Graduate Texts in Mathematics 127

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

Editorial Board S. Axler F. W. Gehring K.A. Ribet

Graduate Texts in Mathematics TAKEUTIIZARING. Introduction to 34 SPITZER. Principles of Random Walk. Axiomatic Set Theory. 2nd ed. 2nd ed.

2 OXTOBY. Measure and Category. 2nd ed. 35 ALExANDERIWERMER. Several Complex 3 SCHAEFER. Topological Vector Spaces. Variables and Banach Algebras. 3rd ed.

2nded. 36 KEUEYINAMIOKA et al. Linear 4 HILTONISTAMMBACH. A Course in Topological Spaces.

Homological Algebra. 2nd ed. 37 MONK. Mathematical Logic. 5 MAc LANE. Categories for the Working 38 GRAUERT/F'RITZSCHE. Several Complex

Mathematician. 2nd ed. Variables. 6 HUGHEs/PiPER. Projective Planes. 39 ARVESON. An Invitation to C*-Algebras. 7 SERRE. A Course in Arithmetic. 40 KEMENY/SNELliKNAPP. Denumerable 8 TAKEUTIIZARING. Axiomatic Set Theory. Markov Chains. 2nd ed. 9 HUMPHREYS. Introduction to Lie Algebras 41 APosTOL. Modular Functions and

and Representation Theory. Dirichlet Series in Number Theory. 10 COHEN. A Course in Simple Homotopy 2nded.

Theory. 42 SERRE. Linear Representations of Finite II CoNWAY. Functions of One Complex Groups.

Variable I. 2nd ed. 43 GILLMANIJERISON. Rings of Continuous 12 BEALS. Advanced Mathematical Analysis. Functions. 13 ANDERSON/FuLLER. Rings and Categories 44 KENDIG. Elementary Algebraic Geometry.

of Modules. 2nd ed. 45 LoEVE. Probability Theory I. 4th ed. 14 GOLUBITSKYIGUilLEMIN. Stable Mappings 46 LoEVE. Probability Theory II. 4th ed.

and Their Singularities. 47 MOISE. Geometric Topology in IS BERBERIAN. Lectures in Functional Dimensions 2 and 3.

Analysis and Operator Theory. 48 SACHS/Wu. General Relativity for 16 WINTER. The Structure of Fields. Mathematicians. 17 ROSENBLATT. Random Processes. 2nd ed. 49 GRUENBERG/WEIR. Linear Geometry. 18 HALMOS. Measure Theory. 2nded. 19 HALMOS. A Hilbert Space Problem Book. 50 EDWARDS. Fermat's Last Theorem.

2nd ed. 51 KLINGENBERG. A Course in Differential 20 HusEMOlLER. Fibre Bundles. 3rd ed. Geometry. 21 HuMPHREYS. Linear Algebraic Groups. 52 HARTSHORNE. Algebraic Geometry. 22 BARNEs/MACK. An Algebraic Introduction 53 MANJN. A Course in Mathematical Logic.

to Mathematical Logic. 54 GRAVER/WATKINS. Combinatorics with 23 GREUB. Linear Algebra. 4th ed. Emphasis on the Theory of Graphs. 24 HoLMES. Geometric Functional Analysis 55 BROWN/PEARCY. Introduction to Operator

and Its Applications. Theory I: Elements of Functional Analysis. 25 HEWITT/STROMBERG. Real and Abstract 56 MASSEY. Algebraic Topology: An

Analysis. Introduction. 26 MANES. Algebraic Theories. 57 CRoWELL/Fox. Introduction to Knot 27 KELLEY. General Topology. Theory. 28 ZARISKIISAMUEL. Commutative Algebra. 58 KOBLITZ. p-adic Numbers, p-adic

Vol.l. Analysis, and Zeta-Functions. 2nd ed. 29 ZARisKIISAMUEL. Commutative Algebra. 59 LANG. Cyclotomic Fields.

Vol. II. 60 ARNow. Mathematical Methods in 30 JACOBSON. Lectures in Abstract Algebra I. Classical Mechanics. 2nd ed.

Basic Concepts. 61 WHITEHEAD. Elements of Homotopy 31 JACOBSON. Lectures in Abstract Algebra II. Theory.

Linear Algebra. 62 KARGAPOWV/MERI.ZlAKOV. Fundamentals 32 JACOBSON. Lectures in Abstract Algebra of the Theory of Groups.

III. Theory of Fields and Galois Theory. 63 BOLLOBAS. Graph Theory. 33 HIRSCH. Differential Topology.

(continued after index)

William S. Massey

A Basic Course in Algebraic Topology

With 57 Illustrations in 91 Parts

'Springer

William S. Massey Department of Mathematics Yale University New Haven, Connecticut 06520

Editorial Board:

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

K.A. Ribet Department of Mathematics University of California Berkeley, CA 94720

F. W. Gehring Department of Mathematics University of Michigan Ann Arbor, MI 481 09

Mathematics Subject Classification (2000): 55-01, 55Nxx

Library of Congress Cataloging-in-Publication Data Massey, William S.

A basic course in algebraic topology I William S. Massey. p. em.- (Graduate texts in mathematics; 127)

Includes bibliographical references and index.

ISBN 978-0-387-97430-9 (alk. paper)

I. Algebraic topology. I. Title. II. Series. QA612.M374 1991 514'.2-dc20 90..46073

CIP

ISBN 978-0-387-97430-9 ISBN 978-1-4939-9063-4 (eBook)

DOl 10.1007/978-1-4939-9063-4

Printed on acid-free paper.

© 1991 Springer Science+Business Media, LLC, part of SpringerNature Originally published by Springer-Verlag New York lnc, 1991

This book is a new edition of the first five chapters of Algebraic Topology: An Introduction (GTM 56) and essentially all of Singular Homology Theory (GTM 70) by WilliamS. Massey.

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 I 0010, 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 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.

Production managed by Dimitry L. Loseff; manufacturing supervised by Jacqui Ashri. Typeset by Asco Trade Typesetting Ltd., Hong Kong.

9 8 7 6 5

SPIN 10866107

Springer-Verlag New York Berlin Heidelberg

Preface

This book is intended to serve as a textbook for a course in algebraic topology at the beginning graduate level. The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory (including cup products and the duality theorems of Poincare and Alexander). It consists of material from the first five chapters of the author's earlier book Algebraic Topology: An Introduction (GTM 56) together with almost all of his book Singular Homology Theory (GTM 70). This material from the two earlier books has been revised, corrected, and brought up to date. There is enough here for a full-year course.

The author has tried to give a straightforward treatment of the subject matter, stripped of all unnecessary definitions, terminology, and technical machinery. He has also tried, wherever feasible, to emphasize the geometric motivation behind the various concepts. Several applications of the methods of algebraic topology to concrete geometrical-topological problems are given (e.g., Brouwer fixed point theorem, Brouwer-Jordan separation theorem, lnvariance of Domain, Borsuk-Ulam theorem).

In the minds of some people, algebraic topology is a subject which is "esoteric, specialized, and disjoint from the overall sweep of mathematical thought." It is the author's fervent hope that the emphasis on the geometric motivation for the various concepts, together with the examples ofthe applica­tions of the subject will help to dispel this point of view.

The concepts and methods which are introduced are developed to the point where they can actually be used to solve problems. For example, after defining the fundamental group, the Seifert-Van Kampen theorem is introduced and explained. This is the principal tool available for actually determining the structure of the fundamental group of various spaces. Another such example

VI Preface

is the cup product. Not only is the cup product defined and its principal properties explained; cup products are actually determined in real, complex, and quaternionic projective spaces, and these computations are then applied to prove certain theorems.

In any exposition of a subject such as algebraic topology, the author has to make choices at various stages. One such choice concerns the class of spaces which will be emphasized. We have preferred to emphasize CW-complexes rather than simplicial complexes. Another choice occurs in the actual defini­tion of singular homology groups: Should one use singular simplices or singular cubes? From a strictly logical point of view it does not matter because the resulting homology and cohomology theories are isomorphic in all re­spects. From a pedagogical point of view, it does make a difference, however. In developing some of the basic properties of homology theory, such as the homotopy property and the excision property, it is easier and quicker to use the cubical theory. For that reason, we have chosen to use the cubical theory. Of course, it is more traditional to use the simplicial theory; the author hopes that possible prospective users of this book will not reject it because of their respect for tradition alone.

The prospective user of this book can gain some idea of the material contained in each chapter by glancing at the Contents. We are now going to offer additional comments on some of the chapters.

In Chapter I, the classification theorem for compact 2-manifolds is dis­cussed and explained. The proof of the theorem is by rather standard "cut and paste" methods. While this chapter may not be logically necessary for the rest of the book, it should not be skipped entirely because 2-manifolds provide a rich source of examples throughout the book.

The general idea of a "universal mapping problem" is a unifying theme in Chapters III and IV. In Chapter III this idea is used in the definition of free groups and free products of groups. Students who are familiar with these concepts can skip this chapter. In Chapter IV the Seifert-Van Kampen theorem on the fundamental group of the union of two spaces is stated in terms of the solution to a certain universal mapping problem. Various special cases and examples are discussed in some detail.

The discussion of homology theory starts in Chapter VI, which contains a summary of some of the basic properties of homology groups, and a survey of some of the problems which originally motivated the development of homology theory. While this chapter is not a prerequisite for the following chapters from a strictly logical point of view, it should be extremely helpful to students who are new to the subject.

Chapters VII, VIII, and IX are concerned solely with singular homology with integer coefficients, perhaps the most basic aspect of the subject. Chapter VIII gives various examples and applications of homology theory, including a proof of the general Jordan-Brouwer separation theorem, and Brouwer's theorem on "In variance of Domain." Chapter IX explains a systematic method of computing the integral homology groups of a regular CW-complex.

Preface vii

In Chapter X we introduce homology with arbitrary coefficient groups. This generalization is carried out by a systematic use of tensor products. Tensor products also play a significant role in Chapter XI, which is concerned with the homology groups of a product space, i.e., the Kiinneth theorem and the Eilenberg-Zilber theorem.

Cohomology groups make their first appearance in Chapter XII. Much of this chapter of necessity depends on a systematic use of the Hom functor. However, there is also a discussion of the geometric interpretation of cochains and cocycles, a subject which is usually neglected. Chapter XIII contains a systematic discussion of the various products: cup product, cap product, cross product, etc. The cap product is used in Chapter XIV for the statement and proof of the Poincare duality theorem for manifolds. This chapter also con­tains the famous Alexander duality theorem and the Lefschetz-Poincare duality theorem for manifolds with boundary. In Chapter XV we determine cup products in real, complex, and quaternionic projective spaces. These products are then used to prove the classical Borsuk-Ulam theorem, and to give a discussion of the Hopf Invariant of a map of a (2n - I)-sphere onto an n-sphere.

The book ends with two appendices. Appendix A is devoted to a proof of the famous theorem of DeRham, and Appendix B summarizes various basic facts about permutation groups which are needed in Chapter V on covering spaces.

At the end of many chapters there are notes which give further comments on the subject matter, hints of more recent developments, or a brief history of some of the ideas.

As mentioned above, there is enough material in this book for a full-year course in algebraic topology. For a shorter course, Chapters I-VIII would give a good introduction to many of the basic ideas. Another possibility for a shorter course would be to use Chapter I, skip Chapters II through V, and then take as many chapters after Chapter V as time permits. The author has tried both of these shorter programs several times with good results.

Prerequisites

As in any book on algebraic topology, a knowledge of the basic facts of point set topology is necessary. The reader should feel comfortable with such notions as continuity, compactness, connectedness, homeomorphism, product space, etc. From time to time we have found it necessary to make use of the quotient space or identification space topology; this subject is discussed in the more comprehensive textbooks on point set topology.

The amount of algebra the reader will need depends on how far along he is in the book; in general, the farther he goes, the more algebraic knowledge will be necessary. For Chapters II through V, only a basic, general knowledge of group theory is necessary. Here the reader must understand such terms as

viii Preface

group, subgroup, normal subgroup, homomorphism, quotient group, coset, abelian group, and cyclic group. Moreover, it is hoped that he has seen enough examples and worked enough exercises to have some feeling for the true significance of these concepts. Most of the additional topics needed in group theory are developed in Chapter III and in Appendix B. Most of the groups which occur in these chapters are written multiplicatively.

From Chapter VI to the end of the book, most of the groups which occur are abelian and are written additively. It would be desirable if the reader were familiar with the structure theorem for finitely generated abelian groups (see Theorem III.3.6). Starting in Chapter X, the tensor product of abelian groups is used; and from Chapter XII on the Hom functor is used. Also needed in a few places are the first derived functors of tensor product and Hom (the functors Tor and Ext). These functors are described in detail in books on homological algebra and various other texts. At the appropriate places we give complete references and a summary of their basic properties. In these later chapters we also use some of the language of category theory for the sake of convenience; however, no results or theorems of category theory are used. In order to read Appendix A the reader must be familiar with differential forms and differentiable manifolds.

Acknowledgments

The author is grateful to many colleagues, friends, and students for their sugges­tions and for the corrections of misprints and mistakes in his earlier two books in this series. Special thanks are owed to Peter Landweber for the correction of misprints in the first printing of this book. The author is also grateful to Keith Conrad of the Harvard Mathematics Department for pointing out many misprints. The secretarial staff of the Yale Mathematics Department has also played an es­sential role, and the author is extremely grateful to them. Finally, thanks are due to the staff of Springer-Verlag New York for their kind help in the production of this and the author's previous books.

William S. Massey New Haven, Connecticut

March, 1988

Contents

Preface Notation and Terminology

CHAPTER I

v XV

Two-Dimensional Manifolds 1

§ l. Introduction 1 §2. Definition and Examples of n-Manifolds 2 §3. Orientable vs. Nonorientable Manifolds 3 §4. Examples of Compact, Connected 2-Manifolds 5 §5. Statement of the Classification Theorem for Compact Surfaces 9 §6. Triangulations of Compact Surfaces 14 §7. Proof of Theorem 5.1 16 §8. The Euler Characteristic of a Surface 26

References 34

CHAPTER II

The Fundamental Group 35

§ 1. Introduction 35 §2. Basic Notation and Terminology 36 §3. Definition of the Fundamental Group of a Space 38 §4. The Effect of a Continuous Mapping on the Fundamental Group 42 §5. The Fundamental Group of a Circle Is Infinite Cyclic 47

X Contents

§6. Application: The Brouwer Fixed-Point Theorem in Dimension 2 50 §7. The Fundamental Group of a Product Space 52 §8. Homotopy Type and Homotopy Equivalence of Spaces 54

References 59

CHAPTER III

Free Groups and Free Products of Groups

§1. Introduction §2. The Weak Product of Abelian Groups §3. Free Abelian Groups §4. Free Products of Groups §5. Free Groups §6. The Presentation of Groups by Generators and Relations §7. Universal Mapping Problems

References

CHAPTER IV

Seifert and Van Kampen Theorem on the Fundamental Group

60

60 60 63 71 75 78 81 85

of the Union of Two Spaces. Applications 86

§l. Introduction 86 §2. Statement and Proof of the Theorem of Seifert and Van Kampen 87 §3. First Application of Theorem 2.1 91 §4. Second Application of Theorem 2.1 95 §5. Structure of the Fundamental Group of a Compact Surface 96 §6. Application to Knot Theory 103 §7. Proof of Lemma 2.4 108

References 116

CHAPTER V

Covering Spaces 117

§1. Introduction 117 §2. Definition and Some Examples of Covering Spaces 117 §3. Lifting of Paths to a Covering Space 123 §4. The Fundamental Group of a Covering Space 126 §5. Lifting of Arbitrary Maps to a Covering Space 127 §6. Homomorphisms and Automorphisms of Covering Spaces 130

Contents xi

§7. The Action of the Group n(X, x) on the Set p-1 (x) 133 §8. Regular Covering Spaces and Quotient Spaces 135 §9. Application: The Borsuk-Ulam Theorem for the 2-Sphere 138

§10. The Existence Theorem for Covering Spaces 140 References 146

CHAPTER VI

Background and Motivation for Homology Theory 147

§1. Introduction 147 §2. Summary of Some of the Basic Properties of Homology Theory 147 §3. Some Examples of Problems which Motivated the Development

of Homology Theory in the Nineteenth Century 149 References 157

CHAPTER VII

Definitions and Basic Properties of Homology Theory 158

§1. Introduction 158 §2. Definition of Cubical Singular Homology Groups 158 §3. The Homomorphism Induced by a Continuous Map 163 §4. The Homotopy Property of the Induced Homomorphisms 166 §5. The Exact Homology Sequence of a Pair 169 §6. The Main Properties of Relative Homology Groups 173 §7. The Subdivision of Singular Cubes and the Proof of Theorem 6.4 178

CHAPTER VIII

Determination of the Homology Groups of Certain Spaces: Applications and Further Properties of Homology Theory

§ 1. Introduction §2. Homology Groups of Cells and Spheres-Applications §3. Homology of Finite Graphs §4. Homology of Compact Surfaces §5. The Mayer-Viet oris Exact Sequence §6. The Jordan-Brouwer Separation Theorem and

In variance of Domain §7. The Relation between the Fundamental Group and

the First Homology Group References

186

186 186 192 201 207

211

217 224

xii

CHAPTER IX

Homology of CW-Complexes

§ 1. Introduction §2. Adjoining Cells to a Space §3. CW -Complexes §4. The Homology Groups of a CW -Complex §5. Incidence Numbers and Orientations of Cells §6. Regular CW-Complexes §7. Determination of Incidence Numbers for

a Regular Cell Complex §8. Homology Groups of a Pseudomanifold

References

CHAPTER X

Contents

225

225 225 228 232 238 243

244 249 253

Homology with Arbitrary Coefficient Groups 254

§1. Introduction 254 §2. Chain Complexes 254 §3. Definition and Basic Properties of Homology with

Arbitrary Coefficients 262 §4. Intuitive Geometric Picture of a Cycle with Coefficients in G 266 §5. Coefficient Homomorphisms and Coefficient Exact Sequences 267 §6. The Universal Coefficient Theorem 269 §7. Further Properties of Homology with Arbitrary Coefficients 274

References 278

CHAPTER XI

The Homology of Product Spaces 279

§1. Introduction 279 §2. The Product of CW-Complexes and the Tensor Product of

Chain Complexes 280 §3. The Singular Chain Complex of a Product Space 282 §4. The Homology of the Tensor Product of Chain Complexes

(The Kiinneth Theorem) 284 §5. Proof of the Eilenberg-Zilber Theorem 286 §6. Formulas for the Homology Groups of Product Spaces 300

References 303

Contents xiii

CHAPTER XII

Cohomology Theory 305

§I. Introduction 305 §2. Definition of Cohomology Groups-Proofs of

the Basic Properties 306 §3. Coefficient Homomorphisms and the Bockstein Operator

in Cohomology 309 §4. The Universal Coefficient Theorem for Cohomology Groups 310 §5. Geometric Interpretation of Cochains, Cocycles, etc. 316 §6. Proof of the Excision Property; the Mayer-Vietoris Sequence 319

References 322

CHAPTER XIII

Products in Homology and Cohomology 323

§ 1. Introduction 323 §2. The Inner Product 324 §3. An Overall View of the Various Products 324 §4. Extension of the Definition of the Various Products to

Relative Homology and Cohomology Groups 329 §5. Associativity, Commutativity, and Existence of a Unit

of the Various Products 333 §6. Digression: The Exact Sequence of a Triple or a Triad 336 §7. Behavior of Products with Respect to the Boundary and

Coboundary Operator of a Pair 338 §8. Relations Involving the Inner Product 341 §9. Cup and Cap Products in a Product Space 342

§10. Remarks on the Coefficients for the Various Products-The Cohomology Ring 343

§11. The Cohomology of Product Spaces (The Kiinneth Theorem for Cohomology) 344 References 349

CHAPTER XIV

Duality Theorems for the Homology of Manifolds 350

§1. Introduction 350 §2. Orientability and the Existence of Orientations for Manifolds 351 §3. Cohomology with Compact Supports 358 §4. Statement and Proof of the Poincare Duality Theorem 360

xiv

§5. Applications of the Poincare Duality Theorem to Compact Manifolds

§6. The Alexander Duality Theorem §7. Duality Theorems for Manifolds with Boundary §8. Appendix: Proof of Two Lemmas about Cap Products

References

CHAPTER XV

Contents

365 370 375 380 393

Cup Products in Projective Spaces and Applications of Cup Products 394

§1. Introduction 394 §2. The Projective Spaces 394 §3. The Mapping Cylinder and Mapping Cone 399 §4. The Hopf Invariant 402

References 406

APPENDIX A

A Proof of De Rham's Theorem

§ 1. Introduction §2. Differentiable Singular Chains §3. Statement and Proof of De Rham's Theorem

References

APPENDIX B

Permutation Groups or Transformation Groups

§ 1. Basic Definitions §2. Homogeneous G-spaces

Index

407

407 408 411 417

419

419 421

424

Notation and Terminology

The standard language and notation of set theory is used throughout. Some more special notations that are used in this book are the following:

Z = ring of integers, Q = field of all rational numbers, R = field of all real numbers, C = field of all complex numbers,

R" = set of all n-tuples (x 1 , ••• , x") of real numbers, C" = set of all n-tuples of complex numbers.

If x = (x 1, ... , x") E R", then the norm or absolute value of x is

!xi = (xf + + · · · + x;)112 •

With this notation, we define the followin'g standard subsets of R" for any n > 0:

E" = {x E R"llxl 1},

U"={xER"Ilxl<1},

S"- 1 = {x E R"llxl = 1}.

These spaces are called the closed n-dimensional disc or ball, the open n­dimensional disc or ball, and the (n- I)-dimensional sphere, respectively. Each is topologized as a subset of Euclidean n-space, R". The symbols RP", CP", and QP" are introduced in Chapter IX to denote n-dimensional real, complex, and quaternionic projective space, respectively.

A homomorphism from one group to another is called an epimorphism if it is onto, a monomorphism if it is one-to-one (i.e., the kernel consists of a single element) and an isomorphism if it is both one-to-one and onto. If h: A-+ B is

xvi Notation and Terminology

a homomorphism of abelian groups, the cokernel of h is the quotient group B/h(A). A sequence of groups and homomorphisms such as

h. -I h. ----+ An -I ----+ An ----+ An+ I ----+

is called exact if the kernel of each homomorphism is precisely the same as the image of the preceding homomorphism. Such exact sequences play a big role from Chapter VII on.