(Lecture Notes in Mathematics 533) Frederick R. Cohen, Thomas J. Lada, J. Peter May (auth.) - The Homology of Iterated Loop Spaces-Springer-Verlag Berlin Heidelberg (1976)

Allowable structures over the Dyer-Lashof algebra I I I on Em spaces generated by the QS and BQS and develop analogs of the notions of unstable modules and algebras over the Steenrod

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Authors

Frederick R Cohen

Department of Mathematics

Northern Illinois University

DeKalb, Illinois 60115lUSA

Thomas J Lada

Department of Mathematics

North Carolina State University

Raleigh, North Carolina 27607lUSA

J Peter May

Department of Mathematics

University of Chicago

Chicago, Illinois 60637lUSA

Library of Congress Cataloging in Publication Data

Cohen, Frederick Ronald, 1946-

The homology of iterated loop spaces

(Lecture notes in mathematics ; 533)

Bibliography: p

Includes index

1 Loop spaces 2 Classifying spaces

3 Homology theory I Lada, Thomas Joseph,

1946- j o i n t author 11 May, J Peter, joint

author 11 May, J Peter, j o i n t author

In Title IV Series: Lecture notes in mathe-

matics (Berlin) ; 533

QA3.L28 vol p 3 l.QA612.761 510' 8s [514' .2J

AMS Subject Classifications (1.970): 18 F25, 18 H10, 55 D35, 55 1340,

55F40,55G99

ISBN 3-540-07984-X Springer-Verlag Berlin Heidelberg - New York

ISBN 0-387-07984-X Springer-Verlag New York Heidelberg Berlin

This work is subject to copyright All rights are reserved, whether the whole

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O by Springer-Verlag Berlin Heidelberg 1976

of i n t e r n a l s t r u c t u r e in, t h e homologies of v a r i o u s s p a c e s of inter& The l a s t studies a n up to homotopy notion of a n a l g e b r a o v e r a monad and t h e r o l e of t h i s notion in the t h e o r y of i t e p a t e d loop s p a c e s I have

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

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

i n the following r e f e r e n c e s , which s h a l l b e r e f e r r e d to b y t h e specified

l e t t e r s throughout t h e volume

[A] A g e n e r a l a l g e b r a i c approach t o Steenrod operations S p r i n g e r

L e c t u r e Notes i n Mathematics Vol 1 6 8 , 1 9 7 0 , 153-231

[GI T h e G e o m e t r y of I t e r a t e d Loop Spaces S p r i n g e r L e c t u r e Notes

i n Mathematics Vol 27 1, 1972

[GI] Ew s p a c e s , group completions, a n d permutative c a t e g o r i e s London Math Soc L e c t u r e Note S e r i e s Vol 11, 1 9 7 4 , 61-93

In addition, the paper I1 h e r e i s a companion piece t o m y book (con-

t r i b u t e d to by F Quinn, N R a y , and J Torneha~e)

[R] E Ring S p a c e s and E Ring Spectra

With t h e s e papers, t h i s volume completes the development of a

t h e o r y of the geometry a n d homology of i t e r a t e d loop

spaces T h e r e a r e no known r e s u l t s in o r applications of this a r e a of

topology which do not f i t n a t u r a l l y into the f r a m e w o r k thus established

However, t h e r e a r e s e v e r a l p a p e r s by o t h e r authors which s e e m t o m e

t o add significantly t o the theory developed i n [GI The relevant

r e f e r e n c e s will be incorporated in the list of e r r a t a and addenda to [A],

[GI, and [GI] which concludes t h i s volume

The geometric t h e o r y of [GI was incomplete in two e s s e n t i a l

r e s p e c t s F i r s t , i t worked well only f o r connected s p a c e s ( s e e [G, p 156-

1581) It was the p r i m a r y purpose of [GI] to generalize the t h e o r y t o

non-connected spaces In p a r t i c u l a r , t h i s allowed i t t o be applied t o

t h e classifying s p a c e s of permutative c a t e g o r i e s and thus t o algebraic

K-theory More profoundly, t h e ring t h e o r y of [R] and fI was t h e r e b y

m a d e possible

Second, the t h e o r y of [GI circumvented analysis of homotopy

invariance ( s e e [G, p 158-1601) It i s t h e purpose of L a d a l s paper V

t o generalize the t h e o r y of [GI t o one b a s e d on homotopy invariant

s t r u c t u r e s on topological s p a c e s in the s e n s e of Boardman and Vogt

[Springer L e c t u r e Notes in Mathematics, Vol 3471 I In Boardman and

' ~ n c i d e n t a l l ~ , the c l a i m t h e r e (p VII) that [GI failed t o apply t o non

2 - f r e e o p e r a d s is b a s e d o n a misreading; s e e [G, p 221

Vogtls work , an action up t o homotopy by a n o p e r a d (or PROP) o n a

s p a c e w a s essentially a n action by a l a r g e r , but equivalent, operad

o n t h e s a m e space In L a d a 1 s work, a n action up to homotopy i s

e s s e n t i a l l y a n action by t h e given o p e r a d o n a l a r g e r , but equivalent,

space In both c a s e s , the expansion m a k e s r o o m f o r higher homotopies

While t h e s e need not be m a d e explicit in the f i r s t approach, i t s e e m s t o

m e that the second approach is n e v e r t h e l e s s technically and conceptually

s i m p l e r (although s t i l l quite complicated i n detail) since the expansion

construction i s much l e s s i n t r i c a t e and s i n c e the problem of composing

higher homotopies l a r g e l y evaporates

We have attempted t o m a k e the homological r e s u l t s of this volume

a c c e s s i b l e t o t h e r e a d e r unfamiliar with the g e o m e t r i c t h e o r y in the

p a p e r s c i t e d above In I, I s e t up the theory of homology operations on

infinite loop spaces This is b a s e d o n actions by E o p e r a d s on s p a c e s

m

and i s used to compute H*(CX; Z ) and H*(QX; Z ) a s Hopf a l g e b r a s

o v e r the Dyer-Lashof and Steenrod a l g e b r a s , where CX and QX a r e

the f r e e -space and f r e e infinite loop space generated by a space X

The s t r u c t u r e of the Dyer-Lashof a l g e b r a i s a l s o analyzed In 11, I s e t

up the t h e o r y of homology operations o n E r i n g s p a c e s , which a r e s p a c e s

shown to determine the multiplicative homology operations of the f r e e

E ring space c(x') and the f r e e E ring infinite loop space Q(x')

and a n exhaustive study i s made of the homology of BSF and of such

related classifying spaces a s BTop (at p> 2) and BCoker J Perhaps

the most interesting feature of these caleulations i s the precise homo-

logical analysis of the infinite loop splitting BSF = BCoker J X BJ a t

odd primes and of the infinite loop fibration BCoker J * BSF - BJ@

a t p = 2

In III, Cohen sets up the theory of homology operations on n-fold

loop spaces for n < a This i s based on actions by the little cubes

operad Cn and i s used to compute H*(C~X; Z ) and H*(Q~z%; Z )

a s Hopf algebras over the Steenrod algebra with t h r e e types of homology

operations While the f i r s t four sections of 111 a r e precisely parallel to

sections 1 , 2 , 4 , and 5 of I, the construction of the unstable operations

(for odd p) and the proofs of a l l requisite commutation formulas between

them (which occupies the r e s t of 111) i s several o r d e r s of magnitude m o r e

difficult than the analogous work of I (most of which i s already contained

in [A]) The basic ingredient i s a homological analysis of configuration

spaces, which should be of independent interest In IV, Cohen computes

H*(SF(~); Z ) a s an algebra for p odd and n even, the remaining

P cases being determined by the stable calculations of 11 Again, the

calculation i s considerably more difficult than i n the stable case, the

key fact being that H*(SF(n); Z ) i s commutative even though SF(n)

P

i s not homotopy commutative Due to the lack of internal structure

on B S F ( ~ ) , the calculation of H*(BSF(n); Z ) i s not yet complete

P

In addition to their original material, I and 111 properly contajh

all work related to homology operations which antedates 1970, while

11 contains either complete information on o r a t least an introduction

to most subsequent work i n this a r e a , the one major exception being

that nothing will be said about BTop and BPL at the prime 2 Up to

minor variants, a l l work since 1970 has been expressed i n the language

and notations established i n I § 1-5 2 and II 5 1

Our tflanks to @ ! a May for preparing the m e x

J P May August 20,1975

IV F Cohen The homology of SF(nt1) 352

The Homology of E Spaces

m

Homology operations on i t e r a t e d loop s p a c e s w e r e f i r s t introduced,

mod 2, by A r a k i and Kudo [1] i n 1956 ; t h e i r w o r k was c l a r i f i e d and ex-

tended by Browder [2] i n 1960 Homology operations m o d p, p > 2, w e r e

f i r s t introduced by Dyer and Lashof [6] i n 1962 T h e w o r k of A r a k i and

n Kudo proceeded i n analogy with Steenrod's construction of the S q i n

t e r m s of V.-products, w h e r e a s that of Browder and of Dyer and Lashof

n proceeded i n analogy with Steenrod's l a t e r construction of the P i n

t e r m s of the cohomology of the s y m m e t r i c group I: The analogy w a s

P'

c l o s e s t i n the c a s e of infinite loop s p a c e s and, in [A], I reformulated t h e

a l g e b r a behind Steenrod's w o r k in a sufficiently g e n e r a l context that i t

could be applied equally well to the homology of infinite loop s p a c e s and

to the cohomology of s p a c e s L a t e r , i n [GI, I introduced the notions of

E o p e r a d and E space T h e i r u s e g r e a t l y simplifies the geometry

r e q u i r e d f o r the construction and analysis of the homology operations and,

i n the non-connected c a s e , yields operations on a wider c l a s s of s p a c e s

than infinite loop spaces T h e s e operations, and f u r t h e r operations on

the homology of infinite loop s p a c e s given by the e l e m e n t s of H*F, will

be analyzed i n section 1

f

t Historically, the obvious next s t e p after introduction of the homology

operations should have been the introduction of the Hopf a l g e b r a of all

homology operations and the analysis of geometrically allowable modules

(and m o r e complicated s t r u c t r u e s ) o v e r this algebra, in analogy withthe

definitions i n cohomology given by Steenrod [22] i n 1961 However, t h i s

s t e p s e e m s not t o have been taken until l e c t u r e s of mine i n 1968-69 The

requisite definitions will b e given i n section 2 Since t h e i d e a that homology

operations should s a t i s f y Adem relations f i r s t a p p e a r s in [6] (although

t h e s e relations w e r e not formulated o r proven t h e r e ) , we c a l l the resulting

a l g e b r a of operations the Dyer-Lashof algebra; w e denote i t by R The

m a i n point of section 2 i s the explicit construction of f r e e allowable s t r u c -

t u r e s o v e r R

D u r i n g m y 1968-69 l e c t u r e s , Madsen r a i s e d and solved a t the p r i m e 2

the problem of c a r r y i n g out f o r R the analog of Milnor's calculation of the

dual of the Steenrod a l g e b r a A His solution a p p e a r s i n [8] Shortly

a f t e r , I solved the problem a t odd p r i m e s , where the s t r u c t u r e of R*

t u r n e d out t o be s u r p r i s i n g l y complicated The details of t h i s computation

(p = 2 included) will be given in section 3

In section 4, w e reformulate (and extend to g e n e r a l non-connected

s p a c e s X) the calculation of H* QX, QX = l i m anxnx, given by Dyer

-C

and Lashof [6] Indeed, the definitions i n section 2 allow u s to d e s c r i b e

H*QX a s the f r e e allowable Hopf a l g e b r a with conjugation o v e r R and A

With the passage of t i m e , i t h a s become possible to give considerably

s i m p l e r details of proof than w e r e available i n 1962 We a l s o compute

the Bockstein s p e c t r a l sequence of QX (for each p r i m e ) i n t e r m s of that

of X

J u s t a s QX is the f r e e infinite loop s p a c e generated by a space X,

s o CX, a s constructed in [G, 921, is the f r e e -space generated by X

(where & i s a n E operad) In section 5, we prove that H*CX i s t h e

3

f r e e allowable Hopf a l g e b r a (without conjugation) o v e r R and A The

proof is quite simple, especially since t h e geometry of the situation m a k e s

half of the calculation a n immediate consequence of the calculation of H*QX

Although the r e s u l t h e r e s e e m s t o be new, i n this generality, s p e c i a l c a s e s

have long been known When X is connected, CX is weakly equivalent t o

QX by [G, 6.31 When X = So, CX = U K(B 1 ) and the result thus

j '

contains Nakaokats calculations [16,17,18] of the homology of s y m m e t r i c

groups We end section 5 with a generalization ( f r o m So to a r b i t r a r y

s p a c e s X) of P r i d d y ' s homology equivalence BEm - aosO [201

In section 6, we d e s c r i b e how t h e i t e r a t e d homology operations of

a n infinite loop s p a c e appear successively in t h e s t a g e s of i t s Postnikov

decomposition

In section 7, we construct and analyze homology operations analo-

gous to the Pontryagin pth powers i n the cohomology of s p a c e s When -

p = 2, t h e s e operations w e r e f i r s t introduced by Madsen 191

Most of the m a t e r i a l of sections 1 - 4 dates f r o m m y 1968-69 l e c t u r e s

a t Chicago and was s u m m a r i z e d i n [12] T h e m a t e r i a l of section 5 dates

f r o m m y 1971-72 l e c t u r e s a t Cambridge The long delay in publication,

f o r which I m u s t apologize, was caused b y problems with the sequel 11

(to be explained i n i t s introduction)

CONTENTS

1 Homology operations: 5

2 Allowable s t r u c t u r e s o v e r t h e Dyer-Lashof a l g e b r a 16 3 The dual of t h e Dyer-Lashof a l g e b r a 24

4 The homology of QX 39

- 5 The homology of CX and the s p a c e s CX 50

6 A r e m a r k on Postnikov s y s t e m s 60

7 The analogs of the Pontryagin pth powers 62

Bibliography 67

51 Homology operations

We first define and develop the properties of homology operations on Em spaces We then specialize to obtain further

properties of the resulting operations on infinite loop spaces

In fact, the requisite geometry has been developed in [GI 51,4,5, and 8.1 and the requisite algebra has been developed in [A, 51-4 and 91 The proofs in this section merely describe the transition from the geometry to the algebra

All spaces are to b e compactly generated and weakly Hausdorff;

'T denotes the category of spaces with non-degenerate base-point [GI p.11 All homology is to be taken with coefficients in Z

P for an arbitrary prime p; the modifications of statements required

in the case p=2 are indicated inside square brackets

We require some recollections from [GI in order to make sense

of the following theorem Recall that an Em space (XI@) is a

'$-space over any Em operad [GI Definitions 1.1, 1.2, and 3.51 ;

@ determines an H-space structure on X with the base-point * E X

as identity element and with O2 (c) : XxX -+ X as product for any

c (2 (2) [GI p, 41 Recall too that the category & [ TI of c-spaces is closed under formation of loop and path spaces [GI Lemma 1.5 I and h a s products and f i b r e d products [G, L e m m a 1-71

Theorem 1.1 Let # be an E, operad and let (XI@ 1 be a $-space

s Then there exist homomorphisms Q : H,X + H,X, s 2 o, which satisfy the following properties:

(1) The QS are natural with respect to maps of %-spaces (2 Q raises degrees by 2s (p-1) [by sl

( 3 ) Q x s = 0 if 2s < degree (x) [if s < degree (x)], x 6H,X

( 4 ) Q x S = xP if 2s = degree (x) [if s = degree (x)] , x EH,X

QS(P = 0 if s > 0, where (P E Ho (X) is the identity element

Q a, = o,Q , where 0,: H, QX -+ H,X i s t h e homology suspen-

s i o n ; i f X i s simply connected and i f x E H X t r a n s g r e s s e s

t o be t h e f o l l o w i n g composite morphism o f a-complexes:

Here Q i s t h e s h u f f l e map; f o r diagram c h a s e s , it s h o u l d be r e -

c a l l e d t h a t n : C,X@C,Y -+ C,(XxY) i s a commutative and a s s o c i a -

t i v e n a t u r a l t r a n s f o r m a t i o n which i s c h a i n homotopy i n v e r s e t o t h e Alexander-Whitney map 5 I n view o f [ G , Lemmas 1 6 and 1 9 ( i ) 1 ,

(C,X,O,) i s a u n i t a l and mod p reduced o b j e c t o f t h e c a t e g o r y

$ (plm) d e f i n e d i n [A, D e f i n i t i o n s 2.11 Moreover, ( x , 0 ) -+ (C,X, O,)

i s c l e a r l y t h e o b j e c t map o f a f u n c t o r from < [ T I t o t h e s u b c a t e - gory p(p,m) of % ( p , m ) d e f i n e d i n [A, D e f i n i t i o n s 2.11 L e t

x E H X A s i n [A, D e f i n i t i o n s 2.21 , we d e f i n e

4

s and we d e f i n e t h e d e s i r e d o p e r a t i o n s Q by t h e f o r m u l a s

s (ii) p = 2: Q x = 0 i f s < q and Q'X = Q ( x ) i f s q ; and

s -q

(iii) p > 2: Q x = 0 i f 2 s < q and Q x = ( - 1 ) ' v ( ~ ) Q (2s-q) (p-1) ( x )

i f 2,s > q , where v ( q ) = (-1)q(q-1)m/2 (m! ) w i t h m = 1 2- (p-1)

The QS a r e homomorphisms which s a t i s f y (1) through (5) by [A,

p r o p o s i t i o n 2.3 and C o r o l l a r y 2.41 Note t h a t [A, P r o p o s i t i o n 2.31

a l s o i m p l i e s t h a t i f p > 2, t h e n f3Q x = (-1) v ( q ) Q(2s-q) (p-l) -1 ( x )

and t h e and p QS account f o r a l l n o n - t r i v i a l o p e r a t i o n s Qi For

( 6 ) , r e c a l l t h a t t h e product of &-spaces ( X I @ ) and ( Y , Q 1 ) i s

$(P)x(XXY) p AXu > & ( P ) x ~ ( P ) x x p 1 X t X i < > r ( p ) x x P X,$(P)XY P > X X Y

( H e r e A, u, and t a r e the diagonal and the evident shuffle and interchange m a p s )

Similarly, the t e n s o r product of objects (K, 8*) and (L, 0;) i n C(p, m ) i s

(K X L, e*), w h e r e @* i s the composite

e*@e;

w@(K@L) >-P W @ W Q ~ K ~ @ L~ lBTQ9' > w @ K ~ @ W @ L ~ - KQL

(Here $, U, and T a r e t h e coproduct on W and the evident shuffle and i n t e r -

change homomorphisms ) Since ( j @ j)$ i s s-homotopic to ( 5 0 C*A)j, a n e a s y

d i a g r a m c h a s e d e m o n s t r a t e s that 11: C*X@ C+Y + C,(XxY) is a m o r p h i s m in the

-A-

c a t e g o r y C ( p , m ) The external C a r t a n formula now follows f r o m [A, C o r o l l a r y

2.71 By [G, L e m m a s 1.7 and 1.9 (ii)], A: X + X X X is a m a p of & -spaces

and (C:,X, 0+) i s a C a r t a n object of $(p, m); the diagonal and i n t e r n a l

C a r t a n f o r m u l a s follow by naturality P a r t (7) i s a n immediate consequence of

[G, L e m m a i 51 and [A, T h e o r e m s 3.3 and 3.41; the simple connectivity of X

2

s e r v e s to e n s u r e that E = H*X@H*OX in the S e r r e s p e c t r a l sequence of

TF: PX -+ X F o r (8), note that the following d i a g r a m is commutative by

[G, L e m m a 1.41:

An easy diagram chase demonstrates t h a t (C,X,O,) i s an Adem o b j e c t ,

i n t h e s e n s e of [A, D e f i n i t i o n 4 1 1 , and ( 8 ) f o l l o w s by [A, Theorem 4.71 P a r t ( 9 ) f o l l o w s by t h e n a t u r a l i t y of t h e Steenrod o p e r a t i o n s from [A, Theorem 9 4 1 , which computes t h e Steenrod o p e r a t i o n s i n

H, ( % ( P ) ~ , r ~ P ) A s explained i n [A, p.2091, our formula d i f f e r s

by a s i g n from t h a t o b t a i n e d by Nishida [ 1 9 ]

L e t xm be t h e c a t e g o r y of i n £ i n i t e loop sequences Recall

t h a t an o b j e c t Y = {yi} i n f,, i s a sequence of spaces w i t h

-

yi - QYi+l and a morphism g = Igi1 i n xm i s a sequence of maps with gi = Ogi+l Yo i s s a i d t o be an i n f i n i t e loop space, go an

i n f i n i t e loop map By t h e r e s u l t s of [ l o ] , t h e s e n o t i o n s a r e equi-

v a l e n t f o r t h e purposes of homotopy t h e o r y t o t h e more u s u a l ones

i n which e q u a l i t i e s a r e r e p l a c e d by homotopies By [G, Theorem 5.11, t h e r e i s a f u n c t o r Wm: f m + c m [ q 1 , w i t h WmY = (YvOm)

and Wmg = g where Gm i s t h e i n f i n i t e l i t t l e cubes operad of

o p e r a t i o n s S i m i l a r l y , p a r t ( 6 ) a p p l i e s t o t h e loop o p e r a t i o n s

s i n c e , by [GI Lemma 5.71, t h e two e v i d e n t a c t i o n s of cm on t h e

product of two i n f i n i t e loop spaces a r e i n f a c t t h e same

The r e c o g n i t i o n theorem [ G I Theorem 1 4 4 ; G I ] gives a weak

CO

an i n f i n i t e loop space BOX; moreover, a s explained i n [G, p.153-

1551, t h e homology o p e r a t i o n s on H,X coming v i a Theorem 1.1 from t h e

given Em space s t r u c t u r e a g r e e under t h e equivalence w i t h t h e loop

o p e r a t i o n s on H B X Thus, i n p r i n c i p l e , it i s only f o r non

* 0 grouplike Em spaces t h a t t h e o p e r a t i o n s of Theorem 1.1 a r e more

g e n e r a l t h a n loop o p e r a t i o n s I n p r a c t i c e , t h e theorem g i v e s

c o n s i d e r a b l e geometric freedom i n t h e c o n s t r u c t i o n of t h e opera-

t i o n s , and t h i s freedom i s o f t e n e s s e n t i a l t o t h e c a l c u l a t i o n s

The following a d d i t i o n a l p r o p e r t y of t h e loop o p e r a t i o n s , which

i s implied by [ G I Remarks 5.81, w i l l be important i n t h e study

of non-connected i n f i n i t e loop spaces R e c a l l t h a t t h e c o n j u g a t i o n

x on a Hopf a l g e b r a , i f p r e s e n t , i s r e l a t e d t o t h e u n i t n ,

Q E = 4 ( 1 ~ X

Lemma 1 2 For Y f xm, ~4 = x QS on H,Yo, where t h e conjugation

i s induced from t h e i n v e r s e map on Y o = QY1

i t s augmentation, a 1 = € ( a ) , and l e t A a c t on t h e t e n s o r product

M @ N of two l e f t A-modules through i t s coproduct,

a (m a n ) = c (-i)deg a"deg ma'mbpa'ln i f +a = Cat e~ a "

A , l e f t o r r i g h t s t r u c t u r e ( a l g e b r a , c o a l g e b r a , Hopf a l g e b r a ,

Hopf a l g e b r a w i t h c o n j u g a t i o n , e t c ) over A i s a l e f t o r r i g h t

A-module and a s t r u c t u r e of t h e s p e c i f i e d t y p e such t h a t a l l of

t h e s t r u c t u r e maps a r e morphisms o f A-modules

t i o n s P: on t h e l e f t , we s h a l l speak of r i g h t A-modules r a t h e r

t h a n of l e f t AO-modules Thus H,YO i s a r i g h t Hopf a l g e b r a w i t h

c o n j u g a t i o n over A , and t h e Nishida r e l a t i o n s g i v e commutation

formulas between t h e A and R o p e r a t i o n s on H,YO

There i s y e t a n o t h e r Hopf a l g e b r a which a c t s n a t u r a l l y and

s t a b l y on H,YO namely H,F where F i s t h e monoid (under compo-

s i t i o n ) of based maps o f s p h e r e s The p r e c i s e d e f i n i t i o n of 8

i s given i n [ G I p.741, and it i s shown t h e r e t h a t composition

of maps d e f i n e s a n a t u r a l - a c t i o n c - YOxF -+ Y o of F on i n f i n i t e

FO- loop spaces The following theorem g i v e s t h e b a s i c ~ r ~ p e r t i e s

' I

i

(BQSx) f = C B Q ~ + ~ ( ~ P ~ ~ ) - c (-1) deg ( X P:B~)

Proof: Part (1) is trivial The maps * + Yo and Yo +* are

infinite loop maps, hence the unit and augmentation of H,YO are

morphisms of H,%-modules The loop product is a morphism of

H,$-modules by a simple diagram chase from [GI Lemma 8.81, and a

similar lemma for the inverse map implies that the conjugation

is a morphism of H,$-modules The coproduct on H,YO is a

morphism of H,$-modules and formula (3) holds because cm, is

induced by a map Formula (2) is an immediate consequence of

[G, Lemmas 8.4 and 8.51 For ( 4 ) , consider the following dia-

gram, in which we have abbreviated ?$ for Sm (p) and X, X(P) , and

duces cm*, and the map induced on s-equivariant homology by

Of c o u r s e , it is the presence of d in the diagram which leads

to the appearance of Steenrod operations in formula (4)

The verification of this formula is now an easy direct cal-

The essential part of the previous diagram is of course

the geometric bottom left square Henceforward, we shall

omit the pedantic details in the passage from geometric dia-

grams to algebraic formulas

We evaluate one obvious example of the operations on

' ,

H,Yo given by right multiplication by elements of H,F

' ,

Lemma 1.5 Let [i]EHOF be the class represented by a map

Proof: Define f: sn + sn by f (sl, ,sn) = (1-sl,s2, ,sn) ,

Recall that QX = lim * 0%"~ and QX = QQZX [ G , p 421

As we shall see in 11 f! 5 , application of the r e s u l t s

N

above to Y = QS', where c reduces to the product on F, completely

rV

d e t e r m i n e s the composition product o n H+F

R e m a r k s 1 6 A functorial definition of a s m a s h product between objects

of x m i s given in [13], i n which a new construction of the s t a b l e homotopy

category i s given (In the language of [13], m is a c a t e g o r y of

coordinatized s p e c t r a ; the s m a s h product i s constructed by passing to the

category & of coordinate-free s p e c t r a , applying the s m a s h product t h e r e ,

a&d then returning to ) F o r objects Y, Z E xQ3 and elements

m

x, y .E HeyO and z E H*ZO, [G, L e m m a 8.11 and a s i m i l a r l e m m a f o r the

i n v e r s e m a p imply the f o r m u l a s

( X * ~ ) A Z = ): (-1) d e g y d e g z ' ( x i u ~ ) * ( y h z ~ ) if +z = z 1 6 z " and (XY)A z = X ( Y A Z ) ,

w h e r e * and A denote the loop and s m a s h products respectively Via a

d i a g r a m c h a s e p r e c i s e l y analogous t o that i n t h e proof of T h e o r e m 1 4 ,

[G, Proposition 8.21 implies the f o r m u l a s

i $2 Allowable structures over the Dyer-Lashof algebra

I

I I on Em spaces generated by the QS and BQS and develop analogs

of the notions of unstable modules and algebras over the Steenrod

algebra The following definition determines the appropriate

I I "admissible monomials"

Definition 2.1 (i) p = 2: Consider sequences I = (sl, ,sk)

such that s > 0 Define the degree, length, and excess of I

j -

k d(1) = C s g(1) = k; and

The sequence I determines the homology operation Q ' = Q

1 / I is said to be admissible if 2sj sjel for 2 2 j 2 k

(ii) p > 2 : Consider sequences I = ( E ~ , s1 , , sk) such

that E = 0 or 1 and s > E Define the degree, length, and

excess of I by

k

The sequence I determines the homology operation Q1 =BEIQS1 Q

I! 1 I is said to be admissible if psj - E~ 2 s ~ for 2 - ~ 2 j 2 k

(iii) Conventions: b(1) = if p > 2 and b(1) = 0 if p = 2

The empty sequence I is admissible and satisfies d(1) = 0,

I/ k(I) = 0, e(1) = m, and b(1) = 0; it determines the identity

i

! relations, with B2 = 0) and by the relations Q = 0 if e(I)< q

Define R(q) to be the quotient algebra F/J(q), and observe that

E

k there are successive quotient maps R(q) -t R(q+l) Let R = R(0) ;

! R will be called the Dyer-Lashof algebra

algebraically The following theorem implies that this defini- tion agrees with that naturally suggested by the geometry

Theorem 2.3 (i) Let i 6 H sqc H Qsq be the fundamental class

is a linearly independent subset of H,Qsq

t (ii) J(q) coincides with the set X(q) of all elements of F

which annihilate every homology class of degree 2 q of every

Em space (equivalently, of every infinite loop space)

(iii) {Q'II is admissible and e(1) 2 q} is a Zp-basis for R(q)

(iv) R(q) admits a unique structure of right A-module such that the Nishida relations are satisfied

(v) R = R(0) admits a structure of Hopf algebra and of un- stable right coalgebra over A with coproduct defined on gene- rators by

I# Q = E Qi @ Q and )8QSt1 = 6 ( BQitlg, Q + Qi s BQ j+l)

Ok and with augmentation defined on R o = PCQO} by E (Q ) = 1, k 2 0 Proof: We shall prove (i) in 54 It is obvious from the Adem

relations that R(q) is generated as a Z -space by the set speci-

P fied in (iii), and J(q) is contained in K(q) by (3) and (8) of

Theorem 1.1 Therefore (i) implies (ii) and (iii) For (iv) ,

the A operations on i f R o(q) are determined by = 1 and Ri (q) =O

I for i < 0 and the A operations on a11 elements Q1 = Q 1 with

$(I) > 0 are determined from the Nishida relations by induction

on $(I) This action does give an A-module structure since

if f (q) : R(q) + H, (Qsq) is defined by T(q) Q = Q iqr then

f(q) is a monomorphism which commutes with the Steenrod opera-

tions Let f(0) = f; since Ji(1) = 1Q)Iand Ji(i 0 ) = i @i 0 and

k

since ~ ( i ) = 1, f commutes with the coproduct and augmentation

0

Here Ji is well-defined on R and R is a Hopf algebra since

J = J(0) is a Hopf ideal, Ji (J) c F 63 J + J QIF, by commutativity

of the following diagram (where a is the quotient map):

Observe that this argument fails for q > 0 since Jii = i LBl+lepi

Since H,QSO is an unstable right coalgebra over A, so is R

Of course, we understand unstable right structures over

A in the sense of homology: the dual object (if of finite

type) is an unstable A-structure of the dual type, as defined

by Steenrod [22,23] We shall study the structure of R it-

self in the next section The remainder of this section will

be devoted to the study of structures over R In order to

deal with non-connected structures, we need some preliminaries

De~ginition 2.4 A component coalgebra is a unital (and aug-

mented) coalgebra C such that C is a direct sum of connected

coalgebras Given such a C, define

aC = { g l g ~ ~ , Jig = g m g and g # 0)

Clearly aC is a basis for Co For gE nC, define C to be the

9 connected sub-coalgebra of C such that g E C and the set of

9 positive degree elements of C is

that & g = I for g e K nC contains the distinguished element $ = tl(i), and

J C = Coker q may be identified with 63 ( 63 Cg) C C

g f $

If X i s a based space, then H*X i s a component coalgebra; the base-point

determines q and the components determine the d i r e c t s u m decomposition Indeed,

t h e r e i s an obvious identification of T X with nH*X As another example, we

0

have the following observations on the s t r u c t u r e of R

Lemma 2.5 R i s a component coalgebra -rrR i s the f r e e monoid generated by

Q and the component ~ [ k ] of (Q ) , k 2 0, i s the sub unstable A-coalgebra of

R spanned by

I {Q I I i s admissible, e(1) 2 0, and l(1) = k}

The product on R sends R[k] @R[l] to R[k+ P ] f o r a l l k and 1 , and the ele-

ments and pQS a r e a l l indecomposable

Definition 2 6 A component Hopf algebra B i s s a i d to be monoidal (resp., group-

like) if nB i s a monoid (resp , group) under the product of B Equivalently, B i s monoidal if a l l pairwise products of elements of nB a r e non-zero

The proof of the following lemma r e q u i r e s only the defining formula

if d e g x > O and + x = x @ g + Z x t @ x " + g @ x w i t h d e g x t > 0 and d e g x N > 0

We can now define allowable structures over R, by which

we simply mean those kinds of R-structures which satisfy the

algebraic constraints dictated by the geometry

Definition 2.8 A left R-module D is allowable if J (q) D = 0

9 for all q 2 0 The category of allowa,ble R-modules is the full

subcategory of that of R-modules whose objects are allowable;

it is an Abelian subcategory which is closed under the tensor

product An allowable R-algebra is an allowable R-module and

s

a commutative algebra over R such that Q x = xP if 2s = deg x

lowable R-module and a cocommutative component coalgebra over

R An allowable R-Hopf algebra (with conjugation) i s a monoidal Hopf

algebra (with conjugation) over R which is allowable both as

an R-algebra and as an R-coalgebra For any of these struc-

tures, an allowable AR-structure is an allowable R-structure

and an unstable right A-structure of the same type such that

the A and R operations satisfy the Nishida relations

Theorem 1.1 implies that the homology of an Em space is

an allowable AR-Hopf algebra Lemma 1.2 implies that the

homology of an infinite loop space is an allowable AR-Hopf al-

gebra with conjugation Observe that a connected allowable

AR-Hopf algebra is automatically an allowable AR-Hopf a l g e b r a with

conjugation

In o r d e r t o take advantage of t h e s e definitions, we r e q u i r e five basic

f r e e functors, D, E, V, W, and G, of which E

apd W are essentially elaborations of D and V in the presence

of coproducts In addition, each of these functors has a

more elaborate counter-part, to be defined parenthetically, in the presence of Steenrod operations The composite functors

WE and GWE will describe H,CmX and H,QX, with all structure in sight, as functors of H,X

We shall describe our functors on objects and shall show that the given Internal structures uniquely determine the re- quired internal structures The verifications (not all of which are trivial) that these structures are in fact well-defined and satisfy all of the requisite algebraic indentities will be left to the reader, since these consistency statements obviously hold for those structures which can be realized geometrically

It is trivial to verify that our functors are indeed free, in the sense that they are adjoint to the forgetful functors going the other way The functor V, which is a special case

of the universal enveloping algebra functor on Abelian restricted Lie algebras, and the functor W occur in many other contexts

in algebraic topology; they are discussed in detail in [ I l l

D: Z -modules (resp., unstable A-modules) to allowable R-modules

P (resp., AR-modules): Given M I define

R acts on DM via the quotient maps R + R(q); thus DM is just the obvious quotient of the free R-module R@M The inclusion

of M in DM is given by m + ICom If A acts on M I then this action and the Nishida relations determine the action of A on

DM by induction on the length of admissible monomia1.s

E: Cocommutative component coalgebras (resp., unstable A-coalgebras)

to allowable R-coalgebras (resp., AR-coalgebras): Given C, de- fine EC as an R-module, and as an A-module if A acts on C, by

EC = DC/IR.Imn = Z @DJC, IR = Ker E and JC = Coker n

P

The inclusion of C in EC is induced by that of C in DC The

coproduct on C and the diagonal Cartan formula determine the

coproduct on EC The unit of C and the augmentations of R and

C determine the unit and augmentation of EC Equivalently,

EC is the obvious quotient component coalgebra of R e C ; thus

0 k

and the component of (no) @ g is the image of R [k] @ C in EC

g

if g f + while the component of I@+ is the image of RQDC +-

V: Allowable R-modules (resp., AR-modules) to allowable R-algebras

(resp., AR-algebras): Given D, define

VD = AD/K

where AD is the free commutative algebra generated by D and K

is the ideal of AD generated by

{xP - nSx12s = deg x if p>2 or s = deg x if p=2)

The R-action, and the A-action if A acts on D, are determined

from the actions on D C V D by the internal Cartan formulas (for

R and A) and the properties required of the unit

W: Allowable R-coalgebras (resp., AR-coalgebras) to allowable

R-Hopf algebras (resp., AR-Hopf algebras): Given E l define

WE as an R-algebra, and as an A-algebra if A acts on E, by

WE = VJE, JE = Coker n

The inclusion of E in WE is given by E = Z 63JE and JECVJE

P

The coproduct and augmentation of WE are determined by those of

E and the requirement that WE be a Hopf algebra (it is a well-

defined Hopf algebra by [11, Proposition 121) The components

of:WE are easily read off from the definition of VOJE

G: Allowable R-Hopf algebras (resp., AR-Hopf algebras) to

allowable R-Hopf algebras (resp., AR-Hopf algebras) with conju- gation :

Given W, define GW as follows aW is a commutative monoid under the product in W and Wo is its monoid ring Let sGW be the commutative group generated by aW and let GOW be its group ring Let + = n(1), let E be the set of positive degree elements of

W, and let k be the connected subalgebra Z + OV of W Define

P

G W = E + ~ P G W o = W B G W

a W 0

as an augmented algebra Embed W in GW as the subalgebra

The coproduct on GW is determined by the requirements that

W and GOW be subcoalgebras and that GW be a Hopf algebra

The conjugation is given by Lemma 2.7 The R-action, and the A-action if A acts on W, are determined from the actions on

W C G W by commutation with X and the Cartan formulas If the product in WG is denoted by *, then the positive degree ele- ments of the component of f€aGW are given by

Observe that GW = W if W i s connected and that, a s a ring, GW i s just

the localization of the ring W at the rnonoid f l

53 The dual of the Dyer-Lashof algebra

Since EH,SO is the allowable AR-coalgebra Z @ R (which

P should be thought of as Z 4 [Ol @R* [l]), a firm grasp on the

P structure of R is important to the understanding of H,cmsO a d

of H,QSO The coproduct and A-action on R are determined by

the diagonal Cartan formula and the Nishida relations, but these

merely give recursion formulas with respect to length, the ex-

plicit evaluation of which requires use of the Adem relations

To obtain precise information, we proceed by analogy with

Milnor's computation of the dual of the Steenrod algebra 1141

In the case p = 2, the analogy is quite close; in the case p > 2,

the Bocksteins introduce amusing complications The structure

of R*, in the case p = 2, was first determined by Madsen [ 8 1 ;

his proofs are closer to the spirit of MilnOr's work, but do

not generalize readily to the case of odd primes

By Lemma 2.5, R = @ R[k] as an A-coalgebra Of course,

k> 0 R[O] = Z We must firsF determine the primitive elements PR[k]

P'

of the connected coalgebras R[k],.k 2 1 To this end, define

P[k] = {I \I is admissible, e(1) O., R(1) = k, and I ends with 11

We shall see that IQ' ]I€ P[k] 1 is a basis for PR[k] Define

(inductively and explicitly) certain elements of P[k] as follows:

(I) lzj~k, p=2: Ill = (1) , Ij,k+l = (2k-2k-j, I ) if j k ,

3 k

k and 1k+l,k+l=(2 , i then d ( ~ 3k )=2k-~k-j, e(xjk)=O if j<k,

k-1-2k-l- j 2k-2 - 2k-Z-j

e (Ikk) = 1; I = (2

k k-j (I1) Ijk, lLjLk, ~ > 2 : I l l =(0,1), Ij,k+l=(Of~ -P , I 3k ) if j<k,

k- k-j and I ~ + ~ , ~ + ~ = ( O ~ ~ ~ , I ~ ~ ) ; thend(1 )=2(p p 1, e(Ijk)=O if j<k,

jk

k k-j (111) Jjk, 1 3 5 P>Z: Jll=(~,l) J ~ , ~ + ~ = ( o P -p , J ) if j<k,

k k-i k-j (IV) Kijk lzi<jk, p>2: Xi, ,k+l=LOfp -p -p , K ~ ~ ~ ) if j<k, -

and Ki , k+l , k+l= (1 P -P Jik) i then d (Ki k) = 2 (p -p -p ,

If we look back at the definition of the in terms of the Qi in the proof of Theorem 1.1, we see that, when acting

on a zero-dimensional class, our four classes of sequences correspond to sequences of operations of the respective forms

Many arguments in this section and the next can be illuminated

by translation to lower indices

Lemma 3.1 P[k] = {I Il~jzk} if p=2; ~ [ k ] = ~ ~ ~ ~ J ~ ~ , K ~ ~ ~ / l ~ j ~ k , l ~ i < j }

3

Proof: Proceed by induction on k, the case k = 1 being trivial

-

Consider I = (E,S,J)EP[~], k>2 Then, since I is admissible and -

e(1) - > 0, JE_P[k-11, p r if J = (6,r,K), and 2s-E 2 d(J) The first part follows inductively from these inequalities by a

trivial examination of cases The second part is an easy calcu-

lation based on the facts that 13'QiQJ=0 if 2i-y<d(J) and that

!3'Q1Q0=0 by the first Adem relation

The computation of R[k]* as an algebra is based on a cor-

respondence between addition of admissible sequences and multi-

plication of duals of admissible monomials We first set up

the required calculus of admissible sequences

Definitions 3.2 The sum I+J and difference I-J of two sequences

(as in Definition 2.1) of length k is defined termwise, under

th the conventions that I+J is undefined if p>2 and the i- "Bockstein

entry" E~ is one in both I and J and that I-J is undefined if

any entry is less than zero Observe that e(I2J) = e(I)te(J)

and d (I+J) = d (1)fd (J) If I and J are admissible, then I+J

is admissible but I-J need not be admissible In order to

enumerate the admissible monomials when p>2, consider all sequences

e = {elf ,e 1 with 1 2 el< <e.<k and define

If p=2, write I[k] = {III is admissible, e(I)LO, and 1?(1)=kl

With these notations, we have the following two counting

Then f is an isomorphism of sets

I Proof: For p>2, omit the irrelevant zeroes corresponding to

/ absence of Bocksteins Then f is given explicitly by

f (sir .,s )=(nlI ,n ) where n.=

k

- C (p sq-sq-l) if j=k q=2

Lemma 3.4 For p>2, k>l,and each non-empty eldefine fe: I[k] + ~,[k]

by fe(I) = I+Lek Then fe is an isomorphism of sets

Proof: Obviously fe must be given by fe (J) = J-Lekl J EIe[k] ,

and it suffices to show that J-Lek is defined, admissible, and has non-negative excess Write Lek = (61,r1, 16kfrk) and

J = (E~,s~; ,E~,s~) Observe that

I J Lemma 3.6 If I is inadmissible and Q =ZAJQ where the J are

admissible, then AJ#O implies J<I If P:Q1=~A p J , where r>O

and the J are admissible, then AJ#O implies'J<I

R* = lT R[k]* as an A-algebra In the dual basis to that

k

0

of admissible monomials, define elements of R[k]* by

To simplify statements of formulas, define Sjk=O if j < O o r

j>k, 0 if j o r k and oijk=O if i<l, j i , or j>k

SOk is ,the identity element of R[k]* and lT EOk is the

k>O identity element of R* The augmentation of R* is given by

E( II AkCok) = Ao Of course, R* is not a coalgebra since Ro

k> - 0

is not finite dimensional (although R is finite dimensional for

q q>O) However, R* does have a well-defined coproduct on positive

degree elements and on finite linear combinations of the SOk;

the latter is evidently given by

n

It is perhaps worth observing that although II R[kl* is a

k= 0 quotient augmented A-algebra of R* and a coalgebra (dual to the

quotient algebra R/Z R[ml of R) such that ,the product is a

m> n

n morphism of coalgebras, lT R[k]* is nevertheless not a Hopf

k= 0 algebra because its unit fails to be a morphism of coalgebras

(dually, (QO)~" = 0 but EQO = 1)

We shall successively compute R[k]* as an algebra, compute the Steenrod operations on generators, and compute the coproduct

R[kl* is determined as an algebra by commutativity and the following relations:

(i) T~~ T~~ = Skkaijk if i<j (and T ~ = 0); ~ T ~ ~

(ii) oijk T~~ = ( T ~ ~ T ~ ~ T ~ ~ ) / S ~ ~ ; and (iii) uijkumk = (T T T 2

ik jk mkTnk)/'kk' (In (ii) and (iii), the right sides are to be evaluated in terms

of the basis monomials by use of (i); the numerators, if non-zero,

2

are divisible by the non zero-divisor Skk or Skk.)

Proof: By the counting lemmas, an admissible monomial I with

I = nlIlk+ .+nkIkk+Lek, n > O and e = {el, ,eJ3,

9-

Let j = 2 i - E , E = O o r 1, and d e f i n e n ( I ) = i f Enq Let he d e n o t e t h e

( 2 ) <Elk Ekk h e , Q ~ > # 0 and J # I imply J > I

Let $: R[k] + ~ [ k ] " ' ~ ) be t h e i t e r a t e d c o p r o d u c t For any J ,

Now ( 1 ) i s immediate from t h e d e f i n i t i o n of t h e Lek Given J

An e a s y d i m e n s i o n a l argument shows t h a t ckkuijk i s t h e o n l y

p o s s i b l e summand of rik -rjk, and t h i s proves ( i ) S i n c e

f o r m u l a s (ii) and (iii) f o l l o w immediately from ( i )

may r e s t r i c t a t t e n t i o n t o t h e g e n e r a t o r s pP and B of A For

d i m e n s i o n a l arguments, it should be observed t h a t R can be

g i v e n a second g r a d i n g by t h e number of B o c k s t e i n s which o c c u r

i n monomials and t h a t a l l s t r u c t u r e ( e x c e p t , of c o u r s e , a c t i b n

by B ) p r e s e r v e s t h i s g r a d i n g Lemma 3 8 The f o l l o w i n g f o r m u l a s a r e v a l i d i n R[k], k > l , -

r and t h e s e f o r m u l a s s p e c i f y a l l o p e r a t i o n s pQJ and PI: QJ, r,O,

on b a s i s e l e m e n t s QJ, which have a summand of t h e form X Q I

Proof: he statements about 6 are obvious For the rest, we

-

first reduce the problem to manageable proportions by a search

of dimensions Observe that

k

k-l<d (I) <2 k -11

(a) ~g P [k] implies zpk" ( ~ ~ - ~ - 1 ) Fd (I) F2 (p -1) C2 - -

Since R[k]* is an unstable A-module, (a) implies that

PpqQ1') = 0 i f r 2 k and i t P[k] F o r r < k and I t P[k], we have

k k - i k-2 2 (b) d(1) t 2pr(p-i) 2(2p - p - 1) < 6p (p - p-l)[d(I) + 2r < 3 ~ 2 ~ - ~ ]

I Clearly (a) and (b) imply that i f P ; P ~ Q ~ has a summand hQ with h # 0 and

I t P[k], then either J t P[k] o r J = J 1 + J " with both J 1 and J1' in ~ [ k ] 1

I

d(I) ~ p ~ ( p - 1 ) < 4p (P - P - i ) , IcP[kl,

if either p > 3 and r ( k-2 o r p = 3 and r 5 k-3; thus the possibility J = J 1 t J n

i s also ruled out in these cases Simple dimensional arguments in the few re- I

maining cases demonstrate that our list will be exhaustive provided that the

following formulas also hold:

k - i I +Kiik

P Q ~k (xi) p>Z: P* = O i f l < i < j S k ,

combinations of admissible monomials QL such that e(L) ( e(J) - 2(p- E ) I

Further, we have the particular Nishida relations

( 7 ) Q ~ ~= B B Q Q ~ ~ if s2.1 ~ Q ~ The proofs of (ii) through (v) and (xii) are similar applications

of ( 4 ) and (5); they are simplified by use of induction on k The following Adem relation is needed in the proof of (xii)

(8) Qps+lQS = 0 if s2O

Because of the change of basis involved in our description

of R[k]*, our formulas simplify slightly upon dualization

Theorem 3.9 The following list of relations specifies all non-trivial actions of the generators p p r , rlO, and 6 of the

Steenrod algebra on the generators (Ejkl T~~~ aijkl of Rtkl*

(i) p>2: Brkk = Ekk and @aikk = -T if l<i<k

k - i - j (iii) p > 2: pp = -7

Tjk j+l,k i f i S j < k

k- 1 -i (iv) p > 2: pp i+I,j,k if 1 5 i < j - 1 < k

k - I - j crijk jk= - u+

(v) p > 2: pp cr ijk - - -ui,j+l,k i f I S i < j < k

(vi) p 2 2: p ~ k - i 'jk = '1k6jk if I c j 5 k

k- 1 (vii) p > 2: pp ijk = Cikijk + Cjkrlk if 1 5 j c k

k- 1 (viii) p > 2 pp qjk = Slkuijk + Cikuljk - ejkrIik if 1 < i < j ( k , u I l k = o

Proof (i) i s trivial since, a s explained in [A, p 2071, the cohomology and

homology Bocksteins a r e related by

<pel a > = (-l)deg fffl < c , @a>

Relations (ii) through (vi) a r e immediate from the corresponding numbered

d

relations of the lemma, since ~ [ k ] * i s one-dimensional in the degrees in which

these relations occur F o r (vii), we can certainly write

k- 1

pP rjk = aglkrjk + bgjkilk (with a = 0 if j = l)

J + I

F o r j 2 2 , I l k + J j k < J l k + J and < 6 lkrjk' Ik jk> = 1, a s can be seen

jk J1k+ljk

by examination of L$ Q By (vii) and (viii) of the lemma, and by

formulas (1) and (2), we find

k-1 I + J

1 = < p p , Q ' ~jk> = a i f 2 5 j 5 k

jk' and

by examinations of coproducts Now (ix), ( x ) , and (xi) of the lemma, to-

k- 1 gether with (1) and (2), imply (viii) by evaluation of PP u on

13k Ilk+ K q k , di*+ K1jk J + K

Q , and Q jk like Note that (viii) can be predicted

k-1 from (vi) and (vii) by application of pZp to the relation rik;k = Skkcrijk

We have the following immediate corollary

Corollary 3 10 If p = 2, ~ [ k ] l' i s generated as an A-algebra by

CIk If p > 2, ~ [ 1 ] * i s generated a s an A-algebra by r and ~ [ k ] * ,

Remarks 3 i1 In order to obtain an upper bound on the spherical classes

of HIQsO by determination of its A -annihilated primitive elements, it would

be desirable to have complete information on the A-module (rather than the

A-algebra) generators of ~ [ k ] * : we can add classes not in R to elements

of R to obtain primitive classes of H*QS': we cannot so obtain A-annihi-

lated classes of HIQSO unless the given class in R was A-annihilated

I have not carried out the necessary calculations Madsen [ 8 ] has obtained considerable information in the case p = 2 and has used this

information to retrieve Browder's results [ 4 ] on the Arf invariant

b

*

It remains to compute the coproduct on generators of ~ [ k ] , and we

I need information about the products in R which hit any of the Q ,

I e P[k] Fortynately, we do not need complete information when I = K ijk'

Lemma 3.12 Let IOk denote the sequence of length k, k 2 0,

with all entries zero Suppose that J and K a r e admissible sequences

I such that QJQK has a summand XQ with X f 0 and I E P[k] Then

either K E P[i] o r K = I for some i < k and, in the latter case,

Proof If (J, K) i s admissible and in P[k], then K E P[i] for

some i; such decompositions of I and J account for the relations

with h = i in (i) and with h = j @ (i) and (ii) and for all relations in (iii)

If ( J , K) i s inadmissible, then K E ~ [ i ] o r K = I O i for some i since the

1 only Adem relations which have Q appearing on the right side a r e

We claim that i f (t, ki) i s inadmissible, 0 I h 2 i , then Qt dhi has no

non-zero summand ending with Q unless t = p and h < i, when

t Iii

dMititi is the only such summand Indeed, Q Q can have no such

summand because, in the Adem relation QrQS = x.Qr+s-j aj for r > ps,

J

A = 0 unless j > s The claim now follows by upwards induction on i

J

and, for fixed i , downwards induction on h, via explicit calculation from

the ~ d e m relations and the inductive definition of the Ihi The essential

Ihi

0 ( h < i Note that, since f3Q = 0 for h < i, it follows that if (J, &)

i s inadmissible, 0 5 h 5 i, and if any Bockstein entry E in J i s non-zero,

(7) i s used to prove the claim when h = i A straightforward bookkeeping

argument from our claims shows that the relations of (i) and (ii) with h < i

J K and h < j g i v e a l l p o s s i b i l i t i e s f o r Q Q t o h a v e a n o n - z e r o summand

h Q j k o r X Q J k when (J,K) isinadmissible

*

In our formulas for the coproduct in R-', the sums a r e to range

over the integers; this makes sense in view of our convention that 6

Jk' Tjk, and o

ijk a r e zero except where explicitly specified otherwise

The formula for 4 o announced in [12 ] i s incorrect; the correct

1Jk formula given here i s in fact somewhat simpler

Theorem 3.13 The following formulas specify the coproduct on

*

the generators of R'

i i-h i-h

= s p - p (i) "jk

(h, i) k-i,k-i j-h,k-i

i i-h i - h (ii) + T = ~ ~ cP - eP

ph- 1

0-

' Sk-h, k-h i-h, j-h, k-h @ 'hh

h

Proof Observe f i r s t that if J = X n I ik + L e k s then e(J) = n k S € ,

w h e r e E = e ( L ) i s z e r o o r one In view of the l e m m a , (i) and (ii) will

ek hold provided that t h e monomials to the left of the t e n s o r signs a r e p r e c i s e l y

J

dual to the corresponding admissible monomials Q By (1) and (2) in t h e

proof of Theorem 3.7, t h i s will certainly hold if the J a r e m a x i m a l among

a l l admissible sequences of the requisite degrees A dimensional argument

i i - h shows that, due to the multiple p - p of Ik-i, k-i which a p p e a r s ,

t h e J actually have m a x i m a l e x c e s s among the admissible sequences of

t h e requisite d e g r e e s We prove (iii) by a trick By the l e m m a , we c a n

c e r t a i n l y w r i t e

(ggh f o r g < h cannot a p p e a r o n the right because, a s noted in the proof

I

*'jk unless

of the l e m m a , Q ~ Q gh cannot have a non-zero summand AQ

( J , i s admissible, when g = h ) We have T Ik T jk = 'kkUijk and

t h e r e f o r e ( +rik)(+rjk) = uijk) After expanding both s i d e s by

u s e of T h e o r e m 3.7 and the f a c t that R[klT R[I 1- = 0 f o r k # P , we find

that t h e r e i s a unique solution f o r the unknowns a f gh' gh' @ and y h '

namely that specified in (iii)

In t h i s section and the next, we s h a l l compute H*QX and H,CX

f o r any space X, where C i s the monad a s s o c i a t e d to an E operad

m [see G, Construction 2.41 We s h a l l a l s o fompute the mod p

Bockstein s p e c t r a l sequences of QX and CX, hence o u r r e s u l t s will

determine the integral homology groups of t h e s e spaces

(M i s the f r e e infinite loop space generated by X in the s e n s e

that if Y r xm and f:X + Y o i s any m a p in 5 , then t h e r e i s a unique m a p g: { Q Z ~ ) + Y in Xrn such that go q = f, where

q : X + QX i s the n a t u r a l inclusion [see G, p 431 Since, for a l l finite

n, the composite

i s the identity, where A i s the evaluation map, q*: H*X -+ H*QX i s a monomorphism It i s t h e r e f o r e reasonable to expect H*QX to be a n

appropriate f r e e object generated by H*X

Similarly, f o r any operad c , (CX, p) i s the f r e e -space generated by X in the sense that if (Y, 8) i s a $ -space and f:X + Y

i s a m a p in 2 , then t h e r e i s a unique m a p g: CX + Y of -spaces such that gq = f , qr X + CX [see G, p 13,16,17] Again, i t i s reason-

able t o expect H*CX t o be a n appropriate f r e e object generated by H*X,

a t l e a s t f o r nice operads

We have constructed c e r t a i n f r e e functors WE and GWE in

section 2 and, by f r e e n e s s , t h e r e a r e unique m o r p h i s m s ?j:, of allow-

able AR-Hopf algebras and ';i* of allowable AR-Hopf a l g e b r a s with conjugation such that the following diagrams a r e commutative:

is a n isomorphism of AR-Hopf algebras with conjugation

The second t h e o r e m i s a reformulation (and generalization) of the

calculations of Dyer and Lashof [ 6 1

contains a concise reformulation of Nakaoka's r e s u l t s [16,17,18] on the

homology of s y m m e t r i c groups An E operad should be thought of

C13

a s a suitably coherent construction of universal bundles f o r s y m m e t r i c

groups; the simple statement that CSO i s a c - s p a c e contains a g r e a t

deal of information that i s usually obtained by more' cumbersome alge-

b r a i c techniques

The elements of H,X C H,CX and of H*X C H*QX play a

r o l e i n the homology of E s p a c e s and of infinite loop s p a c e s which i s

m analogous to that played by the fundamental c l a s s e s of K(a, n ) ' s in the

cohomology of spaces, In p a r t i c u l a r , the following c o r o l l a r i e s a r e

analogs of the statement that the cohomology of any space can be r e p r e -

s ented, via the m o r p h i s m induced by a map, a s a quotient of a f r e e

Corollary 4.3 If (X, B) i s a &-space, where i s a n

Em operad, then 8,: H,CX - H,X r e p r e s e n t s H*X a s a quotient AR-

Hopf a l g e b r a of the f r e e allowable AR-Hopf a l g e b r a WEH,X

Proof 0: CX * X i s the unique m a p of c - s p a c e s such that

C o r o l l a r y 4.4, If Y i s a n infinite loop sequence, then

-$ , H*QYO + H*YO r e p r e s e n t s H,YO a s a quotient AR-Hopf algebra

W T with conjugation of the f r e e allowable AR-Hopf algebra with conjugation

GWEH,X

-,

Proof g m S Y O * Yo i s the unique infinite loop m a p such

that -$ q = 1: ljm i s defined explicitly in [G, p 431

m

Of c o u r s e , T h e o r e m s 4 1 and 4 , 2 a r e not unrelated By

[G, Theorem 4.21, t h e r e i s a m o r p h i s m of monads a : C + Q Thus

Here L i s the n a t u r a l inclusion Since L i s the identity if X i s con-

nected, Theorems 4.1 and 4.2, coupled with the Whitehead t h e o r e m

f o r connected H-spaces, imply the following result

C o r o l l a r y 4.5 a : CmX + QX i s a weak homotopy

m equivalence f o r a l l connected s p a c e s X

unstable A-algebra

The c o r o l l a r y was proven geometrically i n [G, T h e o r e m 6.11

n n

by use of the much deeper fact that cr CnX * S? Z X i s a weak homo-

n *

topy equivalence f o r a l l n and a l l connected X We s h a l l prove

Theorem 4.1 and shall generalize the c o r o l l a r y by obtaining a homology

approximation to QX, f o r a r b i t r a r y X, i n the next section We prove

T h e o r e m 4.2 and compute the Bockstein s p e c t r a l sequence of CX and

QX here

For counting a r g u m e n t s , i t will be useful to have explicit b a s e s

f o r WEH,X and GWEH,X ., L e t tX be a b a s i s f o r JH*X which con-

tains the s e t of components of X, other than the component $ of the

base-point, r e g a r d e d a s homology c l a s s e s of d e g r e e zero Thus

N

tX u { $ } i s a b a s i s f o r H*X Let NnOX and Nn X denote the

?

f r e e commutative monoid and the f r e e commutative group generated

by n X, each subject t o the single relation $ = 1; l e t Z NnOX and

(Recall the conventions, Definition 2.l(iii) ) Then, a s algebras,

(2) WEH,X = ATX @ Z NnOX and GWEH,X = A T X @ Z h O x

Note that the Q x with e(1) = deg x , b(1) = 0, and deg Q x > 0 p r e -

c i s e l y account f o r a l l p-th powers of positive d e g r e e elements Note

also that T h e o r e m s 4.1 and 4 , 2 a r e c o r r e c t i n d e g r e e z e r o by com-

parison of (2) with [G, Proposition 8.141

We need s o m e p r e l i m i n a r i e s in o r d e r to prove Theorem 4 2

f o r rion-connected spaces The following well-known l e m m a c l e a r l y

I implies that T h e o r e m 4.2 will hold provided that i t c o r r e c t l y d e s c r i b e s

the homology of the component X of the base-point of QX

9

L e m m a 4.6 L e t X b e a homotopy associative H-space s u c h

that s X i s a group under the induced product Choose a point a e [a]

0

f o r each component [a] of X, w r i t e a f o r the chosen point i n [a] ,

and l e t X denote the component of the identity element Define

$

I f: X + X$ X sOX by f(x) = (x- a , [a]) if x F [a] Then f i s a homo-

topy equivalence with homotopy i n v e r s e g given by g(y, [a]) = ya If

left translation by any given element of X i s homotopic t o right t r a n s l a -

tion by the s a m e element, then f and g a r e H-maps

To study Q X, which i s the component $2 QXX of the t r i v i a l

I loop in QZX, observe that we m a y a s s u m e , without l o s s of generality,

that a l l connected s p a c e s Y in sight a r e sufficiently well-behaved

locally t o have universal c o v e r s n: UY -+ Y Of c o u r s e , S2s:QUY

* "gY

I !

i s then a weak homotopy equivalence We r e q u i r e two simple l e m m a s

I that X i s connected and IT X i s a f r e e Abelian group Then t h e r e

copy of s1 f o r e a c h g e n e r a t o r of s X, r e s t r i c t e d in the s e n s e that a l l

1 but finitely many coordinates of each point a r e at a chosen base-point i n

product on X f r o m any chosen representatives S' -t X f o r the g e n e r a t o r s

of alX 0 f c o u r s e , if X is a monoid, we can u s e the product directly

r a t h e r than inductively

L e m m a 4.8 L e t (X, 0) be a connected c - s p a c e , w h e r e

& i s any operad Then UX admits a s t r u c t u r e of & - s p a c e s u c h

that n : U X + X i s a m a p o f & - s p a c e s

Proof UX = PX/(N ), w h e r e two paths i n X which s t a r t

at + a r e equivalent if they end a t the s a m e point and a r e homotopic

with end-points fixed, and s i s induced by the end-point projection

It i s t r i v i a l t o verify that the pointwise k -space s t r u c t u r e on P X of

[G, L e m m a 1,5] p a s s e s to t h e quotient s p a c e UX

As a final p r e l i m i n a r y , we have t h e following observation con-

cerning the homology suspension

L e m m a 4.9 Let X b e a space Let x E HOQX and y E H+QX

Then, if & y = 0, the loop product x + y suspends to (EX)(U+YJ

Proof Let a and b be representative cycles i n C+QX f o r

x and y L e t i:QX + PX and s: PX * X b e the inclusion and end-

point projection u*y i s the homology c l a s s of n+c, w h e r e c eC+PX

is a chain such that i+b = d c QX a c t s on the left of PX by composi-

tion of paths, and s(f + g ) = s g f o r a loop f and path g Now

d(i+a * c) = i,(a ., + b) and n,(i,a , 3 + c) = (E a)(s*c) The r e s u l t follows

Proof of T h e o r e m 4.2 If X i s (q-1)-connected, q > 1,

then q+:H+X + H+QX i s an i s o m o r p h i s m i n d e g r e e s l e s s than 2q ,

i

a s c a n easily be verified by inductive calculation of H$ C ~ X f o r i( n

in low d e g r e e s (by use of the S e r r e s p e c t r a l sequence) Indeed, this

i s just the standard proof that q n+X n+QX = ., i s a n i s o m o r p h i s m

i n d e g r e e s 'less than 2 q - 1 and an epimorphism in d e g r e e 2 q - 1 Thus

the t h e o r e m i s t r i v i a l l y t r u e in d e g r e e s l e s s than 2 9 if X i s (q-1)-

connected We c l a i m that if the t h e o r e m is t r u e f o r ZX i n d e g r e e s

l e s s than n, then t h e t h e o r e m i s t r u e f o r X in d e g r e e s l e s s than n-1

This will complete the proof since i t will follow that the t h e o r e m f o r

zqx i n d e g r e e s l e s s than 2q implies the t h e o r e m f o r X i n d e g r e e s

l e s s than q, f o r a l l i n t e g e r s q > 1, We s h a l l prove our c l a i m by con-

structing a model s p e c t r a l sequence {'Er}, mapping it into the S e r r e

s p e c t r a l sequence { E r ) of the path s p a c e fibration o v e r UQZX, and

invoking the c o m p a r i s o n t h e o r e m [7 , X I 11.11 By [G, Proposition 8-14]

( z + x l ) = Q Z*xl l i e in H+UQXX Define WEH,ZX t o be the sub-

a l g e b r a of WEH*ZX generated by the e l e m e n t s of TZX of d e g r e e

1

g r e a t e r than one and, i f p = 2, the s q u a r e s Q E*xl, x E t X Define

0

2 (1f X i s connected, 'E reduces to WEH,ZX@WEH,X ) The

( ( Q X )@ ~Q x a x ] ) = (-1) d(1) 3-1 /3Q s Qx*Ep I P(I)+l ax]

H e r e a x E n X denotes the component i n which t h e homology c l a s s

0

x l i e s ( q x = x @ a x t a x @ x plus o t h e r t e r m s if deg x > 0) and, f o r

a E n X and n E 2, [na] denotes the n-th power of a i n the group

p{z}/(zP) @[E{Tz} @ P { T ( Z P - ~ @ - r z ) H , where E and P denote e x t e r i o r

and polynomial algebras H e r e y r u n s through

{QZ*x1 I I admissible, e(1) > d e g x , deg Q Z , x l > 1 and odd if p > 21

and, i f p > 2, z runs through

I

{ Q% l ' I I admissible, e(1) > d e g x, deg Q E l 1 even)

(Note that, if p > 2, e(1) d(1) mod 2, hence e(1) = deg x S 1 implies

I

t h a t deg Q Z*xl i s even ) Of c o u r s e , to the eyes of {'Er}, the b a s e

N

WEH*ZX looks like a t e n s o r product of e x t e r i o r and truncated poly-

nomial algebras r a t h e r than like a f r e e commutative algebra C l e a r l y

'Em = Z By construction, t h e r e i s a unique m o r p h i s m of a l g e b r a s

P'

f: lE2 _* E 2 such that the following d i a g r a m i s commutative.'

I f

Since Q x = Q x ' i f d(1) > 0, by T h e o r e m 1 1 (5), L e m m a 4.9 implies

that, %for u,: H,QX + H,QZX,

(the sign c o m e s f r o m up = -pu ) By the naturality of u*, the s a m e

f o r m u l a holds for u*: H*SZUQZX -+ H,UQZX, although h e r e the e l e -

m e n t s QSZ,x', x E t X, a r e of c o u r s e not operations because the ele-

m e n t s Z,x' T a r e not p r e s e n t in H*UQZX By T h e o r e m 1.1 (7) and the definition of { ' E r ) , f induces a m o r p h i s m of s p e c t r a l sequences

Since f = f(base) 8 f(fibre), o u r c l a i m and t h e t h e o r e m now follow

d i r e c t l y f r o m the comparison theorem

The following observation on the s t r u c t u r e of HQQX i s some-

t i m e s useful Note that H*Q X i s the f r e e commutative a l g e b r a

-1 a'

generated by i y * (ay) I y c TX) , where a y i s the component

i n which y l i e s This description u s e s operations which o c c u r i n

various components of QX We c a n instead use just those operations

which actually o c c u r i n the component Q X

{x*[-axI I x e t X , deg x > 0 , and a x # ff )

{aJ(# QSx * [-p ax]) I QJpc QSx E TX and a x # f f )

Proof [-P- ax] = [-ax] * *[-ax], and we t h e r e f o r e have

J 5 s l ( J ) t 1

aJ({ QSx 4 ax]) = ( Q p Q X) * [-P ax] ,

modulo decomposable elements of H,Q X, by the C a r t a n formula

When X = S , the f i r s t two s e t s above a r e c l e a r l y empty

We complete t h i s section by computing the Bockstein spectral

sequences of H*CX and of H,QX L e t {ErX} denote the mod p

Bockstein s p e c t r a l sequence of a s p a c e X A slight v a r i a n t (when p > 2

and r = 2) of the proof of [A, Proposition 6.81 .yields the following

lemma

Lemrna 4.11 If (X, 8) i s a -space, w h e r e i s a n E

CKJ

operad, then {ErX) i s a s p e c t r a l sequence of differential a l g e b r a s

s u c h that i f Y E Er-'X, then pryP = yp-lBr-ly if p > 2 o r if p = 2

2q

2 and r > 2, and f3 y = ypy + Q~~~~ if p = 2 and r = 2

2

L e t Y = CX o r Y = QX, and l e t { E ~ A T X } denote the

r e s t r i c t i o n of {Ery} t o ATX; in both c a s e s , we c l e a r l y have

r 2 2, t o be the f r e e s t r i c t l y commutative algebra generated by the

following s e t ( s t r i c t n e s s r e q u i r e s the s q u a r e s of odd degree elements

to be zero):

p r - j

w h e r e S = {y I y eDj, deg y even} and

j r

The proof of the following t h e o r e m i s p r e c i s e l y analogous t o the

computation of the cohomology Bockstein s p e c t r a l sequence of K(Z t, n)

P given i n [A, T h e o r e m 10.41 and will t h e r e f o r e be omitted It depends

2s Q2s-l only on L e m m a 4.11, the fact that f3Q = if p = 2, and counting

arguments

Theorem 4.13 Define a subset SX of TX a s follows:

I ( a ) p = 2 : SX = {Q1x 1 I = (25, J), deg Q x i s even, 1 (I) > 0)

-

$ 5 The homology of CX and the s p a c e s CX

We f i r s t prove T h e o r e m 4.1 and then construct a homology approxi-

Observe that the maps T* of T h e o r e m 4.1 a r e n a t u r a l in k a s well

a s i n X In p a r t i c u l a r , the following r e s u l t holds

L e m m a 5.1 If and &' a r e E operads , then the following

o r d e r t o prove that Tis i s a n epimorphism, we need the following stan-

d a r d consequence of the p r o p e r t i e s of the t r a n s f e r i n t h e (mod p)

homology of finite groups; a proof m a y be found i n [ 5 , p 2551

L e m m a 5.2 If n i s a subgroup of the finite group II and if

the index of a in II is prime to p, then t h e r e s t r i c t i o n

i s a n epimorphism f o r e v e r y Z II-module M

P

We s h a l l a l s o need t h e definition of w r e a t h products

~ e f i n i t i o n 5.3 L e t a be a subgroup of En and l e t G b e any

monoid Then the w r e a t h product n J G i s the s e m i - d i r e c t product of

n

z n t l IG, and Z IG i s defined t o be t h e union of the Z J G f o r finite n

Proof of T h e o r e m 4.1 Consider the monad (C,p, q) a s s o c i a t e d t o

a n E o p e r a d & As i n [G, p 171, we w r i t e p both f o r the f-action

43

on CX and f o r the monad product p: CCX + CX Recall that, by [G, p 13

and 141, CX i s a f i l t e r e d s p a c e such that the product :: and &-action p

s t r u c t u r a l m a p of t h e operad [G, Definition 1.11, then

p,(d; [el, ~ ~ 1 , , [ekt ykl) = [ ~ ( d ; el' • a $ ek); Y ~ , ' a • s ~ k ]

j

f o r d E c ( k ) , ei E & (ji), and y E X We define a corresponding alge-

b r a i c f i l t r a t i o n of WEH*X by giving a l l elements of the i m a g e of

~ [ k ] @ JH*X i n WEH? filtration p r e c i s e l y pk and by r e q u i r i n g WEH*X

t o be a f i l t e r e d algebra Then FOWEH$ i s spanned by fl, F 1 WEH,X - =

H,X, -, and each F k WEH*X i s a sub A-coalgebra of WEH*X Visibly,

t h e r e s t r i c t i o n of y*: WEH,X - + H,CX t o FkWEH_X % f a c t o r s through

H,F CX H*FICX = H*X s i n c e F CX = $ (1) X X and $(1) i s con-

Consider the following commutative d i a g r a m with exact rows and columns:

H e r e L: F CX * F CX i s the inclusion, which i s a cofibration by

k - I k

[ G , Proposition 2.61, and IT i s the quotient map The maps ?, a r e

-8.

known to be monomorphisms and the left m a p 7, , is a s s u m e d t o be a n

epimorphism It follows that L * i s a monomorphism, hence that

a = 0 and n, i s a n epimorphism Define A b y commutativity of the

9-

right-hand square; then A i s a monomorphism by the five l e m m a If

we c a n prove that X i s a n epimorphism, i t will follow that the middle

a r r o w q, is an isomorphism, a s required By [G, p 141, EkCX is

t h e equivariant half-smash product

w h e r e Xfkl denotes t h e k-fold s m a s h product of X with itself By

[A, L e m m a I l ( i i i ) and R e m a r k s 7.21, t h e r e i s a composite chain homo-

topy equivalence

c* C ( k ) @ (H*xlk + C*( C ( k ) x ~ 7 ,

k hence we m a y identify ~ * ( & ( k ) X X ) with H*(%; ( H * x ) ~ Let

of CX Then, since q(x) = [ l , x ] , w h e r e 1 E If(*) is the identity e l e -

ment and since p oCq = 1 on CX, the following d i a g r a m i s commutative:

Of c o u r s e , r' induces a n epimorphism on homology If k < p, then

0 H,E - k CX = HA(* - X = k xLkl) and X i s a n epimorphism since, by the

0

d i a g r a m , H*E CX i s spanned by i m a g e s under n* of k-fold products

k

P

L e t k = p Since i,: H&n; (H*X) ) * H*(Bp: (~$1:) i s a n epimorphism,

w h e r e n i s cyclic of o r d e r p H*(Zp; (HZ)') i s spanned by images

under i* of elements of the f o r m s e @ x @ P

0 1 @ x p and e i @ x , by

0 [A, L e m m a 1.31 By t h e d i a g r a m , H*E CX i s t h e r e f o r e spanned by

The index of Z Z i n Z k i s p r i m e t o p since

F.WEH*F CX i s a n epimorphism since o u r induction hypothesis c a n be

applied t o any space, and i n p a r t i c u l a r to F CX Since n,va = n1 i s

a l s o a n epimorphism, i t follows f r o m the d i a g r a m that X i s a n epimorph-

i s m Fidally, suppose that k i s p r i m e to p L e t p(l): c ( k - l ) + &(k)

b e t h e Zk-l-equivariant m a p defined by p(l)(d) = d a l = y (c; d, 1) and

considel the following commutative diagram:

( p ( i ) X i ) + m a y be identified with the r e s t r i c t i o n i, and i s t h e r e f o r e an

T

epimorphism, i s a n epimorphism by the induction hypothesis,

n,v, i s a n epimorphism, and t h e r e f o r e X i s a n epimorphism The proof

I

a l g e b r a ATX Of c o u r s e , v i a Q x -t Q'X a [-pe(l) ax] on g e n e r a t o r s ,

ATX i s isomorphic a s a n a l g e b r a t o HaQ X

P,

Henceforward i n this section, we r e s t r i c t attention to t h e full sub-

category ?/ of f7 which c o n s i s t s of s p a c e s of the basedhomotopy type of CW-complexes By [GI, C o r o l l a r y A.31, CX 6 ?/ if X 6 ?/ and g i s a suitably nice operad ( a s we tacitly a s s u m e below)

Construction 5.4 L e t be an o p e r a d and l e t X ~ 1 / Construct a

-

s p a c e CX a s follows Choose a point a i n each component [a] of X

i Choose a point c e (i) f o r i 2 1 , with c = 1 , and l e t i a = [ci; a ] 6 CX

1 (Thus, b y abuse, a i s identified with ?(a) = [ l ; a] ) L e t (CX) denote t h e

i a component of CX i n which i a l i e s Define p (a): CX + CX to b e r i g h t

t r a n s l a t i o n b y a , p(a)(x) = x * a Define (CX), to b e t h e t e l e s c o p e of the

sequence of m a p s

*

Define CX t o b e t h e r e s t r i c t e d C a r t e s i a n product ( a l l but finitely many

coordinates of each point a r e * ) of t h e s p a c e s (m) f o r [a] e sox The

a homotopy type of (CX) i s independent of t h e choice of a E [a], and i s

a

t h e object function of a f u n c t o r f r o m t h e homotopy c a t e g o r y of t o i t s e l f

R e m a r k s 5 5 (i) CX i s a functor of Gf a s w e l l a s of X

(ii) If X i s connected, then ??x i s homotopy equivalent t o CX

(iii) If [a] @, then (CX)ia = c(i) X xi , w h e r e X = [a], and

P ( ~ ) : ( C X ) ~ ~ + (CX)(i+l)a i s given by the f o r m u l a

L e m m a 5.6 L e t be a n E operad Then H , ~ X i s naturally

a>

isomorphic t o the connected a l g e b r a HsCX B o c x Z p

Proof Since ( x s a ) +(y + a ) = ( x s y) s a s a f o r x, y e HsCX, e a c h

H * ( ~ x ) ~ , hence a l s o H*Ex, i s a well-defined algebra The r e s u l t i s

obvious f r o m T h e o r e m 4 i and the construction

L e m m a 5.7 L e t 6 b e a n E operad L e t G b e any d i s c r e t e

a

group a n d l e t X = K(G, i ) ', the union of a K(G, i ) and a disjoint b a s e -

- 0 point Then ZX i s a K ( Z ; ~ ~ G , I) In p a r t i c u l a r , with G = { e ) , CS

Proof &(i) XC K ( G , ~ ) ~ is c l e a r l y a K ( z ~ - ~ G , I ) , and EX i s

a, a,

Construction 5.8 Fix n, 1 5 n o r n = m (when D 2 = Q) _

Withthe s a m e notations a s i n Construction 5.4, l e t i a a l s o denote the

image of i a under a;l:CnX + Dnz% and l e t nf C% denote t h e c o m -

l a ponent of DnZ:% i n which i a l i e s Define p(a): Dnz?X + Dnznx by

-n n p(a)(x) = x s a and l e t D Z: X denote the t e l e s c o p e of t h e sequence of

a inclusions

The inclusion of ClnCnx i n xnznx i s a homotopy equivalence (since

e a c h p(a) i s ) ; choose a n i n v e r s e homotopy equivalence

:X"anx + Dnz% Observe that nn p(a) = p(a) an and l e t

l i m i t s E i t h e r by (possibly transfinite) induction and u s e of the o r d i n a r y

loop product o r b y d i r e c t construction i n t e r m s of t h e monoid s t r u c t u r e

L e m m a 5.9 L e t I, : (CnX)ia -* Tnx b e t h e inclusion Then, f o r

X E H,(C _ X) ia r m s L ,(XI = a n s (x) * [-ia] , w h e r e [-ia] i s t h e component

i n v e r s e to [ia] i n t h e group s 0 a%%

Proof The r e s t r i c t i o n of Ba t o DYaz% is homotopic to right

t r a n s l a t i o n b y any chosen point of [-ia]

Our approximation t h e o r e m is now a n immediate consequence of

T h e o r e m s 4.1 and 4.2 and L e m m a s 5.6 and 5.9

m Abelian fundamental group C l e a r l y .ir a .rr -d X

1 co 1 co -.rrlQBX i s the Abelian- ization homomorphism

As explained i n [G', 5 21, T h e o r e m s 4 1 , 4 2 ; and 4.13 imply that

a : CoDX + QX i s a group completion i n the s e n s e of [GI, Definition 1 31

co

T h e o r e m 5.10 i s a reflection of t h i s fact (compare [GI, Proposition 3.91)

This fact a l s o suggests that any n a t u r a l group completion of CX f o r any

E operad $ should yield a homotopy approximation t o QX GCZX i s

03

one example [G, C o r o l l a r y 4.61, and GBDX i s another (but the second r e -

sult labeled T h e o r e m 3.7 i n [GI], about t h e monoid s t r u c t u r e on DX, i s

i n c o r r e c t ; s e e [R, VII 2.71) Yet another example i s B CX, the infinite

0 loop s p a c e obtained by application of the recognition principle to the E

m

s p a c e CX [GI, T h e o r e m 2 3 (vii)] As explained i n [R, VII $41, t h i s l a s t

construction often admits a multiplicative elaboration and yields the m o s t

s t r u c t u r e d v e r s i o n of the B a r r a t t - Quillen t h e o r e m to the effect that QS 0

i s equivalent t o t h e group completion of .h K(Z 1)

f o r all s p a c e s X by proving the analogs of T h e o r e m s 4 1 , 4 2 , and 4.13

Thus his calculations will imply the following unstable analog of T h e o r e m 5.10

n

T h e o r e m 5.11 F o r X r?/ , k : H*Z X - H G Z% i s a n i s o -

n " B

m o r p h i s m of a l g e b r a s

5 6 A r e m a r k on Postnikov s y s t e m s

Infinite loop s p a c e s c a n be approximated by stable Po'stnikov t o w e r s ,

and i t i s n a t u r a l t o a s k whether t h e r e i s a relationship between t h e homology

operations and the Postnikov decomposition of s u c h a space We p r e s e n t

such a r e s u l t h e r e , and we begin with t h e following e a s y (and well-known)

l e m m a

L e m m a 6.1 L e t X b e a K(T, n) f o r s o m e Abelian group IT and

integer n 2 1 Then the Nishida relations and diagonal C a r t a n f o r m u l a

imply t h a t the operations (no m a t t e r how constructed geometrically)

a r e all t r i v i a l on %*x

P r o o f Assume the c o n t r a r y and l e t i b e m i n i m a l such that

a r x # 0 f o r s o m e x E E.x and s o m e r a n d l e t r ' b e minimal s u c h

that Q ~ X # 0 Then a r x i s primitive and i s annihilated by all Steenrod

r operations P* It follows that Qrx = 0, which i s a contradiction

It should be emphasized that t h i s r e s u l t f a i l s far products of

K T n ) The loop operations o n s u c h a product a r e c e r t a i n l y t r i v i a l ,

but such a product c a n a l s o be the zeroth s p a c e of a s p e c t r u m the higher

t e r m s of which have non-trivial k-invariants If we wish t o analyze

infinite loop s p a c e s in t e r m s of Postnikov s y s t e m s , then we m u s t u s e t h e

Postnikov s y s t e m s of a l l of the de-loopings (or p a s s t o s p e c t r a )

T h e l e m m a a d m i t s the following generalization, which m a y a l s o be

Proof A 1-stage s t a b l e Postnikov s y s t e m is a product of K(s, n ) ' s

.rr Abelian and n 2 1 Inductively, a k - s t a g e s t a b l e Postnikov s y s t e m X

is the pullback f r o m t h e path s p a c e fibration o v e r a simply connected

1-stage Postnikov s y s t e m Z of a n infinite loop m a p f:Y * Z, where Y

i s a (k-1)-stage stable Postnikov s y s t e m The n a t u r a l m a p IT:X -2 Y

and the inclusion i: Q Z - X of the f i b r e of a a r e infinite loop m a p s

D i r e c t calculation by Hopf a l g e b r a techniques (see [15, T h e o r e m 6.11)

'demonstrates that E = E i n the Eilenberg-Moore s p e c t r a l sequence

sequence If x e H*X, then sr*Q x = 0 f o r P ( J ) 2 k-1 b y induction

Thus consider QSx where n x * = 0, s a y x = X.Z with 1 1

$7 The analogs of the Pontryagin pth powers

A S was f i r s t exploited by Madsen [ 9 ] (at the p r i m e 2), the homology

of E m s p a c e s c a r r i e s analogs of the Pontryagin pth powers defined and

analyzed by Thomas [25] o n the cohomology of s p a c e s These operations

a r e often useful i n the study of torsion Indeed, i n favorable c a s e s , they

s e r v e t o replace the fuzzy description of t o r s i o n c l a s s e s i n Z(P)

derived by use of t k Bockstein s p e c t r a l sequence by p r e c i s e information

i n t e r m s of p r i m a r y homology o.perations, with no indeterminacy

We r e v e r t t o the g e n e r a l context of $1, except that t h e coefficient

groups i n homology will v a r y , and we f i r s t l i s t the v a r i o u s Bockstein

operations that will appear in t h i s section in the following diagram:

In each row, the notation a t the left specifies the homology Bockstein

derived f r o m the s h o r t exact sequence a t the right All of the homo-

m o r p h i s m s labelled T a r e n a t u r a l quotient m a p s Z = l&m Z and

d e t e r m i n e s a l l of t h e remaining Bocksteins l i s t e d , i t should be thought of

a s the universal Bockstein operation C l e a r l y f3, = = 8 - and

r - 1 (P )*Br = , T : Z + z p , if r > 1 At l e a s t if H*(X; Z ) i s

r.l' survives to d J (x) r E r x ; alternatively, d;r(x) = j r z r ( x )

T h e o r e m 7 1 L e t k be a n E operad and l e t (X, 0) be a & -space

f o r a l l q 2 0 and r ) 1 which s a t i s f y t h e following properties:

(1) The 2 a r e n a t u r a l with r e s p e c t t o maps of -spaces

(4) The following composites both coincide with pth power operations:

H e r e s and t a r e not a s s u m e d to be even; when p = 2 , r > 1 , and s and t

a r e even, the e r r o r t e r m vanishes (since aZq-' = j3aZq and 2 : ~ = 0)

(8) u * 2 x = o if p > 2 and u , b x = (~~),[('lj,*x)(.-,x)] if p = 2,

w

where u*: H*(QX; ? ) -* H*(X; ? ) i s the homology suspension

Proof P r e c i s e l y a s in t h e proof of Theor e m 1.1, except that W

i s h e r e taken a s the s t a n d a r d w-free resolution of Z and C* is taken to

m e a n chains with integer coefficients, 8 induces 8*: WC3) (c*x)' C*X

'In the language of [A,Definition 2.11, (C*X, 8*) i s a unital C a r t a n object

of the category (l= (w,m, Z) The r e s t i s e l e m e n t a r y chain l e v e l algebra, the details of which a r e t h e s a m e f o r the present homology operations a s

f o r the cohomology Pontryagin pth powers Given x E H (X; Z ), l e t x

H e r e e o @ aP f prMe 1 @Jap-lb i s a c y c l e modulo p by explicit

computation (see [25, p 321) T h e salient f a c t s a r e that TM = p - N = MT,

P-1 i where T = a-1 and N = L) a , that dap = prNap-'b if p > 2, and that

of proof of [21,3.1] o r [A, L e m m a 1.11 When p > 2, t h e fact that a, x = 0

if q i s odd depends o n t h e factorization of 8* through c* & (p) 43) (c+x)';

s e e [24, g9-101 f o r details When p = 2, s e e [25, p 421 f o r the verification

of (2) P a r t s (3) and (4) a r e t r i v i a l t o verify, and p a r t (8) i s s t r a i g h t -

f o r w a r d S e e [25, § 91 f o r the verification of (6) and [25, $81 (or [9]

when p = 2) f o r t h e verification of (7)

We have 'chosen the notation 2 since i t goes well with Q' and

s i n c e t h e Pontryagin pth powers a r e often denoted b y @

g e n e r a t e d by pax I n view of (4) a n d (5) of t h e t h e o r e m , t h i s will b e

t h e c a s e if H<.,(X; Z ) i s of finite type o v e r Z and a l l non-trivial

differentials d l , r 1 2, i n t h e mod p Bockstein s p e c t r a l sequence of X

a r e d e t e r m i n e d b y the g e n e r a l f o r m u l a s f o r differentials on pth powers

R e m a r k s 7 3 T h e operations 2 c a n a l r e a d y b e defined o n t h e homology

of 6 , - s p a c e s , w h e r e c2 i s t h e l i t t l e 2-cubes operad, and thus on t h e

homology of second loop s p a c e s All of the p r o p e r t i e s l i s t e d i n T h e o r e m

7 1 a r e v a l i d f o r - s p a c e s , and m o s t of the proper.ties a r e v a l i d f o r

3

C 2 - s p a c e s The exceptions, (2) and (b), a r e t h o s e p r o p e r t i e s t h e proof

of which r e q u i r e s u s e of t h e e l e m e n t e 2 e W , and they have m o r e compli-

c a t e d v e r s i o n s with e r r o r t e r m s which involve the two v a r i a b l e operation

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18 M.Nakaoka Note on cohomology a l g e b r a s of s y m m e t r i c groups

J Math Osaka City Univ 13 (1962), 45-55

19 G Nishida Cohomology operations i n i t e r a t e d loop s p a c e s P r o c

J a p a n Acad 44 (1 968), 104-109

m m

20 S Priddy On S2 S and t h e infinite s y m m e t r i c group P r o c

Symp P u r e Math Vol 22, 217-220 A m e r Math Soc 1971

21 N E Steenrod Cohomology operations d e r i v e d f r o m t h e s y m m e t r i c

group Comment Math Helv 31 (1957), 195-218

22 N E Steenrod The cohomology a l g e b r a of a space

L I E n s e i g n e m e n t ~ a t h & m a t i ~ u e -7 (1961), 153-178

23 N E S t e e n r o d a n d D B A Epstein Cohomology Operations

P r i n c e t o n University P r e s s 1962

24 N E Steenrod and E Thomas Cohomology operations d e r i v e d

The Homolopy of E m Ring Spaces

J P May

+

T h e s p a c e s Q(X ) f o r a n E s p a c e X, t h e zeroth s p a c e s of t h e

w

v a r i o u s Thom s p e c t r a MG, t h e classifying s p a c e s of bipermutative

c a t e g o r i e s and tlie zeroth s p a c e s of t h e i r a s s o c i a t e d s p e c t r a a r e all

examples of E ring s p a c e s The l a s t example includes models f o r

m

BO X Z and BU X Z a s E m r i n g spaces A complete g e o m e t r i c t h e o r y

of such s p a c e s and of t h e i r relationship t o E ring s p e c t r a i s given i n

w [R], along with the examples above (among o t h e r s ) and v a r i o u s applications

On t h e l e v e l of (mod p) homology, the important fact is t h a t a l l of t h e

f o r m u l a s developed by Milgram, Madsen, Tsuchiya, and myself f o r the

I f o r m u l a f o r t h e evaluation of the multiplicative homology operations "QS

25 E Thomas The generalized Pontryagin cohomology operations B

o n elements x * y A p a r t i a l r e s u l t and the b a s i c g e o m e t r i c i d e a a r e due and r i n g s with divided powers M e m o i r s A m e r Math Soc

No 27 (1957) to Tsuchiya [36], but t h e complete f o r m u l a is due to Madsen [15] when