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English Pages 167 Year 1974
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
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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