H-Spaces from a Homotopy Point of View (Lecture Notes in Mathematics 161)

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and Bo Eckmann, Z0rich

161 James Stasheff The Institute for Advanced Study Princeton / NJ / USA

H-Spaces from a Homotopy Point of View

$ Springer-Verlag Berlin-Heidelberg • New York 1970

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar meac~s,and storage in data banks. Under §.54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to he determined by agreement with the publisher. © by Springe~r-VerlagBerlin. Heidelberg 1970. Library of Congress Catalog Card Number 71-154651 Printed in Germany. Title No. 3318

DEDICATION

In fond memory who first

revealed

yet optional nature

of G e o r g e

Yuri Rainich

to m e t h e d e e p s i g n i f i c a n c e of a s s o c i a t i v i t y

Preface

T h e s e n o t e s h a v e t h e i r o r i g i n in a c o u r s e g i v e n a t P r i n c e t o n U n i v e r s i t y in t h e f a l l of 1968; I a m m o s t g r a t e f u l t o P r i n c e t o n f o r p r o v i d i n g t h e o p p o r t u n i t y to g i v e s u c h a c o u r s e a n d to t h o s e i n s t i t u t i o n s w h i c h p r o v i d e d s u p p o r t d u r i n g the p r e p a r a t i o n of t h e s e n o t e s :

P r i n c e t o n , t h e U n i v e r s i t y of N o t r e D a m e ,

The

I n s t i t u t e for A d v a n c e d Study, the A l f r e d P. Sloan F o u n d a t i o n and NSF G r a n t GP-9590.

T h e c o u r s e i t s e l f w a s p r e s a g e d b y a l e c t u r e at t h e M i c h i g a n

C o n f e r e n c e on the T o p o l o g y of M a n i f o l d s [ S t a s h e f f ] .

State

The p r e s e n t a r r a n g e m e n t

of t o p i c s o w e s m u c h t o a s h o r t c o u r s e g i v e n a t B o s t o n C o l l e g e in t h e f a l l of 1969.

F i n a l l y , m y d e e p g r a t i t u d e to M r s .

A n n G o s l i n g of P r i n c e t o n U n i v e r s i t y

a n d M i s s E v e l y n L a u r ~ n t of T h e I n s t i t u t e of A d v a n c e d S t u d y f o r t h e i r f i n e p r e p a r a t i o n of p r e l i m i n a r y

and final t y p e s c r i p t s

of t h e s e n o t e s .

An a t t e m p t h a s b e e n m a d e to b r i n g t h e s e n o t e s up to d a t e , b u t c u r r e n t a c t i v i t y is s u c h t h a t t h e d a t e in q u e s t i o n is at b e s t t h e e a r l y spring of 1970. It is h o p e d t h a t m o r e r e c e n t r e s u l t s w i l l b e c o v e r e d a t t h e C o n f e r e n c e on H - s p a c e s to b e h e l d a t the U n i v e r s i t y of N e u c h a t e l in A u g u s t , 1970, f o r w h i c h t h e s e n o t e s should provide adequate background. Bibliographic references

a r e g i v e n in t h e f o r m [Hopf].

Where a given

a u t h o r h a s m o r e t h a n one e n t r y in the b i b l i o g r a p h y , the v a r i o u s e n t r i e s a r e d i s t i n g u i s h e d by u n d e r l i n i n g s ,

Princeton,

e.g.,

[Hopf] v s .

[_Hopf].

N e w J e r s e y and

Lansdale, Pennsylvania

S p r i n g , 1970

TABLE

OF

CONTENTS

Chapter PREFACE

...............................................

INTRODUCTION

..........................................

THE

HOPF

THE

PROJECTIVE

MAPS

CONSTRUCTION

INTO

iNVERSES,

AN

PLANE H-SPACE:

OTHER

4

ASSOCIATIVITY:

5

H-SPACES

6

THE

7

HOMOTOPY

8

MAPS

9

SPACES

3

.................................

7

ALGEBRAIC

SPACES

ARE

FINITE

CONSTRUCTION

H-SPACES INDUCED

I0

DLFFERENTIAI~

Ii

A

-SPACES

AND

STRUCTURE, ETC ................ TOPOLOGICAL

COMPLEXES.

SPECTRAL

ASSOCIATIVITY

OF

I

................................

MULTIPLICATIONS, LOOP

WHICH

BAR

V

GROUPS

. . 14

.................

SEQUENCE

.............

...............................

.....................................

BY

H-MAPS

IN T H E

BAR

10

................

..........................................

23 27 31

............................. CONSTRUCTION

20

38 44

• • • • 48

n

IZ

MASSEY

13

HOMOTOPY

14

STRUCTURE

15

INFINITE

16

OPERATIONS

IN I T E R A T E D

REFERENCES

............................................

PRODUCTS

AND

GENERALIZED

COMMUTATIVITY ON LOOP

BAR

CONSTRUCTION..

59

..............................

65

B x ...................................... SPACES

................................... LOOP

SPACES

...................

71 75 81 89

H-SPACES

FROM

A HOMOTOPY

POINT OF VIEW

by J a m e s Stasheff

The concept of H-space

e v o l v e d f r o m t h a t of t o p o l o g i c a l g r o u p .

w a s Hopf [Hopf] w h o f i r s t c a l l e d a t t e n t i o n to m a n i f o l d s cations,

and many basic

It

with continuous multipli-

i d e a s i n t h e f i e l d a r e d u e to h i m .

The

H

in H-space

m a y b e t a k e n in h i s h o n o r . The concept of H-space as a significant generalization results

is valuable both because

of its p a r e n t

and because

in t o p o l o g i c a l g r o u p s w h i c h a r e n o t a c c i d e n t s

in the c a s e of L i e g r o u p s , This course

it occurs

in n a t u r e

it h e l p s to e l u c i d a t e

of the e x t r a a l g e b r a

or,

the extra analyticity. is a survey

homotopy point of view.

of the c u r r e n t

s t a t u s of H - s p a c e s

Homology or cohomology,

ordinary

from a

or extraordinary,

w i l l b e u s e d a s a t o o l , b u t w e a r e n o t i n t e n t o n t h e t h e o r y of, f o r e x a m p l e , algebras

per se.

ing s p a c e s

We w i l l b e p a r t i c u l a r l y

and their

as

concerned

with loop spaces,

Hopf

classify-

iterates.

Let us begin.

D e f i n i t i o n 1.

An H-space

that for some point for

e

consists

we have

of a s p a c e

e x = x = x e.

X

and a map

m : X × X-~ X

[Where reasonable,

we write

such x7

m(x,y). ] Several

of "topological and existence

comments

group"

a r e in o r d e r .

That " H-space"

is a generalization

is obvious; we have dropped the conditions on associativity

of i n v e r s e s .

S i n c e w e a r e a d o p t i n g t h e p o i n t o f v i e w of h o m o t o p y t h e o r y ,

it w o u l d

be natural and

to require

m IX × e

only a homotopy unit,

be homotopic

i.e. , require

to the identity rel

has the homotopy extension property, can be deformed

then m'

X

to one with precise

has the homotopy

with exact unit

e'

t y p e of

= I

e.g. , the category

will say means

"X

is

CW"

X'

extending

W e i n t e n d to o p e r a t e nice,

of s p a c e s

For

to indicate

a similar

the s e t of p a t h c o m p o n e n t s

with homotopy unit e

i s c l o s e d in

m

has

e

where

spaces

are at least this

on s p a c e s

of greater

locally is

out elsewhere

in t h e C W - c a s e ,

this

A comparable

generality

we will often assume

[We

Essentially

and r u l e out l o c a l p a t h o l o g y .

can be studied,

X,

as homotopy unit.

X belongs to this category.]

reason,

X)

as a branch X

of

is connected;

as a discrete

set

operation.

From is o f t e n i r r e l e v a n t ,

t h i s h o m o t o p y p o i n t of v i e w , as we shall see,

will play a significant

any such additional structure, general.

(X × X , X v

which can be given a multiplication

to h o m o t o p y t h e o r y b u t is c a r r i e d

point set topology.

homotopy,

if

if

if

m Ie × X

of the h o m o t o p y t y p e of C W - c o m p l e x e s .

s t u d y of c o n t i n u o u s m u l t i p l i c a t i o n s

with a binary

More directly,

-- XeUoI m,

However,

a multiplication

in a c a t e g o r y

we f o c u s on g l o b a l p r o p e r t i e s

less appropriate

unit.

(e,e}.

that the maps

the existence

but associativity,

of h o m o t o p y i n v e r s e s

b o t h s t r i c t a n d up to

role in our development.

Before

considering

we look at what can be said about H-spaces

in

THE HOPF

Geometrically, the Hopf construction. famous

fibratfons=

D e f i n i t i o n 1.1. H

in

the outstanding

S 3 - + S 2, S 7 - ~ S 4 a n d

of t h e m u l t i p l i c a t i o n

l i k e t h i s to p r o d u c e

is

his

S 15-~ S 8.

m : X X Y -* Z, t h e H o p f c o n s t r u c t i o n

is the map given by

join : X × I × Y / R

consequence

H o p f [Hopf] u s e d s o m e t h i n g

Given a map

: XsY-~ SZ

CONSTRUCTION

where

R

(x,t,y)-~

(t,xy).

is t h e r e l a t i o n

[Here

X~Y

is t h e

{(x, 0, y) ~ (x, 0 , y ' ) ,

(x, t , y )

(x, ,1,y)} Theorem

1. 2. [ S u g a w a r a ] .

If (X, m )

i s a CW H - s p a c e ,

then

H

is a q u a s i m

fibration,

i.e.

'

H

: ~ri{X~X, H

m ~

- l ( b ) ) -~ wi(SX, b)

m

is a n i s o m o r p h i s m

for any

be SX. Rather

than prove this theorem,

we study an alternate

f o r m of t h e

Hopf construction. Definition 1.3.

Given

H(ml : X × CY~JZ-~ m Theorem

1.4.

equivalence

SY

Proof.

y, t h e n Since

5.

If p :

If X = Z

H(m)

m(

and

,y}

Y.

is a w e a k h o m o t o p y

Z-~ # are qnasifihrations

follows from fundamental

P I : P - l ( U ) -~ U, p I : P - I ( V ) -~ V sois

and

of q u a s i f i b r a t i o n s

E-~ B

and

Of X ~ Y o n t o

is a q u a s i f i b r a t i o n .

X × CY -~ CY

the theorem

T h o m on t h e c o n s t r u c t i o n Theoreml.

is i n d u c e d b y p r o j e = t i o n

[D_old a n d L a s h o f ] .

for each

fact, bundles),

m : X X Y -~ Z , t h e H o p f c o n s t r u c t i o n

B = U~.JV and

theorems

(in

of D o l d a n d

[Dold and Thorn]: where

p I : P-l(U~V)

U, V

are openin

-~ U ( - ~ V

are

B

andif

q. f. s

p.

Theorem

1.6.

If p : E - - B D A

exist deformations

Dt : E-~ E

and and

p] : p ' l ( A ) dt : B-~ B

: D -~ A such that

is a

q. f a n d t h e r e

then

D 1 : id, D 0(E) C D d I = id, d 0 ( B ) C A

and

d0P = p D 0

DOI . : p ' l ( b ) - . p ' l ( d o ( b ) ) then

p

is a

is a weak homotopy equivalence

q.f. T h e c o n d i t i o n on

m(

e n c e of a r i g h t u n i t b e l o n g i n g to

, y) Y.

follows for connected It a l s o h o l d s if X = Y

Y from the existis CWas

we now

see.

Theorem map:

1.7.

~ugawara].

(x, y) -~ (xy, y) Proof.

If X

is a c o n n e c t e d

H-space,

then the shearing

is a w e a k h o m o t o p y e q u i v a l e n c e . I c l a i m t h e i n d u c e d m a p of h o m o t o p y g r o u p s :

~r.1(X)~ Tri(X) -~ wi(X ) O ~ri(X) is given by 4, ~ -~ = + 6, ~, which is clearly an isomorphism.

To

verify the claim, w e use a lernrna. Lemma

1.8 [Hilton].

If X

is an H-space,

the usual operation in ~r.(X) is 1

induced by the multiplication Proof.

T h i s c a n b e p r o v e d a s in t h e u s u a l p r o o f f o r t h e c o m m u t a -

t i v i t y of lr1 of a n H - s p a c e . l o o p s p a c e of X

in X.

In f a c t , b y r e g a r d i n g

Iri+l(X)

and using the induced multiplication

as

of l o o p s ,

w1 of t h e i - t h the usual proof

applies directly. The restriction group under

"X connected"

shearing Remark.

CW H - s p a c e ,

a n d h e n c e s o is t r a n s l a t i o n

m a p : x, y - ~ x, x y By altering

m(

the shearing

, y).

is a

is c o n n e c t e d

CW.

map is a homotopy

The left unit is relevant

to t h e

in t h e a n a l o g o u s w a y .

t h e t o t a l s p a c e of H ( m )

but without changing the homotopy

type, we can obtain a map with the weak covering X

to " l r 0 ( X )

m.". Thus for a connected

equivalence

can be weakened

This follows from results

homotopy property

of D o l d if

of D o l d on t h e c o n s t r u c t i o n

of

such fibrations

[Dotd].

being c~ntent ourselves Theorem unit

1.9.

and both

X X CYUZ m

Proof.

Both spaces

sequence,

1.10.[S.ugawara].

Proof.

are simply connected.

x-~ (e,t,x). If

X

X X CYUZ

via the Meyer-

t h e f i b r e in

H(m)

is c o n t r a c t i b l e

T h i s i s in f a c t a c h a r a c t e r i z a t i o n .

is connected

CW, then

X

is an H-space

of s p a c e s

if

which are CW

of X

in

of p E

to

b y t h e s u m of ~ p

is g i v e n b y e ¢ X. and

H(m).

Map

For the converse,

X-~ ~B

X-~ ~B.

by

kx(t)=p

If t h e d i a g r a m

kt(x).

of h o m o t o p y

sequences: ,

,r. ( x )

.... Ir.(E) i

z

~ Ir.(B) i

~ ~ri - I(X)

~ i . l ( ~ E ) ~ - " ~ Tri_I{QB)

?

iri.,l(~E × X)--

•i s c o m m u t a t i v e ,

it follows that

~B

p

is a

q.f.,

@[f] = [h I S n] then

h

let

h:CS n-~ E such that

f o r a n y c h o i c e of h.

can be taken to be

In p a r t i c u l a r ,

h(t,x) =kt(x),

sothat

is t h e i d e n t i t y .

Tri_I(~B) -~ ~ri.l(X)

g : S ( S n) -~ B b e t h e a d j o i n t ,

there exists

~i_I(X)

~ri(X) -~ w i ( ~ B ) -~ ~i{X)

The most direct way to see this is to recall how f:S n-~ ~B,

] •

h a s t h e w e a k h o m o t o p y t y p e of ~ E X X.

T h e p o i n t t o c h e c k is t h a t

Given

-~

in h o m o l o g y .

is a n H - s p a c e ,

The existence

~E × X-* ~B

The map

in t h e t o t a l s p a c e .

kt be a contraction

Map

and there is a left

t y p e of X * Y .

i s t h e f i b r e of p : E - ~ B', a q u a s i f i b r a t i o n

with fibre contractible

let

Y are connected

induces an isomorphism

in t h e t o t a l s p a c e ; e. g. , v i a

and only if X

and for the time

m a p is a m a p of t r i a d s w h i c h ,

N o t i c e t h a t if X

Theorem

X and

has the weakhomotopy

induced by the shearing

Vietoris

result

with quasifibrations.

If Y = Z

e • Y, t h e n

X~Y

We will not need this stronger

is defined.

i . e . , g ( t , x ) = f(x) (t).

ph~ g if f

rel

t = 1.

comes

hlSn ;f '.

from [If p

Since

We d e f i n e f':S n-* X isa

Hurewicz

fihring,

the homotopy

then

a

h t ( k ) -- k ( t ) , Now all spaces

of ~ E

X X.

Theorem

Theorem

1.11.

If X

is realized

by a map

starting

f 0 ( ~ ) -- e e X C E . ]

at

being CW,

~ B -~ X

the loop space

1.10 c a n n o w b e c o m p l e t e d is a retract

of a n H - s p a c e

~B

given by a covering

has the homotopy

Use

type

by the following remark. Y, t h e n

X

admits

a multiplica-

tlon with unit. Proof.

of

X X X-~ Y × Y-* Y-~ X

and check on units.

THE

PROJECTIVE

The full significance most

clearly

Definition

in the mapping

2.1.

p i n g cone of H(m), i . e . ,

E.G.

to

Z. 2.

H*(XP(2))

plane

which is isomorphic

XP(2)

of a n H - s p a c e

XP(2) = RP z

X -- S 1

XP(2)

X = S3

XP(2) = QpZ

X = S7

XP(2)

= Cp 2 = SZUe

i s usually e x p l o i t e d

through

are classes

non-trivially, secondary unless

giving a proof,

[Adams].

Proof.

If u

therefore,

or extraordinary,

n = 0,1,3,7.

its cohomology. S u , S v c H v (SX) p u l l b a c k

from

S(uSv) E H*(S(X*X))

H$ ( X P (2), S X ) .

S (u'v)

2.3.

16

such that comes

Su

Theorem

is the map-

8

= Kp 2 = S8Ue

-~ H* (S (X*X) ) --,- H * ( X P ( 2 ) ) - ~

Before

(X,m)

4

S4Ue

( S u ) ~ J (Sv) ~ H # ( X P ( 2 ) ) to

shows up

C ( X x C X U X ) . W=~m )SXm

If u , v e H $ ( X ) then

of t h e H o p f f i b r a t i o n s

H(m).

X = SO

This space Theorem

of t h e e x i s t e n c e

c o n e of

The projective

PLANE

Sn

-*

Su

S u -~ Sv

let us look at the consequences.

is an H-space

generates SuUSu

-~

H * (SX)-~

Hn(sn), # 0.

one shows,

if a n d o n l y if n = 0 , 1 , 3 , 7 . Su

pulls back and

Now using operations, following Adem

S(u*u)

maps

primary,

and Adams,

SuUSu

= 0

Theorem

2.4.

sPt..)eq~.Je p+q

a r e o n e of t h e f a m i l y

(1, 3),

Most cases tions [Adams], and

in conjunction

XP(2)

(3, 5),

with

q>

can be eliminated

with Theorem

Theorem

approximation

can be defined,

2.2.

2. 2.

by using cohomology The remaining

[Hubbuck],

Following

Y -~ Y X Y

up to homotopy,

being given in barycentric

where

as

,x : y - . y × y approximation

coordinates,

can be deformed to the diagonal

on the chain level,

Applying this to

keeping

o2

opera(7, 11)

[Douglas and Sigrist].

We

induces

track

n

maps

to = 0

(t0, t z , x l x 2 )

if

t1 = 0

(t0, t 1 , x l )

if

t2 = 0

if x 1 = e

(t0, t l + t 2 , x l)

if x 2 = e

o

n

For to

on

example,

~

approximation

of t h e i d e n t i f i c a t i o n s ,

we read

note that

points

of

o2

map by taking a nice cellular consider

(first p-face) on

for

the usual

one

@ (last (n-p)-face).

X 2 ' we have induced a

a 2 × X 2 which is compatible

a diagonal

First

a

identifications:

~- ( t 0 + t l , t 2 , x 2 )

g .

we construct

Y = XP(2).

if

to a cellular

on

[Milgram],

we have the

and using the true diagonal

approximation

and hence

(p,q)

cases,

SX~.) o 2 X X 2 w h e r e ,

(t o , t l , t 2 , x I , x 2) ~ (t 1 , t 2 , x 2)

diagonal

if a n d o n l y i f

to this later.

diagonal

which,

p+l

(3, 7).

of u s i n g K - t h e o r y

Now to prove specific

(1, 7),

not listed

(7, 15), a r e d i s p o s e d

will return

is anH-space

XP(2).

with the identifications On the chain level,

off

A~(sZ @ u ® v) = (a2 ® u ® v ) @ * + (el®v)@ (al@ u) + * ® (s2 @ u®v).

w h i c h in t u r n i m p l i e s original representation

Su~.~Sv

is c a r r i e d b y

0.2 0 u @ v, c o r r e s p o n d i n g in t h e

to S(u*v).

T h e a b o v e p r o c e d u r e is t y p i c a l of m a n y r e s u l t s in the t h e o r y of H-spaces.

T h e c o m b i n a t i o n of t o p o l o g y and a l g e b r a in q u e s t i o n is u s e d to

c o n s t r u c t a n o t h e r s p a c e in w h i c h the combination.

h o m o t o p y alone r e f l e c t s the o r i g i n a l

T h i s a u x i l l i a r y s p a c e is t h e n s t u d i e d b y t h e a v a i l a b l e f u n c t o r s of

a l g e b r a i c t o p o l o g y , c o n v e r t i n g the p r o b l e m thus into a p u r e l y a l g e b r a i c one.

MAPS INTO AN H-SPACE: ALGEBRAIC

STRUCTURE,

The fact that X h o m o t o p y classes of m a p s Theorems

INVERSES,

OTHER

MULTIPLICATIONS,

ETC.

is an H - s p a c e is immediately reflected in the set of [K,X]

3.1. [Copeland].

of a space

The functor

K

into X.

[ ,X,x0]

takes values in the category

of sets with binary operation and Z-sided unit w h e n defined on a category including

~X,x0) and

~X X X, x0 z) if and only if X

unit if and only if (ilviz)~:[A × B;X] -~ [AVB;X]

is an H - s p a c e with x 0 as

is onto for all A , B

in the

category. The crucial idea is to realize that the multiplication allows us to add m a p s of A and B tion i t s e l f a p p e a r s If Theorem

into X to get a m a p of A × B and that the m u l t i p l i c a -

a s t h e s u m of t h e two p r o j e c t i o n s

K

is CW,

3. Z. [ J a m e s ] .

[K,X]

of maps

of

binary

operation

K

the set If

K

[K;X]

X

h' and

as a trivial

: K-~ X X X hw~

f hence If

structure

on

Definition

3.3.

homotopic

to

Theorem

3.4.

CW-complex,

X

hu~

X.

an algebraic

loop,

classes

i.e. , has a

identity and left and right inverses.

quasifibration.

such that

on to

t h e s e t of h o m o t o p y

forms

T h e i d e a of t h e p r o o f is to r e g a r d X X X

X X X

structure.

is a CW-complex,

into an H-space

with Z-sided

has more

of

s h' ~

Given (f,u).

the shearing

map

f, u : K -- X , t h e r e T h u s if

h'

= (h,w),

s : X X X -~

is thus a map we have

w~

u

f.

itself is CW, the inverses

come

from

the corresponding

X. A map

I : X -~ X

is a left homotopy

inverse

if

m(l

X 1)

is

e. [Sugawara,

Sibson].

left and right homotopy

If

X

has the homotopy

inverses

always

exist.

t y p e of a c o n n e c t e d If

m

is homotopy

11

associative,

left and right homotopy Proof:

(Theorem q(e,

We know the shearing

1.1); l e t

q : X × X-~ X × X

)_~ I × 1 i s a l e f t h o m o t o p y If m

homotopy

inverses

is homotopy

inverses

map

up to homotopy.

is a homotopy

be an inverse.

inverse

since

associative,

is completely

agree

A map

~

satisfying

m ( ~ × 1) i s h o m o t o p i c

the agreement

analogous

equivalence

to

e.

of l e f t a n d r i g h t

to the agreement

of s t r i c t

inverses

in a monoid. Similarly, Theorem [K,X]

3.5.

If X

is a group,

is a homotopy

natural

Regarding maps

we have the following result.

are fibrewise

associative

with respect

projections

maps.

of

to maps

Hence the shearing

i f a n d o n l y if i t i s a f i b r e h o m o t o p y

translation

(left or right) is a homotopy

X

X

(e.g.,

admits X

a numerable

K

is CW,

then

as fibrations,

map

the shearing

(left or right) is a homotopy

equivalence

equivalence.

[DoLd] w h i c h i m p l i e s

The converse

holds pro-

by sets which are nullhomotopic

in

is CW). In light of these

Theorem

covering

and

K - ~ K ~,

X × X -~ X

equivalence

vided

H-space

remarks

1.4 and in Chapters

definition of H-space

and the importance

of translation

4 a n d U, i t m a y b e a p p r o p r i a t e

to include left and/or

to amend

right translation

in the

being a homotopy

equivalence. Multiplications. Given a space others

X , if i t a d m i t s

which are not homotopic

Theorem

3.6.

multiplications

[Copeland]. is in

Proof: [KVK;X]

1-1

to

If X

is exact for any CW

K.

m,

it may admit

m. i s C W , t h e s e t of h o m o t o p y

correspondence

The sequence

a multiplication

with the loop

-~ [ S K v S K ; X ] - ~ If X

of

[XAX;X].

[K~K;X]--

is an H-space,

classes

(ilV iz)*

[K × K ; X ]

the last three

sets are

L

12

l o o p s , a n d the m a p s a r e m o r p h i s m s t h e r e o f . (ilWi2)* ' l ( x )

It f o l l o w s t h a t the i n v e r s e i m a g e s

a r e in 1-1 c o r r e s p o n d e n c e for a l l x ~ [KVK;X]. If X = K is CW, the m u l t i p l i c a t i o n i n d u c e s a s p l i t t i n g f r o m w h i c h

the t h e o r e m f o l l o w s . T h e e a s i e s t c a s e s to c o m p u t e a r e Theorem 3.7.

K ( ~ r , n ) ' s or s p h e r e s .

Up to h o m o t o p y , K(Ir, n) a d m i t s o n l y one m u l t i p l i c a t i o n .

Up to h o m o t o p y , S 3 a d m i t s

12 d i s t i n c t c l a s s e s of m u l t i p l i c a t i o n .

Up to h o m o t o p y , S 7 a d m i t s

120 d i s t i n c t c l a s s e s of m u l t i p l i c a t i o n s .

[ J a m e s , L e m m e n s ] h a v e s h o w n a l l the c l a s s e s on S 3 and S 7 c a n b e r e p r e s e n t e d in t e r m s of the s t a n d a r d m u l t i p l i c a t i o n s b y u s i n g c o m m u t a t o r s . F o r p r o d u c t s of s p h e r e s ,

[Loibel] gives a f o r m u l a which for

S 3 × S 3 is c o m p u t e d b y [Norman] to be

220 × 316 .

[ N a y l o r ] a n d [Kees] h a v e s h o w n t h a t SO(3) = RP(3) c l a s s e s of m u l t i p l i c a t i o n s .

7 and

220.

768 d i s t i n c t

[Rees] h a s f o u n d 3 0 , 7 2 0 m u l t i p l i c a t i o n s on K P ( 7 ) ,

a n d [ M i m u r a ] h a s f o u n d the n u m b e r s f o r 215- B9 " 5 "

has

B • 55.

SU(3)

a n d Sp(2)

to b e r e s p e c t i v e l y

7.

M o r e g e n e r a l l y A r k o w i t z a n d C u r j e l [A-C] h a v e i n v e s t i g a t e d the f i n i t e n e s s of this n u m b e r for f i n i t e c o m p l e x e s .

T h e y f i n d that a m o n g the

c l a s s i c a l a n d e x c e p t i o n a l L i e g r o u p the n u m b e r is f i n i t e o n l y f o r SO(n)

with

n < 16,

SU(n)

with

n < 5

n # I0, 14

m

Sp(n)

with

G 2, F 4 a n d F i n a l l y , for a space and

n < 7 E 7.

X w i t h j u s t two n o n - t r i v a l h o m o t o p y g r o u p s

~rp, p > n, the s e t of c l a s s e s of m u l t i p l i c a t i o n s c a n b e i d e n t i f i e d w i t h

HP(x~JC;~rp) w h i c h is HP((~rn, n ) A ( ~ n , n ) ;

Tr ). P

A m o r e c l a s s i c a l a p p r o a c h to c l a s s i f y i n g m u l t i p l i c a t i o n s w o u l d

n

~3

r e g a r d a s m ~ a i v a l e n t t h o s e w h i c h c o r r e s p o n d up to h o m o t o p y u n d e r a h o m o t o p y equivalence.

J a m e s s h o w s t h a t t h e r e a r e o n l y s i x s u c h c l a s s e s on S 3, b u t in

g e n e r a l the c o m p u t a t i o n s s e e m m o r e difficult. some reasonable results are available

[Cheng].

For two-stage Postnikov systems,

ASSOCIATIVITY:

Although an important

strict

SPACES

associativity

AND

groups,

be seen to be equivalent homotopy

theory.

Definition

4.1.

s e t of p a i r s

for,

from

to spaces

[Moore].

It characterizes

o u r p o i n t of v i e w ,

of l o o p s ,

the latter

the operation

~X

is an associative

GROUPS

concept,

it plays

loop spaces

topological being more

t'~X : {X:[0, r ] - ~ X I X(0) = k ( r ) = * )

(k, r ) c X R X R .

under

TOPOLOGICAL

is not a homotopy

r o l e i n t h e s t u d y of H - s p a c e s .

hence topological

monoid)

LOOP

H-space

and

groups basic

will

in

topologized

as the

(= t o p o l o g i c a l

m = + given by k , ~ -~ k + / ~

defined by

k + ~ : [0,r+s]-~

x+. In the CW category loops are essentially

the same

That loop spaces groups

was proved

for semi-simplicial Theorem

4.2.

cW complex,

complexes

[Milnor].

Proof: homotopy

If X

then there

type as

X'.

and from

~X

a homotopy

as topological

groups

p o i n t of v i e w ,

spaces

or associative

H-spaces.

of t h e h o m o t o p y

though the result

of

t y p e of t o p o l o g i c a l

was presaged

by a similar

one

d u e to [ K a n ] . has the homotopy

is a topological

First

I [r,r,+s]:t~.(t-r).

are usually

by Milnor,

X by

group

t y p e of a c o n n e c t e d

GO()

has the homotopy

We might as well assume

of the homotopy

t y p e of ~ X .

t y p e of ~ X '

if

then that

is a countable

X

X

countable

has the same

s implic ial complex. Let The equivalence {x1 . . . . .

G(X)

be defined as a quotient

relation

of a s u b s e t

of ~_)X n n

as follows:

is

x n) ~- (x 1 . . . . .

A x i .....

x n)

if x i = x i + 1 o r

x i . 1 = x i + 1.

15

GO<) i s t h e s u b s e t of t h e q u o t i e n t r e p r e s e n t e d x i, x i + 1 b e l o n g to a s i m p l e x of

x.

The group

operation

(x 1 . . . . .

x p ) (Xp+ 1, . . . .

inverse

to

(x 1 . . . . .

wise linear

linearly

shows

principal

bundle

E(X)

p o i n t , iv

The contraction

The unit is

(x 1 . . . . .

from

G(X)

Xn)

x i to

such that

of

*

and

(x n , x n _ 1 . . . . .

x 1) i s

m a y b e t h o u g h t of a s a p i e c e -

x i + 1.

i s C W a n d i s t h e f i b r e of a c o n t r a c t i b l e

obtained by removing

is a weak homotopy

homotopy

Xn)

is juxtaposition:

The n-tuple

loop running

(x 1 . . . . .

x 1 = x n = *, a c h o s e n b a s e v e r t e x

i and

Xn) = (Xl'" " . , x n ).

Xn).

Milnor

lemma

for each

by n-tuples

E(X)

the condition

gives a map

equivalence

"x

n

is the base

G ( X ) -~ f~X w h i c h b y t h e 5 -

and hence,

~X

also being CW, a

equivalence. On the other hand,

CW H-space

is a loop space.

principal

fibration

Theorem

4.3.

translation

associativity

Again the construction

will be very

[Dold-Lashof].

equivalence

of a c o n t r a c t i b l e

important. If

is a weak homotopy

weak homotopy

alone is enough to show a connected

X

is an associative

equivalence,

X-~ fib X

H-space

then there

which respects

such that right

is a space

BX

the multiplication

and a up to

homotopy. Since for any

X

Iro(~Y ) is a group for any

such that

which is not a group. equivalence

If

m(

xy ,x).

is not a group,

lr 0

since for each

ponent) such that we see that

~ro(X)

is a group, x, there

and yx

composed

multiplication;

H-space associativity

e.g. , a discrete

then right translation

is a point

y

lie in the component

On the other hand, of an associative

Y, t h e t h e o r e m

with

m(

,y).

we will see later

need not itself admit is not a homotopy

is obviously topological

e.

com-

Using the associativity,

is the identity in either that a space an equivalent invariant.

monoid

is a homotopy

(any in the "inverse" of

false

order.

of t h e h o m o t o p y associative

type

16

Theorem version

appears

4.3 is the result

in Steenrod'

s Topology

was constructed

by

Dold and Lashof

in the present

Further

[Milnor]

generalizations

of c o n s i d e r a b l e of F i b r e

Bundles.

for arbitrary

A restricted

Such a space

topological

groups

BX

and then by

generality.

are due to [Sugawara]

in detail in the associative

evolution.

and [Stasheff].

casees are due to [Milgram],

Further

variations

[Steenrod] ~nd [Fuchs].

We follow Dold and Lashof. Definition

4.4.

Given a map

[i. e. , p ( x z ) = p ( z )

for

p : E-~ B

and a fibrewise

action

z c E, x ~ X], the Dold-Lashof

D L ( E ) = X X CEI.J

E

DL(B)

B

/~ : X × E - ~ E

construction

is the map

D L (p) CE~9 P

induced by projection Theorem

4. 5.

onto the second

If p : E - ~ B

a weak homotopy

equivalence

is a quasifibration for each

The proof is essentially under

mild restrictions,

Dold' s weak covering

factor.

z, then

a n d if DL(p)

as in Theorem

one should be able to start homotopy

been done by [Fuchs]

under

property)

assumptions

p(

is

is a quasifibration.

1.4.

As remarked

with a fibration

and obtain one. including

,z) : X-~ p'l(p(z))

that

earlier,

{having

This has in fact X

has a continuous

inverse. In p a r t i c u l a r , DL(p)

is just

DL(X)

induced by

of a problem;

H(m).

there

If

if w e s t a r t w i t h ~

is associative,

x, ( y , t , z) -* (xy, t, z). are

several

solutions.

p being the unique map

X - ~ ~, t h e n

w e c a n d e f i n e a n a c t i o n of The continuity

X

on

of t h i s a c t i o n i s s o m e t h i n g

We can restrict

the category

of s p a c e s

17

involved, redefine

struction.

the product topology, or strengthen the topology on the con-

D o l d a n d L a s h o f do t h e l a t t e r .

will be the s a m e

Up to w e a k h o m o t o p y

type,

the spaces

[Stasheff], so w e prefer not to take sides.

If ~

is associative, then, w e can iterate the construction.

be the initial m a p

E-~ B

andlet

Pn+l = DL(Pn)' taking

DL(E

n

Let

) = X × CE

n

P0

~.J E ~n n

into D L ~ B n ) = C E n ~.)P n B n . Inductively, /~n+l (x' (y,t,z)) = (xy, t,z). This is well-defined since

m

is associative a n d the a b o v e r e m a r k s

on continuity continue

to a p p l y . Now let obvious inclusions.

Poo : Eo0

.=~

B~o b e t h e l i m i t of t h e m a p s

Pn

under the

A g a i n D o l d and L a s h o f use a stronger topology on

guarantee continuity of an action of X

on

E

E

00

to

, but this is not relevant to the CO

results w e are after. Theorem

4.6.

For

n <_o0, Pn

is a quasifibration if right translation m ( , x )

is a w e a k h o m o t o p y e q u i v a l e n c e . This follows from Theorem problem

for quasifibrations

4.5 for

since any map

n < Qo. Sn

f :

-~ B

P a s s i n g to the limit is no m u s t lie in s o m e finite

o0

B.. I

E Theorem

CO

4, 7.

has one crucial ~-.(E

attribute.

) = 0 for all

i.

This follows f r o m the fact that E e x a m p l e , b y the h o m o t o p y

T h e total space

h : X-~ ~B X x -~ k

where

is contractible in E n + I, for

z -~ (e,t, z) w h i c h uses the left unit.

N o w let us specialize to the case w e seek.

n

E

0o

w e call E X.

is given by the i m a g e of the m a p

P0 : X - ~ #.

B

oo

is the space

BX

T h e weak. h o m o t o p y equivalence X-~ ~SX

considered before; i.e.,

kx(t) = (t,x) ¢ S X = B I. A s before, w e see

~h~ : ~i(X) -~ wi(X)

is

X

the identity.

fore

h~

O n the other hand, @ is an i s o m o r p h i s m

is a n i s o m o r p h i s m .

since

Iri(EX) = 0, there-

18

That forming

kxy

space on

h

to k

respects the multiplications w e can see b y explicitly de+ k

x

y

within

~ B 2.

This can be done m o r e

SX~-)a 2 × X 2, w h i c h w e considered above.

edge of a 2

and

k

+ k

x

y

Here

along the other two - the h o m o t o p y

k

easily in the loop

xy

runs along one

is obvious.

Loop s p a c e s a r e a p a r t i c u l a r l y s i g n i f i c a n t c l a s s of a s s o c i a t i v e H spaces.

Up to h o m o t o p y , t h e y r e p r e s e n t a l l the e x a m p l e s w i t h h o m o t o p y

inverses.

T h e y h a v e p a r t i c u l a r l y n i c e p r o p e r t i e s w i t h r e s p e c t to o u r c o n -

structions. Theorem 4.8.

If X is CW, t h e n X h a s the h o m o t o p y type of B ~ X .

Proof.

A specific map

B ~ X -+ X w h i c h is a l w a y s a w e a k homo~opy

e q u i v a l e n c e is o b t a i n e d b y e x t e n d i n g the m a p

S ~ X -~ X g i v e n b y

(t, k) -+ k(t).

T h e i d e a is to d e f i n e A n x{~X) n ~ X by mapping

A n - ~ I in such a w a y that then evaluating

w e l l d e f i n e d m a p of B ~ X -+ X.

kl÷... +k n

on

I gives a

S p e c i f i c f o r m u l a s c a n b e g i v e n [Stasheff], b u t the

essentials are conveyed by labeling edges.

J 1

19

This means

if

k i : [0, r i ] - * X, t h e n f o r

r1 [0, rl-~r2] , t h e f a c e

t o = 0 is m a p p e d

n = 2, t h e f a c e

to

t 2 = 0 is mapped

r1 [rl--~r2,1], a n d t h e f a c e

T o s h o w t h i s m a p is a w e a k h o m o t o p y e q u i v a l e n c e , it b y

E~X-~

is g i v e n b y Similar

LX

w h i c h i s t h e i d e n t i t y on ~ X .

(t,k,p)

formulas

(s) = p ( t r s )

where

can be given for

E

p : [ 0 , r ] - * X. -* L X .

n

To begin with,

t 1 = 0 to

to [0,1].

we need only cover I × ~X × ~X-*

This induces

El-* LX.

LX

H - S P A C E S WHICH A R E F I N I T E C O M P L E X E S

T h e m o s t obvious e x a m p l e s of connected topological groups are Lie groups.

T h e positive solution of HilberV s Fifth P r o b l e m by Gleason,

M o n t g o m e r y and Zippin [ G - M - Z ] characterizes a Lie group as being a locally Euclidean topological group.

F o r s o m e time it w a s thought that a h o m o t o p y

version w a s true, i.e. , that if G

w a s a topological group of the h o m o t o p y

type of a c o m p a c t manifold, then G

w a s of the h o m o t o p y type of a Lie group,

although [Slifker] s h o w e d that it w a s too m u c h to ask the h o m o t o p y equivalence to respect the multiplication {cf. Chapter 7 and II). In fact, the conjecture w a s true for all the k n o w n H - s p a c e s except for products with S 7 or quotients thereof.

T h e first e x a m p l e of a simply con-

nected H - s p a c e manifold not of the h o m o t o p y type of a Lie group or a product with S 7 w a s due to [Hilton and Roitberg].

The e x a m p l e w a s the

w

total space of a principal S3-bundle over

$7,

Soon after, [Zabrodsky]developed techniques with which I w a s able to s h o w the Hilton-Roitberg e x a m p l e w a s of the h o m o t o p y type of a toplogical group.

Let P1UP2,

Zabrodsky' s m e t h o d is called " m i x i n g h o m o t o p y types"

P

b e the s e t of p r i m e s in the n a t u r a l n u m b e r s a n d P =

a d e c o m p o s i t i o n into d i s j o i n t s u b s e t s .

L e t CP.~

d e n o t e the c l a s s of

a b e l i a n g r o u p s of o r d e r s n o t d i v i s i b l e b y p r i m e s in P i ÷ l " Let

X, X 0 b e s i m p l y c o n n e c t e d C W - c o m p l e x e s .

T h e o r e m 5.1.

Let

exists a space

X(P1)

f : X-~ X 0 be a r a t i o n a l h o m o t o p y e q u i v a l e n c e . and a factorization X

X{P1)

f i b r e of f. h a s h o m o t o p y g r o u p s b e l o n g i n g to ~ P i "

There

X 0 of f s u c h t h a t the If f is a h o m o m o r p h i s m

1

of t o p o l o g i c a l g r o u p s , X(P1)

h a s the h o m o t o p y type of a t o p o l o g i c a l g r o u p .

21

Theorem

5. 2.

Let

X. b e s i m p l y c o n n e c t e d 1

fi : Xi-* X0 be a rational Z = Z(X1, X2)

and maps

CW-complexes

homotopy equivalence. g i : Z -* X . ( P . )1I

There

for

i = 1, 2.

Let

exists a space

such that the fibre has homotopy groups

b e l o n g i n g to l ~ P i + 1. Z

If the ingredients are topological groups and homomorphisms, has the homotopy type of a topological group. Since f is a rational equivalence, its fibre F type of a product Pe'~PFP

where

has the homotopy

Fp has p-primary homotopy only. X ( P I)

can be thought of as the subfibration of X

in which the fibre is cut down to

pc~-PiF p. A n alternate construction of X ( • I) is given in Chapter 9. If f is a h o m o m o r p h i s m

of topological groups, then X(IP I) has the

homotopy type of ~ B X ( P I) and hence of a topological group. T h e o r e m 5.2 is proved by taking Z to be the fibre product {pull back) of fi : Xi(Pi)"~ X0"

z--~

x I

(~i)

x 2 ( P 2) -- x o

If t h e i n g r e d i e n t s observed

by Browder,

to p r o d u c e

are topological groups

the construction

a classifying

space

and homomorphisms,

can also be carried

Y = Z(BX1, B X 2 ) .

o u t in t e r m s

We t h e n h a v e

Z

as of BX.

of t h e

h o m o t o p y t y p e of ~ Y

a n d h e n c e of t h e h o m o t o p y t y p e of a t o p o l o g i c a l g r o u p .

Proposition

{Xi}

5.3.

If

are simply connected

finite complexes,

l

then

Z has

t h e h o m o t o p y t y p e of a f i n i t e c o m p l e x . Proof: s o is

H (X,Q).

H (X;Z). P

Since

Since

Moreover,

# H (Xi;Q)

H (X.;Z) 1 p

is f i n i t e d i m e n s i o n a l

is f i n i t e d i m e n s i o n a l

the finite dimension

as a Q-vector

for each

has a common

space

p, s o is

finite upper bound for

22

Q

and all

p

simultaneously

(i. e . ,

the maximum

h o m o t o p y t y p e of a f i n i t e c o m p l e x , decomposition

of

for example,

group",

i.e. , the sympletic

in t h e o p p o s i t e o r d e r . classifies

XI, X2).

Thus

X has the

that obtained by a homology

X.

We a r e n o w r e a d y f o r e x a m p l e s . sympletic

for

If co E lr6(S 3)

S 3 - ~ Sp(2) -* S 7, t h e n

lr6(S 3) ~ Z 4 + 7. 3 w i t h g e n e r a t o r s t h e h o m o t o p y t y p e of X 1 = Sp(2)

Let

group

Sp{2)

denote the "opposite

obtained by multiplying

is c h o s e n a s t h e g e n e r a t o r

~pp(2) -~ S 7 i s c l a s s i f i e d v,,a and

2 e P2

and

fied by

-v, + ~ = 7~o. Interchanging

,

We h a v e

X 2 = Sp(2)

over

by

quaternions

which -¢o.

Recall that

¢o = v, + a .

K ( Z , 3) X K ( Z , 7) w i t h

3 E 3F l, the resulting g r o u p is the Hilton-Roitberg e x a m p l e

as a bundle b y

Sp(2)

and

If w e m i x

Sp(2)

classi-

gives a g r o u p classified

5c0.

F o r certain

n ~ +_ 1 (6), w e also obtain H - s p a c e s

exhibit as soon as possible.

O f course the a r g u m e n t

of Lie groups w h i c h are rationally equivalent. later, but first w e study the construction

B X

We

w h i c h w e will

can be applied to any pair

will return to such e x a m p l e s

further.

THE BAR

The space

CONSTRUCTION

BX

SPECTRAL

as constructed

SEQUENCE

is filtered by the subspaces

B .

The

n

e x a c t c o u p l e in h o m o l o g y o r c o h o m o l o g y sequence. sequence spectral

converges sequence

extension same

For example, to

in h o m o l o g y E O H , ( B x }.

e.g.,

if X

a s in T h e o r e m

g i v e s r i s e to a s p e c t r a l

Eln = H , ( B n , B n _ I )

and the spectral

[Milnor] presented

for his construction.

property,

argument

that results

If t h e s p a c e s

is C W , t h e n

the corresponding

involved have the homotopy

H.(Bn, Bn_I) ~

1. 9 s h o w s t h a t if X

H.(SEn_I).

is c o n n e c t e d

The

CW, then

En_ 1

h a s t h e h o m o t o p y t y p e of t h e n - f o l d j o i n , free with coefficients

X * . . . * X. If H , ( X ) is t o r s i o n n in a r i n g A , w e h a v e E 1 • ~ @AH..(X) a n d d 1 c a n b e n

identified with

~(-1) i 1 ®...®

Theorem

If X i s c o n n e c t e d

6.1.

is isomorphic

Corollary

CW a n d

6.2.

Similar

in c o h o m o l o g y

6.3.

E 2 ~ Ext H

generated

in each dimension. There

,

free,

(x)

is u s e d to s h o w

H,(pt))

of t h i s r e s u l t

Corollary

the homology,

of m ,

X

Tor

means

then

E 1, d 1

-,

d l d 1 = 0.

a s in

6.1.

is a n a l g e b r a i c simpler

follow from applying

(H, (pt), H~ (pt))

are many H-spaces

for

is t h a t

to evaluated by algebraic

results

is t o r s i o n

B ( H , (X)).

E2 ~ T°rH,(x)(H*(pt)'

and can sometimes

H.(X)

E 1 c a n b e t a k e n a s t h e d e f i n i t i o n of t h e b a r c o n -

Note that the associativity

The usefulness

H(B).

of

~

1.

with the bar construction,

Our description struction.

m, ®...®

~

provided

than computing Horn

is a n e x t e r i o r

carefully.

H~ (X) is f i n i t e l y ',-

(e. g. , m a n y c l a s s i c a l

with suitable coefficients,

functor

groups) for which

algebra

on odd d i m e n -

sional generators. Theorem

6.4.

dimensional

[Borel].

generators,

If H , ( X ) ~ E ( x 1 . . . . . then

Xn )' a n e x t e r i o r

H ( B x ) ~ P ( T x 1. . . . .

algebra

7Xn ) t h e p o l y n o m i a l

on odd algebra

24

on generators

~rx. which pull back to duals of Sx. on SX. I 1

Proof:

EXtE(xi)(k,k)

for any commutative

k n o w n to be isomorphic to a polynomial algebra Ext

coefficient ring k is

P(yi ) on generators

higher dimension than x.. i

Since

E X t A ® B ~ E X t A ® EXtB"

Thus it is sufficient to s h o w

is a natural functor of algebras, w e have Ext E(x)(k, k) ~ P(y).

The proof amounts to putting a differential on E(x) ® P(y) acyclic, i.e., a derivation

Yi of one

d defined by

dx = y

and

so as to m a k e it

dy = 0 so that d(xy k) =

k+l Y In classes

E 2, t h e g e n e r a t o r s

x. a n d w i l l a p p e a r 1

even total degree.

Yi w i l l b e r e p r e s e n t e d

E 12.

in

Therefore,

Since

b y d u a l s of t h e

d i m x .1 i s o d d ,

the spectral

sequence

E2

is entirely

collapses;

EZ = E

of .

o0

Since the polynomial

algebra

although

to show that the algebra

Recall

it remains

that the algebraic

(*)

a 1®

To complete Theorem

•

"

the proof,

6.5.

multiplication

"

is free,

diagonal

® an-"

for

Z (al ®

there

is no extension

B (A)

"

"

structure

problem;

is correct

E

~ H

(B X)

topologically.

is given by

. ® a p )®

(a p + l

®

"

® a n ).

"

we use the following theorem.

[1.~ilgram].

The diagonal map

on each term

of t h e s p e c t r a l

A : B X -~ B X × B X

sequence,

which on

induces

a

E 1 is given by

(*). Proof: mation. relation

First,

We continue

up to homotopy,

to follow Milgram BX

looks like

to get a nice chain approxi-

nU a n X x n / R

where

given by:

(t o . . . . .

t n , x 1. . . . .

x n) "- (t o . . . . .

t~i. . . . .

x i x i + 1. . . . .

~(to,...,ti_~+ti,...,xi,...,Xn)

x n)

if xi=e.

if

t.=O 1

R

is the

25

with the meaningless symbols u. ~ X, w e denote by I

XoX 1 and XnXn+ 1 to be deleted.

lUll.., lu ] the cell n n

diagonal approximation on G

n

involved.

of B X.

The

On. the chain level w e have

1%1 = EUll.-. of t h e i d e n t i f i c a t i o n s

n

r. (last p-face) X (first (n-p)-face)

given by

induces a diagonal approximation on B X.

because

X uI X . . . X u

Given cells

®

[Up+ll... lu n]

This proves

W e c a n a l s o s e e h o w to c o m p a r e

Theorem

6. 5.

B X with the bar construction

( C , ~X)). Theorem

cation and

6.6.

[Stasheff,

C,(X)

Milgram].

If X h a s a n a s s o c i a t i v e

denotes cellular chains, then

There are examples of such

U(n) with the cell

Ingeneral, a multiplication £an be

deformed to be cellular, but this m a y destroy associativity. h o w to modify

B

multipli-

C,(Bx) -~B(C,(X)).

X, e.g. , 0(n) or

structure described by [M£11er~ Yokota].

cellular

W e will see later

to obtain the analogous result in general.

The spectral sequence can be obtained entirely in terms of B(C,(X)).

This is the approach of ~ilenberg and Moore]. Theorem

6.4. presents us with polynomial algebras, m a n y of which

cannot be realized topologically. Theorem

6.7. [James]. Proof:

If B S 7

p 4 (u8) = u 8 3 # 0 but p 4 is zero.

Thus

B

S 7 admits no associative multiplication. exists, then H

factors as p i p 3

03S 7) ~ P(u8)" which takes

M o d 3, w e have

u 8 through

H 20 which

cannot exist. S7

In f a c t , w e will soon see

S 7 admits no homotopy associative

multiplication. Theorem

6.8. [Clark].

If X

is an associative H-space and

H,(X;Z)

is

26

t o t a l l y f i n i t e l y g e n e r a t e d a n d n o t a c y c l i c t h e n H 3(X) I 0. T h i s r e q u i r e s m o r e c o m p l i c a t e d a n a l y s i s , b u t t h e p r i n c i p a l :is the • same.

I n g e n e r a l , the q u e s t i o n of w h i c h p o l y n o m i a l a l g e b r a s c a n b e r e a l i z e d h a s

not been settled although Clark' s procedure gives many restrictions. have b e e n m a d e explicit in v e r y s p e c i a l c a s e s ,

[Smith] a n d [Ochiai].

These The m o s t

e x t e n s i v e r e s u l t s a v a i l a b l e n o w a r e due to [Hubbuck]. T h e s t u d y of the s p e c t r a l s e q u e n c e w h e n it d o e s n o t c o l l a p s e i s of great interest.

We s h a U r e t u r n to it w h e n we a r e in p o s i t i o n to i n t e r p r e t the

differentials geometricaUy.

ASSOCIATIVITY

HOMOTOPY

Now let us look at "associativity

up to homotopy"

in an elementary

way. Definition and

7.1.

(X,m)

is a homotopy

m { m X 1) a r e h o m o t o p i c

an associating

and homotopy

nected,

7.2.

A specific

homotopy

rn(1 )< rn) is called

[Sugawara].

on

an associating

If

(X,m)

extends

The situation

is homotopy

to a fibration

associativity

X)< CXU X, sowe m

homotopy

ones.

for instance,

on

S 3,

can be analyzed

"geometry".

The lack of strict X

S 7, w h i c h a r e n o t e v e n h o m o t o p y

multiplications,

to associative

the Hopf fibration

action of

X 3 -~ X .

such as

associative

by using a little more

Theorem

if t h e m a p s

as maps

are H-spaces,

which cannot be deformed better

H-space

homotop)r.

There associative

associative

with

over

defeats

work alittle

ho{X, y, z) = ( x y ) z

X X Of X CX[.)X) -~ X X C X h D X

associative

and

is con-

XP(2).

our previous harder.

and

X

Let

definition h

t

of a n

: X 3-~ X

h l { X , y , z) = x ( y z ) .

be

Define

by

x(y,t,z) = (xy, Zt, z) if O < 2 t < l =h2t.l(X,y,z)

T h i s iS w e l l d e f i n e d a n d c l o s e construction. we let

Definition

7.3.

XP(3)

case we let

be the quasifibration

denotes

The projective cohomology.

enough to fibrewise

As in the associative

P2 : E 2 -~ B 2 = X P ( 2 )

if l < Z t < 2 .

space

to apply the Dold and Lashof P o : X - ~ ~, Pl = H ( m )

and now

we have just constructed.

CE2~-)XP(2). XP(3)

This time the non-trivial

is used like cup cubes

XP(Z)

by ar,~lyzing its

are the significant

feature.

28

Theorem

7.4.

S 7 admits no h o m o t o p y associative multiplication.

This w a s originally proved by direct computation involving the obstruction, which is a class in 7rzl(S7) [James] but n o w it follows f r o m the s a m e a r g u m e n t w e used to s h o w

S 7 w a s not associative.

However,

the h o m o t o p y

a r g u m e n t has the advantage of detecting a Z - p r i m a r y obstruction as wet1. T h e existence of such a fibring over

XP(Z)

is equivalent to h o m o t o p y

associativity, at least for C W - s p a c e s . Theorem

7.5. [Sugawara].

multiplication

if t h e r e

If X

is C W ,

is a sequence

then X

admits a h o m o t o p y associative

of f i b r a t i o n s

X-~ El-- E z

• - . B 1 -* B z

with X

contractible in E 1 contractible in E 2. Recall that in this situation X

is a retract of ~ B I. The contraction

of E 1 in EZ. is used to s h o w this retraction respects the multiplication up to homotopy. Theorem

7.5 is in contrast with classical projective g e o m e t r y w h e r e

the existence of projective 3-space is enough to prove associativity of the coordinate r i n g . true

From

in general

but there

Rather Given

o u r p o i n t of v i e w ,

this would mean:

XP(3)

l a c k of n a t u r a l

examples.

is a curious

obvious

candidates

=> B x .

are the exotic multiplications

q : S 3 X S 3 --~ S 3 , t h e q u a t e r n i o n

multiplication,

S3 v $3 by an element of ~6 (s3) = ZIZ.

we can alter

This is not

on

S 3.

it a w a y f r o m

In this way, w e get the IZ different

h o m o t o p y classes of multiplications including quaternions and its opposite

q : x,y~

q(y,x)

29

which differ by a class

plication m we see

is h o m o t o p i c

H(m)

to q + no0 f o r s o m e

is h o m o t o p i c

p1 : w7(S 4)

to

Since

pl

detects

are associative, pI(H(q))

Thus

pI(H(q))

w e have

ZBC~r6(S3),

and

From

Any multi-

the c o n s t r u c t i o n ,

Now a homomorphism in $4~.) e 8 w e h a v e

we have

pl(~co) # 0.

Since

q and

m

pl(zco). In general, for m

-- q + nco,

can be h o m o t o p y associative only if

In fact, J a m e s shows in these cases the multiplication [_s_s

homotopy associative, i.e. , XP(3) XP(n)

[James].

pI(H(q)) = pl(H(q)) + pl(~co) are non-zero.

pl(H(m)) - I + n(3). Thus

these cases

n e ZlZ.

pl(a) = k where

m u s t have the s a m e sign as

n - 0 or I rood 3.

of w6(S 3)

H(q) + n ~ co e ~r7($4).

Z 3 can be defined by

p l ( u 4 ) = ku 8. q

co w h i c h is a g e n e r a t o r

can be formed.

[Slifker] goes on to s h o w in

can be f o r m e d for all n, although by the classification

theorem for Lie groups, only q and cations on the 3-manifold

S 3.

q are homotopic to associative multipli-

Thus there are topological groups of the h o m o -

topy type of S 3, but not multiplicatively equivalent to the unit quaternions in either order of multiplication. The first example of a homotopy associative H-space which is not equivalent to a loop space was given by [Adams]. Example

7.6.

Let

Y

be a M o o r e space

3 but no other primes is possible. Zn-I and for i ~ n, ITi (X) Consider

Y

Z and

has cohomology only in dimension

~r.(sZn'l)p. 3

yC~Z

structions to deforming

Thus

Y(G, Zn-I) w h e r e divisibility by

~Zy

Y X Y -~ 2

1

which is an associative H-space. ~ Z y X ~ Z ~ Z y -, ~ 2 ~ 2 y

into Y

The oblie in

Hi(y A Y; ir.(~Z ~Zy, y)). The relative groups are trivial to at least 5(Zn)-3, i

SO the obstructions vanish.

Similarly, there are no obstructions to deforming

an associating homotopy within ~ Z ~ 2 y

to one in Y.

That

space follows f r o m the decomposability of P n UZn = u2nP Further examples,

even finite complexes,

Y

is not a loop

for p > 3 and

n J-p-h

have been constructed by

30

Zabrodsky.

We w i l l p r e s e n t t h e s e e x a m p l e s w h e n we h a v e d e v e l o p e d the a p p r o p -

r i a t e m a c h i n e r y to e x p l a i n t h e m , n a m e l y , the h o m o t o p y a n a l o g u e s of h o m o morphisms.

MAPS OF H-SPACES

homomorphism

Bya

f ( x y ) = f(x}f{y). morphism.

For

of H - s p a c e s

example,

In a limited

if

sense,

we mean a map

g : W-* Z

then

~

f : X-* Y

: ~W-*

g

f~Z

such that

is ahomo-

we shall see that for associative

H-spaces

this

is the only example. Clearly

a homomorphism

S f : S X -~ S Y , b u t s u c h m a p s Definition 8.1. H-map

if

fm

If

(X,m)

is h o m o t o p i c

H-maps associative H-map

n ( f X f).

generally

are H-spaces,

maps

a map

for H-spaces

X P ( 2 } - ~ YP(2}

an H-map.

If

f: X - * Y

in g e n e r a l ,

but notnecessarily

but,

extending

is an

for An

BX-* By.

have been investigated

(shm) map"

and

n

are

strictly

f(x)f(yz)

exist for all

involving

associative,

or via

if w e h a v e t h e a p p r o p r i a t e

we are led to conditions

If s u c h m a p s

m

in t w o w a y s - v i a

are homotopic

by

map

f(xy)f(z).

f(xyz)and These

I g x X 3 -~ Y.

More

I n ' l x x n -> Y, s u i t a b l y r e l a t e d

on the

n, we have a " strongly homotopy multiplica-

[Sugawara] or an "A

-map"

[Stasheff] or an "H-homomor-

[Fuchs].

Definition 8. g.

an A -map

Let

X

and

Y

b e associative H - s p a c e s .

if t h e r e e x i s t s

a f a m i l y of m a p s

h 1 =f

and

t i _ l , X 1. . . . .

h i { t 1. . . . .

A map

h. : i i-1 ~< X i -~ Y

n

that

YP{n)

t h e y a r e n o t t h e h o m o t o p y a n a l o g of h o m o m o r p h i s m s .

are h0motopic

two homotopies

phism"

to

into

people.

f(x}f(y)f(z}

tive

(Y,n)

XP{n}

generally.

homotopy conditions for the latter

Consider

faces.

and

X-* Y will induce

The necessary several

occur more

are the relevant

H-spaces,

f:

induces a map

for

f : X -~ Y

is

1< i< n

such

1

x i)

t%

= h i - 1 {. . . .

t j. . •. , x. j x. j +. 1,.

= h .j ( t l , . " . , tj_ 1, x 1 . . . .

} if

t.j = 0

x j ) h i _ j (tj+ 1 . . . .

,ti_l, Xj+ 1 .....

x i)

if t.j = 1.

32

Such animals occur naturally.

For any space

m a k e the function space topology nice), let H(K) equivalences of K K r L s K

into itself. If K

and

L

K

(locally compact to

be the space of all homotopy

are two such spaces and

h o m o t o p y inverses, then there is a strongly h o m o t o p y multiplicative

homotopy equivalence

H(K)-~ H(L)

given by ~-~ r ~ s.

homotopy with R(0) = I and R(1) = sr, then the family

If R(t) : K-~ K

is a

{hi}, as constructed by

[Fuchs], looks like: hi(tI..... ti_l,¢l ..... ¢i ) = r #iR(tl ) CZ'''R(ti-1) ¢i s" The composition of A n - m a p s

is again an A n - m a p ,

families m u s t be fitted together, rather than composed.

families are

{h 1 : i i - I × X 1" - Y} and

to be Jlhl and

(jh) z to be

but the

For example,

if the

{Ji : ii-I X yi --* Z} then we define

(jh) 1

Jlhz + jz(hl X hl), m e a n i n g the h o m o t o p y :

Jlhl(xY) _~ Jl(hl(X)hl(Y)) _~ Jlhl(X)Jlhl(Y). For

(jh) 3, Jlh3 , jz(hl × h~), jz(h2 X hl) and J3(hl X h I X hl) a r e f i t t e d t o g e t h e r

as i n d i c a t e d below.

Full details are given (k(jh)).

y [Fuc s]

He points out

although they are homotopic.

a

((kj))1

is not the s a m e as

The a p p r o p r i a t e c a t e g o r y w h o s e o b j e c t s a r e

1

a s s o c i a t i v e H - s p a c e s has as m a p s h o m o t o p y c l a s s e s of s u c h f a m i l i e s

{hi}.

A l t e r n a t i v e l y [ D r a c h m a n ] g e t s a r o u n d the d i f f i c u l t y b y u s i n g h o m o t o p i e s of v a r i o u s lengths.

33

Of c o u r s e a h o m o m o r p h i s m

c o m p o s e d w i t h a n A - m a p is m u c h m o r e n

r e a d i l y s e e n to b e an A - m a p , the o b v i o u s f a m i l y b e i n g h o m o t o p i c to the one n d e f i n e d b y the g e n e r a l p r o c e d u r e u s i n g the " t r i v i a l " h, i . e . ,

h i ( t 1. . . . .

xi) = h(x 1 . . . x i )

family for a homomorphism

= h(Xl)...h(xi).

On the o t h e r h a n d , a m a p h o m o t o p i c to an A - m a p is i t s e l f a n A n

map, though again some fiddling with parameters Theorem 8.3.

[Fuchs].

n

is r e q u i r e d to s h o w t h i s .

A homotopy equivalence

f : X -~ Y is s t r o n g l y h o m o t o p y

m u l t i p l i c a t i v e if a n d o n l y if a n y i n v e r s e i s . Proof:

If f is a n H - m a p a n d g is a n i n v e r s e ,

we h a v e

gn ~ gn(fg X fg) ~ g f m ( g X g) ~ m ( g X g), which shows

g is a n H - m a p .

compatible with gf~

fg ~ id.

1X b y a h o m o t o p y

written as

To p r o c e e d f u r t h e r ,

we n e e d t h i s h o m o t o p y to b e

S p e c i f i c a l l y l e t fg ~ 1y b y a h o m o t o p y k.

If f m ~

n(f × f) b y

- g n ( ! X 1) - g f 2 ( g X g) + k m ( g × g).

I

and

f2' the a b o v e h o m o t o p y c a n b e F o r p u r p o s e s of i n d u c t i o n , w e

w i s h to f i l l in the d i a g r a m

fgfz(g ~

f g n ( l X 1)

[

x

~.

g)

] fz(g X g)

i)

The lower quadralateral

c a n b e f i l l e d in w i t h

l n ( l × 1) a n d t h e n the

u p p e r one b y u s i n g

l f 2 ( g X g) is we h a v e f o l l o w e d F u c h s in m a k i n g t h e c l e v e r

observation that

can be chosen so that

between

fgf a n d

l f.

fk is h o m o t o p i c to

If

as homotopies

34

The induction now proceeds,

constructing

at each state

so that

f n

{(fg)i}

is homotopic

to

(id) i.

Thus A -maps

are a reasonable

class

of m o r p h i s m s .

As expected,

n

A -maps

are nice with respect

to projective

spaces.

n

Theorem

8.4.

t e n d s to

XP(n)-@ YP(n).

Corollary t y p e of

Amap

8.5.

f : X-* Y

The homotopy

t y p e of

n

-map

BH(E)

if and only if Sf : S X - ~ S Y

is an invariant

ex-

of t h e h o m o t o p y

E. The induced map

terms

is a n A

of t h e f a m i l y

that "f respect H-map,

then

SX~)C(X

f P ( n ) : X P ( n ) -~ Y P ( n )

h i by formulas

the identifications fP(2)

X CX~)

m

[S.ugawara] which give

up to homotopy".

can be defined in terms

meaning

example,

of t h e r e p r e s e n t a t i o n

in

to the idea if

f

is an

XP(2) =

0<2s
(t,e,l, hz(Z-Zs,x, y))

1 < 2-S < 2 --

(s,x) ~

For

explicitly

X) by

(t, x, s, y) -~ (t,f(x),Zs,f(y))

and by

is constructed

(min[Zs, 1],x)

To show that

on

fP(n)

on

C(X

X Cx[-Jx)

--

m

SX. implies

the existence

of

h.

for

i < n

requires

1

u s e of

XP(n+l).

Before

doing so, we first

study some

special

A -maps. n

Theorem

8.6.

The usual map

X--

~XP(n)

is an A -map. n

This generalizes same terms

spirit. of

The maps

can be seen from

4.3 and a proof can be given in the

h. : i i - 1 M X i ~ ~ X P ( n ) l

8.6 is best possible,

the fact that

while the generator

(example

of T h e o r e m

can be described

conveniently

I i - 1 X X i × I - * Z~i × X i. Theorem

erator

part

10.7) that

of

H . (f~S 2)

for

is a polynomial

H v ( s 1) h a s t r i v i a l

S 1 --> ~ C P ( n )

S 1 -* ~ S z

is not an H-map. algebra

products.

i s n o t a n A n + l - m a p.

This

on a single gen-

We shall see later

in

35

Theorem

8.7.

If X

is an associative

H-space,

there

is a r e t r a c t i o n

~ X P ( n ) -~X

w h i c h is s t r o n g l y h o m o t o p y m u l t i p l i c a t i v e . Since there 8.6

~ X P ( n ) -~ ~ B x

is a h o m o m o r p h i s m ,

is a s t r o n g l y h o m o t o p y m u l t i p l i c a t i v e X-~ ~B X

it is s u f f i c i e n t to s h o w

equivalence

is s t r o n g l y h o m o t o p y m u l t i p l i c a t i v e ,

~ B x -~ X.

so by Theorem

By Theorem 8.3,

s o is

any inverse. Since we have an A of B x

into

By

corresponds

00

-equivalence

to a n A

between

- m a p of X

X

into

and

~ B X, e v e r y m a p

Y, b u t n o t n e c e s s a r i l y

to

¢O

a homomorphism. morphisms

Consider

f(x) = x

and

[0, 1] w i t h i t s u s u a l m u l t i p l i c a t i o n .

fl(X) = 1 a r e h o m o t o p i c

but not via homomorphisms. To consider A

[X, Y]

(Consider

this relation

denote homotopy

classes

(B I is c o n t r a c t i b l e )

ft(0). )

more

of A

00

via A~-maps

The homo-

fully, we follow [Fuchs] and let

-maps,

i.e. , a homotopy

is a f a m i l y

00

h. t : i i ' l × X i -~ Y

satisfying

the usual conditions for each

t.

Let

H o m ( X , Y]

1

denote homotopy classes ism for each

of h o m o m o r p h i s m s ,

i.e.

,

h t : X -~ Y

is a h o m o m o r p h -

t.

Consider

J : Horn [X, Y]-- A

[X, Y] o0

B :A

CO

[X,Y]-~ [B x , B Y ]

[B x, By] -~ H o m [ e B X, f~By] Let A : A

[•B x , n B Y ] - *

A

o0

{%Y} o {hi} o { A i X}

[X,Y] be given by

{h:nB x-* n B y } - *

¢0

where

{ A i X}

is t h e A

-map we have considered o0

X

~B X

and

{~}

is an inverse.

Let a : [B~x, B ~ y ] -* [X, Y] be given by f-~ a f o -I where G : B ~ X -* X Theorem

is given in T h e o r e m

8.8.

A J ~ B

4.8.

is the identity.

The proof involves showing commutativity

of t h e d i a g r a m

from

36

x!y ~B x

~

~By

f~Bf

a n d the c o r r e s p o n d i n g h i g h e r o r d e r d i a g r a m ,

w h i c h c a n be done in t e r m s

of

specific formulae. C o r o l l a r y 8. 9.

J ~ B

Theorem

~ B J ~

8.10.

Proof:

is

1-1 and onto. is the i d e n t i t y .

Consider X

B~X

f ---*

Y

--~ B~y Bf~f

w h i c h , u s i n g o u r s p e c i f i c d e f i n i t i o n of a, is c o m m u t a t i v e . C o r o l l a r y 8.11.

B J ~

is

l-1

a n d onto.

Combine these results. T h e o r e m 8.12.

B : Aco[X,Y]-* [Bx, By] J ~ : [X,Y]-~ A

[~X,~Y]

is

1-1 a n d onto w i t h i n v e r s e is

A J ~.

1-1 and onto w i t h i n v e r s e

a B.

co

We h a v e s e e n t h a t J

is onto f o r l o o p s p a c e s ,

i. e.

J : Horn [ I , I ] - ~ A ,

co

[I,I]

is n o t

l-l.

Notice that

e v e r y A c o - m a p of l o o p s p a c e s is h o m o t o p i c to

a loop map. T h e s i t u a t i o n f o r c e r t a i n c l a s s i c a l g r o u p s is t a n t a l i z i n g [ A t i y a h and Hirzebruch]. T h e o r e m 8.13.

A [B G, BH] ~ Horn (G, H) w h e r e ~

Notice strict homomorphisms A p p a r e n t l y a l i m i t of h o m o m o r p h i s m s

denotes "completion".

a r e u s e d b u t a c o m p l e t i o n is n e c e s s a r y .

is only an A

-map? CO

A - m a p s c a n be u s e d to c h a r a c t e r i z e n

structure.

H-spaces with additional

We c o n t e n t o u r s e l v e s w i t h the s i m p l e s t c a s e s .

T h e o r e m 8.14.

ACW-space

X is an H - s p a c e if and o n l y if X is a r e t r a c t

of

37

~SX. of

An H-space

XP(2)

is homotopy

associative of

if a n d o n l y i f

X

is a retract

We know if

X

is a CW H-space

associativity homotopy

(X,m)

yields

associative

gives an associating

~XP(2)

_~ X X ~ E 2.

within homotopy

~XP(2) within

X.

s h o w s t h e t w o w a y s of a s s o c i a t i n g

r(x+y+z)

and hence to each other. w e c a n u s e t h e a c t i o n of

associativity. reduced

(x1

product ~i .....

Alternatively, construction x ) i f x. = e n

(XlX?_ {x 3 (- . . (Xn))). • . ))).

1

~SX

then

~SX_~X

The converses

Similarly xyz

X ~ E 1 andhomotopy are easy.

an H-map in

X

are

on

H(m)

X

00

each hornotopic

with the retraction

of

~SX-~

with

given by

(x 1 . . . . .

(x 1 . . . . .

to X is

homotopy

up to homotopy,

=~flxi/---

is

the retraction

which follows from

can be described,

of [3ames]:

X

retraction

To show the usual retraction X

is a retract

via an H-map.

since the map is an H-map:

~SX ~ X

an H-map

~SX

i f a n d o n l y if X

by the x ) "-n

Xn,) -~

SPACES

As H-spaces extensions,

when does there

are those with

p : G --* K formed

are analogues

H-space

of

Theorem

Stasheff].

induced from

p

i
and 9.2.

an H-space

i~p

+q for some

[Copeland]._

criterion,

of m u l t i p l i c a t i o n s Proof

on

on

as fibre.

and

K,

of p a r t i c u l a r N o t i c e t h a t if

on

p-l(e)

G

can be de-

will be a sub-

let

is an H-map

over

Y

then the fibration

admits

a multiplication

is

(p-1)-connected,

p>

0, a n d

w.(Y) = 0 1

q. and

Y = K(G,q),

q>n

then

Ef

is

is an H-map. we will translate

"f

is an H-map"

various

reasons

into a purely

for the existence

K(w, n ) ' s.

of T h e o r e m .

Ef

(xx' , k k ' )

X

If X = K ( w , n )

] xe

is the path space,

multiplication However,

f : X --* Y

and will also consider

Ef = ((x,k) LY

If

i s t r u e if

if a n d o n l y if f

cohomological

H-map,

H

and then the fibre

the path fibration

In the next chapter,

Here

Cases

H

is a homomorphism. The converse

Corollary

with

G?

then the multiplication

is a homomorphism

[Moore,

~ Y -* E f p X

for

given H-spaces

exist such an H-space

and an H-map

to consider

G.

9.1.

such that

p

it is appropriate

In particular

G -* K b e i n g a f i b r a t i o n

is a fibration

so that

BY H-MAPS

of g r o u p s ,

i . e . , H - ~ G - * G / H = K.

we may ask: interest

INDUCED

is

Recall X, kc

LY,

f(x)=

{k : I -* Y ] k(O) =~ }.

An obvious candidate

(x, k) (x' , k' ) = (xx', kk' ) w h e r e

is not in

Ef

unless

h 2 : I X X 2-* Y be a homotopy

h2(1, x , x ! ) = f(x) f ( x ' ).

k(1)), w(x,k) = x.

f

kk' (t) = k(t) k' (t).

is a homomorphism. such that

We now define a multiplication (x,k) (x',k') = (xx',kk'

h2(0,x,x') in

+ H(x,x'))

for a

Ef

by

If

f

is an

= f(xx')

and

39

where

H(x,x')

: I-* Y by

H(x,x')

To see that this multiplication if x

or

x'

satisfying

= e.

It t u r n s

has a strict

homotopies

Notice, ~Y

unit, we need

o u t t h a t if a n y h o m o t o p y

the additional condition.

higher order

fibre

(t) = h 2 ( 1 - t , x , x ' ) . h2(t,x,x'

h 2 exists,

) = f(x) f ( x ' )

t h e n t h e r e is o n e

We wilt often have such comments

respecting

about

the unit.

with this additional condition,

is that induced by multiplication

on

that the multiplication

Y, a l t h o u g h h o m o t o p i c

on t h e

to l o o p

multiplication. The converse obstruction Hi(x~

theory.

The obstructions

X; lri(Y)).

Under

are the obstructions H-map,

p a r t of t h e t h e o r e m

to

fp b e i n g a n H - m a p .

~ri(Y ) = 0 f o r

HP(EfA

E f ; lr) a n d

Ef

involves a little more ~r is a n H - m a p

on

Ef.

By the general on

Ef

which can readily be evaluated Ahomotopy such that

K(G,q).

in

in

f; iri(Y))

f~r is n u l l h o m o t o p i c , f

if

which it is a n

(lr/k Tr)

and otherwise

is

if

on

obstruction

theory.

since the obstruction

If E f

l i e in

is (p-l)-connected.

of m u l t i p l i c a t i o n s

multiplication

Hi(Ef/~

The same follows for

Notice that different choices

classes

are classes

i < p + q with any coefficients

any multiplication,

multiplications

Since

in

exercise

i_> p + q.

The corollary admits

are all zero.

i.e. , for

forward

to f b e i n g a n H - m a p

lr A lr, t h e y m a p t o c l a s s e s

so its obstructions

an isomorphism,

is a s t r a i g h t

is in

of t h e h o m o t o p y result 1-1

h 2 g i v e r i s e to d i f f e r e n t

of T h e o r e m

3. 6, t h e s e t of h o m o t o p y

correspondence

as isomorphic

with

Fixing a homotopy

K{G,q)

and

L

m

is t h e f u n d a m e n t a l

o h 2 and its cochain

E f , El]

b c c q ' l ( K { ~ r , p ) ~ K { I r , p);G)

5b = m S f ~ ( ~ ) - (f X f)~t m ~ : ( L ) w h e r e or

[EfA

H q - 1 (K (Tr, p) A K (~r, p); G).

h 2 gives us a specific cochain

K(lr, p)

with

denotes the c o c h a i n of

b °, the correspondence

40

between multiplications

and the cohomology

E v e n if f is t r i v i a l , example,

if f = ~ t 2 w h e r e

Ef

L

g r o u p is g i v e n b y

can have more

is t h e f u n d a m e n t a l

than one multiplication. class

h a s t h e h o m o t o p y t y p e of S 1 × K ( Z , 2) b u t r e g a r d e d has a Pontrjagin

ring not ring isomorphic

The corollary

of a P o s t n i k o v

Definition 9.3.

A Postnikov

of f i b r a t i o n s

system

system

pnJn = J n - l '

[Suzuki,

fibre homotopy equivalent

to k i l l

systems

is to f o r m

~-ilX) f o r

i > n.

)

result

for a space

X

about

with maps

can be constructed

consists

of

classes Jn : X -* X n for

i
induced by a representative

The inclusion

Ef

[Peterson].

general

and cohomology

X , up to h o m o t o p y , n

fibration by the usual process.

L2' t h e H - s p a c e

( j n ) . : Iri(X) ~- lri(X n)

to a f i b r a t i o n

Such Postnikov obvious possibility

i > n,

Ef

Kahn, Stasheff].

{Xl,k 2,X 2 ....

-~ X n P n X n _ l -~ . . .

lri(Xn) = 0 f o r

~E

c a s e of a m o r e

kn ¢ H n + l {Xn - l ' • wn(X))' c a l l e d k - i n v a r i a n t s , t o g e t h e r that

as

For

of H 2 ( Z , 2 ; Z ) , t h e n

H . ( S 1 × K ( Z , 2))

a b o v e is a s p e c i a l

the k-invariants

a sequence

to

h 2 -~ {b - b ° ) .

in a v a r i e t y

by attaching

such

and

Pn

is

of k n. of w a y s .

c e l l s to

X

One so as

X n ~ Xn_ 1 c a n t h e n b e m a d e i n t o a

T h e f i b r e w i l l h a v e to b e

K(~r ( X ) , n )

and

kn

n

can be chosen as the transgression the obstruction

then

rrn

in t h e s i m p l y c o n n e c t e d

is i s o m o r p h i c

H n + l ( X n . l , X) b y t h e i n v e r s e H n + I ( X n _ I , X ) -- lrn(X) which gives

X

n

with

determines

wn(X)).

or, equivalently,

as

b u t is n o t a n i n v a r i a n t . k n+l

map. a class

That

kn

if Xn_ 1 h a s b e e n c o n -

w h i c h is i s o m o r p h i c

with

Choosing the isomorphism in

H n + l ( X n _ l , X ; ~rn(X))

induces a space satisfying

the

is easy to check.

Note that the Postnikov

of 7r, t h e n

case,

7rn+l(Xn_l,X)

of t h e H u r e w i c z

carefully

k n e Hn+l(Xn_l,

conditions for

class,

to a c r o s s - s e c t i o n .

Alternatively structed,

of t h e f u n d a m e n t a l

and

system

For example, g~ k n + l

determines

if X

determine

n

= K(~r,p)

spaces

t h e h o m o t o p y t y p e of X and

g

is a n a u t o m o r p h i s m

of t h e s a m e h o m o t o p y t y p e .

41

Corollary Theorem

9.4.

X

9.

generalizes.

Z

is an H-space

there

is a m u l t i p l i c a t i o n

k n+l

is r e p r e s e n t e d

Pn

if a n d o n l y if f o r a n y P o s t n i k o v Xn

for each stage

such that

Pn

system

for

X

is a n H - m a p

and

by an H-map.

The multiplication

p

can be constructed

f r o m one on

X

as

n

follows: X (n+l) n

There

is a c r o s s - s e c t i o n

so we map

a l l of X n × X n Similarly i > n+2.

Pn

X

(n+l)

× X

n

to (n+l) n

_in : X - * X n

-~ X × X - ~ X - * X .

It f o l l o w s t h a t The problem

lie inHi(Xn ~

since the obstructions

kn+l

T h i s t h e n e x t e n d s to

n

since the obstructions is a n H - m a p

over the (n+l)-skeleton

is r e p r e s e n t e d

X n ; lr i _ l ( X n ) )

l i e in H i ( X ~

by an H-map

of f i t t i n g t h e m u l t i p l i c a t i o n s

for

X ; "n'i(Xn_l))

by Theorem

p

i > 2n+Z.

together

for

9.1.

to f o r m a

n

multiplication

on

X

the multiplication

is s o l v e d b y c o n s i d e r i n g

"mixing and

however,

X ~1 Y'I"* X0

such that

I H i ( Y . ; Z ). p

the cohomology Yi+l

qi

Let

of Z a b r o d s k y '

X(P 1)~ X 0 by a succession

is a n i s o m o r p h i s m

There

i.

A

s method

of

for

becomes

r < i-1

K(Z

P

,i)

in

PI"

f o r b e i n g so d e t a i l e d .

as

and

and all a

p ~ ]t~1.

in t h e k e r n e l

in t h i s d i m e n s i o n

in t h i s d i m e n s i o n .

enough times

Repeat for all primes

is r e a s o n

in t h e k e r n e l

monomorphic

f : X ~ X0

a far.torization

Y i + l - * Y'I b e i n d u c e d b y a c l a s s

by taking a product with

i s m in d i m e n s i o n

interest.

of f i b r a t i o n s ,

homotopy equivalence

Continue killing classes

morphism

c a n b e a p p l i e d to a

to b e of o n l y t h e o r e t i c a l

is a r a t i o n a l

H r (X, Zp)

p be such a prime. 1

approach

by induction that we have constructed

qi * : Hr (Y.,Z)-~ 1 p

qi

X and constructing

Given a rational homotopy equivalence

we can construct

Assume

system

gives useful applications

homotopy types".

P1CP,

follows:

the Postnikov

H - * E - * K, b u t t h i s a p p e a r s

modification,

of

by induction.

More generally, fibration

the skeleta

to m a k e

qi+l

Let

of until

Now form an isomorph-

42

Theorem map,

9. 5. [Zabrodsky].

then

X{PI)

If f : X - * X 0 is a rational equivalence and an H -

admits a multiplication so that fl' f2 are H - m a p s .

Proof:

If f : X - ~ X'

is an H - m a p

with

f~ : H r ( X '; Z P

being an i s o m o r p h i s m

for

r < n, then ~ ~ K e r f',~ I Hn(X' , Z

)-~ Mr(x; Z ) P

) is represented by P

an H - m a p ,

for f'~ ~ = 0 is represented b y an H - m a p

being represented by an H - m a p

lie in H n ( x ' ~

and the obstructions to

X' ; Z

) w h i c h is m a p p e d

iso-

P morphically resented trivial

to

HncK ~ X; Zp)

by an H-map one) showing

will admit

Corollary H-maps,

The homotopy

showing

so as to map into a homotopy

f$¢~ i s a n H - m a p .

Thus the space

such that

Y

If t h e i n g r e d i e n t s

the space 9. 7.

multiplication

constructed

X'

by

T : X -~ Y.

s construction

are H-spaces

and

will be an H-space. is classified

by

m0, t h e n

M 10 a d m i t s

a

n

if n ~ 2 ( 4 ) . of

can be constructed

S--p(Z) a n d

maps

(e. g. , t h e

f can be lifted to an H-map

of Z a b r o d s k y '

If S 3 - ~ M 1 0 - ~ S 7

The values

or

is rep-

induced over

n

spaces

a

follows by induction.

9.6.

Application

( f : ~ f)~.

can be chosen

a multiplication

The theorem

by

n

not covered

by having

X z = S 3 X S 7 or

Sp(2) -~ K(Z,3)

in Hi(Sp(Z)~'Sp(2))

and for

previously

2 E P1

and

i = 3 or 7.

n -+3,

3 ¢ P2

X 1 = S 3 X S 7 and

Sp(2) -~ K(Z,7)

are

The H-spaces

and taking

X2. = Sp(Z)

are H - m a p s

+4(12).

These

X 1 ; Sp(g)

or Sp(Z).

The

since the obstructions lie

M I0 +3

are definitely not h o m -

m

otopy associative as ~pl : H 3 -~ H 7 is trivial w h i c h contradicts the existence of cup cubes in the 3-projective space.

43

The relation properties

of

f carries

we have studied between over to many

can also be proved

more

Theorem

has the homotop7

9.8.

Postnikov

X

system

k-invariants

of

X

Corollary

9. 9.

X

t y p e of a l o o p s p a c e

has each stage being a loop space

forms

as well as Aoki, H o n m a

structures.

of

Ef

and

The following corollary

directly.

are loop classes Various

other

properties

and the

in' Pn

of t h e t h e o r e m

if a n d o n l y if s o m e such that the

are loop maps.

have been proved:

[Suzuki],

[Iwata],

and Kaneko.

has the homotopy

t y p e of a l o o p s p a c e

if 17.(X) = 0 f o r 1

i
and

i>2p-2. This follows by induction on the stages of the Postnikov s y s t e m since

the only non-trivial k-invariants are in the range w h e r e an isomorphism.

the following result.

Corollary

H-space

rr.(X) = 0 for sented by an

An i
and

A -map" n

is

W e go into this in greater detail in the next chapter w h e r e w e

will be able to verify 9.10.

Hi(y)--~ Hi-I(~Y)

X

i>3p-1.

has the homotopy Todo

in cohomological

t y p e of a l o o p s p a c e

so, we will interpret terms.

if

"ct i s r e p r e -

DIFFERENTIALS

Throughout particularly

this chapter,

nice class

as a cohomology

~K(w,q+l).

w. : K × K - ~ K 1

generator

of

Hq(K;~r).

Alternatively,

the usual addition

in

Hq(;1~)

Hi(K/. as

K; Ir.(K)) 1

u

class,

is a multiplica-

Any other multiplicato the

We shall see later

that

group

Yr.

is represented

u E Hq(x;yr)

by an H-map

if and only if

the equation

into a homotopy

between

repre-

maps.

is primitive

10.2.

of

E 1 in the bar construction

if and only if

of the higher Let

dl[U ] = 0.

Theorem

spectral

sequence

10.1 g e n e r a l i z e s

to an

differentials.

X be an associative

following are equivalent:

q

a

i . e . , m S u = ~rlSu + ~ 2 ~ u .

interpretation Theorem

+ wZ St

u ¢ Hq(X,~r).

Our description shows

admits

~r

One need only translate sentative

q

since the obstructions

which are all zero.

the

K(w, q ) ,

~rl~ t

is the fundamental

A

BB.

Now consider

is primitive,

q

to this one for q >1

K(~r, 0) b e i n g t h e a b e l i a n

A class

the class

Wl~% q + ~Z~t q

"B

10.1.

consider

K(~,q)

by regarding

of

is homotopic

Theorem

for example,

That

#

K(vr, q)

with

ways,

u E Hq(x;~).

AnT representative

tion on

K(zr, n )

class

H-space.

in which case

L

and induces

we can regard

Y = K(lr,q)

and

K{w,q),

belong to

with

are the projections

tion on

homotopy

will denote an associative

occur

can be seen in several

up to homotopy, as where

X

of A - m a p s n

map can be interpreted a multiplication

IN THE BAR CONSTRUCTION

H-space

and

u • Hq(X;~v), t h e n t h e

u

45

l)

d [u] = 0 f o r

r < n

r

2)

Su

3)

u

pulls back to

XP(n)

is represented

by anA

-map. n

The equivalence differential

in a spectral

of

1) a n d

sequence

e x a c t c o u p l e of t h e f i l t r a t i o n . d [u] = 0 r class

is easy.

in

r < n, since A -maps

is a standard

u

proof that

as

f*(L

f being an An-map respect

interpretation

which can be seen most

A direct

If w e r e g a r d

K(lr, q ) , t h e n

2)

u

implies

the differentials

from

being an A -map n

) where

q

easily

of t h e

L

q

the implies

is the fundamental

dr[U ] = (fx...xf)*

dr[L q]

which are defined in terms

for

of

n

XP(s),

s < n. --

resents

L

Now

X-* ~XP(n)

Corollary

if

Su

If X

v ~ Hq+I(Bx;w)

r

since

extends

if a n d o n l y if

reasons,

L

q-r+l

Special

XP(n)-*

survives

q

cases

K(lr, q + l )

in which all maps and

dr[U ] = 0 for

of course,

i.e. , dr[U ] ~

which is zero for

to

is (p-1)-connected

The point is,

Corollary

] for all

-* ~ B K ( y r , q) -* K

10.3.

nectivity

q

to

E

where

it rep-

q+l" Finally

as

dr[t_

then

can be factored

are at least A -maps. n

u ~ Hq(x;~r), then

r + 1 < ¢L+2 -- p+l

that higher

El+r, q- r+l r

u

for

"

differentials

and

u = ~v

vanish

for con-

l+r, q=H ~ E1 r+l q-r+l

r+l

~X)

< (r+l)p. are well known.

10.4.

~ : H q + l (Y;~r) -* H q ( ~ Y ; ~ r )

is onto for

10.5.

~ : H q + l ( y ; l r ) -* H q ( ~ Y ; l r )

maps

q < 2p

where

Y

is p-

connected. Corollary q < 3p

where

Y

is p-connected.

We write "loop class"

greater these

~

rather

(Corollary than

a

content than the over-worked corollaries

yield Corollaries

9.10 is now established.

and refer

rather than as a " s u s p e n s i o n "

onto the primitive

to a class

subspace )

in the image

as a

as w e prefer terminology of

"suspension".

Applied

9. 9 - 1 0 of t h e l a s t c h a p t e r .

to k - i n v a r i a n t s ,

for

46

For class

Y = K(G,n),

it so happens that primitive

implies being a loop

if w = Z , b u t t h i s i s n o t t r u e in g e n e r a l . P

Example

10. 6. [ S t a s h e f f ] . --

a loop class.

In f a c t

Let

~ ~ 0 { HZn(z

d2(~P) ~ 0, s o

, 2n-1;Z).

p

~P

~P is n o t r e p r e s e n t e d

is p r i m i t i v e

but not

by ahomotopy

associ-

ative map. To obtain examples ferent

of m a p s w i t h

d

r

~ 0 for

r > 2, w e h a v e t w o d i f -

sources.

Example

10.7.

An-map

A non-zero

class

a

in

H2n(~cP(n))

is r e p r e s e n t e d

by an

b u t n o t b y a n A n + l - m a P. The class

cannot be a loop class

since

H2n+l(cP(n))

= 0.

Recalling

that ~CP(n) ~ S 1 X ~ S zn+l, the only possible'non-zero differential is d lot] = k[ul.., lu] where n

u ¢ HI(~cP(n)).

It follows that S I-~ COP(n)

is not an

An+l-mapE x a m p l e I0.8. [Zabrodsky]. _

Let X ~ K(Z --

, 2n-l) )< K(Z p

ative multiplication obtained as the loops on the space so that B x ~ Y. Ap_l-map,

The class

, Znp-g) have the associP

u = ~ 2np-Z c HZnP'Z(X;Zp)

Y

with k-invariant

(~ Zn )P,

is represented by an

not an A p - m a p . B y comparing the spectral sequence with that of the product structure

on K(Z

, gn-l) X K(Z P

, 2-np-Z) w e see the only w a y the class

(L Zn )P =

P

it Zn_ll... [~ gn_l ] can be killed is by E x a m p l e I0. 9. [Zabrodsky]. and k-invariant ~ l WZp_z(Y)

restricts to ~ p - 2 in K ~

and

k-invariant ~ l

= 0, there is a class

2p-3

to ~ P - 2 L 4p-6"

be the space with

WZp- 3

Now

u

4p-5"

~Y

where

2p-2 + k(L 2p-2 )2 for any

u c H2P(P'I)-3(X)

(We have -~p-2~pl

k ~ Z p"

which restricts

is not a loop class since there is no class in Y

which

+ ks 2) = -2s P + k z ~ P P - J ~ P J - Z ~

, 2p-2). ) B y the s a m e token, for k = 0, L P goes to P

W4p- 6 ~ Zp

Zp-3' which can be regarded as the loop space

w4 p_5(Y) ~ Zp

Since ]PP-Z~IL

Let X

dp_l[~ 2np_2].

zero

in Y

and

0

47

thus u

[~ I ' ' "

I t ] must

is the first

class

assasin.

Thus

Example

10.10.

u

be killed in the Eilenberg-Moore

in

X

which exists

is represented

[Gheng].

Let

for unstable

spectral

reasons,

by an Ap_l-rna pbut

Y be the space with

~

With respect

back to the fundamental

not an Azi-rna p.

class

is represented

Again

be the

k-invari-

" 2 u ~ H z l - (f~Y) w h i c h p u l l s

a class

K(Zz, zi-z)

must

w2 = w 2 i _ l = Z 2 a n d

to loop multiplication, of

u

Since

not by an Ap-map.

2 i+l ant

sequence.

by an A . -map but 21_i

u kills [~ ii... IL i] in the spectral sequence.

For

i=g,

v ~ H 7 {Y) w h i c h restricts to S g S IL is represented by an A ) - m a p q q 7

the class

but not an A4-rna p for any A4-structure on

Y.

It is possible to give chain formula for using a spectral sequence is to avoid such work.

d , although a m a j o r point of r T h e case

d g is quite m a n a g e -

able and illuminating in t e r m s of our next topic. If

dl[U ] = 0, t h e n

any representative Gq'l{x~:

X;w).

u ~ -For

u

u

we have

any choice

c = (1 × m ) ~F b - {m × 1)~

is primitive.

b

- w1 ~ -u - w/ u

m~

of b , t h e c o m p o n e n t

represents

by a coboundary

On the chain level,

dg[u].

u, w e a l t e r

b

a n d if w e a l t e r

we alter

c

by

dg[u ] is

[(1 X m ) * - ( m X 1)*] H q ' l ( x ~ x ; w )

[(1 X m ) ~: - ( m X 1 ~ ]

= 5b{u)

in

this means

where

our choice

of a c o c y c l e .

b •

G q - l { x ~b X ~ - X ; w )

Notice by altering of b

of

our choice

for a given

of u,

Thus the indeterminacy

which is

dl(Hq-l(x

for

~=X;w))

in

just as

it should be. Example

10. 6 i s w o r k e d

In C h a p t e r sented

by H-maps.

associative

out this way in [Stasheff].

8 we saw that the k-invariants

Similarly

of an H-space

one can show the k-invariants

H - s p a c e are r e p r e s e n t e d

by A3-rnaps,

are repre-

of a h o m o t o p y

so e x a m p l e s I0. 6 and 10. 8

and I0. 9 for p = 3 provide e x a m p l e s of H - s p a c e s w h i c h are not h o m o t o p y associative.

In order

associativity

to generalize

more

fully from

these

results

our homotopy

to A -maps n

we need to study

p o i n t of v i e w .

A

-SPACES n

We h a v e s e e n t h a t t h e e x i s t e n c e H-space

is e q u i v a l e n t to h o m o t o p y a s s o c i a t i v i t y ;

significance

of p r o j e c t i v e

induced by A3-maps natural

of a p r o j e c t i v e

n-space.

In b o t h c a s e s w e a r e l e d to c o n s i d e r

invariant

for an

to i n q u i r e a s to t h e

On the other hand, we have seen that fibrings

to a s k a b o u t t h e s i g n i f i c a n c e

homotopy

it is n a t u r a l

admit homotop7 associative

equation but as a conjery

three-space

again,

it i s

of a f i b r a t i o n b e i n g i n d u c e d b y a n A - m a p . n the associative

of n - v a r i a b l e

characterization

multiplications;

equations.

of s p a c e s

law not as a three variable T h i s in t u r n l e a d s to a

of t h e h o m o t o p y t y p e of a s s o c i a t i v e

H-spaces. Consider determine

five maps

a single application topy as a map

the various

w a y s of a s s o c i a t i n g

of X 4

X, e a c h of w h i c h is h o m o t o p i c

into

of h o m o t o p y a s s o c i a t i v i t y .

h : I-~ X X3, we can construct

Regarding a map

S1 as a pentagon with the five maps as vertices

If t h i s m a p c a n b e e x t e n d e d to a t w o c e l l of p r o j e c t i v e case for

~Ix,

the associating

spaces

four factors.

These

to t w o o t h e r s b y

the associating

homo-

S 1-* X X 4 b y r e p r e s e n t i n g

and the five homotopies

as edges.

,iwx e 2 with boundary

can be extended one stage further.

t h e s p a c e of l o o p s p a r a m e t e r i z e d homotopy can be represented

S 1, t h e c o n s t r u c t i o n

T h i s i s of c o u r s e

by the unit interval.

schematically

by

the

In ~ I x ,

49

s o t h e m a p of

S1 we are looking at is represented

which can be extended to

e 2 by deforming

by

all paths to

(wx) (yz)

in the obvious

way.

To proceed volving maps morphic

to

m.

1

K 2 = ~.

(K r × K s ) k

K . X X i -~ X

:

1

where

w e n e e d a f a m i l y of c o n d i t i o n s

K.

is a s p e c i a l

1

in-

cell complex borneo-

Ii-2.

D e f i n i t i o n U . 1. Let

with this approach

K. d e n o t e s a c o m p l e x 1

Let of

K. = C L . , 1

1

(Kr × K s )

symbols,

e.g.,1

Z ...

responds

to i n s e r t i n g

constructed

t h e c o n e on

L.

(k k + l . . . two pairs

w h i c h i s t h e u n i o n of v a r i o u s

1

corresponding

inductively as follows:

to inserting

k+s-1) ...

i.

of p a r e n t h e s e s

a p a i r of p a r e n t h e s e s

The intersection with no overlap

copies in

of c o p i e s c o r or with one as a

subset of the other: I ...

(k...k+s-l)

...

I...

(k... (j...j+t-l)

(j...j+t-l)

...

r

× K

s

-~ K.

1

(An a l t e r n a t i v e

sense,

is t h e i n c l u s i o n of the c o p y i n d e x e d by indexing by trees

i

k+s+t-Z) ...

Thus the foUowing definition makes K

...

or i.

where

a k ( r , s) :

I . . . (k... k+s-1) . ..

i s g i v e n a t t h e e n d of t h i s c h a p t e r . )

i.

i

50

Definition maps

11.2.

An An-space

(X;{Mi})

M . : K. X X . - ~ X , i < n

x i) = M r (p, x [ . . . . .

M.

exist and satisfy

and a family

of

M s (or, x k . . . .

Xk+ s _ l ) . . . . .

xi)

p ~ K , ~ c K . r

If t h e

X

with unit and

M i ( ~ k ( r , s) (p, a ) , x I . . . . . for

of a s p a c e

such that

1) M 2 i s a m u l t i p l i c a t i o n 2)

consists

these

S

conditions

for all

i > 2, w e s p e a k

of

{X, { M i } )

1

as anAl-space.

Where

Conditions

necessary,

we refer

approximating

The complexes

K.

to the

these were

{Mi}

as an An-fOrm.

first presented

a r e a l s o of i m p o r t a n c e

in [Sugawara].

in category

theory

in

1

relation

to coherence

morphic

to

exhibited

Ii-2

the

of f u n c t o r s

is not obvious.

K.

1

as specific

[MacLane]. Several

That the complexes

are homeo-

ways to see it are available.

convex subsets

of

Ii-2

which are

clearly

I have homeo-

K3 morphic

to the whole cube,

e.g. ,

K2 = *

~0

1~2

/"

1

/

K4

\ Adams

has computed

the homology

and fundamental

group

of

L.

and

1

thus shown

L. f o r 1

i>

shown the cell complex Z~i - 2 .

[Boardman]

5 has the homotopy

t y p e of a s p h e r e .

L . i s t h e d u a l of a c e r t a i n z

has given a cubical

decomposition

subdivision of

K. 1

idea first

suggested

by Adams.

Stallings

has

of the boundary

indexed by trees,

of an

51

Associative fined to have constant

H-spaces value

are

of c o u r s e

Before

description

proving

this,

-spaces

o0

since

M.

can be de-

1

x 1. . . x i.

T h e m a i n p o i n t of t h e d e f i n i t i o n invariant

A

of A

CO

of a s p a c e

of t h e h o m o t o p y

we present

the main theorem

-space

is that it is a homotopy

t y p e of a n a s s o c i a t i v e about A -spaces,

H-space. which is

n

w,h a t o n e s h o u l d e x p e c t . Theorem

11.3.

A connected

CW

admits

X

the structure

of a n A - s p a c e

if a n d

n

only if there

exists

a sequence

of quasLfibrations

E 0 =X-*

1-*

,., with

E.1 c o n t r a c t i b l e

in

En_ 1

. . .

B I-~ ...

Bn_ 1

E i + 1.

The construction

is not iterative,

En

although

En_l~

be

~

inductive.

We let

Kn+ z X X n+l

n

Pn Bn

Bn_l~

Kn+ z X X n n

The attaching ~n{0k(r' s)(P'c)'Xl with the

M

term

s

.....

map for

if

the first

x

factor,

. M. s (. ~ ' X. k . . . .X k. +.s -.1 ). .

k + s - 1 = n + 2.

x

coordinate

By induction we prove first

is given by

n

Xn+l) = ~ r ( P ' X l ' .

deleted

obtained by dropping

E

Pn'

is a quasifibration.

The attaching

map for

B

n

is

consistently.

induced by projection

This

Xn+l)

time

we break

onto all but the

B

into two

over-

n

lapping pieces

by considering

1

crucial

condition

weak homotopy one

onto

in proving

equivalence

and breaking

K. = CL.

Pn

is a quasifibration

occurs

as the fibre

~ -~ ~

occurs

over

over

(~,x_ .....

x ) where n

L..

This map

can be identified with mapping

1

up a cone as before.

The

1

in showing that a

(T,x z .....

is a deformation x

Xn )

is mapped

of a n e i g h b o r h o o d into

x

by

of

to L. 1

52

x-~ M r(p,x,x 2 ..... M

r

(p,x,e .....

e)

As for the limit, Theorem o n l y if

l l . 4. X

Xn) since

for fixed X

p

and

is c o n n e c t e d ,

the arguments A connected

X

admits

the structure

admitting

an associative

before,

an A

00

for some

invariant

multiplication

is not.

associating

if a n d

a s o p p o s e d to

We h a v e m a d e t h i s l a t t e r

follow.

quasifibrations

Thus

multiplicative

as in T h e o r e m

S3 with these particular

the unit. A d a m s

has given m e

-form.

remark

on

S3

chosen

Actually he works by

exotic multiplications

cannot be deformed

T h e p r o o f of T h e o r e m

o0

11. 3 f r o m w h i c h t h e A

h o m o t o p y t y p e of an a s s o c i a t i v e

these multiplications

of m u l t i p l i c a t i o n s

[Slifker] shows that a properly

h o m o t o p y c a n b e e x t e n d e d to a n A

constructing

H-space

=o

-forms

has the homotopy

t h o u g h on t h e s t a n d a r d

to be a s s o c i a t i v e .

11. 4 w h i c h i s i m p l i e d b y o u r e x p o s i t i o n u s e s

an alternative proof of a stronger result w h i c h

no use of units.

Theorem {Mi} ~

-space

b u t l e t us e x p a n d on it now.

only eight are homotopy associative.

makes

00

is a homotopy invariant while

R e c a l l t h a t of t h e t w e l v e h o m o t o p y c l a s s e s

S3

of a n A

case.

Y.

statement

-structure

to

to t h e i d e n t i t y .

t o t h o s e in t h e a s s o c i a t i v e

CW

since admitting

and hence homotopic

similar

N o t i c e t h i s is a h o m o t o p y 4.3,

T h i s in t u r n i s h o m o t o p i c

are

h a s t h e h o m o t o p y t y p e of ~ Y

Theorem

x..1

II. 5. {Adams). satisfying

If X

admits a m a p

2) of II. 2, then X

with an associative multiplication n

M z : X X X-~ X

and a family

is a deformation retract of a space

such that n IX X X

is h o m o t o p i c in Y

Y to

m. T h e proof has b e e n simplified by [Boardman]. while for A

n

-spaces with n

Definition U. 6. Bn_iVKn+

defer the proof

finite, w e again look at projective spaces.

If (X, { M i } ) is an A

Z X Xn

We

n

-space, XP(n)

constructed in proving T h e o r e m

will denote the space

II. 4.

53

Theorem

Ii. 7.

If Y

is a M o o r e

space of type

(G, Zp+l) w h e r e

abelian group in w h i c h division is possible for all p r i m e s prime

p, t h e n

Y admits

The maps

the structure

M.

for

i< p

of a n A p . l - s p a c e

are constructed

G

is an

q less than the b u t n o t of a n A p - s p a c e .

a s in t h e c a s e

p = 5

1

(Example M

P

7.6) by deforming

t h e t r i v i a l o n e s in ~ 2 ~ Z y .

follows from the decomposability

p-fold cup products

in

YP(p)

of ~ p + l

The nonexistence

contrasted

of

with the non-trivial

if i t w e r e to e x i s t .

Given two A n - s p a c e s ,

w e can again consider m a p s

which respect the

structure. Definition II. 8.

If (X;(Mi})

a homomorphism

if

and

(Y, {Ni})

are A n - S p a c e s ,

a map

f : X -~ Y

f M i (7, x 1 ..... xi) = N i (~, fx I..... fx.1)" It is also possible to consider m a p s

of A

-spaces which respect the n

structure

up to homotopy,

pletely here.

but the details are too complicated

F o r example,

to m e n t i o n

respecting a n associating h o m o t o p y

corn-

involves a 2-

cell subdivided as a hexagon, while respecting an A4-structure involves a c o m plex w h i c h looks like

7ill

z_&L

k\ \

is

54

However,

maps

of an A

n

-space into an associative H - s p a c e

or vice v e r s a are

manage able. Definition A map

U . 9.

Let

f : X -~ Y

(X, { M i } ) b e a n A n - s p a c e

is an A -map

if t h e r e

exists

and

Y

an associative

a family

H-space.

of m a p s

n

h i:

Ki+ 1 × X i-~ Y

such that

h 1 = f and h i (Ok ( r , s ) ( p , a ) ,

x 1. . . . .

= h r ( 9 , x 1. . . . .

x .1) =

Ms(a,x k .....

= h r _ l ( p , x 1. . . . .

Xk+s.1) .....

X r . 1 ) h s _ 1 (¢r, X r . . . . .

It is easy to see that an A -map

xi)

x,)l

of a s s o c i a t i v e

for

for

r + s = i+2 k < r

k = r.

H-spaces

is an A -map

n

sense

with respect

Theorem

11.10.

to the trivial If

(X;{Mi})

higher

homotopies

is an An-space,

The proof is a generalization h i : Ki+ 1 X X i-~ ~XP(n)

then

X -~ ~ X P ( n )

conveniently

by defining some

cO

-form.

is an An-map.

8.6.

The maps

in terms

of

reasonable

homeomorphisms

K i + 2. M a n y of o u r r e m a r k s

associative H - s p a c e s

Theorem

used as the A

of t h a t of T h e o r e m

can be described

K i + 1 × X i × I "*- K i + 2 × X i ' * X P ( i ) Ki+ 1 X I~

in this

n

11.11.

X

about H-maps

c a r r y over to A

admits

n

-maps

of H - s p a c e

in this m o r e

and A -maps n

of

general sense.

of a n A - s p a c e if a n d o n l y if e a c h s t a g e

the structure

n

of a n y P o s t n i k o v

system

for

homomorphisms

and the k- invariants

Corollary ll. lZ. (cf. 9. I0). space provided

X

does in such a way that the projections are represented

An An_l-space

~.(X) = 0 for

i< p

and

X

Pn

are

by An-maps.

has the h o m o t o p y

type of a loop

i > np+n-4.

I

Example

11.12.

class

of ExampIe

u

Let

W be the space 10.8

or

constructed

10. 9, t h e n

W

by using as k-invariant

admits

an Ap_i-form

the

but not an

A -form. P These

examples

used the bar

construction

spectral

sequence.

More

55

generally,

for an A -space

X

we have the spectral

sequence

derived

from

the

n

finite filtration Theorem

of

11.14.

XP(n) Let

by

XP(i),

i < n.

(X, { M i ) ) b e a n A - s p a c e

and

u e Hq(x;Tr).

Then

n

If

3)

u

is represented

holds for

1)

d

2)

Su

r

[u] = 0 f o r

r < i

pulls back to

by an A.-map

then

1

i f a n d o n l y if

XP(i)

1) a n d

2)

follow.

The converse

i < n.

The converse know to prove

~XP(i)

is stated

-* X

in this limited

is an A.-map

way because

is to use

the only way I

XP(i+I).

1

Our analysis

of A - m a p s

in terms

of c o h o m o l o g y

classes

also applies

n

to the maps

inducing the succession

Zabrodsky'

s technique.

Zabrodsky's

method

of f i b r a t i o n s

used to construct

T h u s w e f i n d if t h e i n g r e d i e n t s

are A -spaces

and maps,

first

the result

is anA

used the technique

to construct

a homotopy

b u t n o t of t h e h o m o t o p y

sky].

was not an A5-space.

Example

the example

11.15.

Let

P1 = {2,3),

That the resulting

X

~ 1 : H3 ( X z ; Z 5 ) -~ s l l ( X z ; Z 5 ) essentially A3-space marks,

the same if

G

the example

not A -spaces P

is trivial.

can he adapted

several

i d e a of i n d e x i n g b y p l a n a r in the plane,

H-space

t y p e of a l o o p s p a c e

[Zabrod-

follows from Xz

is an A3-space

Adams

divisible

by

the fact that

to show

Z and

3.

to give finite complexes

follows from

Y(G;Zn+I)

is an

As Zabrodsky

re-

which are Ap_ 1 but

p.

Boardman' are

associative

X 2 = (S 3 × S 5 X S 7 × S 9 × s l l ) ( P 2 ) .

That

used by Frank

of r a t i o n a l s

for any prime

There

X 1 = SU(5),

is not an A5-space

argument

consists

-space. n

which was a finite complex In fact,

in

to be mixed by

n

Zabrodsky

X(]t~ 1)

s Proof clever

trees,

so as to keep track

of T h e o r e m

ideas

11. 5.

in the proof.

i.e. , directed

connected

of w a y s of i n s e r t i n g

First, acyclic

parentheses.

there

is Adam's

finite graphs For

example,

56

w{(xy)z)

corresponds Second,

an associative

to t h e t r e e there

operation

m ( m × 1) = m(1 X m )

D e f i n i t i o n 11.16.

~

o

is an idea from categorical not by a multiplication

algebra

of c h a r a c t e r i z i n g

m : X × X -* Y

and a relation

but as follows:

An A-structure

on a s p a c e

X

i s a f a m i l y of c o n t i n u o u s m a p s

n

{k i : X i - ~ X , i > 2} Usually, The other trees

k.

1

s u c h t h a t if k

i

~ m. = m then k o(vk ) = k . 1 n rn. m 1 1 is t o b e t h o u g h t of a s t h e m u l t i v a r i a b l e m a p

can be indexed by the trees

can be obtained by composites The complex

where

T

K.

1

is a tree with

without disconnecting

n

the tree).

For

-//

m

.

and all the

of t h e s e .

will now be represented

branches

edge is subdivided needlessly.

~kk

x 1. . . x

a s a u n i o n of c u b e s

C(T)

(= i n p u t s = e d g e s w h i c h c a n b e r e m o v e d our present

The cube

C(T)

purposes

we will assume

will have parameters

no

indexed by

t h e e d g e s of t h e t r e e w h i c h a r e n o t b r a n c h e s .

Definition II. 17.

WA

with n-branches.

(n, I) is the union with identifications of C(T) over all trees

T h e identifications are that a face

t. = 0

of

C(T)

is to be

1

identified with edge indexing

C(T' ) w h e r e

T'

is obtained f r o m

t.. 1

F o r example, WA(Z,I) =

=

C(V)

W A ( 3 , 1) = ~ ( ~ / ) G ( ~ )

G(h) ~)

T

b y shrinking to 0 the

57

It c a n b e s h o w n t h a t Definition

U. 18.

A

WA(n,1)

WA-structure

on

is a cubical

X

is an A

subdivision

-structure

of

K . n

without units,

i.e. ,

~C

a family

of maps

l~l : W A ( n , 1 ) × X n - ~ X n

Mn(~, x I .....

x n) = lVir (p, x 1 . . . . .

if ~ = (tI ..... tn_Z)

•

C(T)

with

such that

Xk_ 1, M s (c;, x k . . . . .

t. = 0 w h e r e ,

Xk+ s) . . . . .

x n)

if the e d g e indexing

t. is

1

deleted, same

T

as

decomposes

T' [J T "

values to the c o r r e s p o n d i n g

1

while

p e C(T' ), a e C ( T " )

e d g e s as does

7.

T h e f i n a l i d e a of t h e p r o o f i s to u s e t h e nective

tissue

to build something

like a tensor

associate the

WA(n,1)

algebra

complexes

of w h i c h

X

as con-

will be a

retract. First

we let

that now we permit

For

example,

description as

WA(n,1)

trees

WA(Z,I)

be the complex

with an extended

= C(y) = ;

root,

: and

constructed

except

i.e. , "'~'"

WA(n,I)

gives a useful parameterization.

as before

= WA(n,l)

We also let

× I but the tree

WA(1,1) ~ •

regarded

C(1).

Definition MX

If. 19.

Given a

is defined by taking

(T,x I..... • ,p,a

x n)

with

(~)

WA-structure WA(n,1) × Xn (p,x I .....

on

X, the associated

for each

n

Xk_l, Ms(~,x k .....

associative

space

and identifying Xk+s_l) .....

x n)

where

are as above. If

t. = 0

on the edge corresponding

to the extended

root,

then

(•)

1

means

Mn(t I .....

~i .....

The operation

t k - l ' Xl . . . . . on

MX

= ((p,a),x I ..... regarded

as being in

tive involves combinatories

essentially

only the

aptly called tree

is given by

Xr+ s)

C ( T 1 v TZ).

Xn)"

where

(p,x I .....

now for

That the operation WA-parameters,

surgery.

x r ) • (a, X r + I . . . . .

p c C(TI),

ae

X r + s)

C(Tz), (p,a)

is well defined and associaan exercise

in parameterized

is

MASSEY PRODUCTS The differentials were

usefu~ in analyzing

the homology

spectral

AND GENERALIZED in the cohomology

k - invariants

sequence,

BAR (31)NSTRUCTION

Eilenberg-Moore

in terms

of

spectral

A -maps. n

we have in particular

sequence

If we turn to

differentials

of the form

dr[all .. l Ur+ l] represented These

by homology

are closely

ucts originally introduced,

related

of t h e a s s o c i a t i v e

to the Pontrjagin

defined in the cohomology

these homology

the duality; however, algebra,

classes

products

Massey'

ring analogues of a n a r b i t r a r y

were

s procedure

so we will use the term

"Massey

should now expect,

strict

homotopy

we save such generality

analogue;

associativity

As for the differentials

H-space

called

in question.

of t h e [ M a s s e ~ p r o d space.

Yessam

When first

products

product"

generically.

as one

by an appropriate

f o r t h e e n d of t h i s c h a p t e r .

more

generally,

it turns

products

out [ May]

by appropriate

Massey

allow matrices

classes

than single homology

rather

differential

In fact,

that they are all determined of h o m o l o g y

to emphasize

is valid in any associative

can be replaced

dr

X

if w e a r e w i l l i n g t o classes

as

arguments. Until further

notice,

with differential

d.

Definition

Let. u,v,w

product

12.1. ~

Remark.

If defined,

has the larger remarks higher

where

.

u

E H(A)

.



indeterminacy

Massey

be an associative s original

such that

of H ( A )

represents

like this continue order

A

We start with Massey'

is the coset

B

ux(.1)deg u

let

by

uH(A) + H(A)w

.

.

[Uehara

.

triple

by

dy = uv.

Notice that

d2[ulvlw]

this situation,

The differentials are usually

and Massey]-

The Massey

As we generalize

but the latter

algebra

determined

dx = vw,

d2[ulvlw]

represents

to be applicable.

products,

product

uv = 0 = vw.

u, e t c . , a n d

H(A). H(A).

differential

more

are determined delicate,

less

by

60

often defined and with s m a l l e r i n d e t e r m i n a c y .

choice, for of course

T h u s in g e n e r a l i z i n g we h a v e s o m e

d r [ . 1 1 . . , l ur+ 1] can itself be regarded as a generalization.

We w i s h to d e f i n e h i g h e r o r d e r M a s s e y p r o d u c t s < u l. . . . . , u i + s >

a r e defined and z e r o for

l<_i

and

s < r-1.

< u 1. . . . .

Ur> w h e r e

[May] h a s

d e v e l o p e d the f o l l o w i n g n o t a t i o n .

D e f i n i t i o n 12.2.

aij ~ A

The M a s s e y p r o d u c t < a 1. . . . , a t >

for l<_i_< j<_r

excluding the case

is d e f i n e d if t h e r e e x i s t

ij = ir such that a..11 represents

a. ~ H(A) a n d 1

daij = /i) k=i where

aik ak+l. J

aik has signs altered appropriately.

The M a s s e y p r o d u c t < a 1. . . . .

ar>

is the set of all homology classes represented by r-I 1

a_ik a k ÷ l r

for a n y s u c h s y s t e m .

T h e o r e m 12.3.

represent

If < a 1, . . . , a r > is d e f i n e d , t h e n a n y of its r e p r e s e n t a t i v e s

dr'l[all...

]ar].

The p r o o f is s t r a i g h t f o r w a r d , though t e d i o u s .

The d e f i n i n g s y s t e m

aij is used directly to show d s [all " ' " l a r ] = 0 for s < r - 1 a n d to o b t a i n a representative

X i n B'{A) of

[ a l l . . . J a r ] , w h i c h is a c y c l e u n d e r the t o t a l

d i f f e r e n t i a l in B(A).

E x a m p l e lZ. 4.

In H . ( ~ C

P ( n ) ) , if u g e n e r a t e s

Hi, t h e n < u , . ~ n+l

generates

HZn.

S i n c e n - f o l d M a s s e y o p e r a t i o n s a r e e a s i l y s e e n to be n a t u r a l

w i t h r e s p e c t to A - m a p s , t h i s a g a i n shows n

S1-~ ~ C

P(u)

i s - n o t a n A n + l - m a p.

61

T h e c o m p u t a t i o n m a y be done by o b s e r v i n g

H2n+I(CP{n)) = 0 w h i c h c a n b e

a c h i e v e d o n l y if the g e n e r a t o r of H Z n ( ~ C P(n)) The m a t r i c M a s s e y product,

is k i l l e d by

dn.

i n t r o d u c e d b y M a y , is a f a i r l y

s t r a i g h t f o r w a r d g e n e r a l i z a t i o n in w h i c h

a.

V..

is r e p l a c e d by a m a t r i x

l

obtain a reasonable definition, certain conventions about matrices

To

1

will be

observed. If V is a m a t r i x

(v..), t h e n ~

w i l l b e the m a t r i x

((-I) l + d e g v i i v . . ) .

U

An o r d e r e d p a i r of m a t r i c e s n × q and for each

i,j,

matric Massay product Vn

is

q × 1 and

U

(X,Y)

is m u l t i p l i c a b l e if X is

deg Xik + deg Ykj
is c o n s t a n t

for

m × n

and

1 < k < n.

The

V > w i l l be c o n s i d e r e d only if V 1 is n

V 1 . . . V . , j V j+ 1 . . .

Vk

is m u l t i p l i a b l e f o r e a c h

G i v e n the a b o v e c o n v e n t i o n s , D e f i n i t i o n 12. g c a r r i e s d e f i n e the m a t r i c M a s s e y p r o d u c t

"

Y is

1 × P,

j , k _ < n. o v e r v e r b a t i m to

T h e i n d e t e r m i n a c y is the s e t

of a l l p o s s i b l e d i f f e r e n c e s c o r r e s p o n d i n g to d i f f e r e n t c h o i c e s of t h e s y s t e m

A... U

M a y (in p a r t f o l l o w i n g [ K r a i n e s ] ) g i v e s b o u n d s on the i n d e t e r m i n a c y , linearity formulas,

associativity formulas,

Massey products, permutation rules. As b e f o r e ,

" s l i d e " r u l e s and for o r d i n a r y

He a l s o d i s c u s s e s n a t u r a l i t y .

t h e s e M a s s e y p r o d u c t s c a n b e r e l a t e d to the d i f f e r e n t i a l s

in the s p e c t r a l s e q u e n c e .

Matric products are particularly relevant when

dr

is

d e f i n e d on a c o m b i n a t i o n of t e r m s w i t h o u t b e i n g d e f i n e d on the i n d i v i d u a l t e r m s . F i n a l l y M a y s h o w s t h a t m a t r i c M a s s e y p r o d u c t s d e t e r m i n e the s p e c t r a l s e q u e n c e in t h e f o l l o w i n g s e n s e : such that for each element

x

if x

d x P

s u r v i v e s to

<W 0 . . . . .

E , then P

Wp_I,V>.

all

q there are matrices

of E p ' q t h e r e is a c o l u m n m a t r i x is r e p r e s e n t e d

The differentials

of p a r t i a l d e f i n i n g s y s t e m s f o r

For each

drX f o r

<W0 . . . .

x ; on the o t h e r h a n d , t h e y a r e h u g e .

'

W

p-l'

V

such that

by a suitable r e p r e s e n t a t i v e r <_p V>.

W.

of

c a n b e e v a l u a t e d in t e r m s

N o t e t h a t the

W.

i

work for

62

Of course

A -spaces

can be mimiced

on t h e c h a i n l e v e l ,

just as

n

associative

differential

algebras

Definition 12.5 [Stasheff]:

mimic

topological monoids.

( A , m . , 1 < i < n)

is a n A - a l g e b r a

1

--

if m . : A x -* A

n

i

such that 0 = T

(-I)

m r {l@. . . @ m s @ ' ' ' @ I )

k,s

k-1 where for

al@[email protected]

the

(s+l)k + s (i + ~

~ is

d i m a.) 2

/

I

m 1 p l a y s t h e r o l e of d i f f e r e n t i a l . )

(Note:

For an A

-algebra,

there

is a g e n e r a l i z a t i o n

of t h e b a r c o n s t r u c t i o n .

00

Given an A

Definition 12.6 [Stasheff]:

-algebra

(A,m.),

o0

B'(A)

is

@ i=0

A i with the differential

~[all-.-lanl : where

the tilde construction

1

(-1)x [all... [ms(ak®...®%+s_l)l'-" I%1.

k-1 )~ = (s+l)(k+l) + s(i + >

dim aj).

1 A l l of o u r m a c h i n e r y and Massey

products,

is d e f i n e d b y Finally,

connected

C QC P q

uy+xw+_m3(u,v,w)

differential (1~)

Let

including the spectral

for

coalgebra,

JC = C/A

~

Let

x e C

P

For example,

Let

~

P,

and define

E@...®C

n>O ~ --

be a simply

graded,

denote the component

> 0 that

•

(C, ~)

i . e . , if p o s i t i v e l y

%, q

7-Cp

~(C) = T(~) =

sequence

[Stasheff].

the dual situation.

= (&®l)~.

and we shall assume

£~0, p(X) = x@l.

over,

albeit with additional complication.

l e t us c o n s i d e r

associative

C O = A, C 1 = 0 and

carries

n

o(X) = l ~ x

and

of

~

in

63

with

@c = - d c =~'---- (-l)P A

Z__

p,n-q

c for c e C

and e x t e n d m u l t i p l i c a t i v e l y to

n

l
r e g a r d e d as the f r e e a s s o c i a t i v e a l g e b r a g e n e r a t e d by ~ . [Adams] shows

H,(~(C~(X))

is isomorphic to H,(~X)

b u t the p r o o f is a t o u r de f o r c e u s i n g s i m p l i c i a l c h a i n s f o r

as an algebra,

X and c u b i c a l c h a i n s

for f~X. Following a suggestion of Peter May, w e outline a proof which follows m o r e n a t u r a l l y f r o m the p o i n t of v i e w we h a v e e s t a b l i s h e d .

w

T h e o r e m 12.7.

Proof.

T h e r e is an

shin

equivalence

~ C , ( X ) ~--~ C , ( ~ X ) .

We w i l l c o n s t r u c t m u l t i p l i c a t i v e e q u i v a l e n c e s .

B C, (X) ~-~ C, (Bf~X) -- ~ B C , (~X) -~ C, (f~X) The left hand m a p is induced by the h o m o t o p y equivalence multiplicative by naturality of ~ . the standard inclusion

B~X-~

B

and is

The s a m e applies to the middle m a p which is

BC~(Y) -~ C$(BY).

The fact that ~ B

is naturally

equivalent to the identity via shin m a p s should be a basic bit of general nonsense in the appropriate category; at present only a laboriously detailed proof is conceivable. Since ~ B A construct

: BA-~ A

homotopy equivalence.

is the f r e e a s s o c i a t i v e a l g e b r a on B'A, it is e n o u g h to s u c h t h a t its m u l t i p l i c a t i v e e x t e n s i o n We d e f i n e [a] = a = 0 on

A n inverse for given by L

and @ A ®n . n>l

is the obvious (but not multiplicative m a p

L(a) = [a] for which

_~ I ~ A

:~] B A -~ A is a

L = IA.

i : A-~BA

The details of the h o m o t o p y

are given in [Halperin and Stasheff].

It is worth remarking that if A

modules are of finite type then letting

64

(

)

denote

Horn(

, A ) we h a v e 8 ( A )

= (B'A) .

Thus

H~,(~X)

can be

c a l c u l a t e d a s t h e h o m o l o g y of [BC ~ (X)] ~. The spectral sequence derived from filtering n u m b e r of b a r s " h a s

E 1 = BH (X) a n d

(X) b y

E Z = T o r H * (X) (A, A).

a r e now g i v e n in t e r m s of M a s s e y p r o d u c t s w i t h Massey' s original

BC

dz[ul v]w]

The differentials

being represented

.

In general

B and ~

are too large for complete computation but

s p e c i f i c e x a m p l e s in l o w d i m e n s i o n s c a n o f t e n b e done n e a t l y in t e r m s m

"the

of B ~

m

o r ~]B, e . g . ,

Example I0.7.

I t i s t i m e we l e a v e a b s t r a c t d i f f e r e n t i a l a l g e b r a a n d r e t u r n to s o m e geometry.

by

HOMOTOPY We have analyzed view.

associativity

We could do the same

expressed length;

diagramatically.

other examples

X 3 -* X

commutativity

Every

{x,y,z)-~

direct

homotopic

to

An H-space

mT

where

The easiest H-space k + /a

homotopy =/~(s-t)

/~ + k

t<

s<

evaluation

on a space

is homotopy

Moufang.

of t h e o b s t r u c t i o n s .

I, a homotopy

on a sphere

is homotopy

disassociative,

commutativity.

{X, m )

is homotopy

commutative

~X

of a n H - s p a c e . to pointwise

{Here to have

if t

is determined

is

commutative

The point is that loop addition

/~ : [ O , r ] - ~ runs from

by

but homotopy

multiplication

t E I. ) F o r

if m

T{x,y) = yx.

of a n o n c o m m u t a t i v e

k-/~ t w h e r e

t+r.

but we could normalize izedby

example

is homotopic

is given by for

{x(yz))x

T : X × X-* X × X by

is the loop space or

M o u f a n g if t h e f o l l o w i n g

(xy)x.

We now turn to homotopy

13.4.

is homotopy

multiplication

multiplication

x(yx) ~

Definition

X

y = e, we also see

13.3.

we have

at some

{ x , y , z) -~ ( x y ) ( z x )

Every

The proof involves

i.e.,

p o i n t of

We will study homotopy

An H-space

13.2 [Norman].

Theorem

a homotopy

condition which can be

are homotopic:

Taking

from

for any other algebraic

and

Theorem

extensively

have also been handled.

Definition 13.1 [Norman]. maps

COMMUTATIVITY

~Ix

X 0

to

then q

of l o o p s v t (s) = *

where

consisting

12 × ~ X X ~ X - ~

X

k"

k./~ .

The

for

O< s<

[O,q]-~ X

of loops parametergiven by

t

66

(t, s, k,/~) -~ k(t)./~(s).

There

are very few examples

of finite dimensional

homotopy

commuta-

tive H-spaces.

Theorem

13.5 [Hubbuck].

If X

is a connected finite c o m p l e x and

h o m o t o p y c o m m u t a t i v e H - s p a c e , then torus

X

(X,m)

is a

has the h o m o t o p y type of a point or a

S 1 X ... X S I. H u b b u c k ' s proof involves applying c o m p l e x K - t h e o r y to the projective

plane

XP(2).

ways.

T h e h o m o t o p y c o m m u t a t i v i t y of m

H u b b u c k uses the existence of a m a p

is reflected in XP(2)

f : XP(Z) -~ XP(Z)

in two

such that there is

a homotopy commutative diagram S X - ~ SP(2)-~ S X / / S X

tT S X -~ X P (2) -~ S X ~ Z S X where

SX-~ SX

r e v e r s e s the p a r a m e t e r .

T h e ~ e is another reflection of h o m o t o p y c o m m u t a t i v i t y in X P (Z).

Theorem

13.6 [Stasheff].

SX × SX-~ XP(2) associative a map

If

(X, m )

which is the inclusion

and right translation

S X × S X -~ X P ( 2 ) Proof.

is homotopy

on each factor.

is a homotopy

implies

The obstruction

commutative,

the homotopy to extending

If

equivalence commutativity

there

(X,m)

is a map

is homotopy

the existence of m .

S X v S X -~ X P ( 2 )

to the

of s u c h

67

SX X SX

is a generalized

out explicitly,

one sees

a : X-* ~XP(2).

class

Whitehead

product

it is adjoint to

The obstruction

[Pl + P2 ] - [P2 + Pl ] and

Pl

in

[SX z, X P ( Z ) ] .

(a} + P2

to homotopy

(~) - P l

Writing

(a)-

commutativity

a, : iX 2,x]-* ix 2,axp(2)]

P2

in

the map

(~) w h e r e

iX Z,X] is the

takes the one ob-

struction to the other. Now, homotopy

if X

is h o m o t o p y associative and right translation is a

equivalence, w e have the retraction

r : ~ X P ( 2 ) -* X

and hence

a~

is a monomorphism.

Corollary

13.7.

~CP(3)

Proof.

is homotopy

We know

type of a w e d g e o s s p h e r e s . t o a l l of

~Clm(3) ~ S1 ~ ~S 7 so

S~CP(3)

has the homotopy

Thus for

to extend

SX v SX-~ XP(Z)

SX × SX

involves

obstructions

Now all Whitehead

products

in

we at least

have

CP(3)

S X × S X - * B X.

tivity since the obstruction

commutative.

X = ~CP(3),

a l l of w h i c h a r e are trivial

to zero

products.

[Barratt-James-Stein]

This is sufficient

is mapped

Whitehead

in

so

to imply homotopy

[XZ,~Bx]

commuta-

but the map is an

is omorphism. Note u

4

CP(3}

is not an H-space,

= 0 which is not a primitive

further. Since

The familiar f~CP(3)

maps

is homotopy

S I X ~ S 7-* ~ C P ( 3 )

relation.

is again an H - m a p .

ue

HZ(CP(3)}, we have

The discrepancy

S 1 -~ ~ C P ( 3 ) commutative

since for

and

is worth

~ S 7 -~ f ~ C P ( 3 )

considering

are H-maps.

it is not hard to show the product

Now

S 1 )< ~ S 7 ~

~(CP(o0) X S 7)

m

and hence is the loop space of an H - s p a c e so be an A

n

- m a p for s o m e

n.

S l X ~ S 7-* ~ C P ( 3 )

Indeed w e have seen

Higher homotopies related to h o m o t o p y

S I-* ~ C P ( 3 )

m u s t fail to

is not an

A 4 - m a p.

commutativity are of interest (cf.

[Kudo and Araki]) but, as w e shall see, the approximations

do not converge to

68

strict commutativity,

Definition 13.8.

w h i c h is b e t t e r s t u d i e d d i r e c t l y .

An H - s p a c e

(X,m)

is a b e l i a n if m

c o m m u t a t i v e and a s s o c i a t i v e with E x a m p l e s abound.

For each abelian group (v,n).

is

X, d e f i n e d as t h e l i m i t of x n ] ~ { n )

t i o n g r o u p a c t i n g by i n t e r c h a n g i n g c o o r d i n a t e s . t h e b a s e p o i n t and u n i t of SP °°{X).

~r a n d n a t u r a l n u m b e r

n,

S o m e w h a t m o r e g e n e r a l l y , for any

X we c a n c o n s t r u c t the a b e l i a n m o n o i d

p r o d u c t on

If m

u n i t , we s p e a k of an a b e l i a n m o n o i d .

t h e r e is an a b e l i a n m o n o i d of t y p e space

is c o m m u t a t i v e .

SP°°(X), t h e i n f i n i t e s y m m e t r i c where ~(n)

is the f u l l p e r m u t a -

T h e b a s e p o i n t of X b e c o m e s

A n y o t h e r p o i n t in S P °° {X) c a n be r e p r e -

n

sented as a f o r m a l sum

~

/

m(x.)x, 1

1

where

m{x.) 1

is a p o s i t i v e i n t e g e r and

x.

1

i=l i s n o t t h e b a s e p o i n t of X.

Ir.(SP °°{X)) is n a t u r a l l y i s o m o r p h i c to

T h e o r e m 13.9.

1

T h e p r o o f c o n s i s t s in verify~ing t h a t Eilenberg-Steenrod

C o r o l l a r ~ r 13.10.

Ir.(SP°°())

H.(X) 1

if X is

CW.

s a t i s f i e s the

1

axioms.

If X is a M o o r e s p a c e

Y(G,n), then

SP°°{X) h a s the

h o m o t o p y t y p e of K(G, n}.

T h e o r e m 13. iI.

A n a b e l i a n m o n o i d h a s the w e a k h o m o t o p y t y p e of a p r o d u c t of

Eilenberg-MacLane

Proof.

spaces

Let

i n d u c e s an i s o m o r p h i s m

[D__old a n d Thorn].

Y. = Y{H.{X), i). 1 1 in h o m o l o g y .

T h e r e is a m a p

Since

VY. -* X w h i c h i 1

X is an a b e l i a n m o n o i d , t h i s m a p

69

e x t e n d s to

SP°°(VYi)

or

T~SPC°(Yi)

s o a s to r e d u c e

an isomorphism

of h o m o t o p y

1

groups. Leaving aside associativity, ized by a retraction

S P 2 ( X ) -~ X.

e n o u g h to r u l e o u t m a n y s p a c e s multiplication

o n l y in t h e 2 - p r i m a r y

then

Here

strictly

a

of S P Z ( X )

[ e . g . ,if X = S n, t h e n

X

admits

is r i c h

an abelian

Adams

indicates that the obstruction

is e s s e n t i a l l y

If G

is a g r o u p of r a t i o n a l s

with even denominators,

is a n a b e l i a n H - s p a c e .

Associatlvity

illustrated

rood 2 structure

The

is c h a r a c t e r -

part.

13.12 [ A d a m s ] .

Y(G, Zn+l)

either

(X,m)

o n l y if n = 1].

On the other hand,

Theorem

an abelian H-space

and commutativit7

o r up t o h o m o t o p y .

There

can occur independently are

10 p o s s i b i l i t i e s

of e a c h o t h e r ,

to c o n s i d e r

in t h e f o l l o w i n g c h a r t .

and

c

na

ha

ea

nc

1

2

3

hc

4

5

6

ec

7

8

~ ~ / . /[u

abbreviate "associative " a n d c o m m u t a t i v e ,

"equivalent ,Ia n d "not h o m o t o p y " a n d T h e subdivision

9-10

h

e

and

n

abbreviate

stands for " h o m o t o p y but not equivalent".

refers to the alternatives:

9.

equivalent to an associative

multiplication a n d equivalent to a c o m m u t a t i v e

one but not simultaneously;

i0.

multiplication.

equivalent to an associative, c o m m u t a t i v e

means

via a n H - m a p

homotopy

same homotop7 type. indicates

X with

equivalence to a multiplication o n a s p a c e of the

[Adams] provides

~.

H e r e equivalent

the following examples;

killed above dimension

where

X[0, n]

n.

1

Theorem

as

13.13.

There

are examples

of t h e

10 t y p e s of H - s p a c e

above:

70

I.

S7

2.

S 7 w i t h c e r t a i n p - c o m p o n e n t s of lr. (S7) k i l l e d for

i > Zl

1

3.

Q u a t e r n i o n i c m u l t i p l i c a t i o n on S 3

4.

S 7 with the Z - c o m p o n e n t s of ~.(S 7) k i l l e d f o r

i > 14

1

5.

S 7 with c e r t a i n p - c o m p o n e n t s of ~. (S7) k i l l e d for

i > Zl and Z-components

1

k i l l e d for

i > 14

6.

s7

7.

AMoore

space

Y(Z[I/Z],7)

8.

A Moore

space

Y{Z[I/Z,I/3],5)

9.

Y(Z[1/ Z], 5)

10.

[o, lz]

[0,8]

S1 If X is a f i n i t e c o m p l e x , the s i t u a t i o n c h a n g e s r a d i c a l l y .

a r e p o s s i b l e only for

1, Z, 3, a n d 10, of w h i c h a l l b u t

Examples

Z are given above.

For

a n e x a m p l e is p r o v i d e d b y [ Z a b r o d s k y ] : the H - s p a c e w h i c h m o d 2 a n d 3 is SU(6) b u t m o d

a l l o t h e r p r i m e s is

S 3 X S 5 X S 7 X S 9 X S 11.

A t t e m p t s h a v e b e e n m a d e to c h a r a c t e r i z e h o m o t o p y c o m m u t a t i v i t y in t e r m s of a u n i v e r s a l e x a m p l e .

C o n j e c t u r e 13.14.

X a d m i t s a h o m o t o p y c o m m u t a t i v e m u l t i p l i c a t i o n if a n d o n l y

if X is a r e t r a c t of ~ZSZX. [ W i l l i a m s ] h a s g i v e n c o n d i t i o n s w h i c h a r e e q u i v a l e n t to X b e i n g a n A ° ° - r e t r a c t of ~ Z s Z x .

STRUCTURE Since an associative on

BX

implies

homotopy

homotopy

commutative

H-space

X

commutativity

and homotopy

ON

Bx

is essentially

of

X.

~B x,

a multiplication

O n t h e o t h e r h a n d if m

associative,

then

m

is at least

is

an H-map,

since we have

(wx) (yz) _~ w(x(yz)) _~ w((xy)z) _~ w((yx)z) _~ w(y(xz)) _~ (wy)(xz).

Theorem

1 4 . 1 [Su_._gawara].

multiplication

(X,m)

if a n d o n l y i f m

Notice that for x y ..~ y x

If

by taking

m

is strongly

Similarly, regarded

xyz

yzx

Proof B X × By

and

of T h e o r e m . B X X Y"

homotopy

to be an H-map

w = z = e.

fill in the following triangles

is a n a s s o c i a t i v e

m

H-space,

Bx

implies

being an A3-ma p implies of

we can

S 1 X X 3 -* X :

xyz

zxy

The key to the proof is the equivalence

A specific

a

multiplicative.

(i. e . , w x y z _~ w y x z )

as maps

admits

equivalence

called the shuffle map

of is

induced by : A p × (X X e)p × Ziq )< (e × Y)q-~ A p + q )< Of × Y)P+q w h i c h triangulates (e X Y)q

~P X ~q

and shuffles

(X)< e) p

and

together according to w h i c h s i m p l e x of the triangulation is involved. Specific f o r m u l a s are e a s y to write d o w n [Sugawara, Iv[ilgram,

Steenrod] if w e r e p a r a m e t e r i z e • .. < s

< I.

We

An

by n-tuples

(sI ..... s ) s/t

set up the c o r r e s p o n d e n c e so that the face

0 < sI < s 2 <

t. = 0 c o r r e s p o n d s

72

to the f a c e

s i = si+ I.

The m a p ~

can then be w r i t t e n as

(sI..... Sp, x I ..... Xp) (Sp+ I..... Sp+q,

Yl' " " " Yq)

= (s (i)..... s (p+q), z (i)..... z (p+q)) where

z.x = (x.,e)x for

tion such that Although this

i<_p

s (1). . . . .

and z i =

( e , y i p) for

i > p, a n d = is a n y p e r m u t a -

s (p+q) is c o r r e c t l y o r d e r e d ,

i . e . , is i n AP+q.

~ is n o t w e l l - d e f i n e d , it d o e s i n d u c e a w e U - d e f i n e d m a p

XP(p) X YP(q)--

(X × Y) P(p+q).

T h u s if m

is s t r o n g l y h o m o t o p y m u l t i p l i c a t i v e , we h a v e B X X B X_~B

x × X-" B×

w h i c h can easily be checked to be a multiplication.

T h e converse is straight-

forward. N o t i c e t h a t if m

is a n

s h r n m a p , the m u l t i p l i c a t i o n r e s t r i c t s to

X P (p) × X P (q) -~ X P (p+q) just as does map

CP(p) × CP(q) -~ CP(p+q).

S X X S X - ~ XP(2)

(X X X) P(2)

How homotopy

can be seen by going through the above m a p

SX X SX-~

explicitly.

T h e condition involved in constructing reduced to the following.

(in X x 3 ) :

S X X XP(Z) -~ XP(3)

Let h(t,x, y) be a c o m m u t i n g

h(0,x,y) = xy, h(l,x,y) = yx.

triangle

commutativity gives a

T o construct the m a p

homotopy,

can be

i.e. ,

w e m u s t fill in the following

, ~ ~ s ~

h ( t , x , yz) If m

w e r e o n l y h o m o t o p y a s s o c i a t i v e , the f i g u r e w o u l d b e a h e x a g o n .

A

c o r r e s p o n d i n g f i g u r e a p p e a r s i n M a c L a n e ' s s t u d y of c o h e r e n t f u n c t o r s [MacLane]. The symmetric

condition can he c o m b i n e d with this one to f o r m a

73

f i g u r e i n v a r i a n t u n d e r the s y m m e t r i c

group.

y~z

T h e r e c t a n g l e c a n b e f i l l e d in xyz ~

y

z

× by h(t,x, h(s, y, z)).

x z y ~ , z y x

zxy T h i s h e x a g o n a p p e a r s a l s o in M i l g r a m ~ s s t u d y of ~ Z S 2 X

[Milgram].

T h i n g s a r e m u c h s i m p l e r if X is a n a b e l i a n m o n o i d .

The multiplica-

tion B x X B x-~ B X can be described directly as

(s 1 . . . . .

Sp, x 1. . . . .

Xp) (Sp+ 1. . . . .

Sp+q, Xp+ 1. . . . .

Xp+q)

= (sa (1) . . . . # a (p+q)' Xa (1) . . . . . x a (p+q))" One can check directly that this multiplication is again associative and commutative.

By i n d u c t i o n d e f i n e

Theorem 14.3.

B(n)x

as

BB(n-I)X

with

If X is a n a b e l i a n m o n o i d , B ( n ) x

a d i s c r e t e a b e l i a n g r o u p , B(n)~r is a s p a c e of t y p e One c a n a l s o n o t i c e t h a t Now f o r a m o n o i d morphism. group

BSp~(x)

B(1)(X) = B X.

e x i s t s f o r a l l n.

If ~ is

(~r,n).

= SP°o(SX).

X, b e i n g a b e l i a n is e q u i v a l e n t to m

being a homo-

On t h e o t h e r h a n d , t h e r e a r e s p a c e s s u c h a s t h e i n f i n i t e u n i t a r y

U for which

Eilenberg-MacLane of s i g n i f i c a n c e .

Theorem 14.3.

B (n) e x i s t s , spaces.

e v e n t h o u g h the s p a c e s a r e n o t p r o d u c t s of

T h u s the u s e of s h i n

There is, however,

Let

a compromise

m a p s in T h e o r e m 14.1 is c o n d i t i o n of s o m e r e l e v a n c e .

(X, m) b e a n a s s o c i a t i v e H - s p a c e .

t i o n if t h e r e is a h o m o m o r p h i s m

B X admits a multiplica-

n : X X X - * X s u c h t h a t the u n i t of m

is a

74

h o m o t o p y u n i t of n . For a multiplication

example, on

if ~ l X

X

induces a homomorphism

The multiplication is of c o u r s e we have

homotopy

while with

then so is In g e n e r a l ,

structure

Theorem

some

Y

B

n

BX

of l o o p s ~ln

is induced by

on

Bn.

x = y = e

we have

An associative

then

The multiplication so with

n ( w , z ) "-~wz.

and hence the multiplication

on

)~ : [0, 1] -~ X,

~Ix.

n(wx, yz) = n(wy)n(xz)

we can look for additional structure

"up to homotopy"

14.4.

on

commutative:

n ( x , y ) -~ y x

associative,

denotes the space

on

BX on

m

w = z = e If n

is

is also.

BX

in t e r m s

of

X.

H-space

X

h a s t h e h o m o t o p y t y p e of ~ 2 y for K. if there exists a family of shin m a p s M . : X i -~ X 1 satisfying the

conditions for an A - f o r m as shin m a p s . K. i structure and X t h a t i n d u c e d f r o m X. )

{Here

X i has the product H - s p a c e

T h e t h e o r e m is trivial at this point, though it is tediously difficult to write out explicitly w h a t the c o m b i n e d compatibility conditions on higher h o m o t o p i e s are. the f o r m StiU m o r e

C. × X i-~ X J

It is clear, h o w e v e r , where

spaces,

that they can stiU be written in

C. is an appropriate (if foreboding) cell c o m p l e x . J

elaborate conditions are m a n a g e a b l e

homotopy everything

M .i and its

manageable

in B o a r d m a n ' s theory of

because

t h e y do n o t n e e d to be e x p l i c i t .

INFINITE

There

are spaces

are not only loop spaces

D e f i n i t i o n 15.1. i = 1,2 . . . .

such that Of course

X

X

O,

MacLane'

s

have this property,

U, S p

and their analogues

such spaces

is a f o r m i d a b l e PACTs

task.

In a c a t e g o r y

(a)

the objects are

(b)

the morphisms

such as

in t e r m s

[Boardman]

0 , 1 , Z. . . . from

of h i g h e r h o m o t o p i e s

invents a new gadget (modelled on we need.

We follow him

that

has permutations,

morphism

S

n

to

n

form a topological

-~

associative

lre S

m

m

• n= m

+ n ;

we are also given for each

(n,n), S

we impose and

~(m,n),

continuous functor

n

the symmetric

g r o u p on

(We o m i t a n y s y m b o l f o r t h i s h o m o m o r p h i s m . permutations

space

is continuous;

:~..~such

if

on t h e

;

m

(c) w e a r e g i v e n a s t r i c t l y

(i)

F = lira H(sn).

of operators

and composition

if ~

but less trivially we

[B_.oardman a n d V o g t ] .

Definition i5.2.

(d)

X. f o r z

QX = lira f~nsnx ~ f~(lim~n-lsnx).

[MacLane]) to index the structure

throughout this chapter

topology which

X. h a s t h e h o m o t o p y t y p e of f~Xi+ 1. z

the infinite loop space

To characterize

in a l g e b r a i c

loop spaces.

abelian monoids

We a l s o h a v e f o r a n y

importance

is an i n f i n i t e l o o p s p a c e if t h e r e a r e s p a c e s

X = X1 and

have the infinite Lie groups

multiplication

of c e n t r a l

but iterated

A space

LOOP SPACES

p e S

n

two further then

lr @ p

n

a homon

letters.

) In t h e c a s e w i t h

axioms: l i e s in

S

m+n

az~d i s t h e u s u a l

sum permutation; (ii)

given any

r

morphisms

a . : m . -* n. a n d z z L

~r e S , w e h a v e r

76

~r(n)o(aI ~

a z • ... • Ur ) = Ir(al @ aZ • ... ~ a r )

where

:

m

~,m.,1 n = r.n.,1 Tr p e r m u t e s

and the permutation

w(n) E S

o Tr(m),

the factors

is obtained

from

of a 1 @ a 2 @)" " " • a r , lr b y r e p l a c i n g

i by

n

a b l o c k of

n.

elements.

We require

functors

to preserve

all this

1

structure.

Example maps

15.3.

E n d x, f o r a b a s e d

X m -~ X n , w h e r e

This example

Definition

Xn

space

X.

is the n th p o w e r

Endx(m,n) of X.

is the space

T h e functor

of a l l ( b a s e d )

@) is just

X.

has permutations.

15.4.

The category

if we are given a functor

~-~

~

of o p e r a t o r s

End

acts on

X, or

X

is a~-space,

. X

A particularly

Example

A(m,n)

15.5.

A

{I, Z} -~ {I}

element,

A

m

example

encodes

the associative

law.

be the category of operators described as follows:

i s t h e s e t of a l l o r d e r If

map

Let

important

acts on

preserving

X, then

corresponds

is associative.

X

maps

admits

{1,2 .....

the structure

to a multiplication

rn.

m}--> {1,2 .....

n}.

of a m o n o i d .

The

Since

A(3, I) has only one

Since the single e l e m e n t of A(I, I) can be r e g a r d e d

as the composition of l-~ I c {1,2}-~ I or as the composition of I--~ 2 E {l,Z} -~ I, the multiplication has a unit. B e i n g a rnonoid is not a hornotopy invariant. a category action by

WA WA

to construct

of the s a m e

homotopy

is a h o m o t o p y WB

for any

Consider

B

A(m,l).

W e w i s h to replace

A

by

type (as a category of operators) so that

invariant.

Essentially the s a m e

that m a p s

nicely e n o u g h into A.

It consists of precisely one m a p

w h i c h can be r e g a r d e d as factorized in a variety of ways.

method

may

be u s e d

{I ..... n} -~ {I}

T h e s e factorizations

77

c a n be d e s c r i b e d f a i t h f u l l y b y f i n i t e d i r e c t e d p l a n a r t r e e s w i t h a s i n g l e r o o t (no v e r t e x of the f o r m

~ being p e r m i t t e d and I

b e l o n g s to o n l y one edge). unique m a p

(V)

ponding to~

The i d e a i~ c o n s t r u c t i n g and ~

WA(m, n)

~

i.e.,

or

(V) ~ ( V / )

is a c u b i c a l c e l l c o m p l e x i n w h i c h the c e l l s a r e i n d e x e d by

a I ..... ar

such that ~I .... err E A(m,n). less than the n u m b e r

(for n = I) or of the corresponding such a set a copse.

WA is to h a v e d i s t i n c t m a p s c o r r e s -

a n d to h a v e a s p e c i f i c h o m o t o p y b e t w e e n t h e m .

d i m e n s i o n equal to m

For

WA-structure

C(a I ..... ar)

n = I w e have the cell c o m p l e x of K

m

is a cell of

of edges of the corresponding tree

o r d e r e d set of trees for

Chapter II, a cubical d e c o m p o s i t i o n has a

or

d e s c r i b e s the

T h a t t h e s e a r e two f a c t o r i z a t i o n s of the s a m e m a p e x p r e s s e s the

a s s o c i a t i v e law.

sequences

F o r e x a m p l e , in A(3,1), the t r e e ~

w h i c h can be factored as ~

a (~V).

is the o n l y t r e e for w h i c h the r o o t

.

WA(m,I)

T h e category

if Definition Ii. 18 is satisfied,

n > I.

WA

Boardman

calls

described in

acts on

X

or

X

i.e. , w e have compatible m a p s

WA(n,I) )< X n-~ X.

Theorem X

15.6.

If X

if and only if W A

and

Y

acts on

have the s a m e Y.

If W A

a n d a deformation retraction of M X T h e construction of M X deformation retraction. WA-structure

onto

homotopy

acts on

type, then

X, there is a m o n o i d

acts on MX~X

X.

w a s given in Chapter 11 along with the

A s one should expect, the m a p

up to h o m o t o p 7 ,

WA

X-~ MX

respects the

at least in the sense w e n o w define.

Definition 15.7.

Let

L

be the category with two objects

Definition 15.8.

Let

X

and

Y

be W B - s p a c e s .

A map

I and

2 and one m a p

f : X -~ Y

is a h o m o t o p y

79

B-map Y.

if W ( B X L)

By

Endf (re,n}

acts on

Endf

s o a s to i n d u c e t h e g i v e n W B - a c t i o n s

we m e a n the s p a c e of c o m m u t a t i v e

Xm ~fm

example,

Theorem

I

. T so that

is a homotopy A-map

uniquely up to homotopy through a homomorphism

Corollary 15. I0. Bf : B E - ~ B y .

If f : X - ~ Y (Take

f must be an H-map.

into a monoid,

it f a c t o r s

M X ~ Y.

is a h o m o t o p y A - m a p ,

B X = BMX,

diagrams

yn

Vo.ex. V

If f : X ~ Y

and

~fn -~

W ( A × L) (2,1) = !

15.9.

X

_~ X n

ym For

on

then there exists

Bf = BMf.)

T h e real point of B o a r d m a n ' s a p p r o a c h is to be able to iterate the construction of B E

without a s s u m i n g an abelian m o n o l d structure.

will n e e d lots of homotopies. Boardman)

of a c a t e g o r y

contractible

Theorem

X B

for all

15.12.

is an E - s p a c e of o p e r a t o r s

An E-space

map being described

15.13.

if i t if g i v e n a n E - s t r u c t u r e ;

with permutations

such that

i. e . , a n a c t i o n o n B(n,1)

is

n.

The contractibility

Theorem

Rather than give all the details (we await t h e m f r o m

w e will try to outline the theory conceptually.

D e f i n i t i o n 15.11. X

Clearly w e

is a WA-space. of

B(n,1)

is u s e d to m a p

WA~

B, t h e

0-skeleton

specifically.

If X

is an E - s p a c e ,

the operations

A-maps. Thus there are induced operations

on

B X.

: X m ~ Xn

are homotopy

79

Corollar~r 15.14.

is a n E - s p a c e .

BE

T h e a i m of B o a r d r n a n ' examples

the category

{algebraic) dimension

and linear

R °° w i t h o r t h o n o r m a l

{el, e z .....

is to g e t s e v e r a l

~

of r e a l i n n e r - p r o d u c t

isometric

significant

base

finite topology, dimensional

from

A to

of c o u n t a b l e

As examples

{ e l , e2, e 3. . . }, a n d i t s s u b s p a c e

linear maps

which makes

spaces

maps between them.

e n } , w h i c h is a l l t h e r e a r e up to i s o m o r p h i s m .

t h e s e t of a l l i s o m e t r i c

Lemma

s approach

exists.

BB...BX

of E - s p a c e s . Consider

have

By iteration,

R n with base

We topologize~

B, by first giving

we

A

and

(A,B), B

the

e a c h t h e t o p o l o g i c a l d i r e c t l i m i t of i t s f i n i t e -

subspaces.

15.15.

T h e s p a c e ~ (A, R °°)

This is a consequence

(a) ~ i 2 (b) i l

is c o n t r a c t i b l e .

of t w o e a s i l y c o n s t r u c t e d

homotopies:

: A-~ A@A, u :R

°°

-~

R °o

O R °°, for s o m e

S u p p o s e w e have a functor

T

isomorphism

u.

defined on the c a t e g o r y ~

, taking topo-

logical spaces as values, and a continuous natural transformation : T A X T B -~ T { A ~) B) (a) Tf

called W h i t n e y s u m ,

such that:

is a continuous function of f E ~ (A, B);

(b) T R 0 consists of one point; (c) co p r e s e r v e s associativity, c o m m u t a t i v i t y and units;

(d)

Theorem classifying

T R °° is t h e d i r e c t l i m i t of t h e s p a c e s

15.16. space

T R °° is a n E - s p a c e . B T R °° a g r e e s

If

T

h a p p e n s to b e r n o n o i d - v a l u e d ,

with that from Theorem

As a (noncanonical) multiplication T R °° X T R ° ° ~

T R n.

on

15. lZ.

T R °° w e t a k e

T ( R ~ O IR.~ ) - ~ T R °°, Tf

the

80

where

f : R °° @ i~ °° -~ R °°

provides

is any linear

homotopy-associativity,

commutativity,

and all higher

In t h e e x a m p l e s sional

since

the maps

Tf

The Lemma

homotopies.

below we define (d)

embedding.

f o (f $ 1) ~ f " (1 • f), h o m o t o p y -

coherence

A, and note that axiom

each case

isometric

TA

extends

explicitiy

only for finite-dimen-

t h e d e f i n i t i o n t o t h e w h o l e of ~ .

and the Whitney sum

~

are obvious,

In

i n v i e w of t h e i n n e r

products.

Example

1 5 . 1 7 . 1 . T A = O (/%), t h e o r t h o g o n a l Z. T A = U ( A ®

group

C), the unitary

3. T A = B O ( A ) ,

a suitable

of

group

classifying

A.

of

Then

T R °° = O.

A®

C.

Then

space

for

O(A).

T R °° -- U. Then

TRam= B0. 4. T A

= F(A), the space of b a s e d h o m o t o p y

sphere with

Theorem

15.18.

SA, which is the one-point o0 a s b a s e p o i n t .

The following spaces

Then

equivalences of the

compactification

and maps

admit the structure

U-~ 0-~ F and

Top

of A ,

T R o° = F .

loop spaces :

The semi-simplicial spaces PL

A~)oo

c a n also be handled.

of i n f i n i t e

OPERATIONS

For order

IN I T E R A T E D

LOOP

SPACES

X to be an infinite loop space required

homotopies.

Certain

subfamilies

g i v e r i s e to h o m o l o g y o p e r a t i o n s

a r e of i n t e r e s t

of g r e a t s i g n i f i c a n c e .

a whole congery in t h e m s e l v e s Historically

t h e s e o c c u r in t h e [ K u d o a n d A r a k i ] d e f i n i t i o n of H - s q u a r i n g Steenrod

for they t h e f i r s t of

operations,

analogs

operations.

D e f i n i t i o n 16.1. maps

of h i g h e r

An H -space n

consists

of a s p a c e

X together

w i t h a f a m i l y of

0. : i i X X 2 -*" X, i < n, s/t 1

D

Oi(t I . . . . .

ti,x,y ) = Oj.l(t I .....

tj_ I ,

x,y)

t.j : 0

= 8j_l(1-t I..... l-tj_l,Y,X)

[More efficiently,

the family

=x

if y = e

=y

if x = e .

O. c a n b e r e p l a c e d 1 O

n

such that O (T,x,e) = O (T,e,x) = x n n

by switching factors

in

Thus for

where

T ~ 0 E Zz

on

commutative

H-spaces

map

acts antipodally on

is a h o m o t o p y c o m m u t a t i v e

are H

X

-spaces

H-space.

Notice the

n e e d not be a loop space.

Corollary

16.2.

If X

16.3.

If X

Proof~of Theorem. notation,

i.e.,

is a n H n - s p a c e ,

is

~ny,

Represent

then

Sn

a p o i n t is r e p r e s e n t e d

~X

X

as

is a n H n + l - s p a c e .

is a n H n . l - s p a c e .

Z 2 * Z 2 '~ . . . as

t0a 0 •

For

e v e n if n o t h o m o t o p y a s s o c i a t i v e .

cO

Theorem

S n,

X. ]

condition b e a r s no relation to associativity so

example,

by an equivariant

: Sn × X z -"X

X 2 and trivially

n > 0 X

t.j = I

;~ Z 2 w i t h M i l n o r ' s

. . . {~ t a n

n

where

of

82

a i• Z z, (to ..... tn) • A n.

Define

O n + l ( t 0 a 0 • . . . • t n a n • t n + l l ; k 1, kZ) (t) to b e tn o to ~n+l (t)) whe r e k ~ ( t ) = k z ( t - (l-s )(r2)) e n ( l _ - ~ n +1 0 0 e . . . e n l _ t n +1 ; k l ( t ) ' k

w i t h the u n d e r s t a n d i n g

kz(t) = kZ(0)

for

t < 0

= k2(r Z) f o r

Extend by equivariance. an H-space

For

t >r z .

n = 0, this is a standard proof that the loop space of

is homotopy commutative.

Definition 16.4.

Given an H -space

(X, O ), the K u d o - A r a k i

n

Qi : I_Iq(X;Z2) -* H N + i (X;Z z) is d e f i n e d f o r where

square

n

i-n < q _< i by oi([u]) = Or~ (ei_q®U®U)

e. is an i-dimensional cell in the standard equivariant decomposition of I

S n" Before going further w e introduce the rood p analogs [Dyer and Lashof ].

Defintion 16.5.

An Hn-space P

X consists

o n : ~(p)*.

s/t on(~,e ..... e,x,e ..... e) = x

Theorem

Proof.

l6. 6.

If X

We p r o c e e d

tn no

x xP-

map

x

and on(o • ... $ O @

v e r y m u c h a s in t h e c a s e

l'l;xI..... Xn) = x l...x n.

p = 2..

If k i : [0, r i ] - ~ X, d e f i n e

f o r On+ 1 t h e n b e c o m e s to t @ s.id; k I..... k p ) (t) = SniFfs a0~).., l@~.~an;kl(t) ..... kS(t)). p

Now consider TJe.,, 0_< j < p

.. *J(p~

X and an equivariant

is H n, f~X is ~I-F "+I. P P

k s (t) = k i ( t - ( l - s ) ( r 1 + . . . + r i . 1 ) ) .

en+l(t0a 0 @ . . . ®

of a m o n o i d

such that

The formula

W , the standard P 8eZi+l = (T-1)ezi

resolution

of

7.

@ezi -- (1 + T +

P

with generators + T p-l) eZi_l.

83

The i n c l u s i o n "n-skeleton"

Z -~ ~ ( p ) as c y c l i c p e r m u t a t i o n s i n d u c e s c o n s i s t e n t m a p s of the P n of W into C ( ~ ( p ) * . . . * ~ ( p ) ) . Thus we think of 0 . a c t i n g o n P

W ® C (X) @p. P

Definition 16.7.

where

~tO,q)

For

(2j-q)(p-1) <_n and x E H (X), define q OJ(x) = ~ / ( j , q ) ( ~ ) l 8 n, (e(2j_q)(p_l) @ x ® . . .

@ x)

p-12

is the s i g n of q + [j + ( Z - ~ ]

The o p e r a t i o n s defined have the following p r o p e r t i e s ,

s o m e of w h i c h

d e p e n d on this p r e c i s e c h o i c e of ~ (j, q). QJ : Hq(X;Zp)-~

Hq+zj(p_l)(XiZp)

QJ is a h o m o m o r p h i s m

if

for j _> q / Z

and

(3j-q)(p-l)_< n

(gj-q)(p-l) < n-I

QJ is natural with respect to H n - m a p s P QI

is trivial if q = 0

Q q / Z l x ) = xP

if q is e v e n

a . Q j = QJ~, w h e n b o t h a r e d e f i n e d and a , : Hq_I(~X) -~ Hq(X) is the h o m o l o g y " s u s p e n s i o n " . J u s t as S t e e n r o d o p e r a t i o n s give a n i c e d e s c r i p t i o n of H ( Z , n ; Z ) P t h e s e o p e r a t i o n s give a n i c e d e s c r i p t i o n o f i i denote by QI the c o m p o s i t i o n Q 1 . . . Q r. if<2i2-- .... 'ir-i
excess,

is defined on is < q+n.

r

H

of an H n - s p a c e

if the

If e(I) = i = q, then QI

--

r

zr-th power.

T h e o r e m 16.8. ~Kudo and Araki].

For

h a v i n g as g e n e r a t o r s all a d m i s s i b l e Similarly for

k > 0, H , ( ~ n s n + k ; z Z) is a p o l y n o m i a l r i n g

Q~uk of e x c e s s

1.

> k and l a s t d e g r e e

p > 2, we c o n s i d e r s e q u e n c e s £

1

lr), we

q

--

with e. = 0 or

I = (i 1. . . . .

We s a y QI is a d m i s s i b l e if

r

excess of I is > q and the last degree i raises to the

For

e(I) = il - Zi2 + i Z - Zi3 + ... + i r = 2iI- ~i"3

e(I) < i . Notice that QI --

H . (~nsn+k).

so

We let QI denote

S

¢'_

~ 1Q 16 ~ . . . ~

£

I = (e 1, s 1. . . . . S

kQ k, w h e r e

< n+k.

e k ' Sk)

84

: Hq (X) -~ Hq_l(X) 0-~ Z p - ~ Z p 2 - ~ Z the r e l a t i o n

is the B o c k s t e i n b o u n d a r y f o r the s e q u e n c e -~ 0.

[For

~ = 2, ~ n e e d not be u s e d e x p l i c i t l y s i n c e we have

~Qgi+l = oZi. ] [ D y e r and L a s h o f ] d e s c r i b e g e n e r a t o r s f o r

H , {"Ansn+k ~ ; Z p ) in t e r m s of a l l o w a b l e s e q u e n c e s .

May has translated" allowable"

into the following.

k Definition 16.8.

QI is m - a d m i s s i b l e

if PSi+ 1 " ¢ i+l >-- s i > [ m + >

Zsj(p-l)-¢j]l 2.

i+l The following two t h e o r e m s a r e then t r a n s l a t i o n s of t h o s e g i v e n by D y e r and L a s h o f .

T h e o r e m 16. 9.

F o r any

connected

X we have that H.(l~m 6 ~ n s n x ; z ) is f r e e P commutative on {x, Olx I x , basis of H,(X;Zp), Ol is d i m x-admissible}. N o w let O X

denote the base point component of lirn onsnx.

T h e o r e m 16.10.

H , ( Q S 0 ; Z ) is f r e e c o m m u t a t i v e on g e n e r a t o r s y(I) of d i m e n s i o n P e q u a l to deg I w h e r e I r u n s o v e r all 0 - a d m i s s i b l e s e q u e n c e s . [ F o r p = Z, k I = (s 1. . . . . Sk) is 0 - a d m i s s i b l e if 2si+ 1 > s i > ~ ' - - sj. ] The D y e r - L a s h o f o p e r a t i o n s b e h a v e as f o l l o w s : J = (¢l, Sl . . . . .

If

i+l

~k_l, Sk_l), then Q J ~ k y ( s k )

Y(S k) e HSk(QS0;Z z) and if J = (s 1. . . . . The g e n e r a t o r QI

=y(£1, Sl . . . . .

Sk_l)

then

Ck, Sk).

[For

Q J y ( s k) = y(s 1. . . . .

y(I) c a n not be i n t e r p r e t e d as

p = Z, Sk).]

QI(x) f o r a n y x s i n c e

is t r i v i a l on H0(QS 0) a l t h o u g h an i n t e r p r e t a t i o n in t e r m s of QI is p o s s i b l e

if we c o n s i d e r all the c o m p o n e n t s of

lira 0nsn.

Kudo and Araki prove their result by mimicing H*(Z, n;Zz). Theorem

for

The crucial machine is a dual t o ~ o r e ~ s transgression theorem.

16. U.

If H,(X;Z2)

Serre

Let X

be a simply connected, homotopy associative

has a simple system of transgressive generators

x. then I-I,(~X;Z 2) 1

is a polynomial ring on generators

Yi such that Yi £ Tx..1

H-space.

85

Dyer and Lashof proceed somewhat differently. H.(~ksix;Zp)

in the range < 3i- Zk which determines

They compute

H~(lim..f~nsn+kx;Zp).

They do the computation by analyzing the homology as a tensor product of monogenic Hopf algebras, mapping the corresponding tensor product of model spectral sequences into the Serre spectral sequence for ~ k + I s i x - J ~ k s i x - ~ ~ksix

and applying the comparison theorem.

Finally, they identify the various

classes at hand in terms of the operations QJ. ~nsn

The final stage from

~n-Isn

to

requires s o m e special effort. N o w let us return to a space of m o r e geometric interest.

The space

Q S 0 has the s a m e homotopy type as S F = lira SH(Sn), S denoting m a p s of degree I, but the equivalence is not one of infinite loop spaces. multiplications are not equivalent.

In fact even the

A major accomplishment of the last few years

has been the determination of the Hopf algebra H $ ~ ; Z z) and the algebra # H ( B F ; Z ) for p >_.Z. The operations Qi have played an important role in this P development. That H * 0BSF) was not the s a m e as H *(BQS 0) has tong been k n o w n for H ( B S F ; Z 2 ) D Z 2 [w i I i [ 2] w h e r e H~'(BQS0;Zz)

{wi}

is a n e x t e r i o r a l g e b r a .

a r e the S t i e f e l - W h i t n e y c l a s s e s .

While

The c o m p l e t e r e s u l t s c a n n o w b e s t a t e d ,

a l t h o u g h the p r o o f s a r e s o i n v o l v e d a l g e b r a i c a l l y a s to b e i n a p p r o p r i a t e f o r p r e sentation here. Let

o denote composition,

l o o p a d d i t i o n in QS °.

the m u l t i p l i c a t i o n in S F

We h a v e c o r r e s p o n d i n g

operations

t h e f i l t r a t i o n of H~ (SF) b y p o w e r s of t h e a r g u m e n t a t i o n

and l e t

Q~ and Q I. i d e a l and l e t

# denote Consider

E"

denote

the associated graded.

T h e o r e m 16.12. [ M i l g r a m ] . of H~(QS °)

For

p = 2, l e t

o r the i s o m o r p h i c c l a s s e s

I Q,~y(k) b e the D y e r - L a s h o f

in H ~ ( S F ) .

generators

86

y(k)

o y(k) ~ 0 ,

y~)o

y(k) o y e ) .

y(k) : 0

Q y(k) ° Q,y(k) = 0 in E ° (y(k) * y(k)) ° (y(k) * y(k)) : 0

Corollar~" 16.13.

H

(BSF;Z2) ~. Zz[Wi] Q C

E°

in

where

C

is isomorphic to

E(ezi+t Ii > I) ® r(g([)i I is l-admissible of length > 1). Here

eZi+l is dual to sly(i) • y(i)] and g([) is dual to ~y(1). The spectral sequence f r o m

ExtH~(SF)

to H

(BSF) has no choice but

to collapse since, being a spectral sequence of Hopf algebras, only primitive relations can be added.

This m e a n s only 2i-th powers of primitive classes could

be killed, but the only nontrivial ones present in E 2 are in ZZ[wi] which w e k n o w survives untouched. For

p > Z, there is a striking difference which is really a subtle

s i m i l a r ity.

T h e o r e m 16.14. [May].

For

T h e o r e m 16.15. [May].

In the E i ! e n b e r g - M o o r e

E2 ~ EXtH ( S F ; Z ) p If I = (1, j , J )

we h a v e

is 0 - a d m i s s i b l e ,

dp_l[Y(J) I . . . I

p > 2, H , ( S F ; Z

E 2 ~ E p . 1 with

P

)~--H(QS0;Z

P

) as Hopf a l g e b r a s .

spectral sequence for

BSF

with

dp_ t g i v e n as follows:

J odd of d e g r e e Zj-1 and l e n g t h > 1, then

Y(J)] =y(I)-

J.

C o r o l l a r y , 16.16. 1) 2)

H (BSF;Zp) ~ Zp[qi ] O E(~q i) O E O F w h e r e

{qi } a r e the

Wu c l a s s e s

E is a n e x t e r i o r a l g e b r a on p r i m i t i v e g e n e r a t o r s

e. dual to 1

~y(1, Z(p-1) 1, i) in H Z P i ( p - 1 ) - I ( B s F ) a n d

g(I) dual to ~y(I) w h e r e

I r u n s o v e r all 1 - a d m i s s i b l e s e q u e n c e s of e v e n d e g r e e and length

>

1.

87

3)

1~ is a divided polynomial algebra on primitive generators

~e. i

and g(J) where

J runs over all 1-admissible sequences of odd

degree and length> I. [The W u

class qi is dual to ~(l,i) and

~qi to a(0, i). ] Milgram' s and May' s proofs are rather unusual exercises in manipulating Hopf algebras over the Steenrod algebra or its dual. the C a f t a n f o r m u l a e and A d e m r e l a t i o n s for the t h e [ N i s h i d a ] r e l a t i o n s b e t w e e n the operations. structures

Theorem

Qi

Qi

It is important to have

[ D y e r - L a s h o f ] and e s p e c i a l l y

a n d t h e h o m o l o g y d u a l s of t h e S t e e n r o d

An old f a c t in h o m o t o p y t h e o r y c r u c i a l in r e l a t i n g the v a r i o u s a l g e b r a i c _ S n-l, •

involved. 16.17.

(~nsnx, e) is a module over

Corollary 16.18.

He{~nsnx)

Theorem

(~nsnx, e) is a module over

(sn-l,e

is a Hope algebra over

,°).

H e(s n'l,eS

n-I • , ).

S n-l, • 16.17.

(Sn-l,e

, .).

S n-l, Corollary 16.18.

I-Ie {~nsnx)

is a Hopf algebra over

T h e t h e o r e m is a r e s t a t e m e n t

(leg) . ~.a = f • r.c~eg . r.~. ~nsnx

is commutative, corresponding

e).

of the r i g h t d i s t r i b u t i v i t y of c o m p o s i t i o n

if f , g : S n -~ s n x

over track addition, i.e.,

He (Sn'l, •

and

a : S n ' l -~ S n-1

D i a g r a m a t i c a U y we h a v e w i t h

X ~ n s n X × F(n-1) -~ ~ n s n x

then

F ( n ) = Sn , e S n ' e , t h a t

X ~ n ~ n x X F(n)

~ n s n x ~'~F(n)

~ n S n "X - -X ~ n S n~X X F(n) X F(n)

~-~ n x

~'~nsnx X F (n) X ~ n s n x X F (n)

~

at least up to homotopy.

The diagram helps in describing the

c o n d i t i o n in h o m o l o g y .

T h e h i g h e r o r d e r p h e n o m e n a i n v o l v e d in h a n d l i n g t h e o p e r a t i o n s n e c e s sitate studying this distributive t e r n a t e d e s c r i p t i o n s of H n

or

l a w up to h i g h e r h o m o t o p i e s . Hn-structures P

on BO

and

BF

There are also al[Boardman,

T s u c h i y a , M i l g r a m ] w h i c h s h o u l d g i v e the s a m e h o m o l o g y o p e r a t i o n s b u t at the

88

m o m e n t a r e n o t k n o w n to do s o . characterizeable problems,

FinaUT, infinite loop spaces should be

in t e r m s of t h e m a p s J ( n )

*...*

~ ( n ) X X n - ~ X.

In a l l t h e s e

we a r e f a c e d w i t h a n a l T z i n g a f a m i l i a r a l g e b r a i c s t r u c t u r e f r o m a

h o m o t o p 7 p o i n t of v i e w , b u t p e r h a p s t h e s p i r i t of t h a t p o i n t of v i e w is by n o w sufficientl 7 clear.

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