;

Z⊕ Z/n k = 2;

0 otherwise

An elementary spectral sequence argument shows

L〈−∞〉n (ZG)[1/2] ∼= KOk(M ;Z)[1/2] ∼=

Z[1/2] k = 0, 1, 3 mod 4

Z⊕ Z/n k = 2 mod 4

0 otherwise

Hence L〈−∞〉n (ZG) can contain p-torsion for any odd prime p. Recall that for

finite groups G only 2-torsion occurs in L〈−∞〉n (ZG) by Theorem 7.179 (ii).

Exercise 15.18. Let p be an odd prime. Show for any n ≥ 6, k ∈ Z anddecoration j ∈ 2, 1, 0,−1, . . . q −∞ that there is an aspherical closed

smooth manifold M such that L〈j〉k (Zπ1(M)) contains non-trivial p-torsion.

Since we have the decomposition the decomposition away from 2

L〈−∞〉(Z)(2) =∏k∈Z

K(Z(2), 4k)×∏k∈Z

K(Z/2, 4k − 2),

see Remark 7.127, Comment 66: Add a reference to the non-connectiveversion which we need here. Remark 7.127 deals at least with the connectiveversion. we obtain

(15.19) L〈−∞〉n (ZG)(2)∼=∏k∈Z

Hn+4k(BG;Z(2))×∏k∈Z

Hn+4k−2(BG;Z/2).

15.5.5 Cocompact NEC-groups

A calculation of Whn(G), L〈−∞〉n (ZG) and Kn(Cr∗(G)) for 2-dimensional crys-

tallographic groups G and more general cocompact NEC-groups G is pre-sented in [557] (see also [648]). For these groups the orbit spaces G\EFIN (G)are compact surfaces possibly with boundary.

15.5.6 Crystallographic groups

An crystallographic group of dimension n is a discrete group which acts co-compactly, properly and isometrically on the Euclidean space Rn for somen ≥ 0. One does not have a complete calculation of K-and L-groups of in-tegral group rings or reduced group C∗-algebras of crystallographic groups


except in dimension two as mentioned above in Subsection 15.5.5. Computa-tions of the lower and middle algebraic K-theory of the integral group ringof split three-dimensional crystallographic groups are carried out by Farley-Ortiz [284], see also [21].

As an illustration we mention the following result taken from [503, Theo-rem 0.1].

Theorem 15.20 (Computation of the topological K-theory of Zn oZ/m for free conjugation action). Consider the extension of groups1 → Zn → G → Z/m → 1 such that the conjugation action of Z/m onZn is free outside the origin 0 ∈ Zn. Let M be the set of conjugacy classesof maximal finite subgroups of G.

(i) We obtain an isomorphism

ω1 : K1(C∗r (G))∼=−→ K1(G\EFIN (G)).

Restriction with the inclusion k : Zn → G induces an isomorphism

k∗ : K1(C∗r (G))∼=−→ K1(C∗r (Zn))Z/m.

Induction with the inclusion k yields a homomorphism

k∗ : Z⊗Z[Z/m] K1(C∗r (Zn))→ K1(C∗r (G)).

It fits into an exact sequence

0→ H−1(Z/m,K1(C∗r (Zn)))→ Z⊗Z[Z/m]K1(C∗r (Zn))k∗−→ K1(C∗r (G))→ 0,

where H∗(Z/m;M) denotes the Tate cohomology of Z/m with coefficientsin a Z[Z/m]-module M . In particular k∗ is surjective and its kernel isannihilated by multiplication with m;

(ii) There is an exact sequence

0→⊕

(M)∈M

RC(M)

⊕(M)∈M iM

−−−−−−−−→ K0(C∗r (G))ω0−→ K0(G\EFIN (G))→ 0,

where RC(M) is the kernel of the map RC(M)→ Z sending the class [V ] ofa complex M -representation V to dimC(C⊗CM V ) and the map iM comesfrom the inclusion M → G and the identification RC(M) = K0(C∗r (M)).We obtain a homomorphism

k∗ ⊕⊕

(M)∈M

iM : Z⊗Z[Z/m] K0(C∗r (Zn))⊕⊕

(M)∈M

RC(M)→ K0(C∗r (G)).

It is injective. It is bijective after inverting m;


(iii) We haveKi(C

∗r (G)) ∼= Zsi

where

si =

(∑(M)∈M(|M | − 1)

)+∑l∈Z rkZ

((Λ2lZn)Z/m

)if i even;∑

l∈Z rkZ((Λ2l+1Zn)Z/m

)if i odd;

(iv) If m is even, then s1 = 0 and

K1(C∗r (G)) ∼= 0.

The numbers si can be made more explicite, see [503]. For instance, ifm = p for a prime number p, then there exists a natural number k which isdetermined by the property n = (p− 1) · k, and we get:

si =

pk · (p− 1) + 2n+p−1

2p + pk−1·(p−1)2 p 6= 2 and i even;

2n+p−12p − pk−1·(p−1)

2 p 6= 2 and i odd;

3 · 2k−1 p = 2 and i even;

0 p = 2 and i odd.

(15.21)

Exercise 15.22. Consider the the automorphism φ : Z2 → Z2, (a, b) 7→(b,−a− b). Then φ3 = id. Show

Ki

(C∗r (Z2 oφ Z/3)

) ∼= Z8 i even;

Z2 i odd.

Theorem 15.20 in the special case, where m is a prime number, is treatedin [224].

The groups appearing in Theorem 15.20 are crystallographic groups,see [503, Lemma 3.1].

The proof of Theorem 15.20 is surprisingly complicated, it is based oncomputations of the group homology of Zn o Z/n which verify a conjectureof Adem-Ge-Pan-Petrosyan in this special case, see Langer-Luck [502], andon generalizations of the Atiyah-Segal Completion Theorem for finite groupsto infinite groups, see Luck-Oliver [544, 545].

15.5.7 Virtually finitely generated free abelian groups

One does not have a complete calculation of the K-groups and L-groupsof integral group rings or group C∗-algebras of crystallographic groups andhence not of virtually finitely generated abelian groups. The favorite situationis the one occurring in Remark 15.11, when one considers groups G occurring


in an extensions 1→ Zn → G→ F → 1 for finite F such that the conjugationaction of F on Zn is free outside 0 ∈ Zn. Then some general information canbe found in [552].

15.5.8 SL3(Z)

Since SL3(Z) satisfies the Full Farrell-Jones Conjecture 11.20, see Theo-rem 14.1 (id), Theorem 11.52 implies that G satisfies the K-theoretic Farrell-Jones Conjecture 11.1 with coefficients in Z. Using this fact the followingresult is proved in [763] and [796]. Comment 67: Take a look at this pa-per.

Theorem 15.23 (Lower and middle K-theory of the integral group

ring of SL3(Z)). The groups Kn(Z[SL3(Z)]) for n ≤ −2, K0(Z[SL3(Z)]),and Wh(SL3(Z)) are trivial. For an appropriate subgroup C6 ⊆ SL3(Z) whichis cyclic of order six, the inclusion C6 → SL3(Z) induces an isomorphism

Z ∼= K−1(Z[C6])∼=−→ K−1(Z[SL3(Z)]).

15.6 Applications of Some Computations

15.6.1 Classification of some C∗-algebras

Theorem 15.20 is an important input in the classification of certain C∗-algebras associated to number fields by Li-Luck [511]. Here the key pointis the rather surprising result that the topological K-groups are all torsion-free what is not the case for the group homology.

Another application of the computation of the topological K-theory ofgroup C∗-algebras can be found in [262], namely to the structure of crossedproducts of irrational rotation algebras by finite subgroups of SL2(Z).

15.6.2 Unstable Gromov-Lawson Rosenberg Conjecture

We have already discussed in Subsection 12.8.2 that Schick [737] constructscounterexamples to the unstable version of the Gromov-Lawson-RosenbergConjecture with fundamental group π ∼= Z4 × Z/3. However for appropri-ate ρ : Z/3 → aut(Z4) the unstable version does hold for π ∼= Z4 oρ Z/3and dim(M) ≥ 5. This is proved by Davis-Luck [224, Theorem 0.7 and Re-


mark 0.9] based on explicite calculations of the topological K-theory of thereduced real group C∗ algebra of Z4 oρ Z/3.

15.6.3 Classification of certain manifolds with infinite nottorsionfree fundamental groups

Manifolds homotopy equivalent to the total space of certain fiber bundles overlens spaces with tori as fiber are classified by Davis-Luck [225]. Here the keyinput is the calculation of algebraic K-and L-groups of integral group ringsof groups of the shape π = ZoρZ/p for odd primes p, where the conjugationaction of Z/p on Zn is free outside the origine. Notice that π is infinite andnot torsionfree. This is one of the few classification result about a class ofclosed manifolds, whose fundamental group is not finite and not torsionfree.Comment 68: Maybe it is the only one. Comment 69: Shall we say more,when the paper is finished.

15.7 Miscellaneous

The lower and middle algebraic K-theory of integral group rings of certainreflection groups has been computed by Lafont-Ortiz [492] and by Lafont-Margurn-Ortiz [491], of Γ3 := O+(3, 1) ∩ GL4(Z) by Ortiz [636, 637], ofBianchi groups by Berkhove-Farrell-Pineda-Pearson [101], and of pure braidgroups by Aravinda-Farrell-Roushon [33]. Certain extensions of free groupsare considered by Metaftsis-Prassidis [580].


name of texfile: ic

Chapter 16

Assembly Maps

16.1 Introduction

In this chapter we discuss assembly maps and the assembly principle im gen-eral, and establish a universal property which is very useful and has alreadybeen exploited to identify various assembly maps, see Section 16.6.

We recall the homological approach in Section 16.3 which we have used inthis book. We give the version in terms of spectra in Section 16.3. Actuallyin all concrete situation such as in the Farrell-Jones Conjecture or K-and L-theory and pseudoisotopy or the Baum-Cones Conjecture the assembly mapis implemented in terms of spectra. This can easily be identified with theelementary approach in terms of homotopy colimits which nicely illustratesthe name assembly, but works only if we consider classifying spaces of familiesof subgroups only, see Section 16.4. This is possibly the quickest concreteapproach for a homotopy theorist.

The universal property of assembly is explained in Section 16.5. Roughlyspeaking, it says that the assembly map is the best approximation of a(weakly) homotopy invariant functor E : G-CW-COMPLEXES → SPECTRAfrom the left by a (weakly) excisive functorG-CW-COMPLEXES→ SPECTRA,where weakly excisive essentially means that after taking homotopy groupsthe functor yields a G-homology theory. This is very helpful to identify thevarious versions of the assembly maps appearing in the literature with ourhomological approach since the constructions of the assembly maps can bevery complicated and is much easier to use the universal property to estab-lish the desired identifications than to go through the actual definitions. Alsothis universal approach explains the philosophical background of the assem-bly map. It is important to have also the other more geometric definitionof assembly maps in terms of index theory or surgery theory at hand to beable to apply the Farrell-Jones Conjecture and the Baum-Connes Conjectureto geometric questions such as the Borel Conjecture or the existence of aRiemannian metric with positive sectional curvature. The homological or ho-motopy theoretic approach to assembly maps is best suited for computationsand proofs of the conjectures.

479

480 16 Assembly Maps

16.2 Homological Approach

The homological version of assembly is manifested in the Meta-IsomorphismConjecture 13.2. Recall that it predicts for a group G, a family F of subgroupsof G, and a G-homology theory HG∗ in the sense of Definition 10.1 that themap induced by the projection pr : EF (G)→ G/G for EF (G) the classifyingspace of the family F in the sense of Definition 9.18,

(16.1) Hn(pr) : HGn (EF (G))→ HGn (G/G)

is bijective for all n ∈ Z. The various conjectures due to Baum-Connes andFarrell-Jones are special cases, where one specifies F and HG∗ .

16.3 Extension from Homogenous Spaces toG-CW -Complexes

Let E be a covariant Or(G)-spectrum, i.e., a covariant functor E : Or(G)→SPECTRA. We get an extension of E to the category G-CW-COMPLEXESof G-CW -complexes by(16.2)

E% : G-CW-COMPLEXES→ SPECTRA, X 7→ mapG(−, X)+ ∧Or(G) E.

where mapG(−, X) and ∧Or(G) have been defined in Example 10.21 andin (10.22). The projection pr : EF (G) → G/G for EF (G) induces a map ofspectra

(16.3) E%(pr) : E%(EF (G))→ E%(G/G).

After taking homotopy groups we get for all n ∈ Z a homomorphism

(16.4) πn(E%(pr)

): πn

(E%(EF (G))

)→ πn

(E%(G/G)

).

We have constructed a G-homology theory HG∗ (−; E) with the property that

HGn (G/H; E) ∼= πn(E(G/H)) holds for all n ∈ Z and subgroups H ⊆ G in

Theorem 10.24. The G-homology theories relevant for the Baum-Connes andthe Farrell-Jones Conjecture are given by specifying such covariant functorsE. It follows essentially from the definitions that the map (16.1) for HG∗ =HG∗ (−; E) agrees with the map (16.4).

16.5 Universal Property 481

16.4 Homotopy Colimit Approach

Consider a covariant functor E : Or(G) → SPECTRA. Recall that OrF (G)denotes the F-restricted orbit category, see Definition 1.54. If theG-homologytheory HG∗ is given by HG

∗ (−; E), one can identify the assembly map (16.4)with the map

(16.5) πn(p) : πn(hocolimOrF (G) E

)→ πn(E(G/G))

where the map of spectra

p : hocolimOrF (G) E→ hocolimOr(G) E = E(G/G)

comes from the inclusion of categories OrF (G) → Or(G) and the fact thatG/G is a terminal object in Or(G). For more information about homotopycolimits and the identification of the maps (16.1), (16.4), and (16.5) we referto [222, Sections 3 and 5].

This interpretation is one explanation for the name assembly. If the assem-bly map (16.5) is bijective for all n ∈ Z, or, equivalently, the map p above isis a weak homotopy equivalence, we have a recipe to assemble E(G/G) fromits values E(G/H), where H runs through F . The idea is that F consistsof well-understood subgroups, for which one knows the values E(G/H) forH ⊆ G and hence hocolimOrF (G) E, whereas E(G/G) is the object, which onewants to understand and is very hard to access. Comment 70: Are thereother places where we should say something about the name assembly?

16.5 Universal Property

In this section we characterize assembly maps by a universal property. Thisis useful for identifying different constructions of assembly maps.

Lemma 16.6. Let E be a covariant Or(G)-spectrum. Then:

(i) The canonical map

E%(X) ∪E%(f) E%(Y )→ E%(X ∪f Y )

is an isomorphism of spectra, where (X,A) is a G-CW -pair and f : A→ Yis a cellular G-map;

(ii) The canonical map

colimi∈I E%(Xi)→ E%(X)


is an isomorphism of spectra, where Xi | i ∈ I is a directed system ofG-CW -subcomplexes of the G-CW -complex X directed by inclusion andsatisfying X =

⋃i∈I Xi.

(iii) The canonical map

Z+ ∧E%(X)→ E%(Z ×X)

is an isomorphism of spectra, where Z is a CW -complex (with trivial G-action) and X is a G-CW -complex;

(iv) The canonical mapE%(G/H)→ E(G/H)

is an isomorphism of spectra for all H ∈ F .

Proof. One easily checks that the H-fixed point set functor mapG(G/H,−)commutes with attaching a G-space to a G-space along a G-map and withdirected unions of G-CW -subcomplexes. Assertions (i) and (ii) follow fromthe fact that − ∧Or(G) E commutes with colimits, since it has an right ad-joint, see [222, Lemma 1.5]. Assertions (iii) and (iv) follow by inspecting thedefinition of E%. ut

Lemma 16.7. Let E be a covariant Or(G)-spectrum. Then the extensionE 7→ E% is uniquely determined up to isomorphism of G-CW-COMPLEXES-spectra by the properties of Lemma 16.6.

Proof. Let E 7→ E$ be another such extension. There is a (a priori not nec-essarily continuous) set-theoretic natural transformation

T(X) : E%(X) = X+ ∧Or(G,F) E −→ E$(X)

which sends an element represented by (x : G/H −→ X, e) in mapG(G/H,X)×E(G/H) to E$(x)(e). Since any G-CW -complex is constructed from orbitsG/H with H ∈ F via products with disks and disjoint unions, attaching aG-space to a G-space along a G-map, and is the directed union over its skele-tons, and T(G/H) is an isomorphism for H ∈⊆ G by assumption, T(X) isan isomorphism for all G-CW -complexes X. ut

Lemma 16.7 is a characterization of E 7→ E% up to isomorphism. Next wegive a homotopy theoretic characterization.

Definition 16.8 ((Weakly) excisive). We call a covariant functor

E : G-CW-COMPLEXES→ SPECTRA

(weakly) homotopy invariant if it sends G-homotopy equivalences to (weak)homotopy equivalences of spectra.

The functor E is (weakly) excisive if it has the following four properties:

• It is (weakly) homotopy invariant;


• The spectrum E(∅) is (weakly) contractible;• It respects homotopy pushouts up to (weak) homotopy equivalence, i.e.,

if the G-CW -complex X is the union of G-CW -subcomplexes X1 and X2

with intersection X0, then the canonical map from the homotopy pushoutof E(X2)←− E(X0) −→ E(X2) to E(X) is a (weak) homotopy equivalenceof spectra;

• It respects disjoint unions up to (weak) homotopy, i.e., the natural map∨i∈I E(Xi)→ E(

∐i∈I Xi) is a (weak) homotopy equivalence for all index

sets I.

The following result has been proved forG = 1 by Weiss and Williams [828].

Theorem 16.9 (Universal Property of assembly).

(i) Suppose E : Or(G)→ SPECTRA is a covariant functor. Then E% is exci-sive;

(ii) Suppose E : Or(G)→ SPECTRA is a covariant functor. Then we obtain aG-homology theory HG

n (−; E) in the sense of Definition 10.1 from Theo-rem 10.24, and we get for every pair (X,A) of G-CW -complexes (X,A) anatural isomorphism

HGn (X,A; E) ∼= coker

(πn(E%(∅+))→ πn(E%(X/A))

).

If A = ∅, this becomes an isomorphism

HGn (X; E) ∼= πn(E%(X));

(iii) Let T : E → F be a transformation of (weakly) excisive functors E andF from G-CW-COMPLEXES to SPECTRA so that T(G/H) is a (weak)homotopy equivalence of spectra for all H ⊆ G.Then T(X) is a (weak) homotopy equivalence of spectra for all G-CW -complexes X;

(iv) For any (weakly) homotopy invariant functor E from G-CW-COMPLEXESto SPECTRA, there is a (weakly) excisive functor

E% : G-CW-COMPLEXES→ SPECTRA

and natural transformations

AE : E% → E;

BE : E% → E%,

which induce (weak) homotopy equivalences of spectra AE(G/H) for allH ⊆ G and (weak) homotopy equivalences of spectra BE(X) for all G-CW -complexes X.The constructions E%, E%, AE and BE are natural in E.Moreover, E is (weakly) excisive if and only if AE(X) is a (weak) homo-topy equivalence of spectra for all G-CW -complexes X.


Proof. (i) follows from Lemma 16.6.

(ii) There is an obvious G-homotopy equivalence of pointed G-CW -complexesX+ ∪A+

cone(A+)→ X/A. Hence we get from the definitions

HGn (X,A; E) = πn

(mapG(−, X/A) ∧Or(G) E

).

Now the assertion follows from the cofibration sequence of spectra

E%(∅+) = mapG(−, ∅+)+ ∧Or(G) E

→ E%(X/A) = mapG(−, X/A)+ ∧OrF (G) E→ mapG(−, X/A)∧OrF (G) E.

(iii) Use the fact that a (weak) homotopy colimit of homotopy equivalencesof spectra is again a (weak) homotopy equivalence of spectra.

(iv) See [222, Theorem 6.3]. ut

Exercise 16.10. Show that a covariant functor E : G-CW-COMPLEXES→SPECTRA is weakly excisive if and only if the assignment sending a pair(X,A) of G-F-CW -complexes to coker

(πn(E(∅+)) → πn(E(X/A))

)defines

a G-homology in the sense of Definition 10.1.

Definition 16.11 (Homotopy theoretic assembly transformation).Given a covariant functor E : Or(G,F)→ SPECTRA, we call the transforma-tion appearing in Theorem 16.9 (iv)

AE : E% → E

the homotopy theoretic assembly transformation.

Remark 16.12 (No continuity condition E). One may be tempted todefine a natural transformation S : E% → E as indicated in the proof ofLemma 16.7. Then S(X) is a well-defined bijection of sets but is not neces-sarily continuous because we do not want to assume that E is continuous, i.e.,that the induced map from homC(X,Y ) to homC(E(X),E(Y )) is continuousfor all G-CW -complexes X and Y . This is the reason why we have to passto the more complicated construction of E% and only obtain a zigzag

E%BE←−− E% AE−−→ E,

which suffices for all our purposes. The construction of this zigzag uses the(weak) homotopy invariance of E and does not require any continuity condi-tion for E.

Theorem 16.9 implies

Corollary 16.13. Let E : G-CW-COMPLEXES → SPECTRA be a weaklyexcisive functor. Denote by E|Or(G) its restriction to a covariant functorOr(G)→ SPECTRA.


Then we obtain for all n ∈ Z and G-CW -complex X an isomorphism,natural in X,

πn(E(X))∼=−→ HG

n (X; E|Or(G)).

In particular we get for every family of subgroups F and n ∈ Z a commutativediagram with isomorphisms as vertical arrows

πn(E(EF (G)))πn(E(pr)) //

∼=

πn(E(G/G))

∼=

HGn (EF (G); E|Or(G))

HGn (pr;E|Or(G)) // HGn (G/G; E|Or(G))

Remark 16.14 (Universal property of the homotopy theoretic as-sembly transformation). Next we explain why Theorem 16.9 characterizesthe homotopy theoretic assembly map

AE : E% −→ E

in the sense that it is the universal approximation from the left by a(weakly) excisive functor of a (weakly) homotopy invariant functor E fromG-CW-COMPLEXES to SPECTRA up to (weak) homotopy equivalence.Namely, let T : F→ E be a transformation of functors fromG-CW-COMPLEXESto SPECTRA such that F is (weakly) excisive and T(G/H) is a (weak) homo-topy equivalence for all H ⊆ G. Then for any G-CW -complex X the followingdiagram commutes

F%(X)

T%(X)

AF(X) // F(X)

T(X)

E%(X)

AE(X)// E(X)

and AF(X) and T%(X) are (weak) homotopy equivalences. Hence one maysay that T(X) factorizes over AE(X) up to (weak) homotopy equivalence.

In particular we obtain for every G-CW -complex X a commutative dia-gram with an isomorphism as vertical arrow

πn(F(X))

πn(T(X))

++∼=πn(T%(X))πn(AF(X))−1

πn(E(X))

πn(E%(X))

πn(AE(X))

44


Remark 16.15 (The homotopy theoretic assembly transformationand the Meta-Isomorphism Conjecture 13.35 for functors fromspaces to spectra with coefficients). Consider a functor S : SPACES →SPECTRA which respects weak equivalences and disjoint unions. Given agroup G and a free G-CW -complex Z, we get a functor functor

SGZ : G-CW-COMPLEXES→ SPECTRA, X 7→ S(X ×G Z)

whose restriction to Or(G) is denoted in the same way and has already beenintroduced in (13.34). The Meta-Isomorphism Conjecture 13.35 for functorsfrom spaces to spectra with coefficients predicts for a family F of subgroupsof G that the map

(16.16) HGn (pr; SGZ ) : HG

n (EF (G); SGZ )→ HGn (G/G; SGZ )

induced by the projection pr : EF (G)→ G/G is bijective for all n ∈ Z. Thismap can be identified with the corresponding map for the homotopy theoreticassembly map

(16.17) πn(ASGZ

(EF (G)))

: πn((SGZ )%(EF (G))

)→ πn

(SGZ (EF (G))

)by the following argument. Because of Theorem 16.9 (ii) the map (16.16) canbe identified with the map induced by the projection pr : EF (G)→ G/G

πn((SGZ )%(pr)

): πn

((SGZ )%(EF (G))

)→ πn

((SGZ )%(G/G)

),

and hence by Theorem 16.9 (iv) with the map

(16.18) πn((SGZ )%(pr)

): πn

((SGZ )%(EF (G))

)→ πn

((SGZ )%(G/G)

).

We have the following commutative diagram

(SGZ )%(EF (G))(SGZ )%(pr) //

ASGZ

(EF (G))

(SGZ )%(G/G)

ASGZ

(G/G)

SGZ (EF (G))

(SGZ )%(pr)

// SGZ (G/G)

The right vertical arrow is a weak homotopy equivalence by Theorem 16.9 (iv).Since Z is a free G-CW -complex and EF (G) is contractible (after forgettingthe group action), the map id×G pr: Z ×G EF (G)→ Z ×G G/G is a homo-topy equivalence and hence the lower horizontal arrow is a weak homotopyequivalence. Hence we get an identification of the maps (16.17) and (16.18).Thus we have identified the maps (16.16) and (16.17).

Example 16.19 (The Farrell-Jones Conjecture and the Baum-ConnesConjecture in the setting of the homotopy theoretic assembly trans-

16.6 Identifying assembly maps 487

formation). In the sequel Π(X) denotes the fundamental groupoid of aspace X. If we take in Remark 16.15 the covariant functor S : SPACES →SPECTRA to be the one, which sends a spaceX to KR(Π(X)) or L

〈−∞〉R (Π(X))

respectively, see Theorem 10.39, then we conclude from Example 13.33 andRemark 16.15 that the assembly map appearing in the K-theoretic Farrell-Jones Conjecture 11.1 with coefficients in the ring R

HGn (pr) : HG

n (EVCY(G); KR)→ HGn (G/G; KR) = Kn(RG)

or the assembly map appearing in the L-theoretic Farrell-Jones Conjec-ture 11.4 with coefficients in the ring with involution R

HGn (pr) : HG

n (EVCY(G); L〈−∞〉R )→ HG

n (G/G; L〈−∞〉R ) = L〈−∞〉n (RG)

respectively can be identified with the map induced on homotopy groups bythe homotopy theoretic assembly map

πn(SGEG(EVCY(G))%(pr)

): πn

((SGEG)%(EVCY(G))

)→ πn

((SGEG)%(EVCY(G))

).

In the Baum-Connes setting we get an identification of the assembly map

HGn (pr; Ktop) : HG

n (EFIN (G); Ktop)→ HGn (G/G; Ktop) = Kn(C∗r (G))

with the map

πn((SGEG(EFIN (G))%(pr)

): πn

((SGEG)%(EFIN (G))

)→ πn

((SGEG)%(EFIN (G))

).

if we take S = Ktop(Π(X)), see Theorem 10.39, and analogously in the realcase.

16.6 Identifying assembly maps

We have explained in Remark 13.38 the identification of the original for-mulation of the fibered Farrell-Jones Conjecture for covariant functors fromSPACES to SPECTRA, e.g., for pseudoisotopy, K-theory and L-theory, dueto Farrell-Jones[295, Section 1.7 on page 262] with the setting we are using inthe Meta-Isomorphism Conjecture 13.35 for functors from spaces to spectrawith coefficients.

We have discussed the various Baum-Connes assembly maps and theirrelations already in Sections 12.2 and 12.3.

We have explained the relationship between the L-theoretic assembly mapin terms of spectra, which we are using here, and the surgery obstructionmap appearing in the geometric surgery exact sequence in the proof of The-orem 7.160.


Comment 71: Later we have to figure out the details of and to discussthe identification between the controlled approach, which will be used for theproofs of the Farrell-Jones Conjecture, and the homological approach, whichwe are using here.

16.7 Miscellaneous

The Baum-Connes assembly maps in terms of localizations of triangulatedcategories are considered in [413, 414, 415, 581, 582, 583]. A categorial ap-proach in terms of codescent is presented in [57].

Chain complex versions of the L-theoretic assembly map for additive cat-egories are intensively studied by Ranicki [685] and Kuhl-Macko-Mole [485,Section 11] emphasizing the aspect of comparing local Poincare duality andglobal Poincare duality.


name of texfile: ic

Chapter 17

Controlled Topology Methods

Everything which has been written so far in thisand the following chapters is in an experimentalstage and will probably be completely changed.

17.1 Introduction

17.2 Controlled Algebra

17.2.1 Geometric Objects

Let Y be a space Comment 72: Shall we change Y to X? Notice thatequivariant continuous control will be defined for a pair (X,Y ). Or we willthis pair by (X,X). and letA be a small additive category. Define an additivecategory C(Y ;A) as follows.

An object is a collection A = (Ay)y∈Y of objects in A which is locallyfinite, i.e., support supp(A) := y ∈ Y | Ay 6= 0 is a locally finite subset ofY . Recall that a subset S ⊆ Y is called locally finite if each point in Y hasan open neighborhood U whose intersection with S is a finite set.

A morphism φ = (φx,y)x,y∈Y : A = (Ax)x∈Y → B = (By)y∈Y consists ofa collection of morphisms φx,y : Ax → By in A for x, y ∈ Y such that theset x | φx,y 6= 0 is finite for every y ∈ Y and the set y | φx,y 6= 0 isfinite for every x ∈ Y . We define its support supp(φ) = (x, y) | φx,y 6= 0.Composition is given by matrix multiplication, i.e.,

(ψ φ)x,z :=∑y∈Y

ψz,y φx,z.

The category C(Y ;A) inherits in the obvious way the structure of an additivecategory from A.

We will often drop A from the notation. Comment 73: Actually, do itto keep the notation short.

Definition 17.1 (Category C(Y,A) of geometric objects in an addi-tive category A over a space Y ). Let Y be a space and let A be a small

489

490 17 Controlled Topology Methods

additive category. We callC(Y ;A)

the additive category of geometric objects in an additive category A over aspace Y

Remark 17.2 (Dropping the locally finiteness condition on objects).Sometimes we may also consider a larger category C(Y ;A).

An object is any collection A = (Ay)y∈Y of objects in A, we drop thecondition that the support is locally finite. A morphism φ = (φx,y)x,y∈Y : A =(Ax)x∈Y → B = (By)y∈Y consists of a collection of morphisms φx,y : Ax →By in A for x, y ∈ Y such that the set y | φx,y 6= 0 is finite for every x ∈ Y ,we require nothing about the set x | φx,y 6= 0 for y ∈ Y . Composition canstill be defined by the same formula as above.

Then the construction in Example 17.3 assigns to a simplicial complex Ya chain complex C[∗](Y ) in C(Y ;A) without the assumption that Y is locallyfinite. Moreover, if Y is an topological space, its singular chain complex canbe lifted to a chain complex in C(Y ;A), where a singular simplex σ : ∆n → Ysits over the point in Y given by the image of the barycenter of ∆n.

Example 17.3 (Simplicial chain complexes). Let Y be a locally finitesimplicial chain complex. Locally finite means that each vertex belongs onlyto finitely many simplices, or, equivalently, that Y is locally compact. Supposethat we fix an ordering on its vertices. The simplicial chain complex C∗(Y ) isa Z-chain complex such that the d-th-chain module has a preferred basis bi |i ∈ Id, where Id is the set of d-simplices. Each simplex i ∈ Id determines apoint yi in Y , namely its barycenter. Notice that i ∈ Id is uniquely determinedby its barycenter xi.

We obtain an object C[d] = C[d]y in C(Y ;Z-FGF) by putting Ay = Z ify = xi for some i ∈ Id and Ay = 0 otherwise. The support of this object islocally finite since Y is locally finite by assumption.

Next we define a morphism c[d] : C[d] → C[d − 1] in C(Y ), [-FGFR]). Forevery i ∈ Id and j ∈ Id−1 we have the incidence number εi,j ∈ −1, 0, 1which is zero if j is not a face in i, and, in case that j is a face of i, is±1 depending on whether the induced orderings on the vertices of i andof j agree or not. We define c[d]x,y to be εi,j if for some i ∈ Id and somej ∈ Id−1 we have x = xi and Y = yi. This morphism is well-defined sinceY is locally finite. Thus we obtain a chain complex C[∗](Y ) in the additivecategory C(Y ;R-FGF).

Let pr : Y → • be the projection. Given any chain complex D[∗] inC(Y ;Z-FGF), we obtain a Z-chain complex pr∗D[∗] whose d-th chain moduleis⊕

x∈X D[∗]x and whose differential is given by the differential of D[∗] inthe obvious way. Obviously pr∗ C[∗](Y ) is just the simplicial chain complexC∗(Y ) of Y .

17.2 Controlled Algebra 491

Comment 74: Discuss also singular chain complexes, where we will haveto give up the condition that the objects are locally finite. This will becomerelevant when dealing with the transfer.

In the sequel A comes with a right G-action by isomorphisms of additivecategories. If Y is a (left) G-space, we get a right G-action by isomorphisms ofadditive categories on C(Y ;A) by (A ·g)x := Agx ·g and (φ ·g)x,y := φgx,gy ·g.

Notation 17.4. Denote byCG(Y ;A)

the G-fixed point subcategory of the G-category C(Y ;A).

Define CG(Y ;A) analogously

Obviously CG(Y ;A) is an additive subcategory of C(Y ;A). An object inCG(Y ;A) is given by a locally finite collection (Ax)x∈Y of objects in A suchthat Ax = Agx · g holds for all g ∈ G and x ∈ Y . A morphism (φx,y)x,y∈Yin C(Y ;A) between two objects which belong to CG(Y ;A) is a morphism inCG(Y ;A) if and only if φgx,gy · g = φx,y holds for all g ∈ G and x, y ∈ Y .

Exercise 17.5. The additive categories CG(G;A) and∫GA Comment 75:

Check where∫GA has been introduced. We also have to deal with the pas-

sage to∫GA⊕. Maybe we have to replace isomorphic by equivalent. are

isomorphic.

17.2.2 Imposing Control Conditions

Comment 76: Shall this formal definition appear later? Intuitively, it doesnot really help. It is useful, when discussing functoriality and unifies every-thing. But one could also add a list of instances, where certain functor shouldbe induced by maps and list in each case the necessary condition.

We are seeking certain additive subcategories of CG(Y,A), where supportconditions are imposed on the objects and morphisms.

This is formalized by the following definition taken from [68, 2.3], seealso [389].

Definition 17.6 (Coarse structure). A coarse structure (F , E) on a G-space Y consists of a set E of subsets of Y × Y and a set F of subsets of Ysatisfying the following axioms:

(i) For E,E′ ∈ E , there exists E′′ satisfying E E′ ⊆ E′′, where E E′ :=(x, z) ∈ Y × Y | ∃y ∈ Y with (x, y) ∈ E′, (y, z) ∈ E;

(ii) For E,E′ ∈ E , there exists E′′ satisfying E ∪ E′ ⊆ E′′;(iii) The diagonal (y, y) | y ∈ Y belong to E ;(iv) For F, F ′ ∈ E , there exists F ′′ satisfying F ∪ F ′ ⊆ F ′′;


(v) E is symmetric, i.e., if E ∈ E , then τ(E) ∈ E , where τ : X ×X → X ×Xis the flip map; Comment 77: Is this too strong? Do we need it at all?Is it satisfied in all applications? See [68, 2.3].

(vi) F and F and E are G-invariant, where G acts diagonally on Y × Y .

An object in C(Y ;A) is called admissible if there exists F ∈ F which con-tains its support. A morphisms (φx,y) in C(Y ;A) is called admissible if thereexists E ∈ E which contains its support supp(φ) := (x, y) | x, y ∈ Y, φx,y 6=0. The axioms are designed such that the admissible objects together withthe admissible morphisms form an additive subcategory of CG(Y ;A) whichwe will denote by

Notation 17.7. CG(Y, E ,F ;A).

Namely, axiom (i) appearing in Definition 17.6 ensures that the subcate-gory C(Y, E ,F ;A) consisting of admissible objects and admissible morphismsof C(Y,A) is closed under compositions of morphisms, axiom (ii) guaranteesthat it is closed under the sum of morphisms, axiom (iii) ensures that theidentity morphism of any object is again in the subcategory, axiom (iv) guar-antees that the subcategory is closed under direct sums, and (vi) ensures thatthe G-action on C(Y ;A) induces a G-action on C(Y, E ,F ;A).

Let f : Y → Z be a G-equivariant map. In the sequel we will assume thatthere is a functorial model for direct summands for A over index sets ofcardinality < κ for a fixed infinite cardinal. If κ is the cardinal representedby the set of integers, a set has < κ precisely if it is finite.

Comment 78: Explain that one can replace A by an equivalent categorywith this property.

Remark 17.8 (Induced functors). The formula

(f∗(A))z :=⊕

y∈f−1(z)

Ay(17.9)

defines a functor f∗ : CG(Y ;A)→ CG(Z;A) if the preimage of every point ofZ under f has cardinality < κ. It defines a functor f∗ : CG(Y ;A)→ CG(Z;A)if the preimage of every point of Z under f has cardinality < κ, and f sendslocally finite sets to locally finite sets. It defines a functor

f∗ : CG(Y, EY ,FY ;A)→ CG(Z, EZ ,FZ ;A)(17.10)

if the following conditions are satisfied:

(i) The preimage of every point of Z under f has cardinality < κ;(ii) For every compact subset K ⊆ Z and every F ∈ FY the set f−1(K)∩F is

compact again. (This will ensure that the object (f∗(A))z is again locallyfinite without having to require that f sends locally finite sets to locallyfinite sets);


(iii) For every F ∈ FY there exists F ′ ∈ FZ with f(F ) ⊆ F ′;(iv) For every E ∈ EY there exists E′ ∈ EZ with f × f(E) ⊆ E′. Sometimes

we cannot prove the last condition, but only the weaker condition thatfor every E ∈ EY and every F ∈ FY , there exists E′ ∈ EZ satisfyingF × F (E ∩ F × F ) ⊆ E′. This is also sufficient.

Lemma 17.11. Let f, g : Y → Z and g : Z → Y be G-maps

(i) If f and g induce functors CG(Y ;A) → CG(Z;A), then there exists a

natural equivalence T : f∗∼=−→ g∗;

(ii) If f and g induce functors CG(Y, EY ,FY ;A) → CG(Z, EZ ,FZ ;A), then

there exists a natural equivalence T : f∗∼=−→ g∗, provided that for each

object A ∈ CG(Y, EY ,FY ;A) there is an element EA ∈ EZ such that(f(y), g(y)) ∈ EA for all y ∈ suppA.

Proof. (i) The natural equivalence T is given for an object A = Ay | y ∈Y in CG(Y ;A) by the isomorphism T (A) : f∗A

∼=−→ g∗A in CG(Z;A) whosevalue at (z, z′) ∈ Z is the morphism

⊕y∈f−1(z)Ay →

⊕y′∈g−1(z′)Ay′ in A

which sends Ay to zero if y 6∈ f−1(z′) and identically to the summand Ay ify ∈ f−1(z′).

(i) The same definition of T (A) as above makes sense in CG(Z, EZ ,FZ ;A)since we require that for each objectA ∈ CG(Y, EY ,FY ;A) there is an elementEA ∈ EZ satisfying (f(y), g(y)) ∈ EA for all y ∈ suppA. ut

17.2.3 Germs at infinity

It will be important to consider the quotient category of CG(Y×[1,∞), E ,F ;A)given by germs at infinity, where we intuitively ignore everything which hap-pens in a bounded region with respect to [1,∞). Here is the precise definition.

Definition 17.12 (germs at infinity). The germs at infinity quotient cat-egory

CG(Y × [1,∞), E ,F ;A)∞

is the following the additive category. It has the same objects as CG(Y ×[1,∞), E ,F ;A). The set of morphisms from an object A to an objectB is the quotient of abelian groups morCG(Y×[1,∞),E,F ;A)(A,B)/K, whereK ⊆ morCG(Y×[1,∞),E,F ;A)(A,B) is the abelian subgroup of those morphismsφ : A → B for which there exists a real number r(φ) ≥ 1, an object L(φ)with support in G × Y × [1, r(φ)] and morphisms ψ(φ) : A → L(φ) andψ′(φ) : L(φ)→ B satisfying φ = ψ′(φ) ψ(φ).

Exercise 17.13. Show that CG(Y × [1,∞), E ,F ;A)∞ is a well-defined addi-tive category.


17.2.4 Some control condition

Given a G-space Y , we denote by FYGc the set consisting of all cocompactG-subsets of Y . Let EYall be the set of all subsets of Y × Y .

Notation 17.14. Define the additive G-category

T G(Y ;A) := CG(G× Y,FG×YGc , Eall;A)

where G acts diagonally on Y .

Comment 79: Explain the role of the factor G in the definition above. Dowe have to deal with more general setup of resolutions, see [68, Section 3.1]?

Lemma 17.15. Let Y be G-space. Then the projection pr: Y → • inducesan equivalence of additive categories

pr∗ : T G(Y ;A)→ T G(•;A).

Proof. Choose a point y ∈ Y . Define a G-map i : G→ G×Y by i(g) = (g, gy).Then pr and i induce functors pr∗ and i∗ since the conditions appearing inRemark 17.8 are satisfied. We have pr i = id and hence pr∗ i∗ = id. Weconclude from Lemma 17.11 (ii) that there is an equivalence of functor fromi∗ pr∗ to idT G(Y ;A). ut

Let Z be a space equipped with a quasi-metric d. (We remind the readerthat the difference between a metric and a quasi-metric is that in the latercase the distance ∞ is allowed.) Then we define EZd to be the collection ofall subsets E of Z × Z of the form Eα = (z, z′) | d(z, z′) ≤ α with α <∞.A morphism φ ∈ C(Z, EZd ) is said to be δ-controlled if suppφ ⊆ Eδ. Thisterminology will be used in subsection Comment 80: Say where. and wewill often be interested in small δ.

Let Y be a G-space. A subset C ⊂ Y is called G-compact Comment 81:Discuss G-compact versus cocompact. Do these notions agree in the com-pactly generated category. We have used cocompact so far. if there existsa compact subset C ′ ⊆ Y satisfying C = G · C ′. For a G-CW -complex Y ,a subset C ⊆ Y is G-compact, if and only if its image under the projectionY → G\Y is a compact subset of the quotient G\Y . Denote by FYGc the setwhich consist of all G-compact subsets of Y . Comment 82: This set hasalready been introduced above.

Let Y be a G-space. We denote by Gy the isotropy group of a point y ∈ Y .Equip Y × [1,∞) with the G-action given by g(y, t) := (gy, t). The followingdefinition is taken from [68, Definition 2.7].

Let Y be a G-space. Define EY∆ to be ∆Y , where ∆Y = (y, y) | y ∈Y ⊆ Y × Y is the diagonal. Let pY : G × Y → Y be the projection. Definep−1Y (EY∆) = p−1(∆Y ).


Comment 83: In the formulation and in the proof of the followingLemma, we later have later to adjust to the fact that we will have to ex-

tend this to(∫GG(S)

A)⊕

.

Lemma 17.16. For every G-set S, there is an equivalence of additive cate-gories

F (S) :

∫GG(S)

(A pr)'−→ CG(G× S, p−1

S ES∆,FG×SGc , ;A)

It is natural under G-maps S1 → S2 of G-sets.

Proof. We abbreviate in the sequel∫GG(S)

(A pr) by∫GG(S)

A. Recall that

an object in∫GG(S)

A is given by a pair (s1, A1) for s1 ∈ S and an object

A1 in A. The image of such an object under F (S) is the object in CG(G ×S,FG×SGc , p−1

S ES∆;A) whose value at a point in (g, s) in G × S is A1 · g−1 ifg−1s = s1, and is the zero object in A otherwise. One easily checks thatthis object, which a priori lives in C(G × S;A), indeed defines an elementin CG(G × S,FG×SGc , p−1

S ES∆;A), i.e., it is invariant under the G-action onC(G× S;A), and its support is cocompact.

Let (s2, A2) another object in∫GG(S)

A. Recall that a morphism φ in∫GG(S)

A from (s1, A1) to (s2, A2) is given by a formal sum∑g∈G;gs1=s2

g ·φg,where φg is a morphism in A from A1 to A2 · g and the set g | ψg 6= 0 isfinite. Define the image of φ under F (S) by the morphism ψ(g′,s′),(g′′,s′′),where

ψ(g′,s′),(g′′,s′′) = φ(g′′)−1g′ ·(g′)−1 : A1·(g′)−1 → (A2·(g′′)−1g′)·(g′)−1 = A2·(g′′)−1

if s′ = s′′, (g′)−1s′ = s1 and (g′′)−1s′′ = s2, and is trivial otherwise. No-tice that (g′′)−1g′ defines a morphism in GG(G/H) from s1 to s2 since(g′′)−1g′s1 = (g′′)−1g′(g′)−1s′ = (g′′)−1s′′ = s2 holds. For every g ∈ G, wehave ψ(gg′,gs′),(gg′′,gs′′)·g = ψ(g′,s′),(g′′,s′′), and the support of ψ(g′,s′),(g′′,s′′),is contained in p−1

Y (EY∆). This shows that F (S) is a functor from∫GG(S)

(Apr)

to CG(G× S,FG×SGc , p−1S ES∆;A).

Comment 84: Finish the proof. We have to show that it is an equivalence.Check in particular the case where S consists of more than two homogeneousspaces and hence GG(S) is not connected. ut

Comment 85: We are using the control on G with respect to the wordmetric. See the discussion on page 40 in [73]. Shall we elaborate on this? Thenext lemma is relevant.

Lemma 17.17. For any G-space Y , the categories CG(G×Y,FG×YGc ;A) andCG(G× Y, p−1

G (EGdG),FG×YGc ;A) agree.


Proof. Consider a morphism φ : A → B in CG(G × Y,FG×YGc ;A). In order toshow that it also belongs to CG(G× Y, p−1

G (EGdG),FG×YGc ;A), we have to finda real number α ≥ 0 such that ((g, y), (g′, y′)) ∈ supp(φ) implies d(g, g′).

We next examine the support of A = Ag,y. Because of the control con-dition FG×YGc we can find a compact subset K ⊆ G × Y Comment 86:Actually we only have cocompact. Adjust this. with supp(A) ⊆ G · K.Since G is discrete, we can find a finite subset g1, g2, . . . gm of G such thatg × Y ∩ K 6= ∅ implies g ∈ g1, g2, . . . gm. Since gi × Y ∩ K is com-pact, we can find for each i ∈ 1, 2, . . . ,m a compact subset Li ⊆ Y withgi × Y ∩K = gi × Li. Let L ⊆ Y be the compact subset

⋃mi=1 g

−1i · Li.

Then G·K is contained in G·e×L. Comment 87: This argument appearsalready in another proof.

Since supp(A) is locally finite, its intersection with e × L is finite, letus say (e, y1), (e, y2), ·(e, yn). For every i ∈ 1, 2, . . . , n the set (g′, y′) |((e, yi), (g

′, y′)) ∈ supp(φ) is finite. Hence the set (e, y), (g′, y′) ∈ supp(φ)is finite. Choose α such that d(e, g′) ≤ α holds for all ((e, y), (g′, y′)) ∈supp(φ). This implies d(g, g′) ≤ α for all ((g, y), (g′, y′)) ∈ supp(φ), sincewe have d(g, g′) = d(e, g−1g′). ut

Exercise 17.18. Let Y be a G-space. Show that T G(Y ;A) and∫GA are

equivalent as additive categories.

Example 17.19. Let R-FGF be a (skeleton) of the category of finitely gen-erated free R-modules. Comment 88: Check whether this category hasbeen introduced before and whether there is an entry in the notations. ThenCG(G × G/H,FG×SGc , p−1

S ES∆;R-FGF) is a skeleton of the category of finitelygenerated free RH-modules. This follows from Lemma 17.16 and the factthat

∫GG(G/H)

R-FGF is a skeleton of the category of finitely generated free

RH-modules. Comment 89: Add reference to the place where this hasalready been explained.

Comment 90: Maybe one should define this for a pair (X,Y ). We willthen mainly consider the case (Y × [1,∞], Y × [1,∞)).

Definition 17.20 (Equivariant continuous control condition). We de-fine EYGcc to be the collection of subsets E ⊆ (Y × [1,∞)) × (Y × [1,∞))satisfying:

(i) For every y ∈ Y , every Gy-invariant open neighborhood U of (y,∞) inY × [1,∞] there exists a Gy-invariant open neighborhood V ⊆ U of (y,∞)in Y × [1,∞] such that

((Y × [1,∞]− U)× V ) ∩ E = ∅;

(ii) The image of E under the projection (Y × [1,∞))×2 → [1,∞)×2 sends E

to a member of E [1,∞)d where d(t, s) = |t− s|;


(iii) E is symmetric and invariant under the diagonal G-action.

We call EYGcc the equivariant continuous control condition.

Comment 91: The equivariant continuous control condition involves onlyY × [1,∞), but not G × Y × [1,∞). Therefore it is possible to give a refor-mulation of τG(Y ;A) and OG(Y ;A), where the variable G does not ap-pear at all. Because of equivariance an object Ag,,x,t is determined byremembering Ae,x,t. Given two objects Ag,x,t and Bg′,x′,t′, a mor-phism φ(g,x,t),(g′,x′,t′) : Ag,,x,t → Bg′,,x′,t′ is determined by the collec-tion φ(e,x,t),(g′,x′,t′). Hence we can encode objects as collections Ax,t anda morphism φ(x,t),(x′,t′) : Ax,t → Bx′,t′ is given by a collection of mor-phisms φ(x,t),(x′,t′);g : Ax,t → Bx,t · g | g ∈ G which one may write as aformal (twisted) sum

∑g∈G : φ(x,t),(x′,t′);g : Ax,t → Bx′,t′ . This may simplify

the notation. This may in particular be useful when one is studying A-theory.Comment 92: We may change the notation by omitting the projections

and repeating the control conditions if they are given by to different ones.For example, CG(G × Y × [1,∞), p−1EYGcc ∩ r−1EGdG , q

−1FY×Gc ;A) becomes

CG(G × Y × [1,∞), EY×[1,∞)ecc , EGd ,FY×Gc ;A). Maybe we further abbreviate

EY×[1,∞)ecc by EYecc. Also replace Gcc by ecc, and GC by c. Denote metric

control by E?d . One should also have the space in the notation of a control

condition. Then the corresponding map is always clear from the context.One may have to adjust the discussion when a map induces a map betweencontrolled categories. One should also omit Eall from the notation, for instance

CG(G×[1,∞), p−1E•Gcc∩r−1EGdG ,FGall;A) becomes CG(G×[1,∞), E•ecc , EGd ;A),

and CG(G×Y,FG×YGc , Eall;A) becomes CG(G×Y,FG×Yc ;A). Say that E standsfor entourage.

Lemma 17.21. Suppose that Y is locally connected. Then every morphismin the germ at infinity category

CG(G× Y × [1,∞); (pY×[1,∞) × pY×[1,∞))−1EYGcc; p−1

G×Y FG×YGc ;A)∞

can be represented by a morphism φ(g1,y1,t1),(g2,y2,t2) in

CG(G× Y × [1,∞); (pY×[1,∞) × pY×[1,∞))−1EYGcc; p−1

G×Y FG×YGc ;A)

for which φ(g1,y1,t1),(g2,y2,t2) 6= 0 implies that y1 and y2 lie in the same com-ponent of Y .

Proof. Let φ = φ(g1,y1,t1),(g2,y2,t2) : A = Ag1,t1,y1 → B = Bg2,t2,y2 be amorphism in

CG(G× Y × [1,∞); (pY×[1,∞) × pY×[1,∞))−1EYGcc; p−1

G×Y FG×YGc ;A)

By assumption the support of A and B are cocompact. Hence we can find acompact subset K ⊆ Y such that g−1 · y ∈ K holds if Ag,y,t 6= 0 or Bg,y,t 6= 0is valid.


Let C ⊆ Y be a component of y. Since Y is locally connected, C ⊆ Yis an open subset. Obviously C is Gz-invariant for every z ∈ C. For eachx ∈ K∩C there exists an open Gx-invariant neighborhood U(x) of x in Y anda natural number r(x) ≥ 1 such that for

((g1, y1, t1), (g2, y2, t2)

)∈ supp(φ)

satisfying t1 ≥ r(x) and y1 ∈ U(x) we have y2 ∈ C. This follows theequivariant continuous control condition. Since K is compact, we can finda finite subset x1, x2, . . . , xl ⊆ K with K ⊆ ∪li=1U(xi). Put r(C) :=maxr1(x), r2(x), . . . , r(xl). Hence we get for

((g1, y1, t1), (g2, y2, t2)

)∈

supp(φ) satisfying t1 ≥ r(C) and y1 ∈ K ∩ C that y2 ∈ C. Since Kis compact, K meets only finitely many component of Y , let us say, C1,C2, . . . , Cm. Put r := maxr(Ci) | i = 1, 2, . . . ,m. We conclude thatfor every element

((g1, y1, t1), (g2, y2, t2)

)∈ supp(φ) satisfying t1 ≥ r and

y1 ∈ K the elements y1 and y2 must belong to the same component ofY . Since φ(g1,y1,t1),(g2,y2,t2) 6= 0 implies (g1, y1, t1) ∈ supp(A) and hence

g−11 y1 ∈ K, we have φ(g1,y1,t1),(g2,y2,t2) = φ(e,g−1

1 y1,t1),(g−11 g2,g

−11 y2,t2), and y1

and y2 belong to the same component of Y if and only if g−11 y1 and g−1

2 y2

belong to the same component of Y , we conclude that for every element((g1, y1, t1), (g2, y2, t2)

)∈ supp(φ) satisfying t1 ≥ r the elements y1 and y2

must belong to the same component of Y .Now define a new morphism ψ = ψ(g1,y1,t1),(g2,y2,t2) : A→ B by requir-

ing that ψ(g1,y1,t1),(g2,y2,t2) = φ(g1,y1,t1),(g2,y2,t2) if t1 ≥ r, and ψ(g1,y1,t1),(g2,y2,t2) =0 if t1 < r. Then ψ has the desired property that for every element((g1, y1, t1), (g2, y2, t2)

)∈ supp(ψ) the elements y1 and y2 belong to the same

component of Y . By the equivariant continuous control condition, there ex-ists a real number α ≥ 0 such that every element

((g1, y1, t1), (g2, y2, t2)

)∈

supp(φ) satisfies t1−t2 ≤ α. Hence for every element((g1, y1, t1), (g2, y2, t2)

)∈

supp(φ−ψ) we have t1 ≥ r and t2 ≥ r+α. Therefore φ−ψ factorizes throughan object whose support is contained in G× Y × [1, r+α]. This implies thatφ and ψ represent the same element in the germ at infinity quotient cate-gory. ut

On Y × [1,∞) we can define the control condition

Edis := p−1X (EY∆) ∩ p[1,∞)(ε

[1,∞)d ).

So morphisms will not move things in the Y direction and have the standardmetric control in the [1,∞) direction. Notice that the control condition onG×Y × [1,∞) given by pY×[1,∞)(Edis) is stronger than the support condition

pY×[1,∞)(EY×[1,∞)Gee ). In particular,

CG(G× Y × [1,∞); pY×[1,∞)(Edis), p−1G×[1,∞)(F

G×YGc );A)

is a subcategory of

CG(G× Y × [1,∞); p−1Y (EY∆); p−1

G×[1,∞)(FG×YGc );A)


However, after passing to the germ at infinity quotient categories, nothingchanges for a discrete G-space Y , namely:

Lemma 17.22. The two categories

CG(G× Y × [1,∞); pY×[1,∞)(Edis), p−1G×[1,∞)(F

G×YGc );A)∞

andCG(G× Y × [1,∞); p−1

Y (EY∆); p−1G×[1,∞)(F

G×YGc );A)∞

agree, provided that Y is a discrete G-space.

Proof. This is a consequence of Lemma 17.21 since x, y ∈ Y lie in the samecomponent if and only if x = y. ut

Comment 93: Shall we formulate this with taking the control conditionover (G, dG) into account?

17.2.5 The assembly map as a forget control map

Let G be finitely generated group equipped with a word-metric dG. For aG-space Y let Comment 94: Shall we use the convention to denote pro-jection all with p? with a subscript given by its target, e.g.,p = pY×[1,∞),q = pY×[1,∞), r = pG.?

p : G× Y × [1,∞)→ Y × [1,∞);

q : G× Y × [1,∞)→ G× Y ;

r : G× Y × [1,∞)→ G,

be the canonical projections. We will abuse notation and set

p−1EYGcc ∩ r−1EGdG := (p× p)−1(E) ∩ (r × r)−1(E′) | E ∈ EYGcc, E′ ∈ EGdG;q−1FG×YGc := q−1(F ) | F ∈ FG×YGc .

Notation 17.23. Define additive categories

OG(Y ;A) := CG(G× Y × [1,∞), p−1EYGcc ∩ r−1EGdG , q−1FY×GGc ;A);

DG(Y ;A) := CG(G× Y × [1,∞), p−1EYGcc ∩ r−1EGdG , q−1FY×GGc ;A)∞.

where the germs at infinity have been introduced in Definition 17.12.

We remark that in [68, Subsection 3.2] a slightly different definition ofDG(Y ) is given, where the metric control condition EGdG does not appear.Comment 95: Shall we discard this sentence or elaborate on it? UsingTheorem 17.27 below it can be shown that this does not change the K-theory


of these categories. The metric control condition on G will be important inthe construction of the transfer, see in particular Proposition Comment 96:Add reference. Comment 97: It depends on the following text whether thissentence shall stay or shall be discarded. The interested reader may comparethis difference to different possible definitions of cone and suspension rings.Often it is convenient to add finiteness condition to obtain formulas such as(ΛR)G = Λ(RG), compare [68, Remark 7.2].

The following is the so-called germs at infinity sequence

(17.24) T G(Y )→ OG(Y )→ DG(Y ).

Here the first map is induced by 1 ⊂ [1,∞) and the second is the quotientmap.

Lemma 17.25. The category OG(•) is flasque and hence Kn(OG(•))vanishes for all n ∈ Z.

Proof. Notice that by definition

OG(•;A) = CG(G× [1,∞), p−1E•Gcc ∩ r−1EGdG , q

−1FGGc;A)

Obviously q−1FGGc contains G× [1,∞) and hence

OG(•;A) = CG(G× [1,∞), p−1E•Gcc ∩ r−1EGdG ,F

Gall;A)

Consider the map sk : [1,∞) → [1,∞) sending t to (t + k) for k = 1, 2, . . ..One easily checks, see the condition after (17.10), that it induces a functor

(sk)∗ : CG(G× [1,∞), p−1E•Gcc ∩ r−1EGdG ,F

Gall;A)

→ CG(G× [1,∞), p−1E•Gcc ∩ r−1EGdG ,F

Gall;A).

It shifts a geometric module along the [1,∞) direction to ∞ by the distancek. We want to define S =

⊕∞k=1(sk)∗, where we have to check that this is

well-defined, the problem is that the direct sum seems to be an infinite sum onthe first glance. However, it is essentially finite, since for a given t ∈ [1,∞)there are only finitely many integers i with 1 ≤ i and 1 ≤ t − i. Next weformulate precisely, what this idea means, namely, we define the functor ofadditive categories

S : OG(•;A) = CG(G× [1,∞), p−1E•Gcc ∩ r−1EGdG ,F

Gall;A)

→ OG(•;A) = CG(G× [1,∞), p−1E•Gcc ∩ r−1EGdG ,F

Gall;A)

by sending an object A = Ag,t | (g, t ∈ G× [1,∞) to the object S(A) givenfor g, t ∈ G× [1,∞) by


S(A)g,t :=⊕

i∈Z,1≤i≤t−1

Ag,t−i.

Notice that the direct sums are over finite index sets and hence make sense.A morphism φ = φ(g,t),(g′,t′) is sent to the morphism S(φ) whose value at(g, t), (g′, t′) is given by

S(φ)(g,t),(g′,t′) =⊕

i∈Z,1≤i≤t−1,t−1′

φ(g,t−i),(g′,t′−i).

We have to check that the control conditions are satisfied. The support ofS(A) is given by

supp(S(A)) = (g, x) | ∃i ∈ Z, 1 ≤ i ≤ t− 1 with (g, ti) ∈ supp(A).

Since supp(A) is locally finite by assumption, supp(S(A)) is also locally finite.This is the only condition on the support of objects.

Let φ = φ(g,t),(g′,t′) : Ag′,t′ → Bg′,t′ be a morphism in the additive

category CG(G×[1,∞), p−1E•Gcc∩r−1EGdG ,FGall;A). By assumption there exists

a element E ∈ p−1E•Gcc ∩ r−1EGdG with supp(φ) ⊆ E. We have to find E′ ∈p−1E•Gcc ∩ r−1EGdG such that supp(S(A)) ⊆ E′ holds. By definition we canchoose a number α ∈ [0,∞) such that for every (g, t), (g′, t′) ∈ E we have|t − t′| ≤ α, and we can choose for every open neighborhood U of ∞ in[1,∞] an open neighborhood V (U) of ∞ in [1,∞] with the property that for(g, t), (g′, t′) ∈ E with (g′, t′) ∈ G× V (U) we have (g, t) ∈ G× U . Obviously

supp(S(φ)) = (g, t), (g′, t′) ∈ G× [1,∞)×G× [1,∞)

| ∃i ∈ Z with 1 ≤ i ≤ t− 1, t′ − 1 and (g, t− i), (g′, t′ − i) ∈ supp(φ).

We put

E′ = (g, t), (g′, t′) ∈ G× [1,∞)×G× [1,∞)

| ∃i ∈ Z with 1 ≤ i ≤ t− 1, t′ − 1 and (g, t− i), (g′, t′ − i) ∈ E.

Obviously supp(φ) ⊆ E′. It remains to check that E′ belongs to p−1E•Gcc ∩r−1EGdG . Notice that |t− t′| = |(t− i)− (t′ − i)| holds for all t, t′ ∈ [1,∞) andi ∈ Z. Hence for every (g, t), (g′, t′) ∈ E′ we have |t−t′| ≤ α. Consider an openneighborhood U of ∞ in [1,∞]. Choose an integer k ≥ 1 with (k,∞) ⊆ U .For every integer j ≥ 1 we can choose a natural number rj such that for((g, t), (g′, t′)) ∈ E with rj < t′ we get j < t. Choose a real number r suchthat rk−i + i < r holds for i = 0, 1, 2, . . . , (k − 1). Put W (U) = (r,∞).Consider (g, t), (g′, t′) ∈ E′ with (g′, t′) ∈W (U). Then there exists i ∈ Z, i ≥0, i ≤ t − 1, t′ − 1 with (g, t − i), (g, t′ − i) ∈ E. From (g′, t′) ∈ W (U) we


conclude that rk−i < t′− i. This implies k− i ≤ t− i and hence k < t. Hencewe get (g, t) ∈ U . This finishes the proof that the functor S is well-defined.

Now one easily constructs a natural equivalence T : id⊕S∼=−→ S, which is

essentially given by reindexing. Namely, given an object A, define the isomor-

phism T (A) : A⊕S(A)∼=−→ S(A) by the collection of morphism φ(g,t),(g′,t′),

where φ(g,t),(g′,t′) is the identity on⊕

i∈Z,0≤i,t−i≥1Ag,t−i if g′ = g andt′ = t+ 1, and is the zero homomorphism otherwise.

This finishes the proof that OG(•;A) is flasque. The claim about thevanishing of its K-groups follows from Theorem 5.33 (iii). ut

Comment 98: Explain why this does not work in general, if we replace• by a G-space Y .

We will need the following

Lemma 17.26. The sequence (17.24) induces a long exact sequence in K-theory.

Proof. We can replace T G(Y ) by an equivalent category, namely, by the fullsubcategory of OG(Y ) on all objects that are isomorphic to an object inT G(Y ). These are precisely the objects in OG(Y ) whose support is containedin G × Y × [1, r] for some r ≥ 0. Then the first map in (17.24) becomes aKaroubi filtration Comment 99: Shall we elaborate on Karoubi filtrations?and DG(Y ) is its quotient. Now the claim follows because Karoubi filtrationsinduce long exact sequences in K-theory, see for example [159]. ut

Theorem 17.27. The assignment Y 7→ K∗(DG(Y )) is a G-equivariant ho-mology theory on G-CW -complexes. The projection EFG→ • induces theassembly map Comment 100: Add reference.

Proof. This is proven in [68, Section 5, Corollary 6.3], see also [77, Theo-rem 7.3]. As mentioned above a slightly different definition is used in thesereferences, but this does not affect the proof and the arguments can be car-ried over word for word. Comment 101: We will elaborate on this proofand prove it using the universal property of the assembly map following [68,Section 6.1]. ut

Comment 102: The formulation and the proof of the next lemma arefar away from their final form. There will be a zigzag, not just one functor,see [68, Section 6.1].

Lemma 17.28. For every homogeneous space G/H, there is an equivalenceof additive categories

F (G/H) : D1(•;

∫GG(G/H)

A pr

)'−→ DG(G/H;A),

It is natural under G-maps G/H → G/K.


Proof. We first define a functor of additive categories

F (G/H) : O1(•;

∫GG(G/H)

A pr

)'−→ OG(G/H;A),

Recall that an object in∫GG(G/H)

is given by a pair (gH,A) for gH ∈ G/H

and an object A in A. Hence an object in C1(•;

∫GG(G/H)

A pr)

is given

by a collection (gtH,At) | t ∈ [1,∞), where gtH ∈ G/H and A is an objectin A such that the set t ∈ [1,∞) | At 6= 0 is locally finite. The image of

such an object under F (G/H) is the object in OG(G/H;A) whose value at apoint in (g, g′H, t) in G×G/H× [1,∞) is At ·g−1 if g−1g′H = gtH, and is thezero object in A otherwise. One easily checks that this object, which a priorilives in C(G×G/H × [1,∞);A), indeed defines an element in OG(G/H;A),i.e., it is invariant under the G-action on C(G × G/H × [1,∞);A) and itssupport is cocompact.

Recall that a morphism in∫GG(G/H)

from (g1H,A1) to (g2H,A2) is

given by a formal sum∑g∈G;gg1H=g2H

g · ψg, where ψg is a morphismin A from A1 to A2 · g and the set g | ψg 6= 0 is finite. Consider amorphism φ from (gsH,As) | s ∈ [1,∞) to (gtH,Bt) | t ∈ [1,∞)in O1

(•;

∫GG(G/H)

A pr)

. It is given by a collection of morphisms

φs,t,g : As → Bt · g in A, where (s, t) runs through the elements [1,∞) ×[1,∞) and g through the element in G satisfying ggsH = gtH. The col-lection has to satisfy the following properties: For every (s, t) there areonly finitely many g ∈ G with φs,t,g 6= 0, for every s ∈ [1,∞) the sett ∈ [1,∞) | ∃g ∈ G with φs,t,g 6= 0 is finite, for every t ∈ [1,∞)the set s ∈ [1,∞) | ∃g ∈ G with φs,t,g 6= 0 is finite, and the set(s, t ∈ [1,∞) × [1,∞) | ∃g ∈ G with φs,t,g 6= 0 is locally finite. Define

the image of φ under F (G/H) by the morphism ψ(g1,g′1H,s),(g2,g′2H,t), where

ψ(g1,g′1H,s),(g2,g′2H,t)

= φs,t,g · g−11 : As · g−1

1 → Bt · g · g−11 = B · g−1

2

if (g2, g′2H) = (g1g

−1, g2gg−11 g′1H) for some g ∈ G, and is trivial otherwise.

Notice that g defines a morphism in GG(G/H) from g−11 g′1H to g−1

2 g2H sinceg(g1)−1g′1H = g−1

2 g2H, and hence φ(t1,t2,g) is indeed a morphism from As toBt ·g−1 as

∑g∈G g ·φ(t1,t2,g) is a morphism in

∫GG(G/H)

from ((g1)−1g1, As) =

(gsH,As) to ((g2)−1g′2H,Bt) = (gtH,Bt).We have to check that we do get a morphism in OG(G/H;A). Firstly one

has to show that ψ(ug1,g′1H,t1),(ug2,g′2H,t2) ·u = ψ(g1,g′1H,t1),(g2,g′2H,t2) for all u ∈G if (g2, g

′2H) = (g1g

−1, g2gg−11 g′1H) for some g ∈ G. This follows from the

facts that the last equation implies (ug2, ug′2H) = (ug1g

−1, ug2g(ug1)−1ug′1H)and we have

(φt1,t2,g · (ug1)−1

)· u = φt1,t2,g · g−1

1 . Secondly, one has to check

the control conditions. Since φ belongs to O1(•;

∫GG(G/H)

A pr)

, there


exists a subset E ⊆ [1,∞) × [1,∞) such that (s, t) ∈ [1,∞)×]1,∞) | ∃g ∈G with φs,t,g 6= 0 is contained in E, for every neighborhood U of∞ in [1,∞]there is a neighborhood V (U) of∞ in [1,∞] with the property that φs,t,g 6= 0and s ∈ V (U) implies t ∈ V (U). We have for the support of F (G/H)(φ)

supp(F (G/H)(φ)) = (g1, g′1H, s), (g2, g

′2H, t) |

∃g with (g2, g′2H) = (g1g

−1, g2gg−11 g′1H) and φs,t,g 6= 0.

ut

Comment 103: We have to deal with L-theory as well.Comment 104: Shall we first treat K-theory completely and then explain

the differences for L-theory?Comment 105: Shall we give details of the proofs only in the hyperbolic

case and discuss CAT(0)-groups and polycyclic groups and arithmetic groupsafterwards.

The following is now an easy consequence, compare [367, Theorem 7.4].

Proposition 17.29. Suppose there exists an m0 ∈ Z such that for all A andall m ≥ m0 we have

Km(OG(EFG;A)) = 0.

Then the assembly map Comment 106: Add reference. is an isomorphismfor all n ∈ Z and all A.

Proof. If the assembly map is an isomorphism for all m ≥ m0 and all A, thenit is an isomorphism for all n ∈ Z and all A by [77, Corollary 4.7]. Comment107: Elaborate on this. If we apply Lemma 17.26 to the map EFG → •we obtain a map between homotopy fibration sequences

K−∞T G(EFG) //

K−∞OG(EFG) //

K−∞DG(EFG)

K−∞T G(•) // K−∞OG(•) // K−∞DG(•).

By Lemma 17.15 the left vertical map is induced by an equivalence of cat-egories and is therefore an equivalence of spectra. Because the homotopygroups of the lower middle spectrum vanish by Lemma 17.25, the claim fol-lows by considering the long exact ladder of homotopy groups associated tothe diagram above. ut

17.3 Miscellaneous

Comment 108: The following material is taken from bcsfinal and shouldpartially be integrated into the text after some modifications.


Let M be a Riemannian manifold. Recall that an h-cobordism W overM = ∂−W admits retractions r± : W × I → W , (x, t) 7→ r±t (x, t) whichretract W to ∂±W , i.e. which satisfy r±0 = idW and r±1 (W ) ⊂ ∂±W . Givenε > 0 we say that W is ε-controlled if the retractions can be chosen in sucha way that for every x ∈ W the paths (called tracks of the h-cobordism)p±x : I → M , t 7→ r−1 r

±t (x) both lie within an ε-neighborhood of their

starting point. The usefulness of this concept is illustrated by the followingtheorem which follows from [307]. Comment 109: This reference seems tobe right but I could not explicitly find the theorem in it.

Theorem 17.30 (Controlled h-Cobordism Theorem). Let M be acompact Riemannian manifold of dimension ≥ 5. Then there exists anε = εM > 0, such that every ε-controlled h-cobordism over M is trivial.

If one studies the s-Cobordism Theorem 2.44 and its proof one is naturallylead to algebraic analogues of the notions above. A (geometric) R-module overthe spaceX is by definition a familyM = (Mx)x∈X of free R-modules indexedby points of X with the property that for every compact subset K ⊂ X themodule ⊕x∈KMx is a finitely generated R-module. A morphism φ from M toN is an R-linear map φ = (φy,x) : ⊕x∈XMx → ⊕y∈XNy. Instead of specifyingfundamental group data by paths (analogues of the tracks of the h-cobordism)

one can work with modules and morphisms over the universal covering X,which are invariant under the operation of the fundamental group G = π1(X)via deck transformations, i.e. we require that Mgx = Mx and φgy,gx = φy,x.Such modules and morphisms form an additive category which we denoteby CG(X;R). In particular one can apply to it the non-connective K-theoryfunctor K (compare [653]). In the case where X is compact the category isequivalent to the category of finitely generated free RG-modules and henceπ∗KCG(X;R) ∼= K∗(RG). Now suppose X is equipped with a G-invariantmetric, then we will say that a morphism φ = (φy,x) is ε-controlled if φy,x = 0,whenever x and y are further than ε apart. (Note that ε-controlled morphismsdo not form a category because the composition of two such morphisms willin general be 2ε-controlled.)

Theorem 17.30 has the following algebraic analogue [672] Comment 110:Make reference more precise. (see also Section 4 in [651]). Comment 111:Retrieve the manuscript and take a look at this reference.

Theorem 17.31. Let M be a compact Riemannian manifold with fundamen-tal group G. There exists an ε = εM > 0 with the following property. TheK1-class of every G-invariant automorphism of modules over M which to-gether with its inverse is ε-controlled lies in the image of the classical assemblymap

H1(BG; KR)→ K1(RG) ∼= K1(CG(M ;R)).

Comment 112: Mention that this also implies the topological invarianceof Whitehead torsion.


To understand the relation to Theorem 17.30 note that for R = Z suchan ε-controlled automorphism represents the trivial element in the White-head group which is in bijection with the h-cobordisms over M , compareTheorem 2.44.

There are many variants to the simple concept of “metric ε-control” weused above. In particular it is very useful to not measure size directly in Mbut instead use a map p : M → X to measure size in some auxiliary spaceX. For example we have seen in Subsection Comment 113: about boundedh-cobordisms and Comment 114: About negative K-theory and pseudo-isotopy that “bounded” control over Rk may be used in order to define ordescribe negative K-groups.

Before we proceed we would like to mention that there are analogouscontrol-notions for pseudoisotopies and homotopy equivalences. The tracksof a pseudoisotopy f : M × I →M × I are defined as the paths in M whichare given by the composition

px : I = x × I ⊂M × If // M × I

p // M

for each x ∈ M , where the last map is the projection onto the M -factor.Suppose f : N → M is a homotopy equivalence, g : M → N its inverse andht and h′t are homotopies from f g to idM respectively from g f to idNthen the tracks are defined to be the paths in M that are given by t 7→ ht(x)for x ∈ M and t 7→ f h′t(y) for y ∈ N . In both cases, for pseudoisotopiesand for homotopy equivalences, the tracks can be used to define ε-control.

We also have the following result due to Chapman-Ferry [177]

Theorem 17.32 (α-Approximation Theorem). If M is a closed topolo-gical manifold of dimension ≥ 5 and α is an open cover of M, then there isan open cover β of M with the following property:

If N is a topological manifold of the same dimension and f : N →M is aproper β-homotopy equivalence, then f is α-close to a homeomorphism.

Exercises


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Coverings and Flow Spaces

18.1 Introduction

18.2 Miscellaneous

Exercises

18.1. Test


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Chapter 19

Transfer

19.1 Introduction

19.2 Miscellaneous

Exercises

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Chapter 20

Analytic Methods

20.1 Introduction

20.2 The Dirac-Dual Dirac Method

Next we briefly discuss the Dirac-dual Dirac method which is the key strategyin many of the proofs of the Baum-Connes Conjecture 12.9 or the Baum-Connes Conjecture with coefficients 12.11, see for instance [387, Theorem 7.1].

A G-C∗-algebra A is called proper, if there exists a locally compact properG-space X and a G-homomorphism σ : C0(X) → B(A), f 7→ σf satisfyingσf (ab) = aσf (b) = σf (a)b for f ∈ C0(X), a, b ∈ A and for every net fi |i ∈ I, which converges to 1 uniformly on compact subsets of X, we havelimi∈I ‖ σfi(a)−a ‖= 0 for all a ∈ A. A locally compact G-space X is properif and only if C0(X) is proper as a G-C∗-algebra.

The following result is proved in Tu [791] extending results of Kasparov-Skandalis [458, 450].

Theorem 20.1 (The Baum-Connes Conjecture with coefficients forproper G-C∗-algebras). The Baum-Connes Conjecture with coefficients 12.11holds for a proper G-C∗-algebras B.

Theorem 20.2 (Dirac-Dual Dirac Method). Let G be a countable (dis-crete) group. Let F be R or C. Suppose that there exist a proper G-C∗-algebraA, elements α ∈ KKG

i (A,F ), called the Dirac element, and β ∈ KKGi (F,A),

called the dual Dirac element, satisfying

β ⊗A α = 1 ∈ KKG0 (F, F ).

Then the Baum-Connes Conjecture 12.9 and the Baum-Connes Conjecturewith coefficients 20.1 are true over F .

Proof. We only treat the case F = C and the case of trivial coefficients. Theassembly map appearing in Theorem 12.9 is a retract of the bijective assem-bly map from Theorem 20.1. This follows from the following commutativediagram for any cocompact G-CW -subcomplex C ⊆ EG

511

512 20 Analytic Methods

KG∗ (C)

asmbG,C(C)∗

−⊗Cβ // KG∗ (C;A)

asmbG,CA (C)∗

−⊗Aα // KG∗ (C)

asmbG,C(C)∗

K∗(C

∗r (G))

−⊗C∗r (G)jG(β)// K∗(Aor G)

−⊗Aor jG(α) // K∗(C∗r (G))

and the fact that the composition of both the top upper horizontal arrowsand lower upper horizontal arrows are bijective. ut

In order to give a glimpse of the basic ideas from operator theory, webriefly describe how to define the Dirac element α in the case where G actson a complete Riemannian manifold M . Let TCM be the complexified tangentbundle and let Cliff(TCM) be the associated Clifford bundle. Let A be theproper G-C∗-algebra given by the sections of Cliff(TCM) which vanish atinfinity. Let H be the Hilbert space L2(∧∗T ∗CM) of L2-integrable differentialforms on TCM with the obvious Z/2-grading coming from even and oddforms. Let U be the obvious G-representation on H coming from the G-action on M . For a 1-form ω on M and u ∈ H define a homomorphism ofC∗-algebras ρ : A→ B(H) by

ρω(u) := ω ∧ u+ iω(u).

Now D = (d + d∗) is a symmetric densely defined operator H → H anddefines a bounded selfadjoint operator F : H → H by putting F = D√

1+D2.

Then (U, ρ, F ) is an even cocycle and defines an element α ∈ KG0 (M) =

KKG0 (C0(M),C). More details of this construction and the construction of

the dual Dirac element β under the assumption that M has non-positivecurvature and is simply connected, can be found for instance in [797, Chapter9].

20.3 Miscellaneous

Exercises

20.1. Test


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Chapter 21

Cyclic Methods

21.1 Introduction

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Chapter 22

Solutions of the Exercises

Chapter 1

1.7. Check that the homomorphism ψ : K0(R)→ K ′0(R), [P ]→ [P ] is well-defined using the fact that every exact sequence 0 → P0 → P1 → P2 → 0 offinitely generated projective R-modules splits. Obviously ψ is the inverse ofφ.

1.11. Show that the R-R-bimodule(RR

nMn(R)

)⊗Mn(R)

(Mn(R)R

nR

)is

isomorphic as R-R-bimodule to R and that the Mn(R)-Mn(R)-bimodule(Mn(R)R

nR

)⊗R

(RR

nMn(R)

)is isomorphic as Mn(R)-Mn(R)-bimodule to

Mn(R).

1.16. See [704, Theorem 1.2.3 on page 8].

1.24. Consider u ∈ Q[√d]. First one shows by a direct calculation that

u = a + b√d ∈ Q[

√d] is the root of a polynomial of the shape x2 + mx + n

for m,n ∈ Z if and only if u ∈ Z[√d] in the case d = 2, 3 mod 4 and if and

only if u ∈ Z[

1+√d

2

]in the case d = 1 mod 4. Then apply Gauss’s Lemma

which implies that a monic polynomial in R[x] which factors in Q[√d][x] into

monic polynomials already factorizes in R[x] into monic polynomials.

1.30. There exists a nowhere vanishing vector field on Sn if and only ifthere exists F -subbundles ξ and η in TSn such that TSn = ξ ⊕ η and ξ is a1-dimensional trivial F -vector bundle. Now apply Theorem 1.28.

1.33. Let ξ be a vector bundle over Y . It suffices to construct a C0(X)-isomorphism

α(ξ) : C0(X)⊗C0(X) C0(ξ)

∼=−→ C0(f∗ξ).

Given φ ∈ C0(X) and s ∈ C0(ξ), define α(ξ)(φ⊗ s) to be the section of f∗ξwhich sends x ∈ X to φ(x) · s f(x) ∈ ξf(x) = (f∗ξ)x. Since α(ξ ⊕ η) can beidentified with α(ξ) ⊕ α(η) and α(F ) is obviously bijective, α(ξ) is bijectivefor all F -vector bundles ξ over Y .

1.34. Because of the identification (1.32) and the homotopy invariance ofthe functor K0(X) we get

K0(C(Dn)) ∼= K0(Dn) ∼= K0(•) ∼= Z.

515

516 22 Solutions of the Exercises

1.38. This follows from the fact that Z⊗Zπ C∗(X) is isomorphic as Z-chaincomplex to C∗(X).

1.39.

(i) Let f∗ : P∗ → Q∗ be an R-chain homotopy equivalence of finite projectiveR-chain complexes. Let cone(f∗) be its mapping cone. Since f∗ induces anisomorphism on homology, the homology of cone(f∗) trivial. This impliesin K0(R)

0 =∑n≥0

(−1)n · [cone(f∗)n] =∑n≥0

(−1)n · [Pn−1 ⊕Qn] = o(P∗)− o(Q∗).

(ii) By the first assertion o(P∗) = 0 if P∗ is R-chain homotopy equivalent to afinite free R-chain complex.Suppose that o(P∗) = 0. An elementary R-chain complex E∗ is an R-chaincomplex which is concentrated in two consecutive dimensions and its onlynon-trivial differential is given by id: P → P for some finitely generatedprojective R-module P . Since for every finitely generated projective R-module P there exists a finitely generated projective R-module Q suchthat P ⊕Q is free we can find elementary R-chain complexes E∗[1], E∗[2],. . ., E∗[n] such that Q∗ := P∗ ⊕

⊕ni=1E∗[i] has the property that all

its R-chain modules are finitely generated free except the one in its topdimension. Obviously Q∗ is R-chain homotopy equivalent to P∗. Let d beits top dimension. Since [Qd] = o(Q∗) = o(P∗) = 0, the finitely generatedprojective R-module Qd is stably free, i.e., there is a finitely generatedfree R-module F such that Qd ⊕ F is free. By adding the elementary freeR-chain complex whose d-th and (d − 1)-th R-chain module is F to Q∗,we obtain a finite free R-chain complex C∗ which is R-chain homotopyequivalent to P∗.

(iii) This follows from the fact that [Pn]− [Pn] + [P ′′n ] = 0 holds in K0(R) forall n ∈ Z.

1.40. Recall that we have chosen a finite domination (Z, i, r) of X. Constructan extension g : cyl(r)∪Zcyl(i)∪Xcyl(i)→ X of idX

∐F∪XF : X

∐cyl(i)∪X

cyl(i) → X and a homotopy equivalence h : Z → cyl(r) ∪Z cyl(i) ∪X cyl(i).Now the claim follows from the commutative diagram

Solutions of the Exercises 517

X∐

(cyl(i) ∪X cyl(i))j //

f∐f ′

''

cyl(r) ∪Z cyl(i) ∪X cyl(i)

fg

Zhoo

fgh

∅oo

xxY

where j is the inclusion.

1.46. Let (B, b) be a functorial additive invariant for finite CW -complexes.Define a natural transformation T (X) :

⊕C∈π0(C) Z → B(X) by sending

nC | C ∈ π0(X) to∑C∈π0(X) nc ·A(iC)(a(•)), where iC : • → X is any

map whose image is contained in C. Obviously it is the only possible naturaltransformation satisfying T (•)(χ(•)) = b(•). Using the additivity andhomotopy invariance one proves by induction over the number of cells for afinite CW -complex X that T (X)(χ(C) | C ∈ π0(X)) = b(X) holds. Moredetails can be found in [520, Theorem 4.1].

1.48. (i) Fix a finitely dominated CW -complex Y . Define a functor A fromfinitely dominated CW -complexes to abelian groups by A(X) := U(X × Y ).Define a(X) ∈ A(X) to be u(X×Y ). Check that (A, a) is a functorial additiveinvariant for finitely dominated CW -complexes. Hence there exists a uniquetransformation TY : U(?)→ U(?×Y ) sending u(X) to u(X×Y ). Define B(Y )as the abelian group of transformations U(?) → U(? × Y ) and b(Y ) := TY .Show that (B, b) is a functorial additive invariant for finitely dominated CW -complexes. Hence there is a natural transformation S : U → B satisfyingS(Y )(u(Y )) = b(Y ) for all finitely dominated CW -complexes Y . This S givesthe desired natural pairing P (X,Y ).

(ii) If Y is a finite CW -complex with χ(C) = 0 for all C ∈ π0(Y ), then o(C) =0 for every C ∈ π0(C) by Lemma 1.18 and Theorem 1.37. Theorem 1.47implies u(Y ) = 0. We conclude from (i) that u(X × Y ) = P (X,Y )(u(X) ⊗u(Y )) = 0. Hence X × Y is homotopy equivalent to a finite CW -complex byTheorem 1.37 and Theorem 1.47.

1.56. We define a functor F : OrF (G) → SubF (G) as follows. It sends anobject G/H to the subgroup H. Consider a G-map f : G/H → G/K. Chooseg ∈ G with f(1H) = gK. Since f is a G-map, we get hgK = hf(1H) =f(hH) = f(1H) = gK and hence g−1hg ∈ K for all h ∈ H. Hence wecan define F (f) to be the class of the homomorphism c(g−1) : H → K,h 7→g−1hg. The morphism F (f) does not depend on the choice of g since anyother choice of g is of the form gk for some k ∈ K and we have c((gk)−1) =c(k−1) c(g−1) and c(k−1) ∈ inn(K).

Obviously F is bijective on objects and surjective on morphisms.


1.77. Choose an integer n ≥ 0 and a matrix A ∈Mn(FH) such that A2 = Aand im(rA : FHn → FHn) ∼=FH V . We compute for h ∈ G, if lh : V → V isgiven by left multiplication with h

χF (V )(h−1) := trF (lh−1 : V → V ) = trF (lh−1 rA : FHn → FHn)

=

n∑i=1

trF(FH → FH, u 7→ h−1uai,i

)= trF

(FH → FH, u 7→ h−1u

(n∑i=1

ai,i

))

Write∑ni=1 ai,i =

∑k∈H λk · k. Then we get

χF (V )(h−1) := trF

(FH → FH, u 7→ h−1u

(∑k∈H

λk · k

))=∑k∈H

λk · trF(FH → FH, u 7→ h−1uk

)=∑k∈H

λk ·∣∣u ∈ H | u = h−1uk

∣∣=∑k∈(h)

λk ·∣∣u ∈ H | h = uku−1

∣∣=∑k∈(h)

λk · |CH〈h〉|

= |CH〈h〉| ·∑k∈(h)

λk

= |CH〈h〉| ·HSFH(V )(h).

1.81. Suppose that K0(FG) is a torsion group. This is equivalent to the

statement that K0(FG)⊗Z F is trivial. Lemma 1.18 and Lemma 1.79 implythat classF (G)f ∼=F F and hence conF (G)f consists only of one element.Hence every element in G of finite order is trivial.

1.83. Because of the commutative diagram appearing in the proof ofLemma 1.79, it suffices to prove the claim in the case that G is finite. In

this case one computes that HS(P ) evaluated at the unit e ∈ G is dimF (P )|H| .

1.86. Show∑g∈G HSZG(P )(g) = HSZ(Z⊗ZG P ) = dimZ(Z⊗ZG P ).

1.98. The list of finite groups of order ≤ 9 consists of the cyclic groups Z/nfor n = 1, 2, 3, . . . , 9, the abelian non-cyclic groups Z/2×Z/2, Z/2×Z/2×Z/2,Z/2×Z/4 Z/3×Z/3, and the following non-abelian groups S3 = D6, D8 andQ8. Now inspecting Theorem 1.97 gives the answer:


Z/2× Z/2× Z/2, Z/3× Z/3, Q8.

1.100. Theorem 1.99 implies that K0(FD8) is Zn for some n. We concludefrom Theorem 1.75 that n = | conF (D8)f |. A presentation for D8 is 〈x, y |x4 = 1, y2 = 1, yxy−1 = x−1〉. In particular D8 is a semidirect product Z/4oZ/2 if Z/4 is the group generated by x and Z/2 the subgroup generated by y.The elements x2, y, xy, x2y and x3y have order 2, the elements x and x−1 haveorder four. We have one conjugacy class of elements of order 4, namely (x) andthree conjugacy classes of elements of order two, namely (x2), (y) and (yx).Since we also have the conjugacy class of the unit, we see | conC(D8)f | = 5.Since x is conjugated to x−1, we conclude | conR(D8)f | = 5. Since every cyclicsubgroup of order 4 is conjugated to 〈x〉, we get | conQ(D8)f | = 5. This shows

K0(FD8) ∼= Z5 for F = Q,R,C.

1.103. Recall that a hyperbolic group does not contain Z2 as subgroup.Because of Remark 1.102 it suffices to show for a torsionfree hyperbolic groupG that it is cyclic if there exists an element g different from the unit elementwith finite (g). The finiteness of (g) is equivalent to the condition that thecentralizer Cg〈g〉 = h ∈ G | hg = gh has finite index in G. Since 〈g〉is infinite, hyperbolic implies that CG〈g〉 is virtually cyclic. Hence G is atorsionfree virtually cyclic group and therefore cyclic.

1.107. Write n = pn11 pn2

2 . . . pnrr for distinct primes p1, p2, . . . , pr and integersni ≥ 1. Then Lemma 1.12 implies

K0(Z/n) =

r∏i=1

K0(Z/pnii ).

Since Z/pnii is local, the claim follows from Theorem 1.106.

1.110. A counterexample is given byG1 = G2 = Z/3 since K0(Z[Z/3]) = 0and K0(Z[Z/3× Z/3]) 6= 0 by Theorem 1.97.

Chapter 2

2.3. Let f : Rn → Rn be an R-automorphism. This is the same as aK-linear isomorphism V n → V n. Since V is a K-vector space with infi-

nite countable basis, we can choose a K-isomorphism α :⊕∞

k=0 Vn∼=−→ V .

Let u :⊕∞

k=0 Vn∼=−→

⊕∞k=0 V

n be the R-isomorphism given by ⊕∞k=0f .


Let γ : V n ⊕⊕∞

k=0 Vn∼=−→⊕∞

k=0 Vn be the R-automorphism which sends

v ⊕ (v0, v1, v2, . . .) to (v, v0, v1, v2 . . .). One easily checks γ−1 u γ = f ⊕ uDefine an R-automorphism v : V → V by α u α−1. This is the same as anR-automorphism v : R→ R. Now one computes [f ] + [v] = [f ⊕ v] = [v] inK1(R) using the fact that conjugated automorphisms define the same elementin K1(R). This implies [f ] = 0.

2.7. We get from Theorem 2.6 an isomorphism i : H×/[H×,H×]∼=−→ K1(H).

Obviously the collection of maps µn defines a homomorphism µ : K1(H)→ R.The norm of a quaternion z = a + bi + cj + dk is defined by N(z) :=√a2 + b2 + c2 + d2. Let N : H×/[H×,H×] → R>0 be the induced homo-

morphism of abelian groups. Its restriction to R>0 ⊆ H is the identity. Sinceµ1(z) = |z|4 for z ∈ H, it remains to prove that N−1(1) ⊆ [H×,H×].

Since iejθi−1 = e−jθ holds for θ ∈ R and similarly with i, j and k cyclicallypermuted, e2iθ1 , e2jθ2 and e2kθ3 are all commutators. These generate an openneighborhood of 1 in S3 = N−1(1). Since S3 is connected, the claim follows.

2.18. Take the norm on Z[i] sending a + bi to√a2 + b2. It yields an Eu-

clidean algorithm. A direct calculation shows Z[i]× = 1,−1, i,−i. Nowapply Theorem 2.17.

2.22. This follows from Theorem 2.20 and Theorem 2.21.

2.24. The map φ is induced by the composite

K1(Z[Z/5])f∗−→ K1(C)

det−−→ C× | |−→ R>0.

Since (1− t− t−1) · (1− t2− t3) = 1, the element 1− t− t−1 is a unit in Z[Z/5]and defines an element in Wh(Z/5). Its image under φ is (1− 2 · cos(2π/5))and hence different from 1.

2.33. Let ε∗ be a chain contraction for E∗. Choose for any n ∈ Z an R-homomorphism σn : En → Dn satisfying pnσn = idEn . Define sn : En → Dn

by dn+1 σn+1 εn + σn εn−1 en.There are examples of short exact sequences of R-chain complexes whose

boundary operator in the associated long homology sequence is not trivialand hence for which Hn(p∗) is not surjective for all n ∈ Z.

2.38. This is done by the following sequence of expansions. We describe thesimplicial complexes obtained after each step:

(i) The standard 2-simplex spanned by v0, v1, v2;(ii) Three vertices v0, v1, v2 and two edges v0, v1 and v0, v2;(iii) The standard 1-simplex spanned by v0, v1;(iv) The standard 0-simplex given by v0.


2.43. This follows from K0(Z) = 0 (see Example 1.4) and Wh(1) = 0 (seeTheorem 2.17) together with Theorem 1.37 and Theorem 2.42.

2.46. Choose a non-trivial element in Wh(Z/5) (see Exercise 2.24). ByTheorem 2.44 we can find an h-cobordism (W,M0,M1) whose Whiteheadtorsion is x. Hence it is non-trivial. In order to show that (W × S3;M0 ×S3,M1×S3) is trivial, we have to show τ(i0× idS3) = 0 for i0 : M0 →W theinclusion. This follows from Theorem 2.34 (iv) since both τ(idS3) and χ(S3)vanish.

2.54. By definition RP3 is the lens space L(V ) for the cyclic group Z/2, whereV has as underlying unitary vector space C2 and the generator s of Z/2 actson V by − id. The cellular Z[Z/2]-chain complex C∗(SV ) is concentrated indimensions 0, 1, 2, 3 and is given by

. . .→ 0→ Z[Z/2]s−1−−→ Z[Z/2]

s+1−−→ Z[Z/2]s−1−−→ Z[Z/2]→ 0→ . . . .

Hence R− ⊗Z[Z/2] C∗(SV ) is the R-chain complex

. . .→ 0→ R 2·id−−→ R 0−→ R 2·id−−→ R→ 0→ . . .

It is contractible, a chain contraction γ∗ is given by γ0 = γ2 = 1/2·id and γn =0 for n 6= 0, 2. Hence (c+γ)odd : R−⊗Z[Z/2]Codd(SV )→ R−⊗Z[Z/2]Codd(SV )is given by (

2 1/20 2

): R2 → R2

This implies ρ(RP3;V ) = 4.

2.74. We use induction over n ≥ 0. The case n = 0, i.e., the trivial group,follows from Example 1.4. The induction step from n to n + 1 is a directconsequence of Theorem 2.73 (i) since R[Zn][Z] is isomorphic to R[Zn+1].

2.79. Because of Theorem 2.77 (ii) the ring Z[Zn] is regular. Hence we get

from Exercise 2.74 and Lemma 2.82 that K0(Z[Zn]) = 0.To show Wh(Zn) = 0, we use induction over n ≥ 0. The case n = 0, i.e.,

the trivial group, follows from Example 1.4 and Theorem 2.17. The inductionstep from n to n+ 1 follows from Theorem 2.78 since Z[Zn][Z] is isomorphicto Z[Zn+1].

2.90. Obviously (2)∼=−→ (NZ/2) sending 2 to NZ/2 is an isomorphism of rings

without unit.Theorem 2.86 together with Lemma 2.89 yields exact sequences


K1(Z)→ K1(Z/n)→ K0((n))→ K0(Z)→ K0(Z/n);

K1(Z[Z/2])→ K1(Z)→ K0((NZ/2))→ K0(Z[Z/2])→ K0(Z),

since the ring homomorphism Z[Z/2] → Z sending a + bt to a − b induces

an isomorphism of rings Z[Z/2]/(NZ/2)∼=−→ Z. Because of Theorem 2.6 and

Theorem 2.17 the determinant induces isomorphisms

det : K1(Z)∼=−→ ±1;

det : K1(Z/n)∼=−→ Z/n×.

The map Kk(Z[Z/2])→ Kk(Z) is surjective for k = 0, 1 because its composi-tion with Kk(Z) → Kk(Z[Z/2]) is the identity. The map K0(Z) → K0(Z/n)is injective since its composition with the map K0(Z/k) → Z, [P ] 7→ |P | isinjective by Theorem 1.4. This implies

K0((n)) ∼=

0 if n = 2;

(Z/n)×/±1 if n ≥ 3;

K0((NZ/2)) = 0;

K0(Z[Z/2]) = 0.

2.94. Because of Remark 2.93 it suffices to show for each two-sided idealI ⊆ F that E(F, I) = SL(F, I). This is trivial if I = 0. If I = F , this followsfrom Theorem 2.17.

2.98. Consider k ∈ Z with (k, |G|) = 1. Choose l ∈ Z with kl = 1 mod |G|.Choose a generator t ∈ G. Define elements u, v ∈ ZG.

u = 1 + t+ t2 + · · ·+ tk−1;

v = 1 + tk + t2k + · · ·+ t(l−1)k.

Then (t− 1) · (tk − 1) · uv = uv holds in ZG. One easily checks that (t− 1) ·(tk − 1) · w = 0⇔ w ∈ (NG) for w ∈ ZG. Hence u ∈ ZG/(NG) is a unit andmaps to k under the map j1 : ZG/(NG)→ Z/|G|. Now argue as in the proofof Rim’s Theorem in Section 2.8.

2.100. Since K0(Z[Z/2]) = 0 (see Theorem 1.97 (i)) we can assume withoutloss of generality |G| ≥ 3.

Suppose d = 1. Then G\X is a connected finitely dominated 1-dimensionalCW -complex. Since its homology is finitely generated and it is homotopyequivalent to a 1-dimensional CW -complex Y with precisely one 0-cell, theCW -complex Y is finite.


Suppose that d ≥ 2. Then d is odd by Theorem 2.99 (i). The unit sphere Sin C(d+1)/2 with the G-action for which the generator acts by multiplicationwith exp(2πi/|G|) is a free d-dimensional G-homotopy representation suchthat G\S is compact and hence finite. By elementary obstruction theorythere exists a G-map X → S. Now apply Theorem 1.37 (i), Lemma 2.99 (ii)and Exercise 2.98.

2.108. Obviously the image of the map dirlimH∈SubFIN (G×Z)K1(RH) →K1(R[G×Z]) is contained in the image of the map K1(RG)→ K1(R[G×Z]).Theorem 2.69 implies K0(ZG) = 0 if K1(RG)→ K1(R[G×Z]) is surjective.

If R is a commutative integral domain, K0(R) and hence K0(RG) cannotbe zero. Namely, if F is its quotient field, the homomorphism K0(R) →Z, [P ] 7→ dimF (F ⊗R P ) is a well-defined surjective map.

2.115. This follows from Theorem 2.112 and Theorem 2.113 (iv).

2.120. A counterexample is given by G1 = G2 = Z/3 since Wh(Z/3) = 0and Wh(Z/3× Z/3) 6= 0 by Theorem 2.113.

Chapter 3

3.4. Apply Remark 3.3 to the obvious pullback of rings

R× SprR //

prS

R

S // 0

Or, if one does not like the ring 0 consisting of one element, use Lemma 2.9and the Bass-Heller-Swan decomposition 3.2.

3.8. This follows by induction over k using Theorem 3.6.

3.10. The exact Mayer-Vietoris sequence appearing in Example 3.9 yieldsfor a prime p the exact sequence

K1(Z[Zk])⊕K1(Z[exp(2πi/p)][Zk])→ K1(Fp[Zk])

→ K0(Z[Z/p× Zk])→ K0(Z[exp(2πi/p)])→ 0

We have K0(Z[exp(2πi/3)]) = 0 by Theorem 1.90 and Example 1.91. Henceit suffices to show that the map K1(Z[exp(2πi/3)][Zk]) → K1(F3[Zk]) issurjective. Because of the Bass-Heller-Swan decomposition 3.2 it suffices toprove the surjectivity of Ki(Z[exp(2πi/3)]) → Ki(F3) for i = 0, 1. The case


i = 0 follows from the fact that K0(F3) is generated by [F3]. It remains totreat i = 1. Let f : Z[exp(2πi/3)] → F3 be the ring homomorphism whichis uniquely determined by the property that it sends exp(2πi/3) to 1. Be-cause of Theorem 2.17 it suffices to show that for every unit u in F3 wecan find a unit u′ in Z[exp(2πi/3)] which is mapped to u under f . Since± exp(2πi/3) is a unit in Z[exp(2πi/3)] and f(± exp(2πi/3)) = ±1, we con-

clude K0(Z[Z/3× Zk]) = 0 for k ≥ 0.Now all other claims follow from Theorem 3.2.

3.12. The pullback of rings appearing in Example 3.11 yields a pullback ofrings

Z[Z/6× Zk]i1 //

i2

Z[Z/3× Zk]

j1

Z[Z/3× Zk]

j2// (F2 × F4)[Zk]

where j2 = j1. Put j := j1 = j2. We obtain from Remark 3.3 the exactsequence

K1(Z[Z/3× Zk])⊕K1(Z[Z/3× Zk])j∗⊕j∗−−−−→ K1((F2 × F4)[Zk])

→ K0(Z[Z/6× Zk])→ K0(Z[Z/3])⊕K0(Z[Z/3])j∗⊕j∗−−−−→ K1((F2 × F4)[Zk])

→ K−1(Z[Z/6× Zk])→ · · ·

The following facts follow from Theorem 2.77 (i), Exercise 3.4, Exercise 3.10and, Theorem 3.6. We have Kn(Z[Z/3 × Zk]) = Kn((F2 × F4)[Zk]) = 0for n ≤ −1 and K0(Z[Z/3 × Zk]) = 0. The map j∗ ⊕ j∗ : K0(Z[Z/3 ×Zk]) ⊕ K0(Z[Z/3 × Zk]) → K0((F2 × F4)[Zk]) can be identified with themap j∗ ⊕ j∗ : K0(Z[Z/3])⊕K0(Z[Z/3])→ K0(F2 × F4) which in turn can beidentified the map Z⊕Z→ Z⊕Z sending (a, b) to (a+b, a+b). The image ofthe map j∗⊕j∗ : K1(Z[Z/3×Zk])⊕K1(Z[Z/3×Zk])→ K1((F2×F4)[Zk]) canbe identified with the direct sum of the image of the map j∗⊕j∗ : K1(Z[Z/3])⊕K1(Z[Z/3]) → K1(F2 × F4) with (k − 1) copies of the image of the mapj∗ ⊕ j∗ : K0(Z[Z/3])⊕K0(Z[Z/3])→ K0(F2 × F4). In order to prove

Kn(Z[Zk × Z/6]) ∼=

Zk for n = 0;

Z for n = −1;

0 for n ≤ −2,

it remains to show that the map j∗ : K1(Z[Z/3 × Zk]) → K1((F2 × F4)[Zk])is surjective.

Recall that we have used the identification of rings F2⊗ZZ[Z/3] ∼= F2×F4

and because of Lemma 2.9 and Theorem 2.17 the determinant induces anisomorphism K1(F2⊗ZZ[Z/3])

∼=−→ (F2⊗ZZ[Z/3])×. Hence it suffices to show


that for every unit u in F2⊗Z Z[Z/3] we can find a unit u′ in Z[Z/3] which ismapped under the obvious projection pr : Z[Z/3]→ F2⊗ZZ[Z/3] to u. Thereare three units in F2 ⊗Z Z[Z/3] ∼= F2 × F4, namely, 1 ⊗ 1, 1 ⊗ t, and 1 ⊗ t2.Obviously they are images of units under pr.

We conclude NpKn(Z[Z/6 × Zk]) for n ≤ 0, p ≥ 1 and k ≥ 0 fromTheorem 3.2 since K0(Z[Z/6]) ⊕K−1(Z[Z/6])k−1 ∼= Zk ∼= K0(Z[Zk × Z/6]),K−1(Z[Z/6]) ∼= K−1(Z[Zk × Z/6]) and Kn(Z[Zk × Z/6]) = 0 for n ≤ −1holds.

3.14. This follows from Lemma 3.13 since the assumptions imply thatKm(R)→ Km(R[Zn]) induced by the inclusion R→ R[Zn] is bijective.

3.18. Because of Theorem 3.6. it suffices to prove that RG is regular, pro-vided that R is regular, G is a finite group and the order of every elementof G is invertible in R. Since R is Noetherian and G is finite, RG is Noethe-rian. Let M be any finitely generated R-module. Then the RG-module Mis a direct summand in the RG-module M ′ := RG ⊗R M , where g acts onx ⊗ m by gx ⊗ m. So M ′ does not see the G-action on M . The injectionM →M ′ is given by m 7→ 1

|G| ·∑g∈G g ⊗ g−1m and the retraction M ′ →M

by g⊗m 7→ gm. Let P∗ be a finite projective resolution of the R-module M .Then P ′∗ is a finite RG-resolution of M ′. Since M is a direct RG-summandin M ′, it possesses a finite projective RG-resolution as well.

3.23. This follows by induction over k from Theorem 3.2, Theorem 3.21 (i)and Theorem 3.22.

Chapter 4

4.5. This follows from Lemma 2.11, Theorem 2.12 and Definition 4.1.

4.7. This follows from Theorem 4.6 and the fact that H2(E(R)) is the kernelof the universal central extensions φR : St(R)→ E(R) of perfect group E(R).

4.15. Obviously the matrices d1,2(u) and d1,3(v) represent the trivial elementin K1(R). Hence they belong to E(R) by Lemma 2.11 and Theorem 2.12.

Let d1,2(u) and d1,3(v) be fixed preimages of d1,2(u) and d1,3(v) under thecanonical map φR : St(R) → E(R). Then any other lifts are of the form

d1,2(u) ·x and d1,3(v) ·y for elements in the center of St(R). One easily checks

[d1,2(u), d1,3(v)] = [d1,2(u) · x, d1,3(v) · y].

4.18. We get K2(Z) ∼= Z/2 with generator −1,−1 from Theorem 4.16 (vi).


4.21. We obtain Wh2(Z/n) = 0 for n = 1, 2, 3, 4 from Section 4.8.By Theorem 2.112 the Whitehead group Wh(Z/n) vanishes if and only if

n = 1, 2, 3, 4, 6. Theorem 1.97 (i) implies K0(Z[Z/n]) = 0 for n = 1, 2, 3, 4.We conclude Ki(Z[Z/n]) = 0 for n = 1, 2, 3, 4 and all i ≤ −1 from The-orem 3.21 (i) and (v) We conclude K−1(Z[Z/6]) 6= 0 from Example 3.11.Hence the answer is n = 1, 2, 3, 4.

Chapter 5

5.1. Let Z be acyclic. Since H0(Z) is the free abelian group with π0(Z) asZ-basis, Z is path connected. Since the classifying map f : Z → Bπ for π =π1(Z) is 2-connected, it induces by the Hurewicz Theorem an isomorphismH1(Z)→ H1(π) and an epimorphism H2(Z)→ H2(π).

5.7. If P1 and P2 are two perfect subgroups of G, then the subgroup 〈P1, P2〉generated by P1 q P2 is again a perfect subgroup of G.

5.8. Recall that E(R) = [GL(R), GL(R)] by Lemma 2.11. We know alreadybecause of Theorem 4.6 that E(R) = [GL(R), GL(R)] is perfect since onlya perfect group possesses a universal central extension. Since the image ofa perfect subgroup under an epimorphism of groups is perfect and the onlyperfect subgroup of the abelian group GL(R)/[GL(R), GL(R)] is the trivialgroup, every perfect subgroup of GL(R) is contained in E(R).

5.10. Since BGL(R) and hence BGL(R)+ is path connected, this followsdirectly from the definitions in the case n = 0. If n = 1, this follows fromTheorem 2.12, Theorem 5.5 (iv), and Exercise 5.8.

5.24. We conclude from Example 1.4 and Theorem 2.17 that the sequencelooks like

±1 j1−→ Q× ∂1−→⊕p

Z i0−→ Z j0−→ Z→ 0,

where p runs through all prime numbers, that j1 is the inclusion and that j0the identity. Hence the map i0 is the zero map. The map ∂1 sends a rationalnumber of the shape ±pn1

1 · pn22 · · · · · p

nkk for pairwise distinct primes p1, p2,

. . . , pk and integers n1, n2, . . . , nk to the element (np)p whose entry forp = pi is ni for i = 1, 2, . . . , k and is 0 for any prime p which is not containedin p1, p2, . . . , pk.

5.28. From the analogue of the sequence (5.25) for Ktop and the assumptionthat k is odd, we conclude

Ktopn (R;Z/k) ∼=

Z/k n ≡ 0 mod 4;

0 n = 1, 2, 3 mod 4.


We know Kn(R) = 0 for n ≤ −1 from Theorem 3.6. Now the se-quence (5.25) and Theorem 5.27 imply

Kn(R;Z/k) ∼=

Z/k n ≥ 0 and n ≡ 0 mod 4;

0 n ≥ 0 and k = 1, 2, 3 mod 4;

0 n ≤ −1.

5.34. Generators for K0(A) are isomorphism classes of objects. Relationsare [P1] + [P2] = [P1 ⊕ P2] for any objects P1, P2.

The generators of K1(A) are conjugacy classes of objects of A. Relationsare [g f ] = [g] + [f ] for any automorphisms f, g of the same object and[(f1 f0

0 f2

)]= [f1] + [f2] for any automorphisms fi : Pi → Pi for i = 1, 2 and

a any morphism f0 : P2 → P1.The functor S induces homomorphism Si : Ki(A) → Ki(A) for i = 1, 2.

The existence of the natural transformation T implies that the two homomor-phism Si+idKi(A) and Si coincide. Hence idKi(A) is the zero-homomorphismwhich means Ki(A) = 0.

5.35. Let A be the additive category of countably generated projectiveR-modules. Let S be the functor sending an object P to (P ⊕ P ⊕ · · · ).Then we obtain a natural transformation T : id⊕S → S by rebracketing, i.e.(P ⊕ P ⊕ · · · ) = P ⊕ (P ⊕ P ⊕ · · · ). Hence A is flasque and we can applyTheorem 5.33 (iii).

5.43. Because of Conjecture 5.41 it suffices to construct the correspondingsequence for H∗(−; K):

· · · → Hn(BG0; K(R))→ Hn(BG1; K(R))⊕Hn(BG2; K(R))

→ Hn(BG; K(R))→ Hn−1(BG0; K(R))

→ Hn−1(BG1; K(R))⊕Hn−1(BG2; K(R))→ · · ·

One can arrange that BGi is a sub CW -complex of BG and BG = BG1∪BG2

and BG0 = BG1 ∩ BG2. Now the desired sequence above is the associatedMayer-Vietoris sequence.

5.44. Because of Conjecture 5.41 it suffices to construct the correspondingsequence for H∗(−; K):

· · · → Hn(BG; K(R))id−φ∗−−−−→ Hn(BG; K(R))→ Hn(B(Goφ Z); K(R))

→ Hn−1(BG; K(R))id−φ∗−−−−→ Hn−1(BG; K(R))→ · · ·


The automorphism φ induces a homotopy equivalence Bφ : BG→ BG. Themapping torus of Bφ is a model for B(Goφ Z). Now the desired long exactsequence is the Wang sequence associated to the fibration BG → B(G oφZ)→ S1.

5.48. If R is regular, then R[t] is regular. There is an obvious identification(R[t])G = (RG)[t]. Hence we obtain a commutative diagram

Hn(BG; K(R))∼= //

Kn(RG)

Hn(BG; K(R[t])) ∼=

// Kn((RG)[t])

where the vertical arrows are induced by the canonical inclusions R → R[t]and RG→ (RG)[t]. The horizontal arrows are bijective by assumption. SinceR is regular, the left vertical arrow is bijective because of Theorem 5.16 (ii)and the Atiyah-Hirzebruch spectral sequence. Hence also the right verticalarrow is bijective. This implies NKn(RG) = 0 for all n ∈ Z.

5.51. Consider the following commutative diagram

Niln

i∗⊕j∗

Kn(RG0)⊕Kn(RG1)⊕Kn(RG2)

(k0)∗⊕(k1)∗

α

ww0 // ker(pn)

ι //

β

Kn(RG)

∂n

pn // NKn//

0⊕id

0

0 // Kn−1(RG0)f∗(1,−1)//

0⊕(j1)∗⊕−(j2)∗ ''

Niln−1

i∗⊕j∗

i∗⊕π // Kn−1(RG)⊕NKn// 0

Kn−1(RG0)⊕Kn−1(RG1)⊕Kn−1(RG2)

(k0)∗⊕(k1)∗

Kn−1(RG)

Here Niln stands for the n-homotopy group of Nil(RG0;RG1, RG2), NKn

stands for NKn(RG0;RG1, RG2), the letters ι and π denote obvious inclu-sions or projections. The map i∗ is the one induced by i and analogouslyfor f∗, j∗, (j1)∗, (j2)∗,(k0)∗, and (k1)∗. The middle column is the long exactsequence associated to the homotopy cartesian square appearing in Theo-


rem 5.49 (i) with boundary operator ∂n. Theorem 5.49 implies that the twohorizontal short sequences are (split) exact and the diagram (without thedashed arrows) commutes.

Now an easy diagram chase shows that exists dotted arrows uniquely de-termined by the property that the diagram remains commutative.

Define the desired long exact Mayer-Vietoris sequence by the homomor-phism α′ : K0(RG1) ⊕ Kn(RG2) → ker(p) which is the restriction of α,the homomorphism β, and the homomorphism (j1)∗ ⊕ (j2)∗ : Kn(RG) →Kn(RG1)⊕Kn(RG2). We leave it to the reader to check using the diagramabove that this sequence is indeed exact.

5.61. There is an obvious projection pr : R→ R0. Since pr i = idR0for the

inclusion i : R0 → R, it suffices to prove that in prn : KHn(R) → KHn(R)is surjective. Define a map ϕ : R → R[t] by sending rn ∈ Rn to rn · tn. Fork = 0, 1 let evk : R[t]→ R be the ring homomorphism given by putting t = 0for k = 0 and t = 1 for k = 1. Then ev1 ϕ = idR and evk is bijective fork = 0, 1 by homotopy invariance. Hence (ev0)n and ϕn are isomorphisms.Since ev0 ϕ agrees with i pr, the claim follows.

Chapter 6

6.5. The composite of two cofibrations is again a cofibration. The same istrue for weak equivalences. Hence coC and wC are indeed subcategories of C.

Axioms (i), (ii) and (iv) appearing in Definition 6.3 are obviously satisfied.Consider chain maps i∗ : A∗ → B∗ and f∗ : A∗ → C∗ of finite projective

R-chain complexes such that in : An → Bn is split injective for all n ∈ Z.Define D∗ to be the cokernel of the chain map i∗ ⊕ f∗ : A∗ → B∗ ⊕ C∗.Then we obtain a short exact sequence of finite projective R-chain complexes

0 → A∗i∗⊕f∗−−−−→ B∗ ⊕ C∗

pr∗−−→ D∗ → 0 since for every n ≥ 0 the sequence of

R-modules 0→ Anin⊕fn−−−−→ Bn ⊕ Cn

prn−−→ Dn → 0 is split exact because in issplit injective. One easily checks that we obtain a pushout of finite projectiveR-chain complexes

A∗i∗ //

f∗

B∗

pr∗ |B∗

C∗pr∗ |C∗ // D

such that the lower horizontal arrow is a cofibration. Hence axiom (iii) istrue.

Axiom (v) follows from the long exact homology sequences associated toa short exact sequence of R-chain complexes and the Five-Lemma.


6.17. This follows the property of the map i of (6.12) that π:n(i) is bijectivefor n ≥ 1, from Remark 6.13, and from Theorem 6.16 since Kn(Z) vanishesfor n ≤ −1 and is Z for n = 0.

6.22. We obtain from the fibration (6.18) the exact sequence

π1(BG+ ∧A(•))→ π1(A(BG))→ π1(WhPL(BG))

→ π0(BG+ ∧A(•))→ π0(A(BG)).

Since A(•) is connected, the Atiyah-Hirzebruch spectral sequence shows

that π0(•+ ∧ A(•))∼=−→ π0(BG+ ∧ A(•)) is bijective. Since the ho-

momorphism π0(•+ ∧ A(•)) → π0(BG+ ∧ A(•)) is split injective, themap π0(BG+ ∧A(•))→ π0(A(BG)) is injective. Using diagram (6.19), weobtain a short exact sequence

H1(Bπ1(BG); K(Z))→ K1(Zπ1(BG))→ π1(WhPL(BG))→ 0.

Again by the Atiyah-Hirzebruch spectral sequence we obtain an isomor-phism H1(Bπ1(BG); K(Z)) ∼= G/[G,G] × ±1.Hence the image of themap H1(Bπ1(BG); K(Z)) → K1(Zπ1(BG)) is the subgroup of K1(ZG) =K1(Zπ1(BG)) given by the trivial units ±g | g ∈ G. This impliesWh(G) ∼= π1(WhPL(BG)).

6.26. Suppose such M exists. The long exact homotopy sequence of thefibration (6.18) looks like

· → πn(M+ ∧A(•))→ An(M)→WhPLn (M)→ · · · .

The splitting (6.24) yields isomorphisms

An(M) ∼= WhDiffn (M)⊕ πn(Σ∞M).

Rationally the Atiyah-Hirzebruch sequence always collapses. Hence we obtainfrom Theorem 5.22 and Theorem 6.16 isomorphisms

πn(M+ ∧A(•))⊗Z Q ∼= Hn(M ;Q)⊕⊕k≥1

Hn−4k−1(M ;Q).

Hence we obtain long exact sequence of Q-modules

· →WhPLn+1(M)⊗Z Q→ Hn(M ;Q)⊕

⊕k≥1

Hn−4k−1(M ;Q)

→ Hn(M ;Q)⊕WhDiffn (M)⊗Z Q→WhPL

n (M)⊗Z Q→ · · ·

Since by assumption the map WhDiffn (M) ⊗Z Q → WhPL

n (M) ⊗Z Q ⊗Z Q isbijective for n ≥ 0, we obtain for every n ≥ 0 isomorphisms


Hn(M ;Q)⊕⊕k≥1

Hn−4k−1(M ;Q) ∼= Hn(M ;Q).

This implies for every n ≥ 0 and k ≥ 1 that Hn−4k−1(M ;Q) = 0, a contra-diction to H0(M ;Q) = Q.

Chapter 7

7.6. It is straightforward to check that e(P ) is a well-definedR-homomorphism,compatible with direct sums and natural. It remains to show that it is bi-jective for a finitely generated projective R-module P . Let Q be anotherfinitely generated projective R-module. Since e(P ⊕Q) is up to isomorphisme(P )⊕ e(Q), the map e(P ⊕Q) is bijective if and only if both e(P ) and e(Q)are bijective. Since we can find Q such that P⊕Q ∼= Rn, it suffices to considerthe case P = R which follows from a direct computation.

7.14. Let bi(M) := dimR(Hi(M ;R)) be the i-th-Betti number. Poincareduality implies bi(M) = b4k−i(M) for all i ≥ 0. We conclude directly from thedefinition of the signature that sign(M) ≡ b2k(M) mod 2. We get modulo 2

χ(M) ≡4k∑i=0

(−1)i · bi(M)

≡2k−1∑i=0

(−1)i · bi(M) + b2k(M) +

4k∑i=2k+1

(−1)i · bi(M)

≡2k−1∑i=0

(−1)i · bi(M) + b2k(M) +

4k∑i=2k+1

(−1)i · b4k−i(M)

≡2k−1∑i=0

(−1)i · bi(M) + b2k(M) +

2k−1∑i=0

(−1)i · bi(M)

≡ b2k(M)

≡ sign(M).

7.16.

(i) If n is odd, then dim(CPn) is not divisible by four and hence sign(CPn) = 0.If n is even, then the intersection pairing of CPn looks like Z × Z →Z (a, b) 7→ ab and hence sign(CPn) = 1.

(ii) Since STM is the boundary of the total space DTM of the disk tangentbundle, Theorem 7.15 (i) implies sign(STM) = 0.

(iii) This follows from assertions (iv) and (v) of Theorem 7.15.


7.22. Notice in the situation under consideration that ε = 1 and the involu-tion on Z is the trivial involution. Hence the projection pr : R→ Qε(R) is theidentity. We conclude from Remark 7.20 that (P, λ) admits a quadratic refine-ment if and only if there exists a map µ : P → Z such that µ(nx) = n2µ(x)holds for all n ∈ Z and x ∈ P , µ(x+ y)−µ(x)−µ(y) = λ(x, y) is true for allx, y ∈ P and λ(x, x) = 2 · µ(x) is valid for all x ∈ P . Obviously the existenceof µ implies λ(x, x) to be even for all x ∈ P . Suppose that λ(x, x) to is evenfor all x ∈ P . Then we can define µ(x) := λ(x, x)/2 and µ has all desiredproperties.

7.26. Show that the diagonal in M ⊕ M is a Lagrangian for the non-degenerate ε-quadratic form (M ⊕M,φ⊕−φ) and then apply Lemma 7.25.

7.27. This follows from Lemma 7.11, Remark 7.23 and Lemma 7.25.

7.30. A non-degenerate quadratic form on V is a map µ : V → F2 such thatµ(0) = 0, we obtain a non-degenerate symmetric pairing λ : V × V → F2

by λ(p, q) = µ(p + q) + µ(p) + µ(q) and that λ(p, p) = 0 for all p ∈ V (seeRemark 7.20). Fix a basis e1, e2 for V . Then λ(ei, ej) = 1 for i 6= j since λis non-degenerate and we know already λ(e1, e1) = λ(e2, e2) = λ(e1 +e2, e1 +e2) = 0 and λ(e1, e2) = λ(e2, e1). This implies that either µ(e1) = µ(e2) =µ(e1 + e2) = 1 or that precisely one of the elements µ(e1), µ(e2), µ(e1 + e2)is 1. By possibly replacing the basis e1, e2 by the basis e1, e1 + e2 ore2, e1+e2 we can arrange that either µ(e1) = µ(e2) = µ(e1+e2) = 1 or thatµ(e1) = µ(e2) = 0 and µ(e1 + e2) = 1. The first one has Arf invariant 1, thesecond 0. Hence there are up two isomorphism precisely two non-degeneratequadratic forms on V .

7.42. By the definition of the selfintersection number it suffices to showµ(f) 6= 0 in Qε(Zπ). The map Zπ → Z/2 sending

∑g∈π ng · g to

∑g∈G ng

induces a map of abelian groups Qε(Zπ)→ Z/2. Since the set of double pointsconsists of precisely one element, it sends µ(f) to 1 and hence µ(f) 6= 0.

7.43. Consider the inclusion i : S1 → S1×S1 onto the first factor. One easilychanges it locally by adding a kink to an immersion j : S1 → S1×S1 in generalposition with exactly one double point such that i and j are homotopic. ByExercise 7.42 i and j are not regularly homotopic.

7.50. Denote by Cn−∗(X)untw the Zπ-chain complex which is analogously

defined as Cn−∗(X), but now with respect to the untwisted involution.

Its n-th homology Hn(Cn−∗(X)untw) depends only on the homotopy typeof X. If X carries the structure of a Poincare complex with respect tow : π1(X) → ±1, then the Poincare Zπ-chain homotopy equivalence in-


duces a Zπ-isomorphism Hn(Cn−∗(X)untw) ∼= Zw. Thus we rediscover w

from Hn(Cn−∗(X)untw).

7.57. This follows from fact that two embeddings M → Rn+m for largeenough m are diffeotopic.

7.66. It suffices to show that f is l-connected for l = k + 1, k + 2, . . .. Byassumption this holds for l = k + 1. In the induction step f is l-connectedfor some l ≥ k + 1 and we have to show that f is (l + 1)-connected, i.e.,πl+1(f) = 0. By Lemma 7.61 (ii) which applies also to the case, where M

is only a finite Poincare complex, it suffices to show that Kl(M) = 0. ByLemma 7.61 (i) which applies also to the case, where M is only a finite

Poincare complex, it suffices to show Kn−l(M) = 0. Since f is (k + 1)-

connected and n− l ≤ k, Kn−l(M) = 0 vanishes by Lemma 7.61 (ii).

7.73. Let f : M → S4k+2 be any map of degree one. Choose an embeddingi : M → R4k+2+m for large enough m. Then the given stable trivialization ofthe tangent bundle defines a trivialization of the normal bundle. It can beviewed as bundle map f : µ(i) → Rm covering f . Thus we obtain a normalmap of degree one (f, f). It defines a surgery obstruction σ(f, f) ∈ L4k+2(Z).Since L4k+2(Z) is isomorphic to Z/2, this is the same as an element α(M) ∈Z/2. It is independent by the choice of f and f and depends only on the stablyframed bordism class of M since by a theorem due to Hopf the homotopyclass of f is uniquely determined by its degree and the surgery obstructionis a invariant under normal bordism.

7.100. Because of Theorem 7.99 (ii) we can assume that F : W → X ×[0, 1] is a simple homotopy equivalence. We conclude from Theorem iii thatboth inclusions M → W and N → W are simple homotopy equivalence.By Theorem 2.44 there exists a diffeomorphism W → M × [0, 1]. Hence therestriction of this diffeomorphism to N is a diffeomorphism N →M ×1 =M .

7.103. This follows from the various Rothenberg sequence since the Z/2-Tate cohomology of any Z[Z/2]-module is annihilated by multiplication with2.

7.107. Since Wh(Z), K0(Z[Z]) and Kn(Z[Z]) for n ≤ −1 vanish (see Exam-ple 1.4, Theorem 2.17 and Theorem 3.6), the decoration does not matter byTheorem 7.102. We conclude from (7.105) and the computations of Ln(Z) inTheorem 7.28, Theorem 7.31 and Theorem 7.80:

Ln(Z[Z]) ∼= Ln−1(Z)⊕ Ln(Z) ∼=

Z n ≡ 0, 1 mod 4;

Z/2 n ≡ 2, 3 mod 4.


7.110. We conclude from Conjectures 2.107, 3.17, and 7.109 and Theo-rem 7.102 that the decoration does nor matter. If g = 0, π1(Fg) is trivial

and hence L〈−∞〉n (Z[π1(Fg)]) = L

〈−∞〉n (Z). Suppose g ≥ 1. Then Fg itself is a

model for Bπ1(Fg). Because of Conjecture 7.109 we get

Hn(Fg; L〈−∞〉(Z)) ∼= L〈−∞〉n (Z[π1(Fg)]).

Now we use the Atiyah-Hirzebruch spectral sequence to computeHn(Fg; L〈−∞〉(Z)).

This is rather easy since Fg is 2-dimensional, the edge homomorphism which

describes Hn(•; L〈−∞〉) → Hn(Fg; L〈−∞〉) is split injective and L

〈−∞〉n (Z)

is Z if n ≡ 0 mod 4, Z/2 if n ≡ 0 mod 4 and 0 otherwise. The result is

L〈−∞〉n (Z[π1(Fg)]) ∼=

Z⊕ Z/2 n ≡ 0 mod 4;

Z2g n ≡ 1 mod 4;

Z⊕ Z/2 n ≡ 2 mod 4;

(Z/2)2g n ≡ 3 mod 4.

7.136. Because of Poincare duality it suffices to show f∗(L(M) ∩ [M ]) =L(N) ∩ [N ]. But this follows from the Novikov Conjecture 7.131 because ofRemark 7.135 since we can put N = BG.

7.142. This follows from the long exact homotopy sequence associated to afibration.

7.143. Let C ⊆ π1(X) be any finite cyclic subgroup. Since the universal

covering X is a model for Eπ1(X), it is also a model for EC after restricting

the group action. Hence C\X is a finite dimensional CW -model for BC.This implies that the group homology Hn(C) of C is trivial in dimensionsn > dim(X). It is known that the homology of C is C in all odd dimensions.Hence C must be trivial. This shows that π1(X) is torsionfree.

7.145. The top homology group Hn(M ;F2) with F2-coefficients of any closedn-dimensional manifold M is known to be isomorphic to F2. If M is simplyconnected and aspherical it is homotopy equivalent to the one-point-space•. This implies n = 0 and hence M = • for a simply-connected asphericalmanifold.

7.146. See [537, Lemma 3.2].

7.155. See Example 2.59.


7.156. Let k and n be natural numbers such at least one of them is even.Then Sk and Sn are topologically rigid but Sk×Sn is not. See Remark 7.154.

7.165. Let Gk be a nk-dimensional Poincare duality group for k = 0, 1. LetP k∗ be a nk-dimensional finite projective Z[Gk]-resolution of the trivial Z[Gk]-module Z. Then P 0

∗ ⊗ZP1∗ is a (n0 +n1)-dimensional finite projective Z[G0×

G1]-resolution of the trivial Z[G0 × G1]-module Z. The obvious chain mapgiven by the tensor product over Z and the obvious identification Z[G0] ⊗ZZ[G1] = Z[G0 ×G1]

homZ[G0](P0∗ ,Z[G0])⊗Z homZ[G1](P

1∗ ,Z[G1])

∼=−→ homZ[G0×G1]

(P 0∗ ⊗Z P

1∗ ,Z[G0 ×G1]

)is an isomorphism of Z-cochain complexes. Since homZ[G0](P

0∗ ,Z[G0]) is a free

Z-cochain complex whose cohomology is concentrated in dimension nk andgiven there by Z, there exists a Z-chain homotopy equivalence from [nk](Z)which is the Z-chain complex concentrated in dimension nk and having Z asnk-th chain module, to homZ[G0](P

0∗ ,Z[G0]). Hence homZ[G0](P

0∗ ,Z[G0]) ⊗Z

homZ[G1](P1∗ ,Z[G1]) is Z-chain homotopy equivalent to [n0](Z) ⊗ [n1](Z) ∼=

[n0 +n1](Z). This implies that Hn(homZ[G0×G1](P

0∗ ⊗Z P

1∗ ,Z[G0×G1])

)is Z

in dimension (n0 + n1) and trivial otherwise. Hence G0 ×G1 is a (n0 + n1)-dimensional Poincare duality group.

Chapter 8

8.3. Since

χ(X) =∑n≥0

(−1)n · dimQ(Hn(X;Q)) =∑n≥0

(−1)n · dimQ(Hn(X;Q))

holds, this follows directly from (8.1) and (8.2).

8.17. We obtain from (8.16) an isomorphism

K∗G(X)⊗Z Q ∼=∏

C∈C(G)

H∗(XC/G;Q(ζC)).

Now use the fact dimQ(Q(ζC)) = ϕ(|C|).

8.21. Without loss of generality we can assume that KHj (Y,B) is torsionfree

for all j ∈ Z. Now check that we obtain two G-homology theories on pairs offinite proper G-CW -complexes by putting


H∗G(X,A) :=⊕i+j=n

KGi (X,A)⊗Z K

Hj (Y,B);

K∗H(X,A) := K∗G×H((X,A)× (Y,B)).

(When one wants to check the exactness of the long exact sequence of apair for H∗G, we need that assumption that KH

i (Y,B) is torsionfree andhence the functor − ⊗Z KH

j (Y,B) is exact for all j ∈ Z). The exter-nal multiplication defines a natural transformation T ∗G : H∗G → K∗G of G-cohomology theories for pairs of finite proper G-CW -complexes. One checksthat T ∗G(G/H) : H∗G(G/H) → K∗G(G/H) is bijective for all finite subgroupsH ⊆ G. Now prove by induction over the number of equivariant cells usingthe Five-Lemma, the long exact sequence of a pair, excision and G-homotopyinvariance that TnG(X,A) is bijective for all pairs of finite proper G-CW -complexes (X,A) and all n ∈ Z.

8.23. This follows from the long exact sequence of the pair (DE,SE), theThom isomorphisms (8.22) and the commutativity of the following diagramwhich is a consequence of the naturality of the product

K∗G(X)

K∗G(pDE) ∼=

−∪e(p) // K∗G(X)

K∗G(pDE)∼=

K∗G(DE)−∪K0

G(j)(λE) // K∗G(DE)

K∗G(DE)−∪λE∼=

//

id ∼=

OO

K∗G(DE,SE)

K∗G(j)

OO

8.29. If G contains an element g of order ≥ 3, then show ||xx∗|| 6= ||x||2for x = g + 1 − g−1. If G contains an element g of order 2, then show||xx∗|| 6= ||x||2 for x = g + i ∈ L1(G;C). Finally one checks directly thatL1(G;F ) is a C∗-algebra if G is trivial or if G has order 2 and F = R.

8.35. Since K is the colimit colimn→∞Mn(C), we conclude from Moritaequivalence and the compatibility with colimits over directed systems that the

obvious inclusion of C∗-algebras C→ K induces an isomorphisms K0(C)∼=−→

K∗(K). The C∗-algebra B is contractible, i.e., the zero homomorphism ishomotopic to the identity B → B, a homotopy is given by Ft(x) = F (tx).Homotopy invariance implies the vanishing of K∗(B). Now the long exactsequence of the ideal K ⊆ B yields an isomorphism Kn(B/K) ∼= Kn−1(K) forall n ∈ Z. To finish the calculation, one directly proves that Kn(C) is Z forn = 0 and trivial for n = 1 and applies Bott periodicity.


8.45. Since G is by assumption is finite, Hn(BG;Q) is Q if n = 0 and istrivial for n 6= 0. We conclude from the Chern characters (8.1) and (8.7) thatdimQ(K0(BG) ⊗Z Q) = dimQ(KO0(BG) ⊗Z Q) = 1. We have K0(C∗r (G)) ∼=RepC(G) and KO0(C∗r (G)) ∼= RepR(G). Now use the obvious fact thatdimQ(RepC(G) ⊗Z Q) = 1 ⇐⇒ dimQ(RepC(G) ⊗Z Q) = 1 ⇐⇒ G = 1holds.

8.47. Let c : S1 → S1 be the automorphism of S1 sending z ∈ S1 to z−1. LetTc be the mapping torus. One easily checks that Tc is a model for BG. Ele-mentary considerations about homology theories lead to the so called Wangsequence

· · · ∂n+1−−−→ Kn(S1)id−Kn(c)−−−−−−→ Kn(S1)

Kn(i)−−−−→ Kn(Tc)

∂n−→ Kn−1(S1)id−Kn−1(c)−−−−−−−−→ Kn−1(S1)

Kn−1(i)−−−−−→ · · · .

We know that Kn(S1) ∼= Z for all n ∈ Z. Elementary considerations abouthomology theories imply that Kn(c) = − idKn(S1) for odd n and Kn(c) =idK0(S1) for even n. Hence the Wang sequence reduces to

· · · → Z 2·id−−→ Z K1(i)−−−→ K1(Tc)∂1−→ Z 0−→ Z K0(i)−−−→ K0(Tc)→ Z 2·id−−→ Z→ · · · .

This implies

Kn(C∗r (G)) ∼= Kn(Tc) ∼=

Z if n is even;

Z⊕ Z/2 if n is odd.

8.62. Obviously hom1(F, i∗F ) ∼= F . Since i∗F = C0(G,F ), all homomor-

phisms of G-C∗-algebras from i∗F to F are zero and hence homG(i∗F, F )vanishes.

8.71. Put G = Z/p. Since p is an odd prime, we have dimR(V ) =dimR(V G) ≡ 0 mod 2 and hence dim(SV ) = dim(SV G) ≡ 0 mod 2. Sincedim(SV ) = d − 1, we get Kn(SV ) = Kn(SV G) = Kn(Sd−1) for all n ∈ Z.Since RepC(G) ∼= Zp, we get im(θG) = Z[1/p] and im(θ1) ∼= Z[1/p]p−1. Weconclude from Theorem 8.69

Z[1/p]⊗Z KZ/pn (SV )

∼= Z[1/p]⊗Z Kn

(SV G/CGG

)⊕ Z[1/p]p−1 ⊗Z Kn(SV/CG1)

∼= Z[1/p]⊗Z Kn(Sd−1)⊕ Z[1/p]p−1 ⊗Z Kn(SV/G).

The Atiyah-Hirzebruch spectral sequence converges to Kn(SV/G) and has asE2-term E2

r,s = Hr(SV/G;Ks(•)). Since |G| is a p-power, we get a Z[1/p]-


isomorphism Z[1/p]⊗ZHr(SV/G) ∼= Z[1/p]⊗Z[1/p]GHr(SV ). Since p is odd,the G-operation on Hi(SV ) is trivial. Hence we get a Z[1/p]-isomorphism

Z[1/p]⊗Z E2r,s∼= Z[1/p]⊗Z Hr(SV ;Ks(•))

∼=

Z[1/p] if r = 0, d− 1 and s is even;

0 otherwise.

Hence Z[1/p]⊗ZE2r,s is a finitely generated free Z[1/p]-module for each (r, s)

and we conclude from the isomorphism (8.1) for each n ∈ Z∑r+s=n

rkZ[1/p]

(Z[1/p]⊗Z E

2r,s

)= rkZ[1/p]

(Z[1/p]⊗Z Kn(SV/G)

).

This implies that all differentials in the Atiyah-Hirzebruch spectral sequenceare trivial after inverting p and we get

Z[1/p]⊗Z Kn(SV/G) ∼=

Z[1/p] if d is even;

Z[1/p]2 if d is odd and n is even;

0 if d is odd and n is odd.

∼= Z[1/p]⊗Z Kn(Sd−1).

Now the claim follows from

Z[1/p]⊗Z KZ/pn (SV ) ∼= Z[1/p]⊗Z Kn(Sd−1)⊕ Z[1/p]p−1 ⊗Z Kn(SV/G)

∼= Z[1/p]⊗Z Kn(Sd−1)⊕ Z[1/p]p−1 ⊗Z Kn(Sd−1)∼= Z[1/p]p ⊗Z Kn(Sd−1).

8.74. The abelian group K1(C(X)) is not finitely generated because ofTheorem 2.117, whereas as K1(C(X)) ∼= K1(X) is finitely generated.

Chapter 9

9.6. Define the n-skeleton of X to be p−1(Xn). Use the facts that a cover-ing over a contractible space such as Dn is trivial and a covering is a localhomeomorphism

9.7. The Euler characteristic of a compact CW -complex can be computedby counting cells. Each equivariant cell in X − XZ/p contributes p (non-equivariant) cells.


9.12. Choose an irrational number θ. Let φ : S1 → S1 be the homeomor-phism given by multiplication with the complex number exp(2πiθ). The spaceS1 with the associated Z-action is free but not proper.

9.14. Suppose that there is a free smooth Z/p-action on S2n. By Remark 9.13we obtain a free Z/p-CW -structure on S2n. By a previous exercise we getthe contradiction

0 ≡ χ(∅) ≡ χ((S2n)Z/p

)≡ χ(S2n) ≡ 2 mod p.

9.17. This follows from Theorem 9.16 (i).

9.20. Suppose that EF (G) has a zero-dimensional model. Hence it is adisjoint union of spaces of the shape G/H. Since EF (G) is path connected,it must be G/G. This implies G ∈ F .

If G ∈ F holds, G/G is a 0-dimensional G-CW -model for EF (G).An example for L is R.

9.32. We obtain from Subsection 9.6.13 that there is a G-CW -model forEG which is obtained from G/M by attaching free cells of dimensions ≤ 2.Let i : G/M → EG be the inclusion. Consider the map

j = idEG×Gi : EG×G G/M → EG×G EG

Since for a space Y the canonical projection EG ×G (G × Y ) → Y is ahomotopy equivalence, we conclude by a Mayer-Vietoris argument that Hn(j)is bijective for n ≥ 3. The canonical projections EG ×G G/M → EG/M =BM and EG×G EG→ EG×G • = BG are homotopy equivalences sinceEG is (after forgetting the group action) contractible.

9.33. Since hyperbolic groups, arithmetic groups, Out(Fn), mapping classgroups and one-relator groups have a finite dimensional model for EG bySubsections 9.6.7, 9.6.8, 9.6.9, 9.6.10 and 9.6.13, it suffices to show for groupG with a K-dimensional model for EG that Hk(BG;Q) = 0 for k > K.

The cellular Q-chain complex C∗(X) of a proper G-CW -complex X con-sists of projective QG-modules since for any finite subgroup H ⊆ G theQG-module Q[G/H] is projective. Since EG is contractible (after forgettingthe group action), its cellular Q-chain complex yields a dim(EG)-dimensionalprojective QG-resolution of the trivial QG-module Q.

9.47. Since H is infinite and countable, its cardinality is ℵ0. We concludegd(H) = 1 from Remark 9.46. Theorem 9.45 implies that gd(H o Z) ≤ 2.

Since H o Z is finitely generated and does not contain a finitely generatedfree group of finite index, we cannot have gd(H o Z) ≤ 1.


9.54. The universal covering on M is the hyperbolic space and hencecontractible. Therefore N is a model for BG. Since Hn(BG;Zw) ∼= Z forw : G = π1(N) → ± given by the first Stiefel-Whitney class of N , weconclude cd(G) = dim(N) = gd(G). Since G is hyperbolic and hence sat-isfies conditions (M) and (NM), we conclude gd(G) = gd(G) from Theo-

rem 9.53 (iii).

9.60. This follows from Theorem 9.59.

9.63. Apply Theorem 9.62 to X and take Y = H\EG for a torsionfreesubgroup H of G with [G : H] = 2.

Chapter 10

10.4. The desired Z/2-pushout for the 1-skeleton is obvious ad for the 2-skeleton given by

Z/2× S1 pr //

S1

Z/2×D2 // S2

where pr is the projection. Now one easily checks that C∗(S2)⊗Or(Z/2) RC is

given by the Z-chain complex concentrated in dimensions 0,1 and 2

· · · → 0 → 0 → RC(1) c2−→ RC(Z/2)0−→ RC(Z/2)→ 0 → · · ·

where c2 is induction with the inclusion 1 → Z/2. This implies

HZ/2n (S2;RC) ∼=

Z2 n = 0;

Z n = 1;

0 otherwise

10.7. By applying Lemma 10.5 to the skeletal filtration X0 ⊆ X1 ⊆ X2 ⊆. . . ⊆ ∪n≥0Xn = X, the claim can be reduced to finite dimensional pairs.Using the axioms of a G-homology theory, the five lemma and induction overthe dimension one reduces the proof to the special case (X,A) = (G/H, ∅).

10.11. This follows directly from the axiom about the compatibility withconjugation applied in the case X = •.

10.16. The real line R with the obvious action of D∞ = Z/2∗Z/2 = ZoZ/2is a D∞-CW -model for ED∞ (see Theorem 9.25). Up to conjugacy there are


two subgroups H0 and H1 of order two in D∞. One obtains a D∞-pushout

D∞ × S0 = D∞ qD∞pr0 q pr1//

i

D∞/H0 qD∞/H1

D∞ ×D1 // ED∞

where pr0 and pr1 are the obvious projections and i is the obvious inclusion.Hence the associated long exact Mayer-Vietoris sequence reduces to

0→ KD∞1 (ED∞)→ RC(1)⊕RC(1)

f−→ RC(1)⊕RC(Z/2)⊕RC(Z/2)→ KD∞0 (ED∞)→ 0

where f sends (v, w) to (v + w, i∗(v),−i∗(w)) for i∗ the map induced by theinclusion i : 1 → Z/2. This implies

KD∞n (ED∞) ∼=

Z3 n even;

0 n odd.

10.32. One easily checks that for a given group G and every subgroup H ⊆ Gand every n ∈ Z the map HG

n (G/H; t) : HGn (G/H; E)→ HG

n (G/H; F) can beidentified with the map πn(t(tG(G/H))) : πn(E(tG(G/H)))→ πn(F(tG(G/H)))and hence is bijective by assumption. Now apply Lemma 10.6.

10.43. The argument appearing in the solution of Exercise 10.16 yields along exact Mayer- Vietoris sequence

· · · → K0(R)⊕K0(R)→ K0(R)⊕K0(R[Z/2])⊕K0(R[Z/2])

→ HD∞0 (ED∞; KR)→ K−1(R)⊕K−1(R)

→ K−1(R)⊕K−1(R[Z/2])⊕K−1(R[Z/2])→ · · · .

Since the obvious map Kn(R)→ Kn(R[Z/2]) is split injective, we obtain forn ∈ Z isomorphisms

Kn(R[Z/2])⊕ coker (Kn(R)→ Kn(R[Z/2])) ∼= HD∞n (ED∞; KR).

If n ≤ −1, then Kn(R[Z/2]) = 0 for R = Z,C by Theorem 3.15 and Theo-rem 3.21. Hence

HD∞n (ED∞; KR) ∼= 0 for n ≤ −1


The map K0(Z)→ K0(Z[Z/2]) is bijective by Example 1.91 and K0(Z) = Zby Example 1.4. Hence

HD∞0 (ED∞; KZ) ∼= Z.

Since K0(CH) ∼= RC(H) for a finite group H, one easily checks

HD∞0 (ED∞; KC) ∼= Z3.

10.45. Since X/G has no odd-dimensional cells, X has no odd dimensionalequivariant cells. Moreover, if X/G is finite, then X has only finitely manyequivariant cells. We conclude for any coefficients system M that the Bredonhomology Hp(X;M) vanishes, if p is odd, or if p is larger then the dimen-sion of X. If X has only finitely many equivariant cells and M(G/H) is afinitely generated free abelian groups for any finite subgroup H ⊆ G, thenHp(X;M)is finitely generated free abelian for all p ∈ Z. Since KG

q (G/H) = 0for odd q and is a finitely generated free abelian group for even q for everyfinite subgroup H ⊆ G, and all isotropy groups of X are by assumption finite,we conclude for the E2-terms of the equivariant Atiyah-Hirzebruch spectralsequence of Theorem 10.44 that E2

p,q = 0 if p+q is odd. If X has only finitelymany equivariant cells, then E2

p,q is finitely generated free if p + q is evenand vanishes for large enough q. Now the claim follows from this spectralsequence.

10.50. Consider the long exact sequence of the pair (EG×GX,EG×GXG)and of the pair (X/G,XG/G) = (X/G,XG) and the map between theminduced by the projection (EG×GX,EG×GXG)→ (X/G,XG/G), and usethe fact that (X,XG) is relatively free and henceHn(EG×GX,EG×GXG)→Hn(X/G,XG/G) is bijective.

10.54. From Theorem 10.53 we get a natural isomorphism of spectral se-quences from the equivariant Atiyah-Hirzebruch spectral sequence convergingto BHG(X) he equivariant Atiyah-Hirzebruch spectral sequence converging toHG∗ (X).One easily checks that all the differentials in the equivariant Atiyah-Hirzebruch spectral sequence converging to BHG(X) vanish.

10.55. For every finite group H ⊆ G the group WGH is finite and henceQ[WGH] is semi-simple. Therefore every Q[WGH]-module is flat. Because ofTheorem 10.53 it suffices to show that for every finite subgroup H ⊆ G andevery n ∈ Z the map

Hp(CGH\ιHF⊆G ;Q) : Hp(CGH\EF (G)H ;Q)→ Hp(CGH\EG(G)H ;Q)


is bijective. This is obviously true if H 6∈ F . Suppose H ∈ F . Then the claimfollows from fact that both C∗(EF (G)H) ⊗Z Q and C∗(EG(G)H) ⊗Z Q areprojective Q[CGH]-resolutions of the trivial Q[CGH]-module Q which impliesthat

C∗(ιHF⊆G)⊗Z Q→ C∗(EF (G)H)⊗Z Q→ C∗(EG(G)H)⊗Z Q

is a Q[CGH]-chain homotopy equivalence and hence induces after applyingQ⊗Q[CGH] − a Q-chain homotopy equivalence.

10.59. The desired pairing is given by

A(G)×M(G)→M(G), ([G/H], x) 7→ indGH resHG (x).

10.78. Every subgroup F ⊆ SL2(Z) is conjugated to one of the groupsZ2,Z3,Z4,Z6 with generators given by the matrices(

−1 00 −1

) (−1 −11 0

) (0 −11 0

)and

(0 −11 1

).

So we shall restrict from now on to the study of the actions of Z2,Z3,Z4,Z6

given by the actions of the above described generators. The case Z/4 hasbeen carried out in Example 10.77. The computations for the other cases isanalogous. We get in all cases that G\EG is homeomorphic to S2. There areup to conjugacy four non-trivial finite subgroups, which are all isomorphic toZ/2, in G in the case F = Z/2. There up to conjugacy three non-trivial finitesubgroups, which are all isomorphic to Z/3 in G in the case F = Z/3. In thecase F = Z/6 there are up to conjugacy three non-trivial finite subgroups,the first is isomorphic to Z/2, the second to Z/3 and the third to Z/6. Hencewe get in all cases KG

1 (EG) = 0 and

KZ2oZ/20 (EZ2 o Z2) ∼= Z6;

KZ2oZ/30 (EZ2 o Z3) ∼= Z8;

KZ2oZ/40 (EZ2 o Z4) ∼= Z9;

KZ2oZ/60 (EZ2 o Z6) ∼= Z10.

Chapter 11

11.3. If we replace in Conjecture 11.1 the family VCY by FIN , then theConjecture 11.2 for Z reduces to the statement that for any ring R the map


induced by the projection EZ→ Z/Z

HZn(EZ; KR)→ HG

n (Z/Z; KR) = Kn(RZ)

is an isomorphism. Since Z acts freely on Z and (EZ)/Z = S1, we get anidentification

HZn(EZ; KR) = H1(S1; KR) = Kn(R)⊕Kn−1(R).

Under this identification the assembly map above becomes the restriction ofthe Bass-Heller-Swan isomorphism of Theorem 5.16

Kn(R)⊕Kn−1(R)⊕NKn(R)⊕NKn(R)∼=−→ Kn(RZ).

to Kn(RG) ⊕Kn−1(RG). This implies that NKn(R) vanishes for all n ∈ Zand all rings R, a contradiction by Example 2.66.

11.5. If we replace in Conjecture 11.4 the family VCY by FIN , then Conjec-ture 11.6 for Z reduces to the statement that for any ring R with involutionthe map induced by the projection EZ→ Z/Z

HZn(EZ; L

〈−∞〉R )→ HG

n (Z/Z; L〈−∞〉R ) = L〈−∞〉n (RZ)

is an isomorphism. Since Z acts freely on Z and (EZ)/Z = S1, we get anidentification

HZn(EZ; L

〈−∞〉R ) = L〈−∞〉n (RZ)⊕ L〈−∞〉n−1 (RZ)

Under this identification the assembly map above can be identified with theisomorphism appearing in the Shaneson splitting (7.105).

11.11. The structure of an abelian group on each set of morphisms comesfrom the obvious structure of an abelian group on Mm,n(R). The sum of [m]and [n] is [m+ n]. The sum on morphisms is given by taking block matrices.The zero object is [0]. We obtain a natural equivalence from A(R) to theadditive category of finitely generated free R-modules by sending [m] to Rm

and a morphism [m] → [n] given by a (m,n)-matrix A to the R-linear mapRm → Rn given by right multiplication with A.

11.16. We only present the proof of the harder implication. Suppose thatfor every two objects A and B in A the induced map morA(A0, A1) →morB(F (A0), F (A1)) sending f to F (f) is bijective and for each object Bin B there exists an object A in A such that F (A) and B are isomorphic inB. Choose for any object B ∈ B an object A(B) ∈ A and an isomorphism

u(B) : B∼=−→ F (A(B)) in B. Next we define a functor F ′ : B → A of addi-

tive categories. It sends an object B to A(B). A morphism f : B0 → B1 is


send to the morphism F ′(f) : A(B0)→ A(B1) which is uniquely determinedby the property that F (F ′(f)) = u(B1) f u(B0)−1. One easily checksthat F ′(g f) = F ′(g) F ′(f) and F ′(f0 + f1) = F ′(f0) + F ′(f1) holds.Consider two objects B0 and B1. We have to show that for the natural inclu-sions ji : Bi → B0 ⊕B1 for i = 0, 1 the morphism F ′(j0)⊕ F ′(j1) : F ′(B0)⊕F ′(B1)→ F (B0⊕B1) is an isomorphism. This follows from the following di-agram that commutes by definition of F ′ and whose lower left vertical arrowis an isomorphism since F is compatible with direct sums

B0 ⊕B1j0⊕j1=id

∼=//

u(B0)⊕u(B1)∼=

B0 ⊕B1

u(B0⊕B1) ∼=

F (A(B0))⊕ F (A(B1))F (F ′(j0))⊕F (F ′(j1)) //

∼=

F (A(B0 ⊕B1))

id ∼=

F (A(B0)⊕A(B1))F (F ′(j0)⊕F ′(j1)) // F (A(B0 ⊕B1))

Hence F ′ is a functor of additive categories. Natural transformations of func-tors of additive categories S : F F ′ → idB and T : F ′ F → idA are deter-mined by S(B) = u(B) and F (T (A)) = u(F (A)).

11.22. This follows from Theorem 11.21 (iii) since G is virtually cyclic if Qis virtually cyclic.

11.23. Let G be a group. It is the disjoint union of its finitely generated sub-groups. Hence by Theorem 11.21 (iv) the Full Farrell-Jones Conjecture 11.20holds for all groups if and only if it holds for all finitely generated groups.Any finitely generated group can be written as a directed colimit of finitelypresented groups. Hence by Theorem 11.21 (iv) the Full Farrell-Jones Con-jecture 11.20 holds for all finitely generated groups if and only if holds for allfinitely presented groups. Finally notice that group G is finitely presented ifand only if it occurs as the fundamental group of a connected closed orientable4-manifold.

11.24. This follows from Theorem 11.21 (iii) and (vii).

11.31. This follows from Lemma 11.30 by the following argument. SinceKW is finite and the image of φ is by assumption infinite, the compositepW φ : V → QW has infinite image. Since QW is isomorphic to Z or D∞,the same is true for the image of pW φ : V → QW . By assertion (v) ofLemma 11.30 the kernel of pW φ : V → QW is KV . Hence φ(KV ) ⊆ KW

and φ induces maps φK and φQ making the diagram of interest commutative.Since the image of pW φ : V → QW is infinite, φQ(QV ) is infinite. This


implies that φQ is injective since both QV and QW are isomorphic to D∞ orZ.

11.32. Suppose that G admits a proper cocompact isometric action on R.Since the action is cocompact and R is not compact, the group G must beinfinite. Let K be the kernel of the homomorphism ρ : G → aut(R) comingfrom the G-action. Since the action is proper, K must be finite. Let Q ⊆aut(R) be the image of ρ. The group of isometries of R is RoZ/2, where Z/2corresponds to ± id and R to translations lr : R→ R with elements r ∈ R.Let r0 := infr ∈ R | r > 0, lr ∈ Q. Since Q acts properly, we have r0 > 0and Q ∩ R ⊆ R is the infinite cyclic group generated by r0. Now one easilychecks that Q is isomorphic to Z or Z o Z/2. Hence G is virtually cyclic. IfG is virtually cyclic, then it admits an epimorphism with finite kernel onto Zor ZoZ/2 by Lemma 11.30 (i). These two groups and hence G admit propercocompact isometric actions on R.

11.37. Suppose that H is infinite and belongs to HEp ∩ VCYI . Then there

are exact sequences 1→ Z→ Hq−→ Q→ 1 and 1→ P

i−→ H → Z→ 1, wherei : P → H is the inclusion of a finite normal subgroup P , and Q is a finite p-group. The restriction q|P : P → Q is injective since P is a finite subgroup ofH and the kernel of q is infinite cyclic. Hence P is a finite p-group. ObviouslyP is also a subgroup of G. Fix an element t ∈ H whose image under theepimorphism H → Z is a generator. Then t ∈ NGP . Let pm be the order ofQ. Consider any x ∈ P . We have q(tp

m

xt−pm

) = q(t)pm

q(x)q(t)−pm

= q(x).Choose y ∈ P with tp

m

xt−pm

= y. We conclude q(y) = q(x). Since q|P : P →Q is injective, we get x = y and hence tp

m

xt−pm

= x.Suppose H is isomorphic to P oφ Z for some automorphism φ : P → P

whose order is a power of p. Then obviously H belongs to VCYI . The exact

sequence 1 → Z pm·id−−−→ Z → Z/pm → 1 induces an exact sequence 1 → Z →P oφ Z→ P oφ Z/pm → 1. Since P oφ Z/pm is a finite p-group, H belongsto HEp.

11.38. Because of Exercise 11.37 there exists a finite p-group P and anautomorphism φ : P → P whose order is a p-power such that G is isomorphicto P oφ Z. Notice that a model for EFIN (G) is EZ considered as G-CW -complex by restriction with the canonical epimorphism G→ Z. We concludefrom Theorem 5.52 and Remark 5.53 that

HGn (EFIN (G); KR)→ HG

n (G/G; KR) = Kn(RG)

is bijective after applying−⊗ZZ[1/p] for all n ∈ Z if and only ifN±Kn(RP ;φ)[1/p]vanishes for all n ∈ Z. This follows from Theorem 5.54.

11.40. Remark 11.10 and Theorem 11.39 (i) imply that the projection in-duces an isomorphism


HGn

(pr; KF

): HG

n

(EHE∩FIN (G); KF

) ∼=−→ K0(FG).

Since FH is regular for every finite group H, the negative K-groups of FGvanish by Theorem 3.6. Hence the the equivariant Atiyah-Hirzebruch spec-tral sequence of Theorem 10.44 converging to HG

n

(EHE(G); KF

)is a first

quadrant spectral sequence. Therefore the edge homomorphism at (0, 0) isan isomorphism

HG0 (EHE∩FIN (G);K0(F−))

∼=−→ HG0

(EHE∩FIN (G); KF

)where the source is the 0th Bredon homology with respect to the coeffi-cient system given by G/H 7→ K0(FH). It can be identified with the colimitcolimH∈SubHE∩FIN (G)K0(FH). Hence the various inclusions induce an iso-morphism

colimH∈SubHE∩FIN (G)K0(FH)∼=−→ K0(FG).

11.48. The group G satisfies the Full Farrell Jones Conjecture 11.20 byTheorem 11.21 (ii) and (iii). Since every virtually cyclic subgroup of G is oftype I, Theorem 11.47 implies that the projection pr induces for any additiveG-category with involution A and all n ∈ Z an isomorphism

HGn

(pr; L

〈−∞〉A

): HG

n

(EFIN (G); L

〈−∞〉A

)→ HG

n (G/G; L〈−∞〉A ) = πn

(L〈−∞〉A (I(G))

).

Hence we get from Remark 11.15 that the projection pr induces for all n ∈ Zan isomorphism

HGn

(pr; L

〈−∞〉Z

): HG

n

(EFIN (G); L

〈−∞〉Z

)→ HG

n (G/G; L〈−∞〉Z ) = L〈−∞〉n (ZG).

Recall that we have an extension 1 → F → Gf−→ Zd → 1 for a finite group

F . Hence the restriction f∗EZd with f of EZd is a model for EFIN (G).Hence it suffices to construct for any free Zn-CW -complex X an appro-

priate spectral sequence converging to HGn

(f∗X; L

〈−∞〉Z

). Since the assign-

ment sending X to HGn

(f∗X; L

〈−∞〉Z

)is a Zd-homology theory in the sense

of Definition 10.1 and X is assumed to be a free Zd-CW -complex, the equi-variant Atiyah-Hirzebruch spectral sequence of Theorem 10.44 converges to

HGn

(f∗X; L

〈−∞〉Z

)and has as E2-term Hp

(C∗(X)⊗ZdH

Gq (G/F ; L

〈−∞〉Z )

). Us-

ing the induction structure on H?∗(−; L

〈−∞〉Z ) and Lemma 10.12, one can

identify the Zd-modules HGq

(∗G/F ; L

〈−∞〉Z

)and L

〈−∞〉q (ZF ).

11.56. Induction with i : H → G and restriction with f : H → Z induceshomomorphisms i∗ : G0(CH) → G0(CG) and f∗ : G0(CZ) → G0(CH). The


class [C] of the trivial CZ-module C is sent under i∗ f∗ to the class ofC[G/H]. Since there exists a short exact sequence 0→ CZ→ CZ→ C→ 0,we have [C] = 0 in G0(CZ).

Chapter 12

12.10. Equip R with the G = Z× Z/k-action, where Z acts by translationand Z/k acts trivial. There is a G-pushout

Z× 0, 1j //

i

Z

Z× [0, 1] // R

where we think of Z as the G-space G/(Z/k), the map i is the inclusion andj sends (n, 0) to n and (n, 1) to n + 1. Hence R is a G-CW -complex whoseisotropy groups are all finite and whose H-fixed point set is contractible forevery finite subgroup H ⊆ G. We conclude that R is a model for EG. TheMayer-Vietoris sequence associated to the G-pushout looks like

· · · → KGn (Z)⊕KG

n (Z)

1 11 1

−−−−−→ KG

n (Z)⊕KGn (Z)→ KG

n (EG) = KGn (R)

→ KGn−1(Z)⊕KG

n−1(Z)

1 11 1

−−−−−→ KG

n−1(Z)⊕KGn−1(Z)→ · · ·

where we identify KGn (Z × [0, 1]) ∼= KG

n (Z) via the isomorphism induced bythe projection Z × [0, 1] → Z. Since KG

n (Z) ∼= KGn (G/(Z/k)) is RepC(Z/k)

for n even and zero for n odd, we conclude for all n ∈ Z

KGn (EG) ∼= RepC(Z/k) ∼= Zk.

12.15. We only treat the case F = C, the case F = R is analo-gous. Since H and G are torsionfree and satisfy the Baum-Connes Con-jecture 12.9, they also satisfy the Baum-Connes Conjecture for torsionfreegroups 8.44 by Remark 12.14. Hence it suffices to show that the homomor-phism Kn(Bf) : Kn(BH) → Kn(BG) is bijective for all n ∈ Z. This fol-lows from the Atiyah-Hirzebruch spectral sequence converging to Kn(BH)and Kn(BG), since Hn(Bf ;Z) : Hn(BH;Z) → Hn(BG;Z) is bijective forall n ∈ Z by assumption and hence f induces isomorphisms between theE2-pages.


12.27. Take G = Z/2. Consider the Atiyah-Hirzebruch spectral sequenceconverging toKp+q(BG) with E2-term E2

p,q = Hp(BG;Kq(•)). Its E2-termlooks like

......

......

......

Z Z/2 0 Z/2 0 Z/2 · · ·

0 0 0 0 0 0 · · ·

Z Z/2 0 Z/2 0 Z/2 · · ·

0 0 0 0 0 0 · · ·

Z Z/2 0 Z/2 0 Z/2 · · ·

0 0 0 0 0 0 · · ·

......

......

......

Because of the checkerboard pattern and by a standard edge argument ap-plied to the split injection K∗(•) → Kn(BG) coming from the inclusion• → BG, all differentials are trivial and the E2-term is the E∞-term. HenceK1(BG) is non-trivial. (Actually it is Z/2∞ ∼= Z[1/2]/Z). On the other handK1(C∗r (Z/2)) is trivial.

12.31. Since any group is the directed union of its finitely generated sub-groups, it suffices to consider finitely generated free abelian groups andfinitely generated free groups by Theorem 12.30 (iv). Since a finitely gen-erated group abelian group is the direct product of finitely many copies ofZ and of a finite group, Theorem 12.30 (iii) and (vi) imply that it sufficesto prove the claim for Z and any finite group. The Baum-Connes Conjecturewith coefficients 12.11 holds obviously for finite groups. It holds for Z byTheorem 12.30 (vii).


12.32. Let Fg be the surface of genus g ≥ 1. Let F → F be the coveringassociated to the epimorphism π1(Fg)→ H1(Fg). Then Fg is a non-compact2-manifold and hence homotopy equivalent to a 1-dimensional CW -complex.Hence π1(Fg) is free, H1(Fg) is a finitely generated free abelian group andwe have the exact sequence 1 → π1(Fg) → G → H1(Fg) → 1. We concludefrom Theorem 12.30 that G satisfies the Baum-Connes Conjecture 12.9. SinceFg itself is a model for BG, we get Kn(Fg) ∼= Kn(C∗r (G)). Now an easyapplication of the Atiyah-Hirzebruch spectral sequence yields the claim.

12.42. Notice that K0(C∗r (G)) = RC(G). One easily checks by inspecting thedefinition of (8.48) of the trace for a finite dimensional complex representationV that trC∗r (G) : K0(C∗r (G))→ R sends the class of [V ] to |G|−1 · dimC(V ).

12.54. Suppose that F is S2. Since M is spin and hence in particular ori-entable and any orientation preserving selfdiffeomorphism of S2 is isotopicto the identity, M must be S1 × S2 and hence carries a Riemannian metricof positive scalar curvature.

Suppose that F is not S2. Then F and hence M are aspherical. Henceit suffices to show by Lemma 12.53 that the Baum-Connes Conjecture withcoefficients 12.11 holds for π1(M). The Baum-Connes Conjecture with coef-ficients 12.11 holds for all finitely generated free groups and for Z by Theo-rem 12.30 (v). Hence it holds for every free group and every finitely generatedabelian group by Theorem 12.30 (iii) and (iv). Let F → F be the coveringassociated to the epimorphism π1(F ) → H1(F ). Then F is a non-compact2-manifold and hence homotopy equivalent to a 1-dimensional CW -complex.Hence π1(F ) is free. Now apply Theorem 12.30 (ii) to the short exact se-quences 1 → π1(F ) → π1(M) → π1(S1) → 1 and 1 → π1(F ) → π1(F ) →H1(F )→ 1.

Chapter 13

13.3. Let α : H → G be a group homomorphism. Then α∗EC(H)(H) is a G-CW -complex whose G-isotropy groups are of the shape α(L) for L ∈ C(H)and hence all belong to C(G). This implies that there is up to G-homotopyprecisely one G-map f : α∗EC(H)(H) → EC(G)(G). The following diagramcommutes

α∗EC(H)(H)

f

α∗ pr // α∗H/H = G/α(H)

pr

EC(G)(G)

pr// G/G

Now apply HG∗ to this diagram and combine it with the following commuta-tive diagram coming from the induction structure applied to α


HHn (EC(H)(H))HHn (pr) //

HHn (H/H)

HGn (α∗EC(H)(H))

HGn (α∗ pr)

// HG(α∗H/H)

13.7. The restriction of a G-CW -complex X to K with φ is a K-CW -complex φ∗X by Remark 9.3. For a point x ∈ X the K-isotropy group Kx

of φ∗X is φ−1(Gx), where Gx is the G-isotropy group of X. In particularwe get φ(Kx) = Gx and hence every K-isotropy group of φ∗X belongs toφ∗F . Consider a subgroup H ⊂ K. Then (φ∗X)H = Xφ(H). Now apply theseclaims to X = EF (G).

13.15. Let G be any group. Denote by pr: G× Z→ Z the projection. TheFibered Meta-Isomorphism Conjecture 13.8 predicts that the assembly map

Hn(pr; KR) : HG×Zn (Epr∗ FIN (G× Z); KR)

→ HG×Zn (G× Z/G× Z; KR) = Kn(R[G× Z])

is bijective for all n ∈ Z. A model for Epr∗ FIN (G × Z) is pr∗EZ. Since Zacts freely on EZ and (EZ)/Z = S1, the left side of the map above can beidentified with

HG×Zn (Epr∗ FIN (G× Z); KR) = HG×Z

n (pr∗EZ; KR)

= HGn (G/G× S1)

= Hn(G/G; KR)⊕Hn−1(G/G; KR)

= Kn(RG)⊕Kn−1(RG).

Under this identification the assembly map above becomes the restriction ofthe Bass-Heller-Swan isomorphism of Theorem 5.16

Kn(RG)⊕Kn−1(RG)⊕NKn(RG)⊕NKn(RG)∼=−→ Kn(RG[t, t−1]).

to Kn(RG) ⊕ Kn−1(RG). Hence the Fibered Meta-Isomorphism Conjec-ture 13.8 implies that for every group G and n ∈ Z we have NKn(RG) = 0.

13.17. This follows from Lemma 13.16 applied to the inclusion i : H → Gsince C(H) = i∗C(G).

13.30. Put Γ = G ×φ Z. The proof is completely analogous to the one inExample 13.26 but now applied to a 1-dimensional Γ -CW -complex T which isa tree and whose 1-skeleton is obtained from the 0-skeleton by the Γ -pushout


Γ/G× S0 q //

Γ/G

Γ/G×D1 // T

Here q is the disjoint union of identity id : Γ/G → Γ/G and the Γ -mapΓ/G→ Γ/G sending γG to γtG for t ∈ Γ a lift of the generator in Z.

13.36. Put π = π1(X). Conjecture 13.35 yields a weak homotopy equiv-

alence Eπ+ ∧π S(X) → S(X) because of the identifications X = π\X and

ET R(π)+ ∧Or(π) SπX

= Eπ+ ∧π S(X).Suppose that S is of the shape X 7→ X+ ∧ HZ for HZ the Eilenberg-

spectrum of Z. Recall that the homology theory associated to HZ is singularhomology Hn. Then

πn((Eπ)+ ∧π S(X)

) ∼= Hn(Eπ ×π X);

πn((Bπ)+ ∧π S(•)

) ∼= Hn(Bπ),

and Hn(Eπ ×π X) and Hn(Bπ) are not isomorphic in general.

In the sequel we equip S(•) with the trivial π-action. Suppose that Xis contractible or S is of the shape Y 7→ T(Π(Y )) for some covariant functor

T : GROUPOIDS → SPECTRA. Then the projection X → • induces a

π-map f : S(X) → S(•) such that after forgetting the group action f is aweak homotopy equivalence. Hence we obtain a weak homotopy equivalence

Eπ+ ∧π S(X)(idEπ)+∧πf−−−−−−−−→ Eπ+ ∧π S(•) = Bπ+ ∧ S(•).

13.42. We conclude from the assumptions that for two groups H1, H2 ∈ CConjecture 13.35 holds for (H1×H2, C(H1×H2)). Hence Theorem 13.41 (iii)applies. By assumption also Theorem 13.41 (iv) applies. Hence Conjec-ture 13.35 holds for (F o H, C(F o H)) if F is a free group and H is a finitegroup since this is true by assumption for every finitely generated free groupF .

If Conjecture 13.35 is true for (∗i∈IGi, C(∗i∈IGi)), it is also true for(Gi, C(Gi)) for every i ∈ I by Theorem 13.41 (i).

Suppose that Conjecture 13.35 holds for (Gi, C(Gi)) for every i ∈ I. Nowwe can proceed as in the proof of assertion (v) of Theorem 11.21 using The-orem 13.41 (ii) to show that Conjecture 13.35 holds for (∗i∈IGi, C(∗i∈IGi)),provided that I is finite. Now the general case of arbitrary I follows fromTheorem 13.41 (iv).


13.47. The key ingredient is to construct for a group homomorphism φ : K →G and a subgroup H ⊆ K a natural weak homotopy equivalence of spaces

EGK(K/H)×GK(K/H) p∗φ∗Z

'−→ K/H ×K (EK × φ∗Z)

where p : GK(K/H) → GK(K/K) = I(K) is induced by the projectionK/H → K/K. Because of the the third isomorphism appearing in [222,Lemma 1.9], it suffices to construct a map

u : p∗EGK(K/H)×K φ∗Z'−→ K/H ×K (EK × φ∗Z)

where here and in the sequel we consider a K-space as a GK(K/K) = I(K)-space and vice versa in the obvious way. Since (K/H × EK) ×K φ∗Z =K/H ×K (EK × φ∗Z), it suffices to construct for every K-set S a naturalK-homotopy equivalence

v : p∗EGK(S)'−→ S × EK,

since we then can define u = v ×K idφ∗Z for S = K/H. Unravelling thedefinition we see that the source of v is given by

p∗EGK(S) =∐s∈S

K × EGK(S)(s)/∼

for the equivalence relation ∼ given by

(k, x) ∼ (k(k′)−1, EGK(S)(k′ : s→ k′s)(u)).

Define a K-map∐s∈S K × EGK(S)(s) → S × EGK(K/K) by sending the

element (k, x) in the summand K×EGK(S)(s) belonging to s ∈ S to the ele-ment

(ks,EKK(k ·K/K → K/K)(u)

). On easily checks that it is compatible

with ∼ and induces the desired K-map

v : p∗EGK(S) =∐s∈S

K × EGK(S)(s)/∼ → S × EK,

It remains to show that v is a K-homotopy equivalence. Since the source andtarget of v are free K-CW -complexes, it suffices to show that v is a homotopyequivalence (after forgetting the K-action). We obtain a (non-equivariant)homeomorphism ∐

s∈SEGK(S)(s)

∼=−→∐s∈S

K × EGK(S)(s)/∼

by sending the element x ∈ EGK(S)(s) belong to the summand of s ∈ S tothe element represented by (1, s). Hence the both the source and the target


of v have the property that each path component is contractible. Since v isa bijection on the path component, it is a homotopy equivalence.

13.56. We get πn(A(•)) ∼= Kn(Z) ∼= 0 for n ≤ −1 and π0(A(•)) ∼=K0(Z) ∼= Z from Example 1.4, Theorem 2.17, and Theorem 6.16 (i). Nowapply the Atiyah-Hirzebruch spectral sequence to Hn(BG; A(•)) for n ≤ 0.

13.91. Consider the commutative diagram appearing in Remark 13.89. Thetwo vertical arrows are bijective as explained in Remark 13.89. The upperhorizontal arrow is bijective by assumption. The lower horizontal arrow isbijective by Theorem 13.88. Hence the right vertical arrow is bijective.

13.92. Since R[G × Z] = R[Z][G] and R[Z] is regular Conjecture 13.90 istrue for G and G× Z. We obtain from the Bass-Heller Swan decompositionsfor K-theory, see Theorem 5.16, and homotopy K-theory, see Theorem 13.68,the commutative diagram with isomorphisms as horizontal arrows

Kn(RG)⊕Kn−1(RG)⊕NKn(RG)⊕NKn(RG)∼= //h 0 0 0

0 h 0 0

Kn(R[G× Z])

h

KHn(RG)⊕KHn−1(RG)

∼= // KHn(R[G× Z])

where the maps denoted by h are induced by the canoncial map K → KHand bijective.

Chapter 14

14.2. We know that L/K is a smooth manifold, which is diffeomorphic toRdim(L), and, equipped with the obvious left G action, it is a model for theclassifying space for proper G-actions, see Theorem 9.24. Hence G\L/K isan aspherical closed smooth manifold of dimension ≥ 5. Since G satisfies theFull Farrell-Jones Conjecture 11.20 by Theorem 14.1 (id), the claim followsfrom Theorem 7.160.

14.3. We want to apply Theorem 14.1 (iic) to the exact sequence 1 →π1(F )

i∗−→ π1(E)p∗−→ π1(B)→ 1. Consider a cyclic subgroup C ⊆ π1(B). We

have to show that p−1∗ (C) satisfies the Full-Farrell-Jones Conjecture 11.20. If

C is trivial, p−1∗ (C) is π1(F ) and we can apply Theorem 14.1 (ie). Suppose

C is non-trivial. Choose a map f : S1 → B representing a generator of C.Then p−1

∗ (C) is isomorphic to π1(Tf ), where Tf is the mapping torus of fand hence a manifold of dimension ≤ 3. Now apply Theorem 14.1 (ie) again.


14.4. Let G be a group. It is the disjoint union of its finitely generated sub-groups. Hence by Theorem 11.21 (iv) the Full Farrell-Jones Conjecture 11.20holds for all groups if and only if it holds for all finitely generated groups.Any finitely generated group can be written as a directed colimit of finitelypresented groups. Hence by Theorem 11.21 (iv) the Full Farrell-Jones Con-jecture 11.20 holds for all finitely generated groups if and only if holds forall finitely presented groups. Now the claim follow from Theorem 14.1 (iia)since every finitely presented group is a subgroup of U .

14.5. If Q belongs to FJ , we conclude from Theorem 14.1 (iic) that Gbelongs to FJ . Suppose that G belongs to FJ . For every k ∈ K with k 6= 1we can find by assumption a normal subgroup G(k) of G of finite index suchthat k does not belong to G(k). Let H ⊆ G be the intersection of the finitelymany subgroups G(k) for k ∈ K, k 6= 1. Since G is finitely generated, H isa normal subgroup of finite index in G with H ∩ K = 1. The projectionp : G→ H induces an isomorphism H → p(H). Hence p(H),H and G belongto FJ by Theorem 14.1 (iia) and (iif).

14.7. This follows directly from Theorem 14.6 (iic).

14.8. This follows from Theorem 1.68, Theorem 11.52, Lemma 8.51, Lemma 8.53,Theorem 14.1, and Theorem 14.6.

14.15. By Lemma 13.23 (ii) it suffices to prove the injectivity for any finitelygenerated subgroup of G since G is the directed union of its finitely generatedsubgroups. Now use the fact that any finitely generated group is countable.

14.21. The projection pr : G → H1(G) induces an isomorphisms on thegroup homology Hn(G) → H1(G/[G,G]) for all n ∈ Z and G/[G,G] =H1(G) ∼= Z. This follows from Alexander-Lefschetz duality. The Atiyah-Hirzebruch spectral sequence implies that Hn(pr; K(R)) : Hn(G; K(R)) →Hn(G/[G,G]; K(R)) is bijective for all n ∈ Z. Since G satisfies the FullFarrell-Jones Conjecture 11.20 by Theorem 14.1 (ie). and hence the K-theoretic Farrell-Jones Conjecture for torsionfree groups and regular rings 5.41by Theorem 11.52 i, the map Kn(RG)→ Kn(R[G/[G,G]]) induced by pr is abijection. Since G/[G,G] ∼= Z, we get Kn(R[G/[G,G]]) ∼= Kn(R)⊕Kn−1(R)from the Bass-Heller-Swan decomposition for algebraic K-theory, see Theo-rem 5.16.

14.24. Show by induction over i = 1, 2, . . . , d that Gi is torsionfree andsatisfies the Baum-Connes Conjecture 12.11 with coefficients using Theo-rem 14.6 (ie) and (iic).

14.27. We want to show that the Full Farrell-Jones Conjecture 11.20 holdsfor Γ s+1

g,r if it does for Γ sg,r by applying Theorem 14.1 (iic) to the exact


sequence (14.25). Since for every non-trivial cyclic subgroup C ⊆ Γ sg,r we canfind a diffeomorphism f : Ssg,r → Ssg,r whose image under p generates C andfor which p−1(C) is isomorphic to the mapping torus of f , this follows fromTheorem 14.1 (ie).

Analogously we prove using the exact sequence (14.26) and Theorem 14.1 (iic)and (ic) that for s ≥ 1 the group Γ s−1

g,r+1 satisfies the Full Farrell-Jones Con-jecture 11.20 if Γ sg,r does. Now one use induction over r, s.

14.35. We only treat the K-theory case, the argument for L-theory is com-pletely analogous. Let G be a group with a finite model for BG and let Rbe a regular ring. Choose M , i and r as they appear in Theorem 14.34. Weobtain a commutative diagram

Hn(BG; KR) //

Hn(i;K(R))

Kn(RG)

i∗

Hn(M ; KR) //

Hn(p;K(R))

Kn(R[π1(M)])

p∗

Hn(BG; KR) // Kn(RG)

The right vertical arrows are assembly maps. Since R is regular and bothG and π1(M) are torsionfree, G and π1(M) respectively satisfy Farrell-JonesConjecture 11.1 if and only if the upper and the middle respectively horizontalarrow is bijective for all n ∈ Z. The composite of the two vertical arrows of theleft column and the right column are the identity. Since the middle horizontalarrow is bijective, the same is true for the upper horizontal arrow.

14.39. We conclude from Theorem 2.112 that for a natural number n thevanishing of Q ⊗Z Wh(Z/n) = 0 implies n = 1, 2, 3, 4, 6. Now apply Theo-rem 14.38.

Chapter 15

15.2. This follows from Theorem 10.70 and Theorem 11.26.

15.4. This follows from Theorem 3.21 (v) and Example 15.3.

15.6. For every finite cyclic subgroup C ⊆ G we have CGC =⊕

Z F andhence Hp(CGC;Q) = 0 for p 6= 0 and H0(CGC;Q) ∼= Q. Hence we get fromTheorem 15.5 for all n ∈ Z an isomorphism⊕

(C)∈J

Q⊗Q[WGC] ΘC ·(Q⊗Z L

〈j〉n (ZC)

) ∼=−→ Q⊗Z L〈j〉n (ZG).


Now apply Theorem 7.179 (iv).

15.12. This follows from Theorem 9.36.

15.14. This follows from Theorem 2.112, Theorem 2.113 (iv) and Theo-rem 15.13 (ii).

15.18. This follows from the Rothenberg sequence 7.101, the Shaneson split-ting, see Theorem 7.104, and Example 15.17.

15.22. The Z/3-action given by φ on Z2 is free outside the origin. Now applyTheorem 15.20 (iii) together with (15.21).

Chapter 16

16.10. Suppose that E is weakly F-excisive. Theorem 16.9 (ii) and (iv)imply that the assignment sending (X,A) to coker

(πn(∅+) → πn(E(X/A))

)is a G-homology theory.

Now suppose that the assignment sending (X,A) to coker(πn(E(∅+)) →

πn(E(X/A)))

is a G-homology theory. Then we get from Theorem 16.9 (ii)

and (iv) and from Lemma 10.6 applied to E% → E that E is weakly F-excisive.

Chapter 17

17.5. We define an isomorphism F :∫GA∼=−→ CG(G;A) of additive categories

as follows. Recall that∫GA and A have the same objects. Define for an object

A in∫GA its image under F to be the object F (A) = Ag | g ∈ G, where

Ag is just a copy of A. Recall that a morphism from A to B in∫GA is a

formal sum∑g∈g g ·φg, where φg : A→ B · g is a morphism in A and φg 6= 0

only for finitely many g ∈ G. Define F (A) by the collection of morphismsφg1,g2 := φg−1

1 g2· g−1

1 for (g1, g2) ∈ G × G. The inverse sends an object

Ag | g ∈ G to Ae and a morphism φg1,g2 : Ag → Bg to∑g∈G g ·φe,g,

where we view φe,g : Ae → Bg as a morphism Ae → Be using the identityBg = Be.

17.13. The only non-trivial part of the proof is to show that K is indeed anabelian subgroup. Given morphisms φ1 and φ2 from A to B and factorizationsφi : A → L(φi) → B for i = 1, 2, we define L(φ1 − φ2) = L(φ1)⊕ L(φ2) andobtain the desired factorization of φ1 − φ2 by

Aj1+i2−−−→ A⊕A ψ(φ1)⊕ψ(φ2)−−−−−−−−→ L(φ1)⊕L(φ2)

ψ′(φ1)⊕ψ′(φ2)−−−−−−−−−→ B⊕B idB ⊕(− idB)−−−−−−−−→ B


for ji : A→ A⊕A the inclusion onto the i-th summand.

17.18. Apply Lemma 17.16 in the case S = G/G, identify

CG(G,FG×SGc , p−1S E

S∆;A) = T G(•;A),

and apply Lemma 17.15.

Chapter 18

Chapter 19

Chapter 20

Chapter 21


name of texfile: ic

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Notation

An, 21A+, 247A(G), 320A(X), 154A(X), 154Aom G, 244Aor G, 244Arf(P,ψ), 169Aor G, 244Aom G, 244asmbG,CA (X)n, 368

asmbG,RA (X)n, 368

asmbG,C,L1

(EG)n, 373

asmbG,R,L1

(EG)n, 373

asmbG,C,mA (X)n, 372

asmbG,R,mA (X)n, 372BG, 52BG, 326

b(2)n (M), 41C, 21CAT , 153CATCOFWEQ, 153cd(G), 286cdR(G), 286cdR(M), 285Cl(Λ), 49class(G,R), 44class(G,R)f , 44classF (G), 45

colimi∈I Ai, 244colimC F , 40colimi∈I Ri, 54con(G), 44con(G)f , 44conF (G)f , 45cone∗(f∗), 65cone(X), 305cone(A), 250cyl∗(f∗), 65

COr(G)∗ (X), 299

C(X), 30C(X,F ), 30C0(X), 243C0(X,F ), 243Cb(X,F ), 244C∗m(G), 244C∗m(G,F ), 244C∗r (G), 52, 244C∗r (G,F ), 244C∗(X), 375Cn−∗, 176C∗-ALGEBRAS, 310conhomG(H,K), 39Dn, 21Dn, 32D∞, 21D(R, I), 90D∗(X), 375D(ZG), 49

595

596 Notation

Diff(M), 122e(P ), 160EG, 241EG, 274EF (G), 274En(i, j), 59Fpm , 89G, 211(g), 44G-CW-COMPLEXES, 480G0(R), 24G1(R), 58G oQ, 101GL(A), 248GL+

n (A), 247GL+(A), 248GL+(A)0, 248GL(R), 27GL(R, I), 92GLn(R), 27G×α X, 302G o F , 340GROUPOIDS, 307GROUPOIDSinj, 307GROUPSinj, 311gd(G), 288gd(G), 288

H(M), 158HC⊗k∗ (R), 146HH⊗k∗ (R), 145HN⊗k∗ (R), 146HP⊗k∗ (R), 146HSRG, 45HSRG(P ), 44H∗(X,R), 52HG∗ (−; E), 307

H?∗(−; E), 308

HGn (X;M), 299

Hε(P ), 161Hε(P ), 166hdimG(X), 288hofib(f), 126HE, 415HK, 416Idem(R), 27

Idemn(R), 27indαX, 302inn(K), 39In, 59i∗A, 260i∗, 260jGm, 260jGr , 260K(A, i), 213K(R), 127K(C), 153K(P), 134K(A), 135

A(M), 383

Ax(M,u), 383K(P), 136KH(R), 143K∗(A), 246K0(R), 24

K0(R), 25K0(X), 31K1(R), 57

Kf1 (R), 59

K1(R), 64K2(R), 118Kn(R), 127Kn(R;Z/k), 132Kn(A), 135Kn(P), 134, 136K−n(R), 105KHn(R), 144

Kk(M), 182

Kk(M), 182K∗(X,A), 233K∗(X,A), 234KGn (X,A;B), 261

KK∗(A,B), 256KKG

∗ (A,B), 259KO∗(A), 247K∗(A), 258KO∗(X;A), 236KO∗(X;A), 236KOG∗ (X,A;B), 262KR∗(X;A), 237

Notation 597

KtopC , 234

KtopR , 236

Ln(R), 167Ln(R), 167LUn (R), 199,199Lhn(R), 201Lhn(Zπ,w), 200

L〈j〉n (R), 202

Lpn(R), 201Lsn(Zπ,w), 200

L〈0〉n (R), 201

L〈1〉n (R), 201

L〈2〉n (R), 201

l(H), 290L(V ), 78L(t; k1, . . . , kc), 78L⊥, 168L1(G), 244L1(G,F ), 244L2(G), 52, 244L2(G,F ), 244L〈−∞〉(R), 204mapG(−, Y ), 306M(R), 27Mn(R), 27Mm,n(R), 27M∗, 160[M ], 163NG, 93NGH, 269Nil0(R), 82Nil(RG0;RG1, RG2), 141

Nil0(R), 82NKn(R), 81, 129NK−n(R), 105N±Kn(RK,φ), 143nrQG, 100nrCG, 100nrZG, 100O, 211o(X), 33o(X), 33ogeo(X), 35Or(G), 39

OrF (G), 39•, 139Pd(G,S), 278Pd(X), 376pk(M,Q), 214PC0(G), 276P (M), 149PDiff(M), 122, 149PL, 210P(M), 150PDiff(M), 150Proj(R), 25(P, φ), 160(P,ψ), 165Q, 21Qε(P ), 165Qε(P ), 165R, 21(r), 24rad(R), 54rA, 27rF (G), 100RF (H), 45R×, 54RKn(X,A;B), 261SK1(R), 62SK1(ZG), 100RINGS, 310RINGSinv, 310sign(M), 163sign(s), 162signx(M,u), 214Sn, 21S+, 91Sn, 31Sn+, 68SpG, 101SDn, 21SL(R), 102SL(R, I), 93SO, 103SO(n), 103SPACES+, 305SPECTRA, 305Spec(R), 54

598 Notation

St(R), 118Stn(R), 117SU , 103SU(n), 103Sub(G), 39SubF (G), 39supp(A), 489supp(φ), 489swG, 94swG, 97TOP , 210T (P ), 165U(A), 248Un(A), 248U+(A), 248vcd(G), 288VectF (X), 31Wa(Y ), 34WGH, 269Wh(G), 64WhR0 (G), 64WhR1 (G), 64Wh2(G), 122WhDiff(X), 157WhPL(X), 156Xn, 268X+, 305X ∧E, 306X ∧ Y , 305Z, 21Zw, 176Z/n, 21Zp, 234α∗, 260Γg, 453Γ sg,r, 453∆, 153∆k, 150ιF⊆G , 344λE ∈ K+(DE,SE), 240φ∗F , 389π0(X), 32π1(X), 32πi(E), 305η(M), 384

η(2)(M) ∈ R, 384ρ(X;U), 77ρ(2)(M), 384ρG(X), 98ΣA, 249Σn(A), 250ΣC∗, 65σ(f, f), 196,200Swp(G;Λ), 321τ(f), 66τU (f), 197τ(f∗), 66τgeo(f), 72τ (2)(M) ∈ R, 384ΘG, 322θC , 323ΩGn (X,A), 304ALL, 39Aφ[t], 415B, 243BH?∗, 317

B(l2(G)), 52C(G), 365C(Y ;A), 490C(Y ;A), 490CG(Y ;A), 491

CG(Y ;A), 491CG(Y, E ,F ;A), 492CG(Y × [1,∞), E ,F ;A)∞, 493COM, 267CYC, 39FCY, 39DG(Y ;A), 499EYall, 494Edis, 498EZd , 494EY∆ , 494Ep, 354F , 39F|K , 389FYGc, 494FIN , 39FJ , 431H(G), 418HE , 347

Notation 599

HEI , 348HEp, 347GG(S), 307HGn , 298H?n, 302K, 243K(M), 158L(M), 214M, 49N (G), 52Nn(X, ∂X), 208NPLn (X), 210N TOPn (X), 210OG(Y ;A), 499P(M), 149PDiff(M), 149R(X), 152Rf (X), 152Shn(X), 205Ssn(X), 205SPL,hn (X), 210SPL,sn (X), 210STOP,hn (X), 210STOP,sn (X), 210T G(Y ;A), 494T R, 39VCY, 39VCYI , 347Z(N (G)), 53[n], 153u, v, 121∂(P,ψ), 190〈u, v〉, 163


name of texfile: ic

Index

A-class, 383

A-genus

higher, 383

A-regular, 446

A-theory

connective A-theory, 154

acyclic

U -acyclic, 77

map, 126

space, 125

additive category, 133

AF-algebra, 250

Alexander trick, 75

algebra

Roe algebra, 375

algebraic number field, 29

amenable group action, 440

annihilator of a subLagrangian, 168

Arf invariant, 169

Artinian

module, 41

ring, 41

aspherical, 218

assembly

analytic Baum-Connes assembly map,368

assembly map for algebraic K-theory,

334

assembly map for algebraic L-theory,

335

for G-homology theories, 388

homotopy theoretic assembly transfor-mation, 484

asymptotic dimension, 440

balanced product

of C-spaces, 306

of a pointed C-space with a C-spectrum,

307

balanced smash product

of pointed C-spaces, 306

Banach algebra, 242

Banach ∗-algebra, 242

unital, 243

basis

cellular, 66

stable basis, 197

stably U -equivalent, 197

U -equivalent, 197

bordism

normal, 178

Bott isomorphism, 32

Bott manifold, 382

Bott periodicity, 233

boundary

of a non-degenerate ε-quadratic form,

190

bounded geometry, 376

bounded map, 112

Bredon homology, 298

Burnside ring, 320

C∗-algebra, 242

AF-algebra, 250

contractible, 250

Cuntz C∗-algebra, 250

Kirchberg C∗-algebra, 250

maximal complex group C∗-algebra, 244

nuclear, 245

proper, 511

reduced complex group C∗-algebra, 244

separable, 245

stable, 245

unital, 243

601

602 Index

C∗-identity, 242

Calkin algebra, 243

category

additive, 133

additive category of geometric objects in

an additive category A over a space

Y , 490

category of cofibrations, 151

category of weak equivalences, 151

category with cofibrations and weak

equivalences, 151

exact, 134

flasque additive category, 135

of compactly generated spaces, 267

pointed category, 151

small, 40, 133

cell

equivariant closed n-dimensional cell,

269

equivariant open n-dimensional cell, 269

cellular

basis, 66

map, 269

pushout, 37

center

of a group, 118

central extension, 118

universal, 118

chain complex

contractible, 65

dual, 176

finite, 33

finite based free, 65

finitely generated, 33

free, 33

positive, 33

projective, 33

chain contraction, 65

change of rings homomorphism, 24

Chern character

equivariant Chern character for complextopological K-cohomology, 239

for complex topological K-cohomology,234

for complex topological K-homology, 234

for real topological KO-cohomology, 236

for real topological KO-homology, 236

class group of a Dedekind domain, 28

classifying G-CW -complex for a family ofsubgroups of G, 274

classifying space

for a family of subgroups of G, 274

of a group, 52

closed under

directed unions, 449

extensions, 137, 449

coarse structure, 491

cobordism, 73

bounded, 112

diffeomorphic relative M0, 74

h-cobordism, 74

over M0, 73

trivial, 74

cocompact, 304

cohomological dimension

for groups, 286

for modules, 285

virtual, 288

cohomology

complex topological K-cohomology, 233

real topological KO-cohomology, 236

colimit

of a functor to abelian groups, 40

of a functor to rings, 40

of C∗-algebras, 244

of rings, 54

topology, 268

compact

operator, 243

compactly generated space, 268

cone

of a C∗-algebra, 250

reduced cone, 305

Congruence Subgroup Problem, 93

Conjecture

Aspherical closed manifolds carry no

Riemannian metric with positive

scalar curvature, 384

Atiyah Conjecture, 41

Bass Conjecture for fields of char-

acteristic zero as coefficients,

46

Bass Conjecture for integral domains as

coefficients, 47

Baum-Connes Conjecture, 370

Baum-Connes Conjecture for torsionfreegroups, 251

Baum-Connes Conjecture withcoefficients, 371

Borel Conjecture, 220

Borel Conjecture for a group G indimension n, 220

Bost Conjecture, 373

Comparing algebraic and topological

K-theory with coefficients forC∗-algebras, 263

Index 603

Comparison of algebraic K-theory and

homotopy K-theory for torsionfreegroups, 145

Farrell-Jones Conjecture for Wh2(G) for

torsionfree G, 123

Farrell-Jones Conjecture for K0(ZG)

and Wh(G)] for torsionfree G, 99

Farrell-Jones Conjecture for A-theory,

412

Farrell-Jones Conjecture for A-theorywith coefficients, 412

Farrell-Jones Conjecture for K0(R) fortorsionfree G and regular R, 38

Farrell-Jones Conjecture for K0(RG)

and K1(RG) for regular R and

torsionfree G, 99

Farrell-Jones Conjecture for K0(RG) forregular R with Q ⊆ R, 40

Farrell-Jones Conjecture for K0(RG) fora Artinian ring R, 41

Farrell-Jones Conjecture for (smooth)

pseudoisotopy, 412

Farrell-Jones Conjecture for (smooth)

pseudoisotopy with coefficients, 413

Farrell-Jones Conjecture for homotopy

K-theory with coefficients in additiveG-categories, 417

Farrell-Jones Conjecture for negative

K-theory of the ring of integers in an

algebraic number field, 113

Farrell-Jones Conjecture for negativeK-theory and Artinian rings as

coefficient rings, 114

Farrell-Jones Conjecture for negative

K-theory and regular coefficientrings, 113

Farrell-Jones Conjecture for torsionfreegroups and regular rings for K-theory,

138

Farrell-Jones Conjecture for torsionfree

groups for L-theory, 204

Farrell-Jones Conjecture for torsionfreegroups for homotopy K-theory, 145

Fibered Meta Isomorphisms Conjecturefor a functor from spaces to spectra

with coefficients, 409

Fibered Meta-Isomorphism Conjecture,

390

Fibered Meta-Isomorphism Conjecturewith finite wreath products, 428

Finite Models for EG, 294

Full Farrell-Jones Conjecture, 341

Gromov-Lawson-Rosenberg Conjecture

Homological Gromov-Lawson-

Rosenberg Conjecture, 383

Stable Gromov-Lawson-RosenbergConjecture, 382

Homotopy Invariance of the L2-Rho-

Invariant for Torsionfree Groups,

384

Integral Novikov Conjecture, 375

K-theoretic Farrell-Jones Conjecturewith coefficients in rings, 335

K-theoretic Farrell-Jones Conjec-

ture with coefficients in additive

G-categories with finite wreathproducts, 340

K-theoretic Farrell-Jones Conjec-

ture with coefficients in additive

G-categories, 337

K-theoretic Farrell-Jones Conjecturewith coefficients in the ring R, 334

K-theoretic Novikov Conjecture, 357

K-theory versus homotopy K-theory for

regular rings, 428

Kadison Conjecture, 254

Kaplansky Conjecture, 42

Kaplansky Conjecture for prime

characteristic, 42

L-theoretic Farrell-Jones Conjec-ture with coefficients in additive

G-categories with involution, 339

L-theoretic Farrell-Jones Conjecture

with coefficients in additive G-categories with involution with finite

wreath products, 340


with coefficients in rings withinvolution, 335


with coefficients in the ring with

involution R, 335

L-theoretic Novikov Conjecture, 357

Manifold structures on asphericalPoincare complexes, 226

Meta-Isomorphism Conjecture, 388

Meta-Isomorphism Conjecture for

functors from spaces to spectra, 401

Meta-Isomorphism Conjecture for

functors from spaces to spectra withcoefficients, 402

Nil-groups for regular rings and

torsionfree groups, 140

Novikov Conjecture, 214

Passage for L-theory from QG to RG toC∗r (G;R), 423

Poincare Conjecture, 75

604 Index

Strong Bass Conjecture, 47

Strong Novikov Conjecture, 374

The Farrell-Jones Conjecture for the al-

gebraic K-theory of Hecke-Algebras,418

Trace Conjecture for torsionfree groups,254

Trace Conjecture, modified, 381

Weak Bass Conjecture, 47

Zero-in-the-spectrum Conjecture, 255

conjugated

F -conjugated, 45

control condition

equivariant continuous control condition,

497

crossed product

maximal crossed product of C∗-algebras,

244

reduced crossed product of C∗-algebras,

244

crystallographic group of dimension n, 473

Cuntz C∗-algebras, 250

CW -complex

finite, 32

finitely dominated, 32

cyclic homology, 146

cyclotomic trace, 146

Dedekind ring, 28

degree one map, 177

δ-hyperbolic

group, 278

metric space, 278

Dennis trace map, 146

descent homomorphisms

for KK-theory, 260

dimension

minimal homotopy dimension, 288

Dirac element, 511

dual, 511

Dirac-dual Dirac method, 511

direct limit

of a functor to R-modules, 40

of a functor to abelian groups, 40


of groups, 54

of modules, 54

of rings, 54

directed set, 53

directed system


of groups, 54

of modules, 54

of rings, 54

directly finite

module, 42

ring, 42

disk bundle, 240

double of a ring along an ideal, 90

dual chain complex, 176

Eilenberg-MacLane space, 213

elementary (n, n)-matrix, 59

elementary collapse, 69

elementary expansion, 69

equivalence of categories, 308

equivariant

Borel homology, 304

closed n-dimensional cell, 269

continuous control condition, 497

equivariant homology theory, 302

equivariant homology theory over a

group, 330

open n-dimensional cell, 269

smooth triangulation, 271

eta-invariant, 384

Euler class, 240

exact

category, 134

functor, 134

exotic homotopy sphere, 224

exotic sphere, 75

extension

central, 118

closed under extensions, 137

external multiplicative structure

for equivariant topological K-theory, 238

family of subgroups, 39

finite asymptotic dimension, 440

finite decomposition complexity, 441

finite domination, 32

finite propagation operator, 375

finite quotient finite decomposition

complexity, 441

finite resolution, 137

finiteness obstruction, 33

geometric, 35

unreduced, 33

flasque additive category, 135

form

ε-quadratic, 165

ε-symmetric, 160

non-degenerate ε-quadratic, 166

non-degenerate ε-symmetric form, 161

standard hyperbolic ε-quadratic form,

166

Index 605

standard hyperbolic ε-symmetric form,

161

U -based ε-quadratic forms, 198

formation

ε-quadratic formation, 189

stably isomorphic ε-quadratic forma-

tions, 189

trivial ε-quadratic formation, 189

fractional ideal, 28

Franz’ independence Lemma, 79

Fredholm operator, 235

free

free unitary representation, 77

functor

exact functor, 134

of additive categories, 133

functorial additive invariant for finitelydominated CW -complexes, 37

functorial additive invariant for finitelydominated CW -complexes

universal, 37

fundamental class, 163, 177

Fundamental Lemma of HomologicalAlgebra, 87

G-CW -complex, 268

finite, 270

finite dimensional, 271

n-dimension, 271

of dimension ≤ n, 271

of dimensional n, 271

of finite type, 270

G-homology theory, 298

generalized Swan homomorphism, 97

germs at infinite sequence, 500

germs at infinity quotient category, 493

Green functor, 319

Grothendieck group, 24

group

a-T-menable, 449

alternating group of even permutationsof the set 1, 2, . . . n, 21

amenable, 448

CAT(0)-group, 445

crystallographic of dimension n, 473

elementary amenable, 448

finite cyclic group of order n, 21

finite dihedral group of order n, 21

finite dimensional CAT(0)-group, 445

having property (RD), 437

having property (T), 449

having the Haagerup property, 449

hyperbolic, 278, 444

hyperelementary, 347

infinite cyclic group, 21

infinite dihedral group, 21

infinite special orthogonal group, 103

infinite special unitary group, 103

K-amenable, 373

lacunary hyperbolic, 444

p-elementary, 354

p-hyperelementary, 347

perfect, 118

Poincare duality group of dimension n,

224

poly-(P), 85

poly-cyclic, 85

poly-free, 452

poly-Z, 85

quasidihedral group of order n, 21

regular, 141

regular coherent, 141

semidihedral group of order n, 21

sofic, 42

strongly poly-free, 451

symmetric group of permutations of the

set 1, 2, . . . n, 21

Thompson’s groups, 454

virtually (P), 85

virtually finitely generated abelian, 85

virtually free, 85

virtually nilpotent, 85

virtually poly-cyclic, 364

virtualy poly-cyclic, 85

Gysin sequence, 240

h-cobordism

bounded, 112

Hattori-Stallings homomorphism, 45

Hattori-Stallings rank, 44

Hecke algebra, 418

Heisenberg group

three-dimensional, 251

three-dimensional discrete, 251

higher signature

associated to a homology class, 214

Hochschild homology, 145

homology

Borel homology, 304

Bredon homology, 298

cellular, 52

complex topological K-homology, 234

continuous equivariant homology theory,

397

cyclic homology, 146


equivariant homology theory over a

group, 330

606 Index

G-homology theory, 298

Hochschild homology, 145

negative cyclic homology, 146

periodic cyclic homology, 146

proper equivariant homology theory, 302

proper equivariant homology theory over

a group, 330

proper G-homology theory, 298

real topological KO-homology, 236

singular, 52

strongly continuous equivariant

homology theory, 397

topological cyclic homology, 146

homomorphism of C∗-algebras, 242

homotopic, 243

homotopy

pointed regular homotopy, 171

homotopy algebraic K-theory, 143

homotopy fiber, 126

homotopy K-theory functor, 416

homotopy representation

free d-dimensional, 95

homotopy sequence

long exact homotopy sequence of a map,126

homotopy sphere, 224

exotic, 224

homotopy stabilization of a functor from

additive categories to spectra, 415

hyperbolic

group, 278

metric space, 278

ideal

fractional, 28

principal fractional ideal, 28

principal ideal, 24

ideal in a C∗-algebra, 243

immersion

pointed, 171

index pairing, 258

induction

for equivariant KK-theory, 260

for equivariant spaces, 302



of modules with respect to ringhomomorphisms, 24

induction structure, 302

inductive limit of C∗-algebras, 244

integral domain, 24

internal multiplicative structure


intersection pairing

for immersions, 171

for kernels, 184

involution

of rings, 160

w-twisted involution of a group ring, 336

isotopic, 122

Jacobson radical, 54

K-theory mod k, 132

K-theory space

of a category with cofibrations and weak

equivalences, 153

of a ring, 127

of an exact category, 134

K-theory spectrum

complex topological K-theory, 234

non-connective K-theory spectrum of aring, 135

non-connective K-theory spectrum of anadditive category, 135

over groupoids, 311

real topological K-theory, 236

K0-group of a ring, 24



Kn-group of a ring for negative n, 105

Kn-group of a ring for positive n, 127

Kasparov’s intersection product, 258

equivariant, 259

Kirchberg C∗-algebras, 250

KK -equivalence, 258

KK -theory of Kasparov, 256

equivariant, 259

KO-orientation of Spin bordism, 236

Kronecker product, 163

L-class, 214

L-group

decorated quadratic L-groups in evendimensions, 198

decorated quadratic L-groups in odd

dimensions, 199

projective quadratic L-groups, 201

quadratic L-groups in even dimensions,

167

quadratic L-groups in odd dimensions,

191

simple quadratic L-groups, 200

symmetric L-groups in even dimensions,167

L-theory spectrum

associated to a ring with decoration

〈−∞〉, 204

Index 607

over groupoids, 311

L2-Betti number, 41

L2-eta-invariant, 384

L2-Rho-invariant, 384

Lagrangian, 168

complementary, 190

subLagrangian, 168

lattice, 447

length function, 437

having property (RD), 437

length of a finite group, 290

lens space, 78

linearization map, 155

locally compact

operator, 375

locally finite

set, 489

simplicial complex, 490

locally free class group of a Z-order, 49

Mackey functor, 316

manifold parametrized over Rk, 112

map

acyclic, 126

cellular, 269

linearization map, 155

of degree one, 177

mapping class group, 280, 453

mapping cone

of a chain map, 65

of a map of spaces, 70

mapping cylinder

of a chain map, 65

of a map of spaces, 69

marked metric graph, 279

maximal group C∗-algebra, 244

metric space

hyperbolic, 278

proper, 375

uniformly contractible, 376

with bounded geometry, 376

metrically proper, 449

minimal homotopy dimension, 288

module

Artinian, 41

irreducible, 41

simple, 41

stably finitely generated free, 26

stably U -based, 197

U -based, 197

Morita equivalence, 26

Mostow rigidity, 221

Nakayama’s Lemma, 54

negative cyclic homology, 146

Nil group

reduced zero-th, 82

zero-th, 82

Nil-spectrum

non-connective, 141

nilpotent

endomorphism of a module, 82

norm element of a finite group, 93

normal bordism, 178, 196

normal map, 178

of degree one of manifolds withboundary, 195

of degree one, 178

stabilization of a normal map, 178

normalizer, 269

normally bordant, 179

object of finite length, 137

object of length n, 137

operator

compact, 243

finite propagation operator, 375

locally compact, 375

pseudolocal, 375

orbit category, 39

F-restricted, 39

orientation, 177

of a manifold, 163

orientation homomorphism

of a Poincare complex, 177

outer space, 279

spine of outer space, 279

perfect group, 118

perfect radical, 127

periodic cyclic homology, 146

Pimsner-Voiculescu sequences forKK-theory, 259

plus-construction, 126

Poincare duality group of dimension n, 224

Poincare complex

finite n-dimensional, 177

Poincare pair

finite n-dimensional, 195

simple finite n-dimensional, 195

Poincare Zπ-chain homotopy equivalence,177

pointed category, 151

pointed regular homotopy, 171

Pontrjagin class

rational, 214

principal fractional ideal, 28

principal ideal domain, 24

608 Index

principle

separation of variables, 467

Problem

Brown’s problem aboutvcd(G) = dim(EG), 290

Congruence Subgroup Problem, 93

Identification of analytic surgery exactsequences, 426

Identification of transformations fromthe surgery exact sequence to its

analytic counterpart, 426

Relating the dimension of EG and EG,

291

spherical space form problem, 32

projective class group, 24

reduced, 25

projective resolution, 38

d-dimensional, 38

finite, 38

finite dimensional, 38

finitely generated, 38

free, 38

proper


G-C∗-algebra, 511

G-space, 271

metric space, 439

proper equivariant homology theory, 330

proper G-homology theory, 298

proper metric space, 375

pseudoisotopic, 123

pseudoisotopy, 122, 149

pseudoisotopy spectrum, 150

smooth pseudoisotopy spectrum, 150

pseudolocal

operator, 375

pushout

cellular, 37

Q-construction, 134

quadratic refinement, 166

quasi-metric, 494

Question

Isomorphism Conjecture for topologicalcyclic homology, 415

Reducing VCY to VCYI for A-theory,

413

Splitting of the relative assembly maps

for FIN ⊆ VCY for A-theory, 413

Vanishing of the resolution obstruction

in the aspherical case, 227

radical, 54

perfect, 127

reduced group C∗-algebra, 244

reduced K1-group of a ring, 64

refinement

quadratic, 166

regular coherent

regular coherent group, 141

regular coherent ring, 141

Reidemeister torsion, 77

resolution

finite, 137

restriction

for equivariant KK-theory, 259


ring

Artinian, 41

Burnside ring, 320

Dedekind ring, 28

hereditary, 28

integral domain, 24

local, 54

Noetherian, 38

obtained by adjoining a unit, 91

of integers, 29

principal ideal domain, 24

regular, 38

regular coherent, 141

semihereditary, 86

semilocal, 55

semisimple, 51

Swan ring, 321

with involution, 160

Rips complex

of a group, 278

of a metric space, 376

Roe algebra, 375

Rothenberg sequence, 202

selfintersection element, 173

semisimple

object, 137

ring, 51

sequence

germ at infinite sequence, 500

set of normal maps to a compact manifold,

208

signature

higher, 214

homotopy invariance of, 214

higher signature associated to ahomology class, 214

of a closed oriented manifold, 163

of a non-degenerate symmetric bilinear

pairing, 162

simple

Index 609

object, 137

simple homotopy equivalence, 69

simple object, 137

simplicial complex

locally finite, 490

six-term exact sequence of an ideal, 248

skeleton, 268

skeleton of a category, 133

small category, 133

smash product, 305

of a space with a spectrum, 306

smooth pseudoisotopy spectrum, 150

sofic group, 42

space

acyclic, 125

C-space, 306

K-theory space of a category with

cofibrations and weak equivalences,153

K-theory space of a ring, 127

K-theory space of an exact category, 134

of parametrized h-cobordisms, 158

of stable parametrized h-cobordisms, 158

spectrum, 305

Ω-spectrum, 306

complex topological K-theory spectrum,234

homotopy groups of a spectrum, 305

map of spectra, 305

non-connective K-theory spectrum of an

additive category, 135

non-connective K-theory spectrum of a

ring, 135

of a commutative ring, 54

real topological K-theory spectrum, 236

structure maps of a spectrum, 305

weak equivalence of spectra, 306

sphere, 240

spherical space form problem, 32

spine of outer space, 279

stably finite ring, 42

standard k-simplex, 150

Steinberg group, 118

n-th Steinberg group, 117

Steinberg symbol, 121

structure set, 205

simple, 205

subgroup category, 39

F-restricted, 39

support

of a morphism, 489

of an object, 489

surgery kernel, 182

surgery obstruction

even-dimensional, 186

for manifolds with boundary, 196

for manifolds with boundary and simple

homotopy equivalences, 200

in odd dimensions, 194

surgery program, 76

surgery sequence, 210

for the PL-category, 211

for the topological category, 211

suspension, 65

of a C∗-algebra, 249

Swan homomorphism, 94

generalized, 97

Swan ring, 321

Teichmuller space, 280

tensor product of C∗-algebras

maximal, 245

minimal, 245

spatial, 245

Theorem

α-Approximation Theorem, 506

A criterion for 1-dimensional models forEG, 288

A-theory is a homotopy-invariant

functor, 155

Actions on CAT(0)-spaces, 277

Actions on simply connected non-

positively curved manifolds,277

Actions on trees, 277

Additivity Theorem for categories withcofibrations and weak equivalences,

158

Additivity Theorem for exact categories,136

Algebraic L-theory of ZG for finitegroups, 231

Algebraic and topological K-theory mod

k for R and C, 132

Algebraic K-theory and directed

colimits, 129

Algebraic K-theory and finite products,128

Algebraic K-theory mod k of alge-

braically closed fields, 132

Algebraic K-theory of finite fields, 130

Bokstedt-Hsiang-Madsen Theorem, 454

Basic properties of Whitehead torsion,67

Bass-Heller-Swan decomposition

for middle and lower K-theory, 105

Bass-Heller-Swan decomposition for

Wh(G× Z), 83

610 Index

Bass-Heller-Swan decomposition for K1

for regular rings, 86

Bass-Heller-Swan decomposition foralgebraic K-theory, 129

Bass-Heller-Swan decomposition for

homotopy K-theory, 416

Bass-Heller-Swan decomposition for K1,

83

Bass-Heller-Swan decomposition lowerand middle K-theory for regular

rings, 107

Bijectivity of the L-theoretic assembly

map from FIN to VCY afterinverting 2, 354

Bounded h-Cobordism Theorem, 113

Brown’s Problem has a negative answer

in general, 290

Characterization of Dedekind domains,

28

Cocompletion Theorem for the to-pological complex K-homology,

241

Completion Theorem for complex and

real K-theory, 241

Computation of the topological K-theoryof Zn o Z/m for free conjugation

action, 474

Connectivity of the linearization map,

155

Constructing G-homology theories usingspectra, 307

Constructing equivariant homology

theories over a group using spectra,

330

Constructing equivariant homologytheories using spectra, 308

Controlled h-Cobordism Theorem, 505

Devissage, 137

Diffeomorphism classification of lens

spaces, 78

Dimension of EG for elementary

amenable groups of finite Hirschlength, 291

Dimension of EG and extension, 289

Dimension of EG for a discrete subgroup

G of an almost connected Lie group,289

Dimension of EG for countable elemen-tary amenable groups of finite Hirsch

length, 289

Dimensions of EG and EG for groups

acting on CAT(0)-spaces, 292

Dirac-Dual Dirac Method, 511

Dirichlet Unit Theorem, 63

Discrete subgroups of almost connected

Lie groups, 276

Dual of the Green-Julg Theorem, 260

Equivariant cellular Approximation

theorem, 272

Equivariant Chern character for

equivariant K-homology, 261

Equivariant Whitehead Theorem, 273

Estimate on dim(EG in terms of vcd(G),

290

Eventual injectivity of the rational

K-theoretic assembly map for R = Z,

456

Every CW -complex occurs up to

homotopy as quotient of a classifyingspace for proper group actions, 294

Exact sequence of a two-sided ideal for

middle K-theory, 90

Exact sequence of two-sided ideal for

K-theory in degree ≤ 2, 119

exotic universal coverings of asphericalclosed manifolds, 220

External Kunneth Theorem for complex

K-theory, 240

Farrell-Jones Conjecture for torsionfree

hyperbolic groups for K-theory, 471

Finite groups with vanishing Wh(G) or

SK1(ZG), 101

Finite groups with vanishing K0(ZG),51

Finite-dimensional models for EG for

discrete subgroup of GLn(R), 293

Finiteness conditions for BG, 286

Fundamental Theorem of K-theory indimension 1, 84

Geometric and algebraic Whitehead

groups, 73

Grothendieck decomposition for G0 and

G1, 85

h-cobordism theorem, 75

Hilbert Basis Theorem, 85

Homotopy characterization of EF (G),274

Homotopy classification of Lens spaces,

78

Homotopy Groups of Diff(M), 230

Homotopy Groups of Top(M), 230

Homotopy Invariance of τ (2)(M), 385

Hyperbolic groups with Cech-homology

spheres as boundary, 228

Hyperbolic groups with spheres asboundary, 228

Hyperelementary Induction, 347

Index 611

Identifying the finiteness obstruction

with its geometric counterpart, 36

Immersions and Bundle Monomor-

phisms, 182

Independence of decorations, 202

Inheritance properties of the Meta-Isomorphism Conjecture for functors

from spaces to spectra with

coefficients, 404

Inheritance properties of the Baum-Connes Conjecture with coefficients,

377

Inheritance properties of the Farrell-Jones Conjecture, 341

Inheritance properties of the Farrell-

Jones Conjecture for homotopyK-theory, 417

Injectivity of the Farrell-Jones assembly

map for FIN for linear groups, 442

Injectivity of the Farrell-Jones assemblymap for FIN for subgroups of almost

connected Lie groups, 441

K- and L-theory spectra over groupoids,

311

K-and L-groups of Fuchsian groups, 470

Kunneth Theorem for KK-theory, 259

Kunneth Theorem for finite CW -

complexes for topological complexK-cohomology, 235

Kunneth Theorem for topological

K-theory of C∗-algebras, 250

K0(R) of local rings, 54

K0(RG) for finite G and an Artinianring R, 51

K0(ZG) is finite for finite G, 48

K1(B) of a commutative Banach

algebra, 102

K1(B) of a commutative C∗-algebra

C(X), 103

K1(F ) of skewfields, 58

K1(R) equals GL(R)/[GL(R), GL(R)],61

K1(R) = R× for commutative rings with

Euclidean algorithm, 63

118

Karoubi’s Conjecture, 263

L-groups of the integers in dimension

4n, 169

L-groups of the integers in dimension4n+ 2, 170

L-theory and topological K-theory of

complex C∗-algebras, 264

Localization, 138

Localization Theorem for equivariant

topological complex K-theory, 241

Long exact sequence of a two-sided ideal

for higher algebraic K-theory, 128

Low-dimensional models for EG and

EG, 293

Lower and middle K-theory of the

integral group ring of SL3(Z), 476

Mayer-Vietoris sequence for K-theory in

degree ≤ 2 of a pullback of rings, 119

Mayer-Vietoris sequence for middleK-theory of a pullback of rings, 88

Mayer-Vietoris sequence of an amal-gamated product for algebraic

K-theory, 142

Mayer-Vietoris sequence of an amal-

gamated product for homotopyK-theory, 144

Models for EG of finite type, 288

Morita equivalence for K0(R), 26

Morita equivalence for K1(R), 59

Morita equivalence for algebraicK-theory, 128

Negative K-theory of ZG for finite

groups G, 114

130

NKn(R)[1/N ] vanishes for characteristic

N , 129

Noetherian group rings, 85

Non-triangulable aspherical closed

manifolds, 219

Operator theoretic model, 276

p-elementary induction for NKn(RG),

130

Passage from FIN to VCYI forL-theory, 353

Passage from HEI to VCY for K-theory,348

Passage from VCYI to VCY for K-theory,

347

Passage from EG to EG, 281, 284

Poincare duality groups and aspherical

compact homology ANR-manifolds,226

Product decompositions of asphericalclosed manifolds, 229

Properties of the finiteness obstruction,

33

Properties of the plus-construction, 126

Properties of the signature of closed

oriented manifolds, 163

Pseudo-isotopy is a homotopy-invariant

functor, 151

612 Index

Rational algebraic K-theory of ring of

integers of number fields, 131

Rational bijectivity of the relativeassembly map from FIN to VCY for

regular rings R as coefficients, 455

Rational computation of algebraic

L-theory, 465

Rational computation of the sourceof the assembly maps appearing in

the Farrell-Jones and Baum-Connes

Conjecture, 325

Rational computation of topologicalK-theory, 466

Rational computations of Kn(RG) for

regular R, 465

Rational injectivity of

Z ⊗ZG Wh(H) → Wh(G) fornormal finite H ⊆ G, 460

Rationally injectivity of the colimit

map for finite subgroups for the

Whitehead group, 455

Realizability of the surgery obstruction,206

Reducing the family of subgroups for the

Baum-Connes Conjecture, 379

Reductions to families contained in

FIN for algebraic K-theory, 348

Reflection group trick, 459

Regular group rings, 86

Relative Whitehead Lemma, 92

Resolution Theorem, 137

Rim’s Theorem, 49

Rips complex, 278

s-cobordism Theorem, 73

Selfintersections and embeddings, 175

Shaneson splitting, 203

Signature Theorem, 215

Simple surgery obstruction for manifolds

with boundary, 201

SK1(G) = tors(Wh(G)), 101

Slice Theorem for G-CW -complexes, 271

Split Injectivity of the assembly map

appearing in the L-theoretic FarrellJones Conjecture with coefficients in

the ring Z for fundamental groups of

complete Riemannian manifolds withnon-positive sectional curvature, 440

Split Injectivity of the assembly map

appearing in the Baum-ConnesConjecture for fundamental groups ofcomplete Riemannian manifolds withnon.-positive sectional curvature, 438

Split injectivity of the assembly map

appearing in the Baum-Connes

Conjecture with coefficient, 439

Splitting the K-theoretic assembly map

from FIN to VCY, 344

Splitting the L-theoretic assembly map

from FIN to VCY, 344

Status of the Full Farrell-JonesConjecture, 431

Status of the Baum-Conjecture with

coefficients, 434

Status of the Baum-Connes Conjecture,

437

Status of the Coarse Baum-ConnesConjecture, 439

Surgery Exact Sequence, 210

Surgery Exact Sequence for PL and

TOP, 211

Surgery Obstruction for Manifolds withBoundary, 196

Surgery obstruction in even dimensions,

187

Surgery obstruction in odd dimensions,

192

Swan’s Theorem, 30

The K-theoretic Farrell-Jones Con-jecture implies the Farrell-Jones

Conjecture for homotopy K-theory,

427

The p-chain spectral sequence, 314

The Bass Conjecture for integral

domains and elements of finite order,

48

The Baum-Connes Conjecture impliesthe Novikov Conjecture, 375

The Baum-Connes Conjecture implies

the Stable Gromov-Lawson-

Rosenberg Conjecture, 382

The Baum-Connes Conjecture withcoefficients for proper G-C∗-algebras,511

The class groups of a ring of integers is

finite, 29

The dimension of EG, 291

The equivariant Atiyah-Hirzebruch

spectral sequence, 312

The equivariant Chern character, 318

The Farrell-Jones and the BorelConjecture, 222

The Farrell-Jones Conjecture andthe Bass Conjecture for fields of

characteristic zero, 46

Index 613

The Farrell-Jones Conjecture and the

Bass Conjecture for integral domains,47

The Farrell-Jones Conjecture holds for

topological Hochschild homology, 414

The Farrell-Jones Conjecture with

coefficients in additive G-categories(with involutions) is automatically

fibered, 390

The finiteness obstruction is the uni-

versal functorial additive invariant,38

The Full Farrell-Jones Conjecture and

relatively hyperbolic groups, 445

The Full Farrell-Jones Conjecture for

Strongly Poly-surface groups, 451

The Full Farrell-Jones Conjectureimplies all other variants, 357

The generalized Swan homomorphism,

97

The geometric finiteness obstruction, 35

The middle and lower K-theory of RG

for finite G and Artinian R, 111

The reduced projective class group andthe class group of Dedekind domains,

28

The set of normal maps and G/O, G/PL

and G/TOP , 211

The stable parametrized h-cobordismTheorem, 158

The strong Novikov Conjecture implies

the Zero-in-the-spectrum Conjecture,

255

Topological K-theory of the C∗-algebraof finite groups, 265

Topological invariance of rational

Pontrjagin classes, 215

Topological invariance of Whitehead

torsion, 68

Torsion and free homotopy representa-tions, 98

Universal Coefficient Theorem for

KK-theory, 259

Universal Coefficient Theorem for

equivariant complex K-theory, 239

Universal Property of assembly, 483

Vanishing criterion of NKn(RG) forfinite groups, 130

Vanishing of NKn(RK,φ), 143

Vanishing of NKn(/IZG) for n ≤ 1 andG finite of squarefree order, 115

Vanishing of the odd-dimensional

L-groups of Z, 192

Vanishing results for D(ZG), 50

63

Waldhausen’s cartesian squares for

non-connective algebraic K-theory,141

Wang sequence associated to an HNN-

extension for algebraic K-theory,142

Wang sequence associated to an HNN-

extension for homotopy K-theory,144

Whitehead torsion and simple homotopy

equivalences, 70

Whitney’s Approximation Theorem, 178

Thom isomorphism, 240

topological cyclic homology, 146

topological K-theory

K-homology of a C∗-algebra, 258

complex topological K-cohomology ofspaces, 233

complex topological K-homology of

spaces, 234

complex topological K-theory of spaces,

233

equivariant complex topological

K-homology of spaces, 261

equivariant complex topological K-

homology of spaces with compact

support, 261

equivariant real topological K-homology

of spaces, 262


real topological K-cohomology of spaces,236

real topological K-theory of spaces, 236

real topological KO-homology of spaces,

236

twisted topological K-theory, 237

topological rigid, 220

total surgery obstruction, 212

trace

cyclotomic, 146

standard trace of C∗r (G), 253

transport groupoid, 307

tree, 277

triad

manifold triad, 195

Poincare triad, 195

type FPnfor groups, 286

for modules, 286

type FP∞for groups, 286

for modules, 286

614 Index

U -simple, 198U -torsion

of a contractible finite U -stably based

chain complex, 198of a stable isomorphism of stably

U -based modules, 197uniform embedding into Hilbert space, 438

uniformly contractible metric spaces, 376

unit disk, 77unit sphere, 77

universal additive invariant for finitely

generated projective R-modules, 24universal central extension, 118

universal determinant for automorphisms

of finitely generated projectiveR-modules, 58

universal dimension function for finitely

generated projective R-modules, 24universal functorial additive invariant for

finitely dominated CW -complexes, 37

universal G-CW -complex for properG-actions, 274

virtual

cohomological dimension, 288virtual (P) group, 85

weakweak G-homotopy equivalence, 273

homotopy equivalence, 273

weak equivalence of spectra, 306Weyl group, 269

Whitehead group, 64

generalized, 64higher, 468

second, 122Whitehead torsion

of a chain homotopy equivalence, 66

geometric, 72of a homotopy equivalence of finite

CW -complexes, 66

of an h-cobordism, 74wreath product, 101, 340

Z-order, 49zero-object, 151

Chapter 23

Comments (temporary chapter)

Comment 115: This chapter has to be taken out in the final version.

23.1 Mathematical Comments and Problems

17.08.2011 Can one reformulate the proof of the Farrell-Jones Conjecture so that onehas still an honest G-action on the CAT(0)-space, and the difficulties abouthaving only a homotopy action are absorbed in the control conditions?

24.08.15 Do we also have equivariant Chern characters in the case where we lookat equivariant homology over a given group Γ? This is not obvious sincethe compatibility with induction is only available for g in the kernel ofα : G→ Γ .

Let γ ∈ Γ be an element. Conjugation defines a group automorphismc(γ) : Γ → Γ and hence an equivalence c(γ) : GROUPOIDS ↓ Γ →GROUPOIDS ↓ Γ . One can consider for E : GROUPOIDS ↓ Γ →SPECTRA its composite c(g)∗E := E c(g). One has to investigate when

there is a natural equivalence H?∗(,−; E)

∼=−→ H?∗(,−; c(g)∗E) and investi-

gate its properties.

We also have to deal with the observation that conjugation with g induceson Ln(ZG,w) the map w(g) · id. This can be a serious problem.

24.08.2015 Can one also give an example where the K-theory assembly map for thetrivial family is not injective. We have already examples for topologicalK-theory and L-theory in Section 11.8.

24.08.2015 Shall we say something about RO(G)-gradings and Mackey-structures.The sentence at the very bottom on page 301 in [350] is interesting: Itturns out that the Bredon cohomology theory H∗G(−;M) can be extendedto an RO(G)-graded theory if and only if the coefficient system M ex-tends to a Mackey functor, where [510] is cited. We can extend the notionof Mackey structure to infinite groups, what is not so clear for RO(G)-gradings. Maybe for RO(G)-gradings one has to replace representationsby G-vector bundles. See also [236].

15.09.2017 Shall we discuss the Unit Conjecture for torsionfree groups?

19.10.2017 Sollen wir N -F-amenable diskutieren?

615

616 23 Comments (temporary chapter)

23.2 Mathematical Items or Explanations that have tobe Added

• The notion Eilenberg swindle has to be added.

• Show that in the Full Farrell-Jones Conjecture for L-theory one can replacethe family VCY by FIN if one inverts 2. One has to deal with Unil-termsof additive categories.

23.3 Layout and presentation

(i) Shall we add a guide to the introductions of each of the chapters like inthe surgery book?

(ii) We can either write

(see \cite . . . ).

or

, see \cite . . ..

This has to be decided and implemented everywhere.

23.4 Papers which we have cited but could notdownloaded and checked

[125, 179, 191, 377, 404, 406, 472, 796].

The papers by Enkelmann and Pieper and the PhD of land have to beadded.

23.5 Comments about Latex

22.12.04 In labels never let blanks or $-signs appear. Moreover, use sec: the:, pro:,lem:, def:, con: or rem: and so on to indicate whether it is a Theorem,Proposition, Lemma, Definition, Conjecture or Remark and so on. Forinstance\ labelcon:BCC torsionfree intro;

27.08.2011 Use \[ \] and not $$ $$ for display math.

27.08.2011 Do not use CD but always xymatrix for diagrams.

23.6 Comments about the Layout 617

23.6 Comments about the Layout

22.12.04 Except in the main Introduction or in environments like Definition I em-phasize a notion if and only if it appears in the index. For instance

. . . \emphWhitehead group%\indexWhitehead group$Wh(G)$%\indexnotationWh(G) . . . ;

22.12.04 In itemize or enumerate descriptions I use semicolons at the end of anitem except for the last which ends with a point. I have done this also inenumerate environments appearing in Theorems or Lemmas.;

22.12.04 After commutative squares or other diagrams do not insert a point orcomma in contrast to equations or other displayments;

22.12.04 Theorems, Conjectures, Questions and Problems are cited in the index.For instance\indexTheorem!Dirac-Dual Dirac MethodThis is not done for Definitions as new notions regardless whether theyappear in the text or in an definitions have their own entry in the index.This is also not done for Lemmas, Corollaries and Remarks;

27.12.04 I sometimes have used EG and EG for EFIN (G) andEVCY(G). There aremacros \eub# 1 and \edub# 1 to generate EG and EG;

31.12.04 Denote the complex or real representation ring by RC(G) and RR(G);31.12.04 Use always the proof environment.03.01.05 Assembly maps are denote by the letter A with possible subscripts indi-

cating the family, e.g. AFIN or AFIN→VCY . We do not write the groupG or the dimension n in connection with the assembly map. For instance,we write AFIN→VCY : KG

n (EFIN (G))→ KGn (EVCY(G));

04.01.05 Notation for matrix algebras: SLn(R), GLn(R), Mn(R), Mm,n(R);08.01.05 I use . . . except for the beginning or ends or long exact sequence, where I

use · · · . For instance

· · · ∂n+1−−−→ KG0n (EG0)→ KG1

n (EG1)⊕KG2n (EG2)→ KG

n (EG)∂n−→

KG0n−1(EG0)→ KG1

n−1(EG1)⊕KG2n−1(EG2)→ KG0

n−1(EG)→ · · ·

and i = 1, 2, . . . , n;09.01.05 Use everywhere the macro \pt for the one point space•. Then it can

changed later if this is wanted;09.01.05 I have used smooth instead of differentiable everywhere in the text except

for standard notations such as PDiff ;09.05.06 Use the new macro for exercise \exer and for hints \hintex. The exercises

appear now in the text and not in an extra section at the end of a chapter.28.05.06 The titles in chapters, sections and subsections are always capital. Title

of Theorems and Definitions begin with a capital letter but then continuewith small letters.


23.02.07 Theorems and Definitions do get a name, stated in brackets after thebegin-command.

03.06.2012 We must decide whether we take φ or ϕ and ε and ε.24.08.2015 In some chapters we use EG and EG, in others we use EFIN (G) and

EVCY(G). Check that this is at least consistent in each chapter and whetherwe have made the right choice in each chapter.

23.7 Comments about Spelling

22.12.04 I use analogue (British) instead of analog (American);27.12.04 torsionfree (One word);27.12.04 semisimple (One word);28.12.04 pseudoisotopy (One word);29.12.04 prove, proved proven;29.12.04 choose, chosed, chosen;09.01.05 We should agree on a unified use of the words any, each and every; I am

not certain about the rules;09.05.06 Write semigroup, semisimple, semidirect and so on;13.05.06 Before R and S it must be “an”, before h it depends on the pronunciation

e.g., “an h-cobordism”, but “a homomorphism”.13.05.06 Write “generalization” and “generalize”;21.05.06 Write “well-known” and “well-defined”;24.05.06 Write “simply connected” and “path connected”;24.05.06 Insert always a comma after “namely”;25.05.06 Write selfmap, selfdiffeomorphism, and so on;01.03.11 Write finite dimensional and not finite-dimensional.

15.08.2011 Write “aspherical closed manifolds”, i.e., “aspherical” before “closed”24.08.2011 Write “one-relator group”.24.08.2011 Write “hyperbolic group” and not “word-hyperbolic group”.26.08.2011 Write “handlebody” and not “handle body”.

23.8 Reminders

05.09.2013 There is a new preprint by Mole [610] with the title Extending a metricon a simplicial complex on the arXive under arXiv:1309.0981 [math.AT].It should appear somewhere in Part III.

28.10.2013 There is a submission to Bulletin of the LMS by Wu [841] with the titleElementary Amenable Groups and The Farrell-Jones Conjecture. The titleis misleading since it deals only with those of finite Hirsch length which areextentions of locally finite groups with virtually solvable groups. I cannotfind it on the arvive. Has it already appeared?

23.8 Reminders 619

02.10.2014 There is a draft of a book by Weinberger with the title Variations of atheme by Borel. When it has appeared, we shall add a reference.

23.04.2015 There is a new preprint On equivariant asymptotic dimension by DamienSawacki [733] on the arXive under arXiv:1504.04648 [math.GR]. This isrelated to transfer reducibility. It should appear somewhere in Part III.Meanwhile it has appeared.

17.05.2015 There is a new preprint by Kasprowski and Ruping [464] with the ti-tle Long and thin covers for cocompact flow spaces on the arXive un-der arXiv:1502.05001 [math.AT]. It should appear somewhere in Part III.Meanwhile it has appeared.

22.08.2015 Has the preprint by Mole and Ruping [611] appeared. It should appearsomewhere in Part III.

05.11.2015 There is a new preprint with title Stable classification of 4-manifolds with3-manifold fundamental groups by Kasprowski-Land-Powell-Teichner [462]on the arXive under arXiv:1511.01172 [math.GT]. Meanwhile it has ap-peared.

18.11.2015 There is a second version the preprint The isomorphism conjecture forArtin groups by Roushon [724] on the arXive under arXiv:1501.00776[math.KT]. Roushon writes in his comment: It got accepted for publi-cation in the Journal of K-theory. The copyright was signed to A. Bak’scompany ISOPP as was asked to do. Then the journal was discontinuedand now emails to A. Bak are going unanswered

11.12.2015 There is a paper by Gonzalez-Acuna-Gordon-Simon [345, Theorem 5.6 undCorollary 5.7] with the title Unsolvable problems about higher-dimensionalknots and related groups, where they show that the problem, whether theprojective class group or the Whitehead group of a group is trivial, cannotbe decided.

11.06.2016 Roushon has asked to remove his paper K-theory of virtually poly-surfacegroups cite Roushon(2003) about from the text, email from 03.04.2016.

16.08.2016 There is a new preprint with the title Index Theory and the Baum-Connes conjecture by Schick [741] on the arXive under arXiv:1608.04226[math.KT]

09.10.2016 There is a new preprint by Bartholdi-Kielak [83] with the title Amenabil-ity of groups is characterized by Myhill’s Theorem on the arXive underarXiv:1605.09133 [cs.FL]. They deal with the question when a group ringsatisfies the ore condition.

01.11.2016 There is a new preprint Burghelea conjecture and asymptotic dimension ofgroups by Alexander Engel and Michal Marcinkowski on the arXive underarXiv:1610.10076 [math.GT], see [279].

02.11.2016 There is a new preprint The Isomorphism Conjecture for solvable groups inWaldhausen’s A-theory by Farrell-Wu [303] on the arXive under arXiv:1611.00072[math.KT].

27.12.2016 There is a new preprint Note on the injectivity of the Loday assembly mapby Ullmann-Wu [795] on the arXive under arXiv:1509.07363 [math.KT].


09.02.2017 There is a survey article by Reich and Varisco [692] with the title AlgebraicK-theory, assembly maps, controlled algebra, and trace methods on thearXive under arXiv:1702.02218 [math.KT]. Meanwhile it has appeared.

09.02.2017 There is a article by Alexander Engel [278] with the title Strong Novikovconjecture for polynomially contractible groups on the arXive under arXiv:1702.02269[math.KT].

03.04.2017 There is a new preprint with the title Bivariant KK-theory and the Baum-Connes conjecture by Echterhoff [261] on the arXive under arXiv:1703.10912[math.KT]

06.06.2017 There is a new preprint by Cappell-Lubotzky-Weinberger [147] with thetitle A trichotomy theorem for transformation groups of locally symmetricmanifolds and topological rigidity on the arXive under arXiv:1601.00262[math.GR]. It has meanehile appeared.

18.07.2017 There is a new preprint with the title Equivariant K-homology for hyper-bolic reflection groups by Lafont-Ortiz-Rahm-Sanchez-Gracia [494] on thearXive under arXiv:1707.05133 [math.KT]. Meanwhile it has appeared

18.07.2017 There is a new preprint with the title On topological cyclic homology byNikolaus-Scholze [618] on the arXive under arXiv:1707.01799 [math.AT].Meanwhile it has appeared

03.01.2018 Es gibt ein neues preprint von Arthur Bartels [63] mit dem Titel K-theoryand actions on Euclidean retracts auf dem arXive unter arXiv:1801.00020[math.KT]

04.10.2012 The paper by Bartels [61] with the title On proofs of the Farrell-JonesConjecture should appear somewhere in Part III.

19.04.2019 Bestvina-Fujiwara-Wigglesworth have proved the FJC for (finitely gener-ated free)-by-cyclic groups. So far no preprint is available. Xiaolei Wu haswritten note with the title Farrell-Jones Conjecture for normally poly-freegroups, where he extends this to free-by-cyclic groups and to (right-angledArtin group)-by-cyclic. These results shoud be added to the chapter onthe status. Also the list of open cases must be adapted.

25.04.209 Add a reference to the preprint with Kasprowski and Li about computa-tions for graph products.

25.04.2019 The preprint [539], may appear in 2019.

23.9 Additional References

These references may have not yet appeared in the text but we should laterdecide whether they should be incorporated:

[26], [27], [28], [29], [61], [63], [67], [68], [83], [140][141][147], [154], [159], [221],[261],[278], [279], [290], [291], [303], [345], [359], [362], [363], [429], [430], [462],[464], [494],[500], [539],[610], [611], [618], [651], [676], [718], [724], [733], [741],[795], [816],

23.9 Additional References 621


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last edited on 03.05.2019 (master file ic.tex)last compiled on May 3, 2019

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isomorphism conjectures in k and l-theory · 2019-05-03 · and ii are in reasonable but de nitely...

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