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Preface

T h i s i s t h e f i r s t of a s e r i e s of p a p e r s devoted t o t h e s t u d y , o f i t e r a t e d loop spaces.

Our g o a l i s t o develop a simple and

c o h e r e n t t h e o r y which encompasses most o f t h e known r e s u l t s about such s p a c e s .

We b e g i n w i t h some h i s t o r y and a d e s c r i p t i o n o f t h e

d e s i d e r a t a of such a t h e o r y . F i r s t of a l l , we r e q u i r e a r e c o g n i t i o n p r i n c i p l e f o r n - f o l d loop spaces.

That i s , we wish t o s p e c i f y a p p r o p r i a t e i n t e r n a l

s t r u c t u r e such t h a t a s p a c e only i f space.

p o s s e s s e s such s t r u c t u r e i f and

i s of t h e (weak) homotopy t y p e of an n - f o l d l o o p

For t h e c a s e n= 1, S t a s h e f f l s n o t i o n [281 of an A,

i s such a r e c o g n i t i o n p r i n c i p l e .

space

Beck [5] h a s g i v e n an e l e g a n t

proof of a r e c o g n i t i o n p r i n c i p l e , b u t , i n p r a c t i c e , h i s recogni-

t i o n p r i n c i p l e a p p e a r s t o be u n v e r i f i a b l e f o r a s p a c e t h a t i s n o t g i v e n a p r i o r i ' a s an n - f o l d l o o p s p a c e .

I n t h e case n =

v e r y convenient r e c o g n i t i o n p r i n c i p l e i s g i v e n by Boardman and V o g t l s n o t i o n [81 of a homotopy everykhing s p a c e , and, i n [71, Boardman h a s s t a t e d a s i m i l a r r e c o g n i t i o n p r i n c i p l e f o r n < o o

We s h a l l prove a r e c o g n i t i o n p r i n c i p l e f o r n e w i n s e c t i o n 1 3 (it w i l l f i r s t be s t a t e d i n s e c t i o n I) and f o r n

i n section

1 4 ; t h e l a t t e r r e s u l t a g r e e s (up t o language) w i t h t h a t of

AMS Subject Classifications (1970): 55D 35, 55D 40, 55 F 35, 55 B 20

ISBN 3-540-05904-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-05904-0 Springer-Verlag New York . Heidelberg . Berlin 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 means, and storage in data banks. Under tj 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 be determined by agreement with the publisher. @ by Springer-Verlag Berlin . Heidelberg 1972. Library of Congress Catalog Card Number 72-85030. Printed in Germany. Offsetdrudc: Julius Beltz, Hemsbach

Boardman and Vogt, but our proof is completely different.

In the case n

1, this was first obtained by James [151. For nt-,

generalizing the methods of Beck, we are able to obtain immediate

Milgram [22] obtained an ingenious, but quite intricate, approxi-

non-iterative constructions of classifying spaces of all orders.

mation for connected CW-complexes.

Our proof also yields very precise consistency and naturality

'approximation was first obtained by Dyer and Lashof [unpublished]

statements.

and later by Barratt [4], Quillen ~unpublished], and Segal [27]

In particular, a connected space X

which satisfies

homotopy equivalent to an infinite loop space B o XI where spaces +

X are explicitly constructed, but also

the given internal structure on

agrees under this equivalence

with the internal structure on B,x the spaces Bi X.

We shall obtain simple functorial approximations to

our recognition principle (say for n =oo)-is not only weakly

Bi X with Bi X = S L B i

In the case n =oo, such an

derived from the existence of

We shall have various other consistency state-

for all n and all connected X is in section 2).

- n n

in section 6 (a first statement

Our result shows that the homotopy type of fin$")(

is built up from the iterated smash products

xLi'

with

itself and the classical configuration spaces F ( R ~ ; j) of j-tuples of distinct points of R".

Moreover, in our theory the approxima-

ments and our subsequent papers will show that these statements

tion theorem, together with quite easy categorical constructions

help to make the recognition principle not merely a statement as

and some technical results concerning geometric realization of

to the existence of certain cohomology theories, but, far more

simplicia1 topological spaces, will directly imply the recognition

important, an extremely effective tool foy the calculation of the

principle.

homology of the representing spaces.

are the free n-fold and infinite loop spaces generated by

An alternative recognition principle in the case n due to Segal [27] and Anderson [l, 21.

= oo

Their approach starts with

an appropriate topological category, rather than with internal structure on a space, and appears neither to generalize to the recognition of n-fold loop spaces, 1 < n <

M ,

nor to yield the

construction of homology operations, which are essential to the

The second desideratum for a theory of iterated loop spaces

anSn)( and RwS")(

= lim QnsnX e

finsn)(

and

RmSMX

and

should play a central role in any complete theory of iterated loop spaces. The third, and pragmatically most important, requirement of a satisfactory theory of iterated loop spaces is that it lead to a simple development of homology operations.

The third paper in

this series will study such operations on n-fold loop spaces, n < m, and will contain descriptions of H,(Q"s"x)

most important presently known applications.

is a useable geometric approximation to

This is in fact not surprising since

functors of H, (X).

for all n as

The second paper in the series will study

homology operations on Em spaces and infinite loop spaces and will apply the present theory to the study of such spaces as F, F/O, BF, BTop, etc.

It will be seen there that the precise

VII geometry t h a t a l l o w s t h e r e c o g n i t i o n p r i n c i p l e t o b e a p p l i e d t o

proven i n '$13 and 114 and a r e d i s c u s s e d i n g15.

t h e s e spaces i s n o t o n l y w e l l adapted t o t h e c o n s t r u c t i o n of

s t a n d i n g of t h e s e r e s u l t s can be o b t a i n e d by r e a d i n g $1-3 and $ 9

homology o p e r a t i o n s b u t can a c t u a l l y be used f o r t h e i r e x p l i c i t

and t h e n f 1 3 , r e f e r r i n g back t o t h e remaining s e c t i o n s f o r t h e

evaluation.

geometry a s needed.

Statementsof some of t h e r e s u l t s o f t h e s e p a p e r s may

The r e s u l t s of $10 and $11w i l l be used i n [21] t o s i m p l i f y and

b e ' f o u n d i n [20]. Our b a s i c d e f i n i t i o n a l framework i s developed i n s e c t i o n s 1, 2, and 3.

The n o t i o n of "operad" d e f i n e d i n

1 a r o s e simulta-

neously i n Max K e l l y ' s c a t e g o r i c a l work on coherence, and convers a t i o n s w i t h him l e d t o t h e p r e s e n t d e f i n i t i o n .

S e c t i o n s 4 through

8 a r e concerned w i t h t h e geometry of i t e r a t e d l o o p s p a c e s and w i t h t h e a p ~ r o x i m a t i o ntheorem.

The d e f i n i t i o n of t h e l i t t l e cubes o p a d s

i n 5 4 and o f t h e i r a c t i o n s on i t e r a t e d l o o p s p a c e s i n $ 5 a r e due t o Boardman and Vogt [ 8 ] .

The r e s u l t s of 9 4 and $5 i n c l u d e a l l of

t h e geometry r e q u i r e d f o r t h e c o n s t r u c t i o n of homology o p e r a t i o n s and f o r t h e p r o o f s of t h e i r p r o p e r t i e s ( C a r t a n formula, s t a b i l i t y , Adem r e l a t i o n s , e t c . )

The o b s e r v a t i o n s of $8, which s i m p l i f y and

g e n e r a l i z e r e s u l t s o r i g i n a l l y proven by Milgram [23] , Tsuchiya [33] , and myself w i t h i n t h e g e o m e t r i c a l framework developed by Dyer and Lashof

[ l l l , i n c l u d e a l l of t h e geometry r e q u i r e d f o r t h e computa-

t i o n of t h e P o n t r y a g i n r i n g of t h e monoid v a l e n c e s of s p h e r e s .

F of based homotopy equi-

Our key c a t e g o r i c a l c o n s t r u c t i o n i s p r e s e n t e d

i n 5'9, and f a m i l i a r s p e c i a l c a s e s of t h i s c o n s t r u c t i o n a r e d i s c u s s e d i n 310.

A c o n c e p t u a l under-

T h i s c o n s t r u c t i o n l e a d s t o s i m p l i c i a 1 s p a c e s , and a v a r i e t y

of t e c h n i c a l r e s u l t s on t h e g e o m e t r i c r e a l i z a t i o n of s i m p l i c i a 1 s p a c e s a r e proven i n $11 and

312.

The r e c o g n i t i o n theorems a r e

g e n e r a l i z e t h e t h e o r i e s o f c l a s s i f y i n g s p a c e s o f monoids and of c l a s s i f i c a t i o n theorems f o r v a r i o u s t y p e s of f i b r a t i o n s . I t i s a p l e a s u r e t o acknowledge my d e b t t o Saunders Mac Lane .

and J i m S t a s h e f f , who r e a d p r e l i m i n a r y v e r s i o n s o f t h i s p a p e r and made v e r y many h e l p f u l s u g g e s t i o n s .

Conversations w i t h Mike

Boardman and J i m Milgram have a l s o been i n v a l u a b l e .

Contents

Operads and %.spaces

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

. . . . . . . . . . . . . . . . . . . 10 A, and .E operads . . . . . . . . . . . . . . . . . . 19 The little cubes operads gn . . . . . . . . . . . . . . 30 5 . Iterated loop spaces and the cn . .. 39 6 . The approximation theorem . . . . . . . . . . . . . . . . 50 7. Cofibrations and quasi-fibrations . . . 60 8 . The smash and composition products . . . . . . . . . . . 70 9 . A categorical construction . . . . . . . . . . . . . . . 82 10. Monoidal categories . . . . . . . . . . . . . . . . . . 94 11. Geometric realization of simplicia1 spaces . . . . . . . 100 12. Geometric realization and . S . C andfi. . . . . . . . . 113 13. The recognition principle and .A spaces . . . . . . . . 126 14. .E spaces and infinite loop sequences . . . . . . . . . 137 . 3. 4.

Operads andmonads

Remarks concerning the recognition principle

. . . . . . 153

. . . . . . . . . . . . . . . . . . . . . . . . . . 162 Bibliography . . . . . . . . . . . . . . . . . . . . . . . 173 .

Appendix

Operads and $-spaces Our recognition principle will be based on the notion of an operad

acting on a space.

We develop the requisite definitions and give a pre-

liminary statement of the recognition theorem in this section. To fix notations, l e t u denote the category of compactly generated Hausdorff spaces and continuolis maps, and let

denote the category of Base-points

based compactly generated Hausdorff spaces and based maps. will always be denoted by

and will be required to be non-degenerate, in

the sense that (X, *) i s an NDR-pair f o r X r

Products, function

spaces, etc., a r e always to be given the compactly generated topology. Steenrod's paper [ g o ] contains most of the point-set topology required for our work.

In a n appendix, we recall the definition of NDR-pairs and prove

those needed results about such pairs which a r e not contained in [ g o ] . An operad i s a collection of suitably interrelated spaces k(j),the points of which a r e to be thought of a s j-adic operations X'

Pre-

cisely, we have the following definitions. Definition 1 .l.

An operad

with c ( 0 ) a single point t (a)

, together

Continuous functions y : j =

F (j)

consists of spaces

with the following data:

& (k) X 2" (j,,) X ... X

&(jk)

(j),

jsJ such that the following associativity formula i s

satisfied for a l l c r y(y(c; dl'

&(k), ds r c ( j s ) , and e r t '''

' dk); el'

... ' e.)J = y(c;

f,,l.

(it): 2

fk)J

for j 2 0,

... + js-* t 1 , ...,.ejiS ... + js

where f = y(ds; ejIt f =:: (b)

if j

Definition 1.2. of X a s follows.

=O.

An identity element 1 E k ( 1 ) such that y(1; d) = d for d k y(c; 1 ) = c f o r c

(c)

and

k C ( k ) $ 1 = (1,.

.., 1 )

A right operation of the symmetric group C .on j

k (1)

$ (j)

and

& (j)

such that the

following equivariance formulas a r e satisfied f o r a l l c s %(k), ds

&(jS),

U E

Ck. and

Let X Let

Ex(j) be the space of based maps X'

eX(0) i s the inclusion

~ ( f9,. i

(b)

The identity element 1 r

(c)

(f u ) ( ~=) f(uy) f o r f

EX(j). .u

. .. vk) = y(c; d i , ... ,% ) ( T ~63 ... 63 u(jl,. ..,j ) denotes that permutation of j l e t t e r s which k

T ~ ) ,

of j a s u permutes k l e t t e r s , and

in Z An operad morphism

(j)

+ )

...

for

6X(k)

and gs r LX(js).

i s the identity map of X E

and y s ,

xJ, where

. C. acts on J

(XI, 8') of

(X, 6) i s then said to be a

$-spaces

foQ.(c)= e!(c)ofJ f o r all c J J by

... X C

on a space X i s a morphism of operads (r-space.

is a based map frX (j).

The category of

-C

A morphisrn

such that

<-spaces i s denoted

CJ [dl. It should be c l e a r that the associativity and equivariance formulas in

jk the definition of a n operad m e r e l y codify the relations that do in fact hold

i s said to be Z-free if Z acts freely on j

G3 -rk denotes the

under the evident inclusion of Z. X 1

e- g, and the pair

f: (X, 8)

permutes the k blocks of l e t t e r s determined by the given partition

., k )

X; X = n,

and ~ ( c ; d ~ r ~, ,

image of (-rip..

The data a r e defined by

E X(l)

An operation 6 of an operad

where

....gk) = f(gi X ... X gk)

( 1

and define the endomorphism operad

(j) f o r all j.

bf operads i s a sequence of Z.-equivariant maps J

such that

+1(1) = 1

and the following diagram commutes

EX.

The notion of an operad extracts the essential information contained

in the notion of a PROP, a s defined by Adams and MacLane [ I d ] and topologized by Boardman and Vogt [ 8

Our recognition theorem, roughly stated, has the following form. Theorem 1. 3 .

There exist Z-free operads

every n-fold loop space i s a

en,

1 5 n 5 co, such that

-space and every connected

the weak homotopy type of an n-fold loop space.

g n - s p a c e has

In the cases n = 1 and n = m, the second statement will be valid with

and

The spaces

operad and E

replaced by any A m

operad, a s defined in

(j) in the operads of Theorem 1.3 will be (n-2)-

Thus, if n = m, i t i s plausible that there should b e no obstruc-

section 3. tions to the construction of classifying spaces of a l l o r d e r s . Perhaps some plausibility arguments should be given. operad, and let product on X.

(X. 0) r

F (i)

6 [TI.

For c r

Let

6(2), B ~ ( C ) : -X X~

i s connected, then

* i s a two-sided

be any

&(1) connecting 1 to

y (c;

*, 1)

1 < ' n < m, ;he higher homotopies guaranteed by the connectivity of the r n ( j )

defines a a r e o i l y part of the story.

It i s not true that any

c - s p a c e , where

C(j)

homotopy identity

f o r 0(c); indeed, the requisite homotopies a r e obtained by applying paths i n

In the cases

i s (n-2)-connected, i s of the homotopy type of an n-fold loop space.

Thus

to any Theorem 1.3 i s considerably deeper in these c a s e s than in the degenerate

and 1 to y (c; 1, a).

Similarly, if case n = 1 (where commutativity plays no role) o r i n t h e limit c a s e n = co.

e ( 3 ) i s connected, then 0(c) i s homotopy associative since y(c; f , c) can be Since the notion of an action Q of an operad connected to y(c; c, 1).

6 on a space

X is

If c ( 2 ) i s connected, then ~ ( c )i s homotopic to basic to all of our work, it may be helpful to explicitly reformulate this notion

~ ( c T ) where , mutative.

T e

Z2 is the transposition, and therefore Q(c) i s homotopy com-

It should be c l e a r that higher connectivity on the spaces

determine higher coherence homotopies.

Stasheff's theory of A

6 (j)

in t e r m s of the adjoints

-spaces [ 2 8 ]

(i. e., has a classifying space) if and only if it has all possible highor coherence It i s obvious that if X can be de-Loped twice,

then its product must be homotopy commutative.

C- EX

x X'

Lemma 1.4.

An action 8:

maps 0.: c ( j ) X ~+jX , j 2 0

Ex(j);

(eO:*

determines and i s determined by

X), s u c h t h a t

(a)

The following diagrams a r e commutative, where 2 j = j and u denotes the evident shuffle homeomorphism:

Thus higher coherence

homotopies f o r commutativity ought to play a role in determining precisely how many times X can be de-looped.

X of the maps Q ~ : 6 0 )

these adjoints will also be denoted by 0j '

m states essentially that an H-space X i s of the homotopy type of a loop space

homotopies f o r associativity.

C(j)

will

Fortunately, the homotopies implicitly

a s s e r t e d to exist in the statement that a suitably higher connected operad acts on a space will play no explicit role in any of our work. (b).

Bi(i;x)

= X

for

X E

and

7 ~ . ( C V )'

e.(c; vy) f o r c e C(j), a- r Z and y E xJ. j' A m o r p h i s m f : ( X , B ) - + ( X ' , B ' ) in i s a m a p f : X - + X 1 in (c)

C[7]

(+, 0)

Lemma 1.6. Tsuch

C[T],then the unique maps

(X, 8) r

0j i s the t r i v i a l map; are in X and X

Let I: (X,0)

Lemma 1.7.

B c-morphisms. ( x Y )f

We complete this section by showing that, f o r any operad

C , the

)(

Let X X

(8 XB 8 1 ) ~ : (j) X (X x B y ) j

(ex (-spaces.

Y C X X Y denote the fibred B product B

7.Then

fibred product of f and g in the category B

structions and by discussing the product on

(B, 0") and g: (Y, 0') '(B, 0") b e

) of f and g i n

r - s p a c e s i s closed u n l e r s e v e r a l standard topological con-

(XX Y,0X

C[n,

0') i s t h e

where

X X Y i s defined coordinatewise:

...,(x..J Y.)) J = ( 8 j ( ~ ; ~ .1....xj), ej(c; Y ~ . . . ., yj)).

el).(c: J (x~.Y~).

These results will

yield properties of the Dyer-Lashof homology operations in the second paper

In particular, with B = +, (X X Y, 8 X 8 ' ) i s the product of (X, 8)

of this s e r i e s and will be used in the third paper of this s e r i e s to study such

(Y,8') i n the category

spaces a s F/O

thus a

and F / T O ~ . The proofs of the following four l e m m a s a r e

completely elementary and will be omitted. Lemma 1.5. in

Let x('

Let

(X, 8) r

A) :

C (j)X (x"'

C [ r]and l e t

(Y, A) be an NDR-pair

denote the space of maps (Y, A)

* x

(

31,and the.diagona1 map 6-morphism for any (X, 8) [TI].

Lemma18

(X, +), with

Then ( x ( ~ ' ~ 8) (,Y y A1) 6

c [nt

~ i~ s defined ~ ) pointwise:

(2,s)E

are

& -morphisms.

[TI, where Q 8 = 8(1, and PX and end-point projection p: PX * X

X X X is

C [r]by letting +

? =I X X E

[TI

can be

L e i ÂŁ:(X,e)

where g(w) = w(0) f o r w

all In particular. ( Q X , n e ) and (PX, P8) a r e in

[

replaced by a fibration i n

inclusion i: X

, and the inclusion i:QX

A :X

and

P8 = 8"'

[

The previous l e m m a s imply that any morphism in

(non-degenerate) base-point t h e trivial map. where 8 j

$ -morphisms.

that the following diagrams commute:

category of

6[y 1, where each

(Y,B1) b e a m o r p h i s m i n

c[r].

Define

Y I

(Y ) b e the e4fibred product of f and g. Y I and by letting 8 = 0 X ( 0 ) Then t h e

r\J

X, the retraction r: X

X, and t h e fibration f: X

Y are

r - m o r p h i s m s , where i(x) = (x, w ~ ( ~ with )) w f ( 4 (t) = f(x), r(x, W) = rJ

and f(x, w) = w(1).

Finally, we consider the product on a

6-space.

The following lemma

i s the only place in our theory where a l e s s stringent (and m o r e complicated) 6 - m o r p h i s m would be of any service.

notidn of

such a notion i s crucial to

g(d) = y(c; d, d)v, where v

(X X x)' if d

Boardman and Vogt's work precisely because the H-space structure on a

If geometric theory could perfectly well be developed without ever mentioning the product on

explicitly

fixed c (i)

C(2).

Let $2 =

6 (7) and l e t

(j) i s connected and d

0. = $(I x $. ) : X' J 3-1

and

C (j), then

0j . and '& (2j)

~ ( c )X' :

@(d):xl

X f o r some

X f o r j ) 2. +

X is homotopic

to the iterated product (ii)

d ( j ) i s 22.-free J

i s contractible, then the following

diagram is Z.-equivariantly homotopy commutative: 3

Proof.

(i)

Any path in &(j) (ii)

$. = 3

(0 x

(7 x XI'

elj

r x xI x

0(c.), where c = c and c = ytc; 1, c j e l ) f o r j J 2 j

connecting d to c

Define maps f and g from

x.J

and z

provides the desired homotopy.

c ( j ) to

c ( 2 j ) by f(d) = y(d; cJ) and

> 2.

on X2j. E

i'i(c)J then

gives the evident shuffle map 2j An examination of the definitions shows that

x 2 j , then

f + ) ( d ~ z=) eZj(f(d).z)

is embedded in 2

and g are

reason f o r i t to play a privileged role. Let (X, 0) e

6 (j)

2j and g a r e

The product i s only one small part

&-spaces.

of the structure c a r r i e d by an n-fold loop space, and there i s no logical

Lemma 1.9.

X' X X'

In contrast, our entire

&-space plays a central role in their theory.

by cr

and

$ ( 0 ~ 8 ) ~ (r d )= , B2j(g(d), .).

...,2),

~(2,

xj -equivariant.

in the notation of Definition

Our hypotheses guarantee that

-equivariantly homotopic, and the r esult follows.

Operads and monads In this section, we show that an operad

matical structure, namely a monad, and that algebras over the derived monad.

c - s p a c e s can be replaced by

We shall also give a preliminary state-

ment of the approximation theorem. of

and

ccx

determines a simpler mathe-

C'X

The present reformylation of the notion Here squares (and higher iterates) of natural transformations

& -space will lead to a simple categorical construction of classifying

P: C

C' a r e

defined by means of the commutative diagrams

spaces for

Cfn-spaces in section 9.

We f i r s t recall the requisite cate-

gorical definitions. Definition 2.1.

A monad (C, p, q) in a category

(covariant) functor C:

functors p:

7 7 together with natural transformations of +

C and q: 1

mutative for all X

? J consists of a

C sucV that the following diagrams a r e comThus a monad (C, p, q) i s , roughly, a "monoid in the functor category"

with multiplication p and unit q, and a morphism of monads i s a morphism o l "monoidsl'.

Following MacLane, we p r e f e r the t e r m "monad" t o the m o r e

usual t e r m "triple".

operationally, in our theory, the t e r m monad i s par-

ticularly apt; the use of monads allows us to replace actions by operads, which a r e sequences of maps, by monadic algebra structure maps, which a r e single A morphism

:(C, p, q)

tion of functors f o r all X

+: C

(Ct, pt, ql) of monads in

i s a natural transforma-

C t such that the following diagrams a r e commutative

maps.

Definition 2.2. X

An a l g e b r a (X, 5) o v e r a monad (C, p, q) i s a n object

together with a m a p

5: CX

X in

7 such that the following diagrams

a r e commutative:

Construction 2.4.

associated to

Let

be an operad.

a s follows.

For X

Construct the monad (C, p, q)

c(j)X

relation on the disjoint union

denote the equivalence

let

generated by

j20 c , y) c (c, siy) f o r c s (ccr, y)

(ii)

(c, cry) f o r c

C (j), (j),

(I.

0 5 i < j, and y e xj-': E

t:j ' and y

and

xJ.

A morphism f: (X, 5)

(XI, 5') of C-algebras i s a map. f: X

X1 i n

J such

that the following d i a g r a m i s commutative:

Define CX to b e the s e t c ( j ) X x~/(E). L e t FkCX denote the image of j20 k (j) X X' in CX and give Fk CX the quotient topology. Observe that j =O

Fk - lCX i s then a closed subspace of Fk CX and give CX the topology of the union of the FkCX.

r CX'

If c s

point of CX.

FoCX i s a single point and i s to be taken a s the base-

c (j)

and y s

xJ, l e t

[c, y] denote the image of (c, y)

i n CX.

F o r a map f:X +XI

Cf[c; y] = [c; fj(y)].

? ,

define Cf:CX'

Define n a t u r a l maps p:

The category of C-algebras and t h e i r morphisms will be denoted by C [ r ] .

c%-

CX1 by

CX and q: X

CX by the

formulas We now construct a functor f r o m the category of operads to the category

, where

of monads i n

7 i s o u r category of based spaces.

(iii)

In o r d e r to

...

p[c, [dl, yl], for c

[s.

ykI] = [Y(c: dl

c ( k ) , ds

(jS), and ys

js X

dk), Y1,

.. .

yk]

; and

handle base-points, we r e q u i r e s o m e preliminary notation. (iv) Notations 2 . 3 . 0

Let

be an operad.

Define maps rr.:

5 i < j, by the formula u.c = y(c; s.) f o r c s 1

(j)

-c

q ( x ) = [ l , x ] for X E X .

The associativity and equivariance formulas of Definition 1.1 imply both that

c(j-l),

p i s well-defined and that p satisfies the monad identity p. p = p. Cp ; the

j), where

unit formulas of Definition 1.1 imply that p. Cq = 1 = pq. Thus, in the endomorphism operad of X s j-1 and y r X

, where s.:xj-'

, (rr;)(y) = f

i s defined b y

( ~ . ~f o)r f : ~ '

+: 6

a m o r p h i s m of operads, construct the associated morphism 'of monads, also denoted

b y letting

+LC, YI = [+j(c), yl

+: CX c s

C'X be the m a p defined by

(j) and Y r

xJ.

The association

of monads and morphisms of monads to operads and

morphisms of operads thus constructed i s clearly a functor. validate the construction, we should verify that the spaces in

for X E

Of course, to

pactly generated.

CX a r e indeed

Since

(x,*)

i s a n NDR-pair by assumption, t h e r e i s a

~ e A m a A . 4 : Define h . : I X F . C X + J J

We first fix notations for certain spaces, which

a r e usually r e f e r r e d to in the literature a s "e quivariant half -smash products.

h.(t, J z) = z

and

F.CX and u .F.CX+ I by the formulas J j' J

c .J( z ) = 0 for z

Fjel CX,

= u.(y) J f o r z = [c, y], c h J.(t, z ) = [c, h.(t, J y)] and z.(z) J Notations 2. 5.

Let W

any subgroup of 2 j'

and let n act from t h e right on W, where n i s

Let X

7 and observe that the left action of

induces a left action of n on the j-fold smash product x"].

defined by (w, *)

U E

n , and y

X [jl

(w', *) f o r w,wl

Let e[W, n, X]

denote the quotient space W X X[ jI /(%), where the equivalence relation W and (wcr, y) w (w, try) f o r w

com-

representation (h., u.) of (X, *jJ a s a 2 -equivariant NDR-pair by J J j

We shall do this and shall examine the topology of the CX

in the following proposition.

J i s Hausdorff, hence, .by [ 3 0 , 2.61, It follows easily that each F.CX

Then CX

( xJ . ,J~ .represent ) U

(F.CX, J Fj *CX) a s an NDR-pair.

F o r (iii), if ht: X

i s a homotopy, and i t is evident that rJ

t e m s of inclusions in

-t

F ( j ) and y

E(X-*lJ.

By [30,9.2 and 9-41,

Therefore CX

i s a n NDR-pair. and each (CX,F.CX) J

i s now obvious.

and

7. P a r t (ii)

X1 i s a hornotopy, then Cht: CX

CX'

C preserves limits on directed sys-

. We shall s e e in a moment that the CX a r e c - s p a c e s , and our approxi-

The spaces CX a r e built up by successive cofibrations from the spaces e [ C (j),Cj, XI.

mation theorem can be stated a s follows.

Precisely. we have the following result. F o r the operads

Theorem 2.7. Proposition 2.6.

Let

be an operad and let X e

(i)

(FjCX,Fj-lCX)

(ii)

F.CX/F CX i s homeomorphic to e[ J j -1

(iii)

c:T+ Proof.

i s a n N D R - p a i r f o r j Z 1 , and C X r (j), C XI; j'

i s a homotopy and limit preserving functor. It i s immediate f r o m the definitions that

Then

Gn-spaces

i s a natural map of

ctn :CnX

of the recognition principle, t h e r e an.Shx, 1 L n Lm, and an is a

weak homotopy equivalence if X i s connected. In fact, ctn :Cn

nnsn

defines a monad in J

ansn will be

, and the natural transformations

morphisms of monads.

This fact will provide the

essential link connecting the approximation theorem to t h e recognition principle. We now investigate the relationship between where C i s the monad associated to the operad

-spaces and

C-algebras,

Proposition 2.8. monad. 8:

Then there i s a

Let

one-to-one correspondence between

and C-algebra s t r u c t u r e maps

8 correspond to

Henceforward, we shal? u s e the l e t t e r 8 both f o r

be an operad and l e t C be its associated

5: CX

the corresponding C-algebra s t r u c t u r e maps.

c-actions

0 C(j) X X' j'

X defined by letting

X which define a

C (j) X X'

F .CX J

standardlemmaincategorytheory.

Mo=eover,

CX).

-space generated by the space X, i n view of the following

Lemma 2.9. ( w h e r e a i s the evident composite j

We should observe t h a t the previous proposition implies that CX i s

the f r e e -

Thus the maps

-action should now be thought of a s

cokponents of the single map- 8: CX

if and only i f the following diagrams a r e commutative

c-actions and f o r

(CX, p)

Let (C, p, q) be a monad in a category

for X

Then

and t h e r e i s a natural isomorphism

this correspondence defines an isomorpilism between the category of

(: H

O (X, ~ Y)

( ( c x , PI, ( y , 5))

~ HornC[

-spaces and the category of C-algebras. Proof.

By the definition of the spaces CX, a map

5: CX

defined by - ((f) = +

(j) x X' - X

0.(cu, y) = Qj(c,o y ) . 1 .

such that

F o r a given m a p

6: CX

c-action

+ X , the relation

Lemma 1.4(a) f o r the corresponding maps 8 and the relation cq = 1 i s j' Thus a map

QX = (CX,p) a r e adjoint.

on X.

Lemma 2.10. between functors U: q=

X is a

The l a s t statement follows fromihe observation that if

(X,e) and (X',g1) a r e C-algebras and if f:X --XI f . 5 = 5l.Cf

if and,only if f8. = B ! ( ~ f3) x f o r a l l j. 3 J

= g. q.

.--

is a map in J

c[J] defined by

We shall l a t e r need the following converse r e -

sult, which i s also a standard and elementary categorical observation.

4- 1(1QX): X

(: Hom,

Let

UQX and define

uauax

Then (UQ, p, q) i s a monad i n

6=~

then Then (UY,

UQ [

(X, UY)

-+.

Hom

r-, .

and Q:

p = u$(luQx):

C-algebra s t r u c t u r e map if and only if the corresponding maps 0. define a n J action of

-1

$ (g)

The preceding l e m m a s t a t e s that the forgetful functor U: C [ g ] *

5.p = 5.C5 i s equivalent to the commutativity of t h e diagrams given in

equivalent to 8 (1,x) = x f o r a l l x c X. 1

i s given by

defined by U(Y, 5) = Y and the f r e e functor

8j-l(uic, Y) = 0.(c, siy) and J

Since u.c = y(c; s.), the maps 0 given by a j

8 do satisfjr these formulas.

(-'

Cf ;

determines and i s determined (via the stated d i a g r a m s ) by a sequence of maps 8 j'

uax

For Y

x (QX, Y)

For X

be an adjunction

7 ,define

define

( ( 1 UQUY ~ ~ ) --: UY.

3 1,

and

6: UQU

-c

U i s a natural transformation of

functors

given by VY = (Y,

. 5)

Thus there i s a well-defined functor V:

7- u Q [ ~ ]

Am and E03 operads

on objects and Vg = Ug on morphisms. We describe certain special types of operads h e r e and show that the

Of course, V i s not a n isomorphism of categories in general. ever, if the adjunction with

How-

i s derived a s in Lemma 2.9 from a monad

7 = c[S],then it i s evident that the monads

same and that V i s the identity functor.

UQ ahd C a r e the

C ,

constructions of the previous section include the J a m e s construction and the infinite symmetric product.

Most important, we obtain some easy

technical results that will allow us to transfer the recognition principle and approximation theorem from the particular operads

and

Coo to

a r b i t r a r y A and E operads, respectively. m m

We f i r s t define discrete operads f i and

such that an Uf/l -space is

precisely a topological monoid and a n 'R-space i s precisely a commutative topological monoid. Definition 3 .l. (i) Define %'~(j) identity element of Z , e =

Define y(e e. , k' J~

n(j)=

Observe that if

hence any

?fib) constitute

, . . .,f. ) = f.,

constitute a discrete operad

unique functions

by the equivariance formulas of

, a single point. Let

on 1/L(j), and define y(f ;f k ji %(j)

.. . X I:.J k

With these data, the (fjj

I = f

js.

a discrete operad %?k.

let Z. act trivially J With these data, the

i s any operad with each

G (j) non-empty,

(j) + n ( j ) define a rnorphism of operads

2-space is a

j = C j s , and extend the domain of definition of

y to the entire set I: X I: X k ji

(ii) Define

e denote the j

Let 711(0) contain the single element e

. . .,ejk ) = ej'

Definition 1. i(c).

= Z. f o r j 4 1 , and l e t J

$-space.

then the +

and i s determined by the action 9:

rfi-+ & G

be the iterated product and extending permutations in

r)2 s e r v e only to

3 (with - identity element

A topological monoid G in

â&#x201A;ŹI

*) determines

defined by letting 8 .(em): G' J J

to a l l of Z. by equivariance.

-t

The

record the possibility of changing the o r d e r

of factors in forming products in a topological monoid.

Crearly a topological

monoid G i s commutative if and only if the corresponding action

- 6,

9:n

factors through

For X

C(j)

MX and NX a r e called the Jam*

v0 c ( j ) by 6 .(c) = [c], where [c] denotes the path component J

containing the point c. such that a clearly

The data for

i s any discrete operad and i f

morphism of operads, then

E. factors a s the composite

IT & 0

Definition 3.3

An operad over a discrete operad /$ i s a n operad

morphic to the j-fold smash product x[jl and to the orbit space X[jI/X

ism

yield the following proposition. Proposition3.2.

6 ) n , [ g ] =1vl[3]

and r * 2 [ $ ] = N [ $ ]

a r e isomorphic to the categories of topological monoids and of commutative For X

and NX a r e the f r e e

topological monoid and the f r e e commutative topological monoid generated by the space X, subjectto the relation

* = 1.

and let

E'+ = â&#x201A;Ź :

in a sense which we now make precise.

?T, (j) denote the set of path components of

Clearly an operad

i t i s locally connected, and can be augmented over and an augmentation of

@ is

Let

be any operad, (j). Define

19- such that

A morph-

a m o r p h ~ s mof operads

i s locally n-connected if each can be augmented over

(j) i s

if and only i f

5 then admits a unique augmentation. bfl.

- fi

roc

noâ&#x201A;Ź :

fJ.

An operad

if and only if n0C (j) i s isomorphic to X j '

is then a suitably coherent choice of isomorphisms.

We shall say that a morphism of operads

We shall only-be interested in operads which a r e augmented over either

( e l , E') of operads over

such that

n-connected.

& i s called the augmentation of

We shall say that a n operad

Thecategories

topological monoids, respectively.

+: c+ 6'

6 , where

i s a n isomorphism of operads. +

With these notations, we make the following definition.

successive quotient spaces e [ m ( j ) , Xj, XI and e[ n ( j ) , Z j , X] a r e homeo-

+: (C, & )

Cf fi is a

together with a morphism of operads

The arguments above and the results of the previous section

tion and the infinite symmetric product on X; it should be observed that the

respectively.

uniquely determine data for

defines a functor from the category of operads to the category

of discrete operads.

construc-

i s a discrete operad and 6 i s a morphism of operads.

71.

roe :\c

r , the monoids

equivalence, o r a localX-equivalence, if each

+: +

C(j)

F' -+

is a

local

e l ( j ) is a homo-

topy equivalence, o r a X -equivariant homotopy equivalence (that i s , the j requisite homotopies a r e required to be 2 -equivariant). j

Of course, these

a r e n o t equivalence r e l a t i o n s s i n c e t h e r e need be no i n v e r s e morphism of operads C '

We have not defined and shall not need any notion of an A

The following p r o p o s i t i o n w i l l be

e s s e n t i a l i n passing from one operad o v e r m o r n t o another. Let $:C

Proposition 3.4. over m o r n .

Assume e i t h e r t h a t $ i s a l o c a l C-equivalence o r

a s s o c i a t e d maps $:CX

a r e C-ÂŁre-e.

and contractible.

Then t h e

C ' X a r e weak homotopy equivalences f o r

-t

An operad

C ' be a morphism of operads

t h a t $ i s a l o c a l equivalence and C and C '

or E

morphism between A

is an E

operad if and only i f each

H-spaces,

i s a classifying space

f o r 23 i t s homology will give r i s e to the Dyer-Lashof operations on the j' space.

We have required an E

w i l l hold i f t h e maps e [ C ( j ) ,CjIX1

JIj induce isomorphisms on homology. equivalences i f JI

j I f C ( j ) i s C j-freeI

i n o r d e r to have this interpretation of the spaces

C ( j ) / z j and in o r d e r to

have that CX i s weakly homotopy equivalent to

JI induces an isomorphism on

By Proposition 2.6 and t h e f i v e lemma, t h i s

i n t e g r a l homology.

These maps a r e homotopy

dently an infinite loop space.

definition of an E

l o c a l C-equivalence.

i s a local 23-equivalence. nected space.

i s a C-space over any Am operad C.

n such

a l o c a l equivalence.

(X,0)

An Em Space, o r e

i s a C-space over any Em operad C.

t h a t E:C

xis

y everything space, +

Let

be a n operad over

Let (X, â&#x201A;Ź4)

be a

such that E :

Then X i s weakly homotopy equivalent t o

* CX

and

C +a

c - s p a c e , where X i s a conX

n 2i We have the following commutative diagrams:

Proof.

An Am space ( ~ , 8 )

(ii)An Em operad i s a C-free operad over

operad.

Proposition 3.6.

H*(C ( j ) ~ x [ j l ) , hence on ~ , ( e [ C ( j ), P j , ~ 1 ) . j We now d e f i n e and d i s c u s s Am and Em operaas and spaces.

+mis a

The following amusing result shows that,

such t h a t E:C

n - s p a c e i s evi-

for non-triviality, we must not assume & to be a local 2-equivalence in the

converging from E ~ = H , ( H -H, ( C ( j ) x x[j1) ) t o H* ( C ( j ) x x [ j ; ) . j' j Thus i f C ( j ) and C ' ( j ) a r e C - f r e e and $ i s a homotopy equij j valence, then $I+ induces an isomorphism on E ~ hence , on

(i)An Am operad i s a C-free operad over

S X f o r any E

a s an E operad, although a connected co

not to r e g a r d

c l e a r l y a covering map and s o determines a s p e c t r a l sequence

D e f i n i t i o n 3.5.

e C C t ( j ),Cj,X1 d e t e r k n e d by

i s a C - e q u i v a r i a n t homotopy equivalence. j xCjl is then t h e map C ( j ) x x C j l + C ( j ) x j

operad to be 23-free

C t X i s an H-map between connected

it s u f f i c e s t o prove t h a t

C(j) i s X.-free

t(j)/zj

Proof. Since $:CX

spaces o v e r different operads.

Thus the orbit space

homology of an E

a l l connected spaces X.

or E

K(T,(X), n).

By Proposition 3.4, E i s a weak homotopy equivalence. e a s y to prove that q

It i s well-known and

Definition 3 . 8 .

--+

s*(NX) = H*(X) m a y be taken a s the definition

of the Hurewicz homomorphism h.

- i )h,

Thus i = B*q:,

= (8:: E*

monomorphism onto a d i r e c t summand of H*(X).

and h i s a

i s an A

By the proof of

operad if and only if each a

morphic to X . and each component of J i s itseif an A

(T (j). i s contractible.

(a)

Lemma3.7. -+

Any operad c o v e r

Each u

carrying

- i ( r ) homeomorphically

it+ = E

second statement, we may a s s u m e that equation if necessary), and then

-I

E . (ej) J

-i (&j) ( e j )

-I

+.. must 3

Then

X.. J

Gf(j), and

and Em operads.

need be no morphism of operads between two A 03

that the category of operads h a s products.

c ( j s ) X g1(js):

and

i s the product of X

and

&jr' in the categoryof operads. The

will be denoted C X C' (by abuse of notation,

However, t h e r e

o r two Em operads.

X @'.

Since

priate for the study of operads o v e r

! I

a n operad

and an operad over

xi)nl# 8(a3 , the above

Let (

(cv c: .cce1)

over

,& ) and ( );',

T(2 by letting

el)

product i s inappro-

be operads over

%! .

Define

v c' be the fibred product of

i n the category of operads and letting E

mutativity of the following diagram :

fit i s evidently

Observe that the category of

operads has fibred products a s well a s products. Definition 3.9.

the r e s u l t

Fortunately, a l l that i s needed to circumvent this difficulty i s the observation -

xC'

a n operad o v e r

F o r the

In the applications, it i s essential that o u r recognition theorem apply, to a r b i t r a r y A

Cf(i) x c ' ( 1 ) ;

The product of a n operad over

and

03,

<(k) X C1(k) and ds X d i

F ). gory of monads in J

follows.

f o r n = I and n =

since we do not a s s e r t that C X C' i s the product of C and C ' i n the cate-

(redefining 6 by this

and

,...,% x % ) = y ( c ; d i ,...,dk)Xyl(c'; d;, .:. ,%I

for c X c '

monad associated to

r e s t r i c t t o a homotopy equivalence

C (j)

( y ~ y ' ) ( c X c ' ; id X d ;

( c ~ c ' ) c r = c c i . X c ' c rf o r C X C ' E Z ( j ) x c 1 ( j ) and U E Z j

The resulting homotopies c a n be t r a n s f e r r e d by equi-

variance to the remaining components of

C X ~the' follow-

(c)

irfl.

E . (7u) f o r

and giving

In particular,

(j) by permuting components,

onto

C (j) X c ' ( j )

Define a n operad

1= i X i

i s Z - f r e e a n d a n y local equivalence

must act on

be operads.

(b)

C' between operads o v e r 'n;-i s a local Z-equivalence.

Proof.

(c x F 1 ) ( j ) =

by letting

G I

i s iso-

operad. In contrast to the preceding result, we have

the following observation- concerning operads over

C (j)

and

ing data:.

[ 1 8 , Theorem 24.51, this i s precisely enough t o imply the cbnclusion. An operad

Let

V &'

be defined by com-

Corollary 3.i i.

Let

be an E m operad.

i s a local C-equivalence and therefore (X, 0) i s a

C -space,

~ h u esv e r y E

~ V 'Ci s the

Explicitly,

sub operad of

disjoint union of the spaces

- 1(a)X ( E j

(G V ~ ' , E ~ & ' )i s the product of

operads over

? ' such that (C c')(j)

- i (r) f o r El.)

( c ,E )

and

(G', E ' )

i s the

Then

inthe category of

In conjunction with Proposition 3 . 4 , the following result contains a l l the information about changes of operads that i s required f o r o u r theory.

operad over

Let

be a n Am operad and l e t

Then the projection n2:

. C X C'

Proof. -

e 'is a local

Then the pro-

equivalence between C-f r e e operads.

(i)follows from Lemma 3.7 s i n c e E"

nZ:Ej-1

(a) i s contractible

and t h e r e f o r e (0) x ( E ; ) - ~ ( u ) (c!)-l(u) is a j I homotopy equivalence. P a r t (ii)i s immediate from t h e d e f i n i t i o n s . for u

m +?%

i s an A operad. m -space, ni:

C X m+c.

space.

Since A03 spaces a r e of interest solely i n t h e study of f i r s t loop spaces,

obtained by throwing out the permutations by rneans of the following definition and proposition.

Since (ii)depends only on t h e l o c a l c o n f r a c t i b i l i t y (and n o t

We have chosen to describe A spaces in t e r m s of operads w

i n o r d e r to avoid further special arguments, and no u s e shall be made of the theory sketched below.

be any

(vC'-f 6 1 i s a local 8-equivalence.

be an Em operad and let c l p e any 2 - f r e e operad.

(ii) Let jection n

(i)

space i s an A

132

where commutativity plays no role, a s i m p l e r theory of Aw spaces can be

The monad asaociated to ~ ~ c ' w i be l l denoted by C v C 1 .

Proposition 3.10.

then (X, h i ) i s a

Then w2:

Definition 3.12. for j

+ 0,

with

of a n operad.

A non-Z operad

a(0) =

determines a n underlying non-Z operad An action of a non-C operad

i s a morphism of non-C operads category of

?$spaces

i s a sequence of spaces a ( j )

together with the data (a) and (b) in the definition

An operad

neglect of permutations.

(X, 0).

ur$X,

an4

on a space X e

a [ r ] denotes the

By omission of the equivariance relation (ii)

i n construction 2 . 4 , a non-C operad such that the categories

[3]

determines an associated monad

and B[ 9 ] a r e isomorphic.

The notion of

on t h e C-freeness) of C , t h e proof of our r e c o g n i t i o n p r i n c i p l e f o r Em spaces w i l l a c t u a l l y apply t o C-spaces over any l o c a l l y c o n t r a c t i b l e operad C.

a non-C operad over a d i s c r e t e non-C operad i s defined by analogy with Definition 3 . 3 .

The product

a x a' of non-Z operads $3

by analogy with Definition 3 . 8 .

and

i s defined

Let

denote the sub non-Z operad of

The categories

@. [TI

operad over

clearly admits a unique augmentation.

and ?' ?%

determines a n operad Z n trivially on

Let

C ,L ) =

a):

non-I: operad over

be an operad over

hfn,

, L ) i s a non-I: operad over

and to w(

and define

,E ) a r e isomorphic.

W - ~ B= (2% x.??L.

Moreover, w and w

w-'(fi

)

a and of non-I: operads over a.

The first two statements follow immediately f r o m the definitions.

and we must show that

W W - ~

respective identity functors. isomorphic to

W - ~ Ware

Now ww- '(

on morphisms,

naturally isomorphic to the

a)=

i s evidently naturally

, and a natural isomorphism v : ( ~. L )

can be defined by

and

W-~W(C ,&I=(zE-~(G)x

v.(c) = (cu-', u) f o r c

-i

- 1 (u)

givenby v r i ( c , cr) = ccr for c e & (e.) and u J j~

rn,.,)

and u e I:

. v - 1.

The notion of A

space originally defined by Stasheff [ 2 8 ]

Stasheff constructs certain spaces K. for j ) 2; J and K = 1, these K can be verified to admit structure maps

y so a s to form a locally contractible non- Z operad

an equivalence

- 1.

3.14.

with K = o

be a

and to wml(

F o r the l a s t statement, it i s obvious how to define w and w

and the notion of a n A

i s included in our notion.

and the

Let

iiZ).; then

act

and define

operad i s equivalent to the notion

space i s equivalent to the notion of a a - s p a c e over such a non-Z operad

em ark

- 1 a r e the object maps of

between the categories of operads over Proof.

and the monads associated to

i s a n operad over a r e isomorphic.

6 by letting

of a locally contractible n6n-Z operad over

A non-;C, operad

i s isomorphic to

( c ,E )

&-I(then w ( C

monads associated to

;C,

XZfi

It follows that the notion of an A

A non-Z

a r e evidently isomorphic.

such that

a ( j ) . In particular,

Proposition 3.13. W(

[TI

a (j) = {e.).J

such that

i s then

space in Stasheffls sense i s precisely a

%-space.

such that a n A

requirement that the component little cubes of a point of 4.

The Pittle cubes operads

We define the C-free opdrads spaces

this section.

Cn(j)

and discuss the topology of the

. The definition of the cn (in the con-

text of PROP'S) i s due to Boardman and Vogt [ 8 ]. Definition 4. 1. interior.

An (open) little n-cube i s a linear embedding f of J

parallel axes; thus f = f X

. . . X 'fn

where f.: J

f.(t) = ( y i - ~ $ t t x., with 0 5 a. < y. 5 1.- Define 1

j-tuples

<cl,.

. . , c3' >

n n

...,c.> J

.in J , with

J i s a linear function,

c n ( j ) to be fhe.set of those

of little n-cubes such that the images of the c

J". and topologize

of the space of all continuous functions j~~ regard

J ~ Write .

V(c;di,

... ,dk) = c - (dl + ...

for c

c n ( k ) and ds

(b)

(c)

<cis. ..,c.>u = J

tn(0) = n

. .. t

u.<cl,.

. . , c .J > = < c l , . . ., C . , C ~ +..~.,, c J. > ,

< i < j.

The associativity;, unitary, and equivariance formulas required of an operad j

cn cnSl

, 1: J

inherits a structure of C-free operad

rn.

efn(j)

i s convenient f o r continuity proofs

and will be needed in our study of the Dyer-Lashof operations on F in the third paper of this series.

The following .more concrete description of this

topology i s more convenient for analyzing the homotopy type of the spaces c n ( j ) . Lemma 4.2.

. . ,C . >

Let c = < c

cn(j).

Observe that c determines

and is determined by the point c ( a , p)

where cr = (-

Let

, .. ., qi ) E

U denote the topology on

Then

2nj

?,(.

Proof.

and l e t

defined by

. -. cj(a)>cj(B)), 3 3 and p = (q , . .. , q )

c(a*B) = ( ~ ~ (cl(B), 4 , i

2nj

r n ( j ) obtained by so regarding

denote the topology on

. t n ( j ) a s a sub-

(j) defined in Definition 4.1.

. Let w(C, U) denote the v - o p e n s e t consisting of those c such

that c(C) C U, where C i s compact in jsn and U i s open in J

>for U E C j

a r e trivial to verify, and the action of C

Clearly

have disjoint

c m ( j ) denotes the space l i m c n ( j ) , with the

The topology we have given the

s e t of J

jk n

By our functional interpretation of <>, (a) implies that (d)

topology of the union.

The requisite

c n ( j s ) , where t denotes disjoint union;

~ ( 1 )*' ' . "v(j)

< >, and

(1) i s the identity function; and <C

are

data a r e defined by (a)

Each u . i s a n inclusion, and n,j

c n ( j ) a s a subspace

a s the unique "embedding" of the empty s e t in J

ji n

.. , cJ. > = < c l X l , . .., cJ. X l >

a s a map '3"

u .<ci,. nlJ

denote its

pairwise disjoint. i e t j p denote the disjoint union of j copies of J ~ , regard <c.l ,

Define a morphism of operads un:

from the

denote the unit n-cube and let J

Let

images. (e)

I a m indebted to M. Boardman f o r explain-

4.8

ing to me the key result,

Cn (j)

c n ( j ) i s f r e e in view of the \

(resp. f3 ) denote the point ar (resp. p) in the r - t h domain cube 3: r n U and Vr a r e open subsets of 3 , I 5 r 5 j, then

Let cr

'J".

It follows that any

%. -open

set is

v-open.

F a d e l l and Neuwirth have proven the following theorem.

Conve*rsely, consider

We m a y a s s u m e that U is the image of a n open little cube g

W(C, U).

and that C i s contained in a single domain cube J

Let C'

Let M be an n-dimensional manifold, n 2 2.

Theorem 4.4.

n J r be the

Yo =

and Y =

. . . y 1 . 1 5 r < j,

{ yl,.

image of the smallest closed little cube f containing C (f may be

points of M.

degenerate; that i s , some of i t s intervals may be points).< Then, by

a <x l , . . . , ~ > = x 0 5 r < j-1. r j-r I'

linearity, only i f

w(C, U) = W(C1,u). Clearly c =

. . ., CJ. > E

W(C ' , U) if and

crf(0) > g(0) and c r f ( i ) < g ( i ) , with the inequalities interpreted

coordinate-wise and with 0 = ( 0 , . e a s y to verify that w(C', U) i s

.. , 0 )

and 1 = ( I , .

F ( M Yr+i; j - r - I )

where the y. a r e distinct

- Yr

Then a

over the point y

rti'

i s a fibration with f i b r e -

and a

admits a cross-section

if i l i .

It i s now

open.

Let

r n n n S denote the wedge of r copies of S ; since R

homotopy equivalent to

Using this l e m m a , we c a n relate the spaces n spaces of R

n , I ) in I

Define a : F ( M -Yr; j-r) -. M

Let

the theorem gives the following corollary. j- I

c n ( j ) to the configuration

We f i r s t review s o m e of the r e s u l t s of F a d e l l and Neuwirth [12]

r n-I S

- Yr

If n 2 3, then aiF(Rn; j) =

Corollary 4. 5. T~F(R'; j) = 0 for i

2 ni( rSn-1 );

r=i 1 and a l F ( ~ ' ;j) i s constructed f r o m the f r e e groups

r i a t ( S ), i 5 r <.j, by successive split extensions.

on configuration spaces. il

Definition 4.3.

Let M be an n-dimensional manifold.

Define the j-th

configuration space F(M; j) of M by F(M; j) = { < x i ,

. . . , xJ. > l x r ~M,

with the subs pace topology. F(M; 1) = M.

Let

B(R'; j) i s a K(B i ) , where B i s the j' j

braid group on j strings, and t h e r e i s a short exact sequence x r / x S if r / s ) C M ~ ,

F(M; j) i s a jn-dimensional manifold and

. . ,X.> u = <x ~ ( 1 ) ~l x u ( j ) > . J *

This operation i s f r e e , and B(M; j) denotes the orbit space F(M; j)/Z

I. J

B j

1 which i s isomorphic to the homotopy exact sequence of

the covering projection F(R2; j)

Z . operate on F(M; j) by J

The c a s e n = 2 i s classical.

B(RZ;j).

Detailed descriptions of

2 I. = a l F ( R ;j) and of B. m a y be found in Artin's paper [ 3 1; F o x and J J 2 Neuwirth [ i3] have used F ( R ;j) to rederive Artin's description of B. in J t e r m s of generators and relations.

n Let RW = l i m R with respect to the standard inclusions.

i s given by c

Since

r s (t) = (Yrs

- xrs)t + x r s '

We say that c i s equidiameter

-f

o f d i a m e t e r d if yr s - x r s = d f o y a l l r and s ( t h u s e a c h c r i s a c t u a l l y

F(M;j) i s functorial on embeddings of manifolds, we can define

a cube, and a l l c

n F(Rco; j) = l i m F ( R ;j).

t h e r e i s some equidiameter c F(Ra; j) i s Z .-free and contractible. J

Corollary 4.6.

n F ( R ;j) i s isomorphic to Z

Lemma 4.7.

f R e f: F(Jn; j) (ii)

and each component of

i F ( R ;j) i s a contractible space. O

. Let

F = {(xi, 0

(j) such that g(c) = b; we can radially Thus de-

(j)

with maximal diameter subject to the condition g(c) = b.

. . ., x j ) 1

. . . < x.) c F ( R L ;j). J

Fo is

The continuity of f and g i s easily verified by use of Lemma 4.2, and f and

1 . F o i s one component of F ( R ;J )

a s follows.

and i t i s evident that the operation by Z

defines a homeomorphism from F X 2. to F(R ;j). O

Let c

Obviously gf = 1.

Define h:

<(j)

X I

rn(j)

(j) be described a s above, and let d be the diameter

of fg(c). Then define

F o r i I n 5 RI and j

homotopy equivalent to F(Rn; j).

n h ( c , u ) = < X cls(u) s=i

2 1, c n ( j ) i s Z -equivariantly j

ti i s an A

Therefore

operad,

for i < n

a locally (n-2)-connected 2=-free operad over a3

F(.Jn; j),

f(b) = c , where g(c) = b and c i s the equidiameter element of

g a r e clearly Z -equivariant. j

i s an E

(j) by the formula

clearly homeomorphic to the interior of a simplex and i s therefore contractible.

Theorem 4.8.

Obviously, for each b

expand the little cubes of this c until some boundaries intersect.

We shall also need the degenerate case n : = 1.

have the same sizie).

< a , and

cn i s

c r s ( ~ ) ( t=)

(Lo

,..., s X= i c.J S (u)>,

OSu<i,

where

I(~-U)(Y,, - xrS) + udIt f 7 (uyrS + (2-u)~~. - ud)

~ n ' w o r d s ,h expands o r contracts each coordinate interval c

operad. its mid-point to a coordinate interyal of length d.

linearly f r o m

It i s easy to verify that h

The second statement will follow immediately from the f i r s t

Proof.

; statement and the properties of the spaces F ( R ~j).

i s well-defined, Z.-equivariant, and continuous. J

We f i r s t consider the

Since h(c, 0) = c ,

h(c, 1) = fg(c), and h(f(b),u) = f(b), we s e e that I?(Jn; j) is in fact a strong c a s e n < co. a map g:

F o r convenience, we may a s well replace R~ by Jn. Define

( j )

nf i Z.-equivariant deformation r e t r a c t of 6,(j). NOWembed J~ in 3 by J i x (x, -) and let cr .: F(Jn; j) F(sn"; j) be the induced inclusion. Write 2 nlJ

J j) by the formula

(i)

g < cl , . . . ,

For c = <el,.

,...,

c.> = < C ~ ( Y ) 3

. ..!cJ. >

1 c . ( Y ) > , where y = ( -2 J

C,(j),

write c

= c

. . . X crn

,...,

1 n -2 ) â&#x201A;Ź J

, where c r s :J

. +

gn f o r the map g defined in (i).

Then the following diagram commutes:

Z .-equivariantly homotopic to r '. Let

+n: (f,a~")

(sn. e o)

be the relative homeomorphism defined by Toda [ 3 1 , p. 51, where i s the standard n-sphere and e = ( 1 , 0 , . 0

based maps

Thus we can define gOD = l i-+m gn' COD(j) F ( J m ; j). trivial homotopy groups.

Clearly

c_(j)

Toda has observed that, a s

sn,

-1

4JndJn (s19 - -

has

. . ,0).

sn C Rn t i

Sn+i) = ( s i '

S3' S4'

and

'nti ' 2)

&_(j)

It i s tedious, but not difficult, to verify that

nti

i s paracompact and E L C X and therefore has the homotopy type of a CWObviously, these maps lie in the same component of O(n) since they both complex, by Milnor [ 2 5 ,Lemma 41.

Therefore t O D ( j )i s contractible and have degree (-1)

is a C -equivariant homotopy equivalence. co j

and h = i

consequence of the theorem.

( h )( J )

Define rt n-1 ,j.

(n-1)-cube f to the.little n-cube

Cn-,(j)

Cn(j) by sending each little

1 X f, 1; J

Then

LT'

n-1,j

i s C.J

It sufficzs to prove that

1 (x) = (x,-) where u n- 1 2 T, rl: F(Jn;

maps (s,x)

J X J

g-i

n-I' *F(J*-'; j) * F ( J

They a r e thus connected by a path k: I

n- 1

O(n), where

1 n Define h = k(t)an: Jn -+ J ; then h = r t n o

Since each ht i s a tAomeornorphism, the product homotopy

( J ) restricts to give the desired C -equivariant homotopy j on F ( J ~ ;j). +

Barratt, Mahowald, Milgram, and others (see [24 ] f o r a

survey) have made extensive calculations in homotopy by use of the quadratic ;j). e

1 and U ~ - ~ (=X(-) , x ) on points x 2 F(?;

= TI

TI.

Remarks 4.10.

equivariantly homotopic to un-1, j. Proof.

n ~ ( n )acts a s usual on (S e ).

We shall l a t e r need the following technical lemma, which i s an easy

Lemma 4.9.

n- 1

Define

construction e[Sn , Z2 ,X] on a space X (see Nptations 2.5 f o r the definition). Since F(R"";

j) by the following formulas on points

2) i s Z2-equivariantly hornotopy equivalent to Sn ,

e[ Cnti(2), Z 2 , X] i s homotopy equivalent to e[Sn , Z

~ = -Jn:~

,x]. F o r odd primes p,

Toda [3 2 1 has studied the extended p-th power e[wn, Zp, X] on X, where if n i s odd

T(S,X) = (x, S) and r l ( s , x ) =

i s the n-skeleton of (1-s, x)

if n is even acyclic CW -complex.

then

TU'

n-1

= u

n-1

and r1a'

n-1

= u1 hefice it suffices to prove that n-1'

s O Dwith

its standard structure of a regular Z -free P n n tf W clearly maps Z -equivariantly into F ( R ;p) P

Iterated loop spaces and the

and we thus have a map

We h e r e show that e[wn. Zp, XI

~ [ F ( R ~ " p), , E P,XI

" e[ entl(p).

It appears quite likely that the successive quotients e[ the filtered space theory.

CnX

Ep, XI. f n ( j ) , E j , x ] of

will also prove to be useful in homotopy

acts naturally on n-fold loop spaces and that

this action leads to a morphism of monads

n n

n '

S2 S

The f i r s t statement

will yield .the homology operations on n-fold loop spaces and the second statement i s the key to our derivation of the recognition principle from tpe approximation theorem. We must f i r s t specify our categories of loop spaces precisely.

Let

z n , 1 s n 5 m, denote the following category of n-fold loop sequences. objectsof

such that

Y. = S2Y

a r e s e q u e n c e s {Y.I o ~ i ~ n o)r ,{Y.] i 2 0 ) it1

{gil

{ g i l ~ s i ~ on r ) Un:

The morphisms of

i ~ 0 if ) n = co, s u c h t h a t g. 1

The

ifn=m.

a r e sequences

Let

denote the forgetful functor defined by un{yi} = Yo and

Un{g) = go.

For

An n-fold loop space o r map i s a space o r map in the image

n < m , an n-fold loop sequence has the form {S2

n-i

Y), and

n serves only to record the fact that the space S2 Y

U {S~"-'Y) = a n y . n

does not determine the space Y and that we must remember Y in o r d e r to have a well-defined category of n-fold loop spaces.

We shall use the notation

n 52 Y ambiguously to denote both n-fold loop spaces and sequences, on the understanding that naturality statements r e f e r to r

isomorphic to J

qn. Of course, Xn

F o r n = co, it i s more usual to define a n infinite loop space to be the initial space Yo of a bounded a-spectrum

{(Yi. fi)l i 2 0) and to define a n

{g2:{(yi,fi)]

infinite loop map to be the initial map go of a map

of bounded Q-spectra (thus fi: Yi Qgitle f i i s homotopic to f i gi).

{(Y;, f i ) l

QYi 4-1 i s a homotopy equivalence and

The geometric and categorical imprecision

a n action 0 of co on maps in

WnY = (UnY. 8,)

Yo.

The actions 8 , 1 5 n

,..

; precisely, i f W

on objects, where

8 (1, Y) = y i s obvious. n, 1

for all purposes of homotopy theory; we can replace bounded Q-spectra and

diagrams of Lemma-I .4(a) commute,

x_,

P r e c i s e statements and related results

where

may be found in [191.

(sn,*)

a s the space of maps

We regard

n is where S

(X, *),

identified with t he quotient space I ~ / ~ I ~ . Theorem 5.1.

For X r

. .. , c

T,define

en,j : ~ ~ ( j ) x ( Q " x+Q% )j

Let c = < c l ,

. ..

If X = =I,

on Q'k.

en: C

Cn[

U Y n n

i s defined by

UnY i s the C -algebra n

An easy inspection of the definitions shows that the

then

nnx

thus define an action 0 and the Ont j n = Q n ' l ~ l via the correspondence y c r y1

X I; since cr (f) = f X 1 on little n-cubes f,

y(G)(t) = yl(u,t ) f o r (u, t )

en = en f l u n

follows.

If {yi)

rm,

then

therefore 8 = l i m 8 a = * immediate from the definitions.

en =

i s defined.

spaces to produce a morphism of monads C n categorical preliminaries. (1)

g:Hom

Bntl \:

ey0

We next use the existence of the natural

Cn(j) and let y = (yl. yj) E (Q%'-1 n Define 0 . ) to be yrcr on c (J ) and to be trivial on the complement n ,J n of the image of c ; thus, for v E S

follows.

naturally up to homotopy, and via weak

homotopy equivalences on objects.

a r e clearly continuous and 2.-equivariant, and The 0 n,j J

Proof.

that these two notions of infinite loop spaces and maps a r e entirely equivalent

maps by objects and maps of

5 m , a r e natural

structure map determined by the 0 and by Wn(g) = Ung on morphisms, n,j ' then W i s a functor from n-fold loop sequences to C -algebras. n n

I have proven 'in [ 1 9 ]

of this definition i s unacceptable for o3r purposes.

t'_ on

n' -+

Cn * E

and Yo The naturality statement i s

-action

Qn Sn

on-n-fold loop

We require some

We have the adjunction

(X,nY)+Hom

(SX,Y), $ f ( f ) [ x , s ] = f ( x ) ( s ) ,

where SX = X X I/++$ X I u X X a1 defines the suspension. By iteration of

Then the 9

enSj define

= 8 n+lcrn' If

~ ~ " x 1 .

where {yi) r

an action

. n'

- Cn+l

cn and

Tm,then the actions

(2) on Q"x.

enil en

If X = a x ' , then

i s the action of of

en on

n'no

Yo = Q Yn define

$fn:Hom

$, we have the further adjunctions

( x , ~ ~ ~ ) * H o(snx,Y), m

lsn<m.

It i s conceptually useful to reinterpret (2) a s follows. (3)

Qnx ' { Q

~ -/ 0~5 i~5 n)~ E xx n ;

then U Q X = n n

Define

QnsnX.

Since a morphism {g.

xn is determined by

O( i ( n} in

Hom

(s%,Y ) =

gn, we have

The

(x, u ~ { Q ~ - ~ Y } Hom )

(QnX, {an-'Y})

relationships between Q X and the suspension spectrum of X , may be m 'found in.[19

Clearly (4) and (9) state that QnX

$ pass to the limit case n = m. To s e e this, define

J-

(sn,s%),

then

un ( ~ ) = s ~ = ~ * ~ : s ~ A s 'n+l =X s ~= S+ R' X+ ~s S ' , f:sn+s%.

Thus each u

i s an inclusion, and we can define n m m n n QX = Q S X = lirn Q S X , with the topology of the union. -*

We shall use the alternative notations Since a map (8)

QmX= { Q S b l i 2 0 )

If {yil i 2 0}

Y = um{Yi} 0

i s the f r e e n-ford loop

(10)

(11)

pn = Qn$(iQnsnx):

(12)

J , then we

Xn +S2

En=

$'"(1

snx

): X

Vn:

if n < m ;

fins%

n n n s asns"x S

nns%

= lim

$, we have

if n < m ;

n n n = lirn p (which makes sense since Q QX = lirn Q S Q s%). .+ n * n n n Qn$"(lQnY):Q S n Y

any

m w em = lirn Qnp(l ):Q S Yo * nnyn n n By (5),(10), and ( l l ) , each u :Q S n 03 m n n and Q S = lirn Q S a s a monad.

have the commutative diagrams:

n ~ n [ 5 ] ,with VnY = (UnY, 5,)

Explicitly, i n t e r m s of iterates of the adjunction

and functors

are

$n yield monads

By Lernma 2.10, our adjunctions

objects.

Define

i s a map in

m m y m ; then Um QmX = Q S X.

and if f: X

1I n5m

(finsn, pn,

m m QX and Q S X interchangeably.

n n+l X, QQSX = CZX. QSX lands in some Q S

n sequence generated by the space X; i t i s in this sense that the Q S%

Geometrically, i f QnSnX i s identified with Hom

(7)

QX i s the evident inclusion.

$m i s a n adjunction, together with categorical

f r e e n-fold loop spaces.

;en

Therefore ( 2 ) may be interpreted a s defining an adjunction (4)

fgi) = go qm , where q03:X

A pedantic proof that

( QnX, { Q ~ - ~ Y } ) .

Hom

-1

Here

if n < m ; +

for

{Yi)

7, .

nn+l n+l S i s a morphism of monads,

Let (Cn, pn, qn) denote the monad associated to

Cm = lLm C a s a monad. n

n+l n+i+l

n t l n+i+l Q S X

(f)

> Qn+lYn+i+l

$ m : ~ o m , ( ~ , ~ m { Y i } )* Hom

n n+i (I): gm(,f)i = lirn Q $ 4

the definitions. ( Q x,{yi}),

mix

With these notations, we have the following

theorem, which i s in fact a purely formal consequence of Theorem 5.1 and

We therefore have the further adjunction (9)

c n , and observe that

* Yi

where

, i 2 0,

for f: X

Theorem 5.2.

For X

'"'" > nnns%

to be the composite map 'nX an: Cn

and i 5 n 5 m, define an: CnX

Q"S%

A> nns%.

Then

finsn i s a morphism of monads, and the following diagram of *

functors commutes, where a nl ' ( Y , f ) = (Y, Eman):

Moreover, the following diagrams of morphisms of monads a r e commutative for n < m, and a

i s obtained from the a

f o r n < m by passage to

limits :

6 Proof.

nfl

The fact that each pn, f n , and rn for the monad

nnsn

i s an

n-fold loop map and that Bn i s natural on such maps, together with the very definition of a natural transformation and of a n algebra over a monad, immediately yield the commutativity of the following diagrams for X

J :

Proposition 5.4.

The f i r s t diagram gives anqn = qn, the second gives pn an' = anCLn (Cnpn i s inserted solely to show commutativity), the third gives enan =

pn:

& .

a s required for a V v = Wn, and the l a s t gives cr a = a n n n n nttg.

ens

The f i r s t

S2Cn-1S such that a

observation about adjunctions and monads in any category

the natural transformations

.si:niC .si -c n-1 n-1

i2 a

7 implies that

a r e in fact m o r p h i s q s

of monads. Lemma 5.3. and let

Let

PI:

Hom

(C, p, q) be a monad in

where, f o r X

,,'ji and

Horn, (cX, Y ) be an adjunction, J Then (A C 2, p, 9 i s a monad in ,

(X, AY)

Proof.

c = <c

pn:

Define

..., J > E C.

C n (j),

pn.

Therefore a

Q Cn

an-'c1sn-'

. .,xJ. ) E

factors a s

Let

Xj , and l e t t

J and c " :J ~ - ' J"-'.

nnsn .

SX a s follows.

let x = (xl,

order, denote those indices r (if any) such that t

Let r l , c I (J). r

Write

....ri' in

Since the c r

have disjoint images, the little (n-1)-cubes c" 1 ( q ( i, have disjoint r ' 9 images. Thus we can define 6 by the formula n (1)

pn[c,x](t)=*

if t #

\iJ

c:(J),

and

r=l

a'-'(')

> ACZX.

C' i s a morphism of monads, then A@: h C Z

We must still construct morphisms of monads C -+aCn n

i s well-defined and continuous.

For v

n- 1 S ,

formula (1) and Theorem 5.1 give i

t[xr,s,u] (2)

4 { rq] .

AC'Z

i s also a morphism of monads.

for r

(sq) = t , 1 ( q i i, and t {C:(J) 4

It i s easily verified that

+ :C

n-1 S)

a r e the composites

and X

= (Qa

- ... -

CnX

c = c X c where c' :J r r r'.

if c:

Moreover, if

Cn * "n-1

analog in this case; consistency with limits i s c l e a r f r o m the l a s t diagram.

a co&posite of morphisms of monads

two diagrams a r e valid a s they stand f o r n = a,and the third has a n obvious

finsn factor We next show that the morphisms of monads a :Cn n i through niC .S f o r 1 ( i < n. The following elementary categorical n-1

n > 1 , there i s a morphism of monads

For

Q ~ s- al~ n [ c , x I ( t , v )=

, and, by Thus Q% ,S

the lemma, i t suffices to do this in the c a s e i = 1.

= cz :CnX n

*nnshx.

if c ( s , u ) = ( t , v ) r if (t,v)

The fact that

{ ~ r nc. i s a morphism of

monads can easily be verified from the definitions and also follows f r o m the facts that

and

S a r e inclusions for a l l X and that ru and SZa S n n- 1

a r e morphisms of monads.

Lemma5.7.

We conclude this section with some consistency lemmas relating Theorem 5.1 to the lemmas a t the end of section 1.

Let f : X - * B and g : Y - * B b e m a p s i n J .

d ( x xB Y) with s2%

These results will

PB nn Y an^ X

a s subspaces of

be needed in the study of homology operations; their proofs a r e easy veri-

k n -actions

8 and 8 n n

fications and yvill be omitted.

e i x en

nn(xx Y) = a % x n n y .

Lemma 5.5.

~' Let w: ( S 2 ~ x ) (A)

defined by w(f)(v)(y) = f(y)(v) f o r y

S2n(~(Yy

Y and v

be the homeomorphism

sn.

Then w i s a

) 'n(Y*A) on ( S ~ R X ) ( ~ ' *and

morphism with respect to the actions

C n-

Remarks 5.8.

~ (nn(S2X), ) en) i s a

given the product

C n-rqorphism, where

en =

cr on n

nn((;2x). Observe that

s28 corresponds to 8 cr' n ntl n

an-' ( a x x a x )

Similarly, we can give S2% Under the identity map on

agrees with

the inverse map

an-' c

may be

- nnx,

$ i s the standard product. Clearly an-'$ i s then a :S2%

-c

n- 1

-morphism.

Q ~ X , where

n-1 i s the standard inverse, and then s2 c i s a Cn-l-morphisrn.

c:nX

w transfers thk f i r s t coordi-

n+ 1 nate of S2 X (y above) to the l a s t coordinate.

In particular,

Lemma 1,9(ii) i s obviously inappropriate f o r the study of

SF'$: sZk x s2"x

In particular, w: $(a%),

Then the

the product on n-fold loop spaces for n < co. Observe that Q%

Q " ( X ( ~A)). *

where

n a r e identical.

X any.

Identify

and Lemmas 1 . 5 and 4.9 therefore The point is that the product and conjugation on H_(Q%) will commute with T

yield the following result. Lemma 5.6.

For X

7 , the following diagram is commutative,

528 = i s Z -equivariantly homotopic to 8 = n,j 8 n + l , j o c r L , j j nTj

jcrn,

any homology operations which can be derived from the action 8 of n-1 and

nnx.

Coupled with Propositions 3.4 and 3.10, the theorem yields the following

The approximation theorem

corollaries, which transfer our approximations for n = 1 and n = m from

This section and the next will be devoted to the proof of the approximation theorem (2.7) and related results.

The following more detailed state-

For X

to a r b i t r a r y A

and E

operads,

The r e a d e r should recall

on homotopy groups, and that two spaces X and Y a r e said to be weakly and n 2 1 , t h e r e i s a space E X which n

contains C X and t h e r e a r e maps sn:EnX n Zn: EnX

that a map i s said to be a weak homotopy equivalence if i t induces isomorphisms

ment of the theorem contains an outline of the proof. Theorem 6.1.

and

Cn-lSX

homotopy equivalent if there a r e weak homotopy equivalences f r o m some third

and space Z to both X and Y. Thus the following corollary contains the state-

PC2n - l ~ nsuch ~ that the following diagram commutes: ment that the James construction MX i s naturally of the same weak homotopy type a s nSX, f o r connected X; curiously, our proof of this fact uses neither classifying spaces nor associative loop spaces. Corollary 6.2

where, if n = 1, C SX = SX and a i s the identity map. o o

E X i s contractible n

Let X

be connected and l e t

be any A

operad.

Then the following natural maps a r e all weak homotopy equivalences:

for all X and s i s a quasi-fibration with fibre C X for all connected X. n n

h4x <-

CX <

"1 (CctC1)X

Therefore a i s a weak homotopy equivalence f o r all connected X and a l l n, n Corollary 6.3.

1S n S m .

Let X

CIX

be connected and let

> nsx

be any Em operad.

Then the following natural maps a r e all weak homotopy equivalences: We shall construct the required diagram and give various consequences and addenda to the theorem in this section.

The proof that E X i s contractn

ible and that a i s a .quasi-fibration for connected X will be deferred until n the next section, where these results will be seen to be special cases of more

i !

and, if 1 < n < m, (C X C )X n

n Of course, f o r a r b i t r a r y (non-connected) X, we can approximate 0 S%

general theorems. by S2Cn-lSX, since SX i s connected.

Corollary 6.4.

Let X r

and l e t

be any E

operad.

Construction 6.6.

Then the

a closed subspace A of X with

following natural maps a r e all weak homotopy equivalences:

follows.

,n n ~ ln i l

Qn2S Q(C X Cn)SX

QCnSX

f l l : s ~ n - l J ~ - ' ; if n = 1, then f = f . of

C n (j) X xJ

if x r

In these corollaries, all maps a r e evidently given by morphisms of

4 A,

structure i s only one small portion of the total structure preserved.

In [ 4

Barratt has constructed an approximation

m m 51 S 1x1 f o r connected simplicial s e t s X.

that

I I"xI

then we obtain an E

operad

If we define

and i t i s easily verified

03 m and Q S 1x1, f o r connected X.

a l l X, I' X i s a simplicial monoid, and if I'X -I-

generated by I' X, then shall describe

En(j; X, A).

1 I'X 1

For

denotes the simplicial group

i s homotopy equivalent to

qmsm1 x I .

We begin the proof of Theorem 6.1 with the definition of a functor En

denotes the cone on X.

(C;(O),

such that J 1) X :c (Jn-l) and

The equivalence relation

defined on

Define E,(X, A) to be the s e t

Since A i s closed in X, E (x,A) n

A), r in CnX and E ~ ( x

A) i s a filtered space with filtration

% . E,(x,

EnX will be the space En(TX, X), where TX

i s closed

defined by F.E (X,A) = E n ( X , A ) n F . C X J n J n

explicitly in section 15.

from pairs (X,A) to spaces.

topologized a s a subspace of C X. n

Thus Corollary 6.3 displays a n explicit natural weak /I' X

,. .,c.> , xl ,.. ,x.)

homotopy equivalence between

X, A) to be the subspace

j 20

-I-

i s homeomorphic to D 1x1 (where D denotes the monad in ,)

associated to /Q ).

J and

Implicitly, Barratt constructs

a "simplicia1 operad" consisting of simplicia1 sets D *Zj.

19(j) = ID Pj1 ,

j cn(j) X X in the construction, (2.4), of C X r e s t r i c t s to an equivalence n j 20 relation on

Remarks 6.5.

& ,(j;

Define

then the intersection in Jn of the s e t s

c (Jn) i s empty f o r a l l s

Clearly this implies that these maps a r e H-maps, but the H-space

We construct a space En(X, A) a s

consisting of a l l points ( < cl

monads.

F o r a little n-cube f, write f = f1 X f", where fl: J +

and, i f 1 5 n < m,

3 , by which we understand

Let (X, A) be a pair in

and F E (X, A) = o n

C A C E,(X,A). n

If f:(X,A)

Enf: En(X, A)

Clearly +

C n(j) X A'

C En(j; X, A) and thus

(X1,A1) i s a map of pairs, then

En(X1,A') i s defined to be the restriction of C f: C X n n

CnX1

to E (X, A). n The following results, particularly Lemmas 6.7 and 6.10, show that the definition of E (X,A) i s quite naturally dictated by the geometry. n

Observe that

E (X,X) = C X; at the other extreme, En(X, *) i s closely related to C n n n-1 X.

Let X

Lemma 6.7.

1 and let

[c,x]

En(X, *), where c = < c l , .

..,c.>J

(X, A)

- 4)' for some j 2 1. Then j = 1 if n = 1 and 6n-l(j) i f n >l. There is thus a natural surjective based c n = < ci'' ,...,c!I> J

and x

(Y ,*)

map v :E (X, *) n n than

Cn,lX

defined by the following formulas on points other

in the sense that Cn - l g o P n= "1 n oEnf for any such diagram.

(1)

v [c,x]= X 1

(2)

v ~ [ c , x ] = [ c ~ ~ ,Cn-lX x]E if n.1. Proof.

15 s

5 j.

then t

thus r

X = C X;

and

Let x = (xl,.

..,x.), xr J

Lemma 6.9.

- *.

Fix r

the definition of

n J ) and t r (c;(O), l ) , which contradicts and c"

have

Cel(j).

Proof.

Let n: (X, A)

posite map E (x,A) n

En"

embed X in TX by x natural map. N

(Y, *) be a map of pairs in 'j-

E~ (Y, *)

> 'n-1 Y

commutative diagrams

Define

[x, 11; SX = TX/X

q:TX +P S2n-1 s?X

and a: TX

by the formula

n- 1 for [ X , S ] ETX, t ~ 1 , ' a n dV E S

Then the following diagram commutes and the result follows:

is natural on

and

SX denotes the

Then the com-

will be denoted nn.

Since En i s a functor and v is a natural transformation, n

'n-l?

Define the cone functor T by TX = X X I/* X I u X X 0,

qn[x,s](t)(v)=[x,st,v]

Notations 6.8.

E1(j; X, *);

& n(j; X, *). Thus the little (n-1)-cubes c:

disjoint images and c"

If n = 1,

C;(J).

s is impossible if n = 1 and therefore j = 1. If n > 1 and if then ( t ) E (

and n 2 1, there is a natural commutative

C cnx -----------+ E,(TX,

s, 1 5 r (j and

(cr(0),1) n cs(J), which contradicts the definition of

c:(Jn"),

diagram

F o r definiteness, assume that c1(0) 5 c ~ ( 0 ) . Let t

cll(Jn-l). r

For X

Since a factors a s the composite 8 C n h , the lemma gives half of n n the diagram required f o r Theorem 6.1.

v r c"(J"-').

The following sharpening of Lemma

5.6 will lead to the other half of the required diagram and will also be needed

Thus the f i r s t and second parts of the definition of

in the study of homology operations on n-fold loop spaces. For X r

...

. .., J

y = (Y~,

0 .: E ~ ( ~ ; P Q ~ - ' x , Q % ) n2 J Let (c, y) r & n( j ; an-'X,Q%), ~ where c = < c l , ,c j> n-1 y.). F o r t r I and v r S , define

Lemma 6.10. a s follows.

7 , define

yr(s)(u)

i s continuous.

and

for u

n- 1 , and i t follows that a1

By comparison with Theorem 5.1,

c n (j) x (fid. pan-'x

yr(s)(u) =

Of course,

joint domains.

have dis-

en,

en, --

On, j

i s defined in Lemma 1. 5, and the commutativity of the The commutativity of the triangle i s

bottom square follows from that lemma.

immediate from the definitions, since u1 n-1,j

i s given by f

1 X f on little

(n-1)-cubes f.

if c r ( s * u ) = (tsv)

Lemma 6.11.

For X r y

, the

en, N

maps

hn(j,pn"-'~,S2%)

pan-' X

induce a map 0n :En ( P S ~ ~ - ~ X , C ~ % )

pnn-lX

commutative (where, if n = 1, 0 = 1 :X

otherwise

such that the following diagram i s

X):

Then the following diagram i s commutative:

Proof.

en, . ( c r yy) =

.-J

0 .(c, uy) and 0 l ( u i ~ y) , = 0 .(c, siy), in the n, J n, jnyJ rcl

notation of Construction 2.4, and therefore 0 i s well-defined. n . m

1:mma

implies that 0n

on Cnn%.

The previous

Clearly

i f c U ( u ) = v and yr{S2% rJ

pen[c* yl(v) = otherwise Proof. y

# nnX, r

The definition of

& n (j; P S ~ ~ ' ~ X , Q %gives ) that

then no element of c (3") 9

if q # r and

has the form (t,v) with t 2 c1(1) and r

By the definition p = v " E p and by the definition of n n n .-

and of

in Theorem 5.1,

pen

follows.

v n in Lemma 6.7

Define

2n = en.

E,%:

EnX = E (TX, X) n

~ 8 -STIX. l Then the com-

mutativity of the diagram in the statement of Theorem 6.1 results from

Let X

Corollary 6.13.

7 be connected.

Then there i s a natural

commutative diagram, with isomorphisms a s indicated:

Lemmas 6.9 and 6.11. We complete this section by showing that our approximations relate nicely to the Hurewicz homomorphism h and to the homotopy and homology suspensions S*.

Recall that we have morphisms of monads

N, n where NX i s the infinite symmetric product on X; by abuse, if n = 1, E h e r e denotes the evident composite C

we may identify S* =

a -1 :ni(NX)

ni(NX) with H.(X), and then -+

ni+l (NSX), where

of the quasi-'fibration

Nn: NTX

F o r connected spaces X,

h = q*: ai(X)

n.(NX)

:ni(X)

nitn(Sk)

i s the homotopy suspension.

and

denotes the connecting homomorphism ..

NSX with fibre NX ; proofs of these results

Proof. monads.

The triangles commute since

run and E

a r e morphisms of

The upper square commutes by the diagrams of Theorem 6.1 and

the lower square commutes by the lemma applied to X C TX and a: TX

may be found in [lo]. Lemma 6.12.

where a-no

Let n: (X, A)

& denote the composite E (X, A) n

(Y, *) be a map of pairs in J CnX

diagram i s commutative where, if n = 1,

NX. â&#x201A;Ź = q: Y

, and l e t

Then the following

NY.

the augmentation &: CIX

MX;

Gray [14] has made an intensive study of

M(X, A), which he calls (X, A) cn ' The natural map

p: M(X, A)

-c

SX.

Let M(X, A) denote the image of the space E1(X,A) under

Remark 6.14.

al: E1(X, A)

T:X

X/A

induces

X/A, and n 1 clearly factors (via e ) through a map

X/A.

If A i s connected (and if the pair (X, A) i s suitably nice),

then, by Theorem 7.3 and [I+], al and p a r e quasi-fibrations with respective Proof.

The commutativity of the left-hand square i s obvious and the

fibres C 1A and MA, and

commutativity of the right-hand square follows easily from the definition

Proposition 3.4; therefore

F o r n = 1, the crucial fact i s that at most one coordinate x of an r element [< c l , ,c.> ,x l , ,xj] r El (x, A ) i s not in A.

e quivalence.

of n n.

.:.

...

&: C 1A 8:

MA i s a weak homotopy equivalence by

E1(X, A)

M(X, A) i s also a weak homotopy

C o f i b r a t i o n s and q u a s i - f i b r a t i o n s

We prove h e r e t h a t En(X,A) i s a s p h e r i c a l i f (X,A)

i s an NDR-pair

such t h a t X i s c o n t r a c t i b l e and t h a t , f o r a p p r o p r i a t e NDR-pairs t h e maps an:En(X,A)

Cn-l(X/A)

and Cmn:CmX

f i b r a t i o n s w i t h r e s p e c t i v e f i b r e s CnA and CmA.

t (1-vi ( c ) )

,G i s well-defined

Applied t o t h e p a i r s

if 0

vi ( c ) ( 1

i f vi(c) ) 1

(X,A),

a r e quasi-

Cm(X/A)

i f vi ( c ) ( 0

since, a s is e a s i l y v e r i f i e d , vi(c)(l

n- 1 ) n ck (3") ( ~ ; ( 0 ),1)x c i (J

implies t h a t

i (and t h u s t h a t t h e

i s empty f o r a l l k

(TX,X), t h e s e r e s u l t s w i l l complete t h e proof of t h e approximation

ithc o o r d i n a t e i n X i s u n r e s t r i c t e d ) . G s t a r t s a t t h e i d e n t i t y and ends

theorem.

i n Fj-lEn

They w i l l a l s o imply t h a t n,(CmX) i s a homology t h e o r y on

connected s p a c e s X (which, a f o r t i o r i , i s isomorphic t o s t a b l e homotopy t h e o r y )

Theorem 7.1.

L e t ( X I A ) be an NDR-pair i n

7.Then

Proof.

By Lemma A.5 a p p l i e d t o (X,A,*), t h e r e i s a r e p r e s e n t a t i o n

(h,u) o f (XI*) a s an NDR-pair such t h a t h(IxA)

c A.

the representation

(cj .? i (F.C XIFj-lCnX) of j I n

a s an NDR-pair which j was d e r i v e d from ( h . , u . ) i n t h e proof of P r o p o s i t i o n 2.6 r e s t r i c t s t o a I I r e p r e s e n t a t i o n o f ( F .E ( X I A ) ,F E (X,A) ) a s an NDR-pair. The c o n t r a c I n j-1 n t i b i l i t y s t a t e m e n t i s more d e l i c a t e . Indeed, my f i r s t proof was i n c o r r e c t and t h e argument t o f o l l o w i s due t o Vic S n a i t h . g:I x X

4(1,x) =

X be a c o n t r a c t i n g homotopy, g(OIx) = x , g ( t , * ) =

Let

c A.

c;: J -+ J , and d e f i n e j > E C n ( j ) , w r i t e ci = C!1 x c:, vi (c) = 2 max, ( c i (1)- c j ( l l ) / X (c) where X ( c ) = min ( c i (1)-c; ( 0 ) ) kfl: l k <j

x(FjEn(X,A)-Fj-lEn(XIA)) +F.E (X,A) by

G ( ~ I [ ~ I x ~ -, .x j l ) = [ c I g ( t l I x l )

,..., g ( t .I, x j

t h e n any H:I

< 1 suffices.

FI. En (x,A)

0 I f~ X 1i s )

by an e a s y e x e r c i z e i n p o i n t - s e t and

h ( t , C a , s l ) = C j ( t , a ) , s l , and g ( t , C a , s l ) = C a t s - s t ] ,

uCa,sl= v ( a ) . s t

Define a homotopy

F.E I n (x,A)

by H ( t , z ) =

I n

3 n (X,A)-F j-1 En ( X , A ) . f o r C c , y l ~F.E

-u j1 [ 0 ~ 1 ) and, F j-lEn

for z

Fj-lEn(X,A)

by t h e f i r s t p a r t ,

i n F I.En ( X I A )

(x,A)

Then H deforms F.E I n (X,A)

-u j 1 ~ 0 , 1 ) can be

and by

E (X,A) n -+

into

deformed i n t o

. I t f o l l o w s t h a t each FI.En

(X,A)

t i b l e , and t h e argument g i v e n by Steenrod i n C30,9.41

For c = < cl...,c

D e f i n e a homotopy G : I

Now assume

topology; i f X = TA, ( j ,v) r e p r e s e n t s (A,*) a s an NDR-pair,

f:sq

and

C l e a r l y g c a n n o t i n g e n e r a l be s o chosen t h a t g(IxA)

Note,

By Lemma A.4, ( h , u )

determines a r e p r e s e n t a t i o n (h , u . ) o f ( X I * ) ] a s an C - e q u i v a r i a n t j I j NDR-pair. S i n c e any c o o r d i n a t e i n A remains i n A throughout t h e homotopy h

however, t h a t G cannot be extended o v e r a l l o f F..E ( X , A ) . I n t h a t t h e r e e x i s t s E > 0 such t h a t g ( I x U - ~ [ o , E I ) c ~ ~ compact, t h e n t h e r e e x i s t s such an

( i ) (F .E (X,A) IFj-lEn (XIA) i s an NDR-pair f o r j > 1. I n (ii) I f X i s c o n t r a c t i b l e , t h e n En(X,A) i s aspherica1,and E (X,A) i s n c o n t r a c t i b l e i f X i s compact, o r i f X i s t h e cone on A , o r i f n = l .

s i n c e vi ( c ) ( 0 f o r a t l e a s t one i and each c.

(X,A)

i s contrac-

shows t h a t

i s c o n t r a c t i b l e . For a r b i t r a r y c o n t r a c t i b l e X I a map E ~ ( x , A ) h a s image i n F.E I n (Y,A

i s such t h a t g ( 1

n Y ) f o r some j and some compact

u - ~ c o ~ ~c u Y 1[0,1) )

t h e n t h e homotopy

H above deforms FI.En (Y ,Any) i n t o ~ - ~ C 0 , 1 i)n F3.En ( X , A ) t h a t f i s null-homotopic.

Cl(j)

i t follows

Thus En(X,A) i s a s p h e r i c a l . F i n a l l y , i f n = 1,

t h e n we can w r i t e p o i n t s o f E1(X,A) v a l s ci of c

, and

i n t h e form [ c , y l where t h e i n t e r -

a r e a r r a n g e d i n o r d e r (on t h e l i n e ) ; t h e n t h e

r e t r a c t i n g homotopy f o r X

I o b t a i n e d from hj-l

on ~ j - ' and g on X

) 1 , where

by Lemma A. 3 can be used t o deform F .E (X,A) 3 1

i n t o FjelE1 (X,A)

Recall that a m a p p: E

onto and if p*: ni(E, p

-1(x), y)

o r groups) f o r a l l x

B, ye p

t o be distinguished if p: p

ni(B, x.) i s a n isomorphism (of pointed s e t s

- 1(x),

- 1(U)

Proof.

B i s said t o be a quasi-fibration if p i s

and i 2 0.

A subset U of B i s said

U i s a quasi-fibration.

The following lemma,

(i).

The maps n a r e defined i n Notations 6.8. n

n = 1, recall that C~ (X/A) = X/A

and define F ~ ( x / A ) =

F o r the c a s e

and F~(x/A) = X/A.

The proof for n = 1 will be exceptional solely in that we need only consider the f i r s t filtration, j = 1 below, and therefore no special argument will be FOGnm1(x/A) =

i s obviously distinguished, and we must f i r s t show

which results f r o m the statements [ 10 ,2.2, 2.10, and 2.151 of Dold and Thom,

given.

describes the basic general pattern f o r proving that a map i s a quasi-fibration.

that any open subset V of F j C n - l ( ~ / ~ )F j lCn l ( ~ / A ) i s distinguished

Let p: E

L e m m a 7.2.

B be a map onto a filtered space B.

Then

each F.B i s distinguished and p i s a quasi-fibration provided that J (i) (ii)

F O B and every open subset of F . B F B f o r j > 0 i s distinguished. J j-1 F o r each j > 0, t h e r e i s an open subset U of F.B which contains J -1 F B and t h e r e a r e homotopies ht: U U and H :p (u) p-l(~) j-1 t

By use of permutations and the equivalence relation used to define En(X, A), any point y e n - l ( v ) m a y be written in the followand by the definition of n n' n ing form: (2)

y = [ < c , d > , x , a ] , w h e r e c = < c 1' x

(x-A)',

and a

...,c J. > e

Ak; h e r e if c r = c 1 rX

section of ( crl (0), 1 ) X c:(Sn-$

such that B, and h l ( U ) C F B: j-1

(b) Ho = 1 and H covers h, pH = h p ; and t t 1 1 (c) H1 :p (x) p (hl (x)) i s a weak homotopy equivalence f o r a l l x e U.

,..., T c $ , ( k ) ,

, c 1r :S -* S, then the inter-

and ds(S?

i s empty,

..,c!I>J

nn(y) = [cl1,$(x)] e V, where c " = < c l i , . j-1

Cn(j),d=<dl

(a) h o = l , h t ( F j - l B ) C F

1 Define q:n n (V)-).CnA by q ( y ) = [d,a] f o r y a s in (2). that q i s well-defined and continuous.

and n-l(j). I t i s easytoverify

We c l a i m that anXq:an-'(V)

+ V X CnA

i s a fibre homotopy equivalence, and this will clearly imply that V i s disThe notion of a strong NDR-pair used i n the following theorem i s defined i n the appendix, and i t i s verified t h e r e that (M ,X) i s a strong NDR-pair f o r any f map f: X + Y . Theorem 7.3. A i s connected.

Let (X,A) be a strong NDR-pair i n

Let n: X

X/A

be the n a t u r a l map.

tinguished.

nn: En(X, A)

(ii)

Ca, s:CaoX *,,Cm(x/A)

i s a quasi-fibration with fibre C A. 03

Cnl *

n and r-:

t 1 u (f) = g X f on little (n-1)-cubes f, where g ( s ) = $l+s),

(4)

T-(f) = (g- X in-')f

1 on little n-cubes f, where g-(s) = -26 ,

- 1(V)

C n - l ( ~ / ~ i)s a quasi-fibration with fibre CnA ;

(3)

Then define w: V X CnA * rrn (i)

t, 6, +

by the formulas

, and a s s u m e that

Then

Define morphisms of operads

by the formula

g (J) =

1 (?, 1).

1 g (J) = (O,jJ).

(5)

w([c", sJ(x)], [a, a]) = [< u ( c D ) T-(d)>,x, * a], x

J J (P , v ) by s, and l e t (h.,u.)

- A)',

c n ( k ) , and a

The definition of u+ and

T-

(for any k 2 0).

tinuous and fibrewise over V. CnA

Since 1

-1 (V). n

Now ( s X q) w i s the map 1 X n

via the homotopy induced from f

s entdtion of

Clearly w i s con-

C A i s the associated morphism of monads to n T

a s an NDR-pair induced from

T-:

T-,

tion retraction h.: J IX U

on little n-cubes f,

Let (h., J u.) J be the repre-

where

1 where g-(s) = (s Zst), ( s X q)w i s fibre homotopic to the identity map. t n 1 (V) written a s in (2), we have points y E s n

*lJ

( Fj C n - l ~ / ~F,~ - , C ~ - , X / A ) a s an NDR-pair given by Proposition

then ff.(x) J < 1 if and only if x

2.6;

Cn+Gn.

( g ; ~1"-')f

and (1j' v.) J be the representations of (X/A,

j and (X, A) a s NDR-pairs given by Lemma A. 4.

ensures that the little cubes on the right satisfy

the requirements specified in (2) f o r points of a

T-:

Let (h,u) be the representation of (x/A, *) where c" r c n - l ( j ) ,

- 1(U)

I X sn

U, and h.J r e s t r i c t s to a strong deforma-

U of U onto F j - l C n - l ~ / ~ . Define

1 j-1 En(X,A), sn (U) by ?j(t, y) = y for y r F

where

- 1( F j - l C n - l ~ / ~ ) and , by the following formula on points

F ~ - ' En (X, A) = nn

- 1(U) - F ~ - ' E ~ ( x A) ,

y r sn

written in the form (2)

z J ( t , d = [ < c, d>,P J.(t, x), a1

( 6)

w ( s n X ~ ( Y= I[< 0- (c*'),~ - ( d ) >x, , a1 Construct a fibre-wise homotopy 1 s w ( s X q) by deforming d into T-(d) a s n above (without changing c , x , o r a) during the f i r s t half of the homotopy and then deforming c into

+ u (c")

by deforming each c1 linearly to gS (without It i s easily

little cubes of points of T - ~ ( v ) a r e preserved throughout the homotopy. n s X q i s a fibre homotopy equivalence and V i s distinguished.

Thus

Define U to be the union of F C (x/A) j-1 n-1

... ,T(x~)]I x r

1%:

7. 2, it suffices to prove that if x

B f o r a t l e a s t one index r}.

with

Since

i s a homotopy equivalence. trivial for x r F x r u -

C (x/A). j-1 n-1

C (x/A), Fj-l n-1

A, and clearly

(u)

onto F

j-1

c-4

covers h. and J

E ~ ( XA). ,

By Lemma

P jl(xl,.

Thus consider a typical element

say

.. x .J) = (xi . .. x J )

Some of the

x1 lie in A.

mutations and the equivalence relation, we may assume that x ' and x'r E A for i < r

Let

1 U and x' = h (x) , then nn (x') j1 j 1 En (X;A), this i s i s constant on F

7:;

-1 sn (x)

x = [ c n , ~ ( x l ) , . ,T(x.)], where c l *= < c ln" s ' yc!'> a n d x J J

It remains to

C (x/A) in F.C (x/A) and deformaconstruct a neighborhood U of F j-1 n-1 J n-1 -1 (U) which satisfy the conditions of Lemma 7.2(ii). Let tions of U and of T n 1 (1, v ) represent (X, A) a s a strong NDR-pair, and let B = v [0, 1); by

T(X

i s a strong deformation retraction of s -1 n

verified that the disjoint images and empty intersections requirements on the

{[c.,

i s well-defined since P (t, a ) = a f o r a

changing T-(d), x, o r a) during the second half of the homotopy.

definition, P (I X B) C B.

j P'+

5 j (i

may be zero), and then

X-A.

By use of per-

for r

Consider the following diagram:

Eh(X, A) since the contractibility proof of Theorem 7.1 does not apply to Eh(X, A) ; a fortiori, these spaces a r e weakly homotopic equivalent and can be used interchangeably.

We now have commutative diagrams u Eh(X, A)

> E1 nS1 (X, A)

Here the q and w a r e defined precisely a s in the f i r s t part of the proof, and n X q and w a r e inverse homotopy equivalences. We shall construct a n -1 X 1). This will homotopy H: I X (x X CnA) n (xl) from 1:- w to w o(< n jl

and

imply that

2'1

i s homotopic to the composite of homotopy equivalences

w o(h X 1) 0 (nn X q). j1

Since A i s connected, we can choose paths p r '' I

connecting x 1 to + f o r i < r 5 j.

Define H by the formula

-!-

Clearly H i s well-defined, and H = P j w and H = w (h X I ) a r e easily o 1 1 j1 verified from (5) and (6). This completes the proof of (i). Define a subspace t:(j

;X , A )

En(j; X,A) of

= {(< c1,.

where g-!- i s defined in (3).

En(j ; X,A) by

..,c.>, x1, ... .xj) 1 J

= g if xr/

Let E ~ ( x A) , denote the image of

A],

~ h (X,~A) ; j20

i n E ~ ( XA), , and let nr.: E t (X, A) n Eh(X, A). that

Cne1 (x/A) be the restriction of

With a few minor simplifications, the proof of (i) applies to show i s a quasi-fibration.

We have been using E (X, A) rather than n

f, and i i s the

E h + l ( X , ~ )was introduced in o r d e r to ensure that

Cn+ln o i = u

-!-

have that u t n

-!-

inclusion.

nt n+l '

Lemma 4.9 implies that cr

u' :CnX n

naturally in X, and, since u1 (c) = 1 X c on little n-cubes n

(ii).

-!-

where u -!- i s defined by u (f) = g X f on little n-cubes n n

uf' n. CnX

Cn+lX, naturally in X.

CntlX,

c, we evidently

Now pass these diagrams

f observing that cr For n' n+lun = un+l 1 (C n) (x), we have a commutative diagram

to limits with respect to the u x e c ~ ( x / A ) and y

By Proposition 2.6 and the homotopy exact sequence of the

Proof.

quasi-fibrations cone of f:X

* Y,

connected .X.

C X m a*(C

C M c o f

C T , where T f = M ~ / X i s the mapping m f

m X) satisfies the axioms f o r a homology theory on

Since suspension preserves cofibrations and looping preserves

fibrations, a%(QCrnSX) satisfies the axioms f o r all X.

The natural weak

homotopy equivalences of Corollaries 6.3 and 6.4 clearly allow us to t r a n s Clearly st i s still a quasi-fibration; since u a,

composite iu'

and the bottom map uS

on homotopy groups (ox sets). (Cms), right.

on the left.

a s well a s s1 induce isomorphisms ' my

Since a a t *u

Since u

n1 m:: m::

(Cm-ir)*: "*(C,X'

(C,")

uf naturally, both the top n'

about (Can)*

on the

It follows that ( 4 s Y)

s * ( c r n ( x / ~ ) ,x)

i s an isomorphism f o r a l l x and y, which verifies the defining property of a quasi-fibration. The second part of the theorem has the following consequence. Corollary 7.4. theory on connected X X

F o r any E E

tf , -rr,(CX)

operad

defines a homology

and a,(QCSX)

defines a homology theory on a l l

? .

These theories a r e isomorphic to stable homotopy theory, and thp

morphism of homology theories Hurewicz homomorphism.

(Qa! S), 03

i s a monomorphism, so i s

i s an epimorphism, so i s

-1

f e r the result to a r b i t r a r y E

*: P*(CX)

n*(NX) i s precisely the stable

operads

and the maps ( a )* and m

define explicit isomorphisms with neS(x) = s%(aX). The statement

&+ follows immediately from Corollary 6.13.

The smash and composition products

Lemma 8.1.

en: Cnnnx

nnx

and an: CnX

nnsnx

7 ,the following diagram i s commutative:

nmX x nmx x nny

The purpose of this section i s to record a number of observations relating the maps

F o r XI Y

* nmx x a n y

to the s m a s h

and composition products, and to make a few remarks about non-connected spaces.

The results of this section do not depend oen the approximation

theorem and a r e not required elsewhere in this paper; they a r e important

nmx x nnu x i"x x n% a-

in the applications and illustrate the geometric convenience of the use of the little cubes operads. We identify

sn = In/

n (sn, X) of based maps S

X with the space Hom

81n, and we write S for the inclusion

nntlsx

AXA

mtn

(XAY)x nmtn(xA Y) ,

where A i s the diagonal and t i s the switch map. +

given by sus-

Now the loop products in this diagram a r e given by 9

c = < g X1

m-1

, g X1

m-1

> e t m ( 2 ) with g- and gt

(c), where m12 a s defined in formulas

pension of maps. F o r XI Y e

(7.3) and (7.4), and the lemma generalizes to the following computationally

, the s m a s h product defines a natural pairing

important result. $ 2 5

x nny

nmtn(x A Y); explicitly, Proposition 8.2. (f A

for f e

Sf%,

m n g e a n y , s e I , and t e I

f $ : ~X% nmx nmx +

then, for f

g)(s,t) = f ( s ) A g(t)

F o r X, Y e

Jr-

and a l l positive integers m, n, and j,

the following diagram i s commutative: Observe that if m > l

andif

denotes the standard (first coordinate) loop product,

f e n ? X and g e n n y , we have the evident distributivity formula 1' 2

$ ( f l I f 2)" g = f$(f1'\gsf2"g)

Diagrammatically, this observation gives the following lemma.

where A i s the iterated diagonal and u i s the shuffle map.

Proof.

We must verify the formula .(c,xl 'm, J

f o r x. s

5 2 5 , y

m I and t

,...,x.)n y J

= B

Q ~ Y and , c = < c 1'""

(where we have identified S%

.(c;xln Y m,J c

,...,x j ~ y )

(j).

By Theorem 5.1, if

n I , then

sm*sn =

smCn(x

S ~ Y =X

smt, Y A sn

Y) via the map 1 A t

with

1).

We can stabilize the s m a s h products of the previous proposition, up to homotopy, by use of Lemma 4.9 and the following analogous result on change of coordinates. Let X

Lemma 8.4. by letting S t f , f

..., x ~ ) ( s ) A~ ( f ) .

Visibly, this a g r e e s with e m , J.(C.X,,

s n =S'A

'j-

Define St:52n-1 Sn-1 x

ans%,

n 2 1,

n-1 n-1 S X, be the following composite:

sn-l

1Af 1 -----+ S x Asn-'

1 > x4s1t,

sn-l = s%.

An equally trivial verification shows that we can pull back the s m a s h product along the maps Proposition 8.3.

Then S t i s homotopic to S, where Sf = f

cun i n the sense of the following proposition. Define a map

A :CmX X CnY

Proof. -+

Cm+n(X A Y)

Let

T ,T I :

sn sn

and h:

constructed i n the proof of Lemma 4.9. the following formula on points [c, x]

,...,c.>E tm(j), J y = ( y l , . ..,yk) yk:

c = <cl and

x = (x,

and

Cm X and [d, y]

,...,x.)E J

x',

d = < d1'

CnY, with

...,\ > e

Hs(f):

tn(k),

be the composite

h -l s

1~ f ,SIAXASn-l

[c,xl where

[d, yl = [e, 21,

,..., clX \ , . . . , C . JX ~ ~ ~ . . ~ ~ CJ . X \ > z = ( X ~ A Y ~ , ,. x. 1. ^ y ,,..., XJ. A Y1 9 . . . , 3~ . h ~ k )

Then Ho(f) =

e = < c1X d l

1 and H (f) = (t 1 ,

1: sn

T 'J -rt

For f

t~ 1

S~X.

be the maps and homotopy E

n-1 n-1 S X and s

I, let

1 h hs > X A S ~ >xn,sn=s%.

1) o ( l r\ f), a s required.

Of course, it i s now c l e a r that the n suspension maps 52n-l n-1 S X

5 2 " ~ " ~ and Cn 1X

Then the following diagram i s commutative: coordinate a r e all homotopic.

C X obta'ined by the n choices of privileged n

It follows easily that the s m a s h products of

Proposition 8.3 a r e consistent under suspension, up to homotopy, a s m and n vary.

We next discuss the composition product.

sn .-+.sn

of based maps

regarded a s a topological monoid under composition rJ

of maps.

Let ~ ( n denote ) the space

Let F . ( n ) denote the component of F(n) consisting of the maps

of degree i.

As usual, we write

For X e

7 and

F ( n ) = F ( n ) u ? (n) and S F ( a ) = F l ( n ) . 1 1 nl n n F(n) may be identified with Q S , and then, by (5.6), ~ : y ( n ) F ( n t 1 ) I n n ntln+l N agrees with cr :Q S $2 S We write F f o r the monoid lim F(n) n 4

) 1,

define Pcn: PC?% X B(n)

by PC (x. f)(t) = c,(x(t), f) for x e pa%, f e F(n), and t r I. n n+l X X ~ ( n i) s the composite restriction Rc of Pcn to Q n

nnt1x x F ( n )

Lemma 8.5.

nnt'x x F(n+1)

Cntl

PQ%

Then the

,nn+'x

and the following diagram i s commutative:

r.J

and identify F with QSO a s a space.

For X r

, define

to be composition of maps. n on the space Q X.

Then c

is a right action of the monoid F ( n ) The precise relationship between the s m a s h and composition products

The diagram i s given by the following evident interchange formula.

Lemma 8.6.

For x

Qmx, y e #Y,

'iJF(m), and g

c m (x,f ) A cn(y, g) = c m t n ( x ~Y,

i s evidently commutative f o r a l l n L 1.

Therefore, if

[yi) r

xCD,

then

the maps

Lemma 8.7.

QSOX?

:Yo X F + Y o

on Yo.

QSO coincides with the composition product on F.

The

composition product enjoys another stability property, which i s quite analogous to the result of.,Lemma 5.6.

*-r

F o r f e g ( m ) and g r $(n), we have the formulas A

n g = S f 0 (sqmg

m n n since ( s ' ) ~ = 1~ A g and S f = T A 1

Of course,

homotopic, and both products a r e weakly homotopy commutative. Proof.

a right action c

p(n),

The composition and s m a s h products on F a r e weaMy

( ~ ' ) osnf ~ g = f induce

Lemma 8.4, and the result follows.

S and Sf a r e homotopic by

We shall obtain an enormous generalization of this l e m m a in the = a d p a p e r of this series.

There i s an E

operad

such that

acts

03 N

on F (so a s to induce the s m a s h product) in such a manner that the composition product

i s a morphism of

dg -spaces.

'Xu

Cm(j)

and g

Lemma 8.8.

F(n-1).

For X

Diagrammatically, this gives

7 , the following diagram i s commutative

Cm(j) x (a"'%)j

the composition product, namely

. x F(mtnlJ

1 x c','~

-, X

. c +

For X

, let

the following diagram i s commutative:

and all positive integers m, n, and j,

The following reinterpre-

denote the wedge of j copies of

and y = (yl,.

..,y.),J

a%.

Then

X i s the composite

sn-Ljs

ylv...vyj

jx-x, P

where T i s the pinch map defined by F(v) =

n and u, when c(v) = u in the rth copy of S

We next describe C C

m( j ) x ( a m f n x ) j

denote the folding map, the identity on each copy of X.

Proposition 8. 2, simply by writing down the definitions. J

will aid i n the proof. tation of the definition of the maps 8 n, j

.(c, y): sn en, J

but this fact i s slightly l e s s obvious. maps a n'

Let c = c

For X

We can pull back the composition product along the approximation

X and let p: J~

Proposition 8.9.

'm+n

amf%

x F(mtn)j

(am'%

Lemma 8.10.

The following generalized distributive law i s proven, a s was

. nmf % x F ( m t n )

~ X ~ X A ( S ' ) ~

Of course, t h e r e i s a distributive law relating the lpop product jd to

for f ,f

x g(n)

(j) x (nrnf%$

and

%: CnS

unless v = c (u) for some r

n n S ; these maps play a cen-

tral role in the homological applications of o u r theory.

, CS0

Lemma 8.11. F o r any operad joint union of the orbit spaces 0

If S

Proof. 9

and 1, then any point of CS other than

can be wi.itten in the form [c, lJ], c Lemma 8.12.

for j 2 0.

t(j)/xj

has points

(j).

Consider an: CnsO

i s homeomorphic to the dis-

dsn.

For c

t n ( j ) , write

The result follows easily from the definition of y, in 4.1.

Then ac (c) i s the composite n

Note that, in contrast to the s m a s h product, the following diagrams a r e commutative f o r a l l n:

Proposition 8.13.

Define a map c :CnX X CnsO n

k k cn([cyXI, d) = [y(d, c 1, x 1, for c

,(j),

x = (x,,

...,x.)J

xJ1and

diagram i s commutative for a l l n , 1 S n

CnX by

n(k).

Then the following

CnX

and

nnsnxX F(n)

nns%

n C

Cntl

'nflX

1Sn+lX x F ( n f 1 )

'n+lx

< co:

a :CnX n

Of course,

-n

nfl

n4-1Sn+lX

fails to be a weak homotopy equivalence f o r

n n non-connected spaces X, essentially because s (52 S X) i s a group and we 0

have not built inverses into operads.

Conceivably this could be done, but the

advantages would be f a r outweighed by the resulting added complexity.

Proof. q (X )(s) = [xr, n r

Let qn(x) = \(xl) S]

for s

n S

...V qn(x.): jsn 3

Since ac = n

en* C n x ,

's%,

where

be illuminating to compute s (a ):a (C X) o n o o

s (nns%).

It may

Recall that if S is a

based set (regarded a s a discrete space), then MS (resp. NS) denotes the i t suffices to verify f r e e monoid (resp., f r e e commutative monoid) generated by S, subject co the

the commutativity of the following diagram:

* = 1.

relation

,.J

Let MS (resp., NS) denote the f r e e group (resp., f r e e com-

mutative group) generated by S, subject to the relation i: MS

N +

MS (resp.

, j:

r.J +

* = 1,

and let

NS) denote the evident natural inclusions of monoids.

For X

Proposition 8.14.

J , the horizontal arrows

are all

isomorphisms of monoids in the commutative diagrams

U = U*LI{[X,S] g

and For g

and, if n > 1 ,

$ h,

to SX,. 1 .

U n U = U,, g h *

For g

X E

X } g

and s (U,) = 1

for g $

since U* i s homotopy equivalent

a (U ) i s f r e e on one generator, since U i s homo1 g g

topy equivalent to S(X* U X ), and therefore a (SX) = %n0(x) by the g 1 van Kampen theorem.

Here the horizontal arrows a r e induced from the s e t maps

a0(.ln): \(XI

ao(CnX)

2nd

ao(Q:

To(X)

"o(~nsnx)

rc)

by the universal properties of the functors M, N, M, and N. Proof.

Fix b

cn(2)

(with b = (bl, b 2 > where bl(l) & b2(0) if

n = 1); then the product in C X may be taken to be n [c, XI ' id, yl = [y(b; c, dl, x, yl for c e cn(j), x

x',

c n ( k ) , and y

k X

of a (X) generates a (C X) a s a monoid. o o n

It follows easily that the image

Thus the top horizontal arrows

a r e epimorphisms and by the diagrams, i t suffices to prove that the bottom horizontal arrows a r e isomorphisms. of isomorphisms c\r

NT (X) o

n H (X) &2H (S X) o n

For n

lln(snx)

) 1,

we have the evident chain

a (nns"x).

F o r n = 1 , let X denote the component of g, where g runs through a s e t g of points, one from each component of X.

Define open subsets U of SX g

A categorical construction We shall h e r e introduce a v e r y general categorical "two-sided b a r

construction".

When we pass back to topology via geometric realization

of simplicial spaces, this single construction will specialize to yield (1) A topological monoid weakly homotopy equivaIent to any given A

space; (2) The n-fold de-looping of a

& n -space that i s

required for- o u r

recognition principle;

Amap f:X-Y

i s a s e q u e n c e f :X + Y ofmapsin 9 9 9

that 8.f = f a. and s.f1 q = f q+lsi' 1q q-11

(3) Stasheffls generalization [I$]of the Milgram classifying space

of a topological monoid.

maps f, g: X

A homotopy h: f

Y consists of maps h.: X 1

aoho = fq

The construction also admits a variety of applications outside of topology;

and

-t

qfl'

g in

Tsuch between

0 I i 5 q, such that

h = g q+ls s

i n particular, a s we shall show in ยง 10, it includes the usual two-sided b a r constructions of homological a l g e b r a . Throughout this section, we shall work in the category simplicial objects in an a r b i t r a r y category

,---

SiGce verifications of

simplicial identities a r e important, we r e c a l l the definition of simplicial objects and homotopies and then leave such verifications to t h e diligent reader. Definition 9.1.

An object X

q 1 0 , togetherwithmaps

0 I. i 5. q, such that

a.:X + X 1 9 q-1

i s a sequence of objects X and s ' X + X in i ' q qtl

I Thus a purely formal homotopy theory exists i n the choice of

J , and we

deformation r e t r a c t s , etc.

,d Y , regardless of

can meaningfully speak of homotopy equivalences, When

i s o u r category of spaces, these

notions will translate back to ordinary homotopy theory via geometric realization.

We shall need a few v e r y elementary observations about the relation-

7 and A 3 .

ship between

X = X and letting each 9

f:X + X 1 in J

, define

For X r

and s

, define

be the identity map.

(i)

A map

r;(p): X,

p:X

Y in

Let X r -+

,fT .

and l e t Y r

xlao =

?.lal,

(p) =

Then

i s a commutative diagram in

; if

If F:

7 7' +

fined on objects Y r

- 7'

functors

f*r-! J , where

T*(P)

, let

Lemma 9.3.

then

a monad i n B T

X in

i s determinedbythemap

Proof.

denote the functor de-

y: F

G i s a natural transformation between

G*.,- denote the natural transformation lefined