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Chapter II Localization of Homotopy Types
Chapter I1 Localization of Homotopy Types Introduction In this chapter we apply localization methods to homotopy theory. We use the definitions of local groups and localization given in Chapter I, in order to introduce the corresponding notions into homotopy theory; and we prove the basic theorems that relate to localization in homotopy theory. These theorems find many applications in homotopy theory, but we will reserve the applications to Chapter 111. Our definition of a P-local (pointed) space is simply that its homotopy groups should be P-local groups. This definition could be made quite generally for an arbitrary pointed space. However we are concerned to obtain a localization theory and also to obtain useful criteria for establishing when a given map of spaces does in fact P-localize.
It is
therefore necessary for us to work in a restricted category of (pointed) topological spaces. It is also necessary for us to work in a homotopy category (that is, in a category in which the morphisms are homotopy classes of continuous maps), since our procedures for establishing the existence of a localization theory will all operate up to homotopy. A more general treatment, valid in the semisimplicial category, has been given by Bousfield-Kan u 4 ] . We will always suppose that our spaces have the homotopy type of CW-complexes. In Section 1, we present a localization theory in the homotopy category H1
of 1-connected CW-complexes. We establish two fundamental
theorems in H1, namely that every object of the category does admit a P-localization, and that we can detect the P-localizing map
f: X
+.
Y either
through the induced homotopy homomorphisms, which should also P-localize, or through the induced homology homomorphisms, which should also P-localize.
In the course of establishing that there is a localization theory in H1,
Localization of homotopy types
48
we a c t u a l l y c o n s t r u c t t h e l o c a l i z a t i o n of a given CW-complex i m i t a t i n g t h e c e l l u l a r c o n s t r u c t i o n of t h a t of a ZocaZ c e l l .
X
by
X, r e p l a c i n g t h e i d e a of a c e l l by
The f a c t t h a t t h e l o c a l i z a t i o n can b e d e t e c t e d e i t h e r
through homotopy o r through homology h a s t h e immediate consequence t h a t we may l o c a l i z e f i b r e and c o f i b r e sequences i n
H1,
I n S e c t i o n 2 , we d e s c r i b e a broader homotopy c a t e g o r y i n which we w i l l a l s o b e a b l e t o e s t a b l i s h a s a t i s f a c t o r y l o c a l i z a t i o n theory.
It t u r n s
o u t t h a t we would wish t o e n l a r g e t h e c a t e g o r y t o which we apply l o c a l i z a t i o n methods from our o r i g i n a l c a t e g o r y confined t o o b j e c t s of
H1,
even i f our main i n t e r e s t were
For, i n o r d e r t o prove theorems about l o c a l i z a t i o n of
H1.
i t is v e r y u s e f u l t o employ function-space methods, and
H1,
t h e function-space c o n s t r u c t i o n t a k e s u s o u t s i d e t h e c a t e g o r y . i t is t r u e t h a t i f
and i f into
W
X
X
However,
is a niZpotent s p a c e , i n a s e n s e defined i n S e c t i o n 2 ,
is f i n i t e , t h e n t h e f u n c t i o n space Xw is a g a i n n i l p o t e n t .
f a c t s about t h e category
of pointed maps of
S e c t i o n 2 concerns i t s e l f w i t h some b a s i c
NH of n i l p o t e n t s p a c e s , and may be regarded i n
p a r t as propaganda f o r t h e u s e of t h i s c a t e g o r y i n homotopy theory. i t has a l r e a d y been shown by Dror [23] t h a t
homotopy t h e o r y .
W
NH
Indeed,
is a s u i t a b l e c a t e g o r y f o r
Roughly speaking, one may s a y t h a t most of t h e t e c h n i q u e s
of homotopy t h e o r y which have been developed s i n c e t h e p u b l i c a t i o n of S e r r e ' s t h e s i s can a l l be c a r r i e d o u t i n t h e c a t e g o r y
NH
techniques were of course o r i g i n a l l y formulated i n
although many of t h o s e
H1,
The b a s i c theorem
proved i n S e c t i o n 2 is t h a t a space is n i l p o t e n t i f and only i f i t s Postnikov tower admits a p r i n c i p a l refinement. t h e category
Mi
It is t h i s theorem which e x p l a i n s why
is s u i t a b l e f o r homotopy theory; f o r t h e given refinement
of t h e Postnikov tower may be used i n p l a c e of t h e Postnikov tower i n t h o s e arguments i n which t h e c r u c i a l f a c t which is r e q u i r e d is t h a t t h e f i b r a t i o n s which appear i n t h e tower should b e induced o r p r i n c i p a l .
Introduction
49
However, it should be pointed out that the category NH has eertain defects over the category H1.
One of the defects is that it is not closed
under the mapping cone operation. This defect has a serious consequence in Section 3 .
We also describe in Section 2 how to relativize the notion of a
nilpotent space to obtain that of a nilpotent map. In Section 3 we generalize the theorems of Section 1 from the category H1
to the category NH. Formally, we get the corresponding
formulations of the two fundamental theorems of Section 1. However there is an important difference in the way in which we construct the localization of an object. For, whereas in the category H1 we are able to proceed cellularly, since the mapping cone construction respects the category H1, we cannot in the nilpotent case proceed cellularly, since the mapping cone construction would take us outside the category. It is therefore necessary for us to proceed homotopically rather than cellularly in constructing the localization. In this way, of course, we lose much of the conceptual simplicity of the construction in Section 1. Section 4 is a brief technical section in which we introduce the idea of a quasifinite complex in Mi. relative to the category H1.
Here again we see a certain disadvantage
For if X
is a 1-connected CW-complex whose
homology groups are all finitely generated, and vanish above a certain dimension, then X itself has the homotopy type of a finite complex. If we discard the condition of simple-connectivity, we can no longer assert this conclusion. Indeed, we have the obstruction theory of Wall which enables
US
to discuss the question whether a CW-complex X whose homology looks like that of a finite complex in fact has the homotopy type of a finite complex. Thus we are led to introduce the concept of a quasifinite CW-complex, meaning a nilpotent CW-complex X such that the homology of X is finitely
Localization of homotopy types
50
generated i n each dimension and v a n i s h e s above a given dimension.
We prove
t h a t such a q u a s i f i n i t e complex always h a s t h e homology type of a f i n i t e complex. I n S e c t i o n 5 we prove t h e fundamental p u l l b a c k v a r i o u s v a r i a n t s as consequences of t h a t theorem.
theorem and
Here we l e a n v e r y h e a v i l y
on t h e r e s u l t s of Chapter I . The fundamental p u l l b a c k theorem a s s e r t s t h a t t h e p o i n t e d set i s t h e p u l l b a c k of t h e pointed sets
[W,Xl set
1, provided t h a t
[W,X
W
{[W,Xpll
over t h e p o i n t e d
i s a f i n i t e connected CW-complex and
a n i l p o t e n t complex of f i n i t e type.
X
is
T h i s a s s e r t i o n f a l l s i n t o two p a r t s .
The f i r s t p a r t s t a t e s t h a t given two maps
f,g: W
-+
X
such t h a t
= e g: W + X f o r a l l primes p, t h e n f = g . T h i s p a r t of t h e a s s e r t i o n P P P’ does not r e q u i r e t h a t X b e of f i n i t e t y p e . The second a s s e r t i o n s t a t e s e f
that i f
f(p): W
-+
c l a s s of t h e map
X
P
a r e maps, f o r a l l primes
r f(p): W P
-+
Xo
p , such t h a t t h e homotopy
i s independent of
i s t h e r a t i o n a l i z a t i o n map, t h e n t h e r e exists a map
e f P
2
f(p)
f o r a l l primes
t h e condition t h a t
W
p.
p , where
f: W
-+
X
rp: Xp
-+
Xo
such t h a t
We show by an example t h a t we cannot omit
should be f i n i t e .
However, provided t h a t
W
is
n i l p o t e n t , we may i n f a c t weaken t h e hypotheses of t h e p u l l b a c k theorem by simply r e q u i r i n g t h a t
W
be q u a s i f i n i t e .
I n S e c t i o n 6 we make a p r e l i m i n a r y s t u d y of t h e l o c a l i z a t i o n of H-spaces.
Our main r e s u l t i n t h i s s e c t i o n i s a g e n e r a l i z a t i o n of t h e p a r t
of t h e fundamental theorem of Chapter I which t e l l s u s how t o d e t e c t t h e P - l o c a l i z a t i o n of a n i l p o t e n t group i n terms of t h e P - b i j e c t i v i t y of t h e l o c a l i z i n g homomorphism. I n S e c t i o n 7 w e formulate t h e fundamental mixing technique of Zabrodsky w i t h i n t h e c o n t e x t of t h e l o c a l i z a t i o n of n i l p o t e n t s p a c e s .
The
Introduction
51
p a r t i c u l a r r e s u l t which w e emphasize is t h a t , given n i l p o t e n t spaces
X, Y
with equivalent r a t i o n a l i z a t i o n s , and given a p a r t i t i o n of t h e primes
Il
= P
u
Q , then t h e r e e x i s t s a n i l p o t e n t space
2
such t h a t
2
P
= Xp
and
ZQ = YQ. We make very considerable a p p l i c a t i o n of t h e r e s u l t s of t h e l a s t two s e c t i o n s i n Chapter 111.
Indeed, we a r e r a t h e r l i t t l e concerned t o g i v e
e x p l i c i t examples and a p p l i c a t i o n s i n t h i s Chapter i n view of t h e f a c t t h a t Chapter 111 is e n t i r e l y concerned with applying t h e theory of Chapter 11.
Localization of homotopy types
52 1.
Localization of 1-connected CW-complexes. We work i n t h e pointed homatopy category
CW-complexes. X
X C H1,
If
and i f
P
is a family of primes, we say t h a t
is P-zocal i f the homotopy groups of
W e say t h a t
f: X + Y
H1
in
H1 of 1-connected
X
P-localizes
a r e a l l P-local a b e l i a n groups. X
if
Y
i s P-local and*
f*: [Y,Z] z [X,Zl f o r a l l P-local
2 C H1.
Of course t h i s u n i v e r s a l property of
c h a r a c t e r i z e s i t up t o canonical equivalence: both P-localize H1
with
in
H1.
X
hfl = f 2 .
if
fi: X
-+
then t h e r e e x i s t s a unique equivalence
Yi,
f i = 1, 2 ,
h : Y1
PI
Y2
in
W e w i l l prove t h e fallowing two fundamental theorems
The f i r s t a t t e s t s t h e e x i s t e n c e of a l o c a l i z a t i o n theory i n
H1
and the second a s s e r t s t h a t we may d e t e c t t h e l o c a l i z a t i o n by looking a t induced homotopy homomorphismor induced homology homomorphisms. Theorem 1A.
( F i r s t fundamental theorem i n H1.)
Every
X
We T r i t e Theorem 1B.
Let
i n HI e: X
admits a P-localization.
+
Xp
f o r a f i x e d choice of P-localization
of
X.
(Second fundamentaZ theorem i n H1.l f: X * Y
i n H1.
( i ) f P-localizes (ii) nnf: n X (iii) Hn f : HnX
Then the following statements are equivalent:
X;
nnY
P-localizes f o r a l l
n 3 1;
-+ H Y
P-localizes f o r a l l
n 2 1.
-t
We w i l l prove Theorems l A , 18 simultaneously.
*We w r i t e , a s usual, [Y,Z] of maps from Y t o 2.
f o r H1(Y,Z),
W e r e c a l l from
t h e s e t of pointed homotopy classes
Localization of I -connectedCW-complexes
Proposition 1.1.9 that a homomorphism
B
if and only if
is P-local and
$:
A
+
B
53
of abelian groups P-localizes
is a P-isomorphism; this latter condition
@
+
means that the kernel and cokernel of
belong to the Serre class C of
abelian torsion groups with torsion prime to P.
Thus to prove that (ii)
(iii)
in Theorem 1 B above it suffices to prove the following two propositions. Proposition 1.1. Let Y C H1. only if H Y
is P-locat f o r a l l n
Proposition 1.2.
all
Then
f: X
Let
-+
n E 1 if and only if
TI
Y is P-local f o r a l l
Then
nn(f)
is a €'-isomorphism f o r aZZ
for all n E 1 if and only if Hn(Y;Z/p) = 0 disjoint from P.
n 2 1.
P
is a P-local abelian group, so are the homology groups
of the Eilenberg-MacLane space K(A,m).
K(A,m-l)
-+
-+
K(A,m),
with E
if H (A,m-1; Z/p) = 0 n Now let
...
of Y.
is a P-local abelian
1. It now follows, by induction
Hn(A,m)
E
is P-loca?
for all n 2 1 and all primes
Now, by Proposition 1.1.8, if A
group, so are the homology groups H A, n on m, that if A
1 if and
is a P-isomorphism for
Proof of Proposition 1.1. We first observe that HnY
p
?
2 1.
Y in H1,
Hn(f)
n
For we have a fibration
contractible, from which we deduce that,
for all n E 1, then Hn(A,m; Z/p) = 0 -+
Ym
-+
Ymm1-+
...
Thus there is a fibration K(nmY,m)
Thus, if we assume that
nnY
for all n 1 1.
Y2 be the Postnikov decomposition
* Ym
+
Ym-l, and Y2 = K(n 2Y , 2 ) .
is P-local for all n P 1, we may assume inductively
that the homology groups of Ym-l
are P-local and we infer (again using homology
with coefficients in Z / p , with p disjoint from P) that the homology groups of Ym
are P-local.
Since Y
-+
Y,
is m-connected, it follows that HnY
P-local f o r all n 1 1. To obtain the opposite implication, we construct the 'dual' Cartan-Whitehead decomposition
is
Localization of homotopy types
54
There i s then a f i b r a t i o n if we assume t h a t
K(amY,m-l)
i s (m-1)-connected,
Y(m)
*
Y(m+l)
i s P-local f o r a l l
HnY
t h a t t h e homology groups of Y(m)
+
n
T
m
and
Y(2) = Y .
Since
Y Z H Y(m) m
Y(m) = n Y m m
P)
t h a t the homology groups of
s t e p i s complete and
T
Y 2 H Y(n) n
and i s P-local.
to
C
a r e P-local,
p
disjoint
so t h a t the i n d u c t i v e
is P-local.
Proof of Proposition 1 . 2 . mod C , where
Y(m+l)
and
a
Thus we i n f e r (again using homology w i t h c o e f f i c i e n t s i n Z / p , w i t h from
Thus,
2 1, w e may assume i n d u c t i v e l y
a r e P-local.
i t follows t h a t
Y(m)
Since a P-isomorphism
is an isomorphism
i s t h e c l a s s of a b e l i a n t o r s i o n groups with t o r s i o n prime
P, Proposition 1 . 2 is merely a s p e c i a l case of t h e c l a s s i c a l Serre theorem.
We have thus proved t h a t ( i i ) t h a t (ii) * (i). counterimage of
f.
(iii) in Theorem 1 B .
We now prove
The o b s t r u c t i o n s t o t h e e x i s t e n c e and uniqueness of a g: X
-+
under
Z
Now, given ( i i ) (or ( i i i ) ) ,H,f of t h e map
Q
f*: [Y,Z]
C C , where
-+
[X,Z]
H,f
l i e in
H*(f;n,Z).
r e f e r s t o t h e homology groups
Thus (i) follows from the u n i v e r s a l c o e f f i c i e n t theorem f o r
cohomology and Proposition 1.1.8(v).
We now prove Theorem 1 A . of
f: X
-c
Y
in
More s p e c i f i c a l l y , we prove t h e e x i s t e n c e
s a t i s f y i n g (iii). Since we know t h a t (iii) 3 (i), t h i s
H1
w i l l prove Theorem 1 A .
Our argument is f a c i l i t a t e d by t h e following key
observation. Proposition 1.3.
Let
we have constructed
U be a f u l l subcategory of f: X
-+
Y
automatically y i e l d s a functor
satisfying l i i i ) . L: U
transfornation from the embedding U
-+
H1,
5 H1
Then the assignment X * Y
f o r which t o L.
f o r whose objects X
HI
f
provides a natural
Localization of 1-connectedCW-complexes Proof of P r o p o s i t i o n 1 . 3 .
g: X
Let
-+
in
X'
U.
55
W e t h u s have a
diagram
Y in
H1
with
f, f'
Y'
satisfying ( i i i ) .
Since
f
s a t i s f i e s ( i ) and
P-local by P r o p o s i t i o n 1.1, we o b t a i n a unique ( i n
H1)
Y'
is
h 6 [Y,Y'] making
t h e diagram
I t i s now p l a i n t h a t t h e assignment
commutative.
desired functor
X
rf
Y, g* h
yields the
L.
We e x p l o i t P r o p o s i t i o n 1 . 3 t o prove, by i n d u c t i o n on may l o c a l i z e a l l n-dimensional
CW-complexes i n
H1.
n, t h a t w e
.If n = 2, t h e n such a
complex i s merely (up t o homotopy e q u i v a l e n c e ) a wedge of 2-spheres
x = v s2 , a
where
a
runs
through some index s e t , and w e d e f i n e
Y = VM( ZP,2), a
where map
M(A,2) f: X
+
fo: Xo + Y o dim X = n
Y
i s t h e Moore s p a c e having satisfying ( i i i ) . satisfying ( i i i ) i f
+ 1, X
€
H1.
H M = A. 2
There i s t h e n an e v i d e n t
Suppose now t h a t w e have c o n s t r u c t e d dim X
5 n , where
n 1 2 , and l e t
Then w e have a c o f i b r a t i o n
vsn R, xn i . x By t h e i n d u c t i v e h y p o t h e s i s and P r o p o s i t i o n 1 . 3 , we may embed (1.5) diagram
i n the
Localization of homotopy types
56
where fo, fl satisfy (iii) and the square in (1.6) homotopy-commutes. embeds Yo
Thus if j (1.6) by
f: X
+
Y
in the mapping cone Y of h, then we may complete
to a homotopy-commutative diagram and it is then easy
to prove (using the exactness of the localization of abelian groups) that f: X
+.
Y
also satisfies (iii).
(iii) if X
Thus we may construct f: X
+
Y satisfying
is (n+l)-dimensional, and the inductive step is complete.
It remains to construct f: X
-+
Y satisfying (iii) if X
is
infinite-dimensional. We have the inclusions
x2
cx3 2 . .
and may therefore construct
where f", fn+l
satisfy (iii).
We may even arrange that (1.7)
is strictly
If we define Y = UY(n), with the weak topology, n and the maps fn combine to yield a map f: X -+ Y which again
commutative for each n. then Y t H1
obviously satisfies (iii). Thus we have proved Theorem lA in the strong form that, to each X
in H1,
there exists f : X
-+
Y
in H1 satisfying (iii).
Finally, we complete the proof of Theorem 1B by showing that (i) = (iii).
Given f: X
-+
Y which P-localizes X, let fo: X
constructed t o satisfy (iii).
Then fo: X
+
Y
which one immediately deduces the existence of
-+
Y
be
also satisfies (i), from a
homotopy equivalence
Localization of 1-connectedCW-complexes
u: Yo
+.
Y with uf
= f.
It immediately follows that f also satisfies
(iii). Thus the proofs of Theorems l A , 1B are complete. We note that our proof of the first fundamental theorem does much more than establish the existence of a localization theory in H1;
it provides
us with a combinatorial recipe for constructing the localization of a given CW-complex. The Moore spaces S;
=
may be called P-ZocaZ n-spheres,
M(Zp,n)
and a cone on a P-local n-sphere may be called a P-ZocaZ (n+l)-ceZz.
Then,
given a cellular decomposition of X, we may--as shown in the proof of Theorem 1A--construct a P-ZocaZ-ceZZuZar decomposition of Xp by 'imitation'; that is, whenever, in building up X, we attach an n-cell to Y f : Sn-'
of a map to Yp
+.
$, we attach a P-local n-cell
Y, then, in building up
by means of the localized map
sn-l
fp:
(say) by means
P
-+
Yp.
We illustrate this
procedure by means of an example. Example 1.8. Let Sm
-+
E
-+
Sn
be an Sm-bundle over
S",
n
z
m, and let
m
a C T ~ ~ - ~) ( be S the characteristic (homotopy) class of the bundle.
it is well-known that E
Then
admits the cellular decomposition
E
=
sm ua
en
u
emh,
where we will not trouble to specify the attaching map f o r the top-dimensional cell. If m 1 2 we may now localize E Ep = S;
Ua
P
to obtain
nntl-n ep U ep
.
Let us consider,in particular, the Stiefel manifold Vn+l , 2 vectors to
S",
and let us assume that n
over Sn with fibre Sn-', a = 2 6 T~-~(S~-').
so
We obtain
of unit tangent
is even. Then 'n+1,2
that we may take E = V
n+1,2'
m = n
fibres
-
1, and
Localization of homotopy types
58
Suppose now t h a t
P
is t h e f a m i l y of a l l odd primes. Sn-'
s o t h a t i t is e a s y t o s e e t h a t
n ep
2p
Then
is i n v e r t i b l e ,
h a s t h e homotopy type o f a p o i n t .
It t h u s f o l l o w s from (1.9) t h a t
if
E = V n+1,2, n
even, and
f u r t h e r follows t h a t i f [E,Y]
%
P
is t h e f a m i l y of odd primes.
It t h e r e f o r e
is a P - l o c a l space t h e n
Y
[I? ,Y] '2 [Sp 211-1,Y] '2 [ S 2n-1 , y ] =
P
Here t h e most s t r i k i n g f a c t is t h a t t h e s e t
[E,Y]
2n-1
y.
h a s acquired a v e r y n a t u r a l
a b e l i a n group s t r u c t u r e and h a s simply been i d e n t i f i e d w i t h a c e r t a i n homotopy group of
Y.
We have t h e f o l l o w i n g immediate c o r o l l a r i e s of t h e second fundamental theorem. C o r o l l a r y 1.10.
Let
F
i s a f i b r e sequence i n
Proof.
-t
E + B
be a f i b r e sequence i n H1.
Then Fp
+
H1'
Of c o u r s e we a r e o n l y making t h e a s s e r t i o n up t o homotopy,
so t h a t our c l a i m amounts t o s a y i n g t h a t e x a c t sequence of homotopy groups.
Fp
-+
Ep
-t
Bp
induces t h e u s u a l
However t h i s f o l l o w s immediately from
P r o p o s i t i o n 1 . 1 . 7 and t h e e q u i v a l e n c e of (i) and (ii) i n Theorem 1B. S i m i l a r l y , r e p l a c i n g (ii) of Theorem 1B by ( i i i ) of Theorem lB,
we o b t a i n C o r o l l a r y 1.11. Let
5 + Yp
-+
Cp
Ep
X
+
Y
-+ C
be a cofibre sequence i n H1.
i s a cofibre sequence.
Then
+
B
P
Localization of I -connected CW-complexes
59
We now introduce an important definition. Definition 1.12.
A map
f: X
is called a P-equivalence if
in H1
Y
--f
fP
is a (homotopy) equivalence. Theorem 1.13. Let
primes.
f: X
Y in
+
H ,and l e t 1
P
be a non-empty f a m i l y of
Then the following statements are equivalent: (il (iil
f i s a ?-equivalence; f i s a p-equivalence f o r a l l
Q C
(iii) TInf i s a P-isomorphism f o r a l l n ( i v ) H f i s a P-isomorphism f o r a l l n
Proof.
By Whitehead's Theorem f
P; 2 1; 2 1.
is a P-equivalence iff n f n P
is an isomorphism for all n 1 1. By Theorem 1B it follows that
Thus the equivalence (i) (i)
Q
(iv).
(iii) follows from Theorem 1.3.1.
Theorem 1.3.12 ensures that
(nnf)p
TI
f = (nnf)p. n P
Similarly
is an isomorphism iff
(nnf) is an isomorphism for all p C P. Thus the equivalence of (i) P and (ii) readily follows. A further refinement is possible in the case in which
are of f i n i t e t y p e , that is, n
?
1. Notice that, in
H1,
IT
X and nnY
n
X and Y
are finitely generated for all
this is equivalent to asking that H X and
HnY be finitely generated. Theorem 1.14. Let
f: X
-+
Y i n H1 with X , Y
P be a non-empty f a m i l y of primes. f,:
Hn(X; Z/p) zz Hn(Y; Z/p) f o r Proof. Let
Z/p
f: X
+
Y
Then
of f i n i t e type, and l e t
f i s a P-equivalence i f f
p € P.
be a P-equivalence. Then, since, for p C P,
is P-local, it follows from Theorem 1B and PropOsftiOn I.1.8(ii)
we have a comutative diagram
that
Localization of homotopy types
60
Thus
is a n isomorphism; n o t i c e t h a t t h e i m p l i c a t i o n we have proved does
f,
not require t h a t
It is i n t h e o p p o s i t e i m p l i c a t i o n
b e of f i n i t e type.
X, Y
t h a t t h i s condition plays a decisive r o l e . induces type,
Hn(X; Z f p )
f,:
Hn(Y;
We w i l l prove t h a t
H f n
Consider t h e diagram, f o r each
ZIP),
Thus w e suppose t h a t
n 2 1, p 6 P , where
i s a p-isomorphism,
f: X + Y
are of f i n i t e
X, Y
Hnf: HnX
H Y , n 11.
-+
n 5 1,
$
(1.15)
f
*n
t h e v e r t i c a l homomorphisms being induced by
f.
i t f o l l o w s t h a t we must prove t h a t each
is b i j e c t i v e and each :f
It f o l l o w s immediately from (1.15) t h a t e a c h
surjective. and each
f i
surjective.
It a l s o f o l l o w s from (1.15)
Suppose, i n d u c t i v e l y , t h a t we have shown Then
f:
By P r o p o s i t i o n 1 . 1 . 8 ( i i ) ,
Hrf: HrX
--t
fi,
is a p-isomorphism f o r
HrY
It t h u s f o l l o w s from (1.15) t h a t
bijective.
that
..., f C 1 r 5 n f:
-
f;
is injective
fi
is bijective.
bijective,
1, so t h a t
n Z 2.
f:-l
is
is b i j e c t i v e , and t h e i n d u c t i v e
s t e p i s complete. Since
H f
Theorem 1.13, t h a t Remark. if
f: X
i s a p-isomorphism
f
for a l l
p € P , we conclude, from
is a P-equivalence.
Theorems 1.13 and 1 . 1 4 show t h a t we have t h r e e p r a c t i c a l ways t o t e s t +
Y
p-isomorphism
i s a P-equivalence. for a l l
We may t r y t o show t h a t
n Z 1, p € P; we may t r y t o show t h a t
II
f Hnf
is a
is a
Localizationof 1-connected CW-complexes
p-isomorphism for all n 1 I, p t P; or, if X, Y
are of finite type, we
have the potentially most practical procedure, namely to try to show that f,:
Hn(X; Z/p)
-+
Hn(Y; Z/p)
is an isomorphism for all n 1 1, p € P.
In the case when P is empty, we have the following evident modification of Theorem 1 . 1 3 . Theorem 1.16. Let
f: X
-+
Y i n H1.
Then the f o l l o w h q statements are
equivalent: (il (iil
f
i s a 0-equivaZence;
nnf
f i i i l Hnf ( i v ) f,:
i s a O-$somorphism f o r a12 n 2 1; is a 0-isomorphism f o r a l l Hn(X;Q)
2
H (Y;Q)
n 2 1;
for a12 n 2 1.
61
Localization of homotopy types
62 2.
Nilpotent spaces. It turns out that the category
is not adequate for the full
H1
exploitation of localization techniques.
This is due principally to the fact
that it does not respect function spaces. We know, following Milnor, that if X
is a (pointed) CW-complex and W
a finite (pointed) CW-complex, then
the function space Xw of pointed maps W
-+
X
has the homotopy type of a
CW-complex. However its components will, of course, fail to be 1-connected even if X Xw
is 1-connected. However, it turns out that the components of is nilpotent. Moreover, the category of nilpotent
are nilpotent if X
CW-complexes is suitable for homotopy theory (as first pointed out by E. Dror), and for localization techniques [14,83]. Definition 2.1.
A connected CW-complex X
and operates nilpotently on
IT
is nilpotent if nlX
X for every n
is nilpotent
>_ 2.
n
Let NH be the homotopy category of nilpotent CW-complexes. Plainly NH
2
H1.
Moreover, thesimple CW-complexes are plainly in MH; in particular,
NH contains all connected Hopf spaces. It isalsonotdifficult tosee (Roitberg [69])
that, if G
is a nilpotent Lie group (not necessarily connected) then its is nilpotent in the sense of Definition 2.1.
classifying space BG
We prove
the following basic theorem, which provides us with a further rich supply of nilpotent spaces. Theorem 2.2.
Let F
Then F C NH if
Proof.
-i
E
f
B be a fibration of connected CW-compzexes.
E €NH.
We exploit the classical result that the homotopy sequence
of the fibration is a sequence of is
IT
IT
E-modules. 1
E-nilpotent of class icy then 1
IT
n
F
is
IT
We will prove that, if nnE F-nilpotent of class 5c 1
+
1.
Nilpotent spaces
63
(A mild modification of the argument is needed to prove that if
nilpotent of class Zc, then
71
F is nilpotent of class 5c 1
+
IT
E
is
1
1; we will
deal explicitly with the case n 1 2.) We will need the fact that and that the operation of I T ~ Eon
TI
E 1
F
IT
operates on
IT
B
through f,,
is such that
It will also be convenient to write IF, IE for the augmentation ideals of rlF,
IT
E. 1
Then the statement that anE is
IT
E-nilpotent of class 5c 1
translates into
Consider the exact sequence of
... and let S-CC=
,€ €
I;, a
€
a ~ B, c IT,+IB.
(i*n-.l)['a,
=
-IT
F.
Let
n+l B
- - IT
E-modules 1
a
IT
nF
Then i*(S.a) 11 €
i*
=
IT
nE
(i*c)*i,(a)
rlF. Then a((i*n-l).B)
( n - l ) c . c ~ , by ( 2 . 3 ) .
But
... = 0 =
by ( 2 . 4 ) .
(i,n-l).aB
(i*Q-l)*B = (f,i,n-l).B
=
Thus
=
0, SO
0. This shows that IC+'*~ F = (O), and thus the theorem is F n proved. Note that, in fact, our argument shows that, even if F is not (n-1)S-a
=
connected, each component of
F is nilpotent. F e w i l l feel free to invoke
this more general statement. Now let W be a finite connected CW-complex and let X be a connected CW-complex. Let Xw b e the function space of pointed maps W W + X and let Xfr be the function space of free maps. Choose a map W as base point and let g € XW (g€Xfr) of
g.
(XW ,g)((XfrW,g))
be the component
Localization of homotopy types
64
(Compare G. Whitehead [871 , Federer [ 2 6 1 . )
Theorem 2 . 5 .
( i ) (xw,g) i s nilpotent. W
(xfr,g)
liil
x
i s nilpotent i f
Proof. We may suppose t h a t ( i ) , ( i i ) are certainly true i f i n d u c t i o n on t h e dimension of
i s nitpotent. is a p o i n t .
Wo
i s 0-dimensional,
W
Thus t h e a s s e r t i o n s and w e w i l l argue by
We w i l l be c o n t e n t t o prove ( i ) . We have
W.
a cof i b r a t i o n
v where
is a wedge of
V
-+
wn
*
wn+l,
g i v i n g r i s e t o a f i b r a t i o n (where we d i s p l a y
n-spheres,
one component of t h e f i b r e )
where
w"+'-+
g:
Wn (X ,go)
X
and
g
= glw".
Our i n d u c t i v e h y p o t h e s i s i s t h a t
is n i l p o t e n t , so t h a t Theorem 2 . 2 e s t a b l i s h e s t h e i n d u c t i v e s t e p .
Corollary 2 . 6 .
Let
W
be a f i n i t e CW-comptex and
W (X ,g)
X € NH. Then
and
W (Xfr,g) are nilpotent.
Proof. L e t
Wo,
W1,
..., Wd
Then
xw =
x"0
x
;*
b e t h e components of
x
...
x
W, w i t h
o € Wo.
Xd'f r .
Since p l a i n l y a f i n i t e product of n i l p o t e n t s p a c e s is n i l p o t e n t , i t f o l l o w s that
W (X ,g)
is n i l p o t e n t .
Similarly
W (Xfr,g)
is n i l p o t e n t .
C o r o l l a r y 2 . 6 t h u s e s t a b l i s h e s ( i n view of M i l n o r ' s theorem) t h a t we s t a y i n s i d e t h e c a t e g o r y NH when we t a k e f u n c t i o n s p a c e s
X € NH and
f i n i t e , i n t h e s e n s e t h a t each component of
W
Xw
Xw
is i n
with
NH.
W e now proceed t o g i v e an important c h a r a c t e r i z a t i o n of n i l p o t e n t
spaces.
Let
X
be a connected CW-complex and l e t
-
...
(2.7)
- ...
Nilpotent spaces P
xn 4 xn-l
be i t s Postnikov decomposition, so t h a t K(nnX,n).
65
-x1-0 i s a f i b r a t i o n with f i b r e
pn
W e s a y t h a t t h e Postnikov decomposition
n
refinement a t stage
x
(2.8)
qc.
Yc
=
n
where t h e f i b r e of
Let
decomposition of
if and only i f Proof.
TI
pn
- -...
91
Y1
gi: Yi-l + K(Gi,n+l),
Y n o
.
'n-1' K(Gi,n)
x
admits a principal refinement a t stage
X
operates n i l p o t e n t l y on
1
X
71
LK(Gi,n+l),
Since
IT
Y = (O), n o
TI
and
n ? 2
Suppose conversely t h a t W e consider
p :X n n
-t
Xn-l.
IT,X
Suppose
Then we may r e g a r d
i = 1,
X(=nlYi,05i5c) 1
rnYC = rnXn =
(stage 1)
n 2 2.
..., c , o p e r a t e s t r i v i a l l y on
Thus, by r e p e a t e d a p p l i c a t i o n s of t h e proof of Theorem 2.2,
o p e r a t e s n i l p o t e n t l y on
qi
( T I ~ Xi s n i l p o t e n t ) .
We w i l l be c o n t e n t t o g i v e t h e argument f o r
yi
IT
o
Then the Postnikov
f i r s t t h a t we have t h e p r i n c i p a l refinement (2.8).
as a fibration.
Y
1 5 i 5 c.
be a connected CW-compZex.
X
principal
may be f a c t o r e d as a product of f i b r a t i o n s
is an Eilenberg-MacLane s p a c e
qi
i s induced by a map Theorem 2 . 9 .
if
admits a
nlX
X.
IT
is
IT
X-nilpotent of c l a s s 3. 1
Then, by t h e r e l a t i v e Hurewicz Theorem we have
a n a t u r a l isomorphism
where Thus
n+l
(p ) n
Gn+l(pn)
a s a n element of
i s o b t a i n e d from
I T ~ + ~ ( P , )by k i l l i n g t h e a c t i o n of
may be i d e n t i f i e d w i t h Hn+l
2 (pn;nnX/r rnX).
IT
n
2
X/r I T ~ X ,and h-'
Thus
h-l
nlXn.
may b e regarded
g i v e s rise t o a diagram
Localization of homotopy types
66
with
u b - 0.
If
u
induces
ql: Y
1 -+ Xn-1'
then
pn
factors a s
(2.10) with
q1
The homofopy sequence of (2.10) reduces t o
induced a s required.
rl replacing
Thus we may r e p e a t t h e above procedure, with
- -
p
and,
n'
continuing i n t h i s way, we reach
xn
(2.11) each
qi
r yC
&
Yc-l
being induced by a map
...
Yi-l
However, a l l t h e homotopy groups of
-f
r
o
Y2
K(Gl,n+l), where
Gi =
r
vanish, s o t h a t
n-1'
r innX/ri+lnnX.
i s a homotopy
equivalence, and (2.11) is e s s e n t i a l l y t h e p r i n c i p a l refinement a t s t a g e
n
whose e x i s t e n c e we set out t o prove. We would say t h a t t h e Postnikov system of
X
refinement i f i t admits a p r i n c i p a l refinement a t s t a g e
admits a principal
n
f o r every
n 2 1.
We then have the evident Corollary 2.12.
Let
X
be a connected CW-complex. Then X
i s nilpotent i f
and only i f i t s Postnikov system a h i t s a principal refinement. W e p o i n t out t h a t t h e s i m p l e spaces a r e i d e n t i f i e d , by t h e
correspondence i m p l i c i t i n t h i s c o r o l l a r y , with those spaces whose Postnikov system is i t s e l f principal.
Remark.
Once we have obtained a p r i n c i p a l refinement of t h e Postnikov system
of a space, t h e r e i s , of course, no d i f f i c u l t y i n obtaining f u r t h e r refinements,
Nilpotent spaces which w i l l remain p r i n c i p a l .
61
Thus i f , f o r example,
i s of f i n i t e t y p e
X
and n i l p o t e n t we may r e f i n e i t s Postnikov system so t h a t each map
of t h e r e f i n e d system i s induced by some map
ZIP,
or
f o r some prime
Yi-l
+
f: E
i f a l l of IT
+
nlF).
To t h i s end we s a y
a r e connected ( s o t h a t
f,
maps
n E
1
F
onto
n i l p o t e n t l y on t h e homotopy groups of
f.
nlB),
and
( i n c l u d i n g , of c o u r s e ,
W e could a l s o e x p r e s s t h i s l a s t c o n d i t i o n by a s k i n g t h a t
by t a k i n g
A = Z
F , of ( p o i n t e d ) CW-complexes i s nilpotent
o p e r a t e s n i l p o t e n t l y on t h e homotopy groups of
E
1
Yi-l
p.
B, w i t h f i b r e
F, E, B
+
where
K(A,n+l)
We w i l l need a r e l a t i v e form of Theorem 2.9. t h a t a map
Yi
n E 1
operate
Note t h a t w e r e c o v e r D e f i n i t i o n 2 . 1
t o b e a p o i n t , provided w e adopt t h e r i g h t n o t i o n of n i l p o t e n c y
B
f o r t h e o p e r a t i o n of
T
E
1
on
TI
F.
Although we w i l l n o t need t h e g e n e r a l
1
case i n t h i s t e x t , w e now d e s c r i b e t h i s n o t i o n f o r t h e a c t i o n of a group
on a group
Q
With r e s p e c t t o such a n a c t i o n w e d e f i n e a lower central series
N.
as f o l l o w s (see H i l t o n [36]): rlN = N ,
Q
rrh
=
gp{(x.a)ba
W e t h e n s a y t h a t t h e a c t i o n of
YC+lN = 11).
Q
N
N
N
i s commutative.
by c o n j u g a t i o n , t h e n
r% = r 9
nilpotent map, it f o l l o w s t h a t i f
i s nilpotent of class
b€N, i21. 5c
if
f
Note a l s o t h a t i f
i N.
Q = N
and o p e r a t e s
T h u s , w i t h o u r d e f i n i t i o n above of a
i s n i l p o t e n t , t h e n t h e f i b r e of
f
W e a l s o have t h e f o l l o w i n g f a i r l y e v i d e n t p r o p o s i t i o n .
nilpotent.
P r o p o s i t i o n 2.13.
connected f i b r e .
Proof. Q =
on
r iN ,
Note t h a t t h i s d e f i n i t i o n a g r e e s w i t h t h a t g i v e n i n 1.4 i n
t h e case i n which
on
Q
-1 -1 b 1 , x C Q, a €
rlE, and
Let
f: E
+
B
Then, if E , B
be a map of connected CW-complexes with are nilpotent,
f
i s nilpotent.
We have a n e x a c t sequence of groups w i t h Q-action,
F = f i b r e of
f,
where
is
Localizationof homotopy types
68
...
+
IT
n+l
B - +IT F + n
E+ n
IT
...
+
s B
2
-+
TI
F +
1
E+ 1
TI
IT
B. 1
are nilpotent, slE operates nilpotently on nnB, TI E n for all n 2 1. Thus, if n 5_ 2 , the conclusion that IT E operates nilpotently 1 on n F follows from Proposition 1.4.3, since our definition of nilpotent
Now since E, B
action coincides, in the case of a commutative group, with that of 1.4. Thus the case n = 1 remains. We have an exact sequence IT
B-+n F+TIE
2
1
1
of Q-groups and the argument of Proposition 1.4.3 may be adapted to yield the result in this case in view of the fact that the image of n2B lies in the center of
IT
F.
1
It is, of course, necessary to take account of both facts
noted after the definition of a nilpotent Q-action. The relative form of Corollary 2.12 reads Theorem 2.14.
Let
f: E
+
B
be a map of connected CW-complexes inducing a
surjection of fundamental gruups.
Then f i s niZpotent i f and only i f i t s
Moore-Postnikou system admits a principal refinement.
be the Moore-Postnikov system of f .
-
Now if pn may be factored as in ( 2 . 8 ) ,
we obtain a sequence of extensions of n E-modules
Gi>where Gi
nYi
IT
1
nnYi-l, i
*
1,
is a trivial module. Now the fibre of
an Eilenberg-MacLane space K(Hi,n) an extension of
nlE-modules
..., c,
is i = ql’”qi: Yi Y and the relation siqi+l = si+l yields s
-+
69
Nilpotent spaces where Ho = {O}, Hc
=
~ l ~ = + n~ F. p ~ It follows from (2.15) and Proposition
1.4.3, by an easy induction, that
TI
F is a nilpotent
n
~l
1E-module.
(The
case n = 1 is again slightly special, but we will omit the details in this case.) The converse implication is proved exactly as in the absolute case; see the Proof of Theorem 2 . 9 . Before proceeding to discuss how to intr0duce.a localization theory Here we confine
into NH we show how Serre's C-theory may be applied to NH.
attention to the absolute case, since the relative case requires stronger axioms on a Serre class, as is already familiar in the classical case of the Thus we will be considering generalized Serre classes in the
category H1.
sense of Definition 1.5.1.
We prove the one basic theorem which we need in
the sequel. Theorem 2.16.
Let
X E NHand l e t
be a generalized Serre class.
C
Then
the following assertions are equivalent: (il
T I ~ Xf
C for a l l
(ii) HnX E C f o r all n (iii) nlx c cover of
1
n Z
c and H ~ Xc c
1
for a l l
n 2 1, where
X
i s the u n i v e r s a ~
X.
Proof.
We need two lemmas, which are interesting in their own right.
The first is a generalization (to general m?l) of Theorem 1.4.17, though we here only state the result for homology with integer coefficients. Lemma 2.17.
If
~l
acts n i l p o t m t t y on the abelian group
A, then
n
acts
nilpotently on Hn(A,m), n 1 0 .
Proof. n-series of A
Let 0 = rC+'A
5 rCA 5
... 5 I-1A = A
(see Section 1.41, and write Ai = r iA
be the lower central for convenience,
Note that each Ai is a nilpotent a-module, of class less than that of A
Localization of homotopy types
70
if
i 2 2.
Moreover,
a
a c t s t r i v i a l l y on
Ai/Ai+l.
We have a s p e c t r a l
sequence of a-modules,
converging ( f i n i t e l y ) t o t h e graded group a s s o c i a t e d w i t h I f w e assume i n d u c t i v e l y t h a t
filtered.
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 Lemma 2.18.
Proof. 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
Hn(%) where
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
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 infer that
f
E2
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 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.
s+l,O 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
73
Ii-If-
2 (3.2)
Y
Since
X
K(nlX,l)
Y
Jfl K(slY,l)
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
-+
Proof of Proposition 3 . 4 .
X
r+
X
we
Y
NH,f o r which f provides a natural L.
U LNH t o
transformation from the embedding
Let g : X
+
X' in U.
We thus have a
If If
diagram
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
-
be t h e f i b r e of
K(Gp,n+l), where
Gp
If'-
X'
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
-&
K(G,n+l)
h
K(Gp,n+l)
There i s then a map
h.
f: X
-+
Y
rendering t h e
diagram
4f -4f' A X --+
X'
K(G,n+l)
Y
Y'
K(Gp,n+l)
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)
Xi
g
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
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
@Yi.
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.
satisfies (ii).
@ fi
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
- n,
s i n c e i t follows immediately from (3.9). P
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
t o NH; we
H1
k k e w i s e Theorens1.13, 1.14, and 1.16 extend from
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 Section 5. Let X € hkl. We say that X generated for all n 2 1 and that X
is of f i n i t e type if anX is finitely 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 0 P P) P P 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
For each a(p)
*
p C l 7 U {O)
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
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
a x
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. 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
We w i l l d e n o t e by
.
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
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
+
EX
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
nixo
@
... - n p xiX
nn.x 1 P
(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
w X i
&? TT
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 .
-
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
-
mnix)o
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
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
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
ll
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 . f*: [W,X]
2
-
with
For, according to that result, if W
[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
-+
e,
let
P-local rmd
f,:
[W,X]
W.
Then
CW-complexes
Proof. W
f: X
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.
=
$Jcm2
W e now prove
f o r some
e*:
Then
ipym = e*a
that
jam' = 1 f o r some ipe,x
Thus
yml
f o r some
=' : ,e
m
We conclude t h a t
QXP.
-+
[W,%]
P-surjective.
a € [V,X], m C P I .
1
with
e*x = 1.
Thus
m2 6 P' , whence f i n a l l y
[w,x]
= ipym'
= (e*x).($pb),
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,$,], and b = e*c,
It f o l l o w s t h a t
c f [CV,X]
e*
P-injective.
91
C P'.
Thus
It f o l l o w s t h a t
Now
=
e*$c,
m m
(xm)m1m2 = $ a
= 1
[W,%l.
y €
Let
am1
= i x , x C [W,X],
mm 1-power of
bm2 = e*c, f o r some
Thus, by Lemma 6 . 4 below, ynrmlm2= ek(xm2.$c)
ml
e,ja = j Pe*a = 1, so
( f o r a s u i t a b l y bracketed b C vnXp.
e*a
c
<
y).
unX, m2 C P ' .
and t h e a s s e r t i o n i s proved.
(Note t h a t , i n t h e s e arguments, w e have w r i t t e n a l l monoid s t r u c t u r e s i n ( 6 . 3 ) multiplicatively. ) The converse i s t r i v i a l .
f,: II X
-+TI
n 11.
Thus
For i f we s e t
W = Sn
t h e n we know t h a t
i s a P-isomorphism t o a P - l o c a l group and hence P - l o c a l i z e s ,
Y f
P - l o c a l i z e s by Theorem 3B.
The f o l l o w i n g lemma, t h e n , s u p p l i e s t h e one m i s s i n g s t e p . Lemma 6 . 4 .
Let
X
attaching the cone W
*
ZA
induces
be an H-space and l e t CA
to
V
W = V
by means of a map
U
CA, t h e space obtained b y A
-t
V.
The p r o j e c t i o n
Lacalization of homotopy types
92
4 i s central in the monoid
and the image of
Proof.
Let c: W
+
W V CA be the cooperation map, In the terminology
of Eckmann-Hilton. Then, for x € [W,X], a
where € [WVCA,X] Thus
-
[w,x] in the strong sense that
€
[ZA,X], we have
x.$a = c*, [W,X]
(xl.$al) (x,.4ia2)
c* = x1x2. 4 (ala2>
.
[ZA,X].
X
=
(c*)(c*) = c*() =
Remark. Note that we could have proved that
e*
the following stronger sense, namely, that if e,x xn = yn for some nth power with n
€
P'.
in (6.1) is P-injective in =
e*y, x , y € [W,X], then
In the presence of a nilpotency
condition, this sense of P-injectivity in fact coincides with the obvious one (obtained by setting y = 1); and, indeed, it is true, with appropriate definitions, that
[W,X] is a nilpotent loop (non-associative group),
Corollary 4 . 3 enables us to deduce the following modification of Theorem 6.2. Corollary 6.5. Let
CW-complex.
X be a connected H-space mrd W a quasifinite
Then ZocaZization induces
and eg i s P-bijective. The following consequence of this corollary will be very important in the sequel. Theorem 6.6.
Let W be a quasifinite CW-compZex and l e t
H-space such that Wp
9
Xp.
Then there e x i s t s a map
X be a connected
f: W * X such that
Localizing H-spaces
fp:
wp
eY
$.
Proof.
Let
g,
= ng*: R*(Wp)
g: Wp
-+
$
n C P' suchthat
6 . 5 there e x i s t s n
93
-+
n,($)
gn = e l ( f ) , f : W
and t h a t , consequently,
isomorphism of homotopy groups found the required map
b e a homotopy equivalence.
f
.
(n C Zt). Thus
-+
X.
n
g,
g": Wp
By Corollary
But i t i s c l e a r that is,like '\I
$,
g*,
an
so that we have
Localization of homotopy types
94
7.
Mixing of homotopy t y p e s T h e i d e a o f m i x i n g h o m o t o p y t y p e s g o e s b a c k t o Zabrodsky [ 9 3 ] a n d h a s been
e x t e n s i v e l y used t o c o n s t r u c t examples and counterexamples i n homotopy t h e o r y ; see, e . g .
[79,93,96] and Chapter 111 of t h i s monograph.
It seems t h a t l o c a l i z a t i o n
t h e o r y p r o v i d e s t h e r i g h t framework f o r d i s c u s s i n g t h i s i d e a and r e n d e r i n g i t most e a s i l y u s a b l e i n a p p l i c a t i o n s . We b e g i n by d i s c u s s i n g puZZbacks
i n homotopy t h e o r y , a t o p i c of some
independent i n t e r e s t , i n p a r t i c u l a r w i t h r e s p e c t t o l o c a l i z a t i o n .
Given a
diagram
If
X (7.1) Y-B
i n t h e category
g
T of based CW-complexes, we may r e p l a c e
and t a k e t h e ( s t r i c t ) p u l l b a c k
which we c a l l t h e weak
by a f i b r e map
We o b t a i n a diagram
T.
puZZback of (7.1) i n t h e homotopy c a t e g o r y H.
known t h a t t h e homotopy t y p e of
as a diagram i n
in
f
2
depends only on t h e diagram ( 7 . 1 ) , i n t e r p r e t e d
H, and i s symmetric w i t h r e s p e c t t o
we might j u s t as w e l l have r e p l a c e d i n s t e a d of choosing t o r e p l a c e
f.
I t is
g
(or both
Of c o u r s e , i f
f
f , g, i n the sense t h a t and
f
(or
g) by a f i b r e map, g) were a l r e a d y a
f i b r e map, n o replacement would be n e c e s s a r y . I f (7.1) were a diagram i n
To, t h e s u b c a t e g o r y of
T
c o n s i s t i n g of
connected CW-complexes, we would o b t a i n t h e weak p u l l b a c k i n the corresponding homotopy c a t e g o r y base point.
Ho by r e p l a c i n g
W e would t h u s o b t a i n
2
i n (7.7.) by t h e component
Zo
of i t s
Mixing of homotopy types U
1. If
zo (7.3)
in Ho.
x
We are interested in the question of when we may infer that
is, in fact, in NH. Theorem 7 . 4 . i f
zO
W e prove:
Suppose t h a t
X , Y C NH
i n (7.31.
Then
nlZo operates n i l p o t e n t l y on nnB, n 2 2 , v i a
Proof.
fu
Zo
.
E NH i f and onZy
The diagram ( 7 . 3 ) gives rise to a Mayer-Vietoris sequence
of groups with nlZo-action, where
Suppose that nlZo
G
i s the pullback of the diagram of groups
operates nilpotently on TInB, n
operates nilpotently (via u
vo)
nlZo operates nilpotently on
?
2. Then, since n lZ o
on the homotopy groups of X
it follows from Proposition 1 . 4 . 3 that nlZo Now
95
TI
2
and Y,
operates nilpotently on nnZo, n ? 2.
B and hence on Im n2B C n l Z o .
However
here the operation is by conjugation and thus the operation of G on Im n2B induced by the exact sequence
Im n2B is also nilpotent.
-TI
Z
l o
-
Since G, as a subgroup of
we infer from Proposition 1.4.1 that r1Z0 Conversely, suppose that Zo
G IT
X 1
x TI
Y, is nilpotent, 1
is nilpotent.
is nilpotent. Then an immediate
application of Proposition 1 . 4 . 3 to the Mayer-Vietoris sequence
Localization of homotopy types
96
shows that nlZo operates nilpotently on nnB, n 1 2 . Of course, it is most useful to have a criterion for Z
to be
nilpotent which is independet of the maps uo, vo, but depends only on (7.1). Thus we now enunciate Corollary 7.6.
Let ( 7 . 1 ) be a diagram i n NH.
Then, i n the weak pullback
(7.3) i n Ho, Zo C NH.
The following immediate consequence of Theorem 7.4, generalizing Theorem 2 . 2 , 1s also useful. Corollarv 7.7.
Let ( 7 . 1 ) be a diagram i n Ho with X, Y C NH.
If X
OP
Y
i s 1-connected, then Zo C N-l. We now suppose that (7.3) i s a weak pullback in N a n d we localize at the family of primes P.
We obtain
yp
Diagram (7.8) i s a weak pullback in NH.
Proposition 7.9.
Proof.
Form the pullback in To
(7.9)
where we may 8:
Zop+ 2'
assume fp to be a fibration. There is then a map
yielding a commutative diagram
Mixing of homotopy types
in
91
NH, andhence a map of t h e P - l o c a l i z a t i o n of t h e Mayer-V
!to1
3
sequence of
(7.3) t o t h e Mayer-Vietoris sequence of (7.9); h e r e Theorem 1.2.10 p l a y s a c r u c i a l r o l e i n e n s u r i n g t h a t , when we l o c a l i z e
G
i n (7.5) w e o b t a i n
t h e p u l l b a c k of t h e diagram
K1yP
-rB
rigp
1P
I n t h i s map of Mayer-Vietoris sequences a l l groups e x c e p t mapped by t h e i d e n t i t y .
Thus
s
rnZoP
are
i n d u c e s an isomorphism of homotopy groups
and hence is a homotopy-equivalence. Suppose, i n (7.31,
C o r o l l a r y 7.10.
Then Zo
€ Mi
and
Proof.
that
f
is a P-equlvalence and X, Y, B
is a P-equivalence.
vo
We a l r e a d y know t h a t
Z
€ NH
i s an e q u i v a l e n c e so t h a t , by P r o p o s i t i o n 7.9, v
€
by C o r o l l a r y 7 . 6 . vop
Now
fp
is an e q u i v a l e n c e and so
i s a P-equivalence.
Of c o u r s e , t h i s c o n c l u s i o n could more e a s i l y have been drawn w i t h o u t e s t a b l i s h i n g P r o p o s i t i o n 7.9 i n f u l l g e n e r a l i t y . We w i l l b e i n t e r e s t e d i n e s t a b l i s h i n g c o n d i t i o n s under which w e may i n f e r t h a t t h e space
Z
i n (7.2)
is a l r e a d y connected, so t h a t
Z = Zo.
NH.
Localization of homotopy types
98
Obviously this holds if (7.1) is a diagram in To
in which f
(or g)
induces a surjection of fundamental groups. However, we will require a more general criterion. Proposition 7.11. Let ( 7 . 1 ) be a diagram i n nlB i s of the form 17.21 in
H,
f*a.g,f?,
c1
E
Ti
To
i s which every eZement of
Then i n the weak pullback
X, f? E nlY. 1
i s connected.
Z
Proof.
Let us assume f a fibre map, so that ( 7 . 2 )
pullback in T. Given (x,y) E Z, x E X, y E Y, let k o
to x, and m
a path in Y
from o
reverse of m, is a loop in B on p
0 , so
is the strict
be a path in X
to y. Then fII *gi, where
i
from is the
in X,
that there are loops h
in Y with
Thus f(x
* L)
-
-
fk*gm- fh*gp. g(p *m), re1 endpoints,and, since f is a fibre map, we find
L' * h * II, re1 endpoints, that
(II',m')
so
that
f&' = gm', where m' = p *m.
is a path in Z from o
to
It follows
(x,y).
We exploit Proposition 7.11 in the following way.
Let ( 7 . 2 ) be a diagram i n To i n which f,: nlX * TIIB is a
Corollary 7.12.
P-surjection and of the primes.
g,:
TI
Y
1
-+TI
B i s a Q-surjection for some p a r t i t i o n 1
Then, i n (7.21,
Proof.
Z
yn = g,n
P
for m E Q, n € P.
are relatively prime we find integers k, II with km II and then y = f,Sk* g,n ,
+ an =
We are now ready to prove the mixing theorem which is the main objective of this section.
=
i s connected.
Let y E alB. Then ym = f,S,
Since m, n
n
1
u
Q
Mixing of homotopy types
Theorem 7.13.
with Xo
X , Y C NH
Let
of the primes. Then there exists
Proof. h: Y
0
2
X
2
and Let rI
Yo
with
Z C NH
There a r e c a n o n i c a l maps
99
s:
Zp
Xp
+
2
= P U Q
$,
Xo,
ZQ
t: Y
N
Q
Y
-f
be a partition
9' Let
'0'
and c o n s i d e r t h e diagram
0
1
(7.14) Y-%
Q
xO
Form t h e weak p u l l b a c k of ( 7 . 1 4 ) ,
Certainly
s
i s a Q-equivalence and
7.12 e n s u r e s t h a t Corollary Thus
u
7.10 g u a r a n t e e s t h a t induces
up: Zp
2
Xp
u
and
i s a P-equivalence.
Thus C o r o l l a r y
Corollary 7.6 then ensures t h a t
i s connected.
Z
ht
i s a P-equivalence
v
induces
vs:
and
v
2 E NH and
i s a Q-equivalence.
ZQ r- YQ.
The following addendum i s important i n a p p l i c a t i o n s . P r o p o s i t i o n 7.15. Z
(i) Let X ,
Y
in Theorem 7.13 be quasifinite. Then
is quasifinite. lii) Let X , Y in Theorem 7.13 be 1-connected. Then
Z
is 1-connected. liiil Let X,
Y
in Theorem 7.13 have the homotopy type of
a finite 1-connected CW-complex. Then
Z
has the hornotopy type of a finite
1-connected CW-comp Zex.
Proof.
(i) Observe t h a t i f
generated L -module and P by Theorem 1.3.10, is quasifinite.
A
A
Q
A = WiZ
then
is a finitely-generated
%
is a f i n i t e l y -
%-module.
Thue,
is a f i n i t e l y - g e n e r a t e d a b e l i a n group, so t h a t 2
Localization of homotopy types
100
(ii) Observe that
nlZ
is a nilpotent group which l o c a l i z e s t o
the t r i v i a l group a t every prime and hence is c e r t a i n l y t r i v i a l . (iii) This follows from (i) and (ii),using the techniques of
homology decomposition.
It is
therefore necessary for us to work in a restricted category of (pointed) topological spaces. It is also necessary for us to work in a homotopy category (that is, in a category in which the morphisms are homotopy classes of continuous maps), since our procedures for establishing the existence of a localization theory will all operate up to homotopy. A more general treatment, valid in the semisimplicial category, has been given by Bousfield-Kan u 4 ] . We will always suppose that our spaces have the homotopy type of CW-complexes. In Section 1, we present a localization theory in the homotopy category H1
of 1-connected CW-complexes. We establish two fundamental
theorems in H1, namely that every object of the category does admit a P-localization, and that we can detect the P-localizing map
f: X
+.
Y either
through the induced homotopy homomorphisms, which should also P-localize, or through the induced homology homomorphisms, which should also P-localize.
In the course of establishing that there is a localization theory in H1,
Localization of homotopy types
48
we a c t u a l l y c o n s t r u c t t h e l o c a l i z a t i o n of a given CW-complex i m i t a t i n g t h e c e l l u l a r c o n s t r u c t i o n of t h a t of a ZocaZ c e l l .
X
by
X, r e p l a c i n g t h e i d e a of a c e l l by
The f a c t t h a t t h e l o c a l i z a t i o n can b e d e t e c t e d e i t h e r
through homotopy o r through homology h a s t h e immediate consequence t h a t we may l o c a l i z e f i b r e and c o f i b r e sequences i n
H1,
I n S e c t i o n 2 , we d e s c r i b e a broader homotopy c a t e g o r y i n which we w i l l a l s o b e a b l e t o e s t a b l i s h a s a t i s f a c t o r y l o c a l i z a t i o n theory.
It t u r n s
o u t t h a t we would wish t o e n l a r g e t h e c a t e g o r y t o which we apply l o c a l i z a t i o n methods from our o r i g i n a l c a t e g o r y confined t o o b j e c t s of
H1,
even i f our main i n t e r e s t were
For, i n o r d e r t o prove theorems about l o c a l i z a t i o n of
H1.
i t is v e r y u s e f u l t o employ function-space methods, and
H1,
t h e function-space c o n s t r u c t i o n t a k e s u s o u t s i d e t h e c a t e g o r y . i t is t r u e t h a t i f
and i f into
W
X
X
However,
is a niZpotent s p a c e , i n a s e n s e defined i n S e c t i o n 2 ,
is f i n i t e , t h e n t h e f u n c t i o n space Xw is a g a i n n i l p o t e n t .
f a c t s about t h e category
of pointed maps of
S e c t i o n 2 concerns i t s e l f w i t h some b a s i c
NH of n i l p o t e n t s p a c e s , and may be regarded i n
p a r t as propaganda f o r t h e u s e of t h i s c a t e g o r y i n homotopy theory. i t has a l r e a d y been shown by Dror [23] t h a t
homotopy t h e o r y .
W
NH
Indeed,
is a s u i t a b l e c a t e g o r y f o r
Roughly speaking, one may s a y t h a t most of t h e t e c h n i q u e s
of homotopy t h e o r y which have been developed s i n c e t h e p u b l i c a t i o n of S e r r e ' s t h e s i s can a l l be c a r r i e d o u t i n t h e c a t e g o r y
NH
techniques were of course o r i g i n a l l y formulated i n
although many of t h o s e
H1,
The b a s i c theorem
proved i n S e c t i o n 2 is t h a t a space is n i l p o t e n t i f and only i f i t s Postnikov tower admits a p r i n c i p a l refinement. t h e category
Mi
It is t h i s theorem which e x p l a i n s why
is s u i t a b l e f o r homotopy theory; f o r t h e given refinement
of t h e Postnikov tower may be used i n p l a c e of t h e Postnikov tower i n t h o s e arguments i n which t h e c r u c i a l f a c t which is r e q u i r e d is t h a t t h e f i b r a t i o n s which appear i n t h e tower should b e induced o r p r i n c i p a l .
Introduction
49
However, it should be pointed out that the category NH has eertain defects over the category H1.
One of the defects is that it is not closed
under the mapping cone operation. This defect has a serious consequence in Section 3 .
We also describe in Section 2 how to relativize the notion of a
nilpotent space to obtain that of a nilpotent map. In Section 3 we generalize the theorems of Section 1 from the category H1
to the category NH. Formally, we get the corresponding
formulations of the two fundamental theorems of Section 1. However there is an important difference in the way in which we construct the localization of an object. For, whereas in the category H1 we are able to proceed cellularly, since the mapping cone construction respects the category H1, we cannot in the nilpotent case proceed cellularly, since the mapping cone construction would take us outside the category. It is therefore necessary for us to proceed homotopically rather than cellularly in constructing the localization. In this way, of course, we lose much of the conceptual simplicity of the construction in Section 1. Section 4 is a brief technical section in which we introduce the idea of a quasifinite complex in Mi. relative to the category H1.
Here again we see a certain disadvantage
For if X
is a 1-connected CW-complex whose
homology groups are all finitely generated, and vanish above a certain dimension, then X itself has the homotopy type of a finite complex. If we discard the condition of simple-connectivity, we can no longer assert this conclusion. Indeed, we have the obstruction theory of Wall which enables
US
to discuss the question whether a CW-complex X whose homology looks like that of a finite complex in fact has the homotopy type of a finite complex. Thus we are led to introduce the concept of a quasifinite CW-complex, meaning a nilpotent CW-complex X such that the homology of X is finitely
Localization of homotopy types
50
generated i n each dimension and v a n i s h e s above a given dimension.
We prove
t h a t such a q u a s i f i n i t e complex always h a s t h e homology type of a f i n i t e complex. I n S e c t i o n 5 we prove t h e fundamental p u l l b a c k v a r i o u s v a r i a n t s as consequences of t h a t theorem.
theorem and
Here we l e a n v e r y h e a v i l y
on t h e r e s u l t s of Chapter I . The fundamental p u l l b a c k theorem a s s e r t s t h a t t h e p o i n t e d set i s t h e p u l l b a c k of t h e pointed sets
[W,Xl set
1, provided t h a t
[W,X
W
{[W,Xpll
over t h e p o i n t e d
i s a f i n i t e connected CW-complex and
a n i l p o t e n t complex of f i n i t e type.
X
is
T h i s a s s e r t i o n f a l l s i n t o two p a r t s .
The f i r s t p a r t s t a t e s t h a t given two maps
f,g: W
-+
X
such t h a t
= e g: W + X f o r a l l primes p, t h e n f = g . T h i s p a r t of t h e a s s e r t i o n P P P’ does not r e q u i r e t h a t X b e of f i n i t e t y p e . The second a s s e r t i o n s t a t e s e f
that i f
f(p): W
-+
c l a s s of t h e map
X
P
a r e maps, f o r a l l primes
r f(p): W P
-+
Xo
p , such t h a t t h e homotopy
i s independent of
i s t h e r a t i o n a l i z a t i o n map, t h e n t h e r e exists a map
e f P
2
f(p)
f o r a l l primes
t h e condition t h a t
W
p.
p , where
f: W
-+
X
rp: Xp
-+
Xo
such t h a t
We show by an example t h a t we cannot omit
should be f i n i t e .
However, provided t h a t
W
is
n i l p o t e n t , we may i n f a c t weaken t h e hypotheses of t h e p u l l b a c k theorem by simply r e q u i r i n g t h a t
W
be q u a s i f i n i t e .
I n S e c t i o n 6 we make a p r e l i m i n a r y s t u d y of t h e l o c a l i z a t i o n of H-spaces.
Our main r e s u l t i n t h i s s e c t i o n i s a g e n e r a l i z a t i o n of t h e p a r t
of t h e fundamental theorem of Chapter I which t e l l s u s how t o d e t e c t t h e P - l o c a l i z a t i o n of a n i l p o t e n t group i n terms of t h e P - b i j e c t i v i t y of t h e l o c a l i z i n g homomorphism. I n S e c t i o n 7 w e formulate t h e fundamental mixing technique of Zabrodsky w i t h i n t h e c o n t e x t of t h e l o c a l i z a t i o n of n i l p o t e n t s p a c e s .
The
Introduction
51
p a r t i c u l a r r e s u l t which w e emphasize is t h a t , given n i l p o t e n t spaces
X, Y
with equivalent r a t i o n a l i z a t i o n s , and given a p a r t i t i o n of t h e primes
Il
= P
u
Q , then t h e r e e x i s t s a n i l p o t e n t space
2
such t h a t
2
P
= Xp
and
ZQ = YQ. We make very considerable a p p l i c a t i o n of t h e r e s u l t s of t h e l a s t two s e c t i o n s i n Chapter 111.
Indeed, we a r e r a t h e r l i t t l e concerned t o g i v e
e x p l i c i t examples and a p p l i c a t i o n s i n t h i s Chapter i n view of t h e f a c t t h a t Chapter 111 is e n t i r e l y concerned with applying t h e theory of Chapter 11.
Localization of homotopy types
52 1.
Localization of 1-connected CW-complexes. We work i n t h e pointed homatopy category
CW-complexes. X
X C H1,
If
and i f
P
is a family of primes, we say t h a t
is P-zocal i f the homotopy groups of
W e say t h a t
f: X + Y
H1
in
H1 of 1-connected
X
P-localizes
a r e a l l P-local a b e l i a n groups. X
if
Y
i s P-local and*
f*: [Y,Z] z [X,Zl f o r a l l P-local
2 C H1.
Of course t h i s u n i v e r s a l property of
c h a r a c t e r i z e s i t up t o canonical equivalence: both P-localize H1
with
in
H1.
X
hfl = f 2 .
if
fi: X
-+
then t h e r e e x i s t s a unique equivalence
Yi,
f i = 1, 2 ,
h : Y1
PI
Y2
in
W e w i l l prove t h e fallowing two fundamental theorems
The f i r s t a t t e s t s t h e e x i s t e n c e of a l o c a l i z a t i o n theory i n
H1
and the second a s s e r t s t h a t we may d e t e c t t h e l o c a l i z a t i o n by looking a t induced homotopy homomorphismor induced homology homomorphisms. Theorem 1A.
( F i r s t fundamental theorem i n H1.)
Every
X
We T r i t e Theorem 1B.
Let
i n HI e: X
admits a P-localization.
+
Xp
f o r a f i x e d choice of P-localization
of
X.
(Second fundamentaZ theorem i n H1.l f: X * Y
i n H1.
( i ) f P-localizes (ii) nnf: n X (iii) Hn f : HnX
Then the following statements are equivalent:
X;
nnY
P-localizes f o r a l l
n 3 1;
-+ H Y
P-localizes f o r a l l
n 2 1.
-t
We w i l l prove Theorems l A , 18 simultaneously.
*We w r i t e , a s usual, [Y,Z] of maps from Y t o 2.
f o r H1(Y,Z),
W e r e c a l l from
t h e s e t of pointed homotopy classes
Localization of I -connectedCW-complexes
Proposition 1.1.9 that a homomorphism
B
if and only if
is P-local and
$:
A
+
B
53
of abelian groups P-localizes
is a P-isomorphism; this latter condition
@
+
means that the kernel and cokernel of
belong to the Serre class C of
abelian torsion groups with torsion prime to P.
Thus to prove that (ii)
(iii)
in Theorem 1 B above it suffices to prove the following two propositions. Proposition 1.1. Let Y C H1. only if H Y
is P-locat f o r a l l n
Proposition 1.2.
all
Then
f: X
Let
-+
n E 1 if and only if
TI
Y is P-local f o r a l l
Then
nn(f)
is a €'-isomorphism f o r aZZ
for all n E 1 if and only if Hn(Y;Z/p) = 0 disjoint from P.
n 2 1.
P
is a P-local abelian group, so are the homology groups
of the Eilenberg-MacLane space K(A,m).
K(A,m-l)
-+
-+
K(A,m),
with E
if H (A,m-1; Z/p) = 0 n Now let
...
of Y.
is a P-local abelian
1. It now follows, by induction
Hn(A,m)
E
is P-loca?
for all n 2 1 and all primes
Now, by Proposition 1.1.8, if A
group, so are the homology groups H A, n on m, that if A
1 if and
is a P-isomorphism for
Proof of Proposition 1.1. We first observe that HnY
p
?
2 1.
Y in H1,
Hn(f)
n
For we have a fibration
contractible, from which we deduce that,
for all n E 1, then Hn(A,m; Z/p) = 0 -+
Ym
-+
Ymm1-+
...
Thus there is a fibration K(nmY,m)
Thus, if we assume that
nnY
for all n 1 1.
Y2 be the Postnikov decomposition
* Ym
+
Ym-l, and Y2 = K(n 2Y , 2 ) .
is P-local for all n P 1, we may assume inductively
that the homology groups of Ym-l
are P-local and we infer (again using homology
with coefficients in Z / p , with p disjoint from P) that the homology groups of Ym
are P-local.
Since Y
-+
Y,
is m-connected, it follows that HnY
P-local f o r all n 1 1. To obtain the opposite implication, we construct the 'dual' Cartan-Whitehead decomposition
is
Localization of homotopy types
54
There i s then a f i b r a t i o n if we assume t h a t
K(amY,m-l)
i s (m-1)-connected,
Y(m)
*
Y(m+l)
i s P-local f o r a l l
HnY
t h a t t h e homology groups of Y(m)
+
n
T
m
and
Y(2) = Y .
Since
Y Z H Y(m) m
Y(m) = n Y m m
P)
t h a t the homology groups of
s t e p i s complete and
T
Y 2 H Y(n) n
and i s P-local.
to
C
a r e P-local,
p
disjoint
so t h a t the i n d u c t i v e
is P-local.
Proof of Proposition 1 . 2 . mod C , where
Y(m+l)
and
a
Thus we i n f e r (again using homology w i t h c o e f f i c i e n t s i n Z / p , w i t h from
Thus,
2 1, w e may assume i n d u c t i v e l y
a r e P-local.
i t follows t h a t
Y(m)
Since a P-isomorphism
is an isomorphism
i s t h e c l a s s of a b e l i a n t o r s i o n groups with t o r s i o n prime
P, Proposition 1 . 2 is merely a s p e c i a l case of t h e c l a s s i c a l Serre theorem.
We have thus proved t h a t ( i i ) t h a t (ii) * (i). counterimage of
f.
(iii) in Theorem 1 B .
We now prove
The o b s t r u c t i o n s t o t h e e x i s t e n c e and uniqueness of a g: X
-+
under
Z
Now, given ( i i ) (or ( i i i ) ) ,H,f of t h e map
Q
f*: [Y,Z]
C C , where
-+
[X,Z]
H,f
l i e in
H*(f;n,Z).
r e f e r s t o t h e homology groups
Thus (i) follows from the u n i v e r s a l c o e f f i c i e n t theorem f o r
cohomology and Proposition 1.1.8(v).
We now prove Theorem 1 A . of
f: X
-c
Y
in
More s p e c i f i c a l l y , we prove t h e e x i s t e n c e
s a t i s f y i n g (iii). Since we know t h a t (iii) 3 (i), t h i s
H1
w i l l prove Theorem 1 A .
Our argument is f a c i l i t a t e d by t h e following key
observation. Proposition 1.3.
Let
we have constructed
U be a f u l l subcategory of f: X
-+
Y
automatically y i e l d s a functor
satisfying l i i i ) . L: U
transfornation from the embedding U
-+
H1,
5 H1
Then the assignment X * Y
f o r which t o L.
f o r whose objects X
HI
f
provides a natural
Localization of 1-connectedCW-complexes Proof of P r o p o s i t i o n 1 . 3 .
g: X
Let
-+
in
X'
U.
55
W e t h u s have a
diagram
Y in
H1
with
f, f'
Y'
satisfying ( i i i ) .
Since
f
s a t i s f i e s ( i ) and
P-local by P r o p o s i t i o n 1.1, we o b t a i n a unique ( i n
H1)
Y'
is
h 6 [Y,Y'] making
t h e diagram
I t i s now p l a i n t h a t t h e assignment
commutative.
desired functor
X
rf
Y, g* h
yields the
L.
We e x p l o i t P r o p o s i t i o n 1 . 3 t o prove, by i n d u c t i o n on may l o c a l i z e a l l n-dimensional
CW-complexes i n
H1.
n, t h a t w e
.If n = 2, t h e n such a
complex i s merely (up t o homotopy e q u i v a l e n c e ) a wedge of 2-spheres
x = v s2 , a
where
a
runs
through some index s e t , and w e d e f i n e
Y = VM( ZP,2), a
where map
M(A,2) f: X
+
fo: Xo + Y o dim X = n
Y
i s t h e Moore s p a c e having satisfying ( i i i ) . satisfying ( i i i ) i f
+ 1, X
€
H1.
H M = A. 2
There i s t h e n an e v i d e n t
Suppose now t h a t w e have c o n s t r u c t e d dim X
5 n , where
n 1 2 , and l e t
Then w e have a c o f i b r a t i o n
vsn R, xn i . x By t h e i n d u c t i v e h y p o t h e s i s and P r o p o s i t i o n 1 . 3 , we may embed (1.5) diagram
i n the
Localization of homotopy types
56
where fo, fl satisfy (iii) and the square in (1.6) homotopy-commutes. embeds Yo
Thus if j (1.6) by
f: X
+
Y
in the mapping cone Y of h, then we may complete
to a homotopy-commutative diagram and it is then easy
to prove (using the exactness of the localization of abelian groups) that f: X
+.
Y
also satisfies (iii).
(iii) if X
Thus we may construct f: X
+
Y satisfying
is (n+l)-dimensional, and the inductive step is complete.
It remains to construct f: X
-+
Y satisfying (iii) if X
is
infinite-dimensional. We have the inclusions
x2
cx3 2 . .
and may therefore construct
where f", fn+l
satisfy (iii).
We may even arrange that (1.7)
is strictly
If we define Y = UY(n), with the weak topology, n and the maps fn combine to yield a map f: X -+ Y which again
commutative for each n. then Y t H1
obviously satisfies (iii). Thus we have proved Theorem lA in the strong form that, to each X
in H1,
there exists f : X
-+
Y
in H1 satisfying (iii).
Finally, we complete the proof of Theorem 1B by showing that (i) = (iii).
Given f: X
-+
Y which P-localizes X, let fo: X
constructed t o satisfy (iii).
Then fo: X
+
Y
which one immediately deduces the existence of
-+
Y
be
also satisfies (i), from a
homotopy equivalence
Localization of 1-connectedCW-complexes
u: Yo
+.
Y with uf
= f.
It immediately follows that f also satisfies
(iii). Thus the proofs of Theorems l A , 1B are complete. We note that our proof of the first fundamental theorem does much more than establish the existence of a localization theory in H1;
it provides
us with a combinatorial recipe for constructing the localization of a given CW-complex. The Moore spaces S;
=
may be called P-ZocaZ n-spheres,
M(Zp,n)
and a cone on a P-local n-sphere may be called a P-ZocaZ (n+l)-ceZz.
Then,
given a cellular decomposition of X, we may--as shown in the proof of Theorem 1A--construct a P-ZocaZ-ceZZuZar decomposition of Xp by 'imitation'; that is, whenever, in building up X, we attach an n-cell to Y f : Sn-'
of a map to Yp
+.
$, we attach a P-local n-cell
Y, then, in building up
by means of the localized map
sn-l
fp:
(say) by means
P
-+
Yp.
We illustrate this
procedure by means of an example. Example 1.8. Let Sm
-+
E
-+
Sn
be an Sm-bundle over
S",
n
z
m, and let
m
a C T ~ ~ - ~) ( be S the characteristic (homotopy) class of the bundle.
it is well-known that E
Then
admits the cellular decomposition
E
=
sm ua
en
u
emh,
where we will not trouble to specify the attaching map f o r the top-dimensional cell. If m 1 2 we may now localize E Ep = S;
Ua
P
to obtain
nntl-n ep U ep
.
Let us consider,in particular, the Stiefel manifold Vn+l , 2 vectors to
S",
and let us assume that n
over Sn with fibre Sn-', a = 2 6 T~-~(S~-').
so
We obtain
of unit tangent
is even. Then 'n+1,2
that we may take E = V
n+1,2'
m = n
fibres
-
1, and
Localization of homotopy types
58
Suppose now t h a t
P
is t h e f a m i l y of a l l odd primes. Sn-'
s o t h a t i t is e a s y t o s e e t h a t
n ep
2p
Then
is i n v e r t i b l e ,
h a s t h e homotopy type o f a p o i n t .
It t h u s f o l l o w s from (1.9) t h a t
if
E = V n+1,2, n
even, and
f u r t h e r follows t h a t i f [E,Y]
%
P
is t h e f a m i l y of odd primes.
It t h e r e f o r e
is a P - l o c a l space t h e n
Y
[I? ,Y] '2 [Sp 211-1,Y] '2 [ S 2n-1 , y ] =
P
Here t h e most s t r i k i n g f a c t is t h a t t h e s e t
[E,Y]
2n-1
y.
h a s acquired a v e r y n a t u r a l
a b e l i a n group s t r u c t u r e and h a s simply been i d e n t i f i e d w i t h a c e r t a i n homotopy group of
Y.
We have t h e f o l l o w i n g immediate c o r o l l a r i e s of t h e second fundamental theorem. C o r o l l a r y 1.10.
Let
F
i s a f i b r e sequence i n
Proof.
-t
E + B
be a f i b r e sequence i n H1.
Then Fp
+
H1'
Of c o u r s e we a r e o n l y making t h e a s s e r t i o n up t o homotopy,
so t h a t our c l a i m amounts t o s a y i n g t h a t e x a c t sequence of homotopy groups.
Fp
-+
Ep
-t
Bp
induces t h e u s u a l
However t h i s f o l l o w s immediately from
P r o p o s i t i o n 1 . 1 . 7 and t h e e q u i v a l e n c e of (i) and (ii) i n Theorem 1B. S i m i l a r l y , r e p l a c i n g (ii) of Theorem 1B by ( i i i ) of Theorem lB,
we o b t a i n C o r o l l a r y 1.11. Let
5 + Yp
-+
Cp
Ep
X
+
Y
-+ C
be a cofibre sequence i n H1.
i s a cofibre sequence.
Then
+
B
P
Localization of I -connected CW-complexes
59
We now introduce an important definition. Definition 1.12.
A map
f: X
is called a P-equivalence if
in H1
Y
--f
fP
is a (homotopy) equivalence. Theorem 1.13. Let
primes.
f: X
Y in
+
H ,and l e t 1
P
be a non-empty f a m i l y of
Then the following statements are equivalent: (il (iil
f i s a ?-equivalence; f i s a p-equivalence f o r a l l
Q C
(iii) TInf i s a P-isomorphism f o r a l l n ( i v ) H f i s a P-isomorphism f o r a l l n
Proof.
By Whitehead's Theorem f
P; 2 1; 2 1.
is a P-equivalence iff n f n P
is an isomorphism for all n 1 1. By Theorem 1B it follows that
Thus the equivalence (i) (i)
Q
(iv).
(iii) follows from Theorem 1.3.1.
Theorem 1.3.12 ensures that
(nnf)p
TI
f = (nnf)p. n P
Similarly
is an isomorphism iff
(nnf) is an isomorphism for all p C P. Thus the equivalence of (i) P and (ii) readily follows. A further refinement is possible in the case in which
are of f i n i t e t y p e , that is, n
?
1. Notice that, in
H1,
IT
X and nnY
n
X and Y
are finitely generated for all
this is equivalent to asking that H X and
HnY be finitely generated. Theorem 1.14. Let
f: X
-+
Y i n H1 with X , Y
P be a non-empty f a m i l y of primes. f,:
Hn(X; Z/p) zz Hn(Y; Z/p) f o r Proof. Let
Z/p
f: X
+
Y
Then
of f i n i t e type, and l e t
f i s a P-equivalence i f f
p € P.
be a P-equivalence. Then, since, for p C P,
is P-local, it follows from Theorem 1B and PropOsftiOn I.1.8(ii)
we have a comutative diagram
that
Localization of homotopy types
60
Thus
is a n isomorphism; n o t i c e t h a t t h e i m p l i c a t i o n we have proved does
f,
not require t h a t
It is i n t h e o p p o s i t e i m p l i c a t i o n
b e of f i n i t e type.
X, Y
t h a t t h i s condition plays a decisive r o l e . induces type,
Hn(X; Z f p )
f,:
Hn(Y;
We w i l l prove t h a t
H f n
Consider t h e diagram, f o r each
ZIP),
Thus w e suppose t h a t
n 2 1, p 6 P , where
i s a p-isomorphism,
f: X + Y
are of f i n i t e
X, Y
Hnf: HnX
H Y , n 11.
-+
n 5 1,
$
(1.15)
f
*n
t h e v e r t i c a l homomorphisms being induced by
f.
i t f o l l o w s t h a t we must prove t h a t each
is b i j e c t i v e and each :f
It f o l l o w s immediately from (1.15) t h a t e a c h
surjective. and each
f i
surjective.
It a l s o f o l l o w s from (1.15)
Suppose, i n d u c t i v e l y , t h a t we have shown Then
f:
By P r o p o s i t i o n 1 . 1 . 8 ( i i ) ,
Hrf: HrX
--t
fi,
is a p-isomorphism f o r
HrY
It t h u s f o l l o w s from (1.15) t h a t
bijective.
that
..., f C 1 r 5 n f:
-
f;
is injective
fi
is bijective.
bijective,
1, so t h a t
n Z 2.
f:-l
is
is b i j e c t i v e , and t h e i n d u c t i v e
s t e p i s complete. Since
H f
Theorem 1.13, t h a t Remark. if
f: X
i s a p-isomorphism
f
for a l l
p € P , we conclude, from
is a P-equivalence.
Theorems 1.13 and 1 . 1 4 show t h a t we have t h r e e p r a c t i c a l ways t o t e s t +
Y
p-isomorphism
i s a P-equivalence. for a l l
We may t r y t o show t h a t
n Z 1, p € P; we may t r y t o show t h a t
II
f Hnf
is a
is a
Localizationof 1-connected CW-complexes
p-isomorphism for all n 1 I, p t P; or, if X, Y
are of finite type, we
have the potentially most practical procedure, namely to try to show that f,:
Hn(X; Z/p)
-+
Hn(Y; Z/p)
is an isomorphism for all n 1 1, p € P.
In the case when P is empty, we have the following evident modification of Theorem 1 . 1 3 . Theorem 1.16. Let
f: X
-+
Y i n H1.
Then the f o l l o w h q statements are
equivalent: (il (iil
f
i s a 0-equivaZence;
nnf
f i i i l Hnf ( i v ) f,:
i s a O-$somorphism f o r a12 n 2 1; is a 0-isomorphism f o r a l l Hn(X;Q)
2
H (Y;Q)
n 2 1;
for a12 n 2 1.
61
Localization of homotopy types
62 2.
Nilpotent spaces. It turns out that the category
is not adequate for the full
H1
exploitation of localization techniques.
This is due principally to the fact
that it does not respect function spaces. We know, following Milnor, that if X
is a (pointed) CW-complex and W
a finite (pointed) CW-complex, then
the function space Xw of pointed maps W
-+
X
has the homotopy type of a
CW-complex. However its components will, of course, fail to be 1-connected even if X Xw
is 1-connected. However, it turns out that the components of is nilpotent. Moreover, the category of nilpotent
are nilpotent if X
CW-complexes is suitable for homotopy theory (as first pointed out by E. Dror), and for localization techniques [14,83]. Definition 2.1.
A connected CW-complex X
and operates nilpotently on
IT
is nilpotent if nlX
X for every n
is nilpotent
>_ 2.
n
Let NH be the homotopy category of nilpotent CW-complexes. Plainly NH
2
H1.
Moreover, thesimple CW-complexes are plainly in MH; in particular,
NH contains all connected Hopf spaces. It isalsonotdifficult tosee (Roitberg [69])
that, if G
is a nilpotent Lie group (not necessarily connected) then its is nilpotent in the sense of Definition 2.1.
classifying space BG
We prove
the following basic theorem, which provides us with a further rich supply of nilpotent spaces. Theorem 2.2.
Let F
Then F C NH if
Proof.
-i
E
f
B be a fibration of connected CW-compzexes.
E €NH.
We exploit the classical result that the homotopy sequence
of the fibration is a sequence of is
IT
IT
E-modules. 1
E-nilpotent of class icy then 1
IT
n
F
is
IT
We will prove that, if nnE F-nilpotent of class 5c 1
+
1.
Nilpotent spaces
63
(A mild modification of the argument is needed to prove that if
nilpotent of class Zc, then
71
F is nilpotent of class 5c 1
+
IT
E
is
1
1; we will
deal explicitly with the case n 1 2.) We will need the fact that and that the operation of I T ~ Eon
TI
E 1
F
IT
operates on
IT
B
through f,,
is such that
It will also be convenient to write IF, IE for the augmentation ideals of rlF,
IT
E. 1
Then the statement that anE is
IT
E-nilpotent of class 5c 1
translates into
Consider the exact sequence of
... and let S-CC=
,€ €
I;, a
€
a ~ B, c IT,+IB.
(i*n-.l)['a,
=
-IT
F.
Let
n+l B
- - IT
E-modules 1
a
IT
nF
Then i*(S.a) 11 €
i*
=
IT
nE
(i*c)*i,(a)
rlF. Then a((i*n-l).B)
( n - l ) c . c ~ , by ( 2 . 3 ) .
But
... = 0 =
by ( 2 . 4 ) .
(i,n-l).aB
(i*Q-l)*B = (f,i,n-l).B
=
Thus
=
0, SO
0. This shows that IC+'*~ F = (O), and thus the theorem is F n proved. Note that, in fact, our argument shows that, even if F is not (n-1)S-a
=
connected, each component of
F is nilpotent. F e w i l l feel free to invoke
this more general statement. Now let W be a finite connected CW-complex and let X be a connected CW-complex. Let Xw b e the function space of pointed maps W W + X and let Xfr be the function space of free maps. Choose a map W as base point and let g € XW (g€Xfr) of
g.
(XW ,g)((XfrW,g))
be the component
Localization of homotopy types
64
(Compare G. Whitehead [871 , Federer [ 2 6 1 . )
Theorem 2 . 5 .
( i ) (xw,g) i s nilpotent. W
(xfr,g)
liil
x
i s nilpotent i f
Proof. We may suppose t h a t ( i ) , ( i i ) are certainly true i f i n d u c t i o n on t h e dimension of
i s nitpotent. is a p o i n t .
Wo
i s 0-dimensional,
W
Thus t h e a s s e r t i o n s and w e w i l l argue by
We w i l l be c o n t e n t t o prove ( i ) . We have
W.
a cof i b r a t i o n
v where
is a wedge of
V
-+
wn
*
wn+l,
g i v i n g r i s e t o a f i b r a t i o n (where we d i s p l a y
n-spheres,
one component of t h e f i b r e )
where
w"+'-+
g:
Wn (X ,go)
X
and
g
= glw".
Our i n d u c t i v e h y p o t h e s i s i s t h a t
is n i l p o t e n t , so t h a t Theorem 2 . 2 e s t a b l i s h e s t h e i n d u c t i v e s t e p .
Corollary 2 . 6 .
Let
W
be a f i n i t e CW-comptex and
W (X ,g)
X € NH. Then
and
W (Xfr,g) are nilpotent.
Proof. L e t
Wo,
W1,
..., Wd
Then
xw =
x"0
x
;*
b e t h e components of
x
...
x
W, w i t h
o € Wo.
Xd'f r .
Since p l a i n l y a f i n i t e product of n i l p o t e n t s p a c e s is n i l p o t e n t , i t f o l l o w s that
W (X ,g)
is n i l p o t e n t .
Similarly
W (Xfr,g)
is n i l p o t e n t .
C o r o l l a r y 2 . 6 t h u s e s t a b l i s h e s ( i n view of M i l n o r ' s theorem) t h a t we s t a y i n s i d e t h e c a t e g o r y NH when we t a k e f u n c t i o n s p a c e s
X € NH and
f i n i t e , i n t h e s e n s e t h a t each component of
W
Xw
Xw
is i n
with
NH.
W e now proceed t o g i v e an important c h a r a c t e r i z a t i o n of n i l p o t e n t
spaces.
Let
X
be a connected CW-complex and l e t
-
...
(2.7)
- ...
Nilpotent spaces P
xn 4 xn-l
be i t s Postnikov decomposition, so t h a t K(nnX,n).
65
-x1-0 i s a f i b r a t i o n with f i b r e
pn
W e s a y t h a t t h e Postnikov decomposition
n
refinement a t stage
x
(2.8)
qc.
Yc
=
n
where t h e f i b r e of
Let
decomposition of
if and only i f Proof.
TI
pn
- -...
91
Y1
gi: Yi-l + K(Gi,n+l),
Y n o
.
'n-1' K(Gi,n)
x
admits a principal refinement a t stage
X
operates n i l p o t e n t l y on
1
X
71
LK(Gi,n+l),
Since
IT
Y = (O), n o
TI
and
n ? 2
Suppose conversely t h a t W e consider
p :X n n
-t
Xn-l.
IT,X
Suppose
Then we may r e g a r d
i = 1,
X(=nlYi,05i5c) 1
rnYC = rnXn =
(stage 1)
n 2 2.
..., c , o p e r a t e s t r i v i a l l y on
Thus, by r e p e a t e d a p p l i c a t i o n s of t h e proof of Theorem 2.2,
o p e r a t e s n i l p o t e n t l y on
qi
( T I ~ Xi s n i l p o t e n t ) .
We w i l l be c o n t e n t t o g i v e t h e argument f o r
yi
IT
o
Then the Postnikov
f i r s t t h a t we have t h e p r i n c i p a l refinement (2.8).
as a fibration.
Y
1 5 i 5 c.
be a connected CW-compZex.
X
principal
may be f a c t o r e d as a product of f i b r a t i o n s
is an Eilenberg-MacLane s p a c e
qi
i s induced by a map Theorem 2 . 9 .
if
admits a
nlX
X.
IT
is
IT
X-nilpotent of c l a s s 3. 1
Then, by t h e r e l a t i v e Hurewicz Theorem we have
a n a t u r a l isomorphism
where Thus
n+l
(p ) n
Gn+l(pn)
a s a n element of
i s o b t a i n e d from
I T ~ + ~ ( P , )by k i l l i n g t h e a c t i o n of
may be i d e n t i f i e d w i t h Hn+l
2 (pn;nnX/r rnX).
IT
n
2
X/r I T ~ X ,and h-'
Thus
h-l
nlXn.
may b e regarded
g i v e s rise t o a diagram
Localization of homotopy types
66
with
u b - 0.
If
u
induces
ql: Y
1 -+ Xn-1'
then
pn
factors a s
(2.10) with
q1
The homofopy sequence of (2.10) reduces t o
induced a s required.
rl replacing
Thus we may r e p e a t t h e above procedure, with
- -
p
and,
n'
continuing i n t h i s way, we reach
xn
(2.11) each
qi
r yC
&
Yc-l
being induced by a map
...
Yi-l
However, a l l t h e homotopy groups of
-f
r
o
Y2
K(Gl,n+l), where
Gi =
r
vanish, s o t h a t
n-1'
r innX/ri+lnnX.
i s a homotopy
equivalence, and (2.11) is e s s e n t i a l l y t h e p r i n c i p a l refinement a t s t a g e
n
whose e x i s t e n c e we set out t o prove. We would say t h a t t h e Postnikov system of
X
refinement i f i t admits a p r i n c i p a l refinement a t s t a g e
admits a principal
n
f o r every
n 2 1.
We then have the evident Corollary 2.12.
Let
X
be a connected CW-complex. Then X
i s nilpotent i f
and only i f i t s Postnikov system a h i t s a principal refinement. W e p o i n t out t h a t t h e s i m p l e spaces a r e i d e n t i f i e d , by t h e
correspondence i m p l i c i t i n t h i s c o r o l l a r y , with those spaces whose Postnikov system is i t s e l f principal.
Remark.
Once we have obtained a p r i n c i p a l refinement of t h e Postnikov system
of a space, t h e r e i s , of course, no d i f f i c u l t y i n obtaining f u r t h e r refinements,
Nilpotent spaces which w i l l remain p r i n c i p a l .
61
Thus i f , f o r example,
i s of f i n i t e t y p e
X
and n i l p o t e n t we may r e f i n e i t s Postnikov system so t h a t each map
of t h e r e f i n e d system i s induced by some map
ZIP,
or
f o r some prime
Yi-l
+
f: E
i f a l l of IT
+
nlF).
To t h i s end we s a y
a r e connected ( s o t h a t
f,
maps
n E
1
F
onto
n i l p o t e n t l y on t h e homotopy groups of
f.
nlB),
and
( i n c l u d i n g , of c o u r s e ,
W e could a l s o e x p r e s s t h i s l a s t c o n d i t i o n by a s k i n g t h a t
by t a k i n g
A = Z
F , of ( p o i n t e d ) CW-complexes i s nilpotent
o p e r a t e s n i l p o t e n t l y on t h e homotopy groups of
E
1
Yi-l
p.
B, w i t h f i b r e
F, E, B
+
where
K(A,n+l)
We w i l l need a r e l a t i v e form of Theorem 2.9. t h a t a map
Yi
n E 1
operate
Note t h a t w e r e c o v e r D e f i n i t i o n 2 . 1
t o b e a p o i n t , provided w e adopt t h e r i g h t n o t i o n of n i l p o t e n c y
B
f o r t h e o p e r a t i o n of
T
E
1
on
TI
F.
Although we w i l l n o t need t h e g e n e r a l
1
case i n t h i s t e x t , w e now d e s c r i b e t h i s n o t i o n f o r t h e a c t i o n of a group
on a group
Q
With r e s p e c t t o such a n a c t i o n w e d e f i n e a lower central series
N.
as f o l l o w s (see H i l t o n [36]): rlN = N ,
Q
rrh
=
gp{(x.a)ba
W e t h e n s a y t h a t t h e a c t i o n of
YC+lN = 11).
Q
N
N
N
i s commutative.
by c o n j u g a t i o n , t h e n
r% = r 9
nilpotent map, it f o l l o w s t h a t i f
i s nilpotent of class
b€N, i21. 5c
if
f
Note a l s o t h a t i f
i N.
Q = N
and o p e r a t e s
T h u s , w i t h o u r d e f i n i t i o n above of a
i s n i l p o t e n t , t h e n t h e f i b r e of
f
W e a l s o have t h e f o l l o w i n g f a i r l y e v i d e n t p r o p o s i t i o n .
nilpotent.
P r o p o s i t i o n 2.13.
connected f i b r e .
Proof. Q =
on
r iN ,
Note t h a t t h i s d e f i n i t i o n a g r e e s w i t h t h a t g i v e n i n 1.4 i n
t h e case i n which
on
Q
-1 -1 b 1 , x C Q, a €
rlE, and
Let
f: E
+
B
Then, if E , B
be a map of connected CW-complexes with are nilpotent,
f
i s nilpotent.
We have a n e x a c t sequence of groups w i t h Q-action,
F = f i b r e of
f,
where
is
Localizationof homotopy types
68
...
+
IT
n+l
B - +IT F + n
E+ n
IT
...
+
s B
2
-+
TI
F +
1
E+ 1
TI
IT
B. 1
are nilpotent, slE operates nilpotently on nnB, TI E n for all n 2 1. Thus, if n 5_ 2 , the conclusion that IT E operates nilpotently 1 on n F follows from Proposition 1.4.3, since our definition of nilpotent
Now since E, B
action coincides, in the case of a commutative group, with that of 1.4. Thus the case n = 1 remains. We have an exact sequence IT
B-+n F+TIE
2
1
1
of Q-groups and the argument of Proposition 1.4.3 may be adapted to yield the result in this case in view of the fact that the image of n2B lies in the center of
IT
F.
1
It is, of course, necessary to take account of both facts
noted after the definition of a nilpotent Q-action. The relative form of Corollary 2.12 reads Theorem 2.14.
Let
f: E
+
B
be a map of connected CW-complexes inducing a
surjection of fundamental gruups.
Then f i s niZpotent i f and only i f i t s
Moore-Postnikou system admits a principal refinement.
be the Moore-Postnikov system of f .
-
Now if pn may be factored as in ( 2 . 8 ) ,
we obtain a sequence of extensions of n E-modules
Gi>where Gi
nYi
IT
1
nnYi-l, i
*
1,
is a trivial module. Now the fibre of
an Eilenberg-MacLane space K(Hi,n) an extension of
nlE-modules
..., c,
is i = ql’”qi: Yi Y and the relation siqi+l = si+l yields s
-+
69
Nilpotent spaces where Ho = {O}, Hc
=
~ l ~ = + n~ F. p ~ It follows from (2.15) and Proposition
1.4.3, by an easy induction, that
TI
F is a nilpotent
n
~l
1E-module.
(The
case n = 1 is again slightly special, but we will omit the details in this case.) The converse implication is proved exactly as in the absolute case; see the Proof of Theorem 2 . 9 . Before proceeding to discuss how to intr0duce.a localization theory Here we confine
into NH we show how Serre's C-theory may be applied to NH.
attention to the absolute case, since the relative case requires stronger axioms on a Serre class, as is already familiar in the classical case of the Thus we will be considering generalized Serre classes in the
category H1.
sense of Definition 1.5.1.
We prove the one basic theorem which we need in
the sequel. Theorem 2.16.
Let
X E NHand l e t
be a generalized Serre class.
C
Then
the following assertions are equivalent: (il
T I ~ Xf
C for a l l
(ii) HnX E C f o r all n (iii) nlx c cover of
1
n Z
c and H ~ Xc c
1
for a l l
n 2 1, where
X
i s the u n i v e r s a ~
X.
Proof.
We need two lemmas, which are interesting in their own right.
The first is a generalization (to general m?l) of Theorem 1.4.17, though we here only state the result for homology with integer coefficients. Lemma 2.17.
If
~l
acts n i l p o t m t t y on the abelian group
A, then
n
acts
nilpotently on Hn(A,m), n 1 0 .
Proof. n-series of A
Let 0 = rC+'A
5 rCA 5
... 5 I-1A = A
(see Section 1.41, and write Ai = r iA
be the lower central for convenience,
Note that each Ai is a nilpotent a-module, of class less than that of A
Localization of homotopy types
70
if
i 2 2.
Moreover,
a
a c t s t r i v i a l l y on
Ai/Ai+l.
We have a s p e c t r a l
sequence of a-modules,
converging ( f i n i t e l y ) t o t h e graded group a s s o c i a t e d w i t h I f w e assume i n d u c t i v e l y t h a t
filtered.
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 Lemma 2.18.
Proof. 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
Hn(%) where
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
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 infer that
f
E2
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 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.
s+l,O 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
73
Ii-If-
2 (3.2)
Y
Since
X
K(nlX,l)
Y
Jfl K(slY,l)
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
-+
Proof of Proposition 3 . 4 .
X
r+
X
we
Y
NH,f o r which f provides a natural L.
U LNH t o
transformation from the embedding
Let g : X
+
X' in U.
We thus have a
If If
diagram
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
-
be t h e f i b r e of
K(Gp,n+l), where
Gp
If'-
X'
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
-&
K(G,n+l)
h
K(Gp,n+l)
There i s then a map
h.
f: X
-+
Y
rendering t h e
diagram
4f -4f' A X --+
X'
K(G,n+l)
Y
Y'
K(Gp,n+l)
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)
Xi
g
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
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
@Yi.
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.
satisfies (ii).
@ fi
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
- n,
s i n c e i t follows immediately from (3.9). P
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
t o NH; we
H1
k k e w i s e Theorens1.13, 1.14, and 1.16 extend from
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 Section 5. Let X € hkl. We say that X generated for all n 2 1 and that X
is of f i n i t e type if anX is finitely 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 0 P P) P P 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
For each a(p)
*
p C l 7 U {O)
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
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
a x
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. 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
We w i l l d e n o t e by
.
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
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
+
EX
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
nixo
@
... - n p xiX
nn.x 1 P
(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
w X i
&? TT
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 .
-
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
-
mnix)o
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
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
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
ll
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 . f*: [W,X]
2
-
with
For, according to that result, if W
[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
-+
e,
let
P-local rmd
f,:
[W,X]
W.
Then
CW-complexes
Proof. W
f: X
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.
=
$Jcm2
W e now prove
f o r some
e*:
Then
ipym = e*a
that
jam' = 1 f o r some ipe,x
Thus
yml
f o r some
=' : ,e
m
We conclude t h a t
QXP.
-+
[W,%]
P-surjective.
a € [V,X], m C P I .
1
with
e*x = 1.
Thus
m2 6 P' , whence f i n a l l y
[w,x]
= ipym'
= (e*x).($pb),
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,$,], and b = e*c,
It f o l l o w s t h a t
c f [CV,X]
e*
P-injective.
91
C P'.
Thus
It f o l l o w s t h a t
Now
=
e*$c,
m m
(xm)m1m2 = $ a
= 1
[W,%l.
y €
Let
am1
= i x , x C [W,X],
mm 1-power of
bm2 = e*c, f o r some
Thus, by Lemma 6 . 4 below, ynrmlm2= ek(xm2.$c)
ml
e,ja = j Pe*a = 1, so
( f o r a s u i t a b l y bracketed b C vnXp.
e*a
c
<
y).
unX, m2 C P ' .
and t h e a s s e r t i o n i s proved.
(Note t h a t , i n t h e s e arguments, w e have w r i t t e n a l l monoid s t r u c t u r e s i n ( 6 . 3 ) multiplicatively. ) The converse i s t r i v i a l .
f,: II X
-+TI
n 11.
Thus
For i f we s e t
W = Sn
t h e n we know t h a t
i s a P-isomorphism t o a P - l o c a l group and hence P - l o c a l i z e s ,
Y f
P - l o c a l i z e s by Theorem 3B.
The f o l l o w i n g lemma, t h e n , s u p p l i e s t h e one m i s s i n g s t e p . Lemma 6 . 4 .
Let
X
attaching the cone W
*
ZA
induces
be an H-space and l e t CA
to
V
W = V
by means of a map
U
CA, t h e space obtained b y A
-t
V.
The p r o j e c t i o n
Lacalization of homotopy types
92
4 i s central in the monoid
and the image of
Proof.
Let c: W
+
W V CA be the cooperation map, In the terminology
of Eckmann-Hilton. Then, for x € [W,X], a
where
-
[w,x] in the strong sense that
€
[ZA,X], we have
x.$a = c*
(xl.$al) (x,.4ia2)
c*
.
[ZA,X].
X
=
(c*
Remark. Note that we could have proved that
e*
the following stronger sense, namely, that if e,x xn = yn for some nth power with n
€
P'.
in (6.1) is P-injective in =
e*y, x , y € [W,X], then
In the presence of a nilpotency
condition, this sense of P-injectivity in fact coincides with the obvious one (obtained by setting y = 1); and, indeed, it is true, with appropriate definitions, that
[W,X] is a nilpotent loop (non-associative group),
Corollary 4 . 3 enables us to deduce the following modification of Theorem 6.2. Corollary 6.5. Let
CW-complex.
X be a connected H-space mrd W a quasifinite
Then ZocaZization induces
and eg i s P-bijective. The following consequence of this corollary will be very important in the sequel. Theorem 6.6.
Let W be a quasifinite CW-compZex and l e t
H-space such that Wp
9
Xp.
Then there e x i s t s a map
X be a connected
f: W * X such that
Localizing H-spaces
fp:
wp
eY
$.
Proof.
Let
g,
= ng*: R*(Wp)
g: Wp
-+
$
n C P' suchthat
6 . 5 there e x i s t s n
93
-+
n,($)
gn = e l ( f ) , f : W
and t h a t , consequently,
isomorphism of homotopy groups found the required map
b e a homotopy equivalence.
f
.
(n C Zt). Thus
-+
X.
n
g,
g": Wp
By Corollary
But i t i s c l e a r that is,like '\I
$,
g*,
an
so that we have
Localization of homotopy types
94
7.
Mixing of homotopy t y p e s T h e i d e a o f m i x i n g h o m o t o p y t y p e s g o e s b a c k t o Zabrodsky [ 9 3 ] a n d h a s been
e x t e n s i v e l y used t o c o n s t r u c t examples and counterexamples i n homotopy t h e o r y ; see, e . g .
[79,93,96] and Chapter 111 of t h i s monograph.
It seems t h a t l o c a l i z a t i o n
t h e o r y p r o v i d e s t h e r i g h t framework f o r d i s c u s s i n g t h i s i d e a and r e n d e r i n g i t most e a s i l y u s a b l e i n a p p l i c a t i o n s . We b e g i n by d i s c u s s i n g puZZbacks
i n homotopy t h e o r y , a t o p i c of some
independent i n t e r e s t , i n p a r t i c u l a r w i t h r e s p e c t t o l o c a l i z a t i o n .
Given a
diagram
If
X (7.1) Y-B
i n t h e category
g
T of based CW-complexes, we may r e p l a c e
and t a k e t h e ( s t r i c t ) p u l l b a c k
which we c a l l t h e weak
by a f i b r e map
We o b t a i n a diagram
T.
puZZback of (7.1) i n t h e homotopy c a t e g o r y H.
known t h a t t h e homotopy t y p e of
as a diagram i n
in
f
2
depends only on t h e diagram ( 7 . 1 ) , i n t e r p r e t e d
H, and i s symmetric w i t h r e s p e c t t o
we might j u s t as w e l l have r e p l a c e d i n s t e a d of choosing t o r e p l a c e
f.
I t is
g
(or both
Of c o u r s e , i f
f
f , g, i n the sense t h a t and
f
(or
g) by a f i b r e map, g) were a l r e a d y a
f i b r e map, n o replacement would be n e c e s s a r y . I f (7.1) were a diagram i n
To, t h e s u b c a t e g o r y of
T
c o n s i s t i n g of
connected CW-complexes, we would o b t a i n t h e weak p u l l b a c k i n the corresponding homotopy c a t e g o r y base point.
Ho by r e p l a c i n g
W e would t h u s o b t a i n
2
i n (7.7.) by t h e component
Zo
of i t s
Mixing of homotopy types U
1. If
zo (7.3)
in Ho.
x
We are interested in the question of when we may infer that
is, in fact, in NH. Theorem 7 . 4 . i f
zO
W e prove:
Suppose t h a t
X , Y C NH
i n (7.31.
Then
nlZo operates n i l p o t e n t l y on nnB, n 2 2 , v i a
Proof.
fu
Zo
.
E NH i f and onZy
The diagram ( 7 . 3 ) gives rise to a Mayer-Vietoris sequence
of groups with nlZo-action, where
Suppose that nlZo
G
i s the pullback of the diagram of groups
operates nilpotently on TInB, n
operates nilpotently (via u
vo)
nlZo operates nilpotently on
?
2. Then, since n lZ o
on the homotopy groups of X
it follows from Proposition 1 . 4 . 3 that nlZo Now
95
TI
2
and Y,
operates nilpotently on nnZo, n ? 2.
B and hence on Im n2B C n l Z o .
However
here the operation is by conjugation and thus the operation of G on Im n2B induced by the exact sequence
Im n2B is also nilpotent.
-TI
Z
l o
-
Since G, as a subgroup of
we infer from Proposition 1.4.1 that r1Z0 Conversely, suppose that Zo
G IT
X 1
x TI
Y, is nilpotent, 1
is nilpotent.
is nilpotent. Then an immediate
application of Proposition 1 . 4 . 3 to the Mayer-Vietoris sequence
Localization of homotopy types
96
shows that nlZo operates nilpotently on nnB, n 1 2 . Of course, it is most useful to have a criterion for Z
to be
nilpotent which is independet of the maps uo, vo, but depends only on (7.1). Thus we now enunciate Corollary 7.6.
Let ( 7 . 1 ) be a diagram i n NH.
Then, i n the weak pullback
(7.3) i n Ho, Zo C NH.
The following immediate consequence of Theorem 7.4, generalizing Theorem 2 . 2 , 1s also useful. Corollarv 7.7.
Let ( 7 . 1 ) be a diagram i n Ho with X, Y C NH.
If X
OP
Y
i s 1-connected, then Zo C N-l. We now suppose that (7.3) i s a weak pullback in N a n d we localize at the family of primes P.
We obtain
yp
Diagram (7.8) i s a weak pullback in NH.
Proposition 7.9.
Proof.
Form the pullback in To
(7.9)
where we may 8:
Zop+ 2'
assume fp to be a fibration. There is then a map
yielding a commutative diagram
Mixing of homotopy types
in
91
NH, andhence a map of t h e P - l o c a l i z a t i o n of t h e Mayer-V
!to1
3
sequence of
(7.3) t o t h e Mayer-Vietoris sequence of (7.9); h e r e Theorem 1.2.10 p l a y s a c r u c i a l r o l e i n e n s u r i n g t h a t , when we l o c a l i z e
G
i n (7.5) w e o b t a i n
t h e p u l l b a c k of t h e diagram
K1yP
-rB
rigp
1P
I n t h i s map of Mayer-Vietoris sequences a l l groups e x c e p t mapped by t h e i d e n t i t y .
Thus
s
rnZoP
are
i n d u c e s an isomorphism of homotopy groups
and hence is a homotopy-equivalence. Suppose, i n (7.31,
C o r o l l a r y 7.10.
Then Zo
€ Mi
and
Proof.
that
f
is a P-equlvalence and X, Y, B
is a P-equivalence.
vo
We a l r e a d y know t h a t
Z
€ NH
i s an e q u i v a l e n c e so t h a t , by P r o p o s i t i o n 7.9, v
€
by C o r o l l a r y 7 . 6 . vop
Now
fp
is an e q u i v a l e n c e and so
i s a P-equivalence.
Of c o u r s e , t h i s c o n c l u s i o n could more e a s i l y have been drawn w i t h o u t e s t a b l i s h i n g P r o p o s i t i o n 7.9 i n f u l l g e n e r a l i t y . We w i l l b e i n t e r e s t e d i n e s t a b l i s h i n g c o n d i t i o n s under which w e may i n f e r t h a t t h e space
Z
i n (7.2)
is a l r e a d y connected, so t h a t
Z = Zo.
NH.
Localization of homotopy types
98
Obviously this holds if (7.1) is a diagram in To
in which f
(or g)
induces a surjection of fundamental groups. However, we will require a more general criterion. Proposition 7.11. Let ( 7 . 1 ) be a diagram i n nlB i s of the form 17.21 in
H,
f*a.g,f?,
c1
E
Ti
To
i s which every eZement of
Then i n the weak pullback
X, f? E nlY. 1
i s connected.
Z
Proof.
Let us assume f a fibre map, so that ( 7 . 2 )
pullback in T. Given (x,y) E Z, x E X, y E Y, let k o
to x, and m
a path in Y
from o
reverse of m, is a loop in B on p
0 , so
is the strict
be a path in X
to y. Then fII *gi, where
i
from is the
in X,
that there are loops h
in Y with
Thus f(x
* L)
-
-
fk*gm- fh*gp. g(p *m), re1 endpoints,and, since f is a fibre map, we find
L' * h * II, re1 endpoints, that
(II',m')
so
that
f&' = gm', where m' = p *m.
is a path in Z from o
to
It follows
(x,y).
We exploit Proposition 7.11 in the following way.
Let ( 7 . 2 ) be a diagram i n To i n which f,: nlX * TIIB is a
Corollary 7.12.
P-surjection and of the primes.
g,:
TI
Y
1
-+TI
B i s a Q-surjection for some p a r t i t i o n 1
Then, i n (7.21,
Proof.
Z
yn = g,n
P
for m E Q, n € P.
are relatively prime we find integers k, II with km II and then y = f,Sk* g,n ,
+ an =
We are now ready to prove the mixing theorem which is the main objective of this section.
=
i s connected.
Let y E alB. Then ym = f,S,
Since m, n
n
1
u
Q
Mixing of homotopy types
Theorem 7.13.
with Xo
X , Y C NH
Let
of the primes. Then there exists
Proof. h: Y
0
2
X
2
and Let rI
Yo
with
Z C NH
There a r e c a n o n i c a l maps
99
s:
Zp
Xp
+
2
= P U Q
$,
Xo,
ZQ
t: Y
N
Q
Y
-f
be a partition
9' Let
'0'
and c o n s i d e r t h e diagram
0
1
(7.14) Y-%
Q
xO
Form t h e weak p u l l b a c k of ( 7 . 1 4 ) ,
Certainly
s
i s a Q-equivalence and
7.12 e n s u r e s t h a t Corollary Thus
u
7.10 g u a r a n t e e s t h a t induces
up: Zp
2
Xp
u
and
i s a P-equivalence.
Thus C o r o l l a r y
Corollary 7.6 then ensures t h a t
i s connected.
Z
ht
i s a P-equivalence
v
induces
vs:
and
v
2 E NH and
i s a Q-equivalence.
ZQ r- YQ.
The following addendum i s important i n a p p l i c a t i o n s . P r o p o s i t i o n 7.15. Z
(i) Let X ,
Y
in Theorem 7.13 be quasifinite. Then
is quasifinite. lii) Let X , Y in Theorem 7.13 be 1-connected. Then
Z
is 1-connected. liiil Let X,
Y
in Theorem 7.13 have the homotopy type of
a finite 1-connected CW-complex. Then
Z
has the hornotopy type of a finite
1-connected CW-comp Zex.
Proof.
(i) Observe t h a t i f
generated L -module and P by Theorem 1.3.10, is quasifinite.
A
A
Q
A = WiZ
then
is a finitely-generated
%
is a f i n i t e l y -
%-module.
Thue,
is a f i n i t e l y - g e n e r a t e d a b e l i a n group, so t h a t 2
Localization of homotopy types
100
(ii) Observe that
nlZ
is a nilpotent group which l o c a l i z e s t o
the t r i v i a l group a t every prime and hence is c e r t a i n l y t r i v i a l . (iii) This follows from (i) and (ii),using the techniques of
homology decomposition.