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HOMOTOPY THEORY an Introduction to Algebraic Topology HOMOTOPY THEORY An Introduction to Algebraic Topolopy Pure and Applied Mathematics A Series of Monographs and Textbooks Editors =emuel Ellenberg and Hyman Bass Columbia University, New York RECENT TITLES XIA DAO-XING.Measure and Integration Theory of Infinite-Dimensional Spaces : Abstract Harmonic Analysis RONALDG. DOUGLAS.Banach Algebra Techniques in Operator Theory WILLARDMILLER, JR. Symmetry Groups and Their Applications ARTHURA. SACLEAND RALPHE. WALDE.Introduction to Lie Groups and Lie Algebras T. BENNYRUSHING. Topological Embeddings JAMES W. VICK.Homology Theory : An Introduction to Algebraic Topology E. R. KOLCHIN.Differential Algebra and Algebraic Groups GERALDJ. JANUSZ. Algebraic Number Fields A. S. B. HOLLAND.Introduction to the Theory of Entire Functions WAYNEROBERTS AND DALEVARBERG. Convex Functions A. M. OSTROWSKI.Solution of Equations in Euclidean and Banach Spaces, Third Edition of Solution of Equations and Systems of Equations H. M. EDWARDS.Riemann's Zeta Function SAMUELEILENBERG. Automata, Languages, and Machines : Volume A. In pueparation: Volume B MORRISHIRSCH ANU STEPHENSMALE. Differential Equations, Dynamical Systems, and Linear Algebra WILHELMMAGNUS. Noneuclidean Tesselations and Their Groups J. DIEUDONN~.Treatise on Analysis, Volume IV FRANCOISTREVES. Basic Linear Partial Differential Equations WILLIAMM. BOOTHBY.An Introduction to Differentiable Manifolds and Riemannian Geometry BRAYTONGRAY. Homotopy Theory : An Introduction to Algebraic Topology ROBERTA. ADAMS.Sobolev Spaces Iit preparation D. v. WIDDER. The Heat Equation IRVINGE. SEGAL.Mathematical Cosmology and Extragalactic Astronomy WERNERGREUB, STEPHEN HALPERIN, AND RAYVANSTONE. Connections, Curvature, and Cohomology : Volume 111, Cohomology of Principal Bundles and Homogeneous Spaces J. DIEUDONN~.Treatise on Analysis, Volume 11, enlarged and corrected printing JOHNJ. BENEDETTO.Spectral Synthesis 1. MARTINISAACS. Character Theory of Finite Groups HOMOTOPY THEORY An Introduction to Algebraic Topology BRAYTO& GRAY Department of Mathematics University of Illinois at Chicago Circle Chicago, Illinois ACADEMIC PRESS New York San Francisco London 1975 A Subsidiary of Harcourt Brace Jovanovich, Publishers COPYRIGHT 0 1975, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWl Library of Congress Cataloging in Publication Data Gray, Brayton. Homotopy theory. (Pure and applied mathematics series, 64) Bibliography: p. 1. Homotopy theory. I. Title. 11. Series: Pure and applied mathematics; a series of monographs and textbooks. QA3.PI1 IQA612.71 510'.8s [514'.24] 74-5686 ISBN 0-12-296050-5 PRINTED IN THE UNITED STATES OF AMERICA Contents Preface vii List of Symbols xi 0 Preliminaries 1 1 Some Simple Topological Spaces 4 2 Some Simple Topological Problems 9 3 Homotopy Theory 13 4 Category Theory 17 5 The Fundamental Group 22 6 More on the Fundamental Group 29 7 Calculating the Fundamental Group 34 8 A Convenient Category of Topological Spaces 50 9 Track Groups and Homotopy Groups 62 10 Relative Homotopy Groups 70 11 Locally Trivial Bundles 77 12 Simplicia1 Complexes and Linearity 90 13 Calculating Homotopy Groups : The Blakers-Massey Theorem 99 14 The Topology of CW Complexes 113 15 Limits 122 16 The Homotopy Theory of CW Complexes 130 17 K(n, n)’s and Postnikov Systems 157 18 Spectral Reduced Homology and Cohomology Theories 168 19 Spectral Unreduced Homology and Cohomology Theories 183 20 Ordinary Homology of CW Complexes 187 21 Homology and Cohomology Groups of More General Spaces 200 22 The Relation between Homotopy and Ordinary Homology 21 5 V vi Contents 23 Multiplicative Structure 22 I 24 Relations between Chain Complexes 245 25 Homological Algebra over a Principal Ideal Domain : Kunneth and Universal Coefficient Theorems 256 26 Orientation and Duality 269 27 Cohomology Operations 294 28 Adem Relations 310 29 K-Theories 324 30 Cobordism 342 Table 1 Adem Relations Sg" Sgb .for a < 2b 12 358 References 360 Index 363 Preface This book is an exposition of elementary algebraic topology from the point of view of a homotopy theorist. The only prerequisite is a good foun- dation in point set topology. In particular, homology theory is not assumed. Both homology and cohomology are developed as examples of (generalized) homology and cohomology theories. The idea of developing algebraic topology in this fashion is not new, but to my knowledge this is the first detailed exposition from this viewpoint. One pedagogical advantage of a course developed in this way is that it may be studied before or after a course in classical homology theory (e.g. [71] or [28]); alternatively homology and cohomology could first be introduced, as they are here, as examples of a more general theory. The philosophical emphasis here is: to solve a geometrical problem of a global nature, one first reduces it to a homotopy theory problem; this is in turn reduced to an algebraic problem and is solved as such. This path has historically been the most fruitful one in algebraic topology. The first few sections are introductory in nature. These are followed by a discussion of the fundamental group, covering spaces, and Van Kampen’s theorem. The fundamental group serves as a model of a functor that can be calculated. In Section 8 we introduce the category of compactly generated spaces. This seems to be the most appropriate category for algebraic topology. Many results which are most often stated in the category of CW complexes are valid in this generality. The key result we use to make calculations is the Blakers-Massey theorem. This is strong enough to imply the suspension theorem and the Serre exact sequences. The Blakers-Massey theorem is proved by linear approximation vii ... Vlll Preface techniques in Section 13 (after J. M. Boardman) in the case of a pair of relative cells. This allows one to calculate IT,(S").The more general form of the Blakers-Massey theorem is proved in Section 16. In Section 18 reduced " spectral " homology and cohomology theories are defined from arbitrary spectra on compactly generated spaces. They satisfy the usual axioms in this generality. It is more complicated to show that unreduced theories satisfy the usual axioms; this is done in Section 21. Spectral homology theories agree with their " singular approximations." Spectral cohomology theories on paracompact spaces have all the properties usually associated with cech theory. Calculations in the ordinary homology of CW complexes are studied in Section 20. We develop axioms for the chain complex of a CW complex. These are strong enough to make all the usual calculations based on ad hoc decompositions. In particular a proof of the algorithm for the homology of a simplicia1 complex is given. (The algorithm for singular homology follows from the functorial singular complex construction which is included as an appendix to Section 16.) The Hurewicz theorem follows quite easily from the Blakers-Massey theorem. Duality in manifolds in its full generality for an arbitrary ring spectrum follows from the usual inductive approach using " spectral " homology and cohomology. In Section 27 we introduce Steenrod operations geometrically via the quadratic construction. We learned this approach from J. Milgram. The Adem relations are proven in Section 29 by a method due to L. Kristensen. This necessitates calculating H*(K(Z,, n); 2,) for which we state, without proof, the Bore1 transgression theorem. Spectral sequences have been omitted for several reasons : their introduc- tion would increase the length of the book considerably; they are more difficult to write about than to explain; there already exist expositions which we feel we cannot improve upon ([21, 41, 311). In the last two sections we sketch K-theory and cobordism. This serves as an introduction to some of the more powerful functors which have been utilized in algebraic topology. Applications of K-theory to the Hopf invari- ant and the vector field problem are discussed. In the last section x,(MO) is calculated. We have used throughout the symbol 1 whenever we have finished the proof of some theorem, proposition, corollary, or claim made earlier. Sometimes these symbols pile up-for example, if one statement is reduced to another. The exercises are an integral part of the development. No exercise requires outside reading, or the utilization of techniques not previously developed (usually in the relevant section). A * on an exercise indicates that its under- Preface ix standing is necessary for further study-not that it is difficult. Often termi- nology used later is first introduced in such an exercise. Many exercises are used later in only a few places. In such cases it is indicated in parentheses at the end of the exercise where the result is used. A one-quarter introductory course could easily be based on Sections 0-13. Section 8 could be skipped if an independent proof of adjointness is given. Section 8 is used essentially in Section 18 unless one restricts the homology and cohomology theories to finite complexes. Occasionally seminar problems are given at the end of a section. These are topics peripheral to the material in the section and not used elsewhere. They are
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