Algebraic homotopy Algebraic homotopy HANS JOACHIM BAUES Max-Planck-Institut fur Mathematik, Bonn S The right of the University If Cambridge to print and sell a/I manner of books vas granted by Henry VIII in 1534. The University has printed and published continuously since 1584. CAMBRIDGE UNIVERSITY PRESS Cambridge New York New Rochelle MelbourneSydney CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge C132 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521333764 © Cambridge University Press 1988 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1989 This digitally printed version 2008 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Baues, Hans J., 1943- Algebraic homotopy. Bibliography: P. Includes index. 1. Homotopy theory. 2. Categories (Mathematics) I. Title. QA612.7.B385 1986 514'.24 86-32738 ISBN 978-0-521-33376-4 hardback ISBN 978-0-521-05531-4 paperback Fur Barbara and fur unsere Kinder Charis and Sarah CONTENTS Preface xi Introduction xiii List of symbols xvi IAxioms for homotopy theory and examples of cofibration categories 1 0Cofibrations and fibrations in topology 2 1 Cofibration categories 5 laAppendix: fibration categories to lbAppendix: proof of (1.4) 12 2The axioms of Quillen 14 3 Categories with a natural cylinder 18 3aAppendix: categories with a natural path object 27 4The category of topological spaces 29 4aAppendix: the relative homotopy lifting property 31 4bAppendix: the cube theorem 33 5 Examples of cofibration categories 34 6The category of chain complexes 39 7 The category of chain algebras 45 8 The category of commutative cochain algebras 58 9The category of chain Lie algebras 74 IIHomotopy theory in a cofibration category 83 1 Some properties of a cofibration category 83 2Sets of homotopy classes 92 3 The homotopy category of fibrant and cofibrant objects 99 4Functors between cofibration categories 102 5 The groupoid of homotopies 104 5aAppendix: homotopies and functors 110 6Homotopy groups 115 viii Contents 6aAppendix: homotopy groups and functors 118 7 Relative homotopy groups and the exact homotopy sequence of a pair 119 8 Principal cofibrations and the cofiber sequence 125 9Induced maps on cofibers 131 10Homotopy groups of function spaces 135 10aAppendix: homotopy groups of function spaces and functors 141 11 The partial and functional suspensions 142 12The difference operator 145 13Double mapping cones 147 14Homotopy theory in a fibration category 152 15Whitehead products 154 15aAppendix: Whitehead products, co-Whitehead products and cup products in topology 160 16Examples on the classification of maps in topology 164 17Homotopy groups in the category of chain algebras 169 IIIThe homotopy spectral sequences in a cofibration category 179 1 Homotopy groups of homotopy limits 180 2The homotopy spectral sequence 189 3 Based complexes and cohomology 194 4Complexes and twisted cohomology 199 5 Cohomology with local coefficients of CW-complexes 206 5aAppendix: admissible classes of complexes 211 6Eilenberg-Mac Lane spaces and cohomology with local coefficients 213 7 Postnikov towers 215 8 Nilpotency of function spaces in topology 216 9Homotopy groups of function spaces in topology 218 10The relative homotopy spectral sequence 219 11 The relative homotopy spectral sequence for CW-complexes 227 IVExtensions, coverings, and cohomology groups of a category 229 1 Detecting functors 229 2Group actions on categories 231 3 Linear extensions of categories 236 4Linear coverings of categories and exact sequences for functors 239 5 The cohomology of a small category 245 6Classification of linear extensions 251 7 Classification of linear coverings 255 258 V Maps between mapping cones 1 Group actions on the category PAIR 258 2Principal and twisted maps between mapping cones 260 3 A linear group action 263 3aAppendix: the homotopy category of Moore spaces in degree n 267 4 A quadratic group action 269 271 5 The equivalence problem Contents ix 6 Maps between fiber spaces in a fibration category 275 7 The homotopy type of a mapping cone in topology 277 7aAppendix: proof of the general suspension theorem under D 283 8 Example: homotopy theory of the 2-stem and of the 3-stem 286 9Example: The group of homotopy equivalence of the connected sum (S' X S3)#(S2 X S2) 290 10The homotopy type of a fiber space in topology 295 10aAppendix: proof of the general loop theorem over D 302 VIHomotopy theory of CW-complexes 306 1 Crossed chain complexes 306 2The functional suspension and the Peiffer-group for 2-dimensional CW- complexes 315 3 The homotopy category of 2-dimensional CW-complexes under D 319 4The tower of categories for the homotopy category of crossed chain complexes 322 5 Homotopy systems of order n, n >-- 3 336 6The tower of categories for the homotopy category of CW-complexes under D 346 7 Small models and obstructions to finiteness for CW-complexes under D 356 8 n-dimensional CW-complexes and (n - 1)-types 364 VIIHomotopy theory of complexes in a cofibration category 371 1 Twisted maps between complexes 371 2Twisted maps and 1-homotopies 385 3 Examples of complexes in topology 397 4Complexes in the category of chain algebras 401 VIII Homotopy theory of Postnikov towers and the Sullivan-de Rham equival- ence of rational homotopy categories 404 1 The tower of categories for Postnikov decompositions 404 2 The category of K(A, n)-fibrations over D 414 3 The de Rham theorem for cohomology groups with local coefficients 416 4 The tower of categories for commutative cochain algebras under D 424 5 The category of A(V, n)-cofibrations under D 426 6Integral homotopy theory of spaces over D and minimal models 427 IXHomotopy theory of reduced complexes 436 1 Reduced complexes 436 2The tower of categories for reduced complexes 438 3 Functors on reduced complexes and the Quillen equivalence of rational homotopy categories 443 4Whitehead's classification of simply connected 4-dimensional CW- complexes and the corresponding result for reduced complexes 447 Bibliography 455 Index 461 PREFACE This book gives a new general outlook on homotopy theory: fundamental ideas of homotopy theory are developed in the presence of a few axioms so that they are available in a broad variety of contexts. Many examples and applications in topology and algebra are discussed; we consider the homotopy theory of topological spaces, the algebraic homotopy theory of chain algebras, and rational homotopy theory. The axiomatic approach saves a lot of work in the various fields of application and offers a new way of organizing a course of modern homotopy theory. A fruitful interplay takes place among the various applications. This book is also a research monograph on homotopy classification problems. The main new result and our principal objective is the `tower of categories' which approximates the homotopy category of complexes. Such towers turn out to be a useful new tool for homotopy classification problems; they complement the well-known spectralsequences. The theory on complexes is a continuation of J.H.C. Whitehead's combinatorial homotopy. In fact, some of Whitehead's results can be derived readily from the properties of the towers. In a later chapter (Chapter IX) we describe the simplest examples of towers of categories from which nevertheless fundamental results of homotopy theory can be immediately deduced. Most of the material in the book does not appear in any textbook on algebraic topology and homotopy theory. As prerequisites the reader should be familiar with elementary topology and the language of categories. The book can also be used by readers who have only a little knowledge of topology and homotopy theory, for example when they want to apply the methods of homotopical algebra in an algebraic xii Preface context. The book covers the elementary homotopy theory in an abstract way. The nine chapters which comprise the book are subdivided into several sections, §0, § 1, § la, § lb, §2, etc. Definitions, propositions, remarks, etc., are consecutively numberd in each section, each number being preceded by the section number, for example (1.5) or (1a.5). A reference such as (11. 5.6) points to (5.6) in Chapter II, while (5.6) points to (5.6) in the chapter at hand. References to the bibliography are given by the author's name, e.g. J.H.C. Whitehead (1950). I lectured on the material presented in this book in Bonn (1982), Lille (1982), Berlin (1985), Zurich (1986) and on several conferences. There are further applications of the results which cannot be described in a book of this size. In particular, we obtained an algebraic classification of (n- 1)- connected (n + 3)-dimensional polyhedra for n >_ 1. The invariants are computable; for example, there exist exactly 4732 simply connected homotopy types which have the homology groups (n >_ 4) 7L/4 Q+ 71/4 Q+ 7L i = n 7L/8Q+7L i=n+1 H1(X)= Z/2p+71/4p+7L i=n+2 Z i=n+3 0 otherwise. Further details will appear elsewhere. However, the basic machinery for these results is developed in this book. I would like to acknowledge the support of the Sonderforschungsbereich 40 Theoretische Mathematik, of the Max-Planck-Institut fur Mathematik in Bonn, and of the Forschungsinstitut fur Mathematik ETH Zurich. Moreover, I am very grateful to A. Grothendieck for a series of letters concerning Chapters I and II. I especially thank my colleagues and friends S. Halperin, J.M. Lemaire, and H. Scheerer for their interest and for valuable suggestions during the years that I worked on this book; in fact, their work influenced and inspired the development of the ideas; I remember with pleasure the discussions in Bonn, Toronto, Nice, and Berlin and also in Lille and Louvain la Neuve where J.