Universitext

Editors

F.W. Gehring P.R. Halmos

Universitext

Editors: F.W. Gehring, P.R. Halmos

Booss/Bleecker: Topology and Analysis Charlap: Bieberbach Groups and Flat Manifolds Chern: Complex Manifolds Without Potential Theory Chorin/Marsden: A Mathematical Introduction to Fluid Mechanics Cohn: A Classical Invitation to Algebraic Numbers and Class Fields Curtis: Matrix Groups, 2nd. ed. van Dalen: Logic and Structure Devlin: Fundamentals of Contemporary Set Theory Edwards: A Formal Background to Mathematics I alb Edwards: A Formal Background to Higher Mathematics II alb Endler: Valuation Theory Frauenthal: Mathematical Modeling in Epidemiology Gardiner: A First Course in Group Theory Godbillon: Dynamical Systems on Surfaces Greub: Multilinear Algebra Hermes: Introduction to Mathematical Logic Hurwitz/Kritikos: Lectures on Number Theory Kelly/Matthews: The Non-Euclidean, The Hyperbolic Plane Kostrikin: Introduction to Algebra Luecking/Rubel: Complex Analysis: A Functional Analysis Approach Lu: Singularity Theory and an Introduction to Catastrophe Theory Marcus: Number Fields McCarthy: Introduction to Arithmetical Functions Meyer: Essential Mathematics for Applied Fields Moise: Introductory Problem Course in Analysis and Topology 0ksendal: Stochastic Differential Equations Porter/Woods: Extensions of Hausdorff Spaces Rees: Notes on Geometry Reisel: Elementary Theory of Metric Spaces Rey: Introduction to Robust and QuasicRobust Statistical Methods Rickart: Natural Function Algebras Schreiber: Differential Forms Smorynski: Self-Reference and Modal Logic Stanisic: The Mathematical Theory of Turbulence Stroock: An Introduction to the Theory of Large Deviations Sunder: An Introduction to von Neumann Algebras Tolle: Optimization Methods

Leonard S. Charlap

Bieberbach Groups and Flat Manifolds

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo

Leonard S. Chariap Institute for Defense Analysis Communications Research Division Princeton, NJ 08540

AMS Subject Classification: 30FIO, 30F35

Library of Congress Cataloging in Publication Data Charlap, Leonard S.

Bieberbach groups and flat manifolds. (Universitext) Bibliography: p. Includes index. I. Riemann surfaces. 2. Riemannian manifolds.

3. Functions, Automorphic. 4. Bieberbach groups. I. Title. QA333.C46 1986 516.3'62 86-15615

© 1986 by Springer-Verlag New York Inc.

All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names. as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

9 8 765 4 3 2 I

ISBN -13: 978-0-387-96395-2 DOl: 10.1007/978-1-4613-8687-2

e-ISBN -13: 978-1-4613-8687-2

To Ann

Preface

Many mathematics books suffer from schizophrenia, and this is yet

another. On the one hand it tries to be a reference for the basic results on

flat riemannian manifolds. On the other hand it attempts to be a textbook

which can be used for a second year graduate course. My aim was to keep

the second personality dominant, but the reference persona kept breaking

out especially at the end of sections in the form of remarks that contain

more advanced material. To satisfy this reference persona, I'll begin by

telling you a little about the subject matter of the book, and then I'll talk

about the textbook aspect.

A flat riemannian manifold is a space in which you can talk about

geometry (e.g. distance, angle, curvature, "straight lines," etc.) and, in

addition, the geometry is locally the one we all know and love, namely

euclidean geometry. This means that near any point of this space one can

introduce coordinates so that with respect to these coordinates, the rules

of euclidean geometry hold. These coordinates are not valid in the entire

space, so you can't conclude the space is euclidean space itself. In this

book we are mainly concerned with compact flat riemannian manifolds,

and unless we say otherwise, we use the term "flat manifold" to mean

"compact flat riemannian manifold."

It turns out that the most important invariant for flat manifolds is the

fundamental group. In fact, these spaces are "classified" by their funda­

mental groups. Furthermore, these groups turn out to have many inter­

esting algebraic properties. These groups are the Bieberbach groups of the

title. They are torsionfree groups which have a maximal abelian subgroup

of finite index which is free abelian. They a.re groups which are a little

bit more complicated than free abelian groups. In recent years the groups

have come to be studied somewha.t more than the manifolds themselves,

which is why they were awarded the first place in the title.

As you might have already guessed, the main results here are due to

viii Preface

Bieberbach who published his work in about 1910. His work was the so­

lution to Hilbert's 18th problem. The results can be summed up in three

theorems. I'll state them very roughly and not at all the way Bieberbach

thought of them. The first says that the fundamental group of a fiat man­

ifold is a Bieberbach group. You sometimes hear the theorem stated that

a fiat manifold is covered by a torus, which comes to the same thing. The

second says that two fiat manifolds with the same fundamental groups are

the same (e.g. homeomorphic). The third says that up to an appropriate

equivalence there are only finitely many fiat manifolds in each dimension.

Bieberbach's point of view was that of rigid motions, i.e. those trans­

formations of euclidea.n space that preserve distance. Our first chapter is

devoted to these rigid motions and to the proofs of Bieberbach's three the­

orems in this framework. The second chapter gives some background so

we can state these theorems in the framework of fiat manifolds. The third

chapter is concerned with outlining a scheme for classifying Bieberbach

groups and (consequently) fiat manifolds. The fourth chapter carries out

this classification in a special case, namely the case when the index of the

free abelian subgroup of the Bieberbach group is prime. The fifth chapter

tells about the automorphisms of Bieberbach groups and fiat manifolds.

Why do I think that this mathematics is peculiarly suited for a second

year graduate course? First of all, it is relatively elementary. Students with

the standard first year graduate program under their belts should be able

to handle the material contained herein. I have tried to present reasonably

complete versions of any results that are not contained in such a first year

program.

The main reason I think this is "good rich stuff" is because of its

interdisciplinary character. If you glance at the table of contents you can

easily see what I mean. There are sections on differential topology, algebraic

number theory, riemannian geometry, cohomology of groups, and integral

representations. Too often graduate students believe that mathematics

divides up into neat segments labeled "complex variables," or "algebraic

topology," or "group theory," for example. Of course, they get this idea

because they take courses in which the subject matter is divided up in

precisely this fashion. When they start to work on a thesis problem, they

are sometimf'.8 quite surprised to discover that a problem in one area may

lead them into a quite different area. of mathematics. I thought it might be

Preface ix

advisable to show them this phenomenon relatively early in their graduate

studies, and this is the main reason I have written this book.

Another property this book shares with a number of other mathemat­

ics books is that it is "complex" in the sense that it is in two volumes,

the first one (this one) is real while the second one is currently imaginary.

I had hoped to have included material on the Hochschild-Serre spectral

sequence, on the cobordism, cohomology, and K-theory of fiat manifolds,

and on the spectrum of fiat manifolds. After a while, it began to appear

that the book was growing too long to serve its main purpose, namely being

a text, so I decided to leave this more advanced material for a projected

second volume.

Now I suppose I should give the usual admonishment about doing the

exercises. You don't really have to do all the exercises to appreciate the

material. You should at least read them. Some are more important than

others. Some merely point to nice extensions of various concepts while

others are crucial to the proof of important theorems. Concerning the

exercises that are embedded in proofs, I have tried to select those parts of

proofs which would be boring or repetitious and turned them into exercises.

This will, I hope, make them a little more interesting and give you, the

reader, some idea of whether you are really following the argument.

I am glad to be able to thank a number of institutions for their support

in the writing of this book. The State University of New York at Stony

Brook gave me a sabbatical leave during the Fall term of 1982 during which

time I began the manuscript. Harvard University provided facilities and a

congenial atmosphere during the academic year 1982-83. The Communi­

cations Research Division of the Institute for Defense Analyses graciously

allowed me to finish the book in 1984 and 1985.

I would like to thank my collaborators, Al Vasquez and Han Sah, not

only for the usual "important comments and interesting conversations,"

but also for solving many of the problems in the subject that I found too

difficult. In addition some of the material in Chapter V is joint work with

Han Sah and is appearing here for the first time. I would also like to

thank my teacher Jim Eells who did not merely "suggest" that I study

flat manifolds. In a course in differential geometry at Columbia, he "as­

signed" to me the task oflearning all there was to know about flat mani­

folds. (Fortunately less was known then than is known now.) A number of

x Preface

friends and colleagues have read portions of the manuscript in various of

its forms and made comments, some of which ha.ve saved me from making

egregious errors. Among them are William Goldman, Norman Herzberg,

David Lieberman, and David Robbins. Lance Carnes of PCTEX gave in­

valuable assistance in the type setting. In addition, I would like to thank

Ann Stehney for meticulous proofreading, knowledgeable comments, and

comprehensive support. It is unlikely this book would exist at all without

her very considerable contribution. Finally I will take full responsibility for

all remaining obfuscations· and errors (egregious or not).

Leonard S. Charlap

Princeton, Fall 1985

Table of Contents

Chapter I. Bierberbach's Three Theorems

1. Rigid Motions . . . . . . .

2. Examples . . . . .. . . .

3. Bierberbach's First Theorem

4. Bierberbach's Second Theorem

5. Digression - Group Extensions

6. Digression - Integral Repesentations of Finite Groups

7. Bieberbach's Third Theorem and Some Remarks ...

Chapter II. Flat Riemannian Manifolds

1. Introduction ......... .

2. A Tiny Bit of Differential Topology

3. Connections and Curvature

4. Riemannian Structures

5. Flat Manifolds. . . .

6. Conjectures and Counterexamples

Chapter III. Classification Theorems

1. The Algebraic Structure of Bieberbach Groups

2. A General Classification Scheme for Bieberbach Groups

3. Digression - Cohomology of Groups

4. Examples . . . .

5. Holonomy Groups

1

5

10

18

20

34

40

43

43

45

58

60

65

74 77

82

90

99

xii Table of Contents

ChapterIV. Holonomy Groups of Prime Order

1. Introduction

2. Digression - Some Algebraic Number Theory

3. Modules over the Cyclotomic Ring . . . . .

4. Modules over Groups of Prime Order. . . .

5. The Cohomology of Modules over Groups of Prime Order

104

104

117

126

136

139

152

154

163

6. The Classification Theorem

7. ~p-manifolds

8. An Interesting Example

9. The Riemannian Structure of Some ~p-manifolds

Chapter V. Automorphisms

1. The Basic Diagram ........ 167

2. The Hochschild-Serre Exact Sequence 173

3. 9-Diagrams . . . . . . . . . . . . 182

4. Automorphisms of Group Extensions . 199

5. Automorphisms of Bieberbach Groups 204

6. Automorphisms of Flat Manifolds 211

7. Automorphisms of ~p-manifolds . 220

Bibliography . . . . . . . . . . . . . . . . . . . . . . . 233

Index. . . . . . . . . . . . . . . . . . . . . . . . . .. 240

Some Notes on Notation

This book was typeset in my home using PC'IEX on a Tandy 2000

computer. This gave me vastly more freedom to use different notation

than one would ordinarily get with a typewriter. It is quite possible that I

have yielded to temptation and abused this freedom. There are a number

of mat.hematical abbreviations and special symbols that I use in my own

writing, and I have taken the liberty to use them here. Some are quite

standard (V, 3, iff, etc.), and I won't comment on them here. The others

are quite obvious ("s.t.", "w.r.t.", etc.),so I won't say anything about them

either.

I have had a great deal of difficulty (the right word is really "tsoris")

deciding on a method of numbering equations, theorems, exercises, etc.,

and the method I finally chose is far from satisfactory to me. Equation

numbers are in parentheses and are started over at the beginning of each

chapter. All the rest are numbered with a decimal number (e.g. 3.4) where

the part to the left of the decimal point indicates the section, and the part

to the right is numbered consecutively within that section, e.g. Theorem

3.4 refers to the fourth theorem in the third section of the current chapter.

I would have preferred a "Garden State Parkway" system of numbering

in which an object is numbered by the page number on which it appears.

I had to give this up because of the fact the more than one exercise would

frequently appear on a page.