HOPF SPACES
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NORTH-HOLLAND MATHEMATICS STUDIES
22
Notas de MatemBtica (59) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester
Hopf Spaces
ALEXANDER ZABRODSKY Associate Professor, Hebrew University, Jerusalem, Israel
1976
NORTH-HOLLAND PUBLISHING COMPANY
- AMSTERDAM
NEW YORK OXFORD
@ North-Holland Publishing Company - 1976
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X
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Library of Congrai Cataioglng In Publication Data
Zabrodw, Alexmder Hopf spaces.
.
(Hotae de m a t d t i c a ; 59) studies ; 22) BibUography: p. Includes index. 1. H8 spaces. I. Title.
QP6l2.n.z ISBN 0 - R O L 5 5 3 - x
512'.55
(North-Holland mathematics
II.
Series.
76 413%
PRINTED IN THE NETHERLANDS
Table of Contents
IX
Introduction
0.
I.
V
Notations, conventions and preliminary observations 0.1
Spaces and maps
0.2
Homot opi es
0.3
Categories and adjoint maps
0.4
Pullbacks, pushouts and Eckmann-Hilton d u a l i t y
0.5
%spectra, r i n g spectra, generalized cohomology
The category of H-spaces Introduction
8
1.1 Basic properties of H-spaces
9
1.2
Some s p e c i a l classes of H-spaces
1.3 The s t r u c t u r e of
[
, H-space]
19 21
1 . 4 H-deviation and H-homotopy equivalence
25
1.5 Change of H-structures and H-maps
29
11. Homotopy properties of H-spaces
34
Introduction 2.1
H-spaces and f i b r a t i o n s
36
2.2
H-liftings
37
2.3 Postnikov systems
42
2.4
Actions, H-actions and p r i n c i p a l f i b r a t i o n s
47
2.5
HA
2.6
Homotopy s o l v a b i l i t y and homotopy nilpotency
and HC
obstructions
58 63
Table of Contents
111.
The cohomology of H-spaces Introduction
69
3.1 The Hopf algebra H*(X,Zp)
70
3.2
Some r e l a t i o n s between the algebra H*(X,Zp) and t h e coalgebra
IV.
73
H*( CK, Zp)
3.3
Browder's Bockstein s p e c t r a l sequence
84
3.4
High order operations
98
Mod p
theory of H-spaces
Introduction
113
4 . 1 p-equivalence and p-universal spaces
114
4.2
mod p-homotopy
124
4.3
Decomposition of 0-equivalences
128
4.4
A study of
134
4.5
Mod P1 H-spaces
136
4.6
The genus of an H-space
147
4.7
Mixing homotopy types
152
4.8
The non c l a s s i c a l H-spaces and other
157
Ho
spaces
applications
V.
Non s t a b l e
BP
resolutions
Introduction
163
5.1
K i l l i n g homology p t o r s i o n
164
5.2
Wilson's
172
B(n,p)'s
Table of Contents
[
, B(n,p)l
5.3
The groups
5.4
H-maps i n t o B(n,p)
5.5 Examples: Some properties of BU
VII
176 181
187
5.6
Non s t a b l e BP Adams resolutions
190
5.7
Some simple applications
198
Bj 1iograg.y
211
L i s t of symbols
2 19
Index of terminology
222
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IX
Introduction Possibly more than any other f i e l d i n mathematics algebraic topology contains a v a s t amount of r e l a t i v e l y simple f a c t s , c l u s t e r s of small theorems and i n t u i t i v e observations.
Naturally, t h e s e include many
folklore-type theorems which do not appear i n t h e literature.
(It i s
q u i t e l i k e l y t h a t as many theorems i n algebraic topology have appeared verbally i n u n i v e r s i t i e s ' common rooms as have appeared i n p r i n t i n t h e professional l i t e r a t u r e ) .
It i s therefore e s s e n t i a l t o b r i n g some o f
t h e s e fundamentals t o light i n p r i n t from t i m e t o time. The subject of €I-spaces within the f i e l d of algebraic topology i s no exception.
I n t h e last decade some outstnading progress has been
made on t h e subject, a f f e c t i n g r e l a t e d f i e l d s i n homotopy theory such as the theory of cohomology operations , c l a s s i f y i n g spaces, i n f i n i t e loop spaces and l o c a l i z a t i o n theory. These notes t r y t o describe some of these new developments.
m a k e no attempt t o encompass all areas of progress.
We
Instead, we
concentrate only on t h r e e subjects: the s t r u c t u r e of t h e cohomology of H-spaces, t h e r e l a t i v e l y new
mod p
BP
theory of H-spaces and applications of t h e
theory i n the study of H-spaces.
An attempt has been made t o b r i n g a s u b s t a n t i a l p a r t of t h e s e notes t o within the grasp of graduate students and algebraic topologists who do not s p e c i a l i z e i n t h i s p a r t i c u l a r subject. The first two chapters of these notes cover t h e fundamental concepts and hence, are e s s e n t i a l f o r t h e understanding o f t h e last t h r e e .
These
l a s t , however, are f a i r l y independent. The only systematic treatment of the subject of H-spaces i n t h e
literature i s Stasheff's "H-spaces f r o m t h e homotopy point o f view" ( [ S t a ~ h e f f ] ~ ) This . w a s w r i t t e n during a period of r a p i d development
X
Introduction
i n t h e f i e l d and some of t h e newer results were s t i l l unpolished.
There
i s n a t u r a l l y some overlap between t h i s work and S t a s h e f f ' s notes, notably i n t h e f i r s t two chapters of t h i s book.
The p r i n c i p a l d i s t i n c t i o n
between t h e two volumes i s t h a t t h e s p e c i f i c aspects of H-space theory t r e a t e d i n d e t a i l are c l e a r l y d i f f e r e n t :
We do not consider here subjects
such as p r o j e c t i v e planes, c l a s s i f y i n g spaces, homology operations and i n f i n i t e loop spaces. Some r e l a t i o n s h i p can be found between Chapter I V of these notes and [Hilton, Mislin, Roitberg]. While preparing these notes t h e author was p a r t l y supported by a grant from t h e ETH of Zurich and by the B r i t i s h Science Research Council t o whom I would l i k e t o express q y g r a t i t u d e . thank t h e members of t h e Forschungsinstitut
I a l s o would l i k e t o
Mathematik and t h e
Mathematics Department of t h e ETH and the members of the Mathematical I n s t i t u t e of Oxford University f o r t h e i r patience i n discussing with me these notes i n t h e i r various stages of production.
Alexander Zabrodsky The I n s t i t u t e of Mathematics The Hebrew University JERUSAUM
1
Chapter 0
Notations, Conventions and Preliminary Observations
Spaces and maps
0.1.
A l l spaces i n these notes a r e assumed t o be simply connected and of
the homotopy type of CW complexes of f i n i t e type. W e use the notation
base point.
*
image
E
f o r base points of a l l spaces (thus
it
considering a base point as a map
All have a non-singular
from t h e singleton
it
it
to
X
with
x).
Subspaces a r e always assumed t o be NDRs and one can always apply t h e homotopy extension property. All maps are pointed (i.e.: base point preserving).
contain t h e base point.
A l l subspaces
Composition of maps are denoted by juxtaposition:
fog = fg. W e use t h e customsry notations
We denote by
A
the i d e n t i f i c a t i o n map
We use as a standard notation A = AX:
0.2.
X + X x X,
A(x) = x,x
and
A
E
A: X x X
-
X
A
X.
f o r t h e diagonal map: f o r t h e suspension:
EX = S1
A
Homotopies All homotopies are pointed, i . e . :
F(r,t) =
*,
0
5t 5
1.
F: X
x
I -+ X'
always s a t i s f i e s
X.
2
Notations, Conventions, Preliminaries
If F:
x1
x
f : Xo
I
-+
x2
go,gl:
X1,
-+
of
go
and
i s s a i d t o be r e l a t i v e t o
(core1 h) i f If cv
0.3.
gl
fabr.
f
g1 r e 1 Xo
X1
-
gl,
re1 f )
then a homotopy
3
= g,(x),
F(X,E)
if
F(f
E
= 0,1)
1) = g o o , = glQ1
x
F i s s a i d t o be coretative to h
i.e.:
hF = hgOpl = hglpl,
f : Xo c
h: X2 -+ X
2'
(F: go
F ( f ( x ) , t ) = gof(x) = g l f ( x ) .
i.e.
go
X1 -+ X
h F ( x , t ) = hgo(x) = hgl(x).
is an inclusion of a subspace, w e sometimes write
instead of
re1 f.
Categories and adjoint maps We occasionally use categorical notations, but our category theory
never goes beyond t h e phase of a notational system. We work i n t h e category of pointed (homotopy types o f ) CW complexes and continuous maps and not i n t h e homotopy category (where homotopy classes of maps a r e the morphisms).
W e avoid t h e homotopy category
because t h e homotopies themselves are being l o s t i n t h e homotopy category. These homotopies are needed t o obtain i n v a r i a n t s and obstructions throughout these notes. Nevertheless, q u i t e often we i d e n t i f y ambiguously maps with t h e i r homotopy classes and thus mark as equal. homotopic maps. more often i n t h e last three c h a p t e r s . )
(This i s done
Commutative diagrams always
commute only up t o homotopy. As usual we denote by (pointed) maps If
g,:
[X,Y]
h: X ' -+
X +
-+
X,
[X,Y']
Y.
[f]
g: Y
+
[X,Y]
the set of homotopy clasaes of
denotes the homotopy c l a s s of Y'
we w r i t e
h*: [X,Y]
f o r t h e functions induced by
t o the c a t e g o r i c a l notations of
[h,Y]
and
h
[X,g]
-+
[X'
and
f.
,Yl, g
(corresponding
respectively).
Pullbacks, pushouts and Eckmann-Hilton d u a l i t y
equivalence
[X1
-
pointed maps f : X A X2
1
X2
X
h
-+
Y.
(X2
"he equivalence i s given by assigning t o
Y t h e c l a s s of
( )#
We denote by then
A
l o c a l l y compact) t h e r e exists a n a t u r a l X X2, Y] ----* [X,, Y 2 ] where Yx2 i s t h e space of
X ,X ,Y 1 2
Given
( f ) # : X1
+
#' 1 f#,h# as the adjoints of
f and h
Yx2
h: X -,
If
h#(x1,x2) = h ( y ) [ x 2 ] .
i s given by
[f],
(f),(x)[x'] = f(x,x').
Yx2,
t h e inverse assignment as w e l l :
X2 r Y
A
3
We r e f e r t o
(thus omitting t h e l e f t and right
d i s t i n c t i o n of a d j o i n t s ) .
-
We s h a l l Only use t h e s e notions f o r t h e cases
#: [ZX,YI
(where we have
[x,ml
#:
and
X
2
[x,ztll
= I ---*
or
X2 = 'S
[CX,YI).
Pullbacks. pushouts and Eckmann-Hilton d u a l i t y
0.4.
Unless e x p l i c i t l y s t a t e d otherwise, pullbacks and pushouts are homotopy ( o r weak) pullbacks and pushouts:
the pullbaok of
fo,fl
xo,
Y: I
x,Y,y,
xE
i s t h e space -+
Y,
together with the two maps gl(x,Y,y) = y.
x0
x Y
I
x
Yo v X
g
+
Y,
i = 0,1,
of all t r i p l e s
"f,,f,
i: i s t h e one i n h e r i t e d from
Wf0,fl
xl.
I v Y1
x
+
i = 0,1,
Yi,
Y
i
f (XI,*,* 0
5 *,(X,O),*,
*,(
= *, *, * = *
c Y
i s t h e quotient space
Mfo,fl
induced by t h e equivalence r e l a t i o n spanned by
,ti),*
A pushout has two s t r u c t u r a l maps
inclusions
fi: Xi
y E
The topology of
The pushout of f i : X of
If
o v
X x I v Y
1'
*,*,f1(X)
gi: Yi
+
M
fo'fl
E *,(X,l),*
induced by t h e
4
Notations, Conventions
If
then
f: X + Y
t h e cone on
f.
(If
t h e cone on
X.
If
W
i s c a l l e d t h e fiber of
*,f
-t
= C(f)
M
f,
is
*¶f
i s t h e i d e n t i t y map then
lX: X + X
w:
Preliminaries
then
X
C(lX)
= CX
-
= EX.)
M * ¶ *
Pullbacks and pushouts have t h e following semi-universal p r o p e r t i e s : The diagrams
Wf0¶fl
gl +
X
fO
b
x1
1. lfl lfl 1fo
xO
g1
Y
a r e commutative and f o r any space
(2;:
N
Yi + L ,
ghfo
h: L + W
N
zlfl)
( h ' : Mf
fo'fl
O
y1
MfO¶fl
N
L
and maps
g
*
i'
L +
-
xis
fozo
flzl
t h e r e e x i s t s a (non unique!) map +
L) s o t h a t
0) 1
zi
N
g.h 1
(zi
N
h'gi).
One can e a s i l y f i n d d u a l i t y p r o p e r t i e s between pullbacks and pushouts.
This d u a l i t y p r i n c i p a l i s r e f e r r e d t o as the Eckmann-Hilton
d u a l i t y by which one interchanges pullbacks and pushouts, a c t i o n s and coactions
p r i n c i p a l f i b r a t i o n and p r i n c i p a l c o f i b r a t i o n s
MacLane spaces and Moore spaces etc.
homotopy groups and cohomology groups
Some geometric proofs can be dualized t o o b t a i n t h e Eckmann-Hilton
dual statements
0.5.
.
&Spectra, r i n g s p e c t r a , generalized cohomology
An O - S p e c t m i s a sequence of spaces and maps Yn:
Eilenberg-
En
+
SEn+l i s a homotopy equivalence.
y,
E, = {En3Yn3 where
induces a homotopy
5
&Spectra, r i n g s p e c t r a , generalized cohomology
associative and homotopy commutative multiplication {En = [ ,En])
on En.
pn
represents a reduced generalized cohomologY theory.
([X,Enl = En(X)). A r i n g structure f o r
{En,Yn)
are maps $n,m: En
t h a t t h e following four diagrams commute:
on ,m-1
'n+m-l En+m-l
A
Em
-*
En+m
so
Notations , Conventions , Preliminaries
6
En+m
x E I
n+m
R(%
(Y,Z)l
= [(X,Y),
as follows: represented by
+
then
En
u x E En-'[QX]
1.
x E E"(X)
be
i s represented by
g,
nr.
yn-lg
ii:
f: X
Let
x,z)
If
{En,Yn,$n,m}
[X,En]
x [Y,Em]
-
i s an Sa [X
A
r i n g spectrum, then t h e f'unction
Y , En+m]
b i l i n e a r and induces a homomorphism
given by
i(f,g) = $ ,, ( f
A
g)
is
7
52-Spectra, r i n g s p e c t r a , generalized cohomology
a: [X,En] B [Y,Em]
The composition
[X,En] @ [XJ,]
-
[X
a
A
[X
induces a graded r i n g s t r u c t u r e on E*(X)
Y, E
A
n+m
1.
X , En+m 1
En+m(T)
,[ x 'En+m 1
(not necessarily associative
o r commutative o r with unit i n i t s non-reduced version).
8 CHAPTER I
The Category of H-spaces
Introduction This chapter i s devoted t o t h e study of the most elementary properties
of H-spaces. i n details.
With the exception of 1.1.3 and 1.2..3 all proofs a r e given Only the most fundamental Homotopy Theory i s used.
It i s very d i f f i c u l t t o t r a c e t h e o r i g i n of many statements.
Some
references a r e given but t h e r e i s no c e r t a i n t y t h a t t h e s e are t h e earliest. Other statements should be considered as "folklore" and other appear here possibly f o r t h e f i r s t time. Section 1 contains observations which follow d i r e c t l y from t h e d e f i n i t i o n s of H-spaces.
It contains a review of t h e notion of t h e
Moore-Path space which replaces throughout these notes t h e ordinary space of paths and i s used i n describing homotopies. Section 2 i s devoted t o a preliminary study of s p e c i a l c l a s s e s of H-spaces such as homotopy commutative and homotopy associative H-spaces with some examples. The algebraic properties of t h e s e t of homotopy c l a s s e s of maps i n t o an H-space i s studied i n Section 3. sequel a r e established here. Section
4 to
Some of t h e notations used i n t h e
The notions studied i n Section 3 a r e used i n
define the first obstructions i n t h e theory of H-spaces.
In
t h i s s e c t i o n the problem of enumerating t h e €I-structures on a given space i s b r i e f l y discussed.
Section 5 i s devoted t o some analysis of t h e obstructions for a map t o be an H-map and ways f o r i t s a n i h i l a t i o n .
9
Basic properties of H-spaces
1.1. Basic properties of H-spaces
An H-space i s a p a i r
s a t i s f i e s t h e properties Let
where
X,p
F: X v X + X be defined by
Thus, an H-space
X *X
uIX v X = F.
x
so that
11: X x
X *X
u(x,*) = x = p(*,x).
"the folding map'').
u: X
i s a space and
X
F(x,*) = x = F(*,x)
If
multiplication o r an H-structure f o r
i s a space
X,p
X.
X,p
(F i s c a l l e d
X with a map
i s an H-space we c a l l
1-1
a
Thus, an H-space i s a space
together with a continuous multiplication with a u n i t .
From t h e homotopy
theory point of View one m a y replace the unit by a homotopy u n i t , i . e . : i n the d e f i n i t i o n o f an H-space replace t h e property p I X v X = F requirement
ulX v X
-
F.
by t h e
However, with our notion of a space by t h e
homotopy extension property a multiplication with a homotopy u n i t can be homotoped t o a multiplication with s t r i c t u n i t .
The l a t t e r w i l l be t h e
only type of multiplication considered i n these notes. Two examples of H-spaces come i n mind: spaces of loops.
Topological groups and t h e
The f i r s t has a s t r i c t u n i t t h e o t h e r has a homotopy
Later i n t h i s section we s h a l l introduce i t s equivalent
unit.
-
one with
a s t r i c t u n i t , namely the Moore-Loop Space. O u r f i r s t simple observation deals with homotopy-groups type functors
applied t o H-spaces: 1.1.1.
Proposition [Hilton]:
Let
II
be a functor from the category of
spaces and homotopy cZasses of maps i n t o the category o f abelian groups which preserves products, i.e.;
and
IT(*)= If
X,u
0.
is an H-space then
10
The category of H-spaces
coincides with the group addition x,y -+x+y. Proof: i
2
(X)
il: X + X x X,
Let
= *,x.
Then
j,
x -+xx,o, j 2 ( x ) = 0 , x
= a n ( i11: and
(a) X
n(X)
x,y = j,(x)
Lemma (see opela land]
1.1.2.
i2: X
1
-+X x X be given by i ( x ) = x,* 1
-+
a(X) el n ( X )
i s t h e homomorphism
+ j*(Y).
and [Croon1
admits an H-structure if and only if f o r every space
[Y,X]
i.e.
Y
admits a muZtipZication with u n i t in a natural way, [ ,X] is a functor i n t o the category of s e t s with
multipZication with u n i t s . (b) X
a M t s an H-structure if and only if for every pair of
spaces M,L i*:
[M
x
L,X]
-+
[M
is surjective where i : M v L
(el
If
-+
V
L,X] i*
is induced by the incZusion
MxL.
X i s an H-space there e x i s t s a homotopy equivalence
a:
11
Basic p r o p e r t i e s of €I-spaces
f
where
fix =
I
If: I
=
{f: I +
and i
-+X/f(O) = f(l)),
xlf(o)
= f ( l ) = u),
- the inctusion.
X(x) [ t ] = x
E(f) =
a l x = X:
+
= *,A
i,(X)
f1 given by
(And see 1.3.6 in the sequet. )
t.
f o r every
x
f(o),
(d) A r e t r a c t of an H-space is an H-space.
Proof:
-% [Y,X
[Y,X] x [Y,X] 1):
Y
u: X
( a ) If
- + X as a u n i t .
If
[Y,X]
pi: XXX
+
[p11-[p21 E [X x X,X] [*,XI =
a singleton
k = 1,2
([p,]
-
[p21)[ill = [1]
and
M = L =
(c)
v
Let
x
-
*
x
X +X
Plil
= [1], s i m i l a r l y f o r
Put
= 1
P2il
is a multiplication.
?
= p(f,
x fL).
is
If
=
*
and
Let
?
Then
f: M v L +X.
extends
1
k+9
i s a multiplication.
i s a s u r j e c t i o n for every
M,L
then
( i * ) - l [ ~ ] i s a multiplication. b e given by
a(x,cp)[t] = ~ ( x cp(t)) ,
a(x,*)[t] = x
alX = X.
implies
and hence
1'
If pi
[*,XI
i 2'
Ea(x,cp) = a(x,cp)[O] = u(x, c p ( 0 ) ) = p ( x , * ) = x = pl(x"p) Ea = p
and if
i* i s a s u r j e c t i o n .
any element i n a: X x
Y
Indeed, as
X:
E [Y,X] must be t h e u n i t .
i*: [M x L,X] + [M v L,X]
If
where
v: X
*
are the injections
f L = flL.
f: i * [ P ] = [ f ] ,
for
and
1
X x X
f M =f l M ,
with a u n i t f o r every
is a multiplication for
+
Put
-
a r e t h e p r o j e c t i o n s then a map r e p r e s e n t i n g
ik: X
(b) Suppose
i s a m u l t i p l i c a t i o n with
[Y,X]
has a m u l t i p l i c a t i o n
i = 1,2
X
-%
x X]
+*
i s a m u l t i p l i c a t i o n then
x X + X
and
X E E
bX
then
ai2(X)[t] =
u(*, X ( t ) ) = X ( t )
and
a i 2 = i.
As
are f i b r a t i o n s t h e exact sequence of homotopy groups and t h e
f i v e l e m imply t h a t
a(a)
i s an isomorphism and consequently
a
is a
The category of H-spaces
12
homotopy equivalence.
(a)
i: A c X
Let
i s a m u l t i p l i c a t i o n then s o i s
p: X x X + X
1.1.3.
Proof: 1.1.4.
Let
Then
Suppose
u: A +hEA ( )#
If
cr.
Suppose
f*: [B,X]
f : A +B.
-
of
Cf
CA
f-extends
ru
-
ug.
1 and
1.1.4.1.
C B L If
ri
#:
Remark:
t h e customary Sl
h: B
+
ACA,
i s t h e a d j o i n t operation then
u# = 1 and
extension o f
Cr,
has a homotopy l e f t inverse.
can be f-extended, i . e . : t h e r e e x i s t s
i s a homotopy l e f t inverse of
(h) #
has a l e f t inverse
Cf
is s u r j e c t i v e for
i s s u r j e c t i v e f o r a l l H-spaces
f*: [B,X] + [A,X]
6.
cx X
B +X
(ug),:
+
& f = (ag)#Cf
one g e t s
A
A
g: A + X .
Given
-9, X 4 6 C X and t h e a d j o i n t (ug),: CA
is a
u(x)[t] = [x,t].
[A,X]
+
if and only if Zf: CA + C B
X
(h)#(Cf) = ( h f ) #
A
1.e.
be the adjoint
See [James],.
Corollary:
Proof:
-
1 1 1 CX = S AX = S x X / S vX.
where
e v e y H-space
hf
ru(ixi): A x A +A.
l e t a : X +;EX
X
If
is an H-space if and only if a has a homotopy l e f t inverse.
X
X.
For a space
Theorem [James&:
of 1: E X + E X
‘I: X + A .
be a r e t r a c t with a r e t r a c t i o n
EX.
-
extends
We use here
fi
g: ri#f
N
rag
-
CB
+EX
Taking an a d j o i n t
(ag),.
+-= ug
i s an H-space by 1.1.3 t h e r e e x i s t s f
Consider
= (ug)$:
iCx,
B
Zf.
so
*
g#
r : nCX + X
g.
t o denote t h e loop space i n s t e a d of
i n order t o preserve t h e l a t t e r t o denote t h e Moore
loop space which i s going t o replace loops throughout t h e s e notes.
Let A c B and suppose CA is a r e t r a c t 3: be an H-space. Oiven maps go,gl: B + X. If
1.1.5. Proposition [James]
of
CB.
Let
X,v
Basic properties of H-spaces g O I A = gllA
and
a homotopy F: B
go
-
then go
g1
N
gl
13
i.e.:
r e 1 A,
I +
x
There e x i s t s
I
for every
F ( a , t ) = g,(a) = g,(a)
a E A.
To prove 1.1.5 we f i r s t prove:
1.1.5.1.
Lemma:
Let
A cB
a homotopy equivalence u: C ( B U ( A
CA c CB
Proof: and
i
+
Define maps
vl: B U ( A
v3: B U ( A x Y) + Y
x
and Y
B’ ,At
is a r e t r a c t then C ( B U A
Y) c C ( B
x
Y)
+ By
u: C ( B U A
x Y)
A
YI.
uhich
In particutar,
is a r e t r a c t .
Y)
x
Y v CY
v2: B U ( A
x
Y)
+ A A Y
vl(a,y) = a
a r e obviously n a t u r a l with respect t o maps
Define
+
Then there e x i s t s
by:
v (b) = b 1
v
Y)) -%CB v CA
x
is natural v i t h respect t o maps B ,A i f
Y be any space.
and l e t
+ C B v CA
A
Y v CY
B ,A
+
*
9
,
Y
1 -<
[3t-2,v3(x)
3
2
3-
t 53
2<
t 51
3-
Using t h e b a s i c homology computations one can see t h a t H(vl) 13 H(v2) @ H(v
3
isomorphism hence
1:
H(B U A
x
Y) + H ( B )
@ H(A A
u i s a homotopy equivalence.
One has t h e following (induced by
+
by
*
I *
and
B ,A’
B,A c B , B )
Y)
@ H(Y)
i s an
Y I.
14
The category of H-spaces
C(B U A
C(B
x
gOIA = gllA, I Z J I Z I
by p u t t i n g
gl(a,*)
i,: B
x
S1 -+ X.
F
hence
C31A x I
re1 B
x
i
Y ) v CY
i
C[B U A
and
,.
f a c t o r s through
= 1). Fo
(and note t h a t f o r
Define
FIA
p1
: A x S1
go
x
and
Y]
F3: B
=
I
P2(b, e
g
re1 A x x I
I
x
= pl.
-
f
gl.
-+
X
gl.
-+x.
F1: B U A
x
S
1
-+
x
a E A
By 1.1.4 and 1.1.5.1
= g o ( a ) = g,(a)).
-
FO
e a s i l y extends t o
A
i s a homotopy
and
A
,.
*
i3 ( b , t )
Hence
CB v C ( B
F: B x I -+X be any homotopy of
Let
cl(b) = gl(b)
=
Y ) v CY
has a l e f t inverse s o does
$[A x I
= 11,
g,(d
3
w
Ci
Proof of 1.1.5:
( s1 =
A
C ( B x Y).
i s a r e t r a c t of
As
EB v C(A
-
Y)
i: A cB. If
where
U
Y)
x
P,
extends t o
by
bsi(1-t)
k
a E A
For
and by standard homotopy theory F: go
-
gl
re1 A .
F3
-
F
Basic p r o p e r t i e s of H-spaces
15
Using arguments similar t o those of 1.1.5.1 one has
1.1.6.
Examples:
Given spaces
n V
..
j}.
indices
i s a map fplX
-
Then
X y p and
If
fp
A2,.
...%.
k
p'f
n A [a = i=l i j n k C( V Ai) +C( ll Ail i = l , . . ,k i=l
= { ( a l , a2....,%)
i=l,. ,k
as
A1,
f: X V
+
E
a r e H-spaces a
*
so t h a t
X'
X = f F = Ff
fp
-
II Ai i=1
for at least
k
-n
H-map
f : X,p + X ' , p '
I n view of 1.1.5 and 1.1.6,
p'f x f.
f = p'(f x f ) l X v X
V
..
has a l e f t inverse.
- p'
p
k Ai c
i=l,. ,k
.
X',p'
n v
Let
one may always assume t h a t
r e 1 X v X.
x f
The c l a s s of H-spaces and H-maps forms a category Sometimes it i s convenient t o carry t h e homotopy as a s t r u c t u r a l p a r t of an H-map.
H
f'
F: fp
-
p'f x f
I n order t o do t h i s i n a managable way
one should r e c a l l t h e notion of t h e Moore path space (and see e.g: [ Fuchs ] ) :
Let
be t h e set of non negative real numbers.
R+
t h e Moore path space of
PX
We c a l l
i : PX
evaluation a t ( "
and
X
i s t h e space
t h e union (or weak) topology induced by t h e inclusions
W e give
s
X
Given a space
*
R
+
.
s map
L(t,lp)
(0
= t,
the length function.
2 s 2 " ) Es:
Em(t,lp) = c p ( t ) .
Let
kt: X
PX +
*
PX
X
by
Define t h e
Es(t,cp) = q ( s )
be t h e length
path map A
k t ( x ) = ( t , k(x))
k(x)(s) = x
for all
s.
t
if
constant
16
The category of H-spaces
ko = k.
W e write
i s a f u n c t o r from t h e category of spaces and maps i n t o i t s e l f :
P If
f: X +Y
then
P f : PX +PY
a: P(X
homotopy equivalences a(t,lp)
= ( t , plcp), ( t , p2v)
the projections. and
(ab)
for
Phb.
-
h: X x X 1 2
P ~ = X {('Ply
s e t of p a i r s
'pl2
-+
X,
Y we s h a l l w r i t e
n
0
PX x
Add: P X
2
-1 Em (*).
One has
ilp.
b : PX x PY + P ( X x Y ) . p2: X (lpl
x
Y
x
W2)A
Ph: PX
1
+
Y
are
+'
R x PX2
ba = 1
+
P2X c PX x PX
PY
is the
P x I E ~ ( G=~ E) ~ ( ( P ~ ) I . -+
PX
i s d e f i n e d by
i s defined 0
Write
and
= -(tl.t2),
s2X = E-'(r)
'P2) E
The p a t h a d d i t i o n map
and
-+
X x Y
p,:
(t2,w2)1
LX = E,l(s),
Let
Y ) -+PX x PY
where
b[(tlYcp1),
1. I f
x
Pf(t,cp) = t ,
i s given by
Add(Gl,
-
G2) = (P1 + (P2
(S
ft,
then whenever t h e following a r e d e f i n e d
they s a t i s @
and always Let
-
-
-
k ( E O ( G ) ) + (p = cp = cp + k ( E m ( 0 ) ) .
c : PX
+
PX
be given by
c(t,cp) = t,;
where
is defined by
Basic p r o p e r t i e s of H-spaces
and t h a t
Add(QX x LX) c LX
Hence
d e f i n e s an a s s o c i a t i v e m u l t i p l i c a t i o n w i t h unit on Q X
Add
an a c t i o n o f cp + 1,cp'
by
QX
on
< 1 and cp'(s) = c p ( 1 ) s -
cp'(s) = ~ ( s ) f o r
-1
PX + X
The i n v e r s e homotopy equivalence can be given by
{
cpt(s) =
PX +X
I -+
PX
and
X
I
+
2
One has a homotopy equivalence
LX.
PX
dt*s)
l L t
d s )
t (1
-+
2
'
+ PX
given
s ~ 1 .
for
,
and
t,cp +cpt
where
a r e homotopic t o t h e i d e n t i t y v i a
homotopies core1 t h e e v a l u a t i o n s a t t h e endpoints. M x I
A homotopy map
M
-t
PX.
F: M + P X If
E F = fl, 0
with
6:
PX
-+
-f
Y,
n:
M
-+
R
F1: M
+
or i t s adjoint F: fl
i;:
P(PY)
x
satisfy
I
+
X
E,F
G: X
PEs& = GEs Let
X
Thus, a homotopy
gly g2: X
exists
-+
Pgl
N
M + XI
can be r e p l a c e d by a
f2, f : M + X i
i s a map
= f2. -+
N
PY
G: gl
a homotopy
Pg2.
G
-
g2,
then t h e r e
satisfies
and EsEt6 = E GE t s' be a homotopy
n-l(O) = L.
fl
N
f2
r e 1 L c M.
$: M
Consider t h e f u n c t i o n
Let x
given by
t
xti! L,
x
ti! L,
t
(Tl(X)
t ,n(x)
R
+
+ X
18
The category of H-spaces
F: M +PX,
$ i s continuous and corresponds t o a map
Then
I1F = rl
and
FIL = k f IL = kf IL.
2
1
Thus a homotopy a l l homotopies a r e
is
Given
h: X
core1 h
if
f : M +PX, re1
-+
*
-
F: f
is
g
PhF = ke$fo
-+
X.
As
F(L = kf.
if
F(*) = k ( + ) .
we always assume
Y, ' f o , fl: M
re1 L
F:
F: M + PX,
A homotopy
f
0
-
f
1'
= ka#fl.
Now 1.1.5 can be generalized:
1.1.7.
an H-space. gllA
-
A c B, CA
Suppose
Proposition:
Gv e n gl, g2: B
-+
X,
N
Fo: A
Given
+
PX,
.yo: g l ( A -
N
g'2 -
One has
Po
If
+
g2
extends t o
F~
extension p r o p e r t y
N
extends
gl
,
X,p
g2*
g2(A. By t h e homotopy
FO'
F: X x X + PX'
N
E
B +PX,
i s a n H-map one has
X,p + X t , p '
Hence, t h e r e exists
g2
Let
i
0 0
Em $0 = gl2'
= gl,
By 1.1.5 t h e r e e x i s t s
g2[A = g;[A.
and
Pl: gl
Po:
CB.
Then any homotopy
g2.
g2/A can be extended t o a homotopy
Proof:
r e 1 A.
g1
a r e t r a c t of
fp
-
g2
re1 X v X.
p (f x f )
FIX v X = kF,
gb-
EOF =
fp,
EmF = u ( f x f ) .
1.1.8. HfF
Definition:
The precise category o f H-spaces i s t h e category
whose o b j e c t s a r e H-spaces
a r e pairs
f , F,
EmF = p ' ( f
x f).
f: X
+ X I ,
Xyu
and t h e morphism
F: X x X + P X ' ,
FIX v X = k f F ,
The composition i s given by (f',F')(f,F) = f'f,
Pf'F
+
F'(f
X,p
x
f).
-+
XI , u ' E F = fp, 0
be
19
Some s p e c i a l c l a s s e s of H-spaces
1.1.8.1.
Remark:
Throughout t h e s e notes we s h a l l use Moore path spaces
(based paths and loops) r a t h e r than t h e ordinary paths (based paths and loops) and all homotopies w i l l be considered as maps i n t o Moore path spaces.
1.1.9.
The use of t h e p r e c i s e category w i l l be l i m i t e d i n t h e
Remark:
sequel only t o t h e cases where t h e homotopy and we s h a l l use t h e notations
f: Xyp
-t
F
X' , p '
i s s p e c i f i c a l l y needed and
f,F: X,p
-+
XI ,p'
for
H-maps a s t h e case r e q u i r e s .
Some s p e c i a l c l a s s e s of H-spaces
1.2.
It w a s observed i n t h e case of a loop space t h a t a m u l t i p l i c a t i o n may possess some a d d i t i o n a l p r o p e r t i e s , such a s a s s o c i a t i v i t y and a weak form o f an i n v e r s e .
From t h e homotopy point o f view t h e s e a r e defined
as follows:
An H-space
Definition:
1.2.1.
(an HA space) i f then
X,p
x,y,z
-+
i s an
-
HA space if t h e maps
T: X x X
-+
X
x,y
x
X
+.
x-y
i s the twisting
and
x,y
-+
-t
(x.y)z
and
T(x,y) = y , x ,
i.e:
p
-
X,p
i s a map
'r
:
X
cr(ce)
homotopic.
satisfy:
-*
The maps
( ~ ( c ,x
l)A
x -+x*crx ( x
-t
-+
X
*)
C~X-X)
pT
Xyp is
X) s a t i s f y i n g p ( 1 x Cr)A
i.e:
-+
y - x a r e homotopic.
A r i g h t ( 2 e f t l homotopy inverse f o r
x
x,y,z
i s s a i d t o be homotopy commutative (an HC space) i f
HC i f t h e maps
(CQ:
p(x,y) = x
If w e denote
p ( 1 x p l y i.e:
x ( y - z ) are homotopic.
X,p
where
p ( p x 1)
i s s a i d t o be homotopy associative
Xyp
are null
y
The category of H-spaces
20
1.2.2.
regarded as t h e unimodular C8yley numbers i s an H-space b u t not
HA ( [James
I,). S7
Among f i n i t e CW H-spaces we r e f e r t o compact Lie groups and
-
t h e c l a s s i c a l H-spaces.
A l l but
S'
a r e HA.
some of them w i l l be HA
As far as H C one has t h e following i n t e r e s t i n g f a c t
and some w i l l not.
A l l f i n i t e CW H-spaces which are HC spaces 1 n S' x s1 x . , . x s
Theorem [HubbuckI2:
are of the homotopy type o f a t o m s
T
=<
/
n
0 (To = *).
n
Hence, there are m m n c l a s s i c a l f i n i t e CW HC spaces. If
i s an H-space
Xyp
P p : PX x
PX
+ PX,
t h e pointwise
m u l t i p l i c a t i o n of paths, i s defined by
Pd(t,,(P,),
(t2,(P2)1
= maX(tl't2),
'p1"p2
(P1*(P2(s) = !J(cP1(s),cp,(s)). Thus, f o r p a i r s i n operations
-
Pp
1.2.4. Lemma:
&s
I n t h e sequel we s h a l l
introduce some non c l a s s i c a l f i n i t e CW H-spaces,
1.2.3.
i s an
All t o p o l o g i c a l groups are HA.
HA space. S7
A s w a s a l r e a d y observed i n s e c t i o n 1.1 fix, Add
Examples:
P2X
and
(X
Add.
Let wi: PX
-
an H-space) one has two d i f f e r e n t binary
The r e l a t i o n between them i s given by
x
PX
-+
PX
x
PX
i = 1,2 be given by
, H-space]
The s t r u c t u r e o f [
im w
Hence,
corelEi
X
C
i
p2X.
Let
1, 1 X E
al: P X x P X
PX
j = O p ,
i = O p ,
J
-+
PX
x
21
be given by
h a l = Addwl.
and
Similarly i f
a2
-
1 c o r e l Ei
&
-
AddT
On i l X Proof: On
RX
1.3.
On R X
Corollary:
1.2.5.
x
OX On
x
R X = (RX
x
x
Add
c o r e l Ei
LX)
The s t r u c t u r e of
( R X , Add)
w1 = 1, on L X
LX
n
(LX
[
, H-space 1.
A s observed i n 1 . 1 . 2 ( a ) ,
[M,X]
hence
Add- AddT,
RX
-
F$
LX
Ppa2 = Addw2.
and
and on L X
x
RX
i = 0,m.
c o r e l Eiy x
i = 0,-
1, 1 x Ei
x
x
RX)
if
Add
X,p
RX
x
- &
is HC. w
2
= T.
AddT.
i s an H-space f o r any space
has a m u l t i p l i c a t i o n w i t h u n i t p*:
[M,X]
x
[M,X] = [M,X
I t can be e a s i l y seen t h a t i f (commutative).
X,p
is
x
HA
XI
-+
[M,X].
(HC)
p*
is associative
M
22
The category of H-spaces
Theorem ( [James 13: The proof f o l l o w i n g [ S t a s h e f f
1.3.1.
i s an H-space then for any space M
For any pair
Proof: -
s: X x X
Define
n n ( s ) : nn(X) @ n n ( X ) Hence
Df
Let
: M
nn(s) +
+nn(X)
@
-t
X x X
nn ( X )
Df,g
i s given by
i s an isomorphism and
X,p
s o that
E [M,X]
s ( x , y ) = p ( x , y ) , y.
by
Then
n n ( s ) ( a , b ) = a + b,b.
i s a homotopy equivalence.
s
be t h e ( u n i q u e ) element s o t h a t
X
:
i s an algebraic loop, i . e :
[M,X]
there e d s t s a unique
f , g E [M,X]
4) If
Yg
g ) = (Df
(Dryg,
x g)A ¶@;
i s t h e unique s o l u t i o n of t h e l i f t i n g problem
i s t h e unique s o l u t i o n o f t h e equation
Dfyg
1.3.2. Dfyf3
= f-g.
1.3.3.
If
Corollary:
Xyp
i s HA then
[M,X]
If i n addition X i s HC then
Remark:
Even though
[M,X]
The f u n c t i o n
[K,X] x [K,X]
i s a group and
[M,X]
[M,X]
+ f o r t h e o p e r a t i o n , ,+I . ! D:
= f.
i s an abelian group.
i n g e n e r a l i s not a b e l i a n , i n o r d e r
t o d i s t i n g u i s h between t h e o p e r a t i o n i n o p e r a t i o n we use
p,(x,g)
+
[K,X]
and t h e map-composition = f+g.
p,(f,g)
i s r e f e r r e d t o as t h e
d i f f e r e n c e and i s uniquely d e f i n e d by t h e p r o p e r t y The proof of 1 . 3 . 1 h o l d s f o r a g e n e r a l space
f X
-
p(D,
:
p1 'P2
xxx-+x
g)A.
and a CW complex
However i n our case one has a " u n i v e r s a l " d i f f e r e n c e , namely
D = D
x 3Q
M.
, H-space]
The s t r u c t u r e of [
= D(f .g t h e following:
Df
and
x
g)AM f o r
DA = Dlyl: X -+ X
As
p(* x
l)A
i ( x ) = x,* 1
=:
Di
u(*,x) = x,x
- *.
D(x,*) = x,
=
D1,*
il: X +
satisfies
p(D1,*
D(x,x) =
*,
Lemma:
For
respectiveZy.
X
x
1,1 X x X
[K
A
As
u(x,*) = x,*
X, s: X
x
X
-+
X
p ( D x 1)(1 x A)u =
hence,
x
F
X,
and a s
Eu
r e 1 u.
p ( D x 1)(1 x A ) - p l
-+
g : X,p
L,
-+
be a map and an H-map
X',!.I'
f 1h,f2h
L,Xl
- A*
[K
if and only if
i*
L,X]
x
is short e m c t in the folZming sense: i * f = i*g
1.
flyf2 E [L,X]
then for any K , L
-+
-
i s given by
*)A.
x
Lemma. (see [ C ~ p e l a n d ]and ~ [Arkawitz]): If X,p
0
x 1)A
Consequently one may assume
= D
1.3.5.
D implies
e a s i l y implies:
D
h: M
Let
-+
N
Note t h a t i f
S I X v X = u.
then
The uniqueness of
1
u: X v X
1. Now,
has a r i g h t inverse by 1.1.5 w e have
1.3.4.
The uniqueness of
X.
Similarly, i f
i s a cofibration.
s ( x , y ) = p(x,y), y , that
1
-
p ( l x * ) A = 1 D1,*
-+
i s t h e unique map s a t i s f y i n g p ( D
1, DlY1
then
f ,g: M
23
[K v L,X]
is onto,
i*
I\*
is an H-space
-+
is
0
1-1 and
E i m A*.
Df 363
Proof:
i* i s s u r j e c t i v e by 1 . 1 . 2 ( b ) .
for an a r b i t r a r y space [E ( K x
as (by 1.1.6)
L) , X I Ci*
X Ei
*
[E(K
V
L),X]
Consider t h e Puppe sequence
a
[K
h
L,X]
A*
has a r i g h t inverse it i s s u r j e c t i v e and
a
[K =
x
*.
L,X]
The c a t e g o r y of H-spaces
24
=
A*-’(*)
Hence
= Dh
A*Dhlyh2
*.
If
A*hl = A*h2
i s an H-space
Xap
A = D = 1’ 2 hlA sh2A
*.
Hence
Dhl h2
=
implies hl = h 2
* ¶
and
A*
i s 1-1.
Finally
i*Df
i*fl = i * f 2
i f and only i f
-
= Dfli,f
Y
i 2
-
and b y t h e Puppe sequence t h i s i s e q u i v a l e n t t o
1’ 2
Df
f 1’ 2
Dfl af2
=
E im A*.
Here
The f o l l o w i n g i s a more p r e c i s e v e r s i o n o f 1 . 1 . 2 ( c ) .
1
1.3.6.
given by Y = [E
Let
Proposition:
x
a = Pp(i PD(1
X
x
be an H-space.
Xyp
Y:
k)T. Let
Then aY
kE)A]A.
?’
-
X
-+
a: X x
Let
ax -+? be
Q X be given by
x
1 c o r e l E,
Ya
-
1 c o r e l p 1’
Pro of :
aY = [ P p ( i
a(*,x) = (x,x). x
k)T][E kE)
= Pp[PD(l
x
= Pp[PD(l
x 1) x
G : p ( x~ i)(1 x A )
Let
2: P[p(D
x
G:
1)(1 x A
x
x
-
x
-
p1
x
PD(1 x kE)A]A =
kE](1
x
1 ] ( 1 x A ) ( l x kE)A (a: x v
re1 a
+PX yields
Fpl.
A)A =
x
6:
PEo6 = GEo
PX
and
x
x -+x x
PX
PE,k
-+
X, a ( x , w )
PPX
= GFDD imply
1
and
As
G
is
re1 a
it i s
G(1
x kEo)A:
re1 A’
so
Xs
GAE
-+
P?’
i s a homotopy
is a c o n s t a n t p a t h and
= x,*
H-deviation and H-homotopy equivalence
PEe(1 x = ) A :
aY
-
1 corel E.
plya = E P p ( i x k)T = p ( *
x
ay
or
-
and
1
Ya
1.4.
implies
-
-
Ya
1 corel p
p
-
-
Ya
a
As
(p,
i s a homotopy equivalence
p2)~.
x
~a = (p, x
AS
f!J,lJ'(f
X
x
f a c t o r s through
x f)
*.
HD(f,p,p').= p'
1.4.2.
H-map:
f
fp
Thus
f l : X1
-
P,)
i2 p2
-
p'
f: X
be a
-+XI
i s t h e map
h: X
p'(f x f)
x
X
implies -+
X
A
re1 X v X
D
Moreover,
pl)
and
i = *
is
f
and t h i s i s equivalent
i s t h e obstruction f o r
X1,
f)
fPrll'(f
X.
H-map i f and only i f
(Xoy po),
Let -+
-
HD(f,p,p')
is a p
Proposition:
f o : Xo +X1,
f
fpi = p'(f x f)i
then
Let
be H-spaces.
X',p'
H-deviation of
i: X v X C X
D
X,p,
an H-map i f and only i f to
l ) T = pl.
and
1'
Let
Definition:
map of spaces.
and
Ya = ( E x [PD(1 x kE)A])Pp(i x k)T
H-deviation and H-homotopy equivalence
1.4.1.
If
1
25
f
HD(f,p,p')
(X2, p 2 )
t o be a
=
*.
be H-spaces,
maps of spaces.
X2
(a)
If
fo
is a
p o , p1
(b)
If
fl
is a
p
1
- p2
H-map then
HD(flfo) = HD(fl)(fo
H-map then HD(flfo) = flHD(fo)
A
fo)
26
The category of H-spaces Proof: ( a ) A*HD(flfO) = D
”(flfO
flfO!JOY
= HD(fl)Afo
x fo
= A*HD(fl)(fo and as (b)
is
A*
x
A
--
flfo)
= HD(fl)(fo
fo)A =
A
fo)
1-1 ( a ) follows.
i s proven similarly.
Given a space
X.
The existence of an H-structure f o r
i s not
X
e a s i l y detectable and r e l a t i v e l y f e w spaces w i l l admit one (and see Chapter 3 f o r some necessary conditions).
However, i f such an H-structure
e x i s t s it i s seldom unique.
w E [ X A X,X]
Indeed, i f then
= WA
p,
is, =
+
x
x
x+x
i ” w A + i t = wAi + p i = F E [X v X,X]. H D ( 1 , p w y p ) = w.
If
i*-’(F)
p E
given by
a
[X
+
X,X]
1.4.3.
p:
i s a l s o a multiplication.
p
hence,
A
and
IJ
(G)
i s a multiplication
i: X v X c X
If
Obviously
The s e t of H-structures f o r
i s fixed then t h e assignment a :
given by
w
+ pw.
Theorem ( [ C o ~ e l a n d ] ~ If ):
is in
= wA,
i s t h e set
X
(F) * [X
A
XYX]
i s an inverse t o t h e functions
= HD(l,;,p)
H-structures f o r X
PWY P
i*-l
11
i*-l(F)
D
x X
X,p
Hence:
i s an H-space then the s e t of
1-1 correspondence w i t h
[x
A
X,X].
27
H-deviation and H-homotopy equivalence
1.4.4.
Let
Lemma:
be an H-map.
f : X,p + X 1 , p l
equivaZence with a hmotopy inverse
Proof: (f
A
+ = HD(l,p,p)
then g i s an H-map.
HD(g,p',p) = p1
Two H-structures
Definition:
h: X
+
is a
X
p1
- p2
p2
and
for
i+-'(F)
r e l a t i o n i s c a l l e d the set of essent&~Zlydifferent Example:
and as
f)
X
a r e s a i d t o be
h: X,p1
H-map and a homotopy equivalence.
r e l a t i o n i s an equivalence r e l a t i o n and t h e set
1.4.6.
A
+.
H-equivalent if t h e r e e x i s t s an H-homotopy equivalence i.e:
i s a homotopy
f
= HD(gf, p , p ) = HD(g,p',p)(f
f ) * i s an isomorphism
1.4.5.
g
If
-+
X,p2,
This
modulo t h i s
H-StPUCtWe8
for
X.
To i l l u s t r a t e how t h e s e t of e s s e n t i a l l y d i f f e r e n t
H-structures can d i f f e r from t h e set of a l l H-structures consider t h e following example. Let
X = K(Z,2n) x K(Z,4n)
where
i s t h e Eilenberg MacLane
K(G,m)
space with
K(G,m)
may be considered as an abelian topological group ( [Milgram]).
po: X x X -+ X
Let
be t h e product multiplication:
- p0[(klyk2)(kl,k2)]
= kl +
multiplication i n
K(Z,m),
NOW,
connected [X
A
[x
k2 +
is
411-1
+ denotes t h e abelian
where
2
m = 2n, 4n.
X,XI = H ~ ~ A( X x ,Z)
A
X A X
X,X] w 2 .
kl,
e H 4n (x
connected and
The generator u E [X
A
H
A
x,z).
4n
(X
X,X] = Z
A
AS
x
is
X,Z) = Z , i s given by
2n-1
hence
28
The category o f H-spaces
w E [K(Z,2n)
K(Z,2n), K(Z,kn)] = Hkn(K(Z,2n) A K(Z,2n), Z ) =
w =
generator: If
A
1
h E Z
2n
'2n9
@
t h e element i n
is the multiplication
ph
Let
g : K(Z,2n)
+. K(Z,kn)
let
h,,: X
be given by
hv
+.
'2n
X
E H
2n
(K(Z,2n) , Z )
,
i+-l(F)
-
z
is a
t h e fundamental c l a s s .
corresponding t o
hu E [X
A
X,X]
given by
g ( x ) = w(x,x).
be given by
h ( k ,k ) = kly wg(kl) v 1 2
+
For any w E Z k2.
As
h-vh,,
= 1
a r e homeomorphisms.
+ Aw(klykl)
To i n v e s t i g a t e t h e map
H
one should study
2 H 4n (g,Z)
i4n
= i2
2n
i : K(Zy2n) A
-
gpl
-
H*(m,Z)i2n
i2n
thus
L2 K(Z,2n)
+.
K(Z,hn)
(i,Z).
= gm
H*(p2,Z)i2n = 1 0 homomorphisms
4n
+ k2 +
gp2
= i
(m
2n
the addition,
8 1+1 Q
-
H*(plYZ)
i2n
and r e c a l l t h a t f o r every
pi
f
H*(f,Z)
the projections)
i2n
=
12n
8 1
i s a ring
29
Change of H-structures and H-maps
-
It follows t h a t
and
2w
h !J (h-v v x
X
h-")
=
Consequently
t h e r e a r e at most two e s s e n t i a l l y d i f f e r e n t H-structures f o r
by
and
p0
p1
Let
0
# x
E i n H2"(K(Z,2n) ,Z2)
# y E i m H4n(K(Z,4n),Z2) - + H4n ( X , Z 2 ) .
Hence,
p0
H*(X,Z 2 )
and
2
)
!J
and
0
induced
as discussed i n Chapter 3 one can s e e t h a t
H * ( ! J ~ . Z ~ ) # H*(plYZ2): 0
H*(X,Z
(Actually, using t h e Hopf algebra s t r u c t u r e s of
p1,
X:
u1
-+
H 2n ( X , Z 2 )
,
Then
induce non isomorphic Hopf a l g e b r a s t r u c t u r e s on
and are e s s e n t i a l l y d i f f e r e n t ) .
For f u r t h e r study o f t h e enumeration o f H-structures and e s s e n t i a l l y d i f f e r e n t H-structures s e e [Arkowitz and CurjelI1 and [ C u r j e l l .
1.5.
Change of H-structures and H-maps. It i s obvious t h a t t h e possession of an H-structure i s a homotopy
invariant.
Moreover, c e r t a i n homotopy constructions preserve t h i s
property ( s e e Chapter 2 ) :
S t a r t i n g with H-spaces and H-maps t h e outcome
of t h e construction i s another H-space.
Quite o f t e n t h e s p e c i f i c
m u l t i p l i c a t i o n s of t h e given H-spaces a r e not important. ask:
multiplicactions H-map?
X
Given spaces !J
and and
Thus, one m s y
X' and a map h: X -+XI, a r e t h e r e !J' s o t h a t
h: X,p
-+
X'
y
~
'
is a
p
-
!J'
30
The category of H-spaces A useful test is given by the following:
1.5.1.
Proposition: Let
(X,p), (Xl,p') be H-spaces and h: X
+
X'
a map of spaces. so t h a t h is a
( a ) X' ach-its a multiplication
p
- G'
H-map
if and only if HD(h,p,u') E im([X' ( b ) Suppose
X',p'
(XI) = o for n muZtiplication T
A
X',X']
[X A X,X'])
is HA and e i t h e r X is f i n i t e dimensional or n s u f f i c i e n t l y large.
fi so t h a t h is a
if -HD(h,p,p') E im([X
A
X,X]
Then
- p'
hw
[X
x
ahits a
H-map if and only
A
X,X']
Proof: (a) Suppose HD(h,p,p')
-
w(h
= p i = wA + p ' .
Let
A
(or
G'
= p'(wA
E
[x' A
X',X'])
i'=
u;(w
that h is a
p
- 1;'
and HD(h,p,p') = w(h
A
h).
Conversely, if X'
so
h), w E [X'
H-map then
A
X',X'].
x
p')AXl
1. Then:
is an H-structure f o r
31
Change of H-structures and H-maps
(b)
Suppose
c:
p =
tw= WA
in
[ ,X]
If
cw E [ X
X x X
-+
+ j, w
X
E [ X A X,X]
induced by
A
exists so that
d i f f e r e n c e taken with r e s p e c t t o hcw = hD,
,w
=
+
H-map.
denotes t h e operation
j.
i s given by
X,X]
and
- p'
is a
h
D*,hw
cw = D
* ,w
i)
(and D
i s the
then
= -hw = -HD(h,p,u').
The converse is s l i g h t l y more complex and here t h e f i n i t e n e s s condition i n t h e hypothesis i s needed: Suppose Consider
[ ,X]
p
1
-HD(h,p,p')
=
= VA + p
pv
induced by
p).
= hv,
v E [X
(and here
and
one g e t s
Denote
w = HD(h,p,p').
Then:
= WA(VA x p hv + w = 0
X,X].
+ stands f o r t h e o p e r a t i o n i n
hpl = h,(vA + p ) = wA(vA
As
A
)
~
x p)Ax
+ hvA + hp =
+ ~hvA + WA + p ' ( h
x h)
32
The category o f H-spaces
-A x
Let
-
-
*x
x-
x
alA = ( A
w
x:
X
x
AAx
x
x'
A
X + (X x X )
=
V1
: X x X
-+
xA
al:
= w(v
A
u)al = -h(v(v
Il)al = V ( V A l)il(l h
V(V A
p
A
x
f a c t o r s as
smash product o f
vn
and
addition i n n+l
"n+l
A
x
-
CL
Put
yn+l = (yn A
A
(x
X
x)
so that
Put
")al).
-w
n
= -HD(h,pn,p')
= hv n
and
V
AX
where
induced by
1)(1
p,v
(pn)vn = vn A + un
A
pn)al
A
to
v1
(where here
An+%
is the
n+2 f o l d
replace
p
by
pn,v
+ denotes t h e
Then one obtains
pn).
" ' 1 = hvn(vn
= -HD(h, pn+l,
= vn(vn
A
X.
[ ,XI
vn+l
( x A x)
were constructed s o t h a t
n yvn
I n t h e computation l e a d i n g from by
-+
Ax
u)al.
i s a m u l t i p l i c a t i o n with
X
X
-
v n
-w
be induced by t h e diagonal
hence,
Suppose inductively t h a t "n
(X x X )
There e x i s t s
l)rx
= HD(h,ul,p')
1
A
1)(1
A
")al.
can be f a c t o r e d as
l ) Y n y n + l = v ($
n
n
A
1) t o complete t h e i n d u c t i v e step:
Change of H-structures and H-maps
As
AnX
is at l e a s t
[X
A
n-1
X,
connected i f
A2k+1X]
= II
and
dim X = k
a2k-1
=
II
= v2k-1
33
34
Chapter I1
Homotopy P r o p e r t i e s o f H-spaces
Introduction I n t h i s chapter some deeper homotopy p r o p e r t i e s of H-spaces are studied: f i b r a t i o n s , Postnikov (and Moore) approximations and various obstructions i n t h e theory o f homotopy-associative H-spaces.
and homotopy-commutative
The notions of a c t i o n s and p r i n c i p a l f i b r a t i o n s i n t h e context
of homotopy theory and H-spaces a r e considered, as w e l l as t h e notions of homotopy nilpotency and homotopy s o l v a b i l i t y .
As t h e computations become more complex and t h e amount of homotopy theory used i s wider, it i s n a t u r a l l y expected t h a t t h e r e a d e r ' s experience should be somewhat g r e a t e r than t h a t required i n Chapter I.
2.1.
H-spaces and f i b r a t i o n s . Let
pi:
fi,Fi:
w
+
Xi,pi
Xi
-t
i = 1,2
b e H-maps (.in
~
~
be t h e homotopy pullback o f
fl
and
i = 1,2
Y,u
L e~t
1
.
f2:
fl'f2
x
-
Pi(X1¶Y,X2) = xi E
xi
PY x
x2 I
E J 3 = f x ,J,(Y) 1 1
= f x
2 2
1
35
H-spaces and f i b r a t i o n s
i; i s indeed a m u l t i p l i c a t i o n as
Theorem (Moore-Stasheff,
2.1.2.
If f i : Xi,pi
+Yyp
see [ S t a ~ h e f f and ] ~ [StasheffG):
are H-maps
i = 1,2,
then the pullback
Wf
1' 2
of
fl
and
f2
and pi: Wf
a&t
+ Xi
H-stmctures.
1' 2 A s l i g h t l y more general p u l l back theorem i s given by:
Let
2.1.3.
Proposition:
space.
Suppose there e x i s t maps
that
rll(yy*) = y ,
Xiypi
02(*,y) = y.
i = 1,2
y: Y Let
x
be H-spaces and l e t X + Y, 1
hi: Xi
diagrams comute
p2
2
x2
x Y +
Y
be given by
Suppose the following three
h (x = r ~ ( * , x ~ )h2(x2) ~ = q2(x2,*). 1 1
x 2 x x
+Y
"2:
Y be a
x2
SO
36
Homotopy p r o p e r t i e s of H-spaces
h, x 1
x
(3)
x x
*
I.
Y
x1
x
2 /
Y
Then the pullback
and the maps
Wh
Whlyh2
+
Xi
admit H-structures.
1' 2 [Note t h a t i f H-maps, then
admits an H-structure
Y
ql = p ( 1 x hl)'
p
and
hi:
Xiypi
+Y,p
are
r12 = u(h2 x 1) satisfy t h e hypothesis of
t h i s theorem. ]
Proof: We f i r s t show t h a t t h e homotopies of diagrams (1)' ( 2 ) and
( 3 ) can b e chosen t o be a c t u a l l y agree.
el: X1
Let
5
N
Define
If
F1:
x x
r e 1 Xi v X
i s an H-space,
Y
hlul
-
x 1).
nl(hl
?1
J'
t h e homotopic maps
t h i s follows from 1.1.5.)
X + PY be a homotopy 1 X1 +PY by
One can then see t h a t
Xi v X
(On
J'
hlpl
-
ql(hl
x
1).
i s well-defined and i s a homotopy
Now,
N
F1(*,x1) = -?'l(*yx' + F1(*'x')
F1(Xy*) = -F1(*,x) + 1(x,*) - P,(x,*)
N
N
Hence theory
F1: hlpl
FllXl v X1
N
F1 N
N
F1
nl(hl
N
khlF
c o r e l E o y E, x 1) r e 1
c o r e l Eoy E, and
X1 v X1.
+ ?',(*,x)
and by standard homotopy
F IX v X1 = k h F . 1 1
Hence
37
H - l i f ti n @
F2: h2p2
S i m i l a r l y , one has
>3: X2
x
X
1
-
02(1
PY be a homotopy o ( h 2
-+
1)
x
Let
h 2 ) re1 X2 v X 2'
x
- n2(1
Let
hl).
x
and proceed similarly.
If
2.1.4.
Fi
Corollary:
satisfying hp
-
a r e homotopies
q(+,y) = y
rl(l x h).
fiber h + X
Proof:
Let
X, p
J
corresponding t o diagram ( i )
be an H-space,
X
0 :
and p u t t i n g h ( x ) = q ( x , * ) ,
Then the f i b e r of
is a b-p
Put
r e 1 Xi v X
x
Y
-+
a map
supposing t h a t
admits an H-structure
h
Y
;
so t h a t
H-map.
X1 = I,
q l = 1,
X
2
= X,
n2
= 11
and apply 2.1.3.
H-liftings
2.2.
Any l i f t i n g problem i n t h e category o f H-spaces can b e d e a l t w i t h i n
two s t a g e s :
F i r s t , one seeks a s o l u t i o n t o t h e o r d i n a r y l i f t i n g problem,
and t h e n one checks i f one of t h e s o l u t i o n s can be chosen t o s e r v e as an H-map.
I n t h e context o f t h e theory of H-spaces,
it i s t h e r e f o r e n a t u r a l
t o i n v e s t i g a t e t h e o b s t r u c t i o n f o r t h e e x i s t e n c e o f an H - l i f t i n g , provided an ordinary l i f t i n g exists.
More p r e c i s e l y , given H-spaces and H-maps:
38
Homotopy p r o p e r t i e s of H-spaces
f : Xo
If
+
Y
lifts
(as a map)
fo
gHD(f,po,p) = HD(fO,pOypl)
w: X
t o a map
*
jw =
0
Xo +V
A
g
+.
[Xo
Xo, Vg], [x0
A
then
0
Xo
j: V
0
Xo
A
-+
Y,
A
X
0
ax,
fiber j w
lifts
= jw.
HD(f,pO,p)
__*
Y
and t h e
t o be an H-map i s
f
s e t i n i t s simplest form
general,
f
HD(f,p,,,p):
= Fiber g ,
This A
Hence
w l i f t s t o w: X
implies t h a t
o b s t r u c t i o n for
=
-
gf
could be considered as t h e s e t
with t h e denominator s u b s e t i d e n t i f i e d t o a point.
In
x0, ”1
[Wl
i s a group a c t i n g on
[xo
xo’ vgl
A
and
could be considered as an element o f t h e o r b i t s e t . Without f u r t h e r assumptions, it i s hard t o measure t h e e f f e c t t h a t t h e change of t h e l i f t i n g i s the f i b e r
f
w i l l have on
of an H-map
W
flyF1:
[w].
X1,pl
Suppose now and
-+ X2,p2
p
g : Y + X1
i s the
fly*
2 . 1 . 1 m u l t i p l i c a t i o n induced by
The l i f t i n g s
5:
*
-
f f
1 0’
f d X )E Y. G = kgp.
f: X
5: Xo
gf5 = f o
-+
0
Y
-+
pl Y l J 2
and
F 1’
a r e i n 1-1 correspondence with homotopies
v i a t h e assignment
LX2
and as
g
6
-+
f5
f 6 ( x ) = f o ( x ) , c(x)
i s m u l t i p l i c a t i v e , one may assume t h a t
The H - l i f t i n g problem (modulo t h e ordinary l i f t i n g problem) i s
t h a t of f i n d i n g
F: X
0
x
X
0
+ PY
so t h a t
f
5
= f,F
i s an H-map.
Such
N
an
F
e x i s t s i f , and only i f , t h e following o b s t r u c t i o n
D
vanishes.
39
H-liftings
- PF(S v
N
DIXO v Xo = {F
Further, as
D-F
so that
DIX v X =
*
5 ) = SP
*
D N
and i f
-
by 1.1.5,
F
A l t e r n a t i v e l y , one can always assume t h a t i f extends t h e obvious homotopy
FIX
v X
- *,
SF
- *.
N
*,
one can f i n d
-*
D
re1 X v X.
then t h i s homotopy
It follows t h a t one can
N
replace t h e obstruction
D
D
D: Xo
by
A
X0
D = D ( S ,FO,F1).
fK2,
-3
~ 0 y p 2 , f 0 , f l as w e l l as on
a c t u a l l y depends on
E,Fo,F1
b u t it
i s only t h e l a s t t h r e e t h a t we allow t o vary, i n o r d e r t o a n n i h i l a t e t h e D.
obstruction
2.2.1. fi’Fi: + N I
Let
Proposition: XiYPi
-+
xi+ly
pi
i = 0,1,2
i = 0,l
H-Wpa
Xi’
be H-spaces 5 : Xo
3
a homotopy
LX2
flf0.
%0
Put
= PU ( W A x F ~ ) A ~ 1 0 0
0
Then :
(a)
fi,gi
(bl
Let
are H-maps and
Y =W
be the f i b e r of
fl
muZtipZication
p = u(plyu2,Fl).
Let
l i f t i n g of
induced by
*
with the 2.1.1
Yfl
fo
5:
f
5(x)
f
=
f i e n H D ( f , p O y p ) = JD(FOyFl,S) where
5 = f : Xo Y be the f(x) = fo(x), ~ ( x ) , . +
j:
SK2
+Y
Homotopy p r o p e r t i e s o f H-spaces
40
is the inclusion.
Proof:
(a)
fi,pi
a r e obviously H-maps.
Put
By 1.2.4
ji = wiA.
and 1.2.5:
Now,
PflPpl
-
core1 E o y Em.
Q2(Pfl
x
Pfl)
and on ilX x PX
t h i s homotopy i s
Hence
Now again by 1.2.4 and 1 . 2 . 5 if Y
Y E P2(X) 1' 2
A E ilX
t h e n t h e maps
a r e homotopic and as
OX 2'
Add
i s homotopy commutative
X,;
-
an H-space
H - l i fti n g s
41
One can s e e t h a t a l l homotopies a r e constant on may add t h e homotopy
E(EyFOyF1)
-
X
0
v Xo
hence one
D(C'.,FOYF1)A t o complete t h e proof of
(a). (b)
u ( j x 1):ax2 x Y + Y
Let
Corollary:
H-structures
Fi
H-map.
Y
-+
Xl
p
+.
x
DA.
l y ~ l
and
i = 0,l and a homotopy
t o fi
is a
by
fo: X o > P o
Then the f i b e r Y of (i.e.,
and
Y.
2 . 2 . 1 ( b ) follows by r e p l a c i n g
2.2.2.
7
Observe now t h a t t h e following homotopies a l l
coincides w i t h t h e a c t i o n . remain i n
i s given by X,(x,~) +. x,X +
-
fl pl
fl:
5:
admits an H-structure H-map) so that
fo
X1,u1
*
N
p
-+
f f
1 0'
over
lifts to a
Given
X2,u2.
Suppose
X1,pl
po
-
p
42
2.3.
Homotopy p r o p e r t i e s of H-spaces
Postnikov systems Given a space
IXnshnshn ,n-lY (PI)
X
i s a system
k 1 of spaces and maps so t h a t n
hn: X
-+
Xn
nm(xn) =
o
hn,n-lhn
N
m 5 n.
i s an isomorphism f o r
m > n.
for
xn-1 i s + K(n ( X ) , n + l ) n
Xn-l
nm(h ) n
satisfies:
xn
hn,n-l:
('3)
A Postnikov system f o r
X.
+
a f i b r a t i o n induced by a map kn: k-invariant of
c a l l e d t h e n-th
X.
hn-1
In our context of spaces every space admits a Postnikov system. denote
Xn = H t n ( X )
approximation of
X
and r e f e r t o in
hn: X
f a c t o r s uniquely through
nm(Y) = 0
Htn(X),
A cellular structure for
f
-
ci
n
as t h e homotopy
then any map
f: X
+Y
H t n ( f ) h n y H t n ( f ) : H t n ( X ) + Y.
U
. ..
i s t h e mapping cone of
H (C(h ) , G ) = 0 = ?(C(h ),G) m n n
m > n
for
can be given by
Htn(X)
H t n ( X ) = X U ( v e:+2) C(h )
Htn(X)
dim
If Y i s a space with
hence, i f
-+
We
h
n
for
for every group
G
m
< n+l
of c o e f f i c i e n t s .
It follows t h a t
i s an isomorphism for
m
Hm(hn,G): i s an isomorphism f o r
5
n
and an epimorphism f o r
9(Htn(X),G)
m 'n
-+
m = n+l.
e(X,G)
and a monomorphism f o r
m = n+l.
Similarly
43
Postnikov systems
If
hn: X
f i b e r of
-+
H t (X) n
?+' -+ x
h n y Jn+l:
Given a map
f: X * Y
{Yn,fn,gn,hn,kn)
satisfying:
dim 5 n
the
i s c a l l e d t h e n-connective fibering of
a Postnikov decomposition of
X.
i s a system
f
of spaces and maps
x
f,:
i s a homotopy approximation i n
Yn'
-+
gnfn
-
gn: Yn
-+
hnfn
f,
Y = Y 0 % hn: n'
-
gnhn+l
fn-ly
and an epimorphism for
-+'n-l gn+l,
m2n
isomorphism f o r
m > n+l and a monomorphism f o r k :
n
-+
'n-1
m n
) i s an
m = n+l, Tm(gn) i s an
isomorphism f o r
i s t h e f i b e r of a map
II ( f
m = n+l.
hn: Yn -+ Y
n-1
K ( n n ( f i b e r f ) , n+l).
The Postnikov system of a space coincides with t h e Postnikov
X
decomposition of
-+
c.
In case s m ( f i b e r f ) a r e a l l f i n i t e groups, it i s sometimes more convenient t o f u r t h e r decompose hn: Yn -+Yn-l
Yn=Y
where
51
{p
h
-
Y
*
n,J'
n,m
+
n,J
primes,
hn
-+Y
-+
n,ml
Y
-
n,J-1
... - + Y n ,k
a * .
corresponds t o a c o f i l t r a t i o n of a
n n ( f i b e r f ) = Gm
-%
hn,k
am-1 Gm-l
hn,j
, n ' ,k-1
i s t h e f i b e r of
hn , l hn , 2
t o a sequence of f i b r a t i o n s
k : Y n,J n,J-l
*..hnym.
= Y
+
Inym
'n-1-
I
_1
I
...
n
>
...
3
... -
...
+
K(Zp
J
=
'n-1
, n)
--
n (fiber f ) : G
1
G = O 0
hn,k
rk
Y
,
r r -+ +
... Yn,O
This f a c t o r i z a t i o n
This i s done by
Y
-+
-+
K ( G ~ , ~ )(%I
kera
.
= Z
k'
44
Homotopy p r o p e r t i e s of H-spaces
where a l l v e r t i c a l t r i a d s a r e f i b r a t i o n s . Such a system w i l l be r e f e r r e d t o as a primitive Postnikov system. With our notion of spaces every map admits a Postnikov and a p r i m i t i v e Postnikov approximations. For d e t a i l s and proofs, s e e e.g. [Hu] p. 155 or [Spanier] p. 440. An "Eckmann-Hilton dual" o f t h e Postnikov approximation and
decomposition i s t h e Moore f o r homology) approximation and decomposition: f: X
If
,(n)
-+
Y
then t h e Moore decomposition of
,(n) Y g (.n)Y , ( n )
f
i s a sequence
of spaces
Y
g(n): Y(n)
where
and
* Y(n+l)
M(G,n)
C(f)
i s t h e mapping cone o f a map
is the type
i s t h e cone on
The construction of i s constructed l e t
v(n)
is
n-1
Gn
Moore space:
f.
Y(n)
dn)- + Y ( n )
i s done by induction:
If
f ( n ) : Y(n)
-+
b e f i b e r f ( n ) . Assume i n d u c t i v e l y t h a t
connected and H n ( v ( n ) , z ) = Hn,,(c(f),Z).
men
Y
45
Postnikov systems
n n ( v ( n ) ) = Hn+l(c(f),z). isomorphism of
i s then
k(n)
Hn.
M ( ~ )=
Let
,z) ,n) *
(C(f)
M(n)
V(n)
-+
*
v ( ~ )y i e l d an
and
Y(n)
y(n+l) = c(k(n))*
*=
If
f("): Y ( n ) = H e n ( Y )
then
X
Moore) approAmation of
n.
i s a f a i r l y n a t u r a l construction and H t ( f ) i s n
Htn
While
i n dim:
Y
i s c a l l e d t h e homoZqy (or
Y
-+
uniquely defined f o r any map
f
HE
n
does not s h a r e t h e s e conveniences:
There i s no w a y one can choose u n i v e r s a l l y m a p s f:
x
-+
Y
m n ( f ) : mn(X) mn(Y).
w i l l induce m a p s
-+
n+l X = Y = M(G,n) v S HEn(X) = M(G,n)
-+
X,
M(G,n)
-+
cannot be deformed i n t o the
HL
2.3.1.
mn(Y) =
one can f i n d
Y
mn(X)
-+
so t h a t maps
X
e.g.
[GI <
If
m,
f o r any choices
M(G,n)
f: X
-+
so t h a t
Y
f(M(G,n))
Thus one should be very c a r e f u l i n using
M(G,n).
construction.
Proposition (see [Copeland& and [ S t a s h e f f & ) :
Let
f : X,p
isomorphism for
-+
be an H-map.
X',p'
and an epimorphism for
m< n
Given any map
g:
XI
-+
so that
K(A,n+l)
is an
H,(f,Z)
Suppose
m = n. gf
-
t
then
( a ) g is m H-map. r: 'jT
( b ) kt
-+
be the f i b e r of
X'
f o r 'j; so that pt
and some H-structures of
K(A,n+l) (c)
is unique as
For every H-structure of N
-
r is a
f
g.
for
H-map is (2.1.1) induced by
g.
( m e H-structure
f
A
F edsts
f,F
and any l i f t i n g
an H-Zifting of
of
pK
K(A,n+l), K(A,n+l)] = * . )
there e A s t s a unique H-structure
--
y
pt
[K(A,n+l) F
Then any H-structure
f,F
G
for
?: g
X
-+
so that
46
Homotopy p r o p e r t i e s of €I-spaces
Proof:
(a)
*
= HD(*) = HD(gf) = HD(g)(f
HD( g) E [XI
A
The conditions on consequently
Hn+'(f
f, A )
A
HD(g) =
r
r
where
Hm(f,Z)
imply t h a t
f) i s a t least
A
n+l
A
X' , A ) . is
C(f)
n
connected and
connected and hence
n+l H (f
i s a monomorphism and
Choose any H-structure
induced by
As
X' , K(A,n+l) ] = Hn+l(X'
A
f, A)HD(g) = 0
implies
*.
(b)
that
C(f
f) by 1.4.2 ( a ) ,
A
u'
and
- p'
is a
i s both a
po
j : QK(A,n+l)
n-connectivity of connected and
H-map.
- p'
N
C(r)
n H (r
A
choice of
Choose any
?),
so that
p
-
p'
implies t h a t
r, A)
F,G
N
N
X
so
H-map
t h e i n c l u s i o n o f t h e f i b e r of C(r
A
r.
r) i s at least
i s an isomorphism.
Now t h e n+l
Let
i s (2.1.1) induced by
One can e a s i l y check t h a t
(c)
be t h e H-structure
Put
and a
-+? i s
Fo
Let
b e any o t h e r m u l t i p l i c a t i o n i n
Let
G.
for g.
G
and
f,F
6:
g,G
Y
-
gf
(which i s equivalent to t h e
a r e H-maps.
By 2.2.1 ( b )
Actions, H-actions and p r i n c i p a l f i b r a t i o n s
i s induced by
where
isomorphism.
g
(see part (b)).
-w = [Hn(f
Let
A
f , A)]-1
Then by 2.2.1 ( a ) D(E,F,Ppk(-wh
x
Now,
=
*
N
of
If
assume
and
G
G'
G' = PpK(wh
X
both provide an H - l i f t i n g G)AXl
n H (f
2.3.2.
A f,
i s an isomorphism w
A)
Corollary ( [ S t a s h e f f & ) :
A
X',
K(A,n)].
hence t h e e x i s t e n c e N
then one may
f,F
and by 2.2.1 ( a )
0 = D(E,F,G') = D(C,F,G) + w ( f
As
E [XI
N
F.
i s an
H n ( f A f , A)
(D(E,F,G))
G)AXl
47
N
If f :
A
*
f) = w(f A f)
and
G
-
X,p -+ X' , p '
G'
c o r e 1 E o , E,.
i s an H-map then i t s
Postnikov decomposition and primitive Postnikov decompositions c o n s i s t o f H-spaces and H-maps and a22 the H-structures are UniqueZy determined by those o f
2.3.3.
X, X'
Remark:
f.
and
Note t h a t t h e Postnikov decomposition i s not uniquely
defined as a l l intermediate spaces can be replaced by t h e i r homotopy e q u i v a l e n t s , and a l l maps a l t e r e d accordingly.
2.3.2 states t h a t once
t h e decomposition i s chosen, t h e H-structures are unique.
2.4.
Actions, H-actions and p r i n c i p a l f i b r a t i o n s The notions of a c t i o n s and p r i n c i p a l f i b r a t i o n s used i n t h e category
of t o p o l o g i c a l spaces can be adapted t o t h e homotopy category s o t h a t some o f t h e homotopy p r o p e r t i e s of t h e t o p o l o g i c a l models a r e preserved. This can be done i n v a r i o u s ways. can be found, e.g.,
A d e t a i l e d study of such a d a p t a t i o n s
i n [ S t a ~ h e f f ] ~[ S, t a s h e f f ]
5'
[Fuchs] and [ P o r t e r I 2 .
A t o p o l o g i c a l ( l e f t ) a c t i o n of a t o p o l o g i c a l group
i s a map
n:
q ( 1 , x ) = x.
G x X
.+
X
so t h a t
q(gl*g2, x ) = n(g,,
G
on a space
q(g2,x))
and
X
48
Homotopy p r o p e r t i e s of H-spaces
The weakest homotopy analogue of an a c t i o n i s t h e following: 2.4.1.
of
Definition:
X,u
be an H-space and
i s a map 0 : X x Y + Y
on Y
h: X + Y
X,p
Let
i s given by
so t h a t
Y
a space.
rl(*,y) = y
An action
and i f
h ( x ) = q ( x , + ) then t h e following diagram i s
commutative: x x X
2.4.1.1
A
x
Note t h a t 2.4.1.1 coincides with 2.1.3 (2) and hence 2.1.4 s t a t e s t h a t if
X,p
a c t s on
then t h e f i b e r of
Y
s o t h a t t h e p r o j e c t i o n on
If Y,Py
Y
h
admits a m u l t i p l i c a t i o n ,
i s multiplicative.
i s an H-space,
an a c t i o n
rl
of
X,px
on
Y,py
i s said
t o be an H-action i f i n a d d i t i o n
i s commutative.
2.4.2.
Lemma:
If X,p
H-acts on Y,py
then h: X
+
Y,
h ( x ) = rl(x,+)
is an H-map. Proof:
Let
g: X x X
g(xl,x2) = xl.*,h(x2). By 2.4.1.1 and 2.4.1.2 r l ( l x h ) = huX.
-t
Then
X
x
Y x Y
be given by
(1 x uy)g = 1 x h,
(q x
l ) g = h x h.
uy(h x h ) = py(q x 1 ) g = q(l x uy)g =
Actions , H-actions and p r i n c i p a l f i b r a t i o n s
2.4.3.
49
The c l a s s i c a l examples of a c t i o n s are those equivalent
Examples:
t o a group a c t i o n . If
f: X
-+
i s a map, then
Y
a c t s on t h e f i b e r of
bly
f
in the
c l a s s i c a l way. One H-space a c t s on another via an H-map: n(x,x') = p'(h(x),x'). Given any space
If
Y
is
k-1
connected and r m ( Y ) = 0
m > n.
for
II
If H-SpaCe
II 5 n
satisfies
X,p a c t s on
uniquely.
-
k+II
1 > n,
2i? > n
Furthermore,
Y.
h: X
then any -+
11-1 connected
defines t h e a c t i o n
Y
(For t h i s statement not t o be t r i v i a l , one has t o assume
otherwise Indeed, as
[X,Y] = *). [X
X
Y , Y]
b e extended uniquely t o is
X',p',
i s HA t h e n t h i s i s an H-action.
X' , p '
Suppose
Y.
h: X,p ->
[X v Y , Y]
I-I: X x Y
holds.
1-1 (2.4.1.1
-+
-+
i s an isomorphism F ( h v 1) can
and as
Y
Hence for any map
[X
x
h: X
X, Y] +
Y
+
[X v X , Y]
fiber h
i s an
H-space. 2.4.4.
Definitio
:,
A map
h: X
equivalent t o a f i b e r of a map
i s commutative where
N
X -+Y
-+
Y
g: Y
i s a principal f i b r a t i o n i f -+
B.
h
1.e.:
i s t h e f i b r a t i o n induced by
g
from Em.
is
Homotopy p r o p e r t i e s of H-spaces
50
Let
F: Y
be a commutative diagram and l e t Then
fl,fo
= flyy
f(Y,Y)
by
g: Y + B ,
is a pair
and
LfOY
h: X
Let
F: Y
F
and
Y
+
PB'
f = f
PB'
fo
fl
be a homotopy
,F
'
fog
-
B'fl.
f i b e r g + f i b e r g'
FY.
+
and
g': Y'
h' : X'
+
B'
+
Y'
Y
fl,?,fl: -+
induce a map
+
-t
Y' be p r i n c i p a l f i b r a t i o n s induced A map of principal fibrations
respectively.
,
f : X + X'
F: f o g - g ' f l
f o r which maps
f o : B +.
B'
e x i s t so t h a t
P
\ -
X = fiber g
h
ffoyflyF
1
X' = f i b e r g'
Y
h'
+ Y'
i s commutative. If
f,,f
i s a map of p r i n c i p a l f i b r a t i o n
then t h e following i s Commutative
fly?:
(X,Y,h)
-f
(X',Y',h')
51
Actions, H-actions and p r i n c i p a l f i b r a t i o n s
h: X + Y
If
[ M y CiB]
t h e group TI:
QB
X
x
i s p r i n c i p a l induced by
-+
a c t s on t h e s e t
n+
X.
For
w E [My n B ]
One can e a s i l y s e e t h a t if
gl,g2 E[M,X]
so t h a t
j: L
-+
M,
M
then f o r every
[MYXI by an a c t i o n
has t h e property:
and only i f t h e r e e x i s t s
g: Y - + B
induced by
-
hgl
hg2
if
g2 = n*(wYgl).
g: M
-+
X
v: M
-+
QB then
n * ( v J , &I) = n * ( w d J . If h,h'
fl,?:
(X,Y,h)
-+
g: Y
-f
induced by
induced by
;I*(Cifow
B
and
g' : Y'
-+
B'
r e s p e c t i v e l y and
v E [My QB]
g E [M,X],
then f o r any
^fg) = ?*'I*(v,g)
Y
2.4.5.
fo
i s a map of p r i n c i p a l f i b r a t i o n s ,
(X',Y',h')
fly?
one has
*
Example: h: X + Y
( a ) Every map (b)
with f i b e r K(G,n)
i s a principal fibration
Given a commutative diagram f
X'
X
I
Y with
h
p r i n c i p a l induced by
f i b e r h ' = K(G',n)
and
principal fibrations.
Y
Hn(h,G')
Y' B,
B
surjective.
n-connected, Then
f,fl
i s a map of
52
Homotopy properties of H-spaces
’
Proof: ( a ) m 5 n where
By the relative Hurewitz theorem Hm(C(h) ,Z) = 0 for is the cone on h.
C(h)
be the Postnikov approximation in
Let j: C(h) + K(Hn+,(C(h),Z),
dim<_n+l.
n+l)
Let
be the composition
Y * C(h)
+
K(Hn+,(C(h),Z),
N
X =-fiber g and lift h to a : X *
7.
n+l), One may assume that the following
commute :
N
As we assume X,X to be 1-connected by the Serre exact sequence (and see, e.g.: [Spanier] 9.3.4.)
is an isomorphism for m 5 n+2 and as Hm( j) is an isomorphism for m 5 n+l and an epimorphism for m = n+2 it follows that Hm(C(h),Z) + Hm(C(F),Z) epimorphism for m = n+2. m 5n
is an isomorphism for m 5 n+l and an Consequently Hm (a,Z)
and an epimorphism for m = n+l, fiber a
is an isomorphism for is n
connected and so
is the fiber of the map induced by a
Hence, Hn+l(C(h),Z) = G,
.
IX
is a homotopy equivalence and so is a .
Actions , H-actions and p r i n c i p a l f i b r a t i o n s
(b)
53
Consider t h e following:
f
X
X'
B
By t h e S e r r e e x a c t sequence
H~+'(B,G' ) as
6H"+'(Y
and t h e r e e x i s t s
fo: B + K ( G ' ,
F: Y + P K ( G '
, n+l)
h'? = h ' f ,
hence, t h e r e e x i s t s
N
tl*(u,f) = f .
P
As
Hn(h,G')
h: X + Y
q : QB x X +. X
so t h a t
be such a homotopy.
= PuK(v x F)Ay
If
n+l)
u: X
H"+~(x,G'
h*
= Hn+l(h,C')[g'fl] = [ g ' h ' f ] = 0,
h*[g'fl]
Put
,GI
Then
[ g ' f l ] E i m Hn+'(g,G') g'fl
?: =
-
f
fog.
fosF
+ K(G',n) = Q K ( G ' ,
i s surjective
u = vh,
g: Y + B
i s equivalent t o a group a c t i o n .
satisfy n+l)
with
v: Y +QK(G',
t h e n one can e a s i l y s e e t h a t
i s p r i n c i p a l induced by
Let
ffoyF
n+l). N
= q,(vh,
f ) = f.
then the action
As one has a weaker
n o t i o n of an a c t i o n , one can analogously seek a weaker form o f p r i n c i p a l fibrations.
2.4.6.
Definition:
Given a map
f: X - + Y
with
f i b e r j: F +X,
f
s a i d t o be weakZy principal (abr. w - p r i n c i p a z l i f given t h e p u l l back diagrams:
is
Homotopy p r o p e r t i e s of H-spaces
54
.Y
X
t h e r e exists
r: W
+
f
Given f : X
Lemma:
-
1, r X
-
X(x) = x,kx,x E W
obvious cross-section
2.4.7.
r2
F,
Yf
-+
Y
where
c
X: X + W
is the f,f
f,f
c X x PY x X.
with f i b e r
j: F
-+
X.
f
i s weakly
principal if and only i f there e x i s t s a homotopy equivalence u: F
x
X+ W
Proof: v: W
-
-+
F
f,f
rj
1, r X
of type
f,f
If x
-
u: F
x
x
j , ~ ,i . e . :
w
w
u(c,x) = x(x).
u(a,*) = j ( a )
of type
j , ~e x i s t s , l e t
f,f
X be i t s i n v e r s e then
r = p v: 1
W
+
F
satisfies
f,f
it.
Suppose conversely t h a t
r: W
-+
F
exists,
f,f
rj
-
1,
Xj
- +.
Then
one can e a s i l y s e e t h a t F
wf
f
i s a f i b e r homotopy equivalence. (' F
v
X
~
f,f
W
,
1
X
'IA
P2 X
As t h e composition 9
F
x
X
i s homotopic t o t h e i n c l u s i o n ,
Actions , H-actions and p r i n c i p a l f i b r a t i o n s
( r x ?)A
and
u: F x X + W
2.4.8.
v X
i s a homotopy equivalence,
j,X
of type f,f
Corollary:
If
f: X
extends t o
homotopic t o t h e i n v e r s e of
+
55
(r
x ?)A.
i s w-principal, then one has a puZZ back
Y
diagram
of type
with n
Proof: note t h a t
2.4.9.
j,l.
I n t h e n o t a t i o n s of 2.4.7 and i t s proof, p u t
2.u =
p2vu
-
p2
Proposition:
Let
f ' : X' = W
+
and l e t Then f ' :
x'
Proof: -
+
Y'
g,f
and
f:
X
fn = +
fqu
= &u
ru
and
= fp2.
Y be w-principal.
a v e n g: Y'
be the fibration induced from
Y'
q =
f
by
+
Y
g.
i s w-principal.
One can see t h a t i f
z: X'
+
X
covers
g
then one has a
p u l l back diagram
F
X' (This can be e a s i l y seen i f one replaces
f: X + Y
t h e homotopy p u l l backs by t o p o l o g i c a l ones.)
by a f i b r a t i o n and
56
Homotopy p r o p e r t i e s o f H-spaces
Furthermore, i f
r: W
F
-+
f,f
* Wf, ,f,
Xxl : X'
satisfies
rj
-
1, r X
X
N
Y
N
n
Xxg = gXxl.
then
ri:
then
If
Wf',f'
-+
F
satisfies
t h e corresponding p r o p e r t i e s . A j u s t i f i c a t i o n f o r t h e d e f i n i t i o n o f w-principality
2.4.10.
If
Proposition:
f: X
*
n,,:
[M,F] x [M,X]
only i f
[M,F]
properties hold are xo =
*
Proof:
As
fn(1
x
F
N
jglF
V
QY
-+
F
a: F
V
-
a: F
X
F
s o that f o r
-+
X
fgl
N
i f and
u E [M,F].
f o r some
i s not necessarily a group and i t s action on the
lax = x
The only
(u1-u2).xo = u (u - x ) 1 2 0
and
-
fP2(1 x j)
= fjp
~ ( x 1j ) l F v F = ( n I F
--+
fg2
f o r t h e base
Consider t h e following commutative diagram:
j)
V
w
X ) (1 v j )
QY s o t h a t
--
F = i+(a,m(F v F ) .
and i f one r e p l a c e s
V I F V F = n,,(a,m)lF v F = ;,,(;IF
-
m by
v F, ;IF
N
G:
lifts t o
q(l x j)
i s p r i n c i p a l with a c t i o n say
* QY n
M
gl,g2:
E [M,X]. )
X
F
This action induces an action
does not s a t i s f y any a s s o c i a t i v i t y properties.
[M,X]
point
[M,X]
g2 = n,,(u,gl)
(Note t h a t set
-+
i s w-pfincipal, then F = f i b e r f
Y
is an H-space which n-acts on X.
i s given by
F x F
j v j = jF.
;,
there exists a
extends t o
p );= ,;(,,:
v F ) = ;I,,(a,;lF
then
v F) = F.
-+
F.
Actions , H-actions and p r i n c i p a l f i b r a t i o n s
jp =
Obviously 2.4.11.
jz= r l ( l
f: X
Let
Corollary:
and
x j)
-+
Y
i s an action.
q
be a map of spaces.
principal if and only if F = f i b e r f q(a,*) = j ( a ) ,
m [F
v X
-
j: F
is an H-space tl-acting on X with
can be extended t o a homotopy H:
fp21 F v X
is weakly
f
and i n such a w a y t h a t the obvious homotopy
X
-+
57
fq
-
fp2-
(Note t h a t by 1.1.7 if Y i s an H-space, i t s u f f i c e s t o asswne t h a t
is an H-space q-acting on X
F = fiber f
Proof:
If
=
i s w-principal, then
f
- m2).
with
Tu
(with t h e notation of
2.4.6 and 2.4.7) s a t i s f i e s t h e hypothesis of t h i s proposition as can be seen d i r e c t l y from t h e d e f i n i t i o n s of t h e various spaces and maps. If
u: F
x
F x X
q:
X
-+
-+
with t h e desired properties e x i s t s , l e t
X
u ( a , x ) = n ( a , x ) , H(a,x) ,x,
be given by
W
as
>u = p2
f,f
and if a = x,y E X x LY
u(a,*) = j ( a ) , H(a,*),+ hypothesis,
H(a,*) = Y,
equivalence.
u(*,x) = x,
As
H(+,x) = k f ( x )
hence
day*)= j(a)
H(*,x) ,x
u(*,x) = X(x)
and
j ( a ) = x,
by our
u i s a homotopy
and
and again by the hypothesis
u i s of type j , X
and 2.4.11 follows
from 2.4.7. 2.4.12. h
Example:
X,p
r i g h t :-act
on
Y
h(x) = i(*,x).
with
s a t i s f i e s t h e hypothesis of 2.4.11 and hence i s w-pricipal.
F = fiber h
Let
Let
n:
F
x
admits an H-structure
X
-+
X be given by
p
p2
I
X
h
n
x
j: F
-+
X
Indeed,
multiplicative.
p ( j x 1). Then one has a commutative
diagram
F x X
with
xI x
Then
Homotopy p r o p e r t i e s of H-spaces
58
The homotopy hn
i s a homotopy
H1
H2: h j
hj
-
x
N*x
1
N
hp
N
hp2
h(j
can be given by x
1) re1 X v X
HIF
V
X
x
1) + P:H2
where
and
1 = i p ,H = H2 x 1, H2: 2 2 2
One can e a s i l y see t h a t
Y.
H = Hl(j
F -+ PY , t h e obvious homotopy
i s t h e obvious homtopy.
HA and HC obstructions
2.5.
As i n t h e case of t h e study of H-maps where t h e homotopy w a s incorporated as a s t r u c t u r a l p a r t of t h e map t o o b t a i n t h e category
it i s sometimes convenient t o carry t h e homotopy
HfFy
Definition: where
X,u,A
p ( p x 1)
-
dl
x p)
p ( 1 x p)
An HA space f i n the p r e d s e sense) i s a t r i p l e i s an H-space and A : X x X x X
X,P
-
Thus:
as p a r t o f t h e s t r u c t u r e o f an HA space. 2.5.1.
p ( p x 1)
__*
PX
i s a homotopy
which by 1.1.5 and 1.1.6 can be assumed t o be
2
~ ~ ~ x V X V X = V X . 3
An A-map f,F,a: X,p,A a:
x
x
X
x
x
(A3
-
map i n t h e sense of [ S t a s h e f f ] ) i s a t r i p l e 2
-+XX',~',A'
so that
a
i s an K-map
X,p + X ' , p '
~ ( P x ' ) satisfies
(PEo)a = F(p x 1) + P p ' ( F (Pgbda = F ( l
(f,F)
x p)
+
x
kf),
Pp'(kf x F ) ,
can be i l l u s t r a t e d as follows:
Eoa = (Pf)A E,a
= A'(f
x f
x f).
and
f,F: X,p -+
Let Let
?=
0 = P(pX1)
? l X i( X i( X
0A3 =
b e an €I-map,
with
A'(fxfxf)
vXv T'IX v X
X
corel E
,kf(p 6
X
1)
then
X
X = kf(p A
X
-+
- Pp'(kf
,p'
,A'
HA spaces.
be given by F)
X
- F(lXp) - PfA.
i n an obvious way, X
nX'
1)lX
v
X
v
X.
be given by
i s t h e obstruction f o r
t o b e an A-map.
f,F
i s a f f e c t e d by composition i s given by Let
Proposition:
be H-maps where
X,p,A,
e((f',F')(f,F),A,A'') Proof:
6
A
c PX'
__*
+
X
X'
and
X,p,A
kf)
Pp'(F
59
obstructions
6: X x X x X
0 = O(f,F,A,A1): X
%I
The way
,p'
kf(p x l ) l X
corel E Let
2.5.2.
N
+
and
N
?(f,F,A,A'),
N
?-;'
XI
HC
HA
( f , F ) : X,p XtyptyA1
-+
and
= Qf'e(f,F,A,A')
+
The proof is i l l u s t r a t e d by
X',p'
X",p",A"
( f ' , F ' ) : X',p'
+X",p"
are HA spaces.
B(f',F',A',A")(f
A
f A f).
Then
60
Homotopy p r o p e r t i e s o f H-spaces
A" ( f x f x f I ) ( f x f x f )
P d ' ( k f I xF' ) ( f x f x f )
N
9 ( f ' ,F',A',A")(fxfxf)
'IF)
2.5.3.
HA maps.
Proposition:
Then ai ,Ai
Let
f.,Fiyai: 1
XiyuiyAi
----*
X
0
,u
0
,A
0
i = 1.2
induce an HA structure on the p u l l back
be
W flYf2
of
fl
and
Proof: be p o i n t s i n
f 2 with 2.1.1 H-structure.
-
Let
z = (xlYy,x2).
W
Then
fl'f2'
-
z' = (x;,Y',x;)
and
z" = (X~,?',X~)
HA
and
HC
obstructions
Similarly,
The homotopy
(Z*Z')Z''
-
z(z'-z")
i s i l l u s t r a t e d by
61
62
Homotopy p r o p e r t i e s o f H-spaces
The proofs
One can c a r r y out t h e same procedure f o r HC spaces.
involve two r a t h e r than t h r e e v a r i a b l e s , are simpler and hence omitted. (See [.@ski, Kudo] and [ B f o ~ d e r ] ~ . )
An HC space can be regarded as a t r i p l e H-space and that
p
N
i s an
X,p
and one may assume
pT
C ( X v X = kF.
-
An HC map i s a t r i p l e
c: X x X
H-map and
Let
X,p,C
b e an H-map. N
i s a homotopy
C: X x X + PX
where
X,p,C
and
-
P(PX')
X,p,C
-+
X'yp'yC'
Eoc = Pfc
c = FT
E,c
X',p',C'
FT
PfC
which i s t h e obstruction f o r
= C'(f
x
i s an
f,F
f)
be HC spaces, l e t
= T(f,F,C,C'):
-
so t h a t
i s i l l u s t r a t e d by
c = F
Define
$ = F + C'(f x f)
f,F,c:
then f,F
P.
X x X
4 ,kf
-+
XIs1
f,F:
X,p
c PX'
defines a map
t o be an HC map.
-+
X',p'
by
4: X
A
X +GXl
63
Homotopy s o l v a b i l i t y and homotopy nilpotency
2.5.5.
Let
Proposition:
HC maps.
fi,Fi,Ci:
Xi,piyCi
+
XoypoyCo
i = 1,2
be
induce an HC structwle on the p u l l back Wf
Then ci,Ci
1' 2
with the 2.1.1 H-structwe.
2.6.
Homotopy s o l v a b i l i t y and homotopy nilpotency. Given an H-space
[M,X]
X,p
one can look f o r t h e a l g e b r a i c p r o p e r t i e s of
and i n p a r t i c u l a r those p r o p e r t i e s which a r e independent of
and a r e thus i n v a r i a n t s of
M
I n p a r t i c u l a r w e are i n t e r e s t e d i n t h e
X,p.
p r o p e r t i e s of s o l v a b i l i t y and nilpotency. Recall t h a t f o r a group (or an a l g e b r a i c loop) class
< n
i s equivalent t o t h e property t h a t t h e function (on s e t s )
Similarly
Bn: Gn
G
-+
constantly
G
i s n i l p o t e n t of c l a s s
given by
Bn(gly.
< n
if t h e function
.. ,gn) = [.. .[gl.g21y
solvable (or n i l p o t e n t ) of c l a s s i n [Harrison -Scheerer] [ A r k o ~ i t z - C u r j e l ,3,4 ]~
,... 1 ,
Xyp
is:
Lemma:
. ¶ pn )
[ ,XI
= *)
< n
for a l l
M?
g,]
is
, and
[Porter] X
1'
p
i
[M,X]
,
Such a property can b e transformed
as follows:
is solvable (nilpotent) of class
where
When i s
This problem i s s t u d i e d
[Whitehead] , [Berstein-Ganea]
i n t o an i n t r i n s i c property of
(Bn(ply..
g31
1.
The question one may ask f o r a given H-space
2.6.1.
s o l v a b i l i t y of
G
are the projections.
< n
if and only
64
Homotopy p r o p e r t i e s o f H-spaces
Proof: f o r all
M
It i s obvious t h a t implies
a(pl,..
M
f
hand, f o r any
and
f i = pi?.
so t h a t
-
i'
.,p
.. ,pn)
*
) = (B(plY. 2" M - t X there exists
?*:
As
s o l v a b i l i t y ( n i l p o t e n c y ) of
< n
2n [X ,XI
+
?:
M
= *).
8"
-t
(%*: [ f , X ]
[M,X]
-t
[M,X]
On t h e o t h e r
(>: M -~
x")
[M,X]) a r e
homomorphisms
= a(fl,...,f
ba(p,,...,p 2n
) (?*B(pl,...,p n
=
B(fl,...,fn))
2n
We a r e now t o d e f i n e formally
.. ,p
a(pl,.
)
and
B(pl,..
.,pn)
and
2n study t h e i r fundamental p r o p e r t i e s . For an H-space
v
1
= v,(X,p) Let
E [X
AnX
A
X, X I
n
= vn(Xyu): A
2.6.2.
X,u
vn
X + X,
Definition:
fabr: < n H N )
if
2"
if
X,p
v
n
= v (v
1 n-1
X, X]
P YVT
AnX A
+
v
n-1
X,
2
wn(X,u)
-
w
n
and
Lemma:
Ib)
If
f : X,p
2"
+
vm)
X',u'
A n- 1X
1)
< n
it.
by i n d u c t i o n :
~ = +vn(A ~
A
i s s a i d t o be homotopy n i l p o t e n t of class
The following simple p r o p e r t i e s o f
v
n-1
A
).
i s s a i d t o be homotopy solvable of class
fa)
AnX = X
wn = w (w
< n
- *.
2.6.3.
= vlA.
X , A 1X = X ,
A-product o f
wn = w n ( X , p ) :
A
w2A = D
be given by
be t h e n-fold
and d e f i n e i n d u c t i v e l y v
w2 = w 2 ( X , p ) E [X
let
X,p
is an H-map then
v n
(abr: < n
HS)
can be e a s i l y proved
65
Homotopy s o l v a b i l i t y and homotopy nilpotency
(el w 2 ( X , u ) = 0 if and only if X,p
( d ) v ~ =- w ~(1A 1
A
n
implies
< n-1
v1
v2
A
A
v
is HC.
A. . . A
vn-2 )
and hence
HN
< n
HS.
2.6.3 suggests t h e following g e n e r a l i z a t i o n s o f 2.6.2: 2.6.4. < n
Definition: (< n
HN
2.6.5.
HS)
A
1) =
-+
fin(x,p) =
if
An H-map
Definition:
w2(X',p')(f
f: X , u
Let
X'u'
*
be an H-map.
i s s a i d t o be
f
(fvn(X,p) = I).
f : X,p
-+
X' , p '
i s s a i d t o b e central i f
*.
The elementary techniques f o r study of n i l p o t e n c y and s o l v a b i l i t y o f groups can apply t o t h e study o f HN and HS p r o p e r t i e s : 2.6.6.
Lemma:
Let
-
( s e e [Berstein, Ganea] and [Larmore, Thomas])
A
F,uF
E,uE
f
be an H-fibration, i . e . :
B.pB
F -+ E
-+
B
i s a f i b r a t i o n and a l l spaces and maps are H-spaces and H-maps. (a)
If
f
is
< n
HN
(b)
If j
is
< m
HS,
fc) Let
g: B,pB
-t
B
and
0' 'B,
principal H-fibration induced,
pa = Add.
j
f < k
i s central then E HS
then E
be an H-map,
is
is
< m+k
< n+l
HN.
HS.
considering the induced
PBoy u n L E, uE Then j i s central.
f
-B,pB3
pE
2.1.1
66
Hornotopy p r o p e r t i e s of H-spaces
By 1.2.4 and 1.2.5
LU
T BO
j
-
on
Lp
zBo x
BO
LBO ( c o r e 1 E o y Es) hence
is central. Corollary ( see [Berstein, Ganea]):
2.6.7. (a)
If
x
is
< k+l €IN.
w,(x)
is an H-space,
= 0
m # nl, n2,.
(b) Let K be a connect.ed f i n i t e complex dim K = k. any H-space
Proof: a sequence of
Both
k
X
X,
8,pK
is
( i n ( a ) ) and
,n
k
,
then
X
Then for
k+l HI?. ( i n case (b)) can be obtained via
X !
p r i n c i p a l H-fibrations:
( a ) by i t s Postnikov system
and ( b ) by t h e p r i n c i p a l c o f i b r a t i o n s inducing t h e c e l l u l a r s t r u c t u r e of K
(with t h e bottom space
2.6.8.
2,
Proposition (Ganea):
Proof: -
N
N
S = v S1. m
s2n+l
2=
( CKlm i s
HC).
is < 3 HI?.
By [James] and i t s consequences, one has t h e following 1
James f i b r a t i o n :
Hence, one has a f i b r a t i o n
Homotopy s o l v a b i l i t y and homotopy nilpotency
4 ffS2"'2
&34n+3
< 2
2.6.9.
Proposition:
QPJG)i s
men
Proof:
n2s2n+2
is
HC
2.6.8 follows from 2.6.6 ( c ) and ( a ) .
(hence
HN)
and a s
67
Let
Gn = SU(n) or
< 2(n-m)
Put
Sp(n).
f(G)
= Gn/Gm.
HS.
One has a f i b r a t i o n
and by looping one obtains an H-fibration
As
= Sdn-1
$'l(G) n
ns
2.6.8
is
(d = 2
if
G = SU
and
HN, hence < 2 HS,
< 3
d =
4
if
G = Sp)
and by
2.6.9 follows by induction
from 2.6.6 ( b ) .
G = SO,
Now, it i s well known t h a t f o r G(n) + G(2n)
is
< 2
HN
(or
homotopy commutes i n
G(n) C&$n(G)
and
Qv",(G)
+
< 1 HS).
+
G(2n)
Sp t h e i n c l u s i o n
(This property i s s t a t e d as
As t h e f i b e r of
G(2n).)
G(n)
SU o r
G(n) + G(2n)
is
i s an H-fibration with r e s p e c t t o
t h e loops a d d i t i o n and Lie group m u l t i p l i c a t i o n s , 2.6.9 and 2.6.6
(b)
imply: 2.6.10.
Proposition:
SU(n)
any topoZogica1 space Y
compZez1 2.6.11. treatment S2n
[Y, SU(n)l and Remark: &9
2.6.10
a d Sp(n) are < 2n+l HS.
Hence, for
(not necessarily of the homotopy type of a CW
[Y, s p ( n ) l are can be extended t o
s t u d i e d i n Chapter 4:
as w e l l and f o r odd primes
< 2n+1
soZvabZe groups.
SO(2n+l)
via a p r i m i t i v e
The James f i b r a t i o n holds
S0(2n+l)/SO(2n-l) = S4'-l.
mod 2
for
68
Homotopy p r o p e r t i e s of H-spaces
The argument can b e a l s o extended t o some exceptional groups, e.g: Example (Adams): G2
Using t h e f i b r a t i o n -+
Spin(7)
one has an H-fibration:
(as S O ( 7 ) )
and
QS7
-P
G7 is
HC
S7
-+
G2 G2
-+
Spin ( 7 ) .
is
<
As
Spin ( 7 )
improved.
< k
HS
<
15 HS
16 HS.
Note t h a t t h e s e methods do not y i e l d t h e minimum groups a r e
is
k
for which t h e s e
and t h e numbers given h e r e almost c e r t a i n l y can be
69
Chapter I11
The Cohomolow of H-spaces
Introduction The study of t h e cohomology of H-spaces h a s , amongst o t h e r s , t h e following a p p l i c a t i o n s :
The f a c t t h a t a space
X
admits an H-structure
implies t h e existence of c e r t a i n p r o p e r t i e s i n i t s cohomology.
This may
h e l p t o determine t h e cohomology of a space, knowing t h a t it admits a Conversely, i f a cohomology of a space is known, t h e
multiplication.
general theory of t h e cohomology of H-spaces may help t o determine whether or not t h e space admits a m u l t i p l i c a t i o n . I n t h i s chapter we mainly study t h e ordinary cohomology with
= Z/pZ c o e f f i c i e n t s , though t h e r e s u l t s of s e c t i o n 2 are given f o r an P a r b i t r a r y theory. The same i s t r u e f o r t h e p a r t underlying t h e Bockstein 2
s p e c t r a l sequence i n s e c t i o n 3. If with
X,p
H*(p,F)
i s an H-space and and
H*(A ,F)
F
i s a f i e l d then
form a Hopf algebra.
H*(X,F)
together
W e assume t h e r e a d e r
i s f d l i a r with t h e b a s i c notions regarding Hopf algebras and, as usual,
t h e standard reference i s [Milnor and Moore] whose notations we adopt f r e e l y . Section 1 i s devoted t o t h e consequences derived through t h e Hopf algebra theory only. coproduct i n
H*(OX,F)
(Proposition 3.2.3).
I n s e c t i o n 2 we study some p r o p e r t i e s of t h e derived from p r o p e r t i e s of t h e a l g e b r a H*(X,F) As w a s s t a t e d above, t h i s i s done with no added
s t r e s s f o r generalized t h e o r i e s . t h e Hopf algebra
H*(Spin(n), Z2).
I t i s being applied (3.2.4) t o determine
The cohomology of €I-spaces
70
Section 3 i s devoted t o t h e Bockstein s p e c t r a l sequence, and hence
we assume t h e b a s i c acquaintance with s p e c t r a l sequences.
As s t a t e d
before, t h e s p e c t r a l sequence i s constructed f o r generalized t h e o r i e s , though applications are all c l a s s i c a l .
Further r e s u l t s a r e presented
here without p'roof as t h e complete pursuit of t h e proofs requires too
many technical d e t a i l s . In section
4 high
algebra s t r u c t u r e .
order cohomology operations a r e added t o t h e Hopf
The main type i s t h a t of non-stable secondary
some sense t h i s technique has i t s o r i g i n i n [Thomas] 1,233
operations.
though t h e r e t h e secondary operations a r e replaced by primary operations i n the cohomology of the p r o j e c t i v e space of t h e given H-space ( a subject not covered i n these n o t e s ) .
There a r e some l i m i t a t i o n s t o t h i s method:
Not all cohomology classes of t h e given space appear i n t h e cohomology of i t s projective space; not all secondary operations i n t h e cohomology of t h e space have analogues i n i t s projective space; and, f i n a l l y , t h i s method i s e f f e c t i v e mostly f o r t h e treatment of t h e cohomology with coefficients i n
Z2 while f o r odd primes it i s weaker.
The study of the secondary operations techniques requires some knowledge of the s t r u c t u r e of t h e Steenrod algebra and of t h e cohomology of Eilenberg MacLane Spaces.
For t h e d e t a i l s regarding t h e Steenrod
algebra, the standard reference i s [Steenrod and Epstein].
3.1.
The Hopf algebra H*(X, If
X,p
Z ) :
i s an H-space, t h e obvious commutative diagram
X
A
' X X X
71
The Hopf algebra Hy(X,Zp)
implies t h a t f o r a f i e l d form a Hopf algebra.
F H*(X,F)
of f i n i t e type over a f i e l d
where
F is p e r f e c t .
Let
H*(v,F)
be a graded connected Hopf algebra
A
If t h e characteristic of
F.
F
i s not
0
Then as an algebra
i s an e x t e r i o r algebra on one generator
A(xi)
and
Zp) :
H*(X,
Theorem (Hopf-Borel):
asswne that
H*(d,F)
Thus, one can apply t h e fundamental s t r u c t u r e
theorem of Hopf algebras f o r
3.1.1.
together with
xi
and
r1
charac. F = 0
or r =
J
m.)
A s t h e multiplication i n a Hopf algebra i s always assumed t o be
commutative and associative, i f
p
#
dim xi = odd,
2:
dim yi = even.
Recall t h a t t h e augmentation i d e a l of a Hopf algebra A
-
2 = ker
E,
E:
A
The module
If
-4: -A @ A --*A
-+
QA,
F t h e counit.
If
A
of indecomposables of an algebra A
(T
-
4 Hopf 9 + A +QA
Q
A)
A
The
algebra i s s a i d t o be primitively generated i f t h e composition
is s u r j e c t i v e .
dmitively generated, t h e
.
$).
I ) : A + A 8 A.
The Hopf-Bore1 theorem has an extension s t a t i n g t h a t i f
.imi tive
Ai.
@I
iXl
i s t h e module
i s induced by
where
x=
i s t h e cokernel
t h e r e s t r i c t i o n of t h e m u l t i p l i c a t i o n
ubmodule of primitives of a coalgebra A = ker(z
i s connected, then
i s the ideal
xi
and
A
is
yJ of 3.1.1 can be chosen t o be
The cohomology of H-spaces
Recall t h a t A*
A
i s p r i m i t i v e l y generated i f , and only i f , t h e dual
i s a commutative and a s s o c i a t i v e algebra and f o r any
3.1.2.
Remark:
a : E(X) Q E(X)
-
E(X
A
i s not an isomorphism
X)
E = H*(
Hopf algebra (e.g.
yZ)).
E(X)
e.g. :
If
E
is
( s e e 0.5) an element
n
i s c a l l e d primitive i f i t can be prepresented by an H-map
x E E"(X) X,p + E n ,
{En' Y : En -+2En+l)
Q-spectrum
may f a i l t o be a
However, some of t h e notions used f o r
Hopf algebras can be extended t o generalized cohomology. represented by an
= 0.
i s a m u l t i p l i c a t i v e theory s o t h a t
E
If
2
x E A*
Add.
An immediate consequence of t h e Hopf-Bore1 theorem i s t h e following:
3.1.3.
Proposition:
i s an H-space then X is an even dirnensionaZ
If I X
sphere. Proof:
In
H * ( C X , F)
i-: E*(Ix,
F) Q l i w ( z x , F)
+
F) i s
:*(EX,
By 3.1.1. t h e only possible Hopf algebra with t h i s property i s dim x = 2n+l ZX = S
hence
if 2n+l
charac. F # 2. and
2n
x=s
This holds f o r
F = Q
or
A(x), and
F = Z
.
P'
Actually a much stronger statement i s t r u e :
3.1.4.
Theorem ([AdamsIl):
Among spheres onZy
S1, S3
and
S7
admit
H-structures. The various proofs of t h i s theorem c a l l f o r highly non-elementary techniques using e i t h e r high order ordinary cohomology operations or primary operations i n generalized cohomology t h e o r i e s . This observation implies t h a t t h e Hopf-Bore1 theorem, though an excellent f i r s t approximation, i s f a r from being s u f f i c i e n t and more r e f i n e d methods a r e d e s i r a b l e .
0.
H*(X,Z
P
)
and t h e coalgebra
H*(SZX,
Z P
73
Some of t h e s e methods a r e described i n t h e following s e c t i o n s .
3.2.
Some r e l a t i o n s between t h e algebra H*(X,Z coalgebra
H*(QX,
Zp)
Z ) can be derived from P v i a t h e Eilenberg-Moore s p e c t r a l sequence where t h e arguments
Some of t h e r e l a t i o n s i n Z )
H*(X,
) and t h e P -
P are purely algebraic.
H,(OX,
(See, e.g.
, [May
and Zabrodsky]).
We a r e about t o
give a more geometrical argument t h a t by i t s n a t u r e may be applied t o some generalized cohomology t h e o r i e s where t h e Eilenberg-Moore s p e c t r a l sequence may f a i l t o e x i s t and may give some geometric i n s i g h t t o t h e algebraic arguments when t h e s p e c t r a l sequence i s a v a i l a b l e . The following i s somewhat related t o 2.2.1.
3.2.1.
Lemma:
be H-maps.
*
-
flfO
Let
fo,Fo: X o y p o
5,: X
0
respectively.
Define
by:
Let
-+
LX1,
--*
5: Xo
X1,pl +
LX2
and
flyF1:
Xl,pl
be homotopies
*
-----*
-
fo
X2,p2
and
The cohomology of H-spaces
74
Now
$:
*
PF1(CO
-
LflLul(E0
As
RX2
is
x
to) induces a homotopy
x
C0)
+
m1w0 -
+
( j- Rfljo)A
-..
f,F: X,p
EU + F
RU (v 2
x
v) =
x
wo -
= H D ( V , ! - I ~ , R U ~ ) A and
Lfl)(SO
E0).
x
v
@2(
x
v ) + Slf w 1 0
1-1 i m p l i e s 3.2.1.
A*
of 3.2.1 a r e i n v a r i a n t s defined whenever one has an H-map -+
- Lu'(S
5: X
and a homotopy
X',p' x
homotopy induced by
*
N
i
h
5: RX
then i f then f o r
j ( k ( Q h ) A d d , 5) =
~ x ' , 5:
-t
h: X LRX'
-+
-
Y
~ ( F , S ) A=
f:
i: X
and
X'
-+
LX'
is a
i s t h e obvious homotopy
Rh, k(Rh)Add: fX, Add
-+
5:
*
N
fX' , Add
*.
The n o n t r i v i a l i t y of
Proposition:
by the diagonal:
-+
5).
One can e a s i l y check t h a t i f
3.2.2.
- Lp2(Lfl
fo)
x
mx,
HC:
w = vp 0
wo,w
Fl(fo
-
$: Xo x Xo
,. can be
w
For my space
-
A = AA.
natural homotopy given by:
Let
5,:
!LEO =
i l l u s t r a t e d by:
X let sD(
a
-+
h: X
LO(X
A
--+
X),
X
$,:
A
X
-
be induced
* Rh
be t he
Slh
H*(X,Z
Proof: useless.
P
)
H*( mC, Z ) P
and t h e coalgebra
75
The formal proof w i l l be t o o complex t o comprehend, thus
We s h a l l give a less formal proof using t h e c l a s s i c a l p a t h
spaces
8X
Now, it i s q u i t e d i f f i c u l t t o describe
AX.
To overcome t h i s
d i f f i c u l t y consi de r
"-1 -2 Then Yo = !I [Q ( X
and Y1 = T-'(i$X
X)]
A
AX))
can be e a s i l y
described and f o r t u n a t e l y all maps and homotopies i n t o in
OUT
proof f a c t o r through
PX x PX
and
5,
Yi Y
0
and
and p Y = y 2 2
y: I2
+
X
X
QA
Y1:
describes a map i n t o
W e i l l u s t r a t e maps
p Y = Y1 1
Y
A-
Y1.
b(X
A
X)
involved
obviously f a c t o r s through Now:
X by marking on t h e domains
I2 of
t h e values they assume a t various p o i n t s .
are constant along t h e marked l i n e s and a r e constantly t h e base p o i n t
along curled l i n e s .
representative
AX
+
y1
Thus
5,:
bx
as follows:
-+
,%(X
A
X)
is marked by i t s
76
The cohomology o f €I-spaces
S,(Xl
+ X2)
i s i l l u s t r a t e d by
77
A, ( 2 s )
A,( 2s-1)
One can e a s i l y see t h a t t h e images of all t h r e e maps
?X x
fix ->
EO(Al
+ A2)
lies i n
P 2 ( X x X) l i e i n
-
(L Add(S,(A,),
Y1.
The image of
EO(A2))
= w(A1.A2)
i l l u s t r a t e d below
Yo: A,
(4s-1)
A,( 49-31
The cohomology of H-spaces
u i s i l l u s t r a t e d by
One e s t a b l i s h e s t h e homotopy WT i t s e l f v i a deformations
f o r every
t
$i
$,,
t
t
$2
so t h a t
t. deform
N
I2 i n t h r e e s t a g e s :
UA
by deforming
I2 i n t o
H*(X,Z P )
and t h e coalgebra H*(ZW(, Z p )
(1) A deformation r e t r a c t i o n followed by a deformation:
The (deformation) r e t r a c t i o n :
The deformation:
79
The cohomology of H-spaces
80
( 2 ) A simple deformation
(3) A rotation:
H*(X,Z
P
)
and t h e coalgebra
81
H*( IGC, Z ) P
Our main r e s u l t i n t h i s s e c t i o n can now be s t a t e d : Let
and a r i n g s t r u c t u r e
{En,Yn)
3.2.3.
be a cohomology theory represented by an
E*
m.
yi E E ' ( X ) ,
(see 0.5).
'n ,m
Oiven a space
Proposition: 0 < ki,
0 < mi,
&spectrum
X
md cZasses xi E E ki ( X ) ,
k. + mi = n, 1
i = 1, 2 ,...,a.
a C xiyi = 0 then there e x i s t s z E En-2(nX) so that i=l a p*z = c a(oy Q oxi) where ;*: E*( s a ~ ) + E*( sp[ A sp[) is given by
If
-
i
i=l
a : E*(OX) Q E*(RX) + E*(S2X
Proof:
Let
- E*(p2),
- E*(pl)
E*(A)G* = E*(Add) A
as in (0.5).
m) m d o : E*(X) +E*-l(QX)
xi = [ f i ] E [X, E
ki
1,
yi = [gi] E [X, Em.
1.
C X ~ Y is ~
1
then represented by t h e composition:
a +k. ,m. i=l 1 1
,( E
)a
(Add)'
n
equivalent t o
i.e.:
Xo,po
-
fo = R A Y
f
*
-
hh.
= nX, Add, 1
g: *
Let
Xl,pl
= i2-1, FO,Fl
, En
--
hA.
= sl(X
- constant
A
Apply 3.2.1 f o r
X ) , Add,
homotopies,
X2,p2 = RE,,
Add,
6 induced by
and
82
5,
The cohomology of H-spaces
as i n 3.2.2.
Then, as was observed,
w =
by 3.2.2
W,
w0 = UT and
2
-Hd(v,Add,Add) = ( h ) w o = L1 huT One has t h e following commutative diagram
a
nr
Ian,
Hence,
a C a(oyi 0 oxi)
-HD(w,Add,Add) =
and
i=1
-[HD(w,Add,Add) = F*[w].
3.2.4.
Example
Put
-[w] = z .
(see [May and Zabrodsky] f o r another approach):
The Hopf algebra H*(Spin(n), Z 1: P One has a f i b r a t i o n Spin(n) apply t h e Eilenberg-Moore s . s . :
r
SO(n)
H*(SO(n), Z,)
k
K(Z2,1)
and one can
,a(m) Z2[xm]/(xm )
=
m < n m=odd
where
a ( m ) i s given by
H*(K(z,,~),
z,)
= z2[i11
,a(l) Z2"l
]
module.
-
m2a(m) > n > m2a(m)-l. and
H*(k,Z2)i1 = xl.
2a(l) ,411 z2[i11 w z2[i1 1 Q z2[ilI/(il )
as a
H*(X,Z ) P
E
2
= Em = H*(Spin(n),Z2)
and the coalgebra H*(QX,Z
83
P
as an algebra.
The co-algebra of H*(Spin(n),Z 2
will be determined by
is a map of Hopf algebras. Dually, one has an exact sequence of Hopf algebras 0
-+
)
A(;
-+
2a(m)-l
H,(Spin(n),Z2)
-+
im H,(r,Z2)
= Q A(;i) i
-+ 0.
If H , ( Spin( n) ,Z2) is commutative (and it is obviously associative 1, H*(Spin(n) ,Z2) is primitively generated.
As the lowest dimensional commutator of H,(Spin(n),Z2)
is primitive,
the lowest dimension where a commutator can occw is 2a(1)-l d i m < 2'(l)-1
Z2) + H,(SO(n),Z,)
H,(Spin(n),
is surjective) and it has to
involve a commutator of two generators, hence it involves i+j = 2a(l)-1. If n-1 = 2 i # 2k #
3, i+J
(as in
,
.
A
[xi,x,l
the relations i < n, J < n,
= 2a(1)-1, is impossible and H*(Spin(n),Z2)
is
primitively generated. If n-1 > 2
we shall show that
,
.
A
[xi,xJ1 =
24(1)4
#
0
for
84
The cohomology of H-spaces
To see t h a t , consider t h e f i b r a t i o n BSpin ( n )
Br
BSO(n)
Bk
___*
K(Z2,2),
H+(Bk,Z2)i2 = (Bk)*i2 = w2 E ker(Br)*.
Now, H*(BSO(n),Z,) = Z2 w2, w3,..
i s decomposable and i n
hence i n
keru.
ker(Br)*.
H*(Br,Z2)v 2a(l)+1
uBrYwj = xjml,
Br“wi = xi-1
.,wn],
.
n 5 2a(1)
By t h e well-known
= ZBr%
j
Wu
formulas, i f
.Br”wi + CBr”wi*Br*di=O. By 3.2.3
and one may assume
z
= x 2 a W l
3.3.
Browder’s Bockstein Spectral Sequence Given an
X
P
= X
PI”
%spectrum
: En + En
multiplication.
{En, Yn:
FJ
En
mn+l
1 let
be induced by t h e p-th power map with respect t o loop
Let
6 : En-l(Zp)
+ En
gets a Puppe sequence of f i b r a t i o n s
be the f i b e r of
- the
Z
P
X P ,n
.
Then one
c o e f f i c i e n t sequence:
This sequence gives r i s e t o t h e following n a t u r a l exact sequence:
(A (3.3.2)---+En(X)
I*
p*
En(X)
( A )*
6
En(X,Zp)
A
En+’(X)
A_--
85
Broxder's Bockstein s p e c t r a l sequence
As
i s multiplication by
(Ap)*
p
one obtains a "Universal
coefficients" exact sequence. 0 + En(X) Q Zp
(3.3.3)
p:
En(X,Zp)
+
Tor(E"+l(X) ,Zp)
+
0
as well as an exact couple
The s p e c t r a l sequence induced by (3.3.4) first studied by W. Browder i s c a l l e d t h e Bockstein s p e c t r a l sequence.
For simplicity assume t h a t suppose
En(X)
is f i n i t e l y generated (e.g.: sm(E) = 0
i s f i n i t e l y generated, i n p a r t i c u l a r ,
s,(E)
for
m > N(E)). The main properties o f t h e Bockstein s p e c t r a l sequence are given by
3.3.5.
Theorem: fa)
3.3.4
El = E*(X,Zp),
yields a spectral sequence B1 = B*: En(X,Zp)
+ En+l ( X , Z
so that
{Er, 6), P
given
by the e m c t sequence
- ,.
(3.3.5.1)--
-----)
En(X,Z 2 ) P
p*
En(X,Zp)
'*
induced by the top line fibration i n
E
n+l
(X,Zp)
-+
E
n+l (X,Z 2) P
86
The cohomology of H-spaces
(The derivations
B r are
caZZed t h e higher Bocksteim.)
( b ) E: = imI [En(X)/torsion] Q Zp FJ
Proof:
[En(X)/torsion] 8 Z P
-+
En(X,Zp)/p;(torsion
En(X) b ZP
.
(a) By the general theory of spectral. sequences (e.g.: [Hu],
p. 2 3 2 ) ET = E*(X,Z P )
and 8, = p,6,
= ( ~ 6 =) B,.~
As En(X) is a finitely generated abelian group n im(X ) + = # p torsion of En(X) n=l P m
ll
En(X). 6;'(#
E*(X,Z ) P
m
and
U
n=l
ker(X P ):
is a p-torsion group so by 3.3.2 and 3.3.3 one has
p torsion of En(X)) = ker 6, = im p , = im p k .
Hence ,
= p torsion of
Browder's Bockstein spectral sequence
87
and by
0
*I
[torsion E"(X)I Q zP
[torsion E"(X)I Q z P
-1
p:
1 - I
En(X) Q Zp
En(X, Zp)
PI
E~(X,Z
[E"(x)/torsion~ Q z P
1
0
torsion E"(x) Q z
P
0
E", =
is an injection and
[En(X)/torsion] 8 Z P'
Note that En(X,Zp) is a Z vector space only for p-odd. (For P p = 2 take E to be the 2-stage Postnikov system with Sq2 as a k-invariant and let X = M(Z ,n) the Moore space. Then E"(X) = Zq.) 2
However, the above spectral sequence holds for p = 2 and E, then a Z
2
is
vector space.
One obvious application of 3 . 3 . 5 is the following:
is p-torsion free i f and only i f E*(X,Z ) i8 a P and rank En(X,Z ) = rank En(X) = Z -vector space (redundant for p # 2) P P
3.3.6.
= rank
Corollary: E*(X)
(E"(x) Q Q) f o r every n.
Proof:
If E*(X)
is p-torsion free then by (3.2.3)
and for any p-torsion free En(X,Z ) w En(X) Q Zp w (E"(X)/torsion) Q Z P P abelian group G rank G = rank (G Q Zp).
aa
The cohomology of H-spaces
Conversely, if E*(X,Z ) P
is a
Zp
vector space with
r a n k E*(X,Z ) = r a n k E*(X) t h e n rank El = r a n k Em i n t h e B o c k s t e i n P s p e c t r a l sequence, E = Em, E*(X,Z ) w E*(X,Z ) / p i [ ( t o r s i o n E*(X;) 0 Zp], 1 P P [ t o r s i o n E*(X)] Q Z = 0 P
and
i s p - t o r s i o n free.
E*(X)
The f o l l o w i n g i s a g e o m e t r i c i n t e r p r e t a t i o n o f t h e B o c k s t e i n s p e c t r a l sequence:
3.3.7.
Let
Lemma:
&spectrum
{En,Yn: En
survives t o Er fi:
X
+
E (Z ) n P
Br{x1 E Er p,
and
8,
Proof:
X
-
be a space, E* a cohomoZogy theory given by an F3
mn+l~.
i n the Bockstein of
x
S.S.
Then x E E ~ ( x , z ~=) [ x , E ( z n
11
if and only if the representation
can be Zifted t o f r : X
-+ En(Z r ) ,
prfr
P
is then represented by
P
8rfr, :g,
En(Z r) P
-f
-
fl.
En+,(zp).
are given by t h e foZZowing diagram:
By t h e g e n e r a l t h e o r y o f s p e c t r a l sequences
Er = 6;l(im(A
) r-1 * )/p* ker(Ap)f-l
P hence,
to
fry
x E El
survives t o
Ar-l* f = 6f P r 1'
a r e t h e p u l l back o f
Er
if
6,x
r- 1 or, i f
E i m ( A )++
P
I n t h e f o l l o w i n g diagram En(Z r ) , 6 r P 6 f = Pr. 6 and Ar-'P, r r
6fl
and
lifts p,
Browder's Bockstein s p e c t r a l sequence
By d e f i n i t i o n o f
6,
Br{x}
i s represented by
I n t h e remainder of t h i s s e c t i o n we r e s t r i c t our study t o E
n
= K(Z,n).
I n t h i s case t h e Bockstein s p e c t r a l sequence f o r an H-space
i s a s p e c t r a l sequence of Hopf algebras.
B
Here
i s t h e ordinary
(primary) Bockstein operator and it i s w e l l known t h a t
B2 = 0,
B(xYY) = ( B d Y
+
B
is a derivation:
(-l)'x'xBY.
(Fr
N
Br
r,n), Z r) = Z be a generator. is the P P P f i b e r of t h e map K ( Z r , n + l ) +. K ( Z 2r, n + l ) induced by t h e i n c l u s i o n P P = prZ 2r c Z 2 r . ) Then, B1 = 8, prBr = Br and Fr i s a d e r i v a t i o n . "Pr P P Let
I E
Hn+'(K(Z
N
,
For convenience i n t h e sequel, suppose p
.
i s odd.
One has t h e
following:
3.3.8.
Proposition:
Let
2n x E Er (X) i n the Bockstein S.S. of a space
2
and
2-lgrx
X, r > 1. Then
Br+lIxpl = Ixl)-lS,x}.
survive t o Er+,(X)
and
The cohomology of H-spaces
90
By the n a t u r a l i t y of t h e s p e c t r a l sequence and by 3.3.7 s u f f i c e s t o
consider t h e case 2n
E H
2n
(K(Z
P
N
Consider t h e following:
‘2n’
‘iirTn= P
(
~
~ hence ~ t~h e diagram ~ ~ (excluding ~ ~ ) ?el) , N
i,,
e x i s t s t o satisfy BY 3.3.7,
= P
X = K ( Z r , a),x = {;2n} where P a), Zp) = Z i s the reduction of the fundamental class . P
9-1-
~
(
For
~
I
~
F
;
~ =~ F;Br (.;P-lN ~2n + Br~.1 2n~ 1. ~
3.3.9.
A
Such
~
*
N
B Ir ~ 2~n ) ~= {;gi18ri2nl. r = 1 the proof of 3.3.8 needs more care.
for
For t h i s and f o r t h e
(stated for
odd but has an
p
p = 2).
A fundamental lemma:
H * ( s , z ~ )=I .I,:~ ~ ~ fien
any
+
-P ; ? = Br+l{~2nl = P r P^ r+l‘iir+l*rr + l= pr-lprpr+l r+l r+l
next s e c t i o n w e need t h e following lemma analogue
i s commutative,
s
kt 5 : K(Zp,2n)
lifts to
n&~p H D ( i ) = HD(i,Add,Add) =
+ K(Z
P’
2np)
i:K ( ZP ,2n) -,K ( Z
j(V +
-V * d .
be given by
2, 2np)
P
and f o r
91
Browder’s Bockstein s p e c t r a l sequence
v =
where j: K ( Z Zp
+
Z P
2np)
P’
and
1p P k=l k
*;
k
E -(
+
p-k
12n Q 12n
is induced by the obvious injection
K ( Z 2, 2np) P
= H*(p,Z
P
)
- H*(p1 ,Z P ) - H*(p2’Zp).
Consequently, if K ( Z
-
2np+l) P’ $pn: K ( Z p , 2n+l)
fibration induced by X: K(Zp,2n)
m, nrx =1
Add(X
x
80
Qj): K ( Z
is an H-homotopy equivalence, as d o v e rmd p,[(x,y), Proof: -
(Z,7)]
E
K(Zp,
* K ( ZP’
2np+2)
2n+l)
is the
then there e x b t s
that
P
,2n) x K(Zp,2np),
v: K ( Z ,2n) P
pv 4QE,Add
K ( Z ’2x1) P
A
-
----)
K ( Z ,2np) P
= x-x, v(x,T)-y-F. 5
We first show t h a t no l i f t i n g
HD(i,Add,Add) = J(v + z*d + z )
of
5 i s an H-map by
is i n 8 I - t h e A(p) i d e a l i n H*(K(Z ,2n), Z Q H*(K(Z ,2n), Z ) generated B P P P P by all classes of t h e form B ~ Q I ~ and ~ @ Ba12n, a E A(p). showing t h a t
B
where
z
By 2.2 type argument one can see t h a t a l t e r i n g t h e l i f t i n g
dl E [K(Z ,2n), K ( Z ,2np)] H D ( e ) w i l l be a l t e r e d by P P and so one can prove t h e above formula f o r any p a r t i c u l a r
by an element
jT*%
i
choice of
$.
One has t h e following diagram:
of
5
The cohomology o f H-spaces
92
AS
as
BIP = 2n
o
t : K(Z
5 lifts t o
P
,2111
AS
p
factors
n
Y
*
K(Z,2np)
K ( Z 2 y 2np).
--*
K ( ZP ,%p)
one o b t a i n s
P
which need not b e commutative f o r an a r b i t r a r y A . .
plcp
= blp2(?2n)P
so that
p2(zn)
surjective M
dl = dlp
then
However,
dl: K(Z,2n) -+ K ( Z 2np) PY
implies t h e existence o f
= j?
( k e r H*(p,Z
and a l t e r i n g
i p = p 2 ( ~ 2 n ) ~ . AS
If
i.
N
p
+ tp. P
Now,
H*(K(Z
)
i s t h e A(p)
i
by
and p 2
j4
P
,2111, Zp)
H*(K(Z,2n), Z P
i d e a l generated by
is
f 3 1 ~ ~ ) hence ,
i f necessary one may assume t h a t
are ~ - m p s
i = 1,2
pi: K(Z,2n) x K(Z,2n) + K(Z,2n)
H*(Add,Z )y = P 2n
-+
+ H*(p
Z
2' p
)y2n --
are the projections N
Pf12n
N
+
P212n'
Browder's Bockstein spectral sequence
93
Using the commutativity of
K(ZY2np)
K(Zpy2np)
I
I
HD(~)(~A
p) =
P~HD('?&)
= j p Y o = jv(p
Now, the fact that 6 HD(e)p
p = j&(p
A
A
-
- pvo
p)
K(Zpy 2np-1)
- pvo
is an H-map implies that HD(g) = j j ,
has
A
p)
= 8; 1(p
K(Z
PY
A
A
p ) = p v o = v(p A p ) ,
i = v+
i
-
impossibility of -v = U*d + z (;*)":
-
= H*(K(Z
P
6
and altering
p)
To show that for any choice of
Let
N
p v o ] = 0 implies
-
H*(K(Zp,a)
3
2
8'
by
5i1
z
E 1 8
8
A
p,
zP)
if necessary one 3
ker H*(p
A p,
Zp).
HD(i) # 0 one has to show the
0'
zp)
+
g(K(Zp$a)Y
zp)lm
,2n), Z ) 4 F*(K(Zp,2n), 2 ) 8 . . . 0 H*(K(Zp,2n),Zp) P P m
-
-
(r*)m= [(T*)m-l1]r*. - m (x x ... (L*Imbe the dual of (r*)m,then (u,)
(r*)mgiven by Let
-
2np) and by the surjectivity of H*(p
N
6(p
p)
A
factors through fiber j which is
N
&(p
Again, j [ w ( p
&To.
p) =
N
that j ( p
8:
A
A p)
(;*I2
=
u*,
8
8
8 x) = xm.
m
One can see that
( ( ; * ) ' I
((;*)'-'
8 1)[E*(K(Zp,2n))
8 l}(-v) = @
T*(K(Zp,2n))]
0
O...@ I
2n
and as
in dim 2np is contained in
94
The cohomology o f H-spaces
[H2n(K(Zp,2n),Zp)lP, ((p)P-l 9
B 1](I ) = 0
[(;*Ip-'
B
i n t h i s dimension,
= 0.
1)ZB
-
If -v = Ll*d + z
B
y E H2n(K(Z ,2n),Z ) = Z be a dual of P P P
Let
1 =
1 2n
0..
.o 1 a' y B y
1
...0 Jh = <(Ll*)d,
< I ~ ~ , J I=> 1.
then,
2n
y @ . . A Jh =
3 = 0.
contradicting the fact that
i
To prove t h e first p a r t s u f f i c e s t o show t h a t f o r some
HD(g) = jv.
We s h a l l prove t h i s together with t h e second p a r t . One has t h e following p u l l back diagrams of
-
Choose a r b i t r a r y
i : K(Z
P
,2n)
v
+ F*d +
zB
lies in
I
Add(i x
Qj):
B
-
K ( Z 2, 2np). P
i.
HD(i) = G v l
i: K ( ZP ,a)
C1E, O r ; = 1, h i =
H D ( ~ ) = O ~=V j ~[ v +
- v1
P a+
z
B
I.
AS
f a c t o r s through
and one may assume
--loop
j[v +
;*a +
zB
6: K ( Z p a 2np-1)
spaces and maps.
It y i e l d s
and
-
-v
~ =]
o
K(Zp,
2np),
hence
v1 = v + ii*d + ZB.
K(Zp,2n) x K ( Z p , 2np)
---+
nE i s a homotopy equivalence and
95
Browder’s Bockstein s p e c t r a l sequence
if
uv
u
i s given by 1
~ d d ( xi
-
(x,:)]
[(x,y). vl
= x-x,
v , ( x , x ) - y - ~ then
As
n j ) i s a pv a Add H-map hence an H-homotopy equivalence. 1
nE, Add i s HA and HC s o i s
and
pv
1
H*(K(Z
P
,2n)
K(Zpa2np),Zp), pv
x
i s co-commutative and co-associative but as t h e generator
10
1
I
2np cannot be represented by a primitive element it i s not primitively
generated i n
dim
( y 4 1)’ # 0
in
u with
2np.
H
2np
VIY
t h e coproduct of by
(ax x
( K ( Z ,2n) 4 K ( Z ,2np),Z ) P P P
I
18
1
c
2np
#
u;Add
c = 1, and one may assume
H
Let
Proposition:
The f a c t t h a t r > 1 (3.3.8).
i n the case
x =
X = K(Zp,2n),
i: K ( ZP ,2n)
--+
then
i2j =
5,
5v =
L*(Ip-’ 2n
-
1
2
k 1) C3
x
8 l)P-k,
but i n v i e w of
4
d =
HD(X) = n j v ,
HD(i) = j v
r = 1
(p-odd)
8.8.
and
and
.
p-odd.
Then
and xP-l5x
xp
B,rxPj = {xP-15x1.
xP-lgx
survive t o
E2
i s proved as
Again, it s u f f i c e s t o prove 3.3.10 f o r
-
By 3.3.9 one has a l i f t i n g
2n‘
K ( Z 2 y 2np) P
of
6: K ( Z 2 )
j : K ( Z ,2np)
3
K ( Z 2 y 2 n p ) . One can see t h a t P
P
812n).
zB = 0.
homotopy equivalence.
x E H2n(X,Zp),
survive t o E2 i n the Bockstein
Proof:
€3
dl E H2np(K(Zpa2n)rZp)
0,
w i l l then y i e l d a l i f t i n g
d
is a
j)
),
2np
Now one can prove 3.3.8 f o r
3.3.10.
and t h e r e exists a c l a s s
P-1 1 P Z )u= u 0 1 + 1 4 u + Z ( P k=l P k
u = dl 8 1 + c ( 1 B
and
# y E H2n(K(Zpa2n),Zp)
0
-
H*(P
Altering
Consequently, i f
PY
It follows t h a t
K(Zp,2np)
so that
B2i~2np+l = Ipd;1512n +
x,
The cohomology of H-spaces
96
x E PH2nP+1(K(Zp,2n), Zp). completed i f one shows t h a t PHodd(K(Zp, AS
G 2=
{(i?2t)*12np+l 1 = {B21Zn} t h e proof w i l l be
As
x E i m B.
a),Zp)
But on
= ker B
im
~ ( l E i ~= ~ (p-l)iE:*(~i 1 ~ ~ )2n l 2
0,
= 0,
~x =
o
x E i m 6.
and
3.3.8 and 3.3.10 c o n s t i t u t e Theorem 5.4 of [ B r o ~ d e r f] o~r p odd. The main r e s u l t of [ B r o ~ d e r i] s~ then given as follows:
3.3.11.
Definition:
l-impzication of or
#
A
A x E A2". zP s o t h a t e i t h e r "1 = xp
x1 E A2np
i s an element
f o r every
0
be a Hopf algebra over
x E A&
the dual of
(A*
with
A)
0.
m implication of
x = xoy xl,.
3.3.12.
x
Let
.. ,xm
so that
15 m
x E A2ny
5m
i s a sequence
i s a 1-implication of
xi+l
x
i'
Browder's f i r s t implication theorem (6.1 i n [ B r ~ w d e r ] ~ ) :
Let
X,p
be
~n
homology Bockstein
H-space,
8.8.
I f
Ery
Br t h e Bockstein x = B ry
x E PEgn,
s.s,
Er
i t s dual
then x has m
impzication. Outline of proof:
... E Er
xo = x, xl, x2,...,xm, xi
sequence
xi E PE 2np
r
i-1
the
i
,
ro = r,
by a 1-implication of
xi's
to
with
xi
a 1-implication of
i s constructed inductively as follows.
sequence
r
One has t o construct a sequence
/-'= Er
x
0
i-1
5 ri and
- ri-l x
i
i-1'
i s represented i n
One can p u l l b a c k
t o obtain the desired implication sequence.
t h i s inductive argument it s u f f i c e s t o construct
x
1
The
One obtains a
5 1, xi
r = B iyi.
x.
out of
x = x
0'
By
97
Browder's Bockstein s p e c t r a l sequence
If
2=
XP = + + o
<X,]L> #
and suppose
0
x1 =
put
0
y'-$
prove now t h a t
B
3.2.8 and 3.2.10
<{a , Br+l{XP-ly}>
{?I
XP =
r1 = ro,
-x E .:E
Br(xP-'y)
# 0, hence, =
r+l
2,
{p-' - Bry) #
?-$r
computing the
consequently
x1 =
-
x,
in
z,x># 0 B
r
.
implies t h a t
x1 = B
r+l yl,
5
+o
so suppose
= xp = 0.
One can
-x # i m 8 , #
0 and
=
r+l {P-ly}
~,(Xp-'y).
0.
{xP-ly} # 0 that
and by
and and
{XP-ly)
<{;;PI, xi> # o f o r
are primitive.
can be represented by a 1-implieation
y1 = {XP-ly>.
This method y i e l d s f u r t h e r r e s u l t s , a sample of which we present here
( f o r a more complete l i s t and references, see [ B r ~ w d e r ] ~ ) .
3.3.13.
Browder's implication theorems:
Bockstein s. s and
3.3.13.1.
If
X
x E P ( c ) Brx # 0 then x has an
P
m
,Z)
m
implication.
Hence
0.
If X is HC and p = 2 and if x E P ( E F ) then x has an
implication.
If
is f i n i t e then PE?
Hence, if H*(X,Z2)
particular PH*(X,Z2)
3.3.13.3.
be t h e H*(
an H-space.
if H*(X,Z ) is f i n i t e then BrP(EF) = 3.3.13.2.
Let {En,Bn)
X
Hence, if H*(X,Z
cH
odd
(X,Z2)
and
H*(X,Z)
)
is f i n i t e
H*(X,Z)
in
has no 2-torsions.
is HC and HA then x E P E T has an P
= 0,
has no p-torsion.
m
implication.
98
The cohomolojg of H-spaces
3.4.
High order operations. The systematic study of high order operations s t a r t e d e s s e n t i a l l y i n
I n general, l e t
[AdamsIl.
be abelian groups.
G,G'
cohomoZogy operation i s a t r i p l e a = (x,E,y)
u i s i n t h e domain of The value
Given a space
y E H*(E,G').
x E H*(E,G),
a(u)
a
i f there exists
K
where
A (high order) E
i s a space,
and a c l a s s
f: K
-+
E
-+
E,
G-G'
with
u E H*(K,G), H*(f,G)x = u.
i s then defined as t h e set
a(u)
= {H*(f,G')y E H * ( K , G ' ) I f :
K
H*(f,G)x = u).
-
One can e a s i l y replace x by a f i n i t e sequence x = xly...,x m' n. nJ xi E H '(E,G), y = yl,.. .,yn, y E H (E,G') and analogously define t h e
-
J
domain of
a
and i t s values.
arbitrarily.
Suppose
operation as
(x,E,y)
I n practice
A EJ 2
f j : EJ-1
-+
KJ-1'
K
j
are not chosen
-
G = G' = Z One speaks of a Z Z k-order P P P where E i s a generalized k-stage Postnikov system:
1 mj
x,E,y
= K(G,)
P
is the f i b r a t i o n induced by where
G,
i s a graded
Z P
vector space.
99
High order operations
x
i s then chosen t o be t h e image of t h e vector of fundamental classes i n
= H*(K
H*(E1,Zp) all i n
i m H*(r
Z ) 0' P
and y
i s assumed t o be a vector of c l a s s e s not
Z 1. k' P
If all spaces and maps are
-
loop spaces and maps and y
i s an
t h e operation i s then c a l l e d s t a b l e .
-suspension,
I n t h i s s e c t i o n we i l l u s t r a t e a technique used i n t h e study of t h e
.
cohomology of H-spaces involving a secondary operation (i. e : an operation of order two).
We s h a l l give only the most elementary version.
A t the
end of t h i s s e c t i o n w e Bhall l i s t some of t h e stronger r e s u l t s obtained using t h i s method, as well as r e s u l t s concerning t h e cohomology of H-spaces obtained by using operations i n generalized cohomology theories. We s h a l l study only one secondary operation. Let
K(Z
induced by
P'
5 : K(Zp,2n)
x = H*(rl,Zp)i2n.
Proof:
hp-1)
-
Note t h a t
*
El
r
1 K(Zp,2n)
K(Zp,2np) H * ( 5 , Z p ) ~ 2 n p El
i s an
m
=
be t h e f i b r a t i o n Put
loop space.
Consider t h e s t a b l e diagram of induced f i b r a t i o n s :
The cohomology o f H-spaces
100
For m = 2n
one gets a diagram of
loop spaces and maps
lr
K ( Z p y 2n)
By 3.3.9
multiplication
QE
K ( Z p y 2n)
K(Zp,2n)
w
x
K(Zpy2np) ( t h e l a t t e r with a twisted
u,).
I n t h e notations of 3.3.9 one has an H-map g: E1,Add
K ( Z ,%) P t h e desired c l a s s . --*
Note t h a t
y
x
K(Zp,2np),
uV
and
y = H*(g,Zp)(l B
I
%P
)
is
i s not uniquely determined by t h e p r o p e r t i e s of 3.4.1.
It can be a l t e r e d by any element i n
H*(r
Z )[PH*(K(Z ,2n), Z p ) l . 1’ P P
101
High order operations
s a t i s f y i n g t h e 3.4.1 properties and put
Choose an a r b i t r a r y y = ( x , El, y ) .
ci
X,
We would l i k e t o show t h a t under c e r t a i n conditions, i f
u E PH2n ( X , Z p ) ,
i s an H-space,
X,p
Given an H-space
!Jw[(X,Y), where
( ).(
)
and w: X
X x K ( Z ,m) P
multiplication on
then
a(u)
consists of
(hence 0 @ a ( u )1.
u
1-implications of
up = 0
X
A
d K ( Z ,m) l e t
pw
P
be a
given by
(X,Y)1 = !J ( X , X ) ,
W(X,X)*Y.Y
stands f o r t h e abelian group multiplication i n
One can see t h a t any multiplication on
X
x
K ( Z ,m) P
K(Zp,m).
s o t h a t both
K ( Z ,m) + X and i2: K(Zp,m) X x K ( Z ,m) a r e H-maps must be P P of t h i s form. Indeed, i f such i s given, p1p = ppl implies
pl:
X
x
-+
N
For dimension reasons N
SI*(p2!J, Z p ) I m
=
Cl(l
+ c
2;
B lm B 1 0 1) 0 1 0
2;
+ c*(1 0
1 B 1 Q Im)
+
0 1
i As
i
2
i s an H-map,
c1 = c
P,~[(x,Y), satisfies N
p((*,*), N
!J
= lJw.
H*(w,Zp)im ( x , + ) ) = x,*,
2
= 1 and
(X,~)I = %xyY)-y.7;
= Z z ! 0 z;. i 1
Finally,
j:
x
x
x
+ K ( Z ,m) P
N
p ( ( x , + ) , (*,*))
consequently G(x,*) =
*
= W(*,x),
W
= x,* =
WA
and and
102
The cohomology of H-spaces
For any map u: X ? K ( Z ,m) P hu(x,y) = x,u(x).y.
given by
let
hU: X
K ( Z ,m) P
x
-+
X x K ( Z ,m) P
hU a r e homeomorphisms h -1 = hi;'
Then
U
and any s e l f homotopy equivalence hi2
-
h
i2 i s homotopic t o one of t h e
X x K ( Z ,m) P
of
h
isan
i s an
i
i2
2
-
i s an
i2
-
i2
p1
-
p1
-p
- p1 - u
multiplication.
essentially different
i2
- p1
-u
-
p1
equivalences and
m u l t i p l i c a t i o n and
equivalence with homotopy inverse
p1
plh
u
N
p
with
h 's.
W e refer t o such homotopy equivalences a s
it can be e a s i l y seeh t h a t i f
be
g
then
hz(g x g)
Therefore, one can ttilk about multiplications.
One has t h e
following lemma:
3.4.2.
Lemma:
The set of essentially d i f f e r e n t i2 - p1 X x K ( Z ,m) is i n
multiplications for H ~ ( XA X,
P
zP ) / i m J;I.
Proof: -
Let
w: X
A
X
-
-p
1-1 correspondence with
K ( Z ,m), P
u: X
-+
K(Z ,m). P
Consider
As H*(HD(u,u,m) , Z ) =
F*(u)
i t follows t h a t
P
and one has a s u r j e c t i o n of
p ( X
A
X, Z ) / i m
P
*;
huuw(h -1) = P u u F*(U)+W onto t h e s e t of
High o r d e r operations
essentially different
3.4.3.
u: X
Lemma:
structure where
*;:
induced by
pw
- p1
i2
y i e l d equivalent f o r some
i2
+
K ( Z .m)
P
kt
X,p
-
on X
E*(x,z~)
x
- p1
-p
- 1-1
multiplications,
and w
be
multiplications.
-
1
= p*u
HA.
w: X
K(Zp,m)
F*(x,zP
103
pw
1
A
X
-+
K(Z ,m)
Q
F*(x,zP
a r e homotopic o r e q u i v a l e n t l y i f
I n functional notations:
(h
-1 o u
yiekii3 an = (1 Q
;T")w
is the reduced coproduct
pw(pw x 1) if and only if t h e two maps
are homotopic.
P
if and m ~ if y (;i, 8 l)w
p.
-
- h upw
+ w0'
Proof:
p w ( l x pw)
Conversely, i f
w0,w1 -1)
u
104
The cohomology of H-spaces
and
(14 -;.)w
3.4.4.
of
X
=
(T* Q
m e essentiaZly different
Corollary: x
1)w.
K ( Z p y m ) , where
i2
-
p1
-
HA structures
p
is HA, are i n 1-1 correspondence with
X,p
H*(X,Z Cotor
Proof:
elements of
2, a-2
(ZP'ZP)
Such H-structures a r e represented i n view of 3.4.3 by in
dim m
and by 4.3.2 any two such s t r u c t u r e s a r e equivalent i f and only i f they d i f f e r by an element i n the essentially different in
-
u* Q
im
T*:
i2
1-1 correspondence with 1
-
1 0 *;
??*(X,Z )
F*(X,Z
-u
HA s t r u c t u r e s on
P
-
p1
ker(r* 8 1
-
1Q
P
) 0 F*(X,Zp). Hence,
L*)/im
-
X
K(Zpym)
x
IJ*. But
-
p*
are t h e f i r s t two derivations i n t h e co-bar H*(X,Z
construction determining
3.4.5.
Proposition:
H*(X,Z P
Cotor
Let p be an odd prime,
cocomtative.
rf
consists of 1-impZicatiuns o f
Proof:
(ZP'ZP).
2m
x E PH
X,p
(x,zP), 2
=
an HA space with
o then
x.
Consider t h e diagram of H-spaces and H-maps:
a(x)
are and
High order operations
X: X -+ El
105
an arbitrary lifting of x.
Now 5
as an
e(6x) = n(c)e(x)
(m)
loop map is an HA map.
+ e ( c ) ( xA x
A
x).
By 2.5.2
e(5) = o
AS
and
n(5)
-
w
e(o)
=
2mp-1) with the (2.1.1) induced multiplication is an HA P' space. This multiplication being an i2 - p1 p multiplication is of and
X
x
K(Z
-
the form pw,
w
classified by
H*(X,Z Cotor (ZP'ZP) 2, 2mp-3
As H*(X,Z ) P
is co-associative (X,p being HA) and cocommutative
PH*(X,Z 1 o PH*(X,Z 1 P
P
-
H*(X,Z Cotor P 2,*
(z,z)
-
dimensions, hence, w = Cu! O u; + U*d, follows that HD(X) = jlw
-
p*H* X,Zp)y
P P
is an isomorphism in odd
ui, uy E PH*(X,Z ) .
P
It easily
and therefore
-
[H*(X,Z ) 8 H*(X,Z )](Add)*y P P
= Bw.
o
The cohomology of H-spaces
106
Let
zE H
2n
(X,Z ) P
be any element. 2%
= <(;*)'
Then,
H*(X ,ZP)y,
z e
Z Q . . . ~ P
? P
3.4.5 now follows from t h e f a c t t h a t a ( x ) = {H*(x,Z )yIX: X P
3.4.6.
Example:
I n [Harper]
For dimension reasons
dim 2p
2
+
2p.
x
3
i
an H-space
X
-+ E1,rlX
=
XI.
i s constructed so t h a t
are primitive and t h e r e i s no generator i n
3.4.5 implies t h a t
X
cannot be HA.
Another example of an application of t h i s technique i s t h e determination of t h e decomposability o f t h e Steenrod operation
3.4.7.
Proposition:
+1,
80
$2
There exist z
P
t hat
F2t11
+
B$2
-- 9
- zP
p,
p-odd.
st u b t e secondary operations
107
High order operations
Proof: c
CB = 0 = BA.
fA: K(Zp,m)
-
fc: K ( Z p ,
N
it,
m+4p-3)
fCfB
N
m
Let
K(Z
P'
fB: K ( Z p , m + l ) x K ( Z
Then: fBfA
A(p):
(Pp-2' 6)
=
Then
Consider t h e following matrices with e n t r i e s i n
x
1.
Pa
be l a r g e and
m+l)
x
K ( Z p . m + 2p
-
2)
m+2p-2) -+ K ( Z p , e 4 p - 3 ) x K ( Z p y m+2p(p-l))
K ( Z p y m+2p(p-l))
-+
K ( Z p , m+2p(p-l) + 1)
Consider t h e following:
The cohomology of H-spaces
108
f
C
Define :
If
x
E H m ( X y Z ) i s i n domain
PP-'$,(x)
As two l i f t i n g s RscgBAj = Q f c R f B element
P
+
x1
$,
n
domain $,
then
B$2(x) = { H * ( R f C ~ A XZpy11 m+2P(P-l)
-*
and
of
X
t h e set
x
w i l l satis*
Pp-2+,(x)
x1
IX:
X+ E
= jv + X
1'
rx =
XI.
and as
+ B$2(x) c o n s i s t s of the s i n g l e
H*( gcBGAxyZ ) I P e2P(P-l)*
H* ( g m ' ~3 ~'p ) 'm+2p ( p-i Z -generated by P
P p t m y hence,
E f12P(P-1)(K(Zp,m),
H*(g~BXB~' p ' lm+2p (p-1)
The proof w i l l be completed i f one shows t h a t s u f f i c e s t o determine
$i
ZP )
which i s = *PIm,
c = 1. For t h i s it
on any one c l a s s of any chosen space.
c E
z
P
.
High order operations
Fix an a r b i t r a r y fundamental class.
n
Then
and l e t iZn
I
2n E domain
E H2n(K(Zp32n), Z ) be t h e
0, n
domain $2.
P W e evaluate
Consider the diagram
P$"
AS
E
E.
1;
:I
= (n+l)pn+',
pn+l 1 2n+l
= 0
and t h e diagram i s commutative.
Looping t h e above diagram and applying 3.3.9 one obtains a diagram of H-spaces and maps:
J. K(Zp,2np)
J. K(Zp92n)
g
i s an H-map with respect t o a multiplication
pV
on
K(Zp'2n) x K(Zp92np) and t h e loop multiplication on uv[(x,y), v =
C
k=l
(&?)I
-
= x'x, v ( x , x ) - y - y ,
-1 ( pk ) ~k 2 n0
igk.Put
v: K(Zp,2n)
X = gil:
A
K(Zp,2n)
K(Zp,2n) +El.
-f
K(Zps2np),
The cohomology of H-spaces
110
HD(il,Add,pv) K(Zp,2n),
= -i v, 2
p 1 = Add
HD(X,Add,Add) = -jlilv.
hence on
El. Then, as yi
-
S i m i l a r l y , -p*H*(X,Z
P
)y
2
= H*(QfBilv,
Let
p = Add
on
are p r i m i t i v e
Z )(1 Q 1
P
2np+2p(p-l)-l
1
= 0.
Now one checks t h a t
It follows t h a t t h e r e e x i s t primitives that
H*(X,Zp)Y1
To e v a l u a t e
= -do
+
z1 and
"1 E
$l(lL)
z2 consider
zl, z 2 E H * ( K ( Z p , a ) ~ Z
P
so
High o r d e r operations
K(Zp,
2,:
AS
H*(K(Z 211-1)’Z ) P’ P
that
u z l = nj,(zP 2uGl +
tssume uz
i
X
by
1
(-2p t3
u3. - uLi
,2n), Z
+
di
P
+ flP1)uc2)
jl(ilx
i f necessary one may
^U2)AK(zp,2n)
P
.n dim zi one obtains
K ( ZP’ 2np+2p-4)
a r e indecomposable primitives it follows
uzi
As [ k e r u : PH*(K(Z ,en), Z )
= 0.
x
P
decomposable and as
md i f one a l t e r s
K ( ZP’ 2np-1)
----)
i s generated by u H * ( K ( Z
-
di
2n-1)
111
P hence,
z = z2 = 0, 1
1
2n
2n
2n
-t
PH*(K(Z 2n-1), P’
2n
Zp)] = 0
2n
To evaluate t h e l a t t e r one uses t h e f a c t t h a t it equals c
one can make t h e c a l c u l a t i o n s
)dulo t h e i d e a l generated by P ’ I ~ ~ ,i > 1, and t h e r e
1-7
,P-2(P11 2n 2n 12) =
and
(P?2n)P
r The decomposability o f
Pp
c =
1.
r > 1 i s more complex and proofs can be
und i n [Liulevicius I.
Using t h e b a s i c ideas of 3 . 4 . 5 a more r e f i n e d b u t complex technique s l d s s t r o n g e r r e s u l t s ( s e e [Zabrodsky]
192,496
and [Lin]
132
1.
Following
112
The cohomology of H-spaces
i s a sample of t h e s e results:
3.4.8.
Proposition:
If
3.4.8.1.
bH*(X,Zp)
with ker 0 c i m
If
3.4.8.2.
X,p
Let
x , ~ be an H-space. p-odd then there e d s t s a homomorphism
= 0
9.( [ZabrodsQ16) is HC and H*(X,Z2)
dimensional generators then Sq4n: PH4n+1(X,Z2) monornorphism.
-
is an e x t e r i o r algebra on odd is a
PH8n+1(X,Z2)
( [Zabrodsky14)
I n conjunction with t h e discoveries of p i l e s of non-classical f i n i t e CW
H-spaces t h e following was proved using similar methods:
3.4.9. n
Proposition ([Zabrodsky 15):
# 2 j -1 s a t i s f y Hn(f,Z2) # 0 .
and X ( f , X )
be the pullback of
be an H-space,
Let X Let
hi: Sn
-+
-t
S"
be a map of degree A
S"
and hA. I f
f
f: X
A
is even then X ( f , h )
does not a h i t an H-structure. Most r e c e n t l y , a long-standing conjecture was resolved for p-odd.
3.4.10. p-odd.
Proposition ( [ L i n )
Then H*(QX,Z)
1:
Let
be an H-space.
X,p
H*(X,Z
P
)
finite,
i s p-torsion f r e e .
Using K-theory operations, J . Hubbuck proved many cohomology s t r u c t u r e theorems f o r H-spaces.
(See [ H ~ b b u c k ] ~ - ~ One ) of t h e most
s t r i k i n g of h i s discoveries i n t h i s context i s :
3.4.11. Proposition ([Hubbuckh):
Let
X
Then X is homotopy equivalent t o a torus
be f i n i t e dimensional HC space. T" = ( S l ) " .
113
Chapter I V
Mod p theory of H-spaces
Introduction The source of any type of mod p theory i n a l g e b r a i c topology i s S e r r e ' s study of c l a s s e s of abelian groups, where an e s s e n t i a l meaning was given t o maps inducing isomorphisms modulo a given c l a s s of a b e l i a n groups (e.g. :
t h e c l a s s of
#
p-torsion groups).
The a n a l y s i s of S e r r e ' s observations branches i n t o two r e l a t e d b u t d i f f e r e n t methods: [AdamsI2 i g n i t e d t h e i d e a s l e a d i n g t o t h e l o c a l i z a t i o n theory formulated by Sullivan and developed t o become a s u b f i e l d i n a l g e b r a i c topology.
(We s h a l l use [Hilton, Mislin, Roitberg] as a
standard reference t o l o c a l i z a t i o n t h e o r y ) . The o t h e r method i n mod p theory i s t h e one c l o s e r t o t h e o r i g i n a l work of S e r r e : [Mimura, Oneill,Toda] and [Mimura, Toda]
1,2
developed t h e
theory of p-equivalences and p-universal spaces. The main d i f f e r e n c e i n t h e two t h e o r i e s i s t h a t t h e f i r s t c a r r i e s us o u t s i d e t h e category of (homotopy equivalents o f ) CW complexes of f i n i t e type which we t r y t o avoid, on t h e o t h e r hand one has t o concede t h a t t h e mod p equivalence theory i s e f f e c t i v e only f o r a l i m i t e d family of spaces ( t h o s e c a l l e d p-universal) while l o c a l i z a t i o n theory i s u n i v e r s a l l y effective
.
Fortunately, H-spaces a r e p-universal f o r every
p
and f o r t h e m mod p
equivalence theory i s equivalent t o l o c a l i z a t i o n theory and i t i s t h e f i r s t one t h a t w e use here ( t h e only exception i s t h e proof of 4.2.5 where we
refer t o a l o c a l i z a t i o n theory proof i n order t o avoid t h e unnecessary
Mod p
114
theory of H-spaces
development of i t s p-equivalence
equivalent).
This chapter contains a survey of t h e b a s i c f a c t s i n t h e [Mimura, Oneill, Toda] theory ( s e c t i o n 1) as w e l l as t h e notions of mod p homotopy due t o Rector ( s e c t i o n 2 ) . The decomposition of
0
equivalences of s e c t i o n 3 and t h e mixing
technique of s e c t i o n 7 follow [Zabrodsky] t h e non c l a s s i c a l H-spaces i n s e c t i o n
397
and l e a d t o t h e study of
8.
Mod p H-spaces a r e studied i n section 5 with t h e main results being 4.5.2 and 4.5.3.
The first w a s o f t e n assumed t o be t r u e though no
e x p l i c i t proof w a s produced as far as we know.
4.5.3 closes completely
t h e uneasy gap i n mixing H-spaces with nonconsistent r a t i o n a l Hopf algebras (and see remark 4.5.3.1). I n s e c t i o n 6 t h e notion of t h e genus of an H-space i s s t u d i e d and as an i l l u s t r a t i o n t h e genus of
i s given a lower bound.
p-equivalence and p-universal spaces
4.1.
Let
4.1.1.
P denote the set of primes. Definition:
p E P U (0).
Let
p-equivalence if H * ( f , Z p ) , H,(f,Z) for
SJ(n)
Q Zp
, r(f) Q
Zp,
f: X
(equivalently
n,(f,Z
P
*
Y
i s said t o be a
H,(f,Zp),
H*(f,Z)
) ) a r e isomorphisms.
Here
Q Z
Z P
P’ = Z/pZ
p E P, Zo = 8. f
i s said t o be a
p-equivalence f o r every Note t h a t nm(fiber f )
f: X
*Y
P1 equivalence,
P1 c B,
if
f
is a
p E P1.
i s a p-equivalence,
p E P
a r e all t o r s i o n groups (of order prime t o
i f and only i f
p)
and hence
p-equivalence and p-universal spaces
every
p
4.1.2.
equivalence i s a
Definition:
P1 c P U {O}, f = f
q.nO
0-equivalence.
A space
where
P1 Universal,
i s s a i d t o be
X
P1 and n 0 > 0 t h e r e exists i s a P1 equivalence and H n ( f , Z ) = 0 f o r
i f f o r every
: X + X
115
q
f
9
0 < n zno.
4.1.3.
Definition:
A space
i s s a i d t o satisfy a finitenass condition
X
if (f.c.):
Hn(X,Z) = 0
(f-c.H)
n > N(X)
for
The following only s l i g h t l y d i f f e r fromtheorems 2.1 and 3.2 of
[Mimura, Oneill, Toda].
4.1.4.
Proposition:
(a) X
For a space
the f o l l m i n g are equivalent:
X
is P1 universal
( b ) For every P1-equivalence satisfying
(f.c.n)
f: X’
(f.c.n)
X
with f i b e r f
there e x i s t s a P1
( c ) There e x i s t s a P1 equivalence satisfying
-+
and X o
f: X
-+
equivalence X
0
Y,P, P1-equivalence
f i b e r satisfying
and a map
P1 equivalence H
hf
- %I.
f: X
-+
X
n0,q
:
x-+x
so that
X I .
with f i b e r f
3:X
and a map h: X
N
f: Y -+
-+
-+
with
y, there etcist a Y
so that
(el For every integer no and a prime q # P1 there e d s t s f = f
-+
is P1 universal.
( d ) For arbitrary spaces
( f .c . ~ )
g: X
an(f,z ) = 9
o for o
< n :no.
Mod p
116
If)
theory of H-spaces
For every P1 equivalence
f: X +
XI
P1 equivalence
(f.c.H) there e x i s t s a
( g ) There e x i s t s a P1 equivalence f : Xo satisfy i n g
is P
and Xo
f.c.H
1
N
( h ) For any P1 equivalence
f: Y
( f . c. H) and any map h : Y N
and a map h:
f: X + X
Proof:
(a)
I ,
(b).
with cofiber s a t i s f y i n g
.+
-+
XI
X
with cofiber
-+
universal. with cofiber s a t i s f y i n g
there e x i s t a P1 equivalence
X
P .+ X
P
-+
X.
g:
-
so that
fh.
The proof c o n s i s t s of t h e following s t e p s :
Decompose
f:
XI
-+
X
i n t o i t s p r i m i t i v e Postnikov
decomposition (which i s f i n i t e as involves o n l y primes
X' =
qi
xm
Consider:
I
km
t fm
p1
as ~
xm-l
satisfies
fiber f f
i s a P1
- ..
.
(f,.c.n)
equivalence):
and
117
p-equivalence and p-universal spaces
fn f
-
t h e f i b r a t i o n induced by : X
Sll'
-+
Jn j
.
n
If
then as kngn-l?
^f=?
Sl.J1
^r
i s a B1
i < n
gi
N
R 'Jn
fl(f,Z
9
g: X
Bel
a
-+
m
f
... f
S21J2
fng,
equivalence,
N
) = 0
for
flf 2 . . . f .ig i
N
(lidi
gn-lf
fg*-
f: X'
Let
m 5 no.
= gf
9,n0
0 < mLn
for
R
f
gm = g
and
X'.
H ( ,Z ) 9
X' then
) = 0
f
P1 equivalence and f i b e r f
is a
W P
91Jl
exists so that
*,gn
. . . 2G J m i s
f
?
e x i s t with
s2,J2
equivalence
Then
x
,Z
R'Jn'
y i e l d a s u r j e c t i o n of j.
jn-l),
Qn'
R'jn
mL
0 <
g:
f i b e r K( Z
with
P1 equivalence with ?(?
is a
X
kn
0'
X
satisfy
be t h e f i b e r of (f.c.n).
If
P1 equivalence and as
is a
Hm(f 9.n0
+
,Z
P
= 0
O < m c n
0'
(a) * ( c ) t r i v i a l . (c)
r,
Hm(?q
g:
)no
xo
(b)
(a).
*
,Z
W P
Given
9
) = 0
ax.
n
0
0 < m
Put N
(d).
Given
and
q
let
5 no.
f
q 'no
^f
=
2
9 'no
X
= gff.
form t h e i r pull-back:
h
Xo
By ( b ) applied t o
N
f,h
:
N
+Y
RIP
Xo
1 > Xo
satisfy
t h e r e exists
118
Mod p
As
theory of H-spaces
is a P
f i b e r ? = f i b e r flafl
1
equivalence and
U P satisfies
(f.c.n),
f = flgl,
h = hlgl.
(d)
*
Given
(b).
and by
(a) one
If
gl: X
5
?.
X put ? = X ,
fl: X
x
5
ifl =
N
Y=X',h=l,
a. AS
fafl
f = f
are
h : X -+ XI.
follows from t h e following m r e general observation: f: X
m = m(no)
+
X
so that
Indeed, i f Hm(f,Z) €3 Z
9
satisfies p ( f , Z ) = 0 rn(?,Z
) = 0 9
X = K(G,n),
= 0
9
for
for
0 < n < no
Hm(f,Z
) = 0
9
and Tor(Hm-l(f,Z),
n
5 no then t h e r e e x i s t s
- 2.
m 5 n + l 5 no
Zq) = 0 ,
implies
therefore
and T o r ( n n ( f ) , Z ) = 0. But nn(K(G,n), Z = Tor(G,Z 1, 9 9 9 n (K(G,n), Zq) = G Q Z and nm(K(G,n), Z ) = 0 f o r m n-1 9 9 n * ( f , Z 1 = 0. 9 Suppose, inductively, f o r integer
Put
N
f : X'
U P has
equivalences s o i s
(a) * (el
by (b) t h e r e e x i s t s
f i b e r fl
m = m(n)
n < n0
-
nn(f) Q Z
# n-l,n,
9
= 0
hence
1 t h e existence of a p o s i t i v e
s o t h a t i n t h e following commutative diagram
119
p-equivalence and p-universal spaces
Then as
cofiber hn
is
n+l
connected and
( c o f i b e r h ) = 71 (X) n n+l
H n+2
one has
H j ( X,Zq)
(hn)"
I
H ( X , Z 1J n s
H ( c o f i b e r hn,Z )j 9
I
Hj-l(XyZq)
g=0
Hj(XyZq)-
H ( c o f i b e r hn,Z 1j 9
Hj(Xn,Zq)-
= 0,
and hence
j 2 no
Hj(f N2m , Zq) = 0,
H,(?
2m
Htn+2
= 0.
= 0,
Hj(32my
j ( n + 2 z n o y imply
n ((f,)",
Together with
nj((fn+l)4my
, Zq)
(XyZq)
kn+l
Htn+2
implies
J-1
and i n the following
kn+l
IT*(?^, zq)
H
z,)
= 0,
j z n + l < no.
(H*(K(G,k), Z ) 9
Steenrod algebra A ( q )
by
$1.
Z
j
Z ) = 0, 9
= 0,
9
j
j (n+2
5 n+l
and
< no,
~ , + ~-2m ( f,zq) = 0
this
implies
i s generated as an algebra over t h e
Now
f 4m
n
9
4m
fn+l'
ph,
y i e l d a comparison
Mod p
120
theory of H-spaces
of t h e Serre s p e c t r a l sequences of the f i b r a t i o n s
Hj((fn)
4m , Zq) = 0 = H j ( ( 6 ) 4 m , Z
and consequently t h e other by the 0-map. 0 = Eo[(fn+l)
4m
1:
Em
9
j <-no
implies t h a t t h e
Consequently, i n
Eo[H,(Xn+ly Z q ) l
Put
Zq), j 5 no,
m(n+l) = 4m(n).no
+
dim :no, Zq)l
Eo[H,(Xn+l,
t o conclue (e)
*
(a).
j
5 no
The proofs ( e )
( f ) e+ ( h ) are theEckmann-Hilton duals of ( a )
(*
and
4mn0 Hj(fn+l ,Z
z91
j < no-2
Zq)
t o complete t h e inductive s t e p .
n ( f ,Z ) = 0 , j q
E0 i s the
where
H,(Xn+l,
lowers f i l t r a t i o n , hence,
One can dually prove t h a t
terms
of t h e s p e c t r a l sequences are mapped t o each
grading associated with t h e Serre f i l t r a t i o n of Hj(fn+l, 4m
E2
=
imply
**
9
= 0.
Obviously
o
for
j < n0-2.
Hj(fm,Zq)
( f ) , (e)
= 0
( g ) and
( b ) , ( a ) c) ( c ) and ( b )
0)
(d)
respectively where pushouts replace pullbacks, Moore decompositions replace Postnikov approximation and cohomology and homotopy interchange.
4.1.4. 4.1.3
Corollary: StU48
If
x
is P1 uniuersal s o are H t n ( X )
v a l i d if the f i n i t e n e s s conditions imposed on
and H ~ ~ ( x ) . fiber f
in
p-equivalence and p-universal spaces
( b ) , ( c ) and
(a) and those for cofiber
in
f
121
and
( f ) , (g)
are
(h)
repkced by the conditions that a l l spaces i n 4.1.3 satisfy a f i n i t e condition of the same type Proof: -
(f.c.n
o r f.c.H).
The P1 u n i v e r s a l i t y of
IEP. ( X )
d e f i n i t i o n of universal spaces 4.1.2. follows from 4.1.3 [Htn(X), Htn(Y)]
(a). The
follows d i r e c t l y from t h e
n
P1 u n i v e r s a l i t y of H t n ( X )
The
second p a r t follows from the isomorphisms
-% [Ken(X), mn+l(Y)]
and one can move from one t o
another i n the proof of 4.1.3.
4.1.3 (a) e a s i l y implies t h a t any H-space i s P1 universal f o r all But one has a stronger result:
P1.
4.1.6.
Proposition ([Mimura, TodaL, Theorem 1 . 2 ) :
0-equivaZence. SO
is
If
+
Xo
be a
is P1 universal
and Xo
x.
Proof:
Decompose h
x = xm h j : Xj
s a t i s f i e s f.c.n
fiber h
Let h: X
-+
Xj-l
i n t o a primitive decomposition.
hm
xm-l k
induced by
X
*
j'
j-1
+
. ..
K(Z
We s h a l l use induction t o show t h a t
qJ X
ho
9
xo
, n 1. j j
is
lP1
universal hence, it
s u f f i c e s t o prove: Let
h: X
-+
Xo be a K(Z
Pa
k: X + K(Z , n ) . Then, i f P Fix q # P1 and no, n equivalence with
H (f
m "0
,
> n.
z A )
9
p r i n c i p a l f i b r a t i o n induced by
i s P1 universal, s o i s
Xo
0
n-1)
=
If
o
p for
# P1 l e t
fn : Xo 0
= q and f o r
X. -+
Xo
be a P
= p a m ( n 0'
1
Mod p theory of H-spaces
122
Then one has a commutative diagram
2.
X
0
c
xO
is a P
h
equivalence.
equivalence,
1
+ q,
p
If
xO
1.
I
As
r
+ X
a P equivalence hence, 1
f
Hm(l,zq)
=
o
H*(h,Zq)H*(fn ,Z ) = H*(lf,Z )H*(h,Z 1, 0 9 9 9
p(fn ,Zq) =
m 5 n 0'
for
0
0
If
for
f
i s a P1
m 5 n0 : i s an isomorphism and
H*(h,Z ) 9
p = q
A
consider t h e S e r r e s p e c t r a l
sequence f o r hamology i n t h e diagram
-
~ ~ ( 2 . 1E :~ ( x ) 2 -
,z 1 Q
H,(X
, Zp)
E ( f ) = H,(fn
Q H,(r,Z
0 Em (1)= 0 m,n
and
H , ( K ( z ~ , n-11,
zP)
+
E 2 (XI
O P
N
-
H,(f,Z
P
)
P
)
and
E2 m,n
(1) =
0,
lowers f i l t r a t i o n i n
0 < m+n
2 no, n
dim
5 no.
Hence,
- 0
f
123
p-equivalence and p-universal spaces
n
i s a p1
Em(? 0 , zP)
equivalence and
Suppose p E El. n ‘fq,nO
, zq)
r = rank Hn(X
=
o
Let
for
o
<
f
9rn0
: Xo
=
+
o
m 5 no.
for
0 be a P1 equivalence,
X
Hm(fQInO, zP) E
m <-no.
hence t h e r e e x i s t s an i n t e g e r
Z ), 0’ P
GL(r,Zp)
a
so t h a t
n a H (fq,no, Zp) = 1 and one h a s a commutative diagram
and as
is an isomorphism,
H*(h,Z 9 )
Finally, as
.
f
H * ( f aq,nO, Z ) = 0
induces a homotopy equivalence
4
H*(?,
ZQ ) = 0.
f i b e r h -+ f i b e r h
the
upper square of t h e above diagram i s a p u l l back diagram, fiber
4.1.”.
*r
P
= fiber fa
and
Definition:
A space
q+)
P
is a
1
equivalence.
i s c a l l e d an
X
Ho
f r e e ( a s s o c i a t i v e , commutative graded) algebra.
space if H*(X,Q) (e. g:
is a
Every H-space i s
an Ho-space).
4.1.8.
Corollary:
Proof:
An Ho
There e x i s t s a
space 0
X
is P1 universal for every s e t
equivalence
X
+
K(n,[X)/torsion)
l a t t e r is P1 universal, hence, 4.1.8 follows from 4.1.6.
5.
and t h e
124
Mod p
4.2.
theory of H-spaces
Mod p homotopy Most of the notions and properties discussed i n t h i s s e c t i o n are due F i r s t we formulate some simple properties of
t o Rector (unpublished). P1
equivalences some of which w e have already used i n t h e previous
sections. 4.2.1.
Lemma:
( a ) Consider the follaving diagram: h
Y
4.2.1.1
I f 4.2.1.1
+x
is e i t h e r a puZlback or a pushout diagram then
is a P1 equivalence if and only i f f b l I f i n 4.2.1.1
B
- P1
gl,g2
h
ho
is.
are P1 equivalences and h,ho
are
equivalences then 4.2.1.1 is both a puZZback and a
pushout diagram.
Proof: IT
k
(fiber f )
( a ) A map
f
is a
P1 equivalence i f and only i f
are f i n i t e groups of order prime t o
H (cofiber f ) k
are t o r s i o n groups of order prime t o
a pullback (pushout) diagram then
fibers (cofibers)
h
and
ho
(or i f and only i f
P1). If 4.2.1.1 i s
have homotopy equivalent
.
(b) Form t h e pullback of them t o obtain:
P1
ho.g2
(pushout of
h,gl)
and complete
Mod p
h
Y
1
I
1;
being
U
xo
Similarly
f.
p,
f
g2
is a
f
P
fl
-
and as
' xo
P1 equivalence.
i s a homotopy equivalence.
P1
is a
g1
Being a
Similarly
f.
4.2.2. ape
gl
P1 equivalence implies t h a t s o i s
p-equivalence f o r every for
f; V
hO
equivalence s o i s
X
I
___+
g2
k
n
'X
f2
"ho 9 g2
125
hmotopy
Lemma ( R e c t o r ) :
kt
X
fl,f2:
+ Y be maps.
Then the following
equivalent: ( a ) There e x i s t s a P
1
( b ) There e x i s t s a P1
Proof:
(a) * (b)
.
al'a2
-x
equivalence
g : X1
-+
X with flg
equivalence
gl:
Y
+
Y
For any maps
>
Y j
b e an e q u a l i z e r diagram (see e.g.
al,a2:
1
X +Y
with
-
glfl
f g. 2
-
glf2.
let
'E a13a2 [Spanier], p . 406).
Consider t h e diagram
Mod p
126
theory of H-spaces
J,
Y
The cofiber of
^g i s of t h e homotopy type of t h e suspension of t h e
cofiber of
Hence,
g.
g
is a
P1 equivslence.
t o the existence of a l e f t homotopy inverse f o r
flg j,,
N
f 2 g is equivalent
hl: E
flgaf2g
+ Y.
Consider
Y
h.
J,
Y
I" IA
Ig1
with the r i g h t hand square being a pushout diagram.
Then
g1
is a
P1
equivalence and
(b)
( a ) i s dual (using cc-equalizers).
4.2.3.
Definition (Rector)-:
(fl
f2)
flg
N
f2g
equivalence
flaf2: X -+ Y
we s a i d t o be El homotopic
i f one of t h e equivalent conditions of 4.2.2 are s a t i s f i e d :
f o r a P1 equivalence gl'
g
or
glfl
N
glf2
for
B1
Mod p
4.2.4.
-
Lemma:
127
homotopy
i s an equivalence relation respecting composition. i s obviously r e f l e x i v e and symmetric.
Let g2: Y .
.
-+
Y
A
*gi
and as
2
are
g2g2
x1
f;,,fi:
3
f o -P1 fl,
Y,
equivalences,
p1
fl
glfo
N
x2
i s a P1 fo
NP1
equivalence f;,
f l y
f o g - flg.
Y2,
fbfl -pl f i f l
wP1
gl: Y
Let
f2.
g2fl
-
Let
fo,fl:
n
Xo
glfl,
g2f2.
-+
Y
1'
Let
gl,g2
equivalences.
x2,
equivalence
gl:
P1
be
+
complete a pushout diagram f o r
gl,g2,Yo
then
X
fo,fl,f2:
Then
glf;
-
Npl
fo
fi.
fbfog
glfi.
wP1
N
g: Xo
Let
-+
Xo
be a
+
g f'f 1 0 1
5,
P1
and
fhflg
Then
and by t r a n s i t i v i t y
f2.
Let
-
g f'f 1 1 1 and
fAfo -pl f i f l .
Using [Hilton, Mislin, Roitberg] Theorem I1 5 . 3 and t h e f a c t t h a t a
P1 equivalence P l o c a l i z e s t o 1-
4.2.5.
Proposition:
or i f X s a t i s f i e s every p
implies
Given spaces ( f . c . r ) and
fl
a homotopy equivalence ane has:
-
f2.
Y and X . fl,f2:
Y
-+
If Y
x
satisfies ( f . c . ~ . )
then fl
Np
f 2 for
Mod p
128
Decomposition o f
4.3.
0
theory of H-spaces
equivalences
We discuss in this section ideas introduced in [Zabrodsky]
337‘
4.3.1.
Proposition:
P1
$I:
$: X
+
Xo
be a
there e x i s t s a space
o f primes Ply and maps
Let
X
+
X(P,,$),
equivatence and
$I1
+
Given a s e t
unique up t o homotopy type
X(Pl,$)
X(Pl,$)
$I1:
0-equivalence.
x0
s o that
-
is a
$ “ J , ~ $,
is a P
- P1
- a set
o f primes, then there e x i s t s a map
equivalence.
Furthermore, given a
commutative diagram
$,T
0-equivalences,
P1
f ( P 1 y $ y ~ )so y t h a t the following i s commutative:
If both X and Xo f(Pl,$yT)
(f,c.H)
or
xy?o s a t i s f y
(f.c.s)
then
i s unique t ~ ,t o homotopy.
Proof:
x(P,,$),
and
$I1
$A: X + Xn’ $:: 0-equivalences, A$:$ - I+,
Suppose are
satisfy
Xny
are constructed inductively as f o l l o w s :
Xn + X
is a
0
are given so that,
P-P1
$A,$:
equivalence and for
Decomposition of
equivalences
0
p E P1 Hm($AyZp) i s an isomorphism f o r m = n,
or equivalently
prime t o Let
H (cofiber
m
m < n
$A, Z )
129
and an epimorphism f o r
are t o r s i o n groups of o r d e r
P1 f o r m ( n . Vn = c o f i b e r
Consider t h e compositions
$ I .
n
kn+l
P i s t h e sum o f t h e p-primary components of
bn+l(Vn)
Hn+l( Vn , Z ) = Hn + l ( Vn 1, Note t h a t as
i s a 0-equivalence
$'
n
natural.
Obviously
to
x
obviously a
Finally,
-+
xn+l'
kn+l$A Put
Y
$A+1y
z)
i s t o r s i o n and
Hn+l ( V n y Z )
and i f
X = fiber k n+l n+l
= $:hn+l,ns
hn+l,n: Xn+l
P-P1 equivalence, hence so i s
Z) = H (cofiber m
+
X(P1,$)
t h e composition
X(Pl,$)
Lemma:
lifts
Xn* hn+l,n
$A,
Z)
for m 5 n
is
and
= l i m Xn, an
= l i m $I + n
$I
f
-
which completes t h e and
$'I:
X(Pl,$)
f(Ply$,;)
-+ Xo
-
$;
' n'
Xo
(which i s independent of
The uniqueness of homotopy type w i l l
follow from t h e second p a r t of t h e proposition.
4.3.1.1.
is
n+l
$:+l.
b n + l ( c o f i b e r $'n' Z )
a r e t h e d e s i r e d space and maps.
The e x i s t e n c e of
$1:
k
P-P
=
inductive s t e p .
n)
N
Hm(cofiber
Hn+l(cofiber
P E P1.
follows from t h e following
civen a conunutative diagram
Mod p
130
There exists ?: Y1
-+
Proof of 4.3.1.1:
80
theory of H-spaces
that
?g
-
fo, h?:
-
Construct the pull back
fl.
W
of
fl ,h
fl
and
h
and complete:
As
;i
is a P-P1
f is a
?'
(a)) and
g
0-equivalence and by the first part of 4.3.1
exist.
are P
equivalence (by 4.2.1
2."
and
are P-P1
equivalences, hence, so is 1-
equivalence.
One has
is a P1
Y(P,,?),
equivalence and so is n+.
gf"
and
3'
*c
gf".
is a homotopy
equivalence
?*
and and
g
Decomposition o f
equivalences
0
y1
fO
N
ga”
N
M
ry
f =
gf”).
lfl
(Simply choose
1.
?:1a”
a” = ?“u
$1,
h =
-, $11
-
fo = $ I f ,
Y = X,
?:
Finally,
u
i s t h e homotopy inverse of
N
?g = fla”g
N
Y1 = X(PlY$),
fi = f0$”
uniqueness of homotopy type of
and
where
i s t h e desired map:
Now, apply 4.3.1.1 f o r g =
131
X(pl,JI)
t o obtain
fo,
H = y(Pl,T),
-
X1 = Xo,
The
f(Pl,$,T).
follows from 4.3.1.1:
is a P
1 and a P-P1 equivalence, hence a homotopy equivalence. f(P,,$,v) i s unique i n t h e f.c. case by 4.2.5: If given
X
I
JI’
’ xo
I
I
fo
132
pTl
Mod p
- ?C?,
implies
N
P-P1 f 2
fl
-
N
and
f1$'
y1z2, ( n o t e t h a t by uniqueness
and by 4.2.5
X(Pl,$)
N
theory of H-spaces
-
X(Pl,T)
and
they i n h e r i t t h e
N
N
f2$I
implies
N
f l-P1 f 2
o f t h e homotopy type of
f.c.
of
X,Xo
and,
X
M O
Xo
--
X,Xo
respectively).
4.3.2. then
Corollary: If X,Xo
X(Pl,$),
and
$I
$I1
are H-spaces,
$:
is an H-map
acbrrit H-structures.
Proof:
By t h e uniqueness o f t h e 4.3.1 decomposition one has
Y)(P,,$
X
11.1
(X
X
$
an H-map and a
CJ
X(P1,$) x ?(P,,T)
and i f
X,v,
Xo,uo
a r e H-spaces,
0-equivalence one has a commutative diagram
Decomposition of
4.3.3.
Let
Lemma:
.
p a r t i t i o n of P Proof: -
Jl::
X(Pj,$)
$;
and
JI:
x
- 0
0
and l e t
Xo
{€!,I
i = 1,.
Then X i s t he pullback o f t he maps
We prove t h a t t h e pullback of
*
is
Xo
X(Pi U P j y
$1.
133
equivalences
$I!:
1
-+
be a
x(Pi,$)
$;:
X(Pi,$)
..,k)
Xo
-t
x0.
and
Indeed, consider t h e pullback o f
and complete
J
$3
-
J
As
$:
is a
N
equivalence.
P-P a
3
Pi
P-P $!' 1J
j
= $;hj
is a
h
j
j
Pi
equivalence.
and by 4.3.1
equivalent so i s
Pi
is a
(&Pi)
h
P-P
J'
n
xO j
S i m i l a r l y hi
(P-PJ)
equivalence and s o i s
Similarly
ij
is a
P
j
is a
P-Pi
equivalence while as $.:
Consequently
7'
equivalence, one g e t s
id
is
134
Mod p
4.4.
A Study of
theory of H-spaces
H -Spaces. 0
Recall t h a t an
Ho
space i s a space
cohomology algebra.
If
G n
W
QH"(X,Z)/torsion and a n : Gn
a l e f t inverse of t h e projection by
JIn: X - + K(Gn,n).
JI: X
X
If
= nK(G,,n)
+ K(G,)
an isomorphism
i s an
-+
Gn
then
n*(X)/torsion. =
K(C,)
n
n
is a
JI,
Hence,
K(Gn,n)
an
-+
Hn(X,Z)
is
can be r e a l i z e d
space then
Ho
0-equivalence and induces
QH*(JI,Z)/torsion. Note t h a t f o r an
QH*( X,Z)/torsion
i n the form of
Hn(X,Z)
induced by
n
with a f r e e r a t i o n a l
X
where
space
X
spaces have r a t i o n a l models
Ho G,
Ho
= {Gn} i s a f r e e graded
abelian group.
Here a r e some simple observations and notations regarding H
K(G,)
and
spaces. 0There e x i s t s a n a t u r a l monomorphism
+( G,)
Furthermore, i f r i n g generated by
where
i s t h e f r e e ( a s s o c i a t i v e commutative graded)
t h e r e e x i s t s an i n j e c t i o n
G,
stands f o r t h e set of r i n g homomorphisms (one has an obvious
HGm
4: Hom(G,,Gi)
injection
For any
Ho
space
QH*(X,Z)/torsion X
-+
X
H;m($(G,), and
H*(x,z)
$(G&)).
X: QH*(X,Z)/torsion - + H * ( X , Z )
projQH*(X,Z)/torsion
-
so t h a t
is a
monomorphism of m a x i m a l rank t h e r e exists
JI = $(XI: X such a
X
- 0
K(QH*(X,Z)/torsion) r e a l i z i n g
a rational splitting.
Such
X
X.
We s h a l l c a l l
induces a morphism
A study of Ho-spaces
i:
4(&H*(X,Z)/torsion)
H*(X,Z).
-+
Let f: X
-+
135
XI be a map between Ho
spaces and given rational splittings: X: &H*(X,Z)/torsion
-+
X: &H*(X' ,Z)/torsion
-
For a vector X = (A1, A2,
...)
= (Al,- ..,An,. ..)
. a : 0(&H*
('XI
-+
H*(x' ,z)
of integers let I-€ Hom0(G,,G,)
endomorphism (I-)n = Anal: G n x
-
H*(X,Z)
be the
h
-+
Gn. Then one can alwws choose a vector
of integers and a morphism
,Z)/torsion)
+ $ (&H* (
x,z 1/torsion)
The geometric realization of this observation is given by:
4.4.2. Lemma: ( a ) Let X be an €Io-space. QH*(X,Z)/torsion
-+
The rational s p l i t t i n g s
H*(X,Z)
are in 1-1 correspondence with
the rational equivazences $: X ( b ) FOP any map $: $I
x -+xo:
-X =
X'
-+
f: X
-+
X'
-+
Xo = K(&H*(X,Z)/torsion).
and rational equivalences
K(QH*(X,Z)/torsion),
Xb = K(&H*(X' ,Z)/torsion)
(Al, A2,.
.. ,A,,..
.)
there exLsts a vector
of integers and a map
fo: Xo
-+
X' 0
(QH*(fo,Z)/torsion = w*(f ,Z)/torsion) so that
4.4.3. $:
X
-+
Corollary:. Let
X,u
Xo = K(QH*(X,Z)/torsion)
be an H-space.
Suppose
is such that QH*($,Z)/torsion
is an
136
Mod p
isomorphism.
-A =
theory of H-spaces
Then there e x i s t a multiplication m for
(a 1, X2,. . . )
I'(I-)$
of integers so t h a t
and a vector
Xo
is an H-map.
A
Apply 4.4.2 for $ x $ : X x X
Proof: p:
X
x
X
-+
m:
X. Then t h e r e e x i s t s
X
x
-+
X
X
x
-+
Xo
0
Xo,
$:
and
X
x
-+
X
and
0
so that
Consider t h e following:
x0
xO
xo
x
xO
U
imply
W*(Z
E'
Z)/torsion = 1
+nd consequently
i s a homotopy equivalence with a homotopy i n v e r s e
an H-structure for
and
Xo
r(I-)$
is a
YE *
H-map.
p-m
A
4.5.
4.5.1.
Mod P1
Definition:
x
satisfies, i : X
H-spaces
*X
x
-+
X
equivalences )
A mod P
p(x,*)
and
1 H-space i s a p a i r
x
-+
p(+,x)
are the i n j e c t i o n s ,
.
E
are
= 1,2
X,p
where
p: X x
P1 equivalences. then
h
= piE
are
X
-+
X
(i.e. i f
5
Mod P1 H-spaces
4.5.2.
Let
Proposition:
satisfy
X
if and onZy if X is P1-equivaZent Proof: mod P1
H-space
and P1
is
X
satisfies
h: X
as w e l l ) .
-
X' g ' ( h
H-structure f o r
Now suppose t h a t
hE
X
are
* X. Put P1
h
X
X'
' -
an H-space o r i f Hence,
i s an
X
is a
X
KO
space
X
h)
is a
Let
mod P iE: X
X.
mod P
-t
$: X
mod P
H-space with a
X
-f
x
X
E
= 1,2
-
i s an
pl
H-structure f o r
1
1
If
X.
H-structure
1
the injections.
Xo = K(r,(X)/torsion)
be any
Using t h e methods of 4.4.2 we s h a l l show t h a t t h e r e e x i s t
-
A = (A1,
A2,.
..)
and an H-structure
m: Xo
x
X
0
* X
0
so that
Again, t h i s i s a geometric r e a l i z a t i o n o f an a l g e b r a i c observation: G,
Put
= &H*(X,Z)/torsion, a : H*(X,Z)
and a morphism
X'
W P g: X'
and
is a
(and we m a y assume
X'
4.1.4 ( b ) implies t h e existence of
X'
= piE,
equivalences.
0-equivalence.
a vector
X' with
P1 equivalent t o an H-space
(f.c.r)
equivalences
X
to an H-space.
i s a Hopf algebra.
H*(X,Q)
is a mod P1 H-space
X
(f.c.a).
universal.
If
p: X
X-
If e i t h e r
137
8 H*(X,Z)
m*
X: 4 (G,) +H*(X
+
H*(X,Z)
x
X,Z).
t h e morphisms induced by One i s looking f o r a v e c t o r
so t h a t t h e following diagram w i l l commute:
Let
9.
A
138
hr
h* 1 L
H*(X,Z)
8 H*(X,Z)
a
This can be seen by induction:
Let
G
-
were found s o t h a t t h e diagram commutes with
in: $(G,,)if
-p*
+H*(X,Z)
= p*
replacing
- p*h* - p*h* 11 2 2
G,,
-
A , m*
k e r i* @ 1; = im[H*(X 1
im[Hn+l(X
+
A
X,Q)
Hn+'(X
x X,Q)I
A
X, Z]
hl,...,hn,
X
F*(X x
n
m;,
Now,
respectively.
*Xn+l: Gn+l + H"+l(X +
m*
n 1,
n = ( A l,...,X
-
and
Suppose
-X
G
then t h e image of
i s contained i n
Gm.
@
mzn
x
x,
2)
As
X, Z ) ] .
c {H*(hl,Q)(i Q Q)($(I-) Q Q)[rP(G
-
x
11 s o that
Mod P1 H-spaces
Let
$(G
-
-+
= rl Q 1 + 1 Q Now t h a t
O ( G < , + ~ ) 0 $ I ( G < ~ + ~ ) be induced by
-
-
+
mn+l 7
m and
(9: 2
+
.#(G,)
139
m* n
and by
the unit).
were defined one g e t s a geometric r e a l i z a t i o n
IJ
x x x
'X
J,
Put
l'(I-)$ = $-.
x
X(Pl, J,-)
x
x
N
X(E',,
As hg, J,-h
x E
)
E
= 1,2
are
P1 equivalences
and by 4.3.1 one obtains a commutative diagram
140
Mod p
theory of H-spaces
,x
X
x
x
The uniqueness of 4.3.1 and t h e r e l a t i o n s
imply
= 1, u(P1)
p(Bl)iE
4 . 5 . 2 . 1 . : We make proof o f 4.5.2
t h e vector
X
-5
X(B1.
$ 1 X
no minimality claims for the vector
-X
i n the
and one can apply t h e same method of construction i f one i s
JI: X
-X
mod B1
-
given a f i n i t e number of and a fixed
i s an H-structure and
N O
Xo.
H-structures
mi
for
X ( i = 1,
...k)
If such a family i s given one can choose
simultaneously f o r a l l
e x i s t s an H-structure
ui
i
s o t h a t f o r every
i
s o t h a t the following i s commutative:
there
Mod P1 H-spaces
x x x
x x
x0
x
x0
x
only if
Proposition:
"i
x
1
xo
xO
I
I
-1
4.5.3.
141
xo
Let
m
i
X satisfy
xo X is an H-space if and
(f.c.).
X is a mod p H-space f o r every prime
4.5.3.1.
Remark:
cw complexes.
p.
The e a r l y versions of t h i s statement considered f i n i t e
(See [ ~ i s l i n ] ~ Theorem , 2.1).
The proofs rely upon a
preliminary announcement of Curjel (see [Curtis] which was l a t e r r e t r a c t e d ( [CurtisI1, p. 1120).
1'
remark ( e ) , p. 8-9)
The corrected versions of
the proof ( e . g . : [Hilton, Mislin, Roitberg], Theorem I11 1.8) had t o assume some r a t i o n a l consistency condition of the various H-structures of
mod p
X. This r a t i o n a l r e s t r i c t i v e assumption i s removed i n
our proof. A s t o C u r j e l ' s announcement t h a t s t a t e d t h a t every H-space with
f i n i t e dimensional r a t i o n a l cohomoloey admits a multiplication inducing a co-associative (hence primitevly generated) r a t i o n a l cohomolo@;yHopf' algebra.
A s far as we know t h e v a l i d i t y of t h i s statement f o r f i n i t e
Mod p
142
theory of H spaces
However, i f one considers
dimensional H-space i s s t i l l an open question. ( f .c.a)
H-spaces s a t i s f y i n g
and having f i n i t e dimensional r a t i o n a l (See Stasheff ' 6 H-spaces problem 44,
cohomology, t h i s statement i s f a l s e .
Neuchatel, Springer Lecture Notes 196, p. 134).
This can be seen by
observing the following example. Hta(SU)
Let
v: Ht21(SU)
Ht
h
x
(SU) 21
K(Zy23)
-+
H*(v,Z)i = u Q u u
5 13
5
u
a generator of
i
be the
- 'b
- Y5Y9Y13
@
"5"9
2) W Z .
PHi(Ht2,(SU),
10 1
mod 3 reduction of
1 = p3x*3
is primitive and i f
for
= H*(SU(12), Q ) .
The generator i n
&H23( X , Q )
i:
3
23
t h e image of
-
u5%J
"9
@
Then
i s t h e f i b e r of
X
i s an H-space,
nn(X) = 0
cannot be represented by a primitive
i s represented by a primitive, as
E H23 (X,Z )
u5
u.1 Q 1 respectively.
then
regardless of t h e H-structure chosen f o r
23
u5'13
Let
X
K(Z3,27)],
and H*(X,Q)
+
and o f
z E [Ht21(SU) x K(Z,23),
n > 26
uv,
K(Z,23) be given a twisted multiplication
x
23
X.
H*(X,Z)
Indeed, i f t h i s generator is torsion free i n
dim < 23,
i s primitive modulo a decomposable
23 + ;1 E PH ( x , z ~ ) . This implies , . . - . A
x23
+ pl; = y y y
5 9 13
+ P 1 i E PH27(X,23).
As there are no decomposable primitives of odd dimension i n a Hopf algebra 1-
;5$9i13 +P d hold i n
= 0.
One can e a s i l y see t h a t t h i s r e l a t i o n must already
H*(Ht21(SU), Z3)
(or even i n
H*(SU(11), Z 3 ) ) .
But
Mod P1 H-spaces
As (N
3’
X
+ wi
o
g imp1
w5*w9’w13
satisfies
143
E P H ~ ( S U ( U ) , z3).
i s an H-space i f and only i f
(f.c.H)
HtN(X)
s u f f i c i e n t l y l a r g e ) i s an H-space s u f f i c e s t o prove 4.5.3 f o r
satisfying
(f.c.n).
4.5.3.2
X
c a r r i e s t h e main technical. burden of t h e
proof of 4.5.3 and e s s e n t i a l l y shows t h a t t h e r a t i o n a l consistency condition always holds f o r H-spaces s a t i s f y i n g
4.5.3.2.
satisfy
equivaZences,
H-structures
N
pi
yym2
so t h a t
no > n
N
i ,Z P 1 =
Proof:
n
#.
is a
Ti
and
P2 = pi
$i:
1.
be H-spaces,
xi + x o
Suppose
- mi
H-map.
y2 respectively so
Xo
80
Xi
i = 1,2
a&ts
Then Xi
and
t h a t $i
is a
Moreover, for any f i n i t e subset Pi of
can be chosen
pi
Let
P1
Jli
Xo,mo
p2’
is HA.
i = 1’2,
a h i t muZtipZications
- y2 H-map.
pl, X2’
XlY
Suppose xo,mo
(f.c.n).
be P-Pi
x0
Let
Proposition:
( f .c.n
Pi and any
that:
~ ~ ( ‘ i 7 ~f o,r ~ p~ E)
By 1 . 5 . 1 ( a ) t h e existence of
mi
P;
m < n0
so that
$i
is a
pi
-
m
i
H-map i s equivalent t o
Suffices t o f i n d
and W: Xo
i
N
HD($iypiymo) = W(Jli
A
$i)
A
f o r then
X
0
+
Ti
Xo
and
i = 1’2
so t h a t f o r
y2= WA
+ no w i l l be t h e
desired multiplications. Let Xi
and l e t
fibering).
i
8,:
Xi + H t n ( X i. ) i
(n+l)
Yn+l: xi
i = 0,lY2
+Xi
be t h e Postnikov approximations of
be the f i b e r of
8i(Yi+1
-
t h e n-connective
144
Mod p
theory of H-spaces
N
Suppose inductively t h a t multiplications
i = 1,2, were constructed
ai( n ) :
map
Xo( n )
+
xin)
xi
xi
x
+
xi,
that
equivalence so t h e r e e x i s t s an i n t e g e r
P-Pi
X1
SO
*
'i ,n'
so that
Xi
$ln)@in) = Ai
prime t o
1 (n).
and a
P-Pi
P1 tl P2 = k?
As
xO and
(ai
a r e r e l a t i v e l y prime, and t h e r e exist i n t e g e r s
X2
prime t o
- alAl Let
?,a2
P-Pi) so t h a t
+ a2A2 =
1.
jiYn = ji = Yiain)ai '
( + with respect t o
i ,n
).
N
*
As
Put
",n+l
= +i($i
N
A
$,)A
+
in the proof of 1.5.1 ( b ) one has
The first term can be rewritten as
where
a: X
0
A
X
0
+
(Xo
A
Xo) A
(XoiX
X,)
s a t i s f i e s ah = ( A
-
A
l)Ax 0
xo.
Mod P1 H-spaces
then
S u b s t i t u t e ji =
Yi@:n)(ai
*
un)
.
The first term of
W
i ,n+l
'i,n
decomposes as
[$ Yi@(n) i n i
A
11
xO
As
xt) A xo
is n-connected
[x0(n) A
H t (X,)]
Xo,
n
= 0
and
Bn0
annihilates t h e f i r s t term:
It follows t h a t f o r some N N
HD(JI1,
ulYN9
x?)
Wl,n+l =
Y,
- Wa+l "N
= 0,
-- Yn+lun+l. 0 As WlYN =
w2,N
N
no) = HD($2, I J ~ no). , ~put ~
N
Xo
satisfies
and N
ui = i' ,N'
w
= Wi,"
f.c.r
146
Mod p
primes i n
Pi,
- ji
Ai
for
4.5.3.3. then
and
1
Fix $ : X
= %
Replace
0.
Let
FJ
i s properly chosen
P2
4.5.2
i = 1,2
pi,
and m
X(Pi,+-) a
x(P,,$-)
P2
by
*
i'
- P1 be
x
-+
X x Xo 0 (JI-
x
=
dim < no
if necessary s o one m a y assume
mod Pi
-+
Xo
By 4.5.2.1 so that for
k
are
for
X.
there exist
i = 1.2
the
X
admits a m u l t i p l i c a t i o n Xo
H-structures
r(r-)$) x
'i
x x x
:
then i n
i s both a mod Pl and a mod P2 H-space
If X
Xo = K(QH*(X,Z)/torsion).
(Al, A2,...)
,pi
i n the inductive s t e p i s then replaced
i ,N
following i s c o m u t a t i v e
'I JI,
any multiple of
i s a mod P1 U P2 H-space.
X
P fI P2 =
By
^x i i s
H * T ; ~ ~=zH~* )( u ~ , z ~ ) .
Corollary:
Proof: -X =
pi
xi
and i f
p E P;
where
a high enough power of t h e product of primes i n
e.g.
Pi f i n i t e .
Pi c Pi¶ by
Ai*ii
can be replaced by
Xi
Npw,
theory of H-spaces
'(Pi)¶
mi
'(Pi)
H-maps.
so that
By 4.5.3.2
one obtains
The genus of an H-space
N
pi
X(Pl U P2, $-)
t h e i r pullback is a
- m132
i s an H-space.
x
147
H-maps and consequently X
5l
P2 X(P, U P2,
$ I ,
x
mod P1 U P2 H-space. Proof of 4.5.3:
X
Suppose
Then
X
i s an Ho
space.
then
J,
i s a P1
quivalence,
H-space as w e l l as
mod P
Let
is a $: X
By a simple induction 4.5.3.3
Hence
...,k
i = 2,
implies t h a t
H-space f o r every
p.
s Xo = K(QH*(X,Z)/torsion)
P-P1-finite.
= {pi)
i
mod p w o
X
X
where
mod P1
is a
{p2,...,pk} = P-P
i s a mod P
H-space
-
1'
hence an H-space.
The genus of an H-space.
4.6. 4.6.1.
Let
X
be a universal space ( i . e :
Suppose
X
satisfies (f.c.).
Definition:
P1).
every set
P1
!The genus of
universal f o r
X
is the
set G ( X ) = {[Y] A property
[XI E P
4.6.2.
I
Y -p X
for every
p).
P of homotopy c l a s s e s of spaces i s s a i d t o be generic i f
implies
Example:
G ( X ) c P.
By 4.5.3 being an H-space i s generic ( i n t h e f . c .
Combining t h e r e s u l t s of [Zabrodsky]
9
case).
and [Wilkerson] one g e t s a
characterization of t h e genus of an H-space s a t i s f y i n g ( f .c. ) :
4.6.3.
Proposition:
x
xo - Y
X
be an H-space s a t i s f y i n g
if and only if t h e r e ezh?ts an H-space
[Y] E G ( X )
x
Let
x
x0'
Xo
(f.c.n).
so that
Then
148
Mod p
Let
be an H
X
JI: X
0-equivalence
theory of €I-spaces
space s a t i s f y i n g
0
Xo = K(QH*(X,Z)/torsion)
+
d i v i s i b l e by all t o r s i o n of
s a t i s o i n g mA = 0 ) . QH i(X,Z)/torsion
#
L e t .9
Fix an a r b i t r a r y
and l e t
t
be an i n t e g e r
exp r i ( f i b e r
N ( X ) and by
i s t h e smallest i n t e g e r
exp A
be t h e number of i n t e g e r s
ni
If: x
+
XI
f
The main result of [Zabrodsky]
9
is a
p
m
so that
equivalence f o r every
pit}.
states:
4.6.4. Proposition: Let X be an H0 space satisfying
(f.c.).
Then
G(X) admits an abelian group structure w i t h an ezact sequence
n
n
..., l d e t QH ' ( f , Z ) / t o r s i o n l ) .
where a ( f ) = ( l d e t QH l ( f , Z ) / t o r s i o n l , (Note t h a t although
[X,X],
For t h e ( f . c . n )
case,
4.6.5.
dl,...,da
Let I
dl"
n
I
dl'.
..¶d' I &H ry
class of
X
i
**
$1.
Finally, l e t
0.
=
[x,x],
m
$n(X,Z)
(For a f i n i t e abelian group A
n
(f.c.).
i s not a group
im a
is).
6 i s given as follows:
be i n t e g e r s ,
( d i , t ) = 1. Let
: &H*(X,Z)/torsion + &H*(X,Z)/torsion, ¶
(X,Z)/torsion = di
1. Then
given by a pullback diagram
E(d,,
d2,...,dQ)
i s the
The genus of an H-space
149
N
X
4.6.6.
If
X
satisfies
t
t h a t i n t h i s case primes of by
Ilkdim
4.6.7.
for
N
ll
izdim X
exp n i ( f i b e r $ )
and
?
i s replaced
,(PI. !l%e genus of
Example:
SU(n): L e t
( I t i s obvious t h a t i f
(f.c.s).
the same procedure can apply except
i s chosen t o be an i n t e g e r d i v i s i b l e by t h e t o r s i o n and by
H*(X,Z)
(f.c,H)
X
X
be an Ho space s a t i s f y i n g
s a t i s f i e s (f.c.H)
G(X)
-
C(HtN(X))
s u f f i c i e n t l y large s o our observations cover the (f.c.H) case as
w e l l 1. One can choose an i n t e g e r simultaneously f o r function
2 1)'
(Z:/
[X,X], .+
(Zf/
-+
X
t
and H t n ( X )
[Htn(X), H t n ( X ) l t
2 l)',
('ln
t h a t w i l l s a t i s f y t h e 4.6.4 hypothesis f o r all n.
As one has an obvious
and an epimorphism
the number of i n t e g e r s
n < n i -
with
n
&H i (X,Z)/torsion # 0 )
s o t h a t the 4.6.4 exact sequences can be compared
Mod p
150
Hence,
G(X)
X = Sun = H t epimorphism.
2n
Sun
J I : Sun
If
Pt
[SU,,
equivalence,
Pt = { p E PI p l t ) .
-+
If we choose
sum],
n
K ( Z , 2m-1)
- the
t
-+
[SU,,
k = m-1:
Indeed, s u f f i c e s t o show t h i s f o r be a
m,
n
one has even more:
(SU)
-+
f:
i s an epimorphism.
G(Htn(X))
-+
theory of H-spaces
SUklt
i s an
Let
4.6.4 i n t e g e r f o r
By a theorem of Bott ( [ B o t t I 2 )
realizes
Q,H*(SUn, Z ) / t o r s i o n ,
m
r 2m ( f i b e r J I ) = Zm,, Furthermore, t o r s i o n k
H
i n v a r i a n t of
Sum -+
m u l t i p l i c a t i o n by
X
t = II m!,
and
m < n,
Pt =
mcn
2m
Pt
being a
f
H
of torsion
2m
(SU m-1'
P < nl.
and i t s generator i s t h e
Z ) = Z(P1)!
(SUPl,
{PI
equivalence y i e l d s a Z)
(A,t)
= 1. f
induces a
diagram
f
"m-1
+
"m-1
lk a d
?
E [sum, SUmlt
Now 4.6.7.1 f o r h: G(SUm) + G(SUm-l) onto
image f E
with
X = SU
m'
b u t as
ker h = c ( k e r r),
I
1t -
n = 2m-3
Ht--,:
y i e l d s an epimorphism
[Sum, SUmIt
-+
k e r r = {l, l,.. .,l, dl d E Z/:
SUWllt
2 11.
is
We s h a l l
The genus of an H-space
(1, l,...,d)
show now t h a t Indeed, i f
d
?
fl
lw
where
As
u
order u =
(IE-~)!
.
fo = Ht2m-3(f):
sh-l
d
du = u.
-
SUmw SUm with
t h e f i b e r of k < 2m-1,
Hth-l(?)
Sum -+ Sum-,,
bundle
N
Put
f : Sum -+ Sum, a f = (1, l , . . . , d ) .
Then,
-+
= fo.
X: SU m
= Sum -+ Sum and
fd = Ht2m_l(fd)
Hence, -+
.. ,1
a f o = 1,. Corollary
1.14:
?-f = jX,
K ( Z , 2m-1).
hence, by [ Z a b r o d ~ k y ]Corollary ~~ 1.8
d i v i s i b l e by
((m-l)!).
BSU(m-1)
U
homotopy equivalence and by [Zabrodsky] 10
2:
fl
-
SU(m-1)
Conversely, suppose Sum
d f
one has
c l a s s i f i e s the principal
aTa = (l,.. ,1, d ) . Put
E i m a i f and only i f
((m-l)!)
pl-1
151
and
fo
is a
There e x i s t s
j: K(Z,2m-l)
-+
ak(jX) = 0
for
degree
(jX) i s
IT
2m-1
sum
Mod p theory of H-spaces
152
It follows t h a t
ker[G(SUm) -+ G(SUm-l)]
FJ
21 and one
Zrm-l)!/
obtains a decomposition sequence
Z2"l fl I
. Consequently, Cp
G(SU3)
-+
G(SU2) = 0
n
I G(SUn)I
II order ( Z rm-1) I 1
=
m=2 where
-+
t1
i s the Euler function.
G(SU6) = 4.
Finally, t h e epimorphism
(p(120)/2 = 4.16 = 64 e t c .
G(SUb)) -+ G(SUn)
y i e l d s t h e estimate
We dare t o conjecture t h a t t h i s inequality i s indeed an equality.
4.7.
MixinR homotopy types
4.7.1. $i: Xi
Let
Definition: -+
Xo
The r r k h g of
and l e t Xi,
Pi,
pullback o f ' t h e maps of 4.3
1.
Pi (Ji
(J;:
Xi
i = l,...,k
.. ,k
i = 1,.
be a p a r t i t i o n of t h e s e t of primes.
i s t h e space Xi(Pi,
(Ji)
+
be spaces with 0-equivalences
Mix(Xi, Pi, $ i ) obtained as t h e
Xo
(Xi(Pi,$i)
i n t h e notations
153
Mixing homotopy types
4.7.2.
Let
Proposition:
of the same type.
Mix
4.7.2.1.
be universal spaces s a t i s f y i n g
Xi
( f .c .)
Then
(Xi,
Pi, Jli)
-
xi
Xi
i s a mod P
'i
I f for every
4.7.2.2.
Mix(Xi, Pi,
ai)
i
is an H-space.
If there e c i s t s a space
4.7.2.3.
H-space then
i
so t h a t f o r every i
X
Xi E G ( X )
then Mix ( X i , pi, Jli) E G ( X ) .
Proof:
4.7.2.1 P-Pi
N
If
follows from t h e following observations:
f i : Xi
Xo
then t h e pullback of t h e induction f o r
i = 1,...,k,
is
fils
Pi
{Pi)
d i s j o i n t s e t s of primes
Xi.
equivalent t o
Indeed, by
k = 2
J f2
P1 c P N
P-P1
u
replace
-
implies
P2
P2.
For
X1,X2,
as above),
'xo
P-P2
x2
x12 NP1
X
2'
similarly
k > 2,
given
X1,.
. .,%,
fl,f2,
P1,P2
by
X12
f12 = f2gl,
B1 U P2
X12-P2 fly
- .. , f k ,
X
1'
ply
( t h e pullback of
f2gl
is a
- .. ,Bk fl
and apply t h e induction f o r
and
f2
1 54
Mod p
.,$, PIU P2,
X12,
X3,..
f12,
f3,...,fk
theory of H-spaces
. .,Pka
P3,.
i s t h e pullback of
f3,.
f12,
.., f k :
f2a...sfk
fla
X
The pullback
and
x
xi,
i
of 2,
pi
X
wP,
rh, Xi
B2 X12
i = 1,2.
follows from 4.5.3 as
4.7.2.2
X
Xi
3
X
imply
a mod Pi
H-space,
i f o r all p.
mod p
hence 4.7.2.4
4.7.2.3 i s immediate from 4.7.2.1,
follows from 4.3.3.
The following r e l a t e s t h e concepts of mixing and genus of an
Ho
space : 4.7.3,
Let
Proposition:
JI: X
arbitrary 0-equivalence integer
if
i0 80 t h a t
be an Ho
X
-+
X
0
= K(n,(X)/torsion).
Fix an
There e x i s t s an
i s a P1 equivalence f i n
= (X0l)$
$,
space s a t i s f y i n g f f . c . ) .
0
dim ( N ( X )
i n the
case) then
(f.c.H)
e x i s t s a P2 = P-P1 eqtrivalence
4: Xo
[Y] E G(X) -+
Xo
i f and only i f t h e m
of the form
4 = r(IJ A
(in
the notations of 4.41 so t h a t Y w
Proof:
MiX(X,
Obviously
P1’ P2; 4JIX0
JIXo
Mix(X, X; P 1
4.
$,
0-equivalences
x;
Conversely, suppose
[Y] E G ( X ) .
As t h e ( f . c . n ) case i s simpler we
s h a l l prove t h e (f.c.H) case.
As i n 4 . 1 . b f a c t o r s through spheres.
t h e r e e x i s t s an i n t e g e r H t N ( X ) ( S ) , where
= X $ 0
S
ho
so that
X 01: Xo
-b
xo
i s a product of odd dimensional
thus y i e l d s a commutative diagram
Mixing homotopy types
155
S +H t N ( S ) _ _ j X
+O
I
By 4.6.5 and
4.6.6 there e x i s t s 4 = I'(I-) so t h a t
homology approximation i n a
J,
xO
-
Y
x
4 p u l l back diagram.
see t h a t t h i s i s equivalent t o t h e following:
4.7.3.0 -2
1 X-
N
pullback
1'
pullback
. JIY
Consider now:
JI:
Y
f
p2
Y
X
h
i s given as a One can e a s i l y
Mod p
156
theory of H-spaces
By the uniqueness of the 4.3.1 decomposition one obtains:
(OJ, 4.7.3.1
yP2
Y
,
Y U
4.7.3.0 y i e l d s N
Y
h W P
S1
a'
I
a''
i = i,i, X
h
FJP
s-
at
1
Hence : 4.7.3.2
4.7.3.3
J
xO
157
The non c l a s s i c a l H-spaces
.
( @ $ )I' can be replaced by p1
(4.7.3.2).
-4.7.4.
( @ $ )If p2
By 4.7.3.1 one can replace
Corollary:
Hence,
;) 0 2
by
.
By 4.7.3.3
which i n t u r n can be replaced by
$u"
Y = Mix(X, X; Ply P2;
To be (of the homotopy type) of a f i n i t e CW loop
space i s generic, hence the mixing of f i n i t e d CW loop spaces i s a f i n i t e CW loop space.
Proof: Y
FJ
If
(a s a t i s f i e s
[Y] E G(0X)
Mix(X,X; a$, $; P1,P2).
One can a l w a y s choose
= r(I-)
t o be a loop map.
and
A
procedure t o construct
4.8.
-
By 4.7.3
(f.c.H)).
STX
$:
- 0
OXxo
X 1 are obviously loop maps and t h e 0
Mix(nx, QX; Ply P2; $JlX0.
qh0) i s deloopable.
The non-classical H-spaces and other applications Until 1968 t h e only known f i n i t e CW H-spaces were t h e c l a s s i c a l ones:
Those of t h e homotopy type of L i e groups,
S7
and t h e i r products.
I n 1968 Hilton and Roitberg have discovered t h a t bundle over
S7
c l a s s i f i e d by
H-space and a non c l a s s i c a l one. ( [ S t a ~ h e f f ] t~h)a t
E
7w
7w E
3
n6(S. )
-
E
7w
-
the
S
3
Z12 (EW = S p ( 2 ) ) i s a n
It was l a t e r shown by Stasheff
i s of t h e homotopy type of a f i n i t e dimensional
loop space.
observation follows from 4.7.4. discovering an (e.g:
By 4.6.7
-
The l a t t e r i s obviously a source f o r
family of non c l a s s i c a l f i n i t e dimensional loop spaces
G(SU(n)) i n i t s e l f i s a very r i c h source for such
Mod p
158
theory of H-spaces
I n f a c t till now we know of no f i n i t e dimensional loop space
spaces).
The same holds f o r
which i s not i n t h e genus of a L i e group.
mod 2
H-spaces where we know of no non-classical f i n i t e dimensional ones. s i t u a t i o n i s d i f f e r e n t as far as general H-spaces a r e concerned. Sa+'
(see [Adams] ) was t h e first t o observe t h a t 2
The
Adam
i s a mod odd H-space.
(Actually, from t h e point of view of 4.5.2 t h i s follows from t h e much 2[1 , I ] = 0 ,
e a r l i e r observation t h a t and t h e r e e x i s t s a map mod odd H-structure).
S2n+l
s2n+l
E
I
IT^^+^
s2n+l
( S2n+l)
t h e generator,
of type 2,l
which i s a
This i n t u r n yields a huge family of non c l a s s i c a l
H-spaces :
4.8.1.
nl, n2,...,n
, Q)
H*(%
1
... 5 nr
r y n1 - n2 -<
....,xn
= h(xn 1
Z,Q)
x
1
Mix($
1
-
H*(G,Q),
-
x
5
x
5,
-0
m e r e always
1
i
G
i*e:
'(5 , Q).
xn
r
nJ-oddy
n. E H
e x i s t s a f i n i t e product of odd spheres
H*(%
a mod P H-space of type 1
be a s e t of odd primes,
Bel
Let
N
S = VS
a Lie group.
Xo = K(n,(G)/torsion)
2m +1 so that
1
Let
a2:
G
-0
Xo.
Then
G; P1,P-P1; $1,$2) i s a non c l a s s i c a l H-space.
4.8.1 i l l u s t r a t e s how t h e construction of mod p H-spaces leads t o t h e construction of non c l a s s i c a l H-spaces. method t o construct
mod p
We s h a l l present here a
H-spaces which i s a general form of a
construction described i n [ S t a ~ h e f f ] ~ . I n t h e following proposition by a f i n i t e s t a b l e complex w e mean a spectrum
k
= {I%)
connected we w r i t e
where
K
i s a f i n i t e complex.
H ~ ( ; O= H ~ + ~ ( K ) . dim ic
= dim
If
K-S.
K
is
s-1
159
The non c l a s s i c a l H-spaces
4.8.2.
D = C mBm, m
Put 1 < 11
be a f i n i t e stable complex, Hodd(?,Z) = 0
Let
Proposition:
r = EBm
Bm are the B e t t i n d e r s of 2.
where
If
is an odd integer satisfying 2(D +
r e $ )
then there e d s t s an R - 1 a: SX +
k
< pI1
+
p-2
connected f i n i t e
H-space
mod p
X
o f degree zero o f the stable complex SX = IZnX)
with a map
so that the
composition
is an isomorphism.
Proof:
K, N-large.
complex chosen map
Represent
n
k
x Hkk(fr) -+ HI1,,(i)
HE (?) k
-
-
for s u i t a b l y
for some
P
induced by
dimensional
j , = QnZK
The i d e a i s t o show t h a t
w i l l have t h e property
Hak(?)
4N
2N-1 connected
by a
Add:
i
x
3
j,
k.
The
can be
mod p
n
deformed t o a mod p
H-structure
Hllk(?)
The l a t t e r s t a g e i s q u i t e simple, as space
Y,
given
hf;l i s a mod p Ha,(%)
ej HI1
P
2k
h:
Him(
x
a)
Hak(i)
i s p-universal f o r any then i n
H-structure for X = H l l k ( X ) .
(2)
Hkk(i[) = X.
H l l k ( k ) + H!Lk(X),
(depending on t h e choices of
The observation
n
and
k)
is a
Mod p
160
theory of H-spaces
H. (ft,Z
consequence of t h e a l g e b r a i c observation
To see t h i s one has t o analyse Y
Let [Browder]
4'
HS(Y,Z
P
w
) = 0
for
) = 0
-
( a ) o*: Q H ~ ( Y , z ~ )
P
for
k < i
Zp)
H"(QnCK,
be an H-space with theorem 5.14
1
PH'-'(QY,Z
By
s < 2(m-1)p+1.
for
P
s < 2m-1.
A simple Hopf algebra argument y i e l d s
Again by [ B r ~ w d e r ] ~Theorem , 5.14 (c)
w
&H~-'(QY,Z~)
Finally
(a)
-
s-2 QH
w
P H ~ - ~ ( Q ~ zp) Y,
Putting (a)
P H ~ - ~ ( Q ~zY , P
2
( a Y, zP
for s < 2(m-l)p
for
( d ) t o g e t h e r one has
Q H ~ ( Y , Z ~ fil ) H'(K,Z~)
H'(Y,Z H2'+'(Y,Z
P
) = 0
P
for
) = 0.
for
s < 2N,
s <
4N,
s < 2(m-l)p+2
5 2k.
161
The non c l a s s i c a l H-spaces
and
s < 4N
for
s-2n
(or equivalently
s
-
2n-1 < 4N
-
2n-1).
Substitute
2s
for
one o b t a i n s 2s+2n 2s+2n-2N 2s-1 2 n + 4 , Zp) = H (K,Z ) = H (K,Zp) QH (Q P
for
2s-1 < 2(N-n)p-2
and
2s-1 <
4N
-
As
2n-1.
QH2s+1(k,Z
P
) = 0
one o b t a i n s
Q,H2s(Q2n+4, Z ) = 0 P Choose
then
for
4N
-
N
and
n
s < (N-n)p-1
if
and
so that:
2n-1 = 2N + il > (k+l)p-2 = 2(N-n)p-2,
hence,
2s-1 < ( ~ + 1 ) p - 2 . w ~ ~ ( Q z~ += o~ f, o r P Let
A
s < 2N-n
2s < ( ~ + 1 ) p - 2 .
be t h e e x t e r i o r a l g e b r a generated by
QH2s-1(8n+hy Zp)
2s-1 < (E+I.)P-~. Then
2k = 2 dim A < p .(11+1)
By t h e hypothesis A
N
H*(i22n+%,
Zp)
through
X = Hllk(02n+4) -p
1
D + llr < $p(E+l)
dim p(ll+l)
St?"+lY).
-
2)
-
-
2
2
and consequently
and
One can e a s i l y argue t h a t
implies t h a t
dim
fc
< (ll+l)p-2-1
and hence
Mod p
162
4.8.3.
theory of H-spaces
Let
Realizing Hopf algebras:
Suppose
i s odd t o r s i o n free.
H*(G,Z)
P1 and Hopf aZgebra s t r u c t u r e s
6,
there exists an H-space p E B1
H*(6,Z
Proof: Let
give
Si
i
( TI k j ) j
A
-+
s o that
= H*(G,Z)
H*(G,Z)
and for every
N
x2m+l ( II
mod
P1 equivalent t o S
a multiplication
p E El. p '
p E Ply
qP f o r Ap = H*(G,Zp)
i s 0-equivalent t o a product of odd spheres
G
k = .IISi 1=1
X hi+l+
Oiven a f i n i t e s e t of odd primes
as Hopf aZgebras.
) = A P
be an H-space,
algebra,
q1: i
P
be a f i n i t e dimensionaZ H-space.
G
so t h a t
a +1
H*(g,Z
P
S =
k 2mi+l ll S i=1
.
one can e a s i l y
) = A P
as a Hopf
(This i s being done by r e a l i z i n g t h e reduced coproduct
- x2m+1 ' l-l ' x h + l
i j )-+ gi).
v i a a map
One has 0-equivalences
j
a multiplication
i
1
and m u l t i p l i c a t i o n s
H-map and
I),a
u-m
JI,:
= Go y
K(n,(G)/torsion)
G -+ Go.
By 4.5.3.2
one can f i n d
*
S
SO
(for
G)
for LI
that m
H*(C (for
Z
1' P Go)
= H*(j,Zp) SO
that
H-map.
Mix($, G; Ply P-P1;
IJJ~,
J12)
i s t h e d e s i r e d H-space.
JI,
p E P1 is a
i,-
m
163
Chapter V
Non s t a b l e BP resolutions
Introduction The Adams-Novikov s p e c t r a l sequence proved t o be e f f e c t i v e i n sane
computations i n t h e s t a b l e homotopy groups of spheres (see [Zahler]).
In
t h i s chapter we attempt t o advocate f o r t h e use of t h e BP-spectrum as studied by Wilson (e.g. [Wilson] and subsequent papers) f o r some non s t a b l e problem. The connection t o H-spaces l i e s with t h e f a c t s t h a t our model spaces
a r e H-spaces and some H-spaces invariants a r e occasionally used and t h a t some of our applications deal with H-spaces.
We give here a very loose account of t h e prospective theory and our applications are just a scratch on the surface of p o s s i b i l i t i e s and w e dare s a y t h a t already here i t s efficiency compare t o the c l a s s i c a l obstruction theory can be observed.
(It does not imply however, t h a t
some ugly arithmetic computations can be always avoided). Throughout t h i s chapter w e use many of Wilson's r e s u l t s without proof. A s we a r e t o s t a y within t h e category of CW-complexes of f i n i t e type
we adopt t h e Brown-Peterson o r i g i n a l construction of the spectrum BP f o r our nonstable model.
(See [Brown Peterson]).
t h e r e s u l t s obtained t h e r e .
W e use f r e e l y some of
164
Non s t a b l e
5.1.
BP
resolutions
K i l l i n g homology p-torsion
5.1.1.
Let
Proposition:
e x i s t s a space fa)
F(X) = F
H,(F(x), Z )
f b ) n,(fiber h)
fcl
be a space.
X
(PI
For a given prime p
and a map h: F(X)
(X)
there
so that:
+ X
i s p-torsion f r e e .
is torsion f r e e .
There e x i s t s a commutative diagmm
A
fd)
h: F(X)
over
-+
x.
X
does not dominate
i.e.
mod p
any non t r i v i a l space
If in the following commutative diagram
\ Fj: // gl
g2
’
y1
y2
X
g2gl
Proof: hn: Fn(X)
-+
is a mod p equivaZence, then s o are
F( X )
X
gi’
i s constructed i n d u c t i v e l y as follows :
is constructed so t h a t
and t o r s i o n f r e e f o r a l l
m,
am(fiber hn)
H~(F,(X), Z)
and one h a s a commutative diagram:
is
i s p-torsion
0
Suppose for
m 2n
free for
m 5n
165
Killing homology p-torsion
Let
En:Fn(X)
-+
K(Hn+l(Fn(X),
Z), n+l) yield an isomorphism on
N
homology in dim n+l. Put An+l kn : Fn(X)
N
-+
= Hn+l(Fn(X)y Z ) / # p torsion,
K(An+ly n+l) induced by
cn
and the quotient.
N
Let An+1
be a free abelian group, an+l: An+1 -+An+1 a surjection and an+l 8
p an
isomorphism. Let F (X) complete the following into a pullback diagram: n+l
Using the Serre exact sequence one can see (mod # p that Hn+l(fiber kny Z ) = 0 and Hn+l(~n+ly Z ) Hm(hn+l,n, Z )
is an isomorphism for m 5 n
torsion)
is an isomorphism. As
Hm( Fn+l(x),Z )
is p-torsion
free for m 5 n+l. put hn+l
-
- hnhn+l,n and one has a fibration Fiber hn+l
,n
-+ Fiber hn+l
-t
Fiber hn-
Fiber h = Fiber K(an+l, n+l) = K(ker a n+l). n+l’ n+l ,n
ker a cA is n+l n+l
166
Non s t a b l e BP r e s o l u t i o n s
free, hence,
m 5 n,
TI
n+l
n m ( f i b e r hn+l)
can be constructed a l t e r n a t i v e l y by composing
’H,+~.)
N
(Yn induced by any s p l i t t i n g An+l
Fn+l
+ pHn+l((Fn(X), Z ) ,
so that
‘n+l
(X)
n+l
B Z
p
Let
+
n + l ) be any epimorphism of a free group Fn+l (X) i s then t h e pullback
i s an isomorphism.
i s t h e f i b e r of
Ynkn
- K(Gn+l.
n+l): Fn(X)
x
K(in+l, n+l)
n
hence,
Fn+l =
X
hn)
( f i b e r hn+l) = nn+l(fiber hn+l,n).
Fn+l (X)
“n+l: An+l
n m ( f i b e r hn+l) =
i s free:
F,+,(X)
-jp-p
(XIE w P- . F(X)
x K(n,(fiber
Fn(X)
x
hn+l))
x
K(pHn+l(Fn(X),Z),n+l)
n
K(An+l, n+l).
K ( A ~ ,n+l)+ ~
+
Now
X
x
An+l
= n n+l ( f i b e r hn+l ,n
K(r,(fiber hn) x K(in+l,n+l)=
The following commutes:
Killing homology p-torsion
167
X satisfy (a) - (c). F(X) , h = 1Jm Fn(X) , l$.m h n
To see
(a), suppose
one is given
\ I.
g2
F(x)
X
2 y /
a p-equivalence. Decompose h, hi into a Postnikov decomposition:
Hn(hi ,n,n-l'
Z) is surjective (fiber hi,n,n-l being n-1 connected) l
and if one supposes inductively that
gi ,n-1 axe p-equivalences
168
Non s t a b l e BP
Hn(gl,n’
i s a mod p
Z)
isomorphism.
is
mod p
exact it f o l l a i s t h a t
is
mod p
surjective.
Z)
n
l,n,n-1
it follows t h a t fiber hl ,n ,n-1
5.1.1.1.
-+
,Z)
n,n-1
_.
n,n-1
mod p
type of
z). = K(T ( f i b e r h) , n) n
h: F(X) + X
t o r s i o n free b u t not both, otherwise t h e
mod p
mod p
or
is
H (F(X), Z )
n
minimality ( p r o p e r t y ( d ) )
i n v a r i a n t version of property ( c ) i s t h e weaker
\/ X
The c o n s t r u c t i o n of construction
F(X)
h: F(X)
-+
X
i s not unique.
n
:
H ~ ( F ~ (,xz))
However, once a 5.1.1
i s chosen one o b t a i n s i n v a r i a n t s N
~1
,Z)
a r e p-equivalences.
sn(fiber h)
The
n ,n-1
equivalence
concerned we can f i n d models with e i t h e r
w i l l be l o s t .
Hn(fiber h
Z)
fiber h
gl,nS g2,n
and
As f a r as t h e
-
+
l e f t inverse
Hn(fiber h2,n,n-ly
y i e l d s a mod p
fiber h
Remark:
mod p
= K(n;l(fiber h l ) , n),
g l,n
-+
Hn(fiber hl,n,n-l
Z ) “p, Hn(fiber hn,n-ly
H ( f i b e r hl,n,n-l,
fiber h
As
Again, having a
H ( f i b e r hn,n-l, n
As
resolutions
/# P t o r s i o n =
An
-f
An.
K i l l i n g homology p-torsion
5.1.2.
169
Proposition:
( a ) Let
H*(Y,Z) be p-torsion f r e e .
l i f t s t o a map
f: Y
+.
f: Y
-t
X
Moreover, if f o r
F(X).
Hn(Y,
an#: H n ( Y , A,)
Then m y map
xn) (which is onto)
one chooses a l e f t inverse (as a f m c t i o n of s e t s ) rH ~ ( Y An), ,
Jn : 8 ( Y , x n )
an"Jn= 1 then
1 can 1 3
be chosen i n a unique way so t h a t [u ] = J [k n n n n-1 n-1
(p:Y
Fm(X)).
+.
(bl If g: X gh: F(X)
*j; then any map -f
X
* ?) is a
(c) If X is a mod p mod p
Proof: to
Fn(X).
p
g:
F(X) + F(?)
(Zifting
equivalence.
H-space then s o is
F(X)
is a
and h
H-map.
(a) Suppose
i s given.
J
Suppose i n d u c t i v e l y
f
lifts
One has t h e following commutative diagram
K(an+l, n+l) Fn(X)
kn
b
~.( ; ni+~ l) +~,
, which y i e l d s a map and
K(an+l, n + l ) ) :
-
?n+l. Y
-+
F
n+l( X ) ,
(Fn+l(X)
- the
pullback o f
k
n
Non s t a b l e BP
170
Now any l i f t i n g J:
fA+l: Y
-+
F
n+l
[ u ~ + ~ ? A + ~=]Jn+l[kn3n 1
(X)
resolutions
?n
of
w i l l satisfy
induced by
[ u ~ + ~ ? =~ Jn+l[kn?n] +~]
=
[“n+l?n + l 1. fn+l =
Hence, i f =
(Pn
x
A.Y
(p
N
x Y x U ~ + ~ ^ ~ ~ + ~ ) A : Y: ,
A: Y
x
m u l t i p l i c a t i o n o f functions.
AS
-+
H”+’(Y,z)
N
A: Y -+ ~K(A,+~, n+l) l i f t s t o
A: Y
r e p r e s e n t s a self homotopy o f
un+lfn+l
core1 h
n+l ,n
Of
fn+l
nK(Tn+ly
-+
n+l),
PK(An+l, A-Y
i s p-torsion
n + l ) then
- t h e pointwise f r e e any map
and
which induces a homotopy
and
Suppose i n d u c t i v e l y
pullbacks :
+
O K ( A ~ + ~n,+ l )
g
(b)
Y
Fn(X)
Fn (XI).
Given t h e map o f
Killing homology p-torsion
hn+l ,n
h;+l ,n
\ As
an+l Q Z P
p-isomorphism. and so is
8 Zp
and
are isomorphisms Hn+l
gn+l, Z )
( A
is a
Consequently K(Hn+l(bn+l, Z), n+l) is a p equivalence
gn+l*
(c) Let
p:
X
x
X
+
X be a mod p H-structure. By ( a ) one has
BP
Non s t a b l e
172
uii,
As
i s a p-equivalence, hence, by ( b ) F(p)u i s a
p-equivalence, mod p
resolutions
H-structure f o r
mod p
is a
F(p)ui F(X)
and
is a
h
H-map.
5.1.3.
Proposition:
of 5.1.1 i . e . :
h: F(X)
~f fi: Y
-+
-+
X
x satisfy
F(X), h
then there e k i s t s a mod p
and h g -
^h.
Proof:
(a)
-
( d ) with
equivaZence
Y,
g: Y
h replacing
A
5.1.1 ( a ) and 5.1.2 ( a ) imply t h e e x i s t e n c e of
Now, 5 . 1 . 1 ( b ) and ( c ) f o r
Y
imply t h a t t h e
a r e all i n t e g r a l of order power of
-
i s characterized by properties ( a )
p
k
hg
i n v a r i a n t s of
and t h e f a c t t h a t
i s p-torsion f r e e implies t h e e x i s t e n c e of
g,
v : F(X)
-+
Y
F(X)
Y
H,(F(X), $v
-
-
h.
X
-+
Z)
h.
Considering
A
both v
5.2.
and
i s a p-equivalence and by p r o p e r t y 5.1.1 (d) for
gv
by 5.1.2 ( b ) g
Wilson's
a r e p-equivalences.
B( n , p ) ' s
-
5.2.1.
Definition:
B(n,p)
i s t h e space B(n,p) = F ( p ) ( K ( Z , n ) ) = F ( p ) ( K ( Z p , n ) ) .
corresponds t o mod p
Y
Yn
Let
n
be an i n t e g e r and
p
a prime.
Wilson's (B(n,p)
i n [Wilson]) 5.1.1 and 5.1.3 y i e l d t h e following
c h a r a c t e r i z a t i o n of
B(n,p):
(d)
Wilson's
2.2.2.
The mod p
Proposition:
is n-1
( a ) B(n,p)
for m < n, (bl
type of
B(n,p)
connected mod p,
an(B(n,p)) 0 Zp
#
is characterized by:
i.e.:
am(B(n,p))8 Zp = 0
0
is p-torsion f r e e .
H*(B(n,p), Z )
is p - t o r s i o n f r e e .
( c ) a,(B(n,p))
(a)
B(n,p)
is an Ho
(el
B(n,p)
does not dominate
is a mod p
g2gl
173
B(n,p)'s
space. any non t r i v i a l . space, i.e.:
mod p
equivalence and &(Yl,
Z )
P
# 0 then
gi
are mod p equivazences.
Proof: d e f i n i t i o n of
We f i r s t show t h a t
F
(P 1
B(n,p)
satisfies ( a )
and ( d ) .
By 5.1.2 ( c )
B(n,p)
Y1-
g1
As
an(B(n,p))/(# p)torsion = Z
B(n,p)-
g2
n n ( g ) / ( # p ) t o r s i o n = Z. Then t h e r e exists
Y2
g2gl
B(n,p)
i s a mod p
- mod p either
(e).
By t h e
By 5.1.1 ( a ) ,
(5.1.1) ( a ) holds f o r B(n,p).
s a t i s f i e s 5.2.2 ( b ) , by 5.1.1 ( b ) and ( c )
-
B(n,p)
satisfies 5.2.2 ( c )
H-space so i f
M~
Y1
an(Yl)/(# p ) t o r s i o n = Z
or
equivalence
B(n,p)
x
With no l o s s of g e n e r a l i t y assume t h e l a t t e r .
B.
Non s t a b l e BP
174
By 5.1.1
(a)
are
i2,p2
equivalences and z,,(Yl,
p
i: Y
p-torsion f r e e and l e t isomorphism of a2
a1 = 1.
n
n
.
(a2)&
the
i n v a r i a n t s of
a mod p
Now
k
mod p
a
n ( ~ ) / # pt o r s i o n n
9
)Z
a r e i n t e g r a l of f i n i t e order and we m a y assume
pry
as
H,(B(n,p),
i s p-torsion f r e e
Z)
equivalence and by property ( e ) f o r
(al)*h
Proposition:
Let
Then
X
be an Ho
We first show t h a t
u
is a
vu i s
mod p
space with H,(X,Z)
and a,(X)
X w IIB(niy p ) . P i
F i r s t note t h a t being an
i n v a r i a n t s of
Y
(b)
(ai)*).
Ho
space and having p-torsion f r e e
homotopy groups i s equivalent (up t o mod p the
B(n,p)
is
(a2)*(al)* i s a homotopy equivalence so by 5.1.2
p-torsion free.
Proof:
P
( a ) and 5.2.2 (b). By 5.2.2 ( c ) and ( d )
equivalence (and obviously s o a r e
5.2.3.
w
a
I.
l i f t s by 5.1.2
they a r e of order lifts.
Then n n ( Y )
(e).
+ K ( ~ ~ ( Y ) / ( p# ) t o r s i o n , n)
2
1
One obtains
Indeed k
-
a
Let
P
1
satisfying ( a )
Y
The
2 ) = 0.
I/(# p ) t o r s i o n = Z w i l l imply Y
a l t e r n a t i v e assumption n n ( Y Conversely, given
resolutions
equivalence) t o t h e f a c t t h a t
are i n t e g r a l of order power of
X X
t h a t t h e following commutes:
is a
mod p
H-space.
p.
Suppose inductively
.
Wilson's
175
B(n,p)'s
xvx 0
,
I
/ 0 0
hn ,n-1
0 0
0
I I
/ , , -
x x
x
-
mn' X x X
The obstruction f o r the existence of hnF
-
mn i, i s t h e c l a s s
and consequently knmn-l
i s monic
1E
'"n-1
[X
A
f r e e , hence
and mn
exists.
HtnX,
hn,n,l m
K(nnX, n+l)].
X,
has an order power of
too has order power of
,. mn - l - *
-+
p.
But
H*(X
A
lim m * X x X 4 n'
kn
Now,
and as
p
H*(A,
X, Z)
-+
X
= mn-l,
n X) n
i s p-torsion
i s a mod p
H-structure. Suppose inductively
X
2
W
p Then
k
and
i
s a t i s f i e s t h e hypothesis of 5.2.3 as well.
(K(amk, m ) ) = B(m,p) (PI rank ( rm2) = B(m,p) -+ K(nmft, m) l i f t s t o
lifts to
Bk"
x W(ni, p )
F
k
is
m-1
km -+ H t m i
connected. = K(rmk, m)
On t h e o t h e r hand
t o obtain
176
Non s t a b l e
and by 5.1.2
^x
gkn x P
M
(b)
mod p
resolutions
It follows t h a t
equivalence.
rank n m ( i ) f i b e r g2,
i s at l e a s t
f i b e r g2
is
g2gl
BP
X w f i b e r g2 x IIB(ni,p) x B(m,p) i
m
connected.
and
Xw
The p-equivalence
P
llB(ni,p)
follows from a simple limit process. I n [Wilson] it was a c t u a l l y shown t h a t type o f ) --loop
spaces, hence, an
homotopy and homology i s
mod p
Ho
5.2.4.
n
mod p
space.
a r e f r e e algebras
mod 2p-2.
f o r every
B(2,p) = K(Z,2)
--loop
H*(B(n,p), Z ) P
The 5.2.2 c h a r a c t e r i z a t i o n of
Examples:
are (of the
space with p-torsion f r e e
equivalent t o an
It w a s a l s o shown by Wilson t h a t
on generators of dim
B(n,p)
p.
B(3,2) = SU,
B(n,p)
easily yields:
B(4,2) = BSU.
B(3,3) = Spy
B(4,3) = BSp. I n g e n e r a l it w a s proved by Peterson ( s e e [ P e t e r s o n ] ) P-1 t h a t S U M II Xi(p), Xi(p) mod p indecomposable and 2 i connected. P i=l
For 1 1. i 1. p-1 Yi(p)
2i+l
j.3.1.
free.
Lemma:
Y
of maps
-
(M + Y )
BSU
1 1. i
-
P-1 ll Yi(p), P i=l
5 p-1.
, B(n,p) 1.
Let
an Ho
Similarly
B(2i+2, p ) = Yi(p)
connected and
The groups [
5.3.
is
B ( 2 i + l , p ) = Xi(p).
be spaces,
M, Y
space.
Let
(see 4.21.
H,(M,Z)
and
n,(Y)
[MYYIP be the s e t of p-homotopy classes
Then H,(
,Q):
[M,Y]
P
+Hom(H,(M,Q),
1-1.
Proof:
W e replace
fi
p-torsion
by a p-equivalent space
H, ( h ,Z )-tors i o n f r e e : I n d u c t i v e l y c o n s t r u c t
M
with
H,(Y,Q))
l i m Mn = M
..
--ir
4 . .
-
M
T-l - ..
,
hn
n
.A
Mo = M
Ikn
J.
K ( t o r s i o n Hn(Mn-l,
hn
-
t h e f i b r a t i o n induced by
m 5n
for
Y
by a p-equivalent space
JI: Y
JI': Y
2?.
[M,Y]
[MYYIP=
P
-+
implies
-
-
$'fh
To show t h a t
homotopy
Htn(fl)
-
Indeed, a homotopy so t h a t
w
k
Htn-l(f2):
*
Htn(fl)
Htn-l(fl)
-
Z)
is torsion free
similarly, r e p l a c e
torsion free:
$1
= Y({p),
Let
i n t h e terminology
i s i n j e c t i v e and t h e n considering
JI'gh
[h,?]
and
f
which w i l l show
N
P
@;,
i s i n j e c t i v e (and moreover, -+
Hom(H,(h,Q),
Q ) = H,(f2,
H,(fly
+ Htn-l(?)
Htn(f2)
T,(?)
, H(?,Q))
i n j e c t i v e we s h a l l show t h a t i f Htn-l(fl)
Hm(Mn,
M.
is torsion free.
Put
Hom[H,(M,Q), H,(Y,Q)]
[h,?]).
-+
We s h a l l show t h a t
Hom(H,(k,Q)
= H,(g,Q)
%
with
,Q)
H,(
H,(f,Q) that
Y
K(a,(Y)/torsion).
of 4 . 3 ,
[h,Y]
h:
H,(~,z)
and consequently
n)
from t h e p a t h space f i b r a t i o n .
kn
are p-equivalences and so i s
hn
Z),
Q)
Htn(f2).
then any homotopy
can be ( e s s e n t i a l l y ) l i f t e d t o a
and one can pass t o a l i m i t
-
is
H,(?,Q))
Htn,l(f2)
induces a map
fl
w:
h
f
2'
-f
K(nn(i),n)
This homotopy covers t h e homotopy
Non stable BP resolutions
178
Htn-l(fl) = hn,n-l(w (hn,n-l: Htn(^Y)
+
II
-
Htn(fl))
hn,n-1Htn(f2) = Htnml(f2)
Htn-l(i)). A
As
?
and there exists
is an Ho space Y wo K(a,(Y))
n) so that the composition
u: €Itn(?) + K(rn(?),
is a 0-equivalence. One can follow u by an 0-equivalence if necessary so that one may assume yl = h * l ,
If q: K(rn(Y), n) (W Y
Htn(fl) = q(w
H,(Htn(fl),
x
x
0
#
h E
Htn(Y) + Htn(Y) is the principal action one has url = Xpl
f )AA) 1 M
Q) = HY(Htn(f2), Q) H,(dtn(fl),
as H+,(~I,Q), n , ( i )
[k,
Z.
+
up2.
As
are torsion free K(rn?, n)]
Hom(Hn(h,Z), Hn(K(an?, n ) , Z ) = rn?)
-+
+
are injective and uHtn (fl)
Hom(Hn(?,Q),
Hn(K(r
p,
?
uHtn(f2).
N
corresponds to a homotopy w *
Y
Htn(fl)
The homotopy Htn(f2)
homotopy Htn-1(f2)
-
-+
n ) , Q) = nn? 8 Q)
Hence 0 = hw E H"(i, rn?). As the latter is torsion free w
core1 h n,n-1
But
Q) = H*(dtn(f2), Q).
Htn,l(fl).
-
N
-*
and
Htn(fl)
w%tn(fl)
N
Htn(fl)
covers the
3.3.1.1. On:
-
Htn(fl)
H t (f2) t e c h n i c a l l y does not cover
n
-
Qn-l: Htn,l(fl) homotopy
Y
k
-+
L II,(y) < n=O +
It covers t h e l a t t e r followed by a constant
Htn,l(f2).
with non zero l e n g t h
-
SO
n
where n-1
II(n)(y) = L
that i f
= $n-l + ka
II : Phn,n-16n
Q,, t n t o satisfy
However, one can e a s i l y choose
R+.
m
On-1
The homotopy described i n t h e proof of 5.3.1
Remark:
On + k
IIrm(y) then
covers
,(n)
n*-
kII(n-l)
As a d i r e c t consequence of 5.1.2 ( a ) and t h e d e f i n i t i o n of
B(n,p)
one has :
5.3.2. B(n,p)
Lemma: +
Let
0:
[X, B(n,p)l
Proposition:
P
.
If
If
H,(X,Z)
is surjective.
-+ r[
i
For any group
a = rank G Q Z
, Zp)).
.kt X be a space with H,(X,Z)
Then there e ~ s t sa: X Proof: -
be induced by
Zp)
$(X,
(yielding an isomorphism o f Hn(
K(Zp, n)
i s p torsion free then u
5.3.3.
+
H,(X,Z)
B(ni, p )
G
F
(P 1
s o that
p-torsion free.
(K(G,n))mp B(n,p)'
i s p-torsion
is surjective.
H*(a, Z ) P
free
where
i: X
-+
K(H,(X),
Z)
a
a: X
lifts to rank H,(X,Z
P
).
-+
F(p)(KH,(X,Z)) w IIB(n,p) n ,
a
n
Q Z ) = P
i s t h e desired map.
5.3.4.
Proposition:
H,(M,Z)
and n,(Y)
surjection H*(f,Z)
an = r a n k ( H , ( X , Z )
Given spaces
L, M
are p-torsion free.
and an Ho If
f: L
then f o r sane p-equivazent
+
space Y. M
? of
yieZds a
Y
Suppose
180
Non s t a b l e .EP r e s o l u t i o n s
f*: [M,Y] + [L,?]
is s u r j e c t i v e . Proof: -
i s t o r s i o n free and t h e k-invariants of Given
g: L
+
?.
?
P
?,
A s i n t h e proof o f 5.3.1 t h e r e e x i s t s
P
N
Y
so t h a t
a r e of o r d e r power of
IT,(?)
p.
Suppose one has a commutative diagram
n
and a homotopy
$n-l:
L
-+
PHtn-l(?)
r
f r e e , kn
of o r d e r
and
p
H"+l(M,Z)
A
to
If one considers
g.:
be l i f t e d t o a homotopy
w: L
-+
As j: M
-+
K(T,(?).
hn,n-lhng
N
p-torsion
in-lf.A s nY f r e e in-1l i f t s IT
as a f i b r a t i o n t h e homotopy
hn,n-l
$: L - + P H t n ( f ) , 4: hng
-
w w
(iAf)
4n-l
is
can
f o r some
n).
Hn(f,Z)
and consequently
K(IT~(?), n).
as w e l l and
of
4: hng
Replace N
A i
w w (iAf)
Hn(f,
IT^(?))
4= w
1
by
,
are s u r j e c t i v e
w -Gf,
.
w g:
which l i f t s
can be followed by a homotopy
n-1
H-maps i n t o
core1 lifting
*
w
hn ,n-1
-
(iAf)
%f
*
( % f ) = gnf
(and see remark 5.3.1.1).
Qn-l
P u t t i n g together 5.3.1,
181
B(n,p)
5.3.2,
Q : hng- i n f n
t o obtain
One can pass t o a limit
5.3.3 and 5.3.4 and choosing
B(n,p)
t o have p-torsion i n t e g r a l k-invariants one obtains:
5.3.5.
If H,(X,Z)
Corollary: (a)
u : [X, B(n,p)]
fb)
[X, B(n,p) lp
(c)
If f : X' f*:
(d)
+
+
is p-torsion f r e e then
Hn(X,
Hom(H,(X,Q)
-f
is surjective
Z P
, H,(B(n,p) , Q)) is
yields a surjection H*(f,Z)
X
[X, B(n,p)l
[XI, B(n,p)]
+
There exists a : X
-+
IIB(niy p )
injective
then
is a s u r j e c t i o n .
so that
H*(a,
i
zP
is s u r j e c t i v e .
5.4. H-maps i n t o B(n,p) W e study here conditions f o r H-maps
to
X
5.4.1.
-+
B(n,p).
We choose
Proposition:
Given a map
a: X
+
Let
K(Z n ) t o be H-liftable P' t o have i n t e g r a l p-torsion k-invariants.
B(n,p)
X, p
X
-+
be an H-space,
IIB(ni, p ) = B ( X )
so t h a t
H,(X,Z)
torsion free. is s u r j e c t i v e .
H*(a,Z)
i
There ex-ists an H-structure for
Proof:
H*(a,Z)
so that
a
is an H-map.
s u r j e c t i v e implies H*(a
surjective (as
B(X)
H*(X,Z)
A
a, Z):
H*(B(X)
is torsion free).
A
B(X), Z)
By 5.3.4
-f
H*(X
A
X, Z )
182
(a
BP
Non s t a b l e
a)*:
A
[B(X) A B(X), B(X)]
E im(a
HD(a, p ,
3.4.2.
[X
A
X, B(X)]
i s s u r j e c t i v e and
Now apply 1 . 5 . 1 ( a ) .
a)*.
Let
Proposition:
coker *;
A
-+
resolutions
X , 1~ be an H-space.
Suppose H * ( X , Z )
and
are p-torsion f r e e .
Then for any primitive element fx: X + B ( n , p )
80
x E Hn(X, Z ) P
there e x i s t s an H-map
that a [ f x ] = x.
5.4.2 follows from t h e more general proposition.
5.4.3.
N
H-map
Let
Proposition: N
h: Y ,
+Y,
X,
u be as in
Given H-spaces and an
5.4.2.
so that n,(fiber h) are f r e e and suppose there
p0
e x i s t s a commutative diagram of H-spaces and maps:
-
the product muZtiplication.
t o an H-map
Proof:
N
f: X ,
p +
?,
Then any H-map
f: X ,
p +
Y,
uo
Efts
N
p.
One uses a Postnikov decomposition of
h
and induction:
H-maps i n t o
183
B(n,p)
a
Y
K( a,(fiber h ) ,
x
1
x-’
/
x
Pn
,n-1
N
fn-l
N
Y = 0
N
Given an H-map N
t o a map
f:
N
fn-yFn-l:
x, u
N +
yn-y
N
N
un-1,
the l i f t i n g of
X + yn exists by o u r s t a n d a r d k - i n v a r i a n t argument.
N
u , vn) =
jwny
X
A
X
W
K(a ( f i b e r h ) , n )
/j n \
P2an
,,/ -
ln,n-1
N
‘n-1’
X
Fn-l
N
’n-1
K(an(fiber h ) , n )
fn-l
184
Non s t a b l e
TI: n
The homotopy ( c o r e 1 E,,,
E,).
Fn(?A x
jwn +
(We assume
BP
-
?A)
resolutions
covers
n
N
Fn-l*
Phn,n-l
TINN n
Fn-l
t o be a f i b r a t i o n and l i f t i n g s i n t o
hn,n-l
N
t o be exact l i f t i n g s not only up t o homotopy).
Yn
i s an H-map
p2an
As
-
1~*
Q 1: $ ( X , +
A s coker *;
n
N
fn,
Hn(X
(hence,
A
X, Z ) 8 s n ( f i b e r h ) = H " ( X
coker
c*
-
=
(r*Q l ) v ,
N
f' = jv
n
v: X
+ ?n
-+
w E im n
K(an(fiber h ) , n ) .
Alter
furthermore, one has an H-structure
Fn
?A
for
fn
H*(i,Z)
r*0 1, to a lifting N
N
+
X, a n ( f i b e r h ) ) .
HD(yn, 11, u n )
and e s s e n t i a l l y by 2.2.1 w
A
free and
8 1) i s p-torsion
isomorphism i w E i m ( u * Q 1) implies n
P-p w
a n ( f i b e r h ) ) = Hn(X,Z) Q a n ( f i b e r h )
so that
and
=
II
Ph
F n,n-l n
N
N
Fn-l
( c o r e 1 E o , Em). This observation allows us t o pass t o a limit li.m(Yn,
'Fn) = Y, 'F: x
+ lim
f
Hn
N
= Y.
f
To apply 5.4.3 for t h e proof of 5.4.2 t a k e
The e x i s t e n c e of
a
Y = K(Zp.
follows from t h e observatoon t h a t
n),
H*(B(n,p), Q)
p r i m i t i v e l y generated and one has a 0-equivalence and an H-map $: B(n,p)
+
K(a,(B(n,p)).
necessary t o o b t a i n
$'I:
Replace B(n,p)
B(n,p) H Ip-p
by
B(n,p)(Cp),
K(*,(B(n,p))
= B(n,p).
and
$1 i f
is
H-maps i n t o
coker
The condition on
r’f i n 5.4.2
185
B(n,p)
and 5.4.3 can be recognized using t h e
following :
5.4.4.
, J, be a torsion f r e e connected graded
Let A,
Lemma:
algebra of f i n i t e type over coassociative.
h e n coker
if the dual of
A Q Z P
Proof:
Z.
Suppose
A
- - A 0 x) (J, : A -+
Hopf
is c o c o m t a t i v e and
i s p-torsion f r e e if and only
is a f r e e algebra.
F)J. and i m T c k e r ( F Q 1 - 1 Q F) . Cocomutativity implies i m F C ker(1-T). ( l - T ) k e r ( F @ 1 - 1 Q F) c k e r ( F @ 1 - 1 4 F), k e r ( F Q 1 - 1 4 F) = ker(1-T) n k e r ( F Q 1 - 1 Q F) @ W. A Q x = k e r ( F 4 1 - 1 Q F) @ V ( V = i m ( F 4 1 - 1 Q 8) C x Q Z Q which i s f r e e ) .
-A 0
J,
AS
i s coassociative
= [ker(l-T)
ker(FQ 1
-
(F
1)J. = (1 Q
18 F ) ] @ W @ V
and
im
F
As
lies in
t h e f i r s t summand., Coker
F
i s p-torsion f r e e i f and only i f
U = [ker(l-T) n k e r ( F @ 1 - 1 8 F ) ] / i m F i s p-torsion free. every f i e l d ker(TF 4 1
-
F
if
1Q
Now
ker(T Q 1
AF, 6 , , qF = A 0 F,
TF)/im TF =
U 4 F = [ker(l-T)
F = Z P
n
Q Cotor2’*(F,F) = 0
AF
U Q Q = 0 implies t h a t coker
- 18 i ) / i m
U
i s p-torsion f r e e .
6 0 F,
Cotory(F,F)
ker(TF Q 1
CotorE’*(Z,Z) $ Q F
i s t o r s i o n and
then
and
- 1 x TF)]/im -
i f and only i f
and for
%
U B Q = 0.
For
(+I* i s U 4 Z
P
= Q Cotor2’*(F,F).
J,F
= 0
a f r e e algebra. i f and only i f
186
Non s t a b l e BP
resolutions
A s a consequence of 5.4.2 and 5.4.4 one obtains:
Corollary: Let
5.4.5.
X, p
primitively generated and H,(X,
H*(X,Q)
commutative graded) algebra. by an H-mrp
f: X
Proof:
A = H*(X,Z)/torsion
H*(X,Z
As
Then every
P
) = A 8 Z
P
E*(u,Z)
5.4.5.1. H,(X,Z
P
E*(X,Z)/torsion 4 Z*(X,Z)/torsion
f r e e , and as
The non redundancy of t h e condition on t h e algebra
i n 5.4.5 can be seen by t h e following simple example:
$: K(Z,2)
-+ K ( Z p y 2 p ) .
f : K(Z,2) -+ B(2p,p)
H,(B(2p,p), vanish.
+
i s p-torsion f r e e and one can apply 5.4.2.
Remark: )
is
by 5.4.4 i f
r* i s p-torsion
Coker
Z ) a free (associative P x E PHn(X, Z can be r e a l i z e d P
i s coassociative and cocommutative and s o i s
p*: H*(X,Z)/torsion
coker
p-torsion free
i s a Hopf algebra and as A 8 Q
- then
H*(X,Z)
B(n,p) a[?’] = x.
-+
primitively generated A 0 Q A.
be an H-space
Zp)
This H-map cannot be l i f t e d t o an H-map
as t h e l a t t e r i s a mod p H-retract of
i s a f r e e algebra while i n
H,(K(Z,2),Z
P
1
BSU,
hence
p t h powers
P r o p e r t i e s of
5.5.
Examples:
Some p r o p e r t i e s of
w
zp)
and i f
0 # v~~ E PH
Zp),
H*(cP", 2n
(BUY Z
BU
B U ( 1 ) = CP" -+ BU
It i s c l s s s i c a l y known t h a t PH*(BU,
187
BU
induces an isomorphism pH2n+l(BU,Z) = 0
PH2n (BU, Z ) = Z ,
hence,
i n O # c E Z . then P v~~ = ~ ( ~ ) v ~ ~ + ~ ~ P( ~ - ~ )
P
A s a consequence one can e a s i l y see t h a t j
PH*(BU, Z P
5.5.1.1.
i s generated over
A(p)
by
PH2&' -2(BU,
@I
Zp).
a
O P C E Z . P
One has t h e following:
5.5.2.
Lemma:
Z )
H*(X,
P
We prove f o r
is injective, for
0
As
(as H*(X,Z ) P
2n
f: X
-+
(f, Z ) P
BU
an H-map.
If
is i n j e c t i v e f o r some
is i n j e c t i v e f o r a l l k.
Z ) P
p = 2
p-odd.
m = k(p-l)+n,
H*(X,Z
# H*(f.Zp)$(m;+,g
5.5.3.
be an H-space,
PH a+2k(p-1)(f,
Proof:
mod(p-1).
X, p
is a free a l g e b r a and if H
then
n < p
Let
k < ko.
i s similar. Let
Suppose
PHh(f,Zp)
mo = k 0 (p-l)+n = mbp + m,:
i s a f r e e algebra H * ( f , Z ) v 1 ~ 1 1 )# 0 P 2(m0 m: 2m and H O(fyZp)v2mo# 0 . = P H * ( f , Z )v,
P
)
implies
-0
i s a f r e e a l g e b r a ) and
P<-'(f,
Z )v # 0. P 2m0
P
Proposition ([Peterson]):
Proof:
BSU w
Il B(2n,p). n=2
m
Let
qo: BSU
Hence t h e k-invariants of
Ln
K(Z, 2n).
Put
B ~ U= B S U ( { ~ I ,$1.
n=2 BgU
a r e integral. of order power of
p.
188
Non s t a b l e
P Htpp(B^SU) = II K(Z,2n). n=2 and lift
P ll B(2n,p) n=2
B5U
Lift
BP
resolutions
P
II K ( Z , 2n)
+
to
a2: B^SU
-+
n=2
-+
P Ht2pp(BkJ) = ll K(Z,2n) n=2
a
P
II B(2n,p)
A BSU
1
to
al:
P ll B(2n,p) n=2
P ll B(2n,p) n=2
-+
BkJ.
a
n=2
i s a p-equivalence.
by 5.1.2 ( b ) a2a1
Looping twice one obtains
n=2 2
As s2 a1
i s a homotopy equivalence through
injective for
are injective. 2
s2 al,
composition
By 5.5.2 (as o n l y
n < p.
kJ
applies t o
hence
n=2
88
well)
8a28al
mod p
2p
2n 2 PH (a al,
Zp)
arguments a r e used it
2 PH*(Q2al, Z ) and consequently H*(Q a 1’ zp) P p-equivalence implies H*( $a,, zP s u r j e c t i v e ,
consequently a1 BSU +B$U
dim
and
a2
P
a _ ll B(2n,p).
axe p-equivalences and s o i s t h e
As BU w K(Z,2) x BSU,
n=2 P BU w ll B(2n,p) (B(2,p) P n=l P- 1 SU w II B(2n+l, p ) . n=l
5.5.4.
Corollary:
generated, aZgebras.
H,(X,Z)
Let
X,
FJ
K(Z,2)).
S i m i l a r l y one can argue t h a t
u be an H-space.
p-torsion f r e e and H*(X,Z
H*(X,Q) P
)
primitiveZy
and H,(X,Zp)
If x E PH2”(X, Z ) n < p then there e d s t s an H-map P
free
P r o p e r t i e s of
x E i m H*(f,Z
f: X +B(2n,p),
By 5.4.5 an H-map
Proof:
and H * ( f ,
P
exists.
f
189
BU
Z
injective.
P
By
BU w
P
B(2n,p)
and
n=l PH*(f, Z ) P
5.5.2
and consequently
Let X,
Z ) P
H*(f,
be an H-space.
are injective.
5.5.5.
Corollary:
F = Z
and 8 as Hopf algebras i f and only i f X, P
P
p
Then H*(X,F)
w H*(BU,
F)
i s H-p equivalent
m
, t o BU = B U ( { ~ I $1,
Proof: -
(q: BU wO_
II K ( Z , 2n)). n=l
One can obtain an H-map P
By 5.4.5
l i f t s t o an H-map
f
2P
i n j e c t i v e and as
5.5.6.
H*(X,Zp)
Corollary:
f : X +BU.
Z
w H*(BU,
IX,
pi f o r every
h e n X , M E GH(Bu) i f and o n l y i f
Proof: every H*(X,F)
p
H*(X,Z)
there e x i s t s
w H*(BU,Z)
f : X* P
p. W
e x i s t s an
If
f: X
This implies H*(BU,F)
H
for
H*(X,Z
P
H*(X,Z)
FJ
F = Z
or
p-equivalence
f
P
BU}.
P1 there
BU.
-$
X E GH(BU)
w
as a Hopf
Moreover, i n t h i s case f o r every f i n i t e s e t of primes
e&sts an H-map
is
kt
GH(BU) =
a7gebra.
Z ) P
H*(f,
i s a p-equivalence.
f
P
By 5.5.2
-
P'
X
)
FJ
H*(BU,Z
H*(BU,Z).
Q.
P
)
If
as a Hopf a l g e b r a f o r H*(X,Z)
w H*(BU,Z)
By 5.5.5 f o r every BU
and
p
there
X, p E GH(BU).
then
Non s t a b l e BP
resolutions
P1 i s a f i n i t e set of primes,
If
f : X + BU
P
P
(a =
Il
as
i s an
p)
is
BU
H-P1
Let
Corollary:
X
be a CW ecwpZex,
equivalence.
2n+2k(p-l) + 1 rank &H (x,
k
and i f
GH(BU)
H*(X,
Z ) P
i s uncountable.
an exterior
If
algebra on odd d i m e m i o m 2 generators.
is independent of
and HC
KP1
One can e a s i l y argue (and s e e 4.6.8) t h a t
5.5.7.
HA
zP)
= an
P2m: Qi-12m+1(X,Zp)
-+
Qi-Ianp+’(X,Zp)
injective
a ll B(2n+l, p ) n.
P-1 then
X W
n=l Proof: -
Z and H,(QX, Z are f r e e algebras, P P p r i m i t i v e l y generated and one can apply 5.5.4.
5.6.
H*(S2X,
Non s t a b l e BP
Adams r e s o l u t i o n s .
5.6.1. D e f i n i t i o n : Given an %spectrum E, = a strong E,
H*( X, Q)
A h s resolution f o r
X
{E
n’ Y n 1
and a space
X,
i s a sequence o f spaces and maps
Non s t a b l e
BP
19 1
Adam resolutions
so t h a t ( a ) hn:
Bn(x)
+
in-l(x)
(b)
If t h e f i b e r of
(c)
For every
i
i s a p r i n c i p a l f i b r a t i o n induced by a map
is
fn
and
n,
a n
[e,(X), Ei]
::?i
l i m an = n-
connected then
*
[X, Ei]
-.
is
s u r j e c t i v e and
5.6.1.1. spaces
Remark: X with
Note t h a t t h e c l a s s i c a l H,(X,Q)
K(Z P'
)
Adams r e s o l u t i o n f o r
# 0 does not satisfy 5.6.1 ( b ) but some weaker
property holds t o obtain an Adams s p e c t r a l sequence, hence, t h e r e s o l u t i o n i s s t i l l a good estimate f o r
X.
The e f f i c i e n c y of
spectrum f o r an Adams r e s o l u t i o n f o r
X
E,
as t h e underlying
i s measured by t h e r a t e of
Non s t a b l e
192
an
i n c r e a s e of with
H,(
,Z)
i n 5.6.1 ( b ) .
BP
resolutions
e.g.:
b e i n g p-torsion) t h e
monotonic non decreasing sequence
For p-torsion spaces ( i . e . :
,Z
K(
P
{an)
)
spaces
resolution yields a
-+ m.
{ B ( n , p ) ) do not form an a-spectrum as
B(n,p)
SB(n+l, p )
FJ
only f o r
But t h e s e r e l a t i o n s are enough t o e s t a b l i s h Adams r e s o l u t i o n s u s i n g {B(n,p)).
We s h a l l refer t o such a r e s o l u t i o n as a fnon s t a b l e )
BP
Adam8
resolution. We s h a l l o u t l i n e t h e c o n s t r u c t i o n o f t h e r e s o l u t i o n f o r spaces Z ) a free algebra. P c o n s t r u c t i o n can be obtained f o r an H-space X w i t h H,(X,Z)
with
H,(X,Z)
p-torsion f r e e and
H*(X,
X
A similar
p-torsion
free.
5.6.2. H*(X1,
Let
Proposition:
f : X1
-+
X
2'
Z ) / / i m f * are p-torsion f r e e ,
are f r e e algebras.
Let
Y = fiber f.
Suppose H*(Xi.
Z
Z)
H,(Xi,
P
and
Then: H,(Y,Z)
and
H*(X1,
Zp)//im
i s p-torsion f r e e
and the EiZenberg-Moore s p e c t r a l sequence: H*(X2, Tor
collapses f o r F = Z
P
generated by
F) (H*(X1,
and Q.
k e r &H*(f,F)
F ) , F) ==> H*(Y,F)
Moreover, i f
then f o r F = Z
P
M2
is the f r e e algebra
or Q
M
H*(Y,F)
as an H*(%,
F)
FJ
module.
f*
[H*(X1, F)//im H*(f,F)] Q Tor 2(F,F)
Non s t a b l e
Proof:
BP
Adams r e s o l u t i o n s
One can e a s i l y argue t h a t under t h e above hypothesis
W*(Xi, Z )
a r e p-torsion f r e e and i f
by
Z)/torsion
QH*(Xi,
Furthermore,
then
Ai
i s t h e f r e e Z-algebra generated
Ai 0 F wH*(Xi,
F = Z
F)
induces a homomorphism A( f ) : A2
f
A ( f ) Q F = H*(f,F). so that
193
P
-+
(Actually one can replace ;Xi,
Ai = H*(Xi,
Z)/torsion.
A1
SO
and
Q.
that
by p-equivalents
f
This i s done by s t a r t i n g with a
commutative diagram
then use 4.3.1 t o obtain
2.1
are t h e d e s i r e d spaces).
= Xi({p>, lli)
A1 = N1 Q N 2 , Mi
, Ni
i m QA(f ) , N1 w M1. As
rank(Mi Q F ) ,
A2 Q F
Then
Tor
rank(Ni Q F) A2 Q F
t h e same holds f o r
M2
f r e e algebras,
Tor
Now, one has generated by
A2 = M1 0 M2,
k e r QA(f), N1
by
M2 Q F
(A1 0 F, F) w (N2 Q F) Q Tor
a r e independent of
(4B F,
F).
F
( F = ZP
(F,F).
or 61)
Non stable BP resolutions
194
It suffices to show that the Eilenberg-Moore spectral sequence FEtw collapses for F = Q. For then, using a Bockstein spectral sequence argument to obtain rank Hn(Y, Zp) 2 rank Hn(Y,Q) rank Hn(YaQ) =
=
C
rank mE!
C
rank Tor
q+q'=n
q+q ' =n
999'
and one has: =
C
q+q'=n
rank
H* ( x2 ,Q) 9a9'
(H*(X,,
%'9,q' =
Q), Q)
=
= rank Hn(Y, Z ) 2 rank Hn(Y,Q)
P
2
Hence all the (weak) inequalities are actually equalities and
'E2
=
Z 'Em.
Moreover, rank Hn(Y, Z ) = rank H"(Y,Q) implies the collapse of the P Z - Bockstein s.s and H*(Y,Z) is p-torsion free.
P
The collapse of the Eilenberg-Moore spectral sequence f o r can see by comparing
F = Q one
Non s t a b l e
k(Ni)
where
'"Mi
= K(QIVi))
BP
H*(?(Ni,
195
Adam r e s o l u t i o n s
Z),
Z ) / t o r s i o n = Ni
and s i m i l a r l y f o r
1.
k(M2) observe
fiber k ( A ( f ) ) = K(N2) x
To s e e t h a t
The c o l l a p s e of t h e s.s f o r t h e r i g h t hand s i d e f i b r a t i o n o f 5.6.2.1 i s obvious
.
I f one a p p l i e s 5.6.2 f o r
5.6.3.
x1
= w
one g e t s :
Corollary:
(a) Let
X
be a space with H,(X,Z)
a f r e e algebra. F) = Tor
H*(QX,
Ib) I f
X1s
X2s
H*(Y,F)
as an j:
Qx2
Then H*(flX, H"
f, Y
N
Of
P
i s p-torsion f r e e and
z
P
or
Q) as coazgebras.
Q i r n H*(j ,F)
H* X ,F)//im H * ( f , Z ) 1
module and
from the Puppe sequence
Y
Z )
are as i n 5.6.2 then
F)//im H*(f,Z)
Q5
Z)
' y F ) ( F , ~ ) , (F =
H*(r, -+
p-torsion free and H*(X,
ax2 -L Y
-x1
f
x2.
i m H*( j ,F)
comoduZe,
Non s t a b l e BP r e s o l u t i o n s
If i n addition H*(C2X2, F) is a f r e e algebra (always holds for Q) then the above representation of
F =
i s an
H*(Y,F)
algebra representation.
5.6.4.
X be a space,
Let
free algebra. Let
W e construct a BP Adams r e s o l u t i o n f o r
fo: X
obviously of
el:
EO(X)
so t h a t
+
-+
klfO
-
to
-+
Let
kl:
EO(X)
f a c t o r s through H*(C(fO), Z )
-
B.
Lift
can be done as Lift
-+
F(K(A)) which induces
H*(E,(X),
fn
to
en
fn+l: X
-t
can be chosen
H*(En+l(X)y
Z)
t h e cone on
kl
C(fo)
M
exists, Mn
H*(X,
-+
-+
K(A)
f a c t o r s through
i (n
fi
kn:
Let
The cohomology of
to
C(fo).
H*(En(X),
-
En(X)
-+
K(&M,),
knfn
En+,(X)
E
n+l
i s p-torsion f r e e and
-*
(which
= f i b e r kn. (X)
(as H*(F(K(QMn)) = p ( n i , p ) , Z p ) 1
Z)
a f r e e algebra,
so that
F(K(&M,))
is p-surjective).
H*(fo,Z)
As
fo.
i s p - i n j e c t i v e and
Z)
Z ) 4 Mn, P k e r H*(fny Z p ) . Map
En+l(X).
5.6.3.
H*(EO(X),
k 1y
kn: En(X)
H*(fn, Z )
obtained by 5.6.2, algebra).
to
-
One can l i f t
Ei(X),
Zp)
generating t h e i d e a l
cnfn
(and
Lift
FK(A) = llB(mj, p ) , kl
C(fo)
i s p-torsion f r e e .
Suppose a r e s o l u t i o n
Mn
Zp)
Y.
el
p-torsion f r e e ,
zP).
A = ker QH*(fo, -+
X
j
i s a p-surjection
H*(C(fO), Z )
Q)).
H*(X,
yield a surjection H*(fO,
1
K(A)
Indeed,
C(fo)
EO(X) = GB(ni, p )
H*(fo.
Z ) a P as follows:
p-torsion free and
H*(X,Z)
can be is a free
Non s t a b l e
BP
Adams r e s o l u t i o n s
F i n a l l y w e s h a l l o u t l i n e t h e construction of a f o r an H-space
X
generated and
H,(X,
with
p-torsion f r e e ,
H,(X,Z)
Z ) P
197
BP
Adam r e s o l u t i o n primitively
H*(X,Q)
a f r e e algebra:
One s t a r t s with an H-map fo: X
Il
-+
H * ( f o y Z) p-surjection
B(nj, p) = EO(X),
j €Jo
X -+K(PH*(X,Z)/torsion) = KO
This i s done by l i f t i n g
fo:
X
Il
m(BKO) =
-+
B(nj, p )
(BKO
-
t o an H-map
t h e c l a s s i f y i n g space of
KO
XJ0 QBKO = K O ) .
H*(foy Z )
Let
i s p-surjective.
ll
=
E O ( X ) -+$,(X)
A c H*(EO(X),
k e r H*( f o y Z p ) .
algebra generating kl:
Z) i s a f r e e a l g e b r a one can s e e t h a t
H,(i2F(BKO),
AS
B(nj, p ) ,
Realize klfO
J€Jl
x
t o an H - ~ P fl:
fo
is in
im[X
A
-+
X, B1(X)]
Z ) P
A
- *.
be a sub-Hopf
by an H-map
E ( X ) = f i b e r kl 1
Let
G1(x)
E ~ ( x ) . Indeed, if B ~ ( x )=
-+
[X
A
X, E1(X)]
and as
X
A
X
-+
lift
then
E1(X)
E1(X)
A
y i e l d s a mod p s u r j e c t i o n o f i n t e g r a l cohomology by 5.3.5 ( c ) [E1(X)
A
B1(X)]
E1(X),
the multiplication i n that
A1
M
fl
+
i s an H-map and
Z ) P
-+
A
X, B1(X)]
by a map
E1(X)
i s an H-map.
im[H*(E1(X),
[X
i s s u r j e c t i v e and one can a l t e r E1(X)
B1(X)
-+
E1(X)
as Hopf algebras, as
X
+
H*(E1(X),
Z p ) w H*(X,
H*(B1(X),
Zp)]
E (X) 1
E1(X)
A
a f r e e algebra and. one can proceed by r e a l i z i n g
A1
E1(X)
+
B2(X) w
-+
TI B(nj, p ) e t c . J€J*
-+
so
Z ) Q A1, P
is central.
j,:
B1(X)
A
1
M
k e r H*(fly
by
E1(X)
2 )
P
is
Non stable BP r e s o l u t i o n s
5.7.
Some simple a p p l i c a t i o n s . The s i m p l i c i t y r e f e r r e d t o i n t h e t i t l e of t h i s s e c t i o n has t o do
only with t h e f a c t t h a t w e use mostly 3-stage computations.
Let
be an Ho
X
for
= 0
'nn(X)
0,
n < n
0
or
n -2p+3 &H O (x,
Proof:
space,
p-torsion free.
H,(X,Z)
n +1
#
(X)
r e s o l u t i o n s i n our
The a r i t h m e t i c a l c a l c u l a t i o n s however m a y be complex.
5.7.1. Proposition: If 'nn
BP
To:
0
(x, zp) #
0
0
n0-2p+3
zp) #
Map
then either &H
0
X
#
(X,Q).
QH
BO(X) =
-+
II B(ni, p )
so that
H*(fO, Z ) P
i EJ
is
We may assume t h a t f o i s e f f i c i e n t i n t h e sense t h a t t h e n. n n image of t h e composition H '(B(ni, p ) , Z p ) -+ H ' ( B 0 ( X ) , Z p ) -+ H i ( X , Z p )
p-surjective.
n. H '(
i s not i n t h e image of
Il B(nJ,p),Zp)
-+
n H i(Bo(X),Zp)
-+
n H i(X,Z).
j
Note t h a t
QHm(B(ni, p ) , Z p ) = 0
m :n
unless
i
(2p-2) and t h a t
ni +2k(p-1) (B(ni, p ) , Z ) # 0 f o r all k 2 0. L e t mo be t h e minimal &H P m m 0 i n t e g e r with k e r H ( f o , Z p ) # 0. Let A = k e r H O ( f 0 , Z ) / t o r s i o n and let
il: B O ( X )
+K(A, m )
m QH
'(klY
fl: X
-+
'nn(
1
fl
Q ) # 0. E1(X)
n < mO-1.
realize
0
-
ilfo
and
Z ) P
H*(fly
i s an isomorphism f o r
As
1
n < m
0'
n'
= A
o z
P
:
space
i,
fo
then
Obviously
dim
lifts to
5 mo,
'nn(E1(X))
(E ( X ) ) # 0, m -1 1 0
o zp
i s an Ho
i s an isomorphism i n
We s h a l l show now t h a t
(E ( X I ) Pn mO-1 1
X
E (X) = fiber
If
it.
A.
hence = 0
for
as a m a t t e r of f a c t
199
Applications
kl
obviously f a c t o r s as
BO(X)
II
-+
k1
B(ni,P)
K ( A , mo)
5
n <m
i- 0
i
ll B(ni, p ) n.=m 1 0
Now
zero map on
Z
d i v i s i b l e by
1
n
5
n <m i- 0
K(A, m ) 0
k1
B(niy P )
cohomology by t h e e f f i c i e n c y of
P p
n m (klil) Q Zp = 0.
and
must y i e l d t h e
k i is 11
Hence
fo.
Now, f o r any map
0
k i i ) : B(ni, p )
K ( A , mo),
-+
< mo
n
if
n m (kl( i1 ) 0 Zp = 0
then
i
Hurewicz homomorphism a n : nn(B(ni, p ) ) satisfies u
n
8 Z
minimality o f
n
P
(as 0-mod p
= 0
-
B(ni, p )
m
(kl) 8 Zp = n
m
n
mo-1 j
Q Zp @ (
(klil)
n
C
n.<m m~ 1 0
(klil
=
m ‘(k,,
Now, i f for no
i E J
m o - ~ + 2 ( x ,zp) # QH
QH
m 0 QH (klil,
mo
ni
Z ) = 0 P
implies
QH
Let
Proposition: H*( X, Zp)
-
X, p
0
( II B(niy P ) , Q ) iEJ
m0’
If for some
m
5.7.2.
x
(2p-21,
(X, Q).
n +1= m0’ 0
Suppose
ni
-
F i n a l l y observe t h a t
generated,
0
Q ) # 0 i m p l i e s QH
mo 2p +2
#
j E J
m
*I,QH
0
( k ( i ) ) 8 Z ) = 0. l P
E ( X ) = coker n ( k l ) , mO-1 1 m 0
consequently f o r some
as
Hence,
(E ( X ) ) 0 Zp = coker n (kl) 8 Z = A 0 Z 1 m0 P P‘
= no,
n > ni
Hn(B(niy p ) , Z )
0
Now one can e a s i l y s e e t h a t
n
-+
k-invariants w i l l v i o l a t e t h e
5.2.2 ( e ) ) .
0
as t h e
0
(B(moy p ) , Zp)
be an H-space,
j
but then
E J
n
,QH
# o
H*(X,Q)
# 0,
j
= m
o
m O(X,
zp’.
primitively
exterior algebra on odd dimensional generators.
is 2n connected
mod p
(i.e.
H2n+l
(x, zP) #
0
and
Non s t a b l e
200
(XI
Ht
P-1
a(n,p)
Proof:
in
PH*(X, Z
a j E PH
every Put
fo:
X
-+
resolutions
xi
where
satisfy:
i=l
I n t h i s range
PH*(X, 2 P
resolve
xi,
II
w
BP
i s a two s t a g e
X
a(n,p) + 2
dim
P
is a
Zp[P1]
by
fo,j: X
B O ( X ) = JIB(2nj+lY p ) .
resolution.
One can
as follows:
module,
2n .+l ( X , Zp)
BP
+
P1 E A(p)
B(2n +1, p ) , f o Y j - H-mp.
J
Note t h a t
J
i n dim
2
Let
a ( n , p ) + 2.
B(2nj+l, p) + K ( Z
kl,j:
PY
2n
'
+ 2rj(p-l)),
+ 2 r . ( p - l ) + 1) products t a k e n over d l J
N
mod(p-1).
= klyi:
Lift
J
BOyi ( X )
-+
Ki
t o an H-map
j
with
nj
5
i
201
Appli c a t i o n s
2i+1+2k( p-1) f o y Z p ) [PH
one can choose
fo
lifts t o
put
k
so that
l,i
fl: X
xi = Hta(n,p)
-+
i
- *.
klyif0
X ) , Z ) ] = PH2i+1+2k(p-l) (X’ZP) P Put
Ei(X) = f i b e r k l,i
which i s a p-equivalence i n
IIE.(X) 1 i
( E (X))
(Bo,i
Ht
and then
a(n,p)
( X ) mp IIXi
dim
5
then
a(n,p).
i s the
d e s i r e d decomposition,
To i l l u s t r a t e t h e use of t h e
BP
r e s o l u t i o n s for s p e c i f i c
computations we have:
5.7.3.
Proposition:
PlTm(s3)
If 4 5 m < 2p2
=
0
and
#
kl
i f
m = a-1
mod 2p-2
otherwise
Consider t h e r e s o l u t i o n :
have t h e p r o p e r t i e s :
H*(koyZp) (1 8 u4p-1)
(ui E PH1(B(i,p) ki
then
m f 1, 2
Z
ko
2
o if 0
Proof: -
-
, Zp)
-
H*(koy Z p ) ( ~ a + l
1
0 1) = P u3-
E PH4p-1(B(3y~).Zp)y H*(klyZp)u4p-l
a generator).
give r i s e t o t h e following tower:
k o y kl
are H-maps and
= $u2p+l klkO
-
@
0.
1
Non stable BP resolutions
202
s3
E2 = fiber i(. 1
I
sljl = Qkl. is an isomorphism for s 5 2p2-2 hence
HS(h 2,Z P )
isomorphism for m
2
2p -2.
2
To compute coker(a,(k B(3,p)
))
m
(h ) 2
is an
It can be seen that
m $ 1, 2, (2~-2)
0
m
T'
'[coker
T
'[coker
TI
m+l
(kO)]
m
2
(2p-2)
(k ) ]
m :1
(2p-2)
m+2 1
consider k B(3,p) 0,2:
-f
B(4p-1).
is a mod p retract of SU, PH*(kOy2, Zp )u4p-1 =
W
By 5.5.3, 4p-1 # 0 and
Z ) is an isomorphism in dim by &H*(kOy2, Z ) = PH*(k P 02' P 2 2 4p-1 5 b 5 2p + 2p-1. Hence for 2p-1 5 s 5 p + p-1 , s = 1 + k(p-1) , 2 2 k 5 p, one has
Applications
1
U
and degree
'II
2s+l
203
I
2s+l
u2s+1
degree i2s+l] * c,
= [degree
(k 0,2
c E Z,
(C,P) = 1. By a theorem of Bott (and as degree
a
2s+l
degree(a2s+l) Degree n
Now
2s+l
= s!
((k-l)p + p
retract of
SU)
and by the Brown Peterson c o n s t r u c t i o n
. cl,
k-2 = P (k ) 0,2
i s a mod p
B(3,p)
(c,,p)
= 1 and
k
i s given by
s - 1 = k(p-1)
i s then
- k+l)!
= c3(k-l)!p
k-1
1< k
p-1.
Hence,
degree ~ ~ ~ + ~= c( * pk , ~'[coker , ~ )~ ~ ~ + ~ (= kZ ~ , ~ ) 1 P'
To compute degree n2s+l(k0,1)y
isomorphism f o r
a).Hence,
m = 4p-1
H*(ko,ll
(as no
kOs1: B(3,p)
k
2p+l
B(2p-1, p )
I k 5 4p-1
ZP) = €I*( fi-2koy2),
consider
i s d i v i s i b l e by
,
204
Non s t a b l e
degree *2s+l(k0,1)
= degree n
2s+2p-1
BP
resolutions
= pec
(k 0,2
for
s = 1 + k(p-1)
~ ( klyl: k ~ B ()2 p + l , p ) + B ( 4 p - l Y p ) , 1 2 k 2 P-1. To compute ~ ~ ~ + put
.i
= kl
klYl
H*(klYly Zp)u4p-1 = u ~ + ~hence, ,
19
isomorphism for
5
4p-1
i
2 2p2
.
Thus, for
2
QHi(klyl,
Zp)
i s an
2 k 5 p-1
I"' Degree u
k
= c
pk-l,
1
degree .n,(k
091
)*degree a,(k
Consequently; degree II Hence,
'[coker
'[coker
71
2s+l
71
k-2 = c2p
6
degree
2s+l
2s+l
(k 1,2
( k ) ] = Zp 0
(k ) ] = 2 1 P
for
s
,
hence,
) + degree a,(k ) * d e g r e e n,(k 1,1 OY2 L2 = c'sp for
(k1,2:
s 'p,
2p-1
B(4p-1, p ) +B(4p-1, p s
3
1, mod (p-1)
s I 1 (p-1
.
and
5.7.3 follows.
A more t e c h n i c a l computation b u t along t h e same l i n e s leads t o
5.7.4.
Proposition:
Define
j , k, a,
-
O z a < p.
!&en
Given i n t e g e r s n , m, n 2 p-1, by
m = j
+
k(p-l),
0
5
j < p-1,
m 5 p(p-1) n+j = ap
- 1. +
a,
= o
1.
Applications
where
Y (b) i s the highest p m e r o f p dividing b. P [Krishnarao] 1.
where
(Compare t h i s A t h
1,2
Outline of computations: The Adams resolution in this range has just
3 stages and is given by: B
0
= B' x B1, 0
BA =
P-2
n
j=o
SU(n)
B ( a j , p),
fo
P-2 B1 =
-.. Yj = 2n+2j+1+2(p-l)(Z(:(j)+l) where a(j) by
n+j = a(j)p +
;(j),
su j=o
kg
,
BO
B ( Y j , p),
B1'
aj = 2n+2j+lY
is defined for every integer j
0 L c ( j ) < p. kOy1 = plko: SU + B A
a-
is given by
PH '(kOyl, Zp) is an isomorphism, 0 5 j 5 p-2, k0,2 = p2ko: SU +B1 given by PHv j (ko,2, Z ) P
is an isomorphism 0 5 j 5 p-2, kl is given by
YA klil: B + B satisfies PH j(kOy2,Z )u =pa(')+lu 0 1 P Yj klko N
y.
A s in
is
5.7.3 here too we have
aj'
ki H-maps and
Non s t a b l e
206
n'
2n+2s
BP
(SU(n)) = '[coker n
and t o compute these one has t o compute J = 1,2.
degree n,(k
2(p-E1)
times t o obtain maps
, ZP )
PH*(
resolutions
031
)
2n+2s+l(k0 11
degree n,(k
i,j
i s camputed by delooping
8U
+
T
i = 0,1,
SU + B ( a j , p )
B(2n+2J+l + 2(p-z1), p)
an isomorphism i n t h e range of i n t e r e s t .
'[coker
)
2n+2m+l(k0,l 11 = Z rl(n,m) P
with
One obtains
,
A d i r e c t computation y i e l d s
'[coker
n
2n+2m+l(k0,2 11 = Z r2(n,m) P
r2(n,m) = 0
klkO
-
w
if
k
28,
yields for
d = degree n 2n+2m+l(k1 , 2 1: P hence,
Y d = Y (a+k).p. P P
rl(n,m)
- = . pa+l
5.7.4 now follows.
As a f i n a l application of t h e BP resolutions we have:
r2(n,m) Yp(d) 9
Applications
5.7.5.
bundle 5
o
= H 4n+l ( y , z )
dim
>_ 4n.
Then f o r any complex vector
the obstruction f o r 5
of (complex) dimemion 2n+l
4-sections are the 4 top integral &ern classes of
Proof:
Suppose
i s 2 torsion free i n
and H*(Y,z)
and 3 torsion free i n
411-2
dim
Given a 4n+2 dimensional space Y.
Proposition:
H4n-l(y,z) =
207
t o hose
5.
The obstructions i n question axe t h e obstructions i n t h e
l i f t i n g problem:
BU( 2n-3)
(where
f
5).
classifies
To study t h e l i f t i n g problem one prime at a t i m e note t h a t i f i = l,..
hi
.,k,
- a mod Pi
p a r t i t i o n of if
axe spaces s o t h a t f o r every
equivalence i n
P, then
f lifts to
f?:
Let
P1 = I21,
Let
X3
Y
f: Y -+
xi
Then h
5 4n+2
{P.1
and
BU(2n+l) l i f t s t o
i .
f o r every
P2 = (31,
be t h e f i b e r of
t h e Chern classes.
dim
..,k
i = 1,.
is a
i f and only
BU(2n-3)
i.
P3 = P
-
{2,3].
w: BU(2n+l) factors
i’
f a c t o r s as
h
i
X
3
-+
n K(Z, j=o
BU(2n-3)
FJP <, 4:+2
-
4n-4+2j) 1
X3
realizing
-+
BU(2n+l)
208
Non s t a b l e
and t h e o b s t r u c t i o n s o f an
H
4n-4
n
i s odd,
3
l i f t i n g are obviously t h e Chern c l a s s e s .
consider
(ko, Z) = w 4n-4,
surjective.
f
re.solutions
X1 we d i s t i n g u i s h between two cases.
To c o n s t r u c t If
N
BP
koh
- *.
Such a
is surjective i n
H*(ko, Z 2 )
e x i s t s as
ko
dim < 4n+4.
H*(h,Z) kl
is
i s given by A
H*(kl,Z2)u4n+2
i s homotopic t o through
= Sp4’2~4n-4. B(4n-5,2) = QB(4n-4, 2 )
Ql.
d i m 4n+4,
By 5.3.1
hence
h2
klhl is a
- *. mod 2
d i m 4n+2.
For n
even t h e r e s o l u t i o n i s given by
+
El-
kl
B(4n+l, 2 )
QH*(klyZ2) i s a monomorphism equivalence through
Applications
As H*(Y,Z)
is 2-torsion free in d i m > 4n-4 kof
209
-*
if and only
if H*(kof, Q) = 0. As imH 4n-4+2J(ko,Q) are the Cherm classes f lifts 4n-1 4n+l to El and as by hypothesis H (Y,Z) = 0 = H (Y,Z) f lifts further to
"f:
Y+ X 1'
Similar arguments hold for 2n
1
W4n-2a
2n
2
5.
Here we have three cases:
mod 3
imH4n-2a (ko, Z)
mod 3
a = 1, 2.
210
Non stable BP resolutions
211
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Steenrod, N.E.
Studies 50.
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219
L i s t of symbols
The following a r e used as s t a n d a r d n o t a t i o n s throughout t h e s e n o t e s . We i n d i c a t e below t h e page where t h e y a r e f i r s t introduced.
A : X x X -> A : X ->
X X
x
X,
A
X,
t h e i d e n t i f i c a t i o n map
1
1
t h e diagonal
-
A = AA
-
re1 f
2, 18
core1 f
2, 18
[f]
homotopy class of
2
f
[XI homotopy type o f X [X,Y] = r [ f ] I f :
x
-+
Yl
h* = [ h , ~ ] g* = [X&l
( )#
t h e a d j o i n t of t h e homotopy pullback of
fo
t h e homotopy pushout of W
*,f
M
*,f
fo
and
and
fl
fl
3
= fiber f
4
= C(f)
4
{En, Y 1, $ -spectrum l n 'n,m'
'En> 'n'
'n,m
1,
h r i n g &spectrum
5, 6
F t h e f o l d i n g map
9
X,y
9
an H-space
L X t h e loop space of X E: X d
+
X,
t h e e v a l u a t i o n at
H f t h e category of H-spaces
11, 12 t
11 15
L i s t of symbols
220
PX t h e Moore ( f r e e ) p a t h space
LX
t h e based Moore path space
CX t h e Moore loop space
a : PX
+ R+
t h e length of a path
Eo, Et, Em: PX + X kt: X + PX P f , Lf,
t h e e v a l u a t i o n at
the length
t
0, t ,
constant p a t h
16
E
16
P X C P X x PX 2
Add: T2X + PX
16
t h e path a d d i t i o n
18
t h e p r e c i s e category of H-spaces
HfF
an H-map i n
f,F
15
H
18
fF
HA
a homotopy a s s o c i a t i v e H-space
1 9 , 58
HC
a homotopy commutative H-space
1 9 , 62
f+g =
!l*(f,d=
t h e d i f f e r e n c e between
Df,g
22
P ( f x g)A f
and
g
D = D
22 22
p1'p2 t h e H-deviation of
HD(f,p,
p')
K(G,m)
Eilenberg-MacLane space
Htn
homotopy approximation
25
27 42
t h e Moore space
44
homology approximation
45
M( G,n)
Han
f
0 = B(f,F,A,A')
the HA o b s t r u c t i o n
59
KN
homotopy n i l p o t e n t
64
HS
homotopy solvable
64
PA, &A, Pi
p r i m i t i v e s and indecomposables
t h e Steenrod operations
71
,
L i s t of Symbols
a = (x,E,y)
a cohomology operation
P
t h e s e t of primes
M
Pl equivalence
( f . c ) , (f.c.T), f
X(P ,$), 1
r,
mod
P1
P r o p o s i t i o n 4.3.1
MiX(Xi'
h
t h e genus of
X
Pi, $i)
F(X) = F
(PI
f i n i t e n e s s condition
homotopic t o
r(1-1, 4
G(X)
98
114
(f.c.H)
f(Pl,$)
2 21
(X)
B(n,p) = F(p)(K(Zp,n))
g
115 126 128
222
Index of terminology
We i n d i c a t e below t h e page where t h e following notions a r e f i r s t introduced i n t h i s volume.
Action
47, 48
F i n i t e n e s s condition
115
Adams r e s o l u t i o n
190
Generic property
147
BP r e s o l u t i o n
193
Genus
1 47
Homology approximation
44
Adjoint maps
3
Bocks t e i n
Homotopy
operations
86
s p e c t r a l sequence
85
Browder's implication theorems 96, 97
category core1 f re1 f Homotopy approximation
42
Homotopy a s s o c i a t i v i t y
1 9 , 58
Homot opy commutativity
1 9 , 62
Homotopy nilpotency
64
Homot opy s o l v a b i l i t y
64
Hopf a l g e b r a
69, 70
Decomposition of 0-equivalence 128
H-action
48
Eckmann-Hi It on d u a l i t y
H-deviation
35
H-homot opy equivalence
27
H - l i f t ings
37
H-mp
15
Brown Peterson spectrum
163
Category of H-spaces
1 5 , 18
Cohomology
4 98
generalized operations operations of order Connective f i b e r i n g
k
98 43
4
0-equivalence
1 1 4 , 115
Essentially different H-structures
27
Evaluation maps
15
Index of terminology
H-space mod P 1
non c l a s s i c a l
Ho space
9 136 209 157 1239 124
Implications
96
Mixing homotopy types
152
Moore decomposition
44
Moore loop space
16
Moore path space
15
Moore space
44
Pos tnikov decomposition
43
p r i m i t i v e decomposition
44
system
42
Principal fibrations maps of
49 50
Pushout
3
Pullbacks
3
P1 equivalence
114
]El
126
homotopy
P1 universal
115
%spectrum
4
223
Ring spectrum
5
Weakly p r i n c i p a l (w-principal) Wilson's
B(n,p)
53 172
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