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22
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HOPF SPACES
This Page Intentionally Left Blank
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
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
North-Holland ISBN: 0 7204 0553
X
PUBLISHERS :
NORTH-HOLLAND PUBLISHING COMPANY NEW YORK OXFORD AMSTERDAM DISTRIBUTORS FOR THE U.S.A. AND CANADA :
AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017
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
This Page Intentionally Left Blank
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 -
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 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),
dimu = 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
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 = #
and suppose
0
x1 =
put
0
y'-$
prove now t h a t
B
3.2.8 and 3.2.10
{?I
XP =
r1 = ro,
-x E .:E
Br(xP-'y)
# 0, hence, =
r+l
2,
{p-' - Bry) #
?-$r
computing the