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11 0.77- -1 1 AIeb-or raic I opoIoy I m m & of 1- J ti kkl. AN DREW 'H: WA ' LAC li Algebraic Topology Homology and Cohomology Algebraic Topology Homology and Cohomology Andrew H. Wallace University of Pennsylvania W. A. Benjamin, Inc. 1970 New York Algebraic Topology: Homology and Cohomology Copyright © 1970 by W. A. Benjamin, Inc. All rights reserved Standard Book Number 8053-9482-6 Library of Congress Catalog Card Number: 79-108005 Manufactured in the United States of America 12345 M 43210 W. A. Benjamin, Inc. New York, New York 10016 Cop j'right Copyright ©1970. 1998 by Andrew H. Wallace All rights reserved. Bibliographical Note This Dover edition. first published in 2007. is an unabridged corrected republication of the work originally published in 1970 by W. A. Benjamin. Inc.. New York. Librari' of Congress Cataloging-in-Publication Data Wallace. Andrew H. Algebraic topology homology and cohomology / Andrew H. : Wallace. P. cm. Originally published: New York : W.A. Benjamin. 1970. Includes bibliographical references and index. I S BN -13 : 978-0-46-46239-4 ISBN-10: 0-486-46239-0 1. Algebraic topology. 1. Title. QA612. W33 2007 514'.2 dc22 2007010621 Manufactured in the United States by Courier Corporation 46239002 wwnv.doverpublications.com Preface This text is intended as a two-semester course in algebraic topology for first or second year graduate students. It is sufficiently self-contained to be taken as a first course in algebraic topology, although in my own teaching I have usually preferred to precede this material by a more elementary treatment, such as is given in my Introduction to Algebraic Topology (Pergamon Press, 1957, henceforth referred to as Introduction). Prerequisites for this course are introductory courses in general topology and algebra ; but in fact these could be running concurrently since, at the beginning of the study of algebraic topology, all the student needs to know are the definitions of topological spaces, continuous maps, groups, and homomorphisms. The use of any deeper properties does not occur till later. Basically, the aim of algebraic topology is to attach topologically invariant algebraic structures to topological spaces. Here the term " invariant " means that homeomorphic spaces will have isomorphic structures attached to them. The idea behind this, of course, is that if the algebraic structures corre- sponding to two spaces are not isomorphic then it follows that these spaces are not homeomorphic. In other words, topologically invariant structures are tools for distinguishing topological spaces from one another. In this text several algebraic invariants are constructed and studied : the fundamental group, singular and tech homology groups, and various kinds of cohomology groups. The latter are shown to have a ring structure that considerably increases their strength as topological invariants ; that is, it happens that spaces that cannot be distinguished from each other by looking v vi Preface at their homology or cohomology groups can be distinguished by constructing their cohomology rings. As in Introduction, the fundamental philosophy adopted here is that topology is a form of geometry. Accordingly, geometrical motivations and interpreta- tions are given as much emphasis as possible. On the other hand, as soon as the student goes beyond the construction of the singular homology groups it becomes clear that there are certain underlying algebraic patterns running through the whole subject. So it becomes desirable to leave the geometry aside at certain moments (for example, in Chapter 3) to study these patterns as pure algebra. Perhaps the more thoroughly logical approach would be to present all the algebra first, so that the homology and cohomology theory could be started in the same way as, for example, analytical geometry is usually begun, namely, with all the necessary algebraic tools already available. Such an approach appears to be unsuitable for beginners, however, since the algebra involved here has no readily available motivation other than in the homology and cohomology theory to which it is to be applied, and without motivation the definitions would almost certainly appear rather artificial. Briefly, then, the arrangement of the course is as follows. In Chapter 1, singular homology theory is described. In a sense, this is a review of the material of Chapters 5-8 of Introduction, but it differs in two respects from the treatment given there. In the first place, the exposition here moves a bit faster; in the second (more important) place, many of the arguments are carried out in a more algebraic way than in Introduction. This more algebraic approach leads very naturally to the introduction of the axiomatic method of studying homology that is briefly described in Chapter 2. And of course, it also motivates the notion of algebraic complexes, which are described in Chapter 3, and which provide the algebraic foundation of all that follows. Chapter 2 also contains an account of the homology theory of simplicial complexes, with methods of computation. Cohomology makes its first appearance in Chapter 3, where it is treated in the singular and simplicial cases. Chapter 4 describes the ring structure on cohomology groups. Chapters 5, 6, and 7 describe the tech theories. There are two appendixes. Appendix A contains a discussion of the fundamental group. This is not relegated to an appendix because of any inferiority or lack of importance, but simply because it is an independent topic, which does not fall naturally into the sequence of subjects described in Chapters 1-7. Appendix B is a reminder of the various ideas and theorems of general topology which are needed throughout the text. The student is urged to treat the exercises as an integral part of the text. They give practice in the style of thinking involved in algebraic topology; besides, many of them are results that are used elsewhere in the exposition. Philadelphia, Pennsylvania, July 1969 ANDREW WALLACE Contents Preface v Chapter 1 Singular Homology Theory 1-1. Euclidean Simplexes 2 1-2. Linear Maps 4 1-3. Singular Simplexes and Chains 5 1-4. The Boundary Operator 7 1-5. Cycles and Homology 10 1-6. Induced Homomorphisms 14 1-7. The Main Theorems 16 1-8. The Dimension Theorem 16 1-9. The Exactness Theorem 17 1-10. The Homotopy Theorem 23 1-11. The Excision Theorem 29 Chapter 2 Singular and Simplicial Homology 2-1. The Axiomatic Approach 37 2-2. Simplicial Complexes 39 2-3. The Direct Sum Theorem 42 2-4. The Direct Sum Theorem for Complexes 44 vii