Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo

Universitext Editorial Board

(North America):

S. Axter F.W. Gehring

K.A. Ribet

Universitext

Editors (North America): S. Axler, F.W. Gehring, and K.A. Ribet

Aguilar/GiderlPrieto: Algebraic Topology from a Homotopical Viewpoint Aksoy/Khamsi: Nonstandard Methods in Fixed Point Theory Andersson: Topics in Complex Analysis Aupetit: A Primer on Spectral Theory BachmanlNaricilBeckenstein: Fourier and Wavelet Analysis Badescu: Algebraic Surfaces BalakrisbnanlRanganathan: A Textbook of Graph Theory Balser: Formal Power Series and Linear Systems of Meromorphic Ordinary

Differential Equations Bapat: Linear Algebra and Linear Models (2nd ed.) Berberian: Fundamentals of Real Analysis BoltyanskiilEfremovich: Intuitive Combinatorial Topology. (Shenitzer, trans.) BoossIBleecker: Topology and Analysis Dorkar: Probability Theory: An Advanced Course BOttcherlSilbennann: Introduction to Large Truncated Toeplitz Matrices CarlesonlGamelin: Complex Dynamics Cecil: Lie Sphere Geometry: With Applications to Submanifolds Cbae: Lebesgue Integration (2nd ed.) Cbarlap: Bieberbach Groups and Flat Manifolds Chern: Complex Manifolds Without Potential Theory Cohn: A Classical Invitation to Algebraic Numbers and Class Fields Curtis: Abstract Linear Algebra Curtis: Matrix Groups Debarre: Higher-Dimensional Algebraic Geometry Deitmar: A First Course in Harmonic Analysis DiBenedetto: Degenerate Parabolic Equations Dimca: Singularities and Topology of Hypersurfaces Edwards: A FormaI Background to Mathematics I alb Edwards: A Formal Background to Mathematics n alb Farenick: Algebras of Linear Transformations Foulds: Graph Theory Applications Friedman: Algebraic Surfaces and Holomorphic Vector Bundles Fuhrmann: A Polynomial Approach to Linear Algebra Gardiner: A FIrSt Course in Group Theory GhdinglTambour: Algebra for Computer Science Goldblatt: Orthogonality and Spacetime Geometry GustafsonlRao: Numerical Range: The Field of Values of Linear Operators

and Matrices Hahn: Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups Heinonen: Lectures 00 Analysis on Metric Spaces Holmgren: A First Course in Discrete Dynamical Systems HoweITan: Non-Abelian Harmonic Analysis: Applications of S42, R) Howes: Modern Analysis and Topology HsiehlSibuya: Basic Theory of Ordinary Differential Equations HumiIMiller: Secood Course in Ordinary Differential Equations HurwitzlKritikos: Lectures 00 Number Theory Jennings: Modern Geometry with Applications

(continued after index)

Marcelo Aguilar Samuel Gitler Carlos Prieto

Algebraic Topology from a Homotopical Viewpoint

, Springer

Marcelo Aguilar Instituto de Matematicas Universidad Nacional

Aut6noma de Mexico 04510 Mexico, DF Mexico marcelo@math.unam.mx

Editorial Board (North America):

Samuel Gitler Departamento de Matematicas Centro de Investigaci6n y de

Estudios A vanzados A.P.I4-740 07000 Mexico, DF Mexico sgitler@math.cinvestav.mx

Carlos Prieto Instituto de Matematicas Universidad Nacional

Aut6noma de Mexico 04510 Mexico, DF Mexico cprieto@math.unam.mx

S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA

F.W. Gehring Mathematics Department East Hall

axler@sfsu.edu

K.A. Ribet Mathematics Department University of California, Berkeley Berkeley, CA 94720-3840 USA ribet@math.berkeley.edu

University of Michigan Ann Arbor, Ml48109-1109 USA fgehring@math.lsa.umich.edu

Mathematics Subject Classification (2000): 55-0 I

Library of Congress Cataloging-in-Publication Data Aguilar, M.A. (Marcelo A.)

Algebraic topology from a homotopical viewpoint I Marcelo Aguilar, Samuel Gitler, Carlos Prieto.

p. cm. - (Universitext) Includes bibliographical references and index.

ISBN 978-1-4419-3005-7 ISBN 978-0-387-22489-3 (eBook) DOl 10.10071978-0-387-22489-3

1. Algebraic topology. 2. Homotopy theory. I. Gitler, Samuel. II. Prieto, C. (Carlos) m. Title. QA612 .A37 2002 514'2--dc21 2002019556

Printed on acid-free paper. English version by Stephen Bruce Sontz.

© 2002 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1 st edition 2002

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

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Typesetting: Pages created by authors using J4.'IEX. www.springer-ny.com

Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH

To my parents To Danny

To Viola To Sebastian and Adrian

PREFACE

This book introduces the basic concepts of algebraic topology using homoto­py-theoretical methods. We believe that this approach allows us to cover the material more efficiently than the more usual method using homological algebra. After an introduction to the basic concepts of homotopy theory, using homotopy groups, quasifibrations, and infinite symmetric products, we define homology groups. Furthermore, with the same tools, Eilenberg­Mac Lane spaces are constructed. These, in turn, are used to define the ordinary cohomology groups. In order to facilitate the computation, cellular homology and cohomology are defined.

In the second half of the book, vector bundles are presented and then used to define K-theory. We prove the classification theorems for vector bundles, which provide a homotopy approach to K-theory. Later on, K-theory is used to solve the Hopf invariant problem and to analyze the existence of multi­plicative structures in spheres. The relationship between cohomology and vector bundles is established introducing characteristic classes and related topics. To finish the book, we unify the presentation of cohomology and K-theory by proving the Brown representation theorem and giving a short account of spectra.

In two appendices at the end of the book the proof of the Dold-Thom theorem on quasifibrations and infinite symmetric products is given in detail, and a new proof of the complex Bott periodicity theorem, using quasifibra­tions, is presented.

It is expected that the reader has a basic knowledge of general topology and algebra. In any case, the book is mainly aimed at advanced undergrad­uates and at graduate students and researchers for whose work algebraic­topological concepts are needed.

This text originated in a preliminary version in Spanish, which was a joint edition of the Mathematics Institute of the National University of Mexico and McGraw-Hill Interamericana Editores. To both institutions the authors are grateful. The translation of the main body of the text was the excellent

vii

viii PREFACE

job of Stephen Bruce Sontz, to whom we express our deep thanks. Our gratitude goes also to Springer-Verlag, particularly to Ms. Ina Lindemann for her interest in our work, and to the referees for their valuable comments which certainly helped to improve the English version of the book. Its title is, of course, a tribute to John Milnor, from whose books and papers we have learnt many important concepts, which are included in this text.

Last, but not least, we wish to acknowledge the support of Professor Albrecht Dold, who after reading the Spanish manuscript gave various im­portant comments to make some parts better.

Mexico City, Mexico Autumn 2001

Marcelo Aguilar Samuel Gitler Carlos Prietol

IThis author was supported by CONACYT grants 25406-E and 32223-E.

CONTENTS

PREFACE

INTRODUCTION

BASIC CONCEPTS AND NOTATION

1 FUNCTION SPACES

1.1 Admissible Topologies

V'l'/,

x't't't

xV't't

1 1

1.2 Compact-Open Topology . . . . . . . . . . . . . . . . . . . 2

1.3 The Exponential Law . . . . . . . . . . . . . . . . . . . . . 3

2 CONNECTEDNESS AND

ALGEBRAIC INVARIANTS 9 2.1 Path Connectedness ..................... 9

2.2 Homotopy Classes. . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Topological Groups . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Homotopy of Mappings of the Circle into Itself. . . . . . . 15

2.5 The Fundamental Group . . . . . . . . . . . . . . . . . . . 28

2.6 The fundamental Group of the Circle . . . . . . . . . . . . 41

2.7 H-Spaces ........................... 45

2.8 Loop Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.9 H-Cospaces

2.10 Suspensions

ix

50

53

X

3 HOMOTOPY GROUPS

3.1 Attaching Spaces; Cylinders and Cones

3.2 The Seifert-van Kampen Theorem ..

3.3 Homotopy Sequences I

3.4 Homotopy Groups ..

3.5 Homotopy Sequences II . .

4 HOMOTOPY EXTENSION AND

LIFTING PROPERTIES

4.1 Cofibrations.........

4.2 Some Results on Cofibrations

4.3 Fibrations . . . . . . . . . . .

4.4 Pointed and Unpointed Homotopy Classes

4.5 Locally Trivial Bundles . . . . . . . . . . .

CONTENTS

59

59

63

72

80

84

89 89

95

101

119

125

4.6 Classification of Covering Maps over Paracompact Spaces . 138

5 CW..,COMPLEXES AND HOMOLOGY 149 5.1 CW-Complexes ....... .

5.2 Infinite Symmetric Products

5.3 Homology Groups . . . . . .

6 HOMOTOPY PROPERTIES OF

CW-COMPLEXES

6.1 Eilenberg-Mac Lane and Moore Spaces .

6.2 Homotopy Excision and Related Results

149

167

176

189 189

193

6.3 Homotopy Properties of the Moore spaces ......... 201

6.4 Homotopy Properties of the Eilenberg-Mac Lane spaces.. 217

CONTENTS xi

7 COHOMOLOGY GROUPS AND

RELATED TOPICS 227

7.1 Cohomology Groups ..................... 227

7.2 Multiplication in Cohomology . . . . . . . . . . . . . . .. 238

7.3 Cellular Homology and Cohomology. . . . . . . . . . . .. 243

7.4 Exact Sequences in Homology and Cohomology . . . . .. 252

8 VECTOR BUNDLES 259 8.1 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . .. 259

8.2 Projections and Vector Bundles . . . . . . . . . . . . . .. 268

8.3 Grassmann Manifolds and Universal Bundles . . . . . . ., 271

8.4 Classification of Vector Bundles of Finite Type. . . . . .. 276

8.5 Classification of Vector Bundles over Paracompact Spaces. 279

9 K-THEORY 289

9.1 Grothendieck Construction . . . . . . . . . . . . . . . . .. 289

9.2 Definition of K(B) . . . . . . . . . . . . . . . . . . . . .. 292

9.3 K(B) and Stable Equivalence of Vector Bundles .

9.4 Representations of K(B) and K(B) ....... .

295

299

9.5 Bott Periodicity and Applications . . . . . . . . . 302

10 ADAMS OPERATIONS AND ApPLICATIONS 309 10.1 Definition of the Adams Operations. . . . . . . . . . . .. 309

10.2 The Splitting Principle . . . . . . . . . . . . . . . . 313

10.3 Normed Algebras . . . . . . . . . . . . . . . . . . . 315

10.4 Division Algebras . . . . . . . . . . . . . . . . . . . . . ., 317

10.5 Multiplicative Structures on IRn and on sn-l ....... , 319

10.6 The Hopf Invariant . . . . . . . . . . . . . . . . . . . . ., 321

xii CONTENTS

11 RELATIONS BETWEEN COHOMOLOGY AND

VECTOR BUNDLES 331 11.1 Contractibility of §OO. . . . . · . 332

11.2 Description of K(Z/2, 1) . · . . . 334

11.3 Classification of Real Line Bundles . .... . . . . 337

11.4 Description of K(Z, 2) . . . · .. .. . . 340

11.5 Classification of Complex Line Bundles .. . . 343

11.6 Characteristic Classes . . . . .. 345

11.7 Thom Isomorphism and Gysin Sequence . . . . . .. 349

11.8 Construction of Characteristic Classes and Applications 366

12 COHOMOLOGY THEORIES AND

BROWN REPRESENTABILITY 383 12.1 Generalized Cohomology Theories . . . . . . . . . . .. 383

12.2 Brown Representability Theorem. . . . . . . . . . . .. 394

12.3 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . .. 406

A PROOF OF THE DOLD-THOM THEOREM 421 A.1 Criteria for Quasifibrations . . . . . . . . . . . . . . . . .. 421

A.2 Symmetric Products ................... " 431

A.3 Proof of the Dold-Thom Theorem . . . . . . . . . . . . " 434

B PROOF OF THE

BOTT PERIODICITY THEOREM 437 B.1 A Convenient Description of BU x Z . . . . . . . . . . .. 437

B.2 Proof of the Bott Periodicity Theorem

REFERENCES

SYMBOLS

INDEX

440

457

463

467

INTRODUCTION

The fundamental idea of algebraic topology is to associate to each topologi­cal space X a group h(X) and to each map f : X ---+ Y a homomorphism h(f) : h(X) ---+ h(Y) with the property that whenever X and Y are ho­motopy equivalent (in particular, if they are homeomorphic), then h(X) is isomorphic to h(Y). In other words, we consider functors h (both covariant and contravariant) from the category of (pointed) topological spaces to the category of (abelian) groups such that h(f) = h(g) if the maps f, 9 : X ---+ Y are homotopic. The easiest way to construct such a covariant functor is to consider a fixed space Xo and then to define the functor (on objects) by h(Y) = [Xo, Y], where the brackets denote the set of (pointed) homotopy classes of maps from Xo to Y. Similarly, we define such a contravariant func­tor by considering a fixed space Yo and setting h(X) = [X, Yo]. In order to have a group structure on these sets of homotopy classes the spaces Xo and Yo must have certain properties (see Sections 2.7 and 2.9), which are satisfied if Xo = §n or if Yo is an H-group. When Xo = §n we obtain the homotopy groups ll'n(Y) = [§n, Y]. However the homotopy groups of an arbitrary space Y are extremely difficult to calculate due to the fact that they do not satisfy the excision axiom (see statement 5.3.15 and Section 6.2). But one could try to associate to Y another space whose homotopy groups are easier to calculate. It is known (see 6.4.15) that a topological abelian monoid has a simple homotopical structure. So we associate to Y the free topological abelian monoid generated by its points (with the base point of Y acting as the zero element). This monoid is the same as the infinite symmetric prod­uct SP Y. Furthermore, since a topological abelian monoid is completely characterized by its homotopy groups (see 6.4.16), we are led to associate to Y the groups Hn(Y) = ll'n(SP Y). These groups turn out to satisfy the excision axiom and thus are easier to calculate. Similarly, when we study the contravariant functors [-, Yo] with Yo an H -group, we shall consider spaces Yo with a simple homotopical structure, namely spaces K(Z, n) with only one nonzero homotopy group, which is Z in dimension n. These are called Eilenberg-Mac Lane spaces. To construct these spaces we shall also use a suitable symmetric product. Then we set Hn(x) = [X, K(Z, n)].

xiii

xiv INTRODUCTION

The purpose of this book is to introduce algebraic topology from the homotopical point of view. The basic concepts of homotopy theory, such as fibrations and cofibrations, are used to construct singular homology and cohomology, as well as K-theory.

In particular, the presentation of homology, using the homotopy groups of an infinite symmetric product, is nowadays adequate for the purposes of algebraic geometry, specifically for the definition of the Lawson homology theory (see [42, 43]). On the other hand, Voevodsky [79J and others, using the homotopical point of view of this book, translated many concepts of algebraic topology into algebraic geometry. This is the foundation for Voevodsky's proof of the Milnor conjecture, concerning a certain relationship between Milnor's K-theory groups of a field F and the Galois cohomology groups of F. More specifically, Voevodsky constructed a stable homotopy category of schemes in algebraic geometry, analogous to the stable homotopy category in algebraic topology. He defines spectra and the associated cohomology and homology theories. To construct the Eilenberg-Mac Lane spectrum he uses a suitable analogue of the symmetric products. He also constructed spectra for K-theory and cobordism in this setting.

A leitmotif of this book is to pursue the proof of one of the most remark­able results of algebraic topology: J. Frank Adams' theorem solving the Hopf invariant problem, implying that the only spheres that admit a multiplicative structure, converting them into H-spaces, are precisely So, §t, §3, and §7 or, equivalently, that the only real division algebras are the reals, the complex numbers, the quaternions, and the Cayley numbers. Throughout the text many other fundamental concepts are introduced, including the construc­tion of the characteristic classes of vector bundles, to which a full chapter is devoted.

The book is adequate for use in a two-semester course, either at the end of an undergraduate program or at the graduate level. In order to understand its contents, a basic knowledge of point set topology as well as group theory is required. Although functors appear constantly throughout the text, no knowledge about category theory is expected from the reader; on the con­trary, every time categorical or functorial properties appear, the categorical ideas are stressed in order to obtain the functorial properties of the intro­duced invariants.

The design of the text is as follows. We start with a chapter devoted to basic concepts and notation, followed by twelve substantial chapters, each of which is divided into several sections that are distinguished by their double numbering (1.1, 1.2, 2.1, ... ). Definitions, propositions, theorems, remarks,

INTRODUCTION xv

formulas, exercises, etc., are designated with triple numbering (1.1.1, 1.1.2, ... ). Exercises are an important part of the text, since many of them are intended to carry the reader further along the lines already developed in order to prove results that are either important by themselves or relevant for future topics. Most of these are numbered, but occasionally they are identified inside the text by italics (exercise). On the other hand, two important theorems, whose proof somehow goes beyond the horizons of this book (the Dold-Thom theorem on quasifibrations and infinite symmetric products and the complex Bott periodicity theorem) are proved in two appendices. In the appropriate chapters these results are then freely used after some explanation to let the reader understand the scope and meaning of the results and to give their applications.

The chapter on basic concepts and notation, as its name suggests, presents most of the notation used throughout the text as well as some concepts that are not necessarily standard in the regular basic courses on point set topology or algebra.

Chapter 1 deals with the elements of the topology of function spaces, emphasizing the compact-open topology, and discusses the exponential law. Chapter 2 introduces the basic notions of homotopy theory, such as path connectedness and homotopy of maps. The former is, in a way, the basic concept on which all ideas in the book are built. We study the degree of maps of the circle into itself, and introduce the fundamental group. Finally, we define the concepts of topological groups and H -spaces, and the dual concept of H-cospace. As examples of H-spaces and H-cospaces, loop spaces and suspensions are carefully studied.

Chapter 3 contains a study of homotopy groups including the proof of the Seifert-van Kampen theorem. Special emphasis is put on the long exact sequences of homotopy groups. Then in Chapter 4, homotopy extension and lifting properties are analyzed, particularly the concepts of cofibration and fibration.

In order to prepare for the study of cohomology groups, CW-complexes are introduced in Chapter 5, and their homotopy properties are analyzed. The concepts of quasifibrations and infinite symmetric products are also re­viewed. These are used to introduce the homology groups. Further homotopy topics are studied in Chapter 6, among which is the proof of the Blakers­Massey homotopy excision theorem. This is an invaluable tool in the study of homotopy aspects of the Moore and the Eilenberg-Mac Lane spaces.

Cohomology groups are introduced in Chapter 7, and their multiplica­tive structure is defined. After cellular homology and cohomology are intro-

xvi INTRODUCTION

duced, some specific groups are computed. Further on in the same chapter we construct the exact sequences of Kiinneth, of universal coefficients, and of Mayer-Vietoris among others. Later on, in Chapter 8, vector bundles are studied in detail, building up to their classification. For that purpose, Grassmann manifolds and universal vector bundles over them are defined, and some classification results are proved.

Complex K-theory is introduced in Chapter 9 starting from complex vec­tor bundles. Using their classification, various theorems are proved, which allow us to realize the K-theory of a space as a set of homotopy classes of mappings from the space into a classifying space, much in the same spirit as the cohomology groups were defined earlier. In order to exploit K-theoryas much as possible the Bott periodicity theorem in the complex case is pre­sented, but not yet proved. Later on, in Chapter 10, the Adams operations in complex K-theory are introduced to solve the Hopf invariant problem and thereby to study the existence of the structures of normed and division alge­bras in IRn as well as to prove Adams' theorem on multiplicative structures on the spheres §n-l.

In Chapter 11 the relationship between line bundles and cohomology is given, using the fact that the classifying spaces of real and complex line bundles, namely lRJPOO and cr, are Eilenberg-Mac Lane spaces. A simple proof of the existence of the Thom class of an oriented vector bundle and of Thom's isomorphism theorem is given to be used later on to define the Stiefel­Whitney classes of real vector bundles and the Chern classes of complex vector bundles. We finish the main part of the book with Chapter 12, where we present a short account of generalized cohomology and homology and prove the Brown representability theorem. Some remarks on the theory of spectra end the chapter.

The proof of the Dold-Thom theorem on quasifibrations and infinite sym­metric products is postponed to Appendix A, and a topological proof of the complex Bott periodicity theorem is given in Appendix B. In the appendices the sections are doubly numbered (X.l, X.2, ... ), and the items are triply numbered (X.I.l, X.I.2, ... where X is either A or B).

An effort was made to include a very complete alphabetical index; the reader should feel free to use it, even to look for simple concepts. A list of symbols containing much of the notation used in the book is also included.

BASIC CONCEPTS AND NOTATION

In this section we present some of the basic concepts and notations that will be used in the text.

BASIC SYMBOLS

Throughout the text we shall use the following basic symbols, among others. The symbol ~ between two topological spaces means that they are homeo­morphic, ~ between continuous functions or topological spaces means that they are homotopic or homotopy equivalent, and ~ between groups (abelian or nonabelian) means they are isomorphic. The symbol 0 denotes composi­tion of functions (maps, homomorphisms) and will be omitted ocasionally, if doing so does not lead to confusion. The term map invariably means a continuous function between topological spaces, and the term function is re­served either for functions between sets or for those maps whose codomain is lR or C.

And now a final note about some additional notation that will be used in the text. If X is a topological space and A eX, in agreement with the

o special cases mentioned below we shall use the notation A to denote the topological interior of A in X, and the notation 8A to denote its boundary. Xu Y denotes the topological sum of X and Y. On the other hand, if V is a vector space provided with a scalar product (or Hermitian product, if the space is complex), which we usually denote by (-, -), then we use the notation II . II or I . I to denote the norms in V associated to the inner product, that is, IIxli or Ixl = J(x,x). Likewise, if A c V is a subspace, we use AJ. = {x E V I (x,a) = 0 for all a E A} to denote the orthogonal complement of A in V with respect to the inner product.

SOME BASIC TOPOLOGICAL SPACES

Euclidean spaces, various of its subspaces, and spaces derived from these will all play an important role for us.

lR will represent the set (as well as the topological space and the real

xvii

xviii BASIC CONCEPTS AND NOTATION

vector space) of real numbers. ]Ro will denote the singleton set (of only one point) {O} C R Frequently, we shall use the notation * for an (arbitrary) singleton set. ]Rn will be the notation for Euclidean space of dimension n, or Euclidean n-space, such that

Using the equality

we identify the Cartesian product ]Rm x ]Rn with ]Rm+n. Likewise, we identify ]Rn with the closed subspace ]Rn x 0 C ]Rn+1. We give U::'=o]Rn = ]Roo the topology of the union (which is the colimit topology, as we shall see shortly). ]Roo consists, therefore, of infinite sequences of real numbers (Xl, X2, X3, . .. )

almost all of which are zero, that is to say, such that Xk = 0 for k sufficiently large. ]Rn is identified with the subspace of sequences (Xl, ... ,xn , 0, 0, ... ). The topology of ]ROO is such that a set A C ]ROO is closed if and only if An]Rn is closed in ]Rn for all n.

Topologically we identify the set (as well as the topological space and the complex vector space) e of complex numbers with]R2 using the equality X + iy = (x, y), where i represents the imaginary unit, that is i = J=I. Analogously with the real case, we have the complex space of dimension n, en = {z = (Zl>"" zn) I Zi E e, 1:::; i :::; n}, or complex n-space.

In ]Rn we define for every x = (Xl> ... ,xn ) its norm by

Ixl = J x~ + ... + x~ ; likewise, in en we define the norm by

where z denotes the complex conjugate x - iy of Z = x + iy. Up to the natural identification between en and ]R2n, it is an exercise to show that the two norms coincide.

For n 2: 0 we shall use from now on the following subspaces of Euclidean space:

Jl))n = {x E ]Rn I Ixl :::; I}, the unit disk of dimension n.

sn-l = {x E ]Rn Ilxl = I}, the unit sphere of dimension n - 1.

On JI)) = {x E ]Rn Ilxl < I}, the unit cell of dimension n.

BASIC CONCEPTS AND NOTATION xix

r = {x E lRn I 0 :s Xi :s 1, 1 :s i :s n}, the unit cube of dimension n.

aln = {x E In I Xi = 0 or 1 for some i}, the boundary of In in lRn.

I = II = [0, IJ C lR, the unit interval.

On Briefly, we usually call ][»n the unit n-disk, §n-l the unit (n - I)-sphere, ][» the unit n-cell, and In the unit n-cube. It is worth mentioning that all of the spaces just defined are connected (in fact, pathwise connected), except for §o and aI, these being homeomorphic, of course. The disks, the spheres, the cubes, and their boundaries also are compact (but not the cells, except

°0 for the O-cell][» = *).

The group of two elements Z/2 = Z2 = {-I, I} (which can also be seen as the quotient of the group of the integers Z modulo 2Z) acts on §n by the antipodal action, that is, (-I)x = -x E §n. The orbit space of the action, which is the result of identifying each x E §n with its antipode -x, is denoted by l.lP'n and is called real projective space of dimension n.

The infinite-dimensional sphere §oo = u;:,=o §n, where the inclusion §n-l c §n is defined by the inclusion lRn c lRn+1 , is a subspace of lRoo. The action of Z2 in §n induces an action in §oo, whose orbit space is de­noted by l.lP'oo and is called infinite-dimensional real projective space. In fact, the inclusion §n-l c §n induces an inclusion l.lP'n-l c lRlP'n and the union u::o lRpn coincides topologically with lRpoo.

On the other hand, the circle group §l = {( E C I 11(11 = I} acts on §2n+l C Cn+1 by multiplication on each coordinate, namely, ((Zb ... ,Zn+l) = ((Zb ... , (Zn+l). The orbit space of this action, which is the result of iden­tifying Z E §2n+l with (z E §2n+l, for all ( E §\ is denoted by cpn and is called complex projective space of dimension n (in fact, its real dimension is 2n). The action of §l on §2n+l induces an action on §oo, whose orbit space is denoted by Cpoo and is called infinite-dimensional complex projective space. In analogy with the real case, the inclusion §2n-l C §2n+l, defined by the inclusion Cn C Cn+1, induces an inclusion cpn-l C cpn and the union u;:'=o cpn coincides topologically with Cpoo.

The group of n x n invertible matrices with real (complex) coefficients is denoted by GLn(lR) (GLn(C)) and consists of the matrices whose determi­nants are not zero. The subgroup On C GLn(lR) (Un C GLn(C)) consisting of the orthogonal matrices (unitary matrices), that is, such that the matrix sends orthonormal bases to orthonormal bases with respect to the canonical scalar product in I.n (the canonical Hermitian product in cn) or, equiva­lently, such that its column vectors form an orthonormal basis, is called the

xx BASIC CONCEPTS AND NOTATION

orthogonal group (unitary group) of n x n matrices. In particular, 01 = Z2 and UI = §I.

SOME GENERAL BASIC CONCEPTS

If 1 : G -+ H is a homomorphism of groups, then ker(J) = {g E G I 1(g) = I} C G represents the kernel of 1 and im(J) = {J(g) I 9 E G} c H its image. An arrow of the form L..-. represents an inclusion or an embedding of topological spaces, while one of the form >--->- indicates a group monomorphism, and finally, one of the form ---- represents an epimorphism or, possibly, a surjective (quotient) map between topological spaces.

A sequence of homomorphisms (of groups, rings, modules, etc.)

is called exact at B if im (J) = ker(g).

As we have already done in the case of lRn or en for defining lRoo and Coo ,

we shall make frequent use of the general concept of infinte union or colimit. In the case of topological spaces let

be a chain of closed inclusions of topological spaces. We define its union Ui>1 Xi as the union of the sets Xi, and we define its topology by declaring a :rubset C c Ui~l Xi to be closed if and only if its intersection C n Xi is closed in Xi for all i 2:: 1. This topology is called the union topology; frequently it is also called the weak topology with respect to the subspaces. It is an exercise to show that the union has the following universal property. If we have a family {Ji : Xi -+ Y I i 2:: O} of continuous maps such that Ji+1IXi = Ji : Xi -+ Y, then there exists a unique map 1 : UXi -+ Y such that 11Xi = Ji : Xi -+ Y. In a commutative diagram we write this as

It is an exercise to prove that the spaces §oo = U:'=o §n, lRF = U:'=o lRrn, cr = U:'=o ern defined above have the union topology.

BASIC CONCEPTS AND NOTATION xxi

LIMITS AND COLIMITS

In a slightly more general context, given a sequence of closed embeddings, that is, of maps that are homeomorphisms onto their range, which itself is closed,

'1 ·2

Xl' 32 )X2,-' _33_4-)X3,'-_---+) ••. ,

its colimit is a topological space denoted by colim Xi, provided with maps / : Xi ---? colim Xi such that l 0 jk = / : Xi ---? colim Xi, where jk = j!-l 0 . ··0 jf+l : Xi ---? Xk , k > i, and which has the following universal property. If {P : Xi ---? Y I i 2': I} is a family of maps such that fi+1 0 j1+1 =

P : Xi ---? Y for all i ~ lor, equivalently, fk 0 jk = P : Xi ---? Y for all k > i 2': 1, then there exists a unique map f : colimXi ---? Y such that f 0 ji = p. Diagrammatically this says

The space colim Xi can be defined by taking the quotient of the topolog-ical sum

colimXi = (II Xi) / f'V

by the relation Xi 3 x f'V jf+1(X) E Xi+l for all i. The maps ji : Xi ---?

colim Xi are defined as the composition of the canonical inclusion into the topological sum and the quotient map, namely,

/ : X/---+- I1 Xi --- coHm Xi.

It is an exercise to prove that this definition of colimit actually has the universal property. In the book [27] there is a general treatment of the topic of colimits of topological spaces, these being called (as by many other authors) direct limits (see further below).

In the algebraic case we have an analogous situation, namely, given a chain or direct system of abelian groups (or rings, vector spaces, etc.) and homomorphisms

A h~A h~A 1 ) 2 ) 3---)0)···,

we define its colimit as

colim Ai = (EB A) / A' , i2:1

XXll BASIC CONCEPTS AND NOTATION

where A' is the subgroup of E9 Ai generated by the differences h~(ai) -ai E Ai EB Ak C E9Ai' k > i, where h~ = h~-l 0 ht:i 0 •.. 0 h~+1. In other words, we identify each group Ai with its image in Ak. For each i we have homomorphisms hi : Ai ----t colimAi given by the composition of the canonical inclusion in the direct sum and the epimorphism in the colimit

We have, as in the topological case, that

The algebraic colimit also has the following universal property. If {P : Ai ----t B I i ~ 1} is a family of homomorphisms such that fi+l 0 h~+1 = P : Ai ----t B for all i ~ 1 or, equivalently, fk 0 h~ = P : Ai ----t B for all k > i ~ 1, then there exists a unique homomorphism f : colim Ai ----t B such that f 0 hi = fi. Diagrammatically we have the following:

Dually, for an inverse system of abelian groups and homomorphisms

we have a homomorphism

such that

We define its limit as the kernel of d,

lim Ai = ker( d) ,

and its derived limit as the cokernel of d,

liml Ai = coker(d) = (II Ai) / im(d).

In this way we obtain an exact sequence

o ----t lim Ai ----t II Ai ----t II Ai ----t lim 1 Ai ----t 0 .

BASIC CONCEPTS AND NOTATION xxiii

Dually, in the case of the colimit, for each i we have homomorphisms hi: lim Ai ~ Ai given by the composite

lim Ai ~ II Ai

The limit also has a universal property dual to that of the colimit. It is the following.

If {Ii : B ~ Ai I i ~ I} is a family of maps such that h~+l Oli+l = Ii : B ~ Ai for all i ~ lor, equivalently (defining h~ = h~+1 0 h~t~ 0···0 hLl)' such that h~ o!k = Ii : B ~ Ai for all k > i ~ 1, then there exists a unique homomorphism r: B ~ lim Ai such that hi 0 I = /;. Diagrammatically, this is expressed as

As we have already mentioned above, frequently one refers to the colimit as the direct limit, and one denotes it by the symbol lim-+ or dirlim. Like­wise, one often says inverse limit instead of limit, and one denotes it by the symbol lim+- or invlim. In order to avoid confusion between these, we pre­fer the nomenclature of colimit and limit, which is more in agreement with the dual categorical character of both concepts. A systematic treatment of colimits and limits can be found in the book by Mac Lane [46], which is, moreover, an excellent general reference for the categorical concepts (func­tors, natural transformations, etc.) that will be mentioned in this text and briefly described below.

CATEGORIES, FUNCTORS, AND NATURAL TRANSFORMATIONS

Throughout the text we use the concept of functor. This is inherent to the concept of a category, whose definition we now give.

A categoryC consists of a class of objects and, for each pair of objects A, B, a set of morphisms C(A, B) with domain A and codomain B. If I E C(A, B),

one usually writes I : A ~ B or A ~ B. For every triple of objects A, B, C, there is a function

C(A, B) x C(B, C) ~ C(A, C)

associating to a pair of morphisms I : A ~ B, 9 : B ~ C their composite

go/:A~C.

xxiv BASIC CONCEPTS AND NOTATION

Two axioms are satisfied:

Associativity. If f: A --+ B, g: B --+ C, and h: C --+ D, then

h 0 (g 0 f) = (h 0 g) 0 f : A --+ C.

Identity. For every object B there is a morphism 18 : B --+ B such that if f : A --+ B, then 18 0 f = f, and if 9 : B --+ C, then go 18 = g.

Some examples of categories that will be useful in this text are the fol­lowing:

1. The category Set of sets and functions, that is, the category whose objects are all sets, and for sets A, B, Set (A, B) is the set of functions from A to B.

2. The category lOP of topological spaces and continuous maps.

3. The category 9 of groups and homomorphisms.

4. Given a partial order :s; in a set X, there is a category X whose objects are the elements of X and such that the set X(x, y) is either the empty set or the set consisting of one element, according to whether x i y or x :s; y.

There are many other examples, such as the category of pointed sets (i.e., nonempty sets each with a distinguished point called a base point) and pointed functions (Le., functions preserving base points) Set*; of pointed topological spaces and pointed maps IOP*; of abelian groups and homomor­phisms Ab; of modules over a ring R and module homomorphisms ModR ; of vector spaces and linear transformations Ved; etc.

A morphism f : A --+ B in a category C is called an isomorphism if there is another morphism 9 : B --+ A in C such that fog = 18 and go f = 1A . For example, isomorphisms in Set are set equivalences, in lOp are homeomorphisms, and in 9 are group isomorphisms.

Given two categories C and 'D, a covariant functor (or contravariant func­tor) T : C --+ 1) assigns to every object A of C an object T(A) of'D and to every morphism f : A --+ B of C a morphism f* = T(f) : T(A) --+ T(B) (or f* = T(f) : T(B) --+ T(A)) in such a way that

(a) T(lA) = 1T(A),

(b) T(g 0 f) = T(g) 0 T(f) (or T(g 0 f) = T(f) 0 T(g)).

BASIC CONCEPTS AND NOTATION xxv

Some examples are the following:

1. There is a covariant functor from the category of topological spaces and continuous maps to the category of sets and functions that assigns to every topological space its underlying set. This functor is usually called the forgetful functor because it "forgets" the structure of a topological space.

2. There is a covariant functor from the category of sets and functions to the category of topological spaces and continuous maps that assigns to every set the discrete topological space having it as an underlying set.

3. There is a covariant functor from the category of sets and functions to the category of (abelian) groups and homomorphisms that assigns to every set the free (abelian) group generated by the set.

4. There is a contravariant functor from the category of topological spaces and continuous maps to the category of rings and homomorphisms that assigns to every topological space the ring of its continuous real-valued functions.

5. A direct system (or inverse system) in a category C is a covariant func­tor (or contravariant functor) from the category N determined by the ordered set of the natural numbers (cf. example 4 on xxiv).

One can compare functors with each other. This is done by means of a suitable definition of a morphism between functors. Let T1, T2 : C --+ V be functors of the same variance (either both covariant or both contravariant). A natural transformation <p from Tl to T2, in symbols <p : Tl ~ T2, assigns to every object A of C a morphism <p(A) : Tl(A) --+ T2(A) of V in such a way that for every morphism f : A --+ B of C the appropriate one of the following diagrams is commutative

Tl(A) !!i!lT1(B) Tl(A) 1!S!lT1(B)

<p(A) ! ! ",(B) <p(A) ! ! 'P(B)

T2 (A) T;(j) T2 (B) , T2(A) T;(f) T2(B) ,

according to whether T1, T2 are covariant or contravariant.

If <p : Tl ~ T2 is a natural transformation such that <p(A) is an isomor­phism in V for each object A in C, then <p is called a natural equivalence.

xxvi BASIC CONCEPTS AND NOTATION

SMOOTH ApPROXIMATION AND DEFORMATION OF MAPS

We shall need to approximate continuous maps with homotopic smooth maps, that is, maps with continuous derivatives of all orders. We present two results on this. First we approximate functions. This is done using the notion of a smooth bump function. Namely, given A eVe lRn where A is closed and V is open in lRn , a bump function of A in V is a continuous function h : lRn ~ I such that flA = 1 and fllRn - V = o.

Let 0 : lR ~ lR be given by

{ -1/t2

o(t) = ~ if t > 0,

ift :::; o. This function is smooth and can be used to produce a second smooth function

which is such that

0(1 - t) {3(t) = 0(1 - t) + o(t) ,

{{3(t) = 1

0< {3(t) < 1

{3(t) = 0

ift :::; 0,

if 0 < t < 1,

ift :::: 1.

Let A = Dr(a) be the closed ball with center a E lRn and radius r > 0, o

and let V = Ds(a) be a larger open ball; that is, s > r. Then for x E IRn the function

h(x) = {3 (IX - al2 - r2) S2 - r2

is a smooth bump function of A in V, as one may easily check.

Let now U C lRn be open and bounded, and let V c lRn be such that V :) U. Then there exists a smooth bump function h : IRn ~ IR of U in V defined as follows. Since U is compact, it can be covered with a finite number

o 0

of open balls D1, ••• , Dk such that their closures D1, ... , Dk are contained in o

V. Let D;, ... , D~ be balls such that Di C D~ c V and let hi be a smooth o

bump function of Di in D~. Define h by

h(x) = 1- (1- h1(x))····· (1- hk(x)).

We have now the desired smooth approximation theorem, which shows how one can smoothly approximate continuous functions.

Smooth approximation theorem. Let U C ]Rn be open, and let f : U ~ lR be a continuous map that is smooth in an open set W cU. Let moreover

BASIC CONCEPTS AND NOTATION xxvii

W', W" be open sets such that W' c W" and W" is bounded and contained in U. Finally, take e > O. Then there exists a function 9 : U -+ lR that is smooth in W U W' and satisfies

Ig(x) - f(x)1 < e for all x E U and g(x) = f(x) for all x E W - W".

To obtain such a map 9 apply the Weierstrass approximation theorem (see [65]) to find a polynomial function p(x) such that

Ip(x) - f(x)1 < e for all x E U

and take a smooth bump function h of W' in W". Then define

g(x) = h(x)p(x) + (1 - h(x))f(x) for x E U.

Then 9 is smooth in W U W', glW' = pIW' , glU - w" = flU - W", and Ig(x) - f(x)1 < e for all x E W".

We now state the smooth deformation theorem, which shows how one can find smooth maps homotopic to given continuous maps.

Smooth deformation theorem. Let U c IRm and V C IRn be bounded open sets and let <p : U -+ V be a continuous map. Take W, W' c IRm open such that W C W' c W' cU. Then there exists a map 1/J : U -+ V such that:

(1) 1/JIW: W -+ V is smooth.

(2) 1/JIU - W' = <pIU - W' and 1/J ~ <P reI (U - W').

The proof is as follows. Cover the compact set <p(W') by a finite number of open balls contained in V, and let e > 0 be smaller than one-half the smallest radius of the balls. Then use the smooth approximation theorem for each component of <p to obtain 1/J : U -+ lRn such that it is smooth in W, 1/JIU - W' = <pIU - W', and 111/J(x) - <p(x)II < e for all x E U. Then the linear deformation

H(x, t) = (1 - t)<p(x) - t1/J(x)

is a homotopy H : U x I -+ V from <p to 1/J that coincides with <p on U - W'; i.e., it is relative to U - W'. In particular, 1/J(U) c V.

Given a smooth map <p : U -+ IRn , where U C IRm is open, we say that x E U is a regular point if the derivative D<p(x) : lRm -+ lRn is nonsingular.

xxviii BASIC CONCEPTS AND NOTATION

In particular, if m < n, then no point x EU is regular. A point y E lRn is a regular value if all points in cp-l(y) are regular.

The following result holds (see [57]).

Theorem 1. If Y E lRn is a regular value of a smooth map cp : U ---+ lRn ,

where U C lRm is open, then cp-l(y) C U is a smooth manifold of dimension m - n. If, in particular, m < n, then cp-l(y) = 0.

Another theorem that will be useful for us in this text is due to A. B. Brown, and in a sharper form to A. Sardo It states the following (see [57]).

Brown-Sard theorem. Let cp : U ---+ lRn be a smooth map, where U C lRm

is open. Then the set of regular values of cp is dense in lRn.

Combining the smooth deformation theorem with the two previous re­sults, one has the following theorem.

Theorem 2. Let U C lRm and V C lRn be bounded open sets and let cp : U ---+ V be a continuous map. Take W, W' c lRm open such that W C W' C -I W cU. Then there exists a map 'I/J : U ---+ V such that:

(1) 'l/JIW: W ---+ V is smooth.

(2) 'l/JIU - W' = cplU - W' and 'I/J ~ cp reI (U - W').

(3) There is a point y E V such that 'I/J-l(y) is a smooth (m - n)-manifold, and in particular, if m < n, then 1/J-l(y) = 0.

PARTITIONS OF UNITY

We shall now continue with a brief description of a notion that we will find useful, namely, the notion of a partition of unity subordinate to an open cover U = {U>.} of a topological space X. This consists of a family of functions {7J>. : X ---+ I}, indexed with the same index set that the cover U has, such that 7J>.IX - U>. = 0 for all >., and moreover, each x E X has a neighborhood V such that 7J>.IV = 0, except for a finite number of indices .x, and finally, I:>. 17>. (x) = 1 for all x E X. (Note that the sum is always a finite sum.) A partition of unity subordinate to a given open cover is a useful tool, for example, for sets of functions or maps only partially defined and with values in lR, C, or in some vector space. For example, it is an exercise to prove that if {f>. : U>. ---+ lR} is a family of continuous functions, then the function f : X ---+ lR such that f(x) = I:>. 17>. (x)J>. (x) is well defined and is continuous.

A fundamental theorem concerning the topology of paracompact spaces is the following:

BASIC CONCEPTS AND NOTATION xxix

Theorem 3. A topological space X is paracompact if and only if every open cover U of X admits a partition of unity subordinate to it.

The books [60], [27], and [83] can be consulted in order to review this theorem and for general considerations about paracompact spaces.

For subspaces of lRn one can construct smooth partitions of unity making use of the smooth bump functions constructed in the previous paragraph.