Localization of Nilpotent Groups and Spaces 9780720427165, 0720427169, 0444107762, 9780444107763, 9780080871264

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LOCALIZATION OF NILPOTENT GROUPS AND SPACES

LOCALIZATION OF NILPOTENT GROUPS AND SPACES

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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.

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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

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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')


-

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


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

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

- -

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*

,

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