Cech and Steenrod Homotopy Theories with Applications to Geometric Topology

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

542 David A. Edwards Harold M. Hastings

Cech and Steenrod Homotopy Theories with Applications to Geometric Topology

Springer-Verlag Berlin. Heidelberg 9New York 19?6

Authors David A. Edwards Department of Mathematical Sciences State University of New York at Binghamton Binghamton, N.Y. 13901/USA

Harold M. Hastings Department of Mathematics Hofstra University Hempstead, N.Y.11550/USA

Library ot Ceagress Catalegiag ia Pabllcatioa Data E d w a ~ s , David h

1946-

t e ~ and Steenrod hcmotopy theories w~th applications to gecm~tz~ic topology. (Lecture notes in mathematics ; 542) Bibliography: p. Includes index. i. Homotopy theory. 2. Geometry, Algebrale. 3. Algebra, Homological. 4. Algebraic topology. I. Hastings, Harold M., 1946joint author. II. Title. III. Series: Lectume notes in mathematics (Berlin) ; 542. 0~3.ia8 vol. 542 [QA612.7] 510'.8s [514'.24] 76-40180

AMS Subject Classifications 55399

(1970): 14F99, 14G13, 55B05, 55D99,

ISBN 3-540-07863-0 Springer-Verlag Berlin 9 Heidelberg 9 New York ISBN 0-387-07863-0

Springer-Verlag New York 9 Heidelberg 9 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 w 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. 9 by Springer-Verlag Berlin 9 Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.

to Marilyn and Gretchen

CONTENTS

81.

INTRODUCTION

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

1

w

BACKGR~

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

4

w

82.1

Pro - Categories

82.2

Some u s e f u l categories

82.3

Model categories

w

Simpliclal Closed model cateEorles

w

H o m o t o p y theories of p r o - s p a c e s

9

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

. . . . . . . . .

THE M O D E L S T R U C T U R E O N P R O - SPACES

; . . . . . . . . . . . .

41

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

48

.......

53

~ ......

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

56

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

56

83.1

Introduction

w

The h o m o t o p y theo~7 of

w

The h o m o t o p y theory of p r o - C

83.4

S u s p e n s i o n a n d loop functors, c o f ~ r a t l o n

flbration

4

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

sequences

g3.5

Stmplicial

model

w

Pairs

w

Geometric Models

w

InJ - s p a c e s

CJ

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

57

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

71

9 * 9 . . . . . . . . . .

. . . . . . .

structures

.

.

and

.

.

.

.

.

.

.

.

.

.

.

.

.

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

.

94 107

113

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

115

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

125

VI

w

w

T H E H O M O T O P Y INVERSE L I M I T A N D ITS A P P L I C A T I O N S TO HOMOLOGICAL ALGEBRA

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

129

w

Introduction

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

129

w

The h o m o t o p y inverse limit

w

E x~

w

The d e r i v e d functors of the i n v e r s e limit:

w

Results on d e r i v e d functors of the inverse limit

w

A l g e b r a i c d e s c r i p t i o n of

w

T o p o l o g i c a l d e s c r i p t i o n of

w

Strongly

w

The B o u s f i e l d - K a n s p e c t r a l s e q u e n c e

w

H o m o t o p y d i r e c t limits

on pro-Sw

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

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

Mlttag-Leffler

THE A L G E B R A I C T O P O L O G Y O F

lim s

background

- - 9

139 143

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

145

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

152

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

162

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

166

lim s

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

w

Introduction

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

w

H o ( t o w - C,)

w

R e m a r k s on c o m p l e t i o n s

w

Some b a s i c functors

w

W h i t e h e a d and Stability Theorems

w

Strong h o m o t o p y and h o m o l o g y theories

versus

134

.....

pro-groups

pro- C

130

tow - Ho(C,) . . . . . . . . . . . . . . .

169 172 172 172

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

181

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

184

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

187 206

VJ~

w

213

PROPER HOMOTOPY THEORY

w

w

w

w

Introduction

w

P r o p e r h o m o t o p y and ends

w

P r o p e r h o m o t o p y theory of u - compact spaces

w

Whitehead

w

The Chapman Complement Theorem

9 9 9 9 . . . . . . . . . . . . . .

theorems

........

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

GROUP ACTIONS ON INFINITE DIMENSIONAL MANIFOLDS w 7.1

Introduction

w

The theory of s - m a n i f o l d s

w

The S t a n d a r d A c t i o n s

w

P r o o f of T h e o r e m (7.3.4)

. . . . . . . . .

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

STEENROD HOMOTOPY THEORY

and Q - m a n i f o l d s

........

214 220 225 228

233 233 233 237

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

240

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

245

Introduction

w

S t e e n r o d h o m o l o g y theories

w

The V i e t o r i s functor

w

Proofs of Theorems

w

S p e c t r a l sequences

w

Dual~ty

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

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

(8.2.19),

(8.2.20) and (8.2.21)

.....

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

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

SOME OPEN QUESTIONS

213

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

w

REF~ENCES INDEX

- type

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

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

245 246 254 268 269 276 279

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

281

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

294

w

Inverse systems of topological

INTRODUCTION

spaces occur in many contexts in topology.

Some

examples are:

a)

Geometric Topology. inverse systems

One can associate

{U},

{U \X},

over the open neighborhoods then one can associate

to

over compact subsets of

b)

Algebraic Topology.

of X

to any embedding

{(Y \X, U \X)}, X

in Y.

its end,

etc., where

If

X

The Cech construction

completion constructions

associates

C

where

C

varies

to a topological

A similar construction

also associate

to have an appropriate

inverse systems of spaces. C

varies

in

The Postnikov and inverse systems of

to complexes.

It is often important

any category

U

X.

algebraic geometry leads to ~tale homotopy theory.

complexes

the

is locally compact,

e(X) = {X \C},

space)an inverse system of CW-complexes.

pro -finite

X c-->Y

In [Gro -I]

another category

Grothendieck

pro - C

indexed by "filtering categories"

category and homotopy category of showed how to associate

to

whose objects are inverse systems in

and whose morphisms

are so defined as to make

cofinal systems isomorphic.

In [Q -i] Quillen introduced

the notion of a model

category as an axiomatization

of homotopy theory on

and

category is a category tions, fibrations

C

C,

weak equivalences

C.

topology of

in

SS.

together with three classes of morphisms

and weak equivalences

homotopy category of

Top

Ho(C),

and it is natural to consider

is obtained from

C

One has a canonical functor pro -Ho(C)

This point of view goes back to Christie

called cofibra-

which satisfy "the usual properties." by formally inverting

In [A- M] Artin and Mazur developed

pro - Ho(C).

A model

the

the algebraic

pro - C

> pro - Ho(C)

as the homotopy category of [Chr].

The

pro -C.

But Christie also realized that

for some purposes category.

pro -Ho(C)

It can be shown that

model category structure on pro - SS

pro -Ho(C)

pro - C.

is not the homotopy category of a

The second author

[Has -i] has shown that

admits a natural model category structure with homotopy category,

Ho(pro -SS), lences.

was too weak and one really wanted a stronger

obtained from

Ho(pro -SS)

pro - S S

had previously appeared

problem for topological

spaces

model category structure on

[Pot - 2].

and compare

in Porter's work on the stability

Grossman

[Gros - i] has studied a coarser

Towers -SS.

In the first part of these notes Ho(pro -C)

by formally inverting level weak equiva-

(w167

Ho(pro -C)

we develop the algebraic

with

pro -Ho(C).

We also give applica-

tions to the study of the derived functors of the inverse limit. of these notes (w167

contains applications

In w

w

trick," described

In w adjoint

theory of model categories

says that all inverse systems

[Q -i]

is reviewed in w

we show that for nice closed model categories

C,

extending

pro - C

holim:

Ho(pro - C)

> Ho(C)

theorems for

lim s.

The basic algebraic topology of with

various Whitehead

pro -Ho(C), theorems.

C,

CJ

(where

J

is a

inherit natural closed model struc-

[Has -i].

we show that the natural inclusion

obtain vanishing

Ho(pro -C)

in w

to inverse systems indexed over cofinite strongly directed sets.

cofinite strongly directed set) and tures from

We conclude with a brief list of

contains background material about p r o - categories and model

The "Mardesic

are pro - equivalent Quillen's

group

in w

More precisely, categories.

The second part

to proper homotopy theory,

actions on the Hilbert cube, and shape theory. open questions

topology of

Ho(C)

~ Ho(pro - C)

(compare Bousfield and Kan

Ho(pro -C)

is developed

in w

has an [B -K]) and

We compare

discuss homotopy and homology p r o - groups, and prove w

Whitehead and stability theorems,

survey of work of the first author and R. Geoghegan

[E - G -i -5].

includes a

In w

we show that the category of o -compact spaces and proper maps may be

embedded in a suitable category of towers which is a closed model category.

We

then apply pro -homotopy theory to proper homotopy theory and weak -proper -homotopy theory (see [Chap- i] and [C -S] for weak -proper -homotopy theory and its uses in the study of Q -manifolds and shape theory.)

Some of our results are announced in

[m -H - 3 ] . We apply this theory in w manifolds.

In w

w

to the study of group actions on infinite dimensional

represents joint work with Jim West [West-l],

[ E - H -W].

we discuss strong shape theory and develop generalized Steenrod homology

theories using pro -homology.

These theories have found recent applications in the

Brown-Douglas-Fillmore theory of operator algebras Kaminker and Schochet [K -S],

and

[B - D - F -i -2]; see also

with D. S. Kahn,

[K - K - S].

Detailed introductions precede w167

Acknowledgements.

We wish to acknowledge helpful discussions with Tom Chapman,

Ross Geoghegan, and Jim West, and correspondence with A. K. (Peter) Bousfield and Jerry Grossman.

Some of this material was presented at conferences at Syracuse

University (Syracuse, N. Y., U,S.A., December, 1974 and April, 1975), Mobile, Alabama, U.S.A., (March, 1975), New York (March, 1975), the University of Georgia, U.S.A., (August, 1975), Guilford College (Greensboro, N. C., U.S.A., October, 1975), and the Interuniversity Center in Dubrovnik, Yugoslavia (January, 1976).

We wish

to acknowledge the organizers of these meetings for their hospitality.

The second-named author held a visiting position at the State University of New York at Binghamton during the academic year 1974-75 in which much of this work was done.

He wishes to acknowledge their support and hospitality.

He was also

partially supported by N.S.F. Institutional Grants at Hofstra University in 1973-74 and 1975.

We wish to thank Althea Benjamin for typing this manuscript.

w

w

BACKGROUND

Pro -Categories. We need a Category of inverse systems such that cofinal subsystems are isomor-

phic; such a category was first defined by Grothendieck in [Gro -1] and is described in detail in the appendix of [A-M].

(2.1.1)

Definitions.

A non-empty category

J

is said to be left filtering if

the following holds. a)

Every pair of objects

j,j'

in J

can be embedded in a diagram j l

j"

j

b)

If

j'--~> J

is a pair of maps in

such that the compositions

If

C

X:J --~C.

and J

are categories, then a

>j

then there is a map

pro -object

filtering category.

~-dia~ram

over

C

in J

is just a functor

suppressing both

J

and the

X(j) X(F)> X(j').

over The

C

is a

J -diagram over

pro -objects over

C

C

where

J

form a category

defined by (pro - C) ({Xj }, {Yk }) - llmkColim j CC(Xj ,Yk ) }.

(Note:

j"--->j'

are equal.

We will usually use the notation {Xj},

bonding morphisms

A

j"

J,

the indexing categories are not assumed equal.)

is a small left pro - C

with maps

Let

tow - C

be the full subcategory of

indexed by the natural numbers.

pro - C

Objects of

We have defined the set of maps in

consisting of those objects

tow - C

pro - C

are called towers.

from

X

to Y,

but the above

definition is somewhat opaque and it is not obvious how to define composition of maps from the above definition.

Hence, we shall give an alternate definition.

For simplicity, consider inverse systems directed sets

(2.1.2) map

J

some

j

> Yk in J

and

{Yk }

over

C

indexed by

respectively.

Definition.

8:K--~J

fk:Xs(k)

and K

{X.} 3

A morphism

f:X--~Y

in

pro - C

is represented by a

(not necessarily order -preserving) and morphlsms in C with

for each j ~ 8(k)

k and

in K

such that if

j ~ 8(k')

bond

k ~ k'

in K,

the diagram commutes.

then for Two pairs

fk '

Xj

> Xo(k')

~ Yk'

\

bond \ ~

bond fk

X~(k) (8,{fk})

and

k

there is a

in K

(8',{f~})

composite maps

(2.1.3)

factor through

represent the same morphism in

j

in J

fk o bond,

Remarks. pro - C.

with

j ~ 8(k)

fk' o bond : X.3 -'~Yk

and

A pro- object

{X]}

limj{Xj}

pro - C

j ~ 8'(k)

if for each such that the two

are equal.

The inverse limit functors

information than its inverse limit exist.

~"Yk

lim:C J --~C, in pro - C

in C;

The relationship between the pro - object

{X.} 3

if they exist,

contains much more

the latter need not even and its inverse limit

llmj{xj} point

is analogous

p

to the relationship

and its value at

between the germ of a function at a

p.

We shall need the following reindexing results from Artin and Mazur

(2.1.4)

Proposition.

isomorphism

A map

(in Maps (pro -C))

f. -J{X~ --/-~Yj},_

Maps (C).

the following holds.

Proposition.

and suppose that

A

Let

A

be a finite diagram with commutation relations,

has no loops, i.e., that the beginning and end of a chain of

arrows are always distinct.

Let

D

be a diagram in

i.e., a morphism of

A

to pro - C.

{Dj}

C

such that the diagram in

of diagrams of

isomorphic

to

can be represented up to

by a small left filtering inverse system of maps

i.e., by a pro -object over

More generally,

(2.1.5)

f:X---~Y E pro -C

[A -M].

pro - C

of the type of

There is a left filtering inverse system pro- C

determined by

{Dj}

is

D.

Our techniques often require that the indexing categories be cofinite element has finitely many predecessors) implies

A,

a = b).

strongly directed sets

The following relndexing

(a ~ b

(each and

b ~ a

trick was inspired by Mmrdesic

[Mar - i].

Let An object d,

I

be a small loft filtering category and d E D

and for each

objects,

D--~ I

will be called an initial object if d'

in D,

there is a unique map

if they exist, are clearly unique.

diagrams with initial objects over

I.

Let

D

a diagram over

contains no maps into

d --# d' M(1)

We shall call

I.

in D.

Initial

be the set of finite D ! D'

in M(1)

if

D

is a subdiagram of M(1)

D'.

M(1)

Further,

is a directed set. D

the initial objects of

is clearly cofinite.

and

D ~ D'

D'

and

are equal).

Because

D' ~ D

I

is filtering,

implies

Hence,

M(1)

D = D'

(and

is a cofinite

strongly directed set.

Define a rune,or D

in M(1)

D'

its initial object in

from the initial object of

required functor.

as follows.

inlt : M(1) --~ I

Further,

I.

D'

If

Associate

D < D'

to a diagram

there is a unique map in

to the initial object of

D.

This yields

the

the functor inlt is clearly coflnal.

We thus obtain a functor

M:pro - C

9 pro - C ,

with

M{Xi}i E I

= {Xlnlt(D)}D~ M(1)'

and a natural equivalence

init:X

Summarizing,

(2.1.6) equivalent

we have the following theorem.

Theorem.

There exists a funetor

to the identity,

directed set for every

(2.1.7) isomorphic j

in

X

such that

M(X)

An object

pro - C

to an object of

k > j

M:pro - C

~ pro - C,

naturally

is indexed by a eofinite strongly

in pro -C.

Definitions.

there exists a

the diagram

~ M(X).

{5} C.

such that for each

of pro - C {X.} 3 s > k

is called stable if it is

is called moveable

if for each

there exists a filler in

....

->Xs

\,/ X. J

The above description of of direct systems over inj -C

C

pro -C

in which cofinal systems are isomorphic.

inj -C

Morphisms in

are given by the formula

inj - C ({Xj },

As with {Xj}

may be dualized to yield a category

to

pro -C,

{Yk }

J = {j}

and

In this case a map may be represented by a function

fj:Xj

there exists a

we shall give an alternative description of a morphism from

in the case that the indexing categories

are directed sets. and maps

{Yk }) = limj colim k {C(Xj,Yk ) }.

> YO(j) k

with

in C

for each

k > 8(j)

and

j

in J

k > 8(j')

f~

such that if

K = {k} 8:J--~K

j < j'

such that the diagram commutes.

bond

Xj

~ Y8 (J)

" Yk

bond]

/~oL~d f1'

Xj. ~

Ye ( j )

All of the above theory of Mardesic construction, colim: CJ--->C,

(2.1.8) HOM: C ~

x C

pro - C,

including Artin-Mazur reindexing and the

may be dualized to

inj - C.

if they exist, factor through

Proposition. ~ C.

Then

Suppose that HOM

C

The colimit functors,

inj- C.

admits an internal mapping functor

extends to a functor

HOM: (pro- C)~

x C

,inj-C.

Proof.

The required functor is given by

HOM ({Xj},Y) = {HOM (Xj,Y)}.

w

Some useful categories. This expository section is intended as a reference for later sections.

We sug-

gest that the reader omit the proofs the first time through, and refer to this section as needed later.

We shall give brief sketches of the following categories used in the remainder of these notes.

SS,

the category of simplicial sets introduced by D, M. Kan [Kan -2] and

J. C. Moore,

See [May-l]

and [Q-I,

w

D, Quillen's closed model category

[Q -i], a category with sufficient structure to "do homotopy theory," is an abstraction of

CW

SS.

prespeetra and

CW

spectra

(CWSp)

introduced by J. Boardman.

We follow

J. F. Adams [Adams -i] for the "additive structure" and [Has -3 -5] for smash products.

Sp,

the category of simplicial spectra developed by D, M. Kan [Kan-l].

K, S. Brown [Brown] sketched a proof that

Sp

is a closed model category.

We introduce both categories of spectra because homology theories (which involve smash products) are most easily described in

CW

spectra, while the closed model

structure of simplicial spectra is needed to define a suitable homotopy category of pro -spectra.

M. Tierney [Tier] described stable realization and singular functors inducing an equivalence of homotopy categories

Ho(Sp)

~ Ho(CWSp).

D, W. Anderson [An-l,2]

gave a different construction of homology theories using chain functors.

J.P.

May

[May -2] introduced a category of spectra based on the Boardman-Vogt theory of infinite loop spaces [B -V] and thus described the higher structure of ring spectra.

10

While these approaches are more powerful, the "classical" approach we shall follow is simpler and adequate for our purposes.

We shall then describe how the

use

of spectra yields a unified treatment of

generalized homology and cohomology theories, Spanier-Whltehead duality, and homology and cohomology operations, following [Adams - i, 3].

We begin by defining simplicial sets.

(2.2.1)

a)

Definition.

A sequence {X,

A simpliCial Set

n ~0}

X

consists of:

The elements of

of sets.

X

are called n

the ~ - s i m p l i c i e s

b)

Face maps

c)

Degeneracy Maps

of

X.

d~:X--->X_I

for

n ~ i

and

for

n > 0

n

si:Xn--~Xn+l

0 < i < n.

and

0 < i < n.

The maps are required to satisfy the following identities:

did j -~ dj_ I d i

for

i < j,

si s.3 = Sj+l s.m

for

i < J,

sj_ 1 d i

for

i < m,

id

for

i = j

for

i 9 J+l.

~ dis.3 ~ ~

\

sj di_ 1

For example, the singular complex

S(X)

or

j+l,

of a topological space

simplicial set with typical n - simplex a continuous map the standard n - simplex in

Rn+l.

f:An--->X,

X where

is a An

is

11

(2.2.2)

Definition.

sequence of maps

fn-i di = difn

sets.

f:X--+Y n ~ 0,

f :X --->Yn' n n and

Definitions

A map

sets consists of a

which satisfy the identities

Sifn = fn+l s . . 3

(2.2.1) and (2.2.2)

SS

combine to yield the category

SS.

d0x,

Let

x

faces of

x,

example,

dj (djx) = dj (dj+ix) .

extension condition

j = 0,1,''',1,''',n.

A simpliclal

y0,Yl,''',yi,-'',yn,

with

fibration

djx = yj

More generally, b

a map

in B,

n

which satisfy the appropriate

for

for

with in SS

compatibility

and

of spaces induces a Kan fibration

[Kan -2] gave a combinatorial

n

description

[Mil -2] geometric realization

(=

~i(RX)),

functor.

where

(n- i) -

d.x = y. 3 3

for

is called a Kan in E,

conditions and

there is an n - simplex p(x) = b.

also a Hurewiez

S(p):S(E) ---> S(B).

of the homotopy groups of

This definition of the homotopy groups is extended ~i (x) ~ ~i (SRX)

for

to be

Kan complexes and also described a functorial Postnikov decomposition

by defining

The

conditions

(n - i) - simplices

j = 0,1,''',1,''',n,

j = 0,1,"',1,--',n,

X.

conditions,

if given any

x

p:E--->B and

set

of the

is said to satisfy the Kan

the singular complex of a space is a Kan complex;

p:E--->B

D. M. Kan

X

there exists an n - simplex

p(yj) = d3.b

For example,

set

which satisfy the compatibility

fihration if given an n - simplex

in E

satisfy certain compatibility

(or is simply called a Ka____nncomplex)

the faces of an n - simplex,

the requirement

be an n - simplex of a simplicial

dlX,-'',dnX ,

^ ... y0,Yl,-'',yi, 'Yn

simplices

plexes.

of simplicial

The following Kan extension condition is crucial to the development

homotopy theory of

x

of simplicial

of Kan com-

to all simpllcial

R:SS --->Top

sets

denotes Milnor's

12

Call a map (i.e., each

f:X-->Y

fn

of simplieial

is an inclusion),

a weak. equivalence if

and for every choice of basepoints homotopy-extension

sets a cofibration

in

X,

and covering-homotopy

~,(f)

if it is an inclusion

T0(f )

is a bijectlon,

is an isomorphism.

properties

The usual

are combined in the following

theorem.

(2.2.3)

Coverin~ Homotopy Extension Theorem.

Given a commutative

solid-arrow

diagram

9-

A

i>y ~

/

X

in which

i

is a coflbration,

equivalence,

is a flbration,

then there exists a filler

See, e.g. M1

p

>B

[Q-I,

w

[Q-I];

see w

i

or p

is a weak

f.

for a proof.

for a model category

and either

Theorem

(2.2.3) becomes Quillen's Axiom

.

Note that the usual homotopy extension property only holds for maps into Kan complexes.

(2.2.4)

Definitions.

finitely many non-degenerate has no non-degenerate

In w

A simplicial simplices;

simplices

set X

X

is called finite if

is said to have dimension

in dimensions

greater than

we shall discuss function spaces in

SS.

n.

X

has only ~ n

if

X

13

We shall now sketch the basic properties We shall follow J. F. Adams

of

CW

prespectra and

for the "additive structure"

[Has -3 -5] for smash products and function spectra.

evident basepoint.

namely

Let

spectra.

[Adams -i], except for a technical modification made in

[Has -3] to construct smash products,

ture with two vertices,

CW

0

and i.

Let

Give

and follow

[0,i]

the

CW

S 1 = [0,i]/0 ~ i,

S n = S 1 ^ S I ^ "'" ^ S 1

struc-

with the

(n factors), where

K A L = K x L/K v L.

(2.2.5) pointed

CW

Definitions.

A

complexes

X n ^ S 4 -->Xn+ 1

CW

spectrum

{Xnln = 0,1,2,...},

for each

n.

are

^ id CW

S 4"

The category

of

spectra and whose morphisms

A weak spectrum spaces,

CWPs

f :X nn

X = {X } n

CW

~Y

fn : X --->Yn

such that

The category cofinal inclusions

(2.2.6)

f:X-->Y

CWSp

fn+l

of

CW

such that

fn+l

extends

is the category whose objects

are prespectrum maps.

is a sequence of pointed compactly generated Xn ^ S 4---> Xn+ I

for each

n.

consists of a sequence of continuous pointed maps extends

fn ^ ids4

up to homotopy.

spectra is obtained from

[Adams -i].

called cofinal if for each cell

CWPs/{cofinal

consists of a

CWPs

by inverting

of spectra.

Definitions

Io u {basepoint}~

n

f:X--->Y

prespectra

together with continuous pointed maps

weak prespectrum map

consists of a sequence of

together with cellular inclusions

A prespectrum map

sequence of continuous pointed maps

fn

X = {X } n

A S 4k

inclusions}.

X' n+k"

o c X

A subspectrum

n

c X

X'

of a

there exists a

The category of

CW

CW k

spectra,

spectrum

such that CWSp,

is

X

is

14

(2.2.7) a)

Remarks.

The class of cofinal

inclusions

admits a calculus of right-fractlons

in the sense of P. Gabriel and M. Zisman any map

f:X--->Y

of

CW

X'

is cofinal in

such diagrams same map if

X ~ X'

X

and

f' ~ Y

f' = f"

on

This means that

spectra can be represented

X D X'

where

[G -Z].

by a diagram

f' ~ Y

f'

is a prespectrum map.

Two

ft!

and

X m X"

X' N X".

"~Y

represent

Composition

the

is defined as follows.

Consider a solid-arrow d i a g r a m

/

/

X I! \

/I

\ \f!, a Y'

X'

X

Y

Define a subspectrum of X', maps

X'

which

f'

X" c X'

maps into

hence cofinal in X D X'

f' > Y

X. and

This yields a w e l l - d e f l n e d

b)

Adams uses

S!

\

Z

as follows: Y'.

f" = f'IX". K' > Z

y D y'

composition

S4

c)

X"

is cofinal

is given by

and m a k e s

CWSp

of

yield equivalent

X 9 X"

K ' f " ~ Z.

a category.

CW

spectrum.

categories

of

spectra.

By applying realization cylinder

and cylinder

functor to a weak spectrum

functors X

in

The composite of the

in the definition

It is easy to see that both definitions CW

consists of those cells

By construction,

Let

w h e r e we use

X"

and a suitable m a p p i n g

w e obtain a

CW

spectrum

X'

15

X' --->X

and a natural p r e s p e c t r u m m a p

which is a weak homotopy

equivalence on each level.

(2.2.7)

a)

Some examples of

CW

The k -sphere spectrum

s~ -

fs~t

spectra.

(k ~ Z)

is given by

{*'

-

Sk + 4n(space ) ,

I

k+4n

>

I

together with the inclusions Sk + 4 n ( s p a c e ) ^ S4 ( s p a c e )

(sl^

.^s I)^CS I ^

^s I)

S 1 ^ ... ^ S 1

i S k + 4n .

b)

More generally, let

K

be a pointed

CW

complex.

Associate to

K

the spectrum

K

(spectrum) = K

ffi {K A S 4n}

,

together with the evident inclusions. K ^ Sk

for any

k E Z

(spectrum)

(space) ^ S 0

More generally, we may form

(see (2.2.8)).

suspension spectra; for sufficiently large (K A Skln ^ S 4 ~ (K ^ Sk)n+l. a stable finite

CW

complex.

If

K

These spectra are called n,

is finite, we call

K A Sk

E. Spanier and J. H. C. Whitehead

IS -W] first studied stable finite complexes.

18

c)

The Eilenberg-MacLane spectra.

Let

associated with the weak spectrum K(G,4n) ^ S4

~ K(G,4n +4)

cohomology class.

Here

G

K(G)

he the

{K(G,4n)},

maps

d)

The B_~U-spectrum: BU n ^ S 4 BU ^ S 2

~ BUn+ I ~ BU.

equivalences~ BO,

BU

BSO,

n

in X)

= BU

Xn

BSp,

A

The Eilenberg-

CW

> fl4Xn+I

spectrum

X

(adjoint to the

for all

n,

and the maps

are defined by composing Bott periodicity maps

is an ~ -spectrum. and

BSpin

"infinite Lie groups."

BU

BU

> ~2BU

are homotopy

There are similar spectra

associated with the respective classifies complex K -theory,

BO

real K - theory, etc., see (f) below.

e)

The Thom spectra classifying bundles e.g. [Sto]) DU(m)

in

Note that

MU(m) EU(m)

MU,

MO,

MSO,

MSp,

EU(m) ---~BU(m).

M spin.

Consider the

The Thom complex (see,

is the quotient of the unit disk bundle by the unit sphere bundle

MU(1) m D2/S 1 m S 2.

MU(s

is

are homotopy equivalences.

Because the adJoints BU

together with maps

is any abelian group.

if the maps

X n ^ S 4 ----->Xn+ I

spectrum

which classify the fundamental reduced

MacLane spectra are examples of ~ - ~ e c t r a . called a n ~ - s p e c t r u m

CW

^ MU(m) = DU(s

roT(9.

in EU(m).

There are Whitney sum. maps

• DU(m)/Sphl U(s

---> DU(% +m)/sph U(s +m)

=

Sph U(m)

+m).

X

Sph U(m)

17

These yield inclusions

MU(m) ^ S 2

and thus the

MU

spectrum

MU(m+ i),

MU = {MU(2n)}.

yield the other Thom spectra.

MU

Similar constructions

classifies complex cobordism by

a classical result of Thom, see e.g.,

[Sto].

Similarly, the other

Thom spectra classify appropriate cobordism theories. is important because:

Cobordism

(I) some varieties are quite powerful; and

(2) the dual homology theories, bordlsm, have natural geometric definitions.

f)

R. Stong [Sto] is a good source on cobordism.

The examples (c), (d), and (e) above classify well-known cohomology theories:

H*(;G),

various forms of K - theory, and various forms Brown's theorem associates to

of cobordism theory respectively.

E*

any generalized cohomology theory

a spectrum

E = {g } n

with

E4n(x) e [X, En].

(2.2.8)

Definition.

If

K

trum, then their smash product is K ^ Xn ^ S4

~ K ^ Xn+ 1

(2.2.9)

Definition.

there is a map

is a pointed

complex and

K ^ X = {K A Xn},

induced from

Maps

CW

X

is a

CW

spec-

together with inclusions

X.

f,g:X .[ ~Y

H:([0,1] u *) ^ X - - ~ Y

of

CW

such that

spectra are homotopic if HI(0 u ,) ^ X = f

and

H I(l U *) ^ X = g.

Let Ho (CWSp)

Ho(CWSp)

denote the resulting homotop7 category of

CW

spectra.

is essentially the category introduced by Boardman (see [Vogt -2]) as

formulated by Adams [Adams -I].

Caution:

CWSp

is no__~ta model category.

18

The homotopy ~roups of a for

CW

~k x = Ho(CWSp)(sk,x)

spectrum are given by

keZ.

(2.2.10)

Remarks.

full subcategory of

The homotopy category of stable finite Ho(CWSp),

CW

complexes, a

is the classical Spanler-Whitehead category (use

the Freudenthal suspension theorem which states that [X ^ Sn, Y ^ Sn] ~-[X A Sn+l, y A Sn+l] sufficiently large

and YM

X

finite if for some

N,

and Y

and

n).

More generally call a complex and

for finite complexes

Xn = ~

CW

spectrum

A S4 n - 4 N

for

X

n ~N.

If

X

and Y

~

is a finite

are finite, and

are the above complexes, the~

(2.2.11a)

CWSp(X,Y) -= c O I ~ k { C W [ ~

^ Sk - 4 N , YM A sk-4M~},

and

Ho(CWSp)(X,Y) a colimk{[~ ^ Sk - 4 N ,

(2.2.11b)

The category

CWSp

subspectra.

In fact, a

finite subspectra. morphisms of

CW

YM ^ Sk-4M]}"

is then defined so that a spectrum is the colimit of its finite CW

spectrum is the homotopy colimit (see w

J. Boardman (see [Vogt -2]) and A. Heller [Hel -1,2] define spectra by this criterion and (2.2.11a).

Cofibrations and homotopy equivalences behave similarly in complexes.

of its

CW

spectra and

A. Heller formalized these properties by introducing abstract

CW

h-c

categories [Hel -1,2].

We shall need smash products and function spectra in order to discuss generalized homology and cohomology theories.

These constructions are more difficult than the

19

above structure. unit [Has- 5].

In fact, there is no smash product on

Ho(CWSp)

spectra.

SO

as

J. Boardman (see [Vogt 2]) gave the first construction of a

coherently homotopy associative, commutative, and unitary CW

with

D.M.

(S0)

smash product for

Kan and G. W. Whitehead [K -W] described a non-associatiVe

smash product for simpliclal spectra.

Later Adams [Adams- I] and still later the

second author [Has - 3] gave different and simpler constructions for the smash product of

CW

spectra.

J.P.

May [May - 2] and D. Puppe [Puppe] gave a radically

different construction following the Boardman-Vogt theory of infinite loop spaces [B- V].

We shall follow [Has - 3].

(2.2.12)

Th_~einterchanse problem.

a (non-commutative) ring and any

CW

We may regard the sphere spectrum spectrum as a right-module over

Construction of a smash product requires permutations Because

S0

SO

as

SO = {s4n}.

~ of S 4 ^ .,- ^ S 4.

is only homotopy commutative, this requires canonical homotopies

H W

from

~

to the identity.

These are defined as follows.

S 4k _= S 4 A - - -

A S4 & C2.

A "'"

(C 2 x

where

denotes the one-polnt compactificatlon.

factors of

C 2 x 9 -. x C2;

thus

w ~ SU(2k)

"'"

Identify

A C2.

x C2) *

Then

w

Because

simply permutes SU(2k)

netted and simply connected, there is a unique homotopy class of paths SU(2k)

with

r(0)

= w

and

r(1)

H :([0,i] u *)

= e,

A

the identity of

S 4k ! [0,i] x S 4k

W

by K=(t,x)

= r (t)(x).

SU(2k).

- S 4k

is path con-

[r ] Define

in

20 Then

[H]

is the required homotopy class (relative to the endpoints)

of canonical

homotopies.

We shall now define a family of smash products on

CWSp,

and prove that they

are all equivalent and have the required properties.

(2.2.13)

Definition.

Given a sequence of pairs of non-negative integers

{ (in,J n) In => 0, and

{in }

in + i n ffi n,

and

{i n )

are monotone unbounded sequences},

define an associated smash product by

X^Y={(X^Yln}~{Xi

^ Yjn }

n

together with the appropriate inclusions induced from

Then

^

extends to bifunetors on

CWPs,

inclusions is a cofinal inclusion), and

(2.2.14)

Theorem.

smash products on

CWSp

X

and Y.

(the smash product of cofinal

Ho(CWSp).

Any two sequences (2.2.13) yield canonically equivalent

Ho(CWSp).

We shall need the following machinery.

(2.2.15)

Definition.

Let

X

be a

CW

spectrum.

Given a monotone unbounded

sequence of non-negative integers

{Jn[n >__O, define a

CN

Jn < n),

spectrum 4n- 4Jn } DX = ((DX)n} = {Xjn ^ S

together with the appropriate inclusions induced from

X.

21

Then

D

extends to a functors

natural cofinal inclusions

(called destahillzation)

DX--->X,

so that

D

on

CWSp.

There are

is naturally equivalent

to the

identity.

(2.2.16) of DX

Definition.

Let

X

be a

CW

prespectrum.

A permutation

consists of a sequence of maps

{~n = id a ~n:

(DX)n

=

XJn

-'+X.

^ S4 ^ 9

^ S4

A S4 A 9

^ S4

3n

(DX)n} where each

n

S 4 ^ ... ^ S 4.

is a permutation of

A sequence of maps

g = {gn: (DX)n ----~Yn]gn+l

extends

gn

up to permutation}

is called a permutation map.

(2.2.17) homotopies

Proposition.

Permutation maps are weak maps, where the required

Hn:(DX)n A S 4 x [0,i] ---->Yn+l

are induced from canonical homotopies

(2.2.12).

The proof is easy and omitted.

The following lemmas relating weak maps and maps of [Has -3].

The proofs involve construction

CW

spectra are proved in

of suitable mapping cylinders and the

homotopy extension property.

(2.2.18) for

Lemma.

A weak map

f:X-->Y,

together with a family of homotopies

f,

{H:X

^ S 4 • [0,1J

~Yn+l}

22

(see (2.2.5)), induces a strict map

F

depends upon

{f}

and

F:S--->Y.

{H }

up to equivalence in

Ho(CWSp).

n

(2.2.19)

Lemma.

Let

{H } and

{H'}

n

map

f.

If

and

(f,(Hn})

(2.2.20) topies

{H'} n

Hn = H'n

relative to the endpoints for each

induce homotopic maps

Lemma. and

be families of homotopies for a weak

n

Let {H" } n

X--->Y

f':X--->Y

and

respect•

in

n,

then

(f,{Hn})

CWSp.

f":Y--~Z

be weak maps with homo-

Define "composed" homotoples

Hn:Xn ^ S4 x [0,i]

'~Zn+ 1

by

~ H ; ( f n ( X ),s,2t),

0 < t <__89

Hn(X,S,t) t

Let

F',F", and F

(f"f',{Hn}).

Then

be the maps associated with

F "~ F ' F "

in

(f',{Hn})

1.

s

t! (fi! ,~ Hn}),

and

Ho(CWSp).

Lemmas (2.2.18) - (2.2.20) yield the following proposition.

(2.2.21)

Proposition.

A permutatlon-commutatlve diagram of permutation maps

induces a commutative diagram in

Ho(CWSp).

this diagram are defined up to homotopy in

(2.2.22)

Proof of Theorem (2.2.14).

The maps and required homotopies in Ho(CWSp).

There are destabillzatlons (2.2.15) and

natural permutation classes of permutation maps D'(X ^' Y) --~X ^ u

D

The composite mappings

D(X A Y) --~X A' Y~ D'D(X ^ Y) ---~X ^ Y

and

23

DD'(X A' Y) - - ~ X A' Y permutations.

differ

from

the

The conclusion follows.

(2.2.23)

Theorem.

X

respective

>X

a:(X A Y) ^ Z c:X ^ Y

~Y

inclusions by

Q

There are natural maps in

~ X A S0

(cofinal)

Ho(CWSp),

(unit),

~ X ^ (Y A Z) ^ X

(associativlty), (commutativity),

which yield a symmetric monoidal category in the sense of S. Eilenberg and G, M, Kelly

[E -K].

Proof.

There are destabilizatlons

and natural permutation

classes of permuta-

tionmaps

DX

> X ^ S O.

~ X,

a':D((X ^ Y) ^ Z)

~X

^ (Y ^ Z),

c':D(X A Y) ----~y A X

(D

is used generically).

associated maps of

By Proposition

CW

Let the maps

spectra

(2.2.21),

permutation -commutative

DX--~ X ^ S0---~X,

((2.2.17)

a,

and

c

- (2.2.!8)).

it suffices to obtain the coherency diagrams

c

2:D2(X A

[E- K] as

diagrams of permutation maps between destabilizations.

These diagrams include statements that the above maps are isomorphisms, composite map

be the

Y) ---~X ^ Y

certain coherency conditions hold. coherently homotopy associative is

is homotopic For example,

to the identity,

that the

and that

the diagram stating that

A

is

24

D2(a')

D3(((W ^ X) ^ Y) ^ Z)

D2((W ^ X) ^ (Y ^ Z))

L

I

(a' ^ id).

D(a')

i

D2((W ^ (X ^ u

^ Z)

I

D(W ^ (x ^ (Y ^ z)))

D(a')

(id ^ a'),

D(W ^ ((X ^ Y )

^ z)

"~ w ^ (x ^ (Y ^ z)).

These diagrams are readily obtained.

(2.2.24)

Remarks.

able smashproduct on

The approach of Theorems (2.2.22) - (2.2.23) yields a suitHo(CWSp)

for studying homology and cohomology theories and

operations involving maps of spectra.

It does however ignore higher homotopies

which yield the rich structure of infinite loop spaces [May -3]. incorporate this structure in their smash products.

May and Puppe

It would be interesting to

obtain higher homotopies within the above simple framework, perhaps with a suitable operad.

See P. Malraison [Mall.

(2.2.25)

~urther properties o f

^ -

It is easy to verify that

^

commutes

with the suspension

S 1 ^ ?: Ho(CWSp)

^

) Ho(CWSp),

satisfies a Kunneth formula for stable integral homology, and

universal for pairings [Wh-l],

^

is weakly

[K-W].

We shall now define function spectra, and verify that the smash product

^

Ho(CWSp), together with

and function spectra, forms a s/amnetric monoidal closed

category in the sense of Eilenberg and Kelly [E- K].

This includes the usual

2S

exponential law.

(2.2.26)

Construction of function spectra.

Let

For a first approximation to the function spectrum spectrum

Map(X,Y)

(s-4k)n = s4n- 4k

as follows. for

n _> k,

induced by isomorphisms smash product Ho(CWSp)

^'

on

Let and

*

S -4k

X

HOM(X,Y),

be the

be

CW

spectra.

define a weak

-4k - sphere

spectrum:

otherwise; the required inclusions are

s4n - 4 k ^ S 4 ~ S4n -4k + 4 CWSp

and Y

using (2.2.13).

Choose a representative

The eventual function spectrum in

will be independent of this choice.

Define

Map(X,Y) n = CWSp[S -4n A X,Y)

with the topology induced from the compactly generated function spaces

P(IS-4n ^ X')i' S 4 A S -4n -4

where

".s-4n

(S 4

X'

is a coflnal subspectrum of

X.

The maps

is a space, the other terms are spectra) induce the

required maps

Map(X,Y)n A S 4

This yields a weak spectrum

(2.2.27) Map(X,Y),

Definition.

Map(X,Y).

Let

HOM(X,Y)

be the

CW

spectrum obtained from

see Remarks (2.2.7)(c).

We can extend show that

~Map(X,Y)n+l.

HOM

(2.2.28)

HOM

to bifunetors on

CWSp

and

Ho(CWSp).

is the required internal mapping functor.

Theorem

(exponential law) .

Ho(CWSP)(X A Y,Z) = Ho(CWSp)(X, HOM(Y,Z)).

We shall now

26

Proof.

If

X

and Y

(2.2.29)

Because a

CW

are finite spectra, it is easy to show that

CWSp(X ^' Y,Z) = CWSp(X, Map(Y,Z)).

spectrum is the colimit of its finite subspectra (the Boardman-Heller

completion, see (2.2.11) and the following discussion), CW

spectra.

(2.2.29) holds for arbitrary

This also yields an analogous formula in

there are natural weak homotopy equivalences

Ho(CWSp).

HOM(Y,Z)

Finally,

~ Map(Y,Z)

jections of mapping cylinders and the natural transformation

(from pro-

R o Sin--+id).

Thus

Ho(CWSp)(X, Map(Y,Z)) ~ Ho(CWSp) (X, HOM(Y,Z)),

by [Adams -i, Theorem 3.4].

(2.2.30)

Corollary.

The conclusion follows.

HOM( , ) = Ho(CWSp)(

This follows from the definition Ho(CWSp)

is normalized

(2.2.31)

Theorem.

D

, ).

T 0 = Ho(CWSp)(S 0, ).

This also shows that

[E -K, p. 491].

Ho(CWSp),

together with the above structure, is a symmet-

ric monoidal closed category.

The remaining coherence conditions fied.

[E - K, p. 491, Theorem 5.5] are easily veri-

Their precise statement and proof is omitted.

(2.2.32)

Remarks.

Adams defines an internal mapping functor

ing to Brown's Theorem (see [Adams - i]).

HOM

by appeal-

This approach also yields the above

theorem.

This concludes our formulation of the category of

CW

spectra.

We shall now

briefly discuss simpllclal spectra and the equivalence of homotopy categories. the end of this section we shall use

CW

At

spectra to discuss homology and cohomology

theories and operations, following [Adams -1,3].

27

(2.2.33) funetor

Definition [Kan -i]; we follow [May- i].

E:SS,---~SS,

is defined on objects as follows.

simplicial set with basepoint all symbols

(i,x)

identifications

where

i

* .

Then

(EX) 0 = *0'

is a positive integer and

(i, *n_i ) = *n

XK).

basepoint in

The simplicial suspension

(*k

Let

X

and

be a pointed

(EX) n

x ~ Xn_ i,

consists of subject to the

denotes the appropriate degeneracy of the

Face and degeneracy maps are defined as follows:

So(i,x) = (i+l,x)

Si+l(l,x) = (l,six)

~0 (l'x) = *n'

x ~ Xn

~i (l'x) = *0'

x e X0

%i+l(l,x) = (l,~ix),

x ~ Xn,

n > O,

and by the simplicial identities (2.2.1).

Then

EX

is a pointed simplicial set with one non-degenerate

every non-degenerate

(n -i) -simplex

of X

reduced suspension of the realization of the realization

(2.2.34) prespectrum

except the basepoint. E RX

Further, the

is canonically homeomorphic to

[Kan- i], compare Definitions (2.2.5).

A simplicial

consists of a sequence of pointed simplicial sets

with inclusions

EX(n) --->X(n +i).

extends

El(n).

Let

X(n),

A map of simplicial prespectra

consists of a sequence of pointed simplicial maps f(n +i)

simplex for

REX.

Definitions X

X,

n

Ps

f(n):X(n)

> Y(n)

together f:X---~Y such that

denote the category of simplicial prespectra.

28

We could "complete" the category of slmplicial prespectra as in Definitions (2.2.6).

However, Kan introduced a conceptually simpler completion which essen-

tially replaces a simplicial prespectrum by its associated ~ - spectrum (see (2.2.7)(b)).

An

n +k

simplex of

(n + k +i) - simplex of spectrum

SX

X(n)

X(n + i)

associated to

X

(element of under

X(n)n+k)

E.

corresponds to an

Thus Kan defined the simplicial

by taking as stable k -simplices the pointed

(*)

sets

S~

= u X(n)n+k,

k

~

Z,

n

+ k ~ O.

There are induced face and degeneracy maps of pointed sets

di SXk+ 1 --~ SX k si:s

for

1

which satisfy

i > 0

i)

-

the usual simpllcial identities (2.2.1)

and

•

the local finiteness condition:for every simplex there is an integer d.o = *

for

n

o

in SX,

(depending upon o ) such that

i > n.

1

(2.2.35) (*)

sets

~

Definitions

[Kan- i].

(the ~ - slmplices of

A simplieial spectrum X),

k E Z ,

degeneracy maps which satisfy the above conditions.

X

consists of pointed

together with face and A map of slmplicial spectra

29

f:X---~Y

consists of a sequence of maps of pointed sets

commute with face and degeneracy maps.

Let

Sp

{fk:~

~ Yk }

which

be the category o_~fslmplicial

spectra.

The spectrum construction

S

above extends to a functor

admits an adjoint prespectrum functor PX

is defined by letting

with

d.o = * l

for

PX(n)j

i > j.

The categories

Sp

P:Sp --->Ps,

and SS

S

on a simplicial spectrum

consist of those

Further,

S:Ps ---~Sp.

(j -n)

simplices

o

X, of Y

SP = identity:Sp --~Sp.

enjoy many similar properties.

In particular,

there is a Kan extension condition for simplicial spectra, and the homotopy groups of a Kan spectrum admit a combinatorial definition [Kan -i].

More generally,

K. S, Brown proved the following theorem.

(2.2.36)

Theorem

[Brown].

The category of simplicial spectra admits a

natural closed model structure in the sense of D. G. Quillen [Q -i], see w

We shall sketch an independent proof in the spirit of Quillen's proof [Q -i, w

that

SS

is a closed model category.

standard simplices which do not exist within BSp

Sp,

The first task is to define but only in a larger category

of big simplicial spectra.

The definition of a big simplicial spectrum and the category to the definition of simplicial spectra and

Sp

BSp

is analogous

(2.2.35) except that there is n__0_o

local finiteness restriction (li).

Sp

is a full subcategory of

an adjoint

T ; on objects in

BSp.

BSp,

Also, the inclusion T~

J:Sp --~BSp

consists of those simplices in

admits ~

with

30

almost all faces at the basepoint extend the functor

P:Sp---~Ps

(condition to

Following V. K. A. M. Gugenheim

BSp;

(ii) in (2.2.35)). then

Alternatively,

T = SP.

[Gug, p. 36], a simplicial operator

composite of face and degeneracy maps.

Each simplicial operator

~

~

is a

has a unique

standard representative.

~ = s .-- s s d d -'" d mi m 2 mI nI n2 nj

i

with

0 < m I < m 2 < "'" < m i

is defined to be

(2.2.37)

and

0 _5_ n I < n 2 < "'" < nj .

i -j.

Definition.

The standard k - simplex

spectrum with one non-degenerate

k - simplex

Anm= where

~

{r

~k

Ak

disjoint basepoint

U {*},

Proposition.

finiteness

spectrum of

with the maps

Remarks.

For

Definition. Ak

D

the maps

to Condition

(ii).

The boundary of

generated by the faces

spectrum are in

Ak--->Y.

in Sp,

Y

condition analogous

(2.2.40)

together with a

The k - slmplices of a big simplicial

As in [G -Z, Theorem i].

(2.2.39)

m -k,

* .

one-to-one correspondence

Proof.

is the big simpllcial

and whose m - simplices are given by

ranges over all simplicial operators of height

(2.2.38)

The height of

di~ k

Ak, aAk, of A k,

Ak--->Y

satisfy a local

is the big simplicial i ~ 0.

The horn

sub-

V k's

31

is the big simplicial subspectrum of

3A k

generated by the faces

We discuss the extension condition for

BSp

and

Sp,

di~ k

for

i #

and the homotopy groups

of a simplicial spectrum.

(2.2.41) spectrum

X

Definition

(Compare [Q -i,

11.32 for

is said to be Ken if every map

(2.2.42)

Remarks.

SS).

V k'g --+X

A big slmplicial can be extended to

A k,

Ken [Ken -i, Def. 7.3] said that a simplicial spectrum

satisfied the extension condition if for each

n,

the simplicial set

satisfies the Ken extension condition, see, e.g.,~ay-l~)ef. 1.3].

X

PX(n)

It is easy to

check that the definitions are equivalent.

We shall now define the homotopy groups of a Kan simplicial spectrum, and extend the definition to

(2.2.43) o

and o'

Definition.

of X

Sp.

Let

X

be a Ken simplicial spectrum.

(Compare, e.g.,

are homotopic, denoted

[May - i, Def. 3.1]). x ~ x',

Two k - simplices

if for all

i,

d.o = d.o', 1

and if there exists a

(k +i) -simplex

T

with

d 9 = o,

dl~ ~ o',

1

and

O

di~ = diSoO = diSoO'

(2.2.44)

for

Proposition.

spectrum, then

i ~ 2.

(Compare [May -i, Proposition 3.2]).

If

is an equivalence relation on the k - simplices of

X X,

is a Ken for all

k.

Proof.

(2.2.45) spectrum.

As in the proof of [May -i, Proposition 3.2].

Definition. Let

~_

(Compare [May-l,

Def. 3.6]).

Let

denote the set of all of the k - s i m p l i c e s

X

be a Kan a

of X

which

32

satisfy

dlo = *

for all

i.

Define

~k(X) = ~ / ~

By imitating the discussion in [May -i, w

(2.2.46)

Proposition.

~k(X)

,

K

and ~ (K) m

for

we see that

is an abelian group.

Finally, for a pointed simplicial set dition, define

9

(K,*)

D

which satisfies the extension con-

as above for

m > 0.

Wm(K)

is a group

m

m > i.

(2.2.47)

Proof. since

Proposition.

~k(X) = ~k+nPX(n)

A k - simplex

dlo = *

o in ~

for all

i,

for all

can be realized in

hence for

k +n ~i.

PX(n)

for

n+k

> 0

n + k ~ 0, ~

c (PX(n)) k

Similarly, for

"

n +k sO,

(PX(n))

~

c ~

.

Further the simplices needed to define the equivalence relation in realized in (we need

PX(n +k) n +k ~ 1

for

for

n + k ~ 0,

~n+kPS(n)

Hence our definition of

~k(X)

and conversely.

may be

The conclusion follows

to be a group).

agrees with that of Kan [Kan- i, w

The following proposition is essentially contained in [Kan -i, w

We give

an explicit proof because of the variety of equivalent definitions of weak equivalence in

SS

(2.2.49)

(the equivalent conditions in [Q -i, Proposition w

Proposition.

A map

f:X--~y

in Sp

is a weak equivalence if and

only if for

Proof. PX(n)

n h 0,

Let

the maps

f:X---~Y

Pf(n):PX(n)

~ PY(n)

be a weak equivalence.

and for any basepoint

x

in PX(n),

are weak equivalences.

We shall show that for any

and for any

m ~ 0,

the induced

map

w (PY(n),y), n

Pf(n),:~m(PX(n),x)

where

y = Pf(n)(x),

is an isomorphism.

connected subsimplicial set of (the basepoint of

PX(n + i),

To see this, since

EPX(n)

is a

and contains the standard basepoint

X),

m(PX(n)'x) ~ ~m+l(PX(n+l),Ex)

-~ ~m+l(PX(n +i) ,*)

~m+l(PY(n +i),*)

~n+l(PY(n+

by Proposition 4.7)

1),Ey)

! ~ (PY(n),y); m

further, the isomorphisms induced by changing basepoints may be chosen so that the composite is

Pf(n),.

Hence the map on realizations

R(Pf(n)) : R(PX(n))

is a weak equivalence.

~ R(PY)(n))

By [Q -i, Proposition w

so is the map

Proof of the converse follows immediately from Proposition (2.2.47).

Pf(n).

D

34

(2.2.49)

Remarks.

Relative homotopy groups, the homotopy exact sequence for

a pair, the fibre of a flbration, and the homotopy exact sequence of a flbration for Sp,

as in [Kan -3, w167

See, e.g.,

may be obtained by imitating their developments in

[May -i, w167

Cofibrations and fibrations in in

SS

SS.

[Kan-2],

(2.2.50)

Sp

see e.g., [May -i],

Definitions.

are defined analogously with the definitions [Q-I,

w

Cofibrations are injective maps.

A map

E-->B

of

simplicial spectra is a (Kan) fibratlon if given any commutative solid-arrow diagram in

BSp

of the form

~s

~E / .1

#, t /e" / I k

there exists a filler

:B

f.

We can now prove Theorem (2.2.36), that

imitating Quillen's proof that

SS

Sp

is a closed model category, by

is a closed model category [Q -i, w

Details are omitted.

We shall now summarize Adams'

formulation of the foundations of generalized

homology and eohomology theories; see [Adams -1,3].

Let

h,

be a non-negatively

graded generalized homology theory defined on finite complexes and

h

be the dual

cohomology theory.

(2.2.51)

that

Brown's Theorem [Bro].

h4n(x) ~ Ho(CW,)(X,E4n)

(2.2.52) above,

Theorem

There is a

CW

spectrum

for all finite pointed

(G. W. Whitehead

[Wh-2]).

With

CW

E = {E } n

complexes

h,

and

such

X.

E = {En}

as

35

~(X)

= col/m n ___> Ho(CW) IS4 n + k,x A En) .

We use (2.2.51) and (2.2.52) to define

h

and

h,

CW

on all

spectra:

hk(X ) ~ Ho(CWSp)(sk,x A E)

(2.2.53)

E ~k(X ^ E)

hk(x) ~ Ho(CWSp)(X A s-k,E).

(2.2.54)

Alexander and Spanier-Whitehead duality.

hedron linearly embedded in

Sn.

Let

K

be a compact poly-

By Alexander duality there are isomorphisms

HP(K) ~ Hn_p_l(Sn\ K).

E. Spanier and G. W. Whitehead extended Alexander duality to state that determines the stable homotopy type of

Sn \ K

and that

Sn \ K

K

has the homotopy

type of a compact polyhedron.

Spanier introduced the following formulation of duality. compact polyhedra disJointly embedded in K

to L

with

Regarding

Sn

~(0) ~ K,

m(1) E L,

and

as the compactification of

yields disjoint embeddlngs of

(2.2.55)

Sn.

K

~(0,I) Rn

and L

Choose a

PL

path

disjoint from

with

in R n.

~:K • L

Let

~(89

K

and L ~

be

from

K u L.

as the "point at

~,"

Let

>S n-I

be the map

(2.2.56)

p(k,s

=

It is easy to check that the restrictions

~

k-s

I1~-~11

.

~I~(0) x L

and

~[K x ~(i)

are null-

38

homotopic.

Let

~(0)

Then (2.2.37) yields a

and

~(i)

be the basepoints of

and L

respectively.

map

(2.2.55)

u:K A L

Regarding

K

K, L, and

Sn-I

>S n-I

as spectra, and taking the adjoint of

p

in

(2.2.57), yields a map

~,:K

(2.2.58)

~HOM (L,sn-l).

Spanier proved that if the inclusion

L--->S n \ K

~,

K

is a stable homotopy equivalence.

and (2.3.58).

S n-I \ K

is called the

Dn_ID_I

(2.2.60)

play symmetric roles in (2.2.57)

(n -i) -dual

of

K:

Dn_IK ~ s n - l \ K.

(2.2.59)

Then,

and L

is a homotopy equivalence, then

=

- id.

Finally, the natural "composition"

~ HOM

Dn_l K A HOM(Sn-I,E)

(K,S n-l) A HOM (sn-l,E)

>~OM (K,E)

is a stable homotopy equivalence.

(2.2.61)

Definition.

The functional dual of a

CW

spectrum

X

is given by

DX E HOM (X,S0).

There is a natural map X A DX = S A H0M (X,S 0) finite.

X---~D2X SO)

(take the adjoint of the evaluation map

which is a stable homotopy equivalence if

X

is

37

We shall now discuss generalized homology theories represented by ring spectra~ following Adams [Adams -3, especially pp. 60-68].

(2.2.627

Definition.

multiplication map m

A ring spectrum consists of a

m:E ^ E--->E,

and a unit map

be homotopy associative and commutative and that

CW

spectrum

i:S0-->E. i

E,

a

We require that

be a homotopy unit.

J. P. May has gone considerably farther in studying the higher homotopies associated with ring spectra [May -2,3].

(2.2.63)

~

(see (2.2.7) for descriptions).

a)

S 0.

b)

K(R),

c)

BU, BO, ere; the multiplication is induced from the tensor product

where

R

is a ring.

of vector bundles. d)

MU, MO, etc.; the multiplication is induced from the Whitney sum of vector bundles.

e)

bu, bo, etc.; connected versions of

BU,

BO,

etc.;

see

D, W. Anderson [An-l,2].

Module spectra are defined analogously with (2.2.62). over

S~ .

P. E, Conner and E, E. Floyd [Con -FI] showed that

spectrum over

(2.2.64) spectrum

All spectra are modules

E~

E,(S O) = ~,(E)

BU

is a module

MU.

Let

E,

i.e.,

be a generalized homology theory represented by a ring E,(X) = ~,(X ^ E).

The coefficient ring of

with multiplication defined by

E

is

38

m, ~,(Z)

E,(X)

is a right

|

=,(E)

~ ~,(Z ^ E)

E,(S 0) -module

E,(X)

|

. ~,(E).

under the map

Z,(S ~

= ~,(X

^

E)

|

~,(E)

- - ~ , ( X ^ E A E) - - ~ , ( X ^ E).

(2.2.65)

Homology operations.

E,(E)

is a two-sided

E,(S 0) - module,

and

the two actions differ by the canonical involution c:E,(E) = ~,(E A E)

~,(E

A E) = E,(E)

shall now require that the

0 E,(E)

and

~,(s ) right

Then

E,(E)

E,(S 0) - modules

S0.K(Zp).

BO.

E,(E)

BU.

E,(E)

MU.

from right

E,(S O)

R,(S 0) - modules

This requirement holds for at least MSp.

and

though not for

becomes a Hopf algebra over

r

~

E,(E)

K(Z).

E,(S0).

The product map

> E,(E)

is the composite

~,(E

A

E)

~

We

be flat. that is. that the functors

be exact.

MO.

induced by interchanging factors.

~,(E

A

E) ~ , ( E

A

E

A

E

A

E)

(id ^ switch ^ id), > ~,(E ^ E ^ E A E) (m

^

m), . ~,(E

^

E).

to

30

Also,

~

induces a p r o d u c t over

E,(S O) (easy).

The coproduct (or d i a g o n a l map)

= ~E:E,(E)

9 E,(E)

~) E,(E)

is a special case of the coaction map

~X:Z,(x)

~Z,(X)

|

~,(E).

The coaetion map is defined as follows:

~,(X A E) = ~,(X ^ S 0 ^ E)

(id ^ i ^ i d ) , 9

~,(X h

E ^ E)

*~--w,(X ^ Z)

|

(The latter map is an isomorphism because

~,(E)~.(E ^ Z)

w,(E)

is flat [Adams -3, p. 68,

Lemma i]).

It

o f the d u a l o f t h e

i s now e a s y t o e x t e n d t h e t h e o r y o f c o a c t i o n

Sreenrod algebra on

(2.2.6~)

H,( ,Zp)

to a theory of homology "operations" for

Cohomolo~y and eohomoloF~v operations, are defined dually.

becomes a module over

E,(S O)

EP(x)

@

mod-p

as follows:

Eq(s O) ~ IX ^ s - P , E ]

(~

[sq,~]

; ix ^ S -p ^ sq,E ^ E]

IX ^ s-P+q,E] = EP-q(x).

'

E,.

E (X)

40

is also a Hopf algebra, with the product

E*(E)

E (E) ~

E*(E)

~E*(E)

induced by the composition

[E ^ sP,E]

~

[E ^ Sq,E]

>

[E ^ S p ^ S q, E ^ S q]

>

[E ^ S p a Sq,E]

~

[E ^ Sq,E]

= [E ^ SP+q,Ei.

The (right) action of

* E (E)

* E (X)

on

is defined by a similar composition.

The power of the above very general theory is easily demonstrated ing simple proof

(due to Adams

BU (X) = 0.

Recall that

X ^ S p --->BU

may be factored as follows:

(2.2.67)

BU

[Adams-4])

that if

MU,(X) = 0,

is a module spectrum over

X ^ $P

MU.

in the follow-

then Each map

~ X A Sp ^ S O

~xAsP

AMU

-~BU ^ M U

BU 9

But

~,(X A S p ^ MU) -~ ~,(X A MU A S p) ! ~,_p(X A MU) = MU,_p(X)

X A S p ^ MU "- * Hence,

by the Whitehead Theorem for

the composite map (2.2.67)

CW

is null-homotoplc,

spectra

= 0.

(see e.g.,

as required.

Thus [Adams-l]).

41

w

Model categories. We shall describe the basic properties

associated homotopy categories; [Q-I,

w167

of closed model categories and their

this theory is due to D. G. Quillen

1.5].

(2.3.1)

Definition.

An ordered pair of maps

(i,p)

liftin~ property if given any solid-arrow commutative

A

t /

/

I

t

i f

X--

(2.3.2)

~B

f.

Definition.

A closed model category consists of a category

together with three classes of maps in and weak equivalences

MO.

C

MI.

If a map

C,

called the fibrations,

is closed under finite colimits and limits. i

is a cofibration,

pair

i

(i,p)

Any map

f

or p

a map

i

is a fibration,

is a weak equivalence,

then the

has the lifting property. may be ~actored as

f = pi

cofihratlon and a weak equivalence or

p

is a cofibration and

p

and

where p

i

is a

is a fibratlon,

is a fibration and a weak

equivalence. MS.

Fibrations

(resp. cofibrations)

and base change (pullbacks)

are stable under composition

(resp., cobase change

Any isomorphism is a fibration and a cofibration.

C

cofibrations,

satisfying the following axioms.

and either

M2.

diagram

~Y .../1

there exists a filler

is said to have the

(pushouts)).

42

M4.

The base extension

(resp., cobase extension)

is both a fibration

(resp., cofibration)

of a map which

and a weak equivalence

is a weak equivalence. MS.

Let f

X

be a diagram in

C.

g

'~Y

~Z

If any two of the maps

f,g ,

are weak equivalences, then so is the third.

and

gf

Any isomorphism is

a weak equivalence.

M6a.

A map

p

is a fibration if and only if for all maps

are cofibrations

and weak equivalences,

the pair

i

which

(i,p)

has

the lifting property. M6b.

A map

i

is a cofibration

if and only if for all maps

are fibrations and weak equivalences,

the pair

p

(i,p)

which has

the lifting property. M6c.

A map

f

is a weak equivalence

for all cofibrations and

(v,p)

i

if and only if

and fibrations

p,

f = uv

where

the pairs

(i,u)

have the lifting property.

Observe that Axioms M5 and M6 imply Axioms

MI, M3, and M4;

hence to show that

a given category is a closed model category it suffices to verify Axioms M_5 and M6.

(2.3.4)

We shall want the following technical definitions.

Definitions.

A map which is both a fibration

and a weak equivalence

is called a trivial fihration

The initial object of

C

shall be denoted

objects exist by Axiom M0). natural map cofibration.

M0, M2,

X--~*

An object

is a fibration;

(resp., cofibration)

(resp., trivial cofibration).

~ ; the terminal object X

in C

cofibrant

*

(these

is called fibrant if the if the natural map

~ --~X

is a

43

(2.3.4) object

Definition.

X ~

[0,i]

Let

X E C.

(i0,il)

pl 0 = pi I = id X. X~

i

for

product of

i 0 + iI

'--x|

consists of an

[o,i] P~x

is a cofibratlon, the map

We shall frequently write

il(X ). X

X

and a commutative diagram

x_[ix where the map

A cylinder object for

Caution:

with an object

not depend functorially upon

in general, [0,i];

X | X |

p

is a weak equivalence, end

0

for

[0,i]

i0(X)

and

is not the "tensor"

in fact, in general,

X |

[0,i]

need

X.

We may use cylinder objects to form mapping cylinders with the usual properties. For example, we have the following.

(2.3.5~

Proposition.

Let

suitable cylinder objects so that

f:X---~y f

be a cofibration.

Then there exist

induces a trivial cofibration

Y • 0 u X (~) [ 0 , i ]

~-Y ~

[0,i],

and a cofibration

(2.3.6)

If

f

t ion,

Y |

u x @

[0,i] u Y

|

'-Y |

[0,1].

is a trivial cofibration, the induced map (2.3.6) is also a trivial cofibra-

44

Proof.

Consider the commutative diagram

i0 + i1

x | o u x |

i

~

>x |

[o,i]

,Y |

0 u X |

{ I I

JO+Jl

Y @)OuY ~ i

9

[0,i] u Y |

\

1

\,

\,

k

idy +

Y |

\N

[0,11

"

\

q-.

\ \

\.

\

in which the subdiagram i0 + iI X ~

0 u X ~

is a cylinder object for upper left square, and The map

JO + Jl

L

X, g

Y ~

0 u X ~

[0,i]

[0,I] u Y ~

Y,

f

1

is the pushout of the fp

and i ~

Factor

g

as

qk

where

k

+ i ~l .

0 u Y ~

1

induces cofibrations

kJo+kJl;Y

~

[0,i]

i0 + il,

is a cofibration and

We obtain a suitable cylinder

namely

Y ~

so that

*-X

is the pushout (cobase extension) of the cofibration

is a trivial coflhration (dotted arrows above).

object for

P

is the map induced by the maps

hence is itself a cofibration. q

~X ~

q

;Y,

4S

Y |

0 u X |

Y (~) 0 u X |

>Y

[0,1] --

[0,1] u Y |

|

; Y ~

1

[o,i],

[0,i].

The remaining assertions are easily checked by applying Axiom MS; details are omitted.

We shall discuss cocylinder objects (dual to cylinder objects) in

C,

and

loop and suspension functors as well as the induced cofibration and fibratlon sequences in w

below.

In these notes we shall always assume that our closed model categories

C

satisfy the following niceness condition.

Condition N: NI.

Each cofibration is a pushout of a cofibration of cofibrant objects.

N2.

Each fibration is a pullback of a fibration of fibrant objects.

N3.

N4.

At least one of the following statements hold: N3__aa.

All objects are cofibrant.

N3b.

All objects are fibrant.

There exist functorial cylinder objects, denoted by

-~[0,i]

with

io(- ) = - G O

and

il(-) = - ~ i .

The following closed model categories satisfy Condition N.

SS; (D. M. Kan [Kan -3],

all objects are cofibrant, Condition due to J. C. Moore, see [Q -i, w X ~

[0,i] = X x A I,

Since

[Kan -4], see D. Quillen [Q- i, w N(1) For

the usual product.

is trivial. N(3),

let

N(2)

is

46

Top;

the category of topological spaces with the following structure: cofibrations and fibrations are defined by the homotopy-extension and covering-homotopy properties, respectively; weak equivalences are ordinary homotopy equivalences. Condition

CG;

N

This is due to A. Str6m [Str].

is clear.

the category of compactly generated spaces, with a similar structure. See N. E. Steenrod [St -3]; also,

Sing;

[Has -3].

the category of topological spaces with the following singular structure: of

CW

cofibrations are pushouts of inclusions of w

complexes, fibrations are Serre fibrations, weak equivalences

are weak homotopy equivalences

[Q -i,

w

Again, Condition

N

is clear.

SSG (SSAG);

Simplicial groups (resp., simplicial abelian groups)

[Q -I, w

x

•

Here let

[o,1]

=

F(X

•

[O,I])/F(X

• O)

where the products are taken in

SS

abelian) simplicial group functor. Condition

Sp;

N

for

- X

• O,

and F Condition

Sp

• l)

- X • l,

is the free (resp., free N

follows from

SS.

D, M, Kan's simplicial spectra [Kan -i]. proved that

F(X

K. Brown [Brown] first

is a closed model category with a closed model

structure similar to that on follows from condition

N

for

SS.

See w

Condition

N

SS.

We shall now describe the homotopy theory of a closed model category satisfies condition

(2.3.7)

C

which

N.

Definition

[Q- i].

quotient category obtained from

The homotopy category of C

C,

Ho(C),

by inverting all weak equivalences.

is the

47

Quillen proved the following.

(2.3.8)

Proposition

equivalence in

Ho(C)

[ Q - I , Prop. 1.5.1].

if and only if

f

A map

f

in C

becomes an

is a weak equivalence in

C.

The following homotopy theory is required for the proof.

(2.3.9)

Definition.

f,g:X---->Y

~Y

(2.3.10)

with

Definition.

fibrant objects in

(2.3.11)

with

X

is cofibrant and

will be called homotopic (denoted

H:X • [0,i]

f:X--~Y

If

gf = id x

Let

Ccf

and

f = g)

HIXI •

is fibrant, maps if there is a map

= g"

Compare

[Q -i,

w

denote the full subcategory of cofibrant,

C .

Proposition

in Ccf

HIXI x 0 = f

Y

[Q -i, Lemma 1.5.1 and its dual].

A map

is a weak equivalence if sndunly if there is a map and

E:Y-'~X

fg = ~ .

The proof is analogous to that of [Q -i, Lemma 1.5.1], and is omitted.

Proof of Proposition (2.3.8). invertible in

Ho(C),

Given a map

f:X--->Y

I 1

f

' Y

X'

x . . . . .

X'

and Y'

which is

form a commutative diagram

X

where

in C

are cofibrant,

I 1

Y'

[f"k

X"

_

_

.~,,

and Y"

,

are both cofibrant and fibrant,

48

and all vertical maps are weak equivalence. [f"] f"

in Ho(C)

by a map

f"

is a weak equivalence.

w

Simplicial

in C.

Then use the axioms to realize the map

Finally, Proposition

See [Q -i,

w

for details.

SS,

and their generalization

cept df simplicial closed model category given by Quillen

The product in

SS

(or internal mapping functor)

[Q -I,

This product is coadjoint

> An ,

of the adjoint pair

The functors

(i)

~

0 ~ i in.

See e.g.,

[May-l] ~

or [ Q - I ] .

(x, HOM)

we shall write

and

satisfy the following properties.

HOM

for

x .

There is an associative composition HOM (X,Y) x HOM (Y,Z) X, Y, and Z functors

in C,

such that for and

g

u

~HOM

(X,Z),

for all

and a natural isomorphism of

SS(X,Y)

> HOM (X,Y)0,

in SS(X,Y),f

in H O M (W,X),

together with

to the "function

HOM (X,Y) = {HOM (X,Y) n = SS(X x An,y)},

together with the face and degeneracy maps induced form the maps s i :d n+l

to the con-

w167

X x y = {(X x Y)n = Xn x yn} ,

is given by

the induced face and degeneracy maps.

and

implies that

closed model categories.

We shall discuss function spaces in

space"

(2.3.11)

in HOM (u

n,

di:A n-I

~A n

In the context

49

f o (s0)n5 = HOM (u,Z)n(f),

(SO)n f o g

=

and

HOM (W,U)n(g)

r~ SS(X,Y)

for all

(ii)

Y

9 HOM (X,Y)0,

in C.

There are natural maps

u:Y

~ H O M (X,X ~

Y)

which

induce isomorphisms (enriched adJunction [E -K])

HOM (X ~ Y,Z) ----~HOM (Y, H0M (X,Z)),

(iii)

for all

X

and Z

in C.

For all

Y

in C

there are natural maps

B:Y-"~HOM

(HOM (Y,Z),Z)

HOM (x, HOM (Y,Z)) for all

(iv)

X

and Z

which induce isomorphisms

~HOM (Y, H0M (X,Z)),

in C.

There are natural isomorphisms

HOM (*,X) ~

for all

X

X,

in C.

Consequently,

(v)

The composition maps in (i) and

SS

are compatible; i.e.,

the following diagram commutes for all

X,Y,Z

in SS:

50

HOM (X,Y) 0 x HOM (Y,Z) 0

~ HOM (X,Z) 0

ss(x,Y) x ss(Y,Z)

(vl)

~

> ss(x,z) .

is coherently associative and co-~utatlve,

with

*

as coherent unit.

Function spaces and products example,

define

~

on SS,

in

by

CG

have similar properties.

X~Y

is replaced by

- X ^ Y = X x Y/X v Y,

above except that

An

Joint basepolnt.

These ideas have been abstracted

closed symmetric monoidal category

A n*

admits a "singular

HOM (X,Y)

function space."

to be the simpllclal

An

and

HOM

as

by adjoining a dis-

in S. Eilenberg and G.M. KelIy's

[E- K].

The singular closed model structure on w

obtained from

For another

Top,

([Q -i, w

TOPsing

For

X

and Y

in Top,

define

HOM (X,Y) n = Top (X x RA n , y)

set with

see

together

with the induced face and degeneracy maps, see above.

Mere

R

denotes M/inor's

geometric realization functor

For

X

in

K

in SS,

define

X ~

[Mil - 2], see [May -i].

K ffi X x RK.

HOM (X ~

for

K

in SS,

and

X,Y

is a category

Definition C

and

Then

K, Y) = HOM (K, HOM (X,Y))

in Top.

Quillen generalized

ducing closed simplicial model cate~orles,

(2.4.1)

Top

this concept by intro-

described below.

(see [Q -i, Definition

w

together with the following structure:

A simpliclal category

51

(i)

A functor

HOM (-,-)

from

C x C

to SS,

contra-

variant in the first variable and covarlant in the second.

(ii)

X, Y, and Z

For an

in C,

maps in

SS

H0M (X,Y) x HOM (Y,Z) ---~HOM (X,Z)

called composition.

(iii)

An isomorphism of functors

C(X,Y)

where

HOM (X,Y) O,

consists of the O - simpllces of

HOM (X,Y) 0

HOM (X,Y).

These functors are required to satisfy t h e following conditions. (i)

Composition is associative.

(2)

For

u

in C(X,Y),

f

in HOM (Y,Z)n,

and

f - (s0)nQ = HOM (u,Z)n(f),

g

in HOM (W,X) n,

and

(s0)nQ o g = HOM (W,U)n(g).

(2.4.2) K

in SS,

map

u:K

Definition X ~

K

(see [Q -i, Definition w

shall denote an object of

HOM (X, X ~

K)

8:K

K, Y)

shall denote an object of

> HOM (HOM (K,X), X)

X

in C

ROM (K, HOH (X,Y)).

C

and

together with a distinguished

which induces a natural isomorphism

HOM (X ~

HOM (K,X)

C

For

together with a distinguished map

which induces a natural isomorphism

52

ROH

(2.4.3)

(Y,

Examples.

(K,X))

HOM

Clearly

SS,

is a simplicial category (see [Q -i],

HOM (K, HOM (Y,X)).

with its usual symmetric monoidal structure, [May], [E -K]).

TOPslng

(see w

is

also a slmpllclal category.

(2.4.4)

Definition

A closed simp!icial model

[Q- i, Definition w

category consists of a closed model category

C

which is also a simplicial category

satisfying the following two conditions.

SM0.

SMT.

For

X

in C

X ~

K

and

If

i:A---~X

and K

a finite simplicial set, then

HOM (K,X)

exist.

is a coflbration in

is a flbration in

HOM (X,Y)

C,

C

and

p:Y---~B

then

> HOM (A,Y) x HOM (A,B) HOM (X,B)

is a fibratlon in

SS which is trivial if either

i

or p

is trivial.

Recall that for spaces

X

and Y,

say in

CG,

[X,Y] ~ ~o(HOM ( X , Y ) ) , where with

HOM (X,Y) Y

fibrant (i.e., Kan),

[X,Y] ~ ~0(HOM (X,Y)).

in an abstract simplicial closed model category

(2.4.4) C

and Y

Similarly, for

is the usual function space.

Proposition is fibrant in

[Q -i, Proposition w C,

then

X

and Y

in SS

A similar statement holds

C.

If

X

is cofibrant in

53

Ho(C)(X,Y) a ~0(HOM (X,Y)),

the set of path components of

(2.4.6) a)

HOM (X,Y).

D

Remarks.

Our use of S • S, HOM

HOM

in three settings, on

C x C,

C x S,

and

should emphasize the analogy between the functor

of Definition and the usual "function space" (internal

mapping) functors.

b)

All of the above results have pointed analogues; replace SS

by SS,

and

An

by A

n*

,

which is obtained from

An

by adjoining a disjoint basepoint.

c)

In general the function space in

Top

or CG,

HOM (X,Y),

is not homotopy equivalent to the realization of the singular "function space"

R(HOMsing(X,Y)) E R{Top (X x RAn,y), di'si}"

For example, the latter space is always a

w

CW

complex.

Homotopy theories of pro -spaces.

In this section we shall briefly indicate the need for a "sophisticated" homotopy theory of pro -spaces.

M. Artin and B. Mazur took

pro -Ho(Top)

to be the

homotopy theory of

pro -Top.

for some purposes.

For example, Quillen [Q -i, C. II, p. 0.3] observed that the

category

pro -Ho(Top)

pro - Top. to call maps

Unfortunately, this point of view is inadequate

was not the homotopy category of a model structure on

One next attempts to define homotopy ~lobally in f,g:{X i} --~{Yj}

pro- Top,

homotopic if there is a homotopy

that is,

54

H:{Xi} x [0,i] ~ {X i • [0,i]}

~ {Yj}

from

f

to g.

This notion is stronger

than the Artin-Mazur notion which would identify two level maps

if there were homotopies

H .

For example, let

n

Hn:~n = gn

D

S1 z 2

S1

S1 ~

*",

is the degree two map

z t

2 ~z .

many global homotopy classes of maps from a point to Z ~2

...}

such classes).

Then, there is a unique

D,

but there are uncountably D

0

I////////i!

JJJ//J,'JJJJJH X

(more precisely,

The following example shows that the

notion of global homotopy is also too naive.

o

~ Yn}

without any Coherence criteria among the

Artln-Mazur homotopy class of maps from a point to

lim I {Z ~2

fn,gn

denote the inverse system S 1 <2

where

{Xn

P

-> Y

55

here

X = { X n ~ (S 1 v [ 0 , |

Y = {Yn s $1 x {0,1} u [ 0 , 1 ] } .

x {0,1}

u [n,|

The map

x [0,1]},

p = {pn}:X----~Y

e q u i v a l e n c e , b u t t h e r e i s no homotopy i n v e r s e t o t h e b o n d i n g maps o f t h e t o w e r s

X and Y

shall define

p

i s l e v e l l y a homotopy

i n pro - T o p .

are fibrations,

g l o b a l homotopy t u r n s o u t t o be t h e " r i g h t " n o t i o n . Ho(pro - T o p ) ,

and

If,

however,

then the notion of

The " r i g h t " homotopy c a t e g o r y ,

i s d e f i n e d by f o r m a l l y i n v e r t i n g l e v e l homotopy e q u i v a l e n c e s . Ho(pro - T o p )

in w

We

w

w

THE MODEL STRUCTURE ON PRO - SPACES

Introduction.

In this chapter we shall associate to a closed model category fies condition

N

(w

a natural closed model structure on

C

which satis-

pro - C.

This

chapter is organized as fellows.

In w

we discuss the homotopy theory of

is a cofinite

model structure on

(Theorem (3.2.2)) which is natural in the following sense

structure.

= b).

J

(a < b

(Theorem (3.2.4)).

b < a ~a

where

strongly directed set CJ

and

C J,

The constant diagram functor

The inverse limit functor

lim:C J

We shall develop a closed

C--~C J C

preserves the model

preserves fibrations and

trivial fibrations.

In w CJ

to

we shall extend the closed model structure from the level categories

pro - C

(Theorem (3.3.3)) with the same naturality properties as our

closed model structure on

CJ

(Theorem (3.3.4)).

model structure on the full subcategory on

pro - C

w

We also obtain a natural closed

tow -C c pro -C.

Simplicial structures

are discussed in w

is concerned with suspension and loop functors, and cofibration and fibra-

tion sequences.

D. Quillen

[Q - i, w167

developed a general theory of suspen-

sion and loop functors, and cofibration and fibratlon sequences in the homotopy category of an abstract closed model category. context of

Ho(pro - C).

We shall sketch this theory in the

We shall show that an inverse system of fibrations over

57

C

is equivalent in

In w

Ho(pro- C)

to a short fibration sequence.

Maps (pro -C)

we consider the category

(C, pro -C)

A--~X

whose objects are maps

(in

and a full subcategory pro-C)

with

X

stable in

pro -C.

We develop useful (see w167 Ho(tow-Top)

geometric models of

Ho(Top, tow- Top)

and

in w

We shall compare our closed model structure to those of A. K. Bousfield and D. M. Kan [B-K, p. 314] and J. Grossman [Gros-l]

in Remarks (3.2.5).

The above theory of pro - spaces admits an evident dualization to direct systems (in~ -spaces).

w

We shall briefly sketch this theory in w

The homotopy theory of Let

C

J ( = {j})

C J.

be a closed model category which satisfies Condition N (w be a cofinite strongly directed set.

a natural closed model structure from category

Ho(C J)

(3.2.1)

C;

We shall show that

A map

f:X--->Y

cofibrations (resp., weak equivalences)

in CJ

respect to all maps

inherits

(see w

Definitions.

f

CJ

this will yield the required homotopy

( = {fj :Xj -'-~Yj})

cofibration (resp., weak equivalence) if for all

A map

Let

j

in J,

in C J

the maps

is a f. J

are

in C.

is a fihration if it has the rlght-lifting-property with i

which are both cofibrations and weak equivalences.

The main result of this section is the following.

S8

(3.2.2)

Theorem.

C J,

together with the above structure, is a closed model

category.

map

A map

f:X--->Y

in CJ

qj

in the diagram

is a fibration if for each

Xj

.

j

in J,

the induced

.

"'\\

(3.2.3)

> limk < j ~

Pj

', f.'\ \

\

l~mk < jfk

Y. 3

(Pj

-~ i ~ k < jYk

is the pullback) is a fibration.

The special case tow-C

J ffiN

is used to obtain a closed model structure in

in w

(3.2.4)

Theorem.

The constant diagram functor

tions, fibrations, and weak equivalences.

C---~C J

preserves cofibra-

The inverse limit functor

lim:C J "-~C

preserves fibrations, and trivial flbrations (maps which are both fibrations and weak equivalences).

Proof.

I ~ e d i a t e from Definitions (3.2.1).

R

(3.2.5)

Remarks.

model structure on

Bousfield and Kan

CJ

defining eoflbrations

[B -K,

by defining fibrations by the appropriate

structure has the disadvantage tlons and weak equivalences

p. 314] defined a different closed

and weak equivalences

lifting property.

that most of Theorem

(3.2.4) is false:

structure

only fibra-

are preserved by the constant diagram functor; none of

homotopy inverse limit functor

Bousfield-Kan

and

The Bousfleld-Kan

the model structure is preserved by the inverse limit functor.

evident the applications

degreewise,

(w

Consequently,

our

is simpler than theirs; this simplicity makes

to homological

algebra in w167

is natural on direct systems,

see also below.

see w

especially,

The

(3.8.1)-

(3.8.4).

J. Grossman

[Gros -i] also introduced a closed model structure on the category

of towers of simpllclal sets. inverts ~ - i s o m o r p h l s m s

His structure is weaker than ours; essentially,

in the sense of [ A - M ] .

See (5.4.4)-(5.4.5)

he

for the

definition of ~ -isomorphism.

Our definition of fibration was motivated by the definition of a cofibration pairs

(for the inclusion

that the induced map of a cofibratlon of a)

(X,A) --->(Y,B)

X uA B--->y CW

spectra

be a cofibration),

(see [Vogt -2]).

one usually asks

and the analogous definition

Also:

Our definition of flbration is consistent with the definition of a flasque p r o - g r o u p

b)

to be a cofibration

(see w167

4.8).

The associated definition of coflbration means that a proper cofibratlon

E(X) --~e(Y)

X--~y (see w

of

induces a cofibratlon of the ends

60

The proof of Theorem (3.2.2) is contained in Propositions (3.2.6), (3.2.24), (3.2.27) and (3.2.28), below.

(3.2.6)

Proposition.

(Verification of Axiom M0).

CJ

admits finite limits

and colimits.

Proof.

Let

D

be a finite diagram in

have colimits

colim D. J

yield objects

{colim Dj}

colimit and limit

In fact, if

of D,

C

and

limits

and

CJ.

The induced diagrams

llm D. in C J

{lim Dj}

in CJ

respectively.

by Axiom M0 for

Dj over C C.

These

which are easily seen to be the

D

admits more general colimits or limits, so does

In order to verify Axioms M2, M5 and M6

for

CJ

CJ .

we shall give explicit

descriptions of fibrations (Proposition (3.2.7)) and trivial fibrations (Proposition (3.2.17))

in CJ.

(3.2.7) for each

Our descriptions will involve diagram (3.2.3).

Proposition. j

in J

A map

p:Y --->B in CJ

the induced map

qj

is a fibration if and only if

in the diagram

limk < jYk P J ~

PJ

lJmk < jPk

Bj

-

~

i~

k < jB k

81

(Pj

is the pullback) is a fibration in

Proof.

First, let

p

C.

be a fibration in

CJ,

that is, assume that

p

rlght-lifting-property with respect to the class of trivial cofibrations in We shall show that each induced map

qj:Yj ---~Pj

property by constructing suitable "test maps"

has the CJ.

has the same right-lifting-

K--~L

in CJ

which are trivial

cofibrations.

Consider a solid-arrow commutative diagram

A

:Yj

(3.2.8) /

X

in

C

K = ~

in which and

i

L = ~

> Pj

is a trivial cofibration. in C J

Define objects

as follows:

Ii

for k < J, for k = J,

L~

otherwise;

for

k~j,

otherwise . The required bonding maps are induced by

i and id X.

trivial cofibration

Diagram (3.2.8) induces a solid-arrow

commutative

diagrsm

i' :K --~ L

in CJ.

Then there is an induced

82

K

"" '.~ Y / / 1

(3.2.9)

9

i'

g,./

/

p

f / /

L

in

CJ

(the maps

Kk--~Y k

the composite maps

X--~Pj

~B

are induced from the map ~ Yk

for

k < j

g

in diagram (3.2.9).

of diagram (3.2.9), the map (3.2.8).

Hence

qj

qj:Yj --~Pj

j ffi k

p:Y-->B

The right-lifting-property

gj:Lj = X - - > y j

CJ

with the property that the induced Consider a solid-arrow

CJ

H

Y / J

/ J

(3.2.10)

f.-"

p

J 1 9p,

,,

,

h

is a trivial cofibration.

We shall obtain the required

filler f = {fj:Xj ---~Yj}

by induction on

J .

p

is the required filler in diagram

be a map in

A

i

of

Because diagram (3.2.8) is the jth level

(see diagram (3.2.3)) are fibrations.

commutative diagram in

in which the map

and

is a fibration, as required.

Conversely, let maps

for

(see diagram (3.2.8)), the other

maps in diagram (3.2.9) are defined similarly). yields a filler

A--~yj

63

Consider a fixed index fk:~--~Yk

j.

k < j,

Suppose that for all

there exist maps

with the following properties:

fkik = H k , (3.2.11) Pkfk = h k ;

(3.2.12)

f~ o bond = bond ofk

restrictions are vacuous).

(3.2,13)

for

s < k

(if

j

has no predecessors

these

Formula (3.2.12) yields a composite map

Xj

> limk < j Yk

> limk < jBj ;

by formulas O.2/1) this map is equal to the composite map

(3.2.14)

Xj

>

Bj

~ lim k < j B k.

In fact, formulas (3.2.10) - (3.2.14) yield a solid-arrow commutative diagram

H.

]

A,

Yj /

(3.2.15)

J

f

i

i

I lqj

]//

3

/ J

~.

I

,t ~

1.

X. J

in

C.

(This uses the definition of

trivial cofibration and

gj

P. J

in diagram (3t.2.3).) i

is a fibration in

C,

Because

there!exists a filler

i. 3 f.

is a in

3

diagram (3.2.15). for

Further, diagram (3.2.15) yields formulas (3.2.11) and (3.2.12)

f.. 3 By continuing inductively, we obtain the required filler

diagram (3.2.10).

f = {fj}

in

64 (3.2.16)

P!oposition.

level, the map

Proof.

pj:u

Let

--~Bj

For a given

p:X--~B

be a fibration in

is a flbration in

j in J,

Then a t each

C.

consider the commutative-dlagram

< J Yk

"'~

(3.2.3)

CJ .

\

~ '

pj\

I

'j

I~<

II

Bj

JPk

> lim k < jB k

By introducing the indexing category K=

we see that the map Hence

l~<jp

k

{k[k < J) = J,

is a fibration in

p'j, and thus the composite map

(3.2.17)

Remarks.

Trivial fibrations in

(3,2.18) Proposition.

fibrations in

C.

(apply Theorem (3.2.4)). are fibrations in

C.

D

The above proof illustrates the usefulness of cofinite

strongly directed indexing sets: maps in

only if the induced maps

pj = p'jpj

C

CJ

CJ

may be constructed inductively.

admit a similar characterization.

A map

p:@ --~ B i n

qj:Yj --~Pj

CJ

is a trivial

(see Proposition (3.2.6))

flbratlon are trivial

i f and

65

Proof.

First, let

J,

element in

the maps

p:Y--->B

be a trivial fibration.

qj:Yj --~Pj

and

pj:Yj --~Bj

If

j

is an initial

are equal.

a flbratlon by Proposition (3.2.7) and weak equivalence by hypothesis. qj(- pj)

But

pj

is

Hence

is a trivial flbration, as required.

Now suppose that for a fixed (non-lnltlal)

the induced maps

qk

are trivial fibrations.

J

in J,

and for all

We shall show that

qj

k < J, is a trivial

fibratlon.

Consider the commutative diagram

y

qj

~- Pj

'~ l ~ k < jYk

limk< jPk l~mk< js k

Bk

in which

Pj

induced map

is a pullback. Pj --~B k

hence a weak equivalence.

If

l ~ k < JPk

were a trivial fibration, the

would also be a trivial fibration (by Axiom M4 for But then

qj

would be a weak equivalence (since

a weak equivalence by hypothesis, this follows from Axiom M5 for a fibratlon by Proposition (3.2.7), so that required.

fj

C).

But

fj

is

fj

is

would be a trivial fibration, as

It therefore suffices to show that the maps

(3.2.19)

C), and

l ~ k <JPk: llmk <jYk -'-'~llmk <jBk

are trivial fibratlons.

To do this, we consider a commutative solld-arrow diagram of the form

66

K

~"limk < jYk

/

i

(3.2.20)

/

f'~"

i l imk

< jPk

/

i

1 / /

L --

in

C

in which

i

> llmk < jB k

is a cofibration.

We may define a filler

f

in diagram

(3.2.19) by defining maps

fk :L

for

k < j

~ Yk

which make the,diagrams

K

"> limk<

j~k

Yk

(3.2.21) ,

.k L

~

I im k < j I~k

~.-Bk ,

k < j

and fk

> Yk lbon d

(3.2.22~

Ys

~
commute.

This requires a second induction; this time on

fixed

aud all

k,

A < k

k.

there exist the required fillers

predecessors, this condition is vacuous).

K

We obtain a map

; Pk

Suppose that for a fA

(if k

has no

67

(see diagram (3.2.3)) for which the solid-arrow diagram

K

.

.

.

i

.

> Yk

.

/ /

qk

/ / /

(3.2.23)

> l im< < kYs

>Pk

L

I

lim~ < kp s

llm E < kBE

Bk

commutes.

But the map

qk

is a trivial fibration by our inductive assumption.

Hence there exists a filler

fk

for the upper left corner of diagram (3.2.23).

Diagram (3.2.23) immediately implies that the required diagrams (3.2.22) commute.

(3.2.21) and

Continuing inductively yields the required maps

fk

and fille~

f = l~%
in diagram (3.2.20).

Hence, the maps (3.2.19) are trivial flbrations,

as required.

The proof of the converse is similar to the proof of the "if" part of Proposition (3.2.6) and is omitted.

We may now verify t~at

D

C

satisfies Axioms M2, M5, and M6 for a closed model

category.

(3.2.24)

Proposition.

(Verification of Axiom M2)

may be factored as

x i--~ z

P-y

Any map

f:X--~ Y

in C J

68

where

i

is a cofibratlon,

p

is a flbration, and either

i

or p

is a weak

equivalence.

Proof,

We shall factor

f

as pi

with

of the other case is similar and omitted. fj :Xj - ~ Yj

i

a weak equivalence.

To factor

f,

The proof

we~shall factor the maps

as

xj

(3.2.25)

i

zj

p

where: a)

for

k < j,

b)

the maps

c)

the induced maps

i. J

pj:Zj ---~Yj

i.3 and pj

If

J

ik

and Pk'

are trivial cofibrations;

qj:Zj --~Pj

respectively;

and

associated with the maps

(see diagram (3.2.3)) are fibrations.

Suppose for a given factored.

cover

j

and for all

has no predecessors,

form the co-,,utative diagram

k < j,

the maps

fk

have been so

this condition is vacuous.

We may then

69

X.]

> llm k <jXk

(3.2.26)

limk < jZk

Qj

i ~ ~ j Pk

j

YJ

where

Qj

-

i s a p u l l b a c k , and the map

~

gj

ls

is induced from the map

in the proof of P r o p o s i t i o n ( 3 . 2 . 1 7 ) ,

see diagram ( 3 . 2 . 2 0 ) ,

a fibration.

gj

Axiom M3 for

Hence the induced map

< jYk

l ~ k < jPk"

As

limk< jPk

is

is a fibration

(by

the map

(see diagram (3.2.26))

C).

Now, f a c t o r the map

Xj --~Qj

(in diagram (3.2.26)) as the composite

Xj (using Axiom M2 for

C).

~ Zj

q'J ~ Qj

Finally, let

pj = g j q ' j : Z j

"- Qj ~ Y j

9

Then the factorization

Xj

il

~ Zj

Pl

"u

(see diagram (3.2.25)) has the required properties.

The conclusion follows.

El

70

(3.2.27)

Proposition.

(Verification of Axiom M5).

If two of the maps in the

diagram [ X

~Y

Z are weak equivalence,

Proof.

then so is the third.

This follows from Axlom M5 for

C

since weak equivalences

in

CJ

are

defined degreewise.

(3.2.28)

a)

Proposition.

(Verification

of Axiom M6).

A map is a fibration if and only if it has the right-liftlngproperty with respect to the class of trivial cofibrations.

b)

A map is a eoflbration

if and only if it has the left-lifting-

property with respect to the class of trivial fibrations. c)

A map

f

is a weak equivalence

where

v

has the left-llftlng-property

class of fibrations and

u

if and only if

f = uv

with respect to the

has the rlght-lifting-property

with respect to the class of cofibrations.

Proof.

Part a) is contained in Definitions

(3.2.1).

The proof of Part b) is similar to that of Proposition is to associate objects

to an element

E = {E k}

and

j

A = {~}

of J in C J

(3.2.7).

and a fibration defined by setting

Y--->B

The main step in C,

the

71

= ~ Y

for

k ~ j,

Ek

L = ~

*

otherwise,

B

for

*

otherwise~

k ~ j,

Ak

L

with the evident bonding maps and also the fibration

E--~B

then proceed as in the discussion following diagram (3.2.8).

in C J.

One may

Remaining details

are omitted.

For Part c), first let

f

be a weak equivalence.

is a trivial fibration and

v

is a fibration.

Factor

f

By Axiom M5 for v

(3.2.27)) and Propositions (3.2.7) and (3.2.17),

as CJ

uv

where

u

(Proposition

is a trivial coflbration.

The

required lifting properties follow easily.

Conversely, arguments similar to the proof of Proposition (3.2.18) show that maps

u

f = uv

and v

with the given lifting properties are weak equivalences.

is a weak equivalence by Axiom M5 for

C J.

Hence

Details are omitted.

This completes the proof of Theorem (3.2.2).

w

The homotopy theory of Let

C

p r o - C.

be a closed model category which satisfies Condition N (w

shall show that

pro - C

model categories

CJ

(J

required homotopy category

We

inherits a natural closed model structure from the closed is a cofinite strongly directed set); this will yield the Ho(pro -C)

(see w

One of our main tools is the

Marde~i6 trick (Theorem (2.1.6)) which states that any inverse system is isomorphic to an inverse system indexed bY a coflnite strongly directed set.

72

(3.3.1) if

f

Definitions.

is the image in

where

J

A map pro -C

f

in pro- C

of a (level) cofibration

is a cofinite strongly directed set.

cofibrations,

is called a strong cofibration f. J

in some

Strong fibrations,

C J,

strong trivial

and strong trivial fibrations are defined similarly.

A map in

pro -C

is a cofibration

of a strong cofibration.

Fibrations,

in

pro- C

are defined similarly.

if

f = pi

where

(3.3.2)

p

Remarks.

if it is the retract in Maps

trivial cofibrations, A map

f

in pro -C

is a trivial fibratlon and

Clearly,

i

(resp.,

p r o - SS

are both cofibrations

(3.3.13),

below, shows that the converse assertions hold.

By Definitions

(3.3.1)

and trivial fibrations is a weak equivalence

is a trivial cofibration.

trivial cofibrations (resp., fibrations)

(pro -C)

trivial fibrations)

and equivalences.

the classes of cofibrations,

fibrations,

Corollary

trivial cofibra-

tions, and trivial fibrations are each closed under the formation of retracts. need to show that the class of weak equivalences Definitions

(3.3.1) yield an apparently

the composition

Definitions

larger class of weak equivalences

do not know if every weak equivalence equivalence

in some

in the categories

(3.3.1) are essentially

of two weak equivalences in

CJ

than the (Defini-

forced by the requirement

yields a weak equivalence pro- C

We

is also closed under retracts.

class of retracts of the (level) weak equivalences riots(3.2.1)).

that

(Axiom MS).

Theorem.

C J.

model category.

pro -C,

We

is a retract of a (level) weak

In this section we shall prove the following.

(3.3.3)

in

together with the above structure,

is a closed

73

(3.3.4)

Theorem.

cofihrations,

flbrations,

lim:pro -C---~C

Proof.

The constant diagram functor and weak equivalences.

preserves

C

preserves

The inverse limit functor

flbrations and trivial fibratlons.

Immediate from the Definitions

The category of towers, from

C--~pro-

tow -C,

(3.1.1) and Theorem (3.3.3).

D

inherits a natural closed model structure

pro -C.

The proof of Theorem

(3.3.3) involves the following main steps:

Verification

of Axiom M0 (Proposition

(3.3.5));

Verification

of Axlom M2 (Proposition

(3.3.8));

Verification

of Axiom M6 (Proposition

(3.3.9),

Verification

of Axiom MS:

Special cases (Propositions General case (Proposition

(3.3.5)

Proposition

(3.3.15), and (3.3.17));

(3.3.18), and (3.3.26));

(3.3.35)).

(Verification of Axiom MO).

Pro - C

admits finite

colimits and limits.

Proof. a colimit;

Let

A

be a finite diagram in

the construction of a limit for

ing identity maps if necessary,

pro - C. A

is similar and omitted.

we may assume that

the original and new diagrams will be isomorphic). reindexing diagrams of

(see Proposition C

(2.1.5))

which determines

ing the Marde~i6 construction

to

A

We shall show that

A

(Theorem (2.1.6))

By insert-

Applying the Artin-Mazur

pro - C to

has

has no loops (the colimits of

yields an inverse system

a diagram in

A

{A i}

isomorphic

{A i} to

A .

yields a diagram

of ApplyA'

74

over some level category ther

4'

and

X' = colim 4'

A

CJ

indexed by a cofinlte strongly directed set

are isomorphic diagrams over

in C J

(X'

X' = colim A

coherent family of maps from the objects of of

to an object

X', Y

the maps from

to Y

some level category K

with

~,, i ~

Artin-Mazur

Let

CK

restrictions

family of maps from 4, Y,

A

and

~"

in

of

and

in the

Because any dlagram whlch represents

T: K

~".

to Y"

A' e 4' m A

To check the

indexed by a cofinite strongly directed set

in pro -C.

4"

to X'.

As above, we may define a diagram

relndexlng process yields a d i a g r a m w h i c h

the maps from 4" m T*

( m 4')

be the diagram consisting of

in pro - C.

there is a natural coflnal functor appropriate

A

(3.2.6)).

Clearly there exists a

suppose that there exists a coherent

in p r o - C. A

in pro -C. A

fur-

Now let

is defined levelwise by Proposition

We shall now check that

universality

pro -C.

J;

Let

~ J. X"

Now let

required universal property.

4"

be the colimlt of

factor uniquely through X" e T*X' ~ X'

represents

X".

in pro -C.

by restriction,

A

and Y" d"

be the

in C K.

Then

Further, Hence

X'

has the

D

Artin and Mazur give a non-constructive

proof of the following more general

result.

(3.3.6)

Proposition.

universe such that limits.

SS

[A-M,

Propositions

is U -small.

Then

A.4.3 and A.4.4]. pro -SS

Let

U

be a

admits U - small colimits and

D

(3.3.7)

Remarks.

Suppose that

C

infinite diagram over some level category

admits arbitrary colimits. C J,

and

X

Let

be its colimit in

A

be an

CJ

(X

75

It is easy to see that in

is defined levelwise as in Proposition

(3.2.6)).

general

in p r o - C.

X

is n o t the colimit of

A

The statement of the following proposition Axiom MI; see Remarks

(3.3.8)

fibration,

or

Proof.

i

(Verification

f = pi

where

f' f

f'

By Propositions

of Axiom M2).

Any map

is a trivial cofihration and p

p'i'

in

C J.

f' >X' ---~ Y'

CJ

By Axiom M2 for where

complete the proof, let

i

X

p'

and

and p

e

We shall now begin the verification proposition

(3.3.9)

CJ i'

> Z',

(Proposition

Z'

p'

"Y'

f

as the composite

f

J

-.g

pro - C.

solid-arrow diagram

'~Y

~B

Consider

we may factor To

0

p r o - C.

(3.3.10)

X

(3.2.24)),

J.

composite mappings

~'>Y.

of Axiom M6 for

Given any commutative

A

is a

have the required properties.

is a special case of Axiom MI for

Proposition.

p

~Y,

be the respective

i'

"~X'

in pro- C

is a trivial fibration.

(2.1.5) and (2.1.6) we may factor

--

f

is a level map indexed by a cofinite strongly directed set

as a map in as

i

is a coflbration and

X

where

of

(3.3.2).

Proposition.

may be factored as

is a technical reformulation

The following

76

in which either

i

is a trivial cofibration and

cofibration and

p

is a trivial fibration, there exists a filler

Proof.

p

is a fibration, or

We shall only discuss the case in which

i

i

is a

h.

is a trivial cofibration.

The proof of the other case is similar and omitted.

Since the lifting property described in diagram (3.3.10) is preserved under the formation of retracts, we may assume that

i

is a strong trivial cofibration

indexed by a cofinite strongly directed set

J

indexed by a cofinite strongly directed set

K.

and that

p

is a strong fibration

We shall now replace diagram (3.3.10) by a suitable level diagram. merely apply the reindexing techniques of w cofinal functor

T:L--~K

We cannot

since it appears unlikely that a

maps fihrations in

~

into fihrations in

CL

(fibra-

tions are no___tdefined levelwise).

Since

J-"~K

J

is cofinite, we may inductively define an order preserving function

(j +--~k(j))

and commutative solid-arrow diagrams

*j

~(j) (3.3.11)

9

ik (J)i

h'i/// / i / / /

,

Yj iPj

/ /

Xk(j)

"~

' Bj

which represent diagram (3.3.10) (that is, there are maps of diagram (3.3.10) to diagrams (3.3.11) over diagram of diagrams

pro- C

such that if

j' > j

there is a commutative

77

(3.3. i0)

j,

level

(3.3.11),

"

j

level

(3.3.11),

We thus obtain a commutative solid-arrow diagram

{r )

>

u

/ / #

//

(3.3.12)

{ik}

{h'. }/z

{ik(j) }

/

/

/

{pj }

i

/ I

) {Xk(j)}

{xk

with the following properties: f

and g

be cofinal, hence and CJ

the composites along the top and bottom rows are

respectively, and the right-hand square is a level diagram indexed by

(that is, a diagram in

CK

~ {Bj}

CJ).

Note:

{Ak} # {Ak(j)},

are defined levelwlse,

Since, by hypothesis,

{pj}

in general the function etc. {ik(j)}

is a trivial cofibration in

is a fibration in

CJ,

{h'j}

by Axiom M6 for

{h'j} ~{Xk(j)}

~ {Yj}

[]

CJ .

CJ

in diagram (3.3.12).

is clear that the composite mapping

is the required filler in diagram (3.3.10).

need not

Further, since trivial cofihrations in

(Proposition (3.2.28)), there exists a filler

{xk}

J--->K

J

It

78

(3.3.13)

Corollary.

A map is a trivial cofibratlon

a cofibration and a weak equivalence.

if and only if it is both

A similar description holds for trivial

fibrations.

Proof. f:X--~Y

The "only if" assertions hold by definition. be both a cofibration and a weak equivalence.

(3.3.1), write fibration.

f = pi

where

We shall see that

i f

Conversely,

Using Definitions

is a trivial cofibration and is a retract of

i,

let

p

is a trivial

and hence a trivial

cofibration.

i

(3.3.14)

~z

f

p

/ .f

-

Proposition (3.3.14)

;Z

(3.3.9) yields a filler

id.

g

//// I y Y

in diagram (3.3.14).

Rewriting diagram

in the form

X

Y

where

pg = idy,

Verification

shows that

g

f

;Z

P

;Y

is a retract of

of the assertion about fibrations

This answers one of the questions

i.

is similar and omitted.

raised in Remarks

(3.3.2).

D

79

(3.3.15) a)

Proposition.

A map

p

(Verification

is a fibration

cofibrations

of Axioms M6a and M6b).

if and only if for all maps

and equivalences,

the pair

(i,p)

i

which are

has the lifting

property. b)

A map

i

is a cofibration

if and only if for all maps

are fibrations and equivalences,

the palr

p

which

(i,p)

has the lifting

(3.3.13)

(which shows

property.

Proof. a)

The "only if" part follows from Corollary that

i

is a trivial cofibration)

Conversely, a). u

let

p

is a fibration and

u, b)

(3.3.9).

be a map with the lifting property of hypothesis

Use Axiom M2 (Proposition

proof of Corollary

and Proposition

v

3.3.8)

to write

f = uv,

is a trivial cofibration.

(3.3.13),

it follows that

p

where

As in the

is a retract of

and hence a fibration.

The proof is similar to the proof of a) and is omitted.

D

Similar arguments yield the following.

(3.3.16)

Proposition.

equivalence)

A map

i

is a trivial cofibration

if and only if for all f i b r a tions

p,

the pair

(cofibration and (i,p)

has the

lifting property.

A map

p

is a trivial fibration

all cofibrations

(3.3.17) equivalence p,

i,

the pair

Proposition. if and only if

the pairs

(i,u)

and

(fibration and equivalence)

(i,p)

has the lifting property.

(Verification f = uv (v,p)

of Axiom M6c).

if and only if for D

A map

f

is a weak

where for all cofibrations

i

and fibrations

have the lifting property.

80

Proof.

By Proposition

lences is equivalent

(3.3.16),

the above characterization

to that of Definitions

We have completed the verification begin the verification relatively

of Axiom M5:

(3.3.1).

of weak equiva-

Q

of Axiom M6 for

weak equivalence

p r o - C,

and shall now

is a congruence.

This

lengthy process consists of first using the lifting properties developed

above to verify Axiom M5 under the further assumption that all maps are cofibrations or all maps are fibratlons.

Secondly, we use the faetorizations

to verify the general case of Axiom M5 for

(3.3.18) cofibratlons.

Proposition.

pro - C.

Suppose that the maps

If any two of the maps

given by A x l o m M /

f, g, and

f:X--~Y gf

and

g:Y'-~Z

are

are weak equivalences,

then so is the third map.

Proof. Proposition equivalences

Case I: the pairs

There are three cases. (3.3.9)

For Cases I we use Corollary

(3.3.13) and

to characterize maps which are both cofibrations

and weak

by their lifting properties.

Let (f,p)

f

and

and

g

be weak equivalences.

(g,p)

Then for all fibrations

have the lifting property.

composite has the same lifting property;

thus

gf

Consequently,

is a weak equivalence,

P,

their as

required.

Case II:

Let

a weak equivalence

f

gf

be weak equivalences.

by verifying

that for all fibratlons

the lifting property.

and

Consider a commutative

We shall show that p,

the pair

(g,p)

solid~arrow diagram of the form

g has

is

81

fJ . X

(3.3.19)

h

y

/ / / / / / / /

/

in which

p

is a fibration.

We shall show that

required, by constructing a filler

Because the composite map K':Z -'~E

such that

We shall deform

Because category

K'

f

C J,

implies that

k

gf

K'gf = hf

K,

g

is a trivial coflbration, as

above.

is a trivial cofibration, and

pK' = k.

into the required filler

Caution:

f'

J

Hence

f':X' --~Y'

induces a trivial cofibration

f

x [0,i]

induces a trivial cofibratlon

i:Y x 0 u X x [0,1] U Y x 1 --~ u x [0,1]

in

pro -C.

K'g # h.

in some level

is a cofinlte strongly directed set, Proposition (2.3.5)

i':Y' • 0 u X' x [0,i] u Y' x i - - ~ u

in 2.

in general

K:Z---~E.

is a retract of a trivial coflbration where

there is a map

Form the solld-arrow co~,utatlve diagram

82

y x 0 u x •

[0,i]

K'g u h K a proj u h: E

u Y x 1

I"/

/' /

k ~ o proj

Y • 10,11

By Axiom M6 (see Proposition

As above, Proposition

(3.3.99,

(3.2.29)

PB

there exists a filler

K (2)

implies that the cofibration

above.

g:Y--~Z

induces

a trivial cofibration

i':z x 0 u Y x [0,i]

Now form the commutative

3

K' u K (2)

/

~ K(

/

/

K

/ /

/

z ~ [0,1]

Again, as above, there exists a filler

f

/

i'

let

x [0,I].

solid-arrow diagram

Z x 0 u Y • [0,i]

Finally,

~z

K o ~roj

>

K (3) .

be the composite mapping

K(3) Z a Z x 1

Then

K

>Z

is the required filler in Diagram

equivalence,

as required.

x I

(3.3.19)

~E.

(easy check),

so

g

is an

83

Case III:

Let

g

and

gf

be weak equivalences.

Assume,

for now, the

following l~mm~.

(3.3.20)

Lemma.

Suppose that a map

i

in pro -C

has the left-lifting-

property with respect to all fibrations of fibrant objects.

Then

i

is a trivial

cofibration.

Consider a solid-arrow commutative

diagram

~E

X

f. I i I Ikgll/lil/f / !l

P

d (3.3.21)

Y

~

i

g

/

'B

1

k/ ]il i I i I z /

1

i

iI q

r

Z

in which

p

is a fibration,

Because

g

and gf

are weak equivalences

we may successively

fillers

h

aund kg

above.

is a filler in the top square of

Diagram

(3.3.21) above,

cofibratlon

which the pair

B

is fibrant

Because

Hence,

Proof of Lemma 3.3.20. (i,p)

has the following

kg

(that is,

q

is a fibration). construct

the

has the required lifting property to be a trivial

(Lemma (3.3.20)).

(3.3.22)

a)

f

and

>*

f

Let

is a weak equivalence,

i

denote the class of all

has the lifting property.

three properties.

A pullback of a map in

i

is in

L .

as required.

p

It is easy to check that

for [

84

b)

Let

E(J)

be an inverse system of objects in

indexed by a eofinlte directed set O.

J

with a least element

If all the induced maps

E(j)

are in

L ,

is in

E(O)

[ .

To show that in

i

in

Now let directed set.

j 9 J,

is in

i .

C;

we shall show that

By Theorem (3.3.4), by property

N2

i

contains all

i contains all flbrations of

of C

(see w

i

contains all

C.

p:E-'~B Let

an inverse system For

L

is a trivial coflbration,

pro - C.

fibrant objects in fibrations

9 J

then the induced map

A retract of a map in

fibrations

j

~ l i<~j { E ( k ) } , ~

liraj{ E (j) }

c)

pro - C

be a fibratlon in J* = J u {0},

{E(J) Ij 9 J*}

define

E(j)

where over

C J,

where

0 < j pro - c

J

is a coflnite strongly

for all as follows.

j

in J. Set

Define E(O) = B.

by the pullback diagram

E(j)

~ E. 3

(3.2.23)

B

- B.

J

i~

pro-C.

That is, for

k ~ j,

E(J)k = E k,

otherwise,

E(j) k

is a

85

suitable pullback. llm {E(J)} ~ E.

Because

J

is a cofinlte strongly directed set,

We shall see that

{E(J)}

satisfies the hypothesis of

property (b) above.

For now, consider a fixed

j

in J.

Z(J)

is in

To show that the map

> i~

< j{Z(k)}

L , we shall show that given a cc~mutatlve solld-arrow dlagram of the form

h

A

Z

-'; E ( J ) ,f

p

/ / /

It/" / / /

X

in

pro -C,

in

pro - C

h2

there exists a filler

) l ~ k < jE(k)

H.

l i ~ < jE(k)

To do t h i s , form the pullback diagram

llmk < j Ek

~li~k <j{Bk}"

Thus

h2

corresponds

t o maps

h 2':x

~ l ~ k < j{Ek},

h2" :X

~B

and

86

whose images in

l ~ k < j{B k}

pullback diagram (3.2.23) in

agree.

Also, because

pro -C,

hI

E(j)

is defined by the

corresponds to the pair of maps

hI':A---~Ej,

and

hI":A---~B ,

whose images in

Bj

are equal.

Now consider the resulting commutative solid-

arrow diagram

(3.3.25)

i

/

h3 . ' ~

X

P39

Here

~B

, \/IJ

* Bj

~ I ~ k < j { B k}

is a pullback, and the composite mapping

A---~X--~B

the universal property of pullbacks, there is a unique filler (3.2.7), qj

is a fibration in

h4

Diagrams (3.2.23) and (3.2.25) show that the maps

above.

~:X--~Ej

C , hence

induce a unique map

diagram (3.2.24).

H:X---~E(j),

Therefore

L

By

By Proposition

h2":X--->B

and

which is the required filler in E(j)

satisfies the

Hence the induced map

p:E = lira {E(J)} [ .

h 3.

hl".

so that there exists a filler

We have shown that the inverse system

hypothesis of condition (b) above.

is in

qj E i ,

is

~E(O) = B

contains all strong fibrations in

pro- C.

By

87

property

(c) above,

trivial cofibration,

[ contains all flbrations as required.

in

pro-C.

Therefore

i

is a

g:Y--->Z

be fibrations.

If

are weak equivalences,

so is the third.

Q

Similar techniques yield the following.

(3.3.26)

Proposition.

any two of the maps

Let

f, g, and

The proof is analogous

f:X-->Y, gf

and

to the proof of Proposition

(3.3.18) and requires a

lemma "dual" to Lemma (3.3.20).

(3.3.27)

Lem~a.

Suppose that a map

property with respect to all cofibratlons

p

in pro -C

has the right-lifting-

of cofibrant objects.

Then

p

is a

trivial fibration.

The proof is similar to the proof cofibrations

in

CJ

Proposition and L e m a

(3.3.22), and somewhat simpler because

are defined levelwise.

Details of the proofs of the above

are omitted.

We now begin the proof of the general case of Axiom M5 with four preliminary lemmas.

(3.3.28) f = pi,

Lemma.

where

i

Let

f:X--->Y

be a weak equivalence.

is a trivial coflbratlon and

p

Suppose that

is a flbration.

Then

p

is

a trivial fibration.

Proof.

By Definitions

trivial cofibration and arrow diagram

p'

(3.3.1), we may wrlte is a trivlal fibration.

f = p'i',

where

i'

Form ~he commutative

is a solid-

88

x

/ (3.3.29)

Z

.

f

.

4-

.

.

.

.

.

.

-~ Z !

.

. . . . . . . . .

By Axiom M6a (Proposition (3.3.15)), there exist maps

f and f'

(as shown

above) such that diagram (3.3.29) together with either dotted arrow commutes. Hence

f'fl = i

and

pf'f = 0.

Thus there exists a commutative solid-arrow

diagram (see Proposition (2.3.5))

td Z u ieproj Z • 0 u Xx

[0,1]

u Z •

u f'f

1

> Z J I i el J J

p

J I I I

/,

I

~V

As in the proof of Case II of Proposition (3.3.18), Hence there exists a filler which covers

i~.

H,

that isp a homotopy

j

is a trivial coflbratlon. H:Id Z : f'f

A similar construction yields a homotopy

Therefore the flbratlons

p

and p'

relative to

H':idz, = ff'.

are flber-homotopy-equlvalent.

Hence they

have similar lifting properties (use deformations analogous to those in the proof of Case II of Proposition (3.3.18)), so that Proposition (3.3.16).

0

p

X

is a trivial flbratlon by

89

(3.3.30) f = pi,

Lemma.

where

p

Let

f:X--->y

be a weak equivalence.

is a trivial fibration and

i

Suppose that

is a cofibration.

Then

i

is

a trivial cofibration.

The proof is similar to that of Lemma (3.3.28), and is omitted.

At this point we recall Condition N3 for

C:

at least one of the following

statements holds.

N3a.

All objects of

C

are cofibrant.

N3b.

All objects of

C

are fibrant.

We shall assume N3a for the remainder of this section, unless otherwise specified.

If instead N3b holds, replace " f i b r a t l o n " b y

(3.3.31), below, obvious changes.

(3.3.31) a section

"dualize"

"coflbration"

in Lemma

the proof of Lemma (3.3.32), below, and make some other

Details are omitted.

Le-~a. s:B ---~E

Let

p:E--~B

(that is,

be a trivial fibration.

ps = idB) ;

Then there exists

further, any section is a trivial

cofibration.

Proof. (3.3.9))

By Assumption N3,

yields a filler

s

B

is cofibrant,

in the commutative

so Axiom M6 (see Proposition solid-arrow diagram

/ / /

I Jl /

/

/

/

B

'

~B

90

Now, let

s':B---> E

be a section to

there exists a homotopy

H

(3.3.32)

Let

a trivial cofibration.

Proof. i

over

id B

from

s'

to

id E,

As in

f:X--~y

be a trivial fibration and

Then the Composite

p

w

is a trivial cofibration.

gf

g:Y--~Z

be

is a weak equivalence.

By Axiom M2 (Proposition (3.3.8)) we may factor

is a cofibration and

sp

(3.3.18), Case II, or the proof of [Q -i, Lemma 1.5.1,

to show that

Lemma.

As in the proof of Lemma (3.3.28),

H:E • [0,I] --~E

the proof of Proposition we may use

p.

is a trivial fibratlon.

gf

as

pi,

where

Form the commutative solid-

arrow diagram

i

x

(3.3.33)

f

Ii

--~ w

],,:

s

Y

Lemma (3.3.31) yields a section

s

t

$

~ Z

to f .

Because

pis = gfs = g,

there

exists a co~nutative solid-arrow diagram

y

is

9

~.

/

/

/ /

t

(3.3.34)

/

/ / /

Z

Axiom M6 (see Proposition map

t

is a section to

id

9

(3.3.9)) yields a filler p

and satisfies

tg = is

t

in diagram (3.3.34). (see diagram (3.3.33)).

The

91

Axiom M5 for cofibrations

(Proposition

are trivial cofibratlons. (3.3.1)), as required.

(3.3.35)

Hence

(3.3.18))

gf

(= pi)

is a weak equivalence

i

and

(Definitions

D

Proposition.

(Verification

X

be a ~ m ' a m

tg(= is)

implies that

in which two of the maps

of Axiom MS).

f~ Y

~

Let

Z

f, g, and gf

Then

are weak equivalences.

so is the third map.

Proof.

There are three cases.

Case I: write i

Let

f = pi

and j

f

and

and g g = pj,

write

is a trivial cofibration. composite mapping

qr

Case II.

q

where

Let p

and

k

as

(qr) (ki),

are trivial fibrations (3,3.32)

where

By Propositions

f

gf

r

and gf

is a trivial cofibration. where

ki

and

the composite mapping

jp

is a trivial fihration and

is a weak equivalence,

be weak equivalences.

is a trivial flhration, Then

(3,3.1),

k

(3,3.26) and (3.3.18) respectively,

( = qrki)

gf = qJpi.

We may therefore write

is a fibration

and q

By L ~ a

jp = rk,

Hence

is a fibration.

equivalence.

qr

p

As in Definitions

is a trivial fihration and the composite mapping

trivial cofibration.

and

where

are trivial cofibrations.

is a weak equivalence,

g = qj,

be weak equivalences.

i

and j

Write

where

is a

as required.

f = pi

and

are trivial cofibrations,

By Lemma (3.3.32),

Jp = rk,

ki

r

jp

is a weak

is a trivial fibration

We have thus factored the weak equivalence

is a trivial flotation

(by Axiom M6a (Proposition

the

(use Proposition

~3,3.18))

gf and

(3.3.15)) which implies that the class

92

of fibrations is closed under composition). fibratlon. so that

Finally, Proposition (3.3.26) implies that g(=qj)

Case Ill. g

and f

By Lemma (3.3.28),

is a trivial

is a trivial fibration,

is a weak equivaleuce as required.

Let

so that

q

qr

g

and gf

be weak equivalences.

gf = qjpi,

are trivial fibrations, and

i

Lemmas (3.3.32) and (3.3.30).

where

j

is a trivial coflbratlon,

is a coflbration.

pro - C

p

and q

Proceed as in Case If, using

Details are omitted.

This completes the proof that

As in Case II, factor

D

is a closed model category.

We Shall conclude this section by describing cofibrations and trivial cofibratlons up to isomorphism.

This extends J. Grossman's [Gros -i, w

tion of cofibratlon8 in his closed model structure on

(3.3.36)

Proposition.

A map

trivial cofibration) if and only if

The " o n l y

i f tl

cow -5S.

f

in pro -C

f

is isomorphic to a strong coflbratlon

(respectively, strong trivlal coflbratlon)

Proof.

is a cofibration (respectively,

(in some

part is obvious.

CS).

For t h e ttifft p a r t ,

i f n e c e s s a r y ( P r o p o s i t i o n ( 3 . 1 . 4 ) and Theorem ( 2 . 1 . 6 ) ) f ffi {fj}:{Xj} ---~{Yj}, {fj} tion)

j ~ J,

characteriza-

with

J

cofinite.

first

reindex

f

so t h a t By Definitions (3,3.1),

is a retract of a strong cofibration (respectively, strong trivial cofibra{f'k}:{X' k}

c~utative

diagram in

~ {Y'k }, pro -C

k E K,

with

K

cofinite.

Form the following

93

{xj I

id

/ {x'

{fj)

{fj}

(3.3.37)

f'k } id

{Yj } - -

(y.}

3

\ (Y k }

Z

We shall use diagram (3.3.37) to construct a level cofibration level trivial cofibration)

f"

square of diagram (3.3.37).

(3.3.38)

if

isomorphic to

f.

(respectively,

First consider the left front

We shall say that

(f'~:X'~

~Y'~)

<

(fm:Xm ~ )

t h e square (shown i n p e r s p e c t i v e )

X

m

x'

f

(3.3.39) Y

fl

s

m

yl

is a left front square of diagram (3.3.37).

Similarly, use the right front square

94

of diagram (3.3.37) to define relations of the form

(3.3.40)

(fm:Xm

>Ym ) < (f'n:X'n

>Y'n ).

By diagram (3.3.37), relations (3.3.38) and (3.3.40) and their composites yield an inverse system of maps

~

~

= {fk:Xk ~ Y k

indexed by a (cofinite)

Further,

},

strongly directed set

{f'j:X'j

~ Y'j },

J ~ I,

k E K

K.

to~ether with bondin~ maps induced by

diagrams (3.3.37) - (3.3.40) is a coflbratlon (resp., trivial cofibration) and is cofinal in

f,

hence isomorphic to

subsystem which is cofinal in (3.3.40)).

f

Remarks.

defined levelwise.

Also,

{fi:Xi

~ Yi}

admits a coflnal

(the bonding maps agree by diagrams (3.3.37) -

The conclusion follows.

(3.3.41)

f.

D

The above proof used the fact that cofibratlons in

CJ

are

We do not know whether the analogue of Proposition (3.3.36) for

fibratlons holds.

w

Suspension and loop functors, cofibration and fibration sequences. D. Quillen [Q -i, w167

-3] developed a general theory of suspension

functors and cofibration and fibratlon sequences in model category. structures on

Ho(C),

where

C

and loop is a closed

We shall sketch this theory within the context of our closed model pro-C.

95

(3.4.1)

Morphisms i__nn Ho(pro -C).

a fibrant object in

pro - C.

Let

X

[X,Y]

pro - C)

be

= IX,Y],

denotes the set of homotopy classes of maps from

with respect to the cylinder

description of

Y

Then

Ho(pro-C)(X,Y)

where

be a cofibrant object and

Ho(pro - C)(X,Y).

X ~

[0,i].

X

to Y

(in

There is another dual

Factoring the diagonal map

Y --> y x y

as

the composite of a trivial cofibratlon followed by a flbration

y

y[O,l].

yields the cocyllnder functorlally upon maps

Y

f,g:X ~ Y

P0

Now, let

in general

y[O,l]

H

; y~O,l,[]

~

y

>y

x

need not depend

One can easily show that

~

g

y

denote the projections onto the first

y x y~

C,

Caution:

are homotopic if and only if there exists a commutative diagram

and Pl

factors in

>y • y

(compare with Definition (2.3.4)).

X

here

~y[0,1]

(yO)

and second

(yl)

respectively.

be a pointed closed model category, that is, a closed model

category which is also a pointed category.

Then

closed model category (the point

is also the point of

* of C,

shall follow the "usual" conventions and write

v

pro - C,

becomes a pointed pro -C,).

for the sum (coproduct) in

We

96

p r o - C,.

(3.4.2)

Definitions.

shall write

* vx Y

If

f:X--->Y

for the cofibre of

is a cofibration in f

>,

Y

f:X--->Y

fibre of

f

>*

is a fibratlon in

pro-C,,

.

we shall write

* Xy X

for the

.xyX

>X

,

>Y

(level) coflbration

is just the levelwise

similarly the fibre of a strong (level) fibration is the levelwise fibre.

(3.4.3)

Suspensions and loop spaces.

Choose a cylinder object i 0 + if: X v X

~X

u

X x [0,I],

v [0,i].

spaces are defined dually. object

VxY

defined by the pullback diagram

Note that the cofibre of a strong cofibre;

we

defined by the pushout diagram

X

If

pro - C,

and let

Let

and let

We shall call Let

~u

Y c pro - C,

X ~ pro -C. EX EX

be c o f i b r a n t .

be t h e c o f i b r e a suspension of

be fibrant.

be the fibre of the map

o f t h e map X.

Loop

Choose a cocyllnder

(p0,PI):Y [0'I]

>y x y .

97

We shall call

~Y

is flbrant.

a loop-space of

Caution:

(see Definition

Y.

In general

Note that

E

(2.3.4) and paragraph

and

~

EX

above is cofibrant and

need no__~tbe functors on

pro -C,

(3.4.1)).

On the other hand, Quillen proves the following

theorem for arbitrary closed

pointed model categories.

(3.4.4)

Theorem

[Q-I,

pair of functors on all of

w167

X

is cofibrant and

and

~

to an adjolnt

Ho(pro -C,):

Ho(pro -C,)(ZX,Y)

If

E

We may extend

Y

ffi Ho(pro-C,)(X,nY).

is fibrant,

[EX,Y] ~ [X,~Y].

E

and

are defined up to canonical

Ho(pro-C,)(E-,-) (pro -C,) ~

isomorphism.

= Ho(pro-C,)(-,n-)

• (pro -C,)

Also,

as functors from

to groups.

The proof is similar to the proof for the category of pointed spaces, except that somewhat more care is needed because of the choices involved in defining E

and

~.

structure

The group structure on EX

> ~X v EX

the corresponding

H -structure

(3.4.5) Top,.

Let

Ho(pro -C,)(EX,Y)

which makes

group structure on

~y x ~y--->~y

Short cofibration

EX

comes from a

a cogroup object in

Ho(pro -C,)(X,~Y)

which makes

gq

sequences.

Let

Ho(pro

-C,);

comes from an

a group object in

f:A--->X

co-H

Ho(pro -C,).

be a cofibration

in

~Y

98

Mf = A • [0,i]

be the mapping cylinder of

f,

UAX/* x [0,i]

and consider the induced cofibration

i0 A

If

Ci0

denotes the cofibre of

reduced suspension of

A,

>Mf

i0,

CA

the reduced cone on

A,

and

ZA

the

we may form the sequence i0

A

> Mf

> C. 10

and induced diagram

Ci0 <------- CA u A Mf

Further, in

Ho(Top,)

of the cogroup object

~ EAv

the composite mapping

(co -H

needed.

If

A--->X

* --->X is a map in

ZA

is a coaction

on C. 10

to arbitrary cofibrations

the full subcategory of coflbra~t objects in tion means that the map

n:C. ---->ZA v C. 10 10

space with a co-inverse)

We shall extend the above observations

Ci0

pro - C,.

is a cofibratlon.) pro- C,

in

(pro- C,~,

(The "cofibrant" condiThe following notation is

we shall write

A

Q

[0,i] v A X

99

for the cofibre sum (pushout)

A

[O,l]

A |

More generally, il:A

[0,1],

i0:A---->A ~

[0,i].

in

pro -C..

cogroup

A

Let on Z .

-v A A ~

and

A--*X

Now let

A ~

we shall write

>A ~

>A

[0,i] [0,i]

be a cofibration Z

[ 0 , 1 ] vA X .

|

v A-

to denote the cofibre sum with

in

he the cofibre of

(pro -C.) c, A.

(3.4.6)

x |

vA X

,*

[0,i]

[0,I] vA X

>X

A ~

A

is cofibrant

diagram

v A A (~) [ 0 , i ]

, z vx x |

X

in which the cylinder object

that is,

We shall define a coaction of the

To do this, form the commutative

A (~ [ 0 , I ]

X vA A ~

to denote the cofibre sum with

vA X

[0,1]

~Z

[0,I]

is obtained by factoring

as a coflbratlon

the natural map

followed by a trivial fibration

100

X vA A ~

[0,i] vA X

)X ~

[0,i]

)X

(see Proposition (2.3.5)).

By construction both squares in diagram (3.4.6) are

pushout squares.

i

Further,

(see Proposition (2.3.5)) and

coflbratlons, hence so are their respective pushouts (3.4.6) induces the composite mapping

i'

> ZA v Z

c:Z

i0

are trivial

and i0'.

Diagram

Ho(pro- C,)

in

defined

below.

z

[iO]>z

vx x |

[0,1]

) ~ v A A (~) [ 0 , 1 ]

vA X

) * vA A ~) [0,i] vA *

-" E A r

It is easy to check that, in

A

on Z

and that

c

Ho(pro -C,),

c

Ho(pro - C , )

A'

which for some cofibration

constructed above.

is a coaction of the cogroup

is independent of any choices made above (see [Q- i,

coflbratlon sequence in

(3.4.8)

Z

Z

Proposition 1.3.1 and the followlng remarks]).

(3~4.7)

v

) X'

A--*X

A

>X

We therefore define a short

to be a diagram in

9 Z',

in

Z'

) ZA' v Z',

(pro -C,) c

~ Z,

Z

Ho(pro - C,)

is

9 ZA v Z

isomorphic

to the diagram

101

(3.4.9)

Proposition.

induces a short cofibration

A short cofibratlon

sequence

A

~ZAvZ

Z

~Z,

> Z

>ZA

> ZA,

where

ZA

> ZX v ZA,

is the composite

Z

and the coaction

Z

sequence

X

the "connecting map"

>X

> ZA v Z

> ZA,

ZA

> ZX v ZA

is the composite

ZA

> ZA v ZA

- i d v i d > ZA v ZA

-id:ZA--~ZA

is the inverse in the cogroup

> ZX v ZA,

ZA.

The proof is similar to the usual proof of the corresponding Ho(Top,),

and dual to the proof of [Q -1~ P r o p o s i t i o n

1.3.3].

assertion

in

Details are

omitted.

(3.4.10) A--->X-->Z~ (Barratt-Puppe

Lon~ c o f i b r a t i o n Z

>ZA v Z

in

sequences. Ho(pro -C,)

A short cofibration

sequence

induces a lon$ cofibration

sequence)

A f ---> X ---> Z ---> ZA - - ~ 9 9 9 --~ ZnA - - > z n x ---~ znz ---> zn+IA --~ " 9 9 .

Also,

for any object

Y

in p r o - C , ,

the sequence

sequence

~02

(3.4.11)

> Ho(pro -C,)(Enz.Y)

> Ho(pro -C,) (Enx.y)

> Ho(pro -C,)(EnA.y)

)

.~

) Ho(pro -C.) (Z.Y)

> Ho(pro -C,)(EA.y)

> Ho(pro -C,)(X.Y)

; Ho(pro -C.) (A.Y)

has the usual exactness properties:

a)

Sequence (3.4.11) is exact as a sequence of pointed sets, and maps of pointed sets;

b)

Sequence (3.4.11) is exact to the left of

Ho(pro-C,)(EX,Y)

as a sequence of groups and homomorphlsms; c)

Two maps in

Ho(pro - C,)(X,Y)

Ho(pro -C,)(X,Y) the group

d)

Two maps

Ho(pro -C,)(Z.Y) map

h

if and only if they differ by the action of

Ho(pro-C,)(ZA,Y)

gl,g 2

have the same image in

on

Ho(pro -C,)(Z,Y).

Ho(pro - C , ) ( A , Y ) if and only if

have the same image in g2 = ( Z f ) h o gl

for some

in Ho(pro-C,)(EX.Y).

As above, compare the usual exactness properties of Barratt-Puppe sequences and [Q-I, Proposition 1.3.4'] for the proof.

We now summarize the properties of coflbratlon sequences.

(3.4.12)

Proposition.

(dual to [Q-I, Proposition 1.3.5]).

short coflbration sequences in

Ho(pro,-C,)

The class of

has the following properties:

103

a)

Any map

f:X--->Y

in

Ho(pro -C,)

may be embedded in a cofibratlon

sequence

f

X

b)

>Y

~Z,

Z

9 ZX v Z.

Given a commutative solid-arrow diagram

X

>Y

>Z

Z I I

f

(3.4.13)

g

[

h

b

>ZXvZ

i i i i h) ) ) )

X' -->Y' -->Z'

f v h

Z' -->ZX' V Z'

in which the rows are short cofibration sequences, the filler

c)

If the maps

f

is the filler

d)

and g

h

exists.

in diagram (3.4.13) are weak equivalences, so

h.

Proposition (3.4.9) holds.

We omit the proof.

The following straight-forward proposition yields many cofibration sequences in

Ho(pro -C,).

(3.4.13)

Proposition.

Let

cofibratlons of cofibrant objects

{Aj - - ' * 5 -->Zj) Aj --->Xj,

indexed by a cofinite strongly directed set

be an inverse system of

with cofibres J.

Z.,3 over

Then tahere is an induced

cofibration sequence

r

C,

---> (xj} ---> {zj~}, (zj} =-)zr

v (z.}j

104

in

Ho(pro -C,).

S

All of the above theory may be dualized to obtain short and long fibration sequences.

Quillen [Q - i, w

discusses fibrations explicitly.

We shall

summarize this discussion below.

A fibration

p:Y--->B

in

(pro -C,)f

induces a short fibration seqUence in

(3.4.14)

in which

m

Ho(pro -C,).

F

%

n

Pro - C,)

Ho(pro -C,),

P~B,

fib • F m--~->F,

RB

on F

in

There is also an induced Short fibration sequence

RB

~F

~ E,

RE • RB

n > RB,

is the composite map

id • *

RB

and

is fibrant in

B

is a well-defined action of the group object

(3.4.15)

where

~ Y

(that is,

>RB

•

m

F

>F,

is the composite map

RE x RB

> RB x ~B

-id • id

~ RB x RB

> ~B.

This is [Q- i, Proposition 1.3.3], compare Proposition (3.4.9).

Hence, there is

an induced lon~ fibration sequence

(3.4.16)

"'-

>Rn+IB

~n%

>

B

,RnE

~F

~F

,,RnB

9 E -------~

B

> "'"

105

with exactness properties analogous to properties (a) - (d) of long cofibratlon sequences (see Paragraph (3.4.10) above)[Q-l,

Proposition 1.3.4].

We shall need the following d u a l of Proposition (3.4.13).

(3.4.17)

Proposition.

Let

fibratlons of fibrant objects

~ Ej

{Fj Ej

and Bj,

indexed by a cofinite strongly directed set

~ Bj}

be an inverse system of

with fibres J.

Fj

over

* ~ Bj,

Then there is an induced fibra-

tion sequence {F.}3

in

~ {E.}3

9 {Bj},

> {F.}j

Ho(pro -C,).

Proof.

We first replace

replace the map

{Ej}

{B.}3 by a fibrant object by a flbration

{B.}j

(Recall that the natural functor structures.)

Caution:

C, J

recall that fibrations in

is Just the levelwise fibre. first to factor the map

{E'j}

> pro - C,

but the fibre of a strong fibration in

C, J)

~{Sj} x {Fj}

pro- C,,

{B'j},

~ {B'.}] in C, J.

preserves the closed model C, J

are not defined levelwlse,

that is, a fibration in

{Bj} ---~*

as a level trivial cofibration followed

(ij) {Bj} B'.J

fibrant)

C, J ,

Use the proof of Proposition (3.2.24) (Axiom M2 for

by a fibration

( t h i s makes

and n e x t

>,

; {B'S )

and then to f a c t o r t h e composite map

{Sj}

> {Bj}

~ {B'j}

106

as a level trivial cofibration followed by a flbration {i'j} {Ej}

Let

{F'j}

{p'j) > {E'j}

be the (levelwlse) fibre of

tive diagram in

~ {B'j}.

We obtain the following commuta-

{Pj}.

J C, .

(F_)

, (E.) ---~

]

{Bj}

3

(3.4.18)

{i' .} J

}

{p l .}

{F' }5

In diagram (3.4.18)

{i"j}

>{E'j}

-] {B'.}]

is the restriction of

Regard diagram

{iwj}.

(3.4.18) as an inverse system made up of the diagrams

Fj -

> Ej

>B

(3.4.19)

i.

3

F'j

in

C,.

Because

~E'j

pj and p'j

>B'j

are fibrations (a fibration in

C,

J

is a level-

wise fibration, see Proposition (3.2.16), although the converse assertion need not hold), and lences.

ij

and i'j

are weak equivalences, the maps

Hence the diagrams

i"j

are weak equiva-

107

are isomorphic over

(3.4.20)

)

{F'j}

> {E'j}

Ho(pro -C,).

Remarks.

tion sequences in

{Fk}

Diagrams

{Ej}

>

{Bj} ,

> {B'j}

The conclusion follows,

l]

(3.4.19) may be extended to maps of short fibra-

Ho(C,).

F. .)

)E. J

)B. J

~B

E lj

>E'j

> B'j,

J

x F

>F.

J

J

(3.4.21) ~B'j x F'j - - > F ' j

However diagrams (3.4.21) do not yield the required action map

~{B'.} • {F'.} J

in

Ho(pro -C,)

(3.4.22)

(not

Remarks.

pro -Ho(C,)),

w

~ {F'j}

without suitable coherency conditions.

As in Quillen's discussion

Toda brackets and similar constructions applications

J

in

so far of these constructions,

[Q -i, w

Ho(pro - C,).

we may obtain Because we have no

we omit the details.

Simplieial Model Structures. In this section we shall prove that a simpliclal closed model structure (satis-

lying condition N of w

on C

induces such a structure on

results can be readily extended to pointed categories;

pro -C.

These

details are similar and

108

omitted.

For a finite simplicial set

K

and X, Y,

in C,

let

X~K

denote the "tensor product" and "function space" constructions in HOM (X,Y) connecting

(3.5.1)

is a simplicial set), and let SS

and

C;

see w

Definition.

H0M (K,X)

Let

and C

HOM (X,Y)

(recall that

denote the "function space"

{Xj}, {Yk} ~ C.

Let

( 5 } ~ K ffi {Xj ~) K},

HOM (K,{Xj})

and

= {HOM (K,Xj)}

together with the induced bonding maps, and let

HOM ((Xj}, {Yk}) ffi llm k collmj (HOM (Xj,Yk)}.

These constructions extend to functors HOM:

(finite simpllcial sets) ~

HOM:

(pro - C) ~

• pro- C -

• pro - C

) SS,

~

:pro -C • SS 9 ) pro- C,

* pro - C,

respectively.

and

Axiom SM0, and the following

propositions are immediate consequences.

(3.5.2)

Proposition.

For

{X.}

-

0 -slmplices of

and

{Yk } in pro -C,

3

HOM ({Xj}, {Yk}),

R0M ({Xj}, {Yk})0 ! pro =c((xj}, {Yk)), naturally in

{Xj}

and

{Yk }.

the set of

109

(3.5.3) let

Theorem

(Exponentlal Law).

{X.}3' (Yk } E p r o - C .

Let

K

be a finite simpliclal set and

Then

HOM ({Xj} (~ K, (Yk)) ~ HOM (K, HOH ({Xj), {Yk})), naturally in all variables. struction in both

Proof.

pro- C

(HOM and

is used to denote the "function space" con-

SS.)

Because

HOM ((Xj} |

K, {Yk}) ffi HOM ((Xj |

K}, {Yk})

ffi limk(colfmj(,O~ (Xj | ffi lim k {HOM ({Xj) |

and

H0M ( K , ? ) : S S - - ~ S S

c a s e where

{Yk}

preserves limits,

is an object

Y of

K. Yk)))

K. Yk)}.

we may r e d u c e t h e g e n e r a l c a s e t o t h e

C.

In this case,

H0H ({Xj) | K,Y) = colimj{HOH (Xj |

~,Y))

= col~jTHOM (K, RO~ (XS,~))}.

Because

category

{j)

K

has finitely

many n o n - d e g e n e r a t e s i m p l i c e s ,

and t h e i n d e x i n g

is filtering,

colimj~HOM (K, HO~ (Xj,Y)))

ffiHOM (K, dolimj~HOM (Xj,Y)})

ffiHOM (K, HOM ({Xj), Y)),

as required.

Naturallty follows easily.

0

110

(3.5.4)

Corollary.

With

K,{Xj},

and

{Yk }

as above,

pro- C({Xj} ~ K,{Yk}) = SS(K, HOM ({Xj},{Yk})),

naturally in all variables.

(3.5.5)

a)

D

Remarks.

The corresponding assertion for is proven similarly.

b)

examples, use the fact that

MAP (K,-): Top---~Top

(3.5.6)

Theorem

cofibration in

K; to construct counter-

H0M (K,-):SS -->SS

commute with collmits for infinite

for non-compact

(Definition (2.4.2))

Details are omitted.

The above results fail for infinite

space

(K,X)

HOM

K.

does not

Similarly the function

does not commute with expanding colimits

K.

(Verification of Axiom SM7).

pro -C,

and let

p:{Ys

--->{Bm}

Let

i:{Aj} ----->{Xk}

be a fibration in

Then: a)

The induced map

(3.5.5)

HOM ({~},{YA})

is a fibration in

b)

If either

SS

i

q ,HOM ({Aj},{Yg})

(i.e., a Kan fibratlon);

or p

is also a weak equivalence, then the map

above is also a weak equivalence.

be a pro - C .

111 Proof. a)

Consider a solid-arrow commutative diagram in

SS

of the form

HOM

({Xk},{Bm})

(3.5.6) Vn'k

An

in which k th

face.

(3.5.7)

V n'k

f

g

>

HOM ({Kk},{Ys

((Aj},{Ys

;HOM

is obtained from

The maps

g

and q

x

H0M

DA n ,

({Aj},{Bm})

the boundary of

An ,

by deleting the

correspond respectively to pairs of

g,:^n

, Ho~ ((Aj),(Y~)),

g":A n

> HOM ({Xk),{Bm});

({Aj},{Ys

q':HOM ({Xk},{Ys

> HOM

q":HOM ({Xk},{YA})

, HOM (~xk),(Bm));

such that the appropriate composite maps into

HOM ({Aj},{Bm})

Applying the exponential law (Theorem (3.5.3)) to

maps

are equal.

f, g', g", q', and q",

and

assembling the induced maps with the above coherence data (Diagram (3.5.6)), we obtain a commutative solid-arrow diagram

112

(3.5.8)

> (Ys

{Aj} ~ An u {Aj} ~ Vn'k{~} ~ Vn'k

(xk} |

"(B }.

An

m

i:{Aj} "--*{X k}

By Definitions (3.3.1), the map coflbratlon

is a retract of a levelwlse

By Axlom SM7 for

{i'r}:{A' r } --->{X'r}-

C,

(see w

the induced

maps

(3.5.9)

A'

r

~

X'

An U A'

~

vn,k

~

V n'k

> X'

r

~

An

r

r are trivial coflbrations; hence the induced map

(3.5.10) (i'r},:{A'r}~)An

u

.{X' } ~) Vn'k {A,r } Q

is a trivial coflbration.

Because

vn,S

i,

r

is a retract of

i,

is also a trivial cofibratlon; hence Axiom Ml for

h'

in diagram (3.5.8).

yields a map the map required.

h

{X'r}| {i'}, pro - C

by construction, yields the filler

Applying the exponential law (Theorem 3.5.3)) to

h:An---->HOM ({~},{Y~}). makes diagram (3.5.6) commute.

h'

By construction ((3.5.7) - (3.5.10)), Thus

q

is a (Kan) flbration, as

b)

If the map

i

is a trivial cofibration, then

i

induces a

trivial cofibration

(3.5.11)

i#:{Aj} @

An U {Aj} •

by analogues of (3.5.9) with

V n'k

~An { ~ } @

replaced by

DAn

DAn;

analogue of diagram (3.5.6) with the same replacement.

' {~}

@ An

hence a filler in the Thus

q

is a trivial

(Kan) flbratlon, as required.

Finally, suppose that (3.5.11) induced by

i

p

is a trivial flbratlon.

The coflbratlon

has the left-llftlng-property with respect to

i# p,

so the

required fillers in suitable analogues of diagrams (3.5.8) and (3.5.6) (see above) exist.

Thus

q

is a trivial (Kan) flbratlon, as required.

Theorems ( 3 . 5 . 3 )

and ( 3 . 5 . 6 ) ,

a c l o s e d model c a t e g o r y

(w167

together

- 3.3),

with the earlier

imply that

D

proof that

pro - C

pro - C

is a simplicial

model c a t e g o r y .

w

Pairs. We shall use the Bousfield-Kan [B-K] model structure on

More precisely,

Maps(pro -C)

is the category whose objects are maps

A in

pro-C,

Maps(pro-C).

~X

and whose morphlsms are commutative squares.

A map

is

closed

114

(3.6.1)

A

1

X

in

Maps(pro- C)

a fibration if

g

are fibrations,

lifting property is satisfied.

-~Y

g

is a weak equivalence and f

>B

f

if

g

and f

are weak equivalences,

and a cofibration if the appropriate

Explicitly,

(g,f)

is a coflbration if

f

and

the induced map

XJJ_ B A

are cofibrations in

pro -C

(X~

B

>Y

is the pushout obtained from most of

A diagram (3.6.1)).

(3.6.2)

Definition.

Maps(pro -C)

Let

(C,

consisting of maps

pro - C) A-->X

be the full subcategory of with

X

in

C .

It is easy to prove (as in w

(3.6.3)

Theorem.

The closed model structure on

closed model structures on

Maps(pro - C),

It is conven/ent to represent

x-

(%<

%(

---];

and a level map

tow- C)

f0:X0 ----~Yo

consists of a cofinal subtower {Xm } n

pro -C),

as follows.

a morphism~onslsts of amap

together with a compatible map f:{X m) ---> {Yn }

(C,

(C,

>{Yn }.

pro - C

in C.

induces natural and

(C,

Objects are towers

f:X-~Y

in tow-C

Alternatively,

{Xmn } = {xm}

tow -C).

with

a map

x m0

ffi X 0

115

3.7.

Geometric Models.

We shall discuss geometric models of

Ho(Top,

tow -Top)

using filtered spaces and a telescope construction. Ho(Top,

tow -Top)

Ho(tow -Top)

will be used in proper homotopy

and

Ho(tow - Top)

The geometric model of theory in w

The model of

was used in shape theory by the f~rst author and R. Geoghegan

(unpublished).

(3.7.1)

Definitions.

A filtered space

together with a sequence of closed suhspaces Xn c int Xn+ I.

A filtered map

map such that for each number A filtered space

X

consists of an underlying X = X 0 = X 1 D X 2 ~ "'',

(of filtered spaces)

f:X-->Y n ~0

X

there is a number

m ~ 0

is a continuous

Filt

and its associated

under the relation of filtered homotopy)

(3.7.2)

Definition.

is the space

Tel (X) = X 0 x 0 o bond Xl x [0,i] u bond X2 x [1,2] u bond

shown below.

1

(quotient

Ho(Fi~t).

X = {X n}

2

c Yn"

There results

filtered homotopy category

The telescope of a tower

0

f(X)

X • [0,i];

this yields a natural notion of filtered homotopM of filtered maps. a filtered category

X

with each

with

induces a natural filtration on its cylinder

space

"'"

116

Tel (X)

is filtered by setting

Tel (X) n 9, X n x n u

bond Xn+l x [n, n + l ]

u bond "''.

This construction extends to a functor

Tel:

Further,

Tel

(3.7.3) of

Filt

takes a

Top N

"x [0,i] - homotopy"

Definition;

The category

Tel

in Top N

into a filtered homotopy.

of telescopes is the full subcategory

consisting of telescopes.

This should cause no confusion. the telescope category

Tel;

Observe that the functor

also that

that cylinders may be formed within full subcategory of telescopes in

(3.7.4) Top N.

> Filt.

Then

Proof.

Proposition. Tel {f } n

Observe that

Let

Tel { X }

Tel.

Tel

x [0,i] - Tel {X n • [0,I])

We therefore let

Ho(Tel)

{f }:{X } n n

is invertlble in

Tel {X } n

9 {Y } n

be the

be a weak equlvalence in

Ho(Tel).

is a strong deformation retract in

Map (Tel {fn }) = Tel {Map (fn)}

= Tel {Xn x [0,i] u Yn/(X,l) - f(x)}.

{f } n

so

Ho(Fllt).

the mapping cylinder

We thus assume that

factors through

is also a coflbratlon in

Top N.

Tel

of

117

For each Hn:Yn x I

n

choose a retraction

>Yn

with

Hni0 = id

and

~ X,

rn:Y n

and a homotopy Caution:

Hnil ffi fn nr.

bond O r n + 1 # rnO bond,

in general

and

bond o Hn+ 1 # H n o bond.

However,

bond o rn+ I o fn+l = bond

(3.7.5)

ffi r

n

o f

n

bond

= rn o bond o f n+l'

(3.7.6)

bondoHn+ l o ( f n + l x id) ffi HnO bond o ( f n + l x i d ) :

We shall now use (3.7.5)

to define

Xn+ 1 x [0,I]

' Yn"

a filtered map

g;Tel

which will be shown to be a filtered-homotopy-lnverse map

fn+l

is a trivial

a strong deformation retraction.

and

cofibratlon,

retract

The composite

Yn+l

of

Yn+l x [0,I]

map,

to be denoted

to

{Y } n

Tel {f }. n

x 0 u Xn+ I x [0,i] for

n = 0.

Kn+l,

Tel {X }, n Because

u Yn+l x i

Let

Pn+l

the

is be the

118

Yn+l x [0,i] Pn--~->+~Yn+ 1 x O u X

(3.7.7)

n

x [0,i] U Yn+l x i

r n o b o n d u r n = b o n d o fn+l

u bond

o

rn+l~ X n

Yn+l -->

X I]

Xn+l

(see (3.7.5)) yields a homotopy from

r o bond

to

bond o rn+l.

Now define a

n

map

glTel {Y }

-,> Tel {X }

n

Xn+ I x [n, n +i]

to

X

x n

2t - n - l ) ;

according to the formula

g(y,t) ffi (Kn+l(y, 2 t - 2 n ) ,

Y

n

Yn+l x [n + 89 n + i]

n

(3.7.9) maps

maps

g(y,t) ffi (rn+l(y),

maps Yrrbl x [n, n + %]

g

g

according to the formula

(3.7.8)

and

as follows:

n

x n

to

X

x n

n);

according to the formula

n

(3.7.10)

g(y,t) = (rn(Y), n).

Yn+l

n

r n+l

Xn+ I

/

Y

n

r

tl

x n

I

to

119

Then

g

is a filtered map.

Claim 1:

The maps

this, first observe that

(3.7.5).

gf

and

K+llX+I • tO,l]

(3.7.11)

In

~ ~

by

n+~ ! t ! n+l,

~

g~(x,0 = ~I (x' L(~,n),

by (3.7.8) - (3.7.10).

Claim 2:

The maps

fg

and

Xn+ 1 x [ 0 , 1 1 x [ 0 , 1 ] '

tlon retract of

n < = t <= n+%,

Claim i follows easily.

idTe I {yn}

use (3.6.6) to imitate the construction of

interval.)

x+ 1

is t h e projection onto

Thus

2t

As ahove,

To c h e c k

are filtered-homotoplc.

idTel {X } n

g

are filtered-homotopic.

and obtain the required homotopy.

o Yn+l x 2 ( [ 0 , 1 ]

Y.+I • [ o , 1 ] x [ O , l ] ' .

x [0,1]')

([0,i]'

We may therefore define a homotopy

We shall

rn+1

is a strong deforma-

is just a second unit

from

bond rel Xn+ 1 x [0,I] x [0,i]' u Yn+l x a([0,1] x [0,i]')

Kn+ 1

to

schematically,

120

t bond

n

Gluing these maps together yields a filtered h o m o t o p y

r:Tel {Yn } x [0,i]'

from

fg

nares,

"to a map

h:Tel {Y } n

a n d moves t h o s e c o o r d i n a t e s

> Tel {Yn x [0,i]')

Tel {Yn }

which changes only vertical coordl-

a t m o s t 89 u n i t

analogous to those in Claim 1 and are omitted.

(3.7.12) Ho(TopN)

Corollary.

The functor

9 Tel (Yn }

Tel:Top N

as in

(3.7.11).

Details

Claim 2 follows.

> Ho(Tel)

are

0

factors through

to induce

Te1:Ho(Top N)

Proof. by a diagram

9 Ho(Tel).

By the model structure, each map in

Ho(TopN)

may be represented

121

{Xn}

(Yn)

{z' } I1

where

{Y' } n

is fibrant in

(levelwise) weak equivalence. Any two such m p s

Top N

(i.e., a tower of flbrations) and

(Note:

all objects in

-x[0,1].

We shall now extend

(3.7.13)

(Xn}

with

Tel {X } n

Top N

Tel

-x[0,1].

Further,

admits a homotopy inverse

~o Ho(Top, tow- Top).

Suppose that

(X

%

)

is a coflnal subtower of

Then there is a natural equlvalence

~ Tel {X } in Ho(Tel). nk

The r e q u i r e d map to

illustrated.

Xnk x k

Tel {X}

is a

are cofibrant).

The concluslon follows by Proposition (3.7.4).

Proposition.

Xno = x O.

Proof. Xnk x nk

N

are homotoplc with respect to the 6yllnder

a weak equivalence between fibrant objects in with respect to

Top

{in )

~ Tel (Xnk}

is d e f i n e d by mapplng

under the identity and extending to

Tel {Xn }

as

[]

122

X

~+1

x

• nk+ 1

X

•

This map is easily seen to be a filtered homotopy equivalence.

(3.7.14) a)

• (k+l)

k

[]

Proposltlon.

The functor

Tel:Top N

> Ho(Tel)

factors through

(Top, towyTop).

b)

This functor factors through

Ho(Top, two-Top),

Tel:Ho(Top, tow-Top)

Proof.

a map in

> Ho(Tel).

Part (a) follows from Proposition (3.7.13) because each map in

may be represented by a diagram

(Top, tow-Top)

where

to induce

{Xnk }

~s cofinal in

{Xn} , {Xno}= XO,

and

{fk }

is a level map (i.e.,

TopN).

Part (b) follows from Part (a) and the observation that

Y

is fibrant in n

123

(Top,

tow-Top)

if

{Y } n

is a tower of flbratlons by using the proof of

Corollary (3.7.12).

We are ready to show that

> Ho(Tel)

Tel:Ho(Top, tow-Top)

ylelds an

equivalence of categories.

(3.7.15) ~(X) = (X } n

Definition. in

The end of a filtered space

(Top, tow-Top).

Then

e:Tel

The definition of

(3,6,16)

Ho(Tel)

proposition,

c

Definitions.

be the map on

TopN

k > n.

extends to a functor

9 (Top, tow-Top).

induces a functor

Let

9 Ho(Top, tow-Top).

X ffi {Xn} e Top N.

given by lettlng

x [k-l, k]

for

,~

pnl~

Let

x [k-l, k]

bond o . . .

o bond

p:e o Tel (X) be the composite

~X n

Let

q ffi Tel (p):(Telo e)(Tel (X))

(3.7.18)

is the tower

implies the following.

~:Ho(Tel)

(3.7.17)

~

X

9 Tel (X).

Proposltlon,

a)

The maps

p

are natural weak equlvalences in

b)

The maps

q

are natural weak equlvalence in

Ho(Top,

Ho(Tel).

tow-Top).

9X

124

Proof.

Part (a) follows i~mediately from the definitions of

(note that each

Pn is an equivalence in

Ho(Top)).

Tel,

e,

and

p

Part (b) follows from

Part (a), the definition of the telescope category and Proposition (3.7.14b).

Propositions (3.7.14b),

(3.7.19)

Theorem.

(3.7.16), and (3.7.18) imply the following:

The categories

Ho(Top, tow-Top)

and

Ho(Tel)

are

naturally equivalent.

A geometric model for assume that all towers (Top, tow-Top) Tel,

ConTel

X = {Xn}

(regard

X

as

satisfy (*,X)),

We may

may be obtained as follows. X0 ~ * .

This embeds

tow-Top

in

and gives rise to a full subcategory of

(contractible telescopes) consisting of those telescopes

Tel (* = X0+--- ~ <

(3.7.20)

Ho(tow-Top)

.--).

Corollary.

The categories

Ho(tow-Top)

and

Ho(ConTel)

are

naturally equivalent.

(3.7.21)

Remarks.

categories of diagrams,

~x0 +-- xl<

...~.

In R. Vogt's [Vogt -I] approach to the homotopy theory of Tel {Xn}

is the homotopy collmlt of the diagram

In this setting our homotopy category

Ho(Tel)

represents

"coherent pro - hom~topy," that is, a version of pro - homotopy where maps and homotopies are required to satisfy various coherency conditions.

The development of

coherent pro -homotopy theory (Vogt only works with level maps) is an interesting problem whose solution should have applications to proper homotopy theory (see w and shape theory, especially in alternative proofs of the Chapman [Chap - i]

125

complement theorem (see w

Inj

w

spaces.

-

The above t h e o r y o f p r o - s p a c e s inj-spaces

Let

(direct

C

systems).

SS J~

(3.8.1)

C

indexed by

to

theory without proofs.

> Yj})

f

f. 3

in C J~

respect to all maps

(3.8..2)

C J~

is the category of

The closed model structure on the level

TheOrem

[ B - K , p. 314] (dual to (3.2.1)).

in C J~

the maps

A map

J.

Then

Let

is due to Bousfield and Kan [B- K, p. 314].

Definitions

(= {fj:Xj in J,

sketch this

be a cofinite strongly directed set.

direct systems over

j

We s h a l l

dualization

be a closed model category which satisfies Condition N of w

J ( = {j})

category

admits a straight-forward

A map

f:X--~Y

is a fibration (resp., weak equivalence) if for all

are fibrations (resp., weak equivalences) in

C.

is a cofibration if it has the left-lifting property with p

which are both fibrations and weak equivalences.

(dual to (3.2.2)).

C J~

together with the above structure

is a closed model category.

(3.8.3)

Proposition

cofibration if for each

(dual to (3.2.3)). j

in J

A map

the induced map

f:X--->Y qj

in C J~

in the diagram

is a

126

>X. ] \

1 c o l i ~ < J fkl

c~

< j Yk

9-

>Pj~,\\~ \

J (Pj

is the pushout) is a cofibrat!on.

This result, in the ease CW

J = N,

is the usual definition of cofibration of

spectra (see [Vogt -2]).

(3.8.4)

Theorem

(dual to (3.2.4)).

The constant diagram functor

preserves cofibrations, flhrations, and weak equivalences. functor

colim:C J~

>C

C---->CJ~

The direct limit

preserves cofibrations and trivial cofibrations (maps

which are both cofibrations and weak equivalences).

We shall now discuss the homotopy theory of

(3.8.5)

stron$ cofibration if in some

(dual to (3.3.1)).

Definitions

cJ~

where

f J

is the image in

inJ -C.

A map Inj -C

f in inj -C

is called a

of a (level) cofibration

is a cofinite strongly directed set.

{fj}

Strong fibrations,

stron$ trivial cofibrations, and stron~ trivial flbratlons are defined similarly.

127

A map in

inj -C

is a cofibratlon

of a strong cofibration.

Fibrations,

in

inj -C

are defined similarly.

if

f = pi

where

(3.8.6) structure,

Theorem

p

if it is the retract in

trivial cofibrations, A map

f

in inj - C

is a trivial fibration and

(dual to (3.3.3)).

i

inJ -C,

Maps(inj -C)

and trivial fibrations is a weak equivalence

is a trivial coflbration.

together with the above

is a closed model category.

(3.8.7) C--->inj - C

Theorem

(dual to (3.3.4)).

preserves cofibratious,

direct limit functor

eolim:

The constant diagram functor

fibratlons,

inj -C -->C

and weak equivalences.

preserves cofibrations

The

and trivial

cofibrations.

The results of w167 note for later reference

- 3.7 may also be dualized to that the full subcategory

of spaces indexed by the natural numbers)

contained

inj - C.

of direct towers in

Ho(inj -C)

We shall simply (direct systems admits a

geometric model along the lines Of w

(3.8.8) direct tower

Definition

(Milnor

X = {X } n

is the space

Dir tel (X) = X 0 x [0,i] u bond

0

[Mil -3], dual to (3.7.2)).

X 1 x [1,2] u bond "'"

I

2

The telescope of a

128

Dir tel (X) tion).

admits a natural increasln~ filtration

D

We shall now consider those direct towers tractlble direct telescopes inj -C

(we call this a cofiltra-

to a direct tower

(3.8.9) equivalence

contractible

Theorem

(agaln,

direct

X 0 = *).

X ffi {X } n

with

(dual to (3.7.20)).

of homotopy categories

X ffi {X n}

from

with

X 0 ffi *,

and con-

Any direct tower is equivalent in

X 0 ffi *.

The direct telescope functor induces an Ho(inJ - C)

t e l e s c o p e s and c o f i l t e r e d

maps.

tO the homotopy category of

4.

THE HOMOTOPY INVERSE LIMIT AND

ITS APPLICATIONS TO HOMOLOGICAL ALGEBRA

w

Introduction. In this chapter we shall define a homotopy inverse limit functor

holim:Ho(pro - C)

~ Ho(C)

adJolnt to the inclusion functor

for suitable closed model categories study of the derived functors

lim s

C

(w

Ho(C)

~ Ho(pro- C)

and obtain applications to the

of the inverse limit.

Bousfield and Kan

[B -K; Chapter XI] gave a less intuitive construction of a (somewhat different) homotopy inverse limit on

pro -SS J

(see w167

4.9) j and suggested the study of

homotopy inverse limits on other pro - categories.

In w

w

is an appendix to w

we survey some of the results of other authors, notably Z. Z. Yeh,

J. -E. Roos, J.- L. Verdier, C. Y. Jensen, and B. Osofsky, on for our own work. sections: limS;

w

w

Our results are stated in w Algebraic description of

Vanishing theorems for

In this chapter, Condition

N

(w

shall discuss abelian groups).

C

limS;

as background

and proved in the following w

Topological description of

lim s.

shall denote a closed model category which satisfies

and admits arbitrary (inverse) limits.

C = SS,

llm s

SSG

(simplicial groups) and

SSAG

In particular, we (simplicial

130

w

The homotopy inverse limit. In this section we shall define a homotopy inverse limit functor

holim:

adJoint to the inclusion {Yj}

in

Ho(C) "

Ho(pro -C)

~Ho(C)

9 Ho(pro -C).

That is, for

X

in C

pro-C,

(4.2.1)

Ho(pro-C)(X, {Yj}) i Ho(C) (X, holim {Y.}).j

We begin by considering formula (4.2.1) in the case that and hence in

(4.2.2)

pro -C,

and

Proposition.

(Yj}

is fibrant in

X

is cofibrant in

pro -C.

The inverse limit functor

lim:(pro -C)ef

'C

induces a functor on the homotopy categories

So(C).

lim: Ho((pro -C)cf)

Proof.

To define

lim

on maps in

Ho((pro -C)cf),

recall that for

{Xj}, {Yk } E (pro-C)cf,

(4.2.3)

Ho((pro -C)cf)({Xj},{Yk}) = [{Xj},{Yk}],

the set of homotopy classes of maps from cylinder pro -C, Hi I = g.

and

{Xj} ~ and let

[0,i]. H:{X.} ~ 3

Let [0,i]

f,g:{Xj } ~Y

{Xj}

to

~ {Yk }

{Yk }

with respect to a

be homotopic maps in

be a homotopy with

Applying the inverse limit to the diagram

Hi 0 = f

and

C,

131

{X.} 3

[X.} |

(4.2.4a)

3

[0,1]

H > {yk}

{x.} 3

yields the diagram

lim {Xj }

k~'~O

lim {Xj

.} ~S

i

Now observe that the maps trivial fibration

lira ({Xj} ~) [0,I])

lim H) lim {Yk }.

I

i0 and iI

p:{Xj} ~

[0,i]

in diagram (4.2.4a) are sections to the ~ {Xj}.

Hence,

lim i0

and

lim iI

are

sections to the induced (lim preserves trivial fibrations, see Theorem (3.3.4)) trivial fibration the maps

lim i0

lim p: lim ({Xj} | and

lim iI

[0,I])

> lim {Xj}.

are weak equivalences.

This implies that

By diagram (4.2.5), the

maps lim f = lim H o l i m

i0

and lim g = lim Ho lim i1 become equivalent in

(4.2.5)

Ho(C).

Proposition.

D

The inverse limit functor

a functor on the homotopy categories,

lim:(pro -C)

>C

induces

132

llm:

Proof.

Let

Ho((pro -C)f) ----~Ho(C).

{Xj) ~ pro -C.

Factor the natural map

~ --~{Xj)

a cofibratlon followed by a trivial flbratlon.

~o(pro-c)({xj},-) = flbratlons,

Ho(pro -C)({~k},-);

also, because

Ho(C)(lim { 5 },-) ~ Ho(C)(lim {~},-).

lim

as

Then

preserves trivial

The conclusion now follows

from Proposition (4.2.27.

(4.2.6)

Proposition.

For

X

cofibrant in

C

and

{Yj)

flbrant in

pro - C,

(4.2.7)

Proof.

Ho(pro -C) (X, {Yj}) e Ho(C) (X, llm (Yj}).

Formula (4.2.7) follows easily from adjolntness of

llm on pro - C,

p r o - C (X, {Yj}) = C (X, lim (Yj}),

and Proposition (4.2.2).

[]

We are now ready to define the homotopy inverse limit. model categories (see w Ho((pro- C)f)

the homotopy theory of flbrant objects in

is equivalent to the homotopy theory of

Following Quillen, for each object a trivial eoflbratlon

ix

(4.2.8)

(If

X

is flbrant, choose

By Qu/llen's theory of

X

in pro - C,

pro- C p

factor the map

pro- C~

Ho(pro - C). X---~*

followed by a fibratlon

X iX

Ex~X = X).

Ex~X

9 *

This construction induces a functor

as

133

(4.2.9)

Ex~:Ho(pro- C)

9 Ho((pro -C)f);

is defined on morphisms by applying Axiom M6 to obtain fillers in the diagrams

f

X

> Y

Ex X ............ ~ Ex=Y

The homotopy class of

Ex~f

depends only on the homotopy class of

f

because

[Ex~f] = [iy] [f] [ix ]-1.

We therefore define the homotopy inverse limit, holim, to be the composite functor

(4.2.10)

holim ~ llmo Ex ~ : Ho(pro -C)

(4.2.11) (4.2.8).

Remarks.

Hence,

Propositions

(4.2.12) the inclusion

If

> Ho((pro -C)f)

X ~ pro- C

holim X = lim X

on

> Ho(C).

is fibrant, we may take

Ex~X = X

in

(pro -C)f.

(4.2.5) and (4.2.6) im~edlately yield the following.

Theorem.

The functor

Ho(C)

~ Ho(pro - C).

holim: Ho(pro -d)

~ Ho(C)

is adjoint to

134

(4.2.13)

Remarks.

Bousfield and Kan [B -K, Chapter XI] defined a different

"homotopy inverse limit" functor that for a tower of fibrations fibrations

X,

we may take

ours on suc h systems.

holimB_ K : Ho(SS J) X,

* Ho(SS).

holimB_KX ~ llm X.

Ex=X = X;

They observed

But, for a tower of

hence their definition is equivalent with

In general our definitions differ except on fibrant objects.

For example, Bousfield and Kan only obtain the following analogue of Theorem (4.2.11):

R holimB_ K

R holimB_ K

is adjoint to the inclusion

Ho(SS) ----*Ho(SSJ) j

is Quillen's total right deriVed f u n c t o r [ Q

In fact, for our

holim,

holim = R holim = R lim,

-i, w

where

lim

of

where

hol~_

K .

is the ordinary

inverse limit.

w

Ex ~

o_~n pro- S~S.

In this section we shall describe an explicit to

Ho(pro -SS)f,.

X--*Ex~X

in

(4.3.1)

together with natural

pro -SS.

Ex |

o~n objects,

MX=

{Xj~

maps

Zj

X. --** J

He(pro -S$)

trivial cofibrations

Let

X ~ pro- SS. X

First apply the Mardew

(Zj} = Ex~{X.) J

con-

functorially by a naturally isomor-

where

J = {j}

J = {J}.

is a cofinite strongly

We shall proceed inductively.

First, suppose that cial sets

Ho(pro ,SS))

indexed by a cofinite strongly directed set

We thus need only define directed set.

functor from

Compare (4.2.8) - (4.2.9).

struction (Theorem (2.1.6)) which replaces phic object

(in

Ex |

J

is an initial object of

and a trivial cofibration as in [Q-I~

w

X.J --*Zj

J.

Define a flbrant simpll-

by functorlally factoring the

135

Next, assume inductively that for a given simplieial sets brations

a)

~,

~--->~

for

bonding

maps

j

in J, for

Zk ----+ZA

and all

s < k,

k < J,

and trivial cofi-

have been defined so that:

s < k

the diagrams

> Yk

XA --

> Ys

commute;

b)

for

m < s < k

the diagrams

Zk - -

commute;

c)

; Zg.

and

The maps

Zk

> lims < k{ZA}

induced by (b) are fibrations.

Apply the Quillen factorization to the composite mapping

xj to obtain

a diagram

>lirak <S {Xk}

~ lirak < j{zk}

that

136

Xj

, Zj

> lisk< j{Zk}

The map

consisting of a trivial cofibration followed by a fibration. Zj

> iimk< j{Zk}

induces bonding maps

Zj --+Z k

k < j.

for

It is easy to

Condition (c) above is

see that these maps satisfy conditions (a) and (b) above. satisfied by construction.

Continuing inductively yields a fibrant pro- (slmplicial set) By construction,

Ex~

extends to a functor on

In order to define

Ex =

on morphisms in

S$J.

pro- SS,

{Zj} = Ex={Xj}.

D

we need the following

cofinality lemma.

(4.3.3)

Lemma.

directed sets. EX={~:}__~

Let

T:J --->K

Then for any

> Ex~{X~(j)}z

in

{~},

be a cofinal functor on cofinlte strongly T

induces an isomorphism

Ho(pro -SS),

SS K

in other words, the diagram

T

~ SSJ

S K

SSJ

\

/ tto (pro - SS)

commutes up to natural equivalence of functors.

Proof. assume that

Let

{Yj}

J and K

denote

Ex={XT(j)}

and

have initial elements

{Zk} J0

denote

and k 0

with

Ex={Xk}. T(J0) = k 0,

We may and'

'i37

also that

~ 0 ffi*"

j' in J,

and assume that the natural maps

j < j'

This yields a natural map

ZT(J0 )

ZT(j) ---->Yj

+ Yk0.

Now fix

have been defined for

such that the diagrams

(4.3.4)

ZT(j)

, Z.3

XT(j)

> ZT(j) ",.

and

ZT(j,,)

commute.

:- zj,,

Y. 3

Consider the induced commutative diagram

(4.3.5)

> l ~ k < T(J'){~}

~ limk < T(J'){Zk}

limT(j)
xr(j,)

Since

ZT(j, )

II

> limj <j,{~(j)}

limT(j) < T(j'){ZT(j)}

is defined by applying the Quillen factorlzatlon to the composite

along the top row Of diagram (4.3;5), namely the map and

Yj,

1

>limj< j,{Yj)

~(j,) ---+ limk< T(j,){Zk},

is defined by applying the Quillen factorization to the composite along

the bottom row f (4.3.5),

~(j,)

) limj < j,{Yj},

there is an induced map

138

(4.3.6)

ZT( j ,) ---->Yj, .

By diagram (4.3.5); the map (4.3.6) satisfies the conditions of diagrams (4.3.4). Continue inductively to define maps

ZT(j)

~ Yj

(4.3.4) then yield a commutative diagram in

(4.3.7)

{~(j)}

for all

j

in J.

Diagrams

SS J

~ {ZT(j)}

=

T*Ex~{~}

3

Since for each

j,

~(j)

= ZT(j)

and

~(j)

= Yj,

level weak equivalence, hence an isomorphism in

the map

z

in (4.3.7) is a

Ho(pro -SS).

Since the constructions which led to diagrams (4.3,7) were natural, we have defined the required natural isomorphism

~~

~ -" ~ ~ 1 7 6

=

We have thus proven that the construction

' ~~176

X

*.

) MX

D

~ Ex=MX,

denotes the Marde~id construction, extends to a functor from Ho((pro - SS)f)

and that there are trivial coflbratlons

which are natural in Ho(pro -SS),

(4.3.8)

Ho(pro -SS).

By construction,

pro- SS

X _z MX ExaM

where to

>Ex MX, factors through

as required.

Remarks.

The only special property of

existence of functorial factorizatlons in Axiom M2 (w

SS

which we used is the

139

w

The derived functors of the inverse limit:

background.

In this section we shall briefly summarize vanishing theorems and cofinality theorems for right-derived

functors

lim s

lim:

(where

AG

of the inverse limit

(AG) J

> AG

is the category of abelian groups).

Because

no___~t,in general, right exact, vanishing theorems for of

lim.

Cofinality theorems are used to extend

lim

lim s lim s

is left-exact, but measure the exactness

to

pro -AG.

There

is a close relationship between these results and the homological dimension of modules which we shall not discuss here; see B. L. Osofsky

[Osof] for a good survey.

We refer the reader tO [Mit -2], for example, for the basic theory of derived functors due to H. Cartan and S. Eilenberg

[C -E].

Z. Z. Yeh [Yeh] gave the first vanishing theorem in 1959.

His results were

extended first by J. -E. Roos [Roos] in 1961 and later by C. U. Jensen [Jen] in 1972.

(4.4.1) directed set

Definition I

(4.4.2) limS{G i} = 0

set

J

> limi~ j{G i}

Proposition for

[Roos]).

is called flasque

necessarily directed) limi E l{Gi }

([Yeh],

s > 1.

An inverse system

(or star-epimorphic)

contained in

I,

{G i}

indexed by a

if for each ordered

the natural map

is surjective.

([Yeh],

[Roos]).

If

{G i}

is flasque,

then

(not

140

Roos obtains this result by showing that complex obtained from

{Gi}.

(4.4.3) set

I

I

CW

complexes.

([Jen]).

An inverse system

lim i ~ l{Gi}

Proposition

l ims{C i} = 0

for

~ lim i ~ j{G i}

([Jen]).

lim

If

indexed by a directed J

contained in

I,

the

is surjective.

{G i}

is weakly flasque, then

s ~ i.

Jensen first proves that showing that

liraI

and

In fact, Roos only

{Gi}

is called weakly flasque if for each directed set

(4.4.4)

lim

be ordered, not necessarily directed.

Definition

natural map

is the homology of a

Compare J. Milnor's [Mil -3] use of

in axiomatizing the cohomology of infinite requires that

lim s {G i}

lim s

applied to a weakly flasque system is

is rlght-exact on such systems.

Vanishing of

lim s

0

by

then fol-

lows by a suitable iteration.

(4.4.5)

a)

Remarks.

Clearly flasque implies weakly flasque.

Jensen observed that

the converse is false.

b)

Roos and Jensen actually worked in categories of the form where

A

A I,

is an abelian category which satisfies suitable

exactness axioms of A. Grothendleck [Gro -2].

One may extend the domain of follows.

First, a cofinal functor

lim

and

lim s

T:J --->K

from

(AG)J

to

pro- AG

induces an isomorphism

~S

141

T*:{G k}

~ {GT(j)}

in pro ~ AG,

hence we need the following result of Roos

(1961), Jensen (1972) and B. Mitchell [Mit -2] (1973).

(4.4.6)

Theorem [Mit-2].

A coflnal functor

T:J--~K

between filtering

categories induces commutative diagrams

K

l~KS

(AG

AG

T*I (Aa J Secondly, although Artin and Mazur [A -M, w representation of a map {Gj(s

---~(s

{Nilj E J}

in some

level representatives.

(AG)L,

~ { ~ I k c K}

J (~)

where

commutative diagrams

in pro -AG

a given map in

}

pro -AG

by a level map may have many

~ {~(~) }

{Gj(m) } "'"

pro- AG,

above) gave a natural

Therefore we need to show that commutative diagrams

{G.

in

(see w

L = {s

and

" {Hk(m) }

M = {m}

are filtering categories, induce

142

I~Ls(cj (~) }

.

~

~ li-Ls{Hk(~) }

S

IL'~N {C3(m) }

- 1~s{~(~)

}-

J. -L. Verdier [Ver] announced this result in 1965, and hence extended the inverse limit functor and its derived functors to

pro - AG.

limj (AG) J

In particular, the diagrams

s

o.

> AG

pro - AG

commute, where

limproS

is the s th right derived functor of

limpr ~ .

We

shall include independent proofs of these results, see w

(4.4.7)

lim and HOM functors.

the category of R - m o d u l e s . R -Mod,

that is,

underlying set. the functors R ~

J

Let

HOM (X,Y)

Let HOM

R

be a commutative ring, let

R-Mod

be

denote the internal mapping functor on

is the natural R -module with

R - Mod (X,Y)

as

In 1973 B. L. Osofsky [Osof] gave the following representation of limS:(R-Mod) J

~ R - Mod.

such that the categories

(R- Mod) J

She defined a "tensor product" ring and

(R~

This isomorphism induces natural equivalences of functors

J) - Mod

are isomorphic.

143

(R -Hod) J

lira:

~ R -Hod

(~ HOM(R _Mod)J(R, lim(-)):

(R -Mod) J

i HOM,R~ (9 j _Mod,(Rj (9 I,-):

> R - Mod)

R (9 G -Mod

9 R- Mod,

hence also

liras:

(R -Rod) J

9 R - Mod

s EXt(R (9 J ) - Mod (R (9 J , - ) :

(R (9 J-MJod)

7R-Mod.

Later in this chapter we shall use pro- categories to give more natural results See w

w

Results on derived functors of the inverse limit.

We shall prove the following three theor~s in w

Theorem A. HOHpro,

Extj

Let and

J

be a cofinite strongly directed set, and let

EXtpr ~

be the appropriate

a)

limj - HOMj(Z,-) : AG J - - A G .

b)

limpr ~ = HOMpro (Z ,-) : pro -AG --~ AG.

c)

limj s ~ ExtjS(z,-):

d)

llm

pro

Theorem B.

s =

Ext

Let

pro

J

HOH

and

Ext

HOHj,

functors.

Then

AGJ--->AG.

S(z,-):

pro -AG--~AG.

be a cofinite strongly directed set.

Then the diagrams

144

AGJ

~

limjs~

>

pro

~ i ~

-

AG

pro

s

AG commute up to natural equivalence.

Theorem C.

Let

{G.} be a stable pro- group.

-

Then

3

{Gj}, " s = 0, llmjS{Gj } = limproS{Gj} -= limpr Os l i m {Gj}) ~ f[ lim 0, s r 0 .

In w

we shall use the relationship between the topological and algebraic

structures on

Theorem D. {~_} ~ p r o - AG.

pro -SSAG

Let

to prove the following.

{Gj} be an inverse system of free abellan groups and

Then

Ext s({Gj }, {Hk}) = Ho(pro - SSAG) ({Gj }, WSHk})

where

is the W-constructlon of Ellenberg and MacLane (see (4.7.9)).

Theorema A and D imply the following.

Theorem E.

limS{Gj} = Ho(pro-SSAG)(Z, {~Gj}) = w0{~Gj}

An analogous formula holds on

Theorem F.

lims{cj }

of slmpllclal spectra and

=

pro-G

for

s = 0

Ho(pro -Sp)(sO,{KGj}), KGj

and

where

on

pro-AG.

1.

Sp denotes the category

the simpllclal Ellenberg-MacLane spectrm~ with

145

Gj,

n = O,

0,

n # 0 .

~n(KGj) =

Bousfield and Kan [B - K, w

In w

give an analogous formula.

we describe our [ E - H -5] strong-Mitta~-Leffler

groups, a natural extension of the Mittag-Leffler [E - H -5] we proved that a pro -group if

G

is pro -isomorphic

directed set.

Let

a) liml{Gj }

= O;

c)

w

If

{Gj}

condition for towers.

is strongly-Mittag-Leffler

{Gj}

is abelian,

lim {G.} = O, 3

then

then

limS{Gj} = 0

{G.} ~ 0 3

in

for

pro - group.

limpr ~

pro - A G

extends

derived functors (4.4.7).

limj, Ext s

Then:

s > 0;

pro - G.

o_ff lim s.

In this section we shall interpret the inverse limit functors lim : pro

if and only

proof of the following in w

be a strongly-Mittag-Leffler

Algebraic description

and

In

to a flasque pro -group indexed by a cofinite strongly

We give a "topological"

Theorem G.

b) If

G

condition for pro -

> AG

as suitable

HOM

functors.

> AG

This will-show that

and also identify the derived functors of HOM.

llmj:AG J

lim s

as the

Compare the results of B. Osofsky described

in

Our results can be extended easily to categories of modules over a tom-

mutative ring with identity.

We shall need the following structure.

146

(4.6.1)

Theorem.

a)

AG J

is an abelian category.

b)

pro- AG

is an abelian category.

Part (a) is well-known.

For part (b), see [A -M, w

Some of the abelian

structure is described below.

(4.6.2)

Definition.

is exact if for each

j

A sequence in J,

0

~ {Aj} ~

the sequence

0

{Bj} >A.3

>{Cj} ~Bj

~0 ~Cj

in AG J ~ 0

is

exact.

(4,6.3) pro - AG

Proposition.

A sequence

0

~ {Aj}

> {Bk}

~ (C A)

0

in

is exact if and only if there exists a commutative diagram

0

~ {Aj } --~ {Sk}

--- {C~} - ~ 0

0

> {A m} ---~ {Bk(m) } --> {C' m} --~ 0

(4.6.4)

in which the bottom row is a short exact sequence i n the appropriate level category

A~,

M = {=}.

Proof.

The "if" part is clear.

For the "only if" part, first relndex the

given short exact sequence to obtain a diagram

147

0 --> {Aj }

---> {Bk}

--'>{C A}

--~0

(4.6.5)

0 --->{Aj (m) } --> {Bk(m) } --> {CA(m)} ---->0

in which the rows are exact, and the bottom row consists of level maps. M = {m},

and let

{A'm}--->{Bk(m)}

A'm

be the kernel of the map

is a kernel of the map

Bk(m) ~

CA(m).

{Bk(m)}--->{CA(m)}

Let Then the map

in pro-AG,

hence

we may replace the bottom row of diagram (4.6.5) by the isomorphic short exact sequence

(4.6.6)

0---> {A'm} ---~{Bk(m)}--~{Cg(m)}--'~0.

Similarly, let

C'm

be the cokernel of the map

short exact sequence in

A'm --~ Bk(m)"

There results a

AGM,

0 --> {A'm} -'->{Bk(m) } --->{C'm} ---~0,

which is isomorphic to (4.6.6), via reindexin8 in the middle term, as required.

(4.6.7)

Remarks.

Proposition (4.6.3) can easily be extended to finite dia-

grams o f s h o r t e x a c t sequences w i t h o u t

(4.6.8)

Definitions.

define direct sums in J

and K

0

Direct sums in

pro - AG

respectively; let

loops;

AG J

as follows. {Xj} ~

see w

are defined degreewlse. Given

{Yk } ffi {Xj

~

{Xj} Yk },

and

{Yk }

We may indexed by

indexed over the

148

product category if

J x K.

and

j ~ j'

J

If

and

(J,k) ~ (j',k')

HOM

on

AG

also extends to

pro-AG.

(4.6.9)

Definitions.

Given

HOMj({Xj},{Yj}) = AGJ({xj},{yj}), Similarly, given

{Xj}

HOMpro({Xj},{Yk})

Then limit

are directed sets,

k ~k'.

The internal mapping functor AG J

and K

the incluslon HOM (Z,X) m X

(4.6.10)

for

X

in AG,

b)

l i m p r o = HOMpro(Z,-):

(4.6.11)

Corollary.

(4.6.12)

Corollary.

limpro:

A G - - ~ p r o - AG),

limj s - ExtjS(z,-):

b)

lira

s = Ext

pro

AGJ --->Ag. pro - A G - - > A G .

B

AGJ--*AG.

AGJ--->AG.

S(z,-):

pro-AG--~AG.

{Yk }-

Because the i n v e r s e

pro -AG--~AG)

we have the following.

llmj = llmpr ~ . w:

a)

{Yj}.

let

Theorem.

l i m j ffi HOMj(Z,-):

define

with group operations induced from

(respectively,

a)

pro

in AG J,

may be extended to f u n c t o r s .

(respectively,

AG--~AG J

{Yj}

{Yk.} in pro -AG,

HO%r ~

' ~ AG

and

With group operations induced from

= pro -AG ({Xj),{Yk)),

HOMj and

limj:AG J

and

{Xj}

and

is adjoint to

149

Proof. isomorphic functors

In each case, functors.

llm s

and

[Mit -i,

p. 193D.

The above theorem and corollaries natural transformation

of connected

sequence) AG J

in

AG J

of

form Theorem A.

T:{ExtjS(z,-)}

~ .

into an extension in

natural transformation

in

Let

J

9

T:{Extj s}

~ {EXtproS},

induces an isomorphism

---->{EXtproS(Z,-)}.

they map an extension

p r o - AG,

p r o - AG.

be a coflns

AG J

follows.

We shall now define a

~

(long exact

and send a map of extensions

Therefore,

in

T induces the required

of connected sequences of functors

Theorem.

of derived

Because the natural quotient functors

preserve abellan structures,

into a map of extensions

(4.6.14) diagrams

The conclusions

is a coflnite directed set, then

COnstruction

~ pro -AG

characterization

of connected sequences of functors

sequences of functors

(4.6.13). ~:AG J

J

are the s th derived functors of

Now apply D. Buchsbaam's

[Buch] (see also

and show that if

Ext s

:{Extjs}

strongly directed set.

,

{tproS}

Then the

> pro - A G

ExtjS(z,-)~ ~XtproS(Z,-) AG commute up to natural equivalence.

Proof.

For

s = 0,

now follows by Buchsbaum's

this follows from (4.6.10) - (4.6.13). characterization

of derived functors

The conclusion

[Buch], or by the

150

following alternative direct proof for

We shall first show that

s > 0.

T:ExtjI(z," {Gj})

epimorphlsm for each cofinite strongly directed set Consider an extension in

(4.6.15)

replace the monomorphiem > {H'~},

we may assmne that

and L

for each

{J(~(j))}

{Gj} --->{Hk}

in

AG J .

j

Now use the maps

(4.6.18)

Finally, the map

as well as

J

L ffi {s

is a First,

{C~}

to obtain

{Cs

is a coflnite strongly directed set.

are eonflnite strongly directed sets, we may choose elements in J

is cofinal in

(4.6.17)

K

by an inverse system of monomorphisms

O--->{Gs163

Again, we may asstmse that

s

{Gj}

We shall relndex (4.6.15) several times.

and take the levelwise cokernel

(4.6.16)

J

and

o --~ {G~ } - ~ {R k} --+ Z - ~ O.

coflnite strongly directed set.

Because

J

pro- AG

By the Marde~id construction, w

(Gj(~)}

is an

> EXtprol(Z, {Gj})

so that J.

j ~J'

implies

~(j) ~ s

and

We now replace (4.6.16) by

0 --+ {G.j(~(j))--~ {H'~(j) }-~

Gj(~(j)) ---~Gj

{C~(j)}-->0.

to push out (4.6.17) and obtain

O--->{Gj}--+{H"j}--+{C'j}--->O.

Z a {Z.3 = Z}

~ {C'j}

is a pro-isomorphlsm, so pulling back

(4.6.18) by this map yields the required extension

151

(4.6.19)

0 --+ {G.} --+ {H"'j) --->{Z.} --+ 0 3 3

isomorphic to (4.6.15). epimorphiem. set

J

and

T:ExCjI(z, {Gj}) --+EXtprol(z, {Gj})

is an

Similar techniques imply that for each cofinite strongly directed (Gj} in AG J,

isemorphism for all system.

Hence,

s.

T:ExtjS(z, {Gj}) ---->EXtproS(Z, (Gj }) The crucial

point

is

that

Z

is

is

a constant

an

inverse

D

Because~ we have already identified

lim s

as

Ext s,

TheOrem (4.6.14)

implies the following.

(4.6.20)

Theorem B.

Let

J be a cofinite strongly directed set.

Then the

diagrams AGJ

~

limj~s~

+ pro - AG /improS

AG

commute up to natural equivalence.

Hence we shall write

lims

(4.6.21) Definition. it is isomorphic in

If pro- G

the

by functoriality of

ing vanishing theorem.

limjs = llmpr ~ s

An inverse system of groups

pro- G

{G.}3 is stable,

for

(If

{Gj}

is called stable if

to a group.

natural map lim.

lim {Gj} --+~G.}3

is an isomorphism in

Theorem B then immediately implies the follow-

{G.} i s not abelian, everything works for 3

152

s = 0

or

i.

non-abelian groups.

(4.6.22)

[B -K]

See, for example, In this case,

Theorem C.

Let

lim I

{Gj}

for

Topological description of

of an inverse system of

is only a pointed set.)

be a stable p r o - group.

limaS{Gj} = l~mproS{Gj} = 1 r o S ( l i m i m

w

llm I

{G.}) ~ 3

{

li=

Then

{Gj},

e = 0

O,

s#O.

llm s.

In the last section we showed that

= Extl(z, {Gj })

lim I Gj

where

J ffi {j)

is a cofinite strongly directed set.

Because

ExtI(z, {Gj})

is the set of short exact sequences

0 ---* {Gj } ~

and a short exact sequence in

{Sj } ~

pro - AG

{Z} ---~ 0,

is a fibration sequence in

pro -SSAG

(see (4,7.1) - (4.7.13), below), one is led to ask whether

Extl(z, {Gj}) = Ho(pro-SSAG)(Z,B {Cj}) for a suitable classifying space in this section.

abelian group

We shall carry out the above program

The first step is to relate the abeliau structure of

and the closed model structure of

(4.7.1)

B{Gj}.

Definition. SG

with

pro - SSAG.

Associate to an abellan group (SG) n = G

p r o - AG

for all

n ~ O,

G

the dlscrete simplicial

and all face and degeneracy

153

maps the identity. TH = H0,

Associate

to a simplicial abelian group

the group of 0 - simplices of

(4.7.2)

Proposition.

Then

S

and T

coadjoint to

T,

S

extend to functors

9 SSAG,

T: SSAG

S

the abelian group

H.

S: AG

with

H

9 AG,

a full embedding,

and

TS = lAG.

The proof is easy and omitted.

Now prolong

S

and T

to functors

(4.7.3)

S: pro -AG----~pro - S S A G T : pro - SSAG

(4.7.4) TS = i

Proposition.

is c o a d j o i n t to

S

Immediate from Proposition

We shall frequently identify

A

T,

S

iS a full embedding and

p r o - AG.

Proof.

S.

9 pro - A G

Artin and Mazur

[A -M,

is an abellan category.

w

(4.7.2).

pro - AG

with its image in

showed that

See w

pro - A

for the case

pro - SSAG

under

is an abelian category if

A = AG.

We are in the fol-

lowing situation.

(4.7.5)

kernels,

SSAG

is an abelian category.

c o k e r n e l s , and d i r e c t

The required structures,

sums a r e d e f i n e d d e g r e e w i s e .

namely

Addition in

0,

154

SSAG (G,H)

is defined degreewlse.

The functors

S and T

(4.7.2) preserve

abelian structures.

(4.7.6)

The normalizatlon

is the chain complex

NG

of a simplicial abellan group

NG = {NnG,dn}

I NnG =

G - {Gn,dnl,snl}

with

G 0, (dni:Gn 0 ker i>0

n = 0 n = 0

~ Gn_l),

dn = dnoINnG.

Then

N

extends to a functor on

SSAG.

Moore showed that

w,G = H,{NnG,dn),

the homology of the chain complex

(4.7.7)

Call a simpllclal abelian group

S:AG--->SSAG, 0.

If

dnl = 1G0

G

NG.

is a discrete simplicial abelian group, for all

n and i;

G

G

n

= GO

is in the image of G

have dimension

for all

u,

and

hence

I G0,

G

discrete if

that is, if the only non-degenerate simpllces of

NnG

if

G

n ffi0

=

O,

n#O

is discrete.

Proposition (4.7.6) ylelds the following relatlonship between the abelian and closed mode I structures of

SSAG.

~S5

(4.7.7)

PropOsition

a flbratlon in

(Quillen [ Q - I , Proposition II.3.1]).

SSAG

(hence in

SS)

if and only if

N f n

A map

f:G --~H

is surjectlve for

> 0.

(4.7.8)

Remarks.

We shall need two special cases:

a)

An 7 map of discrete simpllclal abelian groups is a fibratlon;

b)

Any (levelwlse) surJection of simpllclal abellan groups is a flbratlon.

The fibre of a fibration

(4.7.9) SSG

f

in SSAG

is Just the (levelwlse) kernel of

f.

S. Eilenberg and S. MacLane defined a classlfying space construction

(see [May - 1, p. 21] for a description).

To a slmpllclal abellan group

their construction associates a flbratlon sequence in

SSAG

G--->wG--->WG;

is a contractible (in Lct

WG

SSAG) simpllclal abellan group, and

is always a slmpllcial group, even if

G

WG ~ SSAG.

is not abelian.

In

Their con-

:ruction Immediately yields the following.

(4.7.10) l

SSAG

W

takes a short exact sequence

into a short exact sequence

(4.7.11) ~ly if

Proposition.

Wf

Proposition. is a flbration.

A map

0 -~K

f:G--~H

0 --~ K --~ G --~ H --~ 0

--~WG --~WH --> 0

in SSAG

in SSAG.

is a surJectlon if and

156

Proof.

The "if" part follows from Proposition (4.6.10) and Remarks (4.7.8).

For the "only if" part, the map

(WG) 0

NWf:NWG--~NWH

W.

(~)0

each consist of trivial group

0,

is a degreewise surjection by Proposition (4.7.7).

[Q -I, Lemma II.3.5], the map the construction of

and

Wf

is a surJectlon.

Hence,

f

so By

is a surJectlon by

0

We shall now extend the above discussion to

pro - SSAG.

Call a pro-

(slmplicial abelian group) discrete if it is pro - isomorphic to one in the image of S:pro -AG

~ pro -SSAG.

(4.7.12)

Proof.

Proposition.

If

{G.} 3

by Remarks (4.7.8).

(4.7.13) in

Proof.

pro- SSAG.

All discrete simplicial abelian groups are fibrant.

is discrete, the maps

Gj --->i ~ k

<j

{Gk}

are fibrations

D

Proposition.

pro -SSAG

Prolong the W -construction levelwise to

W

takes a short exact sequence

into a short exact sequence

0---+K--~H-->G--~0

0--->WK--->WG--->WH--->0

Use Propositions (4.6.3) and (4.7.10).

in pro- SSAG.

0

The analogue of Proposition (4.7.11) is difficult to state; the ideas will be used in the latter part of this section.

We shall now use

pro- SSAG

and

Ho(pro -SSAG)

to classify extensions in

pro -AG.

(4.7.14) isomorphism

Proposition.

For

G

and H

in p r o - A G ,

S

induces a natural

157

~

Proof. because

S

Because

o _ AGS(G, H)

S

is full, TS = 1

is full and

~

is a monomorphism.

pro -SSAG,

HOMpr ~

o _ AG(G, H)

To show that

where

G

o

AG(G,H) = HOMpr ~ - SSAG(G,H).

Also,

the induced natural transformation

pro -AG'

E:0 - ~ G

in

>EXtpr o _ SsAGS(G,H) 9

EXtpr o _ SSAG(G, H)

is an epimorphism{ consider an extension

--->X --~ "'" --->X' -->H --> 0

a~d H

are discrete.

Applying

T

to E

yields an

extension

TE:0 --->G --+ TX ---~ "'" --->TX' --->G --->0

in

pro -AG.

Because there is a (natural) map

on

G

E ~ STE ~ Im o ,

and H,

as required.

We shall henceforth simply write

(4.7.15) and for{Q}

Theorem D.

for

which is the identity

[]

Ext

pro - A G

and

Ext

pro- SSAG"

For a levelwise free abelian pro- (abelian group)

in p r o - AG,

ExtS({Gj}, { ~ } )

Proof.

Ext

STE--~E

= Ho(pro-SSAG)({%},

First, consider the case

Ext0({Gj}, { ~ } )

= HOM ({Gj}, { ~ } ) .

s = O,

where

Because

{Gj}

{wS~}).

is free, hence coflbrant

{G S },

158

(see w

w

and [Q- I, w

and

{~)

is discrete, hence flbrant

(Proposition (4.7.12)),

Ho(pro- SSAG)({Gj), {H~}) = [{Cj}, {Hk}] (hamotopy classes of maps with respect to the cocylfnder Because

~)

is discrete,

{~[0,l]} ~ (~).

{Hk}[O'l] - {Hk[O'l]}).

Hence,

Ho(pro-SSAC)({Gj}, {Hk}) = pro-SSAG ({Gj}, {Hk}), as

required. For

s > 0,

we shall use one of B. Mitchell's characterizations of derived

functors [N_it- 1, p. 198, case III]. 0 - - ~ { A i} --~{BI}--~{C i} ---~0 pro -SSAG~ SSAGJ.

Suppose first that

is a short exact sequence of level maps in

that is, a short exact sequence in the appropriate level category Consider a fixed

J

in J.

Then there are fibre sequences

Aj ~ Bj "--~Cj,

and

WAj "-~ ~Bj ~ WCj .

We may obtain a connecting morphlam in

Ho(SSAG),

~:Cj --~ WAj

and thus a

co-Puppe (flbratlon) sequence (each map is the fibre of the next map)

(4.7.16)

as follows. f and g

Aj

~ Bj ~

The homomorphlsms in the diagram

Cj

Bj ~

~WAj

0

and

~WBj ---~WCj

WAj --~ 0

induce the flbrations

159

WA. 3

B.3

l

>WAj x AjBj

f'>WAj Aj x O -= WA.3

O x A.Bj -= Cj 3

Further,

g

i s an e q u i v a l e n c e .

Cj e 0 x AjBj

Then

6

Let

6

[g]-1 ~WAj

be t h e c o m p o s i t e i n

if]

x AjBj

Ho(SSAG)

~WA.3 x A"O3 ~ WAj.

is the required connecting morphism, and it is easy to show that sequence

(4.7.16) is fibration (co- Puppe) sequence.

Therefore, the sequence

(4.7.17)

is an inverse system of long fibration sequences, hence a long fibration sequence in

Ho(pro -SSAG)

by Proposition (3.4.17).

For any

G i n Ho(pro -SSAG),

(4.7.18)

[G,{Ai} ]

(4.7.17) induces a long exact sequence 6,

o f a b e l i a n g r o u p s , where

More generally, if pro -SSAG,

> [G,{Bi}]

[-,-]

[G,{Ci}]

> ~_.[G,~NAi)I

E Ho(pro-SSAG)(-,-).

0--->A--->B--+C--+0

, ...

Compare ( 3 . 4 . 1 6 ) .

is a short exact sequence in

Proposition (4.6.3) and the above techniques yield a long exact

160

sequence

(4.7.19)

[G,A]

analogous to (4.7.18).

9 [G,B]

9 [G,C]

(H,H I) " ([@,-],[@,W'(-)I)

is a connected pair of exact functor on

(H,H I)

pro -SSAG.

If

G

is discrete

and free, and we restrict the functors (4.7.20) to is a connected pair of functors with

pro -AG(G,-). WB = WB i

> ".-

Thus

(4.7.20)

pro -AG)

6 * 9 [G, WA]

Further,

H = Ho(pro -SSA~(G,-)

Ho(pro -SSAG)(G,W(-))

then

=

vanishes on objects of the form

since such objects are levelly contractible

given any short exact sequence

pro - AG,

( in

(see (4.7.9)).

0---~A---~B--->C---~O

in pro- SSAG,

Finally, there is a

diagram

(4.7.21)

O-->A--~

0 ""~ A ~

in which the bottom row is exact. O--->A--~WB--~WB/A--~0 0 ---~A---~B --->C ---~0. that is,

B--~

C

i

WI3 -----~~ / A

--~0

~

0

Thus, sequences of the form

are coflnal in the directed set of all sequences Since

HI(A) = Ext (G,A),

HI(wB) - 0,

H1

is the derived functor of

by M/tchell's criterion.

By iterating the above construction, we obtain the required isomorphisms

ExtS({Gj }, { ~ } )

- Ho(pro - SSAG) ({Gj }, { w S ~ } )

H,

161

for {G.} levelwise free in 3

pro- AG and

The above results hold on

{~}

in pro-AG.

pro- G for s = 0 and i.

We may use the homotopy inverse limit (w

D

Details are omitted.

to reformulate the above theorem

(for (a) and (b) see Bousfleld and Kan [B -K, w

Part (c) is our Theorem F

(w (4.7.22) Theorem. a) Let {Gi} be an inverse system of groups.

Then

f l i m I {Gj} if n ffi0,

w11 (hollmWGi) ffi ~ i i m {Gj}

if nn ffi>I.i'

b) Let {G.} be an inverse system of abelian groups. 3

Then

{wsGj}) = ~0~llmS-n{Gj} if 0 < n < s, wn (holim if

c)

Let

n>

s.

{Gj} be an inverse system of abelian groups, and let

KG. be the simpliclal spectrum obtained from the simplicial 3 prespectrum

{wsGjIs ~ 0}.

Then using stable homotopy groups,

Sn (hollm {KGi}) - If0im-n(G•

n>o.n <__0,

162

Proof.

(a) and (b) follow from the observation

w

(holimWScj)

n

= Ho(pro- SS) (S n, {~Gj})

= Ho(pro-SS)

(S O , {WS-nGj})

= Ho(pro-SSAS)

together with Theorem E. spectra

Sp

w

For (c), use analogous computations with simpliclal

and simplicial abelian group spectra

and consider each

KGj

(Z, {WS-nG.}). 3

SpAG

[Kan- i, 2, 3], or use (b)

as a simpliclal prespectrum.

Strongly Mitta~-Leffler

pro- groups.

We shall give an appropriate generalization

to inverse systems indexed by

uncountable indexin~ sets of the following well-known results.

(4.8.1) surJections.

(4.8.2)

Let Then

{Gn}

be a tower of groups such that the bonding maps are all

limS{Gn } = 0

Suppose further that

for

s > 0.

lim {G n} = 0,

then

A pro -group is said to satisfy the Mitta~-Leffler

{G n} m 0

(M-L)

in pro - G.

condition if it is

pro- isomorphic to an inverse system of groups whose bounding maps are surjections.

Keesling

[Kee -2] has exhibited a M - L pro - (abelian group)

an uncountable directed set, such that

lim {G i} = 0

Thus (4.8.2) fails in general for M - L pro - groups. structed a movable

(Definition

but Keesllng

{Gi},

{G i} ~ 0

indexed by in pro - G.

[Kee- 2] also con-

(4.8.5), below) inverse system of long exact

sequences of abelian groups such that the inverse limit sequence is not exact.

We

18S

shall use this example to prove (Proposition (4.8.6), below) that (4.8.1) also fails, in general, on M -L pro -groups.

In a positive direction, we suggest the following definition as the appropriate generalization

(4.8.3)

of the M - L condition to uncountable inverse systems.

Definition.

Leffler (S - M - L ) that

G.3 ---> l ~ k <

G

J{%}

pro -group

is

j

{Gj}

such

in J,

the

are surjections.

implies M - L .

Proposition

if ~nd only if

is said to satisfy the stron~-Mittla~-

is a cofinite strongly directed set, and for all

Clearly, S - M - L

(4.8.4)

pro -group

condition if it is pro- isomorphic to a

J = {j}

natural maps

A

Also, for towers, M - L

[E - H - 5 ] .

A pro-group

pro -isomorphic

G

to a flasque

implies S - M - L .

is strongly Mittag-Leffler pro -group

indexed by a

is flasque and

J = {j}

cofinite strongly directed set.

We sketch a proof for completeness.

If

{Gj}

cofinite strongly directed set, the natural maps tions by definition.

Conversely,

directed set and the natural maps be an ordered subset of {gj lJ ~ J'} {Gj}

{G.} 3

are surjec-

is indexed by a cofinite strongly

Gj --->lim k <j{Gk}

~re surJections.

Let

J'

It is easy to extend an inverse system

to an inverse system indexed by

is flasque.

(4.8.5)

J.

suppose

Gj --->l~k< j{G k}

is a

J

by induction over

J \ J'

Thus

D

Theorem G.

Let

{Gj}

be a strongly-Mittag-Leffler

pro - group.

Then

164

a)

liml{Gj} ffi0 ;

b)

If

c)

If

(Gj}

llm {Gj} = 0,

Proof.

j

the induced maps

(4.7.11).

{Gj} a 0

J = {J}

the induced map Wp:WGj

W(I~

W

p:Gj --~ l ~ k < j{G k}

is a surjection.

Then,

are fibrations by Proposition

is an adJolnt functor (see, for example, [May - 1, Hence the inverse system

that is, the induced maps

Q'Gi ( in

s > 0;

is a cofinite strongly directed set and

< j {Gk} ) ~ I ~ k < j{WGk}.

is fibrant (w

are fibrations

for

in pro-G.

> W (l~k<j{Gk})

Further, since

Theorem 27.1]), {WGj}

then

We may assume that

that for each

limS{Gj} = 0

is abelian, then

SSG

,- l~bS~< J {WGk}

or SSAG).

By Theorem E (g4.6),

llml{cj } ffi Tfo{W'G j } " "~o(hol~m{W'G}) j (see w

for holim),

Because

{WGj}

is fibrant,

holim {WGj} = lira {WGj}

because

is an adjolnt.

Hence,

(see (4.2.11))

~0(hollm {WGj}) = O.

Part (a) follows.

165

The proof of part (b) uses the formula

limS{Gj} = ~ 0 { ~ G j }

in a similar way.

Details are omitted.

For part (c), assume that for some

j

in J,

Gj

# O.

Choose

g # e

0

(the identity) in

Gj .

For

k ~ Jo

in J,

let

gk

Otherwise use induction on the number of predecessors of

element

{gj}

in lim {Gj}

with

gJ0 = g ~ e.

Hence,

be the image of

j

in J

g.

to define an

lim {Gj} # 0.

This

contradiction establishes part (c).

Recall that an object there exists a

k 9 J

{X.} 3

of pro

- C

is said to be movable if for each > k

such that for all

J

there exists a filler in the

diagram

x~

9

xj

\ \ \ \ k N

It is easy to check that movable

xk

pro- groups

satisfy the Mittag-Leffler

con-

dition.

(4.8.6) Leffler

Pr__oposltlon

pro- groups.

[E -H - 5].

In general,

llm I

need not vanish on Mitta8-

168

We sketch the proof for completeness. pro - groups

{Gj},

If

liml(Gj} = 0

for all

M- L

then any short exact sequence

0 --->{Aj } --~ {Bk} --~ {C~} --->0

of

M -L

pro- groups

would yield a short exact sequence under the functor

In [Kee -2] Keesling constructs a movable pair,

(Xj,Aj)

in

llm.

pro-ANRpairs,

hence a movable system of long exact sequences

~j (4.8.7)

{--.

~HI(Xj) ------>HI(Xj,Aj)

>H0(Aj)

9 ..-} ,

such that the induced sequence

(4.8.8) lim {Hl(Xj)}

is not exact at the middle term.

lim (~j) lim {HI(Xj,Aj)}

> lim {H0(Aj) }

Since the kernels and images in sequence (4.8.7)

are movable, and hence M -L, sequence (4.8.8) would then be exact, contradicting Keesling.

Therefore,

w

lim I

cannot vanish on all

M-L

pro- groups.

The Bousfield -Kan Spectral seqUence In this section we discuss the Bousfield -Fan spectral sequence for the homotopy

groups of the homotopy inverse limit of a pro- (slmplicial set) [B - K, Chapter XI]. Although their model structure [B- K, p. 314]

on SS J

differs from ours, the

167

resulting homotopy categories are isomorphic (see w object in our model structure, then

Let replace

X E SS JX

assume that

Because our

by a fibrant object X

is fibrant.

limit, temporarily denoted

X

X

is a fibrant

~s fibrant in the Bousfield - K a n structure.

holim X'

Also, if

preserves weak equivalences, we may

with

hollm X = holim X'.

Hence, we may

In this case, the Bousfield -Kan homotopy inverse h~

satisfies

Ho(pro -SS)(W,X) m Ho(SSJ)(cW,X) = ~o(SS)(W,

(cW

denotes the constant diagram w~th

Proposition~XI.8.1]. holim X ~ holim B _KX.

cW. = W 3

holi~_KX) for all

J

in J)

[B- K,

By uniqueness of adJoints (see, e.g.~ [Mit-l]), A direct proof also exists, compare Milnor's use of tele-

scopes in [Mil -S].

Hence the following result of Bousfield and Kan holds for our homotopy inverse limit.

(4.9.1)

Theorem

[B-K,

there is a spectral sequence

w

For a pro - simplicial set

w {Er(X),dr} ,

with

E2(X) = {E2P'q(x) ffi limPj {~q(Xj)}},

Under suitable conditions,

{Er(X)}

bidegree

d

r

= (r, r - l ) .

converges completely to

There are two important special cases.

X,

w, holim X.

D

168

(4.9.2)

If

X

in pro - SS,

is equivalent in

pro - Ho(SS,)

to a slmpli-

clal set, then

E~'q(x)

=

E~'q(x) = {~q hollm X, O,

(4.9.3)

If

{Xn, n ffi 0 , i , " ' } ,

X = {Xj) then

in pro -SS, E2(X) = E (X)

p

=

0

p~O

is equivalent to a tower and the spectral sequence collapses to

the short exact sequences

0

~ llmlj{Irq+iXj}

~ q hollm X

; limj{~qXj}

"10.

We shall need the following extensions in order to discuss strong homology (5.6.7) - (5.6.8) and Steenrod homology (w

First, we may replace slmplicial sets by simplicial spectra in the Bousfield Kan spectral sequence.

Their proof of convergence still applies if suitable care

is taken with smash products.

In particular, for

{5}

there is a spectral sequence with

EP'q = limPj{~qS(Xj ^ E)} 2 = limPj{hq(Xj)}

which converges under suitable assumptions to

~ pro - SS,

and

E E Sp,

169 S

~, holim {Xj ^ E)

Sh,{Xj)

(see (5.6.8))

( ~ h, (holim {xj }))

Second, suppose

X = {Xj)

(slmpllcial set) or a

where each

pro - (simplicial

Xj = { X j , k }

spectrum).

is itself

Definition

a pro -

(4.2.10)

implies

that

hollm {Xj, k} = holimj {hollm k {Xj,k}}.

Hence there is a spectral sequence w i t h

(4.9.4)

E~ 'q = limPjwq (hol~ k {Xj,k} )

( ~ llmPj {,q{X~,k>k},

w

see (5.6.1))

Homotopy collmits We shall review the history of homotopy collmits and give a construction dual

to our construction of homotopy limits in w

J. Milnor [Mil -3] introduced the homotopy collmlt of a direct tower in his work on axlomatlzlng additive homology theories on countable

x - {x0 - ~

be a direct tower.

CW

complexes.

-~x 2 -~--.)

Milnor introduced the (direct) telescope of

X

Let

170

Dir tel X = X 0 x [0,i] u X I x [1,2]

u X2 •

[2,3]

--',

shown below

0

I

There is a natural map a tower of cofibrations in

inj -Top,

see w

2

Dir tel X

(i.e.,

X

3

>colim X.

Milnor proved that if

X

is

is cofibrant in the category of direct towers or

then this map is a homotopy equivalence.

Bousfield and Kan used a dual construction

(replacing the infinite mapping

cylinder of a direct tower by the infinite path fibration construction of an (inverse). tower) in their geometric unstable Adams' spectral sequence They extended

this construction

to obtain their

(see [B -K]).

holim.

J. Boardman and R. Vogt's work on homotopy everything H - spaces required the homotopy collmit of a diagram; This construction

an explicit description is given in [Vogt -i].

is dual to the Bousfield -Kan description.

There is also an evident duality between the Bousfield - Kan spectral sequence and the Segal spectral sequence for the classifying

We obtain a homotopy colimit functor

hocolim:

space of a category

Ho(inj -C)

[Seg].

> Ho(C)

171

coadjoint to the natural inclusion

Ho(Top)

~ H o ( i n J -C)

model category ) by dualizing the construction in w is obtained by replacing with

X = X'

in

X

Ho(Inj -C)

by a coflbrant object

(C

a nice closed

Explicitly, X'

in

inj - C

hocolim

X

(see w

and defining

hocolim X ~ colim X'.

Because the colimit functor

collm:

inj - C ---~C

preserves coflbrations and

trivlal coflbratlons (Theorem (3.8.7)),

Ho(C) (hocollmX,Y)

for

X r inJ-C

conclude that for

and X

Y ~ C.

a Ho(inj -C)(X,Y)

Milnor's theorem on telescopes can be extended to

coflbrant in

inj - C 9

from the Vogt homotopy collmit to our hocolim.

ther~ is a natural weak equivalence

w

w

pro - ~.

Introduction.

In w w

THE ALGEBRAIC TOPOLOGY OF

we showed that if

C

is a nice model category then so is

we developed the theory of homotopy inverse limits for

pro -C.

section our main goal is to describe the algebraic topology of compare

Ho(pro -C)

In w

pro - C

In this and to

pro -Ho(C).

we prove comparison theorems which relate maps and isomorphisms in

Ho(tow -C)

w

with

pro -C.

to maps and isomorphisms in

tow -Ho(C).

contains some remarks about completions.

In w

w

we review the Artin-Mazur theory of

pro - Ho(SS,).

is concerned with Whitehead and stability theorems and counter-examples.

In w

we introduce strong homotopy and homology theories and prove a Brown

theorem for cohomology theories.

w

Ho(tow-C,)

Let

C,

tow-Ho(C,).

be a pointed nice simplicial closed model category which satisfies

Condition N of w functor.

versus

and let

~:Ho(tow -C,)

~tow -Ho(C,)

denote the natural

In

173

(5.2.1)

Comparlson Theorem.

There is a natural short exact sequence of

pointed sets:

(5.2.2)

0 ---~liml k colimj {Ho(C,)(E X~,Yk)}

~Ho (tow -c.) ({xj),{Yk}) 9 tow - Ho (C,) ({Xj }, {Yk ))

(5.2.3) Remarks. Ho(tow -C) ({Xj},{Yk})

~0.

It is easy to see that the natural map , tow -Ho(C) ({Xj },{yk} )

is always surJective; we need base

points only to make a more precise statement.

(5.2.4)

Remarks.

J. Grossman [Gros -2] obtained the above sequence in his

coarser homotopy theory of

tow- SS,.

Proof of Theorem (5.2.1).

By Axiom M2 for

C, N,

ment, we,.can replace any tower by an equivalent (in i.e., tower of flbrations of flbrant objects. is fibrmnt.

Define

Y-I = * ;

or an easy inductive arguHo(tow -C,))

fibrant tower,

We may therefore assume that

then the map

Y0---~Y 1

is a flbration.

Consider the following function spaces (all of which take values in w

HOM (Xj,Yk) , J ~ O,

(5.2.5)

k ~-i;

HOM ((Xj},Yk) -- colimj (HOM (Xj,Yk)},

HOM ({Xj},(YkD m i ~

{Yk }

k >__0;

(~OM ({Xj),Yk)},

SS,;

see

174

The fibrations

Yk--->Yk_l

the simplicial structure on HOM ({Xj},Yk)

induce fibrations C,,

HOM (X.3,Yk)

> HOM (Xj,Yk_I)

by

and also fibrations

9 '>HOM ({Xj},Yk_l)

by the simplicial structure on

pro-C,

or a

direct argument (the colimit of a sequence of Kan fibrations is a Kan fibration by the "small-object argument" [Q-i, w

Because (5.2.5) expresses

H0M ({Xj},{Yk})

as the limit of a tower of

fibrations, the Bousfield-Kan spectral sequence for a tower ((4.9.3)) yields an exact sequence of pointed sets

(5.2.6)

0

> limlk{~l(HOM ({Xj},Yk))}

~ ~0(HOM ({5},{Yk}))

---> I ~ { ~0 (HOM ({Xj },Yk)) }

>0.

By the above simplicial structures, ~I(HOM ({Xj},Yk)) = [S I, HOM ({5},Yk)] -~ ~0(HOM (SI, HOM ({Xj},Yk)))

-= ~0(HOM ((Xj} ^ SI,Yk))

-_-~0(HOM ({Xj

~ SI},Yk))

a ~0(HOM ({ ZXj},yk) )

-= Ho(pro -C,) ({ 7.Xj},yk)

colimj [ Z Xj,Yk ]

175

([-,-]

denotes

Ho(C,)(-,-);

extension property).

the last isomorphism follows by the homotopy

Also,

9o(~OM ({Xo }, {Yk }) = Ho (tow - C,) ({Xj }, {Yk}),

and

.o(HOM ({Xj},Yk)) = colim j {[Xj,Yk]} ,

as above.

Because the above isomorphisms are natural (w

tow -Ho(C,) ({Xj},{Yk}) -: limk{colim j { [Xj ,Yk] } }, (5.2.6).

(5.2.7)

and

the conclusion follows from

D

Remarks.

In the above proof, one can replace the Bousfield - Kan "lim I argument"

spectral sequence by a direct argument dual to Milnor's

[Mil -3,

Lemma i]; see, for example [B- K, p. 254].

Theorem (5.2.1) suggests the following.

(5.2.8)

Question.

tow -Ho(C), ~f,

Let

f

is invertible.

be a map in Is

f

Ho(tow -C)

invertible in

whose image in Ho(tow -C)?

This question appears quite difficult, and its analogue in proper homotopy theory (see w

has attracted recent interest (Chapmann and Siebenmann [C -S]).

At present we can only offer a partial answer (Theorem (5.2.9)). (5.2.9) in proper homotopy theory [E -H -3], (see w [C = S ] .

The analogue of

answers another question in

176

(5.2.9) image

~f

Theorem. in

tow -Ho(C)

g:{Xj} --->{Yk }

Proof. {Xj}

and

both

~f

Let

in

f:(Xj}

) (Yk }

is Invertlble.

Ho(tow-C)

with

be a map in

Ho(tow -C)

whose

Then there is an isomorphism

~f ffi~g

in

tow-Ho(C).

As in the proof of Theorem (5.2.1), we may assume that both {Yk }

are fibrant.

and its inverse in

diagram over

By relndexing if necessary, we may then realize tow "Ho(C)

in the following homotopy-eommutatlve

C.

Xk+l

) Yk+l

(5.2.10)

> Yk

1 Let

Z

I

be the tower

hl X0<

1

fl Y1 <

fk

~

hk+l

Yk+1 '

fk+l

~+i"

hk+l

""~ 9

177

Form the homotopy c o m m u t a t l v e d i a g r a m

(5.z.iD X j_ '''~-

'~

bond

(

Xk+ 1

id

hk+l Zi " ' ' < -

~.-,,

fk+l Yk+l ~

hk4-2 Xk+l . . . .

Yk+2 <

" ""

Yk+2 ~

"" "

bo t-~l Yk+l ~

Diagram ( 5 . 2 . 1 1 ) Id'~

--~

and

factors

~f

i d : Y k ---~Yk

in

tow-Ho(C).

By i n d u c t i v e l y

a b o v e , we may o b t a i n a s t r l c t l y

d e f o r m i n g t h e maps commutative diagram

178

(5.2.12)

hood ~+I

i

l=

i

7. "'" 9

f +l

X~<

~-

Yk+l ~

~+i

Yk+l <

posite map

"""

u

"""

bond

y ....

such that diagrams

Yk+2 <

<

(5.2.11) and (5.2.12) are equivalent over

g:X--->Y --->Z

in diagram

(5.2.12)

Ho(C).

is invertlble

in

because it is the composite of two levelwlse weak equivalences, ~g

=

~f,

as required.

(5.2.13) isomorphic

in

isomorphism

Proof. in

Corollary.

in

If either

tow-He(C)

X

or

to an object of

Y

and satisfies

is stable in Ho(C)),

then

tow -Ho(C) f

(i.e.,

above is an

Ho(tow -C).

say isomorphic

of the theorem shows that to objects

X'

and Y'

Consider the composite map

(5.2.14)

Ho(tow - C)

0

A first application

Ho(tow -C),

The com-

X'

--" X

f

Y

-= Y' .

X of C,

and Y

are stable

respectively.

179

For any

Z

in C,

Ho(C)(Z,X') = tow -Ho(C)(Z,X') tow -Ho(C)(Z,Y') Ho(C)(Z,Y').

Thus the composite (5.2.14) is an isomorphism in Ho(tow -C).

f

Remarks.

If

Ho(C,)

is an isomorphism in

We uowmake

(5.2.16)

Ri~Idification.

{Xj}

is abellan, t h e Comparison Theorem implies

Ho(tow -C,).

the following observation.

image of an object in replace

hence in

0

(5.2.15) that

Ho(C),

Each object in

Ho(tow -C).

tow -Ho(C)

To see this, given

by a tower of flbrant objects

{X'j},

is equivalent to the {Xj}

in tow -Ho(C),

and choose representatives

for the bonding maps of the latter tower.

(5.2.17) tow -Ho(C)

Proof.

Corollary. and

The isomorphism classification is the same in

Ho(tow-C).

Use (5.2.9) and (5.2.16).

D

The above results give a usable relationship between the weak and strong homotopy theories of towers,

See w

especially w

proper homotopy theory of o -compact spaces, and w the shape theory of compact metric spaces.

where towers are used in the for a similar application to

180

The Comparison Theorem (5.2.1) is replaced by

Off towers, little is known.

the following extension of the Bousfield -Kan spectral sequence (see w

(5.2.16)

For

Theorem.

{Xj}

and

{Yk}

in pro -C,,

there is a spectral

sequence with

E2p,q = limP k {colimj { [ z q x j , Y k]}},

which is closely related to

Proof.

We may assu|

Ho(pro -C,)({Xj},{Yk} ).

that

{Yk}

is flbrant.

{J}

Because

is filterlng,

collmj {[Z qS'Yk] } " Ho(pro-C,)({ Eqxj},Yk)

(each map on the left may be represented by a map By the slmplieial closed model

structure

on

Z qxj ---~Yk

pro - C,,

Ho(pro-C,)({T. qxj}.Yk) ~ ITqCHOM({Xj}.Yk)). and

Ho(pro -C,)({Xj}.{Yk} ) ~ ~o(HOM ({5}.{yk})). where

HOM

is the "function space" of w

HOM ( ( 5 } , { Y k } )

Finally,

= i ~ k {HOM ({Xj},Yk)}

(essentially by "enriched adJunctlon," w

= h o l ~ k HOM ({Xj},Yk) (because

holim ~ lim

flbrant objects).

on

f o r some

j).

181

Applying the Bousfield -Kan spectral sequence to desired result.

HOM ({Xj},Yk)

gives the

D

Unfortunately, we cannot conclude that ~:Ho(pro -C,)({Xj},[Yk} )

9 pro -Ho(C,)({Xj},(Yk})

The ri~idification question - which objects of Ho(pro -C) -

pro -Ho(C)

is unanswered and appears quite hard.

question about slmplicial objects over

Ho(SS)

is onto.

"come from"

Alex Heller asked a similar

and over

SS.

isomorphism classification question (compare (5.2.9) - (5.2.15))

Also, the is unanswered

off towers.

Thus, our present knowledge suffices for the proper homotopy theory of o -compact spaces (see w

and the shape theory of compact metric spaces (see w

but not for more general spaces.

Similarly, there are good comparison theorems relating the strong and weak homotopy theories of direct towers. Ho(inj -C)

w

and

inj -Ho(C)

Also, similarly, the relation between

remains obscure.

Remarks on Completions. In 1965, Artin and Mazur [A -M, Chapter 3] introduced the profinite completion e pro -Ho(CWo)

of an object

theorems in ~tale homotopy theory.

X ~ Ho(CWo)

in order to prove comparison

The inverse system

X

category which has as objects based homotopy classes of maps X

is indexed by the X--->X

where each

has finite homotopy groups, and has as morphisms homotopy commutative triangles

182 X

/'\ Then the association

(X--+X)

---->X=

yields an inverse system

Sullivan's

X.

G/PL

work on the Adams conjecture and the homotopy type of spaces such as to study the functor ing the funetors

lim~{[-,X ]}

[-,~],

[Sul -i],

[Sul - 2].

led him

By suitably topologiz-

Sullivan showed that the functor

lim {[-,X ]}

satisfies the Mayer - Vietoris (exactness) axiom as well as the wedge axiom, and hence

lim { [-,X ]}

is representable by Brown's Theorem [Bro].

concentrated on the complex

X

which represented

Sullivan then

llm [-,X ],

[-,~] ~ Lim {[-,~ ]);

(5.3.1)

being a simpler and more familiar object than

In 1972, Bousfield and

{X }.

Kan [B - K], motivated by their work on the Adams spectral sequence, defined for every commutative ring R X.

They obtain

{R X}. s

R

R X

and pointed simplicial set

X

a functorial R - completio%

as the simplieial inverse limit of a tower of fibrations

In this situation it is n_oolonger true that the funetors

lim s {[-,RsX ] }

are naturally equivalent.

of pointed sets [B-K]

(5.3.2)

[-,RX]

and

Instead, one has a short exact sequence

(see w

0 --+ iimls{ [ E W,RsX]} --* [W,R=X] --->liras{ [W,RsX]} --*0.

We shall briefly compare (5.3.1) and (5.3.2), modulo rigidification problems (see the end of w

In (5.3.2), we always have

183

[W,R X] e Ho(pro-SS,)(W,{RsX)) this sense,

because

R X = holims{RsX}

term of (5.3.2).

If

In

because

[-,R X] # lims{[-,RsX]}

and this difference is measured by the

Ho(pro -SS,)(-,{RsX}) # lims{[-,RsX]} , llm I

(see w

R = Z,

and

X

is a simply-connected (or even nil-

potent [B -K, Chapter 3]) finite complex, then [B- K]

{Z X}

is cofinal in

S

{Xa}.

Also, for

lim 1 {[ ~ W,ZsX]} S

(5.3.3)

W

finite, the groups

vanishes.

{[ E W,ZsX]}

are finite, so

This s u g g e s t s the f o l l o w i n g .

Proposition.

{X } be a tower of pointed, connected

Let

(SS or

n

CW)

complexes.

If

is representable and

limn{[-,Xn]}

limln{[ E -,Xn]}

vanishes, then

[ - , h o l i m {Xn }] i limn { [ - ' X n ] }

on all pointed complexes.

Proof.

Let

Q

represent

limn{[-,Xn]}.

[-,Q]

~ lira n { [ - , x n]}

\

(5.3.4)

Consider the diagram

x \

/ \ \

\ [ - , h o l i m {Xn}]

Evaluation of the top row on

holim {X }

yields the filler which makes (5.3.4)

n

commute.

Vanishing of

~i(holim {Xn})

limln{[ E -,Xn]}

> limn{~i(Xn)} ,

i ~ 1,

implies that the group homomorphisms are isomorphisms (see w

also

184

(5.3.2)). w i ( h o lml is

Then diagram (5.3.4) yields isomorphisms {Xn} ) e wi(Q),

connected,

holim {X } n

the Whitehead Theorem.

w

i ~ i. and

Because vanishing of Q

lim I

implies

holim {Xn}

have the same weak (singular) homotopy type by

The conclusion follows.

D

Some basic functors. Artin and Mazur [A -M.

of an object in

w167

pro -Ho(SS0) ,

introduced the homology and homotopy pro - groups as well as Postnikov decompositions,

Theorem, and a type of Whitehead Theorem.

the Hurewicz

In this section we shall review the

above results, except for the Whitehead Theorem.

The Whitehead Theorem will be

discussed in w

Recall that any covariant functor pro -T:pro -C

", pro -D.

pro -homology functors on

T:C--->D

We may therefore define the pro - homotopy and pro -Ho(SS,)

(5.4.1)

pro

A

is an abelian group.

homology theory. on

pro -SS,

structure. on

by the formulas

-.i({xj)) ~ {.i(xj)},

pro-Hi({Xj};A)

where

may be prolonged to a functor

m {Hi(Xj;A)}

A similar formula holds for any generalized

These functors induce pro - homotopy and pro - homology functors which satisfy the usual properties with respect to the closed model

(Note that fibre sequences and related constructions are no___~tfunctorlal

Ho(SS,),

pro -Ho($S,).)

hence it is difficult to describe homotopy and homology theories on Artin and Mazur even define homology with twisted coefficients; we

185

shall not need these formulas in our work.

Because cohomology is contravariant, the analogue of formula (5.4.1) for cohomology takes values in the category groups.

Because

colim: inj -AG

inj - A G

> AG

of direct systems of abelian

is exact, Artin and Mazur define the

cohomology groups by

fii({xj};A) ~ colim.3 {Hi(Xj ;A)}"

(5.4.2)

The category

K0

of pointed, connected Kan complexes (= fibrant simpllcial

sets) admits functorlal Postnikov -type resolutions (see, e.g., [May -i], or [A -M;w

We shall describe these resolutions and the induced Postnikov - type

resolutions on

pro-Ho(SS0).

Let

An

denote the p-skeleton of the standard P

simplieial n - simplex

An.

By analogy with the formula

set of n -simpllees of a simplicial set

X,

for

X

in K 0

X

= Ss(An,x)

n

let

for the

cosk X P

be

the simpllcial set whose n - simplices are given by

(coskpX)n = SS(Anp,X),

together with face and degeneracy maps induced by the coface and codegeneracy maps d i :An-I

) An P

and

is obtained from to maps from

si:A n+l

P

>A n P

X

Anp

compatible maps are not fibrations.

for

0 ~ p ~ n.

by adjoining additional n -cells for to X. coskpX

The inclusions >

Roughly,

P

coskp_iX

and

A np --->Anp+l X

> coskpX.

n > p and

cosk p X P

corresponding Anp --->An

Caution:

induce these maps

We may define the coskeleta of an arbitrary simplicial set

X

186

by the formulas

cosk Ex~X P

weakly equivalent to

X).

(recall that For

X

Ex X let

in K 0,

is a Kan complex naturally X (p)

and X(p)

be the

homotopy -theoretic fibres of the maps below:

X (p)

> coskpX,

>X

(5.4.3) X(p)

Because

SS 0

admits canonical factorizations of maps as trivial cofibrations fol-

lowed by fibrations ( [ Q - I , in

X.

> cosk X. P

~ coskp+iX

Further,

w

see w

cosk X P

MacLane space of type

the sequences (5.4.3) are functorial

is (p -i) - connected, and

K(~p(X),p).

X.

In fact, the above constructions are functorial on

(i)

(ii)

X

Ho(SS0);

the

pth

co-

is characterized by the properties:

~i (coskpX) = 0

for

The canonical map

i ~ p;

X---> cosk X P

to maps into objects

Similarly, the fibre X (p) --->X ---> cosk X P properties.

is an Eilenberg -

We therefore regard the sequences (5.4.3) as

the canonical Postnikov resolution of

skeleton of

X(p)

X "p"

Y

is

with

is universal with respect ~.(Y) = 0 i

(p - i)

is trivial, and the map

for

i ~ p.

connected, the composition X (p) --+X

is universal for these

187

Following Artin and Mazur, we define the POstnikov system of an inverse system X = {Xj}

in either

p r o - SS 0

or

(5.4.4)

to be the inverse system

X # = {coskpXj}

indexed by

{(p,j)}.

pro -SS 0

pro -Ho(SS0)

and

Clearly

pro -Ho(SS0)

to

@

extends to functors from pro -Ho(SS0).

(5.4.5)

If

{Xn}

pro - SS 0

is a tower, then

X ~ a {cosknXn} ,

so we may restrict (5.4.4) to a functor from towers to towers. f:X-->Y

to

in

pro -Ho(SS0)

(respectively,

Ho(pro -SS0))

A map is called a

-isomorphism if it induces an isomorphism on Postnikov systems

f#:X#--~Y ~ .

By using the above machinery and a spectral sequence argument, Artln and Mazur proved the following.

(5.4.6) i < n,

Hurewicz Theorem for

where

n

is an integer

pro -Ho(SS0). > i.

Let

pro -~i(X) = 0

for

Then the canonical map

pro -Wn(X)

) pro -Hn(X)

is an isomorphism of pro -groups.

w

Whitehead and Stability Theorems. In this section let

SSAG,

Sp

Then the Whitehead Theorem (5.5.1) holds in the category

C0

objects in

C.

C

be any of

SS,

SSG,

(simplieial spectra). of pointed, connected

188

(5.5.1)

Whitehead TheQrem in

induces isomorphisms

f,:~i(X)

Ho(Co). > wi(Y )

A map for

f:X--->Y

i ~ 1

in

Ho(C 0)

which

is an isomorphism in

Ho(C0). A natural question is whether (5.5.1) can be extended to Ho(pro -C O )

if the homotopy groups

homotopy pro -groups

{~i(Xj)}

when is an object of

pro -Ho(C0)

~o(C0))

~i(X)

(X E CO)

({Xj} E pro - CO). or

pro -Ho(C 0)

and

are replaced by the The stability problem (i.e.,

Ho(pro -C O )

isomorphic to an object of

will also be studied using homotopy pro - groups.

The following example

shows that additional hypotheses are needed for a Whitehead Theorem in pro -homotopy.

(5.5.2) all

Example.

i ~ i,

because

hut

Let

S~

for infinitely many

there is an essential map

f:X--+Y for in

i ~i.

pro -Ho(C0) Then

pro -Ho(C0).

isomorphism in

f

i,

(see e.g.,

pro- ~i(S ) = 0 pro -Ho(SS0).

for In fact,

[Spa, Corollary 9.7.6]),

S ~ --->S 3 .

Whitehead ~heorem in

in

Then

is not equivalent to a point in

w.(S 3) # 0 i

(5.5.3)

S~ s {V i > n Si }n> 0

pro-Ho(C0).

induces isomorphisms induces an isomorphism

Suppose that a map f,:pro -~i(X) f~:X+--->Y #

> pro -~i(g) of Postnikov systems

Under either of the following additional conditions, pro - HO(Co):

f

is an

189

(a)

supj,k{dim (Xj), dim (Yk)} < ~;

(b)

For each

J,

dim ( 5 ) < |

dim (Yk) < ~,

Proof.

For

C = SS,

Corollary (4.4).].

f

k,

is movable.

the first part is due to Artln and Mazur [A-M,

Their proof uses a spectral sequence argument and easily

ex~ends to the other

Similarly, for

and

for each

C.

C = SS,

the second part is due to the first author and

Ceoghegan [E -G-1,211 see also [A-M~ Theorem (12.5)], [Mos -i], [Mar -i], [Mor ~I]. that

f

(a), let such

chaL

By relndexing as in w

We shall sketch the preef. is a level map

we may assume

indexed by a directed set.

{fj:Xj --->Y.}j

n ffisupj{dim (5), dim (Yj)}.

Consider a fixed

j.

For case Choose

Jl

the diagram

>~)(Yjl )

~l(Xil )

/ / / / / / / / /

1

> ~l(Yj)

~l(Xi ) admits ~ filler.

Then tile map

Convert

fj

and f. Jl

' X~l ' ----,=l(Y~,~) Wl(YJI'

into cofibrations

f'. J

is

0.

and

f

'

Jk

~en the composlte~p

.

190

y~ i ~---->y '. Jl 31

(Y~i IJ

X~ Jl

denotes the i - s k e l e t o n

into X~ 2"

of

Y~ ) i

similarly for all

k

-~y,. 3

may be "deformed" relative tO

with

k > Jl" =

Now choose

" J3 ~ J2'" 9"'in = > 3n-i

so that similar results hold for

conclude that the map

y! -->Y~ 3n 3

(because

dim (Yjn) ~ n ) .

a homotopy inverse to

(5.5.4)

Lemma.

f

~2,~3,

can be deformed relative to

This argument is due to Marde~id

"'"'~n" X~ 3n

Let

be inverse systems in a category

X = {Xl,pll,,A}

C

We

into X~

[Mar -i] and yields

in case (a) by using the following l ~ a

[Mor-l].

J2 > Jl" =

and

over the same directed set

of K. Morlta.

Y = {Yl,qll,,A}

A ,

and let

f% {X 1

> YI}

be a level morphism in

pro -C

iff for any

exists

~l :Y~

~X l

I 9 A

pro - C .

there is some

for which

~I~ f

Then

p 9 A

= Plp

and

f

is an isomorphism in

such that

fl~l~ = ql~;

filler exists in the following solid arrow commutative diagram

f X

P '.J

lP>'l~r

XA

",Y /

/ :~'I I.~/... / /

I~

// Iqx1~ ~ Y1

i ! p

and there

i.e., a

191

Case (a) follows.

In case (b), consider a fixed ~ k,

j.

There exists a

k ~ j

such that for all

homotopy fillers exist in the diagram

f

ond

\bond

Xj

~

Now use the above argument with map

Yk--+Yj

into

An object

X E pro -C

to an object of

C.

which imply that

X

X 9 pro -C O If

y. 3

n = dim ( Y k , ~ )

relative to

~.

As above, the conclusion follows.

is said to be stable if it is isomorphic in

is stable.

If

X e pro-Ho(C0)

and let

an isomorphism if

pro - C

h:holim (X) --->X

is stable for all X

X

is stable, then so are its

pro -~i(X).

Th__eeStability Th.eorem i__n_n pro - HO(Co)

pro -~i(X)

to obtain a deformation of the

The stability ~roblem is the problem of giving criteria on

homotopy pro - groups

(5.5.5)

X.3

bond

[E - G - i].

be the canonical map in

i ~ I,

then

h~

Let pro - HO(Co).

is an isomorphism,

satisfies either of the following conditions:

h

is

192

(a)

supj{dlm (Xj)} < ~;

(b)

X

is dominated in

We sketch the proof. for w

s > 0 h

(w

pro -Ho(C0)

Because

pro

by an object of

-

~i(X)

is stable,

limS{~i(Xj) } = 0

By the Bousfield - K a n spectral sequence (see

Theorem C).

induces isomorphisms

~i(holim (X)) ~ pro -=I (holim (X))

Therefore

Ho(C0).

h#

pro -~i(X).

is an isomorphism by Theorem (5.5.3).

To show that

h

is an isomorphism in case (a), the first author and Geoghegan

applied Wall's finlte-dlmensionallty criterion [Wall] to the homology and cohomology groups of

holim (X)

(which are isomorphic to the homology pro - groups and

cohomology groups of the finite-dimenslonal system phism by [A -M]).

Th~s shows that

dlmensional slmpliclal set.

In case (b), let Y

in C O ,

and

homotopy idempotent

because

h~

is an isomor-

has the homotopy type of a finite-

The conclusion then follows by Theorem (5.5.3).

u:X-->Y

du = i x.

hollm (X)

X

and

d:Y--->X

be the domination maps with

One then applies Brown's Theorem [Bro] to split the

ud:Y-->y

((ud) 2 = u(du)d = ud)

through

Z.

One easily

checks that

X-" { Y <

Y<

Y<

"''} = Z.

Then the ordinary Whitehead Theorem implies that the composite map is an isomorphism.

The conclusion follows.

holim X - - + Z

193

(5.5.5.a)

Remarks.

assumed to be in

The argument given in part (b) shows that if

pro -Ho(C0) ,

it still follows that

X

is stable.

X

is only Dydak has

recently shown that the same conclusion holds in part (a).

So far, we have only been able to prove the following strong tower versions of Theorems (5.5.3) and (5.5.5).

(5.5.6)

The Whitehead Theorem i__n_n Ho(tow -Co).

induces isomorphisms

f~:pro -~i(X)

induces an isomorphism

f#:X#--->Y #

phism in

if

Ho(tow -Co)

f

~ pro -~i(Y) in

Suppose

f:X-->Y

in tow - C O

for all

i ~ i.

Then

Ho(tow -Co).

f

f

is itself an isomor-

satisfies either of the following additional condi-

[ions:

a)

sup {dim (Xj),

b)

f

dim (Yk)} < =;

is movable.

We sketch the proof. Grossman [Gros -2].

For

C = SS

For other

C,

the first part of this theorem is due to

the proof is similar and omitted.

of the second part follows from w

The proof

and appropriate filtered Whitehead theorems,

which are proved in an identical manner to the proper Whitehead Theorem occurring in [Br].

(5.5.7)

The Stability Theorem in

h:holim (X) --->X

Ho(tow-C0).

be the canonical map in

stable for all

i ~ i,

then

isomorphism in

Ho(tow -Co)

h# if

Let

Ho(tow - CO).

is an isomorphism in X

X ~ tow-C O If

and let

pro - ~i(X)

Ho(tow -CO).

h

is is an

satisfies either of the following conditions:

194

a)

sup (dim (X.)} < ~; 3

b)

X

is dominated in

Proof.

pro -Ho(C 0)

by an object of

The first part follows from the first part of (5.5.6).

second part, Theorem (5.5.5) implies that an object Q

Q

of Ho(C0).

in Ho(tow -Co).

an isomorphism in

(5.5.8) if

h

Ho(tow - CO).

Remarks.

canonical map

h:holim (X) --->X

X e pro -Ho(C)

is dominated by

in pro -Ho(C)

tow -Ho(C)

by P.

Y E tow -C dominated by

with

holim,

X

to

is isomorphic to

now implies that

h

is

by an object in

and hence shows that if by an object of

is an isomorphism in P E Ho(C)

Y

Ho(C),

Ho(pro - C).

then the If

(i.e., we are given m~ps

in pro -Ho(C)),

...}.

Ho(SS0),

In fact, Porter's argument depends

Ho(pro -C)

du = i X

y = {p
then

X

is isomorphic in

is easily seen to be dominated in

Recent work of Chapman and Ferry [C -F] seems to imply that if

is dominated in P

Ho(pro -SS0)

Ho(pro -SS0).

is dominated in

to

tow-Ho(C0)

D

is dominated in

is an isomorphism in

pro -Ho(C)

holim

For the

Porter [Por -2] has given a simple argument which shows that

X ~ Ho(pro -C)

d ~X u

is isomorphic in

Theorem (5.2.9) then implies that

only on the functorial properties of

P <

X

The properties of the functor

X E Ho(pro -SS0)

then

Ho(C0).

tow -Ho(C)

in Ho(tow -C),

will also be stable in

by a

P ~ Ho(C),

then

Y

and hence stable by Porter's argument.

pro -Ho(C).

Thus, an object of

if and only if it is dominated by a stable object.

pro -Ho(C)

is also Thus, is stable

X

195

A map

f

f:P --+ P

is said to be a homqtopy idempotent

is said to split through

Q

if there exists maps

P <

if d>

f2 = f

Q

in Ho(C).

such that

u

du = IQ

splits.

and

ud = f

in Ho(C).

The above shows that every homotopy idempotent

This may be used to show that every homotopy idempotent on an s

is homotopic

to a strict idempotent and every homotopy Idempotent

Q -manifold

M

is homotopic

to a strict idempotent

f

-manifold

on a compact

if and only if a certain Wall

+ obstruction

W(f) ~ KO(~I(M))

The development Theorems

vanishes.

of coherent prohomotopy

(5.5.6) and (5.5.7)

to

theory should enable one to extend

Ho(pro -Co)

(see [Pot -3]).

Below we present a number of examples which show the precision of the above results.

(5.5.9)

Ex___ample.

Example

(5.5.2) showed the need of some extra condition,

such as conditions a) or b) of (5.5.3), phism.

in order that a # -isomorphlsm

In this example we show that it is insufficient

be movable.

to require

be an isomor ~ X

and Y

Such an example was first constructed by Draper and Keesling in

[D -El.

Let

S

n

=k/

i>n

Si

and

i :S ---> n n Sn-i

be the natural inclusion.

+These observations were obtained in a conversation with T. Chapman.

Let

to

196

Xn = H k < n S k

= Yn"

Let

Pn:Xn~Xn_l

,

bn:Yn ~ Y n _ l

,

and

fn n:X ~ Y n

be

defined by the following diagram

1 x ... x i x 0. S ] x --- x Sn_ I x S . n

S I x --- x Sn_ I x Sn

i ... S1

Then

X = {Xn,Pn}

, . .

and

Ix --

Sn_l

Y = {Yn,bn}

# -isomorphism (easy), but

f

(5.5.10)

~SI • - -

are movable and

S~--->X

x Sn_ I

f = {fn }

is not an isomorphism in

there is an obvious essential map S~--->xf-~Y

, ...

.-- x O

is a

pro - Ho(CW0);

in fact,

such that the composition

is inessential.

Examples.

We will show in the examples below that the phenomena

exhibited by examples (5.5.2) and (5.5.9) can he realized by inverse systems of finite complexes. J. F. Adams.

These are much deeper examples, though all the depth comes from

Recall Adams' essential map of Moore spaces

finite complexes of the form

Sk o

D k+l)

q

Y

(these are pointed

[Adams - 2]

A: [ 2ry -->y ;

this map is detected by the isomorphism

* ~ A :K(Y)

where

K

denotes reduced K - theory.

=

'K(I

2ry)

# 0,

Hence, the composite maps

197

A

are all of

A

essential.

the inverse system

Thus

is not equivalent to a point in

for all S~

1 2 r m - 2r A : 1 2 r m

.....

i > i.

y

.> [2rny

Z ~ { I 2rny}

pro -Ho(CWo)

bonded by suspensions

even though

Applying the construction of Example (5.5.11) to

pro- ~i(Z) = 0 Z

in place of

yields a map between movable towers of finite complexes which is a

# -isomorphism but not an isomorphism in

pro -Ho(CW0).

The following example provides counter-examples

to many conjectures

(see

[E -H -4]).

Let n > i, n- 1

be the inverse system with

and with bonding maps S 2r

Let

Fn,

{Bn }

Bn

.--->

B0

n 2r B n = Hi=IS

a point, and

for

given by projection onto the last

Bn_ 1

factors.

E n = Y x Bn,

the fibre of

fn:En--->En_l

yet no composite

Pn'

and let

is

Y.

Pn:En = y x B n

> En-k

be the projection;

thus

We shall define "twisted" bonding maps

so that the restrictions

En

~B n

fn Fn:Fn

factors through a

Form the commutative solid-arrow diagrams

~Fn-l

Bn_ I.

are null-homotopic,

198

Ef n

En

y x (S2r~ n

On > y x (s2r) n

(s2r) n

(E'

En-i

,

(s2r) n

>y x (s2r)n-1

....

>

[S2r) n-I

II

{t

(l

Bn

Bn

Bn_ I

p'

is the pullback, and n

is the projection).

Define the filler

n

requiring

P t non - Pn'

that

On

and letting the composite mapping

y x (s2r) n - [ ~ u

x (s2r) n

>y

be the composite mapping

y

here

Wn

•

(s2r) n

id•

n

y • s2r '

is ~he projection onto the first

Finally, let

bonding map

the

fn:En

> y A S2r

$2 r

> En-i

A

~ y;

factor.

be given by the composite

On En

{Fn

~ E'n

9 En-l"

This yields the tower

{En}

and

tower

of flbratlons

Pn > En------>Bn}.

Claim i.

fn[Fn:F n ~

The tower

Fn_ 1

F

is contractible; since the bonding maps

a r e g i v e n by the c o m p o s i t e s

by

189

F

= y

, y • , .. > y • S 2r -..>y A S 2r

A>y

,

n

{F }

is isomorphic in

Pro-Top

to * .

We may use the basepoints in the

{Sn:B

n

>En}.

Since

Fn( = Y)

{pn}{S n} ffi id _

{Bn}

,

to define a section

to show =hat

{pn }

is not inverti-

ble, it suffices to verify the following.

Claim 2.

arbitrarily

{Sn}{Pn} # id 9 {E n}

large

n

in Pro - Ho(Top).

and s u i t a b l e

m

Ass,-ne otherwise, then for

(depending upon

n,

but with

n-m ~0)

the diagram

E

=

y • (s2r)n

Pa

~

(S2r)~

sn

>y

•

(s2r) n

n

y x (s2r) n-m

Pn-m> (s2r)n-m

bond

/

/

E

n-m

would commute up to homotopy.

s

/

x (s2r) n-m

200 Consider the subdiagram

y x (s2r)n

bond > y x (s2r)n-m

l

Pn-m

(s2r) n-m

bond

> y x (~2r)n~m

y x is2r)n-m

Note that all of the above maps are products with

id(s2r)n_m.

fact, and projecting the lower right corner onto

Y,

Hence, by this

we obtain a homotopy commuta-

tive diagram

Y • (s2r) m =

>Y

I g

i =

Y

where

g

is induced from the bonding maps.

Y • (sZr) TM -

Since g*:K(Y)

K(YA

(s2r)m)

~ K(y x (s2r)m)

9 ~A

I

;y

By construction,

(s2r) m = X 2~5~

is a direct summand in

g

Am~ Y 9

K(y x (s2r)m),

is non-zero, hence Claim 2 holds.

Q

is the composite

201 Let

~, ~

(5.5.10a)

~ w. ~ [ ~ si,-]. i= I i i=l

Proposition.

pro - ~,{pn}:{~,(En)}

9 {~,(Bn)}

is a p r o -

isomorphism.

Proof.

This follows from chasing in the commutative solid arrow diagram

s ~,(F n)

) ~,(E n)

01

n*

~ -pn,_ ~ ~,(B n)

i

J Pn-l*>

~,(Fn_ I)

(5.5.10b)

> g,(En_ I)

Proposition.

~ ~*-i (Fn)

1 ~,(Bn_ I)

0 ) > ~,_l(Fn_l) 9

An infinite dimensional Whitehead Theorem fails in

shape theory.

Proof. metric

Let

spaces

E = lim En,

B = lim Bn;

then

p:E -->B

is a map of compact

which is an isomorphism on Cech pro- homotopy groups

(pro-~,)

but not a shape equivalence.

(5.5.10c)

Remarks.

p

(5.5.10d)

Proposition.

is even a C - E map, see J. L. Taylor [Tay- i].

An infinite dimensional Whitehead Theorem fails in

proper homotopy theory.

Proof.

Let

Tel E

be the infinite mapping cylinder (telescope)

Tel (,<---- E0~--- EI~--- ...);

define

Tel B

and

Tel p:

Tel E

>Tel B

202

similarly.

Tel p

Then

isomorphism at {pn}:{E n } ~

|

is an ordinary homotopy equivalence and a

(E(Tel p):r

E)

up to homotopy where

{Bn }

not a proper homotopy equivalence

(5.5.11)

Example

[E -H -&].

simplicial sets such that not know whether

X

> ~(Tel B) ~

pro - ~,

is given by

is the ends functor) but

Tel p

is

(at = ).

We will construct below an inverse system

pro -~ (X) = 0,

but

X

is not contractible.

X

of

We do

can be chosen to be an inverse system of finite complexes or

even the end of a locally finite complex.

Let

K(Z2,n )

denote the simplicial Eilenberg -MacLane space (see [May]).

The direct system

K(Z2,1 )

Sq I > K(Z2,2)

Sq 2 > K(Z2,4)

Sq 4 ) -..

: K(Z2,2n)

~ K(Z2,2n+k)

i

has the property that all composite maps

2n+k-i = Sq

are essential (evaluate the generator of

2n+l ..- Sq

~

on the class

HI(K(Z2,1);Z2) ;

but each bonding map kills

Form the inverse system

2n Sq

x

in

H 2n

(K(Z2,1);Z2)

where

x

is

see N. E. Steenrod and D. B. A. Epstein [S -E])

*

X

2n

shown below.

203

\\\ X 3 = K(Z2,1 ) x K(Z2,2 ) x K(Z2,4) x "'"

\ \\ "\ \ \

X2 = K ( Z 2 , . ) x K ( Z 2 , 2 ) x K ( Z 2 , 4 )

kxl = K(Z2,1) • K(Z2,2) •

Then

X

Example.

m {K(w ,i)}

pro -group

~ .

contractible)

If

~ = {~ }

An Eilenberg-MacLane

such that

X

to

K(Z 2,1)

(5.5.10).

If

K(~,I)

K(Z 2,1),

= ~ n

K(Z2,2m)

using Example

X

such that

for all

n, let

X

= F n

X n --+ Xn_ I

is

Below

(5.5.11) as the

the bonding maps then there does

pro - ~I(X) = ~

and

can come from the end of a locally

question in infinite dimensional

m=l

and define twisted bonding maps

pro - spaces!

and twisting

X

{Xn }

in pro -Ho(SSo)

is not finitely dominated,

Whether such an exotic

finite complex is an~important

(i.e.,

{K(~I(Xn),I)}

by (roughly)

not exist an inverse system of finite complexes

F

X = {X n}

There exist exotic Eilenberg-MacLane

fibre of a fibration over the standard

is contractible.

we shall call

pro - space with fundamental

pro - space

is not equivalent

we shall construct an exotic

as in Example

is a p r o - g r o u p ,

the standard Eilenherg-MacLane

will be called exotic.

Let

K(Z2,~) . . . .

has the required properties.

(5.5.12) K(~,I)

x -..

topology

x K(Z2,1)__ n

so that the diagrams

(see w

for all

n,

204

Xn

K(Z2,1 )

> Xn-i

. . . .

.

Xn

id ) K ( Z . 2 , 1 )

'

pt..__

-.>

Xn-I

>

K(Z2,2

)

and

Xn

,

Xn_ I

(5.5.13)

211~ ] K(Z2,2m )

x

y

K(Z2,1 )

x

x

> K(Z2,2m+I )

(m >= 1)

co~ute.

In diagram (5.5.13);

x

in HI(K(Z2,1);Z2)

and

y

in H 2 (K(Z2,2m ;Z21

2m are the generators, and the map 2m y x X

y x x

represents the cohomology class

H 2m+l

in

(K(Z2,2m) x K(Z2,1);Z21.

As in Example (5.5.10), we obtain an inverse system of fibratlons

hE,rid Xn

-

-

->

Xn-1.

> "'"

(5.5.14)

> K(Z2,1'~..

•

> KfZ2,1"~..

> - . .

205

(5.5.15)

Theorem

[E- H - 4].

exotic Eilenberg-Maclane

The tower

diagram (5.5.14) induce isomorphisms on Further, the bohding maps of

{X }

By construction ~i'

and

(5.5.13), the bonding maps of

{pn}

{Xn }

{Fn}

pro -space

so that

{F } n

n

If

o

that class

y r H*(Xn+k,Z 2)

in

n.

By diagrams {X } n

is an

Z2 .

K(Z2,n) .

then the map

..,

o

bond

--> X n

K(Z 2, l)

H*(Xn,Z2)

Pn

This would yield homotopy commutative diagrams

bond

cohomology

for all

Hence

pro - grouv

were isomorphic to the constant tower

and for suitable

Xn ~ Fn

are null-homotopic.

Xn+ k

n

the fibrations

are compatible.

with fundamental

would be an isomorphism.

for each

constructed above is an

pro - space.

We sketch the proof for comvleteness.

Eilenberg-MacLane

X = {Xn}

id

k

"~

depending upon

> H*(Xn+k,Z 2) maps to

~onstruction in diagram (5.5.13).

K(~ 2 ,1)

n.

Thus the induced map on

factors through 0

H*(K(Z 2,1),Z2),

in H*(Xn;Z2).

Hence,

X = {X n}

\so

This contradicts the

is exotic.

D

n

(5.5.16)

Ill

i ~ I.

Example.

Let

Hence, the map

Xn =

~ S i. i=l

h:holim [Xn}

Then

~ {X n}

pro-~.{X } i n

~s stable for

is a # - isomorphism.

206

Also,

X = {X } n

that if

X

is movable.

is movable and

But

X

is not stable.

pro - ~,(X)

Dydak [Dyd- i] has shown X

is stable.

We shall define strong homotopy and homology theories on

pro -SS,.

w

is stable, then

Strong homotopy and homology theories. At

present our main application is the development of generalized Steenrod homology theories on compact metric spaces (see w

(5.6.1)

{x.}

set)

3

Definition.

The strong homotopy groups of a pro - (pointed simplicial

are given by

~i{Xj} ~ Ho(pro-SS,)(S i, {Xj}) = Ho(SS,)(S i, holim {Xj})

~i(holim {X.}). 3

These satisfy the usual properties, in particular a fibration sequence in pro -SS,

yields a long-exact-sequence of homotopy groups.

(5.6.2) {Xj},

Strong stable homotopy groups.

For any

pro - (simplicial spectrum)

define

~'S{x'}z 3 -Ho(pro-Sp)(S i, {Xj}) -= Ho(Sp)(Si, hollm {Xj})

~iS(holim {Xj}).

(5.6.3)

Proposition.

{=.s} i

pro -Sp.

forms a generalized homology theory on

207

Proof.

Clearly the functors

7. 1

s

on pro - Sp

begin by verifying the exactness axiom.

are homotopy invarlant.

Because each cofibration

isomorphic to a levelwise cofibration

{Aj} --->{X.}3

A--->X

We is

(Proposition (3.3.36)) it

suffices to show that an inverse system of coflbration sequences {Aj --> X.3 --->Xj/Aj} > ~iS{xj }

~iS{Aj } fibrant

induces three-term exact sequences ' ~i s {Xj/Aj}.

pro - (slmplicial spectrum)

{X.s} --> {Y'3}

by a fibration

(levelwise) fibre

{Fj},

{Xj/Aj }

We may functorially replace {Yj}

(Xj/Aj

{X'j.} ~

{Y'J}

> Yj),

by a

replace the map

( % ---->X'j)

in pro - Sp

with

and form compatible commutative diagrams

A. 3

> X. J

~ x./A. 2 3

1 (5.6.4)

F_

~

X'

J

-

(The left weak equivalences arises because fibration sequences in "the same" in ~. s {A.} z

>

Ho(Sp);

Ho(Sp).)

~iS(xj }

>

~i s

3

is exact on both coflbration and

Therefore the sequences

~is(xj/Aj}

and

=.S{F.) z 3

>~. s {X , .} z 3

>~.S{y.) i 3

are

But the latter sequence is isomorphic to the sequence

~iS(holim {Fj }) -->~iS(holim

] }) ---->~iS(holim {Y. J }) '

{X'.

an inverse system of fibrations is a fibration sequence in holim ~ lim

Y.

loosely, cofibration and fibratlon sequences are

j

isomorphic.

>

on the fibrant objects

(Fj},

{X'j}

preserves fibration sequences (use Theorem (3.3.4)).

and

which is exact because Ho(pro - Sp), {Y.} 3

(w

and

lim

The exactness axiom follows.

208

We may iterate this process to obtain a long-exact sequence.

For the suspension axiom, consider a cofibration sequence of the form {xj}

~ {cxj}

>{~xj}.

By regarding this sequence as a fibration sequence ,

we obtain an exact sequence

0 = ~i+l s {CXj} ---->~i+l S{ EXj}

hence the suspension axiom holds.

(5.6.5)

~ > ~'S{x'}l 3

~ ~iS{c Xj} = O;

The conclusion follows.

D

Strong homotop~ groups and homotopy pro-groups.

Both unstably and

stably these are related by the Bousfield-Kan spectral sequence (w

(5.6.6)

Strong (ordinary) homology groups.

with identity.

*

X

in SS,,

(RX) n

is the hasepoint of

RX = {(RX)n,di,si} , cofibration sequence RA--->RX--->R(X/A),

R

is the free R -module with X.

be a commutative ring S~,(;R)

on pro -SS,.

a free simpliclal R - m o d u l e Xn

functor;

as basis, mod

R*

where

There results a simpllcial R " module,

which depends functorially on A--~X--->X/A the functor

in SS, ~,(R-)

We prolong the Bousfield-Kan functor

X.

R

maps a

is a (reduced) homology theory on

R

[D - T],

to pro - SS,

and define

SH,({Xj};R)

Because

into a fibration sequence

As with Dold and Thom's infinite symmetric product

R{Xj} = {RXj},

R

We shall develop a strong homology theory

Bousfield~ and Kan [B - K] associate with for

Let

= w,(R{Xj}~.

~,(R-) ~ fi, C-;R). by defining

SS,.

209

Observe that

R

takes an inverse system of cofibration sequences into an

inverse system of fibration sequences, which is a fibration sequence in Ho(pro -SS.)

by Proposition (3.4.17).

This yields the exactness axiom for

S~,( ;R)For the suspension axiom, first observe that natural map (induced by the identity of E

R)

yields maps

{EXj }

is the simplicial suspension (see e.g. [May]).

{Xj}

> R{Xj }

9> {ERXj } ---->{WRXj }

But

{WRX.}

3

where

is a

pro -(simplicial R -module) so we obtain a map

R{EXj } - {REXj } ---->{WRX.j}.

It is easy to check that this map is a level weak equivalence.

Hence

S}{,({Xj};R) _--~.{%}

,~+!{WREj 9

(use the flbration sequence {RX.}

> {WRX.}

3

> {laX.}

3

3

(3.4.17), and (5.6.1)).

Therefore,

SH,(-;R)

is a homology theory on

pro -SS,.

On

SS,,

Sfi.(-;R) ~ h.(-;R). (5.6.7)

Strong (generalized) homology groups.

reduced homology theory on

CW,,

Let

h,

which is represented by a

be a generalized CW

spectrum

E.

210

That is,

E = {Enln ~ 0},

together with cellular inclusions

E En---->En+l,

and

h,(X) = ~,S(x A E) E ~,S{x A En}

on

CW,.

pro- Sp

We prolong the smash product

- AE

to a functor from

pro- SS,

to

by defining

{X.}3 ^ E = {Sin ((RXj) ^ E)}, and set Sh,{Xj} = ~,S({xj} ^ E).

Because

- ^E

sequences over Sh,{Xj}

takes cofibration sequences over Ho(pro -Sp),

and similarly,

Ho(pro -SS,) - AE

into cofibration

preserves suspensions,

is a generalized reduced homology theory on

pro -SS,.

As above,

S~. ~ ~, on SS,. It is now easy to see that any map of spectra

E--+F

transformation of strong generalized homology theories E and F

represent

h,

and k,

and products associated with

(5.6.8)

Proposition.

h,

For

respectively. extend to

{X.} 3

s~.

in SS,,

induces a natural

S~, ____>S~,

where

Thus all homology operations (see [Adams -i, 3], also (2.2.65~.

there is a Bousfield-Kan spectral

sequence

E2p,q = limPj{hq(Xj)},

which converges under suitable conditions to

Sh,{Xj}.

If

{Xj}

is isomorphic

211

to a tower in

Ho(pro -SS,),

the spectral sequence collapses to the short exact

sequences

0

Proof.

~ limi{hn+l(Xj )}

Use (5.6.6),

' > StUn(X)

> lira {hn(Xj)}----->

O.

(5.6.7) and the Bousfield-Kan spectral sequence (4.9.~.

We conclude this section with several remarks about cohomology theories.

(5.6.9) theory on

Cohomology groups. SS,,

then

h

If

~* h

is a generalized,

induces a cohomology theory on

reduced cohomology p r o - SS,:

~, ~~ h {Xj} -- colimj{h (Xj)},

because

colim

is exact.

Further, if

En

represents

~n

on SS,,

then

~* h {Xj} = pro-Ho(SS,)({Xj},En)

-= Ho(pro -SS,) ({%},En).

The latter isomorphism exists because

(5.6.10)

Representable

E

theories.

is stable.

n

For

{Yk } E p r o - S S ,

there is an

associated generalized cohomology theory defined by

~n(_) = Ho(pro- SS,)(-,~-n{Yk}),

Conversely, Alex Heller [Hel -i, w on an

h- e

showed that any group-valued cohomology theory

category (an abstraction of

categories as abstractions of

SS,)

n ~ 0.

CW,

analogous to pointed closed model

is the colimit of a directed system of

212

representable theories. stable category (such as

Because Heller's proof only uses factorization through a pro -Sp),

cofibrations, his result also holds for

and properties of the homotopy relation and pro -SS,

and

pro- gp.

w

w

PROPER HOMOTOPY THEORY

Introduction. In this section we shall use pro -homotopy theory to study the proper homotopy

theory of locally compact, a -compact Hausdorff spaces via a functor (the end) into pro -Top.

This functor describes proper homotopy theory at

~

We obtain

proper homotopy theory by combining proper homotopy theory at

~

with ordinary

homotopy theory.

w

contains the basic definitions of proper homotopy theory and the end

functor.

In w

we shall prove that the end functor from locally compact, o -compact

Hausdorff spaces to

pro -Top

yields a full embedding of the proper category

at

We discuss proper Whitehead Theorems in w

In particular we sketch a proof

of L. Siebenmann's finite-dlmensional proper Whitehead Theorem [Sieb -i] and show that an infinlte-dimenslonal analogue (claimed by E. M. Brown [Br] and

F. T. Farrell, L. R. Taylor, and J. B. Wagoner [F - T -W]) fails in general.

We discuss the Chapman complement theorem and an analogous "strong" complement theorem in w

214

w

Proper homotopy and ends. We shall show how T. Chapman's

[Chap -I] formulation

leads to an embedding of the proper category at

(6.2.1) f:X--+Y

Proper homotopy

A c X

This is just a reformulation f,g:X-->Y

with

theory

into a closed model category.

theory following Chapman.

o f locally compact Hausdorff

there is a compactum

~

of proper homotopy

Call a continuous map

spaces proper if for each compactum

f(cl(X \A))

c cl(Y \ B)

B c Y

(cl denotes closure).

of the usual notion of proper map.

Proper maps

are called properZy homotopic if there is a proper homotopy

H:X x I

~Y

with

Hi0 = f

and

HJl = g.

Proper-homotopy-equivalences

are now

defined in the obvious way.

Chapman also introduced weak proper homotopy

theory for the complement

theorem.

See w

(6.2.2)

Definition.

Let

spaces and proper maps, and Let

Po

and

Ho(Pc)

P

be the category of locally compact Hausdorff

Ho(P)

be its associated proper homotopy category.

be the restrictions

We shall study proper homotopy

to o -compact

theory by associating

spaces.

to each proper map its

ordinary homotopy class and its proper homotopy class at homotopy

theory at

=

by introducing

the following

the sense of P. Gabriel and M. Zisman

(6.2.3)

The proper category at

=

We describe proper

category of right fractions in

[G-Z].

~

cofinal if the closure of the complement of

We shall call an inclusion A,

cI(X\A),

j:A r

X

is compact; we shall

215

also sometimes say that

A

is cofinal in

inclusions in the proper category

P.

X.

Let

-=P\Z

will be called the proper category at a~erm

at

We shall sometimes call a morphism in

of a proper map.

~

P \ Z

It is easy to prove that means that each morphism from

X

admits a calculus of right fractions. to Y

X<

where

A

diagrams f, = f,,

is cofinal in X 9

~ A'

X

f' > Y

and

f

and

class of cofinal inclusions

Z

in P \ Z

f

-~A

~ A"

X.

~

>Y,

f" ~ Y

Two such

represent the same morphism if The

clearly contains identity maps and is closed under

of proper maps.

X 9

9A

~ >Y

and

Y<----=B

Form the solid-arrow diagram

c /

x

P).

Composition is defined as follows.

Suppose that the diagrams

represent germs at

This

can be represented by a diagram

is a proper map (morphism in

X (

on a cofinal subspace of

composition.

be the class of all cofinal

The quotient category

P

P

E

2

\ \ flC

Y

Z

~ >Z

216

Let

C = f-l(B).

cofinal in

X.

Because

f

is proper,

C

is cofinal in

A,

hence also

The required composite morphism is represented by

X<

~C

g ~ flC" Z.

It is now easy to make the connection between proper homotopy theory at

and

pro- homotopy theory.

(6.2.4) of

X

Definition.

Let

X

be a locally compact, Hausdorff space.

The end

is the inverse system

e(X) = {cI(X\A) [ A

a compactum in

X} ,

bonded by inclusion.

It is now easy t o

check that a map

spaces is proper if and only if

f

f:X--->Y

of locally compact Hausdorff

induces a map

~(f) :c(X)

> ~(Y)

which makes

the diagram

c(x)

E(f)

, E(Y)

L co-,-ute. pair

(X,r

More generally, the following result holds, where

(l,c)(X)

is the

217

(6.2.5)

Proposition.

The end construction yields the following funcrors and

commutative diagrams:

P,=

r

~ pro - Top

Ho(P )

e

~ Ho(pro-Top),

p

(l,c)

Ho(P)

(1,e)

:'(Top, p r o - T o p )

> Ho(Top, pro -Top)

The proof follows i m e d i a t e l y from the definitions.

(6.2.6) (I,E):P

Proposition.

~ (Top, pro -Top)

The functors

E:P

9 pro - Top

and

are full embeddings.

Again, this is clear from the definitions.

In the next section we shall prove the following.

(6.2.7) (l,e):Ho(Po)

Proposition.

The restrictions

- ~Ho(Top, p r o - T o p )

e:Ho(Pa,~)

are full embeddings.

> Ho(pro -Top)

and

218

(6.2.8)

Corollary.

Let

f:X-->Y

a - compact Hausdorff spaces. if

f

at

=

Then

f

be a proper map of locally compact, is a proper homotopy equivalence if and only

is an ordinary homotopy equivalence and a proper homotopy equivalence D

(6.2.9) (a)

Remarks.

The above results summarize our approach to proper homotopy theory. We use pro -homotopy to study proper homotopy theory at then blend proper homotopy theory at

=

= ,

and

and ordinary homotopy theory

to obtain proper homotopy theory.

(b)

The proof of Proposition (6.2.7) relies heavily on the telescope of a tower described in w towers.

i.e., on the coherent homotopy theory of

A suitable theory of coherent pro- homotopy should yield

an extension of Proposition (6.2.7) and its corollary to all of

(6.2.10)

Definitions.

weakly properly homotopie at in

pro -Ho(Top).

wHo(P)

We shall call germs at =

of proper maps

if the induced maps

e(f), e(~)

f,g:X

P

~Y

are equivalent

The corresponding weak pKoper homotopy category at

is then obtained by identifying maps in

P.

~ ,

which are weakly properly

homotopic at

T. Chapman introduced the following weak proper homotopy category in order to prove the second Dart of the complement theorem [Chap -i], see w

Call proper

maps

B

f,g:X---~Y

weakly properly homotoplc if for each compactum

in Y

219

there is a compactum B)

with

H!0 = f,

A

in X

HI1 = g,

and a homotopy and

concern was with contractible

x I --->Y

H:X

(depending upon

H(cI(X \A) x I) c cl(Y \B).

spaces.

Chapman's main

In this case, a straightforward

application

of Urysohn's lemma yields the following.

(6.2.11) Hausdorff

Proposition.

spaces.

pact and

X

to Y

and

be contractible,

classes

category to

wHo(P).

For

first let the diagram

~A

the germ of a proper map from

X

cl(X \A)

g:B

locally compact,

wHo(P )(X,Y).

X <

cl(X \ A) c i n t function

and Y

There is clearly a functor from Chapman's

the converse,

represent

X

Then there is a bijection between weak-proper-homotopy

of proper maps from

Proof.

Let

B .

f

>Y

to Y.

is compact there is a eompactum Because

> [0,i]

tracting homotopy for

B

g(aB) = 0

Y~ H :Y • [0,i] f:X-->Y

,:f(• ~(•

B

is compact and Hausdorff,

with

We may define a proper map

Because

=

and >Y

g(cl(X\A)) with

X

in X

is locally comwith

hence normal, = i.

HIo = id

and

there is a

Choose a conHII = * ~ Y.

by setting

• ~ X \

~ ~ ( f ( x ) , g(x)), x ,: A ,, I

L*,

It is easy to check that

f = f

in

x ~ X \A

wHo(P )(E,Y)

and that this construction

220

yields a well-defined weak-proper-homotopy A similar a r g ~ e n t wHo(P~

class of proper maps from

X

to Y.

shows that germs of proper maps which are equivalent in

yield the same weak-proper-homotopy

class of maps.

The conclusion

follows.

In the next section we shall prove that

> pro -Ho(Top)

c:wHo(Pc, ~)

is a

full embedding.

w

Proper homotopy

theory of c - c o m p a c t

spaces.

We shall relate the proper homotopy theory of c -compact homotopy

theory of towers, and thus prove Proposition

(6.3.1) c -compact,

The end of a c -compact Hausdorff

space.

Let

space.

spaces to the strong

(6.2.7).

X

be a locally compact,

Suppose that co

X=

where X

K 0 = @,

= cl(X \Kn)

each

Kn

for each

is compact, n

U K n n=0 and each

K n = int(Kn+l).

Define

and let

n

~'(x) ={x 0 ~ x l ~ x 2 .... }.

Then

~' (X)

is a cofinal subtower of the end of

cofinal subtowers of loosely regard

g(X)

are canonically

e' as a functor from

P

X,

isomorphic

a(X). in

to tow -Top

Of course, any two

tow - Top. and call

E'(X)

We may thus the end

C

of

X

when there is no chance of confusion.

regarded as a funetor from

Pc

Similarly,

to (Top, tow -Top),

(I,E') may be loosely

or as a functor from

Pc

to

221

the category

Filt

of filterd spaces, see w

We shall associate to a space telescope

Tel (e(X)),

(6.3.2)

see w

X

in P

with end

E'(X)

(as in (6.3.1)) the

and projection

Px:Tel (E'(X))

> X,

Px(X,t) = x.

< Px

X

Then

Px

Tel (e(X))

is a filtered map

(X

is filtered by

e~(~)).

We shall need a suitable notion of naturality for c'(X) and

a"(X)

be two eofinal towers in the end of

~'(X)

e"(X)

are mutually cofinal,

and

Tel (e-V(X)) z Tel (E"(X)) is true.

Maps

in

Ho(Tel)

H0 = f0'

homotopy.

If

and

pHt = f

f0'

and

H

fl"

Somewhat more are called

px f ~= Pxfl = f:W--->X H = {Ht}:W x [0,i]

for all

t.

We call

are also filtered maps,

called f iltered-ver tically-homo topic.

Let

Because

~s in Proposition (3.7.13).

fo,fl:W ----+,Tel (E'(X)) with

HI = fl'

X.

PX"

there is natural equivalence

vertically homotopic if there is a homotopy with

Tel (~'(X)) and

f0

)Tel (E(X)) H

a vertical

and f l

It is easy to prove the following.

are

222

(6.3.3) a)

Le~mm.

Tel %'(X))

and

are canonically equivalent up to

Tel (c"(X))

filtered vertical homotopy. b)~ The map

PX

is natural in

X.

For part (a) use the proof of Proposition (3.7.13).

Proof.

Part (b) follows

immediately.

(6.3.4) s:X

Definition.

> Tel ~'(X))

(6.3.5) maps

with

hn

(~Kn) = n -i

map

h:X--->R +

for

there results a map

9 [n- 2, n -i] n => 2.

c [n-2, n-l] s:X

s'

Filt.

hn( Kn_ I) = n - 2

and

denotes the set of non-negative real numbers) such for

n ~i.

) Tel ~'(X)),

Because

(Xn_l\Xn)

c [Kn\Kn_lJ ,

given by the formula

s(x) = (x, h(x)).

is a proper section for e'(X).

for E'(X)

(6.~.7)

with

Urysohn's Lemma yields

We may glue these maps together to obtain a proper R+

(6.3.6)

s

is a filtered map

ps = id X.

(notation:

h(Kn\Kn_l)

Clearly

e'(X)

Construction of proper sections [E - H - 5].

hn:Cl(Kn\ Kn_ I)

that

A proper section for

In fact, each proper section

comes from a suitable proper map

Proposition.

X

h:X--->R +

and formula (6.3.6).

is a strong deformation retract of

Tel ~'(X))

in

223

Proof.

The required retraction and inclusion are given by

Px:Tel (e'(X)) topy from

;X

and any proper section

idTel (e~(X))

to

SPx

s

for

The required homo-

~'(X).

is given by

H(x,t,t') = (x, ( i - t ' ) t + t'' h(x)),

where

s(x) = (x, h(x)).

The arrows in the figure 'below represent

H :

I j s(X)

For each

n

choose

m > n

so that

h(Xm) c [n,~).

Then

H(Tel (E~(X))m x [0,i]) c Tel (E'(X))n,

sQ that

H

(6.3.8)

is a filtered homotopy, as required.

Proof of Proposition (6.2.7).

(l,e):Ho(P a)

> Ho(T0p , pro -Top)

corresponding assertion about Let

X,Y E Pa"

is a full embedding.

e'(X)

H

is even vertical. D

We shall show that the functor

c:Ho(Pa,~)

Choose 'tends"

Note that

The verification of the

> Ho(pro -Top) and

~'(Y)

in

is easier and omitted.

tow -Top

By Proposition (6.3.7), proper homotopy classes of proper maps from

as in (6.3.1). X

to Y

are

224

in bijeetive correspondence with filtered-homotopy classes of filtered maps from Tel (e'(X))

to

Tel (E'(Y)).

By Proposition (3.7.19), which states that fil-

tered maps of telescopes yield a geometric model of

Ho(Top, tow -Top),

the

latter class of maps is in bljectlve correspondence with

Ho(Top, tow-Top)(e'(X),e'(Y))

see (6.3.1).

(6.3.9)

! Ho(Top, pro-Top)((X,

The conclusion follows.

Proposition.

E(X)),(Y, E(Y))),

D

The functor

e:wHo(P) ----*pro - Ho(Top)

is a full

embedding.

Proof. sider

e

is an embedding by construction.

e(X), ~(Y) e pro -Top.

e(X), e(Y)

as in (6.3.1).

Choose "ends"

e

e'(X), e'(Y)

Then

= tow-Ho(Top)(e'(X),e'(Y)),

Ho(pro -Top)(e(X),e(Y))

= Ho(tow -Top)(e'(X),e'(Y)).

Ho(tow -Top)

(5.2.3)), we may realize any map in map in

Ho(tow -Top).

above.

Q

Proposition.

is full, con-

cofinal in

pro-Ho(Top)(e(X),e(Y))

Because the functor

(6.3.10)

To show that

~ tow -Ho(Top) tow-Ho(Top)

is surjective on maps (see from

e'(X)

to e'(Y)

by a

The conclusion follows from Proposition (6.2.7) proved

The natural functor

a bijection on isomorphism classes of objects.

HO(Po, ~)

> wHo(Po,~)

induces

225

Proof.

Use the above propositions

pro-homotopy

(Corollary

(5.2.17)).

and the corresponding

D

The above result is close to a "Whitehead-type" theory;

w

Theorem in proper homotopy

see w

Whitehead - type TheOrems. L. Siebenmann

[Sieb] gave various convenient criteria for a proper map of

finite-dimensional,

one-ended

(see (6.4.1))

be a proper homotopy equivalence. "homotopy groups at

~ ."

three positive results:

(6.4.1)

Basepoints.

criterion implicitly

we showed that Siebenmann's

restriction

Siebenmann's;

involving weak equivalences

locally finite simplicial

Siebenmann's

In w

without a flnite-dimensional

(see (5.5.10d)).

follows.

Let

so basepoints embedding.

X for

be a non-compact

e(X,m) =

pro -Top,.

Call

X

to

(cs

Note that

one-ended

space in

are irrelevant).

We associate

to

involves

We shall discuss here and a result

at

We shall need basepoints

c(X)

complexes

criterion fails

a result involving movability;

in order to obtain the p r o -

homotopy groups of the end of locally compact space.

in

result for towers in

X

and m

P Let

(if

We introduce "basepoints" X

were compact,

m:[0,=)

~X

~(X) = ~ ,

be a proper

the inverse system

u ~[0,~),m(0))[A

E(X,m) ~ e(X)

in

a compactum in

as

X}

Ho(pro - Top).

if there is a unique proper homotopy class of proper maps

226 [0,=) --~X

(6.4.2)

in

Ho(P),

(equivalently, in

Homotopy "groups" at

=

Ho(P )).

These are the pro - groups

pro -wi(e(X,~)),

i = 1,2,''',

and

pro - ~,(e(X,~)),

where

~,(-) - H

~i (-) - [V i=l

si,-]; i=l

see w

In (5.5.10d) we constructed a proper map

p

of infinlte-dimenslonal,

one-ended

countable, locally finite simplicial complexes which was an ordinary homotopy equivalence and induced an isomorphism on

p r o - ~,,

but was not a proper homo-

topy equivalence.

However, by introducing suitable dimension restrictions or movability assumptions, one obtains the following positive results.

(6.4.5)

Theorem.

Let

f:X--->Y

be a proper map of one-ended, connected,

countable, locally finite simpllcial complexes which is an ordinary homotopy equivalence and induces isomorphisms

pro - ~,e(X)

~ pro- ~,e(Y).

Then

f

a proper homotopy equivalence if either of the following additional conditions holds. a)

[Sieb]

b)

f

dim X < =

is movable.

and

dim Y < | ;

is

227

Proof.

By Corollary

is invertible e(X)

and

in

(6.2.8),

Ho(pro -Top).

e(Y)

it suffices Because

X

admit cofinal subtowers

Theorem (5.5.6), with basepoints

and Y

E'(X)

Q -manifolds,

~(f)

are countable,

E'(Y).

and

defined as above.

We now describe a useful substitute work on compactifying

to verify that the induced map

Now use

D

for a true Whitehead

Chapman and Siebenmann

Theorem.

In their

[C - S] asked the follow-

ing questions. i)

Is every weak-proper-homotopy

equivalence

a proper homotopy

equivalence? 2)

Is every weak-proper-homotopy homotopic

equivalence weakly-properly-

to a proper homotopy equivalence?

Chapman and Siebenmann confirmed a special case of (2), namely the case of Q-manifolds

with tame ends.

We obtained

the general case in [ E - H -3].

It is

much easier to obtain the following result, which in fact suffices for the applications in [ C - S ] .

(6.4.6) invertible

Theorem. in

Let

f:X--->Y

pro -Ho(Top)

a proper homotopy equivalence and

g = f

Proof. such that

in

and

f

be a map in

P

is invertible

in

g:X--->Y

such that

o

such that

e(f)

Ho(Top).

is

Then there is

e(g) = e(f)

in

pro -Ho(Top)

Ho(Top).

By Theorem (5.2.9) g' = e(f)

Use Theorem (6.3.8)

in

there is a map

pro -Ho(Top)

to realize

g'

and

as a map

g':E(X) g'

> E(Y)

is invertible

g" : (X \ K0)

~Y

in in

p r o - Top Ho(pro-Top).

for some

228

compact,-,

K 0 cX:

Ho(Top).

Let

--->[0,I]

E(g") = g'.

H:(X \Ko) x [0,i]

choose a compactum h: 5

i.e.,

with

K1 c X

with

9Y

For suitable

be a homotopy from

K 0 cint K 1

h(K O) = 0

and

r

E(x),

K0,

g" = fI(X\K0) f

to g".

in

Then

and Urysohn function

h(bd KI) = I.

The required map

g

is given

by

for

x e K0,

g(x) = ~H(x,h(•

for

xr

],, k g C:4)9

for

x ( X \ K I.

The required properties are easily verified.

KI,Ko,

D

The following chart smmnarizes our use of pro- homotopy theory.

Proper homotopy

Pro- homotopy

theory at

Strong

Weak

theory

Ho(P )

Ho(pro-Top)

wHo(P|

pro- Ho(Top)

We shall see further connections in the next section.

w

The Chapman Complement Theorem. In the late 1960's Borsuk sparked an avalanche of interest in the study of the

global homotopy properties of compacta (see [Mar -3], surveys of shape theory).

[Ed -i] and [E -H -2] for

Borsuk's original formulation of the shape theory of

229

compact subsets of Hilbert space [Bor- 2] lacks the flexibility of the approach to be described in w

but it has the advantage of being more geometric.

This

added geometry was quickly capitalized upon by Chapman in [Chap -i].

Let

s = H

(-l/n, l/n)

be the psuedo-interior of the Hilbert cube

n=l Q = ~

[-l/n, l/n].

One defines the fundamental category or shape category,

n=l Sh,

as follows.

The objects of

are compact subsets of

a sequence of maps

V

of Y

in Q

n,

V.

Note that

near

Y.

and

then a fundamental Sequence

fn:Q--->Q

n' ~ n o

fn(X)

U

fn, IU

of X

in Q

fnlU

and

the restrictions

f,

f':X ~ Y

s

and Y

is defined as

and an integer

are homotopic in

V.

V

no

no

are homotople in

Y ; it only has to be

are considered homotoplc,

of Y

in Q

such that for

The morphisms in

homotopy equivalence classes of fundamental sequences.

contained in

X

and an integer

f~IU

does not have to be contained in

in Q

If

f:X--+Y

of X

provided that for every neighborhood

U

s.

with the property that for every neighborhood

Two fundamental sequences

neighborhood

fnIU

are compact subsets of

there exists a neighborhood

such that for

f = f',

s,

Sh

Sh

there exists a

n ~ no,

are now taken to be

Two compacta

X

and Y

are said to have the same shape if they are isomorphic in

In [Chap -i] Chapman proved the following beautiful theorem.

Sh.

230

(6.5.1) X

and Y

Chapman Complement TheqKem.

If

X

and Y

are compacta in

have the same shape if and only if their complements

Q \X

s,

and

then

Q \Y

are homeomorphic.

Chapman then extended the association

X ~ - > Q iX

to a functor

shape category to the weak proper homotopy category of complements in pacta in

T

from the Q

of com-

s.

(6.5.2)

Definition.

Let

PQ,~

Q

of

compacta in

s

and germs at

Because

Q

is contractible, Chapman's weak proper homotopy category above is

isomorphic to

wHO(PQ ~)

|

be the category of complements in

of proper maps.

(use Proposition (6.2.11)).

Chapman then proved the following categorical version of the Complement Theorem, stated in our language.

(6.5.3)

Theorem.

Outline of proof. U = {U.} J tively, in

and Q.

V = {Vk}

There is a category isomorphism

Let

K

and L

be compacta in

T:Sh

s.

be bases of open neighborhoods for

Chapman showed that

K

and L

inclusions

{Uj \ K} ~ ~{Uj} and

{Vk\ L} ~ { V k}

> wHO(PQ,~).

Let K

and L,

respee-

are Z -sets, hence the natural

231

are levelwise homotopy equivalences. Sh(X,Y) a pro -Ho(Top)(U,V).

It is easy to show that

We then obtain a string of isomorphisms

Sh(X,Y) ~ pro-Ho(Top)(U,V)

a pro -Ho(Top)({Uj \ K},{V k \ e})

wHO(PQ,~)(Q\ K, (the complements of

(6.5.4)

V. 3

and U k

are compact in

Th___~estrong shape category.

easy to check that the complement countable basis of open sets

Uj

Q \K with

Let

s

in

Q;

K

Q).

The conclusion follows.

be a compactum in

s.

is ~ - compact, and hence that cI(Uj)

therefore define the strong shape category in

Q \ L)

c5_

s - Sh

1

K

(j = 0,i,2,-'').

has a We may

to be the category of compacta

and coherent homotopy classes of maps of their associated neighborhood bases namely

(6.5.4)

s - Sh(K,L) E Ho(Tel)(Con Tel {U.}, Con Tel {Vk}) 3 m Ho(Tel)(Con Tel {cl(Uj)}, Con Tel {cl(Vk)})

where

It is

Con Tel

see w

(6.5.5)

is the contractible telescope

By Proposition (3.7.20),

s-Sh(K,L)

~ Ho(pro-Top)({Uj},{Vk}).

232 By following the

proof of Theorem (6.5.3), we obtain the following strong

categorical version of the Complement Theorem.

(6.5.6)

Theorem.

There is a commutative diagram of categories and functors

s-

Sh(K,L)

Sh(K,L)

(6.5.7)

Remarks.

9 HO(PQ ~)

i

> wHO(PQ|

E

Wlth reference to the orlginal Chapman Complement Theorem

(6.5.1), recall that the functor

HO(PQ~)

> wHO(PQ|

isomorphism classes of objects (Proposition (6.3.10)).

induces a bljection on

w

w

GROUP ACTIONS ON INFINITE DIMENSIONAL MANIFOLDS.

Introduction. We shall discuss the classification

s =

H (-i/n, I/n) n=l

In w

and on

Q =

of actions of compact Lie groups on

~ [-l/n, l/n]. n=l

we shall review the theory of s - m a n i f o l d s

Standard group actions will be constructed

and Q -manifolds.

in w

We shall also show that all principal actions on

s

following Jim West

[West -i].

are standard up to equivariant

homeomorphism.

w

contains a classification

actions of a finite group on interesting,

w

theorem

Q.

Whether such actions are unique is an open,

and deep question.

Basic theory of s - m a n i f o l d s An s -manifold

homeomorphic

(7.2.1) Anderson's

(largely due to West) for semifree

(respectively,

to an open set in

s -manifolds.

and Q - manifolds.

Q - manifold) s

is a separable metric space locally

(respectively,

Q).

Work of Kadec, Bessaga,

and Pelczynski

proof that all separable Frechet spaces are homeomorphlc

The classification

theory of s -manifolds

is due to Henderson

culminated

in

(see [A -B]).

[Hend].

First,

This section represents joint work with Jim West and much of the material taken from his unpublished notes [West-i].

is

234

every s- manifold simplicial

M

complex

can be trlan$ulated, K

such that

homotopy equivalence

M

f:M 1 --->M2

that is, there is a locally finite

is homeomorphic

to

K x s.

between s- manifolds

Second, every

is homotopic

to a homeo-

morphism.

Torunczyk X

is an

[Tow -1,2] has shown that

ANR

(7.2.2)

~ -manifolds.

[Kel] showed that

[West -2,3].

Q

West

Q

complexes

K.

Chapman

K x Q

If a map

to a homeomorphism

f:K--->L

thus every compact

ANR

to

K x Q

ANR's;

f x IdQ:K • Q

type of a locally finite simplicial

complex.

is

is

[Chap- 3]

torsion.

ANR

has finite homotopy

every such

~L x Q

The converse also holds

of Whitehead

in particular,

K

for some locally

is the CE -image of a type.

Chapman

that West's result can be used to extend = - simple homotopy

locally compact

if

of locally finite simplicial

then

[West - 2].

invariance

[Chap -1 - 7] and

is a Q - m a n i f o l d

i.e., homeomorphic

[West - 3] showed that every locally compact

Q -manifold;

(see w

[Chap - 4] proved the converse:

is an ~ -simple homotopy equivalence,

and implies the topological

observed

complex.

manifold is trian~ulable,

properly homotoplc

West

is due to Chapman

[West -2] showed that

finite simplicial complex

if

is homogeneous.

theory of Q - manifolds

a locally finite simplicial every

if and only if

Many of the above results hold for Q - manifolds

is replaced by ~ -simple homotopy equivalence

The classification West

is an s - manifold

(separable, metric).

homotopy equivalence Keller

X x s

ANR

[Chap - 7]

theory to

has the ~ -simple homotopy

Hence, a proper map

f:MI--->M 2

235

between Q - m a n i f o l d s

is properly homotopic

to a homeomorphism

if and only if

f

is

-simple.

Bob Edwards only if)

X

[Edw] has recently shown that

is a locally compact metric

X • Q

(7.2.3)

[West -i]

treatment of

theory after Siebenmann. ''~

The basic idea of (finite)

simple homotopy

theory

[Cohen],

single-out and study m a p s of finite cell complexes which are homotopic compositions

of maps which are of the form of an inclusion

.n-i on K = L u e u e

with

e

n-i

an n - c e l l

locally finite and n o t necessarily i

above by an inclusion

of disjoint complexes (K~ n L)

> Ki

Ki

S(K)

each of which collapses

S(K).

In his treatment the group

is an equivalence

with domain

K,

complexes, K \L

to

Kin

where

this notion to

one replaces

is the union L,

the u

K. i

i.e., such as

i,

and

to proper mappings and proper homotopies.

Siebenmann introduces of

in which

to finite

or are of the

is the result of a finite sequence of inclusions

one restricts oneself

(7.2.4)

j:L --~K

n e ,

In generalizing

finite-dimensional

is to

i:L--->K,

which is a face of

form of a homotopy inverse to such an inclusion.

map

if (and

ANR.

We shall conclude this section by giving West's "infinite simple homotopy

is a Q - m a n i f o l d

where

class

g:K--->M

S(K)

[Sieh] of infinite simple homotopy of simple structures on

[f:K-->L]

Each element

of proper homotopy equivalences

is in the class of

This survey is a quote from [West -i].

K.

theory,

f

whenever

there is a

236

simple homotopy equivalence tion on

S(K)

s:L ---~M

such that

[f] and [g]

tops (domains) of mapping cylinders, and then

K.)

(The group opera-

need not concern us here, but it is (essentially) geometrically

defined in [Sieh] by representing

inclusion of

g = sf.

K

with inclusions, say, into the [f] [g]

is represented by the

into the result of identifying the two spaces along the copies of

In particular, if there is only one simple structure on

K,

then all proper

homotopy equivalences are simple.

(7.2.5)

Wh,

head functor

Wh

K0,

and limits.

K.

is that these are functors.) K

S(K),

Siebenmann uses the White-

and the projective class group functor

constructions at the end of

end of

To examine

K0

in several limiting

(In essence, all that is needed for this paper The limiting constructiens are as follows.

Let the

be

e(K) = {K = W 0 ~ W I m W 2 ~ --'},

(see (6.2.14)), where the

W

are subcomplexes of

K

whose complements have

n

pact closure.

Choose a proper base ray

e(K,~) = {(Wn u m[O,=),m(O))}.

a:[0, =) -

~K

as in w

Now consider the inverse system

Tie(K) m ~iS(Kl,m) ~ {nl(Wn, m(0))}.

Now define

K 0 ~IE(K) E limn{K 0 ~l(Wn, ~(0)), K0(i,)},

Wh ~le(K) 5 limn{Wh ~l(Wn, ~(0)), Wh(i,)},

Let

com-

237

and the attenuation

Wh ~le'(K) E limln{Wh ~l(Wn, m(0)), Wh(i,)}.

Observe that the attenuation is zero if

~le(K)

is stable, i.e., p r o -

isomorphic to a group.

Exact sequences. S(k).

Siebenmann gives two exact sequences to aid in computing

They are as follows:

(7.2.6)

0

(7.2.7)

(Sb(K)

~Sb(K)

Wh TIe(K )

> S(K)

~ W h ~I(K)

> K 0 ~le(K)

~Sb(K )

~ K 0 ~I(K);

~ W h ~I~'(K)

> 0.

is the group of equivalence classes of proper homotopy equivalences defined

analogously to

S(K)

but where one allows the inclusion

K~ n L

>K.

to be any

inclusion of finite complexes which is a homotopy equivalence.)

w

The Standard Actions. Following West [West -i], we shall construct standard principal actions of any

compact Lie group

G

on Q0

that all principal actions of

left translation.

(Q

G

~ C(G) i=l

on s

are standard.

Let

G

s.

We shall show

act on itself by

This principal action extends to a semi-principal action with

unique fixed point on the cone of

action on

with a point deleted) and on

G,

C(G) E G x [0,i] / G x {0}.

The product

is also semi-principal with unique fixed point, the infinite

238

cone point.

Q.

aG

H C(G) iffil

is homeomorphic to

Removing the unique fixed point yields the standard principal action

of G

Q0 • s

PG

But, it follows from [West -2] that

on Q0"

Since

Q0

is homeomorphlc to

of G

on s.

is contractible, and

s.

Q0 x s

is an s -manifold,

Thus, we also obtain a standard action

Any principal action of

G

on Q0

or s

which is not con-

jugate to the standard action will be called exotic.

Let

p

and p'

be two principal actions of

the actions are nice if the quotient spaces (e.g., if s/~

G

are both classifying spaces for

map such that the induced bundle

s/p

on s.

(s,p)

and s/p'

(see [Hus]).

f (s,p') Since

f

are both S -manifolds,

G.

over

We will say that

and s/p'

is finite, then this is always the case).

and s/p'

bundle

s/p

G

are s -manifol~s

In any ease, Let

s/p

f:s/p

~s/p'

be a

is isomorphic to the

is a homotopy equivalence, if

then

f

is homotopic to a homeomorphlsm

g 9

One thus obtains the diagram of principal G -bundle isomorphisms

S

)

S

~

S

(7.3.1)

slp

in which hence

h=~or p

and O'

> s/p

$

"-slp'

is a G -equivariant homeomorphism from are equivalent actions of

G

on s.

(s,p)

to (s, p');

Summarizing, we have the

239

following theorems.

(7.3.2)

Theorem.

are standard.

(7.3.3)

The

G

on s

0

Theorem.

standard.

All nice principal actions of a compact Lie group

All free actions of a finite group

G

on s

are

0

Qo

case is much more subtle~ since one must show that

q0/p

and

Q0/p '

have the same = - simple homotopy type, and not just the same homotopy type, before one can conclude that they are homeomorphic.

The main result of this section is

the following theorem, which will be proved in w

(7.3.4) G

Theorem.

on Q0"

Let

p and p'

be free actions of a finite group

Then the following statements are equivalent:

i)

p

is equivalent to

2)

Q0/p

is homeomorphic

3)

Qo/p

is ~ -simple homotopy equivalent to

4)

Q0/p

is proper homotopy equivalent to

5)

The end of c(Q0/p')

(7.3.5)

in

Remarks.

to West [West -i]. c(Q0/p)

Qo/p,

p'; to

Q0/p';

E(Q0/p)

Qo/p'; Q0/p';

is homotopy equivalent to

pro -Ho(Top).

The equivalence of (1) - (4) for The end

~(Q0/p)

G

is a quotient of

is a pro -space analog of the classifying space

showed that the natural inclusion

~(Q0/p)

~ Q0/p

a finite group is due ~(Q0 ) = pt.;

hence,

BG = Q0/p.

West

is always a~ - isomorphism;

240

even an isomorphism on if and only if exotic

p

is standard.

K(Z2,1)'s;

actions on

Q0"

pro -~,.

By Theorem 7.3.4, it is a homotopy equivalence In w

but we still do not know of any exotic compact Lie group On the other hand, work of Tucker [Tuc -2] shows that there are

uncountably many different actions of

w

we showed the existence of uncountably many

Z

on QO"

Proof of Theorem (7.3.4). The following preliminary lend,as, as well as the equivalence of statements (i)-

(4) in (7.3.4) are taken from [West -i].

Our machinery (pro - spaces) is used to

simplify some of the statements and arguments.

Let point

G

be a fixed finite group acting semifreely on

q.

Q

with unique fixed

Let

a:C • Q

(g,x) ~ a~ gx

9 Q,

denote the action.

(7.4.1) tractible (in

Proof.

Lepta.

Q \ {q}

is contractible, and its end

g(Q \ {q})

is con-

Ho(pro -Top)).

Represent

Q

as the product

[0,i] i.

Because

Q

is homo-

i=l geneous (see 7.2.2), we may assume that vex, hence contractible.

q = (0,0,0, ....).

Q\{q}

is then con-

Also,

~(q\{q}) ~ {ui ~

(O,l/i]i ~

n

(o,1]jliil},

j >i

bonded by inclusion.

Because each

U. l

is convex, hence contractible,

r

\{q})

241

is contractible in (5.2.17).

pro -H0(Top),

hence in

Ho(pro -Top)

0

(7.4.2)

Lemma.

There is a commutative diagram of covering maps in

G

. . . .

(7.4.3)

I

e(q \ {q}) - -

~ Q \{q}

1

1

e((Q \{q})/~)

Proof.

' (Q\ {q})/u.

First, choose a representative

U 0 = Q\ {q}

and

pro - Top

G

I

Let

by Corollary

{Ui[i ~ i}

tower for

E(Q \{q})

be as in (7.4.1).

For each

as follows.

n gU i

i,

gcG (the intersection of translates of

Ui

under

u)

contracts

in

u. \ {q}.

Let

l

(7.4.4)

~ ( Q l { q } ) _-- {vo = v I = v 2 . . . .

be a subsequenee of

that each

V.

{

0 gEG

gUi}

chosen so that

Vi

}

contracts in

is invariant under

1

Next, rewrite diagram (7.4.3) as

-

(7.4.5)

l

c(Q \ {q})

l

-

G

i

Q \ {q}

~(p)

e((Q \ {q})/e)

> (Q \ { q } ) / a,

Vi_ I.

Note

242

where

p

,

is the covering map induced by

and

c(p)

is the levelwise covering

map

E(Q\ {q}) = {V.} l

yielding the conclusion.

(7.4.6)

D

Corollary.

pro -Wl ffiG;

{V.la} ~ e((Q\ {q})l=), I

E((Q\ {q})/=)

is an Eilenberg-MacLane pro - s p a c e with {(Vi/~ ) }

i.e., the tower of universal covers

is contractible in

Ho(pro -Top).

Proof.

It remains only to observe that

easy exercise involving covering spaces.

(7.4.8)

Proof of Theorem (7.3.4).

(i) ~ ( 2 )

{(Vi/~ ) } ~- ~ ( Q \ { q } ) ;

this is an

0

The following implications are easy:

by covering space theory;

(2) ~=~ (3) by Chapman and West's classification of Q - manifolds (see (7.2.2)); (3) ~

(4) by definition; and

(4) = ~ (5) by definition.

To verify (4) --~ (3), we shall show that for any semi-free action finite group

G

on Q

with unique fixed point

any proper homotopy equivalence with domain late

(Q\{q})/=

as

K • Q

Because the projection map

(7.4.9)

q,

S((Q \ {q})/a) = 0,

(Q \ {q})/a

is | - simple.

for some locally finite simplicial complex (Q \ {q})/a

>K

of a so that TrianguK.

is a proper homotopy equivalence,

~le(K) ~ ~i K ~ G,

243

via the inclusion. (7.2;6).

Henc~

Also,

K 0 ~IE(K) = K 0 =I(K),

Wh TIC(K) a Wh ~I(K)

(7.2.5)), so that

Sb(K) = 0

To verify (5) ~

and

by (7.2.7).

so that

S(K) a Sb(K)

Wh ~la'(K) = 0 Hence

by

(by (7.4.9), see

S(K) = 0,

as required.

(4), first consider the diagram

holim e(K)

hollm K

1

(~)

induced by (7.4.5) (the homotopy inverse limit, holim, is developed in w167 and w

where

unless

i = i,

K

is as above.

in which case

spectral sequence to

hollm E(K)

By Corollary (7.4.6), pro -~l(C(K))

= G.

t0 holim e(K)

= 0

Applying the Bousfleld-Kan

yields

holim e(K) = ~ G ,

(note that

pro-~i(E(K))

is pointed and connected).

i = i,

i~l

Hence the map

holim e(K) --->K

is a homotopy equivalence by the ordinary Whitehead theorem

(K

by Lem~as (7.4.1) and (7.4.2)).

e(K)

is a in

K(G,I)

Ho(pro - Top)

by the diagram

Therefore

K

is a retract of

244

holim s(K)

) holim K \

".~

"

e(K)

p

Now, for two semi-free actions points

q

and q',

assume

\

_% "N \

> K

and p'

(Q \ {q})/p

1

\

of G

and

on Q

with unique fixed

(Q \ {q'})/p'

equivalent at

Then

E((Q \{q})/p) ~ E((Q\{q'})/p')

by Theorem (6.3.4).

This equivalence extends to the diagram

IK ~ '

E((Q \ {q})10) r

(7.4.12)

Remarks.

~

The implications

must verify that the maps covered by equivariant maps

(Q\ {q'})/p'

'> ( Q \ [ q ' } ) / o '

(Q \ {q))/0

(globally), as required.

for arbitrary compact Lie groups

and

> (Q \ {q})Ip

Theorem (6.3.3) now implies that

are proper homotopy equivalent

Ho(pro-Top)

ll l l

e((o. ,, { q ' ) ) / ~ ' )

by (7.4.10).

in

-,%

l

(7.4.11)

are proper homotopy

G

(2) <==~ (3) ~=~ (4) ~=~ (5)

with a similar proof.

(Q\{q})/p Q \ {q}

(I) ~

"- (Q\{q'})/p' > Q \ {q' }.

D

For

(2) ~

(i)

of statement (2) are

hold we

w

w

STEENROD HOMOTOPY THEORY

Introduction. This chapter is concerned with shape theory, shape functors, and Steenrod

homology theories.

In w

we discuss the Steenrod homology theory [St -i], the Kaminker-Schochet

[K- S] axioms for generalized Steenrod homology theories, and the Vietoris construction [Por -i]. homotopy theory. Sh,

and h,,

plexes.

We then use the Vietoris construction to discuss Steenrod In particular, we define canonical Steenrod and

of any generalized homology theory,

Proofs of the properties of

Sh,

h,,

6eeh

extensions,

defined on finite

CW

com-

occupy the next four sections.

D. S. Kahn, J. Kaminker, and C. Sehochet have independently obtained Steenrod extensions by different methods.

In w

we verify useful properties of the Vietoris functor.

In w

we prove that

metric spaces,

Sh,

that is, that

is a homology theory on the category Sh,

CM

is a homotopy invariant functor which satisfies

the first two Kaminker-Schochet axioms for a Steenrod homology theory. show that products and operations associated with

We give two spectral sequences converging to spectral sequence to prove that axiom.

of compact

Sh,

h,

Sh,

extend to

in w

We also

Sh, .

and use the first

satisfies the remaining Kaminker-Schoehet

246

In w

we prove functional and Steenrod duality theorems for

duality states that for a compactum

X

in Sn,

Sh,.

Steenrod

Sh (X) = hn-p-l(s n \X). P

Func-

tlonal duality is due to D. S. Kahn, J. Kaminker, and C. Schochet [K -K -S].

w

Steenrod homology theories. Let

h,

complexes.

be a generalized homology theory defined on the category of finite In this section we shall define a canonical Steenrod extension

(see (8.2.3)) of w167

-8.5.

h,

to the category

CM

of compact metric spaces.

CW

Sh,

Proofs occupy

See (8.2.1) and (8.2.2) for ordinary Steenrod homology and Steenrod

K - homology.

The problem of constructing generalized Steenrod homology theories with "good properties" was posed by M. F. Atiyah and others at the Operator Theory and Topology Conference held at the University of Georgia in April, 1975.

By "good properties"

Atiyah meant that products and homology operations extended from

h,

to Sh,.

Our extension enjoys these properties.

(8.2.1)

Duality and homotopy theories.

cohomology of a compactum

X

in Sn

Alexander duality states that the Cech

is canonically isomorphic to the ordinary

homology with compact supports of the complement defined a homology theory the ordinary eohomology of homology.

SH,

such that

Sn \ X.

SH,

SH,(X)

Sn \ X.

is canonically isomorphic to

is now called (ordinary)

Milnor [Mil -i] (see also [Sky]) showed that

Eilenberg-Steenrod

[E -S] axioms on

CM,

In [St - i] Steenrod

and further that

SH, SH,

Steenrod

satisfies all of the is characterized by

247

these axioms together with two additional axioms; namely, invariance under relative homeomorphism (generalized excision) and the strong wedge axiom (see (8.2.3)).

(8.2.2)

Steenrod K-homology.

In a series of papers [Brow], [ B - D - F -

L. Brown, R. Douglas, and P. Fillmore defined a functor for

Ext

on CM

i- 2],

by taking

, unitary equivalence classes of C - algebra extensions of the compact

Ext (X)

operators by the C - algebra of complex-valued functions on Schochet [K-S]

X.

Kaminker and

then set

SEn(X) ~ I E x t

(X),

LExt (EX),

for

n

odd,

for

n

even,

( E is the unreduced suspension) and showed that

SE,

for a generalized reduced Steenrod homology theory.

satisfies the axioms (8.2.3) L. Brown, Douglas, and

Fillmore have shown that on the category of finite complexes,

SE,

is reduced

K- homology.

(8.2.3)

The Kaminker-Schochet

[~T S] axioms.

homology theory consists of a sequence invariant functors from the category AG

A seneralized (reduced) Steenrod

h, = {hnln c Z} CM

of covariant, homotopy

of compact metric spaces to the category

of abelian groups which satisfy the following axioms.

(E)

Exactness:

if

(X,A)

is a compact metric pair, then the

natural sequence

hn (A) is exact for all

n.

> hn(X)

> hn (X/A)

248

(S)

Suspension:

there is a sequence of natural equivalences

On:hn

called suspension, where

(W)

Strong wedge:

if

E

> hn+ 1 ~ E

is unreduced suspension.

X = v Xj E lim N { v N Xj} n=l n=l

is the strong wedge of a sequence of pointed compact metric spaces, then the natural projections

X~X.

induce an J

isomorphism

h,(X) -= ~jh,(Xj).

(8.2.4)

Remarks.

but not pointed. k,

by the formula

The homology theory

A reduced theory

k,

kn(X) = ker (kn(X)

h,

is reduced (see axiom (E) above)

may be obtained from any unreduced theory

kn(*)).

We break the problem of construction Steenrod extensions into two parts.

The

first part involves approximating a compactum by an inverse system of simplicial complexes, an idea which goes back to Alexandroff

[Alex].

The second part involves

prolonging a generalized homology theory defined on the category of finite plexes to a generalized homology theory defined on ble analogues of the Kaminker-Sehochet axioms.

pro -SS

CW

com-

and satisfying suita-

This was done in w

We shall carry out the first part of this program using a Vietoris functor (see (8.2.7); this construction was first introduced by T. Porter [Por-l]) Steenrod's original construction for motivation.

after giving

The Vietoris functor also yields

249

a Steenrod homotopy type for any compact metric space.

We can construct Steenrod theories in a wide variety of other settings, example, algebraic geometry and algebraic K -theory,

(~tale homotopy

for

theory), proper homotopy theory at

by applying suitable functors into

pro - SS

~ ,

and then

following the second part of our program.

(8.2.5)

Regular cycles followin~ Steenrod

metric space,

K

and let

f:VK--~X

VK

be the set of vertices of

such that for each

many simpliees have diameter less than coefficients complex

K

Let

X

be a compact

an abstract countable locally finite simplicial complex

simplicial complex), is a map

[St- I].

in an abelian group and a regular map

f

G,

s > 0,

e .

A regular map

the images of all but finitely

A regular R - chain on

(K,f,Cq),

as above,

K.

consists of a

elf

c

C~(X;G)

based upon regular chains, and reduced Steenrod homology

on K

with coefficients

S~q(X;G)

(8.2.6) space

Remarks.

X.

Let

Lebesque number of x

U

itself)

be an open cover of

(depending upon X

f

X.

Also,

(K,f,c) q

Then there is an x

in X,

U c U .

of the vertices of

A) e U .

simplicial

One then obtains a chain complex

such that for any point

the image under

to

G.

Consider a regular q- chain

U)

with

E Hq+ I[C~(X;G)].

is contained within a single open set of K,

set

U

in

X

together with a (possibly infinite)

q -chain

q

(clf

Hence, K

X.

~ > 0

(the

the c -neighborhood

of

for almost all simplices

is contained within some open

the fact that

conceals the "local pathology" of

on a compact metric

f

maps

VK

(and no___~t K

250

(8.2.7)

The Vletoris construction [Por -i].

topological space

X.

The Vietoris nerve of

Let U ,

U E U.

(n + i) - tuple

U ~ V).

The identity map of

and V X

is the (x0,xl,-.',x n)

and

s i ( x 0 , x l , ' " , x n) = (x0,xl,"',Xi_l,Xl,Xl,Xi+l,'",Xn),

U

VN(U),

Faces and degeneracies are given by

di(x0,xl,''',Xn) = (x0,xl,'",Xi_l,Xi+l,"',Xn),

Now consider open covers

be an open covering of a

denoted

slmplieial s e t in which an n - simplex is an ordered of points contained in an open set

U

of X

for

where

V

0 ~ i ~ n.

refines

induces a canonical inclusion

U

(notation:

VN(V) e--->VN(U).

We may therefore associate to a topological space its Vietorls complex.

(8.2.2)

VX = {VN(U) IU

cal inclusions

(8.2.9)

VN(V) c

>

Proposition.

an open cover of VN(U)

when

U

<

=

X} .

VX

is bonded by the canoni-

V .

The Vietorls construction extends to a functor

V:Top----->pro - S S .

Proof.

We define

and open cover

U

f-l(u) = {f-I(uIIU VN(f-I(u)) whenever

of Y c

> VN(U) U ~ V

U)}

V

on morphisms as follows:

to a continuous map

f:X--->y

we associate the open cover of X.

Then

((x0,Xl,-'-,Xn) ,

(and hence

f

induces an obvious map ~ (f(Xo),f(xl),-..,f(Xn))

f-l(u) ~ f-l(v)),

the diagrams

such that

251

vN(f-l(v))

> VN(V)

f

I

9

i

VN(f-I(u))

commute. Clearly

>VN(U)

These diagrams induce the required morphism V

preserves

identity maps and

Theorem

[Dow].

Let

U

theorem of Dowker.

be an open covering of a topological

Then the realization of the Vietoris nerve of

X.

homotopy equivalent

We need Vietoris

in p r o - SS.

V(fg) = Vf o Vg.

We shall frequently use the following

(8.2.10)

>VY

Vf:VX

U,

RVNU,

to the realization of the Cech nerve of

space

is canonically

U,

RCNU.

nerves rather than Cech nerves in order to obtain a funetor

from compact metric spaces to

pro - spaces.

An interesting

problem is the con-

struction of a nerve that is "small" like the Cech nerve and "rigid" like the Vietoris nerve.

(8.2.11) generalized plexes. i.e.,

V

Steenrod and Cech extensions of hom01osy theories.

Let

(reduced) homology theory defined on the category of finite

By G. W. Whitehead h,(-) ~ ~,s((_)

and Cech extensions of following formulas.

[Wh] there is a

^ E). h,

See also w

to the category

[We write

CW

Sh,

(5.6.7) and the Steenrod extension;

spectrum

and (5.6.7). CM

E

h, CW

be a com-

which represents Define the Steenrod

of compact metric spades by the

for both the extension of

h,

to pro - S S

the usage should be clear from the context.)

he,

252

(8.2.12)

Steenrod extension:

Sh.(-) ~ Sh, o V

= ~,S(hollm (Sin (RV(-) ^ E)) = Ho(pro -Sp)(S , Sin (RV(-) ^E)).

(8.2.13)

~ech exs

~, (-) = i~ {h,V(-)j} = li.j{=,S(sln (Rv(-)~^ z)} = p r o ~Ho(Sp)(S

Here,

V(-) = {V(-)j}

, Sin (RV(-) ^ E)).

denotes the Vietoris functor and

simplicial spectra.

For ordinary homology these formulas become

(8.2.14)

Sfi,(-;R) ~ Sfi,(V(-);R) = w,(holimRV(-)) = Ho(pro-SS,)(S

, RV(-));

w

(8.2.15)

.,(-;R) = limj {fi,(V(-)j;a)} = lim.j {~,(RV(-)j)} = p r o - H o ( S S , XS , R V ( - ) ) .

Sp

the category of

253

In (8.2.14) and (8.2.15),

R

denotes any commutative ring with identity as well as

the free R - module functor of Bousfield and Kan [B -K -i], not the geometric reallzation functor.

(8.2.16)

Remarks.

Ken Brown [Brown] defined generalized Sheaf cohomology

theories with a similar use of slmplicial spectra and smash products.

(8.2.17)

Proof.

Theorem.

h,

is the ~ech extension of

h,.

This follows from Dowker's Theorem (see (8.2.10)).

Alternatively,

follow the proofs of Theorems (8.2.18) and (8.2.21).

(8.2.18)

Theorem.

Sh,

(8.2.19)

Theorem.

Sh, a h,

(8.2.20)

Theorem.

Products and operations associated with

(8.2.21)

Theorem.

Sh,

is a homology theory on the category

CM

of compact

metric spaces.

on finite

CW

complexes.

h,

extend to

Sh, .

is a Steenrod homology theory on the category of com-

pact metric spaces.

(8.2.22)

Theorem

(see also [ K - K -S]).

Sh,

satisfies functional and Steen-

rod duality on compact metric spaces.

We begin the proofs by verifying properties of the Vietoris funetor in w We shall prove Theorem (8.2.18) in w

using strong homology theories on

pro- SS,

254

(5.6.7).

Theorems

a Bousfield-Kan

spectral sequence for

Theorem (8.2.21)

w

(8.2.19) and (8.2.20)

there.

The Vietoris

Proposition

topic maps of spaces

i0

X x [0,1].

(announced

f,g:X ~ Y

H:X x [0,i]

and i I

K

systems, hence

RVi 0

K 0 = RVi 0

and

K I = RVi I

> Y

X

and limits, at

in

we obtain a diagram

[Por - I]).

Homo-

p r o - SS,

induces a functor

Hi 0 = f

and

Hi I = g,

as the ends of the cylinder

Vi 0 = Vi I

in Ho(pro -SS).

in p r o - Top,

We shall

i.e., a map

>RV(X x [0,i])

(the realization

Because adjunction morphisms

V

be a homotopy with

to RVi I

wise). SS,

suspensions,

of compact metric

> Ho(pro -SS).

to show that

from

CM

induce homotopic maps in

K:RVX x [0,i]

with

in w

independently by T. Porter

are the inclusions of

It suffices

define a homotopy

[Brown]) and prove

Ho(pro -SS).

V:Ho(Top)

where

(8.2.22)

We shall develop

functor on the category

in

of their Vietoris

Let

(compare

cofibration sequences,

least up to canonical equivalence

Proof.

in w

see w

functor

spaces preserves homotopies,

Vf,Vg:VX ~ V Y ,

Sh,

We shall prove Theorem

We shall prove that the Vietoris

(8.3.1)

follow easily,

id--->Sin R

functor

R

is applied level-

are natural weak equivalences

255 (8.3.2)

vx

[0,I] - > S l n

x

(RVX x [0,I])

9 Sin RV(X x [0,i])

~--

v(x x [0,i]).

Because the "wrong-way" arrow in diagram (8.3.2) is invertible in we shall see that

Vi 0 a Vil,

Call an open covering

U

of families of open sets open covering of

[0,I]

Ho(pro- SS),

as required.

of X x [0,i]

U x Va

where

depending upon

U U

a stacked covering if

U

is an open set in

and V a

X

is a union is an

(see [E- S]).

~'//,,~ X

Let U be a covering of

X • [0,i]

U = {Us x V IU~

by basic open sets, i.e.,

open in

X,

V~

open in

[0,I]}.

Such coverings are clearly cofinal in the inverse system of all coverings of X x [0,i]. U

For each

x

in X,

of x x [0,1] c X x [0,I].

consider the induced covering Because

[0,i]

is compact,

U

x

admits a finite X

n X

subcover, say

{Ux, i x Vx,ili = l ' 2 ' " " n x } "

the stacked covering

Let

Ux = i=la U ,i,

and form

256

u'

=

Clearly each open set of

{u x v x , i l i

U'

= 1,2,-",n

K,

let

U

~ ~

x}

is contained in an open set of

coverings are coflnal in all coverings of

To define

,

X x [0,i].

be a stacked covering of

(8.3.3)

U=

uu ~ U , { u

•

X x [0,i],

U'

is an open covering of

X.

Ku:RVN(U')

to

RVi 0

RVi I

• [0,i]

>RVN(U),

form the r e q u i r ~

with

= ((x0,t),(Xl,t),'-',(Xn,t))

homotopy

K = {Ku:RVN(U' ) x [0,i]

> RVN(U) IU

a stacked covering of

in

p r o - Top

K I = RVII:RVX

(8.3.4)

with

~ RV(X x [O,i]).

Proposition.

Then the induced map

Proof. appropriate

K 0 = RVio:RVX

Let

VA--->VX

> RV(X x [0,i])

SS J

X,

U'

as in (8.3.3)}

and

[]

A

be a closed subset of a topological is a cofibration

We may represent the map level category

[0,i]}

Then the homotopies

Ku((x0,Xl,-.-,Xn),t)

from

say

valfV~}

is an open covering of

where

Hence, stacked

U .

VA--->VX

as follows.

in

space

X.

pro - SS.

as a levelwise cofihration

Each open covering

U

of X

in an

257

induces an open covering

UIA

of A,

(8.3.5)

namely

UIA = {U n AIU E U },

and an i n c l u s i o n o f V i e t o r i s nerves

w(u IA) Because each open covering of adjoining the open set X

to A,

A.

{U]A}

A

> VN(U).

can be extended to an open covering of

X \ A,

X

by

the set of restrictions of open coverings of

(see (8.3.5)), is cofinal in the set of all open coverings of

We obtain the required representation

VA ! {VN(UIA) IU

(8.3.6)

VX/VA

Proposition.

an open covering of

Given

A c X,

X} ~

> {VN } E VX.

D

there is a natural map

> V(X/A).

Proof.

In the solid-arrow diagram

VA

> VX

,,,,

>VX/VA i

l +

V(X/A),

the composite mapping map.

VA-->VX--->V(X

/A)

is trivial.

This yields the required

D

(8.3.7)

Proposition.

Let

A

Then there is a natural equivalence sequence

VA--->VX--+V(X/A)

be a closed subset of a compact metric space VX/VA

> V(X/A)

in

is a cofibration sequence in

Ho(pro - SS), Ho(pro -SS).

X.

hence the

258

The following lemma about "shape cofibrations" is a key tool in the proof. is analogous to the statement that a map N

retract of

We state it in somewhat greater generality than is needed

[St -3].

A

in X

is a cofibration if and only if

there is a neighborhood N

of

A--->X

It

such that

now; we use the extra generality in Proposition

(8.3.8) let

U

Lemma.

Let

A

refines

N

is a strong deformation

(8.3.22), below.

be a closed subset of a compact metric space

be an open covering of

and a neighborhood

A

of A

X.

Then there is an open covering

in X

a)

V

b)

For each open set

c)

For each neighborhood

V

X,

and

of X

with the following properties:

U ; V

of V, N'

either

of A

V o A = ~

in N,

or

V 9 N;

the inclusion of Vietoris

nerves

VN(V]A) ---~W(VIN') is an equivalence in

Proof of Le~na. open cover, indexed by

in X

X

is compact, we may assume that

U = {Uili = 1,2,'--,n}. i,

UIA ~ {U i n A}. U'i

Because

Ho(SS).

i

ranges over

{l,2,o--,n}.

U i n A = U' i n A,

Conslder the restriction

[Kur, p. 122], there are open sets and if

v

Ui n A ~

> U' i

is a finite

From now on, in all constructions and sets

By a result of Kuratowski

such that

U

induce an isomorphism on Cech nerve~

U' E {U'.}I '

the inclusions

259 To do this, let

(8.3.9)

U'.l = {xld(x'Ui n A) < d ( x , A \ U i ) } ,

Next, perform the following constructions. U" = {U"i} , set with

let

N" = u U"i,

A c N c N r B".

(8.3.10)

Let

U"i = U i N U'i,

and, using normality of

X,

let

let

N

be an open

Next, let

V. 1 = U". n N, i 1 V. 2 = U". n (N" i i

A),

and

V. 3 = U. n ( X \ N ) , i l

N" \ \

N "'--..,

) -.

and let

VI = {vil},

_

V 2 = {Vi2},

Vi 2

~

and

V 3 = {Vi3}.

v" 3

Finally, let

V = V1 u V2 u V3.

Clearly our construction yields an open covering of Because the open sets are disjoint from

A

vll

are subsets of

N,

X

which refines

and the open sets

by construction, property (b) holds.

Vi2

U .

and V. 3 i

260

To check property (c), first note that the open sets

N,

hence for any open set

N'

with

(8.3.11)

3

are disjoint from

A c N' c N,

VIN' = (V 1 u V2)IN'.

Next, observe that each open set of

Vi

VIIN' ,

namely,

(8.3.12)

Vi2 n N'

vil N N'.

of V21N'

is contained in an open set

Hence,

VN(VIIN') = VN((V 1 u V 2) IN') = VN(VIN'),

and similarly,

(8.3.13)

Finally,

VN(VIIA) = VN(V 1 u V2) IA) = VN(VIA).

VIA = VIIA = VIA,

applied within

N'

and a look at Kuratowski's construction (8.3.9)

yields

(8.3.14) CN(VIA ) = CN(VIIA ) = CN(UIA ) ~ CN(U'IN ) a CN(VIIN) = CN(VIN),

via the inclusion

A c N.

the desired equivalence

(8.3.15) X V2

and V

VN(VIA) = VN(VIN) ,

Proof of Proposition

any open covering of

and V 3

Because

By Dowker's Theorem (8.2.10), formula (8.3.14) yields

N

as above, let

V'

is a neighborhood of

are disjoint from

A,

(8.3.7).

X

as required in (c).

Let

U

D

be an open covering of

constructed by Lemma (8.3.8) above.

be the open covering A

the projection

(in

X)

V 2 u V 3 u {N}

and the open sets of

~:X--->X/A

With

of X. V2

and V 3

induces an open covering

261

V" E wV'

of

X/A.

Because

V refines

V',

we obtain a co=m~utatlve diagram

VN(VIA) ~v~(v) . vN(v)/w(VIA) \\ i

(8.3.16)

,~.~ ~

bp

vt:cv"). We shall show that

p

is a weak equivalence.

Form the following commutative diagram.

VN(VJA) '"

)VN(V)

> VN(V)/VN(V IA)

q, (--)

(--) W(V]~)

,

w(v)

9 nICV)IVNCVlm

q (--) v.(v' I")

. vNCV')lW(V'

9 VN(V')

IN)

q"l(-')

(--3

t

v.(v' ]A)

p'

9 vs(V')IW(V'

, v~(V')

\

-,...

P"](~-) VN(V")

(8.3.17)

]A)

262

In diagram (8.3.17) the composite map

p"p'

and the rows are cofibration sequences in equivalences

a)

(or isomorphisms)

RVN(V'IN) = RVN({N}) (8.2.10)) = * the maps

so

is the map SS.

(because

The indicated maps are weak

N e V') = CN({N}) similarly,

~VN(V')/VN(V'IN)

weak equivalences.

in diagram (8.3.16),

for the following reasons.

VN(V'IN) = *

VN(V')

p

Hence

q"

and

(by Dowker's Theorem

VN(V'[A) = *.

VN(V')

Therefore

-->VN(V')/VN(V'IA)

is a weak equivalence

(by Axiom M5

for

SS).

b)

The map of

VN(VIA) ----->VN(VIN)

V and N.

equivalence

is a weak equivalence by the construction

Hence, the induced map of the cofibres,

(by [Q -i, Prop. 1.3.5] for

SS,

q',

is a weak

compare Proposition

(3.4.12)(c)).

c)

To show that

p"

is an isomorphism in

V' = {V 2} u {V3} from point

A

and a

N

N

recall that

where the open sets of

is a neighborhood

in A .

SS,

of

A,

V 2 and V 3

and

V" = ~,V'.

Then the required inverse of

p"

formula

p"-l(y0,Yl,-..,yn)

= (x0,xl,-..,Xn)

where

l a xi =

if

Yi = [A] s X/A

if

Yi e ~ ( X \ A )

-1 Yi

c X/A.

are disjoint Choose a

is given by the

are

263

d)

To show that

q

is a weak equivalence observe that

(8.3.18)

VN(V') : VN(V) UVN(VIN)VN({N})

(and recall that VN({N})

N E V'

so that

VN(V'IN) = VN({N})).

is contractible, the diagram of geometric realizations

RVN(VIN)

-> RVN({N})

1"

(8.3.19)

RVN(VIN ) -.

commutes up to homotopy (C

>CRVN(VIN )

denotes the unreduced cone).

the homotopy extension property to find a map

RVN({N})

which makes diagram (8.3.19) strictly commute. RVN(V')

Because

> RVN(V) uRVN(VIN)(RVN(VIN )

We may use > (RVN(VIN)

This yields a map

( R preserves eofibrations,

cones, and quotients by [Mil -2]) which extends the identity map of RVN(V)

and is a weak equivalence by an easy argument involving the

homotopy extension property. (see (a), above), the map

q

Because

VN(V') =

VN(V')/VN(V'IN)

of diagram (8.3.17) is a weak equivalence,

as required.

It follows that the map

p

in diagram (8.3.16) is a weak equivalence.

easy to check that open coverings of cofinal in all open coverings of

X/A.

X/A

of the form

V"

Therefore, the map

factors as a level weak equivalence (use the maps

p

It is

constructed above are VX/VA -

> V(x/A)

of diagrams (8.3.8)) followed

264

by a eofinal inclusion

{VNCV") } = V(S/A).

Thus

VX/VA -" V(X/A)

as

required.

(8.3.20) E VX

Proposition.

9V Z X

in

There are natural weak equivalences

Ho(pro -Top),

where

E

denotes the appropriate

unreduced

suspension.

Proof.

We use geometric realizations

(8.3.1), we may define a map the Vietoris complex of of

X)

X

as in Proposition

CRVX----~RVCX

(8.3.1).

Following

(from the cone of the realization of

to the realization of the Vietoris complex of the cone

which yields a commutative

solid arrow diagram

RVX

+ CRVX

> CRVX/RVX I I q

(8.3.21)

I I i

RVX

> RVCX

RVCX/RVX

in which the rows are cofibration sequences and the vertical maps are equivalences in

Ho(pro -Top).

CRVX/RVX But

> RVX/RVCX

Proposition in diagram

CRVX/RVX E E RVX a R E VX

RVCX~RVX = R(VCX/VX) Proposition follows.

(8.3.7) D

(R

(3.4.12) yields a filler (in

(8.3.21) which is also an equivalence (R

there.

preserves suspensions by [Mil - 2]) and

preserves quotients by [Mil - 2])

a RV Z X .

Ho(pro -Top))

Naturality

= RV(CX/X)

is easy to check.

(by

The conclusion

265

(8.3.22)

Propositlon.

spaces, and let

{X.} 3

X = limj{Xj}.

natural equivalence

Proof.

Let

VX

~ {VX.} 3

be an inverse system of compact metric

Then the projections in

X---> Xj

induce a

Ho(pro - SS).

By first applying the Marde~ic trick (Theorem ~.i.6)) if necessary,

we may assume that the indexing category

J = {j}

is a coflnite strongly

directed set.

Let

{Uj,klk~ Kj }

be the inverse system of all finite open coverings of

where the k - index is assigned so that for

U.],k

to

assume that each indexing category

Because each

Vietorls system of

ujs j{j}xKj

(8.3.23)

Uj,,k

is the pullback of

Again, by applying the Marde~ic trick if necessary, we may

Xj',k"

directed set.

j' > j,

Xj,

5,

where

j E J,

X.3 is compact,

hence isomorphic to it.

as follows:

(j,k) ~ (j',k')

is a cofinite strongly

__{VN(Uj,k)}

if

is cofinal in the

Assign a partial order to

j ~j'

and

k ~k'.

Then

limj{VXj} -= limj{{VN(Uj,k)}k~K } = {VN(Uj,k)}j c j, kE K] 3

(an inverse system is its own limit in any pro- category).

otherwise stated,

(J,k)

ranges over

We shall now write the natural map

From now on, unless

uj ~ j{j} • Kj.

VX---->limj VXj

as a composite of

several maps which will be later shown to be weak equivalences. X'. 3

Xj,

denote the image of

X

in X.. 3

To do this, let

Apply Leana (8.3.8) inductively to obtain

286

open coverings

Vj,k of Xj

and neighborhoods

Nj, k of X'.3 in Xj

with the

following properties: a)

Vj,k refines

{~j,k}k~Kj

for j ~ J;

b)

Vj,k refines

Uj,k,

c)

Vj, k refines

Vj,,k

{vj,k} d)

yielding inverse systems

Vj,k, for k' < k,

so

that

for j' < J,

whi~ is cofinal in

N.3,k c Nj,k,

{Vj,k}keK. 3

for k' < k,

is cofinal in

yielding an inverse system

{uj~,k} by (b); yielding inverse systems

for

3 e) Nj, k

is contained in the pullback of

yielding an inverse system

f) The inclusions in

Ho (SS),

N.3, k

to Xj, k

for j' < j

{VN~Vj,kIS,k)};

VN(Vj,klX'j)-r > VN(Vj,klNj ,k )

are equivalences

hence the levelwise inclusion

{VN(Vj,klX'j) } ' > {VN(Vj,klNj,k) }

Factor the natural map

VX ~

llmj{VXj}

is an equivalence in

as follows:

Ho(pro-SS).

267

(8.3.24)

~,- {w{uj,~]~j)}

VX

--" {wIs,~[~}

(by ( c ) ,

=> {v~5,ki~j,k)}

above)

(by (f), above)

i ) {VN(Vj,k)}

-

li%{{v~(Vj,k)}k~K.} .]

lim.{VX.} 3

(by (b), above).

3

We shall complete the proof by observing that the maps isomorphisms.

For

~

X~,

in Xj

for

For

j' > j

is contained in the open set the image of Vj, k

and i

above are pro-

this is an easy consequence of compactness and properties

of the product topology. of

~

Xj, in Xj

i,

consider any fixed index

(j,k).

The images

form a family of compact sets whose intersection Nj, k.

Hence we may choose a

is contained within

N.3,k.

j' > j

Because

w(vj,~l~j,k) ~ 'w(~j,,k)

Hence,

i

i s a pro - i s o m o r r h i s m , a s r e q u i r e d ,

such that

Vj,,k

(property (c), above), we obtain a commutative diagram

the conclusion follow&

X'j

refines

268

w

Proofs of Theorems

(8.4.1)

(8.2.19),

(8.2.20), and (8.2.21).

Proof Of Theorem (8.2.19).

(Proposition

Because

(8.3.1)) and the strong homology

homotopy invariant (5.6.7), the composites

preserves homotopies

V

Sh,

theory Sh

oV

on pro- SS

are homos

is

invariant.

n

Similarly, because exactness of

V

preserves cofibration sequences

Sh ~ V n

(8.2.3)(E) holds.

Sh n ~ V

follows from exactness Of

Let

RCN {Un+ I}

(by (5.6.7))

m

(by (5.6.7) and Proposition

>

Sh

n+l

{RCN {U }} n ~ RCN {U } n

{RVN {U }} = {RCN {U }} n n in

n

isomorphism

(8.4.2)

V o

> X

{VN(%)}

Let

(8.3.20)).

be a finite complex. Un

be the tower of Cech nerves. to rigidify in

tow- Ho(Top).

Then

VX

is a finite open covering of Choose bonding maps By [Dow], (see (8.2.10),

{RCN {U }}. n

But, because

X

is a complex,

But, by (5.2.13), the composite mapping

is an isomorphism in

~:Sh,--->h,

X

where each

Ho(tow- Top).

on finite complexes.

Proof Of Theorem (8.2.21).

of maps of spectra (see [Adams- i] or w our formula.

Thus, Axiom

Q

tow-Ho(Top).

n

{RVN {U )}

o

Proof of Theorem (8.2.20).

{RCN {U }} m X

(5.6.7).

= > Shn+ I o [ o V

admits a cofinal subtower X.

n

(8.3.7)),

For Axiom (S), the required natural equivalences are given by

The conclusion follows.

(8.4.2)

Sh

(Proposition

This yields a natural

D

Interpret products and operations in terms The conclusion follows easily from

269

w

Spectral sequences. We develop Bousfield-Kan and Atiyah-Hirzebrueh type spectral sequences which

converge to

Sh,.

Ken Brown [Brown] developed similar spectral sequences for

sheaf cohomology;

Kaminker and Schochet [K - S, Theorem (3.10)] obtained the

second spectral sequence using fundamental complexes. Ken spectral sequence to verify that pletes the proof that

(8.5.1)

Sh,

Theorem.

Sh,

We shall use the Bousfield-

satisfies Axiom (8.2.3(W)~

This com-

is a generalized (reduced) Steenrod homology theory.

(Bousfield-Kan spectral sequence).

Let

{X.}

be an

3

inverse system of compact metric spaces and let

X = lim {Xj}.

Then there is a

spectral sequence with

EP,q = limPj{Shq(Xj)}, 2

which converges completely under suitable circumstances to

Proof.

"Recall that

VX ~ {VXj}

spectral sequence (4.9.4) for "pro - (pro - SS)".

for

limP.{Shq(Xj)}j = 0

r > n +I

Now apply the Bousfield-Kan

applied to the inverse system

Xj

unless

the differentials

end at a 0 -group.

case.

pro- SS.

Compare Proposition (5.6.8).

In particular, suppose that

that

Shq

in

Sh,(X).

d

{VXj}

in

The conclusion follows.

D

is an inverse system of cardinality

=nH<

0 =< p =< n + l .

because

r

(of bidegree

This is complete convergence.

Then

Ep'q = EP~ q n+2

(r, r - i))

either begin or

We cite an important special

so

270

(89149

Corollary.

metric spaces and

0

X = lim {Xj}.

9 liml'3 {Shq+I(Xj)}

(8.5.3) Corollary.

Proof.

(Compare (4.9.3)).

Let

Then there are short exact sequences9

>Shq(X)

9 llmj {Shq(Xj)}

For compact metric spaces

~2''"

liraN { v

J =i

Apply Corollary (8.5.2) to this tower. Shq+i(Vj=l Xj) ---->Shq (Vj=l Xj)

Sh,.

|

Xj} = v

j =i

limI

D

consider the tower

'

N

xj IN = 1,2,--'} with

surjections, the

>0.

The strong wedge axiom ((8.3.3)(W)) holds for

N

{v

{Xj} be a tower of compact

Xj,

the stron 8 ~edge.

j =l

Because bonding maps

in the towers

{Shq+l(Vj=l Xj

terms vanish in this case.

are clearly

The conclusion follows.

D

(8.5.4) Remarks. a)

This completes the proof of Theorem (8.2.21).

b)

If a compact metric space of a tower of polyhedra

X

is represented as the limit

{Xj},

Corollary (8.5.2) yields

short exact sequences 0 9 > liml'3 {hq+l(X3 )}

relating

Sh, to h,.

Sh (X) q

~ limj {hq(Xj)}

>0

Compare Milnor's characterization of ordinary

(reduced) Steenrod homology [Mil -i].

Uniqueness does not follow in our

case because of possible extension problems; however, any natural transformation of 8teenrod extensions of h,

is an isomorphism by the above

271

short exact sequences.

(8.5.5)

Theorem.

(Atlyah-Hirzebruch

a compact metric space of dimension

spectral sequences)

d < ~.

[K- S].

Let

X

be

Then there is a spectral sequence

with E2p,q = SHp(X;hq(S0))

and

dr

of bidegree

(-r, r - i)

Sh,(X )

which converges to

in the sense that

E d+l = E~"

Our proof is contained in (8.5.6) -(8.5.21), below. X

of dimension

d < ~

there is a cofinal tower

For a compact metric space

{U }

in the Vietoris system

n

VX

such that each

U

is a finite open cover (use compactness) with

n

dim CN(U n) ~ d

(use the definition of covering dimension).

We therefore begin by

proving the following.

(8.5.6)

Theorem.

Let

X = {X

}

be

a tOWer of finite simplicial

complexes

n

and simplicial maps.

Then there is a spectral sequence with

E2p,q = SHp(X;hq(SO))

which converges dim X ~ d < ~ ,

Proof.

to

Sh,(X)

if

dim X

is finite.

More precisely,

if

Ed+l = E .

The proof is broken up into several steps:

(8.5.7) -(8.5.20), below.

272

(8.5.7)

(Compare

Construction of the spectral sequence.

FpX - {FpXn}

be the levelwise p- skeleton of

[K- S]).

Let

Following Massey, we define

X.

an exact couple

i

D I.

i

\/ i

where E1 = ..Shp+q(FpX/Fp- 1 X ) p,q

D lp,q = Shp+q(FpX),

degree

i ffi (i,-i),

degree

yields a spectral sequence

(8.5.9)

J ffi (0,0),

,

and

and degree

k ffi (-i,0).

This

{E;,q(X)}.

Description of

E 1 and

E 2.

For each

p and n,

(FpX/Fp_iX)n = (FpXnlFp_lX) = v Sp,

a finite wedge of p - spheres, one for each p -cell of

X

n

Hence~

E~,q = Shp+q{(VSP)n}.

A L s o , . b e c a u s e t h e b o n d i n g maps

Sp

(where

i

Xn - ~ X l

i> ( v SP)n+l

is a typical injection and

w

are simplicial,

> (v SP)n

t h e c o m p o s i t e maps

= ~ Sp

a typical projection) have degree 0 except

273

for at most one Sh,(S0)

~

for each

i;

in which case the degree is

+i.

Hence, if

is a graded ring, we may choose bases for the free modules

Shp+,( v SP)n -= (9 finite ~Sh*(S0))

(8.5.10) Shp+,((V SP)n+I)

so that the maps

~ Shp+,((v SP)n )

are represented by matrices of

the form

(8.5 .ll)

\

o

.

Hence, the towers

(8.5.12)

{ h p + q + l (vSp)nln ~ 0 }

are M_ittag-Leffler, so that

(8.5.13)

for all

p,q.

liml{hp +q +i ( v SP)n } = 0

(If

Sh,(S0)

is not a ring, analogues of (8.5.10) -(8.5.11)

still hold, but are more difficult to describe. (8.5.13) yields the following useful calculation.

Thus, (8.5.13) holds in general.)

274

(8.5.14)

Shp..~ { ( v SP)n} = hp+q{ ( v SP)n} = limn{hp+q( v SP)n}

= li.

({.p( ~ sP)n}; hq(S0))

= ~p({(~sP)n}; hq(S0)) = SHp ({(

Now, for each fixed with the

q = q0

v SP)n} ; hq(SO))

consider the exact couple (8.5.8) associated

generalized Steenrod homology theory

(8.5.15)

Skn(-) SHn_q0(,hq0 =

-

(sO))

~h

(so)

.=qo, \)

Skn(S0 ) = J q 0

In this case, the resulting

Ei = 0 P,q

n

unless

#

qo"

q = qn, %2

so that in the result-

ing spectral sequence

= Sfip(X; hq(SO)), E2p,q(X) =

E'p,q(X) = I 0Skp+q(X) otherwise.

This shows that

275

E2 (X) m SHp(X; hq(S0)), P,q

(8.5.16)

in the original spectral sequence, as required.

(8.5.17)

0

Convergence.

q ~ d,

so that

(8.5.18)

If

dim X = d < ~,

dd+ 2 = dd+ 3 = "'" = 0

Naturalit~.

in

unless

E

E d+l P,q

P,q

X--->Y

of towers of

Then there is a diagram

• ~Z<

Tow- SS,

and

E1 = 0 P,q

Consider a weak equivalence

simpllcial sets of bounded dimension.

(8.5.19)

then

Y,

hence an isomorphism

E~,q(X) a Sfip(X; hq(S0))

(8.5.20)

~>SHp(Z, ~ .

hq(SO))

<----SHp(Y, a ~ 9 hq(S0) )

E 2 (y). P,q

It is easy to see that the isomorphism (8.5.20) is independent of in (8.5.19).

Hence naturality follows from naturality of

Z

and the maps

S~,(;h,(80)).

This

concludes the proof of Theorem (8.5.6).

(8.5.21) sion

d,

that each

.Proof of Theorem (8.5.5).

With

X

a compact metric space of dimen-

choose a cofinal tower of open coverings of U n

is finite and satisfies

X,

dim CN(U ) ~ d . n

{Un},

in

V(X)

such

Next, choose bonding

276

maps to rigidify the Cech system

By Dowker [Dow], see (8.2.10),

CX E {CN(U )}. n

RVX = RCX

in

Hence, by (5.2.9),

tow-Ho(Top).

RVX = RCX

in

Ho(tow- Top).

Applying Theorem (8.5.6) to

RCX = (RCN(Un)}

required spectral sequence.

Naturallty follows from naturallty of the Vietorls

construction; replace

w

RCX

by

RVX

with

X = llmn{RCN(Un )}

in the formula for

E ,,,(x). 2

yields the

[]

Duality. We shall prove a Steenrod Duality Theorem (8.6.1) for our generalized Steenrod

homology theories and a more general theorem (8.6.2) involving functional duality. See w

(2.2.36) - (2.2.43) for Alexander and Spanier-Whltehead duality and

functional duality.

Kahn, Kaminker and Schochet [K - K -S] took the formula in

Theorem (8.6.2) as their definition of Steenrod homology; hence the two definitions agree up to (non-canonlcal) isomorphism.

Our present proof of (8.6.2) and its

application to (8.6.1) was motivated by a letter from Kaminker.

(8.6.1)

Theorem.

Let

X

he a compactum in

Sn.

Then

Sh (X) a hn-p-l(sn \X), P and the isomorphism is natural with respect to inclusion maps.

We shall first prove the following.

(8.6.2)

Theorem.

functional dual of

X.

Let

X

he a compact metric space, and let

Then

Sh (X) ~ h-P(Dx). P Theorem (8.6.1) will follow from (8.6.2).

DX

he the

277

Proof.

Let

E

be the spectrum which represents

inverse limit of finite complexes, CWSp

be the functional dual of

DXj ^ Xj

9 S0 .

X - lim {Xj}. X.. 3

h, . Let

Write

X

as an

DXj ~ HOM(Xj,SO)

in

Consider the evaluation maps

These induce maps

DX.j ^ Xj ^ E

~S O ^ E

9

hence by adjointness

DXj Xj A E ----~E

which are weak equivalences because

,

Xj = DDXj

for finite complexes.

Thus

Shp(X) = ~pS{Xj ^ E}

~pSIEDXj }

We therefore need to show that the natural map

hocolim {DXj }

(see w in a map

~DX

for homotopy colimits) is a weak equivalence.

~,sDX

represented by a stable map

X ^ Sn ~ S

O

or

X

Sn

First, consider a class

> HOM (X,SO) ,

>HOM (sn,s 0) -" S-n.

or by adJointness,

Because each map from

X

to

a polyhedron factors through a nerve (up to homotopy), the map w,S(hocolim {DXj}) ----~,sDX X x I --~Y,

Y

is an epimorphism.

Similarly, because each homotopy

a polyhedron, factors through a nerve of

=,S(hocolim {DXj}) ~

w,sDX

is a monomorphism.

X,

Therefore the natural map

278

hocolim {DXj}

is a weak equivalence,

> DX

so

s{Dxj}-~ ~ s P

{EDXj}

holim

P hocolim {DXj } __ ~ s E P (E

is stable)

-~ w SEDX P = h-P(Dx),

as required.

(8.6.3) (n- I) -dual

Q

Proof of Theorem (8.6.1). Sn \ X

the functional dual a sequence X.

{Xj}

DX.

were a polyhedron, (n- i) st

Theorem (8.6.1) would then follow.

of polyhedral neighborhoods {sn\x.} J

hence in

("dual" to (5.2.9)). Further,

X

would be stably equivalent to the

The direct systems

inj -Ho(Sp),

If

and

Ho(inj - Sp)

of

{En-~xj }

X

in S n

D

Instead we choose whose intersection is

by the comparison theorem for direct towers = h-P{Dx.}j

as required.

in Theorem (5.2.9) and its "dual" can be

carried out for pairs, so the isomorphism respect to inclusion maps.

suspension of

are then equivalent in

hn-p-l({s n \ Xj}) e hn-p-l{zn-~xj}

the required constructions

then the

Sh (X) e hn-p-l(s n \X) P

is natural with 1

w

SOME OPEN QUESTIONS

We shall give a brief list of some open questions.

(9.1)

For proving Whitehead Theorems and defining general profinite

comple-

tions, it would be useful to have a tractible description of coherent prohomotopy theory.

R. Vogt

[Vogt - i] gave a geometric description

coherently-homotopy-commutative between such diagrams.

of a category of

diagrams and coherent-homotopy

In [Por - 3], T. Porter has defined a coherent prohomotopy

theory eopro- Top, and shown that the natural functor from copro -Top

(9.2)

Ho(pro - Top)

to

is a natural equivalence.

Off towers,

is still mysterious. from an object of

(9.3)

classes of morphisms

If

the relationship between In particular,

does every object of

and

pro -Ho(Top)

p r o - Ho(Top)

come

pro - Top?

f:X~Y

~(f) E pro -Ho(Top)

Ho(pro - Top)

is a morphism of is invertible,

then is

Ho(pro- Top) f

Invertible?

such that This problem is

probably very delicat e .

(9.4)

Now that the homotopy

systematically obtained

study completions

to the usual one (e.g.,

theory of as

pro - SS

pro - objects

[B- K]).

is well de~eloped,

one should

and compare the theory thus

280

(9.5)

we expect applications of the machinery of these notes to the problem

of classifying open principal G-flbrations (G

a com~)act topological group) (see

[Coh]) and to the problem of defining the continuous algebraic K-groups of a topological ring (see [Wag]).

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[Adams- i]

J. F. Adams,

Stable homotopy and generalized cohomology, Notes by R. Ming, U. Chicago Math Lecture Notes9 1973.

[Adams - 2 ]

,

On the Groups

[Adams- 3]

,

Lectures on generalized cohomolgoy, in Category theory,

J(X) - IV, Topology 5, (1966), 21-71.

homology, and their applications III, Lecture Notes in Math. 99, Springer, Berlin-Heidelberg -New York, 1969, 1-138. [Alex]

P. Alexandroff,

Untersuchungen [roer Gestalt und Lage Abgeschlossener Mengen, Ann. of Math. 30 (1929), 101-187.

[An- 1]

D. W. Anderson,

Chain functors and homology theories, in Symposium on Algebraic Topology, Lecture Notes in Math. 249, Springer, Berlln-Heidelberg- New York9 1971, 1-12.

[An- 2]

9

Simplicial K -theory and generalized homology theories, (mimeographed notes).

[And]

R. D. Anderson,

Homsomorphisms on Infinite-Dimensional Manifolds, Actes, Congres Intern. Math., (1970), Tome 2, 13-18.

[A - B]

and R. Bing,

A Complete Elementary Proof that Hilbert Space

is Homeomorphic to the Countable Infinite Product of Lines, Bull. Amer. Math. Soc., 74 (1968), 771-792. [A- M]

M. Artin and B. Mazur,

Etale Homotopy, Lecture Notes in Math. i00,

Springer, Berlin-Heidelberg - New York, 1969. [Ball]

B. J. Ball,

Alternative Approaches to Proper Shape Theory, in Studies in Topology, Ed., N. Stravrakas and K. Allen, Academic Press, New York, 1975.

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[B - S - i ]

B. J. Ball and R. Sher,

A theox? of proper shape for locally compact

metric spaces, Bull. Amer. Math. Soc. 79 (1973), 1023-1026. ,

[B - S - 2 ]

A theory of proper shape for locally compact

metric spaces, Fund. Math. 86 (1974), 163-192. [B -

v]

J. M. Boardman and R. M. Vogt,

Homotopy-everything H-spaces, Bull. Amer.

Math. Soc. 74 (1968), 1117-1122. [Bor- i]

K. Borsuk,

Theory of Retracts, Monografie Matematyczne 44, Warszana, 1967.

[Bor - 2 ]

,

Concerning Homotopy Properties of Compacta, Fund. Math. 62 (1968), 223-254.

[Bor- 3] [B - K]

,

Theory of Shape, Monografie Matematyczne 59, Warszawa, 1975.

A. K. Bousfleld and D. M. Kan,

Homotopy Limits, Completions and Localiza-

tions, Lecture Notes in Math. 304, Springer, BerlinHeidelberg-New York, 1973. [Brown]

K. S. Brown,

Abstract Homotopy Theory and Generalized Sheaf Cohomology, Trans. Amer. Math. Soc. 186 (1973).

[B-G]

__

and S. Gersten,

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Index

Adams map 196 Artin-Mazur reindexing 6 Atiyah-Hirzebruch spectral seql~nce

271

basepoints in proper homotopy theory 225 Borsuk shape 229 Bousfield-Kan spectral sequence 166 (C, pro- C) 114 Ccf 47 Cech extension 251 C~apman complement theorem closed model category 41 cocylinder cofibrant 42 cofibration 41 in CJ 57 in pro- C 72 short ... sequence 97 trivial 42 cofinite 6 coherent pro -homotopy 124 comparison theorem 173 completions 181 condition N 45 CW (prespectrum, spectrum) cylinder 43 discrete 154 Dowker's theorem 251 duality Alexander 35 functional 36 Spanier-Whitehead Steenrod 276

230

13

35

Eilenberg-MacLane pro -space 203 Eile~berg-MaeLane spectrum 16 end 123, 216 of o -compact space 220 Ex~ 132 fibration 41 in C J 57 in pro - C 72 long ... sequence 104 short ... sequence 104 trivial 42 fibrant 42 filtered category (map, space, 35c.) filtering category 4 finite simplicial set 12 flasque 139

115

295

germ of proper map

214

hocolim 170 holim 130, 133 HOM 48, 108 HOMj 148 HOM 148 pro homology operations 38 see also pro-homology, homotopy idempotent 195 Hurewicz Theorem 187

inj-C

Steenrod homology, etc.

8

J -diagram K0 236 Kan complex

4

ii

lim (of a pro -abelian group) limS 139 loop -space 97

139

Maps (pro- C) 113 Mardew construction 6 Mittag-Leffler 162 strongly ... 163 model category 41 closed ... 41 simplicial closed ... 48 moveable 7 N

(condition N)

45

Postnikov system 187 pro- C 4 pro -homotopy groups 184 pro- homology groups 184 proper c a t e g o r y a t ~ 214 homotopy theory 214 section 222 weak . . . h o m o t o p y t h e o r y Q

233

reindexing 6 ring spectrum

37

S 233 Shape 226 category 226 theory 229 strong ... 231 Simple homotopy theory

235

218

296

simplicial closed model category 48 perspectra 27 sets i0 spectra 28 smash product of spectra 20 sphere spectrum 15 stable 7 stability problem 188, 191, 193 standard action 235 Steenrod extension 251 homology groups 247 homotopy groups 245 Strong homotopy groups 206 Strong homology groups 208 strongly directed 6 suspension 96 Tel 115 telescope 115 direct ... 169 trivial (cofibration, fibration)

42

V 250 van~shlnE theorem 139 vertical homotopy 221 Vletorls (construction, functor, etc.) W,W 155 weak equivalence

41

i n CJ 47 i n p r o - C 72 Wh 236 Whitehead Theorem 188, 193 countereza~ples 201 i n p r o p e r homotopy t h e o r y

- isomorphism

187

226

248, 250