Shape Theory: An Introduction [1978 ed.] 3540089551, 9783540089551

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

688 Jerzy Dydak Jack Segal

Shape Theory An Introduction

Springer-Verlag Berlin Heidelberg New York 1978

Authors Jerzy Dydak Institute of Mathematics Polish Academy of Sciences ul Sniadeckich 8 Warszawa/Poland Jack Segal Department of Mathematics University of Washington Seattle, WA 98195/USA

AMS Subject Classifications (1970): 5 4 C 5 6 , 5 4 C 5 5 , 55 B05

ISBN 3-540-08955-1 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-08955-1 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

to

Arlene

and B a r b a r a

CONTENTS

Chapter

I.

Introduction

Chapter

II.

Preliminaries

III.

Theory

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

8

Category

Theory

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

ii

The

Shape

Some

of the S h a p e

properties the

shape

Representation

§4.

Borsuk's

§5.

Chapman's

General

Inverse

§3.

Fox's

theory

Shape

. . . . . . .

26 30

. . . . . . .

32

. . . . . . . . .

36

Category

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

Limits

Theorem

in the

Shape

in S h a p e

§4.

Shape

Properties

§5.

Space

of C o m p o n e n t s

Shape

category

Theorem

of t h e

20

Functor

Continuity

§2.

shape

. . . . . . .

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

to s h a p e

ComFlement

Properties

Category

of s h a p e m o r p h i s m s

approach

the S h a p e

§l.

of the

functor

§3.

and

V.

Category

Definition

and

Chapter

6

Homotopy

§2.

IV.

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

Topology

§2.

§i.

Chapter

1

§i.

§3. Chapter

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

Category

Theory

of S o m e

.....

. . . . . . . . .

Decomposition

Spaces

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

47 51 53 56 60

Invariants V

§i.

Cech

homology,

and homotopy

Chapter

VI.

§2.

Movability

§3.

Deformation

§4.

Shape

Algebraic §i.

§2. VII.

§i.

VIII.

Chapter

IX.

Associated

with

Shape

64 64 68 74

Theory

condition

of l i m I . . . . . . . . . . . . . .

idempotents

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

77 81

1-movability

Definition their

of p o i n t e d properties

§2.

Representation

§3.

Pointed

Whitehead §i.

. . . . . . . . . .

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

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

Properties

the u s e

and

Chapter

Dimension

retracts

Homotopy

Pointed

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

and n-movability

The Mittag-Leffler and

Chapter

cohomology,

pro-groups

1-movability

§2.

The Whitehead

§3.

The Hurewicz

Characterizations

on curves

and

Preliminary

§2.

Characterizations

. . . . . . . .

in S h a p e

in s h a p e in s h a p e

Properties

results

continua

theory theory

of P o i n t e d

ANSR's

98

103

.....

106

.....

109

ANSR's

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

of p o i n t e d

88 93

Theory

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

Theorem Theorem

§i.

1-movable

Theorems

results

continua

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

of p o i n t e d

and Hurewicz

Preliminary

1-movable

......

iii 114

VI

Chapter

X.

§3.

The

§4.

ANR-divisors

Union

Cell-like

XI.

Bibliography List Index

of

for

ANSR's

. . . . . . . . .

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

Preliminary

§2.

The

§3.

Examples

Smale

definitions Theorem

of

§4.

Hereditary Open

in

cell-like

equivalences

Some

shape

Problems

and

shape maps

results theory

119

which

. . . . . .

123

. . . . . . .

127

are

not

. . . . . . . . . . . . . . equivalences

. . . . . . . . .

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

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

Symbols

116

Maps

§i.

shape

Chapter

Theorem

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

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

129 132 141 143 148 150

Chapter In 1968,

K. B o r s u k

of c o m p a c t

type but w h i c h

coincides

idea was

metric

to take

spaces

metric with

and n e g l e c t

the theory

it on absolute

properties

Shape

retracts

to h o m o t o p y

~ech h o m o l o g y

(ANR's).

of c o m p a c t

can be thought

type and its r e l a t i o n s h i p between

as a than h o m o t o p y

neighborhood

the global

the local ones.

to the r e l a t i o n s h i p

of shape

spaces w h i c h was coarser

into a c c o u n t

a sort of ~ e c h h o m o t o p y analogous

Introduction

[i] i n t r o d u c e d

classification

His

I.

of as

type

is

and s i n g u l a r - h o m o l o -

gY. Consider and let

Y

the f o l l o w i n g

denote

example.

the P o l i s h

Let

circle,

i.e.,

the graph of

1 y = sin ~,

1 0 < x ~ ~,

1 ({, 0)

is d i s j o i n t

from the graph

Then

which X

and

of the same because Y

Y

are of d i f f e r e n t

shape.

These

continuous be an a r c

image of in

Y

local

of X

Y

fail

than

prevent

X

and

(0, -i)

type but will

functions)

connected. connected

Y

Borsuk

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

to

Since

into

any

continuum,

it must

trivial.

alike

remedied

sequence

X

which

type

In other

of the

(e.g.,

same

they both

this d i f f i c u l t y is more

general

that of mapping. v S. Mardesic

theory b a s e d on inverse are d e f i n e d systems

systems

for a r b i t r a r y

are d e f i n e d

homotopy

classes

and J. Segal

relation

are called

of ANR's.

Hausdorff

as well

[i] and

as a n o t i o n

classifies

shape maps.

[2] d e v e l o p e d

shape

In this approach,

compacta.

of

turn out to be

of

from being

are very much

into two components).

the n o t i o n

of the closure

to be of the same h o m o t o p y

to be locally

S1

at its end points.

and so any such map is h o m o t o p i c a l l y

the plane

In 1970,

This

except

must be a locally

difficulties

by i n t r o d u c i n g

the 1-sphere

the union

(continuous

h o m o t o p y type even though globallythey divide

denote

and an arc from

homotopy

there are not e n o u g h maps

due to the failure

words,

spaces

X

Maps

of h o m o t o p y

between

shapes such

of such maps.

maps b e t w e e n A N R - s y s t e m s

and these

Since any m e t r i c of an inverse sequences

continuum

sequence

instead

of ANR's

metric

topological

spaces

every map

not true

in the

shape

theory,

metric

There

with

(Note:

i.e.,

which

the shape

functor

theory

or pairs

is not c o n t i n u o u s

approaches compact

agree

metric

two. theories

spaces.

while

defined

of

compact

X

of

important

V.~

(Mardeslc-Segal functors

such

of nice

abelian shape

[i]).

inverse

systems

theory

with

situation,

X.

this

yields taking

is

a inverse

for a single

that Borsuk's

shape

the two

on pairs

of

geometrical

of the

approach

is

any A N R - s y s t e m

the space

X

itself

is

are d e f i n e d by

as an inverse

manifold-like

spaces

limit or

groups).

are ~ech h o m o l o g y

topological of inverse

with

solenoids

(e.g.,

f

categorical.

because

(e.g.,

spaces

is a map,

the A N R - s y s t e m

cases,

of

in the h o m o t o p y

So while

or can be o b t a i n e d

invariants

associated

shown

is more

X

It is also p o s s i b l e

limits

to the limit in this

In many

is any map

they differ

categorical,

to

fixed and sends

is true

is the more

approach

topological

for an a r b i t r a r y

by taking

has

spaces,

a sequence

of circles)

sequence

connected

Two

inverse

can be used.

This

of compaeta.

the shape of a space

sequence

of an inverse

Borsuk's

to being m o r e

by means

an inverse

on pairs

the A N R - s y s t e m

in studying

expansion

Mardeslc

on compact m e t r i c

In a d d i t i o n useful

V.~

of compacta.

theory

approach

commutes

just as in the case of ~ e c h homology.

compactum

spaces

The A N R - s y s t e m

and shape maps

from the c a t e g o r y

while

has a r e p r e s e n t a t i v e

category.)

shape

representative

f.

limit

one can use ANR-

spaces

is a functor

c a t e g o r y w h i c h keeps

associated

every m o r p h i s m

continuous limits

spaces.

as an inverse

case,

[3] g e n e r a l i z e d

into the shape map whose

of A N R - s e q u e n c e s category,

Compact

Mardesi6

to the shape

f

in the metric

of A N R - s y s t e m s .

form the shape category. arbitrary

can be r e p r e s e n t e d

space

systems

to describe X

new continuous

such as the shape

of h o m o t o p y

Furthermore,

one obtains

and c o h o m o l o g y

groups

groups

of

if one does not pass

the h o m o t o p y

pro-groups

which

are more

introduced

delicate

shape

invariants.

an i m p o r t a n t

shape

invariant

far-reaching looked.

generalization

Mardeslc

systems.

its i m p o r t a n c e

[3] r e d e f i n e d

movability

stems

from the

to the limit of an inverse about

the

X

and

contained Y

Recall

have

that

of

This

categorical a version

same.

with

information

spaces

W i t h each

however,

system.

it allows the ~ech

s y s t e m w i t h a given

the direct system

Then

and

methods

are

using

metric

space.

various

the

formal

of h o m o t o p y

inverse

notes,

inverse

of shape

we

system, advantages

theory.

For

pro-groups.

a particular

This

on

of a shape map

In these

a single

associates

is b a s e d

the A N R - s y s t e m

has d i s t i n c t

introduction

also d e v e l o p e d

are e s s e n t i a l l y

he a s s o c i a t e s

development

for t o p o l o g i c a l

of shape map and is very

spaces w h i c h

space

approach

approach

compact

Q.

homotopy

category

that the notion

space,

topological

to the t h e o r e t i c a l

cube

are homeomorphic.

Chapman's

common properties.

This

Y

[i] independently,

can also be d e s c r i b e d

with each

be two metric

the two theories

[i] showed

certain

and

manifolds

the shape

for t o p o l o g i c a l

topological

character-

theory.

G. K o z l o w s k i

K. M o r i t a

elegant

Q - Y

equivalent.

manifold

of shape t h e o r y

the ~ech

Moreover,

algebraic

X

and

~-dimensional

are often

in nature.

respect

iff Q - x

is b a s e d on the n o t i o n

possessing

example,

and

one may pass

of the Hilbert

approach

associate

called

Let

[3] d e s c r i b e d

In 1975,

systems

losing

following

compacta:

shape

transformations,

approach.

of ANR-

in any p r o - c a t e g o r y

v. Mardes±~

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

will

the same

~-dimensional

spaces.

the

in the p s e u d o - i n t e r i o r

problems

In 1973,

natural

of m e t r i c

in the theory of

homeomorphism those

is a

been over-

in terms

in its presence,

system without

[I] has o b t a i n e d

of the shape

compacta

can be d e f i n e d fact that

This

had p r e v i o u s l y movability

[3]

system.

T. A. C h a p m a n ization

Borsuk

called movability.

of A N R ' s w h i c h

and Segal

Actually

In addition,

inverse

is p r e c i s e l y

the

historical

approach

Borsuk movabi l i t y ,

[4] also i n t r o d u c e d

an n - d i m e n s i o n a l

called n-movability.

The p o i n t e d

special

interest

locally

connected

connected

to us because continuum

continuum

be c h a r a c t e r i z e d homotopy

by w h i c h Cech dealt with homology.

by a purely

strong

arbitrary

local

Let

(X, x),

f

for

fact that

and

"correction" groups

spaces

has been

[2],

in shape Recall

can

first

theory was

theory

or spaces for

Whitehead's

class-

CW-complexes,

(Y, y)

be a map

such that

algebraic

developed

in the

also

cylinder

introduced

Mardeslc theorem

and

a shape

for more

[2-4]. the

reduces

to a shape

cylinder"

pro-groups

the

of

to the pair

X.

version

but w h i c h we refer to as A N S R ' s

theorem

showed that

"mapping

of A N R ' s

(absolute

lies

into

is to replace

V.p

of the h o m o t o p y

theorem

in s u c c e s s i v e l y

shape v e r s i o n

the

k = nO,

information

[4] and M o r i t a

pro-groups.

for

of this

of the W h i t e h e a d

of the W h i t e h e a d

the exactness

y)

importance

Mardesi~

by the h o m o t o p y version

The

version

required

of this m a p p i n g

Borsuk

of shape

and is an e p i m o r p h i s m

of the Fox t h e o r e m by c o n s i d e r i n g

and by a p p l y i n g

FANR's

f: (X, x) ÷

strictly

A shape

by M o s z y £ s k a

of the shape

composed

of their

for C W - c o m p l e x e s

~k (x' x) ÷ ~k(Y,

it t r a n s l a t e s

topological

generality

version

such c o n t i n u a

in terms

be c o n n e c t e d

equivalence.

information.

connected

only

"corrections".

1 s k < nO

is a h o m o t o p y

homotopy

proof

locally

homomorphism

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

homotopy

[2]) and every

in the d e v e l o p m e n t

(Y, y)

fk#:

The

of some

Moreover,

should be true

certain

+ dim X, dim Y)

the induced

in the

is of

shape

property

theory valid

properties

spaces with

n O = max(l

ideas

in h o m o t o p y

theorem:

then

algebraic

of

pro-group.

that theorems

ical

1-movable.

case

has the

(see K r a s i n k i e w i c z

One of the m o t i v a t i n g

with

1-movable

such a c o n t i n u u m

is p o i n t e d

stratification

c a l l e d by him

neighborhood

shape

f

retracts). the

An important

property

shape of CW complexes.

complexes

to i n v e s t i g a t e

Since

these notes

we are unable interested have more ences

covers,

theory

(4) Hu's

the U n i v e r s i t y time

support

of W a s h i n g t o n

these notes were written.

and hospitality.

supported

during

by a N a t i o n a l

Sibe M a r d e ~ i ~

The

Science

for a very helpful

June van L e y n s e e l e

partitions

held

and C h r i s t i n a

Foundation

referfor

on A l g e b r a i c

of unity,

numerable of ANR's,

of ANR's.

year

position

1977-78,

to a c k n o w l e d g e

at

at their

author was p a r t i a l l y grant.

conversation. Ignacio

also

Topology"

a visiting

He wishes

The

for the g e o m e t r y

the academic

second-named

[3] w h i c h

"Lectures

of Retracts"

author

of the theory.

"Algebraic

for its t r e a t m e n t

first-named

theory of shape,

use of four basic

(2) Dold's

"Theory

to the

[6] or Segal

We make

on polyhedra,

"Theory of Retracts"

The

spaces.

aspects

(i) Spanier's

of polyhedra,

(3) Borsuk's

Acknowledgments.

w hich

They are

for its a p p e n d i x

etc.,

to Borsuk

is that they have

the use of the theory of CW

of these

all the various

bibliographies.

in this work.

Topology"

the g e o m e t r y

is r e f e r r e d

extensive

the h o m o t o p y

This permits

ANSR's

are only an i n t r o d u c t i o n

to m e n t i o n

reader

of p o i n t e d

We wish

to thank

We also wish

for typing

to thank

this manuscript.

Chapter §i. be

Topology.

dealing

spaces

with

are

class

are

closely

The

notion

of ANR's

to

subset

a metric

set

in

can

be

ANR.

X

the

We

take

as

Y

of

in

r(x)

Tietze

for

and

for

all

we

2.1.2. absolute space

Y An

property

in

is

have

r:

called

an

the

be

the A

X

it one

there i.e.

a polyhedron.

is an o p e n If a s p a c e

is c a l l e d used

by

an

Borsuk.

an a b s o l u t e

any

metrizable

a neighborhood

a map

two

is a c l o s e d

for

is

the

following

is c a l l e d

provided

subset

of

there

then

space

but

extended.

following

U + X,

if

objects.

such

U

that

X as

play

absolute

the

role

retract

of

(AR) .

A metrizable

space

in n o t a t i o n )

provided

a closed

property

Theorem.

can

R1

in t h e

Tietze

As

a formal

following

(X c A R

is a l s o

of

a closed

from

then

in n o t a t i o n )

which

Definition.

important

subset

as

type

namely,

can

the

polyhedra

arising

corollary,

A metrizable

space

containing

which

this

f

of

we will

These

topological

homotopy

f: A ÷ S I,

definition

X

class

as

(ANR's).

purely

the

of

which

(X ~ A N R

any

retract

2.1.3. closed

S1

the

spaces

x c X.

Theorem

definition,

over

are

has

of m e t r i c

retracts

Theorem,

and

a retraction

Similarly, Extension

ANR

thought

X

a formal

classes

and

Extension

A

containing

= x

be

space

retract

Y

basic

contains

each

Definition.

neighborhood space

can

containing

2.1.1.

X

since

substituted

the

neighborhood

property

an ANR

corollary

Preliminaries

axiomatically

related of

of

is a b s o l u t e

described

The

of

One

II.

known

of A N R ' s

f: X

is

as B o r s u k ' s

(Homotopy

a metrizable

subset

x {0}

there

the

X. u AxI

is for

called any

an

metrizable

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

following

homotopy

r:

Theorem).

Then

any

÷ Y

c ANR

map

Let

A

Y÷X.

extension

Theorem.

Extension

space

X

be

a

7 has

an e x t e n s i o n For

f' : X×I

the p r o o f

2.1.4.

of T h e o r e m

Notation

t h e n we d e n o t e

÷ Y.

the

If

2.1.3,

A

by

inclusion

2.1.5. (possibly {s} the

i(A,

map

infinite)

of f i n i t e condition If

s

subsets that

any

subset

~s

be the c o m p l e x

all

a non-empty

functions

p

{v~KIp(v)

(ii)

for any

p,

[ p(v) veK

There

is a m e t r i c

topology

K

a finite

set

U

IKI on

consists

s < K)

X).

v

So

K,

# 0}

of

faces

faces

of

IKI to

is a s i m p l e x

a set satisfy

a simplex

of all

K

and which

will

let

of a

{v},

s

of all p r o p e r

also of

of

K.

be s,

and

s.

be the I

such

of

K,

set of that

= i.

(p(v)

by

-q(v))2]

determined

with

n Isl

number on

the m e t r i c

IKI c o n s i s t s

is l o c a l l y

topologies

K

is a l s o

consisting

IKI d e f i n e d [ [ vcK

IKI

i(A,

space

½

by this

metric

is c a l l e d

the

strong

topology.

topology

for w h i c h

on

=

by

simplexes,

K

on the v e r t i c e s

on

X

vertices,

called

complex

p,

The

case

complex

for any

the

complex

(written:

(i)

(or metric)

weak

K

simplicial

q)

into

called

consisting

defined

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

of a s i m p l e x

of

simplicial

p(p,

and

A simplicial

is a s i m p l e x

94.

xcA.

of the v e r t i c e s ,

as the

For

A

for e a c h

set of o b j e c t s ,

regarded will

from

[5], p.

X) : A ÷ X

X) (x) = x

Definition.

see B o r s u k

is a s u b s p a c e

i(A, is d e f i n e d

we

is o p e n

in

finite

(i.e.

of s i m p l e x e s IK[ c o i n c i d e

topology

precisely ISld

is d e n o t e d

of those

for e a c h

for e a c h

containing (see S p a n i e r

simplex

vertex

v) b o t h [i],

by

subsets s

vcK

of

there

strong

Theorem

IKId. U

The

of K.

In

is o n l y

and w e a k

8 on p.

IKI

119).

8

For each {pc

IKI:

that

vertex

2.1.6. with

the o p e n

star

st(v)

of

v

is the

strong

and w e a k

set

st(v)

is o p e n

in b o t h

topologies

v£K.

Theorem.

the m e t r i c

The

topology

For

the p r o o f

If

K

the w e a k

v£K

p(v) > 0}.

Observe for e a c h

vertex

space

of a s i m p l i c i a l

complex

K

(Theorem

11.3).

equipped

with

is an ANR.

of T h e o r e m

is a s i m p l i c i a l

topology

IKld

is c a l l e d

2.1.6,

see Hu

complex,

then

a simplicial

[i], p.

the

set

space

106 IKI

and

is d e n o t e d

by

IKI,

too.

§2.

Homotopy

2.2.1. on

Definition.

X' c X

provided

Theory. Two

are h o m o t o p i c

there

that

H(x,

t) = f(x)

H(x,

a homotopy "relative

2.2.2. homotopy g: and

= g(x)

X'"

(X, A)

X'

+

(Y, B)

(f = g

which

rel.

X'

agree

in n o t a t i o n )

For

definition

Lundell -~eingram

H(x,

xeX'

such

A map

and

t~I. f

for

Such

and

f:

f.g

g.

x~X

and

a map If

and

÷

X'

Given

(Y, B)

provided

~ l(y,

properties

[i] or S p a n i e r

Definition.

(X, A)

equivalence)

that

and basic

(K, K')

(Y, B) i) = g(x)

joining

(homotopy

(X, A)

÷

H

is c a l l e d

- @,

then

be o m i t t e d .

Definition.

= I(X,A)).

subcomplex)

to

AxI)

and

X'

will

g.f

2.2.3.

(X×I,

for

to

domination

(Y, B) ÷

relative

0) = f(x)

relative to

f,g:

is a m a p H:

such

maps

B) = i(Y,

of

CW

is c a l l e d there Y)

a

is a map (f'g = l ( y , B )

complexes,

see

[i]. CW

(L, L')

pairs a map

(i.e. f:

a CW

(K, K')

complex ÷

(L, L')

and a is

said

to b e

denotes

cellular

the

that

cellular

of

K.

Theorem.

If

f:

flK'

a proof

2.2.5. let

such

is

cellular

of

Theorem

c L (n)

(K,

K')

then

for

all

n,

(L,

L')

is

÷

f

is

where

a map

homotopic

tel.

K (n)

of

K'

CW to

pairs

a

map.

For

and

f ( K (n))

n-skeleton

2.2.4. such

if

Theorem.

(K,L)

be

2.2.4,

Let

a

A

see

be

CW

pair.

H(K

×

Lundell-Weingram

a subspace If

there

of

is

[i],

p.

a topological

a homotopy

H:

72.

space

X

K×I ÷X

that

then

there

G(x,

i)

is

• A

a homotopy for

{i}

G:

xeX,

u L×I)

K×I

c A

÷ X

such

that

0)

for

(x,

such

that

G(x,

t)

= H(x,

÷ Z

is

a homotopy

G(x,

0)

t)

= H(x,

0)

and

• L×I.

Proof.

for f :

Claim.

If

(x,t)

• YxI,

y x I ÷ Z, Proof

of

F:

YxI then

where

= F(x,0)

F' (x,t,s)

= F(x,

F':

if

rel.

for

YxIxI and

if

s _< 2t

s -> 2 - 2t

if

1 1 - t - is)

if

yx{0,

(x,t)

÷

s _> 2t

1 t - ~s)

F(x,0) = F(x,

homotopic

= F(x,0)

Define

F' (x,t,s)

F' (x,t,s)

is

f(x,t)

Claim.

F' (x,t,s)

F

Z

1 -< ~ and

and

i}

to

Now

F'

is

a homotopy

for

map

, t

_
1 -

F

and

f rel.

Kx{l}

u Lxl)

÷

÷ A

such

that

Y

x

2

"

{0,i}.

suppose

a map. (x,t)

a retract

the

t -> i,

joining

H: (KxI, is

= F(x,

follows:

-

Then

t)

• Y x I.

as

t

F(x,

Take

a map

• K×{0} of

KxI).

H':

u LxI Define

KxI (such

a map

a map

exists

(X, A) H' (x,t)

because

= H(x,

Kx{0}

l-t)

u LxI

is

l-t)

10

F: as

follows:

F(x,t)

1 t ~ 2"

for

F I LxI f(x)

Then

= H(x, F(x,t)

is h o m o t o p i e

= F(x,

0)

F':

KxIx{0}

F' (x,t,0)

F' (x,l,s)

for FILxI

is a r e t r a c t

of

there

u

F'.

(x,t)

Then

c KxI

u

KxIxI,

there

2.2.6. following

Theorem.

conditions

fxl:

andbythe

Claim

÷ XxI,

where

LxI

(x,t)

i})

x I ÷ X

~ KxI,

F' (x,0,s)

is a h o m o t o p y

tel.

= F(x,0)

(L x I

u Kx{0,1})xl

is an e x t e n s i o n

K×I×I

÷ X

defined

by

the r e q u i r e d shall

For a

need

G(x,t)

= F"(x,t,l)

for

conditions.

the

following

pointed

topological

space

(X, x)

is h o m o t o p y

dominated

2.

(X, x)

is h o m o t o p y

equivalent

to a p o i n t e d

CW

3.

(X, x)

is h o m o t o p y

equivalent

to a p o i n t e d

ANR,

4.

(X, x)

is h o m o t o p y

equivalent

to a p o i n t e d

simplicial

5.

(X, x)

is h o m o t o p y

equivalent

to

Proof: 3.8 o n p.

by a pointed

(IKld,

CW

k)

complex, complex,

for

space,

some

K.

This

is p r o v e d

in L u n d e l l - W e i n g r a m

[i]

(Theorem

127).

2 ÷ 4. 126)

1 ÷ 2.

the

are e q u i v a l e n t :

(X, x)

complex

=

Lx{0,1}

i.

simplicial

p.

G: KxI ÷ X

we

~ L×I

Since

Kxlx{0}

sequel,

= H' (x, 2t-l)

is a m a p

u Kx{0,

for

F(x,t)

(x,t)

to t h e m a p

F'IL×I×I

f.

satisfies

In the

(LxI

and

for

Hence

F": of

l-t) i}

scI,

and

1 t s ~

for

L×{0,

= F(x,t)

x6K,

+ X

: F(x,

xeL.

such that

joining

2t)

rel.

for

Kxl

that

t i o n of its

It is p r o v e d (X, x) singular

in L u n d e l l - W e i n g r a m

is h o m o t o p y complex,

equivalent

which

admits

[i]

(Corollary

to the g e o m e t r i c

a simplicial

3.5 on realiza-

subdivision

11

(see T h e o r e m fact

6.1 on p.

is c o r r e c t e d

simplizialer

i00

there).

in the paper:

Mengen

I,

II,

An e r r o r

in the p r o o f

R. F r i t s c h ,

Math.

Z. 108

of

the

Zur U n t e r t e i l u n g

(1969),

329-367;

last

semi-

109

(1969),

131-152. 4 ÷ 5. (]K I, k)

ThiS is a c o n s e q u e n c e

are h o m o t o p y

Lundell-Weingram

[i],

5 ÷ 3. ~ i s i s

equivalent 4.6 on p.

p.

from

by a p o i n t e d

that

(]Kid,

simplicial

k)

and

complex

K

(see

131). of T h e o r e m

the

CW

fact

for e a c h

a consequence

3 ÷ i. This f o l l o w s is d o m i n a t e d

of the

fact

complex

that

2.1.6. if

Y e ANR,

(see Hu

then

[i], C o r o l l a r y

(Y, y) 6.2

on

211).

§3. For C(X,Y)

Category

any c a t e g o r y be the

By

defined For

and

for

(resp.

functor

we

f 6 C(X,

X')

C(',Y)f(g) the

of

some

of m o r p h i s m s 2.3.1.

let

ObC

C(-,Y)) from

C

following

X e ObC

of o b j e c t s set

(resp.

in the

Consider

C

be the

set of all m o r p h i s m s

C(Y,-)

contravariant)

Ens

Theory.

set

class

denote

to the

of all

in this

C(Y,-)

to the

Y

in

of

and

let

C.

covariant

category

category l~m

f(g)

sets

(resp. and

functions

C(X

C(-,Y)(X) = C(X,Y))

for e a c h

in

C.

It is the

the p r o - c a t e g o r y

defined , YB)

geC(Y,

X)

Y)).

systems

called is

(resp.

= f-g

g c C(X',

inverse

pro-C

= l~m

of its o b j e c t s

way:

for e a c h

category

pro-C(X,Y)

we

X

C(Y,-) (X) = C(Y,X)

we h a v e

= g.f

from

class

of

class

C.

The

, A)

and

by

if

X =

(X , p~

B = of

(Y~' qB C,

, B).

Thus

if

the

is a o n e - p o i n t

set,

i.e.,

Y

is an o b j e c t

then pro-C(X,Y)

Then

B

composition

g'f

= l i m ( C ( X , Y),

C(-,Y)p~

of two m o r p h i s m s

, A).

f: X =

(X , p~

, A)

÷ YcObC

12

and

g: Y ÷ Z e ObC

g-f

~ C(X

for some

, Z),

of p r o - C

where

f

is the m o r p h i s m

c C(X

, Y)

whose

representative

is a r e p r e s e n t a t i v e

is

of

~ e A.

N o w each m o r p h i s m f: X = of p r o - C and

q

(X , p~

can be r e g a r d e d .~,

So we

= ~

for

define

the

, A)

÷ Y =

as a f a m i l y

(Y , q~

, A)

{~}BcB'

where

f~epro-C(X,Y~)

B -< 8'. composition

g. f

f =

{~}BeB:

(Xc~' Pa

' A)

--g =

{g--{0}w(D: (Ys' q~

of

÷

two m o r p h i s m s

(Y[{, q

, B)

and !

as follows:

take

for e a c h

weD

gwB(~) of

~w

and

!

' B)

÷

(Zw~

r~

a representative

~ C

(YB(w)'

Z)

let g.f =

{g~(W) "~B(w) }w~D

" a

The

identity

{P--~}~cA' £

where

: X ÷ X

, D)

morphism

the

ix

identity

for e a c h

eeA.

of

X =

morphism The

t

(X , p~ 1X

morphism

, A)

is the

family

is a r e p r e s e n t a t i v e p~

is c a l l e d

of

the p r o j e c t i o n

morphism. Observe category An = the

inverse

order is

B

BeB

(X , pe

system

, A)

on

2.3.2. =

a category

C

can be c o n s i d e r e d

as a full

sub-

of pro-C.

(X , p~

there

that

[ =

(YB'

provided

B

is i n d u c e d ~ ~ B),

Theorem.

If

, A),

then

Y =

, B)

is a c o f i n a l

is a c o f i n a l

from

with

q

the

order

on

and

q

YB = XB (Y~, q

the m o r p h i s m

p = {pB}B(B:X

subset

÷ _X,

, B)

A

of and

= PB

subsystem

of

A

B c A,

(i.e.

for e a c h for

is a c o f i n a l

~6A

B ~ B' subsystem

of

19

where

ps: X ÷ Y8 = X~

isomorphism

of

Proof.

is the p r o j e c t i o n

morphism

for

8(B,

is an

pro-C.

Let

g = {g_a}A:

that a representative

of

g_~

Y ÷ X

be the m o r p h i s m

p~B(~)

is

for some

of p r o - C

B(~)eB

with

such 8(~) > ~

T hen

~ ' E = {P~(~) " P B ( e ) ) ~ e A : { ~ ] ~ ( A

= IX

and p-g = {IxB.~B}Bc B : {qs} = iy where

~8:

~ ÷ YB

is the p r o j e c t i o n

The n e x t r e s u l t

2.3.3.

characterizes

Theorem.

A morphism

f = {f^}~cA:-~ X = -

-

there

Proof.

morphism.

isomorphisms

(X , p ~

, A) ÷ Y =

(YB' q~

if a n d o n l y if for any m o r p h i s m

f

' B)

i

is a u n i q u e m o r p h i s m

If

in p r o - c a t e g o r i e s .

of p r o - C

- -

is an i s o m o r p h i s m pro-C

,

g':

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

Y ÷ Z

g: X + ZcObC

with

of pro-C,

of

g = g'.f.

then

~ = g'.f

iff

g.f-1.

~, =

SO s u p p o s e ~': ~ ÷ Z Take

that

with

for any

g: X ÷ ZeObC

there

is a u n i q u e

~ = ~'-f.

for e a c h p r o j e c t i o n

morphism

p_~: X ÷ X a morphism is

p~

g_ : [ ÷ X e

-g_~,-f = p~

= {~}~eA Now

"~,

such that

= p~ = g_< .fj

is a m o r p h i s m ~'~ = {~'~}~eA

from

~8: ~ + Y~

[

to

then

Since pe

.g_~, = g_~

and if

(£'~)'£= {h~'~}B~B ,

is the p r o j e c t i o n

for

~ ~ ~'

there

i.e.

X.

= {P-~}~cA = IX

{~B'~} = £ = where

g_ .f = ~ .

morphism.

f.g = {hB}~eB,

then

14

Consequently, If is the

over

F:

~8 = h8

C + ~

functor

Let

X =

the

same

of m o r p h i s m s

is a c o v a r i a n t

naturally (X , p~

, A)

directed such

induced

that

~ < ~'.

whose

representative

from

X

by

{f }

to

Y.

is

(A, ~).

pro-F:

pro-C

, A)

be two

Suppose

that

{fe}~eA

, Y )

÷ pro-~

eeA

we

take the

Such a morphism i s

inverse

systems

is a f a m i l y

and

= f -pe

then

then

~ . ~ = iy.

(Y , qe

-f,

f ,

and

F.

Y =

fe ~ C(X

If for e a c h

8(B,

functor, by

and

set

q~

for

for e a c h

the m o r p h i s m family

called

f E pro-C(X, __~

{f~}~A

a special

Y )

is a m o r p h i s m

morphism induced

~cA

There

is a u s e f u l

criterion

for a s p e c i a l

morphism

to be i s o m o r -

phism. 2.3.4.

Theorem•

Let

f: X =

be the s p e c i a l

morphism

ing

are e q u i v a l e n t :

conditions

such

i.

f

2.

for e a c h

induced

by a f a m i l y

÷ ~ =

{f~}~cA"

(Y , q~ Then

, A)

the

follow-

~eA

there

exist

B z ~

and a m o r p h i s m

g: Y$ ÷ X

that

3.

for e a c h

, X

)

such

~eA

Proof. Then ~eA

and

Take

is e q u a l for

exist:

some

= q~B

hand,

= p~

g'f~

,

B z ~, w z a,

and

g = -f-i =

geC(Y~,

X )

and

{g--~}~A

representative

g''fB'

h'f w = p~

to the p r o j e c t i o n

f.g' On the o t h e r

and

that

1 ÷ 2" f "~

= q~B

there

f~'g

each

r A)

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

f.g

hcC(Y

(X , p~

g':

and

let

morphism ¥B'

÷ X

~cA. ~: of

_Y + Y g~

we

for have

= q~

is a r e p r e s e n t a t i v e

of

g''~B'

= P-~

and

15

there

is

B a ~' w i t h B , = P~ •

g''fs,'P Thus

for

g = g'-q~, g'f8

we h a v e

= g'.q

,

= g'-fB," p

, = pe

and B' = qe "q~

f -g = f .g'-q~

= a~

2 ÷ 3 is o b v i o u s . 3 ÷ i. its

Suppose

representative

that

~ a e

g: X ÷ Z c ObC g

÷ Zo

: X

~'-f

= g,

Suppose

where

g'-f

g'

of

g"

there

such

Ym ÷ X

and

be a m o r p h i s m

take

such

are

= p:

. by

h.

where

g":

Y ÷ Z e ObC.

representatives

h':

Y

÷ Z

of

g'

and

h" : Y ÷Z

that h' .f

Take

h:

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

= g"-f, g',

Then

Let

of pro-C

and h-f

Then

is a m o r p h i s m

a morphism

= h".f

g: YB ÷ X

such

f~" g = g~

that

B a ~

and

•

Then h''q~ i.e.

g'

= g".

Sometimes special

morphism

By T h e o r e m

-g = h " - f 2.3.3.

it is c o n v e n i e n t

morphism.

2.3.5.

= h''f

of pro-C,

If then

the m o r p h i s m

to r e p l a c e

A possibility

Theorem.

of that

f: X = there

-g = h"-q~B

is

(X n, p~)

is a s p e c i a l

g: X'

÷ Y

f

is an i s o m o r p h i s m .

a morphism shown ÷ X =

of p r o - C

by a

in the

following

(Yn' q~)

is a

morphism

16

s u c h that

X'

is a c o f i n a !

i

E: ~ ÷ ['

is the n a t u r a l

Proof.

Let

of

X

and

~ ' E = f,

where

isomorphism.

£ = { ~ n } n c N : X ÷ ~.

By i n d u c t i o n , {nk}ke N

subsequence

we can f i n d a s t r i c t l y

of n a t u r a l

numbers

increasing

sequence

and r e p r e s e n t a t i v e s

fk: Xn k ÷ Yk of

~k

such t h a t nk+ 1 k+l = fk+l'qk fk'Pnk --

Then

let

It is o b v i o u s 2.3.6.

n m) Pnk

,

X' =

(Xnk

and let

that the d e s i r e d Definition.

is s a id to be m o v a b l e that for e a c h

~" ~ e'

Theorem.

conditions

An i n v e r s e

provided there p~

2.3.7.

g: X ' ÷ Y

~eA

is a m o r p h i s m .r = p~

If an i n v e r s e

{fk}keN •

are s a t i s f i e d .

system

for any

be i n d u c e d by

X =

(X , p~

there r: X

is

, A)

in

e' a ~

C

such

, ÷ X ,, w i t h

. system

X =

(Xe, pe

, A)

is

B'

dominated X

i

in p r o - C

by a m o v a b l e

inverse

system

[ =

(Ys' q8

' B)

then

is movable. Proof.

Take morphisms

f = {~B}BeB:

X ÷ Y

and

g = {g_ nk + m k

Let =

c U k n U{

= gp J U{'

m (U{', (i'')k)

pro- W

of

pro-HT

and

f' : U ÷ V,

~i " !''

(~,,)-1.£,, (i)-I

.m (Vk,3k),

which

~2 . : U''

pro-W

and

~,

: Y ÷ Z' =

satisfy,

by T h e o r e m

, , m (Vk, ( J ) k ) ,

3.3.4,

the

÷ U'

i n d u c e d by m a p s

v2 : V''

i(V~',

V k)

÷ V~'

÷ ~'

i ( U { ' , U k)

and

and

i(V k

!

, Vk)

= ~

and

(Z,,) . .= ({)-l.zl.f,,. . . . :

(i)-I

f'' : U'' ÷ V''

be m o r p h i s m s

gmk J U{ : U{ ÷ V{

~' " ~i = Z 1 " ~''

(~'')-- : -~

(Ul)-l Z =

" ~' " A-

of

respectively.

respectively.

~i ° (~'')-- = ~' V l "

"i

I!

fnk J Uk : U k ÷ Vk,

= i, V l • ~''

" i' " ~i " (~i)-I

be s p e c i a l m o r p h i s m s

and

g' : U' ÷ V'

i n d u c e d by m a p s

Consequently

=

of

÷ V

flk J U~' : U~' Then

and .

i n d u c e d by m a p s

of pro- W

(U~, (i') mk ), i'' : X ÷ U''

respectively.

Z1 : ~''

Let

P > Zk'

condition.

be s p e c i a l m o r p h i s m s

Let

for

k.

_J : Y ÷ V_ =

~i : U'' . ÷. U

i(U''k 'Uk)'

V{'

(V{', (j' ')~) be n a t u r a l l y

defined morhpisms

Let

in

(Uk,i ~ ), i' : X ÷ U' =

and

~'' : Y ÷ V''_ =

k,

for all

i : X ÷ U =

continuity

for all

and

=

35 Thus both m o r p h i s m s morphism

f''

and s i m i l a r l y

and

f''

f'

and

g'

represent represent

the

same shape

the same

shape

morphism. Thus

~ ( [ { f n } ~ = l ])

Let us show that the c a t e g o r i e s

and

is a f u n c t o r

and

if

establishing

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

of

Sh(Q).

fn = idQ

for each

n,

then we can take

U =

! = i" Hence

=

~

Sh B

F i r s t of all,

is w e l l - d e f i n e d .

f'

is the i d e n t i t y

morphism

and

f =

(I)-1f,

. _~ =

(~)-i_ _f=

and

{gn } =i

i x-

If

{fn}n=l

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

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

we may

take

sequence

Y

Z,

then

~ ( [ { f n } n = l ]) =

(I) -1

f'

~

~([{gn}n=l] ) :

(~)-l. g , . ~.,

])

~ ( [ { g n } n = l ])

Now

~ : Sh B ÷ Sh(Q) suppose

sequences

to

Y

in the f o r m u l a s and

=

(I) -lg'

=

(i,)-i • g_, • _I' • (~) - 1 • _f' • _I =

and

is a c t u a l l y

{gn}~=l

~ ( [ { f n } : = l ]) we can take the same b a s e s

U

• f'

• _I =

• ~([{fn}n=l]). a functor.

~([{fn}~=l ] ) = ~([{gn

{fn}~=l

X

and c o n s e q u e n t l y 9([{gnfn}~=l

Thus

from

to

i' = ~

from

sequence

n=l ])

from and

and

X

to

for two f u n d a m e n t a l Y.

Then,

when

defining

~ ( [ { g n } : = l ]) V

for both these

fundamental

sequences. Then morphism

(~)-l _ " f, " _~ = corresponding

corresponding

to

see that this

implies

(~)-i v _ " _g, " l,

to

{gn}n=l .

{fn}:=l

and

Consequently

the h o m o t o p y

of

where

f, : ~ ÷ Z

g' : U ÷ V f' = g' {fn}~=l

and

is the

is the m o r p h i s m and it is easy {gn}n=l .

to

36

So it r e m a i n s the f o r m

to s h o w t h a t any shape m o r p h i s m

~ ( [ { f n } ~ = l ~)

Take a b a s i s basis

{Vn}~= 1

Let

for some

{Un}~= 1

of o p e n n e i g h b o r h o o d s

If{'] : Unk ÷ V k

of

numbers

~'k'

where

and the m a p s

fn : Q ÷ Q

Y

in

(~)-i : U ÷ Z {nk}k= 1 f{'

fk+''l -~ ~k='' I U n k + l Let

of

be any e x t e n s i o n

Vk

of

X

in

Q

and a

Q.

and take r e p r e s e n t a t i v e s is some i n c r e a s i n g

satisfy in

is of

sequence.

of o p e n n e i g h b o r h o o d s

f, = { f , n } ~ = 1 = ~. f .

of n a t u r a l

fundamental

f : X ÷ Y

the f o l l o w i n g

for all

of the m a p

sequence

condition:

k.

f{',

where

nk_ 1 < n -< n kT hen

{fn}n=l

Thus

~ : Sh B ÷ Sh(Q)

§ 5.

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

establishes

Chapman' s C o m p l e m e n t

The H i l b e r t

cube

Q

sequence

and

f = ~([{fn}n=l]).

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

of c a t e g o r i e s .

Theorem.

w i l l be r e p r e s e n t e d

by the c o u n t a b l e

infinite

product oo

Q =

where

each

I

is the c l o s e d

~ Ii , i=l

interval

I-l,1].

It is w e l l - k n o w n

1

that

Q

is a s t r o n g l y

infinite-dimensional

and t h a t e v e r y c o m p a c t u m d e n o t e d by

q =

(qi) ,

can be e m b e d d e d

where

qi e Ii'

in

compact Q.

absolute

Points of

retract Q

a nd we use the m e t r i c

will be on

Q

d e f i n e d by d((qi), The p s e u d o - i n t e r i o r

of

(ri)) Q

=

~ i=l

lq i - ril • 2-i"

is o

S =

Ii, i=l

where

I.

is the o p e n

l

n -> i

we let

interval

(-i,i)

and

Bd(Q)

= Q - s.

For e a c h

37

in =

----

Qn In

general

3.5.1. Z-set

in

that

h(X) An

in

use

Definition.

A

Q

there

provided

0

to

compact

X

In+ 1

represent

subset

is

.... (0,

X

of

0,

Q

a homeomorphism

...)

is h :

c Qn •

said

to

Q ÷ Q

be

x Q

a such

c s x {0}. f ~

maps

f,g :

g(x))

< ~

3.5.2. f :

any

X ÷ Q

X ÷ Q

e > 0 f

is

g [A

The

Suppose Take

A

there

and

X ÷ Q for

Lemma.

Proof.

is

a

Z-embedding

said

x

~ X.

each

compact such

is

Z-embedding

a

to

subset

a map

=

if

f(X)

is

a

Z-set

be

A

that

6-near

of

f IA

a

is

g :

provided

Z-set a

X

is

a

Z-embedding,

X ÷ Q

which

is

Z-set. then

e-near

X

assertion

÷ Q

is

is

a map

trivial.

such

that

f IA

is

a

Z-embedding.

a homeomorphism

such

that

e > 0

h.

there

h • f

and

Represent

f(A) is

c s x {0}.

a

intervals

It

× Q

suffices

g :

X ÷ Q

to

prove

× Q

which

= h ° f I A.

as

Q1

Take

Q ÷ Q

Z-embedding

g IA Q

for

f I A.

first

f :

are

h:

to

X

In

In ,

Q.

d(f(x),

to

always

embedding

Two

If

we

I 1 × ...×

Q1 x

=

~ i=l

[an,bn3

Q2'

I2i-i

c In

where

and

Q2

=

such

that

~ i=l

I2i"

that is

for

any

e-near

38

oo

h.

f(A)

c

For

z i=l

[ a i , b i] i > 0

=

Take

m

{0}

and

1 - b


m :

P2i-i

Q1

and

÷

is

I2i-i

projection. Then

and

g :

g [A Thus

the

a map

such

each

e > 0

Z-set

in

and

g

is

XUhB

for

x

Lemma

Corollary.

Let

f(A)

there

a

Z-embedding

g Ix - A :

Let ÷ Q

such

A.

the

to

f'

It

the

required

to

h =

By

which

is

e-near

to

h.

f

c X

concluded.

be

where

B

compacta is

a map

g :

- A

÷ Q - B

X

f IA : that

3.5.2 map,

that

properties.

A ÷ B.

f' I B

Lemma

clear

A

is

a

and

Z-set

X ÷ Q

such

is

an

let

in

f :

Q.

that

g

embedding,

X + Q

Then (X)

be

for

u B

g [A

is

a

= f IA

f.

inclusion

is

3.5.2

c B,

exists

e-near

~ Xis

is

of

Q,

f' :

x Q

proof

that

Proof.

IB

X ÷ Q

= h • f I A.

3.5.3.

g'

that


~, a n d

, x ), p~

for a n y

e e A

for any h o m o t o p y

is an n - d i m e n s i o n a l (K, k) ÷ P~

Similarly

((X

as in the

(X ,, , X "g = p~

case

CW

,,)

, A)

there class

complex,

of p r o - H T exists f:

~'

(K, k)

there

a ~, ÷

exists

a

with

"f .

of T h e o r e m

2.3.7,

one

can p r o v e

the

following 5.2.3.

Theorem.

an n - m o v a b l e

object

Observe 5.2.4.

that

If

(X, x)

(Y, y)

Theorem

Proposition.

is an o b j e c t

of p r o - H / ,

2.2.4 Let

implies

of p r o - H T

then the

(X, x)

dominated

by

is n - m o v a b l e .

following

((K , k ), [p~'], A)

be an i n v e r s e

system

!

of

CW

complexes.

any

~ E A

map

r:

there

(K~n)

5.2.5. n-movable

,

Then exists

k

!

a'

~(X,

, k

~ e

) + (K ,,, k

Definition.

provided

((K

,,)

), such

with

A pointed x)

[p~ ], A) that

for any

p~~,, -r

topological

is n - m o v a b l e .

is n - m o v a b l e

~

e"

> e

Pe~, I K(n) e, space

iff there

.

(X, x)

is

for is a

68

5.2.6. then

Proposition.

(X, x) Proof.

implies

If

(X, x)

is

n-movable

and

d i m X ~ n,

is m o v a b l e . 5 !

V.

Let

C(X,

x) =

((K , k ), pe

, A).

Then

dim X

n

that B = {e~A:

dim K

~ n} !

is a c o f i n a l n-movable

s u b s e t of

A.

By T h e o r e m

a n d this implies,

5.2.3

((K , k ),

in v i e w of P r o p o s i t i o n

5.2.4,

[p~ ], B)

is

that

~w

((Ks, k ), completes

[p~ ], B)

X

m

Theorem.

Let

(X, x o)

(finite or infinite)

and a d e c r e a s i n g

the f o l l o w i n g

~(X,

x)

X =

2.

(Z k, x o)

3.

Zk

4.

dim(Z

Proof. simplicial

is m o v a b l e

which

compactum.

Then

{Zk}ke N

a compactum

Z

of s u b c o m p a c t a

of

for

containing Z

such t h a t

are s a t i s f i e d :

n Zk, k=l is m - m o v a b l e

is a r e t r a c t - X)

Let

of

and of

for e a c h

for e a c h

k,

k,

~ m. ~k ), Pkn),

where

Lk

is a f i n i t e

Let (Yn' Yn ) =

(for m - i n f i n i t e

Z

(movable)

(X, x o) = l~m((Lk,

space.

Define

be a p o i n t e d

there exist

sequence

conditions

i.

k < n,

Thus

the proof.

5.2.7. each

is m o v a b l e .

we take (Ln,

n-i V k=l

(L~ m) , i k) v

L k(m) = Lk )

(L n,

be the w e d g e

£n ) of

in).

n+l qn : (Yn+l,

Yn+l ) ÷ (Yn' yn )

qnn+l (x) = x

for

x c L i m)

n+l, . qn ~x) = x

for

(m) xeL n ,

n+l (x) n+l (x) qn = Pn eLn

for

as follows:

k s n-I

X e L n + I.

(L~ m)

~k ) ,

67 Observe

n+l

((Yn' Yn) '

that

[qn

Yn(m) c Y n + l for e a c h

i s m-movable (movable).

and

y(m) n

qn+l

Indeed,

= id

n. Zk c Z

Define (Ln,

])

n-i V i=k

£n ) v

as

qnn+l) '

lim (Zk,n'

where

(Zk, n,

yn ) =

(L (m) %i . ) i ' " co

Then Z.

it is c l e a r

Hence

_< m

each

which 5.2.8.

is

(Z k,

completes

that x o)

n k=l

Zk

and each

is m - m o v a b l e

Zk

(movable).

is a r e t r a c t

Moreover

of

dim(Z

- X)

the proof.

Corollary.

(n+l)-movable

X =

If

X

for e a c h

is a

LC n

compactum,

then

(X, x o)

x e X. o0

Proof.

Suppose

of

subcompacta

k

and

dim(Z

of

Z

- X)

Sh(X,

for Xo).

X

such

U

of

Observe

of

Z

of the

5.2.3,

is

(n+l)-movable

in B o r s u k

[5]

a retraction

retraction

i~+l),_

r

induces

(X, x o)

component

is

(p.

sequence for e a c h 80)

there

r: U ÷ X.

Then

Sh(Z k, x o)

->

(n+l)-movable.

of a c o m p a c t u m

of a c o m p a c t u m

where

in the H i l b e r t

inclusion

X

is m o v a b l e ,

map

{Un}ne N cube

i ( U n + I, U n)

Q

Z

is e q u i v a l e n t

is a b a s i s and

i n+l n

(see T h e o r e m

of o p e n

is the h o m o t o p y 3.3.4

and

3.1.6).

So s u p p o s e neighborhood

and

and

that movability

neighborhoods

V

the

Z

is a d e c r e a s i n g

is m o v a b l e .

(U n,

hood

in

If e a c h

of

Then

9.1

by T h e o r e m

to m o v a b i l i t y

Corollary

By T h e o r e m

Thus,

Proof.

class

Zk

x o)

k

Theorem.

where (Zk,

X

and

Zk,

that

some

5.2.9. then

n k=l

-< n+l.

is a n e i g h b o r h o o d Zk c U

X =

of

that X

in

for e v e r y of

for e v e r y

X

X

in

is a s u b s e t

of

Q

and

let

U

be an o p e n

Q.

component U

neighborhood

such W

X that of

of

X

there

its b o u n d a r y X

in

U

exists

an o p e n

is d i s j o i n t

there

neighbor

from

is a h o m o t o p y

X

68 ~,W: such

that

a finite

o)

¢ ~ , W (x' system

of

= x

V

× I ÷ U

and

¢ ~ , W (x,

indices

~i'''''

l)

u

c W

such

for

that

x

~ V

V = V

n contains

X.

Then

for

any

.

Take

u...u

V

~i

neighborhood

W

of

X

in

U

~n we

define

n

~W : by

~w(X,

t)

Then are

in

x I :

= ~i,w(X,

~W

is

t)

(V -

for

a homotopy

u i=l

Bd

x c V

joining

V

i)

i - cl(

i(V',U)

× I ÷ U u j ~

O

< n

" 'K ~(n) , K it

["K(n) ~ , k

)

in

K.(n) a

and

let

)r

-- q

a e A. .

Now

g ' q ~ o -- r

with

take

aO

-> a

.

o

, K~)-g-q~

_ q~

__ p O . q ~

O

an

lies

and, c o n s e q u e n t l y , O

with i( K(n) ~ , K

) - g ' p ~~'

-- p ~~ o .p~~'

O

O

-- p ~~'

there

is

x)

89 l

Putting

!

q:

l

: i'K ( ne) ( , K )"g'p~

we get

a map

homotopic

to

p:

O

whose

range Now

lies

in

suppose

(n)

K

that

for e a c h

~ e A

there

(X !

is an

e'

_> e

and

a map

(~w

q~

: (K ,, ks,) Let

f:

(K, k).

_~

f "q~

(K , k ) h o m o t o p i c

(X, x) +

Then

f -" f~'qe"

÷

there

is

We m a y

f

and

def-dim(X,

x)

(K, k)

be a m a p

a • A

assume

the r a n g e

to

whose

from

(X, x)

and a m a p

that

f

f -qe

range

lies

to a

CW

f : (K , k ) ÷

is c e l l u l a r . C% !

of

pe

-q~,

lies

in

x)

s d i m X.

in

K (n)

complex

(K, k)

Then

f "q~

K (n)

Thus

with .q~,

-< n.

5.3.3.

Corollary.

Proof.

Let

def-dim(X,

!

implies

that

~(X,

the

Proposition

set

5.3.2

x) = B =

we

((K , k ), p~

{~eA:

dim

K

completes 5.3.4.

dim(X,

x o)

is

dim X ~ n

cofinal

in

A.

Then

by

x)

m.

where

subcomplexes

Xk'

Zk

are

Yk

and

Xk+l:

Xk+l

have

X = l i m ( X k, p~)

A = l i m ( A k, p k ) ,

and

y = l i m ( Y k, Pk ). Let

n = max(def-dim

Then

we m a y

and

Pkk+l

Yk+l:

and

pkk+l Ak+l:

assume Yk+l

X, that

+ Yk

def-dim for e a c h

Y,

def-dim k

are h o m o t o p i c

Ak+ 1 ÷ A k

is h o m o t o p i c

A + i). Pkk+l

the m a p s to m a p s

whose

range

lies

to a m a p w h o s e

range

lies

+ Xk Zl n) in

A~n-l) Thus ~n Then

we

A k(n-l) using

such that k+l qk (Ak+l) Take

can h o m o t o p and e x t e n d Theorem

p ~I + l-l A k + this

2.2.4

we

homotopy can

find

k+l k+l qk X k + l = Pk iXk+l' ~(n-l) c Ak " a cellular

to a c e l l u l a r

homotopy

first

on

a cellular

k+l qk IYk+l

map

Xk+ 1 map

whose

and then

on

k+l qk : Zk+l

k+l ~ Pk IYk+l

H: Xk+ 1 × I ÷ X k

range

joining

lies Yk+l"

÷ Zk

and

k+l qk

Xk+l

and

74

a map whose

lies

in

Xl n)

. k+2 x i) x I) H (n-l) H- ~qk+ 1 (Ak+ 2 c (Ak+ 1

Then Theorem

range

2.2.5, the m a p

map whose

range

is h o m o t o p i c

lies

rel.

q kk+2 + l Xk+2

q~+l

X k(n) .

in

Ak+ 2

is h o m o t o p i c Thus As we get

to a m a p w h o s e

def-dim(X

the

X,

5.3.10.

and

Take

4.5.5

§4.

compacta.

(A, x o)

such

lies

in

q~+l

lies

we

Ak+ 2 q kk+2 +l

in

X~ n)

get

that

to a Yk+2

k+l qk

k+2 qk+l

Z~n)-

concludes

of T h e o r e m

that

the proof.

5.3.8

and Corollary

4.4.2

are

compacta,

then

def-dim(X/A)

Let

X

be a c o m p a c t u m .

def-dim

X = 0,

5.3.6

def-dim

equivalence. X

of

o

= S h ( X o)

rX

If

X.

By T h e o r e m

(see T h e o r e m

is a shape

X

Then

4.3.1).

equivalence.

retracts.

that

X

is c a l l e d

x

o

r:

r " S[i(A,

X)]

retract

((X,

Let

retraction

Definition.

Y

A c X

a component

says

a shape

containing

compactum

range

homotopies,

which

is a shape

A shape

retract

(X, x o)

the m a p

rel

by

A + i).

Definition.

5.4.2.

Y

~ n

Sh(point)

Shape

5.4.1.

morphism

range

If

Theorem.

therefore

Theorem

shape

def-dim

X ÷ A(X)

Proof. 0

these

consequence

Corollary.

max(def-dim

rx:

Sim~arly,

Hence,

following

5.3.9.

then

u Y)

an i m m e d i a t e

is h o m o t o p i c

to a m a p w h o s e

By c o m b i n i n g

x I) c X~ n)

x o)

there

of

where

(X, x o) +

A

(A, x o)

= I(A ' Xo ) •

A pointed e ASR

compactum

in n o t a t i o n )

is a s h a p e

X

,

and

X

is a n y

In this

case,

are

shape we call

(X, Xe).

an a b s o l u t e

containing

c A c X

there

provided

retraction

neighborhood

(X, x o)

is an a b s o l u t e

for any

compactum

r:

(Y, x o)

+

shape

retract

if for

is a c l o s e d

neighborhood

(X, Xo).

U

any of

X

o

=

75

in

Y

and

a shape

5.4.3. Sh(X)

Theorem.

[:

A pointed

(U, x o)

÷

(X, Xo).

compactum

(X, x O)

is an A S R

iff

= Sh(point). Proof.

Suppose

of the H i l b e r t Since

Sh(Q)

[.S[i(X,

Q.

= i x.

p:

any

compactum

(X

x )

infer

Sh(X)

({Xo},

= Sh(Q)

is a shape

induced

Q)].r

where

q:

5.4.4. following

unique

map

Theorem.

conditions

from

Let are

Y

to

(X, x)

and

4.4.6,

r: p-i

(Y

the

x o)

÷

Now,

. ~:

for

(y, x° ) ÷

({x o]

'

--

by the

= IQ

equivalence.

the m o r p h i s m

retraction,

subset

be a r e t r a c t i o n .

by C o r o l l a r y

is a shape

O

as a c l o s e d

= Sh(point).

Then,

X

X

(X, x o)

S[i(X,

x o)

containing

and embed

(Q, x o) ÷

= Sh(point).

(X, x o) ÷ Y

~:

we

Thus

Sh(X)

c ASR

Let

= Sh(point),

Q)]

morphism

(X, x o)

cube

Suppose

'

retraction

x O)

is

'

{Xo}.

be

a pointed

Then

compactum.

the

equivalent:

i.

(X, x)

c ANSR

2.

(X, x)

is shape

dominated

by

(Y, y)

for

some

3.

(X, x)

is s h a p e

dominated

by

(K, k)

for

some

complex

Y • ANR, finite

CW

(K, k).

Proof. 1 ÷ 2.

Embed

X

in the

Hilbert

(X, x)

be a shape

retraction

in

Then

is an A N R - s p a c e

Q.

r.S[i(Y, (X, x)

U)] : (Y, x) is shape

2 ÷ 3. ANR

there

(Y, y)

(K, k)

This

(X, x)

dominated

f:

Y

(X, x) ÷

(K, k)

and

with

let

r:

(U, x) ÷

neighborhood X c y c U.

retraction,

U

of

X

Hence

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

(Y, x).

dominated

a pointed

Q

closed

is a shape

by

(this can be d e r i v e d Take

some

is a c o n s e q u e n c e

is h o m o t o p y

3 ÷ i. morphisms

÷

for

cube

of the

by a p o i n t e d

from Theorem finite and

fact

CW g:

that

each pointed

finite

CW

compact

complex

2.2.6).

complex (K, k)

÷

(K, k) (X, x)

and with

shape l ( x , x ) = ~-f.

76

Suppose with

S[h] Take

closed

X c y.

3.2.1

there

is a m a p

h:

(X, x) +

(K, k)

= f. an extension

neighborhood

Then

By T h e o r e m

for

h':

of

X

r=g-S[h']

~.S[h'].S[i(X,

U)]

retraction.

in

(X, x)

(K, k)

of

h,

where

U

is a

Y.

: (U, x)

= g.S[h]

Thus

(U, x) +

÷

(X, x)

we

: g.f = l(x,x ) is an

have

i.e.

r-S[i(X,

~

U)]

=

is a s h a p e

ANSR.

NOTES The also

notion

of n - m o v a b i l i t y

was

Bogatyi

[4], K o z l o w s k i - S e g a l

Theorem

5.2.7

introduced

b y K. B o r s u k

[i] a n d K o d a m a - W a t a n a b e

is a g e n e r a l i z a t i o n

of a result

[4]

(see

[i]).

from Overton-Segal

[i]. Corollary Theorem

5.2.8

5.2.9

The notion Theorem

5.3.5

5.3.6

and

5.3.7

is due

is due

to B o r s u k

of deformation

is due

5.3.8

is due to B o r s u k

to W.

are proved

to O l e d z k i

In o u r e x p o s i t i o n

[4].

[3],

dimension

Holsztynski in N o w a k

[i],

the

of s h a p e

[6]. was

introduced

(see N o w a k

[i].

second

retracts,

The one we

by D y d a k

[i]).

first part to N o w a k

[2].

Theorems of T h e o r e m

[2]. V..

follow Mardeslc

[2].

Chapter Algebraic

§i.

Properties

The M i t t a g

6.1.1.

Mittag-Leffler

Associated

- Leffler

Definition.

qI

condition

A pro-group

condition

with

provided

Shape

a n d the

G =

use of

(G , p~,

for any

~ e A

for a n y

6.1.2.

Proof. ~ e A.

Then

for e a c h

G

the

satisfies is

~'

z

(G ,,) = p ~

G =

(G ,)

(G , p~,

is m o v a b l e

A)

satisfies

as an o b j e c t

the

of the

of p r o - s e t s .

Suppose

let

iff

A)

~

pe

A pro-group

condition

pro-Ens

~,,

~ ~' w e h a v e

Proposition.

Mittag-Leffler category

~"

l i m I.

there

~"

such that

Theory

Take a"

that e'

G

z e

~ ~'

satisfies such that

we

can

p

the Mittag-Leffler

condition

~

for

,,)

"(G

= pea' (G ,)

find a function

r: G

~"

, ÷ G ,,

and

z ~'.

with

~ w

p~

r = p~

is m o v a b l e

Thus

G

as a p r o - s e t

is m o v a b l e and

let

as a p r o - s e t .

~ c A.

Take

Now

~'

suppose

a ~

that

such that

for

~,,

any

~"

~ ~'

there

I

p~

pe

,) = p~

(G ,) c ps

6.1.3.

G

We m a y

n+l

Theorem

6.1.4. groups.

2.3.3 we

Definition.

~ n=l

Gn

G =

(Gn,

with

~ i

p~,

= Pn+l(Gn+3) for e a c h

get that ~ =

G

p~)

.

,

we

get

be an

inverse

condition

condition. sequence

a n d l i m G = Or

group.

n+2

= Pn+l(Gn+2)

for

for

m e n + i.

each

n

and

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

m)

(Gn' P n

provided

is a s e q u e n c e

• r = p~

the M i t t a g - L e f f l e r

m G n+l Pn ( m ) = Pn (Gn+l)

n+3

~ l

p~

~,,

= p~

the M i t t a g - L e f f l e r

that

Let

p~

satisfies

to the t r i v i a l

lim I G = * there

since

G

Let

(Gn+ I) = 0

We write c

Thus

assume

n+3

Pn

and

satisfies

Pn+ltPn+2(Gn+3))

therefore

{~n}~=l

G

is i s o m o r p h i c

n+2.

By

(G ,,)

(G ,,).

If

Proof. Hence

, ÷ G ,, ~,,

Proposition.

of g r o u p s . then

r: G

~,,

Hence (G

is a f u n c t i o n

be an i n v e r s e

for a n y {bn} ~

to the t r i v i a l

=i

sequence

sequence c

~ n=l

G

n

with

group. of

78

=

n+l

an

bn

Pn

6.1.5. groups such

(b

i)

Lemma.

such

that

Let

that

G :

l i m I _G

qn+l

Proof•

for each

. fn+l

(G n, p nm)

*

=

pn+l, ~ n=l

~

and

If t h e r e

= fn

{an} n = i

Let

n. H =

exist

then

Hn .

(Hn,

qm)

be

epimorphisms

pro-

fn:

Gn

÷

H

n

l i m I _H = * a n' c f -n l ( a

Take

n) "

Since

co

lim I G = * , there a' = b' n n i.e.

exists

{b'} ~ n n=l

" n+l(bn+l))-l, tpn

Take

b

c

= f

n

K n=l n

G

with n

(b') . n

Then

a

= b

n

n+l(b-i ) qn n+l

•

n

l i m I H = *. 6.1.6.

Lemma.

lim I G = *

and

Let

G =

m Pn

each

(Gn,

is

an

p~)

be

a pro-group

inclusion

such

that

homomorphism.

Then

integers

is

for

any

co

increasing

sequence

where

(Hk'

for

~ =

each

{nk}k: 1

q km) '

Hk = G nk

oo

{bk}k= 1 ~

Let ~ k=l

Then

is t h e

inclusion

l i m I H = *,

homomorphism

groups.

Thus

condition.

a

nk

:

bk

Let

G =

P kk + l ( b k l l )

k+l b n-I

(Gn, pm)

satisfies

the

are

countable

groups

and

c

Gk

and

= bk

this

there

" b k-i +l

is

-

implies

be

an

Mittag-Leffler and

inverse

sequence

condition,

l i m I G = *,

then

then

G_

of l i m I G = *.

satisfies

condition.

Suppose Take

•

a k e Gnk

lim I H = *

G

the Mittag-Leffler Proof.

•

Then

ak = b k

that

-.. •

Theorem.

If Gn

such

ak+ 1

H k.

6.1.7.

co

{ a k } k = 1 c k=lZ Hk"

Gk

ak

b k c Gnk

P nm ( G m )

m qk

and

there

k.

Proof.

If a l l

of p o s i t i v e

an

G =

(G n,

p nm )

increasing

= P na(n) ( G (n) )

satisfies

function

for

m

G

Let

the Mittag-Leffler

~:

N ÷ N such

that

> - ~ (n) "

o0

Let B(1)

= 1

{an}n= 1 c and

~ n:l

B(k+l)

.

B: N ÷ N

be

the

function

defined

by

n = a ( ~ (k)

+ i).

Define

by

induction

a sequence

79 co 0o

{Ck}k= 1 c k=l

such t h a t

G8 (k)

for

p~(k+l) (a B )....p~(k+2)-l(a (k) (k+l " (k)

k z 1

B

(k+2)

_l).p~

(k+2) (Ck+2) (k)

=

p~(k+l) ) (k) (Ck+l N O W for b

m

= a

satisfying

m+l, , Pm tam+l)

m

Hence, i.e.

m

=

am

if

p m+

_< m < m+l

ibm+ l 1 .

m+2 Pm+l(am+2)

b m + l = am+l

s m
- i.

are is

-m

=

obvious.

going

an

Since

~(blk.

= elk.

b2 • b k)

to

show

that

isomorphism.

homomorphism.

It

~ :

H + G

is

easy

a2 • a k = a k + 2

for

to

defined see

that

k >_ i,

we

by, ~

~ ( b i) is

infer

= a i,

a

that

epimorphism.

Now n = r 2 (b 2) (bik"

and

~. rl(c2)

B " r ~ ( e 2)

all

we

i = 1,2,

an

3 Cl).

r 2 • B = B • r I.

m

that

B • r l ( c 2)

is

(c13 . c2 .

consequently

= bil • b 2 • b I

• c2 • c k+l, I ; =

induction

holds

=

c H.

NOW For

= 1

:

b l-n • b 2 • b nI =

B (cln • c2 • C l ) =

B . r l (c 2)

= r 2 . B(c2)

=

and

b 2 • bk) -I " ( b l m .

= r2k(b21 ) . r 2 ( b 2 )

b 2 • b~)

• (blk.

b 2 • b k)

. r k ( b 2 ) = r2k+l"%ml-l') . r2( b 2 ) m

k+l m-k-i r2 (bl I • r 2 (b 2) • b I)

k+i m-k = r2 • r2 (b 2)

=

. r2k+l (bl)

m+l = r2 (b 2)

=

= blm-i

. m+l . b2 • mi

88

for

1 s k < m.

Thus

H =

.(bi, i ~ 1 :

bk+ 2 =

bk I • b m • b k = bm+ 1

=

(bi,

Thus

• bm.

isomorphic

the

In the r2:

for

i >_ 1 :

is n a t u r a l l y

b~k

proof

k

• b2 • bI

1 s k < m)

to

we

k ~ 1

and

=

b k = bm+ 1

for

1 -< k < m)

and

the

last

group

G I.

of Theorem

sequel,

for

shall

6.2.6 need

is c o n c l u d e d .

the

following

property

of homomorphism

H ÷ H. 6.2.7.

and

qn+l

Proposition.

= r2

for

The

each

n,

(Hn, qn+l)

pro-group does

not

satisfy

where

r

Hn

=

H

the Mittag-Leffler

condition. Proof. (see

Let

4:

H + Go

be

defined

by

~(bi)

= gi'

i = i,

2

6.2.1). Then

h.~

~ r2

=

(see

"

6.2.2)

Suppose

"

that

(H n

'

qn

n+l.

)

satisfies

the Mittag-Leffler condition. T h e n t h e r e is no > 1 such that n no-i m-i r2 (H) = r 2 (H) for m -> n o • ql°(H ) = q m () H± for m _> n i.e. n o o n -i n -i no-i m-i r2 (H) = ~ • r ° (H) = Hence h o (G o ) = h o • ~ (H) : hm-l-@(H) n -i h o (Go) by (no,

gi'

= hm-l(Go )

m

-> n

.

Since

h ( g i)

we

= gi+l'

infer

that

o is g e n e r a t e d

i ~ m.

i)

for

for

by

gi'

Hence we have i > no

and

i ~ no a

gn

(no,

and

h m - l ( G o)

contradiction i)

=

(no,

2).

is g e n e r a t e d

because Thus

gi(no,1) the

proof

= is

o concluded. NOTES For see

a definiticn

Bousfield-Kan Other

proofs

Theorem form).

6.1.8

of

lSm I G

for

any

inverse

sequence

G

of groups

[i]. of Theorem is d u e

6.1.7

can

to Keesling

be [2]

found

in G e o g h e g a n

(in a s l i g h t l y

less

[i]. general

87 The example of a n o n - s p l i t t i n g P. Minc and J. Dydak A. Heller

(see Dydak

(unpublished).

homotopy idempotent is due to

[7]) and independently

to P. Freyd and

C h a p t e r VII.

§i.

Definition

of p o i n t e d

7 .1.1 D e f i n i t i o n . provided

(X,x)

Observe

p r o - ~ 1 (X,x)

pointed

satisfies

Let

(X,x)

and

= lim

÷

CW

complexes.

is

condition

1-movable.

CW

(Xk,Xk)

f : i (X m(I) 'Xm)

f :

complex, with

k _> m

which

Mittag-Leffler

iff

(L,Z) ÷

continua

1-movable

iff

implies

that

Take

Let

one-dimensional

where

(Xn,Xn)

(~i (Xn,X n) ' ~i (pro)

pro-~l(X,x)

does.

n -> 1

(Xm,Xm), k >_ m

are

there e x l s t s

where

L

is a

there e x i s t s

By t a k i n g

(L,i)

a map

=( X m(I) 'Xm)

w e get

that

c ~i (pk) (zI (Xk,Xk)) (~i (Xn'Xn) ' r~l (pro))

(71(Xn'Xn) ' ~i (pro))

no > _ 1

and let

m _> n o (

k _> m.

connected

condition.

T h e n for e a c h

pk . g _- p m . f.

~i (pro) (z] (Xm'Xm)) o for

1-movable

satisfies

the

condition.

Now suppose condition.

is

Then

and for any

~i (Pn m) (~i (Zm,Xm) for

pointed

x { X.

(X,x)

((Xn,Xn) , pm ) ,

such that for any m a p

(L,£)

and their p r o p e r t i e s .

says t h a t l o c a l l y

the M i t t a g - L e f f l e r

finite

(X,x)

one-dimensional g :

is c a l l e d

continuum

the M i t t a g - L e f f l e r

Suppose m -> n

continua

for e a c h p o i n t

5.2.8

A pointed

connected

satisfies

1-movable

X

1-movability

1-movable.

Lemma.

Proof.

1-movable

A continuum

that Corollary

are p o i n t e d

7.1.2.

is

Pointed

f : CW

(L,i)

÷

complex.

satisfies

the M i t t a g - L e f f l e r

be a n u m b e r

such that

k

= Zl Pn ) (~l(Xk'Xk)) o

(Xm,Xm)

be a m a p such that

(L,Z)

is a

89 We m a y to each

assume

component

homomorphism Hence k

Pno

•

~

g

m

g :

Theorem.

(L,%)

÷ ~i (Xk'Xk) ÷

(X,x)

is a free

group

~i ( P % ) h o

with

~

apply

the

and

= ~

argument

there

(p~

is a

• f). o

(g) = h

we h a v e

1-movable.

(Y,y)

1-movable,

= lim

we

with

and

is

otherwise

(Xk,Xk)

is

(X,x)

(X,x)

(X,x)

~

(L,i)

Let

and

Let

Then

~i (L,%)

a map

= Sh(Y)

is c o n n e c t e d ,

L.

• f, i.e.,

Pno

Proof.

L

of

h :

for

7.1.3 Sh(X)

that

be p o i n t e d

then

((Xn,Xn), p~)

continua.

Sh(X,x)

If

= Sh(Y,y).

and

+

(Y,y)

= lim ÷

((Yn,Yn) , q~) ,

where

Xn

and

are

Yn

compact

connected

ANR's. Since

Sh(X)

Xn + Y n

fn:

and

n+l fn " gn = qn achieve

~

n

~n-homotopic that

such on p. and

an

and

[i]

to

has

the

to shape

exist

maps

n+l ~ Pn

gn " fn+l

Theorem

and

2.1.3

we c a n

n+l qn

that

a homotopy

= f n " gn (x)

[i] ' p. is

379).

Take

hng n

is

such

that

H : Yn+l

x I ÷ Yn

and

~n-h°m°t°pic

that

is

a map

to

fn.

hn : Xn ÷ Yn Then

~n-homotopic

Lemma

to

fng n

Yn+l"

Xn

p~+l

such

exists

H(x,l)

hn

Take

of

there

By a p p l y i n g

Yn

there

implies

in

p~+l

that

at

n

(see S o~ a n i e r

be a loop

~n-homotopic

(Y,y)

Y

n+l = qn (x),

= Yn

such

÷ Xn

n+l h n " gn = qn rel.

~n-homotopic is

i.e.,

in S p a n i e r

that Let

to

fngn,

that

g n ( Y n + l ) = x n-

in

h n ( X n)

380

and

assume

2.3.4).

be a loop

= ~ n (t)

that

Yn+l

(see T h e o r e m

H(x,0)

H(Yn+l,t)

we m a y

gn :

fn(Xn ) = Yn

Let

such

= Sh(Y)

at

xn

a map Then

(Z,z)

gn" hn+l

s~ +I : X n + 1 ~ X n gn" hn+l

= lim

~ Snn+l

((Xn,Xn),

such

rel.

s n+l )" n

is that

Xn+l" Since

sn+in Thus

2

90

. n+l,

(~l(Xn,Xn),~l~Pn

))

satisfies

, n+l,

liml(zl(Xn,Xn),

~itPn

))

the Mittag-Leffler

= *

(see T h e o r e m

condition,

6.1.7).

Hence

we

there

infer exists

+

a loop

Bn

in

Xn

at

an

Let

Un :

Un

to

• sn+l n

~n-homotopic n+l Pn

id X is

to

Since

be

a map

Sh(X,x)

equivalence.

As

an

7.1.5. Sh(X,x)

As we

some

1-movable

for

x n.

U n ( X n)

= Xn

and

Un

is

have

observed

B n-I

* e n -homotopic

to

infer

Sn

rel. each

X n + I. h

is a p o i n t e d

n

homotopy

is c o n c l u d e d .

X

of T h e o r e m

X

7.1.3

we

is p o i n t e d

any

is a p o i n t e d two

X

then

locally going

points

and

= Sh(Y) ,

are

is

to

get

the

following

1-movable

iff

(X,x)

x e X.

Let

Sh(X)

we

proof

If

and

Now

rel.

we

because

A continuum

Corollary.

1-movable.

that

n+l

consequence

= Sh(X,x')

1-movable

Y

continuum,

then

x, x' E X.

be Y

continua.

If

is p o i n t e d

connected

to p r e s e n t

1-movable

continua another

X

is p o i n t e d

1-movable.

are

class

pointed of p o i n t e d

continua.

7.1.7. then

the

Corollary.

7.1.6.

M,

for

such

that

1),

~ Un"

= Sh(Z,z)

Corollary.

1-movable

(~nll)

such

n+l * e n ~- P n (B

immediate

7.1.4. is

Thus

n

un • sn n+l

Hence

n+l Pn " Un+l Hence

each

. Then p~+l • Un+ 1 is p~+l(Bnil)-homotopie n B-l_homotopic to s n+l and s n+l is n n n

p~+l

B-I n

for

~ Bn , p n + l

X n ÷ Xn

Bnl-homotopic n+l Pn '

xn

Theorem. X

If

is p o i n t e d

X

is a p r o p e r

1-movable

and

subcontinuu/n def-dim

X

~ i.

of

a 2-manifold

91

Proof. positive of a

With

integer

e(N)

closed

curves

in

2-manifold

with

these

polyhedral

curve

cobounding

that an

annulus

there

properties:

simple bounds

N

closed

a disk in

N

each

curves

in

N,

(i.e.

is

disjoint

in or

associated

N

collection

either

it c o n t a i n s

a pair

of

a

contains a pair

parallel

curves

N). Take

a

2-manifold

Let

{ H n } ~I=_

such

that

empty

be X =

and

let

We

- Int(N)

c M

- Int(N) not

by

attaching

are

some

are

to

show

Hn

to

for

is

show

X

such

of

neighborhoods

that

of

curves

then

all

components

m

be

D. 3 that

So

2-mainfold

whose

boundaries

~(M')

< e(M).

of

component

in with

M non-

of

N Hm

is

not

n H i + I.

If

- int(Hm+l)

suppose

obtained are

X

i.

in

of

is m i n i m a l .

2-manifold

large

lying

> i + i.

a

e (M)

H i - I n t ( H i + I)

the

for

each

sufficiently N

that

a compact

be

M'

disks

each

component

disks

Let

sequence

going

a disk

a disk,

a disk.

going

and

1 ~ j ~ k, is

is

are

is

that

Sj,

containing

a decreasing

boundary.

Suppose

M

n Hn n=l

H i - I n t ( H i + I)

M

compact

e(N)

or m o r e

simple

of

each

S. 3

that

M

from for

a disk

c - Int(N)

M-

Int(N)

1 ~ j ~ k.

We

!

Suppose curve

that

bounds

Then

S I,

a disk

there

are

S p'

...,

in

M'

are

and

points

xj

simple

no

pair

curves

of

c Int(Ds),

in

curves

M'

such

cobounds

1 ~ j ~ k,

such

that

an

no

annulus.

that

J

x. ~ M' 3

-

~ S' n=l n

xj

for

each

has

the

same

morphism

(n,j)

we

that such

each

j.

such

that

k ( u D~ ) n j=l 3

position

h :

Thus clear

j

for

M'

may no

+ M'

S' n

M'

with

assume

curve

that

as

-

and

bounds S. 3

small

k u D~ j=l 3

h(M'

S' c M n

S' n

Take

-

disks

D~ c D. ~ 3

( ~ S' ) = @. n=l n in

- Int(N) a disk

in

cobound

an

Then

M' , i.e., t h e r e

k u D[ ) = M j=l 3

- Int

for

each

M

and

annulus

n

containing

M

is

- Int(N)

a homeo-

N.

< p.

Now

it

no

pair

there

is

in

(otherwise

M

is

92

S' n

would

bound

Thus

~(M)

Hence large

i :

a disk

in

> ~(M')

which

H i - I n t ( H i + I)

contradicts

is a d i s j o i n t

our method union

of c h o o s i n g

of d i s k s

for

of

M.

sufficiently

i.

By Lemma

8 in S p a n i e r

(Hi+l,X)

÷

Thus

M').

(Hi,x)

pro-~l(X,x)

x c X

and

def-dim

X

[i]

induces

satisfies

is p o i n t e d

X ~ 1

(p.

because

146)

inclusion

an epimorphism

~!(i)

the M i t t a g - L e f f l e r

1-movable

each

the

H

by Lemma

collapses

for a n y

condition 7.1.2.

to some

x E X.

for e a c h

Also

subset

of

its

1

1-skeleton.

7.1.8. then

Corollary.

(X,x)

is m o v a b l e

Proof. suppose have

that

X

(X,x)

is a s u b c o n t i n u u m

that

is a p r o p e r ~ 1

is a p o i n t e d

Sh(Y,y)

X

for e a c h

It is o b v i o u s

def-dim

there

If

and

Then

(M,x)

is m o v a b l e

subcontinuum (X,x)

(Y,y)

2-manifold

M,

x e X.

1-dimensional

= Sh(X,x).

of a

is

of

M.

7.1.7

By Theorem

(Y,y)

1-movable

x e X.

By Theorem

1-movable.

continuum is

for e a c h

So we

5.3.5

with

and by Proposition

5.2.6

it is m o v a b l e . Thus

(X,x)

7.1.9. but

is m o v a b l e .

Example.

not pointed Let

We construct

be

the w e d g e

we can consider

Zl(Sl

generated

(represented

and

b2 Let

2-cells sents

the

bI

v S2 )

(represented (K,k o) D1

X

being

1-movable

1-movable.

S1 v S2

by

a continuum

=

and

element

of two c i r c l e s . to b e a f r e e

(S 1 v S 2 uf D 1 Ug D 2, k o) D2

to

non-Abelian

by a homeomorphism

by a homeomorphism

S 1 v S 2,

where

It is w e l l - k n o w n

from

from S1

group

f :

F

S1

onto

onto

$2~.

be obtained

that

S I)

be attaching

~D 1 ÷ S 1 v S 2

repre-

93

and

g :

~D 2 ÷ S 1

v S2

b 2- 1 .

of

F.

there

Then

represents

exists

2 • b2 bI

b12b2"

~l(K,ko)

= H

that

r 2 ( b I)

n+l Pn

Zl(h)

= r 2,

(X,Xo)

= h Let

for us

W l ( h 2 • i)

show

X

is and

~l(h.

[i]

(Chapter

7)

h2 • i

and

if

f: L ÷ X n +

1

L ÷ K (I)

for

each

pointed

The

is a m a p , up

p~+2. n.

§2.

that

is d e f i n e d

by

where

(Xn,Xn) = (K,k O)

and

i)

to h o m o t o p y

L

By

are

Theorems

are

freely

is

a

as

observe

= r 2 • ~l(i)

map.

h • i

where

First

that conjugate,

3 and

8 in S p a n i e r

homotopic.

1-dimensional

f = i . fl

for

Consequently, CW

some

where

complex,

map

= S 1 v S 2. f = h 2 " f = h 2 " i • fl = h .

Thus

It f o l l o w s not

H ÷ H

1-movable.

inclusion

Hence

it is c l e a r

(K,ko)

n+l, ((Xn,Xn) , P n )'

~l(i)

factors

÷

r2 :

is t h e

fl :

and

n.

that

= r~.

where

S1 v S2 ÷ K

f

6.2.3)

= b l I • b 2 • b I.

= lim

each

(K,ko)

i :

then

3 b 2- 1 • b I

• b13.

a map

= b 2, r 2 ( b 2)

Let

element

(see T h e o r e m

h :

such

the

X

from

is

i • fl = h f

= p~+l.

f

1-movable.

Proposition

6.2.7

and

Lemma

7.1.2

that

X

is

1-movable.

Representation aim

of

this

of p o i n t e d

section

1-movable

is to p r o v e

continuum

X

has

the

shape

of

First

we

need

the

following

some

that

locally

continua. a pointed connected

1-movable continuum.

94

7.2.1. of

the

tains end

Proposition.

Hilbert

a neighborhood

V

of

be

shrunk

in

can

of

Proof. in

X

such

{Un}~= 1 that

~(i)

m

E X

BI(0)

> n. take

=

B2(1),

B1 * ~ *

B2 :

loop

in

w

Um

put

V It

for

Y

is

* w U

n

= im

in

path

in

V

with

inside

U

into

Q

conits

any

at

that

*

is

I ÷ Um

a loop

is

neighborhoods

of

have

a path

(m > n)

in

~(0), that

Then

U n + I.

= B1 * e *

a path

with such

= ~(i).

x° w

we

of

~l ( i ( U n + l ' U n ) )

I ÷ Un+ 1

such

Hence

there

B 2 rel.{0,1}

in

Um

Q

we

such

is

in

a U n-

that

.

neighborhood

to

see

that

Proposition.

Q

cube. such

~-homotopic

Proof.

U

of

maps

V

V

X

in

Let

be

each

any

map

f ~

polyhedron

rel.

Lo

Let

U

are

~-homotopic

be

Y

for

that

of

satisfies

Then

finite

star-refinement Let

X

basis

c X

= ~(0) , B2(0)

is

xO

in

easy

1-dimensional

U-near

of

each

{0,i}

xo

BI,B 2 :

BI(1)

7 =

any

Hilbert in

U

subcontinuum

find

Un

c U

and

= Un+ 1 .

7.2.2. the

that

rel.

point

e :

paths

at

= e rel.{0,1} Now

such

~i ( i ( U m ' U n ) )

I ÷ Un+ 1

Consequently,

1-movable

neighborhood

a decreasing

some

Suppose

and

= xo

each X

be

for

im

each

a pointed

X.

Let

Q

then

is

Q,

points

for

X

cube

neighborhood

X

If

in

be

Q

an

to

open

the

a locally

~ > 0 (L,L o) and

is

a

subcontinuum

a neighborhood where

L

is

subpolyhedron,

V a is

L ÷ Y.

of

[i],

is

(V,Y),

g :

covering Hu

÷

condition.

connected

there

Lo

a map

(see

desired

Q p.

such iii).

that

any

two

Let

U'

be

a

U.

a refinement

of

U'

covering

Y

such

that

any

two

of of

95

points

x, y

element

of

Let V=U

~ W U'

V'

{W

~ V' :

W

simplex g

the

same

we

can

extend

an

connected

and

is

Lo

pair

V

and

in

V'

onto

L

some

its

by

an

arc

in

some

let

such

g I Lo

such

that

g(s)

n Y,

Then

e-homotopic

rel.

a finite Take

image

a

f(s)

of

V'.

each is

and

and

g(v)

vertex an

where

g

is

the

= f I LO

for

U'

L

that

of

f(v)

of

where

subpolyhedron.

element

by

s.

For

is

as

element

1-simplex

a map,

(L,L o)

u I T(°) I

of

are

of

(V,Y)

lies

Theorem.

there

connected

exists

arc

so f

v

c T.

sI

lies

Then

joining

and

are

6 Y

g ( s o) are

U-near

and,

con-

L o.

a continuum

a decreasing

continua

deformation

such

retract

b.

X

has

c.

X

is

Proof.

the

of

and

by

that X. ]

shape

pointed

a ÷ b.

isomorphism

X

the

following

conditions

of

X for

a

sequence =

n j=l

X. 3

each

locally

Xl, X2, and

...

X

of is

locally

a strong

j+l

j ~ i, connected

continuum

Y

1-movable.

For

each

Theorem

j

the

morphism

S[i(Xj+i,Xj)]

4.1.6

the

continua

X1

and

X

is have

an the

shape. b ÷ c.

1-movable

It and

c ÷ a. the

be

equivalent: a.

same

÷

the

Lo

some

they

7.2.3. are

g in

of

sequently,

on

element

g(sl)

vertices

of

s { T

in

can

~ ~}.

(L,L o)

T

Define

and

n Y

polyhedron

triangulation each

~ V,

a star-refinement

f :

1-dimensional

W

n Y.

be

Suppose

n Y,

standard

follows Corollary

Assume Cantor

X

by

Corollary

7.1.6 is

ternary

5.2.8

implies

a pointed set

on

that

that

X

1-movable the

unit

Y is

is

pointed

pointed

continuum.

interval

I =

1-movable. Let [0,1].

C

be

96

Adjoin we

I

get

by

to

X

by means

a locally

A I, A 2, ...

connected

the

arcs

corresponding

to t h e

An

n z m.

~ Am

for

Setting

Yn = X

of

locally

Y1

= I uf X Using

sequence of open

as a

u

(or s i m p l e

Am,

Z-subset 7.2.1

of

of the

and

that

for

j a 1

Yn. c Vj 3

c Uj

2.

any

in

3. is a

each

U. 3

path

(2/j)-homotopy

and moving

the

contiguous

we

obtain

I uf X

X.

Denote

I uf X to

C.

limiting

can

Thus

We

a decreasing

cube

we

X.

on

assume

sequence

X.

Consider

Q.

find

inductively and

two

a sequences

path

the

;

U 1 = Q , U 2 , U 3, ... ,

following

conditions

are

satisfied:

c Vj_l, with

Vj_ 1 rel.{0,1} for

of

n o = n I = 1 < n 2 < n 3 < ...,

i.

inside

containing

curves)

Hilbert

onto

Q

each

path

C

I

7.2.2

V 0 = Q, V l , V 2, ... such

closed

n ~ i,

of

sending I uf X

of

subcontinua

integers,

f

continuum

u man

Propositions

subsets

a map

subintervals

connected

of

of

its

to a p a t h in

Vj_ 1

of

this

to a path

endpoints in

with path in

Yn' the

in for

X each

terminal

keeping

the

can

be

shrunk

n a i,

point

in

terminal

X

point

there fixed

Y nj- 1

Let j ~ 1

wn : with

Using induction,

I ÷ An, nj

n a i,

that

the

properties

i- 3 and

Corollary

Q -

determined

by

An .

Fix

3.5.3

we

can

construct,

by

maps

h(t,0)

I

2

= ~n(t)

and

÷ Vj-i hn(t,l)

= Wn(1)

(YI u n -ui k=l

for

(hk(I2)

each

gn :

E Y

'

hn(l,s)

path

s n < nj+l.

hn : such

be

12 ÷ Q

, hn(0,s) nj+ 1

t,s

u gk(I2))),

e I, Y1 u

(0) n

h n I I2 - $I 2 un k=l

= w

(hk(I 2)

'

is a n e m b e d d i n g u g k ( i 2 ) ) is a

into

97

Z-set and

in

Q,

diam

g n ( I × 0) = h n ( I 2 ) ,

g n ( t × I)

is an e m b e d d i n g

of

retract

(YI u

g n ( I × {0,i}

g n ( I 2)

and

of

Let

for a l l

t,s

= ~n(1) ,

c 12,

g n ( I × i)

gn I 12 -

c y

(I x {0,i}

nj_ 1 '

u {i} × I)

into

Q Then

< ~2 3

gn(l,s)

n-i u k=l

(hk(I 2)

u {i}

u gk(I2))

x I)

h n ( I × {i}

u hn(I2)).

is a s t r o n g

u {0,i}

x I)

deformation

is a s t r o n g

retract

deformation

h n ( I 2) .

Xj

= Yn. u u h n ( I 2) 3 n_>nj

u

u g n ( I 2) k>_nj+ 1

for

j >_ i. 0o

It is c l e e r Observe

that

X. 3

are

compact

and

X =

n X . j=l 3

that

Xj = X j + 1 u

Now,

the

for

u hn(I2) njsn0

such

t h a t an

c-neighborhood

of x

in

Y

is c o n t a i n e d n.

3 in

V

(recall

that

Y

is l o c a l l y

connected).

n.

]

Then

any

point

less

than

3'

less

than

3"

Since

can be

:

:

nj

connected x

continuum,

connected

at the r e m a i n i n g the p r o o f

7.2.4.

in

and

n. ]

X.. ]

7.2.3

a decreasing

sequence and

X3 of

Y

sequence

that

of

nj+ l ~ k < n(~)}

Thus

X. ]

that

X =

n j:l

for e a c h

locally

Pointed

infer

1-movability

a i m of this

section

curves,

i.e.,

we

the

need

we

con-

is l o c a l l y is l o c a l l y

X. 3

X2,

and

...

of

X

j > _ i.

Y

is to d i s c u s s

following:

locally

is a s t r o n g

Then

continua

Y].

=

such

retract

is p o i n t e d

of

X].

uf

properties

continua.

Y

is

that Y. 3

for e a c h

1-movable.

on curves.

1-dimensional

If

j+l

deformation that

continuum.

1-movable.

Xl,

connected

is a s t r o n g 7.2.3

U

X.

1-movable

j+l

By T h e o r e m

of

is a

is c o n c l u d e d .

is p o i n t e d

retract

and

points.

Y

such

~ k}

it is c l e a r

then

Take

is

x

of d i a m e t e r

the c o m p o n e n t

is a map,

n Y. j=l 3

First

Y

x

:

that

be a p o i n t e d

a decreasing

1-movable

n(~)

X

deformation

The

of

infer

of

of T h e o r e m

continua

§3.

:

Let

Proof.

j a i.

we

to

nj ~ n < nj+ I} u

Corollary.

f : X ÷÷ Y

connected

:

distance

by a s u b c o n t i n u u m

u {gk(I2)

is a n e i g h b o r h o o d at the p o i n t s

Y =

V

whose

~ n < nj+ I} u u{gk(I2)

connected

Thus

to

nj+ 1 ~ k < n(~)} u

Yn. u u {hn(12) ]

taining

joined

n ~ n(~),-

Xj = Yn. u u {hn(I2) ]

J u{gk(I2)

locally

y c gn(I2),

of p o i n t e d

99

7.3.1

Lemma.

a positive

Let

integer.

sequence

of

G

be

Then

subgroups

a free

there

of

is

the

rank

of

each

no

group

infinite

and

properly

let

R

be

increasing

G

U1 c U2 c where

non-Abelian

U.

...

is

c Un

less

. ..

c

than

or

,

equal

to

R.

1

Proof.

Suppose,

on

the

contrary,

that

each

U.

is

a proper

1

subgroup

of

U i + I.

Let

U = u{U.

:

i -> i}.

1

It be k

follows

contained ~ i).

[i],

p.

x I,

x 2,

and

by

that in

U

Ui

is

for

not

some

i ~ 1

By

the

Nielsen-Schreier

95)

the

group

...,

Xn,

abelizing

7.3.2

... U

is

free.

of

U.

Then

If

get

X

generated which

theorem

U

we

Theorem.

finitely

So

would

(see let

Xl,

us

...,

a

1-movable

imply

U

would

U = Uk

for

Karras-Magnus-Solitar take

generators

XR+ 1 ~ U m

a contradiction.

is

(otherwise

The

for

result

curve,

then

some

m

follows.

X

is

pointed

1-movable. Proof. x

c X

By

and

Proposition

(X,x)

polyhedron

is

gn :

gn(Xn+l) n+2 Pn " gn

and

Then Zl(Xn,Xn)

A1

each

+ Xn+2

So

get

that

let

n.

such Hn :

We

where

may

that

X

is m o v a b l e . Xn

assume

n+2 Pn " gn

Xn+ 1 × I + X n

is

that n+l = Pn

be

an for

Let 1-dimensional

each

n

there

and

a homotopy

joining

n+l Pn

= Let

for

Xn+l

= X n + 2.

we

n+l, = lim((Xn÷ 'Xn) ' P n )'

connected a map

5.2.6

im

A1 c A2

71

( n+2 Pn ) and

determined

and

A 2 = im

t - A 2 • t -I by

the

c AI,

, n+l, zltPn ). where

loop

{Hn(Xn+l,S)

:

0 -< s -< i}

t

is

the

element

of

100

directed We

from now

0

to

i.

have: -i A1 c A2 c t

A 1 c t -I

where

each

7.3.1

now

The

of

A 1 • t c t -2 • A 1 • t 2 c

the

groups

...

t -k • A 1 • t k

and

c t -k • A 1 • t k

has

the

same

c

...

finite

,

rank.

Lemma

-i

implies

result

• A 1 • t,

A1

=

t

• A 1 • t,

so

that

A1

=

A 2.

follows. oc

By

the

plane,

Hawaiian

where

radius

1 n

Sn+ 1

is

7.3.3 the

the

onto

earring

the

A

the

x

assume

c X

a i.

By

95)

the

each

n.

point.

r

By

the

circle

of

n n+l. l i m ( v S k, r n ), ÷ k=l

to

is

Theorem

1-movable shape

and

im

• n+l~ zl%Pn

for

is

the

a retraction

such

that

7.1.7

the

Hawaiian

of

curve

X

a finite

has

wedge

either

of

the

circles,

shape

or

a point.

that

p.

is h o m e o m o r p h i c

circles,

pointed

We

[i],

( y _ n)l 2 = n21 }

of

1-movable.

polyhedron

n

of

base

1-dimensional

for

u A n n--i

(0 i) ' "

wedge

earring,

Let

may

x2 +

subcontinuum

n

Theorem.

Proof. an

the

pointed

of

center

the

n+l

Hawiian

shape

mean

E R2 :

Hawaiian

is

is m a p p e d

earring

we

{ (x,y)

the

that

n v Sk k=l

where

of

An =

about

Observe

Earring

(X,x) for ~

n+l, = lim((Xn÷ 'Xn) ' P n ),

each

(p~)

the

P n + l ))

= im

~ 1 (Pnn+ l )

Nielsen-Schreier

group

im

, n+l,

~i tpn

for

m

( n+2, =

)

Xn

is

n.

( n+2, ) (im 71

where

im

71

is

i.

Then

(n+l)

Pn+l ) =

theorem

> n +

(see

a finitely

im

71

Pn

Karras-Magnus-Solitar generated

free

group

• n+l,

Suppose some

n.

Xn,l,Xn,2,

..., X n , k n

• n+l, im 71 tPn )

Since

by the F e d e r e r - J o n n s o n

are g e n e r a t o r s

(see F e d e r e r - J o n n s o n

Xn+l, I, Xn+l, 2,

k n + 1 a kn,

71 (Pn n + l ) (Xn+l, i)

( n+l. 71 Pn ) (Xn+l, j) = 1 Starting above

from

condition

..., X n + l , k n + l

= Xn, i

for

n = 1

for

[i])

of

)

for

we infer

t h a t t h e r e are such that

and

i ~ kn

j > k n.

w e can find sets of g e n e r a t o r s

for e a c h

),

, n+2 im ~i tPn+l )

•

generators

im ~l tpn

. n+2 im 71 tpn

is the i m a g e of

theorem

of

satisfying

the

n. (n+l)

N o w the c h o i c e naturally

of g e n e r a t o r s

Xn,l,

..., X n , k n

of

im 71 Pn

induces maps k

fn :

n v S. ÷ X i=l z n

n+l kn+l Pn " fn+l ~ f n " rk n

s u c h that

for

n ~ i.

k

Also

there

are m a p s

gn : X n + l ÷

n v Si i=l

such that • kn+l,

71 (fn , gn ) = 7 1 t,P nn+l,)

n+l f n " gn = Pn

Hence

Thus

X

and

of c i r c l e s

or it is h o m e o m o r p h i c

result

~i (gn " fn+l ) = ~i trn

has the shape of

to the f i n i t e w e d g e m,

and

follows.

kn+l g n " fn+l -~ rn

)

for

for

n >_ i.

kn rkn+l) lim( v S i, k , which ÷ i=l n if

km = km+ 1

to the H a w a i i a n

n > i.

is h o m e o m o r p h i c

for s u f f i c i e n t l y

earring,

otherwise.

large The

102

Notes

Theorem

7.1.3

is due to Dydak

Theorem

7.1.7

is due

Krasinkiewicz McMillan

[i].

[7].

to M c M i l l a n

Our proof

[1,2]

of this

theorem

independently uses

a trick

to

from

[2].

Example

7.1.9

is due

Theorem

7.2.3

is due to K r a s i n k i e w i c z

Corollary Krasinkiewicz

Theorem McMiLlan

and

7.2.4

is due

Strok

and J. Dydak

to M c M i l l a n

(see Dydak

[5]).

[2].

[i] and i n d e p e n d e n t l y

to

[i].

7.3.2

is o b t a i n e d

[i] for more

Theorem

to M.

7.3.3

general

is due

by A. Trybulec results).

to Trybulec

[i].

in his

thesis

(see also

Chapter Whitehead

In this analogous

chapter,

to the

VIII

and H u r e w i c z

Theorems

in S h a p e

we

to p r o v e

results

are

classical

going

Whitehead and H u r e w i c z

Theory

in

shape

theorems

theory

in h o m o t o p y

theory.

§I.

Preliminary

8.1.1. of p o i n t e d

results.

Lemma.

Let

connected

complex

of

X n) .

trivial

group

for

k _< m,

qn:

(Xz, A£

pairs

If

of

CW

(Zk(Xn,

An,

in p r o - G r then

((Xn' An'

Xn)'

complexes

u XZ (m) , x Z) ÷

n

be an i n v e r s e (i.e.

, n+l) x n) , n k ~ P n )

(the t r i v i a l

for e a c h

n+l, Pn )

there

element exist

(X n, A n , x n)

is a sub-

is i s o m o r p h i c

pro-Ens

of

Z > n

such

An

sequence

if

to the

k = i)

a n d a map

that

qn

Ji -- Pn'

where ji: is the

inclusion

Proof. Obviously, for m=

CW

(Xi, Ai,

xl)

÷

(m) u XZ , xl)

(X£, Ai

map.

We are

going

it is true complexes).

for

to p r o v e m = 0

So s u p p o s e

Lemma

8.1.1 by i n d u c t i o n

(it s u f f i c e s it is true

for

m.

Theorem

m = k _> 0

and

2.1.3 take

k + i. Let

to the

n o ~ N.

trivial

Since

(~k+l(Xn ' An'

Xn)'

of p r o - G r (pro-Ens n1 such t h a t ~ k + l ( P n ) is trivial. nI > no o there exist n2 > nI and a m a p

q: such

to a p p l y

on

that

element

(Xn2

q'3n2

:

if

Then

~k+l(Pno

(n+l) Pn

, i

q)

if

(k)) ~: (Bk+l , S k) ÷ (X(~+l) , Xn2

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

k = 0),

By the

(k) An2 u Xn2 , Xn2 ) ÷ (X nl n2 n1

-~ pn I.

Zk+l

nl

, x

there

inductive

nl

exists

assumption

)

is trivial.

Consequently,

104

is a c h a r a c t e r i s t i c is h o m o t o p i c Ano

rel.

n1

m a p of some

Sk

(k+l)-cell

to a m a p

Thus there exists

joining

"ql X (k+l)

Pn o

a homotopy

we g e t

that

there

exists

O

O

such t h a t

8.1.2.

values

qno:

of p o i n t e d

(Xn2

.

qn

Let

connected

lie in

i n d u c e d by

((Xn' An' CW

then

((An' Xn)'

{[i(A n, Xn) ]] ~ n=l

which

n+l Pn )

Xn)'

(Zk(Xn, A n , Xn),

pro-Gr

(to the t r i v i a l

Proof. exists

completes

÷

the proof.

sequence

If the m o r p h i s m

((Xn' Xn)'

isomorphisms

is i s o m o r p h i c of p r o - E n s

t h a t for e a c h

])

pro-~k(i)

for

to the t r i v i a l

if n

[Pn

k < i)

for

and for each

k s m,

group

in

k sm. k s m,

there

a homomorphism hn,k:

such t h a t Thus

zk(i(An, im ~

k e r ~k (qnn+l )

Z k ( X n + l , Xn+ I) ÷ Zk(An, , n+l,

Xn)) .hn, k = ~ k ~ P n

( n+l. k Pn ) cim

for

We are g o i n g

n >- 1

~k(i(An, and

to p r o v e

, n+2 ~k~Pn ) : ~k(Xn+2,

is the t r i v i a l

homomorphism

In the s e q u e l by boundary

)

n+l h n , k . Z k ( i ( A n + l , X n + l ) ) = Zk~ n ).

and

Xn))

x n)

and

ker ~ k ( i ( A n + l ,

Xn+l))

k 1

a nd

A n , x n)

k < m.

A, x) ÷ Zk_l(A,

x)

we d e n o t e

the

homomorphism.

Suppose 8(a)

))

element

We m a y a s s u m e

, Xn2)

r n+ll

]) ÷

induces

zktPn

By e x t e n d i n g

n

be an i n v e r s e

complexes.

qn

, n+l,

lie in

O

[ n+l~

~:

A

o u X (k+l) An2 n2

t

n2

3n 2 = Pn

p a i r s of

a

Xn2

O

Lemma.

values

q

no

a map

O

Pno

× I + X

n2 (Xn , A n , Xn )

then

t

whose

n2 H

n2

n2

(k)

rel.

and a m a p w h o s e

n2

X

~': B k+l ÷ X

H: X (k+l) n2 n1

in

a ~ Zk(Xn+2,

~ ker ~k_l(i(An+2,

An+2,

Xn+ 2) ,

k -< m.

Xn+2) ) a n d t h e r e f o r e

Then

~(a)

, n+2 e ker ~k_l%qn+l ) .


n

i.

Assume

n.

By L e m m a t a

and a map

gnl (Xe(n) ' x

(n))

Hence

(X, x) +

if

g:

Xn

gn:

-- pn~(n)

is a s u b c o m p l e x 8.1.1

and 8.1.2

of

(Y~(n)'

i(Xn' Y~(n) )

and

for each

y (m+l) (X (n) u e(n) , x and

Yn

(n)) ÷

n

fn : i(Xn' there

(Xn, x n)

exist such that

Yn ) "gn _- q~(n) IX (n) u Y is a map

where

Yn )

(m+l) (n)

def-dim(X,x)

n.

"q~-~(n)

= q~(n)"

Then

_a(n')

fa(n)'Pa(n)

= q~(n

"gn'

Ct ( n ' ) -- q a ( n ) " f a ( n ' ) ' g n

ct ( n ' ) ' -- q a ( n ) " q ~ ( n ' )

= qa(n)

and, c o n s e q u e n t l y ,

n' 0t!n') Pn " (Pn "gn') (see C o n d i t i o n

2 of C o r o l l a r y

By T h e o r e m such that

or(n) -" Pn "gn

4.1.6,

S[Pn]. ~

S[qn].f.g

=

exists

S [ p : ( n ) - g n]

4.1.6,

"q~ (n)

is a right The result 8.2.2.

n.

inverse

Theorem.

and

] : S[qn ]

(Y,y)

÷

(X,x)

Then

= S[q:(n).f

for each

(n)-g n] =

n

to

(Y,y)

,

f.

follows.

continua.

k s m + 1

for each

g:

we have f-g = S[id

pointed

a shape m o r p h i s m

= S[fn].S[Pn]- ~ = S[fn.p:(n)-gn]

and by T h e o r e m

n

8.1.3).

there

S[qn~(n)

i.e,

for each

Let

f:

If pro-~ k

max(def-dim

(X,x) (f)

÷

(Y,y)

be a shape m o r p h i s m

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

X, d e f - d i m

Y)

~ m,

then

is a right

shape

of p r o - G a f

of

for

is a shape

isomorphism. Proof. Then pro-~k(g)

By T h e o r e m

8.2.1

there

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

for

k < m + 1

inverse

and t h e r e f o r e

~

of g

has

108

a right

shape

inverse

h.

Then

f = h

and, c o n s e q u e n t l y ,

(X, x)

÷

f

is a shape

isomorphism. 8.2.3.

Corollary.

of p o i n t e d

continua

If

and

(X,x)

then

f

(Y,y)

that

~k(f)

are m o v a b l e

By T h e o r e m

of p r o - G r

for

By T h e o r e m 8.2.4. pointed all

such

f:

(Y, y)

be a shape m o r p h i s m

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

and

max(def-dim

X,

for

k ~ m + i.

def-dim

Y)

~ m,

is an i s o m o r p h i s m .

Proof. phism

Let

8.2.2

k.

the

Theorem.

continua

6.1.8

one

can

get

that

is an i s o m o r -

pro-~k(~)

k ~ m + I.

Let

such

If d e f - d i m

result f:

that

X

follows.

(X,x)

÷

pro-~k(~)

is f i n i t e

(Y,y)

be a shape

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

and

Y

is m o v a b l e ,

morphism

of

of p r o - G r

then

f

for

is an

isomorphism. Proof.

Let

(X,x)

= l i m ( ( X n, Xn),

lim( (Yn' Yn ) ' qnn+l) '

where

connected

CW

and

finite

By T h e o r e m

5.3.5,

(X n

x n)

n+l Pn ) and

and

(Yn'

(Y,y)

Yn )

=

are p o i n t e d ,

complexes.

we m a y

assume

dim X ~ n

< + ~

and

O

dim X n ~ no

for all

by a s p e c i a l

morphism

~': induced a map

by the gn:

Let such with

that

÷ Yn+2

8.2.2

with

n c N.

assume

÷

and

((Yn' that

that

Yn )

_f

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

, [ n+l] ) qn

for e a c h

n

there

exists

q ~ + 2 . g n ~ q~+l.

By C o r o l l a r y g: Y n + l

÷ Y

m

8.1.3, there

there

exists

exists a map

m > n + 2 h: Y n + l

÷ X

n

m = qn.g.

us take

g = gm-2"''''gn : Yn+l

m fn "h = q n ' g m - 2 " ' ' ' ' g n map whose

we may

{[fn]} ~n=l ,

for any m a p

fn-h Let

fix

Also,

r n+ll J) ((Xn ' Xn) ' [Pn

family

Yn+l

us

n.

values

n+l = qn

lie

the m o r p h i s m

in f

÷ Ym"

and t h e r e f o r e

n+l qn

(n o ) . Hence def-dim n is an i s o m o r p h i s m .

Y

Then is h o m o t o p i c Y ~ no

to a

a n d by T h e o r e m

109

8.2.5. of pointed If

X

Corollary. continua

is movable Proof.

g: (Y,y) ÷

and def-dim Y

with

Then pro-~k(~) the m o r p h i s m finishes

§3.

g

f:

(X,x) ÷

(Y,y)

such that pro-~k(~)

By T h e o r e m

(X,x)

Let

be a shape m o r p h i s m

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

is finite,

then

f

for all

k.

is an isomorphism.

8.2.1 there exists a shape m o r p h i s m f . ~ = S[id(x,x)].

is an isomorphism

is an isomorphism.

for all

Thus

k

f

and by T h e o r e m

8.2.4

is an isomorphism which

the proof.

The Hurwicz Theorem in shape theory.

Recall the classical

result by W. Hurewicz

(see Spanier

[i],

T h e o r e m 5 on p. 398). 8.3.1. If

~m(K,

Hn+I(K)

Theorem.

k) = 0

for

Let

(K, k)

m < n

be a pointed connected

(nzl),

then

complex.

~n+l(K,k) : ~n+l(K,k)

÷

is an isomorphism.

We are going to prove an analogous 8.3.2. pro-~k(X,

CW

Theorem.

x)

Let

is isomorphic

(X, x)

result in shape theory.

be a pointed continuum.

to the trivi~ group for

prO-~m+l(X,

x) : prO-~m+l(X,x)

k ~ m

If (m~l),

then

÷ pro-Hm+l(X)

is an isomorphism of pro-G~. Proof.

Let

pointed, connected is isomorphic Lemma

8.1.) Let

over

and finite

CW

complexes.

to the trivial group for

that

p~+llXn+ (m)1

(Yn' Yn ) =

Xn(m)

Xn) , Pnn+l) '

(X,x) = l~m((Xn,

Since

pn-n+llx(m)l n+l

of

p~+l

k s m,

Xn) ,

(Xn

x n)

for each

where

• n+l,

C(kl m) )

(see

n is the cone

there is a map

Yn+l ) ÷ (Yn' Yn )

such that

are

(~k(Xn, Xn) , zktPn

we may assume

is null-homotopic,

n+l qn : (Yn+l, being an extension

Since

is null-homotopic

(Xn u C(X~m)),

where

q~+l(Yn+ I) c X n.

Hence

))

110

n+l) (X, x) = lim((Y n, yn ) , qn Since

Yn(m) = C ( X ~ m))

we infer

Zk(Yn, By T h e o r e m

8.3.1

yn ) = 0

the H u r e w i c z

for

k s m.

homomorphism

~ m + l ( Y n , Yn ) : ~ m + l ( Y n , Yn ) ÷ H m + I ( Y n) is an i s o m o r p h i s m

for each

prO-~m+l(X,x): is an i s o m o r p h i s m 8.3.3. If

which

n.

Hence

prO-~m+l(X,x) completes

Corollary.

Let

÷ Hm+I(X)

the proof.

(X, x)

be a m o v a b l e

pointed

continuum.

v

v

~k(X,

x) = 0

for

k s m

(m>_l) ,

then

v

~m+l(X'x) : ~m+l(X'x)

+ Hm+ I(x)

is an isomorphism. Proof.

By P r o p o s i t i o n

to the trivial

group

for

6.1.3 we get that pro-zk(X,x)

k s m.

prO-gm+l(X,x) : prO-~m+l(X,x) By a p p l y i n g

the limit v

By T h e o r e m

÷ prO-Hm+l(X)

functor

l~m:

pro-Gr

8.3.2,

is i s o m o r p h i c

the m o r p h i s m

is an i s o m o r p h i s m + Gr

we infer

of pro-Gr.

that

v

~m+l(X,x) : Zm+l(X, x) ÷ ~m+l(X) is an isomorphism. NOTES Lemma

8.1.1

V .

is due to M a r d e s i c

Theorem

8.2.1

is due to D y d a k

Theorem

8.2.2

is due to M o s z y ~ s k a

[2] and C o r o l l a r y

and C o r o l l a r y

are due to D y d a k

to K e e s l i n g

[2] and i n d e p e n d e n t l y

to M o r i t a 8.2.3

8.2.4

For a d i s c u s s i o n

is due

8.2.5

of the W h i t e h e a d

theorems

[2].

in shape

theory,

see

[i]. Theorem

Artin-Mazur

[4].

[2].

Theorem

Dydak

[4].

8.3.2

is a special

case of a more

[i].

Corollary

8.3.3

is due to

Kuperberg

[i].

general

result

from

Chapter Characterizations

and

of P o i n t e d

§i.

Preliminary

In this properties

where

section

Proof. [i],

pp. Let

(Kn,k n) there

If

which

If] :

(K,k)

÷

connected

we

are

going

CW

complex,

to use

and

for

~" h = C

o

sets

p~+l

=

[f]

then

idempotent, [f]

splits.

from Spanier

functors.

each

of p r o - H T

n.

defined

By P r o p o s i t i o n

÷

l(K,k)

(K,k)

and

be the h o m o t o p y base

and

h •g = category

points

a contravariant

2.3.14

h :

(K,k)

(K,k)

If]. of p a t h - c o n n e c t e d

(.see S p a n i e r

functor

÷

H

[i],

from

Co

p.

pointed

spaces

406).

to the c a t e g o r y

as follows: H(X,x)

H(u] (~) = v . u

for

= pro-HT((X,x),

any h o m o t o p y

(K,k))

class

u :

(X,x)

÷

(Y,y)

and

any m o r p h i s m :

In o t h e r

words

(see S e c t i o n

by

of p r o - H T

(K,k)

nondegenerate

to i n v e s t i g a t e

the r e s u l t s

be an o b j e c t

(K,k)

we n e e d

is a h o m o t o p y

n+l, ((Kn,kn) , Pn )

=

homotopy

(K,k)

(K,k)

We d e f i n e

and

results

concerning

that

pointed

some

406-412

exist morphisms

having

ANSR's

ANSR's.

In the p r o o f

=

Let

prove

is a p o i n t e d

g :

such

we

Theorem.

(K,k)

Properties

results.

of p o i n t e d

9.1.1

IX

H

(Y,y)

÷

(K,k)

is the r e s t r i c t i o n

3 of C h a p t e r

II) .

of pro-HT.

of

p r o - H T ( . , (K,k))

to

Co

of

112

We [i],

are

p. So

going

to p r o v e

that

H

suppose

that

(A,a)

(X,x)

H ( [ f 0 ] ) (u) Then

÷

[j] : and

(X,x) let

h • u.

÷

u

= H ( [ f l ] ) (u) , i . e . ,

there

suppose

il :

:

Thus

H

for

homotopy means

2.3.9).

By

splits

9.1.2

CW

(see L e m m a

complex

finite

CW

complex.

Proof. Replace

x

has

becomes

u

(Z,z)

÷

(K,k)

with

indexed

family map,

of o b j e c t s

where

and is

v ( X l , x I)

that

and

pp.

410-411,

a morphism

(X,x)

+

+

(L,i)

(K,k)

is

~:

there

(L,i)

of p r o - H /

with

limit

(K,k)

[1],

÷

there

(K,k) is a

~ = ~ • v'.

of

(K,k)

stable

(see D e f i n i t i o n

which

implies

that

2.3.15).

Let

X

K.

be

Then

the mapping

f :

[j'] :

[j]

is a n e q u i v a l e n c e .

inverse

2.3.11,

by

a map

because

~ = H ( [ j ] ) (g • [j']).

in S p a n i e r

(X,x)

is a n

Theorem

same

and

it is c l e a r

v :

By

the

(K,k)

inclusion

(L,Z)

v' :

Proposition

CW

the

14

complex

that

÷

functor.

ll a n d

class

Theorem.

be Then

any morphism

finite

which

[f0 ] ,

[fl ] .

class

is a n

= ~ H ( X l , x I)

by Theorems

that

This

(Xl,xl).

is a n h o m o t o p y

a pointed

(A,a)

i.e.,

{ (Xl,xl)} 1

H(v(XI,xl))

Hence,

[f]

that

of a l l

{H[il]}l

unique

of

[j'] " [j].

[j'] • [j],

( X l , x I) ÷ v ( X l , x I)

the wedge

such

Spanier

satisfy

u_- [f0 ] : u "

is a h o m o t o p y

~ = g • h • u = g.

exist

(see

is a n e q u a l i z e r

~ H(X,x)

h " u =

Now

(Z,z)

[f0 ] = h - u • [fl ] :

is a n e q u a l i z e r ,

let

functor

407).

[fl ] :

Then

is a h o m o t o p y

2.2.6

X

× S1

we may

cylinder

homotopy X ÷ X

a topological

type

whose

has

space the

dominated

homotopy

assume

that

X

of

given

cellular

as image

the

X.

Then

lies

in

type

is a

by

a

of

a

CW

complex.

map

K ÷ X, X ÷ K

the

cellular

map

K

embedded

in

X,

and

113

which

is h o m o t o p i c

Define

the m a p p i n g

(x,l)

with

T ( i d x)

= X × S1

p.

to the

torus

(f(x),0)

T(f)

for e a c h

have

the

by

taking

x c X.

same

X x I

Since

homotopy

type

and

f ~ id x

identifying

, T(f)

and

(see L u n d e l l - W e i n g r a m

[i],

122). Define

s+t~l

a homotopy

and

Then hl(T(f))

ht(x,s)

type.

But

type

Lemma.

K0,

Let

If

type

Take f0 :

K1

gl :

Then

ht(x,s)

=

(x,s+t)

for

s+tal.

c T(flK)

= X × S1

X,Y

and CW

and

of a f i n i t e CW

for

0 ~ t ~ 1

and

K2

K0 ÷ K1

CW

have

complex,

n Y

are

complex,

the

so the

compact then

same

homotopy

theorem

ANR's

X u Y

is proved.

each having has

the h o m o -

equivalences

X n Y ÷ K0,

and

X

T(f;K)

complex~

homotopy

fl :

are and

X + K1

finite g2 :

and

CW

f2 :

K0 ÷ K2

be c e l l u l a r

=

[fllX n Y] • If0 ]-I

[g2]

=

[f21X n Y] • [f0]

gl " f0 = fl IX n y

and

Y ÷ K2'

complexes.

[gl]

maps

such

that

and

-i

g2 " f0 = f2 IX n y,

and,

consequently,

are m a p s hl :

such

by

for

ht(T(flK))

is a f i n i t e

of a f i n i t e

Proof.

there

÷ T(f)

(f(x),s+t-l)

T(f)

T(flK)

the h o m o t o p y

where

=

T(f)

c T(flK).

9.1.3.

topy

ht :

h O = idT(f) ,

Consequently,

that

homotopy

hl(X)

h(y)

X ÷ M ( g I)

= f0(x)

= h2(x)

and

h2 :

Y ÷ M ( g 2)

for

x e X n y

and

hl,

h2

are

equivalences.

Define and

identity.

h : X = h2(Y)

u y ÷ M(gl) for

y c Y.

u M(g2)

by

h(x)

= hl(X)

for

x • X

114

We tract

are of

going M(h)

Since

infer

= M(hl)

X n Y

M ( f 0) = M ( h I) we

to p r o v e

Hence

X u Y u M ( f 0) X u Y

implies

Thus

§2.

is a s t r o n g

deformation

deformation

retract

retract

deformation

of

retract

of

M(hl),

M(hl). of

M(h2).

retract

of

M(h).

is a s t r o n g

deformation

retract

of

X u Y u M ( f 0)

is a s t r o n g u M ( g 2)

deformation

have

are

for e a c h

Let

the

retract

of

same homotopy

M(h).

type

and

2.

(X,x)

is a n A N S R

3.

X

has

the

4.

X

× S1

5.

X

X

is a p o i n t e d

ANSR

provided

c X.

be a continuum.

Then

the

following

has

1 ÷ 2

the

of a shape

and

X

By Theorem

and

we may

for

some CW

x

{ X,

complex,

of a f i n i t e

CW

is p o i n t e d

complex,

1-movable.

is o b v i o u s .

2 ÷ 3.

(X,x)

ANSR,

shape

is a n A N S R

÷

ANSR's.

equivalent:

is a p o i n t e d

(K,k)

x

X

X

Proof.

of p o i n t e d

A continuum

i.

connected,

of

deformation

M(gl)

Theorem.

conditions

(K,k)

retract

is a s t r o n g

Definition.

9.2.2.

~ :

re-

follows.

is a n A N S R

complex

deformation

deformation

is a s t r o n g

X u Y

and

is a s t r o n g

is a s t r o n g

Characterizations

9.2.1 (X,x)

that

X u Y

the r e s u l t

X

Y u M ( f 0)

Moreover, which

and

X u M ( f 0)

Analogously

X u Y

u M(h2).

is a s t r o n g

n M ( h 2)

that

that

two such

assume

5.4.4

there

shape morphisms that that

exist

f :

(X,x)

g • f = S[id(x,x)]. K

a finite

is c o n n e c t e d .

÷

pointed

(K,k)

Since

X

and is

CW

115

Take

a map

Corollary

CW

S [ h 2]

Theorem

complex

such

9.1.1

each

u-

Lemma

v

3 ÷

4.

lim

(Xn,

Theorem type

that

consequently

maps u

(K,k)

~- id

(X,x)

where

h

[h]

u :

v"

both

S[h]

= f.

~- h 2

g

(see

Corollary

i.e.,

÷

and

(L,£)

there v :

3.2.2. exists

(L,Z)

÷

a (K,k)

"

(L,£) =

by

splits,

(L,£)

and

(Kn,kn)

are

(K,k)

x S1

isomorphic n+l Pn

and

in

pro-Sh

= S[h]

for

4 ÷

Let

By

Since

Sh(X,x)

5 ÷

f : a

X ÷ L

r

(X n

is

finite n+l

×

L

a

is

finite

(e.g.,

(Xn' [Pn

that

lim(Xn+

Sh(X

Corollary

that

a finite

×

K

L

I.

× S I,

× S I)

the L

By homotopy

isomorphic

to

' lds1 ])

isomorphic

pn+l,~ x i d s l

7.1.6

= Sh(K),

the

where

space

_< S h ( X

x S I, (x,a))

(X,x)

is

Let K

is

5.4.4).

assume

is

subcomplex

of

is

Let

Then

equivalence).

complex

is

S 1, [pn+l

by

complex.

complexes.

a shape

CW ])

CW

CW

in to

it

follows

that

X for

1-movable

for

× S1 any

K

is

is

pointed

two

each

x

a

finite

CW

com-

1-movable.

points

x

• X,

• X.

Thus

X

a

e S I,

is

1-movable.

that

Theorem

complex

exists

x S1 =

plex.

pointed

are

= Sh(L).

= Sh(K).

5.

infer

Xn

where

Sh(X)

pro-HT.

X

× S I)

CW

Since

infer

in

Since Sh(X

x S I.

we

the

where

there

implies

= Sh(L),

where

a finite f(X),

L

which

Sh(X)

n+l, Pn ),

9.1.2

of

= Sh(L,£)

Suppose

by

pro-HT,

may

such

idempotent

and

2.3.15

containing

such

and

and

-~ h

Sh(X,x)

dominated

we

(K,k)

n. Thus

X =

÷

the

n+l, ((Kn'kn) ' Pn ),

to

L

= S[h]

(L,~)

that By

(K,k)

3.2.2) .

Then By

h :

that

f : a

X ÷ K

finite

and CW

Let

h :

K ÷ K

h(k)

= k

for

g :

K ÷ X

complex

be

an 4

two

shape

morphisms

g • f = Sh[idx]

be

a map

such

some

point

k

that

• K (0)

S[h] .Let

(see =

f • g.

We

116

Y = l ~ m ' K n , P nn+l ), By L e m m a

2.3.15

by C o r o l l a r y Hence i m ~ ( h m)

the

h 2m

at

to

Consequently,

h 2m

loop

that

such

Let

(K,k)

u :

h 4m

is

•

(h 2 m " B -1)

= u

Sh(Z,k)

*

§ 3.

Sh(Y)

= Sh(X).

Hence,

1-movable. m ~ 1

to

h 3m

hm

such

is

and

that

e-homotopic

h 3m

is

(h 2 m " ~ ) - h o m o t o p i c

to

to

to

h 2m.

(hm " ~)

*

h 4m.

Take

(h 2m" ~)

a

(this

= im Z l ( h 2 m ) ) .

be a m a p w h i c h to

to

= Sh(Z),

is

B-homotopic

h 2 m - u, u

u 2,

is

which

B *

i.e.,

h 2m.

(h 2 m " B) *

u ~ u 2 rel.

2.3.14

to

is

k.

Z = l~m( Z n , q nn+l ),

where

By P r o p o s i t i o n

= Sh(Z,k)

for

is an A N S R

for

the p r o o f

of T h e o r e m

The

Theorem

Union

Lemma.

polyhedron.

Let

If

X

then

there

exists

that

each

Un

of

Proof. X

that

Consequently,

n.

Sh(X,x)

9.3.1

Let

such

*

(B-l)-homotopic

(X,x)

and

retract

Sh(Y)

n.

Zn = K

and

we h a v e

~ Sh(K,k).

infer

ANSR

integer

is h o m o t o p i c

(K,k)

u 2.

for e a c h

Since we

(h m " ~)

im ~ l ( h m) ÷

to

Of course, n+l qn

is p o i n t e d

(h 2 m " B - l ) - h o m o t o p i c

(B-l)-homotopic

we h a v e

for e a c h

h 4m.

h 2m" B

because

= h

n ~ m. k e K

is

is p o s s i b l e

Then

Y

(hm - ~ ) - h o m o t o p i c

(h2m-e)-homotopic

p~+l

4.1.6

a positive

for

~

is

B

space

exists

a loop

and

and T h e o r e m

= i m ~ ( h n)

Take Then

7.1.6

there

Kn = K

where

and

U

X

x c X.

(see T h e o r e m Thus

X

7.1.3),

is a p o i n t e d

is c o m p l e t e d .

be a c o m p a c t u m

a basis

Suppose P

9.2.2

is e m b e d d e d

for e a c h

each

x c X

for A N S R ' s .

as a

{Un}~= 1

is a c o m p a c t n

each

ANR

and

having

Z-set

the

in the H i l b e r t

of n e i g h b o r h o o d s Un+ 1

shape

of

is a s t r o n g

X

of a c o m p a c t cube in

Q, Q

such

deformation

n.

Sh(X)

be e m b e d d e d

= Sh(P), as

where

Z-sets

in

P Q.

is a c o m p a c t Take

polyhedron.

a homeomorphism

117

h :

Q - P ÷ Q - X

basis to

of

and,

and

moreover, Hn : Vn

for

x

each Let of

By each

for

exist

x I ~ Vn Hn(X,t)

= x

for

its

Hanner's

open

(see

Borsuk

= x,

being

[5],

is

admits

of

Vn

H-I(P)n

- P)

for and

H n (x,l)

and

a

homeomorphic

V n ÷ Vn+ 1

rn :

= h(Vn

each

Vn

P

retract

e Vn+ 1

Un

that

each

Hn(X,0) x

Then

subsets,

Theorem

that

a retraction that

two

Observe

deformation

such

u h ( V n - P).

of

such

a strong

Un = X

each

a

= r n (x) = P

u I n t ( U n)

× I. is

the

an ANR.

p.

97)

U

is

n

an ANR

for

n. Define

Gn :

Un

Gn(x,t)

= h.

Hn(h-l(x),

x

x I + Un+l,

as

t)

follows:

for

x

~ Un - X

and

Gn(X,t)

= x

e X.

Then x

is

there

~ Vn,

3.5.6).

{Vn}~= 1

Vn+ 1

homotopy

union

Theorem

neighborhoods

P × Q

n

(see

G

e Un+ 1

is w e l l - d e f i n e d

n

and

a strong

Gn(X,1)

deformation

for

e Un+ 1

retract

each

for of

U

n.

Since

x c Un, for

n

G

(x,t)

n

we i n f e r

each

n.

= x

that

Thus

the

for

Un+ 1

is

proof

is

concluded.

9.3.2

Theorem.

polyhedra,

then

X u Y

By

applying

Proof. { U n } n~= l is

a

for

of

strong each

retract

compact

X,Y has

Then Un

u X

an ANR

of

we

of

get

Un

for

Sh(X)

X n Y having

by the

each

= Sh(X u UI),

U1 shape

we of

that

X n Y =

(U n

u Y)

the

a compact

( U n + 1 u Y)

= Sh(X u Y u UI).

is

9.3.1

have

Un+ 1 u X

S h ( X u Y)

X n Y

shape

retract

= Sh(UI),

replacing

the

X n Y

A N R 's c o n t a i n i n g

S h ( X n Y)

So

and

Lemma

deformation

n. of

If

Un

is n.

Sh(Y)

reduce

the

a finite

compact

exists such

Un

n X = Un

a strong By

of

polyhedron.

there

n n=l

and

shape

a

sequence

that n Y = X

Un+ 1 n Y

deformation

Theorem

4.1.6

we

= S h ( Y u U I)

and

proof

case

to

polyhedron.

the

get

where

Analogously,

118

we

can

reduce

having has

the

the

the

shape

shape

finite

CW

proof of

of

finite

the

case

CW

are

homotopy

Corollary.

X0,

X1

and

has

the

shape

Proof.

X2

have

of

f :

the

shape

a compact

This

X 1 n M(f)

of

9.3.4 X u Y

is a p o i n t e d

Proof. (X n Y)

simplicial

shape

of

9.3.5

Y

to

ANR's

the

space

to n o t i c e

finite

X u Y that

simplicial

is a m a p

and

polyhedra,

compacta

then

X 1 uf X 2

Theorem

9.3.2

because

= S h ( X 1 u M(f))

4.4.1).

and

X n Y

n

the

are

CW

spaces

(y x S I)

complexes

9.3.2

a finite

a pointed

are

CW

complexes

By Theorem

X,

9.2.2

(X x S I)

finite

it r e m a i n s

also

pointed

ANSR's,

then

ANSR.

By Theorem

x S1 =

(because

If

9.2.3

are

140).

compact

of

(see T h e o r e m

Theorem.

p.

Y

polyhedron.

fs a c o n s e q u e n c e

= X0

so

X0 c X1 ÷ X2

S h ( X 1 uf X 2)

and

and

equivalent

[i],

If

X

By Lemma

complex,

(see L u n d e l l - W e i n g r a m

9.3.3

where

polyhedra.

a finite

complexes

complexes

to

have

have

the

X

x S I,

Y

the

shape

of

homotopy [i],

the

× S1 =

X x S1 u Y

complex.

Therefore,

by

p.

and

finite

type

(see L u n d e l l - W e i n g r a m space

× S1

of

polyhedra

finite

140). (X u Y)

Theorem

× S1

9.2.2,

has

the

X u Y

ANSR.

Corollary.

pointed

ANSR;s,

Proof.

The

If then

proof

f :

X0 c X1 ÷ X2

X 1 uf X 2

is a n a l o g o u s

is a m a p

is a p o i n t e d

to

that

and

X 0,

ANSR.

of C o r o l l a r y

9.3.3.

X I,

X2

is

119

9.3.6 of

Example.

a finite Let

L

a finite CW

be CW

complex

let

g :

of

L.

complex (for

the

By

Lemma

this

a pointed

X

CW

K

and

h :

is n o t

pnn+l')'

does

ANSR

which

not

having

the

shape

Theorem

that

the L

such

Kn = K 4.1.6

shape

dominated

equivalent

complexes

be maps

have

is h o m o t o p y

homotopy

such

where

and

not

imply

of

K ÷ L

2.3.15

would

complex

existence

and

Hence

because CW

a connected

X = l~m(Kn'

n.

construct

complex.

L ÷ K

Put each

CW

We

see

that

the

space

is h o m o t o p y

[I]).

So

h • g ~ id L. p~+l

any

to a f i n i t e

Wall

and

of

by

= g. h X

finite

has CW

equivalent

for the

shape

complex,

to a f i n i t e

complex. By Theorem

§ 4.

provided

Definition. for

any

quotient

9.4.2 Q/X

is an ANR,

subset.

can

neighborhood

W

infer W/

then

W/

X ÷ Y/

9.4.3 Y

A compactum

Y/X

Let

ANSR.

X

e ANR

is c a l l e d containing

an ANR-divisor X

as

a closed

subset

be

a

Z-set

in

the

Hilbert

space

Q.

If

is an A N R - d i v i s o r .

Y

is a c o m p a c t

~ ANR

Y

and

as

Y/

If

is a n A N R - d i v i s o r .

a

ANR

Z-set

a retraction

Observe

Hence

Corollary.

X

is an ANR.

consider

X E ANR. X.

Y

X

then

Suppose

We

:

is a p o i n t e d

compactum

space

Lemma.

Proof.

we

X

ANR-divisors.

9.4.1

the

9.2.2,

that

X c ANR

Sh(X)

r

containing in

Q.

r :

W + Y.

induces which

= Sh(Y)

Take

as

a closed

a closed Since

Q/

X E ANR,

a retraction

concludes

and

X

X

the

proof.

is a n A N R

divisor,

120

Proof.

Embed

Since their also

Q - X

one-point

X

and

and

Y

Q - Y

as

Z-sets

are

homeomorphic

compactifications

in

Q/x

Q.

and

(see T h e o r e m

3.5.6),

Q / Y,

respectively,

space

Y

are

homeomorphic. Hence

Q/ Y ~ ANR

and

by

Lemma

9.4.2

the

is a n A N R -

divisor.

9.4.4 If

X0/

Lemma.

X ~ ANR,

Proof. (see B o r s u k C(X)

Let

X

then

X

Embed

X0

[5],

Theorem

and Borsuk

n

[5],

a 6.1

Z-set

in

on

90).

p.

Q.

(Q u CiX0))

/ C(X)

/ C(X)

6.1

of

a compactum

=

( Q / X)

= X0/

on p.

Then By

Sh(C(X))

(C(X 0) / C ( X ) )

Theorem

subset

because

(Q u C ( X 0 ) )

( Q / X)

a closed

X 0 c ANR.

is an A N R - d i v i s o r .

as

is a n A N R - d i v i s o r

Since

be

Corollary

is a n A N R

9.4.3

the

space

Hence

~ ANR.

u

By

u C ( X 0)

= Sh(point) .

(C(X 0) / C ( X ) )

X • ANR,

90).

Q

we

Lemma

infer 9.4.2

Q/X the

• ANR

space

(see

X

is

an A N R - d i v i s o r .

9.4.5. a. divisor b.

Theorem.

If

X c Y

iff

and

Y /X

If

X,

Let X

X

and

Y

be

compacta.

is an A N R - d i v i s o r ,

then

Y

is a n A N R -

is a n A N R - d i v i s o r .

Y

and

X n Y

are

ANR-divisors,

then

X u Y

X u Y

and

X n Y

are

ANR-divisors,

then

X

is an

ANR-divisor. c.

If

and

Y

are

ANR-divisors.

Proof. a. 9.4.4,

Embed

Observe Y

X u Y

that

as

(Q/X)

is an A N R - d i v i s o r

a

Z-set /

in

(Y/X)

iff

Y/

Q.

= Q / Y. X

By Lemma

9.4.2

is an A N R - d i v i s o r .

and

121

b.

By

Statement

ANR-divisors. is

an ANR

c. Embed

=

is

(M /

(M/

by

Lemma

(X n Y)

n

u

ANR's

and

by

ANR-divisors.

9.4.6. compactum

space

Y /

(M u N )

(X/

n Y)))

( X n Y)) 9.4.4

Statement

and

both a

the

Y

is

is

in

an ANR

(X n Y)

are

(Y/

(X n Y) X

X

an ANR-divisor.

N

/

• ANR

(X n Y))

such

that

=

and

a one-point

spaces

Y,

(X n Y)

((X u Y)

is

X /

an ANR-divisor

suppose

Y /

an A N R - d i v i s o r .

(X n Y)

N /

If

First

and

is

/

/

is

Theorem.

Proof.

XuY

(X u Y)

and

Then

(N / (Y / ( X

then

(X n Y)

(N / (Y / (X n Y)))

Lemma

By

space

• ANR

set.

M / are

M

X /

= ((Q/(XnY)/(X/XnY))/Y/(XnY)) the

the

in

(X n Y)))

(X / ( X n Y ) ) )

spaces

9.4.2

a

a one-point

(X/

the

Q/(XuY)

Statement

X /

M n N

Hence

and

By

a

set.

Therefore,

( X n Y)) and and

Y / Y

homotopy

( X n Y)

are

are

ANR-divisors.

dominates

a

an ANR-divisor.

that

Y

is

× {0}

u Y

a retract

of

X.

Then

there

is

a retraction r : Embed f : and

X

X

× {0} f(z)

Take

as u Y

a

an

Z-set

x I

= r(z)

X

u Q

for

in

× {i} z E X

f.

Let

÷ Q

x {i}

be

u Y

x I.

×

{0}

Since X

×

{0}

u Y

Thus (see

Lemma

defined

by

f

Q × {i}

extension

Then h :

× {i}.

Q.

g .~ Q of

× I + Y

g

is

a retraction

(Q × I)

/

X

u Y

x × I

(Q ×

{0} is {i})

9.4.2) .

× I ~ Q

(x × {0}

an /

× I

which

u Y

has

the

ANR-divisor (Y × {i})

× I)

× {i} induces ÷

(Q × {i})

shape (see

• ANR

a retraction

of

X,

Corollary and

Y

is

/ we

(Y × { i } ) . infer

that

9.4.3). an ANR-divisor

= id

122

Now Y.

suppose

Let

f :

Then

Y

that

M(f)

X

is an A N R - d i v i s o r

Y ÷ X

and

is a r e t r a c t

ANR-divisor

9.4.7.

g :

of

X ÷ Y

M(f).

homotopy be maps

Since

(see C o r o l l a r y

by

in o u r

first

Theorem.

step

If

X

such

Sh(M(f))

is a n A N R - d i v i s o r the

dominating that

a compactum

g • f = idy.

= Sh(X),

9.4.3)

and

we Y

infer

is a n

proof.

is a p o i n t e d

ANSR,

then

X

is an A N R -

divisor.

Proof. finite the

By Theorem

CW

complex.

space By

X × S1

Theorem

9.2.2

the

Hence,

by

space

X × S1

Corollary

9.4.3

has

the

shape

and

Lemma

of

a

9.4.4

is an A N R - d i v i s o r .

9.4.6

the

space

X

is

an A N R - d i v i s o r .

Notes

Theorem

9.1.1

belongs

Theorem

9.1.2

is d u e

Lemma West

9.1.3

[i] w h i c h

a finite

CW

Theorem Geoghegan

The 9.4.5

9.3.1

9.3.4

that

by

each

[i].

using

the

powerful

compact

ANR

has

9.3.6

Theorem

9.3.2

the

result

due

homotopy

is d u e

to

type

are

it

due

to H y m a n

is p r o v e d

in E d w a r d s -

and

Geoghegan

are

due

to D y d a k - O r l o w s k i

[i].

[i].

[i].

to E d w a r d s - G e o g h e g a n

was

proved

[i]

to D y d a k - N o w a k - S t r o k

of A N R - d i v i s o r

[8]

of r e s u l t s

Dydak-Orlowski

and

9.4.6

In D y d a k

obtained

is a c o m b i n a t i o n

is d u e

notion

and

to M a t h e r

folklore.

complex.

9.2.2

Example

be

asserts

[1,2,4],

Lemma Theorem

can

to m a t h e m a t i c a l

introduced

[i].

by

Hyman

[i].

Theorems

[i].

that

each

ANSR

is a n A N R - d i v i s o r .

of

Chapter

X

Cell-like

i.

Preliminary

If

U

is

U-homotopy, H({y}

a cover

of

provided

× I) If

definitions

U

and

V

are is

finite

simplicial

complex

for

subset sets

Lemma. any

y = in

and

N

for

any

open

in

U

Y

are

two

maps,

g' :

dim P ÷

P

f-l(u)

U1

that

Let

U'

two maps compact (K,L)

=

such

Y

c U

(P,R)

that is

× I ÷ X

is

a

with

for

then

of

! n + i)

a

extends

g'l

a star

let

g ~

(with that

and

dim

of

n Y

for

there

U

open

sets

and

h :

±s

< n +

i).

each

simplex

exists g

a

N

s

P

are

U-near.

Take

covering holds,

U2

e A. R ÷

f-l(u')

subpolyhedron Take

in

a map

uV = U.

, f-l(v))

P ÷ U'

Y

polyhedron

in

~

of

R ÷ f-l(u')

with

for

R

U

and

~ @

and P

of

such

a neighborhood

h :

f • g'

(Ln(f-l(Ul)

and

V

a compact

then

open

e A

exists

of

}

U

a closed

and

of

and

covering

neighborhood

refinement

U1

spaces

a collection

P + U'

R = h

) , f-l(v))

metrizable

N

there

f • h = g I R,

that

such

X,

realization IKI

of

in

any U

g :

f •h = g IR P

a map

subpolyhedron

U 2 = {U

of

a space

L(f-I(v),f-I(u))

of

if a

and

and

in

dim

exists

that

U

be

u U ~A that

polyhedron of

U

partial

sets

there

covering

L(f-I(uI

refinement

N,

be

open

Then

such

collections

a star

is

H :

sets

(with

÷ N

of

such

R

open any

V

holds.

such

V

M

U

Y

~ n + i)

Let

such

there

of

in

f :

of

where

Proof.

Y

e Y

a map

U.

collection

f-l(u))

of

K

Let

covering

U'

(with

y

then

statement:

in

f(f-l(y))

(Ln(f-l(v), N

the

realization

10.l.1. that,

X,

collections

(Ln(V,U))

a full

each

results.

c U.

L(V,U)

to

and

a space

for

Maps

of

be a

a triangulation of

K,

Isl

is

is

124

contained

in

g-l(u

is c o n t a i n e d v

~ K (0)

gl(oX)

in

- L

= h(x)

n

g(Isl) gl

g- f

for

that

in

is a s t a r

g' :

f - g'

U

in

g

are

of

Y

f-l(u)-homotopic

vertex

c U

.

Let

IL u K(0) I ÷ f - l ( u ' )

Indeed,

if

s < K,

then

where

of

K

h ( I s I)

U I,

we

infer

Therefore in

f-l(v).

of

compacta

that

gl

extends

It is e a s y

U-near.

f : X ÷÷ Y

U

gl :

f-l(Ul). of

each

g(v)

n U 1 ~ ~},

P ÷ f-l(u)

and

Let

that

s < L,

for

where

refinement

K

each

Choose

f-l(Ul).

) ~

of

for

n Y),

us verify

K

realization

covering

being

of

U2

Theorem.

open

Let

Then

~ ( A.

e f-l(u

u L) I ) c u { f - l ( u Since

e £ A.

some

gl(v)

realization

10.i.2.

some

x ( R.

is a p a r t i a l

any

)

realization

c U i.

to check

for

f-l(u

for

(K (0)

to a full

for

a point

is a p a r t i a l gl(IS

)

be

there

to

a map

exists

a map

g :

such

that

Y ÷ X

id x.

If

X

is a n A N R ,

U

of

Y

there

with

then

Y

is

an ANR.

Proof. Claim

i.

covering

For

V

Proof

of

X × I ÷ X

of

X

such

that

element

covering

U2

of both tion

open

such

i.

for of

of

112).

Let

g • f

V

and

and

be

realization

~' :

]K I ÷ X

]K I × {0}

set

L ( U 2 , U I)

K

ILl

~ :

in K

is

U 2, in

an

U1

be

so

an open

Y

is a n o p e n

and

(see B o r s u k a refinement

is a p a r t i a l

Then

covering

is c o n t a i n e d

satisfied of

an open

f-l(u)-homotopy

× I)

there

]L 1 ÷ X

f-l(v).

exists

holds.

and

H(U

covering

in

K of

u

Let

X ~ ANR,

Suppose

of

Y ÷ X

the

an open

complex

f-l(u))

id X.

Since

that

U.

realization

g :

U c U1

such

is a p a r t i a l

G :

L(f-I(v),

a map

f-l(u) .

of a s i m p l i c i a l

Define

that

each

X

g - l ( u 2)

covering

Take

joining

in s o m e

p.

Y

of C l a i m

H ;

[5],

any

g • f • ~ :

it e x t e n d s

realiza]L 1 ÷ X

to a full

U I.

x I ÷ X

by

G(k,0)

= ~' (k)

for

125

k £

[K]

and

Since any

G(k,t)

Is]

= H(~(k),t)

× {0}

s < K - L,

u Is n L]

we can

G'(Isl for each

Claim

2.

P ÷ Y,

4 :

IK]

Proof

such

Un,

of

P

× {0}

0

]K]

× I.

of

Isl

× I ÷ X

Is n ~I

× I

for

such

that

× ~)

2.

i,

Un+ 1

in

by

f-l(u)

covering

Take

is

...,

of

Y

U-homotopic

£

E < 0

is a s t a r r e f i n e m e n t

exists

that any

~.

a map

in s o m e

subset

element

of o p e n c o v e r i n g s

n,

for e a c h

of

IKI

h.

such

of

k e

for a n y m a p

there to

is c o n t a i n e d

for

an extension

and

polyhedron,

be a sequence

holds

= G'(k,l)

being

U

a number

than

~(k)

of

of

of Y

U. such

Un ,

and mesh

U n