more concise algebraic topology ~ may

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More Concise Algebraic Topology:

Localization, completion, and model categories

J. P. May and K. Ponto


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Contents

Introduction 1Some conventions and notations 7Acknowledgements 10

Part 1. Preliminaries: basic homotopy theory and nilpotent spaces 11

Chapter 1. Cofibrations and fibrations 131.1. Relations between cofibrations and fibrations 131.2. The fill-in and Verdier lemmas 161.3. Based and free cofibrations and fibrations 191.4. Actions of fundamental groups on homotopy classes of maps 221.5. Actions of fundamental groups in fibration sequences 25

Chapter 2. Homotopy colimits and homotopy limits; lim1 292.1. Some basic homotopy colimits 292.2. Some basic homotopy limits 342.3. Algebraic properties of lim1 372.4. An example of nonvanishing lim1 terms 392.5. The homology of colimits and limits 41

2.6. A profinite universal coefficient theorem 43

Chapter 3. Nilpotent spaces and Postnikov towers 453.1. A-nilpotent groups and spaces 453.2. Nilpotent spaces and Postnikov towers 473.3. C ocellular spaces and the dual Whitehead theorem 483.4. Fibrations with fiber an EilenbergMac Lane space 523.5. Postnikov A-towers 56

Chapter 4. Detecting nilpotent groups and spaces 634.1. Nilpotent Actions and Cohomology 634.2. Universal covers of nilpotent spaces 654.3. A-Maps ofA-nilpotent groups and spaces 664.4. Nilpotency and fibrations 684.5. Nilpotent spaces and finite type conditions 70

Part 2. Localizations of spaces at sets of primes 75

Chapter 5. Localizations of nilpotent groups and spaces 775.1. Localizations of abelian groups 775.2. The definition of localizations of spaces 795.3. Localizations of nilpotent spaces 82

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5.4. Localizations of nilpotent groups 845.5. Algebraic properties of localizations of nilpotent groups 875.6. Finitely generatedT-local groups 91

Chapter 6. Characterizations and properties of localizations 956.1. Characterizations of localizations of nilpotent spaces 956.2. Localizations of limits and fiber sequences 996.3. Localizations of function spaces 1016.4. Localizations of colimits and cofiber sequences 1036.5. A cellular construction of localizations 1036.6. Localizations of H-spaces and co-H-spaces 1056.7. Rationalization and the finiteness of homotopy groups 1076.8. The vanishing of rational phantom maps 109

Chapter 7. Fracture theorems for localization: groups 1117.1. Global to local pullback diagrams 112

7.2. Global to local: abelian and nilpotent groups 1157.3. Local to global pullback diagrams 1177.4. Local to global: abelian and nilpotent groups 1197.5. The genus of abelian and nilpotent groups 1207.6. Exact sequences of groups and pullbacks 124

Chapter 8. Fracture theorems for localization: spaces 1278.1. Statements of the main fracture theorems 1278.2. Fracture theorems for maps into nilpotent spaces 1298.3. Global to local fracture theorems: spaces 1328.4. Local to global fracture theorems: spaces 1348.5. The genus of nilpotent spaces 1358.6. Alternative proofs of the fracture theorems 139

Chapter 9. Rational H-spaces and fracture theorems 1439.1. The structure of rationalH-spaces 1439.2. The Samelson product andH(X;Q) 1459.3. The Whitehead product 1489.4. Fracture theorems for H-spaces 149

Part 3. Completions of spaces at sets of primes 153

Chapter 10. C ompletions of nilpotent groups and spaces 15510.1. Completions of abelian groups 15510.2. The definition of completions of spaces atT 16010.3. Completions of nilpotent spaces 163

10.4. Completions of nilpotent groups 166

Chapter 11. Characterizations and properties of completions 17111.1. Characterizations of completions of nilpotent spaces 17111.2. Completions of limits and fiber sequences 17311.3. Completions of function spaces 17511.4. Completions of colimits and cofiber sequences 17611.5. Completions ofH-spaces 177


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11.6. The vanishing ofp-adic phantom maps 177

Chapter 12. Fracture theorems for completion: Groups 179

12.1. Preliminaries on pullbacks and isomorphisms 17912.2. Global to local: abelian and nilpotent groups 18112.3. Local to global: abelian and nilpotent groups 18512.4. Formal completions and the adelic genus 186

Chapter 13. Fracture theorems for completion: Spaces 19113.1. Statements of the main fracture theorems 19113.2. Global to local fracture theorems: spaces 19313.3. Local to global fracture theorems: spaces 19513.4. The tensor product of a space and a ring 19913.5. Sullivans formal completion 20113.6. Formal completions and the adelic genus 203

Part 4. An introduction to model category theory 207

Chapter 14. An introduction to model category theory 20914.1. Preliminary definitions and weak factorization systems 20914.2. The definition and first properties of model categories 21414.3. The notion of homotopy in a model category 21814.4. The homotopy category of a model category 223

Chapter 15. Cofibrantly generated and proper model categories 22715.1. The small object argument for the construction of WFSs 22815.2. Compactly and cofibrantly generated model categories 23215.3. Over and under model structures 23315.4. Left and right proper model categories 236

15.5. Left properness, lifting properties, and the sets [X, Y] 238Chapter 16. Categorical perspectives on model categories 243

16.1. Derived functors and derived natural transformations 24316.2. Quillen adjunctions and Quillen equivalences 24716.3. Symmetric monoidal categories and enriched categories 24916.4. Symmetric monoidal and enriched model categories 25416.5. A glimpse at higher categorical structures 258

Chapter 17. M odel structures on the category of spaces 26117.1. The Hurewicz orh-model structure on spaces 26117.2. The Quillen orq-model structure on spaces 26417.3. Mixed model structures in general 26717.4. The mixed model structure on spaces 273

17.5. The model structure on simplicial sets 27517.6. The proof of the model axioms 280

Chapter 18. Model structures on categories of chain complexes 28518.1. The algebraic framework and the analogy with topology 28518.2. h-cofibrations andh-fibrations in C hR 28718.3. The h-model structure onC hR 29018.4. The q-model structure onC hR 293


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18.5. Proofs and the characterization ofq-cofibrations 29618.6. The m-model structure onC hR 300

Chapter 19. Resolution and localization model structures 30319.1. Resolution and mixed model structures 30419.2. The general context of Bousfield localization 30619.3. Localizations with respect to homology theories 31019.4. Bousfield localization at sets and classes of maps 31319.5. Bousfield localization in enriched model categories 315

Part 5. Bialgebras and Hopf algebras 319

Chapter 20. Bialgebras and Hopf algebras 32120.1. Preliminaries 32120.2. Algebras, coalgebras, and bialgebras 32420.3. Antipodes and Hopf algebras 327

20.4. Modules, comodules, and related concepts 329

Chapter 21. Connected and component Hopf algebras 33321.1. Connected algebras, coalgebras, and Hopf algebras 33321.2. Splitting theorems 33521.3. Component coalgebras and the existence of antipodes 33621.4. Self-dual Hopf algebras 33821.5. The homotopy groups ofM O and other Thom spectra 34221.6. A proof of the Bott periodicity theorem 344

Chapter 22. Lie algebras and Hopf algebras in characteristic zero 34922.1. Graded Lie algebras 34922.2. The Poincare-Birkhoff-Witt theorem 351

22.3. Primitively generated Hopf algebras in characteristic zero 35422.4. Commutative Hopf algebras in characteristic zero 356

Chapter 23. Restricted Lie algebras and Hopf algebras in characteristicp 35923.1. Restricted Lie algebras 35923.2. The restricted Poincare-Birkhoff-Witt theorem 36023.3. Primitively generated Hopf algebras in characteristicp 36223.4. Commutative Hopf algebras in characteristicp 363

Chapter 24. A primer on spectral sequences 36924.1. Definitions 36924.2. Exact Couples 37124.3. Filtered Complexes 37224.4. Products 374

24.5. The Serre spectral sequence 37624.6. Comparison theorems 37924.7. Convergence proofs 380

Index 385

Bibliography 395


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INTRODUCTION 1

Introduction

There is general agreement on the rudiments of algebraic topology, the things

that every mathematician should know. This material might include the fundamen-tal group, covering spaces, ordinary homology and cohomology in its singular, cellu-lar, axiomatic, and represented versions, higher homotopy groups and the Hurewicztheorem, basic homotopy theory including fibrations and cofibrations, Poincare du-ality for manifolds and manifolds with boundary. The rudiments should also includea reasonable amount of categorical language and at least enough homological alge-bra for the universal coefficient and Kunneth theorems. This material is treated insuch recent books as [3, 34, 36, 59, 91] and many earlier ones. What next? Possi-blyK-theory, which is treated in [3] and, briefly, [91], and some idea of cobordismtheory [36, 91]. None of the most recent texts goes much beyond the material justmentioned, all of which dates at latest from the early 1960s. Regrettably, only oneof these texts, [34], includes anything about spectral sequences, but [78, 97, 122]help make up for that.

The subject of algebraic topology is very young. Despite many precursors andearlier results, firm foundations only date from the landmark book of Eilenberg andSteenrod [45], which appeared in 1952. It is not an exaggeration to say that eventhe most recent published texts do not go beyond the first decade or so of the seriousstudy of the subject. For that reason, people outside the field very often know littleor nothing about some of its fundamental branches that have been developed overthe past half century. A partial list of areas a student should learn is given in thesuggestions for further reading of [91], and a helpful guide to further developmentof the subject (with few proofs) has been given by Selick [122].

It seems to us that the disparity between the lack of accessibility of the pub-lished sources and the fundamental importance of the material is nowhere greaterthan in the theory of localization and completion of topological spaces1. It makeslittle more sense to consider modern algebraic topology without localization andcompletion of spaces than it does to consider modern algebra without localizationand completion of rings. These tools have been in common use ever since they wereintroduced in the early 1970s. Many papers in algebraic topology start with theblanket assumption that all spaces are to be localized or completed at a given prime

p. Readers of such papers are expected to know what this means. Experts know thatthese constructions can be found in such basic 1970s references as [ 21, 62, 131].However, the standard approaches favored by the experts are not easily accessibleto the novices, especially in the case of completions. In fact, these notions can andshould be introduced at a much more elementary level. The notion of completionis particularly important because it relates directly to mod p cohomology, which isthe invariant that algebraic topologists most frequently compute.

In the first half of this book, we set out the basic theory of localization and

completion of nilpotent spaces. We give the most elementary treatment we know,making no use of simplicial techniques or model categories. We assume only a littlemore than a first course in algebraic topology, such as can be found in [3, 34, 36,59, 91]. We require and provide more information about some standard topics,such as fibration and cofibration sequences, Postnikov towers, and homotopy limitsand colimits, than appears in those books, but this is fundamental material of

1These topics are not mentioned in [3, 36, 59].


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independent interest. The only other preliminary that we require and that cannotbe found in most of the books cited above is the Serre spectral sequence. Thereare several accessible sources for that, such as [34, 78, 97, 122], but to help makethis book more self-contained, we shall give a concise primer on spectral sequencesin Chapter 24; it is taken from 1960s notes of the first author and makes no claimto originality.

The second half of the book is quite different and consists of two parts that canbe read independently of each other and of the first half. While written with alge-braic topologists in mind, both parts should be of more general interest. They aredevoted to topics in homotopical algebra and in pure algebra that are needed by allalgebraic topologists and many others. By far the longer of these parts is an intro-duction to model category theory. This material can easily be overemphasized, tothe detriment of concrete results and the nuances needed to prove them. For exam-ple, its use would in no way simplify anything in the first half of the book. However,its use allows us to complete the first half by giving a conceptual construction and

characterization of localizations and completions of general, not necessarily nilpo-tent, spaces. More fundamentally, model category theory has become the centralorganizational principle of homotopical algebra, a subject that embraces algebraictopology, homological algebra, and much modern algebraic geometry. Anybodyinterested in any of these fields needs to know model category theory. It playsa role in homotopical algebra analogous to the role played by category theory inmathematics. It gives a common language for the subject that greatly facilitatescomparisons, and it allows common proofs of seemingly disparate results.

The short last part of the book is something of a bonus track, in that it isperipheral to the main thrust of the book. It develops the basic theory of bialgebrasand Hopf algebras. Its main point is a redevelopment of the structure theory ofHopf algebras, due originally to Milnor and Moore [103] but with an addendumfrom [84]. Hopf algebras are used in several places in the first half of the book, and

they are fundamental to the algebra of algebraic topology.We say a bit about how our treatments of these topics developed and how they

are organized. The starting point for our exposition of localization and completioncomes from unpublished lecture notes of the first author that date from sometime inthe early 1970s. That exposition attempted a synthesis in which localization andcompletion were treated as special cases of a more general elementary construction.The synthesis did not work well because it obscured essential differences. Thosenotes were reworked to more accessible form by the second author and then polishedto publishable form by the authors working together. There are some new results,but we make little claim to originality. Most of the results and many of the proofsare largely the same as in one or another of Bousfield and Kan [21], Sullivan [131],and Hilton, Mislin and Roitberg [62].

However, a central feature of the subject is the fracture theorems for the passage

back and forth between local and global information. It is here that the treatmentsof localization and completion differ most from each other. The relevant materialhas been reworked from scratch, and the treatment in the first authors 1970s noteshas largely been jettisoned. In fact, the literature in this area requires considerableclarification, and we are especially concerned to give coherent accounts of the mostgeneral and accurate versions of the fracture theorems for nilpotent spaces. Theseresults were not fully understood at the time the primary sources [21, 62, 131]


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were written, and there are seriously incorrect statements in some of the importantearly papers. Moreover, generalizations of the versions of these results that appearin the primary sources were proven after they were written and can only be found inrelatively obscure papers that are known just to a few experts. We have introducedseveral new ideas that we think clarify the theorems, and we have proven someresults that are essential for full generality and that we could not find anywhere inthe literature.

The first half of the book is divided into three parts: preliminaries, localizations,and completions. The reader may want to skim the first part, referring back to itas needed. Many of the preliminaries are essential for the later parts, but masteryof their details is not needed on a first reading. Specific indications of material thatcan be skipped are given in the introductions to the first four chapters. The firstchapter is about cofibrations, fibrations, and actions by the fundamental group. Thesecond is about elementary homotopy colimits and homotopy limits and lim1 exactsequences. This both sets the stage for later work and rounds out material that was

omitted from [91] but that all algebraic topologists should know. The third chapterdeals with nilpotent spaces and their approximation by Postnikov towers, givinga more thorough treatment of the latter than can be found in existing expositorytexts. This is the most essential preliminary to our treatment of localizations andcompletions.

The fourth chapter shows how to prove that various groups and spaces arenilpotent and is more technical; while it is logically placed, the reader may want toreturn to it later. The reader might be put off by nilpotent spaces and groups at afirst reading. After all, the vast majority of applications involve simply connectedor, more generally, simple spaces. However, the proofs of some of the fracturetheorems make heavy use of connected components of function spaces F(X, Y).Even whenXand Y are simply connected CW complexes, these spaces are rarelysimple, but they are nilpotent when X is finite. Moreover, nilpotent spaces provide

exactly the right level of generality for an elementary exposition, and the techniquesused to prove results for nilpotent spaces are not very different from those used forsimple spaces.

We say just a bit about the literature for spaces that are not nilpotent andabout alternative constructions. There are several constructions of localizationsand completions of general spaces that agree when restricted to nilpotent spaces.The most important of these is Bousfield localization, which we shall constructmodel theoretically. These more general constructions are still not well understoodcalculationally, and knowledge of them does not seem helpful for understanding themost calculationally important properties of localization and completion, such astheir homotopical behavior and the fracture theorems.

We construct localizations of abelian groups, nilpotent groups and nilpotentspaces in Chapter 5, and we construct completions of abelian groups, nilpotent

groups, and nilpotent spaces in the parallel Chapter 10. We characterize local-izations and describe their behavior under standard topological constructions inChapter 6, and we do the same for completions in the parallel Chapter 11. Weprove the fracture theorems for localizations in Chapters 7 and 8 and the fracturetheorems for completions in the parallel Chapters 12 and 13. In many cases, thesame results, with a few words changed, are considered in the same order in thecases of localization and completion. This is intentional, and it allows us to explain


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and emphasize both similarities and differences. As we have already indicated, al-though these chapters are parallel, it substantially clarifies the constructions andresults not to subsume both under a single general construction. We give a fewresults about rationalization of spaces in Chapter 9.

We say a little here about our general philosophy and methodology, which goesback to a paper of the first author [89] on The dual Whitehead theorems. Asis explained there and will be repeated here, we can dualize the proof of the firsttheorem below (as given for example in [91]) to prove the second.

Theorem 0.0.1. A weak homotopy equivalencee: Y Zbetween CW com-plexes is a homotopy equivalence.

Theorem0.0.2. An integral homology isomorphisme : Y Zbetween simplespaces is a weak homotopy equivalence.

The argument is based on the dualization of cell complexes to cocell complexes,of which Postnikov towers are examples, and of the Homotopy Extension and Lift-ing Property (HELP) to co-HELP. Once this dualization is understood, it becomesalmost transparent how one can construct and study the localizations and com-pletions of nilpotent spaces simply by inductively localizing or completing theirPostnikov towers one cocell at a time. Our treatment of localization and comple-tion is characterized by a systematic use of cocellular techniques dual to familiarcellular techniques. In the case of localization, but not of completion, there is adual cellular treatment applicable to simply connected spaces.

We turn now to our treatment of model categories, and we first try to answeran obvious question. There are several excellent introductory sources for modelcategory theory [42, 53, 65, 66, 114]. Why add another one? One reason is that,for historical reasons, the literature of model category theory focuses overwhelm-

ingly on a simplicial point of view, and especially on model categories enriched insimplicial sets. There is nothing wrong with that point of view, but it obscures es-sential features that are present in the classical contexts of algebraic topology andhomological algebra and that are not present in the simplicial context. Anotherreason is that we feel that some of the emphasis in the existing literature focuseson technicalities at the expense of the essential conceptual simplicity of the ideas.

We present the basic general theory of model categories and their associatedhomotopy categories in Chapter 14. For conceptual clarity, we offer a slight re-formulation of the original definition of a model category that focuses on weakfactorization systems (WFSs): a model category consists of a subcategory of weakequivalences together with a pair of related WFSs. This point of view separatesout the main constituents of the definition in a way that we find illuminating. Wediscuss compactly generated and cofibrantly generated model categories in Chap-

ter 15. We will describe the difference shortly. We also describe proper modelcategories there. We give essential categorical perspectives in Chapter 16.

To make the general theory flow smoothly, we have deferred examples to theparallel Chapters 17 and 18. On a first reading, the reader may want to skip directlyfrom Chapter 14 to Chapters 17 and 18. These chapters treat model structures oncategories of spaces and on categories of chain complexes in parallel. In both, thereare three intertwined model structures, which we will call the h-model structure,theq-model structure, and them-model structure.


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The h stands for homotopy equivalence or Hurewicz. The weak equivalencesare the homotopy equivalences, and the cofibrations and fibrations are defined bythe HEP (Homotopy Extension Property) and the CHP (Covering Homotopy Prop-erty). Such fibrations were first introduced by Hurewicz. The qstands for Quillenor quasi-isomorphism. The weak equivalences are the weak equivalences of spacesor the quasi-isomorphisms of chain complexes. The fibrations are the Serre fibra-tions of spaces or the epimorphisms of chain complexes, and the cofibrations inboth cases are the retracts of cell complexes.

Themstands for mixed, and them-model structures, due to Cole [33], combinethe good features of the h and q-model structures. The weak equivalences are theq-equivalences and the fibrations are the h-fibrations. The m-cofibrant objects arethe spaces of the homotopy types of CW complexes or the chain complexes of thehomotopy types of complexes of projective modules (at least in the bounded belowcase). We argue that classical algebraic topology, over at least the mathematicallifetime of the first author, has implicitly worked in the m-model structure. For

example, the first part of this book implicitly works there. Modern approaches toclassical homological algebra work similarly. We believe that this trichotomy ofmodel category structures, and especially the precise analogy between these struc-tures in topology and algebra, gives the best possible material for an introductionto model category theory. We reiterate that these features are not present in thesimplicial world, which in any case is less familiar to those just starting out.

The m-model structure can be viewed conceptually as a colocalization modelstructure, which we rename a resolution model structure. In our examples, it codi-fies CW approximation of spaces or projective resolutions of modules, where theseare explicitly understood as up to homotopy constructions. Colocalization is dualto Bousfield localization, and this brings us to another reason for our introductionto model category theory, namely a perceived need for as simple and accessiblean approach to Bousfield localization as possible. This is such a centrally impor-

tant tool in modern algebraic topology (and algebraic geometry) that every studentshould see it. We give a geodesic development that emphasizes the conceptual ideaand uses as little special language as possible.

In particular, we make no use of simplicial theory and minimal use of cofibrantlygenerated model categories, which were developed historically as a codificationof the methods Bousfield introduced in his original construction of localizations[16]. An idiosyncratic feature of our presentation of model category theory is thatwe emphasize a dichotomy between cofibrantly generated model categories andcompactly generated model categories. The small object argument for constructingthe WFSs in these model structures is presented in general, but in the most basicexamples it can be applied using only cell complexes of the familiar form colim Xn,without use of cardinals bigger than . Transfinite techniques are essential tothe theory of localization, but we feel that the literature focuses on them to an

inordinate extent. Disentangling the optional from the essential use of such methodsleads to a more user friendly introduction to model category theory.

Although we make no use of it, we do describe the standard model structureon simplicial sets. In the literature, the proof of the model axioms is unpleasantlylengthy. We sketch a new proof, due to Bousfield and the first author, that is shorterand focuses more on basic simplicial constructions and less on the intertwining ofsimplicial and topological methods.


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The last part of the book, on bialgebras and Hopf algebras, is again largelybased on unpublished notes of the first author that date from the 1970s. Sincewe hope our treatment has something to offer to algebraists as well as topologists,one introductory remark is obligatory. In algebraic topology, algebras are alwaysgraded and often connected, meaning that they are zero in negative degrees and theground field in degree zero. Under this assumption, bialgebras automatically haveantipodes, so that there is no distinction between Hopf algebras and bialgebras.For this reason, and for historical reasons, algebraic topologists generally use theterm Hopf algebra for both notions, but we will be careful about the distinction.

Chapter 20 gives the basic theory as used in all subjects and Chapter 21 givesfeatures that are particularly relevant to the use of Hopf algebras in algebraictopology, together with quick applications to cobordism and K-theory. Chapters22 and 23 give the structure theory for Hopf algebras in characteristic zero andin positive characteristic, respectively. The essential organizing principle of thesetwo chapters is that all of the main theorems on the structure of connected Hopf

algebras can be derived from the PoincareBirkhoffWitt theorem on the structureof Lie algebras (in characteristic zero) and of restricted Lie algebras (in positivecharacteristic). The point is that passage to associated graded algebras from theaugmentation ideal filtration gives a primitively generated Hopf algebra, and suchHopf algebras are universal enveloping Hopf algebras of Lie algebras or of restrictedLie algebras. This point of view is due to [103], but we will be a little more explicit.

Our point of view derives from the first authors thesis, in which the citedfiltration was used to construct a spectral sequence for the computation of the co-homology of Hopf algebras starting from the cohomology of (restricted) Lie algebras[83], and from his short paper [84]. This point of view allows us to simplify theproofs of some of the results in [103], and it shows that the structure theorems aremore widely applicable than seems to be known.

Precisely, the structure theorems apply to ungraded bialgebras and, more gener-

ally, to non-connected graded bialgebras whenever the augmentation ideal filtrationis complete. We emphasize this fact in view of the current interest in more generalHopf algebras, especially the quantum groups. A cocommutative Hopf algebra (overa field) is a group in the cartesian monoidal category of coalgebras, the point beingthat the tensor product of cocommutative coalgebras is their categorical product.In algebraic topology, the Hopf algebras that arise naturally are either commutativeor cocommutative. By duality, one may as well focus on the cocommutative case. Itwas a fundamental insight of Drinfeld that dropping cocommutativity allows veryinteresting examples with quantized deviation from cocommutativity. These arethe quantum groups. Because we are writing from the point of view of algebraictopology, we shall not say anything about them here, but the structure theoremsare written with a view to possible applications beyond algebraic topology.

Sample applications of the theory of Hopf algebras within algebraic topology

are given in several places in the book, and they pervade the subject as a whole. InChapter 9, the structure theory for rational Hopf algebras is used to describe thecategory of rational H-spaces and to explain how this information is used to studyH-spaces in general. In Chapter 22, we explain the Hopf algebra proof of Thomscalculation of the real cobordism ring and describe how the method applies to otherunoriented cobordism theories. We also give the elementary calculational proof ofcomplex Bott periodicity. Of course, there is much more to be said here. Our goal


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SOME CONVENTIONS AND NOTATIONS 7

is to highlight for the beginner important sample results that show how directlythe general algebraic theorems relate to the concrete topological applications.

Some conventions and notations

This book is perhaps best viewed as a sequel to [91], although we have tried tomake it reasonably self-contained. Aside from use of the Serre spectral sequence,we assume no topological preliminaries that are not to be found in [91], and weredo most of the algebra that we use.

To keep things familiar, elementary, and free of irrelevant pathology, we workthroughout the first half in the category Uof compactly generated spaces (see [91,Ch. 5]). It is by now a standard convention in algebraic topology that spaces meancompactly generated spaces, and we adopt that convention. While most resultswill not require this, we implicitly restrict to spaces of the homotopy types of CWcomplexes whenever we talk about passage to homotopy. This allows us to definethe homotopy category HoUsimply by identifying homotopic maps; it is equivalent

to the homotopy category of all spaces in U, not necessarily CW homotopy types,that is formed by formally inverting the weak homotopy equivalences.

We nearly always work with based spaces. To avoid pathology, we assume onceand for all that basepoints are nondegenerate, meaning that the inclusion Xis a cofibration (see [91, p. 56]). We write T for the category of nondegener-ately based compactly generated spaces, that is nondegenerately based spaces inU.2 Again, whenever we talk about passage to homotopy, we implicitly restrictto spaces of the based homotopy type of CW complexes. This allows us to definethe homotopy category HoT by identifying maps that are homotopic in the basedsense, that is through homotopies h such that each ht is a based map. The cate-gory T, and its restriction to CW homotopy types, has been the preferred workingplace of algebraic topologists for very many years; for example, the first author hasworked explicitly in this category ever since he wrote [85], around forty years ago.

We ask the reader to accept these conventions and not to quibble if we do notrepeat these standing assumptions in all of our statements of results. The conven-tions mean that, when passing to homotopy categories, we implicitly approximateall spaces by weakly homotopy equivalent CW complexes, as we can do by [91,10.3]. In particular, when we use Postnikov towers and pass to limits, which arenot of the homotopy types of CW complexes, we shall implicitly approximate themby CW complexes. We shall be a little more explicit about this in Chapters 1 and2, but we shall take such CW approximation for granted in later chapters.

The expert reader will want a model theoretic justification for working in T.First, as we explain in 17.1, the category U of based spaces in U inherits an h-model, or Hurewicz model, structure from U. In that model structure, all objectsare fibrant and the cofibrant objects are precisely the spaces in T. Second, as weexplain in17.4, U also inherits an m-model, or mixed model, structure, from U.In that model structure, all objects are again fibrant and the cofibrant objects areprecisely the spaces in Tthat have the homotopy types of based CW complexes.Cofibrant approximation is precisely approximation of spaces by weakly homotopyequivalent CW complexes in T. This means that working in T and implicitly

2This conflicts with [91], where T was defined to be the category of based spaces in U(denoted U here). That choice had the result that the nondegenerately based hypothesisreappears with monotonous regularity in [91].


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approximating spaces by CW complexes is part of the standard model-theoreticway of doing homotopy theory. The novice will learn later in the book how verynatural this language is, but it plays no role in the first half. We believe that toappreciate model category theory, the reader should first have seen some serioushomotopical algebra, such as the material in the first half of this book.

It is convenient to fix some notations that we will use throughout.

Notations0.0.3. We fix some notations concerning based spaces.

(i) Spaces are assumed to be path connected unless explicitly stated otherwise,and we use the word connected to mean path connected from now on. Wealso assume that all given spaces Xhave universal covers, denoted X.

(ii) For based spaces X and Y, let [X, Y] denote the set of maps X Y inHoT; equivalently, after CW approximation ofXif necessary, it is the set ofbased homotopy classes of based mapsXY.

(iii) Let F(X, Y) denote the space of based maps X Y. It has a canonical

basepoint, namely the trivial map. We write F(X, Y)ffor the component ofa map f and give it the basepoint f. When using these notations, we canallowYto be a general space, but to have the right weak homotopy type wemust insist thatXhas the homotopy type of a CW complex.

(iv) The smash product X Y of based spaces X and Y is the quotient of theproductX Yby the wedge (or one-point union)X Y. We have adjunctionhomeomorphisms

F(X Y, Z)=F(X, F(Y, Z))

and consequent bijections

[X Y, Z] =[X, F(Y, Z)].

(v) For an unbased spaceK, let K+ denote the union ofKand a disjoint base-

point. The based cylinder X(I+) is obtained from XI by collapsingthe line through the basepoint ofXto a point. Similarly, we have the basedcocylinder F(I+, Y). It is the space of unbased maps I Y based at theconstant map to the basepoint. These specify the domain and, in adjointform, the codomain of based homotopies, that is, homotopies that are givenby based mapsht : XY fort I.

We also fix some algebraic notations and point out right away some ways thatalgebraic topologists think differently than algebraic geometers and others abouteven very basic algebra.

Notations0.0.4. LetTbe a fixed set of primes and p a single prime.

(i) Let ZT denote the ring of integers localized at T, that is, the subring ofQ

consisting of rationals expressible as fractions k/, where is a product ofprimesnotin T. We let Z[T1] denote the subring of fractionsk/, where isa product of primes in T. In particular, Z[p1] has only p inverted. Let Z(p)denote the ring of integers localized at the prime ideal (p) or, equivalently, atthe singleton set{p}.

(ii) LetZp denote the ring ofp-adic integers. Illogically, but to avoid conflict of

notation, we write ZT for the product over p T of the rings Zp. We then

writeQTfor the ringZTQ; whenT ={p}, this is the ring ofp-adic numbers.


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SOME CONVENTIONS AND NOTATIONS 9

(iii) Let Fp denote the field with p elements and FTdenote the product overp Tof the fields Fp. Let Z/n denote the quotient group Z/nZ. We sometimesconsider the ring structure on Z/n, and then Z/p= F

pfor a primep.

(iv) We write A(p) and Ap for the localization at p and the p-adic completion of

an abelian group A. Thus Zp is the underlying abelian group of the ring Zp.

(v) We writeAT and AT for the localization and completion ofA at T; the latter

is the product overp Tof the Ap.(vi) Let Ab denote the category of abelian groups. We sometimes ignore the

maps and use the notation Ab for the collection of all abelian groups. Moregenerally, A will denote any collection of abelian groups that contains 0.

(vii) We often write , Hom, Tor, and Ext for Z, HomZ, TorZ1 , and Ext

1Z. We

assume familiarity with these functors.

Warning 0.0.5. We warn the reader that algebraic notations in the literatureof algebraic topology have drifted over time and are quite inconsistent. The reader

may find Zp used for either our Z(p) or for our Fp; the latter choice is used ubiq-uitously in the early literature, including most of the first authors papers. Infact, regrettably, we must warn the reader that Zp meansFp in the book [91]. The

p-adic integers only began to be used in algebraic topology in the 1970s, and oldhabits die hard. In both the algebraic and topological literature, the ring Zp is

sometimes denoted Zp; we would prefer that notation as a matter of logic, but thenotation Zp has by now become quite standard.

Warning 0.0.6 (Conventions on graded algebraic structures). We think ofhomology and cohomology as graded abelian groups. For most algebraists, a gradedabelian group A is the direct sum over degrees of its homogeneous subgroups An,or, with cohomological grading,An. In algebraic topology, unless explicitly statedotherwise, when some such notation as H is often used, graded abelian groupsmean sequences of abelian groupsAn. That is, algebraic topologists do not usuallyallow the addition of elements of different degrees. To see just how much differencethis makes, consider a Laurent series algebra k [x, x1] over a field k , where x haspositive even degree. To an algebraic topologist, this is a perfectly good graded field:every non-zero element is a unit. To an algebraist, it is not. This is not an esotericdifference. With k= Fp, such graded fields appear naturally in algebraic topologyas the coefficients of certain generalized cohomology theories, called Morava K-theories, and their homological algebra works exactly as for any other field, a factthat has real calculational applications.

The tensor product A B of graded abelian groups is specified by

(A B)n =

p+q=n

Ap Bq.

In categorical language, the category Ab of graded abelian groups is a symmetric

monoidal category under , meaning that is unital (with unit Z concentratedin degree 0), associative, and commutative up to coherent natural isomorphisms.Here again, there is a difference of conventions. For an algebraic topologist, thecommutativity isomorphism: A B B A is specified by

(a b) = (1)pqb a

where deg(a) = p and deg(b) = q. A graded k -algebra with product is commu-tative if = ; elementwise, this means that ab = (1)pqba. In the algebraic


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10 CONTENTS

literature, such an algebra is said to be graded commutative or sometimes evensupercommutative, but in algebraic topology this notion of commutativity is andalways has been the default (at least since the early 1960s). Again, this is not anesoteric difference. To an algebraic topologist, a polynomial algebrak [x] where xhas odd degree is not a commutative k -algebra unless k has characteristic 2. ThehomologyH(Sn; k),n even, is an example of such a non-commutative algebra.

The algebraist must keep these conventions in mind when reading the materialabout Hopf algebras in this book. To focus on commutativity in the algebraistssense, one can double the degrees of all elements and so eliminate the appearanceof odd degree elements.

Acknowledgements

Many people have helped us with this book in a variety of ways. Several gener-ations of the senior authors students have had input during its very long gestation.Notes in the 1970s by Zig Fiedorowicz influenced our treatment of nilpotent spaces

and Postnikov towers, and Lemma 3.4.2 is due to him. The mixed or resolutionmodel structures that play a central role in our treatment of model structures onspaces and chain complexes are due to Mike Cole. Part of the treatment of modelstructures comes from the book [95] with Johann Sigurdsson. Notes and a 2009paper [117] by Emily Riehl influenced our treatment of the basic definitions inmodel category theory. She and Mike Shulman, as a postdoc, made an especiallythorough reading of the model category theory part and found many mistakes andinfelicities. Anna Marie Bohmann, Rolf Hoyer, and John Lind read and commentedon several parts of the book. Mona Merling went through Part I with a meticulouseye to excesses of concision. Many other students over the past thirty plus yearshave also had input.

We have many thanks to offer others. John Rognes texed 21.4 for his own usein 1996. Kathryn Hess suggested that we include a treatment of model category

theory, which we had not originally intended, hence she is responsible for the exis-tence of that part of the book. Bill Dwyer helped us with the fracture theorems.We owe an especially big debt to Pete Bousfield. He gave us many insights andseveral proofs that appear in the chapters on the fracture theorems and in the sec-tions on Bousfield localization. Moreover, our treatment of the model structure onthe category of simplicial sets arose from correspondence with him and is primarilyhis work.

We also thank an anonymous reviewer for both complimentary words and co-gent criticism.

January 1, 2010


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Part 1

Preliminaries: basic homotopytheory and nilpotent spaces


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CHAPTER 1

Cofibrations and fibrations

We shall make constant use of the theory of fibration and cofibration sequences,and this chapter can be viewed as a continuation of the basic theory of such se-quences as developed in [91, Chapters 68]. We urge the reader to review thatmaterial, although we shall recall most of the basic definitions as we go along. Thematerial here leads naturally to such more advanced topics as model category theory[66, 65, 95], to which we will turn later, and triangulated categories [92, 110, 136].

However, we prefer to work within the more elementary foundations of [91] in thefirst half of this book. We shall concentrate primarily on just what we shall uselater, but we round out the general theory with several related results that are offundamental importance throughout algebraic topology. The technical proofs in3and the details of4 and5 should not detain the reader on a first reading.

1.1. Relations between cofibrations and fibrations

Remember that we are working in the category Tof nondegenerately basedcompactly generated spaces. Although the following folklore result was known longago, it is now viewed as part of Quillen model category theory, and its importancecan best be understood in that context. For the moment, we view it as merely aconvenient technical starting point.

Lemma1.1.1. Suppose thati is a cofibration andpis a fibration in the followingdiagram of based spaces, in whichp g= f i.

Ag

i

E

p

X

fB

If either i orp is a homotopy equivalence, then there exists a map such that thediagram commutes.

This result is a strengthened implication of the definitions of cofibrations andfibrations. As in [91, p. 41], reinterpreted in the based context, a map i is a

(based) cofibration if there is a lift in all such diagrams in which p is the mapp0 : F(I+, Y) Ygiven by evaluation at 0 for some space Y . This is a restate-ment of the homotopy extension property, or HEP. Dually, as in [91, p. 47], a map

p is a (based) fibration if there is a lift in all such diagrams in which i is the in-clusioni0 : Y Y I+ of the base of the cylinder. This is the covering homotopyproperty, or CHP. These are often called Hurewicz cofibrations and fibrations todistinguish them from other kinds of cofibrations and fibrations (in particular Serrefibrations) that also appear in model structures on spaces.

13


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14 1. COFIBRATIONS AND FIBRATIONS

The unbased version of Lemma 1.1.1 is proven in Proposition 17.1.4, using nointermediate theory, and the reader is invited to skip there to see it. The basedversion follows, but rather technically, using Lemmas 1.3.3 and 1.3.4 below. Thededuction is explained model theoretically in Corollary 17.1.2 and Remark 17.1.3.

One can think of model category theory as, in part, a codification of the no-tion of duality, called Eckmann-Hilton duality, that is displayed in the definitionsof cofibrations and fibrations and in Lemma 1.1.1. We shall be making concreterather than abstract use of such duality for now, but it pervades our point of viewthroughout. We leave the following dual pair of observations as exercises. Theirproofs are direct from the definitions of pushouts and cofibrations and of pullbacksand fibrations. In the first, the closed inclusion hypothesis serves to ensure that wedo not leave the category of compactly generated spaces [91, p. 38].

Exercise 1.1.2. Suppose given a commutative diagram

Y

X

f

i

Z

Y X

f

iZ

in whichi and i are closed inclusions and andare cofibrations. Prove that theinduced map of pushouts

Y XZY XZ

is a cofibration. Exhibit an example to show that the conclusion does not hold fora more general diagram of the same shape with the equality X=Xreplaced by acofibrationXX. (Hint: interchange i and = in the diagram.)

Exercise 1.1.3. Suppose given a commutative diagram

Yf

X Zp

Y

fX Z

p

in which andare fibrations. Prove that the induced map of pullbacks

Y XZY XZ

is a fibration. Again, the conclusion does not hold for a more general diagram ofthe same shape with the equality X= Xreplaced by a fibration XX.

We shall often use the following pair of results about function spaces. The firstillustrates how to use the defining lifting properties to construct new cofibrationsand fibrations from given ones.

Lemma 1.1.4. Let i : A X be a cofibration and Y be a space. Then theinduced mapi : F(X, Y)F(A, Y)is a fibration and the fiber over the basepointisF(X/A, Y).


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1.1. RELATIONS BETWEEN COFIBRATIONS AND FIBRATIONS 15

Proof. To show that i : F(X, Y) F(A, Y) is a fibration it is enough toshow that there is a lift in any commutative square

Z f

i0

F(X, Y)

i

Z I+

h

F(A, Y).

By adjunction, we obtain the following diagram from that just given.

Ah

i

F(I+, F(Z, Y))

p0

X

f

H

F(Z, Y)

.

Hereh(a)(t)(z) = h(z, t)(a) and f(x)(z) = f(z)(x) where a A, z Z, x X,and t I. Since i : A X is a cofibration there exists a lift H. The mapH: ZI+ F(X, Y) specified by H(z, t)(x) = H(x)(t)(z) for x X, z Zand t I, gives a lift in the original diagram. Thereforei : F(X, Y)F(A, Y)is a fibration. The basepoint ofF(A, Y) sends A to the basepoint ofY, and itsinverse image in F(X, Y) consists of those maps X Y that send A to thebasepoint. These are the maps that factor throughX/A, that is, the elements ofF(X/A, Y).

For the second, we recall the following standard definitions from [91, pp 57,59]. They will be used repeatedly throughout the book. By Lemmas 1.3.3 and 1.3.4below, our assumption that basepoints are nondegenerate ensures that the terms

cofibration and fibration in the following definition can be understood in either thebased or the unbased sense.

Definition 1.1.5. Letf : XY be a (based) map. The homotopy cofiberCf off is the pushout Y fCX off and i0 : X C X. Here the coneCX isX I, where Iis given the basepoint 1. Since i0 is a cofibration, so is its pushouti : Y Cf [91, p. 42]. The homotopy fiberF f off is the pullback XfP Yoff and p1 : P Y Y. Here the path space P Y is F(I, Y), where Iis given thebasepoint 0; thus it consists of paths that start at the basepoint ofY . Sincep1 isa fibration (by Lemma 1.1.4), so is its pullback : F fX [91, p. 47].

We generally abbreviate homotopy cofiber to cofiber. This is unambiguoussince the word cofiber has no preassigned meaning. Whenf: XYis a cofibra-tion, the cofiber is canonically equivalent to the quotient Y /X. We also generally

abbreviate homotopy fiber to fiber. Here there is ambiguity when the givenbased map is a fibration, in which case the actual fiber f1() and the homotopyfiber are canonically equivalent. By abuse, we then use whichever term seems moreconvenient.

Lemma 1.1.6. Letf: X Y be a map andZ be a space. Then the homo-topy fiberF f of the induced map of function spacesf : F(Y, Z) F(X, Z) ishomeomorphic to F(Cf,Z), whereCfis the homotopy cofiber off.


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Proof. The fiber F f is F(Y, Z)F(X,Z) P F(X, Z). Clearly P F(X, Z) ishomeomorphic to F(CX,Z). Technically, in view of the convention that I hasbasepoint 0 when defining Pand 1 when defining C, we must use the homeomor-phism II that sendst to 1 tto see this. Since the functor F(, Z) convertspushouts to pullbacks, the conclusion follows.

1.2. The fill-in and Verdier lemmas

In formal terms, the results of this section describe the homotopy category HoTas a pretriangulated category. However, we are more interested in describingprecisely what is true before passage to the homotopy category, since some easybut little known details of that will ease our later work.

The following dual pair of fill-in lemmas will be at the heart of our theoriesof localization and completion. They play an important role throughout homo-topy theory. They are usually stated entirely in terms of homotopy commutativediagrams, but the greater precision that we describe will be helpful.

Lemma 1.2.1. Consider the following diagram, in which the left square com-mutes up to homotopy and the rows are canonical cofiber sequences.

Xf

Y i

Cf

X

X

fY

iC f

X.

There exists a map such that the middle square commutes and the right squarecommutes up to homotopy. If the left square commutes strictly, then there is aunique = C(, ) such that both right squares commute, and then the cofiber

sequence construction gives a functor from the category of maps and commutativesquares to the category of sequences of spaces and commutative ladders betweenthem.

Proof. Recall again thatC f=Y XC X, where the pushout is defined withrespect tof: XYand the inclusioni0 : XCXof the base of the cone. Leth : X IY be a (based) homotopy from f to f . Define(y) =(y)fory Y Cf, as required for commutativity of the middle square, and define

(x, t) =

h(x, 2t) if 0 t 1/2((x), 2t 1) if 1/2 t 1

for (x, t) CX. The homotopy commutativity of the right square is easily checked.When the left square commutes, we can and must redefine on CX by(x, t) =

((x), t) to make the right square commute. For functoriality, we have in mind theinfinite sequence of spaces extending to the right, as displayed in [91, p57], andthen the functoriality is clear.

Exercise 1.1.2 gives the following addendum, which applies to the comparisonof cofiber sequences in which the left hand squares display composite maps.

Addendum1.2.2. IfX=X, is the identity map, the left square commutes,andis a cofibration, then the canonical map : C fCf is a cofibration.


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1.2. THE FILL-IN AND VERDIER LEMMAS 17

It is an essential feature of Lemma 1.2.1 that, when the left square only com-mutes up to homotopy, the homotopy class of depends on the choice of thehomotopy and is not uniquely determined.

The dual result admits a precisely dual proof, where now the functorialitystatement refers to the infinite sequence of spaces extending to the left, as displayedin [91, p. 59]. Recall that F f=XYP Y, where the pullback is defined with respecttof : XY and the end-point evaluation p1 : P Y Y.

Lemma 1.2.3. Consider the following diagram, in which the right square com-mutes up to homotopy and the rows are canonical fiber sequences.

Y

F f

Xf

Y

Y

F f X

fY

There exists a map such that the middle square commutes and the left squarecommutes up to homotopy. If the right square commutes strictly, then there is aunique = F(, ) such that both left squares commute, hence the fiber sequenceconstruction gives a functor from the category of maps and commutative squares tothe category of sequences of spaces and commutative ladders between them.

Addendum1.2.4. IfY =Y, is the identity map, the right square commutes,andis a fibration, then the canonical map : F fF f is a fibration.

The addenda above deal with composites, and we have a dual pair of Verdierlemmas that encode the relationship between composition and cofiber and fibersequences. We shall not make formal use of them, but every reader should seethem since they are precursors of the basic defining property, Verdiers axiom, inthe theory of triangulated categories [92, 136].1

Lemma 1.2.5. Let h be homotopic to g f in the following braid of cofibersequences and letj = i(f) (g). There are mapsj andj such that the diagramcommutes up to homotopy, and there is a homotopy equivalence: CgCj such

1In [136], diagrams like these are written as octagons, with identity maps inserted. For

this reason, Verdiers axiom is often referred to in the literature of triangulated categories as

the octahedral axiom. In this form, the axiom is often viewed as mysterious and obscure.Lemmas 1.2.1 and 1.2.3 are precursors of another axiom used in the usual definition of triangulated

categories, but that axiom is shown to be redundant in [92].


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thatj i(j) andj =(j) .

X

f

h

Z

i(h)

i(g)

Cg

(g)

j

Cf

Y

g

i(f)

Ch

j

(h)

Y

i(f)

Cf

j

(f)

X

f

The square and triangle to the left of j and j commute; if h = g f, then there

are unique mapsj andj

such that the triangle and square to the right ofj andj

commute.

Proof. Let H: g f h. The maps j and j are obtained by application ofLemma 1.2.1 to Hregarded as a homotopy g f h idX and the reverse ofHregarded as a homotopy idZh g f. The square and triangle to the left ofj and

j are center squares of fill-in diagrams and the triangle and square to the right ofjandj are right squares of fill-in diagrams. The diagram commutes whenh = g fandj andj are taken to be

j = g id : Y fCXZhCX and j = id Cf : ZhCXZgC Y,

as in the last part of Lemma 1.2.1. Define to be the inclusion

Cg = ZgC Y (ZhCX) jC(Y fCX) = Ch j C Cf=Cj

induced byi(h): ZChand the map C Y C(Cf) obtained by applying thecone functor to i(f): Y Cf. Then j = (j) since (g) collapses Z to apoint, (j) collapses Ch = ZhC Xto a point, and both maps induce i(f) onCg/Z = Y. Using mapping cylinders and noting that j and j are obtained bypassage to quotients from maps j : M f M g and j : M g Mh, we see bya diagram chase that is an equivalence in general if it is so when h = g f. Inthis case, we claim that there is a deformation retraction r : C j Cg so thatr = id and r i(j) = j . This means that there is a homotopyk : C j ICjrelative toC g from the identity to a map into C g. In effect, looking at the explicitdescription ofC j, k deforms C CX toC XCY. The details are fussy and left tothe reader, but the intuition becomes clear from the observation that the quotientspaceCj/(Cg) is homeomorphic to the contractible spaceCX.

Remark1.2.6. There is a reinterpretation that makes the intuition still clearerand leads to an alternative proof. We can use mapping cylinders as in [91, p. 43]to change the spaces and maps in our given diagram so as to obtain a homotopyequivalent diagram in which f and g are cofibrations and h is the composite cofi-bration g f. As in [91, p. 58], the cofibers off, g , and h are then equivalent toY /X,Z/ Y, andZ/X, respectively, and the equivalencejust becomes the evidenthomeomorphismZ/ Y =(Z/X)/(Y /X).


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1.3. BASED AND FREE COFIBRATI ONS AND FIBRATIONS 19

Lemma1.2.7. Letfbe homotopic tohgin the following braid of fiber sequencesand letj =(g) p(h). There are mapsj andj such that the diagram commutesup to homotopy, and there is a homotopy equivalence : F j F g such that

j p(j) andj (j).

F h

p(h)

j

F g

j

p(g)

Z

g

f

X

Y

(g)

h

F f

p(f)

j

Y

h

X

(f)

(h)

F h

p(h)

Iff=h g, then there are unique mapsj andj such that the diagram commutes,and thencan be so chosen thatj = p(j) andj = (j).

1.3. Based and free cofibrations and fibrations

So far, we have been working in the category Tof based spaces, and we shallusually continue to do so. However, we often must allow the basepoint to vary,and we sometimes need to work without basepoints. Homotopies between maps ofunbased spaces, or homotopies between based maps that are not required to satisfyht() = , are often called free homotopies. In this section and the next, we areconcerned with the relationship between based homotopy theory and free homotopy

theory.Much that we have done in the previous two sections works just as well in thecategory Uof unbased spaces as in the category T. For example, using unreducedcones and suspensions, cofiber sequences work the same way in the two categories.However, the definition of the homotopy fiber F f, f: XY, requires the choiceof a basepoint to define the path space P Y. We have both free and based notionsof cofibrations and fibrations, and results such as Lemma 1.1.1 apply to both. Itis important to keep track of which notion is meant when interpreting homotopicalresults. For example, we understand free cofibrations and free fibrations in thefollowing useful result. It is the key to our approach to the Serre spectral sequence,and we shall have other uses for it. Recall that a homotopyh : X IXis saidto be a deformation ifh0 is the identity map ofX.

Lemma 1.3.1. Let p : E B be a fibration and i : A B a cofibration.Then the inclusionD = p1(A)E is a cofibration.

Proof. As in [91, p, 43], we can choose a deformation h of B and a mapu : B I that represent (B, A) as an NDR-pair. By the CHP, we can find adeformationH ofEthat covers h, p H=h (p id). Define a new deformationJ ofEby

J(x, t) =

H(x, t) ift u(p(x))H(x, u(p(x))) if t u(p(x)).


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ThenJ andu p represent (E, D) as an NDR-pair.

One of the many motivations for our standing assumption that basepoints are

nondegenerate is that it ensures that based maps are cofibrations or fibrations inthe free sense if and only if they are cofibrations or fibrations in the based sense.For fibrations, this is implied by the case (X, A) = (Y, ) of a useful analogue ofLemma 1.1.1 called the Covering Homotopy and Extension Property.

Lemma 1.3.2 (CHEP). Let i : A X be a free cofibration andp : E Bbe a free fibration. Letj : M i= X {0} A IX Ibe the inclusion of the

free mapping cylinderMi in the cylinderX I. For any commutative square

Mifh

j

E

p

X I

H

hB,

there is a homotopyH that makes the diagram commute.

Proof. Hereh : A IEis a a homotopy of the restriction off: XEtoA, andhis a homotopy ofpfwhose restriction toA is covered byh. The conclu-sion is a special case of the free version of Lemma 1.1.1 proven in Proposition 17.1.4since j is a cofibration and a homotopy equivalence by [91, p. 43].

Lemma 1.3.3. Letp : EB be a map between based spaces. Ifp is a basedfibration, thenp is a free fibration. Ifp is a free fibration andYis nondegeneratelybased, thenp satisfies the based CHP with respect to homotopiesY (I+)B.

Proof. For the first statement, we apply the based CHP to based homotopies(Y I)+ =Y+ I+ B to obtain the free CHP. For the second statement, wemust obtain lifts in diagrams of based spaces

Yg

i0

E

p

Y I+

fB

whenYis nondegenerately based, and these are supplied by the case (X, A) = (Y, )of the CHEP, with h constant at the basepoint ofE.

The following result, like the previous one, was stated without proof in [ 91, pp.56, 59]. Since its proof is not obvious (as several readers of [91] have complained),we give it in detail. Unfortunately, the argument is unpleasantly technical. 2

Lemma 1.3.4. Let i : A X be a map between based spaces. If i is a free

cofibration, theni is a based cofibration. IfA andXare nondegenerately based andi is a based cofibration, then i is a free cofibration.

Proof. The first statement is clear: since the basepoint is in A, free lifts inLemma 1.1.1 are necessarily based when the given maps are based. Thus assumethati is a based cofibration andA and Xare nondegenerately based. The problemhere is that the maps in a given test diagram for the HEP (as in [91, p. 41]) need

2The proofs of Lemmas 1.3.3 and 1.3.4 are due to Strm [129, p. 14] and [130, p. 440].


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1.3. BASED AND FREE COFIBRATI ONS AND FIBRATIONS 21

not preserve basepoints, and the based HEP only gives information when they do.One might try to deform the unbased data into new based data to which the basedHEP applies, but we shall instead check a slight variant of the NDR-pair criterionfori to be a free cofibration [91, p. 43].

Just as for free cofibrations, the fact that we are working with compactly gen-erated spaces ensures thati is a closed inclusion since the based mapping cylinderM i is a retract ofX I+. It suffices to prove that (X, A) is an NDR-pair. Thismeans that there is a map u : X Isuch that u1(0) = A and a deformationh ofX relative to A (so that h(x, 0) =x and h(a, t) =a) such that h(x, 1) A ifu(x) < 1. Inspection of the proof of the theorem on [91, p. 43] shows that if westart with a free cofibration i, then we obtain a pair (h, u) with the stronger prop-erty that h(x, t) A ifu(x) < t. It follows that the characterization theorem forfree cofibrations remains true if we redefine NDR-pairs by requiring this strongercondition, and we use the stronger condition throughout the proof that follows. Weproceed in three steps.

Step (i). Letw : X Ibe any map such that w1

(0) = . Then there isa deformation k : X I X relative to A and a map w : X I such thatw(x) w(x), A= (w)1(0), and

(1.3.5) k(x, t) A if w(x)< min(t, w(x)).

Proof. LetM+i= XA(A I) X Ibe the free mapping cylinder ofi,whereXis identified with X {0}. Define

M(w) = X {(a, t)|t w(a)} M+i.

The basepoint ofA Xgives M(w) a basepoint. The reduced mapping cylinderM i is obtained from M+iby collapsing the line{} Ito a point. Define a basedmapf: M i M(w) byf(x) =x for x Xandf(a, t) = (a, min(t, w(a))). Since

i is a based cofibration, fextends to a based map g : X I+ M(w). Define

k(x, t) =1g(x, t) and w(x) = sup{min(t, w(x)) 2g(x, t) | t I}

where 1 and 2 are the projections from M(w) toX and I. Clearly k (x, 0) =x,k(a, t) = a for a A, w(x) w(x), and w(a) = 0 for a A, so that A (w)1(0). To see that this inclusion is an equality, suppose that w (x) = 0. Thenmin(t, w(x)) 2g(x, t) for all t I. Ifk (x, t) /A for anyt, then2g(x, t) = 0, sothatt = 0 or w(x) = 0. Ifw(x) = 0, then x = and k (x, t) = A. We concludethat ift >0, then k(x, t) A. Since A is closed in X, it follows thatk (x, 0) =xis also in A. Finally, (1.3.5) holds since k (x, t) /A implies 2g(x, t) = 0 and thusw(x) min(t, w(x)).

Step (ii). There is a representation (, z) of (X, ) as an NDR-pair such that(A I) A.

Proof. Let (kX , wX) and (kA, wA) represent (X, ) and (A, ) as NDR-pairs,where kX(x, t) = ifwX(x)< t and kA(a, t) = ifwA(a) < t. Sincei is a basedcofibration, we may regard kA as taking values in Xand extend it to a based map

kA : X I+ X


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such thatkA(x, 0) = x. Construct (k, w) from wX as in Step (i) and extend wAto wA : XIby

wA(x) = 1 w(x)wX(x)

wA(k(x, 1)) + w(x).We interpret this as w(x) when w(x) = wX(x); in particular, since wX() = 0,wA() = 0. The definition makes sense since k(x, 1) A if w(x) < wX(x), by(1.3.5). We claim that w1A (0) = . Thus suppose that wA(x) = 0. Thenw

(x) = 0,so that x A, and this implies that k(x, 1) = x and wA(x) = 0, so that x = .The required pair (, z) is now defined by

(x, t) =

kA(x,t/wA(x)) if t wA(x)

kX(kA(x, 1), t wA(x)) if t > wA(x)

and

z(x) = min(1, wA(x) + wX(kA(x, 1))).

To see that (A I) A, recall thatkA and wA extendkA andwA and let x A.ClearlykA(x, t) A for all t. Ift > wA(x), then wA(x)< 1 and thus kA(x, 1) = .ThereforekX(kA(x, 1), t wA(x)) = A. Note that(x, t) = ifz (x)< t.

Step (iii). Completion of the proof. Constructh : XIXand z : XIby applying step (i) to z : XI. Then define

h(x, t) =

(h(x, t), min(t, z(x)/z(x))) if x = ifx =

and

u(x) =z (x) z(x) + sup{z(h(x, t)) | t I}.

Then (h, u) represents (X, A) as an NDR-pair (in the strong sense).

1.4. Actions of fundamental groups on homotopy classes of mapsFor based spacesXandY, let [X, Y] denote the set of based homotopy classes

of based maps XY. For unbased spacesXandY, let [X, Y]free denote the setof free homotopy classes of maps X Y . If we choose basepoints for X and Yand use them to define [X, Y], then we obtain a function [X, Y] [X, Y]free byforgetting the basepoints. There is a classical description of this function in termsof group actions, and we describe that and related material on group actions inhomotopy theory in this section.

We will be considering varying basepoints in a given space, and we want them allto be nondegenerate. Therefore, we tacitly restrict attention to spaces which havethat property. This holds for locally contractible spaces, such as CW complexes.

Definition 1.4.1. Let X and Y be spaces. As usual, we assume that they

are connected and nondegenerately based. Leti : Xbe the inclusion of thebasepoint andp : Y be the trivial map. A loop based at Yand a basedmap f: X Y , give a map f : M i = X I Y . Applying Lemma 1.3.2to i, p, and f , we obtain a homotopy h : X I Y such that h0 = f andh(, t) = (t). Another use of Lemma 1.3.2 shows that the (based) homotopy classof h1 depends only on the path class [] and the homotopy class [f]. With theusual conventions on composition of loops [91, p. 6], the definition [][f] = [h1]gives a left action of1(Y, ) on the set [X, Y] with orbit set [X, Y]/1(Y, ). It is


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1.4. ACTIONS OF FUNDAMENTAL GROUPS ON HOMOTOPY CLASSES OF MAPS 23

clear that two based maps that are in the same orbit under the action of1(Y, )are freely homotopic.

Lemma 1.4.2. The induced function[X, Y]/1(Y)[X, Y]free is a bijection.Proof. By HEP, any mapXYis freely homotopic to a based map. If two

based mapsf andg are freely homotopic via a homotopyh, then the restriction ofh to {} Igives a loop , and [g] = [][f] by the definition of the action.

An H-space is a based space Y with a product : Y Y Y, written x yor by juxtaposition, whose basepoint is a two-sided unit up to based homotopy.That is, the mapsy y and y y are both homotopic to the identity map.Equivalently, the composite of the inclusion Y Y Y Y and the product ishomotopic to the fold map : Y Y Y, which restricts to the identity mapon each wedge summand Y. Using our standing assumption that basepoints arenondegenerate, we see that the given product is homotopic to a product for whichthe basepoint ofY is a strict unit. Therefore, we may as well assume henceforward

thatH-spaces have strict units.

Proposition 1.4.3. For an H-space Y, the action of 1(Y, ) on [X, Y] istrivial and therefore [X, Y]=[X, Y]free.

Proof. For a map f: X Y and a loop based at Y, the homotopyh(x, t) = (t) f(x) satisfies h1 = f. Using this choice of homotopy in Defini-tion 1.4.1, as we may, we see that [][f] = [f].

The following definition hides some elementary verifications that we leave tothe reader.

Definition 1.4.4. Take X =Sn in Definition 1.4.1. The definition then spe-cializes to define an action of the group 1(Y, ) on the groupn(Y, ). Whenn = 1,this is the conjugation action of1(Y, ) on itself. A (connected) space Y is simple

if1(Y, ) is abelian and acts trivially on n(Y, ) for all n 2.

Corollary 1.4.5. AnyH-space is a simple space.

In the rest of this section, we revert to the based context and consider extrastructure on the long exact sequences of sets of homotopy classes of (based) mapsthat are induced by cofiber and fiber sequences. These long exact sequences aredisplayed, for example, in [91, pp. 57, 59]. As observed there, the sets [X, Y]are groups ifXis a suspension or Y is a loop space and are abelian groups ifXis a double suspension or Y is a double loop space. However, there is additionalstructure at the ends of these sequences that will play a role in our work. Thus,for a based map f : XYand a based space Z, consider the exact sequence ofpointed sets induced by the canonical cofiber sequence off:

[Y, Z] (f)

[X, Z]

[Cf,Z] i

[Y, Z] f

[X, Z].

Lemma 1.4.6. The following statements hold.

(i) The group[X, Z] acts from the right on the set[Cf,Z].

(ii) : [X, Z] [Cf,Z] is a map of right[X, Z]-sets.

(iii) (x) =(x) if and only ifx= (f)(y) x for somey [Y, Z].

(iv) i(z) = i(z) if and only ifz = z x for somex [X, Z].


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(v) The image of [2X, Z] in[Cf,Z] is a central subgroup.

(In (iii) and (iv), we have used the notation to indicate the action.)

Proof. In (i), the action is induced by applying the contravariant functor[, Z] to the coaction Cf Cf X of X on Cf that is specified bypinching X {1/2} Cfto a point (and of course linearly expanding the halfintervals of the resulting wedge summands homeomorphic to Cf and X to fullintervals). Then (ii) is clear since the quotient map Cf X commutes withthe pinch map. To show (iii), observe that since [X, Z] is a group, (ii) impliesthat(x) = (x) if and only if(x (x)1) = . By exactness, this holds if andonly ifx (x)1 = (f)(y) and thus x = (f)(y) x for some y [Y, Z]. For(iv), ifi(z) = i(z), then the HEP for the cofibrationi implies that the homotopyclassesz and z can be represented by mapsc and c fromC f toZthat restrict tothe same map onY. Thenz = z x, wherex is represented by the map XZthat is obtained by regarding Xas the union of upper and lower cones on X,

using c|CX with cone coordinate reversed on the lower cone, and using c|CX onthe upper cone.

Finally, for (v), let G = [Cf,Z] and let H G be the image of [2X, Z].The suspension of the pinch map C fC f Xgives a right action ofH onG, a priori different from the product in G. We may obtain the product, gg say,on G from the pinch map defined using the suspension coordinate of Cf. Thenthe usual proof of the commutativity of [2X, Z] applies to show that hg = gh forh H and g G. In detail, if 1 H is the identity element, then, for g, g Gandh, h H,

1 h= h, g 1 = g, and (g h)(g h) = (gg ) (hh).

Therefore

gh = (g 1)(1 h) = (g1) (1h) = g h= (1g) (h1) = (1 h)(g 1) = hg.

Dually, consider the exact sequence of pointed sets induced by the canonicalfiber sequence off : XY:

[Z, X](f) [Z, Y]

[Z , F f ] p [Z, X]

f [Z, Y].

This is of greatest interest when Z = S0. Since [, ] refers to based homotopyclasses, [S0, X] = 0(X) and the sequence becomes

1(X) f 1(Y)

0(F f) p 0(X)

f 0(Y).

Lemma 1.4.7. The following statements hold.

(i) The group[Z, Y] acts from the right on the set [Z , F f ].

(ii) : [Z, Y][Z , F f ] is a map of right [Z, Y]-sets.

(iii) (y) = (y) if and only ify= (f)(x) y for somex [Z, X].

(iv) p(z) = p(z) if and only ifz = z y for somey [Z, Y].

(v) The image of [Z, 2Y] in [Z, F f] is a central subgroup.


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1.5. ACTIONS OF FUNDAMENTAL GROUPS IN FIBRATION SEQUENCES 25

1.5. Actions of fundamental groups in fibration sequences

It is more usual and more convenient to think of Lemma 1.4.7 in terms of

fibrations. That is, instead of starting with an arbitrary map f, in this section westart with a fibrationp : EB. We give some perhaps well-known (but hard tofind) results about fundamental group actions in fibrations.

Observe that a fibration p : EB need not be surjective,3 but either everypoint or no point of each component ofB is in the image ofE. For nontriviality,we assume that p is surjective. We may as well assume that B is connected,since otherwise we could restrict attention to the components ofEover a chosencomponent ofB . If we were given a general surjective map p : EB , we wouldreplaceEby its mapping path fibrationN p(which depends on a choice of basepointinB) to apply the results to follow. That is, we can think of applying the results tothe fibrationN pB as applying them to the original map p up to homotopy.

Although we have been working in the based context, we now think ofp as afree fibration and let the basepointb B vary. We choose a point e Esuch that

p(e) = band we letFb= p1(b). Also letEe be the component ofEthat containseandFebe the component ofFbthat containse. We view these as based spaces withbasepoint e (as recorded in the notation), retaining our standing assumption thatbasepoints are nondegenerate. As in [91, p. 64], we then have the exact sequence

(1.5.1)

n(Fe, e) n(Ee, e)

p n(B, b) n1(Fe, e)

1(Ee, e) p 1(B, b)

0(Fb, e) 0(E, e).

Notice that we have replaced E and Fb by Ee and Fe in the higher homotopygroups. Since the higher homotopy groups only depend on the component of thebase point this doesnt change the exact sequence.

By [91, p. 52], there is a functor = (B,p) : B HoUthat sends a pointbof the fundamental groupoid B to the fiber Fb. This specializes to give a grouphomomorphism1(B, b)0(Aut(Fb)). Here Aut(Fb) is the topological monoidof (unbased) homotopy equivalences ofFb. We think of1(B, b) as acting up tohomotopy on the space Fb, meaning that an element 1(B, b) determines awell-defined homotopy class of homotopy equivalencesFb Fb. (Ifp is a coveringspace, the action is by homeomorphisms, as in [91, p. 29].) As we shall use later,it follows that 1(B) acts on the homology and cohomology groups ofFb.

Observe that we can apply this to the path space fibration Y P Y Yof a based space Y. We thus obtain an action of 1(Y, ) on Y. Since Y issimple, by Proposition 1.4.3, [Y, Y]=[Y, Y]free and we can view 1(Y, ) asacting through basepoint preserving homotopy equivalences of the fiber Y. Thereresults another action of1(Y, ) on n(Y, ). We leave it to the reader to check

that this action agrees with that defined in Definition 1.4.4.We will need a more elaborate variant of the functor = (B,p) : B HoU.

Here, instead of starting with paths inB , we start with paths in Eand work withcomponents of the fibers and the total space, regarded as based spaces. Thus let : IEbe a path frome to e. Letb = p(e) andb =p(e), and let= p bethe resulting path from b to b . Consider the following diagram, in which F = Fe

3In [91, p. 47], fibrations were incorrectly required to be surjective maps.


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1.5. ACTIONS OF FUNDAMENTAL GROUPS IN FIBRATION SEQUENCES 27

The naturality and homotopy invariance statements proven for (B,p) in [91,pp. 52-53] apply with obvious changes of statement to (E,p). We write r for thetrivial fibrationr : Y for any spaceY. Whenp= r in the construction above,the resulting action of1(Y, ) on n(Y, ) agrees with that discussed earlier. Ineffect, in the based context, taking X=Y in Definition 1.4.1 we see that we usedthis construction to give our original definition of fundamental group actions. Themore concrete construction Definition 1.4.4 is then obtained via mapsSn Y.

Proposition1.5.4. The long exact sequence (1.5.1), ending at1(B, b), is anexact sequence of1(E, e)-groups and therefore of1(F, e)-groups. In more detail,the following statements hold.

(i) Forj 1(Fb, e) andx n(Fb, e), jx = (j)x.

(ii) Forg 1(E, e) andz n(B, b), gz = p(g)z.

(iii) Forg 1(E, e) andx n(Fb, e), (gx) = g(x)

(iv) Forg 1(E, e) andy n(E, e), p(gx) = gp(x).

(v) Forg 1(E, e) andz n(B, b), (gz) = g(z).

Proof. We define the actions of1(E, e) on n(Fb, e), n(E, e) and n(B, b)to be those given by (E,p), (E,r) and(B,r) p, respectively. We let1(F, e) acton these groups by pull back along : 1(Fe, e)1(E, e). By (i) and inspection,this implies that its actions are given by (Fe,r), (E,r) , and the trivial action,respectively. The maps in the exact sequence are maps of1(E, e)-groups by (iii),(iv), and (v).

By restricting the construction of (E,p) to loops : I Fe, we see that(Fb,r)[] =(E,p)[ ]. This implies part (i), and part (ii) is immediate from thedefinition of the action of1(E, e) on1(B, b). Part (iv) holds by the naturality ofthe action of1 on n, which applies to any map, not just a fibration such as p.

To prove (iii), we use the notations of the diagram on the previous page. Since : F Eis a cofibration, by Lemma 1.3.1, we can apply the CHEP to the diagram

M

id E

p

E I 2

k

I

B

to obtain a deformation k : E I E. Since 1 : Fb Fb E represents

(E,p)([]) and this map is k1 , it also represents(E,r) . This implies (iii).To prove (v), recall from [91, p. 64] that : n(B, b) n1(F, e) is the

composite of the inverse of the isomorphism p : n(E , F , e) n(B, b) and theboundary map of the pair (E, F), which is obtained by restricting representativemaps (Dn, Sn1) (E, F) to Sn1 F. Let be a loop at e in E thatrepresentsg and let p(e) = b and p = . Let h : B IB be a deformation(h0 = id) such thath(b, t) = (t). As in Definition 1.4.1, for a based map f: XB, [][f] = [h1]. To describe this action in terms of the pair (E, F), use the CHEP


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to obtain lifts j and k in the following diagrams.

Fe {0} {e} I

E

p

F I

j 2

I

B

and Mid j

E

p

E I

k pid

B Ih

B.

Then (k, j): (E, F) I (E, F) is a relative homotopy that restricts to j andcovers h. Precomposing with a map z : (Dn, Sn1) (E, F) that representsz 1(B, b), the compositeh1zrepresents []z, which isgz = p(g)z. Its restrictionto Sn1 represents(gz) and, by the first of the above pair of diagrams, it alsorepresentsg(z).

Remark1.5.5. Consider (1.5.1). Lemma 1.4.7 implies that0(Fb, e) is a right1(B, b)-set and : 1(B, b) 0(Fb, e) is a map of1(B, b)-sets and thereforeof1(E, e)-sets. However, we will be especially interested in the next step to the

left. Assume that the image of p : 1(E, e) 1(B, b) is a normal subgroup.Then cokerp is a group contained in the based set0(Fb, e) of components ofFbwith base component Fe (later sometimes denoted [e]). We denote this group by0(Fb, e) and have the exact sequence

1(E, e) p 1(B, b)

0(Fb, e) i .

It is an exact sequence of 1(E, e)-groups and thus of 1(Fb, e)-groups, where1(Fb, e) acts trivially on the last two groups.


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CHAPTER 2

Homotopy colimits and homotopy limits; lim1

The material of this chapter is again of general interest. We describe the mostbasic homotopy colimits and limits, with focus on their precise algebraic behavior.In particular, we develop the dual homotopical lim1 exact sequences. We shall notgo into the general theory of homotopy colimits, but the material here can serveas an introduction to such more advanced sources as [44, 65, 126]. In this book,homotopy limits, and especially homotopy pullbacks, will play a central role in the

fracture theorems of Chapters 8 and 13.In 3 and 4, we describe the algebraic properties of the functor lim1 and give

a concrete topological example where nontrivial lim1 terms appear. In5 and 6,we give some observations about the homology of filtered colimits and sequentiallimits, and advertise a kind of universal coefficient theorem for profinite abeliangroups. While1 and2 are vital to all of our work, the later sections play a moreperipheral role and need only be skimmed on a first reading.

2.1. Some basic homotopy colimits

Intuitively, homotopy colimits are constructed from ordinary categorical col-imits by gluing in cylinders so as to give domains for homotopies that allow us toreplace equalities between maps that appear in the specification of ordinary colim-

its by homotopies between maps. There is always a natural map from a homotopycolimit to the corresponding ordinary colimit, and in some but not all cases there isa convenient criterion for determining whether or not that natural map is a homo-topy equivalence. Since homotopies between given maps are not unique, not evenup to homotopy, homotopy colimits give weak colimits in the homotopy category,in the sense that they satisfy the existence but not the uniqueness property of or-dinary colimits. We shall spell out the relevant algebraic property quite preciselyfor homotopy pushouts (or double mapping cylinders), homotopy coequalizers (ormapping tori), and sequential homotopy colimits (or telescopes). We record theanalogous results for the constructions Eckmann-Hilton dual to these in the nextsection. We work in the based context, but the unbased analogues should be clear.

Definition2.1.1. The homotopy pushout (or double mapping cylinder) M(f, g)of a pair of maps f: A Xandg : A Y is the pushout written in alternativenotations as

(X Y) AAA I+ or Xf(A I+) gY.

It is the pushout defined with respect to f g : A AX Yand the cofibration(i0, i1): A AA I+.

Explicitly, with the alternative notation, we start with X (A I+) Y andthen identify (a, 0) withf(a) and (a, 1) withg(a). In comparision with the ordinarypushoutXAY, we are replacing A by the cylinder A I+. Except that the line

29


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30 2. HOMOTOPY COLI MITS AND HOMOTO PY L IMITS; lim1

Ithrough the basepoint should be collapsed to a point, the following pictureshould give the idea.

f(A)

X

g(A) YA I

Proposition2.1.2. For any spaceZ, the natural map of pointed sets

[M(f, g), Z][X, Z] [A,Z][Y, Z]

is a surjection. Its kernel is isomorphic to the set of orbits of [A, Z] under theright action of the group [X, Z] [Y, Z] specified by

a (x, y) = (f)(x)1 a (g)(y)

fora [A, Z], x [X, Z], andy [Y, Z].

Proof. The pullback in the statement is the set of homotopy classes ([], [])in [X, Z] [Y, Z] such that [] and [] have the same image in [A, Z], which meansthat fis homotopic to g.

We have an evident cofibration j : XY M(f, g). As in [91, p. 57], itgives rise to an exact sequence of pointed sets

[X, Z] [Y, Z] [Cj,Z]

i [M(f, g), Z] j [X, Z] [Y, Z].

If : XZand : Y Zare such thatfg, then any choice of homotopyA I+ Z determines a map : M(f, g) Z that restricts to and onX and Y . Thus j induces a surjection onto the pullback in the statement of theresult. That is, for homotopy classes [] and [] such that f[] = g [], there isa homotopy class [], not uniquely determined, such thatj [] = ([], []). This iswhat we mean by saying that the homotopy colimit M(f, g) is a weak pushout off and g in HoT.

By Lemma 1.4.6(iv), ker j = im i is the set of orbits of [A, Z] under theaction of [X, Z] [Y, Z] specified there. Sincej is a cofibration, the canonical

quotient map : C

more concise algebraic topology ~ may

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