LOCALIZATION OF NILPOTENT GROUPS AND SPACES LOCALIZATION OF NILPOTENT GROUPS AND SPACES This Page Intentionally Left Blank NORTH-HOLLAND MATHEMATICS STUDIES 15 Notas de Matematica (55) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester Localization of Nilpotent Groups and Spaces PETER HILTON Battelle Seattle Research Center, Seattle, and Case Western Reserve University, Cleveland GUIDO MlSLlN Eidgenossische Technische Hochschule. Zurich JOE ROITBERG Institute for Advanced Study, Princeton, and Hunter College, New York 1975 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM * OXFORD AMERICAN ELSEVIER PUBLISHING COMPANY INC. - NEW YORK @ NORTH-HOLLAND PUBLISHING COMPANY, - AMSTERDAM - 1975 All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior'permission of the Copyright owner. Library of Congress Catalog Card Number: ISBN North-Holland: Series : 0 7204 2700 2 Volume: 0 7204 2716 9 ISBN American Elsevier: 0 444 10776 2 PUBLISHER: NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NORTH-HOLLAND PUBLISHING COMPANY, LTD, - OXFORD SOLE DISTRIBUTORS FOR THE USA. AND CANADA: AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017 PRINTED IN THE NETHERLANDS Table of Contents Introduction v11 Chapter I. Localization of Nilpotent Groups Introduction 1 1. Localization theory of nilpotent groups 3 2. Properties of localization in N 19 3. Further properties of localization 23 4. Actions of a nilpotent group on an abelian group 34 5. Generalized Serre classes of groups 43 Chapter 11. Localization of Homotopy Types Introduction 47 1. Localization of 1-connected CW-complexes 52 2. Nilpotent spaces 62 3. Localization of nilpotent complexes 72 4. Quasifinite nilpotent spaces 79 5. The main (pullback) theorem 82 6. Localizing H-spaces 90 7. Mixing of homotopy types 94 Chapter 111. Applications of Localization Theory Introduction 101 1. Genus and H-spaces 104 2. Finite H-spaces, special results 122 3. Non-cancellation phenomena 133 Bibliography 14 7 Index 154 V This Page Intentionally Left Blank Introduction Since Sullivan first pointed out the availability and applicability of localization methods in homotopy theory, there has been considerable work done on further developments and refinements of the method and on the study of new areas of application. In particular, it has become quite clear that an appropriate category in which to apply the method, and indeed--as first pointed out by Dror--in which to study the homotopy theory of topological spaces in the spirit of J. H. C. Whitehead and J.-P. Serre, is the (pointed) homotopy category NH of nilpotent CW-complexes. Here a pointed space X is said to be nilpotent if its fundamental group is a nilpotent group and operates nilpotently on the higher homotopy groups. For a given family P ofrationalprimes, the concept of a P-local space is based simply on the requirement that its homotopy groups be P-local. Thus a localization theory for the category NH requires, or involves, a localization theory of nilpotent groups and of nilpotent actions of nilpotent groups on abelian groups. This latter theory could be obtained as a by-product of the topological theory (this is, in fact, the approach of Bousfield-Kan) but we have preferred to make a purely algebraic study of the group-theoretical aspects of the localization method. Thus this monograph is devoted toanexposition of the theory of localization of nilpotent groups and homotopy types. Chapter I, then, consists of a study of the localization theory of nilpotent groups and nilpotent actions. It turns out that localization methods work particularly well in the category N of nilpotent groups, in the sense that we can detect the localizing homomorphism e: G + Gp by meansof effective properties of the homomorphism e, and that localization does not destroy the fabric of a nilpotent group. For example, the nilpotency . class of Gp never exceeds that of G, and G embeds in HG Chapter I P P also contains some applications of localization methods in nilpotent group theory. v11 Vlll Introduction Chapter I1 takes up the question of localization in homotopy theory. We first work in the (pointed) homotopy category of 1-connected CW-complexes, H1 and then extend the theory to the larger category NH of nilpotent CW-complexes. This extension is not only justified by the argument that we bring many more spaces within the scope of the theory (for example, connected Lie groups are certainly nilpotent spaces); it also turns out that even to prove fundamental theorems about localization in H1, it is best to argue in the larger category . NH One may represent the development of localization theory as presented in this monograph--as distinct from an exposition of its applications to problems in nilpotent group theory and homotopy theory--as follows; here Ab is the category of abelian groups. Thus we start from the (virtually elementary) localization theory in the category Ab of abelian groups. The arrow from Ab to N represents the generalization of localization theory from the category Ab to the category N of nilpotent groups. The arrow from Ab to H represents 1 the application of the localization theory of abelian groups to that of 1-connected CW-complexes. The remaining two arrows of the diagram indicate that the localization theory in NH is a blend of application of the localization theory in N and generalization of the localization theory in The diagram (L) which, as we say, representsschematically our approach H1. to the exposition of the localization theory of nilpotent homotopy types, is, of course, highly non-commutative! Introduction 1x In Chapter 111, we describe some important applications of localization methods in homotopy theory. Naturally, our choice of application is very much colored by our particular interests. We have concentrated, first, on the theory of connected H-spaces, and, second, on non-cancellation phenomena in homotopy theory. Localization methods have proved to be very powerful in the construction of new H-spaces and in the detection of obstructions to H-structure. We give a fairly comprehensive introduction to the methods used and obtain several results. Again, it has turned out that there is a close connection between concepts based on localization methods and the situation,already noted by the authors and others, of compact polyhedra exhibiting either the phenomenon X V A N Y V A , X+Y, or the phenomenon X x A = Y x A , X$Y; we describe this connection in some detail. Given a localization theory in some category C (and a reasonable finiteness condition imposed on the objects under consideration, for reasons of practicality), one can introduce the concept of the genus G(X) of an object X of C. Thus we would say that X, Y in C belong to the same genus, or that Y € G(X), if X is equivalent to Y for all primes p. P P It turns out that in the category Ab (confining attention to finitely- generated abelian groups), objects of the same genus are necessarily isomorphic; however, no such corresponding result holds in the categories N, H1, NH. (In N we again confine attention to finitely-generated groups; in H1 and NH, we confine attention to spaces with finitely-generated homotopy groups in each dimension.) Thus localization theory naturally throws up questions of the nature of generic invariants; we embark on a study of these questions in this