Hopf spaces (North-Holland mathematics studies 22)

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 163

Introduction 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 +

#' 1 f#,h# as the adjoints of

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

( f ) # : X1

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

3

We r e f e r t o

(thus omitting t h e l e f t and right

f and h

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

O

y1

MfO¶fl

N

a r e commutative and f o r any space

(2;:

N

Yi + L ,

ghfo

h: L + W

N

zlfl)

( h ' : Mf

fo'fl

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

n+m

I

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

x

X *X

11: X x

X *X

u(x,*) = x = p(*,x).

F(x,*) = x = F(*,x)

Thus, an H-space

so that

i s a space and

X

F: X v X + X be defined by

"the folding map'').

u: X

where

X,p

uIX v X = F.

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: Proposition [Hilton]:

1.1.1.

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)

Let

i: A c X

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

and l e t

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

uhich

In particutar,

YI.

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.

+ C B v CA

A

Y v CY

B ,A

+

*

9

,

Y

1 -
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: X.

k-invariant of

c a l l e d t h e n-th

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),

dim 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 m

m

n im(X ) + = # p torsion of En(X) n=l P ll

En(X). 6;'(#

E*(X,Z ) P

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

87

Browder's Bockstein spectral sequence

and by

[torsion E"(X)I Q zP

[torsion E"(X)I Q z P

0

-1

*I

p:

En(X) Q Zp

En(X, Zp)

1 - I PI

[E"(x)/torsion~ Q z P

E~(X,Z

1

torsion E"(x) Q z

P

0

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:

3.3.6.

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

= 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

i s p - t o r s i o n free.

E*(X)

and

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

S.S.

Then x E E ~ ( x , z ~=) [ x , E ( z n

-+ En(Z r ) ,

prfr

P

is then represented by

11

if and only if the representation

can be Zifted t o f r : X

x

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

p

odd but has an

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

*;

+

p-k

k

E -(

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’

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

showing t h a t

K(Zp,

HD(i,Add,Add) = J(v + z*d + z )

of

5 i s an H-map by

where

z

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).

B

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

p

AS

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

= j?

( k e r H*(p,Z

and a l t e r i n g

i p = p 2 ( ~ 2 n ) ~ . AS

If then

However,

dl: K(Z,2n) -+ K ( Z 2np) PY

implies t h e existence o f

M

dl = dlp

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'

93

Browder's Bockstein spectral sequence

Using the commutativity of

K(ZY2np)

K(Zpy2np)

I

I

HD(~)(~A

p) =

= j p Y o = jv(p

P~HD('?&)

Now, the fact that 6 HD(e)p

p = j&(p

A

A

-

- pvo

is an H-map implies that HD(g) = j j ,

&To.

p) =

A p)

Again, j [ w ( p

p)

A

-

N

p v o ] = 0 implies

N

that j ( p

8:

A

p)

K(Zpy 2np-1)

- pvo

factors through fiber j which is K(Z

PY

2np) and by the surjectivity of H*(p

N

&(p

has

A

p)

= 8; 1(p

A

N

6(p

A

p ) = p v o = v(p A p ) ,

i = v+

i

To show that for any choice of

-

impossibility of -v = U*d + z Let

(;*)":

-

= H*(K(Z

P

6

and altering

p)

-

H*(K(Zp,a)

3

2

8'

by

5i1

z

E 1 8

8

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

A

(;*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