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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 H I L T O N Battelle Seattle Research Center, Seattle, and Case Western Reserve University, Cleveland GUIDO M l S L l N 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 U S A . A N D C A N A D A : AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017 PRINTED IN THE NETHERLANDS Table of Contents v11 Introduction Chapter I. Chapter 11. Localization of Nilpotent Groups Introduction 1 1. Localization theory of nilpotent groups 3 2. Properties of localization in N 19 3. 23 Further properties of localization 4. Actions of a nilpotent group on an abelian group 34 5. 43 Generalized Serre classes of groups 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 6. 82 90 Localizing H-spaces 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. 133 Non-cancellation phenomena 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 . embeds in HG Chapter I P P also contains some applications of localization methods in nilpotent group theory. class of Gp never exceeds that of G, and G v11 Introduction Vlll Chapter I1 takes up the question of localization in homotopy theory. We first work in the (pointed) homotopy category H1 of 1-connected CW-complexes, 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, NH it is best to argue in the larger category . 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 H1 represents 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 H1. The diagram (L) which, as we say, representsschematically our approach 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 XVANYVA, X+Y, XxA=YxA, X$Y; or the phenomenon 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) object X of C. Thus we would say that X, Y of an 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, N we again confine attention to finitely-generated groups; in H1 NH. (In 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 X Introduction menograph. We do not describe explicitly any algebraic invariants (beyond the fundamental group) capable of distinguishing homotopy types in NH of the same genus. We remark that all known examples of the non-cancellation phenomenon referred to above concern spaces X, Y of the same genus; this explains the connection with localization theory to which we have drawn attention. Each chapter is Surnished with its own introduction describing the purpose and background of the chepter, and detailing its contents. We will therefore not need to offer a more comprehensive description of the section contents in this overall introduction. It is a pleasure to acknowledge the encouragement of Professor Leopoldo Nachbin, who first proposed the writing of this monograph; the excellent cooperation which we have received from the editorial staff of the North-Holland Publishing Company; the assistancereceived from many friends working in or close to the area covered by the monograph; and, last but certainly not least, the truly wonderful assistance of Ms. Sandra Smith, who succeeded in converting a heterogenous manuscript reflecting the many divergences of style and handwriting of its three authors into a typescript which could be transmitted with a clear conscience to the publisher. Battelle Seattle Research Center and Case Western Reserve University, Cleveland Peter Hilton Eidgenlissische Technische Hochsahule, Ztirich Guido Mislin Institute for Advanced Study, Princeton and Hunter College, New York Joe Roitberg June, 1974 Chapter I L o c a l i z a t i o n of N i l p o t e n t Groups Introduction Our o b j e c t i n t h i s c h a p t e r is t o d e s i r i b e t h e t h e o r y of l o c a l i z a t i o n of n i l p o t e n t groups a t a s e t of primes P. This t h e o r y was f i r s t developed i n an important s p e c i a l c a s e by Malcev [52] and was l a t e r reworked and extended by Lazard [ 5 0 ] and o t h e r s (cf. Baumslag I 6 1, H i l t o n [34 1 , Q u i l l e n [66 1, Warfield [ 8 6 1 ) . With t h e advent of S u l l i v a n ' s t h e o r y of l o c a l i z a t i o n of homotopy t y p e s [ 8 3 ] , i t was observed by t h e a u t h o r s [ 4 2 , 431 and independently by Bousfield-Kan [12, 1 3 , 1 4 1 , t h a t t h i s a l g e b r a i c t h e o r y of l o c a l i z a t i o n of n i l p o t e n t groups e n t e r e d q u i t e n a t u r a l l y and s i g n i f i c a n t l y i n t o c e r t a i n q u e s t i o n s of homotopy t h e o r y . Our approach i n t h i s c h a p t e r i s , i n f a c t , i n s p i r e d by t h e homotopy-theoretical c o n s i d e r a t i o n s of [ 4 3 ] , and f o l l o w s r a t h e r c l o s e l y t h e s y s t e m a t i c t r e a t m e n t of H i l t o n [ 3 4 , 3 5 1 . It should b e mentioned t h a t Bousfield-Kan have a l s o given a t r e a t m e n t of t h e t h e o r y of l o c a l i z a t i o n of n i l p o t e n t groups from a homotopy-theoretical p o i n t of view, b u t t h e i r approach rests on t h e t h e o r y of l o c a l i z a t i o n of n i l p o t e n t homotopy t y p e s , whereas i n our approach a l l t h e i n e s s e n t i a l topology h a s been s t r i p p e d away, and w e make u s e only of s t a n d a r d homological a l g e b r a t o g e t h e r w i t h elementary group t h e o r y ; s e e Hilton-Stammbach [47]. The c h a p t e r i s organized a s f o l l o w s . In Seetionlweintroduce thebasic n o t i o n s and terminology and prove t h e e x i s t e n c e of a P - l o c a l i z a t i o n f u n c t o r on t h e c a t e g o r y of n i l p o t e n t groups, where ( r a t i o n a l ) primes. is an a r b i t r a r y c o l l e c t i o n of P Our proof proceeds by i n d u c t i o n on t h e n i l p o t e n c y c l a s s of t h e group and i s based on t h e c l a s s i c a l i n t e r p r e t a t i o n of t h e second cohomology group of a group. I n c o r p o r a t e d i n t o t h e e x i s t e n c e theorem is t h e v e r y c r u c i a l f a c t t h a t a homomorphism iff K is P-local and $ $: G -f is a P-isomorphism; K of n i l p o t e n t groups P - l o c a l i z e s s e e D e f i n i t i o n s 1.1 and 1.3 below. Localization of nilpotent groups 2 Section 2 contains some immediate consequences of t h e methods and r e s u l t s of Section 1, t h e most notable a s s e r t i o n s being t h e exactness of P - l o c a l i z a t i o n and t h e theorem t h a t a homomorphism 0: G + K of n i l p o t e n t groups P-localizes i f f t h e corresponding homology homomorphism g,($) : H*(G) + fi,(K) P-localizes. I n Section 3 , we prove a number of r e s u l t s on l o c a l i z a t i o n of n i l p o t e n t groups which t u r n out t o be t h e a l g e b r a i c precursors of corresponding r e s u l t s on t h e l o c a l i z a t i o n of n i l p o t e n t homotopy types. f i n i t e l y generated n i l p o t e n t group its localizations G G may be i d e n t i f i e d with t h e pullback of over i t s r a t i o n a l i z a t i o n P For example, we show t h a t a G o , The homotopy-theoretical counterparts of t h e r e s u l t s of Section 3 w i l l be discussed i n t h e l a t t e r p a r t of Chapter 11. I n Section 4 , we present r e s u l t s concerning n i l p o t e n t a c t i o n s of groups on a b e l i a n groups, which play an important r o l e i n t h e c o n s t r u c t i o n , i n the f i r s t p a r t of Chapter 11, of t h e l o c a l i z a t i o n f u n c t o r on t h e category of n i l p o t e n t homotopy t y p e s . F i n a l l y , i n Section 5, we introduce a generalized version of t h e notion of "Serre c l a s s " , which provides t h e c o r r e c t a l g e b r a i c s e t t i n g f o r general Serre-Hurewicz-Whitehead theorems f o r n i l p o t e n t spaces. A s mentioned e a r l i e r , we s h a l l follow, f o r t h e most p a r t , t h e exposition i n [ 3 4 , 3 5 1 . I n f a c t , much of Chapter I is a revised and somewhat condensed version of [ 3 4 , 351, f o r t h e f i r s t time. though some m a t e r i a l appears here Since our primary concern i n t h i s monograph is r e a l l y with t h e l o c a l i z a t i o n of n i l p o t e n t homotopy types, we have l a r g e l y r e s t r i c t e d ourselves i n Chapter I t o a discussion of those a s p e c t s of t h e theory of l o c a l i z a t i o n of n i l p o t e n t groups which a r e r e l e v a n t t o homotopy theory. reader may consult [ 3 4 , 351 l o c a l i z a t i o n theory. The f o r purely g r o u p - t h e o r e t i c a l a p p l i c a t i o n s of Localization theory of nilpotent groups 3 1. L o c a l i z a t i o n t h e o r y of n i l p o t e n t groups G , we d e f i n e t h e lower c e n t r a Z s e r i e s of For a group G, by s e t t i n g r 1(G) Recall t h a t c G = G, r i+l(G) is nilpotent i f r j ( G ) = {l} i s t h e l a r g e s t i n t e g e r f o r which class c and w r i t e nil(G) = c. = [G,r i (G)], i 2 1. f(G) for j sufficiently large. # {l}, we s a y t h a t If hasnizpotency G (Dually, we d e f i n e t h e upper c e n t r a l series of G, by r e q u i r i n g t h a t zi+l i (G)/Z (G) = c e n t e r of so t h a t 1 Z (G) i s t h e c e n t e r of Zc(G) = G , ZC-'(G) # G.) G. G/Z i Then (G) , G has nilpotency c l a s s i 2 0, The f u l l subcategory of t h e c a t e g o r y N groups c o n s i s t i n g of a l l n i l p o t e n t groups i s denoted by subcategory of Nc. G In particular, G c of a l l and t h e f u l l c o n s i s t i n g of a l l n i l p o t e n t groups w i t h n i l (G) 5 c N 1 = iff by Ab, t h e c a t e g o r y of a l l a b e l i a n groups. W e s h a l l be concerned w i t h c o l l e c t i o n s of r a t i o n a l primes and s h a l l denote such c o l l e c t i o n s by c o l l e c t i o n of a l l primes. denote by P' P , Q , e t c . ; we r e s e r v e t h e n o t a t i o n I n general, i f P for the i s a c o l l e c t i o n of primes, w e t h e complementary c o l l e c t i o n of primes. a product of primes i n Il If the integer PI, we (somewhat a b u s i v e l y ) w r i t e n is n C P'. *It would seem t o b e more r e a s o n a b l e t o r e n o r m a l i z e and w r i t e ro(G) = G , e t c . b u t we f o l l o w t h e convention most f r e q u e n t l y employed i n t h e l i t e r a t u r e . Localization of nilpotent groups 4 Definition 1.1. A group G is said to be P-local if x - xn, x € G, is If H is a full subcategory of G, then a bijective for all n € homomorphism e: G Gp in H is said to be P-universal (with respect to -+ P'. H), or to be a P-localizing map if Gp i s P-local and if e*: Hom(G ,K) 2 Hom(G,K) provided K € H , with K P-local. P instead of If P = &, we speak of 0-local, 0-universal, ..., ..., and write &local, &universal, Go instead of G4. We also sometimes speak of rationalization instead of 0-localization. If P = fp), we speak ..., and write G instead of G {PI. P Assume now that each group in H admits a P-localizing map. Then, of p-local, p-universal, for any $: G + K in H, we have a unique map ": Gp -+ 5 rendering the diagram comutative. Thus we have a functor L: H functor and we may view e as a -+ H which we call a P-localizing natural transformation e: 1 -+ L having the universal (initial) property with respect to maps to P-local groups in H. We regard the pair H. (L,e) as providing a P-localization theory on the category It is clear that if a P-localization theory exists on H, it is essentially unique, Our main goal in this section is, in fact, to construct such a theory on the categories Nc, N. We note, for later use, the following Proposition. Proposition 1.2. Let G' >-b G % G" be a central extension of groups. Then G is P-local if G', Proof. Let x E y € G. Thus x = ynu(y'), GI' G, are P-local. n E PI. Then y' € G'. x = ynu(x')" EX * yttn= ~y~ for some y"€ G " , But y' = xtn for some x' € G' 5 (YP(X'))", so Localization theory of nilpotent groups since is central i n uG' Suppose now t h a t ex = in G, x q n = 1, SO xn = yn, x , y € G , n € P ' . XI Then xn = ynu(x'"> Then cxn = eyn, so s i n c e uG' is c e n t r a l = 1, x = y . A homomorphism D e f i n i t i o n 1.3. ker G. x = yu(x'), x' € G'. EY, 5 G @: -f is s a i d t o be P - i n j e c t i v e K if c o n s i s t s of P I - t o r s i o n elements; and is s a i d t o b e P - s u r j e c t i v e if, C$ given any y € K, t h e r e e x i s t s P-isomorphism or n € P' with yn € i m @.A homomorphism is a P - b i j e c t i v e i f i t is b o t h P - i n j e c t i v e and P - s u r j e c t i v e . It i s p l a i n t h a t a composite of P - i n j e c t i v e is again P-injective (P-surjective). (P-surjective) homomorphisms I n addition, the following r e s u l t s w i l l be u s e f u l i n t h e s e q u e l . Lemma 1.4. L e t a: G + G2, B : G2 + G3 be group homomorphisms. (il If Ba is P - s u r j e c t i v e , fii) If Bci is P - i n j e c t i v e and (iii) If Ba is P - i n j e c t i v e , i s P-surjective; B then i s P-surjective, a then B is P-injective; (ivl G 2 € N, then Since x1 € G1. Then a is P-surjective, Baxl = 1, s o , xy = 1. Then x; of ax^ for some = (ax,) y2 is P - i n j e c t i v e and if, i n a d d i t i o n , B (i) and ( i i i ) a r e t r i v i a l . Ba with x1 € G1, n € P ' . and n € P' being P-injective, mn x2 = 1, mn € P ' , so y2 € G2 To prove (ii), l e t there exists F i n a l l y , t o prove ( i v ) , l e t ~x: = i s P-injective; a i s P-surjective. a Proof. with @a i s P - s u r j e c t i v e , If Bx2 = 1. then y: B with x2 € G2, xn = ax 2 1' there e x i s t s m € P' is P - i n j e c t i v e . x2 € G2, Since Then, s i n c e = 1, m € P ' . B Ba is P-surjective is P-injective, But i t i s a consequence P. H a l l ' s t h e o r y of b a s i c commutators (see [34]) t h a t we t h e n have 6 Localization of nilpotent groups mc .yC provided = (ax,) a nil(G ) C c. 2 Since nmc € P ' , i t follows t h a t is P-surjective. Lemma 1 . 5 . Y: G1 Let be a homomorphism between P-local groups. G2 (il If y is P - i n j e c t i v e , then (iil If y i s P-surjective, Proof. (i) Since there e x i s t s y is i n j e c t i v e ; then y is s u r j e c t i v e . ker y is PI-torsion and has no PI-torsion, G1 ker y = {l), proving ( i ) . To prove ( i i ) ,l e t is c l e a r t h a t with + n € P' yy = xl. with Thus Proposition 1 . 6 . (yyl) x; n = yxl, n = x2 so t h a t and t h e r e e x i s t s yyl = x 2 , i . e . y y1 then Then if Q', Q" a r e P - i n j e c t i v e (resp. 0 i s a l s o P - i n j e c t i v e (resp. P - s u r j e c t i v e ) . $"EX=;Qx = 1 so t h e r e e x i s t s x ' € G ' , and $ ' X I = 1. mn E P ' , and Q so yn = so Q Q', with ($x0)(;y'), y T m= Q ' x ' . n € P' But then with EX)^ = 1. Thus xIm = 1 f o r some Then xn = ux', m € P ' , so xmn = 1, is P-injective. Assume now n E P' G1 is s u r j e c t i v e . Proof. Assume Q', Q" P - i n j e c t i v e and l e t x C ker Q. x" C G", Then Let be a map of c e n t r a l extensions. p-surjective), x1 C G1 x2 € G2. it Q" P - s u r j e c t i v e and l e t y C K. ;yn = @"x". Let x" = E Xo , xo E G . y' € K ' . Then, s i n c e is P-surjective. FK' But now there exist is c e n t r a l in K, There e x i s t Then x' € G', Eyn = EQxo, m € P' with Localization theory of nilpotent groups The preceding discussion, with the exception of Lemma 1.4(iv), of a rather general nature. was We now concentrate our attention on nilpotent groups and state the main result of this section. Fundamental Theorem on the P-Localization of Nilpotent Groups. There mists a P-1ocaZization theory to a P-localization theory nil LG S on the category N. (L,e) Moreover, Nc, for each c (Lc,ec) on ? L restricts 1. In particular, G € N. nil G if Further, $: G -+ K in N P-ZocaZizes iff K is P-local and $I is a P-isomorphism. The proof of the Fundamental Theorem is by induction on c = nil(G). More precisely, given Lc-l: Nc-l -+ Nc-l desired properties, we construct Lc: Nc and -+ e * c-1' Nc and 1 theory on Ab. c = Lc-l having the e : 1 the desired properties and such that moreover LclNc-l To start the induction at -+ = -+ L also having Lc-l, ec/Nc-l = ec-l. 1, we must construct a P-localization In addition to doing this, we shall consider the interrelationships between P-localization on Ab and the standard functors arising in homological algebra, which information will be required both in the inductive step and in Chapter 11. Recall that the subring of with R € P'. Q Zp is the ring of integers localized at P, that is, consisting of rationals expressible as fractions k/L Note that %= 72, Zo = Q. For A € Ab, we define L ~ =A ~p and we define el: A -+ % = A a zP to be the canonical homomorphism Note that a P-local abelian group is just a Zp-module. 8 Localization of nilpotent groups It is e v i d e n t t h a t el: A + Ap is P-universal w i t h r e s p e c t t o Ab, so t h a t we have c o n s t r u c t e d a P - l o c a l i z a t i o n t h e o r y on prove s h o r t l y ( P r o p o s i t i o n 1 . 9 ) t h a t el ”: % -+ Bp We w i l l is P - b i j e c t i v e , from which we immediately deduce, u s i n g Lemmas 1.4 and 1 . 5 , t h a t P - b i j e c t i v e i f and only i f Ab. 6: A -+ is an isomorphism. B i n Ab is Localization theory of nilpotent groups 9 Before v e r i f y i n g Proposition 1 . 9 and t h u s Ab, we e s t a b l i s h t h e following v a l i d a t i n g t h e Fundamental Theorem f o r Propositions. Proposition 1.7. The f u n c t o r + Ab i s exact. It is only necessary t o n o t e t h a t Proof. (flat) L1? Ab Zp is t o r s i o n f r e e . We now c o l l e c t together i n t o a s i n g l e p r o p o s i t i o n a number of u s e f u l f a c t s about . (Ll, el) P r o p o s i t i o n 1.8. If (i) Tor(el,l): A , B f Ab, then: e 1 a91: A @ B + A p @ B ; e l m e l : Tor(A,B) -+ Tor(%,B), Tor(el,el): A@B+%@Bp; Tor (A,B) + Tor(%,BP) all P-localize. (ii) A p-isomorphism $: A Conversely, a homomorphism $ : A and Tor($, Z / p ) -+ B --t B, induces isomorphisms $ O Z f p , Tor($, Z l p ) . such t h a t $ @ Z/p i s an isomorphism is a s u r j e c t i o n , is a p-isomorphism provided t h a t A, B f i n i t e l y generated. (iii) fi*(el): g,(A) -+ H*(%) P-localizes, where H* i s reduced homology with i n t e g e r c o e f f i c i e n t s . (iv) e f : Ext(kp,B) (v) If %' B i s P-local, Ext(A,B) If A then e 1x : Hom($,B) %' Hom(A,B), . is PI-torsion and B is P-local, then Hom(A,B) = 0 , Ext(A,B) = 0. Proof. (i) The f i r s t two a s s e r t i o n s a r e obvious and t h e f o u r t h follows from t h e t h i r d , which we prove a s follows. Let R >- F -> A are Localization of nilpotent groups 10 be a free abelian presentation of A. - Localizing this short exact sequence s- yields, by Proposition 1.7, a short exact sequence Fp % and Fp is flat, Thus we have a commutative diagram Tor(A,B) >------t R 8B I I - F C4 B >- A M, B I I and we invoke Proposition 1.7 together with the fact that el 69 1 P-localizes to infer that Tor(el,l) P-localizes. (ii) Since Z/p from (i), and the factthat is p-local, the first statement follows immediately isan isomorphism, Toprove the converse,consider $p the sequence K >- A ---f> obtained from I$, where K = ker $, L L >- = im B $, - C = C, coker $. We thus infer a diagram where the horizontal and vertical sequences are exact. I( c sz/p We want to prove that and C are p'-torsion groups. However, since A, B, and hence K, C, are finitely generated, it suffices to prove that K @Z/p = 0, C S Z / p = 0. Now C S Z / p = 0 since + @ Z / p is surjective. Thus C is p'-torsion and hence Tor(C, Z/p) = 0. Reference to the diagram then shows that Tor(A, Z/p) >- Tor(L, Z/p) and A @Z/p >-> L O Z / p , from which the conclusion K @Z/p = 0 immediately follows. Note that the converse certainly requires some restriction on A, B. For the homomorphism Tor( $, Z/p) $: Q + 0 certainly has the property that $ @Z/p are isomorphisms, without being a p-isomorphism. and Localization theory of nilpotent groups (iii) The a s s e r t i o n i s r e a d i l y checked i f A i s a c y c l i c group. Use of t h e Kunneth formula t o g e t h e r w i t h ( i ) and P r o p o s i t i o n 1 . 7 shows t h e a s s e r t i o n t o be t r u e f o r f i n i t e d i r e c t sums of c y c l i c groups, hence f o r a r b i t r a r y f i n i t e l y g e n e r a t e d a b e l i a n groups. and H, Finally, since both l o c a l i z a t i o n commute w i t h d i r e c t l i m i t s , t h e a s s e r t i o n is t r u e f o r a r b i t r a r y a b e l i a n groups. The f i r s t isomorphism simply r e s t a t e s D e f i n i t i o n 1.1. (iv) t h e second, l e t ZP-module. I -+-> B >- be an i n j e c t i v e p r e s e n t a t i o n of J S i n c e Zp is f l a t , i t f o l l o w s t h a t an i n j e c t i v e p r e s e n t a t i o n of B B >--f a s an a b e l i a n group. I -> J B For as a is a l s o Thus we have a c o m u t a t i v e diagram - Hom(A,J) Hom(A,I) Ext(A,B) >- S i n c e t h e f i r s t two v e r t i c a l arrows a r e isomorphisms, so i s t h e t h i r d . (v) Clearly, - i n ( i v ) , then Hom(A,J) Hom(A,J) Ext(A,B) Hom(A,B) = 0. - 0 as J shows t h a t Now, i f J h a s t h e same meaning a s is P - l o c a l and t h e s u r j e c t i o n Ext(A,B) = 0. We now r e t u r n t o t h e proof of t h e Fundamental Theorem and complete t h e i n i t i a l s t a g e of t h e i n d u c t i o n by means of t h e f o l l o w i n g P r o p o s i t i o n . P r o p o s i t i o n 1.9. P-local and $ Proof. el If is a 0: A + B i s in Ab, then $ P-localizes iff B is P-isomorphism. We f i r s t show t h a t embeds i n t h e e x a c t sequence el: A + Ap i s a P-isomorphism. In fact, 12 Localization of nilpotent groups Tor(A, Zp/ Z) and since, plainly, Z / Z P A @Zp/Z @ Z P /Z is a PI-torsion group, it follows that both and Tor(A, Zp/ Z) Conversely, i f % *A A B are PI-torsion groups. is P-local and $: A + B is a P-isomorphism, we have a commutative diagram and the proof of Proposition 1.9 is completed by means of Lemma 1.4 (i), (ii) and Lemma 1.5. Assume now that we have defined appropriately. Our objective is to extend Lc-l extend e c-1 sequence +. Nc and to correspondingly, to have the universal property in NC Proposition 1.10. Lc-l: Nc-l GI to Lc: Nc Nc-l i s an exact functor. If, further, i s a central extension i n Nc-l, then so i s the localized EP G -Z, GI' PP + G' >- G >- Proof. We will write e for ec-l. P . We prove: P .G; Consider, then the diagram (1.11) in Nc-l. Assuming the top row short exact, we must show that the bottom row is likewise short exact. We rely on the (inductive) fact that the vertical arrows are P-isomorphisms. First, E~ is surjective. For E P e = eE is Localization theory of nilpotent groups P-surjective, so that, by Lemma 1 . 4 ( i ) , Lemma 1.5(ii), 13 is P-surjective. Hence, by E~ is surjective. E~ Second, up up by Lemma 1.4(ii), u Pe is injective. For = eu is P-injective, so that, is P-injective. The conclusion now follows from Lemma 1.5(i). Third, ker ker E~ C im up. Let E~ = E P im up. Since clearly = 0, we must prove Then, for some n C P', yn y = 1, y € Gp. n y = 1. Thus = ex, m x C G , so eEx = E ex = E EX = 1 for some m € P', so that P P m x = ux', x' C G ' , whence ymn = upex', and mn € PI. One now argues as in the proof of Lemma 1.5(ii) that, since then y € im up. ymn C im u p , This completes the proof of the first statement of the proposition. Notice that the proof that up normality of are P-local and Gp G;, uG'. is ihjective made no use of the Thus we may say that l o c a l i z a t i o n respects subgroups and normal subgroups. Assuming now that the top extension in (1.11) is central, we show that the same is true for the bottom extension. Let x' C G I , y C Gp. x C G , n C P'. Thus (UX')-~X(~X') = has unique nth roots, (upex')-'y(p Gp center of x, Then yn = ex, -1 n (upex') y (upex') = yn. so ex') = y, so P Since upex' belongs to the GP' Then ylm = ex: x' € G I , m t P'. Thus, upyIrn -1 m m belongs to the center of G ; , so that, for any y € Gp, y (upyl) y = (upy') -1 Since Gp has unique mth roots, y (upy')y = upy', so upy' belongs to Now let y' the center of Gp. Theorem 1.12. i K If C G;. Thus ppGi G 6 Ni, i is central in 5 c - - 1 by Proposition l.a(iii). € Ni-l, 2 5 i 5 c - Gp. 1, then G,(e) Proof. We argue by induction on : H,(G) + H,(Gp) P-localizes. i, the theorem being true for Suppose the theorem true for all 1 and let G € Ni. . If 2 - center of G , then Localization of nilpotent groups 14 nil(2) = 1, nil(G/Z) 5 i - 1 and by Proposition 1.10, we have a map of central extensions (1.13) Then (1.13) induces a map of the Lyndon-Hochschild-Serre spectral sequences {EZtI + I E z J , where the coefficients being trivial i n both cases. It now follows from the inductive hypothesis, together with Proposition 1 . 8 ( i ) , taken i n conjunction with the natural universal coefficient sequence i n homology, that (1.13) induces z2 e2: E2 -+ which P-localizes provided that s + t > 0. Applying St st Proposition 1.7 allows us to infer that em: Ett + 6Lt also P-localizes provided that s + t > 0. Finally, since for any n, Hn(G)(Hn(Gp)) has a finite filtration whose associated graded group is @Eit(Ezt) with it follows once again from Proposition 1.7 that Hn(e): -+ provided Hn(G) s + t = n, H (G ) P-localizes n P n > 0. Corollarv 1.14. Let G C Nc-l and Zet t r i v i a l G-action. Then e : G -+ A be a P-ZocaZ abeZian group with Gp induces e*: H*(G P ;A) E H*(G;A). (The conclusion of Corollary 1.14 holds, more generally, if G acts nilpotently on A; see Section 4.) Proof. The homomorphism e induces the diagram and it follows from Theorem 1.12 and Proposition 1.8(iv) that e' and e" Localization theory of nilpotent groups 15 are isomorphisms. Thus e*, too, is an isomorphism. Let now G E Nc. We then have a central extension with nil(r) C 1, nil(G/r) 5 c corresponds t o r -+ 1. 5 an element Then, applying e: - rp, we By the cohomology theory of groups, (1.15) E H2(G/r;r) obtain e,S € with G/r acting trivially on HL(G/r;r,) there exists a unique element 5, E H2((G/r)p;rp) (1.16) is induced by rp >- (1.17) correspond to e: G + and, by Corollary 1.14, such that e*Sp = e,S, where now e* 5,. e: G/r -+ Gp (G/T)p. -J Let the central extension (G/r), It follows from (1.16) that we can find a homomorphism Gp yielding a commutative diagram (1.18) In fact, the general theory tells us that given two central extensions of (arbitrary) groups, together with homomorphisms then there exists (1.19) T: G1 + G2 r. p: A1+ A,, yielding a commutative diagram U: Q, -+ Q,, Localization of nilpotent groups 16 p r e c i s e l y when (1.20) Moreover, i f then T T and (1.21) T' and T' a r e two maps y i e l d i n g commutativity i n (1.191, a r e r e l a t e d by t h e formula T'(x) = T(X).II 2K E 1 ( x ) , x € G1, f o r some Q, K: + A2. Returning t o our s i t u a t i o n , we see,from (1.17) and t h e i n d u c t i v e hypothesis that r rp = {l}, Gp € N that G/r and, f u r t h e r , u s i n g P r o p o s i t i o n 1.2, = G , and we n a t u r a l l y t a k e t h e same P - l o c a l i z a t i o n i f we d e f i n e P r o p o s i t i o n 1 . 6 , e : G + Gp is P-local. - C' e is P-loca1,and t h e f a c t t h a t e* and t o prove t h e n a t u r a l i t y of e. G w i l l f o l l o w d i r e c t l y from t h e e Lc Let + Hom(G,K) is s u r j e c t i v e i f is i n j e c t i v e follows immediately is P - s u r j e c t i v e and Thus i t remains t o d e f i n e Then we have By is a n a t u r a l t r a n s f o r m a t i o n of f u n c t o r s . For we then r e a d i l y i n f e r t h a t ex: Hom(G ,K) P e Gp = (G/rIp, p r e s e r v i n g L G = Gp, a s we propose t o do. Then t h e u n i v e r s a l p r o p e r t y of from t h e f a c t t h a t then is a P-isomorphism and hence an isomorphism i f f a c t , s t i l l t o be proved, t h a t K € Nc G € Nc-l, We also remark t h a t i f , i n f a c t , is P-local. Gp = {l}, (G/I')p C Nc-l, K is P - l o c a l . on morphisms of 0: G +c in Nc a s a functor, Nc, and l e t r = rc(c). Localization theory of nilpotent groups 17 5: 5, : (1.22) and o u r object is to define i s clear that any functoriality of 4, "front face" of ( 1 . 2 2 ) . e*: H2 ((G/r),; to make ( 1 . 2 2 ) commutative. Gp T: G , - + Gp It is uniquely determined so that i s automatic once a suitable $, W e shall first find But + $,e = e$ yielding Lc = Cp "p: is defined. yielding commutativity in the To this end, we compute e*$&S,. r,) S H2 (G/r;yp) ,$I"*? P P = by Corollary 1.14 so that, i n fact, 4 - Thus, by ( 1 . 1 9 ) , ( 1 . 2 0 ) , we may find 1: Gp + Gp so that Lmalization of nilpotent groups 18 as claimed. However T need n o t satisfy the equation re = e$, so we now modify T, preserving (1.23), so that the last equation also obtains. Consider the diagram, obtained from ( 1 . 2 2 ) , (1.24) where JI' = e+' = $ie, JI" = e$" = $Fe. Clearly (1.24) commutes if we set JI = e$ or J, = re so that, using (1.211, there exists 0 : G / r Let Bp: (G/rIp * Fp be given by Ope of e in Nc-l. = 8 ; Bp + rp such that exists by the P-universality Define From (1.23), it follows that and also $pex = (rex).(i Pe PEPex) = (Tex).(L,ecx) = e$x, x c G. It remains to verify the final assertion of the Fundamental Theorem for We already know that e: G + Gp i s a P-isomorphism and Gp Nc. is P-local. The converse i s proved j u s t as for Proposition 1.9, making use of Lemma 1.4(i), and Lemma 1.5. (ii) Properties of localization in N 2. 19 Properties of localization in N In this section, we deduce a number of immediate consequences of the methods and results of §I. G C N and If Theorem 2.1. Q i s a coZlection of primes, then the s e t consisting of the Q-torsion elements i n G i s a (normal) subgroup of Proof. Since e Let P = Q' and consider the P-localization e: G T Q G. + Gp. is a P-isomorphism and Gp is P-local, it is clear that T = ker e. Q Suppose Theorem 2 . 2 . G C N has no Q-torsion. Then i f xn = yn, x, y € G, n C Q , i t f o l l m s t h a t ~x = Y. Proof. Again, let P = Q' e: G -+ Gp. Then e ex = ey, so and consider the P-localization is injective and (ex)" = (ey)". Since Gp is P-local, x = y. Corollarv 2 . 3 . G € N i s P-local i f f it has no PI-torsion and i s surjective f o r a l l n c x - xn, x € G, P'. We now turn to results which make explicit mention of P-localization. Theorem 2.4. The P-localization functor L: N -+ N i s an exact functor. Proof. This follows from Proposition 1.10, in conjunction with the Fundamental Theorem. A s immediate corollaries, we have the following assertions, of which the first is the definitive version of Proposition 1.2 and the second -- is related to Proposition 1.6. Corollarv 2 . 5 . Let Then if any two of Corollary 2 . 6 . G' G', G, G" G G" be a short exact sequence i n N. are P-local so i s the third. Let be a map of short exact sequences i n N. so does the third. Then i f any two of $', $, 4'' P-localize, Localization of nilpotent groups 20 Theorem 2.7. ri($): ri(G) -+ ?(K) Proof. and l e t $: G P-localizes ri(G) G C N Let -+ Then K P - l o c a l i z e G. f o r a21 i 2 1. It f o l l o w s from C o r o l l a r y 2.6 t h a t i t is s u f f i c i e n t t o prove t h a t t h e homomorphism W e argue by i n d u c t i o n on i G/T (G) $i: K/ri (K) induced by i , t h e a s s e r t i o n being t r i v i a l f o r following from Theorem 1.12 f o r i = 2. i ? 2 , and prove t h a t -+ $i+l Thus we assume t h a t P-localizes. P-localizes. $ i = 1 and $ P-localizes, A second a p p l i c a t i o n of C o r o l l a r y 2.6 shows t h a t i t is s u f f i c i e n t t o prove t h a t t h e homomorphism 5: ri(G)/rifl(G) -+ ri(K)/I"+'(K), induced by 6, P - l o c a l i z e s . We apply t h e 5-term e x a c t sequence i n t h e homology of groups t o t h e diagram t o obtain where t h e s u b s c r i p t by Theorem 1.12 and Theorem 1.12. ab ,,$, refers to abelianization. Oiab Then I$,, bab P - l o c a l i z e P - l o c a l i z e by t h e i n d u c t i v e h y p o t h e s i s and It f o l l o w s from P r o p o s i t i o n 1.7 t h a t P-localizes and t h e proof of t h e theorem is complete. There is a d u a l theorem t o Theorem 2 . 7 concerning t h e upper c e n t r a l series of G which, however, r e q u i r e s more d i f f i c u l t t o prove. G t o be f i n i t e l y generated and is We c o n t e n t o u r s e l v e s h e r e w i t h a s t a t e m e n t of t h e r e s u l t , r e f e r r i n g t o [34 ] f o r d e t a i l s . Properties of localization in N Theorem 2.8. i z ( e ) = el z (G) i e: G -f is P-localization, Gp i i z (G) i n t o z ( G ~ ) . Moreover, carries if G Z (G) P-localizes and G € N If i 21 then t h e r e s t r i c t i o n z i ( e l : z i (GI + z i ( G ~ ) is f i n i t e l y generated. Our n e x t r e s u l t i n t h i s s e c t i o n i s t h e d e f i n i t i v e v e r s i o n of Theorem 1 . 1 2 . Theorem 2.9. Let $: G K -+ N. be i n $ P - l o c a l i z e s iff H,($) H,($) P-localizes i f Then P-localizes. Proof. Theorem 1 . 1 2 asserts t h a t We n e x t prove t h a t i f e: K + % H*(K) P-localize. i s P-local, so is P - l o c a l , Then H,(e) H*(e) : H*(K) commutes. that Now l e t 8,($) P-localize. Thus f a c t o r s as $ But s i n c e isomorphism. + i s an isomorphism. Stammbach Theorem ( s i n c e K , I$€N) P-local. then G fi*($), i , ( e ) H,(Kp) P-localizes. For l e t P-localizes; but &(K) I t f o l l o w s from t h e S t a l l i n g s - e Then Gp is P - l o c a l . K t$ i s an isomorphism. H*(K) K i s P-local, SO K and both P-localize, HA($) is an Thus t h e S t a l l i n g s - S t a m b a c h Theorem a g a i n i m p l i e s t h a t J, an isomorphism, so t h a t @ is is P-localizes. Our f i n a l r e s u l t i n t h i s s e c t i o n p l a y s a c r u c i a l r o l e i n Chapter I1 when we come t o s t u d y (weak) p u l l b a c k s i n homotopy t h e o r y . Theorem 2.10. Localization c o m t e s with pullbacks. Proof. Suppose g i v e n Localization of nilpotent groups 22 in N, and form p u l l b a c k s G a > H Cm 4 K-M Of course y: G -f E, UJ G € N, being a subgroup of c h a r a c t e r i z e d by ~y and we show t h a t h a s pth r o o t s , a C Hp, b € (a,b) € 5 Next, then and y € P' Let ($,a)' and - Since (x,y) € G E, 5 Hp Mp x' It f o l l o w s t h a t i s P-injective. y = For i f Kp. Then i s P-local, x = a', y = bp, $ a = $ b, P SO P y(x,y) = 1, x C H , y E K , (x,y) € G , 1, m,n C P ' . is P-surjective. = e a , y' $a i t s u f f i c e s t o show t h a t (x,y> = (a,b>'. = = eb, a € H , b € K. ($b)u, u € M, and C So For l e t xm = e h , yn = ek, m,n € PI, h E H , k € K . and Kp, x x € Hp, y E Since = (UJ,b)'. ex = 1, ey = 1, xm = 1, yn Finally, Then i s P-local. G p € PI. Kp, = e a , isy = eB, i s P-universal. y - First, that There i s then a homomorphism H x K. us (x,y) € E, Thus (with Now = ( x , ~ =) 1, ~ ~ mn € P ' . x C Hp, y € k = mn) we have $px = Jlpy, so 1, s € P'. Kp. e$a = e$b. We deduce t h a t i f C (see t h e proof of Lemma 1 . 4 ( i v ) ) . Thus n i l M 5 c t h e n $aS = $bs c c c c ( a s ,bS ) , w i t h Ilsc E P I . T h i s shows t h a t (aS ,bs ) € G and ( x , ~ ) '=~e ~ y i s P - s u r j e c t i v e and t h u s , i n view of t h e Fundamental Theorem, completes t h e proof of Theorem 2.10. Further properties of localization 3. 23 F u r t h e r p r o p e r t i e s of l o c a l i z a t i o n I n t h i s s e c t i o n , we prove a number of r e s u l t s i n v o l v i n g t h e l o c a l i z a - t i o n functor i n t h e category N. A s mentioned i n t h e I n t r o d u c t i o n , w e a r e s p e c i f i c a l l y concerned w i t h r e s u l t s which have f r u i t f u l homotopy-theoretic analogs. W e f i r s t examine more c l o s e l y t h e n o t i o n of P-isomorphism introduced i n 81. Theorem 3.1. Let P-localization. $: G be i n K N and l e t $p: Gp 5 + lil $ i s P - i n j e c t i v e iff $p i s i n j e c t i v e ; fiil $ i s P - s u r j e c t i v e iff $p i s s u r j e c t i v e . (i) If $ is P-injective, t h e n so i s i s P - i n j e c t i v e and t h e composite of P - i n j e c t i o n s i s P-injection. be i t s Then: Proof. e -+ Thus $pe = e$ is P-injective, we may apply Lemma 1 . 4 ( i i ) t o deduce t h a t $p and s i n c e e+: G + 5 since of c o u r s e , a e is P-surjective, is injective. The converse is proved s i m i l a r l y , u s i n g Lemma 1 . 4 ( i i i ) . (ii) If $ is P-surjective, then so is e$ since e and t h e composite of P - s u r j e c t i o n s i s , of c o u r s e , a P - s u r j e c t i o n . $pe = e$ is P - s u r j e c t i v e and we may apply Lemma 1 . 4 ( i ) t o deduce t h a t $p is s u r j e c t i v e . is P-surjective Thus and Lemma 1 . 5 ( i i ) The converse i s proved s i m i l a r l y , u s i n g Lemma 1 . 4 ( i v ) and Lemma 1 . 5 ( i ) . Remark. The f a c t t h a t proved by i n d u c t i o n on $p surjective implies $ P - s u r j e c t i v e may a l s o b e n i l ( G ) , making use of Theorem 2 . 7 . We may t h u s avoid u s i n g Lemma 1 . 4 ( i v ) which, w e r e c a l l , was based on P. H a l l ' s commutator calculus. Localization of nilpotent groups 24 As a corollary of Theorem 3.1, we have the following definitive version of Proposition 1.6. Theorem 3 . 2 . Let be a map of short exact sequences i n N. my Then i f any t u o of $', $I' are P-isomorphisms, then s o i s the t h i r d . ,. Theorem 3 . 3 . Let G, K E N . li) Gp and Then the following assertions ore equivalent: Kp are isomorphic; (ii) There e x i s t M moremer, M C N and P-isomorphisms a: G may be chosen t o be f i n i t e l y generated i f + My B: K and G -+ M; are K f i n i t e l y generated; l i i i ) There e x i s t M' C N and P-isomorphisms moreover, M' may be chosen t o be f i n i t e l y generated i f f i n i t e l y generated. (In the special case P = that G and K 4, the equivalence (i) - 3 (ii), let B M, 6: K -+ K; K are (i) follow directly M to be the maps defined by are P-isomorphisms, since -a , -B - and set M - K p , Kp to be the composite G %- Gp % % and $: K e. We then define M to be the subgroup of M -r - Gp 2 Kp w: - a: G -+ and K are finitely generated and The implications (ii) = (i), (iii) from Theorem 3.1. To prove (i) -P and G , 6 : M' (iii) amounts to the assertion torsion-free.) E: G G -+ have isomorphic rationalizations iff they are commensurable (in the senseof [ 6 , 6 5 ] ) , at least when G Proof. M' y: -+ Kp to be simply generated by aG U E K ., B. It is clear that areP-isomorphisms, and that M is and a and Further properties of localization f i n i t e l y generated i f G K and 25 a r e f i n i t e l y generated. F i n a l l y , t o prove ( i i ) * ( i i i ) , w e c o n s t r u c t t h e p u l l b a c k diagram M, a G x B and having t h e i r p r e v i o u s meanings. M' € N K, c e r t a i n l y f i n i t e l y generated. and i s f i n i t e l y generated i f t h e argument f o r We prove t h a t a Gp E and K are n € P', y € G 5 but $: G + K, JI: K $p: Gp is a P-isomorphism; + Hom(K,G) = 0. $: G Kp G. i s a P-isomorphism, 6 By t h e p u l l b a c k p r o p e r t y , Now l e t 6. with 6(y,xn) = xn, so t h a t and example of a P-isomorphism (so t h a t so i s Note t h a t i t i s n o t a s s e r t e d t h a t of P-isomorphisms then is P-injective, there exist (y,xn) C M ' Remark. being p e r f e c t l y symmetric. y ker a ; a s is P-surjective, then G (We u s e h e r e t h e f a c t , coming from t h e aforementioned a r e themselves f i n i t e l y g e n e r a t e d . ) S i s a subgroup of P. Hall, t h a t subgroups of f i n i t e l y g e n e r a t e d n i l p o t e n t groups t h e o r y of ker 6 M' Since 6 Gp G x € K. Since Bxn = ( 6 ~ =) a~y . a But i s P-surjective. Kp implies the existence For example, i f G = Z and K = Zp, I n f a c t , Milnor h a s even c o n s t r u c t e d an + of f i n i t e z y generated n i l p o t e n t groups K by Theorem 3 . 1 ) w i t h t h e p r o p e r t y t h a t no map JI: K + see R o i t b e r g [70]. A q u i t e analogous phenomenon a r i s e s i n t h e homotopy c a t e g o r y , a s h a s been shown by Mimura-Toda [57 I. (Compare [70].) We t u r n now t o a new s e r i e s of r e s u l t s descrjhine r e l a t i o n s between t h e o b j e c t s and morphisms i n A s e t of primes P N and t h e i r v a r i o u s l o c a l i z a t i o n s . i s c a l l e d cofinite i f P' is f i n i t e . G 26 Localization of nilpotent groups If Lemma 3 . 4 . G C N s e t of primes i s f i n i t e l y generated, then there e x i s t s a c o f i n i t e such t h a t P r a t i o n a l i z a t i o n of Proof. G Gp -c i s i n j e c t i v e , where Go land hence also of The t o r s i o n subgroup Go i s the Gp). ( c f . Theorem 2 . 1 ) of T f i n i t e l y generated and hence, a s is r e a d i l y seen, f i n i t e . has p-torsion s e t , then P Theorem 3 . 5 . cofinite, Let $ Proof. Let e ...,x {xl, } yi C K , m Now choose a c o f i n i t e subset factorize generate i and $ is injective. 4: G + P KO be a such t h a t of K 415, Q e: K G, l e t Q + so that KO such t h a t such t h a t mi C P', 1 5 i C n, and as is P-local, we have If G € N the r a t i o n a l i z a t i o n maps elYi = zi i 2KO, e 2 1' , so i s f i n i t e l y generated, then G = e e 5. l i f t s uniquely i n t o Theorem 3 . 6 . z 0 P m Kp Since Go 5. and then f i n d K -+ By Lemma 3 . 4 , we f i r s t choose a c o f i n i t e s e t is injective. rationalize Gp P +Go, p C II. G T i s t h e complementary Then there e x i s t s a c o f i n i t e s e t of primes has a unique l i f t i n t o KQ + K O P be f i n i t e l y generated and l e t G, K € N given homomorphism. i s t o r s i o n - f r e e and Gp is I t follows t h a t f o r only f i n i t e l y many primes, s o t h a t , i f is G i s the puZZback of Further properties of localization Proof. We argue by induction on if G C Ab. G = Z/pk nil(G), 21 the theorem being easily proved For, in this case, the assertion is obvious if G = 52 or and then we infer it for any finitely generated abelian group by remarking that, if the assertion is true for the abelian groups A , B, it is plainly true for A Ci; B. To establish the inductive step, we consider the short exact sequence with nil(G') < nil(G), nil(G") theorem is true for GI, G". = 1, so we may assume inductively that the Write e : G+ G for the localization, P P r : G + G for the rationalization. We want to prove that, given x € G P P 0 P P' with r x = x for all p, there exists a unique x € G with e x = x P P 0 P P Now E x € G" and P P P . r"E x = P P P Hence there exists a unique € x" P EX. Then E x = P P E e P P x = (e ,IX): P P !J, up, uo E x all p. P P' - x, so that P where x' € G' P P' being regarded as inclusions. Moreover, x where all p. x 00' G" with e"x" = Let x" = E r: G = (rG)(r'x'), P P all pI Go is the rationalization, so that the elements x' have a P common rationalization. Hence there exists a unique x ' < G' with e'x' = x' P P' + Then x = e (xx'). P P Localization of nilpotent groups 28 Uniqueness is clear, even in the event that G is not finiteZy generated, since, e being p-injective, the map P component is e is always injective. PS &: G -+ IIG whose p P’ th Remarks. A particularly simple consequence of the injectivity of 6: G -f i7G P that nil(G) is that G = - 11) iff Gp=fl), all p. Another consequence is max nil(G ). We also note that we could generalize this theorem P by considering any infinite partition of II into disjoint families of primes. There is a stronger statement (compare Theorem 3.9) i f the partition is finite. Theorem 3 . 7 . where o If G C N is finitely generated, L)e have a pullback diagram and T is the rationaZization P G is abelian, then the diagram is als o a pushout. is the rationaZizatioB map of IlG of 6. If,further, Proof. We proceed as in the proof of Theorem 3 . 6 . first the case that G Thus we consider is finitely generated abelian. Since localization commutes with finite direct sums, we may assume G cyclic. For G finite cyclic, the conclusion is obvious. For G = Z, we have a map of short exact sequences leo 9 (n Ep)o-- >- C1, C2 being the respective cokernels. Since e so too is the induced map y: C1 +. C2. c‘ c2 and u are rationalizations But it is readily seen that C1 = II L / Z P Further properties of localization is torsion-free, divisible, that is, 0-local, Hence 29 y: C1 e C 2 , which is equiva- lent toour assertionthat, inthis case, the diagramis apullback anda pushout. We now easily complete the proof of Theorem 3 . 7 by induction on following the pattern of proof of Theorem 3 . 6 . nil(G) It is certainly not true that localization commutes with infinite Cartes an products, even where the product is nilpotent. We do have the following special result, which will be of use to us later. Theorem 3 . 8 . If G C N i s f i n i t e l y generated f o r , more generally, i f the p-torsion subgroup T (G) = 11) f o r p s u f f i c i e n t l y large), then the map P @ : (nGp)o rIG G e!Go, induced by the map 8 : rIG -+ IIG P,O’ P90 P P90 which rationaZizes each component, i s i n j e c t i v e . If, f u r t h e r , G is abelian, -+ then $ admits a l e f t inverse. Proof. Of course, $ But since 0 = IIr r : G P’ P P + is injective iff ker 8 is a torsion group. G the rationalization, we have PSO ker 8 = II ker r P = nTP(G) and this is a torsion group if (and only if!) Tp(G) = {1} for p sufficiently large. The final assertion follows because (TIGp)o and IIG P9 0 both rational vector spaces and we may invoke the Basis Theorem. are It is possible to formulate a version of Theorem 3 . 6 in which an arbitrary decomposition of I7 into mutually disjoint subsets is given. If the number of subsets in the decomposition is infinite, as in Theorem 3 . 6 , then we must impose the condition that G be finitely generated, as in Theorem 3.6. On the other hand, if the number of subsets in the decomposition is finite, it is unnecessary to impose a finiteness condition on G. Since, in the sequel, we shall be particularly concerned with the case in which rI Localization of nilpotent groups 30 is decomposed i n t o two d i s j o i n t s u b s e t s , we s t a t e t h e r e s u l t i n t h i s form, while r e c o r d i n g t h e f a c t t h a t t h e g e n e r a l i z a t i o n t o a f i n i t e decomposition of Il is v a l i d . If Theorem 3.9. G E N, then we have a puZZback diagram eP GG I IP r p , r p l denoting the rationaZization maps. Proof. Consider f i r s t t h e c a s e t h a t G is a b e l i a n . Since t h e a s s e r t i o n is c l e a r f o r c y c l i c groups, i t is t r u e a l s o f o r f i n i t e l y generated a b e l i a n groups, a s i n t h e proof of Theorem 3.6. I n general, G may be expressed as t h e d i r e c t l i m i t of i t s f i n i t e l y generated subgroups G E But I& GaQ = (liln Ga)Q %' +G a y GQ G" f i n i t e l y generated f o r any c o l l e c t i o n of primes Q , and I& p r e s e r v e s p u l l b a c k diagrams, so t h e a s s e r t i o n is v e r i f i e d f o r a r b i t r a r y a b e l i a n groups. Again, a s i n Theorem 3.6, we argue by i n d u c t i o n on nil(G) t o prove t h e theorem f o r a r b i t r a r y n i l p o t e n t groups. Remark. It is e a s i l y proved t h a t t h e diagram of Theorem 3.9 is a l s o a pushout i n Ab if G is abelian. f o r an a r b i t r a r y n i l p o t e n t group as This remark g e n e r a l i z e s t o t h e s t a t e m e n t , G , t h a t every element of r p ( x ) r p l ( x ' ) , x E Gp, x ' € G p l . Go is expressible A s i m i l a r remark a p p l i e s t o Theorem 3.7. While, i n Theorem 3.9, no f i n i t e n e s s c o n d i t i o n is imposed on G, i t i s n e v e r t h e l e s s u s e f u l t o know when such a c o n d i t i o n can b e deduced from analogous c o n d i t i o n s on groups. Gp, G p r . We prove t h e f o l l o w i n g r e s u l t f o r a b e l i a n Further properties of localization Theorem 3.10. 4, If A C Ab then A are f i n i t e l y generated Z Proof. We assume t h a t i s a f i n i t e l y generated abeZian group i f f P -, Zpl- %, modules, respectively. a r e f i n i t e l y generated Z p l - modules, r e s p e c t i v e l y , and prove t h a t A f i n i t e l y generated R-modules. IS1, A a9 R ...,6 II 1 Let $ @%I Moreover, be a s e t of R-generators 5 j B for B >-A which i s t o r s i o n - f r e e , w e A S A b e t h e r i n g Zp d Z p l . A @ R a s R-modules, R and w r i t e ij C = A/B. A@R-- C A. a i j ’ we get a Tensoring w i t h R , C@R, t h e i n d i c a t e d isomorphism f o l l o w i n g from (3.11). clearly implies t h a t Let g e t a n e x a c t sequence B@R>-w C = 0, so t h a t Thus C @ R = 0 , which A = B , which is f i n i t e l y g e n e r a t e d . Theorem 3.10 admits an obvious g e n e r a l i z a t i o n , i n which we have a decomposition of i 7 i n t o f i n i t e l y many m u t u a l l y d i s j o i n t s u b s e t s . g e n e r a l i z a t i o n f a i l s f o r an a r b i t r a r y ( i n f i n i t e ) decomposition of f o r example, but -, g e n e r a t e d by t h e C, with -> Qo rij E R, a = Z(aij@l)rij, i t o b e t h e subgroup of s h o r t e x a c t sequence Remark. R is f u r n i s h e d w i t h t h e n a t u r a l R-module s t r u c t u r e . (3.11) I f we d e f i n e P i n h e r i t n a t u r a l R-module s t r u c t u r e s and, as such, a r e and where Z is a f i n i t e l y generated abelian group, t h e converse i m p l i c a t i o n b e i n g t r i v i a l . Then 31 @ Z/p P Theorem 3.12. : G + K @P P P The ll since, (eP Z / P ) ~ i s a f i n i t e l y g e n e r a t e d Z -module f o r a l l primes q q is n o t a f i n i t e l y generated a b e l i a n group. Let Q: G + K be i n N. Then Q i s an isomorphism i f f is an isomorphism f o r a l l p . Localizationof nilpotent groups 32 Proof. We assume 0 is an isomorphism for all p. Thus, by the P Fundamental Theorem, 4 is a p-isomorphism for all p . Since ker 0 is a torsion group, and all primes are forbidden, ker @ = {l}. Now let y t K . ynp Then, for each p, we have x(~) t G, n prime to p, and = 4x(,). P Since gcd(n ) = 1, we may find integers a almost all 0, such that P a P' Ca n = 1. Set x = llx It i s then plain that y = ox. P P (PI ' Theorem 3.13. Let Proof. map &: K $,$I: G + K be i n N. Then + = $I iff 0 P - $I P for aZZ p. This is an immediate consequence of the injectivity of the noted in the proof of Theorem 3 . 6 . P' The assertion of Theorem 3 . 1 3 , whose homotopy-theoretical counterpart -+ nK is of considerable significance, is that the morphisms in N are completely determined by their localizations. It is fundamental to note, however, that Thus, if we define the genus this is not true of the objects in N. G(G) of to be the set of isomorphism classes a finitely generated nilpotent group G of finitely generated nilpotent groups K satisfying K S G for every prime P P p, it is not necessarily the case that K % G, when K belongs to the genus of G. The following specific examples, to some extent inspired by similar examples in the homotopy category, were pointed out to us by Milnor: For let N be the nilpotent group of nilpotency r1 s class 2 which is generated by four elements xl, x2, yl, y2 subject to the mutually prime integers r, 8, defining relations that all triple commutators are trivial and [x1,x21r = [Y,,Y,I. Nr/s Nr'/s Then N iff either r ~ and / ~ Nrt,s 3 2' (mod s) [Xl.X2IS - 1, are in the same genus but or rr' :21 (mod s ) . Thus, for example, NlIl2 $ N7/12 although these groups have isomorphic p-localizations for every prime p. p-isomorphisms (In fact, for every prime p, it is easy to construct N1/12 +. N7/12' N7/12 N1/12)' Further properties of localization 33 Subsequently, f u r t h e r examples have been d i s c o v e r e d by M i s l i n [ 611. I t should b e noted t h a t , i n d e f i n i n g t h e genus, we have r e s t r i c t e d o u r s e l v e s t o f i n i t e l y generated groups. s i z e d genus sets. For example, i f A T h i s i s done i n o r d e r t o avoid over- is t h e a d d i t i v e subgroup of c o n s i s t i n g of elements e x p r e s s i b l e as f r a c t i o n s A(n) f r e e " by with square-free A 2 Z f o r e v e r y prime p . More g e n e r a l l y , P P i s d e f i n e d i n t h e same way a s A e x c e p t t h a t w e r e p l a c e "square- L, t h e n denominator if k/k Q A'$ "nth-power-free", Z but we o b t a i n i n f i n i t e l y many m u t u a l l y nonisomorphic a b e l i a n groups w i t h p - l o c a l i z a t i o n s isomorphic t o Z f o r e v e r y prime p . P With o u r d e f i n i t i o n of genus, t h e genus of a f i n i t e l y g e n e r a t e d a b e l i a n group A c o n s i s t s of ( t h e isomorphism c l a s s o f ) A a l o n e . We s t a t e t h i s a s a theorem, even though i t i s e l e m e n t a r y , s i n c e we w i l l wish t o r e f e r t o it l a t e r . Theorem 3.14. abelian and Let B C G(A). Proof. be f i n i t e l y generated nilpotent groups with A A, B Then B A. The n i l p o t e n c y c l a s s of a n i l p o t e n t group is an i n v a r i a n t of t h e genus ( s e e t h e Remark f o l l o w i n g Theorem 3 . 6 ) . The s t r u c t u r e theorem f o r f i n i t e l y generated a b e l i a n groups shows t h a t any f i n i t e l y g e n e r a t e d a b e l i a n group i n t h e genus of A must c e r t a i n l y b e isomorphic t o A. More g e n e r a l l y , i t i s known t h a t t h e genus of a f i n i t e l y g e n e r a t e d n i l p o t e n t group is a f i n i t e s e t . [ 651. T h i s f a c t f o l l o w s from r e s u l t s of P i c k e l ( P i c k e l ' s u s e of t h e term "genus" d i f f e r s from o u r s . ) The homotopy-theoretical c o u n t e r p a r t of t h e f i n i t e n e s s of t h e genus i s as y e t unsolved i n g e n e r a l , a l t h o u g h p a r t i a l r e s u l t s a r e known. Localization of nilpotent groups 34 4. Actions of a nilpotent proup on an abelian group Throughout this section, we denote by A w: Q an arbitrary group, and by -f Aut(A) an abelian group, by Q an action of Q on A . We adopt x E Q , a € A. the customary abbreviation x - a for w(x)(a), Define the lower central w-series of A , ... by setting 1 = A, rw(A) rF(A) = Observe that if I Q group generated by {x-a-alx € Q , a E T,(A)), i i 3 1. is the augmentation ideal of the integral group ring Z Q , then ri+'(A) i in particular, each rw(A) We say that Q j sufficiently large. we say that operates nilpotently on A If c A w on A. A >- if rA(A) = 111 is the largest integer for which w has nilpotency class Proposition 4.1. Let (IQ)i.A; is a submodule of A. Proposition is easily proved. Q-action = G - Then G E N i f f c and write nil(,) Q = c. f(A) for # Ill, The following be an extension giving r i s e t o the Q € N and Q operates nizpotently on through w. Indeed, max{ nil ( Q ) .nil (0)1.5 nil (GI 5 nil ( 9 ) + nil (w) In the situation of Proposition 4.1, we may define $(A) of A by setting . a subgroup Actions of a nilpotent group on an abelian group 35 I t i s t h e n clear t h a t A r e s u l t c l o s e l y r e l a t e d t o P r o p o s i t i o n 4 . 1 , w i t h almost i d e n t i c a l proof, i s t h e following. Let Proposition 4.3. A' respect t o the Q-actions w', then are n i l p o t e n t . w'l w', w" B. a b e l i a n group Notice t h a t and + w', w, A" w" be an exact sequence of Q-modules w i t h respectively. Then i s niZpotent i f w If the sequence is short exact and i f w is n i l p o t e n t , R on t h e are n i l p o t e n t , and L e t now homomorphism * A A(R,B) b e t h e s e t of a c t i o n s of t h e group The l o c a l i z a t i o n map Aut(A) + e: A (pw)' % e v i d e n t l y induces a A u t ( % ) , which i n t u r n g i v e s rise t o a map v r e s p e c t s submodules; t h u s , i f w' = wIA', -t = IJW~G for A' is a submodule of A w F A(Q,A), then By analogy w i t h Theorem 2 . 7 , we now prove Theorem 4 . 5 . Let Proof. w < A(Q,A). Then I n view of ( 4 . 4 ) i t s u f f i c e s t o prove t h i s f o r s i n c e a n e a s y i n d u c t i o n then completes t h e argument. Q-module map i f we f u r n i s h + w i t h t h e Q-action pw Now i = 2 e: A + + and p l a i n l y i s a e induces Localization of nilpotent groups 36 eo: r:(A) + 2 rllw(Ap) by restriction. By the Fundamental Theorem, we need to prove that is P-local and that eo is a P-isomorphism. To prove of 2 I';w(Ap) P-local, it is sufficient, by the commutativity to show that any generator x*b - b, x E Q, b t r,,(fh), %, may be n rtw(Ap).Now divided by n, n t P', in b 5 nb', b' C Ap, since Ap is P-local, so x.b -b Since e = x'(nb') - nb' = n(x*b') - nb' = n(x*b' - b'). i s obviously P-injective, it remains to prove eo P-surjective. But the argument here i s very similar to that in the previous paragraph. Namely, given any element of the form x'b n t P', a C A with nb = e(a) then there exist n(x+b - b) = e (x-a It follows from the commutativity of ri(A), Let AV(Q,A) 5 A(Q,A) - and - b, x t Q, b € Ap, so a). 2 rpw(fh) that eo is P-surjective. consist of the nilpotent Q-actions on A. Then the following is a direct consequence of Theorem 4.5. Corollarv 4 . 6 . ?.IA~(Q,A) 5 AV(~,%). nil(pw) 5 niliw) . If now A >-+ Q-action w, G - Indeed, if w € Av(Q,A), then Q i s any extension corresponding to the we have a commutative diagram (4.7) where the Q-action induced by the lower extension is pw. By ( 4 . 2 ) , we have Moreover, if G E N, then by Proposition 4.1 and Corollary 4.6, applied to the lower extension in ( 4 . 7 ) , we conclude that G ' t N, f i s a P-isomorphism. Actions of a nilpotent group on an abelian group Q E N Now assume and l e t e: Q --t 37 P-localize Q, Q. Then e induces and, o b v i o u s l y , Theorem 4.8. Let be P-ZocaZ, A e*: A,,(Q,,A) Proof. Let w Then Q C N. s A,,(Q,A). E A,,(Q,A) and l e t e x t e n s i o n corresponding t o A Q >Gt--j> By P r o p o s i t i o n 4 . 1 , w. G C N be t h e s p l i t s o we may l o c a l i z e to obtain (4.9) Let Since h w C A(Q,,A) be t h e a c t i o n o b t a i n e d from t h e lower e x t e n s i o n i n ( 4 . 9 ) . Gp F N, hw C Av(Qp,A) and s i n c e t h e r i g h t hand s q u a r e i n (4.9) i s a pullback, e*Aw = w so t h a t satisfies e*h = 1. But i f we s t a r t w i t h s p l i t extension A >- E-> Q, C A (Qp,A) f o r this action and form t h e i, t h e n C N, by P r o p o s i t i o n 4.1, and is P-local by C o r o l l a r y 2 . 5 , s o t h a t e s s e n t i a l l y t h e same diagram ( 4 . 9 ) shows t h a t h e * = 1. Thus h is i n v e r s e t o e*. Localization of nilpotent groups 38 Let C o r o l l a r y 4.10. w C AV(Q,A) be any extension corresponding t o % >+ is G P Q, -> with Q C N and Zet W. A -G LocaZizing yieZds an extension and hence an action o f on Ap. Qp Then t h i s action X ~ W . independent of the original choice of extension. Proof. W e f i r s t assume A P-local. Then ( 4 . 9 ) , where t h e e x t e n s i o n s a r e no longer assumed s p l i t , again shows t h a t t h e a c t i o n Q, on A , given by t h e lower e x t e n s i o n , s a t i s f i e s Now c o n s i d e r t h e g e n e r a l c a s e . f Q is a P-isomorphism. where ef Xuw. Theorem 4.11. Let Proof. A W. Thus T = of Xu. We r e v e r t t o (4.7) and r e c a l l t h a t W e t h u s may amalgamate (4.7) and (4.9) t o o b t a i n is P-localizing. extension i s e*r = T Thus t h e a c t i o n of be P-local. i Then r,(A) Q, on % given by t h e lower i = rXw(A). Reverting t o (4.9), w e s e e t h a t ri, ( ~ )= i rG(A), i rxw (A) = ri (A). GP W e now claim For i = 2 , t h i s may b e proved by an argument s i m i l a r t o t h a t of Theorem 2.7 (apply t h e 5-term homology sequence t o t h e diagram Actions of a nilpotent group on an abelian group i , we u s e an e a s y i n d u c t i o n . and,for general On t h e o t h e r hand, Theorem 4.5 i m p l i e s t h a t Thus i rG(A) = r 39 i i s P-local. Ti(A) (A). GP We are now i n a p o s i t i o n t o g e n e r a l i z e Theorem 2 . 9 and C o r o l l a r y 1 . 1 4 . Theorem 4.12. w, Q C N. Let A be a n a b e l i a n group equipped w i t h a n i l p o t e n t Then t h e n a t u r a l homomorphism induced by P - l o c a l i z i n g b o t h Proof. n = 0 Q-action A and Q, P-localizes. 2 Ho(Q;A> = A / r w ( A ) , Ho(Qp;Ap) = 2 kp/rAuw(Ap). Thus * t h e case f o l l o w s from Theorem 4.5 and Theorem 4.11. W e suppose w e a l s o have n 2 1 and a r g u e by i n d u c t i o n on n i l ( p w ) = 1, n i l ( h p w ) = 1 Now write e x t e n s i o n of Theorem 2 . 9 . and so n i l w. For nil w e* P - l o c a l i z e s by an e a s y A2 = r L ( A ) , s o t h a t w e have a s h o r t e x a c t sequence of Q-modules A (4.13) Suppose nil w 5 c, where Q-actions of n i l p o t e n c y 5 c we s e t w2 = wIA2. 2 A >- A/A2. c 2 2 , and t h a t t h e theoeem i s demonstrated f o r - 1. Then w e have n i l ( w 2) 5 c Moreover, t h e induced a c t i o n of Q on W e may t h u s a p p l y (4.13) and Theorem 4.5 t o o b t a i n a diagram - 1 where A/A2 is t r i v i a l . = 1, Localization of nilpotent groups 40 where w e know t h a t Let Theorem 4 . 1 4 . Q-action 0, ek2, e , 4 , e,5 e,l, P-localize. e*3 P-localizes. Thus be a P-local abeZian group equipped with a nilpotent A Then Q E N. e*: H " ( Q ~ ; A ) H"(Q;A), n 2 0. Proof. Ho(Q;A) = A" = {a E A1x.a = a , a l l Referring t o ( 4 . 9 ) , we see t h a t Ho(Qp;A) = A'". A" where ) Z( A = n z ( G ) , "'A = A n z(G~), denotes, a s u s u a l , t h e c e n t e r . (or Theorem 2.8) an i n c l u s i o n e : G+Gp that A" 5 A'". sends But p l a i n l y "'A We suppose Z(G) We know from Proposition 1.10 to C Ae*'w Z(Gp). = Thus ( 4 . 9 ) induces Am. Thus A" = ."'A n ? 1 and again argue by induction on = 1, t h i s is p r e c i s e l y Corollary 1 . 1 4 . nil(") x C Q}; s i m i l a r l y , from Theorem 4.5 t h a t A2 nil(w). For Referring t o ( 4 . 1 3 ) , we see is P-local and hence a l s o A/A2. Thus, invoking ( 4 . 1 3 ) , we o b t a i n a diagram .. . + H n-1 ( Q ~ ; A / A+ ~H)" ( Q ~ ; A+~H) " ( Q ~ ; A-)+ H " ( Q ~ ; A / A+ H~n+l ) ( Q ~ ; A ~ ).. . + 1.* g n-1 . .. -+H & ' e* g/e* /e* Rt-1 (Q;A/A~) + H " ( Q ; A ~ )-+H"(Q;A) + H ( Q ; A / A ~ )-+H ( Q ; A ~ ) -+ J, ... and t h e Five Lema completes t h e proof. Remark, that For a r b i t r a r y (Aw), 5 <". Q , a r b i t r a r y (abelian) If further Q A and w E A(Q,A), i t i s t r u e is f i n i t e l y generated, then (Am), = <". For t h e proofs, we r e f e r t o Hilton [351. Consider f i n a l l y a n i l p o t e n t a c t i o n +J C Av(Q,A), where Q i s not Actions of a nilpotent group o n an abelian group 41 necessarily nilpotent. We have, of course, an induced action of Q on the homology (and cohomology) of A axd we wish to assert the nilpotency of this action. First we state a general Proposition. Proposition 4.15. Given F: Ab + Ab. w € AV(Q,A) as above and a haZf-exact functor Then the induced action Fw of Proof. Q on FA i s nilpotent. Fw is simply the composition Q Thus the conclusion is clear for nil(w) = Aut (A) +Aut (FA). 1. We proceed as usual by induction on nil(w). Applying F to (4.13) yields an exact sequence by the half-exactness of F. The proof is then completed with the aid of Proposition 4.3. Corollary 4.16. Note that, in fact, nil (Fo) Let w € AV(Q,A) Then the induced actions of C nil w. and l e t B be an arbitrary abeZian group. Q on A 8 B, Tor(A,B) and H,(K;A), K any group & t h t r i v i a l action on A, are nilpotent. Theorem 4.17. If w € Av (Q,A) , then the induced action of Q on H,(A;C) ,c triu-ial A-moduZe, is nilpotent. (A similar statement holds for H*(A;C) .) Proof. In case nil(w) = 1, the result is clear. The inductive step is carried out by applying the Lyndon-Hochschild-Serre spectral sequence to (4.13). We have A/A2 acting trivially on H,(A2;C). It follows from the inductive hypothesis 2 together with Corollary 4.16 that the induced action of w on Ers is nilpotent. Localization of nilpotent groups 42 By t h e f i n i t e convergence of t h e s p e c t r a l sequence, we conclude, by r e p e a t e d a p p l i c a t i o n of P r o p o s i t i o n 4.3, t h a t t h e induced a c t i o n of also on w on E;s, hence is nilpotent. H,(A;C), F i n a l l y , we draw a t t e n t i o n t o t h e f o l l o w i n g u s e f u l addendum t o Theorem 4.14. Let Theorem 4.18. Q be a finitely-generated n i l p o t e n t group operating nilpotentZy on the abelian group H " ( Q ; A ) ~ =H"(Q;+) Proof. Then A. = H"(Q~;$). The second isomorphism was proved i n Theorem 4.14. The f i r s t i s proved by t h e u s u a l i n d u c t i o n on t h e n i l p o t e n c y of t h e a c t i o n , once i t is proved f o r t h e c a s e of t r i v i a l Q-action. from t h e n e x t s e c t i o n , t h e f a c t t h a t t h e homology of I n t h a t c a s e , we t a k e , Q is finitely-generated i n each dimension ( P r o p o s i t i o n 5 . 4 ) , when t h e r e s u l t f o l l o w s from t h e following elementary p r o p o s i t i o n . P r o p o s i t i o n 4.19. Let A, B be abelian groups with B finitely-generated. Then Hom(B,AIp = Hom(B,$), Ext(B,A)p = Ext(B,Pp), P r o p o s i t i o n 4.19 h a s obvious i m p l i c a t i o n s f o r t h e cohomology groups of t o p o l o g i c a l spaces. Indeed, i t e n a b l e s us (once we have e s t a b l i s h e d an a p p r o p r i a t e l o c a l i z a t i o n theory) t o i n f e r t h e analog of Theorem 4.18 when is r e p l a c e d by a t o p o l o g i c a l space generated (and A X whose homology groups a r e f i n i t e l y - i s merely an a b e l i a n c o e f f i c i e n t group). Such an analog i s i m p l i c i t l y invoked, f o r example, i n t h e p r o o f s of Theorems 111.1.7 and 111.1.14. Q Generalized Serre classes of groups 5. 43 G e n e r a l i z e d S e r r e c l a s s e s of groups The n o t i o n of " c l a s s of a b e l i a n groups" goes back t o t h e fundamental paper of S e r r e [73]. For t h e t o p o l o g i c a l a p p l i c a t i o n s we have i n mind, i t is n e c e s s a r y t o c o n s i d e r n o n a b e l i a n groups, so w e f i n d i t c o n v e n i e n t t o extend S e r r e ' s theory accordingly. D e f i n i t i o n 5.1. We b e g i n w i t h t h e d e f i n i t i o n . A generaZized Serre class i s a c o l l e c t i o n C of groups satisfying: (i) That i s , i f c l a s s i n t h e o r d i n a r y sense. e x a c t sequence of a b e l i a n groups, t h e n are a b e l i a n groups, t h e n then H.(A) GI, G" € C iff A ' >-A A ' , A" C C A @ B , Tor(A,B) C C ; i f € C , i > 0 ; and (ii) - The s u b c o l l e c t i o n of a b e l i a n groups i n G ' >- A C C G forms a S e r r e A" i s a short A € C; i f iff 0 C C. Given a c e n t r a l e x t e n s i o n C - A, B C C i s an a b e l i a n group, of g r o u p s , t h e n G" G € C. The f o l l o w i n g are examples of g e n e r a l i z e d S e r r e classes: (a) The class G of a l l groups. (b) The c l a s s N , resp. FN, of a l l n i l p o t e n t , r e s p . f i n i t e l y g e n e r a t e d n i l p o t e n t groups. (c) The class G(p) , r e s p . FG(p) , of a l l p-groups, resp. f i n i t e p-groups. (Observe t h a t FG(p) W e remark t h a t i f v i r t u e of D e f i n i t i o n 5 . 1 ( i i ) , C i s , i n f a c t , a s u b c l a s s of FN.) i s any g e n e r a l i z e d S e r r e c l a s s , t h e n , by t h e n i l p o t e n t groups i n C form a s u b c l a s s . Another consequence of D e f i n i t i o n 5 . l ( i i ) i s t h a t an o r d i n a r y S e r r e c l a s s w i l l almost n e v e r be a g e n e r a l i z e d S e r r e c l a s s . The f o l l o w i n g p r o p o s i t i o n i s e s s e n t i a l l y proved by S t a m b a c h [78 1 , i n t h e c o n t e x t of t e n s o r i a l cZasses of groups. Localization of nilpotent groups 44 G iff Gab C € C be generalized Serre class and l e t Let Proposition 5 . 2 . € N. Then C C. Proof. Assume G T 3 G € C. Setting r i = r i (G) and letting c = nil(G), have central extensions from which we conclude, by Definition 5,1(ii), - Conversely, if we assume Gab ri/ri+' € C, 1 P i i c. € that Gab = G / r 2 € C. C, then observe first that Indeed, the function i factors G X . . . X G - ~ I which sends an i-tuple of elements of G into the corresponding i-fold commutator clearly induces a surjective homomorphism G~~ 8 .. . 8 G~~ ->r i/r i+l, It follows from De'finition 5 . 1 ( i ) that Gab 0 belongs to C . .. . 63 Gab and hence ri/I?+' But then, by appealing to \the extensions ( 5 . 3 ) (in reverse order), we conclude that G € C. We have also the following variant of Proposition 5 . 2 . Proposition 5.4. Let iff HI(G) € C, i > Proof. C be a generalized Serre class and l e t G € N. 0. We assume G € C and argue by Induction on c = nll(G). If c = 1, then the conclusion is Incorporated into Definition S . l ( i ) . c ? 2, Then G If we apply the Lyndon-Hochschild-Serre spectral sequence to the central € C extension Tc Generalized Serre classes of groups - G -> 4s G/rC and obtain the desired result. The converse is a special case of Proposition 5.2 in view of H (G) 1 % G ab' If Corollarv 5.5. then H*(G) % G, K are f i n i t e l y generated nilpotent groups i n the same genus, H*(K). Proof. We apply Theorems 1.12 and 3.14, since, by Proposition 5.4, we know that the homology groups of G, K Theorem 5.6. Let that the relation are finitely generated. C be a generalized Serre class, l e t G G € N and operates nilpotently on the d e l i a n g r m p € C A and suppose further through W. We have : (i) i f A C , then Hi(G;A) € C, i € ? 0; (ii) i f A f C , then Ho(G;A) f C . Proof. (i) Referring to (4.13) for the definition of A2, we have Ho(G;A) = A/A2, hence Ho(G;A) C C. (not nil(G)!), 5.4. For i z 0, we argue by induction on nil(w) the assertion for nil(w) = 1 being a consequence of Proposition The short exact sequence (4.13) gives rise to the exact sequence Hi(G;A2) - Hi(G;A) Hi(G;A/A2), from which the conclusion evidently follows. (ii) We again argue by induction on nil(w), clear since then, Ho(G;A) = A. the case nil(w) = 1 being There are two possibilities: Case 1: A/A2 # C: As the induced action of G on A/A2 is trivial, we have Ho(G;A/A2) = A/A2, thus Ho(G;A/A2) # C. But we have a surjection Ho(G;A) + Ho(G;A/A 2 1, Case 2: so we conclude Ho(G;A) f C. A/A2 € C: In this case, A2 f C, otherwise A € C. Thus, 46 Localization of nilpotent groups by the inductive hypothesis and (i), we infer Ho(G;A2) C, H1(G;A/A2) f c. From the exact sequence H1(G;A/A2) ----f Ho(G;A2) + Ho(G;A), 6 C. we thus conclude Ho(G;A) We conclude by using Proposition 5.2 to prove a basic theorem on generaffzed Serre classes. Theorem 5 . 7 . Let be a generalized Serre class and l e t C be an arbitrary (not necessarily central) extension i n N. G G' G -> G" Then G ' , GI' € C i f f € C. Proof. Gab € C . Assume G', G" € C. By Proposition 5.2, it suffices to show But the given short exact sequence induces the short exact sequence of abelian groups Moreover, G'/G' n r 2 belongs to C since it is a quotient group of Gib which belongs to C by Proposition 5.2. follows that Gab As Gib € C, again by Proposition 5.2, it € C. Now assume G € C. Then, as previously noted, Gab maps surjectively and two applications of Proposition 5.2 allow us to conclude that G" € C . tob''G To show G' € C, we argue by induction on c = nil(G), the result being obvious for c = 1. Since nil(G/rC) = c Girc/rc c C. But - 1, we have, by the inductive hypothesis, that ~'r~/r'G G'/GlnrC and abelian, it is clear that G' n c'I with the two extreme groups in C . € rc E c by (5.3). Since rc C. We thus have a central extension It follows from Definition 5.1(ii) that G' € C , thereby completing the induction. is Chapter I1 Localization of Homotopy Types Introduction In this chapter we apply localization methods to homotopy theory. We use the definitions of local groups and localization given in Chapter I, in order to introduce the corresponding notions into homotopy theory; and we prove the basic theorems that relate to localization in homotopy theory. These theorems find many applications in homotopy theory, but we will reserve the applications to Chapter 111. Our definition of a P-local (pointed) space is simply that its homotopy groups should be P-local groups. This definition could be made quite generally for an arbitrary pointed space. However we are concerned to obtain a localization theory and also to obtain useful criteria for establishing when a given map of spaces does in fact P-localize. It is therefore necessary for us to work in a restricted category of (pointed) topological spaces. It is also necessary for us to work in a homotopy category (that is, in a category in which the morphisms are homotopy classes of continuous maps), since our procedures for establishing the existence of a localization theory will all operate up to homotopy. A more general treatment, valid in the semisimplicial category, has been given by Bousfield-Kan u 4 ] . We will always suppose that our spaces have the homotopy type of CW-complexes. In Section 1, we present a localization theory in the homotopy category H1 of 1-connected CW-complexes. We establish two fundamental theorems in H1, namely that every object of the category does admit a P-localization, and that we can detect the P-localizing map f: X +. Y either through the induced homotopy homomorphisms, which should also P-localize, or through the induced homology homomorphisms, which should also P-localize. In the course of establishing that there is a localization theory in H1, Localization of homotopy types 48 we a c t u a l l y c o n s t r u c t t h e l o c a l i z a t i o n of a given CW-complex i m i t a t i n g t h e c e l l u l a r c o n s t r u c t i o n of t h a t of a ZocaZ c e l l . X by X, r e p l a c i n g t h e i d e a of a c e l l by The f a c t t h a t t h e l o c a l i z a t i o n can b e d e t e c t e d e i t h e r through homotopy o r through homology h a s t h e immediate consequence t h a t we may l o c a l i z e f i b r e and c o f i b r e sequences i n H1, I n S e c t i o n 2 , we d e s c r i b e a broader homotopy c a t e g o r y i n which we w i l l a l s o b e a b l e t o e s t a b l i s h a s a t i s f a c t o r y l o c a l i z a t i o n theory. It t u r n s o u t t h a t we would wish t o e n l a r g e t h e c a t e g o r y t o which we apply l o c a l i z a t i o n methods from our o r i g i n a l c a t e g o r y H1. confined t o o b j e c t s of H1, H1, even i f our main i n t e r e s t were For, i n o r d e r t o prove theorems about l o c a l i z a t i o n of i t is v e r y u s e f u l t o employ function-space methods, and t h e function-space c o n s t r u c t i o n t a k e s u s o u t s i d e t h e c a t e g o r y . i t is t r u e t h a t i f and i f into W X X However, is a niZpotent s p a c e , i n a s e n s e defined i n S e c t i o n 2 , is f i n i t e , t h e n t h e f u n c t i o n space Xw is a g a i n n i l p o t e n t . f a c t s about t h e category S e c t i o n 2 concerns i t s e l f w i t h some b a s i c NH of n i l p o t e n t s p a c e s , and may be regarded i n p a r t as propaganda f o r t h e u s e of t h i s c a t e g o r y i n homotopy theory. i t has a l r e a d y been shown by Dror [23] t h a t homotopy t h e o r y . W of pointed maps of NH Indeed, is a s u i t a b l e c a t e g o r y f o r Roughly speaking, one may s a y t h a t most of t h e t e c h n i q u e s of homotopy t h e o r y which have been developed s i n c e t h e p u b l i c a t i o n of S e r r e ' s t h e s i s can a l l be c a r r i e d o u t i n t h e c a t e g o r y NH techniques were of course o r i g i n a l l y formulated i n although many of t h o s e H1, The b a s i c theorem proved i n S e c t i o n 2 is t h a t a space is n i l p o t e n t i f and only i f i t s Postnikov tower admits a p r i n c i p a l refinement. t h e category Mi It is t h i s theorem which e x p l a i n s why is s u i t a b l e f o r homotopy theory; f o r t h e given refinement of t h e Postnikov tower may be used i n p l a c e of t h e Postnikov tower i n t h o s e arguments i n which t h e c r u c i a l f a c t which is r e q u i r e d is t h a t t h e f i b r a t i o n s which appear i n t h e tower should b e induced o r p r i n c i p a l . Introduction 49 However, it should be pointed out that the category NH has eertain defects over the category H1. One of the defects is that it is not closed under the mapping cone operation. This defect has a serious consequence in Section 3 . We also describe in Section 2 how to relativize the notion of a nilpotent space to obtain that of a nilpotent map. In Section 3 we generalize the theorems of Section 1 from the category H1 to the category NH. Formally, we get the corresponding formulations of the two fundamental theorems of Section 1. However there is an important difference in the way in which we construct the localization of an object. For, whereas in the category H1 we are able to proceed cellularly, since the mapping cone construction respects the category H1, we cannot in the nilpotent case proceed cellularly, since the mapping cone construction would take us outside the category. It is therefore necessary for us to proceed homotopically rather than cellularly in constructing the localization. In this way, of course, we lose much of the conceptual simplicity of the construction in Section 1. Section 4 is a brief technical section in which we introduce the idea of a quasifinite complex in Mi. relative to the category H1. Here again we see a certain disadvantage For if X is a 1-connected CW-complex whose homology groups are all finitely generated, and vanish above a certain dimension, then X itself has the homotopy type of a finite complex. If we discard the condition of simple-connectivity, we can no longer assert this conclusion. Indeed, we have the obstruction theory of Wall which enables US to discuss the question whether a CW-complex X whose homology looks like that of a finite complex in fact has the homotopy type of a finite complex. Thus we are led to introduce the concept of a quasifinite CW-complex, meaning a nilpotent CW-complex X such that the homology of X is finitely Localization of homotopy types 50 generated i n each dimension and v a n i s h e s above a given dimension. We prove t h a t such a q u a s i f i n i t e complex always h a s t h e homology type of a f i n i t e complex. I n S e c t i o n 5 we prove t h e fundamental p u l l b a c k v a r i o u s v a r i a n t s as consequences of t h a t theorem. theorem and Here we l e a n v e r y h e a v i l y on t h e r e s u l t s of Chapter I . The fundamental p u l l b a c k theorem a s s e r t s t h a t t h e p o i n t e d set i s t h e p u l l b a c k of t h e pointed sets [W,Xl set 1, provided t h a t [W,X W {[W,Xpll over t h e p o i n t e d i s a f i n i t e connected CW-complex and a n i l p o t e n t complex of f i n i t e type. X is T h i s a s s e r t i o n f a l l s i n t o two p a r t s . The f i r s t p a r t s t a t e s t h a t given two maps f,g: W -+ X such t h a t = e g: W + X f o r a l l primes p, t h e n f = g . T h i s p a r t of t h e a s s e r t i o n P P P’ does not r e q u i r e t h a t X b e of f i n i t e t y p e . The second a s s e r t i o n s t a t e s e f that i f f(p): W -+ c l a s s of t h e map X P a r e maps, f o r a l l primes r f(p): W P -+ Xo p , such t h a t t h e homotopy i s independent of i s t h e r a t i o n a l i z a t i o n map, t h e n t h e r e exists a map e f P 2 f(p) f o r a l l primes t h e condition t h a t W p. p , where rp: Xp f: W such t h a t -+ X -+ Xo We show by an example t h a t we cannot omit should be f i n i t e . However, provided t h a t W is n i l p o t e n t , we may i n f a c t weaken t h e hypotheses of t h e p u l l b a c k theorem by simply r e q u i r i n g t h a t W be q u a s i f i n i t e . I n S e c t i o n 6 we make a p r e l i m i n a r y s t u d y of t h e l o c a l i z a t i o n of H-spaces. Our main r e s u l t i n t h i s s e c t i o n i s a g e n e r a l i z a t i o n of t h e p a r t of t h e fundamental theorem of Chapter I which t e l l s u s how t o d e t e c t t h e P - l o c a l i z a t i o n of a n i l p o t e n t group i n terms of t h e P - b i j e c t i v i t y of t h e l o c a l i z i n g homomorphism. I n S e c t i o n 7 w e formulate t h e fundamental mixing technique of Zabrodsky w i t h i n t h e c o n t e x t of t h e l o c a l i z a t i o n of n i l p o t e n t s p a c e s . The Introduction 51 p a r t i c u l a r r e s u l t which w e emphasize is t h a t , given n i l p o t e n t spaces X, Y with equivalent r a t i o n a l i z a t i o n s , and given a p a r t i t i o n of t h e primes Il = P u Q , then t h e r e e x i s t s a n i l p o t e n t space 2 such t h a t 2 P = Xp and ZQ = YQ. We make very considerable a p p l i c a t i o n of t h e r e s u l t s of t h e l a s t two s e c t i o n s i n Chapter 111. Indeed, we a r e r a t h e r l i t t l e concerned t o g i v e e x p l i c i t examples and a p p l i c a t i o n s i n t h i s Chapter i n view of t h e f a c t t h a t Chapter 111 is e n t i r e l y concerned with applying t h e theory of Chapter 11. 52 Localization of homotopy types 1. Localization of 1-connected CW-complexes. H1 of 1-connected We work i n t h e pointed homatopy category CW-complexes. X X C H1, If and i f P is a family of primes, we say t h a t is P-zocal i f the homotopy groups of W e say t h a t f: X + Y P-localizes H1 in X a r e a l l P-local a b e l i a n groups. X if Y i s P-local and* f*: [Y,Z] z [X,Zl f o r a l l P-local 2 C H1. Of course t h i s u n i v e r s a l property of c h a r a c t e r i z e s i t up t o canonical equivalence: both P-localize H1 with in H1. X hfl = f 2 . if fi: X -+ then t h e r e e x i s t s a unique equivalence Yi, f i = 1, 2 , h : Y1 PI Y2 in W e w i l l prove t h e fallowing two fundamental theorems The f i r s t a t t e s t s t h e e x i s t e n c e of a l o c a l i z a t i o n theory i n H1 and the second a s s e r t s t h a t we may d e t e c t t h e l o c a l i z a t i o n by looking a t induced homotopy homomorphismor induced homology homomorphisms. Theorem 1A. ( F i r s t fundamental theorem i n H1.) Every X We T r i t e Theorem 1B. Let admits a P-localization. i n HI e: X + Xp f o r a f i x e d choice of P-localization of X. (Second fundamentaZ theorem i n H1.l f: X * Y ( i ) f P-localizes (ii) nnf: n X (iii) Hn f : HnX Then the following statements are equivalent: i n H1. X; nnY P-localizes f o r a l l n 3 1; -+ H Y P-localizes f o r a l l n 2 1. -t We w i l l prove Theorems l A , 18 simultaneously. *We w r i t e , a s usual, [Y,Z] of maps from Y t o 2. f o r H1(Y,Z), W e r e c a l l from t h e s e t of pointed homotopy classes Localization of I -connectedCW-complexes Proposition 1.1.9 that a homomorphism B if and only if is P-local and $: A + B 53 of abelian groups P-localizes is a P-isomorphism; this latter condition @ + means that the kernel and cokernel of belong to the Serre class C of abelian torsion groups with torsion prime to P. Thus to prove that (ii) (iii) in Theorem 1 B above it suffices to prove the following two propositions. Proposition 1.1. Let Y C H1. only if H Y is P-locat f o r a l l n Proposition 1.2. all Then f: X Let -+ n E 1 if and only if TI Y is P-local f o r a l l Then nn(f) is a €'-isomorphism f o r aZZ for all n E 1 if and only if Hn(Y;Z/p) = 0 disjoint from P. n 2 1. P is a P-local abelian group, so are the homology groups of the Eilenberg-MacLane space K(A,m). K(A,m-l) -+ -+ K(A,m), with E if H (A,m-1; Z/p) = 0 n Now let ... of Y. is a P-local abelian 1. It now follows, by induction Hn(A,m) E is P-loca? for all n 2 1 and all primes Now, by Proposition 1.1.8, if A group, so are the homology groups H A, n on m, that if A 1 if and is a P-isomorphism for Proof of Proposition 1.1. We first observe that HnY p ? 2 1. Y in H1, Hn(f) n For we have a fibration contractible, from which we deduce that, for all n E 1, then Hn(A,m; Z/p) = 0 -+ Ym -+ Ymm1-+ ... Thus there is a fibration K(nmY,m) Thus, if we assume that nnY for all n 1 1. Y2 be the Postnikov decomposition * Ym + Ym-l, and Y2 = K(n 2Y , 2 ) . is P-local for all n P 1, we may assume inductively that the homology groups of Ym-l are P-local and we infer (again using homology with coefficients in Z / p , with p disjoint from P) that the homology groups of Ym are P-local. Since Y -+ Y, is m-connected, it follows that HnY P-local f o r all n 1 1. To obtain the opposite implication, we construct the 'dual' Cartan-Whitehead decomposition is Localization of homotopy types 54 There i s then a f i b r a t i o n if we assume t h a t K(amY,m-l) i s (m-1)-connected, Y(m) * Y(m+l) n i s P-local f o r a l l HnY t h a t t h e homology groups of Y(m) + T m and Y(2) = Y . Since Y Z H Y(m) m Y(m) = n Y m m P) t h a t the homology groups of s t e p i s complete and T Y 2 H Y(n) n and i s P-local. to C a r e P-local, p disjoint so t h a t the i n d u c t i v e is P-local. Proof of Proposition 1 . 2 . mod C , where Y(m+l) and a Thus we i n f e r (again using homology w i t h c o e f f i c i e n t s i n Z / p , w i t h from Thus, 2 1, w e may assume i n d u c t i v e l y a r e P-local. i t follows t h a t Y(m) Since a P-isomorphism is an isomorphism i s t h e c l a s s of a b e l i a n t o r s i o n groups with t o r s i o n prime P, Proposition 1 . 2 is merely a s p e c i a l case of t h e c l a s s i c a l Serre theorem. We have thus proved t h a t ( i i ) t h a t (ii) * (i). counterimage of f. (iii) in Theorem 1 B . We now prove The o b s t r u c t i o n s t o t h e e x i s t e n c e and uniqueness of a g: X -+ under Z Now, given ( i i ) (or ( i i i ) ) ,H,f of t h e map Q f*: [Y,Z] C C , where -+ [X,Z] H,f l i e in H*(f;n,Z). r e f e r s t o t h e homology groups Thus (i) follows from the u n i v e r s a l c o e f f i c i e n t theorem f o r cohomology and Proposition 1.1.8(v). We now prove Theorem 1 A . of f: X -c Y in H1 More s p e c i f i c a l l y , we prove t h e e x i s t e n c e s a t i s f y i n g (iii). Since we know t h a t (iii) 3 (i), t h i s w i l l prove Theorem 1 A . Our argument is f a c i l i t a t e d by t h e following key observation. Proposition 1.3. Let we have constructed U be a f u l l subcategory of f: X -+ Y automatically y i e l d s a functor satisfying l i i i ) . L: U transfornation from the embedding U -+ H1, 5 H1 Then the assignment X * Y f o r which t o L. f o r whose objects X HI f provides a natural Localization of 1-connectedCW-complexes Proof of P r o p o s i t i o n 1 . 3 . g: X Let -+ in X' U. 55 W e t h u s have a diagram Y in H1 with f, f' Y' satisfying ( i i i ) . Since f s a t i s f i e s ( i ) and P-local by P r o p o s i t i o n 1.1, we o b t a i n a unique ( i n H1) Y' is h 6 [Y,Y'] making t h e diagram I t i s now p l a i n t h a t t h e assignment commutative. desired functor X rf Y, g* h yields the L. We e x p l o i t P r o p o s i t i o n 1 . 3 t o prove, by i n d u c t i o n on may l o c a l i z e a l l n-dimensional CW-complexes i n H1. n, t h a t w e .If n = 2, t h e n such a complex i s merely (up t o homotopy e q u i v a l e n c e ) a wedge of 2-spheres x = v s2 , a a where runs through some index s e t , and w e d e f i n e Y = VM( ZP,2), a where map M(A,2) f: X + fo: Xo + Y o dim X = n Y i s t h e Moore s p a c e having satisfying ( i i i ) . satisfying ( i i i ) i f + 1, X € H1. H M = A. 2 There i s t h e n an e v i d e n t Suppose now t h a t w e have c o n s t r u c t e d dim X 5 n , where n 1 2 , and l e t Then w e have a c o f i b r a t i o n vsn R, xn i . x By t h e i n d u c t i v e h y p o t h e s i s and P r o p o s i t i o n 1 . 3 , we may embed (1.5) diagram i n the Localization of homotopy types 56 where fo, fl satisfy (iii) and the square in (1.6) homotopy-commutes. embeds Yo Thus if j (1.6) by f: X + Y in the mapping cone Y of h, then we may complete to a homotopy-commutative diagram and it is then easy to prove (using the exactness of the localization of abelian groups) that f: X +. Y also satisfies (iii). (iii) if X Thus we may construct f: X + Y satisfying is (n+l)-dimensional, and the inductive step is complete. It remains to construct f: X -+ Y satisfying (iii) if X is infinite-dimensional. We have the inclusions x2 cx3 2 . . and may therefore construct where f", fn+l satisfy (iii). We may even arrange that (1.7) is strictly If we define Y = UY(n), with the weak topology, n and the maps fn combine to yield a map f: X -+ Y which again commutative for each n. then Y t H1 obviously satisfies (iii). Thus we have proved Theorem lA in the strong form that, to each X in H1, there exists f : X -+ Y in H1 satisfying (iii). Finally, we complete the proof of Theorem 1B by showing that (i) = (iii). Given f: X -+ Y which P-localizes X, let fo: X constructed t o satisfy (iii). Then fo: X + Y which one immediately deduces the existence of -+ Y be also satisfies (i), from a homotopy equivalence Localization of 1-connectedCW-complexes u: Yo +. Y with uf = f. It immediately follows that f also satisfies (iii). Thus the proofs of Theorems l A , 1B are complete. We note that our proof of the first fundamental theorem does much more than establish the existence of a localization theory in H1; it provides us with a combinatorial recipe for constructing the localization of a given CW-complex. The Moore spaces S; = may be called P-ZocaZ n-spheres, M(Zp,n) and a cone on a P-local n-sphere may be called a P-ZocaZ (n+l)-ceZz. Then, given a cellular decomposition of X, we may--as shown in the proof of Theorem 1A--construct a P-ZocaZ-ceZZuZar decomposition of Xp by 'imitation'; that is, whenever, in building up X, we attach an n-cell to Y f : Sn-' of a map to Yp +. $, we attach a P-local n-cell Y, then, in building up by means of the localized map sn-l fp: (say) by means P -+ Yp. We illustrate this procedure by means of an example. Example 1.8. Let Sm -+ E -+ Sn be an Sm-bundle over S", n z m, and let m a C T ~ ~ - ~) ( be S the characteristic (homotopy) class of the bundle. it is well-known that E Then admits the cellular decomposition E = sm ua en u emh, where we will not trouble to specify the attaching map f o r the top-dimensional cell. If m 1 2 we may now localize E Ep = S; Ua to obtain nntl-n ep U ep P . Let us consider,in particular, the Stiefel manifold Vn+l , 2 vectors to S", and let us assume that n over Sn with fibre Sn-', a = 2 6 T~-~(S~-'). so We obtain of unit tangent is even. Then 'n+1,2 that we may take E = V n+1,2' m = n fibres - 1, and Localization of homotopy types 58 Suppose now t h a t P is t h e f a m i l y of a l l odd primes. Sn-' s o t h a t i t is e a s y t o s e e t h a t n ep 2p Then is i n v e r t i b l e , h a s t h e homotopy type o f a p o i n t . It t h u s f o l l o w s from (1.9) t h a t if E = V n+1,2, n even, and f u r t h e r follows t h a t i f [E,Y] % P is t h e f a m i l y of odd primes. It t h e r e f o r e is a P - l o c a l space t h e n Y [I? ,Y] '2 [Sp 211-1,Y] '2 [ S 2n-1 , y ] = P Here t h e most s t r i k i n g f a c t is t h a t t h e s e t [E,Y] 2n-1 y. h a s acquired a v e r y n a t u r a l a b e l i a n group s t r u c t u r e and h a s simply been i d e n t i f i e d w i t h a c e r t a i n homotopy group of Y. We have t h e f o l l o w i n g immediate c o r o l l a r i e s of t h e second fundamental theorem. C o r o l l a r y 1.10. Let F i s a f i b r e sequence i n Proof. -t E + B be a f i b r e sequence i n H1. Then Fp + H1' Of c o u r s e we a r e o n l y making t h e a s s e r t i o n up t o homotopy, so t h a t our c l a i m amounts t o s a y i n g t h a t e x a c t sequence of homotopy groups. Fp -+ Ep -t Bp induces t h e u s u a l However t h i s f o l l o w s immediately from P r o p o s i t i o n 1 . 1 . 7 and t h e e q u i v a l e n c e of (i) and (ii) i n Theorem 1B. S i m i l a r l y , r e p l a c i n g (ii) of Theorem 1B by ( i i i ) of Theorem lB, we o b t a i n C o r o l l a r y 1.11. Let 5 + Yp -+ Cp Ep X + Y -+ C be a cofibre sequence i n H1. i s a cofibre sequence. Then + B P Localization of I -connected CW-complexes 59 We now introduce an important definition. Definition 1.12. A map f: X is called a P-equivalence if in H1 Y --f fP is a (homotopy) equivalence. Theorem 1.13. Let primes. f: X Y in + H ,and l e t 1 P be a non-empty f a m i l y of Then the following statements are equivalent: (il (iil f i s a ?-equivalence; f i s a p-equivalence f o r a l l Q C (iii) TInf i s a P-isomorphism f o r a l l n ( i v ) H f i s a P-isomorphism f o r a l l n Proof. By Whitehead's Theorem f P; 2 1; 2 1. is a P-equivalence iff n f n P is an isomorphism for all n 1 1. By Theorem 1B it follows that Thus the equivalence (i) (i) Q (iv). (iii) follows from Theorem 1.3.1. Theorem 1.3.12 ensures that (nnf)p TI f = (nnf)p. n P Similarly is an isomorphism iff (nnf) is an isomorphism for all p C P. Thus the equivalence of (i) P and (ii) readily follows. A further refinement is possible in the case in which are of f i n i t e t y p e , that is, n ? 1. Notice that, in IT X and nnY n X and Y are finitely generated for all this is equivalent to asking that H X and H1, HnY be finitely generated. Theorem 1.14. Let f: X -+ Y i n H1 with X , Y P be a non-empty f a m i l y of primes. f,: Hn(X; Z/p) zz Hn(Y; Z/p) f o r Proof. Let Z/p f: X + Y Then of f i n i t e type, and l e t f i s a P-equivalence i f f p € P. be a P-equivalence. Then, since, for p C P, is P-local, it follows from Theorem 1B and PropOsftiOn I.1.8(ii) we have a comutative diagram that Localization of homotopy types 60 Thus is a n isomorphism; n o t i c e t h a t t h e i m p l i c a t i o n we have proved does f, not require t h a t X, Y It is i n t h e o p p o s i t e i m p l i c a t i o n b e of f i n i t e type. t h a t t h i s condition plays a decisive r o l e . induces Hn(X; Z f p ) f,: Hn(Y; We w i l l prove t h a t type, H f n Consider t h e diagram, f o r each ZIP), Thus w e suppose t h a t n 2 1, p 6 P , where i s a p-isomorphism, f: X + Y are of f i n i t e X, Y Hnf: HnX H Y , n 11. -+ n 5 1, $ (1.15) f *n t h e v e r t i c a l homomorphisms being induced by f. i t f o l l o w s t h a t we must prove t h a t each is b i j e c t i v e and each :f It f o l l o w s immediately from (1.15) t h a t e a c h surjective. and each f i surjective. It a l s o f o l l o w s from (1.15) Suppose, i n d u c t i v e l y , t h a t we have shown Then f: By P r o p o s i t i o n 1 . 1 . 8 ( i i ) , Hrf: HrX --t fi, is a p-isomorphism f o r HrY It t h u s f o l l o w s from (1.15) t h a t bijective. that ..., f C 1 r 5 n f: - f; is injective fi is bijective. bijective, 1, so t h a t n Z 2. f:-l is is b i j e c t i v e , and t h e i n d u c t i v e s t e p i s complete. Since H f Theorem 1.13, t h a t i s a p-isomorphism f for a l l p € P , we conclude, from is a P-equivalence. Remark. Theorems 1.13 and 1 . 1 4 show t h a t we have t h r e e p r a c t i c a l ways t o t e s t if + f: X Y p-isomorphism i s a P-equivalence. for a l l We may t r y t o show t h a t n Z 1, p € P; we may t r y t o show t h a t II f Hnf is a is a Localizationof 1-connected CW-complexes p-isomorphism for all n 1 I, p t P; or, if X, Y are of finite type, we have the potentially most practical procedure, namely to try to show that f,: Hn(X; Z/p) -+ Hn(Y; Z/p) is an isomorphism for all n 1 1, p € P. In the case when P is empty, we have the following evident modification of Theorem 1 . 1 3 . Theorem 1.16. Let f: X -+ Y i n H1. Then the f o l l o w h q statements are equivalent: (il (iil f i s a 0-equivaZence; nnf f i i i l Hnf ( i v ) f,: i s a O-$somorphism f o r a12 n 2 1; is a 0-isomorphism f o r a l l Hn(X;Q) 2 H (Y;Q) n 2 1; for a12 n 2 1. 61 Localization of homotopy types 62 2. Nilpotent spaces. It turns out that the category is not adequate for the full H1 exploitation of localization techniques. This is due principally to the fact that it does not respect function spaces. We know, following Milnor, that is a (pointed) CW-complex and W if X a finite (pointed) CW-complex, then the function space Xw of pointed maps W -+ X has the homotopy type of a CW-complex. However its components will, of course, fail to be 1-connected even if X Xw is 1-connected. However, it turns out that the components of are nilpotent if X is nilpotent. Moreover, the category of nilpotent CW-complexes is suitable for homotopy theory (as first pointed out by E. Dror), and for localization techniques [14,83]. Definition 2.1. A connected CW-complex X and operates nilpotently on IT is nilpotent if nlX X for every n is nilpotent >_ 2. n Let NH be the homotopy category of nilpotent CW-complexes. Plainly NH 2 H1. Moreover, thesimple CW-complexes are plainly in MH; in particular, NH contains all connected Hopf spaces. It isalsonotdifficult tosee (Roitberg [69]) that, if G is a nilpotent Lie group (not necessarily connected) then its classifying space BG is nilpotent in the sense of Definition 2.1. We prove the following basic theorem, which provides us with a further rich supply of nilpotent spaces. Theorem 2.2. Let F Then F C NH if Proof. -i E f B be a fibration of connected CW-compzexes. E €NH. We exploit the classical result that the homotopy sequence of the fibration is a sequence of is IT IT E-modules. 1 E-nilpotent of class icy then 1 IT n F is IT We will prove that, if nnE F-nilpotent of class 5c 1 + 1. Nilpotent spaces 63 (A mild modification of the argument is needed to prove that if nilpotent of class Zc, then 71 F is nilpotent of class 5c 1 + IT E is 1 1; we will deal explicitly with the case n 1 2.) We will need the fact that and that the operation of I T ~ Eon TI E 1 F IT operates on IT B through f,, is such that It will also be convenient to write IF, IE for the augmentation ideals of rlF, IT E. 1 Then the statement that anE is IT E-nilpotent of class 5c 1 translates into Consider the exact sequence of ... and let S-CC= ,€ € I;, a € a ~ B, c IT,+IB. (i*n-.l)['a, = -IT F. Let n+l B IT E-modules 1 - - a IT nF Then i*(S.a) 11 € i* IT = nE (i*c)*i,(a) rlF. Then a((i*n-l).B) ( n - l ) c . c ~ , by ( 2 . 3 ) . But ... = 0 = by ( 2 . 4 ) . (i,n-l).aB (i*Q-l)*B = (f,i,n-l).B = Thus = 0, SO 0. This shows that IC+'*~ F = (O), and thus the theorem is F n proved. Note that, in fact, our argument shows that, even if F is not (n-1)S-a = connected, each component of F is nilpotent. F e w i l l feel free to invoke this more general statement. Now let W be a finite connected CW-complex and let X be a connected CW-complex. Let Xw b e the function space of pointed maps W W + X and let Xfr be the function space of free maps. Choose a map W as base point and let g € XW (g€Xfr) of g. (XW ,g)((XfrW,g)) be the component Localization of homotopy types 64 (Compare G. Whitehead [871 , Federer [ 2 6 1 . ) Theorem 2 . 5 . ( i ) (xw,g) i s nilpotent. W liil x i s nilpotent i f (xfr,g) Proof. We may suppose t h a t ( i ) , ( i i ) are certainly true i f i n d u c t i o n on t h e dimension of i s nitpotent. is a p o i n t . Wo i s 0-dimensional, W Thus t h e a s s e r t i o n s and w e w i l l argue by We w i l l be c o n t e n t t o prove ( i ) . We have W. a cof i b r a t i o n v is a wedge of V where -+ wn * wn+l, g i v i n g r i s e t o a f i b r a t i o n (where we d i s p l a y n-spheres, one component of t h e f i b r e ) where w"+'-+ g: Wn (X ,go) X and g = glw". Our i n d u c t i v e h y p o t h e s i s i s t h a t is n i l p o t e n t , so t h a t Theorem 2 . 2 e s t a b l i s h e s t h e i n d u c t i v e s t e p . Let Corollary 2 . 6 . W be a f i n i t e CW-comptex and W (X ,g) X € NH. Then and W (Xfr,g) are nilpotent. Proof. L e t Wo, W1, ..., Wd b e t h e components of W, w i t h o € Wo. Then xw = x"0 x ;* x ... x Xd'f r . Since p l a i n l y a f i n i t e product of n i l p o t e n t s p a c e s is n i l p o t e n t , i t f o l l o w s that W (X ,g) is n i l p o t e n t . Similarly W (Xfr,g) is n i l p o t e n t . C o r o l l a r y 2 . 6 t h u s e s t a b l i s h e s ( i n view of M i l n o r ' s theorem) t h a t we s t a y i n s i d e t h e c a t e g o r y NH when we t a k e f u n c t i o n s p a c e s X € NH and f i n i t e , i n t h e s e n s e t h a t each component of W Xw Xw is i n with NH. W e now proceed t o g i v e an important c h a r a c t e r i z a t i o n of n i l p o t e n t spaces. Let X be a connected CW-complex and l e t - ... (2.7) Nilpotent spaces - ... P xn 4 xn-l be i t s Postnikov decomposition, so t h a t K(nnX,n). 65 -x1-0 i s a f i b r a t i o n with f i b r e pn W e s a y t h a t t h e Postnikov decomposition refinement a t stage x (2.8) n qc. Yc = n where t h e f i b r e of Let decomposition of if and only i f Proof. TI pn may be f a c t o r e d as a product of f i b r a t i o n s - -... 91 Y1 gi: Yi-l + K(Gi,n+l), Y n o . K(Gi,n) x admits a principal refinement a t stage X operates n i l p o t e n t l y on 1 X 71 LK(Gi,n+l), Since IT Y = (O), n o TI and n ? 2 Suppose conversely t h a t W e consider p :X n n -t Xn-l. IT,X Suppose Then we may r e g a r d i = 1, X(=nlYi,05i5c) 1 rnYC = rnXn = (stage 1) n 2 2. ..., c , o p e r a t e s t r i v i a l l y on Thus, by r e p e a t e d a p p l i c a t i o n s of t h e proof of Theorem 2.2, o p e r a t e s n i l p o t e n t l y on qi ( T I ~ Xi s n i l p o t e n t ) . We w i l l be c o n t e n t t o g i v e t h e argument f o r yi IT 'n-1' Then the Postnikov f i r s t t h a t we have t h e p r i n c i p a l refinement (2.8). as a fibration. o 1 5 i 5 c. be a connected CW-compZex. X Y is an Eilenberg-MacLane s p a c e qi i s induced by a map Theorem 2 . 9 . if principal admits a nlX X. IT is IT X-nilpotent of c l a s s 3. 1 Then, by t h e r e l a t i v e Hurewicz Theorem we have a n a t u r a l isomorphism where Thus n+l (p ) n Gn+l(pn) a s a n element of i s o b t a i n e d from I T ~ + ~ ( P , )by k i l l i n g t h e a c t i o n of may be i d e n t i f i e d w i t h Hn+l 2 (pn;nnX/r rnX). IT 2 n X/r I T ~ X ,and h-' Thus h-l nlXn. may b e regarded g i v e s rise t o a diagram Localization of homotopy types 66 with u b - 0. If u induces ql: Y 1 -+ Xn-1' then pn factors a s (2.10) with q1 The homofopy sequence of (2.10) reduces t o induced a s required. rl replacing Thus we may r e p e a t t h e above procedure, with p and, n' continuing i n t h i s way, we reach xn (2.11) each qi & r yC Yc-l - - being induced by a map ... Yi-l However, a l l t h e homotopy groups of -f r o Y2 Gi = K(Gl,n+l), where r vanish, s o t h a t n-1' r innX/ri+lnnX. i s a homotopy equivalence, and (2.11) is e s s e n t i a l l y t h e p r i n c i p a l refinement a t s t a g e n whose e x i s t e n c e we set out t o prove. We would say t h a t t h e Postnikov system of X refinement i f i t admits a p r i n c i p a l refinement a t s t a g e admits a principal n f o r every n 2 1. We then have the evident Corollary 2.12. Let X be a connected CW-complex. Then X i s nilpotent i f and only i f i t s Postnikov system a h i t s a principal refinement. W e p o i n t out t h a t t h e s i m p l e spaces a r e i d e n t i f i e d , by t h e correspondence i m p l i c i t i n t h i s c o r o l l a r y , with those spaces whose Postnikov system is i t s e l f principal. Remark. Once we have obtained a p r i n c i p a l refinement of t h e Postnikov system of a space, t h e r e i s , of course, no d i f f i c u l t y i n obtaining f u r t h e r refinements, Nilpotent spaces which w i l l remain p r i n c i p a l . 61 Thus i f , f o r example, i s of f i n i t e t y p e X and n i l p o t e n t we may r e f i n e i t s Postnikov system so t h a t each map of t h e r e f i n e d system i s induced by some map ZIP, or f o r some prime Yi-l + f: E i f a l l of IT + nlF). To t h i s end we s a y a r e connected ( s o t h a t f, maps n E 1 F onto n i l p o t e n t l y on t h e homotopy groups of f. nlB), and ( i n c l u d i n g , of c o u r s e , W e could a l s o e x p r e s s t h i s l a s t c o n d i t i o n by a s k i n g t h a t by t a k i n g A = Z F , of ( p o i n t e d ) CW-complexes i s nilpotent o p e r a t e s n i l p o t e n t l y on t h e homotopy groups of E 1 Yi-l p. B, w i t h f i b r e F, E, B + where K(A,n+l) We w i l l need a r e l a t i v e form of Theorem 2.9. t h a t a map Yi n E 1 operate Note t h a t w e r e c o v e r D e f i n i t i o n 2 . 1 t o b e a p o i n t , provided w e adopt t h e r i g h t n o t i o n of n i l p o t e n c y B f o r t h e o p e r a t i o n of T E 1 on TI F. Although we w i l l n o t need t h e g e n e r a l 1 case i n t h i s t e x t , w e now d e s c r i b e t h i s n o t i o n f o r t h e a c t i o n of a group on a group Q With r e s p e c t t o such a n a c t i o n w e d e f i n e a lower central series N. as f o l l o w s (see H i l t o n [36]): rlN = N , Q rrh = gp{(x.a)ba Q W e t h e n s a y t h a t t h e a c t i o n of YC+lN = 11). Q N N i s commutative. by c o n j u g a t i o n , t h e n r% = r 9 nilpotent map, it f o l l o w s t h a t i f i s nilpotent of class b€N, i21. 5c if f Note a l s o t h a t i f i N. Q = N and o p e r a t e s T h u s , w i t h o u r d e f i n i t i o n above of a i s n i l p o t e n t , t h e n t h e f i b r e of f W e a l s o have t h e f o l l o w i n g f a i r l y e v i d e n t p r o p o s i t i o n . nilpotent. P r o p o s i t i o n 2.13. connected f i b r e . Proof. Q = N r iN , Note t h a t t h i s d e f i n i t i o n a g r e e s w i t h t h a t g i v e n i n 1.4 i n t h e case i n which on on -1 -1 b 1 , x C Q, a € rlE, and Let f: E + B Then, if E , B be a map of connected CW-complexes with are nilpotent, f i s nilpotent. We have a n e x a c t sequence of groups w i t h Q-action, F = f i b r e of f, where is Localizationof homotopy types 68 ... + IT n+l B - +IT F + n E+ n IT ... + s B 2 -+ TI F + 1 E+ 1 TI IT B. 1 are nilpotent, slE operates nilpotently on nnB, TI E n for all n 2 1. Thus, if n 5_ 2 , the conclusion that IT E operates nilpotently 1 on n F follows from Proposition 1.4.3, since our definition of nilpotent Now since E, B action coincides, in the case of a commutative group, with that of 1.4. Thus the case n = 1 remains. We have an exact sequence IT B-+n F+TIE 2 1 1 of Q-groups and the argument of Proposition 1.4.3 may be adapted to yield the result in this case in view of the fact that the image of n2B lies in the center of IT F. 1 It is, of course, necessary to take account of both facts noted after the definition of a nilpotent Q-action. The relative form of Corollary 2.12 reads Theorem 2.14. Let f: E + B be a map of connected CW-complexes inducing a surjection of fundamental gruups. Then f i s niZpotent i f and only i f i t s Moore-Postnikou system admits a principal refinement. be the Moore-Postnikov system of f . Now if pn may be factored as in ( 2 . 8 ) , we obtain a sequence of extensions of n E-modules Gi>where Gi nYi IT - 1 nnYi-l, i * 1, is a trivial module. Now the fibre of an Eilenberg-MacLane space K(Hi,n) an extension of nlE-modules ..., c, is i = ql’”qi: Yi Y and the relation siqi+l = si+l yields s -+ 69 Nilpotent spaces where Ho = {O}, Hc = ~ l ~ = + n~ F. p ~ It follows from (2.15) and Proposition 1.4.3, by an easy induction, that TI F is a nilpotent n ~l 1E-module. (The case n = 1 is again slightly special, but we will omit the details in this case.) The converse implication is proved exactly as in the absolute case; see the Proof of Theorem 2 . 9 . Before proceeding to discuss how to intr0duce.a localization theory Here we confine into NH we show how Serre's C-theory may be applied to NH. attention to the absolute case, since the relative case requires stronger axioms on a Serre class, as is already familiar in the classical case of the Thus we will be considering generalized Serre classes in the category H1. sense of Definition 1.5.1. We prove the one basic theorem which we need in the sequel. Theorem 2.16. Let X E NHand l e t be a generalized Serre class. C Then the following assertions are equivalent: (il T I ~ Xf C for a l l (ii) HnX E C f o r all n (iii) nlx c cover of 1 n Z c and H ~ Xc c 1 for a l l n 2 1, where X i s the u n i v e r s a ~ X. Proof. We need two lemmas, which are interesting in their own right. The first is a generalization (to general m?l) of Theorem 1.4.17, though we here only state the result for homology with integer coefficients. Lemma 2.17. If ~l acts n i l p o t m t t y on the abelian group A, then n acts nilpotently on Hn(A,m), n 1 0 . Proof. n-series of A Let 0 = rC+'A 5 rCA 5 ... 5 I-1A = A (see Section 1.41, and write Ai = r iA be the lower central for convenience, Note that each Ai is a nilpotent a-module, of class less than that of A Localization of homotopy types 70 if i 2 2. Moreover, a a c t s t r i v i a l l y on Ai/Ai+l. We have a s p e c t r a l sequence of a-modules, converging ( f i n i t e l y ) t o t h e graded group a s s o c i a t e d w i t h filtered. I f w e assume i n d u c t i v e l y t h a t i t o p e r a t e s n i l p o t e n t l y on n i l p o t e n t l y on Hn(%) where K(nmX,m) + that TI x E2 whence i t r e a d i l y f o l l o w s t h a t P4' X € NH and Zet a = nlX. i s the universaZ cover of Proof. o p e r a t e s n i l p o t e n t l y on operates X. m Z 2 , where X 1 = 0. %. We o p e r a t e s n i l p o t e n t l y on t h e homology of have a f i b r a t i o n Thus we may suppose i n d u c t i v e l y o p e r a t e s n i l p o t e n t l y on t h e homology of IT Hq(Ai+l,m), Then a operates n i l p o t e n t l y on Consider t h e Postnikov system of + Xm-l, IT suitably completing t h e i n d u c t i v e s t e p . Hn(Ai,m), Let Lemma 2.18. a Hn(Ai,m), %m-l and, by Lemma 2.17, K(nmX,m). We ?iave a s p e c t r a l sequence of n-modules converging ( f i n i t e l y ) t o t h e graded group a s s o c i a t e d w i t h filtered. We s e e immediately t h a t i t r e a d i l y follows t h a t the inductive step. IT Sihce a o p e r a t e s n i l p o t e n t l y on o p e r a t e s n i l p o t e n t l y on k + Hnk, imi s m-connected, Hnim. suitably EL P4' whence This completes the c o n c l u s i o n of t h e lemma f o l l o w s . We now r e t u r n t o t h e proof of Theorem 2.16. (i)0 (iii) is c l a s s i c a l , s i n c e t h e a b e l i a n groups i n c l a s s i n t h e o r i g i n a l sense. (ii) 0 (iii). Of c o u r s e , t h e e q u i v a l e n c e c constitute a Serre Thus we may complete t h e proof by showing t h a t For t h i s we invoke t h e s p e c t r a l sequence of t h e covering I n t h i s s p e c t r a l sequence we have k* X. Nilpotent spaces 71 and t h e s p e c t r a l sequence converges t o t h e graded g oup a s o c i a ed w i t h a ( f i n i t e ) f i l t r a t i o n of HnX. By Lemma 2 . 1 8 and Theorem 1 . 5 . 6 Assume, t h e n , t h a t ( i i i ) h o l d s . EL C C u n l e s s p + q = 0. I t t h e r e f o r e q u i c k l y follows t h a t P4 n 2 1. Assume now, c o n v e r s e l y , t h a t ( i i ) h o l d s . By P r o p o s i t i o n we i n f e r t h a t c, H X f 1.5.2 we know t h a t TI = II X C C. 1 ( i f s u c h e x i s t s ) such t h a t E2 infer that f Pq c H q q c s if q = s 2 2 Let 2 fC . b e t h e s m a l l e s t v a l u e of By Lemma 2.18 and Theorem 1.5.6 we (unless p + q = 0) 2 and t h a t Eos f C. Consider t h e diagram, e x t r a c t e d from t h e s p e c t r a l sequence, I Es+l s+l,O Then, by t h e axioms of a S e r r e class, each of c , while belongs t o 3 EoS, ..., Eoss+l, EEs E2 0s r' c. 2 E2,s-l, 3 ..., Es+l s+l ,0 E3,s-2, We t h u s deduce, s u c c e s s i v e l y , t h a t do n o t belong t o C. But E:s i s a subgroup of HsX, which b e l o n g s t o C , s o t h a t w e have a r r i v e d a t a c o n t r a d i c t i o n . Theorem 2.16 w i l l , i n p a r t i c u l a r , be a p p l i e d i n t h e s e q u e l t o t h e c a s e i n which Remark. (2.19) c i s t h e class of f i n i t e l y g e n e r a t e d n i l p o t e n t groups. It is e a s y t o see t h a t t h e converse of Lemma 2.18 h o l d s . Let X be a connected CW-complez. Then X f NH i f nlX That i s , is nilpotent and operates nilpotently on the homology groups of the universal cover of X. However, no use w i l l b e made of (2.19) i n t h e s e q u e l . q Localization of homotopy types 12 3. Localization of nilpotent complexes. In this section we extend Theorems 1A and 1B from the category H1 to the category NH. To do so we need, of course, to have the notion of the localization of nilpotent groups, which was developed in Chapter I. We are thus able to make the following definition. Let X ENH. Then X Definition 3.1. all n 2 1. A map X is P-local for n in Ni P-localizes if Y is P-local and f: X + Y is P-local if TI f*: [Y,Z] s [ X , Z ] for all P-local in NH. Then the main theorems of this section extend the enunciations of Theorems lA, 1B from H1 to NH. Theorem 3A (First fundamental theorem in NH.)EVery X in NH admits a P-localization. Theorem 3B (Second fundamental theorem in NH.) Let f: X -f Y in NH. Then the following statements are equivalent: li) f P-localizes X; (iil vnf: snx+nnY (iii) Hnf: HnX +. P-localizes f o r all n P 1; HnY P-localizes for all n P 1. The pattern of proof of these theorems will closely resemble that of Theorems lA, 1B. However, an important difference is that the construction of localization in NH does not proceed cellularly, as in the 1-connected case, but via a principal refinement of the Postnikov system. We first prove that universal covers of X, Y so (ii) =3 (iii) in Theorem 3 B . that we have a diagram Let X, i! be the Localization of nilpotent complexes 2 73 -X K(nlX,l) Y Jfl K(slY,l) Ii-If- (3.2) Y Since induces localization in homotopy, it induces localization in homology by Theorem 1B. Moreover, we obtain from ( 3 . 2 ) a map of spectral sequences 2 which i s , at the E -level, (3.3) By Lemma 2.18 n X operates nilpotently on H and alY operates nilpotently 1 q on H 9 . We thus infer from Theorem 1 . 4 . 1 2 , together with Theorem 1 . 2 . 9 q if q = 0, that ( 3 . 3 ) is localization unless p = q = 0. Passing through the spectral sequences and the appropriate filtrations of HnR, Hna, we infer that Hnf localizes if n 2 1. Now let ( i ' )be the statement: f*: [Y,Z] Z in Zi [X,Z] f o r aZZ P-zOCUZ NH. - Note that this statement differs from (i) only in not requiring that Y be P-local. We prove that (iii) (ii)=a (i'). This will, of course, imply that (0. If Z i s P-local nilpotent, then we may find a principal refinement of its Postnikov system. Moreover this principal refinement may be chosen that the fibre at each stage is a space K(A,n), where A so is P-local abelian. For, as we saw i n the proof of Theorem 2 . 9 , we may take A = riITnZ/ri+'anZ for some i , and we know (Theorem 1 . 2 . 7 ) that P-local. Given g: X + r iB i s P-local if B is Z, the obstructions to the existence and uniqueness of a counterimage to g under f* will thus lie in the groups H*(f;A) and, as in the corresponding argument in the 1-connected case (note that we have trivial coefficients here, too), these groups will vanish if f induces P-localization in homology. Localization of honiotopy types 14 Next we proceed to prove Theorem 3A, via a key observation playing the role of Proposition 1.3. Proposition 3 . 4 . Let U be a f u l l subcategory of have constructed f: X -+ Y s a t i s f y i n g (ii). Then t h e assignment automatically y i e l d s a functor L: U transformation from the embedding diagram -+ X r+ X we Y NH,f o r which f provides a natural U LNH t o Proof of Proposition 3 . 4 . L. Let g : X + X' in U. We thus have a If If in NH,where x X' Y Y' f, f' satisfy (ii). ,fi (3.5) satisfies (i) and Y' Then f P-local, so that there exists a unique h commutes. NH, f o r whose o b j e c t s Y is in NH such that the diagram If' Y' It i s now plain that the assignment X I + Y, g * h yields the desired functor L. We now exploit Proposition 3 . 4 to prove Theorem 3A. consider spaces X in We first NH yielding a f i n i t e refined principal Postnikov system and, for those, we argue by induction on the height of the system. Thus we may assume that we have a principal (induced) fibration where G is abelian even if n = 1, and we may suppose that we have constructed f ' : X ' + Y' satisfying (ii). (The induction starts with X ' = 0.) Since Localization of nilpotent complexes 75 ( 3 . 6 ) i s induced, we may, i n f a c t , assume a f i b r a t i o n - X Now we may c e r t a i n l y l o c a l i z e i s t h e l o c a l i z a t i o n of X' K(G,n+l); we o b t a i n - X' be t h e f i b r e of K(Gp,n+l), where Gp K(G,n+l) If'- Y' Y K(G,n+l) and s o , by P r o p o s i t i o n 3 . 4 , w e have a diagram G x Let -& h K(Gp,n+l) There i s then a map h. f: X -+ Y rendering t h e diagram X --+ X' K(G,n+l) Y Y' K(Gp,n+l) 4f -4f' A commutative i n NH and a s t r a i g h t f o r w a r d a p p l i c a t i o n of t h e exact homotopy sequence shows t h a t f satisfies ( i i ) . It remains t o consider t h e case i n which t h e r e f i n e d p r i n c i p a l Postnikov system of has i n f i n i t e height ( t h i s i s , of course,the ' g e n e r a l ' c a s e ! ) . X Thus we have p r i n c i p a l f i b r a t i o n s ... (3.7) - g Xi 4-XiWl and t h e r e i s a weak homotopy equivalence - ... X * 0 Lim Xi. Now w e may apply t h e reasoning already given t o embed ( 3 . 7 ) i n t h e diagram, commutative i n NH, ... -xi gi. I - ... 0 (3.8) where each fi satisfies (ii). Moreover, w e may suppose t h a t each hi is Localization of hornotopy types 16 a f i b r e map. of @Yi. Let Y be t h e geometric r e a l i z a t i o n of the s i n g u l a r complex Then t h e r e is a map is homotopy-commutative. f: X -+ such t h a t t h e diagram Y Moreover, t h e construction of (3.8) shows t h a t t h e Y -sequence is again a r e f i n e d p r i n c i p a l Postnikov system, from which i t i r e a d i l y follows t h a t is i n NH. @ fi satisfies (ii). So t h e r e f o r e does f , and f Thus we have completed t h e proof of Theorem 3A i n t h e s t r o n g e r form t h a t t h e r e e x i s t s , f o r each The proof t h a t (i) t h e category H1. X =) i n NH, a map f: X -+ Y in NH s a t i s f y i n g (ii). ( i i ) proceeds exactly a s i n t h e e a s i e r case of Thus we have e s t a b l i s h e d t h e following s e t of i m p l i c a t i o n s , r e l a t i n g t o Theorem 3B: (3.9) (ii) = (iii), ( i i i ) * (if), (ii) =) (i), (i) =a (ii). All t h a t remains is t o prove t h e following p r o p o s i t i o n , f o r then we w i l l be a b l e t o i n f e r t h a t , i n f a c t , (iii) =. (i) Proposition 3.10. is P-local f o r every n 1 1, then n Y If Y C NH and HnY is P-local f o r every n ? . n 1. To prove t h i s , we invoke Dror's theorem, which we, i n f a c t , reprove s i n c e i t follows immediately from (3.9). P - n, where n Thus we consider t h e s p e c i a l case is t h e c o l l e c t i o n of a l l primes. Then a homomorphism of ( n i l p o t e n t , abelian) groups Il-localizes i f and only i f i t i s an isomorphism. Moreover, every space i n NH is II-local, so t h a t , i n t h i s s p e c i a l c a s e , t h e d i s t i n c t i o n between (if) and (i) disappears. t h e equivalence of (ii) and ( i i i ) f o r P = n, Thus (3.9) implies, i n p a r t i c u l a r , which i s Dror's theorem. Localization of nilpotent complexes We construct f: Y Now we prove Proposition 3.10. (ii). I1 It thus also satisfies (iii); but HnY i s P-local, so + Z satisfying that f induces an isomorphism in homology. By Dror's theorem, f induces an isomorphism in homotopy. However, the homotopy of Z i s P-local, so that Proposition 3.10 is proved, and, with it, the proof of Theorems 3A and 3B is complete. Remark. Of course, we do not need the elaborate machinery assembled in this section to prove Dror's theorem. In particular, Theorem 3A is banal for P - IT, since, then, the identity X -r X n-localizes! The fact that we have both the homotopy criterion (ii) and the f homology criterion (iii) of Theorem 3B for detecting the localizing map enables us to derive some immediate conclusions. For example we may use (ii) to prove Theorem 3.11. If X i s nilpotent and W connected f i n i t e a d i f localizes, then fw : (Xw ,g) (Yw,fg) localizes, where W and (X ,g) i s the component of xW containing g. +- f: X -+ Y w x g i s any map -t Proof. We argue just as inthe proofs of Theorem 2.5 and Corollary 2.6, using Theorem 3.12 below. A similar result holds for W Xfr (Roitberg [ 6 9 ] ) ; thus we may remove the condition that W be connected in the theorem. We also note that - the theorem implies that H(Fp) = E(F)p - where F € NH is finite and H is the identity component of the space of (free or pointed) self-homotopy-equivalences. Theorem 3.12. Let F +- E + B be a f i b r e sequence i n NH. Then Fp + Ep -+ is a f i b r e sequence i n tti. Theorem 3.13. Then % + Yp -+ Let X + Y -+ C be a cofibre sequence i n NH. With c Cp is a cofibre sequence i n NH. These two theorems are proved exactly in the manner of their counterparts in H~ (Corollaries 1.10, 1.11). Our reason for H1- Bp Localization of homotopy types I8 imposing i n Theorem 3.13 t h e condition C proof t h a t , i n general, t h e c o f i b r e of 5 If 7 i s t h a t w e have given no E H1 -t is necessarily nilpotent. Yp were t h i s c o f i b r e , we would, of course, have a homology equivalence H1 k k e w i s e Theorens1.13, 1.14, and 1.16 extend from t o NH; we w i l l f e e l f r e e t o quote them i n t h e sequel in t h i s extended context. Given k X C NH l e t component of t h e loop space of be t h e supension of a l l belong t o X. b e the u n i v e r s a l cover of X X, l e t ZX be t h e containing t h e constant loop, and l e t k , PX It i s , of course, t r i v i a l t h a t NH ( f o r t h i s we do not even need t h a t X and CX EX i t s e l f be n i l p o t e n t ! ) . We then have Theorem 3.14. (i) N ($) ru (k)p; (ii) E ( X p ) = Proof. To prove (i) that B we l i f t e: X 3 (zX)p; ( i i i j to Xp E: s a t i s f i e s c r i t e r i o n (ii) of Theorem 3B (or 1B). follow immediately from Theorems 3.12, 3.13 r e s p e c t i v e l y . k Z(%) 3 rr/ 3 (ZX),. and observe (X,) P a r t s (ii) and (iii) Notice t h a t Theorem 3.14(i) has t h e following g e n e r a l i z a t i o n ; r e c a l l t h a t l o c a l i z a t i o n preserves subgroups (Theorem 1.2.4). Theorem 3.15. Let covering space o f of 3 X E NH and Let be a subgroup of X corresponding t o Q and l e t corresponding t o Q,. P-ZocaZizes. Q Then e : X + 5 2 nlX. Let Y be the be the covering space l i f t s to e: Y -+ Z which Quasifinite nilpotent spaces 19 4. Quasifinite nilpotent spaces. In this short section we present a result which will enable us to prove an important modification of the main theorem (The Pullback Theorem) of is of f i n i t e type if anX is finitely Section 5. Let X € hkl. We say that X generated for all n 2 1 and that X is q u a s i f i n i t e if X is of finite type and moreover H X = {O} for n and H X = {O} N, we will say that X has homological dimension for n and may write dim X (iil x 3-l N. i s of f i n i t e type; H X i s f i n i t e Z y generated f o r n (iii) X is quasifinite X € NH. Then the following statements are equivalent: Theorem 4.1. Let (i) 5 sufficiently large. If X N Y, where Y n 1 1; i s a CW-complex with f i n i t e skeleta. Proof. The equivalence of (i) and (ii) follows from Theorem 2.16. That (iii) implies (ii) is trivial. We prove that (i) implies (iii). Since nlX is finitely-generatednilpotent, the integral group ring Z[alX] is noetherian. Moreover, if x Is the universal cover of X, Hi? is certainly finitely-generated over Z [ n X I , being, in fact, finitely-generated as abelian 1 group. Thus (iii) follows from Wall's Theorem (p. 61 of [ S S ] ) . From Theorem 4.1 we deduce the result in which we will be interested in the next section. Theorem 4.2. Let f: x -+ x X 6 NH. Then X i s q u a s i f i n i t e i f f there e x i s t s a map o f a f i n i t e CW-complex i n t o X inducing isomorphisms i n homology. Proof. It is obvious (in the light of the equivalence of (i) and (ii) in Theorem 4.1) that the existence of such a map quasifinite. Suppose conversely that X f implies that X is is quasifinite. By Theorem 4.1 we Localization of homotopy types 80 may assume that each skeleton of X is finite. If dim X 5 N, we will show that we can attach a finite number of (N+l)-cells to XN to obtain a finite complex X such that the inclusion XN 5 X extends to f: .+ X inducing - x homology isomorphisms. We have a diagram where the vertical arrows are Hurewicz homomorphisms. Now %+l(X,X N) , as a N subgroup of H# , is free abelian and finitely-generated. Let B be a basis N N for s+l(X,X ) and let be a subset of K~+~(X,X) mapped by h bijectively to B. Attach (N+l)-cells to XN by maps in the classes ab, b € B, to form X. It is then obvious that the inclusion XN 5 X‘ extends to a map X -+ X. - Let f: x + X be any such extension. N It is plain that f induces an isomorphism s + p , x ) It follows almost immediately that %+lX isomorphism HN”; % €$X. Corollary 4 . 3 . 4.2. Let - = ?2 5 N + p , x 1. {O}, and that f induces an This completes the proof of the theorem. X € NH be q u a s i f i n i t e and l e t f: x .+ X be a s i n Theorem Then f*: [X,Y] for all Y E [X,Y] NH. -.Proof. Construct a principal refinement ... -Yi & Yi-l - * *. of the Postnikov tower of Y. Then, if the fibre of gi is K(Gi,ni), nil 1, the pbstructiomto the existence and uniqueness of a counterimage, Quasifinite nilpotent spaces under i = f*, of an arbitrary element of 1, 2, ..., r = ni + 1 or ni. these cohomology groups all vanish. [x,Y] Since f will all be in Hr(f;Gi), induces homology isomorphisms, 81 Localization of homotopy types 82 5. The Main (Pullback) Theorem. We will denote by X the p-localization of the nilpotent CW-complex P X; by e the canonical map X + X where p E II, or p = 0; by r : X X P P) P P 0 the rationalization, p E n , and by can ('canonical map') the function -+ [W,Xp] [W,Xo] induced by -+ r P . We also denote by g P the p-localization of a map g. Theorem 5.1. (The Pullback Theorem). and Let W be a connected f i n i t e CW-complex X a n i l p o t e n t CW-complex of f i n i t e type. pullback of the diagram of s e t s {[W,Xpl Then the 3et [w,x0i IP E [W,X] i s the ni. It will follow that, under the conditions stated, X is determined r by the family {X a Xolp E n). Indeed, X is the unique object in the P homotopy category of connected CW-coriplexes which represents the functor {[W,X 3 + [W,X ]Ip E II} from the category of connected finite P CW-complexes to the category of pointed sets. W I+ pullback Our main theorem also implies, in the light of Corollary 4 . 3 , that, for X as in Theorem 5.1 and W now quasifinite nilpotent, a map g: W+ X is completely determined (up to homotopy) by the family of its p-localizations {gplp E rI), and, conversely, a family of maps a unique homotopy class g: W the maps g(p) -+ rationalize to X with a {g(p): X Ip E n) determines P for all p, provided that We -t g N g(p) P common homotopy class not depending on p. Therefore the situation is analogous to that in the theory of localization of finitely-generated nilpotent groups (Theorem 1.3.6). Indeed, this algebraic fact provides one with an easy proof of Theorem 5.1 in case W or X is a suspension a loop space, in view of Theorem 3B. The method we use to prove Theorem 5.1 is the localization of function spaces (Theorem 3.11), which enables us to prove the result by induction on the number of cells of the CW-complex W. The main (pullback) theoreni Definition 5.2. g,: g: X A map -f Y 83 i n NH i s an F-monomorphism i f i s i n j e c t i v e f o r a l l connected f i n i t e CW-complexes [W,X]+ [W,Y] 2: X W. IIX t h e map with components { e p l p C rI]. We P prove one h a l f of Theorem 5.1, b u t , f o r t h i s h a l f , remove a f i n i t e n e s s Denote by r e s t r i c t i o n on Theorem 5.3. 2: X IIX -+ P + (Compare Theorem 1.3.6.) X. Then t h e canonical map be a n i l p o t e n t CW-complex. X Let is an F-monomorphism. Proof. W e have t o show t h a t f o r an a r b i t r a r y f i n i t e CW-complex [W,X] If W. the cofibration Sn-l + Z l[W,X ] P i s injective i s a f i n i t e wedge of s p h e r e s , W i:W Given 3 W = V U en W , and assume V + W. P Hence we can proceed by induction t h e theorem follows from Theorem 1.3.6. on the number of c e l l s of [WJX -+ -+ X, l e t n 2 2. with We consider g = g l V ; we g e t a f i b r a t i o n , up t o homotopy, ( i n which we e x h i b i t one component of t h e f i b r e ) (x',E) + (xv,g) + p-1 (X ,o), n 2 . 2 , giving r i s e t o a diagram with exact rows Here and l a t e r where h [W,X]g [ , Ih s e r v e s a s basepoint f o r t h i s s e t . 6 i and, by exactness, the o r b i t of which a r e homotopic t o that denotes t h e s e t of (based) homotopy c l a s s e s of maps g Notice t h a t i m $' when r e s t r i c t e d t o g' i m $J g X o p e r a t e s on c o n s i s t s p r e c i s e l y of t h o s e maps i s i n j e c t i v e , and we have t o show t h a t i n j e c t i v e and s i n c e 71 V. By induction we may assume y-'(Yp) = g. Since a r e t h e i s o t r o p y subgroups d r e s p e c t i v e l y , i t follows t h a t the s e t Y -1 (YE) i, 6 is Ispi) i s i n one-one correspondence Localization of homotopy types 84 with the set ker (coker $ localize their domain and g -+ so, coker J, ) . The components of B clearly all g too, do those of a by Theorem 3.11. Therefore the cokernel of J, splits into a product of p-local groups and the map g coker 0 + coker J, has components which p-localize. Hence g ker (coker 0 g required. + coker J, ) = I01 by Theorem 1.3.6, and y -1 g (yp) = 9, as Notice that no finiteness conditions on X were needed for this argument. But if the space X is of finite type then, by following the lines of the proof of Theorem 5.3,we obtain the following corollary. Corollary 5.4. Suppose W is a connected f i n i t e CW-complex and X a nilpotent CW-complex of f i n i t e type. Let S 5 T denote s e t s of primes. Then: a) The canonical map [W,XT] b ) The canonical map f i n i t e l y many primes c) map + [W,Xs] is finite-to-one. [W,Xp] + [W,Xo] i s one-one f o r a l l but p. There e x i s t s a c o f i n i t e s e t of primes Q such that the canonicaZ [W,XQl -+ [W,X I i s one-one. Notice also that we may replace the partition of I7 into singleton sets of primes, in the enunciation of Theorem 5.3, by any partition of lT. We now illustrate, by means of an example, the fact that, even when A X is a sphere, the map X A n X is not a monomorphism in the homotopy P category of a l l CW-complexes . Proposition 5.5. Let W = (Si V S;)UAen+l non-empty complementq s e t s of primes, and Then there is an essential map primes p. K: w + sn+' where n 1. 2, R and A = (1,l) C nn(S; such t h a t K. P T are v $j = o for a l l The main (pullback) theorem Proof. Let W K: + Sn+' 85 be the collapsing map and consider the Puppe sequence Then, for all primes p, E X (CS;)p or wP "4. sP n But, were K cA has a left homotopy-inverse since either P is homotopy equivalent to S*l. From the cofibration P = 0 for all p. P = 0, this would imply that ZX had a left homotopy-inverse and + 1 4 (CS; V Cs;Ip we conclude that K. hence, by taking homology, Z would be a direct summand of ZR @ZT which rISn+l i s not a monomorphism. P We now complete the proof of Theorem 5 . 1 . i s absurd. Thus Theorem 5 . 6 . Let Sn+' -+ W be a connected f i n i t e CW-complex and X a nilpotent CW-complex of f i n i t e type. p C I'l U {O], such that i s the canonical map. that e g P = g(p) Proof. Suppose given a f a m i l y of maps g(p): W -+ xP' n. g(0) f o r a21 p f Il where r : X + X P P P 0 Then there i s a unique homotopy class g: W - + X such r g(p) for all p. Uniqueness has already been proved in Theorem 5.3, have only to prove the existence of g. If W then the theorem follows from Theorem 1.3.6. so we is a finite wedge of spheres, Hence we proceed again by induction. Let W = VUXen, n 1 2 , and assume that we have constructed g': V such that epg' = g(p) IV for all p. such an extension exists since by Theorem 5.3. (g'k)p Let i:W +X be an extension of g'; 0 for all p and hence g'A = 0 Now consider the diagram + X Localization of homotopy types 86 7 U {O) p C l For each a(p) * t h e r e i s a unique * epp = g ( p ) , t h e on t h e set 0 -1 (epg'). d e n o t i n g t h e f a i t h f u l a c t i o n of Note t h a t used t o prove Theorem 5 . 3 . action x a ( p ) C coker $ g l ( p ) coker $ , ( p ) e' (coker $ g g Further, since eog(p) is f a i t h f u l , i t f o l l o w s t h a t each C= g(0) ,p such t h a t coker $ (p) by t h e argument P C g' n, and the n, r a t i o n a l i z e s a(p), p C to coker $ i s f i n i t e l y g e n e r a t e d , i t f o l l o w s from Theorem 1.3.6 g t h a t t h e family {o(p)} C n(coker $ ) d e t e r m i n e s a unique element g' P a C coker $ which p - l o c a l i z e s t o a ( p ) f o r a l l p . By n a t u r a l i t y w e g' a(O), Since conclude t h a t a x h a s t h e p r o p e r t i e s r e q u i r e d of g. P u t t i n g t o g e t h e r Theorems 5 . 3 and 5.6 w e o b t a i n our main r e s u l t , Theorem 5.1. One can, of c o u r s e , g e n e r a l i z e Theorem 5.1, w i t h o u t changing a n y t h i n g e s s e n t i a l i n i t s p r o o f , t o t h e case of a n a r b i t r a r y p a r t i t i o n of mutually d i s j o i n t famil ies Pi Il i n t o of primes. I n o r d e r t o deduce t h e e x i s t e n c e of certain g l o b a l s t r u c t u r e s on o u t of given s t r u c t u r e s on t h e X Is, as w e w i l l w i s h t o do i n Chapter 111, P i t i s p a r t i c u l a r l y u s e f u l t o know how t o c o n s t r u c t i n a " t o p o l o g i c a l " way. We w i l l d e n o t e by s i n g u l a r complex of . map, and p: Xo w i l l assume t h a t -r l7X P by Exp r: X o u t of t h e maps xp +. xo t h e geometric r e a l i z a t i o n of the EXP - + ~ X p ) ot,h e rationalization EX l o c a l i z e d a t 0. We P are f i b r a t i o n s (without changing t h e n o t a t i o n ) , t h e c a n o n i c a l map r p by a l t e r i n g t h e domains of Theorem 5 . 7 . There are maps &p)o, and X r and p X + in t h e u s u a l way. Suppose X i s a nilpotent CW-complex of f i n i t e type. 51 the topologiaal pullback of Xo ex ) PO &EX P Denote , Then t h e canonical The main (pullback) theorem map X + x a7 is a homotopy equivaZence. Proof. Consider t h e p u l l b a c k - square The "Mayer-Vietoris" sequence i n homotopy g i v e s a n e x a c t sequence ... r n iTi (5.8) where @ ... - n p xiX - - <px,-r*> nixo nn.x 1 P mnix)o (n X ) x (rrn X ) l o 1 P i s f i n i t e l y generated. The maps .1 - n,rr nn are a l l p u l l b a c k diagrams. a g a i n by Theorem 1.3.7. x mnlxp)o, r* <p*,-r*>, i 2 2, defined f o r n x io i l l (Trn.x ) I P O i P But s o are t h e diagrams The map X + TI which i s t h e i d e n t i t y on t h r e e c o r n e r s . w X i P* Hence i t f o l l o w s from (5.8) t h a t t h e are a l l s u r j e c t i v e by Theorem 1 . 3 . 7 . diagrams - ... i n d u c e s a map of p u l l b a c k diagrams I t t h u s i n d u c e s isomorphisms &? TT X, and s o is a homotopy e q u i v a l e n c e . i Of c o u r s e t h e r e i s a l s o a form of Theorem 5.7, mutatis mutandis, for an a r b i t r a r y p a r t i t i o n of II into m u t u a l l y d i s j o i n t f a m i l i e s of primes. I f t h e p a r t i t i o n i s f i n i t e , i t i s e a s y t o see t h a t we need no l o n g e r i n s i s t Localization of homotopy types 88 that be of f i n i t e t y p e . X Theorem 1.3.7, W e may a l s o e x p l o i t Theorem 1.3.9 i n s t e a d of ll t h a t we a r e concerned w i t h t h e c a s e of a p a r t i t i o n of so i n t o two d i s j o i n t s u b s e t s . Theorem 5.9. Let partition of n. n be a nilpotent CW-complex and l e t X Denote by rp: Xp -+ Xo rO: X a d . canonical m a p s , which we assume t o be fibrations. equivalent t o the topological pullback of X Q -+ = P U Q 0 be a the Then X i s homotopy rp and rQ' The proof e x p l o i t s Theorem 1 . 3 . 9 j u s t a s t h e proof of Theorem 5.7 e x p l o i t e d Theorem 1.3.7. W e omit t h e d e t a i l s . Often one reduces a problem i n v o l v i n g i n f i n i t e l y many primes t o one i n v o l v i n g o n l y f i n i t e l y many by means of a c o r o l l a r y which is i n some s e n s e T h i s c o r o l l a r y f o l l o w s a t once by u s i n g t h e diagram d u a l t o C o r o l l a r y 5.4. X used i n t h e proof of Theorem 5.6, r e p l a c i n g C o r o l l a r y 5.10. by Suppose W i s a connected f i n i t e nilpotent CW-complex of f i n i t e type. Given a map a ) For a l l but a f i n i t e nwnber of primes There e x i s t s a c o f i n i t e s e t of primes b) f € QI im([W,X + X Q and X P by CW-complex and f: W + Xo, X a then: p , f E im([W,X ] P Q Xo. -+ [W,Xo]) such t h a t [W,Xol). Combining t h i s w i t h C o r o l l a r y 5.4 w e g e t and f be as i n Corollary 5.10. e x i s t s a c o f i n i t e s e t of primes Q such that C o r o l l a r y 5.11. where w g: -+ Let xQ, I n case W, X and rQ: xQ -+ X W f Then there factors uniquely as f - i s the canonical map. i t s e l f is n i l p o t e n t , we can r e f o r m u l a t e Theorems 5 . 3 and 5.6 u s i n g t h e u n i v e r s a l p r o p e r t y of l o c a l i z a t i o n , namely, t h e f a c t t h a t e : W P -+ W P induces a b i j e c t i o n e*: [W ,X ] P P P + [W,Xp]. W e get rQg The main (pullback)theorem Let W be a nilpotent f i n i t e CW-complex and X an arbitrary Corollary 5.12. nilpotent CW-compZex. Given t u o maps g, h: W i f gp hp f o r aZZ primes n. 89 + X, then g n. h i f and o n l y p. This is immediate from Theorem 5.3. In case h = 0 this answers a conjecture of Mimura-Nishida-Toda [ 5 3 ] affirmatively. From Theorem 5 . 6 we get Let Corollary 5.13. W be a nilpotent f i n i t e CW-complex and CW-compZex o f f i n i t e type. such t h a t cZass g: g(p), w e x g(p'), Given m y f m i Z y o f maps f o r aZI p, p' c n, {g(p): a niZpotent X Wp -+ n) Xplp € there is a unique homotopy g n. g(p) f o r a11 p. P However, we may further improve on Corollaries 5.12, 5.13 by exploiting -+ Corollary 4 . 3 . with For, according to that result, if W - f*: [W,X] 2 [W,X], where f: w -+ W is quasifinite, then is a map of a finite CW-complex W Thus Theorems 5.3, 5.6 remain valid if the assumption that W replaced by the assumption that W is quasifinite (nilpotent). to W. is finite be Thus we conclude Theorem 5.14. The conczusions of Corozlaries 5.12, 5.13 remain valid, i f i s supposed q u a s i f i n i t e instead of f i n i t e . W Localization of homotopy types 90 6. Localizing H-spaces I n t h i s s e c t i o n we prove a theorem which w i l l be c r u c i a l i n our study of t h e genus of an H-space i n 111.1, and which provides a n a t u r a l analog of t h e b a s i c recognition p r i n c i p l e i n t h e l o c a l i z a t i o n theory of n i l p o t e n t groups. X Let be a connected H-space. so may be l o c a l i z e d . -+ Xp i s an H-map. Then, f o r any CW-complex For any monoid M and any element x in M x € M, and we w i l l f o r such an nth power, even though t h e r e i s , i n general, no unique n t h power. It i s thus c l e a r what we should understand by t h e claim t h a t a homomorphism $: M Theorem 6 . 2 . The map -+ f: X e, let P-local rmd f,: [W,X] W. Then CW-complexes Proof. W of monoids i s P - i n j e c t i v e (P-surjective, N Conversely, true i f property of W, t h e induced map we may, in an obvious way, speak of an n t h power of xn i s n i l p o t e n t and may be endowed with an H-space s t r u c t u r e such t h a t Xp i s a homomorphism of monoids? write X Moreover, i t i s p l a i n , from t h e u n i v e r s a l l o c a l i z a t i o n , t h a t each e: X Then c e r t a i n l y -+ (6.1) i s f i n i t e connected. be an H-map of connected spaces such that Y -+ i s P - b i j e c t i v e if W P-bijective). [W,Y] f We prove i s P - b i j e c t i v e f o r a l l f i n i t e connected P-localizes. e, (6.1) P-bijective. This a s s e r t i o n i s c l e a r l y is 1-dimensional, by t h e Fundamental Theorem of Chapter I. t h e r e f o r e argue by induction on t h e number of c e l l s of of Theorem 5 . 1 ) . We assume is Y W We (compare t h e proof W = V U en, n 2 2 , and t h a t w e have a l r e a d y proved t h a t e,: P,XI * [V,%l is P - b i j e c t i v e f o r a l l connected H-spaces X. W e consider t h e diagram (of monoid-homomorphisms) *By monoid, we understand a s e t endowed with a m u l t i p l i c a t i o n with two-sided unity Localizing H-spaces We prove e*: [W,X] + [W,Xp] e*ix = 1, s o Then $pexa = e*$a = 1. ix" = 1, f o r some f o r some respect t o flX m 1 so t h a t and is P-injective. $Jcm2 f o r some e*: Then ipym = e*a that jam' = 1 f o r some Thus yml with e*x = 1. Thus We conclude t h a t QXP. e*a ml = e*$c, m m = W e now prove ipe,x m € P'. x € [W,X] h e r e we invoke t h e i n d u c t i v e h y p o t h e s i s w i t h and i t s l o c a l i z a t i o n m m a 1 2 and € P'; Thus l e t xm = $ a , a C TI X , and m e*a = Jipb, b € [ZV,$,], and b = e*c, It f o l l o w s t h a t c f [CV,X] e* P-injective. 91 f o r some =' : ,e m [w,x] -+ [W,%] P-surjective. a € [V,X], m C P I . 1 = ipym' = (e*x).($pb), m2 6 P' , whence f i n a l l y C P'. am1 Now = i x , x C [W,X], mm 1-power of ( f o r a s u i t a b l y bracketed b C vnXp. bm2 = e*c, f o r some Thus, by Lemma 6 . 4 below, ynrmlm2= ek(xm2.$c) [W,%l. y € Let = 1 e,ja = j Pe*a = 1, so Thus It f o l l o w s t h a t (xm)m1m2 = $ a c < y). unX, m2 C P ' . and t h e a s s e r t i o n i s proved. (Note t h a t , i n t h e s e arguments, w e have w r i t t e n a l l monoid s t r u c t u r e s i n ( 6 . 3 ) multiplicatively. ) The converse i s t r i v i a l . f,: II X -+TI n 11. Thus For i f we s e t W = Sn t h e n we know t h a t i s a P-isomorphism t o a P - l o c a l group and hence P - l o c a l i z e s , Y f P - l o c a l i z e s by Theorem 3B. The f o l l o w i n g lemma, t h e n , s u p p l i e s t h e one m i s s i n g s t e p . Lemma 6 . 4 . Let X attaching the cone W * ZA induces be an H-space and l e t CA to V W = V by means of a map U CA, t h e space obtained b y A -t V. The p r o j e c t i o n Lacalization of homotopy types 92 4 i s central in the monoid and the image of Proof. Let c: W + [w,x] in the strong sense that W V CA be the cooperation map, In the terminology of Eckmann-Hilton. Then, for x € [W,X], a € [ZA,X], we have x.$a = c*<x.a>, where <x,a> € [WVCA,X] Thus - [W,X] (xl.$al) (x,.4ia2) c*<x1x2,a1a2> = x1x2. 4 (ala2> [ZA,X]. X = (c*<xl,al>)(c*<x2,a2>) = c*(<xl,a1><x2,a2>) = . Remark. Note that we could have proved that e* the following stronger sense, namely, that if e,x xn = yn for some nth power with n € P'. in (6.1) is P-injective in = e*y, x , y € [W,X], then In the presence of a nilpotency condition, this sense of P-injectivity in fact coincides with the obvious one (obtained by setting y = 1); and, indeed, it is true, with appropriate definitions, that [W,X] is a nilpotent loop (non-associative group), Corollary 4 . 3 enables us to deduce the following modification of Theorem 6.2. Corollary 6.5. Let CW-complex. X be a connected H-space mrd W a quasifinite Then ZocaZization induces and eg i s P-bijective. The following consequence of this corollary will be very important in the sequel. Theorem 6.6. Let W be a quasifinite CW-compZex and l e t H-space such that Wp 9 Xp. Then there e x i s t s a map X be a connected f: W * X such that Localizing H-spaces fp: wp eY $. Proof. Let g, = ng*: R*(Wp) g: Wp -+ $ n C P' suchthat 6 . 5 there e x i s t s n 93 -+ n,($) gn = e l ( f ) , f : W and t h a t , consequently, isomorphism of homotopy groups found the required map b e a homotopy equivalence. f . (n C Zt). Thus -+ X. n g, g": Wp By Corollary But i t i s c l e a r that is,like '\I $, g*, an so that we have 94 7. Localization of homotopy types Mixing of homotopy t y p e s T h e i d e a o f m i x i n g h o m o t o p y t y p e s g o e s b a c k t o Zabrodsky [ 9 3 ] a n d h a s been e x t e n s i v e l y used t o c o n s t r u c t examples and counterexamples i n homotopy t h e o r y ; see, e . g . [79,93,96] and Chapter 111 of t h i s monograph. It seems t h a t l o c a l i z a t i o n t h e o r y p r o v i d e s t h e r i g h t framework f o r d i s c u s s i n g t h i s i d e a and r e n d e r i n g i t most e a s i l y u s a b l e i n a p p l i c a t i o n s . We b e g i n by d i s c u s s i n g puZZbacks i n homotopy t h e o r y , a t o p i c of some independent i n t e r e s t , i n p a r t i c u l a r w i t h r e s p e c t t o l o c a l i z a t i o n . Given a diagram X If (7.1) Y-B i n t h e category g T of based CW-complexes, we may r e p l a c e and t a k e t h e ( s t r i c t ) p u l l b a c k which we c a l l t h e weak T. by a f i b r e map We o b t a i n a diagram puZZback of (7.1) i n t h e homotopy c a t e g o r y H. known t h a t t h e homotopy t y p e of as a diagram i n in f 2 depends only on t h e diagram ( 7 . 1 ) , i n t e r p r e t e d H, and i s symmetric w i t h r e s p e c t t o we might j u s t as w e l l have r e p l a c e d i n s t e a d of choosing t o r e p l a c e f. I t is g (or both Of c o u r s e , i f f f , g, i n the sense t h a t and f (or g) by a f i b r e map, g) were a l r e a d y a f i b r e map, n o replacement would be n e c e s s a r y . I f (7.1) were a diagram i n To, t h e s u b c a t e g o r y of T c o n s i s t i n g of connected CW-complexes, we would o b t a i n t h e weak p u l l b a c k i n the corresponding homotopy c a t e g o r y base point. Ho by r e p l a c i n g W e would t h u s o b t a i n 2 i n (7.7.) by t h e component Zo of i t s Mixing of homotopy types U x zo 1. If (7.3) in Ho. We are interested in the question of when we may infer that zO is, in fact, in NH. Theorem 7 . 4 . i f W e prove: Suppose t h a t X , Y C NH i n (7.31. Then nlZo operates n i l p o t e n t l y on nnB, n 2 2 , v i a Proof. fu Zo E NH i f and onZy . The diagram ( 7 . 3 ) gives rise to a Mayer-Vietoris sequence of groups with nlZo-action, where Suppose that nlZo G i s the pullback of the diagram of groups operates nilpotently on TInB, n operates nilpotently (via u vo) nlZo operates nilpotently on ? 2. Then, since n lZ o on the homotopy groups of X it follows from Proposition 1 . 4 . 3 that nlZo Now 95 TI 2 and Y, operates nilpotently on nnZo, n ? 2. B and hence on Im n2B C n l Z o . However here the operation is by conjugation and thus the operation of G on Im n2B induced by the exact sequence Im n2B is also nilpotent. -TI Z l o - Since G, as a subgroup of we infer from Proposition 1.4.1 that r1Z0 Conversely, suppose that Zo G IT X 1 x TI Y, is nilpotent, 1 is nilpotent. is nilpotent. Then an immediate application of Proposition 1 . 4 . 3 to the Mayer-Vietoris sequence Localization of homotopy types 96 shows that nlZo operates nilpotently on nnB, n 1 2 . Of course, it is most useful to have a criterion for Z to be nilpotent which is independet of the maps uo, vo, but depends only on (7.1). Thus we now enunciate Corollary 7.6. Let ( 7 . 1 ) be a diagram i n NH. Then, i n the weak pullback (7.3) i n Ho, Zo C NH. The following immediate consequence of Theorem 7.4, generalizing Theorem 2 . 2 , 1s also useful. Corollarv 7.7. Let ( 7 . 1 ) be a diagram i n Ho with X, Y C NH. If X OP Y i s 1-connected, then Zo C N-l. We now suppose that (7.3) i s a weak pullback in N a n d we localize at the family of primes P. We obtain yp Diagram (7.8) i s a weak pullback in NH. Proposition 7.9. Proof. Form the pullback in To (7.9) where we may 8: Zop+ 2' assume fp to be a fibration. There is then a map yielding a commutative diagram Mixing of homotopy types in 91 NH, andhence a map of t h e P - l o c a l i z a t i o n of t h e Mayer-V !to1 3 sequence of (7.3) t o t h e Mayer-Vietoris sequence of (7.9); h e r e Theorem 1.2.10 p l a y s a c r u c i a l r o l e i n e n s u r i n g t h a t , when we l o c a l i z e G i n (7.5) w e o b t a i n t h e p u l l b a c k of t h e diagram K1yP -rB rigp 1P I n t h i s map of Mayer-Vietoris sequences a l l groups e x c e p t mapped by t h e i d e n t i t y . Thus s rnZoP are i n d u c e s an isomorphism of homotopy groups and hence is a homotopy-equivalence. Suppose, i n (7.31, C o r o l l a r y 7.10. Then Zo € Mi and Proof. that f is a P-equlvalence and X, Y, B is a P-equivalence. vo We a l r e a d y know t h a t Z € NH i s an e q u i v a l e n c e so t h a t , by P r o p o s i t i o n 7.9, v € by C o r o l l a r y 7 . 6 . vop Now fp is an e q u i v a l e n c e and so i s a P-equivalence. Of c o u r s e , t h i s c o n c l u s i o n could more e a s i l y have been drawn w i t h o u t e s t a b l i s h i n g P r o p o s i t i o n 7.9 i n f u l l g e n e r a l i t y . We w i l l b e i n t e r e s t e d i n e s t a b l i s h i n g c o n d i t i o n s under which w e may i n f e r t h a t t h e space Z i n (7.2) is a l r e a d y connected, so t h a t Z = Zo. NH. Localization of homotopy types 98 Obviously this holds if (7.1) is a diagram in To in which f (or g) induces a surjection of fundamental groups. However, we will require a more general criterion. Proposition 7.11. Let ( 7 . 1 ) be a diagram i n nlB i s of the form 17.21 in H, f*a.g,f?, c1 E Ti To i s which every eZement of Then i n the weak pullback X, f? E nlY. 1 i s connected. Z Proof. Let us assume f a fibre map, so that ( 7 . 2 ) pullback in T. Given (x,y) E Z, x E X, y E Y, let k o to x, and m a path in Y from o reverse of m, is a loop in B on p 0 , so is the strict be a path in X to y. Then fII *gi, where i from is the in X, that there are loops h in Y with - Thus f(x * L) - fk*gm- fh*gp. g(p *m), re1 endpoints,and, since f is a fibre map, we find L' * h * II, re1 endpoints, that (II',m') so that f&' = gm', where m' = p *m. is a path in Z from o to It follows (x,y). We exploit Proposition 7.11 in the following way. Let ( 7 . 2 ) be a diagram i n To i n which f,: nlX * TIIB is a Corollary 7.12. P-surjection and of the primes. g,: TI Y 1 -+TI B i s a Q-surjection for some p a r t i t i o n 1 Then, i n (7.21, Proof. Z yn = g,n P for m E Q, n € P. are relatively prime we find integers k, II with km II and then y = f,Sk* g,n , + an = We are now ready to prove the mixing theorem which is the main objective of this section. = i s connected. Let y E alB. Then ym = f,S, Since m, n n 1 u Q Mixing of homotopy types Let Theorem 7.13. with Xo X , Y C NH of the primes. Then there exists 2 0 X and Let rI Yo with Z C NH There a r e c a n o n i c a l maps Proof. h: Y 2 99 s: Zp Xp + 2 = P U Q $, Xo, ZQ t: Y N Q Y -f be a partition 9' Let '0' and c o n s i d e r t h e diagram 0 1 (7.14) Y-% Q xO Form t h e weak p u l l b a c k of ( 7 . 1 4 ) , Certainly s i s a Q-equivalence and 7.12 e n s u r e s t h a t Corollary Thus u 7.10 g u a r a n t e e s t h a t induces up: Zp 2 Xp u and i s a P-equivalence. Thus C o r o l l a r y Corollary 7.6 then ensures t h a t i s connected. Z ht i s a P-equivalence v induces vs: and v 2 E NH and i s a Q-equivalence. ZQ r- YQ. The following addendum i s important i n a p p l i c a t i o n s . (i) Let X , P r o p o s i t i o n 7.15. Z Y in Theorem 7.13 be quasifinite. Then is quasifinite. lii) Let X , Y in Theorem 7.13 be 1-connected. Then Z is 1-connected. liiil Let X, Y in Theorem 7.13 have the homotopy type of a finite 1-connected CW-complex. Then Z has the hornotopy type of a finite 1-connected CW-comp Zex. Proof. (i) Observe t h a t i f generated L -module and P by Theorem 1.3.10, is quasifinite. A A Q A = WiZ then is a finitely-generated % is a f i n i t e l y - %-module. Thue, is a f i n i t e l y - g e n e r a t e d a b e l i a n group, so t h a t 2 100 Localization of homotopy types (ii) Observe that nlZ is a nilpotent group which l o c a l i z e s t o the t r i v i a l group a t every prime and hence is c e r t a i n l y t r i v i a l . (iii) This follows from (i) and (ii),using the techniques of homology decomposition. Chapter 111 A p p l i c a t i o n s of l o c a l i z a t i o n t h e o r y Introduction I n t h i s c h a p t e r , we p r e s e n t some a p p l i c a t i o n s of t h e t h e o r y developed i n the previous chapters. I t would seem t h a t t h e r e are two main d i r e c t i o n s a l o n g which t h e a p p l i c a t i o n s of l o c a l i z a t i o n t h e o r y should proceed. First, w e may w i s h t o s t u d y a problem concerning ' i n t e g r a l ' s p a c e s and maps by p a s s i n g t o t h e corresponding l o c a l i z a t i o n s . The l o c a l i z e d s p a c e s and maps o f t e n have much s i m p l e r s t r u c t u r e , t h e r e b y making t h e l o c a l i z e d problem more t r a c t a b l e . As a s i m p l e example, r e c a l l from Chapter I1 (Example 1.8) t h a t i f t h e S t i e f e l manifold of 2-frames i n 7-space, X P is homotopy e q u i v a l e n t t o S", P t h e n f o r any odd prime X = V 792 p, t h e p - l o c a l i z a t i o n of t h e 11-sphere. example arises i n o u r s t u d y i n S e c t i o n 3 of n o n - c a n c e l l a t i o n is phenomena. This As a n o t h e r example, in connection w i t h o u r s t u d y of f i n i t e H-spaces i n S e c t i o n 2, we are a b l e t o show, by p a s s i n g t o t h e l o c a l i z e d s i t u a t i o n , t h a t c e r t a i n c a n d i d a t e s i n f a c t f a i l t o admit H-space s t r u c t u r e s . L o c a l i z e d s p a c e s are n o t o n l y s i m p l e r t h a n t h e i r a n c e s t o r s , b u t are, in a s e n s e , more ' f l e x i b l e ' . The r i c h e r symmetry o f l o c a l s p a c e s stems from t h e f a c t t h a t Z*, t h e group of u n i t s of P Z? P' which a c t s on t h e p - l o c a l s p h e r e s , and on t h e homology and homotopy groups o f p - l o c a l s p a c e s , is v e r y l a r g e , whereas E* = {tl}. A v e r y s u b t l e example of t h i s symmetry, which w i l l n o t b e d i s c u s s e d i n d e t a i l i n t h i s monograph, is S u l l i v a n ' s theorem t h a t if p is a prime and k a number which d i v i d e s p - 1, t h e n t h e S2k-1 admits a loop space s t r u c t u r e . Actually, P i2k-1 S u l l i v a n proves t h i s f i r s t f o r t h e p - p r o f i n i t e completion of t h e P p-localized (2k-l)-sphere, (2k-l)-sphere by making j u d i c i o u s u s e of t h e s t r u c t u r e of t h e group of u n i t s of t h e p-adic i n t e g e r s , and t h e n u s e s t h e r e l a t i o n s h i p between l o c a l i z a t i o n and p r o f i n i t e completion t o deduce t h e r e s u l t f o r S2k-1 P Applications of localizationtheory 102 The second type of application of localization theory, and one in which we are particularly interested, derives from the fact (see, e.g., Example 1.2) that a space X in NH is not determined uniquely in general by the family of its p-localizations X although X can be reconstructed from P) the rationalizations Xp -+ Xo in case X is of finite type ( s e e Theorem 11.5.1 and subsequent remark). We are thus able to construct many new examples of spaces exhibiting various types of phenomena. Particularly noteworthy in this regard is the construction of several sorts of exotic finite H-spaces, a program pioneered by Zabrodsky. The organization for the rest of the chapter is as follows. Section 1 discusses the concept of the genus of a space of finite type X in NH, which by definition is the collection of all homotopy types Y in NH which are of finite type, such that the p-localizations of X are homotopy equivalent to the corresponding p-localizations of Y. We illustrate and give some general theorems concerning this notion and, in particular, study the possibility of furnishing a space with a structure which is present in all of its localizations. In Section 2, we are concerned with the theory of finite H-spaces. While we do not present an exhaustive study of the construction of new finite H-spaces --we do not enter into some of the more technical aspects of the theory, such as An-structures--we do discuss in some detail the rank 2 case, where the classification problem is essentially solved, and give a sampling of the sorts of strange behavior which finite H-spaces, in contrast with Lie groups, may exhibit. In Section 3 , we discuss the relationship between localization theory and the non-cancellation phenomena first discovered in Hilton-Roitberg [451. The existenceof sucharelationship is not surprising inview of the connection between the non-cancellation phenomena and rank 2 H-spaces. Indeed, as we attempt to show, localization theory sheds considerable light on Introduction the examples of non-cancellation, and conversely our main theorem concerning non-cancellation offers an excellent opportunity for application of the fundamental Theorem 11.5.1. 103 Applications of localization theory 104 1. Genus and H-spaces We have seen in Chapter I1 that a space X C NH determines a family {X Ip € l-l} of p-local spaces, its p-localizations, together with a P X Ip € II}, the rationalization maps. Moreover, family of maps {rp: X P O we have observed in case X is of finite type that X may be reconstructed + in a suitable way out of these two pieces of data (TheorenrsII, 5.1, 11.5.7). However, if one is given a collection of rationally equivalent p-local - n} such that H (Y 72 ) is a finitely generated i P’ P Z -module for all i and all p, one cannot deduce in general that there is P only one ‘integral‘ space X (i.e., a space X of finite type) whose spaces {Yplp C are homotopy equivalent to Y P P the following fundamental definition. p-localizations X Definition 1.1. The genus G ( X ) for all p. This prompts of a space of finite type X € NH, is the collection of all objects of finite type Y ENH such that X P % Y P’ for all p € II. Further, a homotopy-theoretic property is said to be a generic property if it is shared by all or none of the members of a genus. Our definition of the genus G(X) differs from the one originally given in Mislin[59] in thatwe require afiniteness condition. The definition chosen in this way in order that the genus sets should not be too big. Actually all presently known examples of genus sets G(X) complex are finite sets. For instance, G(S1) = IS1} with X a finite whereas there exist infinitely many different homotopy types X € NH with X “1 S1 for all P P primes p, because there are infinitely many non-isomorphic abelian groups A with A E Z for all primes p (cf. examples following Theorem I. 3.13). P P We observe (using Theorem I. 3 . 1 4 ) that members of the same genus Genus and H-spaces 105 have abstractly isomorphic homology and higher homotopy groups, so that we may describe these groups as generic, or genus invariants. However, their fundamental groups are not necessarily isomorphic, unless they are abelian. For instance, let X = K(N 1) and Y = K(N7/12,1) where and 1/12' N1 112 N 7/12 are the groups described following Theorem 1.3.13. Then Y C G(X) but rr Y * n X. 1 1 From the results proved in Chapter I1 concerning the localization of products, wedges, suspensions, loopspaces and mapping spaces, it is immediate that for X, Y and 2 spaces of finite type in NH and W a not necessarily connected finite complex, the equality of genus sets G(X) = G(Y) G(x (i) x G(Xk) (ii) 2) = G(Y x 2) = G(Yk) (iii) G(ZX) = G(cY) (iv) G ( k ) = G(EY) (v) G ( x W , o ) In case X, Y v k 2 implies that = G(Yw,o) and 2 are in addition 1-connected and if we denote by the k-fold wedge of a space 2, then one can also conclude that (vi) G(X V 2) = G(Y V 2) (61.0 G ( v k = G(v% It is not known whether being of the homotopy type of a finite complex is a generic property; but certainly quasifiniteness is a generic property. Example 1.2. or if X If X is a sphere Sn, or more generally, a Moore space K'(A,n), is an Eilenberg-MacLane space K(A,n) abelian group and n Theorem 1.3.14. 2 with A a finitely-generated 1, then we have G(X) = {X). This follows from 106 Applications of localization theory Let Example 1.3. element € n LY n-1 be t h e mapping cone X p. Cka if p r 2 2, of a homotopy In t h i s c a s e given by that, uLYe n , S r, which we suppose f o r s i m p l i c i t y t o be i n t h e s t a b l e range and of prime o r d e r so Ca = Sr - k denotes an odd prime, t h e c a r d i n a l i t y of t h e s e t G(CLY) i s (p-1)/2. To prove t h e s e a s s e r t i o n s , we begin by showing t h a t i f then Y has t h e r e q u i r e d form. t h e obvious f a c t t h a t Y Y € G(CLY) I t i s c l e a r from homology c o n s i d e r a t i o n s and must be 1-connected, t h a t Y may be put i n t h e form Y for a suitable rr cB = sr uB en, For every prime B € T,-~S~. h = h ( q ) : (C ) a q Assuming h + q, we have a homotopy equivalence (CB)q. c e l l u l a r , and invoking a c l a s s i c a l argument ( H i l t o n [ 3 3 1 ) , we deduce a commutative diagram with u = u(q) and v = v(q) of (1.4) r e a d i l y i m p l i e s t h e desired conclusion Bq homotopy e q u i v a l e n c e s . = 0 for The l e f t hand s q u a r e q # p; and, t a k i n g q = p , we o b t a i n Genus and H-spaces since then u and v may be viewed as elements of 107 Z* P and we are in the stable range. The same argument, applied in the unlocalized situation, proves the assertion that C =C iff k a ka 3 51 (mod p ) . It remains to show that any G(Ca). Now the composite sn-l of ka -ska r -c and the inclusion map is trivial, Sn-l sr ka kl 11 a sn-l Since Cka, (k,p) = 1, actually belongs to sr ~ so there is a commutative diagram @ I k a , ca (k,p) = 1, it follows by the Five Lemma that @*:H,(Cka; ZP) H,(C,; + ZP) is an isomorphism and hence that : (cka)p -+ (CaIp @P is a homotopy equivalence. Since for q # p (C ) aq = sr v sn = (C 9 q we have ) kaq our assertion is established. For more results on the genus of complexes with two cells the reader is referred to Molnar [631. Remark. One gets similar examples in the dual situation, i.e. using 2-stage Postnikov systems instead of spaces with two non-vanishing homology groups. It is to be expected, of course, that many homotopy-theoretic properties are In fact generic properties. We will illustrate this with a few examples. Recall the following definition. Applications of localizationtheory 108 D e f i n i t i o n 1.5. A space n > 0 i s c a l l e d reducible provided t h e r e e x i s t s an X and a map f : Sn i s an isomorphism f o r a l l i 2 n. integer A space + such t h a t X We s a y i n t h i s c a s e t h a t i s c a l l e d S-reducible i f f o r some X Hi(Sn; Z) -+ Hi(X; Z) f,: k 1 0, C k x reduces f X. is reducible. F o r i n s t a n c e , t h e Thom space of t h e normal bundle of a c l o s e d m a n i f o l d embedded i n some Euclidean space i s r e d u c i b l e . then is S-reducible (Browder-Spanier X Let Theorem 1 . 6 . If i s a f i n i t e H-complex, X [16]). X € NH be of f i n i t e type and Then Y € G(X). ( i ) X i s reducible i f f Y i s reducible, (iil X i s S-reducible i f f Proof. Hi(X; Z) = 0 Further, that X TI Suppose for i > n. can n Y f: Sn Hence -f X reduces Hn(Y; Z) E L HI(Y; Z) and H (X; Z) is s u r j e c t i v e , and t h e r e f o r e Hn(Y; Z) i s s u r j e c t i v e by Theorem 1 . 2 . 1 ( i i ) . t o b e a counterimage of a g e n e r a t o r of ( i n t h e homological s e n s e ) and m Recall that X - 0 Y € G(X) p. k C X d u a l i t y w i t h (untwisted) i n t e g r a l c o e f f i c i e n t s . i > n. implies Hence Choosing sn g: g -f Y reduces i s S-reducible i f f X is r e d u c i b l e f o r is a Poincare? complex i f and for Hn(Y;Z), i t i s c l e a r t h a t To g e t t h e second a s s e r t i o n , w e observe t h a t dim X < H (X; Z) G Z Then X. Hn(Y ; Z) i s s u r j e c t i v e f o r a l l primes P n P nnY Y. i s S-reducible. Y X k - dim X s a t i s f i e s Poincar6 A Poincard complex n e c e s s a r i l y h a s finitelygeneratedhomology groups (Browder [ 1 5 ] ) , a n d h e n c e i s q u a s i f i n i t e . Theorem 1.7. Y Let X E NH be a Poincar6 complex and l e t Y € G(X). Then i s a Poincare? complex. Proof. cap product Let p € Hn(X; + 1. Z) G Z be a fundamental c l a s s so t h a t t h e Genus and H-spaces i nu: H (X; Z) is an isomorphism for all i. tl E H (Y; Z) Z Z Hn-,(X; Z) If Y E G ( X ) , we pick a generator and attempt to show that is also an isomorphism for all i. equivalence -f h ( p ) : X, + Y 0 . For each prime p choose a homotopy From this we deduce a commutative diagram !J is a *P for some w(p) € Zt. This which proves that nh(p),pp is an isomorphism. Hence h(p) generator of Hn(Yp; Z ) and v = o(p)h(p),pp P P proves that nv is an isomorphism for all primes p. Since nv = (flvIp, P P this proves that nv is an isomorphism and hence Y is a Poincar6 complex. Remark. It may well be conjectured that the properties 'having the homotopy type of a closed manifold' and 'having the homotopy type of a closed rr-manifold' are generic properties. The latter conjecture may indeed be verified in certain cases, e.g.when X isl-connected and dim X is odd or of the form 4k with k > l , by using the Browder-Novikov Theorem in conjunction with Theorems 1.6 and 1.7. We are going to give much attention to the question whether the property of admitting an H-structure is generic. For aconnectedH-space X with multiplication u: X f o r which e :X P from u . x X, there is a unique H-structure p(p) : X x X -+ X P P P is an H-map. We call p(p) the H-structure induced X + X P It follows from the universal property of p-localization, + on X P that ~ ( p ) is homotopy-associative (homotopy-commutative) if p is. Applications of localization theory 110 N Similarly, i f is a c o n n e c t e d l o o p s p a c e , s a y f : X-QY, X loop s t r u c t u r e s f ( p ) : X P 7ClYP thenthere existunique e :X P f o r which t h e maps -+ X are l o o p maps. P If c o n v e r s e l y one wants t o p r o d u c e a n H-space s t r u c t u r e o n X H - s t r u c t u r e s on t h e p - l o c a l i z a t i o n s one n e e d s a c e r t a i n ' r a t i o n a l c o h e r e n c e ' P' More p r e c j s e l y we h a v e condition. Let Theorem 1.8. X C NH Lie of f i n i t e type and suppose t h a t each X equipped with an H-structure H*(X;Q), X from g i v e n induced from e : X P H-structure f o r which p furthermore, ~ ( p ) such that the Hopf algebra structure on -+ X i s independent of P' ep: X X -+ (homotopy-commutative) then so i s a Loop space, then the for Proof. By Theorem 1 1 . 5 . 7 If, p. ~ ( p ) i s homotopy-associative FinaZZy, i f i n addition each p. same i s t m e Then X admits an p. i s an H-map f o r a l l primes P i s q u a s i f i n i t e and each X is P X P is X. X is homotopy e q u i v a l e n t t o - X, t h e (weak) p u l l b a c k of nxP liX P is a n H-space and induced by r. r (flxp)o is a n H-map, A xo; i f we equip b e induced by t h e p r o j e c t i o n X H - s t r u c t u r e on P,O liX P + i n d u c e d from -+ . X . P X P (XpIo -+ X = xP Clearly 10 +(p) is a n H-map f o r t h e The c o n d i t i o n s t i p u l a t e d f o r e q u i v a l e n t t o t h e e x i s t e n c e of a n H - s t r u c t u r e r :X w i t h the H - s t r u c t u r e Let +(p): o x p ) o maps fiXp)o ~ ( 0 )o n Xo X is f o r which a l l t h e are H-maps o r , e q u i v a l e n t l y , f o r w h i c h a l l t h e c a n o n i c a l P P 0 homotopy e q u i v a l e n c e s X(p): X p,o -+ Xo are H-maps (for a more g e n e r a l s t a t e m e n t c o n c e r n i n g t h e r e l a t i o n s h i p between H-maps and induced maps i n homology compare Lemma 1.15). Now t h e c o m p o s i t e map Xo --t @Xp), can -t ""P ,O Genus and H-spaces h a s components {A(p)-'lp € n); 111 hence, by.assumption, it is an H-map. Since, for a finitely generated abelian group G, is a split monomorphism (Theorem I.3.8), we deduce, using the fact that X is of finite type, that the canonical map are the maps $(p) above, is a (split) monomorphism in the homotopy category. is an H-map, and therefore Clearly $ flXp)o -+iiXp,,, whose components $: must be an H-map. The first p assertion of the theorem then follows. is quasifinite and each ~ ( p ) is homotopy-associative In case X (homotopy-commutative), we obtain the corresponding property for v : X by applying Theorem 11.5.1 (ps, p: X x X + S is the switching map). In case one has in addition H-homotopy equivalences ~(p): Xp J/ * 5 IIK(P), Y + W(p) a prime or 0, one gets a commutative diagram rnY(p) with + to X respectively; here for certain Y ( p ) C NH, p X x induced by - + @ and -c A 3(i=iY(p))o X = fiY(0) Yyp~(o)-~, It remains to prove that A is a loop map; then the conclusion of the theorem is evident. Certainly, A is an H-map, since K ( o ) , k if nY(o) = II K(Q, ri) i=1 . Hence, A p and Y being an H-map, are. Notice that X Applications of localization theory 112 [A] = [A1] + ... + [ A k ] E H*(QY(o);A) is a sum of p r i m i t i v e elements i n hA(xl , . . . , ~ k ) % H*(nY(o) ;A). Thus l i e s i n the [A] image of t h e cohomology s u s p e n s i o n , and i t f o l l o w s t h a t is a loop map. A Remark. I f we a r e only i n t e r e s t e d i n Theorem 1 . 8 i n t h e q u a s i f i n i t e c a s e , then we can prove a l l b u t t h e f i n a l a s s e r t i o n of t h e theorem simply by applying Theorem 11.5.1 t o get the H-structure unique map which induces p ( p ) : Xp x X P -+ X P 1~: X x X a t each prime coherence c o n d i t i o n s t i p u l a t e d i n Theorem 11.5.1 maps r : (XP,p(p)) P + (Xo,p(0)) + X as the p. The n e c e s s a r y is f u l f i l l e d , s i n c e t h e a r e H-maps. I n t h e q u a s i f i n i t e c a s e Theorem 1.8 may b e g e n e r a l i z e d t o t h e e x t e n t t h a t one r e q u i r e s t h a t t h e Hopfalgebra s t r u c t u r e s on ep: X + Xp, a r e isomorphic, r a t h e r t h a n e q u a l . H*(X;Q), induced by This g e n e r a l i z a t i o n is c r u c i a l t o deduce t h a t a q u a s i f i n i t e H-complex admits an H-space s t r u c t u r e i f admits a homotopy-associative H-space s t r u c t u r e f o r a l l primes X P p ( s e e Theorem 1 . 7 ) . It aeemsnot t o b e knownwhether thehomotopy-associativityconditioncould bedropped i n t h i s assertion. However one can prove t h a t being of t h e homotopy type of a q u a s i f i n i t e H-complex is a g e n e r i c p r o p e r t y , and t h i s i s our n e x t o b j e c t i v e . For t h i s , we w i l l f i r s t need some lemmas, which a r e of g e n e r a l use i n t h e s t u d y of f i n i t e H-complexes. R e c a l l t h a t f o r a n a r b i t r a r y connected s p a c e is a coalgebra w i t h d i a g o n a l F, H,(X;F) w i t h c o u n i t induced by that for x E G*(X;F) A,x The elements - x 8 1 X -+ 0, A* X induced by and a r b i t r a r y f i e l d A: X -f t h e projection onto t h e basepoint. X x X , and It f o l l o w s one has + 1-3 x + Exi x E H,(X;F) 63 yi; such t h a t A,x deg xi, deg yi < deg x = x 8 1 + 1 8x a r e , naturally, called Genus and H-spaces primitive elements. a map f: X + so t h a t , i f They form a l i n e a r subspace and an element Y x x € H,(X;F), is p r i m i t i v e , is a l s o p r i m i t i v e . by 113 A,f,x W e d e n o t e by = Pf,: f,x 0 wehave 1 PH,(X;F) + 5 H,(X;F). PH,(X;F) = (f,@f,)A,x, A,f,x 1 8 f,x and hence PH,(Y;F) -+ For f,x t h e map induced f,. Let Lemma 1.9. a map. X and Then f,: be arbitrary connected spaces and Y H,(X;F) -+ i s one-one if Pf,: H,(Y;F) f: X PH,(X;F) + Y -+ PH,(Y;F) i s one-one. Proof. Now every element i n t h a t w e have e s t a b l i s h e d t h a t let w C Hi+l(X;F). then = 0 f,w A,w If f, is p r i m i t i v e . Suppose, t h e n , is one-one i n dimensions 3,i P 1, and = to 8 1 implies t h a t H1(X;F) A,f,w + 1 8w + Zv = (f,@f,)A,w j Qw j' 8 1 = f,w , deg w j 5 i, j 1 Q f,w deg v + + Zf v 8 f w = 0. Thus Zf v 8 f w = 0 and t h e r e f o r e Zv a 9 w = 0 , s i n c e "j * j * j "j j j f, i s one-one i n dimensions 3 . T h i s means t h a t w is p r i m i t i v e and hence w = 0. If Y is a n H-space w i t h m u l t i p l i c a t i o n u: Y x Y -+ Y, then H,(Y;F) is a (not n e c e s s a r i l y a s s o c i a t i v e o r commutative) H o p f a l g e b r a , w i t h d i a g o n a l A,, u, and m u l t i p l i c a t i o n Lemma 1.10. tmo maps. Let u. be an arbitrary space, X Define induced by fg: X + Y Y an H-space, and f ,g: X by ( f g ) a = f ( a ) - g ( a ) 6 Y , a C X, using the muZtipZication of Then, i f Y. (fg)*x = f*(x) Proof. f g = u(fxg)A: X (fg),x -+ If !J: Y. Hence f o r = u,(f,@g,)A,x Y x Y -+ = u,(f,x@l Y x E PH,(X;F), + g,(x). d e n o t e s t h e H-structure map, t h e n x E PH,(X;F) + 1 one h a s = f,(x> + g,(x). + Y 1 I4 Applications of localization theory A s an immediate consequence we have Lemma 1.11. Let inductively by one has Y be an H-space and $(k)(y) = $(k): Y y*$(k-1) (y), $‘(y) -+ = y. Y t h e k-power map, defined Then, f o r x E PH,(Y;F), $ik)x = kx. N Define, by abuse of language,the k -power map inductively by ~$(~”)(y) = $ (k) ( p N - 1 ) (y)), Y $(k’N): + Y $(kJ) (y) =. $(k) ( y) . Let Y be a connected H-complex of f i n i t e homological dimension Then the pN-power map $(pBN) = $(p) , Lemma 1.12. N. is a {PI’-equivalence and induces Proof. The map $(p) induces, in homotopy, multiplication by pN and is therefore certainly a {el’-equivalence. We prove the second statement by induction. If x 6 H1(Y; Z / p ) , then $ip)x = px = 0, since all elements of H1 are primitive. Suppose $ ( p s i ) 3 , where i 2 1, and let y € Hi+l(Y; Z / p ) . Then A*y = y Hence A*$ipyi)y = $ie’i)y 8 1 deg Yj’ deg $ip*i)(y) induces zero on H*(Y; Z/p) in dimensions = 0 1 + 1 8y + 1 8 $!psi)y j i* is primitive. Lemma 1.11 shows that $(*psi+l)(y) = p*$2yi)(y) = 0. Thus $* €9 in dimension i $iP) €9 = it is certainly 0 in dimensions 3 . This establishes the inductive hypothesis and the result follows immediately. From this lemma we can deduce the following consequence. Lemma 1.13. Let Denote hy P Y be a q u a s i f i n i t e connected H-compzex and l e t a f i n i t e s e t of primes. h(P): X + X € G(Y). Then there i s a P-equivalence, Y. z and therefore ($*(p’i)(y)) + 1; but + Cyj j’ Genus and H-spaces Proof. p -equivalence i P = {pl, . . . , p m} Let f(i): X g ( i ) = $(p,) where $(p ) 1 0 + and choose, f o r each ( s e e Theorem 11.6.6). Y $(p2) ... $A ( P J . ... 0 N p -power map denotes t h e 115 j is a p -equivalence and, f o r i Y p C P-{pij, 0 + pi C P , a Define $(pm) o f ( i ) Y, a s i n Lemma 1.12. Then g(i) one h a s Define where t h e product is performed in t h e monoid We claim t h a t h(P) is a P-equivalence. i t is enough t o prove t h a t , f o r p [X,Y], u s i n g some f i x e d b r a c k e t i n g . Because of t h e f i n i t e n e s s assumptions C P , h(P),: j h*(X; Z / p j ) +H,(Y; B y L e m m a 1 . 9 i t i s e n o u g h t o check t h i s on p r i m i t i v e s . be a p r i m i t i v e element. u s i n g Lemma 1.10. x = 0. Hence .i x E H,(X; Z / p ) j Then h(P),x = Pg(i)),x Since g(j) h(P) So l e t Z / p ) i s one-one. = zg(i),x = g(j),x, is a p -equivalence j h(P),x = 0 implies t h a t is a P-equivalence. We a r e now ready t o prove t h a t b e i n g of t h e homotopy t y p e of a q u a s i f i n i t e H-complex is a g e n e r i c p r o p e r t y . Theorem 1.14. Let (i) X (ii) If Y be a quasifinite H-complex and l e t X € G(Y). i s a quasifinite H-complex. G(Y) then there e x i s t s a x x W".Y x z. w such that Then Applications of localization theory 116 Proof. Choose a quasifinite H-complex V E G(Y), for instance V and choose a rational equivalence B: X integral homology B, X + K = nK( Z,m,) (see Theorem 11.6.6). V has a finite kernel and cokernel, B for T a cofinite set of primes. A: -+ = Y, Since in is a T-equivalence Further choose a rational equivalence such that A//: nlX + II 1K is surjective; such a X exists since H*(X;Q) h(x l,...,x ) , an exterior algebra on odd-dimensional n generators. Clearly X is a nilpotent map, whose fibre has finite homotopy as a composite of u and inducedfibrations A groups. Therefore we can factor A j’ where A has fiber K( L/p n ) and a is (2N-1)-connected, N being the j j’ j homological dimension of Y. Let S = T’ u {p 10 5 j < r} and let a = h(S) : X j Further let $j = $ ( p j ) : V be an S-equivalence (see Lemma 1.13). N p -power map. Form + V be the j $ = ... o $ r - l ~$ r - 2 ~ $o and X Y = $j-lo$j-20...o $oy 1 5 j < r. We will prove that {cr,B}: + V x has a homotopy left inverse. Notice that a $6 is an S-equivalence and is an S’-equivalence. Hence {a,$B}*: H*@ XV; L) -+ H*(X; L) is surjective, since, for trivial reasons, H*(Y XV; Lp) H*(X; L ) is P surjective for all primes p . We conclude that there is a map p: Y x -+ such that - !.I o ~ a , $ B ~ A: X + To solve our lifting problem X I X ,*l I K. v +K + Y Genus and H-spaces I17 it is enough to prove that we can find a lifting in the following typical diagram (i 11) : Here ui = Xi-l o Xi-* 0.. .o X ou ui+l and may be assumed inductively to exist such that and ur = P. We have an exact cohomology sequence for the map (where we write the ‘relative’group in traditional notation), which breaks up into short exact sequences 0 - H*(YxV.,X; Z/pi) -% <a*’, ($i8)*> H*(Y; Z/pi) @ * H*(X; H*(V; Z/pi) z/pi) --t 0 ni+l x Tf C H (YxV,X;Z/pi) then it follows from (Di) lies in But im(184Jt). denotes the obstruction to the factorization pi, and the naturality of the obstruction that 6x $* = 0 on reduced cohomology with by construction. Hence 6x = y @J is a pi-equivalence we infer that From this we conclude that Z/pi coefficients, i Pi retract of the H-space Y x V. 1 and <a*,($ B)*>6x = 0 = a*y. i Since a y = 0 and, 6 being one-one, x exists in Thus X (Di) and hence that X = 0. is a certainly admits an H-structure. For part (ii) of the theorem observe that Z is an H-space by (i). Following up the proof of (i) with V = Z, we see that of Y x Z, with retraction map p, then certainly X x W p: Y x Z + X. Y x Z, since Y x 2 If W X is a retract denotes the fibre of is an H-space. Applications of localization theory 118 Remark. It is not hard,to prove that the W constructed above actually (See Mislin [60]; see also Wilkerson [921). also belongs to the genus of Y. To be able to construct specific maps between rational H-spaces we will need the following lemma. Lemma 1.15. Let X, Y C NH and assume that Yo i s an H-space whose rational, homology i s finitely-generated over Q i n each dimension. Consider the canonical map Then lil (iil e i s a bijection. Xo i s an H-space whose rational homology If, i n addition, i s finitely-generated over Q i n each dimension, then a b i j e c t i o n betueen H-maps Xo H*(Yo;Q) + H*(Xo;Q) +. induces by r e s t r i c t i o n 6 Yo and Hopf-algebra homomorphisms * Proof. The assumptions on Y imply that H*(Yo;Q) is a free graded commutative and associative algebra, on free generators {yil i C I). If deg yi = mi, then Y -17K(Q,mi) Hence the elements of in NH. [Xo,Yo] are in one-one correspondence with families of homotopy classes {Xo+.K(Q,mi)} which themselves are in one-one correspondence with algebra maps H*(Yo;Q) + H*(Xo;Q) by the freeness of H*(Yo;Q). To prove (ii), first observe that an H-map induces a morphism of Hopf algebras H*(Yo;Q) + Xo H*(Xo;Q) +. . Yo certainly Conversely, if Genus and H-spaces f* i s a map of Hopf a l g e b r a s , then t h e f o l l o w i n g diagram commutes f o r every K(Q,n): r a t i o n a l Eilenberg-MacLane space Therefore t h e same diagram commutes w i t h t o t h e i d e n t i t y map Xo x Xo px: f 119 + X Yo and -t Yo py: K(Q,n) this yields Yo x Yo + r e p l a c e d by Yo. Applied f o u x = p Y o ( f x f ) , where denote t h e H - s t r u c t u r e maps. Yo Hence is an H-map. w e announced We can now prove t h e e x t e n s i o n of Theorem 1 . 7 earlier. Theorem 1.16. p, X P Let X E NH be q u a s i f i n i t e and suppose t h a t , f o r every prime i s equipped with an H-structure such t h a t the maps e : X P -+ X P Then X admits an induce isomorphic Hopf-aZgebra structures on H*(X;Q). H-space structure. Proof. We w i l l produce an H-space f o l l o w s from Theorem 1.14. c a n o n i c a l map r 2 : X2 + Xo corresponding s t r u c t u r e map X Equip w i t h t h e H - s t r u c t u r e induced by t h e p : Xo x Xo e xe xxx- Q -+ that rQ: XQ. (+p(Q)) + (Xo,p) and d e n o t e by u(p,6) e ve Xo. X2; c a l l the By C o r o l l a r y 11.5.11 t h e r e such t h a t t h e two maps xo x xo -bxo xo v xo and have unique l i f t s t o Then t h e r e s u l t from t h e g i v e n H - s t r u c t u r e on e x i s t s a c o f i n i t e s e t of primes xvx- E G(X). Y 7 xo Hence w e o b t a i n a n H-structure is an H-map. u(p) Denote by u(Q) 0 XQ such t h e €I-structure t h e induced H-structure on (X ) = X P on P,O on X . The assumption on P Applications of localization theory 120 the X ' s implies that there are abstract isomorphisms of Hopf algebras P with respect to the Hopf algebra structures induced by u and p(p,o) respectively. By the previous lemma we can realize these isomorphisms by H-maps K(P) : (XP,,,u(p,o)) orp = K(p): K(P) (Xp,u(p)) H-may. Define Y + (Xo,u). + (Xo,u) It follows that is a rational equivalence which i s an to be the weak pullback of the finite family of H-maps {rQY;(p);p CQ'). It is now not difficult to prove that Y E G(X) py: Y (Y,uy) x Y .* + Y such that the canonical maps (Xp,p(p)), and there is an H-structure (Y,uy) + (XQ,u(Q)) p € Q', are H-maps. Then X € G(Y), so and that X is an H-space. An important special case is the following, for which the rational coherence condition is automatically fulfilled. Theorem 1.17. Let X be quasifinite in NH and suppose that, f o r each p, is equipped with a homotopy-aesociative H-space structure. Then X admits P an H-space structure. X Proof. By the Hopf-Samelson Theorem there is, up to isomorphism, only one coassociative Hopf algebra structure on H*(X;Q). Therefore the X * X induce isomorphic Hopf algebra structures on H*(X;Q) P P the result follows from the previous theorem. maps e : and There are many ways to combine the mixing theorems of Chapter I1 with the results of this section. We will only mention one theorem, which will be used later on. Genus and H-spaces Theorem 1.18. by P and Q Let X and Y be q u a s i f i n i t e complexes i n t u o complementary s e t s of primes. equipped w i t h H-structures such t h a t as Hopf algebras. 121 H*($;Q) xp Suppose and NH and denote H*(Y ;I)) Q Then there e x i s t s a q u a s i f i n i t e H-complex and Y Q are are isomorphic 2 and H-homotopy equivalences zp “ x p ’ z Q “.Y Q‘ Proof. Choose (by Lemma 1.15) an H-homotopy equivalence Then the (weak) pullback Z in the diagram Z-Y 1 S f Q T ‘Q,o is quasifinite by Proposition 11.7.15. Since r and prp are H-maps, admits an H-structure such that the canonical homotopy equivalences Zp -%Xp and Z %- Y Q Q are H-maps. Z 122 2. Applicationsof localizationtheory Finite H-spaces. special results We begin this section with a discussion of H-spaces of rank 2. Recall that if X is an H-space of the homotopy type of a quasifinite complex (or, as we say, a quasifinite H-space), then the exterior algebra on odd-dimensional generators xl, deg(xi) = ni and assuming n1 5 ..., x . Setting ... 5 nr, we define rank(X) = r, type(X) = (nl,. ..,nr). From the point of view of applications of localization theory, the only interesting cases for rank 2 H-spaces occur when so we shall restrict attention to these cases. In fact, we shall impose the further condition that X be 1-connected;hence X will be a finite complex. The classical examples of 1-connected H-spaces of type (3,7) are, of course, S 3 x S7 and the 2-dimensional symplectic group Sp(2). homotopy structure of Sp(2) 3 The has been long known and can be described as follows. The inclusion S = Sp(1) + Sp(2) gives rise to a principal SJ-bundle s3 -+ Sp(2) which is classified by an element in as an element of TI + s7 3 7(BS ) G TI (S 6 3 ) 2 2/12. 'When viewed r 6 ( S J ) , this Blakers-Massey class u with the class of the map induced by the commutator map may be identified Finite H-spaces,special results 123 and is known to be a generator. We now form the principal S3-bundle S3 classified by ko, SO + Eku+ S 7 , 0 C k 11, C that Sp(2) = Ew, and state our first theorem, which summarizes results from various sources [18,45,46,75,79,93,941. Theorem 2.1. The space Ekw a h i t s an H-space structure i f f Conversely, any 1-connected f i n i t e H-space equivalent t o some Proof. Assuming k # 2, 6 , 10, we show that Eko only with the cases k = 3, 4 , 5 = o f type (3,71 i s homotopy Eku. H-space structure. This is clear for k k X k # 2, 6, 10. = admits an 0 or 1 so we need concern ourselves since, plainly, Ekw 9 E (12-k)w' For 3, we use the commutative diagram where the vertical maps are the bundle projections, the lower horizontal maps are maps of the indicated degrees and the upper horizontal maps are bundle maps covering the maps on the bases. Localizing the right hand square of (2.2) at any prime p # 3 gives a homotopy-commutative diagram w i t h the columns fibrations by Theorem 11.3.12. homotopy equivalence for p # 3 so, But 3 is obviously a P by the Five Lemma, we conclude (E3Jp = (Ew)p, P # 3. Similarly, localizing the left hand square of (2.2) at any prime p # 2 124 Applications of localization theory shows that 3 7 (E ) = (S X S )p, p # 2. 3w P Thus, (E3w)p admits an H-space structure for all p C P and since the condition stipulated in Theorem 1.16 on the various comultiplications on is evidently satisfied for dimensional reasons, we may conclude H*(E3w;Q) our result for E3, from Theorem 1.16. Very similar arguments, using the diagrams show that Ekw and E5, also admit H-space structures. The proof that EL and Eh do not admit H-space structures is quite complex and will not be reproduced here. The result is due to [94] and uses the theory of higher order mod 2 cohomology Zabrodsky operations. See also Sigrist-Suter 1771 for a proof of this result which i s based on K-theory. (EZwIp and (E6w)p a8y we ~~t~all , can see from the diagrams admit H-space structures for all odd primes problem of showing that E2w and E6- so the do not admit H-space structures is, as indicated above, a purely mod 2 problem. Also, the diagram shows that E2w admits an H-space structure iff the same is true of E6w' We now sketch a proof of the converse, emphasizing those points relevant to localization theory. To begin with, we must invoke certain general Finite H-spaces, special results results on the homology o f H-spaces [ 4 6 ] 12.5 to infer that an H-space X satisfying the hypotheses of the theorem has no homological torsion. It then follows that X possesses a cellular structure o f the form AS f o r the second attaching map and, of course, a = kw, 0 5 k C 6 . various techniques " 4 6 , 621) satisfies 1(B) = tu,'31; (2.4) here, j: ng(Ca) and B may be used to show that B, l3 € -+ 3 ng(CapS ) T3(S 3 ) = L Whitehead product of u is the natural homomorphism, are generators, and and i3. [u,i3] E h9(C ,S (J 3 6 a7(Ca,S ) = L 3) is the relative The condition ( 2 . 4 ) may be used to show that X has the homotopy type of an S3-fibration over 7 S ; indeed, close examination of the set of homotopy types o f the form ( 2 . 3 ) with 6 as in ( 2 . 4 ) reveals that there are ten such in number and that each one has a representative which is an orthogonal S 3 -bundle over S7. We compare X with the corresponding principal S3-bundle Ekw Since j(@) = j ( B ' ) , = S3 . Ukwe7 UB, e10 the exact sequence shows that Now i f ( k , 3 ) = 1, then i = 0 because S g ( S 3 ) Thus, X %,EL i f ( k , 3 ) = 1. is generated by woZ 3 W. Applications of localization theory 126 On the other hand, if k is a multiple of 3 element, (Ch)3 N- 7 and S3 V S 3 , injective. Localizing X where t3 [ I ~ , I ~is ] the ~ 3 E n3(S ), i7 at 3 , then kw i s a 2-primary i, as well as its localization i3’ is 3 , we get 3-localization of the Whitehead product of generators 7 In particular, the fibration C a7(S ) . s3 3 + x 3 + s 37 admits a section S 7 3 + Combining this section with the fibre inclusion X3. and utilizing the H-space structure on X3 induced from that on X, we obtain a map 4 : s33 x s37 + x3 which induces a homotopy isomorphism and is hence a homotopy equivalence. Assuming $ cellular and applying [ 3 3 ] , we obtain a commutative diagram (see (2.5)) 3 7 s3vs I 3 with u and v homotopy equivalences. Straightforward calculation using the left-hand square of (2.6) now shows that i3(e3) = 0, hence i(e) = 0, so that X=E kw 3, if k is a multiple of and the proof of the theorem is completed. Remarks. 1. Our proof shows that E5w E G(sp(2)) so that, by Theorem 1.8, E g w is a loop space. 2. The techniques used in the second part of the proof of Theorem 2.1 Finite H-spaces, special results 127 l e a d t o a d e t e r m i n a t i o n of t h e genus s e t s of t h e s p a c e s G(Ew) = {Eu,E5w}, G ( E ~ J = {Ekw} Indeed, f o r any k, Ekw we have Ekw: ( k , l 2 ) # 1. if i s a n S-reducible Poincar6 complex (it i s i n f a c t e a s i l y checked t h a t t h e t o t a l s p a c e of a p r i n c i p a l G-bundle, where G is a L i e group, w i t h b a s e s p a c e a n-manifold, thesame is t r u e f o r any of t h e form (2.3) w i t h by Theorems 1 . 6 , 1.7. X € G(Eku) B is stably parallelizable),So Hence w i l l be X a s i n (2.4) and t h e proof of our a s s e r t i o n i s e a s i l y completed. We t u r n now t o H-spaces of t y p e (3,11) and r e c a l l t h a t t h e c l a s s i c a l example h e r e i s t h e e x c e p t i o n a l L i e group w h i l e n o t an H-space, In fact, i f Note a l s o t h a t G2. S3 x Sll, becomes an H-space when l o c a l i z e d a t any prime p # 2. S2k+1 is any odd-dimensional s p h e r e , t h e n t h e s o l e o b s t r u c t i o n S2k+1 t o p u t t i n g an H-space s t r u c t u r e on [ I ~ ~ + ~ , Iof~ a~ g + e n~ e r a] t o r c l a s s i c a l l y t o have o r d e r is t h e Whitehead s q u a r e 2k+l) 1 2k+l € T ~ ~ + ~ ( S 52, it follows t h a t , i f . S i n c e t h i s is known p i s odd, S2 k+l P admits an H-space s t r u c t u r e . One may t h e n apply Theorem 1.18 t o t h e s p a c e s G2 and S3 x S l l t o c r e a t e new examples of f i n i t e H-spaces of type (3,ll) and, i n f a c t , t h i s was done i n [ & I , u s i n g some knowledge of t h e homotopy s t r u c t u r e of A G2. c l o s e r a n a l y s i s of t h e s i t u a t i o n w a s c a r r i e d out i n Mbmura-Nishida-Toda [54] and can b e s u m a r i z e d as f o l l o w s ; we u s e t h e n o t a t i o n of [411. G2 3 may b e d e s c r i b e d a s t h e t o t a l s p a c e of a p r i n c i p a l S -bundle over t h e S t i e f e l manifold 792 = S0(7)/S0(5), c:S3+G (2.7) Since V V 792 2 + V 7,2' has t h e c e l l u l a r s t r u c t u r e v7,2 = S5 U2 e 6 U e 11 , 128 Applications of localization theory it follows that 11 = sp , (v7,2)p where P (see Example 11.1.8) denotes the family of odd primes. Hence if classifies (2.7), we may view the localization ap Moreover, since all(BS 3 II lo(G2) = 0, it follows that ap Let B be the generator of nll(BS 3 ) ap - V + BS3 7Y2 as an element of a: is a generator of corresponding to ap , i.e., B 8 1. Now form the principal S 3 -bundle Sk: S3 -f \ + V7,2, 0 5 k 5 14, which I s classified by the composition - v7,2 c '7,2 <a V S1' ,(k-1) B> BS , where c is the cooperation map (In the terminology of Eckmann-Hilton) arising from the attachment of el1 to form V7,2. S1 = 6. We may now state one of the main results of 1541. Theorem 2.9. (%=Xa X Note that X1 = G 2 , Each of the spaces Xk a h i t s an H-space structwle iff k -&t(mod 15) .) Conversely, any 1-connected f i n i t e H-space of type (3,111 such that H*(X;22/2) equivalent t o some Proof. i s primitively generated is homotopy- 5. We sketch a proof of the fact that each % admits an H-space structure. The proof of the converse Involves extensive homotopy calculations and so I s omitted here. Finite H-spaces,special results 129 The structure of the classifying map (2.8) for Sk (XkI2 = (G2j2, (SIP = (Eka)p if P where EkB is the principal S3-bundle over S1’ assert that each P = shows that {21’, classified by kB. We (EkB)p, p € P, admits an H-space structure. In fact, according to [54], so that certainly statement for any (Ea)p, p C P, admits an H-space structure. The corresponding (EkB)p, p € P, now follows from the type of reasoning used in the first part of the proof of Theorem 2.1. structure for any p € n, Thus, (X,)p admits an H-space and since the condition stipulated in Theorem 1.8 on the various comultiplications on H*(X,;Q) is clearly satisfied for dimensional reasons, we conclude from Theorem 1.8 that % does have an H-space structure, as claimed. Remarks. 1. Clearly \ € G(G2) iff (k, 15) = 1. In these cases, % is, by Theorem 1.8, a loop space. 2. The genus sets of the X, are computed as follows: Theorems 2.1 and 2.9 suggest that a fruitful source for further examples of finite H-spaces is the collection of total spaces of principal G-bundles over spheres and related spaces. There are in fact a number of papers in the literature (see, for example [18,32,96J ) describing results along these lines, We content ourselves here with stating one of the more comprehensive results in this direction, due to Zabrodsky [961. Applications of localization theory 130 If dn - ahits an H-space structure iff 1 z 7 , then Xm(n,d) m <s odd. In general, it is not reasonable to expect a homotopy classification of finite H-spaces of a given type since the homotopy calculations involved rapidly become prohibitively difficult. For a slight indication,of what seems to occur for H-spaces of arbitrary rank, see Mislin-Roitberg [62]. Our next result suggested by an old result in Lie group theory, I s according to which the integral homology of a Lie group has p-torsion, p a prime, i f f it has q-torsion for all primes q 5 p. The following result of Mislin [58] shows that this result does not generalize to arbitrary finite H-spaces . Theorem 2.11. There exists a finite H-space Z such that H,(Z; Z) possesses 3-torsion but no 2-torsion. Proof. Y = Sp(6) Let X = F 4 x S7 x S1', F4 the exceptional Lie group, and let P = {31, Q = {31'. As F4 has type (3,11,15,23), it is clear that X and Y are rationally equivalent. Moreover Xp and Y Q evidently admit H-space structures and the resulting Hopf algebra structures H*(ILp;Q) and H*(Y Q ;Q) have to be isomorphic for dimensional reasons. It follows that the space Z guaranteed by Theorem 1.18 Since Sp(6) F4 Is Is a finite H-complex. has both 2- and 3-torsion in its integral homology and since homologically torsionfree, we conclude from zp % $, ZQ that 2 has the advertised properties. CX YQ 131 Finite H-spaces, special results 5 S i n c e , by a cohomology argument, Remark. a s s o c i a t i v e s t r u c t u r e , n e i t h e r can 2. cannot s u p p o r t a homotopy- Thus, i t may s t i l l b e c o n j e c t u r e d t h a t t h e pathology of Theorem 2.11 does n o t occur f o r q u a s i f i n i t e homotopya s s o c i a t i v e H-spaces. Thus f a r , a l l of t h e examples of f i n i t e H-spaces which we have c o n s t r u c t e d have t h e p r o p e r t y t h a t each of t h e i r l o c a l i z a t i o n s i s homotopy e q u i v a l e n t t o t h e corresponding l o c a l i z a t i o n of a ' c l a s s i c a l ' example. As our f i n a l a p p l i c a t i o n i n t h i s s e c t i o n , we show, f o l l o w i n g Mimura-Toda [561 ( s e e a l s o S t a s h e f f 1811 and Zabrodsky [96] f o r independent t r e a t m e n t s ) t h a t t h i s need n o t b e t h e c a s e i n g e n e r a l . Theorem 2.12. There e x i s t s a f i n i t e H-space such t h a t Z 3 is not 2 homotopy equivalent t o the 3-localization of the product of a Lie group and odd-dimensional spheres. Proof. over S9 Let X' 5 be t h e t o t a l s p a c e of any o r t h o g o n a l S -bundle 5 (Such bundles c e r t a i n l y e x i s t s i n c e 3. T ~ ( S) = Z / 2 4 h a s o r d e r whose c h a r a c t e r i s t i c elqment i n r 8 ( S 0 ( 6 ) ) 8 Z3 5 + n8(S ) 8 Z 3 s u r j e c t i v e ; a l s o , i t may be v e r i f i e d , u s i n g t h e f a c t t h a t n is 5 13(S ) 8 Z 3 = 0, t h a t t h e 3 - l o c a l i z a t i o n s of any two such bundles a r e fibre-homotopy e q u i v a l e n t . ) It f o l l o w s from a s t u d y of t h e a c t i o n of t h e mod 3 Steenrod a l g e b r a on t h e Z/3-cohomology of where Eu SU(5) ( s e e Oka [ 6 4 ] as w e l l a s [561) t h a t h a s i t s p r e v i o u s meaning. Hence Xi, and so a l s o X3, where w e set x admit H-structures. = x' x s3 x SJ, W e may now apply Theorem 1.18 w i t h Y = SU(5), P = { 3 1 , Q = {31' t o o b t a i n a f i n i t e H-space which s a t i s f i e s i n p a r t i c u l a r X Z a s above, of t y p e (3,5,7,9) Applications of localization theory 132 By comparing the effect of the Steenrod reduced power P i (small) list of all the classical examples of type on Z and on the (3,5,7,9), we easily conclude that Z does indeed satisfy the conclusion of the theorem. The argument of Theorem 2.12 may be suitably generalized so as to provide similar examples for other odd primes, but the prime 2 appears to be exceptional Non-cancellation phenomena I33 3. Non-cancellation phenomena Our first general theorem on non-cancellation phenomena concerns total spaces of what we call quasiprincipal bundles (seeHilton-Mislin-Roitberg We begin with the definition. 1401). bundle with structural group G, and g: X principal G-bundle associated with 5 . provided g of rr- + BG ---f the classifying map of the We say that 5 is quasipAn&pal 0. Note that if 5 gof f X a fibre Let G be a topological group, 5 : F*E Definition 3.1. is a principal G-bundle, then the composition is trivial since it factors through the contractible total space EG of the universal bundle over BG; this justifies the terminology 'quasiprincipal' In a similar way to that of our previous studies of spaces at individual primes, it is plainly very useful to study fibrations (over a fixed base space) by p-localizing the fibration (see Theorem 11.3.12). We will introduce the notion of the genus of a fibration which, in case the base space consist of a point only, reduces to the definition of the genus of a homotopy type, the total space of the fibration in question. The notion of the genus of a fibration turns out to play a central role in the theory of non-cancellation phenomena. Definition 3 . 2 . Two fibrations Si: Fi + E i + X, i = 1, 2, with Fi, Ei, X C N H are called p-fibre homotopy equivalent, if the localized bundles (Si)p: (Fi)p We say that (Ei)p -r c1 and are fibre-homotopy equivalent for all primes p. Xp -r belong to the same genus, if all the spaces C2 Fi, Ei, X 6 NH are of finite type and if, in addition, C1 and C2 are p-fibre homotopy equivalent for all primes p. Definition 3.3. spaces F, E If c: F -f E+X is a fibration in M-l with all the and X of finite type, then the genus of 6, G ( C ) , consists Applications of localization theory 134 NH of a l l f i b r e homotopy equivalence c l a s s e s of f i b r a t i o n s i n over X, 5. which belong t o t h e same genus a s We now prove t h e f o l l o w i n g , which e x t e n d s a r e s u l t of [40]. Let Theorem 3.4. Fi NH which are p-fibre El and E2 Ei -t -t be quasiprincipaZ G-bundZes i n X, i = 1, 2 homotopy equivaZent f o r a22 primes are q u a s i f i n i t e rmd that Assume p. (For example, BG C NH. G that may be a connected topoZogicaZ group, or an arbitrary niZpotent topoZogical group [69 ]I. Then El F2 x rmd Proof. with 6,: Let F +E i i f i --t gi: p € Il X c l a s s i f y t h e p r i n c i p a l G-bundles a s s o c i a t e d BG -+ - X; t h u s we a r e g i v e n t h a t f i r s t aim is t o show t h a t f o r each are homeomorphic. E2 x F1 g2 o f l and 0 giofi g of2 1 N 0. = 0, i = 1, 2. To t h i s end, choose a homotopy equivalence O(P) : (Ellp + such t h a t (f Then, s i n c e (g2)p o ( f 2 I p (g*)p 0 5 1 1 P = (f2)p OO(P). 0 , we o b t a i n (fl)p = (g21p 0 (f21 P 0 @(PI = and so,by Theorem 11.5.14, g2 Ofl Similarly glof2 CI 0. = 0. Next, form t h e p u l l b a c k i" X g2 * BG Our 0, p n, Non-cancellation phenomena Since g2 ofl".O, t h e product 135 i t f o l l o w s t h a t t h e i n d u c e d p r i n c i p a l G-bundle over Thus, t h e a s s o c i a t e d bundle w i t h f i b r e E l x G. F2 El is El is X F 2' S i n c e induced bundles commute w i t h passage t o a s s o c i a t e d bundles, we i n f e r that E12 = El where e q u a l i t y means homeomorphism. X F2 Similarly, using t h e f a c t t h a t glo f2 = 0, we f i n d E12 = E2 X F 1' so t h e theorem i s proved. I n o r d e r t o make use of Theorem 3.4 t o manufacture n o n - c a n c e l l a t i o n examples i n v o l v i n g s p a c e s of f i n i t e t y p e , we w i l l b e e s p e c i a l l y i n t e r e s t e d i n t h e case t h a t F = F = F 1 2 is a s p a c e of f i n i t e t y p e . An immediate consequence of t h e p r e v i o u s theorem is t h e n t h e following. Theorem 3.5. Suppose F + Ei + X, i = 1, 2, are quasiprincipal G-bundlesof the same genus with BG € NH, and suppose t h a t the t o t a l spaces quasifinite. Then El F x and E2 x F x F and E2 x F and E2 me are homeomorphic. Moreover, i f a l l spaces involved are differentiable manifolds, and i f then El El i s a Lie group, G are diffeomorphic. W e w i l l now develop a c r i t e r i o n , which w i l l e n a b l e u s , i n t h e c a s e of p r i n c i p a l G-bundles over s p h e r e s , t o deduce t h a t two f i b r a t i o n s belong t o t h e same genus. More g e n e r a l r e s u l t s , i n v o l v i n g p r i n c i p a l bundles over suspensions, may b e found i n [ 4 4 ] , Remark 2. L e t then (F,o) be a p o i n t e d CW-complex, (under composition) of self-homotopy e q u i v a l e n c e s of e: H(F) + F H(F) F t h e H-space and Applications of localization theory I36 the evaluation map - h e It i s c l e a r t h a t is a f i b r a t i o n with f i b r e self-homotopy equivalences of when h ( o ) , h € H(F). F. Ho(F), the space of pointed Moreover, i n t h e important s p e c i a l case o F = G , a topological group, and i s t h e i d e n t i t y of G, t h e r e i s a canonical s e c t i o n s: G -+ H(G), given by which together with t h e f i b r e i n c l u s i o n of H(G) Now t h e a c t i o n of on Ho(F) ho.h.hil, I-+ n n-1 (H(F)). on H(F), given by ho € Ho(F), h € H(F), Ho(F) ( i n f a c t , of no(Ho(F)) Furthermore, t h e r e i s an a c t i o n of n n- 1(F) given by (ho,a) ho.a, I-+ yields a representation H(G) H(G) = Ho(G) x G. c l e a r l y induces an a c t i o n of no(Ho(F))) -+ as t h e t r i v i a l f i b r a t i o n (ho,h) on Ho(G) ho C Ho(F), and, by d i r e c t c a l c u l a t i o n , we have: C H (F) n n-1 (F), ( i n f a c t , of Non-cancellationphenomena Turning again to the case where F we note that, i n general, I37 is a topological group is not a no(Ho(G))-module s* G, map. However we may prove (see [ 4 0 ] ) : If the 'Scheerer' diagram Lemma 3.7. Sn-' x G where ho C Ho(G), c1 € TI and v : G (G), n-1 map, i s homotopy-comtative, lJ (ax11 ____t x G G + G i s the muZtipZication then Sx(ho.a) = ho*s*(a) y in V n-1 @(GI). Proof. (t,x) The hypothesis asserts that the maps + ho.(a(t)*x), (t,x) are homotopic. Now, s,(ho*a) t E Sn-', ho(cx(t))'ho(x), + is represented by the map u: x € G, Sn-l + H(G) given by u(t) (x) = ho(a(t)) and ho.s,(a) *x, is represented by the map v(t)(x> = v: S"-' -t H(G) given by ho(a(t)hol(x)), where hi1 is a homotopy inverse of ho. The hypothesis therefore implies that the adjoints of u and v are homotopic, and so therefore are u and v. is connected; then, clearly, G €NH and we We assume now that G of the s&ace G. ( I n [ 4 1 ] it is may speak of the p-localizations G P implicitly assumed that G is connected. That is why we spoke there of the no(Ho(G))-actions. Since H(G) when G is connected.) - Ho(G) x G, no(Ho(G)) = Localizing Lemma 3.7, we obtain: no(H(G)) precisely Applications of localization theory 138 I f the diagram Lemma 3.8. where ho(p) C Ho(G ) P is (ho(p) p-localization of an element not assumed t o be homotopic t o the ho € Ho(G)), i s homotopy-comutative, then are then fibre-homotopy equivalent. Proof. Only t h e l a s t s t a t e m e n t remains t o b e proved. i t is e v i d e n t t h a t t h e elements same o r b i t of (sp)*(ho(p)*a ) P T ~ - ~ ( H ( G ~ under )) t h e a c t i o n of But, by c l a s s i c a l t h e o r y , t h i s means t h a t Scr P and T~ (sp),(ap) From ( 3 . 9 ) , l i e i n the (Ho (Gp) 1 = r0 (H(Gp) 1 and 'ho(p) 'ap . are fibre- homotopy e q u i v a l e n t . We a r e now i n a p o s i t i o n t o e n u n c i a t e one of our c e n t r a l r e s u l t s . Let Theorem 3.10. Suppose t h a t k G be connected and l e t i s an integer prime t o the order of sn-l x G sn-jx K: G + G p(kax1); 1. and that t h e diagram , the k t h power map, i s homotopy-commutative. G-bundles c l a s s i f i e d by a G I X K with a € T ~ - ~ ( G be ) of f i n i t e order. a , ka belong t o the same genus. Then the principal Non-cancellationphenomena 139 Remark. Under the hypothesis of Theorem 3.10, it is in fact possible to prove directly that ka’ of (a’ the adjoint of a) 3 0 and this was the manner in which non-cancellation examples were first constructed. (See [45] for the case G = S3 and [441 for the general case. See also Sieradski [75] for related results using yet a different approach.) Thus, in case the base spaces of the fibrations involved are spheres, it is possible (though not necessarily desirable) to bypass the theory of localization altogether. However, Theorem 3.10 has an obvious local version and, as we show later, there are fibrations Si, i = 1, 2, over a space X which is not a sphere but such that the local version of Theorem 3.10 applies to Xp Sp” for some family of primes P, while the for p f P (cl)p, (C,) equivalence of 3 follows from other, more obvious, considerations. Proof of Theorem 3.10. then (La), and (€,ka)p If p € Il is prime to the order of are fibre-homotopy equivalent since they are both (p,k) = 1, then clearly fibre-homotopy trivial. On the other hand, if G + G KP: P P is a homotopy equivalence and we may apply Lemma 3.8 with ho(p) = K P’ Applying Theorems 3.5, 3.10 in the case G = S3, we get: Theorem 3.11. Let 3 a C T~-~(S ) have order m, l e t k be an i n t e g e r prime t o m, and suppose k(2k-l)w o z3a (3.12) Then Ea x S 3 and Eka a, x S3 Furthermore, Ea & Eka = o 3 6 +n+2(~ ) . are diffeomorphic. unless k ! +1 (mod m). 140 Applications of localization theory Proof. A s t a n d a r d homotopy c a l c u l a t i o n shows t h a t t h e diagram 3 of Theorem 3.10 (with is homotopy-commutative i f f (3.12) h o l d s . G = S ) Thus t h e f i r s t p a r t of t h e theorem f o l l o w s from Theorems 3.5 and 3.10. To prove t h e second p a r t , n o t e t h a t so t h a t a homotopy equivalence t i o n , a homotopy equivalence h: E 6: Ca a + + E i n d u c e s , by c e l l u l a r approxima- ka Cka. By a s l i g h t m o d i f i c a t i o n of t h e argument used i n Example 1 . 3 (a need not b e s t a b l e , b u t S3 is an H-space), it follows t h a t ' k 5 +1 (mod m), a s claimed. A s a c o n c r e t e example i l l u s t r a t i n g Theorem 3.11, l e t a = w , k = 7 . 3 Then c o n d i t i o n (3.12) is m e t because 3 i n t h e course of proving Theorem 2 . 1 , Eo & E70, Ew has o r d e r 3; indeed, a s noted w o C w w OC w S3 = E7w x generates x S 3 3 ng(S ) = 1213. Thus, . A second a p p l i c a t i o n of Theorem 3.5 and ( t h e l o c a l v e r s i o n o f ) Theorem 3.10 is provided by t h e s p a c e s $ s t u d i e d i n S e c t i o n 2 ( s e e Theorem 2.9). Theorem 3.13. x We have diffeomorphisms s3 = x1 x4 x s3 = x7 x s3 = X13 x s3, x3 x s3 = X6 x s3 . Proof. C,, Consider, i n t h e n o t a t i o n used i n Theorem 2.9, C 1 and 3 P. = 4 , 7 , 13. Since .rrll(BS ) = 12/15, i t is c l e a r from (2.8) t h a t (El), and and (C2lp p = 5. map of 5 have, f o r a are fibre-homotopy e q u i v a l e n t e x c e p t p o s s i b l y f o r p = 3 For t h e s e primes, we may i d e n t i f y t h e l o c a l i z e d c l a s s i f y i n g with !Lap =&Bp. P. = 4 , 7, 13, Now, i f 6' € II (S 3 ) 10 is a d j o i n t t o 6 , w e Non-cancellation phenomena 141 31 (a-1) , 12w = 0, 156' = 0. since Thus, we may apply Theorem 3.10 and t h e succeeding remark w i t h P = 13,5> t o conclude t h a t equivalent f o r p = 3 (c,), c3 a r e a l s o fibre-homotopy P p = 5. and Similarly, (5,) and c6 and belong t o t h e same genus and t h e proof i s completed by a p p e a l i n g t o Theorem 3 . 5 . Thus f a r , a l l o u r examples have been p r i n c i p a l G-bundles f o r G A s a f i n a l a p p l i c a t i o n of a L i e group and hence a t most 2-connected. Theorem 3 . 2 , we p r e s e n t examples of n o n - c a n c e l l a t i o n w i t h t h e s p a c e s involved being a r b i t r a r i l y h i g h l y connected manifolds ( s e e 1 4 0 1 ) . It is t o o b t a i n t h e s e examples t h a t we have made t h e g e n e r a l i z a t i o n ( D e f i n i t i o n 3.1) from p r i n c i p a l t o q u a s i p r i n c i p a l bundles. Theorem 3 . 1 4 . p > q es and l e t C IT n-1 be a f k e d odd number, q + Denote by 1 mod p, where e : SO(q+l) q = 2p e x i s t s , since s (Sq) - 2. n has degree such an a Let q 1 3, let s: Sq + Sq + be an element of order p and l e t a map such that SO(q+l) denotes the evaZuation map; i s a regular prime f o r p be a prime, p sO(q+l). a = s a E IT n-1 Let (SO(q+l)). Consider nka: the orthogonal Sq-bundle over i s c l a s s i f i e d by (il liil (iiil Sq IG(na) If Sn 3 {nkal(k,p) = I 2 (p-1) / 2 (k,p) = 1 then f i v ) If k 2 +1 (mod p) Proof. Bka Sn + whose associated principal SO (q+l) -bundle Then ka. G(na) + Since 11 Ba x Sq and Bka x Sq are diffeomorphic then Ba $ Bka. a = sa, and a , a a r e of o r d e r p, i t f o l l o w s t h a t 142 Applications of localization theory - a = e a , and t h e n = sq uka-en u Bka en+q, Thus ( i v ) is proved i n t h e same way a s t h e l a s t s t a t e m e n t of Theorem 3.11, a using t h e f a c t t h a t has order p. Next, we want t o show t h a t # 1 then if (k,p) if (k,p) = 1, t h e n equivalent, i f case. q qka I (qa)q qka E G(n,) iff s i n c e t h e n even G(qa) and (nka)q (k,p) = 1. (Bkalp + (Balp. Clearly, Conversely, a r e c e r t a i n l y fibre-homotopy # p , s i n c e they a r e both fibre-homotopy t r i v i a l i n t h i s It remains t o s t u d y t h e s i t u a t i o n f o r q = p. To t h i s end, c o n s i d e r t h e diagram Here, 1, is induced by t h e obvious i n c l u s i o n , r e s p e c t i v e evaluationmaps, (ep)* E* e* by t h e and by l o c a l i z a t i o n and (gp) * l o c a l i z a t i o n ' (see t h e d i s c u s s i o n f o l l o w i n g Theorem 11.3.11). by ' f i b r e w i s e Setting a' = G p ) * i * ( a ) we w i l l show t h a t under t h e a' and ka' l i e i n t h e same o r b i t of ao(Ho(Sq))-action, thereby proving t h a t P a r e fibre-homotopy e q u i v a l e n t . n-1 (Ho (Sq)) p 'TI i t follows from t h e c h o i c e s of Since nn-l+q(Sq) p, q and 0 Ep n that n n-1 (Ho (Sq)) = 0 p so t h a t ?'* is injective. Hence, i f w e now s e t a" = ? * ( a ' ) (qJP IT n-1 (H(Sjf)) and (nkcr)p Non-cancellation phenomena we a r e reduced, by Lemma 3 . 6 , t o showing t h a t same o r b i t of IT n-1 under t h e (Sq) p a" 143 and ka" l i e i n the (H ( S q ) ) - a c t i o n induced by l e f t - IT O O P T h i s l a t t e r a s s e r t i o n i s c l e a r because composition. k o a" = ka" as i s an H-space; Sq P and k C E* = no(Ho(Sq)) P P (k,p) = 1. T h i s completes t h e proof of ( i ) . as The a s s e r t i o n ( i i ) f o l l o w s (Bka)p $ (BRa)p i f now from ( i ) by observing t h a t k $kR (mod p ) . We g e t (iii) from (i) by v i r t u e of Theorem 3 . 5 , once w e have v e r i f i e d t h a t t h e bundles nka nka: Sq gk: Sn and l e t -+ p r i n c i p a l bundle. is a A: - are a l l quasiprincipal. Sq+l + BSO(q+l) jk Sn B kcr be t h e c l a s s i f y i n g map f o r t h e a s s o c i a t e d To prove t h a t BSO(q+l) Let gkofk N 0 i t s u f f i c e s t o show t h a t there making t h e f o l l o w i n g diagram commutative uZkti,Jka>- $q+l / //A (3.15) / i! i where t h e t o p l i n e i s t h e Puppe sequence of X t h e a d j o i n t of is adjoint t o X oJka? 0 and s: Sq order SO(q+l). Now gk jk ( s e e [ 4 0 ] ) . We t a k e f o r is a d j o i n t t o ka and AoZka Hence (3.15) w i l l b e commutative, provided t h a t s o k a = ka. g k o x rr 0. Our c h o i c e of n (S") n+q + BSO(q+l) n, q i s z e r o , and t h a t p, we i n f e r t h a t g u a r a n t e e s t h a t t h e p-primary component of x g ox k i s a suspension. = Thus, s i n c e gk i s of 0. On t h e o t h e r hand, w r i t i n g 'n f o r t h e p-primary component of 'TI, Applications of localization theory 144 By our choice of n , q , t h e second summand is zero; f o r 2p - 3 < n - 1 < 4p Thus (S2q+1) Jka = [i,i] o y , y C and Thus, i s a suspension. y A o Now ’TI n+q C [I ,I] TI 2q+l (BSO(q+l)) is an element of f i n i t e o r d e r prime t o h o[I,I] It follows t h a t A o J k a = X o[I,I] oy N 2 azq(SO(q+l)). p y since p > q. This completes t h e proof of t h e 0. theorem. We consider now another type of non-cancellation phenomenon, which i n v o l v e s H-spaces . Let Theorem 3.16. Ekw and XI1 have themeanings given i n Section 2 . Consider the bijections G(G2) = IX1,X2,X4,X71 % Then one has, for spacesin E~~ x E~~ Xk x = Xk = Proof. G(Sp(2)) - ( 2/15)*/I&l) -i; and G(GZ) respectivezy, E ~ , x E~~ iff IZ = iii i n (z/’iz)*/ +I xm iff iz = i; x Xn in (2/15)*/ +1 We w i l l o n l y consider t h e c a s e of t h e spaces i n o t h e r c a s e i s similar and a c t u a l l y s i m p l e r . f: 3 x?, a x m x xn Suppose given G ( G ~ ) ;t h e k, a , m y n and - 145 Non-cancellation phenomena Denote by P so the set of odd primes. Now we can assume a homotopy commutative diagram where A , B are localizations ofhomotopy equivalences. Since we may regard C and D as 2x2-matrices with entries in The map B TI 10(S3) = 22/15 22/15. Thus is represented by the P-localization ofa unimodular integral matrix. The axes of A are P-localizations of integral vectors (al,a2), (a3,a4), such that is is unimodular. The.homotopy commutativity of the left-hand square in (*) expressed by the matrix equation KC = DB over L/15. Taking determinants yields W E +mn (mod 15). given k, E, m, n - certainly Xm, Xn with W x Xi of type (units mod 15) with c F = < G(X,) Xm x Xn. in Conversely, (22/15)*/{+1}, then and by Theorem 1.14 there exists an H-space W It follows that W (3,11), and hence W c" is a 1-connected finite H-complex Xi for some i relatively prime to 15. -- -- From the first part of the proof we conclude that ill = mn and hence i = Applications of localization theory 146 in ( Z/15)*/{+1}. I t follows that - Xi $ and hence 3 x x t = x m x x . n A s our f i n a l r e s u l t , we have the following corollary. Let Theorem 3 . 1 6 . Eko and Then the powers of Ekw, XQ are related by: 2 fi) E w (ii) X: Proof. 4 Hence X 1 c= X4 - 4 - have t h e meanings given them in Section 2. Xt 'u E c1 X: 2 50 + X: c1 2 x7, X; Theorem 3 . 1 5 g i v e s X2 2 X X2 - - 7 - X4 2 - X4 7' 2 X; E 'u 2 X: 2 c1 E50 - x47 ' and Bibliography 1. J. F. Adams, The sphere considered as an H-space mod p , Quart. J. Math. 12 (19611, 52-60. 2. D. W. Anderson, Localizing CW complexes, Illinois J. Math. 16 (1972), 519-525 3. M. Arkowitz, The generalized Zabrodsky theorem, Lecture Notes in Math., Springer (1974). 4. M. Arkowitz, C. P. Murley and A. 0. Shar, The number of multiplications on H-spaces of type (3,7), preprint. 5. M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Math. 100, Springer (1967). 6. G. Baumslag, Lecture Notes on Nilpotent Groups, Amer. Math. SOC. Regional Conference Series No. 2 (1971). 7. M. Bendersky, Localization of mapping spaces, preprint. , A functor which localizes the higher homotopy groups of 8. an arbitrary CW complex, Lecture Notes in Math., Springer (1974). 9. R. Body, Homotopy types whose rational cohomology has a regular set of relations, Lecture Notes in Math., Springer (1974). 10. A. K. Bousfield, The localization of spaces with respect to homology, preprint. 11. , Homological localizations of spaces, Lecture Notes in Math., Springer (1974). 12. A. K. BousfiePd and D. M. Kan, Homotopy with respect to a ring, Proc. Symp. Pure Math. her. Math. SOC. 22 (19711, 59-64. 13. , Localization and complet on in homotopy theory, Bull. Amer. Math. SOC. 77 (1971), 1006-1010. 14. , Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer (1972). 148 15. Bibliography W. Browder, Poincar6 s p a c e s , t h e i r normal f i b r a t i o n s and s u r g e r y , Inv. Math. 17 (1972), 191-202. 16. W. Browder and E. S p a n i e r , H-spaces and d u a l i t y , P a c i f i c J . Math. 12 (1962), 411-414. 17. A. H.Copelandand A. 0. S h a r , F i n i t e n e s s and l o c a l i z a t i o n , p r e p r i n t . 18. M. L. C u r t i s and G. M i s l i n , H-spaces which a r e bundles over S 7 , J . Pure and Appl. Alg. 1 (19711, 27-40. 19. A.Deleanu, 20. A . h l e a n u , A. F r e i and P. H i l t o n , Idempotent t r i p l e s and completion, Existence of t h e Adams completion f o r CW-complexes, p r e p r i n t . preprint. , Generalized 21. 22. Adams completion, p r e p r i n t . A. Deltanu and P e t e r H i l t o n , L o c a l i z a t i o n , homotopy and a c o n s t r u c t i o n of Adams, Trans. h e r . Math, SOC. 179 (1973), 349-362. 23. E. Dror, A g e n e r a l i z a t i o n of t h e Whitehead theorem, L e c t u r e Notes i n Math. 249, S p r i n g e r (1971), 13-22. 24. , The p r o - n i l p o t e n t completion of a space, Proc. h e r . Math. SOC. ( t o a p p e a r ) . 25. 26. , Acyclic s p a c e s , Topology ( t o a p p e a r ) , H. F e d e r e r , A s t u d y of f u n c t i o n s p a c e s by s p e c t r a l sequences, Trans. h e r . Math. SOC. 82 (1956), 340-361. 27. H. H. Glover and Guido M i s l i n , M e t a s t a b l e embedding and 2 - l o c a l i z a t i o n , Lecture Notes i n Math., S p r i n g e r (1974). , Immersion 28. i n t h e m e t a s t a b l e range. Proc. h e r . Math. SOC. 43 (21, 1974, 443-448. 29. J. Harper, Quasi-regular 30. , The primes f o r H-spaces, mod 3 homotopy type of Springer (1974). preprint. F4, L e c t u r e Notes i n Math., Bibliography 149 31. J. Harrison and H. Scheerer, Zur Homotopietheorie von Abbildungen in homotopie-assoziative H-Uume, Archiv der Mathematik, Vol. XXIII (1972), 319-323. 32. J. Harrison and J. Stasheff, Families of H-spaces, Quart. Jour. Math. 22 (1971), 347-351. 33. P. Hilton, Homotopy theory mzd duaZity, Gordon and Breach, New York (1965). , Localization and cohomology of nilpotent groups, Math. Zeits. 34. 132 (19731, 263-286. , Remarks 35. on the localization of nilpotent groups, Corn. Pure and Appl. Math. XXVI (1973), 703-713. , Nilpotent actions on nilpotent groups, Proc. Austr. Summer 36. Institute (1974). 37* , On direct limits of nilpotent groups, preprint. 38. , Localization in 39. , On topology, her. Math. Monthly (to appear). direct limits of nilpotent groups, Lecture Notes in Math., Springer (1974). 40. P.-$ilton, G. Mislin and J. Roitberg, Sphere bundles over spheres and non-cancellation phenomena, J. Lond. Math. SOC. (2) 6 (1972), 15-23. 41. , H-space of rank two and noncancellation phenomena, Inv. Math. 16 (19721, 325-334. 42. , Topological localization and nilpotent groups, Bull. her. Math. SOC. 78 (19721, 1060-1063. 43. , Homotopical localization, Proc. London Math. SOC. 26 (1973), 693-706. 44. , Note Illinois J. Math. 17 (4) (1973), 680-687. on a criterion of Scheerer, Bibliography 150 45. P. Hilton and J. Roitberg, On principal S 3 -bundles over spheres, Ann. of Math. 90 (1969), 91-107. , On the classification problem for H-spaces 46. of rank two, Comm. Math. Helv. 46 (19711, 506-516. 47. P. Hilton and U. Stambach, A course i n homological algebra, Graduate Texts in Mathematics No. 4, Springer Verlag (1971). 48. P. Hoffman and G. Porter, Cohomology realizations of 91x1, Quart. J. Math. 24 (1973), 251-255. 49 * P. G. Kumpel, On p-equivalences of mod p H-spaces, Quart. J. Math. Oxford (11) 23 (1972), 173-178. 50. M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. sci. Ecole Norm. Sup. 71 (1954), 101-190. 51. G. Lieberman and D. Smallen, Localization and self-equivalences,Duke Math. Journ. 41 (1974), 183-186. 52. A. L. Malcev, Nilpotent groups without torsion, IZV.Akad. Nauk. SSSR, Math. 13 (1949), 201-212. 53. M. Mimura, G. Nishida and H. Toda, Localization of CW-complexes and its applications, J . Math. SOC. Japan 23 (19711, 593-624. , On the classification of H-spaces 54. of rank 2, Jour. Math. Kyoto Univ. 13 (1973), 611-627. 55. M. Mimura, R. O'Neill. and H. Toda, On p-equivalence in the sense of Serre, Japanese Jour. Math. 40 (1971), 1-10. 56. M. Mimura and H. Toda, Cohomology operations and the homotopy of compact Lie groups I, Topology 9 (1970), 317-336. , On 57. p-equivalences and p-universal spaces, Comm. Math. Helv. 46 (1971), 87-97. 58. G. Mislin, H-spaces mod p I, Lecture Notes in Mathematics 196, Springer, (1971), 5-10. Bibliography 59. G. Mislin, The genus of an H-spacc, Lecture Notes in Mathematics 249, Springer (1971), 75-83. , Cancellation properties of H-spaces, Comm. Math. Helv., 49 60. (2) (1974), 195-200. , Nilpotent groups with finite commutator subgroups, Lecture 61. Notes in Math., Springer (1974). 62. G. Mislin and J. Roitberg, Remarks on the homotopy classification of finite-dimensional H-spaces, J. London Math. SOC. XXII (19711,593-612. 63. E. A . Molnar, Relation between wedge cancellation and localization for complexes with two cells, J. Pure and Appl. Algebra 3 (1972), 73-81. 64. S. Oka, The homotopy groups of sphere bundles over spheres, Jour. Sci. Hiroshima Univ. 33 (1969), 161-195. 65. P. F. Pickel, Finitely generated nilpotent groups with isomorphic finite quotients, Trans. Amer. Math. SOC. 160 (1971), 327-341 66. D. Quillen, Rational homotopy theory, Ann. of Math. 90 (1969), 205-295. 67. D. L. Rector, Subgroups of finite dimensional topological groups, J. Pure and Appl. Algebra 1 (3) (19711, 253-275. , Loop structures on the homotopy type of S3, Lecture Notes 68. in Math. 249, Springer (19711, 99-105. 69. J. Roitberg, Note on nilpotent spaces and localization, Math. Zeits. (to appear). , Rational Lie algebras and p-isomorphisms of nilpotent groups 70. and homotopy types, Comm. Math. Helv. (to appear). 71. , , Nilpotent groups, homotopy types and rational Lie algebras, Lecture Notes in Math., Springer (1974). 72. H. Scheerer, On principal bundles over spheres, Indag. Math. 32 (1970), 353-355. 151 Bibliography 152 73. J. P. Serre, Groupes d'homotopie et classes de groupes abeliens, Ann. of Math. 58 (1953), 258-294. 74. A. Shar, Localizations and the homotopy groups of certain complexes, Quart. Jour. Math. 23 (1972), 249-257. 75. A. Sieradski, An example of Hilton and Roitberg, Proc. her. Math. Soc. 28 (L971), 247-253. , Square roots up to homotopy type, her. J. Math. 94 76. (19721, 73-81. 77. F. Sigrist and U. Suter, Eine Anwendung der K-Theorie in der theorie der H. R'gume, Corn. Math. Helv. 47 (1972), 36-52. 78. U. Stammbach, Homology in group theory, Lecture Notes in Math. 359, Springer (1973). 79. J. D. Stasheff, Manifolds of the homotopy type of (non-Lie) groups, Bull. Amer. Math. Soc. 75 (1969), 998-1000. , H-spaces 80. in Math. 81. from a homotopy point of Vim, Lecture Notes 161, Springer, (1970). , Sphere bundles over spheres as H-spaces mod p > 2, Lecture Notes in Math. 249, Springer (1971) , 106-110. 82. , The mod p decomposition of Lie Groups, Lecture Notes in Math., Springer, (1974). 83. D. Sullivan, Geometric topology, part I: Localization, periodicity and Galois symmetry, MIT, 1970 (mimeographed notes). a4. J. Underwood. Finite dimensional associative H-spaces and products of sphere, preprint. 85. C. T. C . Wall, Finiteness conditions for CW-complexes, Ann. of Math. 81 (1965), 56-59. 86. R. B. Warfield, Localization of nilpotent groups, Forschungsinstitut f u r Mathematik, ETH ZUrich, 1973. Bibliography 153 87. G. Whitehead, On mappings into group-like spaces, Comm. Math. Helv. 28 (1954), 320-328. 88. C. Wilkerson, Spheres which are loop spaces mod p, Proc. her. Math. SOC. 39 (3) (1973), 616-618. 89. , K-theory operations in mod p loop spaces, Math. Zeits. 132 (1973), 29-44. 90. , Rational maximal 91. , Genus and cancellation, preprint. 92. , Genus and cancellation for H-spaces, Lecture Notes in tori, preprint. Math., Springer, (1974). 93. A . Zabrodsky, Homotopy associativity and finite CW-complexes. Topology 9 (1970), 121-128. 94. , The classification of simply connected H-spaces with three cells I, 11, Math. Scand. 30 (1972), 193-210 and 211-222. 95. , On the homotopy type of principal classical group bundles over spheres, Israel J. Math. 11 (3)(1972), 96. , On 315-325. the construction of new finite CW H-spaces, Inv. Math. 16 (1972), 260-266. 97. , On the genus of finite CW H-spaces, Comm. Math. Helv. 49, 1 (1974), 48-64. 98. ,p equivalences and homotopy type, Lecture Notes in Math., Springer (1974). 99. , O n rank 2 mod odd H-spaces, New developments in topology (ed. G. Segal), C.U.P., Lond. Math. SOC. Lecture Note Series 11 (1974). Index Abelianization 20 F-monomorphism 83 Action (of nilpotent group) 34 Function space 48, 63, 77 Fundamental Theorem 7 Basic commutators 5 Bousfield-Kan 1, 47 Generalized Serre class 43, 69 Browder-Novikov Theorem 109 Generic property 104, 115 Genus 32, 102, 104, 133 Class (of abelian groups) 43 Genus invariant 105 Cofibre sequence 48, 58, 77 Cofinite (set of primes) 26,84 Half-exact functor 41 Cohomology of groups 1, 15 €!all 5, 25, 26 Commensurable 24 Hilton 1, 40, 67 1-Connected 47, 52 H-map 109 Cooperation map 92, 128 Homological dimension (of quasifinite space) 79, 114 CW-complex 47 Homology decomposition 100 Homology of groups 20, 39 Decomposition (= partition) 29, 31 Homotopy-associative H-structure 120 Dror 48, 76 Homotopy category 47, 94 Hopf algebra structure 110, 112, 119 0-Equivalence 61 H-space 50, 90, 104, 119, 128 Evaluation map 136 Induced H-structure 109 Fibre sequence 48, 58, 77 0-Isomorphism 61 Finite H-space 122, 130, 131 Finite type 59, 79 Lazard 1 Five-term exact sequence 20 0-Local 4 Index Local cell 48 155 Orthogonal bundles 141 Localization 1 functor 2 p-fibre homotopy equivalent 133 of connected CW-complexes 52 p-local 4 of homotopy types 47 p-universal 4 of nilpotent complexes 72 P-bijective 5, 92 of nilpotent groups 3, 19 P-equivalence 59 Lower central series 3, 20, 67 w-series 34, 67 Lyndon-Hochschild-Serre spectral sequence 14, 41, 44 P-injective 5, 23 P-isomorphism 1, 5, 24 P-local abelian group 7 cell 57 group 1, 4 Malcev 1 space 41, 52, 72 Mayer-Vietoris sequence (in homotopy) 87, 95 sphere 57 Milnor 26, 32 P-localization 1, 47 Mislin 33, 104, 130 theory 4, 7 Mixing homotopy types 94 P-localizing functor 4 Moore-Postnikov system 68 map 4, 41, 52, 1 2 P-surjective 5, 23 Nilpotency class 3, 34 P-universal 4 Nilpotent action 2 , 34 Partition (of the set of primes) 28, 51 complex 62 Pickel 33 homotopy types 1 Poincar6 complex 108 group 1, 3 Postnikov decomposition 65 Lie group 62 map 67 k-Power map 114 N k -Power map 114 space 48, 62 Primitive elements (of coalgebra) 112 Non-cancellation phenomena 102, 133, 144 Index 156 P r i n c i p a l refinement 65 Tensorial c l a s s 43 P r o f i n i t e completion 101 Thom s p a c e 108 Pullback 21, 2 6 , 28, 3 0 , 8 6 , 8 8 Type (of H-space) 122 i n homotopy t h e o r y 94 Theorem 79, 82 0-Universal 4 Pushout 28 Upper c e n t r a l s e r i e s 3 , 20 Q u a s i f i n i t e (space) 4 9 , 79, 89 Weak pullback 9 4 , 96 Q u a s i p r i n c i p a l (bundle) 133 Zabrodsky 50, 9 4 , 1 0 2 , 129 Rank 122 Rank 2 H-space 1 0 2 , 122 R a t i o n a l i z a t i o n 2 , 4 , 2 4 , 26 Reducible (space) 1 0 8 Roitberg 2 6 , 6 2 , 77 Scheerer diagram 137 S e r r e c l a s s 2 , 43 Simple (space) 62 S p l i t e x t e n s i o n 37 S-reducible (space) 1 0 8 Stallings-Stammbach Theorem 2 1 S t m b a c h 43 S t i e f e l manifold 57 Symplectic group 122