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

1367 Manfred Knebusch

Weakly Semialgebraic Spaces

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Author Manfred Knebusch Fakult&t fL~r Mathematik, Universitgt Regensburg 8 4 0 0 Regensburg, Federal Republic of G e r m a n y

Mathematics Subject Classification (1980): 1 4 G 3 0 , 5 4 E 9 9 , 5 4 E 6 0 , 5 5 Q 0 5 , 5 5 N 10, 5 5 N 2 0 , 5 5 P 0 5 , 5 5 P 10 ISBN 3 - 5 4 0 - 5 0 8 1 5 - 5 Springer-Verlag Berlin Heidelberg N e w York ISBN 0 - 3 8 7 - 5 0 8 1 5 - 5 Springer-Verlag N e w York Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9. 1966, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 214613140-543210

Introduction

This

is the second

an e x p l i c a t i o n arbitrary preface

in a chain of

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

real closed

of the

(hopefully)

of s e m i a l g e b r a i c

field R. We refer

first v o l u m e

topology"

Let us r o u g h l y

as a basis of real

algebraic

recall what has been a c h i e v e d

over

reader

papers

in mind w i t h

devoted

topology

the u n i n i t i a t e d

[LSA] I and some other

to get an idea of the p r o g r a m we have braic

three volumes

cited

the term

to an

to the there

"semialge-

geometry.

in the

first v o l u m e

and

w h e r e we stand now. As we e x p l a i n e d which

in

fortunately

lar p a r a c o m p a c t spaces w h i c h locally

[LSA],

seem to suffice

ones.

are p r e c i s e l y

M u c h more

result

in

applications, those

volume.

locally

spaces,

are the regusemialgebraic

and II.4.4) 2. Moreover,

semialgebraic (II.4.4).

sets

This

any

in such a space

fact seems

to be the

[LSA].

has been

can p r o b a b l y

[y] of Verona,

in the p r e s e n t

(I.4.8

semialgebraic

less work on the t r i a n g u l a t i o n

Here our m a i n

the book

locally

for m o s t

simultaneously

for m a n y proofs

We a c c o m p l i s h e d

(II.6.13).

These

family of locally

can be t r i a n g u l a t e d

maps.

"good"

can be t r i a n g u l a t e d

finite

key result

the

the t r i a n g u l a b i l i t y

be done,

over

semialgebraic

of finite maps

as is to be e x p e c t e d

but we do not pursue {Verona w o r k s

of locally

~

this

by

line of i n v e s t i g a t i o n

and uses

transcendental

techniques.}

cf.

the r e f e r e n c e s

This refers to E x a m p l e 4.8 in Chapter I and T h e o r e m 4.4 in C h a p t e r II of [LSA]. The m a i n body of this volume starts w i t h Chapter IV. The signs I, II, III refer to the c h a p t e r s of [LSA].

IV

On the other hand we o b t a i n e d in Chapter II of

[LSA] a fairly detail-

ed picture of the v a r i o u s p o s s i b i l i t i e s how to "complete" a regular paracompact

space M, i.e. to embed M d e n s e l y into a p a r t i a l l y complete

regular p a r a c o m p a c t

space. Partial c o m p l e t e n e s s

is a typical notion

of s e m i a l g e b r a i c t o p o l o g y w h i c h has no c o u n t e r p a r t logy,

cf.

in c l a s s i c a l topo-

I, §6.

In Chapters

I and II of

[LSA] we also o b t a i n e d basic results on the

structure of locally s e m i a l g e b r a i c maps. and c o v e r i n g maps

(= Uberlagerungen)

But the theory of fibrations

had to be d e l a y e d since a certain

amount of h o m o t o p y theory is needed here,

not yet a v a i l a b l e in the

first two chapters.

Some of that h o m o t o p y theory has been p r e s e n t e d in the last Chapter III of [LSA]. Our central result there are the two "main theorems" v a r i o u s versions

(III.3.1,

4.2,

5.1,

in

6.3, 6.4). As a c o n s e q u e n c e of

these theorems all the h o m o t o p y groups and various h o m o t o p y sets in the c a t e g o r y of regular p a r a c o m p a c t topy groups

(resp. sets)

spaces over ~ .

spaces over R are "equal" to homo-

in the classical t o p o l o g i c a l

sense of such

This opens the p o s s i b i l i t y to t r a n s f e r a c o n s i d e r a b l e

amount of c l a s s i c a l h o m o t o p y theory to the locally s e m i a l g e b r a i c

sett-

ing, as has been illustrated in Chapter III by several examples.

The h o m o t o p y theory in [LSA] seems to be s u f f i c i e n t for s t u d y i n g (ramified)

c o v e r i n g s of regular p a r a c o m p a c t spaces. To some extent it

also gives access to the theory of fibrations and fiber bundles for such spaces

(although here something remains to be desired,

see below).

N e v e r t h e l e s s this h o m o t o p y theory has serious d e f i c i e n c i e s c o m p a r e d with c l a s s i c a l

(= topological)

h o m o t o p y theory,

the c o n t e n t s of the p r e s e n t volume.

and this brings us to

V

The m a i n d e f i c i e n c i e s I) In the c a t e g o r y spaces

over

2) In LSA(R) spaces

One main ciency

are the following.

LSA(R)

of regular

R we do not have

goal

as for e x a m p l e

in the p r e s e n t

can be overcome.

We will

3] w h i c h

is a s u i t a b l e

gether w i t h a cell called

"weak pol[topes".

end of III, complete fied

§6 and in

affine

since

these

are i s o m o r p h i c finite

We have

closed

w e a k polytopes.

restrictive

then our weak

become

different Working

too w i l d

from ~

continuous

functions

trol on c o n n e c t e d n e s s

limits will

generally

the basic

over

R. This

all ringed

spacec

R, tobe

at the

R we simply m e a n a terminology spaces

is justi-

over R w h i c h

space of some closed

isomorphic

systems

will

to the u n i o n

(Recall

spaces

we admit on a given geometric

of these

problem.

of

in d a n g e r

we admit

On the other

that our i n d u c t i v e closed

in the t o p o l o g i c a l over

field sense.)

R gives us control

space,

and this

properties

in

If we are too

that every real

disconnected

and other

of p o l y t o p e s

not be useful.

then we are

of ringed

properties

to these

semialgebraic

Once we have d e f i n e d w e a k p o l y t o p e s lished

alluded

is a d e l i c a t e

spaces.

in the c a t e g o r y

space over R [LSA,

inductive

polytopes

is t o t a l l y

CW-complexes

in some R n.

if we are too p e r m i s s i v e

limits

how the first defi-

over

inductive

This

sub-

"polytopes"

over R, hence

which

and p r o m i n e n t

limit of

space over

simplices

at our disposal.

, at our disposal.

R is a ringed

are p r e c i s e l y

complex

to be careful

building

hand,

spaces

semialgebraic

"semialgebraic"

By a p o l y t o p e

to the u n d e r l y i n g

simplicial

finite l y m a n y

over

(We b r i e f l y

semialgebraic

~X

is to e x p l a i n

construct

Such

[DK6].)

CW-complexes

loop spaces

inductive

structure.

locally

spaces Map(X,Y)

volume

over the field R. A C W - c o m p l e x p.

infinite

we do not have m a p p i n g

of them,

paracompact

which

gives us con-

implicitly.

in the right way and have estabspaces

it will be an easy m a t t e r

V~

to d e f i n e plexes.

cell

structures

Then the door

classical

homotopy

on some of them,

is open

theory

spectra,

ized h o m o l o g y

and c o h o m o l o g y

Although

as e a s i l y

CW-complexes category, These

as in c l a s s i c a l

the c a t e g o r y

of weak

are

WSA(R)

inductive

stead of just polytopes. of locally space,

ringed

but u s u a l l y

exclude WSA(R)

open

the c a t e g o r y

semialgebraic

spaces

The m o r p h i s ~ s

between

semial~ebraic

maps.

properties

of w e a k l y

later

use

glued

to another

ly proper" analogous

seems

over

weakly

In Chapter

But the class

than the class of proper maps below).

map

Most

important,

if the

also A is a weak p o l y t o p e is p a r t i a l l y

proper.

spaces

R.

in-

semialgebraic

be c u m b e r s o m e

to

paracompact

locally

subcategory.

spaces

any closed

semialgebraic

over

(in the sense

is a w e a k l y

of regular

spaces will

and maps. WSA(R) subspace

f : A ~ N

in If,

of p a r t i a l l y

space M above

(cf.

for

a space M can be A of M by a "partialIV.8.6).

the c a t e g o r y

p r o p e r maps

and more useful

and basic

The key result

(Theorem

§10 w i t h i n

and every w e a k l y

be called w e a k l y

the d e f i n i t i o n

in the c a t e g o r y

space N along

broader

from our c o n s i d e r a t i o n s .

IV we give

semialgebraic

spaces"

subspace

VI).

infinite

in a s l i g h t l y

It w o u l d

semialgebraic

result had been p r o v e d

for p r o p e r maps.

polytope.

LSA(R)

to deal w i t h

semialgebraic

an open

polytopes

and general-

(cf. C h a p t e r

semialgebraic

of affine

R as a full

weakly

to be that

to work

of a weak p o l y t o p e

of weak

In p a r t i c u l a r

topology,

theory

~uffices

"weakly

limits

is not a weak

subspaces

contains

of

of

R, and we can work with

homotopy

advisable

For example,

spaces)

over

polytopes

it is t e c h n i c a l l y

the c a t e g o r y

spaces

setting.

in the sense of a l g e b r a i c theories

be our CW-com-

a really big amount

to the s e m i a l g e b r a i c

we can d e f i n e

them n e a r l y

to t r a n s f e r

w h i c h will

LSA(R)

is m u c h b i g g e r

I, §5-§6

and IV,

is a w e a k p o l y t o p e

semialgebraic

An

map

§5

then

f :A ~ N

VII

In general a w e a k l y s e m i a l g e b r a i c

space M cannot be triangulated.

M still is isomorphic to a "patch complex".

But

This is a very weak sub-

stitute of a s i m p l i c i a l complex w h i c h n e v e r t h e l e s s

is s u f f i c i e n t for

some h o m o t o p y considerations.

Roughly one obtains a patch complex if one work w i t h a r b i t r a r y affine semialgebraic

spaces instead of simplices.

The theory of patch complexes

and their use in h o m o t o p y theory is d i s p l a y e d in Chapter V. A l s o some a p p l i c a t i o n s to open c o v e r i n g s

(= Uberdeckungen)

of w e a k l y semialge-

braic spaces are given in V, §3.

C h a p t e r V reveals that w e a k l y s e m i a l g e b r a i c a h o m o t o p y viewpoint.

For example,

sets from Chapter III in [LSA]

spaces are b e a u t i f u l from

the two m a i n t h e o r e m s on h o m o t o p y

extend to these spaces

(V, §5) and

there holds a strong

"Whitehead theorem",

homotopy equivalence

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

It is this chapter w h e r e the reader,

stating that every w e a k (Th. V.6.10).

having m a s t e r e d the foundational

labours of Chapter IV, will find out that w e a k l y s e m i a l g e b r a i c spaces are easy to handle and in some sense better natured, than t o p o l o g i c a l

since "tamer",

spaces.

On the other hand,

from a more g e o m e t r i c viewpoint, w e a k l y semialge-

braic spaces can be ugly. We shall d e m o n s t r a t e this in IV, §4 and A p p e n d i x C by rather simple examples. V a r i o u s nice geometric properties we are a c c u s t o m e d to from locally s e m i a l g e b r a i c instance the curve s e l e c t i o n lemma,

spaces,

as for

fail for these spaces. We do not

k n o w w h e t h e r a w e a k l y s e m i a l g e b r a i c space M can be completed,

i.e.

d e n s e l y embedded into a weak polytope. We do not know either w h e t h e r M c o n t a i n s a weak p o l y t o p e w h i c h is a strong d e f o r m a t i o n retract of M.

In c o n t r a s t to locally s e m i a l g e b r a i c spaces there does not always

exist a space N over the field R ° of real a l g e b r a i c numbers such that

VIII

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

to the base

of IV,

still we

§4).

But

lent

to s u c h

ter,

in C h a p t e r

a closed

Under

the m i l d

are e v e n space

a weak

It has

the

R,

Then

there

polytope.

The

The

P(M)

is the

in M.

space

M

braic

and

(simplification

defined

in I, §7.

Rn which

is not

of N

(cf.

t h a t M is h o m o t o p y over

end

equiva-

R o. M u c h

la-

equivalent

to

a "finer"

limit

then

the

of the

in R n t h e n

structure

If M is l o c a l l y with

P(M)

§9.

than

of M.

"simplification"

coincides

IV,

of all p o l y t o p e s

the

if M is a s e m i a l g e b r a i c

closed

~ M with

in C h a p t e r

identity

natural

purposes). P(M)

: P(M)

space

system

things

semialgebraic

be d e f i n e d

PM is just

some

already

locally

will

to be a v e r y

complete But

P(M)

PM

i.e.

to zero,

weakly

equivalence

set as M but

for

R is s e q u e n t i a l ,

converging

for e v e r y

homotopy

level,

field

elements

exists,

inductive

It seems

locally

N(R)

N a CW-complex

the b a s e

space

underlying

set t h e o r e t i c

contained

(V, §7)

§2)

see t h a t M is h o m o t o p y

of p o s i t i v e

M. On the

space

that

a canonical

same

shall

IV,

complex.

restriction

better.

(cf.

even with

§7, w e

a sequence

M over

can p r o v e

N(R),

VII,

simplicial

R contains

P(M)

a space

extension

semialge-

space

subset

is not

of the

Mlo c of some

locally

semialge-

braic.

More

generally,

given

define

in IV,

weakly

semialgebraic

the

§10 a w e a k l y

following

and e v e r y proper

factors = P(M) .

These

spaces

map

universal

weakly

Pf(M)

a weakly

pf

semialgebraic : Pf(M)

property.

semialgebraic

uniquely

Pf(M),

and

semialgebraic

through

~ M

space

q

pf.

in p a r t i c u l a r

f : M ~ N, we

Pf(M)

together

(if R is s e q u e n t i a l )

The m a p

map

map

f o pf

is p a r t i a l l y

: L ~ M with

spaces

with which

a has

proper,

f o q partially

If N is the o n e - p o i n t

the

shall

P(M),

space

will

do

then

IX

good service in h o m o t o p y theory at various places.

They are typical for

the somewhat d i f f e r e n t flavour of s e m i a l g e b r a i c h o m o t o p y theory compared with c l a s s i c a l h o m o t o p y theory.

A p a r t i c u l a r l y good instance to see how the spaces Pf(M)

and similar

ones can be used and how the various t e c h n i q u e s we have d e v e l o p e d C h a p t e r s IV and V fit t o g e t h e r is the proof of T h e o r e m equivalences

in

V.6.8 on d-

(instead of just w e a k h o m o t o p y equivalences)

w h i c h pre-

cedes and implies the W h i t e h e a d t h e o r e m m e n t i o n e d above. The reader cannot do better than trying to obtain an i m p r e s s i o n of the m a i n lines of this proof at an early stage in order to get a good feeling for the subject.

Of course, we try to p r o c e e d in s e m i a l g e b r a i c h o m o t o p y theory as m u c h as p o s s i b l e theory,

in a way parallel to the classical t o p o l o g i c a l h o m o t o p y

as long as this is advisable.

Here there comes up a d i c h o t o m y

of goals and m e t h o d s e v e r y o n e w o r k i n g in this area will face.

On the other hand,

one would like to obtain results in the semialge-

braic theory by transfer from the t o p o l o g i c a l theory,

as already exer-

cised in Chapter III. One wants to have a v a i l a b l e the enormous body of results of t o p o l o g i c a l h o m o t o p y theory in the s e m i a l g e b r a i c setting w i t h o u t m u c h further labour.

On the other hand,

there is a more radical viewpoint,

my k n o w l e d g e first e x p r e s s e d by B r u m f i e l in his book

to the best of [B]: One should

do a l g e b r a i c t o p o l o g y from scratch over an a r b i t r a r y real closed field in such a way that the field ~

does not play any special role.

This is an a m b i t i o u s program. W h i l e w r i t i n g this volume I somewhat o s c i l l a t e d b e t w e e n the two viewpoints.

W h e n e v e r the s e m i a l g e b r a i c

g e o m e t r y was easy I avoided transfer principles. ference to the first view point,

When not I gave pre-

but often I also tried to indicate

how things can be done in the spirit of the second one.

Long p a s s a g e s in Chapter V may nourish the c o n v i c t i o n that a h o m o t o p y theory in the sense of B r u m f i e l is already at hands.

But there are still

problems to be settled. As a testing ground I have chosen here - as already in [LSA], Chapter III Blakers and Massey.

- the h o m o t o p y e x c i s i o n t h e o r e m of

In t o p o l o g y there exists an e l e m e n t a r y proof of

this t h e o r e m going back to Boardman,

cf.

[DKP, p. 211ff].

(as well as the proof of Blakers and Massey)

This proof

strongly uses the axiom

of A r c h i m e d e s in the field of real numbers. We are able to prove the analogue of the theorem for w e a k l y s e m i a l g e b r a i c

spaces

(V, §7), but

for that we need the B l a k e r s - M a s s e y t h e o r e m for topological CW-complexes and t r a n s f e r techniques.

The h o m o t o p y theory d e v e l o p e d in Chapter V suffices for studying g e n e r a l i z e d h o m o l o g y and c o h o m o l o g y groups of pairs of w e a k l y semialgebraic spaces.

{The word

"generalized" means that we do not insist on

the E i l e n b e r g - O t e e n r o d d i m e n s i o n axiom.}

In Chapter VI we define gener-

alized h o m o l o g y and c o h o m o l o g y theories on the c a t e g o r y ~(2,R)

of

w e a k p o l y t o p e s over R in full analogy to the d e f i n i t i o n of such theories on the c a t e g o r y ~(2) (or [W],

of pairs of t o p o l o g i c a l C W - c o m p l e x e s

[W 2]

[Sw], etc.). We then e x p l i c a t e how every t o p o l o g i c a l h o m o l o @ y

theory h~ or c o h o m o l o g y theory h ~ on4@(2)

leads in a natural way to a

h o m o l o g y theory r e s p e c t i v e l y c o h o m o l o g y theory on ~(2,R) w h i c h we denote again by h. resp. h ~. We thus obtain a bijection, valence,

up to natural equi-

between the h o m o l o g y and c o h o m o l o g y theories on ~(2)

and on

~(2,R) for R fixed (VI, §2-4). We extend these theories i n V I , § 5 from ~(2,R) to the category W S A ( 2 , R ) o f pairs of w e a k l y s e m i a l g e b r a i c spaces over R, and we prove in VI,§6 a fairly general excision t h e o r e m for the groups hn(M,A)

Xl

and hn(M,A). We also d e s c r i b e the theories h~ and h ~ by spectra as one does in t o p o l o g y

(VI, §8).

In this w h o l e b u s i n e s s it is important that we have w e a k l y semialgebraic spaces at our d i s p o s a l

instead of just locally s e m i a l g e b r a i c

spaces. We m e n t i o n e d already the need for infinite CW-complexes. even suspensions pose a problem. alized h o m o l o g y theory,

They play an e s s e n t i a l role in gener-

of course. U n f o r t u n a t e l y we do not have suspen-

sions for a r b i t r a r y w e a k l y s e m i a l g e b r a i c topes.

This turns out to be sufficient.

algebraic

But

spaces but only for w e a k polyBut if M is a locally semi-

(pointed) w e a k p o l y t o p e then u s u a l l y the s u s p e n s i o n

SM will

not be locally semialgebraic.

If h~ is one of the p r o m i n e n t h o m o l o g y or c o h o m o l o g y theories logy, as s i n g u l a r homology, or symplectic K-theory,

singular cohomology,

orthogonal,

one of various c o b o r d i s m theories,

in topounitary,

then there

remains the i m p o r t a n t task to attach a g e o m e t r i c m e a n i n g to the elements of hn(M,N) spaces.

or hn(M,A)

for

(M,A) a pair of w e a k l y s e m i a l g e b r a i c

{In t o p o l o g y usually such a m e a n i n g is inherent in the defini-

tion of these groups.}

In the next v o l u m e m e n t i o n e d above. H~(-,G)

[SFC] we shall solve this p r o b l e m for the K-theories

In the present one we solve it for o r d i n a r y h o m o l o g y

and o r d i n a r y c o h o m o l o g y H~(-,G) w i t h c o e f f i c i e n t s in some abe-

lian group G. These are those h o m o l o g y and c o h o m o l o g y theories w h i c h fulfill the E i l e n b e r g - S t e e n r o d d i m e n s i o n axiom.

They arise from topo-

logical singular h o m o l o g y and c o h o m o l o g y theory with c o e f f i c i e n t s in G.

We prove in VI, groups Hn(M,A;G)

§3 that if

(M,A)

and Hn(M,A;G)

and cochains as in topology.

is a pair of C W - c o m p l e x e s then the

have a d e s c r i p t i o n by cellular chains

It is then easy to c o n c l u d e that for

(M,A) a pair of locally s e m i a l g e b r a i c spaces,

these groups coincide

×If

w i t h the groups Hn(M,A;G) [D],

[DI] , [DK3].

and Hn(M,A;G)

defined e s s e n t i a l l y by Delfs

{We d e s c r i b e d the groups Hn(M,A;G)

in III, §7.}

Here our theory reaches a r e m a r k a b l e point. To understand, why, us recall the a p p r o a c h of Delfs to the h o m o l o g y groups, single p o l y t o p e M.

let

say, of a

{We take A = @.} The polytope M can be triangulated.

C h o o s i n g an i s o m o r p h i s m ~ : rKtR-L~M w i t h K a finite abstract simplicial complex we "know" a priori what Hn(M,G) w i t h the a b s t r a c t h o m o l o g y Hn(K,G)

should be:

of the simplicial complex K. The

p r o b l e m is, to prove that the groups Hn(K,G) choice of the triangulation.

It should c o i n c i d e

do not depend on the

Delfs solves this p r o b l e m in an ingenious

way. He looks at the simplicial c o h o m o l o g y groups Hn(K,G)

for the tri-

a n g u l a t i o n s of M. He proves that they all are n a t u r a l l y isomorphic to the c o h o m o l o g y g r o u p s H n(M,G M) of the constant Knowing that the Hn(K,G) cludes that the Hn(K,G)

sheaf G M w i t h stalk G.

are independent of the t r i a n g u l a t i o n he conalso are independent of the triangulation.

In the course of this a p p r o a c h Delfs has to cope with some tedious g e o m e t r i c problems.

{The m a i n task is to prove the h o m o t o p y invariance

of the groups H n ( M , G M ). In [D I] Delfs solves this p r o b l e m b r i l l i a n t l y by using sheaf theory on abstract locally s e m i a l g e b r a i c spaces.}

The

r e m a r k a b l e fact now is that we obtain the independence of the groups Hn(K,G)

from the choice of the t r i a n g u l a t i o n in a m u c h easier way.

Once we have the h o m o t o p y theory of Chapter V at hands, w h i c h is a s t r a i g h t f o r w a r d matter, we define the o r d i n a r y h o m o l o g y groups Hn(M,G) almost by general categorial nonsense, the s t a n d a r d way

and prove Hn(M,G)

~ Hn(K,G)

in

(cf. VI, §3). Thus one m a y say that it is p o s s i b l e

to c i r c u m v e n t the labours of Delfs by e n l a r g i n g the c a t e g o r y of affine s e m i a l g e b r a i c spaces over R to a c a t e g o r y of spaces w h i c h is more comfortable for h o m o t o p y considerations,

namely WSA(R).

{But notice that

our a p p r o a c h does not give a c o n n e c t i o n of o r d i n a r y c o h o m o l o g y with

XlII

sheaf cohomology.}

How about an i n t e r p r e t a t i o n of the elements of Hn(M,A;G) singular

simplices,

as in topology? Of course,

m e a n s a s e m i a l g e b r a i c map V(n)

(= morphism)

in R n+1 to M. For any pair

by chains of

a singular simplex here

from the closed standard simplex

(M,A) of w e a k l y s e m i a l g e b r a i c spaces

over R we can define the singular chain complex C. (M,A;G) as in topology. The p r o b l e m is to prove that the groups Hn(C. (M,A;G))

fit together

to an o r d i n a r y h o m o l o g y theory and that Ho(C. (*,~;G)) & G, w i t h • denoting the one point space.

This would imply a natural i s o m o r p h i s m

from this h o m o l o g y theory to H~(-,G).

Dells and I have tried for years in vain to find such a proof in a direct g e o m e t r i c way.

The d i f f i c u l t y was always to prove an e x c i s i o n

t h e o r e m for the groups Hn(C. (M,A;G)) not archimedean.

in the case that the field R is

We could not prove e x c i s i o n even for a triad of poly-

topes. As in classical theory one would like to m a k e a given singular chain

"small" w i t h respect to a given finite open c o v e r i n g

open s e m i a l g e b r a i c singular s i m p l i c e s

sets)

by a p p l y i n g some iterated s u b d i v i s i o n to the

in the chain. But the trouble is that, as long as

one tries b a r y c e n t r i c subdivision,

(with two

s u b d i v i s i o n or some other sort of finite linear

the simplices have no reason to become small if R is not

archimedean.

The last Chapter VII of the present book contains a solution of the p r o b l e m - along very d i f f e r e n t lines. This solution is perhaps the most convincing

single issue, up to now, to d e m o n s t r a t e that w e a k l y semi-

a l g e b r a i c spaces are really useful.

We proceed roughly as follows. set = s e m i s i m p l i c i a l complex,

Every simplicial

set K

in other terminologies)

(= s e m i s i m p l i c i a l can be "realized"

XIV

as a weak p o l y t o p e

iK1R over R in m u c h the same way as this is known

in t o p o l o g y

[Mil]. The space

complex.

(K,L) is a pair of simplicial

If

iKi R carries a natural structure of a CWsets

(of course, with L a

simplicial subset of K), then it follows from the cellular d e s c r i p t i o n of o r d i n a r y h o m o l o g y m e n t i o n e d above that the o r d i n a r y h o m o l o g y groups Hn(IKIR, iLIR;G) can be identified with the well known [May])

(cf.

[La] or

"abstract" h o m o l o g y groups Hn(K,L;G).

If M is a w e a k l y s e m i a l g e b r a i c space over R we can form the singular simplicial

set Sin M

c o n s i s t i n g of the singular simplices of M. The

realization

ISinMl R

JM : iSinML

~ M. We prove that JM is a h o m o t o p y e q u i v a l e n c e

f o l l o w i n g the book

comes w i t h a canonical w e a k l y s e m i a l g e b r a i c map

[LW] of Lundell and Weingram.

only a weak h o m o t o p y equivalence.

(VI, §7)

{In topology JM is

In m o s t texts on simplicial methods

- but non in [LW] - this is proved by already using the fact that the t o p o l o g i c a l singular h o m o l o g y groups form an o r d i n a r y h o m o l o g y theory.}

M o r e generally,

if A is a subspace of M, then JM gives a h o m o t o p y equi-

v a l e n c e from the pair

Hq(M,A;G)

(ISinMiR,

~ Hq(ISinMiR,

ISin Ai R ) to

(M,A). Thus

iSin AiR; G) ~ H q ( S i n M , Sin A; G),

and this group is Hq(C. (M,A;G)) by definition.

Since we k n o w that the canonical maps JM are h o m o t o p y e q u i v a l e n c e s the door is now wide open for the use of simplicial geometry. simplicial

Thus,

sets in s e m i a l g e b r a i c

finally, we can abolish our previous v e r d i c t

sets, only simplicial complexes"

"no

[DK3, p. 124].

S i m p l i c i a l sets have proved to be enormously useful in many b r a n c h e s of topology,

in p a r t i c u l a r in the theory of fibrations. M u c h

of this m a t e r i a l can now be used in s e m i a l g e b r a i c geometry.

Some

×V

applications

to the theory of s e m i a l g e b r a i c

in the next v o l u m e

But one n e e d s more. cial

sets.

in WSA(R), over

between

simplicial

Roughly the

simplicial

i.e.

(Xn[n£No)

a sequence

(VII,

spaces

for future

weakly §I).

Simplicial

proper.

instead

of just simpli-

R we m e a n a s i m p l i c i a l semialgebraic

object

spaces

face and d e g e n e r a c y

sets m a y be r e g a r d e d

last C h a p t e r

application

VII

is d e v o t e d

spaces

maps

as d i s c r e t e

will

arise

the r e a l i z a t i o n

Fortunately

to an e x p l i c a t i o n

and their realizations.

discrete

of

Difficul-

from the fact that we are only

IXl R of a p a r t i a l l y

space X. By this we m e a n a simplicial tially

spaces

of w e a k l y

semialgebraic

of s i m p l i c i a l

able to c o n s t r u c t

be given

over R.

half of our

fundamentals

ties

One n e e d s

space X over

them

will

[SFC].

By a s i m p l i c i a l

R with various

fibrations

space

all w h o s e

simplicial

spaces

proper

simplicial

face m a p s

are par-

are p a r t i a l l y

proper.

A reader having worked spaces

and maps

similar

stuff

in C h a p t e r

about

fort we i n d i c a t e

then meant

it is clear

simplicial

We now pose

fundamentals feel b o re d

spaces.

semialgebraic

g r o u p over

G-fibre

bundle

the f o l l o w i n g

stuff

in C h a p t e r VII

R.

some com-

is r e a l l y useful.

{For instance

think of

semialgebraic

of s e m i a l g e b r a i c ~

semialgebraic

such a reader

If M is an affine

of course with

semialgebraic

of w e a k l y

to m e e t

To give

that this

from the b e g i n n i n g s

as in topology,

of M by open

IV m a y

g r o u p O(n,R).}

by a p r i n c i p a l

exactly

the

n o w by an e x a m p l e

Let G be a c o m p l e t e some o r t h o g o n a l

through

geometry

space, what

is

: E ~ M over M. The d e f i n i t i o n a finite

trivializing

is

covering

subsets.

problem.

Let S be a real

closed

overfield

of

XVI

R and let ~ : F ~ M(S) exist a p r i n c i p a l ~S

: E(S)

~ M(S)

It seems solve

G-bundle

[SFC]

Example V I I . 1 . 2 . v

i~Gl.

One

complete finds

pal bundles elements topy

(V.5.2.i;

the c a n o n i c a l B(G(S)).

correspond over M(S)

At first

that

by b=se

semialgebraic

with

bijection classes

the i s o m o r p h i s m

extension.

the present

geometry

to d e s t r o y

topy theory the c a t e g o r y mains

WSA(R),

to

classes

The answer

book might

[M,BG]

of p r i n c i p a l classes

to

above.

the second one

While

By

(BG) (S) =

G-bundles

of p r i n c i p a l above

over M

G(S)-bundles is "Yes".

that

in

theory at hands w h i c h

But this

the

is bi-

[M(S)B(G(S))].

impression

it I come back to the two d e f i c i e n c i e s listed

on homo-

that

the i m p r e s s i o n

one now has a h o m o t o p y

the

from III.3.1).

it is e v i d e n t

convey

with

theorem

to the q u e s t i o n

one.

of G - p r i n c i -

[M(S), (BG) (S)]

clear

from

cf.

is deceptive. of the homo-

first one d i s a p p e a r s

(existence

of m a p p i n g

is

spaces)

in re-

serious.

One w o u l d

like to have good

not ex i s t i n g define

in [LSA]

of ~ G

the nerve

the r e a l i z a t i o n

correspondence

is already

shall

since G is c o m p l e t e

isomorphism

[M,BG]

of the d e f i n i t i o n a natural

proper

By the first m a i n

from

as good and easy as the t o p o l o g i c a l In order

as in topology,

one-to-one

this

the i s o m o r p h i s m

uniquely

glance

map

essentially

nature

space built

Let BG denote

[M,BG].

extension

Thus we have

We con c l u d e

set

there

We

Let ~G denote

that the

in natural

of the h o m o t o p y

sets the base

jective

suffice).

way.

as follows.

is p a r t i a l l y

as in t o p o l o g y

over M are

Does

the base e x t e n s i o n

in a direct g e o m e t r i c

is a simplicial

would

over M(S).

to ~ over M(S)?

this p r o b l e m

below. ~ G

G(S)-bundle

: E ~ M over M such that

in the a f f i r m a t i v e

of the group G. This

(partially

~

is isomorphic

hard to solve

it in

be a p r i n c i p a l

mapping

spaces

"pseudo-mapping

substitutes and their

spaces"

and

(or "models")

prominent

"pseudo-loop

of the p r e s u m a b l y

subspaces. spaces"

In VI,

which

§7 we

do some

XVll

of the service

one expects

based on an a n a l o g u e struction better

the r e a l i z a t i o n presumably

mapping

are not s u f f i c i e n t

equivalence

[DKP,

is a case

5.3])

p.

spaces

then we can

17] M a p ( S i n M, S i n N )

and choose

But this

of h o m o t o p y

turns

a given m a p

is to e s t a b l i s h

big

a space

of w e a k l y

for some purposes).

from the o r i g i n a l fibrations

I intended

much

stage.

In the last

WSA(R).

plan,

to i n t r o d u c e

In the m e a n t i m e

can be done

I also r e a l i z e d

duce w e a k l y

for M = N =

in Chapter

of arguments,

and c o v e r i n g s

become

amply

the n o t i o n The

space"

f : M ~ N into a fibra-

semialgebraic

maps

section

the unit

from M to of Chapter

interval

in the p r e f a c e

of

of

[LSA],

IV and w i t h coverings (= U b e r l a g e r u n -

weakly

in the next v o l u m e semialgebraic

I realized proofs

in the same way

spaces

[O,1],

announced

since m a n y

how well

on a s u f f i c i e n t l y

semialgebraic

V. We shall do this only

Originally

duplication

and similar

equivalence.

structure

the field R is sequential.

later

as will

"path m a p p i n g

R, in the case that

in Chapter

substitute

of the

There one has to w o r k w i t h

like this

gen)

substitute

for all purposes,

VII we do s o m e t h i n g

to deal w i t h

This con-

Chapter VII we are

as a c a n o n i c a l

instead

subset of the set Map(M,N)

I deviated

[Bn].

is

in point.

strategy

(sufficiently

theorem

Using

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

which

Our c o n s t r u c t i o n

semialgebraic

space Map(M,N).

for a s u b s t i t u t e

7.2],

Another

N

[May,

in the theory of fibrations.

question

big

set

[ M a p ( S i n M , Sin N) IR

of fibre h o m o t o p y

tion,

only up to homotopy.

not e x i s t i n g

constructions

([W,

representation

If M and N are any w e a k l y

form a s i m p l i c i a l

clear

of Brown's

is c a n o n i c a l

off.

from such substitutes.

it pays as early

[SFC].

spaces

that this w o u l d

at a

cause

a

in the t h e o r y of f i b r a t i o n s

for the c a t e g o r i e s in m a n y

other ways

as possible.

LSA(R)

and

to intro-

XVIII

I thank

Professors, Ronnie

Rainer Vogt I further Schwartz here.

for useful

thank

advice

Hans Delfs,

these persons

Roland

in pre v i o u s

versions

to my s e c r e t a r y

March

with

J. Peter May,

simplicial

Claus

C is e n t i r e l y

Finally

search

special

for a v e r y e f f i c i e n t

losing p a t i e n c e

in c r i t i ca l

Manfred

Knebusch

I also

thank

for m i s t a k e s

thanks

are due

typing of all these

situations.

1988

and Niels

to be listed

due to Huber.

and a successful

and

homotopy.

Scheiderer

of the proofs too n u m e r o u s

reading

Franke

Fritsch,

Huber,

of this volume.

Marina

without

Regensburg,

Appendix

for proof

Rudolf

in t a c k l i n g

for help with details

In particular,

versions

Brown,

TABLE

OF

CONTENTS

page CHAPTER

§1

IV - B a s i c

- Definition

theory

and

of w e a k l y

construction

semialgebraic

of w e a k l y

spaces

semialgebraic

spaces

I

§2

- Morphisms

§3

- Subspaces

§4

- Spaces

of countable

§5

- Proper

maps

§6

- Polytopic

spaces;

§7

- A theorem

on

§8

- Strong

§9

- The weak

15 and products

23

type

36

and partially

polytope

spaces

pA(M)

§11

quotient

proper

the one-point

inductive

quotients;

§10 - The - The

limits

gluing

maps

42

completion

49 54

of s p a c e s

60

of spaces

71

P(M)

86

a n d Pf(M)

by a partially

proper

equivalence

relation

CHAPTER

I

99

V - Patch

complexes,

§I

- Patch

§2

- Some deformation

and homotopies

again

decompositions

106

106

retractions,

and

related

equivalences

homotopy 114

§3

- Partially

finite

open

§4

- Approximation

§5

- The two main

theorems

§6

- Compressions

and n-equivalences

§7

- CW-complexes

125

coverings

of spaces

by weak

polytopes

on homotopy

sets

133 147 152 165

×X page CHAPTER

VI - H o m o l o g y

182

and cohomology

§I

- The basic

categories;

§2

- Reduced

§3

- Cellular

homology

§4

- Homology

of p a i r s

of w e a k

§5

- Homology

of pairs

of s p a c e s

§6

- Excision

and

§7

- Representation

§8

-

cohomology

suspensions

of w e a k

and cofibers

194

polytopes

209 214

polytopes

224 233

limits theorems,

pseudo-mapping

spaces

VII

- Simplicial

The basic

244 252

~-spectra

CHAPTER

183

spaces

260

260

definitions

§I

-

§2

- Realization

§3

- Subspaces

280

§4

- Fibre

292

§5

- Quotients

§6

- Semia!gebraic

§7

- The

§8

- Simplicial

§9

- A group

APPENDIX open

of

some

268

spaces

products

303 realizations

s p a c e l S i n MI a n d

C

simplicial

singular

homotopy,

and

of a u t o m o r p h i s m s

(to C h a p t e r

of s i m p l i c i a l

IV) : W h e n

s e t s of M t o p ?

homology

again

[0,1]

is ~(M)

311 320

homology

singular of

sets

331 341

a basis

of 352

References

355

Symbols

359

Glossary

363

Contents

of C h a p t e r s

I -III

375

Chapter

IV - B a s i c

§I - D e f i n i t i o n

R is a f i x e d topological missible)

chain

and

real

open

I,

Definition every

sets

U 1 DK

Example

1.1.

Definition

and

a subset

I running

space

}(M)

with

spaces

a generalized

the

set of

(admissible)

such

a space

to the

set U n K

(ad-

open

M we g i v e

definition

(Definitions

a

of a w e a k -

6,7 b e l o w ) .

in M if,

for e v e r y

U 6 ~(M)

is a l r e a d y

the u n i o n

of the

a suitable

semialgebraic

of M is s m a l l

means

set of

polytope

through

M equipped

semialgebraic

K of M s m a l l

the

2. a) A f u n c t i o n

topological

Here

leading

a weak

spaces

§I we c o n s i d e r

Cov M the

examples

6COVM(U),

subset

in I,

I. S t a r t i n g

If M is a l o c a l l y

semialgebraic

As

of M and

and

I. We call

with

of w e a k l y

field.

Def.

space

(UIII6A)

semialgebraic

(M,~(M),COVM).

§I,

of d e f i n i t i o n s

ly s e m i a l g e b r a i c

and

M =

subsets

cf.

of w e a k l y

construction

closed

space

coverings,

theory

finite

space

subset

over

of A.

R then

every

in M.

ringed

space

M over

with

a sheaf

~M

R is a g e n e r a l i z e d

of r i n g s

of R - v a l u e d

M,N

R is a c o n t i n u o u s

functions. b) A m o r p h i s m

between

function

map

(in the

sense

f : M ~ N

that h-f

for e v e r y :

V £ ~(N)

the

Example

Every

ringed

and h £ ~N(V)

1.2. space

semialgebraic

over

category

locally

spaces

of g e n e r a l i z e d

f-1 (V) ~ R is an e l e m e n t

c) We d e n o t e

ringed

the

topological

composite

spaces),

such

function

of @ M ( f -I (V)).

of f u n c t i o n

ringed

semialgebraic

R. The m o r p h i s m s b e t w e e n

maps.

over

spaces

space such

over

over

R by S p a c e ( R ) .

R is a f u n c t i o n

spaces

are the

locally

Henceforth a small ringed U DK

let M b e a f u n c t i o n

subset space

with

of M t h e n M i n d u c e s

over

R as

U 6~(M).

follows.

If

(VIII6A)

(VIII6A) 6 C o v K if a n d o n l y such that I 6A'

the

set V

{As u s u a l ,

we

~K° d e f i n e d

is an e l e m e n t

follows.

U nK

D V a n d h = g IV.

that

U DK

= V.} T h u s

tions for

iff t h e r e

gi 6 ~ M ( U i )

h

of all V 1 w i t h

The

axioms

associated

some U 6~(M)

can make

many

sets

i-viii

to the

: V ~ R o n s o m e V 6 ~(K) and

U a little

: V ~ R, w i t h V 6 ~(K),

finitely

such that

h

A' o f A

some

g 6~M(U)

smaller

such

is an e l e m e n t

U I .... ,U r 6 }(M)

and

func-

K n (U I U ... U U r) = V and g i l K D U i = h l K D U i

1 < i < r.

Definition Notice

3. W e c a l l

that we have

function

ringed

Definition structure spaces

a space

K equipped

space

such

4. Of c o u r s e , as a f u n c t i o n

1.3.

semialgebraic algebraic

that

also

(K,~ K)

with

the

the

ringed

a s_mall s u b s p a c e

"coarsest

inclusion

every

4.5].

If M is a l o c a l l y subset

subspace

from Tietze's

K has

such

U 6 }(M)

space

over

structure"

map

has

of

K ~

of a

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

a natural

R. T h e s e

(M,©M).

are

induced

the o p e n

sub-

of M.

Example

Th.

exist

exists

{We t h e n

a function

sheaf

A function

iff t h e r e

with

of OK(V)

©K is the

intersections

subset

the union

(VIII6A) 6 C O r K ( V ) .

If K is

then

a finite

is a l r e a d y

R).

of a f u n c t i o n

set of all in ~(K)

exists

over

structure

is t h e

if t h e r e

fulfilled.}

of ~gK(V)

(always

is a f a m i l y

then write

as

space

on K the

~(K)

:= U (VIII6A)

in I, §I a r e c l e a r l y presheaf

ringed

of M t h e n

of M

extension

semialgebraic

K, w i t h

(I, §3), theorem

If in a d d i t i o n ,

an open

semialgebraic

its u s u a l

is a s m a l l for a f f i n e

M is r e g u l a r

affine

semialgebraic

subset

of M is a f f i n e

space

structure

subspace

of M.

semialgebraic

(I, §3),

neighbourhood [R].

a n d K is a c l o s e d as a s e m i This

spaces

then O o K = OK in M.

follows [DK 5,

since

Indeed,

now

every

Important space

convention.

always

algebraic

means

space

Definition

From now

an a f f i n e

means

in M,

in the w h o l e

semialgebraic

a regular

5. A s u b s e t

K is c l o s e d

on,

locally

K of M is c a l l e d

i.e.

M~K

6 ~(M),

space,

closed

(K,~ K) of M is a s e m i a l g e b r a i c

space.

i__nn M if,

in a d d i t i o n ,

space

i.e.

complete

[DK2,

of M is d e n o t e d by

1.4.

ordered

ready

family

sets w i t h

space

as in

the

semi-

space.

in M,

and

(K,~ K)

in M if if the

K is c a l l e d

closed

small

a polytope

is a p o l y t o p e ,

semialgebraic

set of all p o l y t o p e s

subsets

in M is d e n o t e d

semialgebraic M =

E2

If

E3

For

every

E4

For

any

a ~

~c(M)

have

of a set X we m e a n

index

space

an o r d e r e d

5)

such

book.

By an

a family

of

sub-

set.

semialgebraic

(Def.

of the w h o l e

that

over

R is a f u n c t i o n

family the

(M Is6I)

following

ringed

of c l o s e d

properties

hold.

B then M a c M B.

Y~

s 6I

two

index

there

indices

6, and M

I is d i r e c t e d ,

of

and

U(Mala6I).

Y~,

The

(XlII6A)

contains

subsets

El

definition

ordered

R which

t h e n ~(M)

[LSA].

of s u b s e t s

6. A w e a k l y

M over

semialgebraic

for the m a i n

a partially

Definition

E6

and

set of all

If M is l o c a l l y

same m e a n i n g

N o w we are

E5

by ~(M)

The

a locally

~c (M) .

Example the

§9].

and

semialgebraic

if K is small

semialgebraic

a semialgebraic

semialgebraic

subspace

the

book,

y 6I w i t h

function

only

a,~ 6I

there

finitely exists

many

8 6I with

an i n d e x

~ <s.

y 6I with

D M B = My.

i.e.

for any

~¥,

~ ~y.

ringed

semialgebraic

exist

space

spaces

two

indices

M is the (Mala6I)

~,~ 6 I

inductive

in the

there

limit

category

exists

of the

Space(R).

an

family This

means

the

following:

a) A s u b s e t every the

U of M is an e l e m e n t

a 6 I.

set of

b) A f a m i l y

{N.B. all

such

for e v e r y

If h

such

{The

letter

tope

over

polytope

For

the

in w e a k

"E"

II,

which

) = ~(M

) =

of Cov M iff

a finite

s 6I.

subset

This A(s)

means of A

= U(UI n M a l l E A ( a ) ) .

in the

of

labels

Def.

§4.}

closed of M.

refers

space

class

we h a v e

M over

in m i n d

of w e a k l y

paracompact

i.e.

then

s 6 I. an e x h a u s t i o n

of M.

"exhaustion".}

R is c a l l e d such

space

that

a weak

every

I denote

a locally

the

ordered

poly-

M a is a

R.

If M is p a r t i a l l y

Indeed,

with

to M.

inclusion. complete

M over

a weak

exists

locally

(~(Y) IY 6 I) ~(Y)

of a even

finite,

subcomplexes

the

R

tri-

isomorphism

{There

Then then

space

choose

X strictly finite

to w o r k

spaces.

semialgebraic

(I, §2)

interested

it is n e c e s s a r y

semialgebraic

set of all by

are m a i n l y

semialgebraic

over

complex

we

reasons

locally

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

in X,

to

(Mal~EI)

for t e c h n i c a l

simplicial

Let

for e v e r y

U 6 ~(M),

5).

which

: X ~M,

on some

of M is c a l l e d

above

ringed

semialgebraic

i.e.

(U n M s)

an e x h a u s t i o n

but

Every

~

function

subsets

(cf.

finite

exhaustion

~(M

for

of M s. }

for e v e r y

exists

(M IsEI)

general

1.5.

are

there

6 7~(Ma )

[LSA],

subsets

of C o v M

6~M

polytopes,

triangulation, cf.

of

is an e l e m e n t

iff hlU n M

applications

is a w e a k l y

locally

II6A)

R if M has in M

angulation

s EI,

7. A f u n c t i o n

in the m o r e

Example

in ~(M)

: U ~ R is an R - v a l u e d

family

Definition

semialgebraic

iff U D M

that

h 6©M(U) Any

~(M)

the n o t a t i o n

is an e l e m e n t

u(ul n M

c)

open

(Ulll6i)

(Ul D M a L I 6 A ) that,

Using

of

Y of X is an

are p o l y -

a

5

topes,

hence

M is a w e a k

We n o w d e s c r i b e This

method

algebraic

polytope.

a method

is b a s e d

spaces,

for

constructing

on the r e s u l t s

which

will

weakly

in II,

be e x p l o i t e d

semialgebraic

§I on g l u i n g here

only

spaces.

locally

for

semi-

semialgebraic

spaces.

We

start

ordered Def.

with family

6) h o l d

E4 ~) For

the

the

space

M with

B ~ a, t h e n relation. J nK the

like

B 6J.

limit

enjoy The

of the

For

exist

space

family

of

of

subset

any

I. M o r e o v e r

E2,

¥1,...,yr

... U M y r

every R and

M s is e q u i p p e d that

for s <

of M B.

spaces

(M Is6I).

the

over

6

I we d e n o t e

set of all If a 6 J

ordered

by the

elements

I has

holds.

in I

space

J of

is an (cf.

condition

in D e f i n i t i o n

two

E3

ringed

property:

set I is p a r t i a l l y

are e l e m e n t s

that over

as g i v e n

for

weaker

subspace

following

since

El,

indices

assume

By I we d e n o t e

the

(MsLa6I)

and M a N M B = M y 1 U

We

any

and

conditions

of a f u n c t i o n

description

s 6J.

is a set,

following

semialgebraic

It is a lattice,

and J U K

E5.

structure

same

of all M a w i t h I which

the

(I ~ i ~ r)

I is not d i r e c t e d ) .

J of

that

of a s e m i a l g e b r a i c

that

the

M

s,B 6 I t h e r e

anything

inductive

has

of E4,

B

M a is a c l o s e d

structure

sets

indices

assume

M the

union

two

situation.

of M such

instead

structure

We e q u i p

perhaps

of s u b s e t s

Yi ~ ~' Yi ~

We do not with

following

and,

any

with

makes

the

R which This

(although by M j the

finite and

sub-

B 6 I,

inclusion

J , K of I also

a smallest

element,

set ~.

Theorem

1.6.

Every

ringed

space

M,

family

(MjIJ£1)

space.

If the

set M

is c l o s e d

and M i n d u c e s

on M

is an e x h a u s t i o n

family

(Msla6I)

semialgebraic the g i v e n

of M.

fulfills

Thus E4

in the

space

function

structure.

M is a w e a k l y

and E5 t h e n

The

semialgebraic

already

this

family

is an e x h a u s t i o n

Before

proving

that weakly

the

of M.

theorem

semialgebraic

than

paracompact

full

strength

spaces

locally

o f the

we give

simple

form a more

semialgebraic

theorem

examples

is n o t

which

general

spaces.

For

already

class

these

show

of s p a c e s

examples

the

needed.

~) Example over

1.7

R. As

ordered let

(Simplicial in

simplices. braic

we denote

b y E(X)

face

relation.

L e t M be the u n d e r l y i n g

be the

Every

space.

subcomplexes

i.e.

contains

Example

1.8

x I and we set

symbol every

closed

by the

every

obtain

in X. W e

letter

open

being

face

X.

in M of a l l open

denote

If t h e

of e v e r y

of a s e m i a l g e -

structure

given

of a w e a k l y

b y the

finite

this weakly

complex

~ 6E(X),

of X,

set of X a n d

structure

a natural

an e x h a u s t i o n

are

semialgebraic

identify

all

I 6A.

spaces.

these

after

"O" a n d w e o r d e r

this the

No two elements

the

obtain

space,

we

simplices

of c l o s u r e s

a natural

(Wedge of s e m i a l g e b r a i c

U(MIII6A)

Applying we

again

set of o p e n

complex

semi-

X is c l o s e d , then

the

space X

polytope.

of d i s j o i n t

the

o n X,

of X w h i c h

space

is a w e a k

theorem

space

the

(~ Q M I o 6 E(X))

set ~ D M c a r r i e s

algebraic E(X)

family

By o u r

semialgebraic

L e t X be a s i m p l i c i a l

[LSA]

by the

(M Is6I)

complexes).

theorem

on the

such that

to t h e

spaces).

We choose

base

points

I := i U {O}

of A are family

set M a n a t u r a l

(M LaEI)

structure

be a f a m i l y

in e v e r y M 1 a b a s e p o i n t x o.

We add

to t h e

as f o l l o w s .

comparable.

e v e r y M 1 - in its g i v e n

(MIIIEA)

into one

identification. set

Let

of

L e t M be set A a

0
We put M 0

structure

for

:= {Xo}.

semialgebraic

of a w e a k l y

point

spaces

semialgebraic

- is a s m a l l

sub-

~) In t h i s means

chapter

- and

a geometric

the next one

simplicial

- a simplicial

complex,

as d e f i n e d

complex in I,

§2.

always

space of M. We call this space M, w i t h base point x O, the wedge of the pointed spaces

(Ml,x I) and w r i t e M = V(MIIIEA),

or more p r e c i s e l y

(M,x o) = V ( ( M I , x I) ilEA) The sets U(MIII6K), w i t h K running through the finite subsets of A, form an e x h a u s t i o n of M. If the M 1 are p o l y t o p e s then M is a weak polytope.

Example

1.9. We c o n s i d e r the family

(IPn(R) InE]N o)

points of the s t a n d a r d p r o j e c t i v e spaces is a polytope.

We embed IPn (R)

into IPn+ I (R)

h y p e r p l a n e Xn+ I = O. Let IP (R) theorem,

IP (R)

~Pn,R

of sets of real

over R. Each IPn(R)

in the usual way as the

be the u n i o n of all ~ n ( R ) . By the

is a w e a k p o l y t o p e in a natural way, w i t h e x h a u s t i o n

(IPn(R) I nEIN o) .

Example

1.10

(Direct sums). For every family

(MIIIEA)

of function

ringed spaces over R there exists the direct sum U(MIII6A)

= M in the

c a t e g o r y of function ringed spaces over R. The set M is the d i s j o i n t union of the s e t s M I. The elements of ~(M) are the unions U(UIII6A) families

(UIIIEA) w i t h U 1 E ~(MI),

and also COVM,

©M are d e f i n e d in

the obvious way. A s s u m e now that every M 1 is w e a k l y semialgebraic. choose an e x h a u s t i o n

of

(Ml,slaEI I) of M I. Let I := U(IIIIEA)

We

be the

direct sum of the o r d e r e d sets I I. For every ~ 6 I we define M s := MI, s if a £ I I. The o r d e r e d family fulfills the conditions of T h e o r e m

(M Ja6I) of s e m i a l g e b r a i c

1.6. Clearly the function ringed

space M is the inductive limit of this family. M is a w e a k l y s e m i a l g e b r a i c

spaces

Thus, by T h e o r e m 1.6,

space.

We start out to prove T h e o r e m 1.6. The o r d e r e d family subsets of M filfills El -E5,

(MjIJ6I) of

as is e a s i l y checked. We equip e v e r y

Mj w i t h the f u n c t i o n ringed space s t r u c t u r e as the inductive limit of

the

family

that

the

of s e m i a l g e b r a i c

space Mj

t h e n M K is,

spaces

(Msla6J).

For

is s e m i a l g e b r a i c

a n d that,

whenever

space

structure,

a closed

a 6 I we denote

b y m(s)

the n u m b e r

of e l e m e n t s

For any J 6 I we denote

b y m(J)

the m a x i m u m

space

For

in its g i v e n

every

any

the

f r o m E4*

that

direct

claim the

finitely

the

m(L)

hand,

since

semialgebraic

M L a n d M~.

Thus we

are

a closed

B 6 I with

= O,

with

then

sub-

M

B <s.

s 6 J.

it f o l l o w s

disjoint.

spaces

two different

Mj

is the

(I, 2.4),

s

and

the

learn, the same.

the unique

such that

as a b o v e w e

the

there

exists

that

Theorem

structure,

for

and e v e r y

in M s.

It n o w

that

is c l o s e d

II.I.3

e v e r y Ms,

in J then,

w i t h m(y) = n .

tells

space

any K C I with

K cJ,

semialgebraic

y £ L,

of M L. On t h e

subsets

of M a

subspace

structure

same

holds

semialgebraic

us,

that

our

on M j

from

for in M s

respect soace~ M j

is

semialgebraic argument

s p a c e M K in its

subspace

to

is in f a c t

same uniqueness the

is,

say

is a c l o s e d

By the

induction

follows,

The

structure

s £ J,

By

on M s n M B w i t h

structure.

is a c l o s e d

the

structure.

structures

My,

subspace

semialgebraic

II.I.3,

the given

in its g i v e n

see that,

closed

given

some y £L

s % B- T h u s m(L) < n .

that M s NM~

subspace

and

of M j

in Th.

with

semialgebraic, one

are

indices

t h a t y ~ s, ¥ ! B for e v e r y

semialgebraic

semialgebraic

statement

in M B a n d t h a t and M~

B are

Suppose

these My

of M L in M a c o i n c i d e s

subspace

If m(J)

s p a c e M L is s e m i a l g e b r a i c ,

set M L is c l o s e d

given

semialgebraic

of all m(s)

pairwise

some L £ I such

< n.

structure,

the u n i q u e n e s s

Ma

s 6 J are

f o r c e ¥ = s and y = B- B u t

in its g i v e n

and

on m(J).

many

= n > I. If s a n d

Of c o u r s e ,

hypothesis,

the

induction

M a NM B = M L with

This would

other

K c J a n d K 6 I,

is o b v i o u s .

L e t n o w m(J) b y E4~,

by

sets M s w i t h

s u m of the

claim

y 6L.

claim

of Mj.

We prove

the

J 6 I we

in Mj,

and our

claim

is p r o v e d .

Of course, the n e w

the

family

inductive

limit

(MjIJ6I)

is the

space same

structure as w i t h

on M w i t h

respect

respect

to the

old

to

family

(Mala£1).

For

any

two

subsets

J and K of

we h a v e M j ~ M K 6 ~(Mj), M\M

K 6 ~(M)

that,

for e v e r y

for e v e r y

Certainly that

for e v e r y

J 6 I

prove

with

of all

1.11o

some

evident

set M j

the

For

c)

Given

such

a) has

b)

U

is c l o s e d

in M,

©o Mj

some

:=

been

(M~A)

semialgebraic

U 6 ~(M). consider

Clearly the

spaces.

I, and

induced

the

by ~ M on

on Mj.

something

the

We

more.

let K d e n o t e

the

set

set A : = M K = M R

of the o r d e r e d

Then

and

some

a function

family

following

holds.

U 6 T(M)

with

= V.

h 6 ©A(V)

U NA

there

exists

some

.

= M~

above. (A\V).

in M U DA

this

For

, since

every

s 6I

the

the p r o p e r t i e s

set

(A~V)

AM

E3 and E4 • hold.

= V.

set of all p a i r s

flA = g. We o r d e r

and

We e q u i p

structure

exists

verified UV

B 6K.

if we v e r i f y

functions

fact

of

the d e f i n i t i o n s

M ~ A £ ~(M).

there

glV = h

this

subset

space

i.e.

sets V , U

with

semialgebraic

contains

B for

limit

V 6 ~(A)

Proof. Let

of

of s e m i a l g e b r a i c

every

g £~M(U)

which

a <

inductive

a) A is c l o s e d b)

sheaf

Thus

in M.

be f i n i s h e d

of f u n c t i o n s

infinite,

L 6 I, L c J.

from

is small

1.6 w i l l

L e t K be any

s 6 I with

(MjIJ£I,JcK)

and

J c I, the

the p r e s h e a f

the

a proposition

Proposition

c) We

= M L with

K perhaps

'

Mj coincides

Thus

J 6 I but

It is a l s o

of T h e o r e m

'

with

M K AMj

K c I.

finite

the p r o o f

since

I, w i t h

set

(J,f)

with

in the o b v i o u s

K cJ cI, way:

(J,f)

f 6©Mj(U <

DMj)

(J',f')

10

iff J c J' ordered

and

set,

f'IU AMj

then

the

= f.

If

((Jl,fl) l16A)

fl a u t o m a t i c a l l y

with J = U(JIII6A) , and we have by

Zorn's

that The

lemma,

L = I and

our

set M L N M s N U

extension tion

theorem

u on M a NU

is c l o s e d

and

[DK5,

4.5]

which

Th.

extends

contradiction

to t h e m a x i m a l i t y

takes

values

subset

This

cf.

is p r o v e d for

[DK5,

Remark the

then

1.13.

(F)

For

every

This

that, \ the

of

and

functions

Then

Thus

I.]I

that

By Tietze's

be

g takes

with

in q.e.d.

improved.

If h

semialgebraic

values

Tietze's

functions

g glue

(L,g) < (S,v)

(= c o n v e x

using

u and

L = I.

can

func-

in L.

extension

values

in L,

situation

of T h e o r e m

1.6 a s s u m e

that

in a d d i t i o n

exist

only

finitely

many

B 6 I such that

.

semialgebraic

s p a c e M is a p a r a c o m p a c t

locally

semi-

space.

follows

gluing

The

prove

some a6I ~L.

a semialgebraic

L c R

as a b o v e

spaces

a 6 I there

the w e a k l y

algebraic

such

We choose

We

holds.

M a N M~ # ~

Then

(L,g).

(L,g).

in M a D U .

exists

I £ i. Thus,

4.5].

In t h e

following

of

interval

same way

semialgebraic

L %I.

S = L U {a].

g c a n be c h o s e n

in t h e

Thm.

there

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

in a g e n e r a l i z e d

of R)

theorem

c)

Suppose

f 6 © M j ( U N M J)

for e v e r y

element

semialgebraic

NM S ) with

Assertion

(J,f)

glM L NM s NU.

v 6~Ms(U

1.12.

~

in t h i s

to a f u n c t i o n

a maximal

be done.

to a function

Remark

(Jl,fl)

set c o n t a i n s

then will

glue

is a c h a i n

from Lemma

locally

for every

semialgebraic

1.2 a n d

semialgebraic

a 61, space

the

family

M . c~

This

Theorem

1.3

spaces.

There

(M s Q M s I B £ 1 ) means

just

in C h a p t e r one

has

islocally (F).

II a b o u t to a s s u m e finite

in

11

We close

this

given weakly have

some

section

with

semialgebraic

redundancy

Definition

some

easy observations

space

which

M.

on exhaustions

An e x h a u s t i o n

(Msls6I)

of a

of M m a y

c a n be e l i m i n a t e d .

8. An e x h a u s t i o n

(Mala£I)

of M is c a l l e d

faith-

*) ful

if,

E2')

instead

For

of E2,

any two

Proposition

indices

1.14.

If

space M then there (MsDs£I')

Proof. with

y
In t h i s

check

following

For

any two

E8)

For

the

that

of c l o s e d

Such being

I' of

s 6 I such

check

holds.

of a w e a k l y

I such that

that

properties

the properties

with

indices

E3 a n d

the

semialgebraic

family

there El, E4

exists

E2',

some y 6 I

E3-E6

for t h e o l d

for t h e n e w family

play

then

set

Every

there

~ 6 I we

{Mala6I} subsets

(M~Is£I)

of M

indexed

by

faithful

be r e g a r d e d itself.

properties.

is an e x h a u s t i o n

exists

have M

some y 6 I with = ~

hence

exhaustion

M s UM~ =My.

.

is a s u b l a t t i c e

of M,

s p a c e M c a n be e n l a r g e d

a family may

of M

formal

properties.

s,B 6 I

index

the

better

exhaustion

smallest

1.15.

even

additional

semialgebraic

Proposition algebraic

E2'

~ ~ M a c M B.

is an e x h a u s t i o n

a subset

index

9. A l a t t i c e

E7)

~)

every

property

exhaustion.

find exhaustions

the

Notice

exists

stronger

role.

Definition with

s,B 6 I, s ~

and My = M s , and

a crucial

We can

out

following

(Msls6I)

is a f a i t h f u l

Throw

family!

the

of t h e

lattice

~(M)

t h e name.

(Mala6I)

to a f a i t h f u l

of a w e a k l y

lattice

as a s e t of s u b s e t s

semi-

exhaustion

of M,

each

set

12

(MsLs6I),

with

ordering

Proof.

by

elements

also

E7,

if

containing

use

of H,

given The

exhaustion

by

inclusion,

lead

(Msla6I)

to w o r k

in a c r u c i a l

always

would

insist

way

(NBIB6J)

are

El,

two

on IxJ b e i n g

J { (s,B)

~

(y,6)

(NsIB6J)

are f a i t h f u l ,

In l a t e r

sections

be r e g a r d e d

iff

given

of M,

of f i n i t e l y

E2',

E3-E6,

and

exhaustions.

(We

such

a subscript,

by

faithful

exhaustions,

of

For

we since

example,

as w i l l

of the even

as has

orderings

if

a space

(Msls6I)

like

exhaustions

clear

on I and and

is n o t

faithful.

M will

most

often

done

above.

already

something

become

been H, H,

are

H',

or H

considered

at

time.

We n o w d e s c r i b e exhaustion

a partition

of M into

"patches"

starting

from

a given

in some

set M a,

(M la6I).

Lemma

1.16.

there

exists,

a 6 I with

of M,

an e x h a u s t i o n

if s e v e r a l

of M,

But,

exhaustion

hand,

of M t h e n

(M s N NBI (a,~) £ IxJ)

as a set of s u b s e t s

denote

same

perhaps

lattice

product

~ ~ 6}.

On the o t h e r

complications.

an e x h a u s t i o n

s ~ y and

a faithful

or

exhaustions

the

lattice

§8.)

notational

in §2,

the o r d e r i n g

faithful

faithful

is a g a i n

the

the

subsets

of all u n i o n s fulfills

in §7 and

to use

to u n n e c e s s a r y

and

with

(M a N NBI (s,B) 6 IxJ)

with

inducing

as a set H of

set H c o n s i s t i n g

ordered

comfortable

them

not

We t h e n

I and

E8.

It is o f t e n

this

the

inclusion.

many

should

set

on I.

We r e g a r d

ordered

shall

I an o r d e r e d

For

F cM

any

in the

subset

F of M, w h i c h

ordered

, and F c M y .

set

I, the

is c o n t a i n e d infimum

y of all

indices

13

This

follows

Definition given

easily

10.

The

exhaustion

f r o m E3 a n d E4.

index

q(x)

(Mala£I),

of a p o i n t

is t h e

x 6M,

infimum

with

of all

respect

indices

to the

s 6 I with

x 6Ms.

Notice

that

namely

for e v e r y

M

M°

that

11. F o r

every

a' w i t h sets M ° respect

The

(M~Is6I °) Clearly,

given

the exhaustion,

primitive,

if M ° is n o t e m p t y ,

of p r i m i t i v e

through

I °,

are

indices called

which means

is d e n o t e d the patches

by

I O. T h e

of M w i t h

exhaustion.

is a p a r t i t i o n for every

(1.17)

describes

a 6 I we define

set ~(M)

a running

to t h e

: M ~ I fully

= a}.

a £ I is c a l l e d

a 6~(M).

~

< a}.

:= {x 6 M 1 q ( x )

index

function

a 6 I,

= {x 6 M I ~ ( x )

Definition

An

the index

of the

set M i n t o n o n e m p t y

disjoint

subsets.

a 6 I,

M°a = M a ~ U ( M B I B E I , B

and

(1.18)

M a = U (MyLy o _< a,T E I °)

This partition

Example

1.19.

algebraic finite

of M w i l l

be v e r y

useful

L e t X be a s i m p l i c i a l

space

(Ex.

subcomplexes

1.7). Y which

We

in t h e

complex,

consider

are closed

following.

regarded

as a w e a k l y

the exhaustion

of X b y t h e

in X,

every

Y being

indexed

semi-

by

14

itself.

Here

the patches

(x) of a p o i n t

are

just

the o p e n

x in a g i v e n

open

simplex

simplices

of X.

The

~ of X is t h e c o m p l e x

index ~ AX.

§2 - M o r p h i s m s

In t h e fixed

following exhaustion

another ringed

weakly

M is a w e a k l y

semialgebraic

of M.

an u n d e r s t a n d i n g

over

2.1.

Assume

be a m a p

from

the

R

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

b)

There

exists

f(L)

Proof.

the

between

the

then

2).

space

The

(Mala6I)

category

crucial

fact

L is s e m i a l g e b r a i c .

set M.

that

The

following

f(L) c

of the

semialgebraic

the map

and

M

is a

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

L to M in t h e

Def.

a £ I such

and

range, spaces

of

from

function

is

Let

are

f : L ~ M

equivalent.

the m a p

f

: L ~ M

is s e m i a l g e b r a i c

,

(i.e.

L a n d Ms).

fa is s e m i a l g e b r a i c

for e v e r y

a 6 I

c M a.

Every map

f with

property

map M a ~ M

is a m o r p h i s m

now that

: L ~ M

f

then we

c M

ture

of M a as a s m a l l

that

indices

I °. F o r

D(f(L))

point

Ya

every

y 6 I the patches

M~S

6 ~[M).

L. F o r t h e

since

subspace

index

subspace

If w e h a v e

Ma

found

it is e v i d e n t

some

fa is a g a i n

subset

J of t h e

choose

set S D M

Y

i.e.

is f i n i t e ,

reason

x

6L

A

:= f-1(S)

struc-

§I,

Def. 10).

set of p r i m i t i v e such that

the

Ya EM°.a L e t S := { y a l a E J } . since

this

M

is t h e u n i o n

¥

implies

is a c l o s e d

the preimage

space

(cf.

(M ~ S) D M y 6 ~(My) , a n d w e c o n c l u d e

f is a m o r p h i s m

6 ~(L). T h u s

a point

Assume

a morphism.

is an i n f i n i t e we

inclusion

a 6 I with

from the

exhaustion

a 6J

the

of M to M.

of o u r

M°a" T h u s

same

small

since

function

every

Since

is a m o r p h i s m

of M t h a t

:= f ( x a) h a s i n d e x a,

many

L ~f-1(S)

from the

are done,

n : M ~ I be t h e

Suppose

b)

is a m o r p h i s m .

f(L)

Let

§I,

f by restriction

If f is a m o r p h i s m with

that

some

from

a morphism

(cf.

space

set L to the

a)

obtained

seek

semialgebraic

spaces

Theorem

We

space

f

-I

(S')

that

of

f-1(M~S)

subset

finitely

that

semialgebraic

of a n y

For

= subset

S' of

of

S is

16

a closed f-1(S')

The

semialgebraic ~

since

f-1(S')

semialgebraic

We choose

of L, h e n c e

contains

s p a c e A has

of A.

the points

a finite

If S' % @

x a with

number

then

also

Ya 6 S'.

r of c o n n e c t e d

components•

a partition

S = S I U S2 U of S i n t o r+1 finite.

subset

... U Sr+ I

disjoint

non empty

sets.

This

is p o s s i b l e

since

S is in-

Then

A = f-1 ($I) U f-1 ($2}

is a p a r t i t i o n subsets•

But

components.

of A i n t o

U

r+l

.. • U f-1 ( S r + I ) disjoint

such a partition This

by properties

E2

non

empty

closed

is i m p o s s i b l e s i n c e

contradiction

proves

that

a n d E5 of the

exhaustion

A has

semialgebraic only

J is f i n i t e . of M,

that

r connected

It f o l l o w s ,

f(L) c M a for s o m e

a 6 I.

q.e.d.

Definition

I. In v i e w

from a semialgebraic algebraic

Here

map

of c o n d i t i o n space

f r o m L to M

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

Proposition

2.2.

Let A 6~(M)

structure

on A as a c l o s e d

Proof.

The

as a s m a l l we obtain

L to a w e a k l y

of T h e o r e m

A c M a. F o r

structure

every

as a s m a l l

inclusion subspace

the result.

(cf.

such

theorem

we call

semialgebraic

a morphism

space M a semi-

2.1.

§I,

index

Def.

subspace

there

A 6~(Ma),

exists

some

and the

of M a coincides

with

of M.

Applying

Ex.

Then

subspace

M is a m o r p h i s m

to M.

(Recall

5

a we have

semialgebraic

map A ~ of M,

in t h e

(over R).

a 6 I with

the

b)

1.3

f r o m A, w i t h

Theorem

for the

2.1

last

its

structure

to t h i s m o r p h i s m

statement).

17

Theorem

2.3.

(NB]B6J) to t h e

L e t N be a s e c o n d

be an e x h a u s t i o n

set N.

The

f is a m o r p h i s m

b)

For

every

that

a 6 I there

the map

of N.

following

a)

fsB

weakly Let

are

exists

semialgebraic

space

f : M ~ N be a m a p

and

let

f r o m the

set M

equivalent.

some

B 6 J such

: M a ~ NB o b t a i n e d

from

that

f ( M s) c

N~,

f by restriction

and

is s e m i -

algebraic. If f is a m o r p h i s m , then

All

and

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

this

a map

fsB

is e v i d e n t

inductive

limit

if s 6 I, : M

B 6 J are

~ N B is a l w a y s

from Theorem

space

indices

structure

f : M ~ N is a m o r p h i s m

2.1.

Indeed,

of the

iff all

with

f ( M s) c NB,

semialgebraic.

since

function

M carries

ringed

restrictions

the

spaces

f[M s

Ms ,

: M s ~ N are

semialgebraic.

Definition

2.

In v i e w

between

weakly

Example

2.4.

between

simplicial

algebraic

of f

semialgebraic

Let

Example

2.5.

complexes cf.

Ex.

simplicial

(loc.cit.)

spaces

X and

1.7.

(cf.

is a g a i n

A map

ly s e m i a l g e b r a i c

b)

in the

a weakly

f : X ~ Y be a w e a k l y

spaces,

If f is e v e n

of c o n d i t i o n

[LSA,

a weakly

regard

24]),

then

semialgebraic

map

(cf.

for e v e r y

s 6 I, t h e

a morphism

map

(over R).

[LSA, p. 23])

semialgebraic the

closure

semi-

map.

f :X ~

map.

f : M ~ N f r o m M to a s e m i a l g e b r a i c

iff,

call

X a n d Y as w e a k l y

f is a w e a k l y p.

we

semial~ebraic

simplicial

Y. W e

Then

theorem

restriction

space

N is w e a k -

f [ M s : M s ~ N is

semialgebraic.

In p a r t i c u l a r , R are p r e c i s e l y 2.3

is a l r e a d y

the weakly the

semialgebraic

elements

evident

of I~M(M).

maps This

from the definition

f r o m M to t h e special

real

line

case of Theorem

of an e x h a u s t i o n .

18

Definition

3. F r o m

semialgebraic

functions

the elements justified braic

Remark

of O M ( U )

since

with

U,

In t h e

semialgebraic

<(a)

map.

as an o p e n (U N M

this

remark

If

6 I, a n d N~ c M

i.e.

to the

on U.

is a g a i n w e a k l y

2.3,

let

< : I ~ J as

s < ~ implies

in M ~(B)"

Corollary

2.8.

are

N = M,

f

This

is

semialge-

: M ~ N be a w e a k l y

follows

<(a)

< K(B).

f = idM,

"comparable".

is a s e c o n d

K : I ~ J and ~

If M

every

case

space

is c l o s e d

(cf. L e m m a 1.16).

Moreover,

we see t h a t

More

for

exhaustion

any

two

precisely,

of M,

then

: J ~ I with M a c N

there

for e v e r y

~ 6J.

semialgebraic

is a w e a k

in N < ( s ) ,

polytope

and

a n d N~

(N8]~6J)

is c l o s e d

semi-

is a n y e x h a u s t i o n

N B is a p o l y t o p e .

L e t n o w N be a l o c a l l y

semialgebraic

the m o r p h i s m s

f r o m M to N a n d

ringed

over

subsets.

of all Ul w i t h

space

over

f r o m N to M in the

R. W e c h o o s e

semialgebraic

by U j t h e u n i o n

functions

U £ ~(M),

) cNB)

(B) for e v e r y

algebraic

by o p e n

of M,

for e v e r y

[s£I).

(NBIB6J)

exist monotonic maps

spaces

we call,

the weakly

) c NK(S)

2.7.

then

subspace

a map

Proposition

of M,

of © M ( M )

semialgebraic

f induces

of a g i v e n

Ms

generally

of T h e o r e m

exhaustions

Of course,

elements

situation

< is m o n o t o n i c ,

Applying

the

on M. M o r e

:= i n f ( B 6 J [ f ( M

e v e r y s 6 I, f ( M

call

the w e a k l y

exhaustion

2.6.

This map

n o w on w e

an a d m i s s i b l e For

every

category

covering

finite

~ 6 J. N o t i c e

R. W e w a n t

subset

to a n a l y s e of

function

(UILI£A) J of

t h a t N is t h e

of N

I we denote

inductive

19

limit

of the

Proposition

family

2.9.

the r e s t r i c t i o n [LSA,

This

is e v i d e n t

: M

since

call

semialgebraic

conflict

(UjLJ c A , J

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

iff,

i.e.

finite).

for e v e r y

s £ I,

a semialgebraic

map

9]).

4. We

f r o m M to N.

spaces

f : M ~ N is a m o r p h i s m

M is the

it in a s p e c i a l

Definition weakly)

semialgebraic

A map fiM a

(cf.

observed

p.

of

This

with

case

inductive already

limit

in Ex.

of the

spaces

f r o m M to a l o c a l l y

space

the w e a k l y

is j u s t i f i e d

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

Definition

2 if N is at the

The w e a k l y

se~ialgebraic

. (We

2.5.)

the m o r p h i s m s N again

M

(instead

semial~ebraic 2.9.

same

(N.B.

time

maps

There

weakly

of

and

is no locally

semialgebraic.)

Example p.

2.10.

15, p. 81])

M which

are

Proposition

are p r e c i s e l y

bounded

2.11.

Proof.

The

morphisms. every flUj the

I 6A.

on e v e r y

A map

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

flUl

condition Assume Then

now also

is a m o r p h i s m

that

ftUl

finite

f r o m Uj to M.

Definition

5. We

call

that

since

the

[LSA,

functions

Since

for e v e r y

on

subset

i.e.

UI~

N are

a morphism,

J of A the

N is the

inductive

from

semialgebraic

a locally

maps

for

restriction limit

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

the m o r p h i s m s

I 6A,

map.

inclusions

is s e m i a l g e b r a i c ,

for e v e r y

conclude

iff,

is a s e m i a l g e b r a i c

is n e c e s s a r y

locally

semialgebraic

(cf.

Ma.

: Ul ~ M

Uj we

N to M the

weakly

f r o m M to Rlo c

f : N ~ M is a m o r p h i s m

spaces

space

those

maps

of q.e.d.

semialgebraic

f r o m N to M.

20

If M

is w e a k l y

semialgebraic

semialgebraic maps

maps

f r o m M to N,

just

the m o r p h i s m s

spaces map

over

as c o n s i d e r e d

from

2.12.

this

in

remark

that M

be an a d m i s s i b l e of

J of A w i t h M a c Uj.

same

since

locally

both of

then

and

2.11

of m a p s

function

the

the w e a k l y semialgebraic

kinds

to the c a s e M = N and

is a l s o

For

as the

category

2.9

covering

[LSA]).

semialgebraic

[LSA],

Propositions

Assume

(in the t e r m i n o l o g y subset

the

f r o m M to N in the

id M we o b t a i n

(UIII6A)

locally

f r o m M to N are

R. A p p l y i n g

Proposition

and

are

ringed the

identity

following

fact.

locally

semialgebraic

and

of M by o p e n

semialgebraic

subsets

every

s 6 I there

Then M s 6~(Uj).

For

exists

every

some

finite

let

finite

subset

J

o

of A t h e r e

exists

Definition weakly

semialgebraic

of f u n c t i o n

over

Let

maps

by WSA(R)

spaces over

Uj c M~.

R.

over

Then

Uj 6 ~(M~).

the c a t e g o r y R and w h o s e

It is a full

ringed

spaces

over

the c a t e g o r y

LSA(R)

of p a r a c o m p a c t

R, and

whose

objects

morphisms

are

subcategory

it c o n t a i n s , locally

of the

are

the

the w e a k l y category

as a full

subcate-

semialgebraic

spaces

R.

S be a r e a l

useful

functor

and p.

309]

this

closed "base

and,

functor

Given,

(Msla£I)

(Ms(S)Is6I)

for

overfield

extension"

of

inclusion

MB(S)

~ Ms(S)

[DK3,

p.

142].

of R. from

semialgebraic

to a f u n c t o r

as b e f o r e ,

haustion

the

B 6 I with

6. We d e n o t e

semialgebraic

gory,

some

spaces,

semialgebraic

by b a s e

spaces

m a p M B ~ M s gives,

Let N denote

the

over

by b a s e

into Ms(S)

exists

a natural

to L S A ( S ) , [DK3,

§4].

cf.

and v e r y

[LSA,

We w a n t

p. 19f

to e x t e n d

to W S A ( S ) .

semialgebraic

of M we o b t a i n

of MB(S)

LSA(R)

f r o m WSA(R)

a weakly

There

space M o v e r extension S. F o r

inductive

family

indices

B < s

an e m b e d d i n g

semialgebraic

limit

an ex-

an o r d e r e d

any t w o

extension,

as a c l o s e d

R and

of the

subspace

family

of sets

21

(Ms(S) la£I), We r e g a r d

with

the Ms(S)

of N f u l f i l l s ture

of the

ringed its

call

M(S)

S. We

as s u b s e t s

EI-E5.

We e q u i p

N over

space

(Ms(S) la6I)

Definition

7. We d e n o t e the

space

from

on the

of the

choice

follows again

space

easily

locally

We c h o o s e

a monotonic

: Ms(S)

Let

(cf.

f

M

struc-

function (S)

subspace

semialgebraic

f r o m M by b a s e

is,

of N,

in

and

that

field

tlhe s p a c e

(MaJs6I).

is the

space

N by M(S).

We

extension

from

R t__oo

M(S)

not depend

of M(S).

2.7

same 2.12.

semialgebraic map

Remark

field

does

If M is also

space

as d e f i n e d

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

exhaustions

8. We

f r o m R to S.

call

map

2.6).

~ N<(s) (S). All fs

space

fs

< : I ~ J such

extension

choice

sion

space

every

N.

(Ms(S) Is6I)

1.6 this

and

semialgebraic

: M ~ N be a w e a k l y

map

Definition

limit

By T h e o r e m

semialgebraic

semialgebraic of the

subsets

into

of N.

exhaustion

weakly

a weakly

(fs)s

of

injects

a locally

semi-

in

This

[LSA].

t h e n M(S)

is

semialgebraic.

We o b t a i n

by b a s e

Ms(S).

semialgeb:~aic,

from Proposition

of N.

yield,

inductive

Proposition

exhaustion

c NK(a)

the

the w e a k l y

t h e n M(S)

Let N be a s e c o n d

f(Ma)

N with

M as a s u b s e t

It is e v i d e n t

Ma(S)

family

a closed

obtained

regard

Every

The

S is w e a k l y

structure,

maps.

of N.

spaces

is an e x h a u s t i o n

always

algebraic

transition

semialgebraic

space

given

these

: M(S)

The from

over

R and

semialgebraic

: M(S) that,

restrictions R to S,

~ N(S) for fa

fit t o g e t h e r

~ N(S).

This

map

obtained

over

: Ma

from

choice

over

S as

R.

follows.

~ N<(a) maps

to a w e a k l y

fs n e i t h e r

on the

an

a 6 I,

semialgebraic

maps

fs the m a p

map

every

these

of M and N nor

(NBIB6J)

depends of

f by b a s e

on the

~.

field

exten-

22

If

N

way,

is

a locally

from

fs

: M(S)

in

[LSA]

any

morphism

4 N(S) if

semialgebraic

both

(resp. M

and

f

: M ~ N

space

over

(resp.

g

R we

obtain,

: N ~ M)

N(S)

~ M(S))

over

S.

This

N are

locally

semialgebraic.

over is

the

in

a similar

R a morphism same

map

as

§3 - S u b s p a c e s

As before, fixed

and p r o d u c t s

M is a w e a k l y

exhaustion

Definition

I. A s u b s e t s 6 I, the

space

i.e.,

weakly

This as

X of M is c a l l e d

in the

notation

subsets

of

weakly

[LSA],

does

not depend

on the

is e v i d e n t

from

Proposition

2.7.

M a 6 T(M)

for e v e r y

X ~Y

are a g a i n

semialgebraic

in M.

that

the p r e i m a g e

f-1(X)

f : N ~ M is w e a k l y

semialgebraic

as

follows.

of any X 6 ~ ( M ) semialgebraic

sets

Recall

that

(MsLs6I).

of M s

hence

Remarks

1.17),

3.1.

algebraic

subset

is c o n t a i n e d that,

for

i) A s s u m e

that,

in X B. Let X d e n o t e

every

M°s is an

for e v e r y

X a of M a such

exhaustion

c ~(M).

It is a l s o

X UY, easily

a weakly

~(M) c ~(M).

and

verified

by

"collecting" the

(open)

semialgebraic

family

of p a t c h e s

semialgebraic

subset

space.

a 6 I, t h e r e

whenever

the u n i o n

subset

X DY,

semialgebraic

B
is g i v e n the

of all X a. U s i n g

of M a. Thus

a semi-

set X a n M B

s 6 I,

is a s e m i a l g e b r a i c

(Msls6I),

Also

X n M a = U(XBIB<_a).

By E3 this

set of

in N.

a semialgebraic

that,

semialgebraic

of M t h e n

(M~Is6I °) d e n o t e s Every

in M if,

T(M).

~(M)

under

can be o b t a i n e d

of the e x h a u s t i o n (cf.

is a

a 6 I.

subsets

weakly

in the

of the

Clearly

semialgebraic

sets

(M Is6I)

semialgebraic

by

choice

If X and Y are w e a k l y

Weakly

R and

X D M a 6 ~(M ). The

of M is d e n o t e d

set Y(M)

map

over

set X N M s is s e m i a l g e b r a i c

semialgebraic

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

space

of M.

for e v e r y Ms,

semialgebraic

X 6 T(M).

E4 we

see

24

ii)

Assume

that,

semialgebraic

for e v e r y

subset

Z

primitive

index

s 6 I O,

o f M°.~ L e t X d e n o t e

the

l y X A M ~° = Z ~ for e v e r y ~ 6 I °. T h u s w e h a v e ,

there

union

is g i v e n

of all

for e v e r y

a

Z~. C l e a r -

a 6 1'

X A M s = u(zBIB<_s,~6I°)

This

is a s e m i a l g e b r a i c

If X is a w e a k l y s 6 I, t h e subspace spaces

semialgebraie

of M

. We c o n c l u d e

subset

of M s . A p p l y i n g (X N M s l s 6 I )

we

obtain

on X w h i c h

is t h e

space

X N M s is a c l o s e d

is an e x h a u s t i o n structure

Theorem

of X.

a structure limit

semialgebraic It is e v i d e n t

on X d o e s

not

space,

1.6 to the

inductive

depend

t h a t X 6 V(M).

of M t h e n w e

set X N M s as a s e m i a l g e b r a i c

space

space

subset

regard,

namely

a semialgebraic

set X a n d

of a w e a k l y of the

choice

family

of

X NM s . Every

of X a n d

from Proposition on the

the

semialgebraic

spaces

subspace

for e v e r y

(X A M s l s 6 I )

2.7

that

the

of the e x h a u s t i o n

(M~Ls6I).

Definition

2. W e c a l l

such

a space

X a

(weakly

semialgebraic)

subspace

of M.

This

terminology

immediately spaces

[DK2,

Proposition

from Theorem p.

b y the

2.3 a n d

the

following subspace

fact, theory

which for

follows

semialgebraic

186].

3.2.

semialgebraic. space

is j u s t i f i e d

L e t X 6 T(M).

If g

The

inclusion

: N ~ X is a m a p

N to X t h e n g is w e a k l y

from

map

j : X ~

a weakly

semialgebraic

iff

jog

M is w e a k l y

semialgebraic is w e a k l y

semi-

algebraic.

Also

the

algebraic

following spaces.

fact

is r a t h e r

obvious

from the

theory

of semi-

25

Proposition iff

3.3.

Y 6 ~(M).

X and

3. A

subset

the

subspace

all

semialgebraic

that

Proposition iff

there

Proof.

some

By X,Y

semialgebraic

s 6 I with

X c M s.

and

X cM a for

some

of M s, equal.

is an

and Thus

Then

X n M s = X,

it X UY,

2.3

and

image

f(X)

any

X 6 ~(M)

We

introduce

Definition for

every

Ma,

as

a)

a 6 I,

defined

Then

Y with

Y 6 T(X)

respect

to

the

in M

We

if X 6 T(M)

denote

the

s 6 I then

for

every

X of M

is

semialgebraic

X = X n M s is

structures

a semi-

on X with

respect

in M.

Conversely,

is an e x h a u s t i o n

of X.

But

i.e.

X.

By

Proposition

2.7

also

there

exists

X c M a.

q.e.d.

that

X ~Y

are

for

any

again

semialgebraic under

of

a 6 I.

semialgebraic

of

set

semialgebraic

semialgebraic

theory

a weakly

two

we

also

semialgebraic

sets

in M.

see

that

map

f

the

: M ~ N

in N.

on M

4.

is

{X}

X nY,

Theorem

semialgebraic

X

space.

subset

subspace

is o b v i o u s

Using

is

the

exhaustion

sets

of

on

X.

semialqebraic

M a 6 ~(M)

some

X 6 ~(M).

the

of

~(M).

A weakly

proposition

in M

of M b y

3.4.

s 6 I with

this

a subset

structures

is c a l l e d

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

to M a n d M s a r e

(X n M s l s 6 I )

Y be

subspace

~(M)

subset

that

let

is a s e m i a l g e b r a i c

subsets

If X 6 V(M)

assume

the

X of M

X of M

exists

algebraic

case

and

equal.

and

Notice

X 6 ~(M)

In t h i s

to M a r e

Definition

Let

the

strong

A

subset

the

set

in

topology,

U of M

is o p e n

U n M a is o p e n

[DK2,§7].

defined

In o t h e r

in t h e

as

strong

in M s in t h e words,

follows.

topology

strong

U N M s is

if,

topology

a - perhaps

of

in-

26

finite

- union

b) We d e n o t e This

of open

the set M t o g e t h e r

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

ductive

limit of the

logical

spaces,

gical

spaces.

(M la6I).

Caution. basis

space

sets

3.5.

is open

in Mtop,

i) ~(M)

topology.

ii) ~(M)

is the

topology. now,

ringed

space M

Proof.

Both a s s e r t i o n s

general

(cf.

the

with

case.

X 6 ~(M) the

(cf.

the

hold

but

the

Def.

which

subspace

as a small

tained.

is not a

are open

in

in M,

subspace

in M in the in M,

subspace

if M is semialgebraic. it is easy

of

strong

as d e f i n e d

of the

(Recall

to e x t e n d

function

1.3 and

t h e m to the

for ii.)

"closure"

to the strong semialgebraic

"closed

as an open

structure

"continuous",

terminology

~(M)

structure

are closed

"closed",

"open w e a k l y

in general

4).

topological

refer

topolo-

of the e x h a u s t i o n

subspace

F r o m now on,

usually

of g e n e r a l i z e d

C.

A new terminolo@y. "dense",

of topo-

3).

from that

(Recall2.2

in-

that M t o p is Hausdorff.

structure §I,

structure

§I, Def.

Starting

Mto p is the

is the set of all X 6 ~(M) w h i c h

space M

ringed

1.4 for ii.)

3.12)

set of all X 6 ~(M)

with

sense.

by Mto p.

in the c a t e g o r y

cf. A p p e n d i x

For e v e r y X 6~(M)

coincides

topology

on our choice

in Mtop,

For every

as d e f i n e d now, c o i n c i d e s the fu n c t i o n

(Cor.

of M s.

strong

in the c a t e g o r y

M t o p does not d e p e n d

of open

the

((M)topla£I)

in fact even

Every U 6~(M)

strong

with

subsets

in the c l a s s i c a l

family

We shall see soon

Proposition the

semialgebraic

topology. sets"

semialgebraic"

The

instead

notions and

like

"interior"

sets U £ ~(M) of

"open

for the sets

"open",

sets"

in ~(M)

of a set,

are now called Our p r e v i o u s can be m a i n -

27

Proposition again

The c l o s u r e X of a s e m i a l g e b r a i c

semialgebraic

Proof. also

3.6.

This

[DK2,

Th.

(in the

is d e n o t e d

is the

also that ~(M) n ~(M)

~(M)

a 6 I and since X is known

to be s e m i a l g e b r a i c

closed weakly

strong

topology)

6. ~c(M)

denotes

set of c o m p l e m e n t s = ~(M)

the

semialgebraic

and X £ ~(M).

The

by ~(M).

M~U

of all U 6 ~(M).

and

= {X 6T(M) IX N M a 6 ~ ( M a) for every

Definition

a 6I}.

set of all X 6 ~(M)

such that the

X of M is a w e a k polytope.

Proposition

3.7.

X N M a £ ~c(Ma)

Proof.

in M

sets

3.5.i, ~(M)

subspa c e

is well

5. A subset X of M is called

set of all these

Notice

X c M a for some

of X in Ma, w h i c h

if X is closed

By Prop.

since

7.7].

Definition in M,

of M is

in M, h e n c e X £ ~ ( M ) .

is evident

the c l o s u r e

subset

A subset X of M is an e l e m e n t

for every

Let X £~c(M)

X DM a 6~(Ma).

Since

(X n M a l a £ I )

space X D M a is a p o l y t o p e

Thus X 6 ~ ( M ) .

X D M a 6 ~ c ( M a) for every

iff

a 6 I. Then X 6 ~ ( M ) .

be given.

of the space X, e v e r y

of ~c(M)

Conversely,

is an e x h a u s t i o n in M a. In particular,

if X is a subset

a £ I, then X 6 ~(M)

of M w i t h

and the space X is a w e a k

polytope.

Remark locally

3.8.

Assume

that our w e a k l y

semialgebraic.

coincides

Then

w i t h the strong

semialgebraic

the strong

topology

topology

defined

space M is also in M d e f i n e d

here

in I, §3. The set

~(M)

28

of w e a k l y

semialgebraic

semialgebraic space

structures

follows 7(M), Now

~(M),

Remark space

~(M),

and

3.9.

~(M),

~(M)

Let X 6 T(M).

holds

4.8.e

in I, here

§3.

~c(M)

2.12. the

below.

if X 6 ~ ( M ) .

But

Indeed,

this

A M s is c l o s e d

in

(Ms)to p for e v e r y

symbols

sets

is o p e n

All

sub-

this

T(M),

as in

for the

holds

the

~(M),

[LSA].

strong

topology.

a topological

obviously

in M t o p,

sub-

if X 6 f(M).

of X w h i c h since

a 6 I. C l e a r l y

is

(M \ U ) U AX

NMa =

= V.

U £ ~(M).

3.10

("Tietze's

semialgebraic

interval

Our

let V be a s u b s e t

(M~X)

Proposition

X 6 ~(M)

are equal.

X t o p is not

:=

then

UV

set of l o c a l l y

same m e a n i n g

of o p e n

In g e n e r a l ,

the

every

and t h e r e

have

bases

with

For

U

If V £ T(X)

weakly

coincides

Proposition

are b o t h

in X t o p. T h e n

(X ~ V )

g

from

of M t o p, cf.

It also

defined

on X d e f i n e d

immediately

~(M)

open

subsets

subsets

extension

function

theorem").

Let A 6 ~ ( M ) .

f : A ~ L with values

L c R can be e x t e n d e d

to a w e a k l y

Every

in a g e n e r a l i z e d

semialgebraic

function

: M ~ L.

This

follows

from Proposition

1.11.c

and

Remark

1.12

if we h a v e

proved

Lemma exists

3.11.

Given

a closed

an e x h a u s t i o n

weakly

(AIlI6A)

semialgebraic

of M and

a subset

subset

A of M t h e r e

J of A such

that

A = U(A~I~6J).

Proof.

We

start

with

direct

product

second

set b e i n g

A

the

given

:= I x {0,I} ordered

with

the

of the o r d e r e d

by 0
if t = I a n d A ( ~ , t ) = A N M s exhaustion

exhaustion

desired

For

if t = O. property.

any Then

(M la6I). sets

We c o n s i d e r

I and

(s,t) 6 A

{0,1},

we d e f i n e

(A(~,t) I (a,t) £ A )

the

the A

(~,t)

is an

=M

29

Corollary

3.12

semialgebraic function

("Urysohn's

g : M ~

g-1

[O,1] w i t h

sets g - I([o, ¼ [) and g - I ( ] ~3, 1 ] ) .

Remark

3.13.

tion g : M ~

Starting

g-1(O)

the f o l l o w i n g

procedure

observation.

of M and g l , g 2 : X 1 U X 2 ~

[O,1]

g~1(O) D X j

-I gi ( I ) N X j

for

m B DXj,

is a s e m i a l g e b r a i c -I

function

(O) = A n (X I U X 2 ) ,

Definition covering

Notice

case

1.6]

(1,2)

of M if e v e r y B £ ~(M)

subsets

functions with

(I) D X i = B n X i ,

and =

(2,1),

I then g :=~(g1+g2 ) in

[O,1]

and

(X I U X 2 ) .

in ~(M)

is c a l l e d

is c o n t a i n e d

it s u f f i c e s

1.11.c, u s i n g

semialgebraic

are s e m i a l g e b r a i c g~1

func-

this can be p r o v e d

to the p r o o f of Prop.

= B N

(XIII£A)

semialgebraic

= B.

[DK I, Th.

D A NXj, =

semialge-

A and B as one p o i n t

on X I U X 2 w i t h v a l u e s

g-I(I)

7. A f a m i l y

m a n y X I. Of course, through

(i,j)

= B.

semialgebraic

*)

on M is H a u s d o r f f .

If X I , X 2 are c l o s e d

-I g i (0) N X i = A N X i ,

a weakly

to find a w e a k l y

similar

closed weakly

by the o p e n w e a k l y

= A and g-I(I)

f r o m the s e m i a l g e b r a i c

by an i n d u c t i v e

exists

Choosing

topology

It is e v e n p o s s i b l e [O,1] w i t h

two d i s j o i n t

(0) m A and g-I(I)

A and B can be s e p a r a t e d

sets we see that the s t r o n g

g

Given

s u b s e t s A and B of M t h e r e

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

Lemma").

to c h e c k

an a d m i s s i b l e

in the u n i o n of f i n i t e l y

this p r o p e r t y

with B running

the sets M a.

that,

admissible

for any U 6 ~(M),

coverings

(XIil6A)

w i t h Xl 6 ~(U)

for e v e r y

~) AS always,

[0,1]

the e l e m e n t s of U,

of COVM(U)

are just the

in the s e n s e of t h i s d e f i n i t i o n ,

I 6A.

denotes

the c l o s e d

unit

interval

in R.

30

Admissible

coverings

Proposition (XIiI6A)

3.14.

behave well under

If f : N ~ M is a w e a k l y

is an a d m i s s i b l e

admissible

covering

Proof.

Let B 6 ~(N).

A with

f(B)

Admissible

covering

implies

of M by c l o s e d

semialgebraic

sets,

This

by the f o l l o w i n g

theorem,

weakly

Let

(AIII6A)

(resp. ~(AI) , resp. and

If U 6 ~(M), for e v e r y

d)

iff,

into a f u n c t i o n

for w e a k l y

ringed

I 6 A, the r e s t r i c t i o n

properties Then

Let

E2-E5.

(AIiI6A)

of ~(M)

for e v e r y

I 6 A, the f u n c t i o n

3.16.

or e v e n by

T h e y are less special

which

do the sames

service.

can be v e r i f i e d

covering

in

of M by c l o s e d

(resp. ~(M),

resp.

X D A 1 is an e l e m e n t

is a f a m i l y of s u b s e t s

then a function

(Gluing p r i n c i p l e

Corollar[

sets,

q.e.d.

~(M))

iff,

of ~ ( A I)

~(AI) )-

(XBiB6J)

(XBi~6J) 6 COrM(U) c)

s u b s e t J of

subsets.

I 6 A, the i n t e r s e c t i o n

If U 6 ~(M)

is an

a finite

sometimes

be an a d m i s s i b l e

a) A s u b s e t X of M is an e l e m e n t

b)

if

manner.

semialgebraic

for e v e r y

exists

can be useful. they

3.15.

(f-1(Xl) IiEi)

semialgebraic

and n e v e r t h e l e s s

Theorem

m a p and

t h a t B c U(f-1(Xl) I16J).

than exhaustions,

a straightforward

semialgebraic

of M then

T h e n f(B) £ r(M). T h e r e

coverings

is i n d i c a t e d

preimages.

of N.

c U ( X l i l 6 J ). This

closed weakly

taking

(X~ D A I I B 6 J )

f : U ~ R is w e a k l y

6COVAI(U

flU ~ A 1 on A 1 is w e a k l y semialgebraic

DAI).

semialgebraic

maps).

iff

semialgebraic.

A map

space N o v e r R is a m o r p h i s m

f : M ~ N

iff,

for e v e r y

fJA l : A 1 ~ N is a m o r p h i s m .

(AIII6A)

Assume

I 6 A,

of U, t h e n

be an o r d e r e d

t h a t e v e r y B 6 ~(M)

is an e x h a u s t i o n

of M.

family

in ~(M)

is c o n t a i n e d

(Again it s u f f i c e s

with

the

in some A Ito let B run

31

through

the

s e t s M0.)

Indeed,

Theorem

s p a c e M is the

Definition val

[O,1]

3.15.d) inductive

8. A p a t h in R to M.

of x in M

is t h e

y

~ M with

: [0,1]

{as w e l l limit

For

any point

s e t of all y 6 M y(O)

open weakly

semialgebraic

as s u b s p a c e s

Proof. union

Every

different

It is e a s i l y

seen

disjoint

empty

tion

3.17

nentsof

that

The

components

s 6 I, the

that

there

the

(AIII6A).

from the unit component

exists

is c l o s e d

M is t h e

of M,

inter-

C(x,M)

a path

and

direct

considered,

of t h e

) N~(Ma).

that

a path

semialgebraic claim

also

sum

(cf.

of c o u r s e ,

the path

space M

X is n o t

semialgebraic

such a space

C(x,M)

n M

is a

. Thus

follows.

component

open weakly

9. W e c a l l we call

The

intersection

components

the u n i o n

of t w o

subsets.

X connected.

Justified

of M a l s o

the

by Proposi-

connected

compo-

M.

semialgebraic

exhaustion

to c o n s t r u c t

with

map

C(x,M)

space

L e t N be a s e c o n d w e a k l y

spaces

of s p a c e s

us

= y.

component

in M.

path

components ~(M

Definition

path

for e v e r y

C(x,M) N M a £

non

such

tells

of M.

Clearly, of p a t h

family

x of M the path

= x a n d y(~)

3.17.

of t h e

of t h e

3.15.a)-c)}

in M is a s e m i a l g e b r a i c

Proposition

1.10)

as Th.

of N. W e w a n t

M a n d N. W e e q u i p

the

inductive

semialgebraic

limit

spaces

the

cartesian

space

space the

over

direct

product

structure

(M a xN~b (~,B) 6 IxJ),

of

R and

(NBL~6J)

product

of t h e

MxN

of the

the o r d e r e d

where

IxJ

an

sets M,N

family

of

is t h e d i r e c t

32

product tions the

of t h e o r d e r e d sets

in T h e o r e m

space MxN

1.6 a n d

is w e a k l y

(M s x NBI (a,B) 6 IxJ) cides

with

braic

spaces

algebraic product spaces

maps,

pr]

Caution.

strong

product

topology

of t h e

3.18.

semialgebraic

strong

L e t X £ T(M)

in the

coincides

spaces

X a n d Y of M a n d N.

semialgebraic. XxY

is c l o s e d

All

this

F(f)

subspace

(resp.

3.19. of

finally

2.3

assump-

by T h e o r e m

of M a x N B in M x N

product

of t h e

it is e a s i l y

1.6,

coin-

semialge-

checked

that

p r 2 : M x N ~ N are w e a k l y

these

projections,

WSA(R)

of weakly

semi-

is t h e d i r e c t

semialgebraic

have more

topologies

on M a n d N.

and Y 6 T(N). and t h e

Then

the

subspace

closed

open)

than

set X x y

product

semialgebraic

(resp.

sets

structure

as t h e d i r e c t

If X a n d Y are

open

the

is w e a k l y on XxY

of t h e

sub-

then XxY

is

in M a n d N t h e n

open) in MxN.

from

Let

the d e f i n i t i o n s .

is e a s i l y

Using

Prllr(f) to M.

construct

fibre

Theorem

2.3

also

the

verified.

f : M ~ N be a w e a k l y

of MxN

f : M ~ L and g

on M x N m a y

structure

f is a c l o s e d

projection F(f)

the

If X a n d Y are

proposition

Proposition

natural

with

is o b v i o u s

following

We

with

s p a c e MxN,

in M x N

graph

Theorem

Thus,

the

R.

The

Proposition

structure

: M x N ~ M,

t h a t MxN,

fulfills

exhaustion

as the d i r e c t

of M a n d N in t h e c a t e g o r y over

direct

and

with

subspace

M s a n d N B. U s i n g

family

E4 and E5.

semialgebraic

structure

projections

This

in a d d i t i o n

and the

the given

the n a t u r a l

I a n d J.

weakly from

semialgebraic

F(f)

products

: N ~ L be w e a k l y

semialgebraic

t o M is a n

in t h e

subset

of MxN.

isomorphism

category

semialgebraic

map.

maps

WSA(R). over

R.

The The

of the

Let Then

33

fxg

: MxN ~ L×L

XLN

M

is again a w e a k l y

:= { (x,y) 6 M x N l f ( x )

is a c l o s e d

weakly

of t h e d i a g o n a l

with

subspace

the

in a s t r a i g h t f o r w a r d

Theorem

3.20.

The

in MxN.

commutative

of L x L u n d e r The

fxg.

following

is t h e

We equip M XLN

c a n n o w be v e r i f i e d

, L

f

p a n d q the n a t u r a l

Caution.

We do not

g o r y of f u n c t i o n

Remark

3.27.

claim

ringed

projections,

is again l o c a l l y semialgebraic

product

M XLN

in

is r a t h e r

objects

tension

(cf.

that spaces

If M a n d N a r e

locally

All

since M XLN

diagram

a n d pr 2 to M X L N , is a c a r t e s i a n

This

of MxN,

g

M

M xL N

and

manner.

P

with

subset

A L = F ( i d L)

structure

map

= g(y) }

semialgebraic

preimage

semialgebraic

square

this

then our

restrictions

of pr I

is c a r t e s i a n

in the

cate-

R.

locally

semialgebraic.

the

in W S A ( R ) .

diagram

over

also

i.e.

semialgebraic

If all

s p a c e M xL N

three is t h e

spaces,

spaces same

M,N,L

as t h e

then are fibre

[LSA].

evident

defined

from

in t h i s

the definitions

section

Def. 7 a n d Def. 8 in §2)

behave to some

and Proposition

well real

under

base

closed

2.12.

field

field

ex-

S m R.

34

Remarks

3.22.

a)

L e t X £ ~(M).

set X n M s of M a y i e l d s (X N M s ) (S) of Ms(S) sets The

in M ( S ) . subspace

base

field

notion X(S)

by base

[DK3,

p.

every

field

142].

W e h a v e X(S) N M s ( S ) structure

extension

X(S)

For

has

on X(S)

of the

(resp.

open)

X(S)

semialgebraic

a semialgebraic

denote

the union

(X ~ M s ) (S), h e n c e

in the

s p a c e M(S)

subspace

no a m b i g u i t y .

is c l o s e d

extension

Let =

s 6 I, t h e

structure

If X is c l o s e d

in M ( S ) .

same

on X in M.

If X 6 ~(M)

subset

of t h e s e

X(S) 6 ~(M(S)).

is t h e

(resp.

sub-

as t h e

Thus

open)

in M,

the then

t h e n X(S) 6 [(M(S)).

W e h a v e X(S) n M = X. b)

If

(XIII6A)

subsets, c)

If

then

The

weakly

the two

(M x LN)(S) sion

map,

same

then

semialgebraic

subsets

WSA(R)

M(S)

proofs

maps

components

of M,

components

then

of M ( S ) .

If f

F(f s)

o f a),

b), c)

e)

: M ~ N is a

and

F(f) (S) of

is a l s o

This means

easy

the

be given.

Then x 6Ma(S)

for

~ M with

some

q 6 M s . (This

6(O)

given

fs,gs ) and

that

the base

field

cartesian

squares.

similar

statements

once we know

following

that

the

may

exten-

safely

spaces

claim:

is c o n n e c t e d .

as f o l l o w s :

to a simplicial

: N ~ L are

from the maps

and of

c a n be p r o v e d

f r o m x to s o m e p o i n t

(coming

preserves

d),

t h e n M(S)

f : M ~ L, g

We conclude

claim

6 : [O,1]

of M ( S ) .

subsets

xL(S) N(S)

~ WSA(S)

are connected.

morphic

covering

as M ( S ) x N ( S ) .

the

semialgebraic

equal.

If M is c o n n e c t e d

The

of c o n n e c t e d

is the

l e f t to the r e a d e r ,

Cl(S)

of M by weakly

of c o n n e c t e d

of (MxN)(S) are equal.

functor

The easy be

(MxN) (S)

If t w o w e a k l y

then

family

family

semialgebraic

M(S) x N(S) a r e e)

is t h e

is the

space

covering

(XI(S) il6A ) is an a d m i s s i b l e

(CILI6A)

(Cl(S) ll6A) d)

is an a d m i s s i b l e

We

fix

a point

p 6M.

Let x 6M(S)

a 6 I. T h e r e

exists

a p a t h y in Ma(S)

is e v i d e n t

since,

say,

complex

over

R.)

= q and

6(I)

= p.

There Then

exists

M s is iso-

a path

6 S is a p a t h

in M(S)

35

f r o m q to p.

The

composite

path

y~6 S c o n n e c t s

x to p.

§4 - Spaces

In this

of c o u n t a b l e

small

introduce

a class

able type", spaces

and,

up to some examples, of w e a k l y

which

behaves

admit

over

following

R. Also,

its natural

Definition covering able

down the

means

set ~

(XIII6A)

the

section we "spaces

exhaustions.

in this paper.

a weakly

class

Unfortunate-

semialgebraic

numbers

of

topology.

family of sets

of natural

of count-

This

in s e m i a l g e b r a i c

an o r d e r e d

space

(Xnln6~)

is e q u i p p e d

ordering.

I. A space M is of c o u n t a b l e

index

Remarks

that

total

constructions

always

trivial

spaces,

simple

for all purposes

if we write

then we always mean with

particularly

a "space"

almost

semialgebraic

well under most

ly it does not suffice

In the

type

by s e m i a l g e b r a i c

type

if M has an a d m i s s i b l e

(cf.

§3,

sets

Def.

7) with

count-

set A.

4.1.

i) Of course,

every

semialgebraic

space

is of c o u n t a b l e

type. ii)

If M is of c o u n t a b l e countable

iii) The direct

product

MxN of two type.

coverings

K countable,

The direct spaces

v)

then

sum

If S is a real

spaces

Indeed,

if

(X 1 x yKI (I,K) 6 AxK) sets with

U(MIII6A)

of c o u n t a b l e

countable

also every

subspace

of M is of

M,N of c o u n t a b l e (XI{16A)

and

of M and N by s e m i a l g e b r a i c

of M x N by s e m i a l g e b r a i c iv)

then

type.

again of c o u n t a b l e admissible

type,

type

closed

of a family

is of c o u n t a b l e

overfield

type then M(S)

(YK{<6K) sets,

is an a d m i s s i b l e

countable

index

(MIII6A) type

type

are

with

A and

covering

set A x K. of non empty

iff A is countable.

of R and M is a space over

is again of c o u n t a b l e

is

type

(cf.

R of

3.22.b).

37

Proposition

4.2.

an e x h a u s t i o n with

s(n)

We

of M.

~ s(n+1)

exhaustion

Proof.

contains

of s u b s e t s

for e v e r y

algebraic

indices

X n and

s(n+1)

~.

that

Let

(Mals6I)

(s(n) I n 6 ~ )

(Ms(n)

a faithful

n6~)

Assume

be

in I, is an

By Prop.

I such

that

(Ma(n) t n 6 ~ )

exists

some

n 6 ~

this

and

4.5.

able

iff

with

union

Let the

that

there

implies

their

Corollary type

4.2

The

ordered

E2-E5,

3.16,

just

and

such

family

that

(Ms(n) I n 6 ~ )

it is an e x h a u s t i o n

type

a family

sets.

it is an a d m i s s i b l e of M.

iff M has

(Mnln6~)

M n c M n + I for e v e r y

M is of c o u n t a b l e Then

exists

q.e.d.

an e x h a u s t i o n

of c l o s e d

n 6 ~ , such

semi-

that

an i s o t o n i c

M B c Ms(n). B ~a{n).

The

is I. Thus

type

and

(Mals6I)

is

I is c o u n t a b l e .

is an e x h a u s t i o n

(MalaEI) set

semialgebraic

in some M n.

of M.

Proof.

by

in I for n = 1,2,...

~ a(n).

means

of M, w i t h

exhaustion

(Xnln6~)

M is of c o u n t a b l e

is c o n t a i n e d

4.4.

a(n)

trivially

This

subsets

Proposition

finite

type.

a sequence

n £ ~ , such

by Cor.

A space

set

A £~(M)

faithful

exists

covering

find

Thus,

4.3.

index

every

there

of M f u l f i l l s

of M.

Corollary with

Then

an a d m i s s i b l e

successively

covering

t h a t M is of c o u n t a b l e

of M.

M has

Ma(n)

Assume

Since sets

sequence

of M. the

J(n)

For

(a(n) In6D~)

every

exhaustion

B 6 I there (Mals6I)

:= {a611s ~ ( n ) }

I is c o u n t a b l e ,

be an e x h a u s t i o n

I ° of p r i m i t i v e

indices

of M. (cf.

in

is

are q.e.d.

T h e n M is of c o u n t end

of

§I)

is

countable.

Proof.

The

family

M by s e m i a l g e b r a i c

of p a t c h e s sets.

Thus,

o s£1 o) (Mal

is an a d m i s s i b l e

if I ° is c o u n t a b l e

covering

of

t h e n M is c e r t a i n l y

38

of c o u n t a b l e the

type.

set of all

B
indices

with M B = M

proof

of

1.14)

and this

and

4.6.

now that M

a 6 I such

. Then

implies

Proposition

Assume

that

Assume

that

We

assume

choose

an e x h a u s t i o n

without

of M b y o p e n

UK(n)

for e v e r y since tained

X

union

§4),

of g e n e r a l i t y of M a n d

semialgebraic.

and every

connected

subsets.

finite

For

covering

any

finite

~ £ K. W e

subsets

We

subset

K

inductively

of A and a s e q u e n c e

that

start with

(as a l r e a d y

:= U(UK(n) I n 6 ~ ) = U ( M

K(1)

= @.}

observed

in 2.12).

conclude

from the

S i n c e M is c o n n e c t e d

t h a t M is of c o u n t a b l e clude by Proposition and

type

3.6

and Lindel6f

(which s t a t e s

1.7.15

is p o s s i b l e set U K is c o n -

Let

( n ) I n 6 ~ ).

6) t h a t X n M s = M 8 for s o m e

open.

This

in s o m e U K and e v e r y

E4

M is taut)

set J is c o u n t a b l e ,

an a d m i s s i b l e

then we

[LSA],

(cf.

c UK(n+I)

in s o m e M B

B u t X is a l s o

the

of M

t h a t M is c o n n e c t e d .

of all Ul w i t h

set M a is c o n t a i n e d

Def.

an i n d e x

type.

(K(n) I n 6 ~ ) of

n £ ~ . {We m a y

§I,

exist

exhaustion

4.4,

If s 6 I is g i v e n (cf.

L e t J be

q.e.d.

in I,

semialgebraic

the

c Ma(n)

every

loss

(M Is6I)

(s(n) I n 6 ~ ) in I s u c h

not

s p a c e M is l o c a l l y

(as d e f i n e d

Proof.

a sequence

is a f a i t h f u l

the

of M is of c o u n t a b l e

choose

does

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

component

of A let U K d e n o t e

there

type.

I ° is c o u n t a b l e ,

T h e n M is p a r a c o m p a c t

(UIlI£A)

that

(Mala6J)

jo = I o.

is of c o u n t a b l e

exhaustion

that,

E3 a n d

~ ~ s. T h u s X 6 ~ ( M ) .

we have (cf.

axioms

X = M.

I, §4,

This

Def.

proves

3).

We con-

in the t e r m i n o l o g y

t h a t M is p a r a c o m p a c t ,

of

q.e.d.

39

Example

4.7

(The c o u n t a b l e

is the u n i o n

of M o

{~} x ]O,I]

("the

n E ~,

set M n

the

Theorem such

1.6 we

that

M and

b)

This

are m e t w i t h

and

Let M be the the

comb",

countably

k 6~)

subset many

the

set M w i t h

in its

given

the u n i q u e

structure, of M.

of R 2 w h i c h

sets

. We r e g a r d ,

:= M ° U U I U ... U U n as a p o l y t o p e

is an e x h a u s t i o n

M is u s e f u l weak

since

algebraic. (~,O),

of the

a)

Uk

:=

for e v e r y

in R 2. U s i n g

structure

of a s p a c e

is a c l o s e d

subspace

of

T h e n M is a w e a k

polytope

of

type.

space

of ~(M),

teeth

M n,

(MnlnEIN)

countable

:= [O,1]x{O}

equip

every

comb),

polytopes.

The

U = M ~ M o. But

Indeed,

with

to e x h i b i t

a semialgebraic

union

the

subset

U of all

the

which

teeth

commonly

U n is an e l e m e n t

U of U is not w e a k l y

of the

of M o. The

M o in M is t h e c o m p l e m e n t

pathologies

closure

U NM ° consists

n £ ~ , and p e r h a p s

some

infinitely

point

xo

interior

of U N M o in Mo,

:=

(O,O),

of the and

many

semi-

points

and this

xn

is not

semialgebraic

this

:=

set

set is a g a i n

not

semialgebraic. c)

The

space

simplicial tion

M cannot

complex

hoods

such

that,

spaces lies

in the

a path

s

selection e)

The

and

Thus,

: [0,1] lemma

The

is n o t

open

in this

sum

that

U of the

fails

set

the

M o of M c a n n o t

~ M with

X

simplices

Indeed,

have

1.10)

set U in Mtop, c U and

case,

of the

in Xto p.

subspace

Xto p is n o t

Every

but s(1)

direct

many

a topological

But

the

vertices.

the p o i n t

there

does

= x o. Thus

x°

not exist the

curve

spaces.

sum of the

of X w i t h

connected

semialgebraic

If R is a r c h i m e d e a n

topology

point

neighbour-

three

Then

semialgebraic

:= U U {x o} of M is the

in the

~I.

infinitely

to a

a triangula-

triangulation.

R is a r c h i m e d e a n .

is o p e n

assume

dimension

contain

(cf.

s([O,1[)

isomorphic

semialgebraic

of the

in M for w e a k l y

{x o}

is n o t

set N ~ { x n} has

be a v e r t e x

Assume

closure

as a space).

N 6~,

U is the d i r e c t

subspace

{Xo}.

{x o}

x n must

Un(n E ~).

i.e.

s y s t e m ~ of c o n n e c t e d

subset

space

all

for e v e r y

Thus

semialgebraic The

Then

a fundamental

components.

d)

(regarded

of M is given.

x n has

be t r i a n g u l a t e d ,

respect

subspace

spaces

Un

then to M t o p.

of M t o p.

40

Example

4.8

( U n c o u n t a b l e c o m b s ) . Let,

subset

of

of R 2

Further

Mj

:=

equipped space

]O,1],

let,

the

R 2. U s i n g

ture,

by Proposition

Remark

A'

Let We

Assume over

([O,1] x {O}) U (Ax]O,1])

J of i

in the the

that

of M and

semialgebraic

set M w i t h

e v e r y Mj,

(MjIJ c A ,

standard

the unique

in its g i v e n

J finite)

struc-

is an ex-

t h e n M is n o t of c o u n t a b l e

contains

O in its

described

set i in E x a m p l e

a space

R o. W e We

choose shall

a closed

M{I } =

N over

closure,

in 4 . 7 . d

and

e.g.

type

A = ]O,1],

4.7.e

above

the

let ~

also

: M-~N(R)

~([O,I]

~(A' x {O}) c A .

to N(R).

with N a

x {O}) c A ( R ) .

This will

Let

be the

:= ~ ( I , 0 ) . subset

We have

to v e r i f y

X of N such that

that

the

~ £A.

image

set

in X ( R ) .

U

({l} x [O,I]R)

N o w X is a p o l y t o p e

y of a one-dimensional some

with

~

there

set A is c o u n t a b l e .

semialgebraic

( [ O , 1 ] R X {O})

is c o n t a i n e d

that

then

t h a t M is i s o m o r p h i c

s o m e A 6 ~(N),

since

is u n c o u n t a b l e

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

prove

and

4.8

R o such

is g i v e n

contradiction

~ ( M { I }) of t h e

if y h a s

such

the phenomena

I 6 A' b e g i v e n ,

a point

polytope

If A is u n c o u n t a b l e ,

there

:= A ~ {I}.

choose

structure

subspace

If the

not exist

desired

subset

infinite

archimedean.

4.9.

Proof.

finite

1.6 we e q u i p

A as a set w h i c h

for R n o t

the s u b s e t

i be a n y

4.4.

we can observe

space

Theorem

of a w e a k

of M.

Choosing

for a n y

subspace

is a c l o s e d

haustion

l e t ~[ d e n o t e

generally,

([O,I] x {O}) U J × ]O,I],

with

structure

does

and

more

semialgebraic

polytope

of d i m e n s i o n

Y a branchin~

neighbourhood

I. W e point

V o in Y s u c h t h a t

call of Y, for

41

every

semialgebraic

least

three

Choosing

components.

see t h a t

Example

A

does

not

over

decreasina

exist

Suppose

R o. As

R d. o

inequality

field

choosing

R is n o t

there

exists

and we

[0,I] R. W e

L e t li II d e n o t e

elements

s p a c e M in E x a m p l e We claim

We

identify

norm

in R d

with

Mo =

[0,I] R × {O}

square

[O,I] R × [O,I] R, t h e r e

li~(x)-~(y)ll

since Thus

points

[O,I] R. L e t

in R d w i t h o

for n l a r g e

(x,y)

~

tx-yl

and

exist

that

< CIx-yl

N

for a l l x a n d y in rent

such

subset

By the Lojasiewicz

(x,y) ~ 11~(x) - ~ ( y ) ll o n t h e u n i t

in R a n d N in ~

4.8

N a space

of A as a s e m i a l g e b r a i c

euclidean

converg-

to N(R).

with

s o m e A £ ~(N)

con-

that there

functions

C >0

39],

still

to the

constants

p.

but

applied

some

[BCR,

the

think

of X

q.e.d.

: M-~N(R)

proof we choose

see t h a t ~ ( A x {0}) c A .

interval

c A.

type.

~

see t h a t

a triangulation

t h a t M is i s o m o r p h i c

isomorphism

at

of ~ ( M { I } ) .

( 6 n l n 6 ~ ) of p o s i t i v e

Ro such

an

point

arehimedean

is o f c o u n t a b l e

a space N over

{y} h a s

a n d ~ ( M { I }) w e

~ 6 X N A(R)

sequence

in t h e p r e c e d i n g

the unit

in s o m e

Then

that

the

of X(R)

(cf.§9, D e f . 3 below). W e c o n s i d e r t h e

~(Mo) c A ( R ) , with

that

space V~

~ is a b r a n c h i n g

of X(R).

:= { 6 n l n 6 ~ }. T h e n M

Proof.

V c V ° the

triangulation

We conclude

Assume

tains a strictly ing to z e r o

point

~ 6X.

4.10.

with

Clearly

a simultaneous

is a b r a n c h i n g we

neighbourhood

In

11~n- [mliN < C~

C~ n is s m a l l e r

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

:= ~ ( 6 n )" T h e

~ as a b o v e

than

does

n

if m > n every

"

In a r e p a i r w i s e But this

positive

not exist,

diffe-

is i m p o s s i b l e ,

element

of R Oq.e.d.

§5 - P r o p e r

In t h i s and

maps

section

a "map"

else

means

following

exhaustions of M w i t h

a weakly

a weakly

semialgebraic

semialgebraic

to the

I. A m a p

(Y) of e v e r y

spaces

i)

through

the

sets

3.16,

given

map

between

It s u f f i c e s N B.

subset

space spaces,

every

iv)

pull

subspace back

f'

by an a r b i t r a r y

(cf.

this

then

semialgebraic

: M XNN' (weakly

end

property then,

over

R,

if nothing

are

family

the

subset

with

f(X)

in M.

Y running 3.14

and

of M.

of e v e r y

of N. map

X~

M

of a s e m i a l g e b r a i c

semialgebraic)

of patches

if the p r e i m a g e

b y Prop.

image

fixed

of §I).

is an e x h a u s t i o n

inclusion

~ N'

is the

semial@ebraic

(f-I(NB) IB6J)

X of M the

(NBI~£J)

Y of N is s e m i a l g e b r a i c

to c h e c k

family

is a w e a k l y

iii) F o r

(M~Is6I °)

If f is s e m i a l g e b r a i c

the o r d e r e d

X £ ~(M)

(Mala6I),

exhaustion

If f : M ~ N is s e m i a l g e b r a i c

The

and

f : M ~ N is c a l l e d

semialgebraic

5.1.

Cor.

M and N are

respect

Remarks

ii)

means

maps

of M and N r e s p e c t i v e l y .

Definition -I

a "space"

proper

is said.

In the

f

and p a r t i a l l y

map

g :N'

is s e m i a l g e b r a i c . map

f : M ~ N

~ N is s e m i a l g e -

braic.

We c a l l crete,

a set A £ ~(M) i.e.

means

that

point

sets.

for a m a p

Proposition a)

all

is t h e the

discrete

direct

connected

Analogously

to be

5.2.

fibres

of

sum

(in M) (cf.

1.10)

components to Prop.

if the

(cf.

1.5.4

subspace

of o n e - p o i n t §3,

Def.

we h a v e

the

f : M ~ N be a map.

f are

semialgebraic,

Assume

spaces.

9) of A

that

This

are

following

semialgebraic.

Let

A of M is d i s -

one

criterion

43

b)

every

closed

weakly

discrete

semialgebraic

image

Then

f is s e m i a l g e b r a i c .

N.B.

Conversely,

Remark

5.1.ii

The proof

prove

fact

A 6~(M).

discrete

in M.

set

that

:= f(A)

(cf.

[LSA,

A of M has

and b)

A'

of A.

For

goes

p.

every

hold

57]).

For

a

(cf.

every

of the d i s c r e t e set,

subset

and we

space i.e.

[LSA,

the A,

of A.

of the

hence

to

a point every

with Thus

Moreover,

set

56f.]

T h e n we h a v e

A'

conclude

of the

p.

intersection

subset

y 6B,

proof

s 6 I ° we c h o o s e

f(A) £ r(N).

every

A is a f i n i t e

to the

spaces

a finite

for any

b),

Thus

analogous

N is s e m i a l g e b r a i c .

is s e m i a l g e b r a i c ,

subset that

same

By a s s u m p t i o n

B

conclude

a)

semialgebraic

:= { x y I y 6 I °} has The

subset

algebraic

then

is f u l l y

locally

assume

set A

M s . Thus

space

5.2

is s e m i a l g e b r a i c .

x s 6M°a " The

for e v e r y

for

We m a y

that M

subset

f(A).

if f is s e m i a l g e b r a i c

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

easier.

semialgebraic

for b).

corresponding but

weakly

A is

f(A') 6 ~(N)

semialgebraic that

f-1(y)

B is a f i n i t e NA

a finite

I ° is finite.

This

is a semiset.

We

means

that

M is s e m i a l g e b r a i c .

Definition g

: N'

2. A m a p

~ N,

the pull

property:

The

image

algebraic

in N'.

The

f : M ~ N is c a l l e d back

f'

: M XNN'

f' (X) of e v e r y space

one p o i n t

space

is proper.

The

following

facts

are

Remarks the

5.3.

usual

i)

formal

Proper

easily

maps

properties,

~ N'

X 6~(M

M is c a l l e d

the

proper,

if,

for e v e r y

by g e n j o y s xN N ' )

complete,

the

is c l o s e d

map

following weakly

if the m a p

semi-

f r o m M to

verified.

between cf.

weakly

1.5.5.

semialgebraic

spaces

enjoy

44

ii)

If M

is s e m i a l g e b r a i c ,

is c l o s e d spaces [DK2, iil)

Let

in N a n d the m a p

is p r o p e r

in t h e

subsets

f : M ~ N is p r o p e r

iv)

(B l) ~ B~ of

then

is p r o p e r

Theorem

iff,

a map in the

consider

space.

N o w M is a c o m p l e t e

for e v e r y

This means b)

In t h e

braic.

every

that

subset

A 6~(M),

3. A m a p

the

partially

special

case

Then

a map

Thus

semialgesense

flA

to p r o v e x

6M~.

semialgebraic

(I, §5).

p.

59],

and M is i n d e e d that

all

The

set and the

discrete

space.

these

of M.

We set.

semialgebraic.

fibres

of

partially

f are

proper

The

semialgeq.e.d.

if,

for e v e r y

s p a c e M is c a l l e d

from M to the one point

to c h e c k

same

t h a t A is a f i n i t e

: A ~ N is p r o p e r .

A 6~(M)

point

t h a t M is s e m i -

in M,

A is a c o m p l e t e

cf. [LSA,

that every

exhaustion

iff f

spaces

5.2 t h a t M is s e m i a l g e b r a i c ,

if t h e m a p

it s u f f i c e s

locally

t h a t N is t h e o n e

a point

f : M ~ N is c a l l e d

Of c o u r s e ,

and also

a n d we w a n t

and weakly

b y Prop.

This means

s e t s M a of o u r

of N).

weakly

~ 6 A, the r e s t r i c t i o n

locally

space,

argument,

ly p r o p e r .

the

cf.

f : M ~ N is s e m i a l g e b r a i c .

the

of A.

restriction

complete

of

case we now know

We conclude

Definition

map

I ° is f i n i t e ,

general

for e v e r y

s 6 I° we choose

is c l o s e d

b y an e a s y

an e x h a u s t i o n

semialgebraic

first

:= { x a l a 6 I °}

of N b y c l o s e d

category

a) W e

For

covering

in the p r e s e n t

Proof.

conclude

spaces,

f : M ~ N is p r o p e r

proper

holds

f(M)

semialgebraic

semialgebraic

semialgebraic

Every

A

of

between

iff

f is p r o p e r .

5.4.

algebraic.

f : M ~ N is p r o p e r

: M ~ f(M)

category

(e.g.

If M a n d N a r e w e a k l y braic

"f"

be an a d m i s s i b l e

semialgebraic

f

a map

§9].

(Blll6A)

-I

then

space

is p a r t i a l -

is c o m p l e t e .

properties

Notice

that

for A r u n n i n g the p a r t i a l l y

through complete

45

spaces topes

(resp. (resp.

Partially describe

their than

5.5.

spaces)

polytopes)

proper

is e a s i e r

Remarks

complete

maps

formal

defined

will

be

the

in t h e

ii)

If g ° f

is p a r t i a l l y

proper

then

iii)

If g . f

is p a r t i a l l y

proper

and

and

All

this

ties

poly-

of o u r

interest.

Prop.

{The

Thus we

theory

here

3.6.}

proper

proper.

then

gof

is p a r t i a l l y

f is p a r t i a l l y f is s u r j e c t i v e

{It s u f f i c e s

(cf. Def.

proper. and

to a s s u m e

3 in §8)

proper.

instead

semialgebraic that

f is

of surjective

semialgebraic.}

follows

of p r o p e r

Remark

as the w e a k

: N ~ L be m a p s .

If f a n d g are p a r t i a l l y

surjective"

objects

explicitly.

we c a n u s e

i)

"strongly

center

rather

f : M ~ N and g

t h e n g is p a r t i a l l y

same

in §I.

properties

in I, §5 s i n c e

Let

are

5.6.

immediately

from the definitions

and

formal

proper-

maps.

A map

f : M ~ N is p r o p e r

iff

f is p a r t i a l l y

proper

and

semialgebraic.

This

is e v i d e n t

Definition y

We

: [D,I[

are

from the definitions,

4. A n

incomplete

~ M from

interested

completed,

i.e.

c a n be at m o s t

the h a l f

whether

extended

Remark

5.3.iii

and Theorem

path

in M is a s e m i a l g e b r a i c

open

unit

a given

incomplete

to a p a t h ~

one completion

~

.

interval

: [O,1]

5.4.

map

in R to M.

p a t h y in M c a n be ~ M.

Notice

that

there

46

Theorem the

5.7

(Relative

following

f is p a r t i a l l y

b)

If y is an in N,

completion

criterion).

For

a map

f

: M ~ N

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

a)

ed

path

proper.

incomplete

path

in M,

such

t h e n y c a n be c o m p l e t e d

[0,1[

Y

I-

that

6

:= foy c a n be c o m p l e t -

in M.

, M

z

f

/ /

[0,I]

, N

Z

Proof.

a) = b):

(Prop.

3.6).

The

Thus

closure

A of y ( [ O , l [ )

f(A) 6 ~ ( N )

and the map

A by restriction

is p r o p e r .

follows

f r o m the

semialgebraic

(I.6.8,

[DK4,

b) ~ a) : T h e that

flA

in M is s e m i a l g e b r a i c h

: A ~ f(A)

The path 6 runs relative

path

in f(A).

obtained The claim

completion

in M from b) n o w

criterion

2.3]). semialgebraic

is p r o p e r

relative

for e v e r y

path

A 6~(M).

completion

This means

criterion

that

implies

f is p a r t i a l l y

proper.

Corollary weak

5.8

(Absolute

path

polytope

iff e v e r y

incomplete

ProDosition_ map

f

This

The pull

back

: M ~ N b y an a r b i t r a r y

c a n be p r o v e d

criterion. nitions

We

5.9.

completion

as

In c o n t r a s t

in I, to

f'

path

: N'

§5 b y u s i n g also

The

space

M is a

in M c a n be c o m p l e t e d .

: M XN N' ~ N'

map g

[LSA]

criterion).

of a p a r t i a l l y

~ N is p a r t i a l l y

the

a proof

relative

path

directly

proper

proper.

completion

from the defi-

is p o s s i b l e .

indicate

this

second proof.

L e t M'

:= M xN N '

and

let g' :M'

~ M

47

denote the

the p u l l

closure

back

of g by

of g' (A)

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

in M by B and

3.6 we h a v e

and B X D C 6 7 ( M ' ) .

We h a v e

B XDC

f{

obtained

is proper.

f{ IA is proper.

Proposition

B 6~(M)

Since

Since

5.10.

from

C 6~(N)

For

a map

proper.

b)

If P is a w e a k

polytope

If Q is a p o l y t o p e

Proof.

completion

5.11.

algebraic.

and

this

f' (A) Then

of

D

in N'

by C.

:= f(B) 6 ~ ( N )

semialgebraic

by r e s t r i c t i o n ,

semialgebraic

implies

f : M ~ N the

in N t h e n

f

dim Q

defined

sense

of

a) = b) = c)

its C o r o l l a r y since,

that

maps

following

-I

(P)

is proper, also

is proper,

q.e.d.

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

is a w e a k f

fl

in B XDC,

f'IA

< I then

Assume

that

the

are

5.8

-I

(Q)

polytope

in M.

is a w e a k

poly-

iff

path

c) ~ a) and

follows

absolute

6 in N,

the

set

path 6([0,1])

spaces

M,N

f : M ~ N is p a r t i a l l y

here

trivial,

(relative

for e v e r y

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

A map

sense

[LSA]

5.7

criterion),

is a p o l y t o p e

in

f,g,f',g'

in N w i t h

implications

from Theorem

Indeed,

of

We d e n o t e

in M.

The

Remark

square

A is c l o s e d

f is p a r t i a l l y

tope

closure

be given.

and C 6 ~ ( N ' ) .

a cartesian

a)

c)

the

A 6~(M')

~ D

fl

fl,gl,f~,g{

hence

some

, C

B

with

f. Let

f is p a r t i a l l y

are

paracompact

proper

proper

(resp. (resp.

locally

proper)

proper)

semi-

in the in the

[LSA].

partially by the

proper

same

maps

relative

from M to N are

characterized

path

criterion

completion

(Th.

here 5.7,

and

48

Th.

1.6.8).

Evidently

the same here "proper" (Remark

We finally of this

[LSA,

spell

p.

out

complexes.

Let

proper

"semialgebraic"

also holds

for

terms what

are

similar

to those

theories

f as a m a p b e t w e e n iff,

for every

maps

between

(cf.

simplex

simplicial

p. 71f].

map between

spaces

open

the m a i n d e f i n i t i o n s

in [LSA,

f : X ~ Y be a simplicial

a) f is s e m i a l g e b r a i c

in both

since

70]).

in the case of simplicial

We r e g a r d

for f means

"proper",

and s e m i a l g e b r a i c "

in c o m b i n a t o r i a l

The proofs

5.12.

the p r o p e r t y

Thus this

"partially

section mean

complexes.

Examples

and there.

means 5.6,

also

simplicial

2.4).

p 6Z(Y)

the c o m p l e x

f-1 (p) is finite. b)

f is p a r t i a l l y

proper

If T is an open c) Thus

f is p r o p e r

in a) and b).

iff for every

o 6Z(X)

face of ~ w i t h f(~) 6 Z ( Y ) , iff f fulfills

the f o l l o w i n g then

holds:

T 6Z(X).

the c o m b i n a t o r i a l

properties

stated

§6 - P o l y t o p i c

In t h i s

spaces;

section,

as

semialgebraic

space

semialgebraic

map.

L e t M be a s p a c e

Definition

the one-point

in t h e p r e c e d i n g over

and

R, and

(Msls6I)

I. M is c a l l e d

braic

s p a c e M s is l o c a l l y

point

x 6M a has

this

property

Remarks ii)

6.1.

Every

closed

and

every

a "map"

a "space"

between

if,

complete.

This means

on t h e

polytope open

spaces

for e v e r y

in M s w h i c h choice

a weakly

means

s 6 I, the

(I, §7)

a weakly

of t h e

semialge-

that

is a p o l y t o p e .

is a p o l y t o p i c

subspace

means

of M.

polytopic

not depend

i) E v e r y w e a k

one,

an e x h a u s t i o n

a neighbourhood

does

completion

every Clearly

exhaustion

of M.

space.

of a p o l y t o p i c

space

is a g a i n

polytopic. iii) T h e d i r e c t

product

M I x M 2 of two p o l y t o p i c

spaces

M I , M 2 is a g a i n

polytopic.

Examples

6.2.

a) L e t

(Xl,X2,X3,...) nates the

x i ~O.

subset

structure

space

point also tions

We

identify,

of a space

of R ~.

b) F o r

= x of e l e m e n t s

{x 6 R ~ I x i

R~ a n d e a c h

every

R n, w i t h This

has

set of all

of R w i t h

R~

i >n}

such

infinite

sequences

finitely

many

coordi-

space

Rn with

standard

Theorem

(Rnln6~)

space

1.6 g i v e s

us a

is an e x h a u s t i o n

structure,

is a c l o s e d

of sub-

R ~ is p o l y t o p i c . coordinate

the value

function

xi,

F 6R[TI,T2,...]

If F I, .... ,Fr,

then we conclude

only

of R ~.

that

its s t a n d a r d

space

all p o l y n o m i a l s

nomials

on

the

for a n y n £ ~ , the

:: 0 for

i £ ~ , the

(Xl,X2,...)

o n R ~.

R~ d e n o t e

is w e a k l y

semialgebraic

finitely

6.1.ii

on a

semialgebraic.

are weakly

G I , . . . , G s are

from Remark

T i o n R ~, w h i c h

that

many the

Thus func-

such poly-

subspace

50

{x 6 R ~ I F I ( X )

h O ..... Fr(X) h O ,

of R ~ is p o l y t o p i c . families

in R [ T I , T 2 , . . . ]

T n occurs ly m a n y

then

Indeed,

weakly

the

that

every

(More

the

space

semialgebraic). in I,

interior

U

and we c o n c l u d e complete

are

two

the v a r i a b l e

in G<

for o n l y

finite-

M is l o c a l l y

semialge-

is p o l y t o p i c . )

semialgebraic

(and,

iff M

is l o c a l l y

§7).

set M

is an e l e m e n t

covering

of M

(cf.

2.12).

complete.

Thus

also

every

that

then

of R ~,

X is w e a k l y

T h e n M is p o l y t o p i c

of e v e r y

M a is l o c a l l y

precisely,

as a s u b s p a c e

is an a d m i s s i b l e

is l o c a l l y

M is l o c a l l y

every

M

complete.

is l o c a l l y

of

~(M),

and

If M is p o l y t o p i c , U a is l o c a l l y Conversely

complete

which

com-

if M

means

M is p o l y t o p i c .

Definition

2. A c o m p l e t i o n

a weak

of P,

Open

of R ~.

regarded

(as d e f i n e d

(Uals6I)

M'

subspace

Assume

(G
set

K 6 K}

6.3.

and

>O}

n £ ~,

I 6 A and

> 0 for e v e r y

complete

into

the

many

every

GK(X)

as b e f o r e ,

that

for

I 6A,

in R ~ and,

plete,

such

that,

(FIII6A)

~ 0 for e v e r y

Example

then

if

:= {x 6 R ~ I F I ( x )

is a p o l y t o p i c braic

generally,

in F 1 for o n l y f i n i t e l y

< 6K

X

More

G1(x) > 0 ..... Gs(X)

polytope

followed

Question

In II,

A.

§I we have

semialgebraic

P,

i.e.

by the

Does

of

every

(even

an

space

M is a d e n s e

isomorphism

inclusion

constructed

space

the

space

m a p M' ~

have

of M o n t o

embedding a dense

~ :M ~P subspace

P.

a completion?

a completion

in the c a t e g o r y

for any p a r a c o m p a c t LSA(R)).

Also,

in I,

locally §7, we

51 + have

constructed

semialgebraic struct

the one-point

space

N.

a completion

M~

completion

Starting

N

from this

M + of o u r

space

of a n y

special

M under

locally

complete

case we now con-

the

assumption

that

M is p o l y t o p i c o

Let M + denote

the

further

point

~.

M + with

the

set w h i c h For

every

structure in I,§7.

polytope

By T h e o r e m

set M + s u c h t h a t an e x h a u s t i o n M as an o p e n dense

of M.

If B < s

1.6 w e h a v e

The

subspace.

the

subset M +

completion

a unique

is n o t

yet

:= M

U {~} of

s p a c e Ms,

subspace

of t h e

structure

of M + a n d

polytope,

a weak

set M a n d o n e

of the

space

subspace

s p a c e M + is a w e a k If M

of the

+ t h e n M B is a c l o s e d

e v e r y M+a is a c l o s e d

3. W e c a l l

M + the

Remark

6.4.

direct

sum

case,

equip

union

(M:I~6I)

which

polytope,

on t h e is

contains

then M

is

in M +.

Definition j : M~

~ 6 I we

of t h e o n e - p o i n t

as d e s c r i b e d M:.

is the d i s j o i n t

the

one-point

s p a c e M + or, completion

If M is a l r e a d y (cf.

1.10)

the definition

a weak

of M a n d

the

is an a b u s e

more

precisely,

the

inclusion

of M.

polytope

then

one-point

the

space

s p a c e M + is t h e

{~}.

Thus,

in t h i s

of l a n g u a g e .

+ In I, §7 w e g a v e in t h e c a s e a similar

6.5.

U 6 ~(M)

b) A f a m i l y

(UILI6A)

If U 6 }(M),

of M + if M

a) A s u b s e t

or U =

is an e l e m e n t c)

description

t h a t M is s e m i a l g e b r a i c

description

Proposition either

an e x p l i c i t

(M ~ K ) in ~(M +)

and

of the locally

space

structure

complete.

This

of M implies

is p o l y t o p i c .

U of M U {~} U {~} w i t h

is an e l e m e n t

some

is an e l e m e n t

of

~(M +)

iff

K 6 Tc(M) . of C O V M +

iff

(U 1 ~ {~}LX6A)

of C o y M.

then OM(U)

= OM+(U).

6 + If U 6 ~ ( M ) a n d ~ 6 U

then

a func-

52

tion

f

: U ~ R

and

f(~)

{i.e., M ~K

is

for

cU

and

let

with

map

Then

We

<

R there

-f(~)

an

fB

= ~

for

M

is

a partially to

a

iff

f(x)

exists every

x 6QB

\ {~})

x 6U \{~}

K 6~c(M)

such

that

Let

map

Q be

from

an

semialgebraic)

a weak

open map

polytope

subspace

g

V of

: Q ~ M+

.

(Qs]B6J)

~ M<(B)

spaces.

6©M(U

x ~ ~,

some

of

f(V NQB ) c M<(B)

: V NQ8

for

polytopic.

(weakly

flU ~{~}

x 6M~K}.

proper

x 6Q ~V

that

for

values

exhaustion

semialgebraic g~(x)

[ <~

that

every

I such

restrictions

with

in

Assume

choose

: J ~

plete

~ >O

kf(x)

for

of © M + ( U )

the

f extends

= ~

element of

: V ~ M be

g(x)

Proof.

an

limit

every

6.6.

f

Q to M.

the

and

Proposition

is

are

Thus

for

proper

they

\ (V N Q B ) ,

Q.

every

maps

extend

cf.

There

to

1.7.6.

exists B 6 J

a monotonic

(cf.

2.6).

The

between

locally

maps

gB

: QB

com+ ~ M<(B)

maps

fit

together

These

+ to

the

desired

Corollary spaces

6.7.

f

tially

From

to

if

g

a map

proper.

have

existence the

6.8.

hood

a given is

is

of

Assume

f+

a weak

map

: N + ~ M + with

is

a map

with

proper

f

: N ~ M between

f+ (~)

g-I (~) since

g

polytopic

= ~.

= -I

{~} (M)

then

the

= N and

g

restricis p a r -

= g.

a completion of

that

point

proper

partially

structure

Theorem

in U w h i c h

f+

: N+ ~ M+

We

.

partially

g

on

of

: Q ~ M

: N ~ M of

the

quence

g

Every

extends

Conversely, tion

map

M

x 6M.

we

can

a polytopic

is

polytopic.

Then

polytope.

there

draw

an

important

conse-

space.

Let exists

U 6 ~(M)

be

a neighbour-

a neighbourhood

K of

x

53

Proof.

Since

f : M+ ~ K

U 6 ~(M +)

[O,1]

:= f - l ( [ o , ½ ] )

with has

there

f(x) the

exists

= 0 and required

a weakly

f(M + ~ U )

semialgebraic

= {I}.

properties.

The

set

function

§7 - A t h e o r e m

In t h i s that

section

spaces

nothing

The

else

theory

(and a p a r t

technical will

1.6.

subsets

of M f u l f i l l i n g

over

ture R,

o f M,

that,

spaces

(Msls6I) there

§I,

8, Def.

exists 9)

{X 6 H ( M ) IX c M s}

P2.

If X 6 H ( M )

faithful

convention

semialgebraic

carries

Def.

will

if

such

the

(Mals6I) EI-E3

the

M with

6. T h e n

its g i v e n

is a l a t t i c e

of

in the e l e m e n t a r y

occupy nearly the whole

of

the

b e an o r d e r e d a n d E4.

the

inductive

limit

lattice

ringed

(cf.

the

exhaustion

§3,

H(M)

with

strucover

semialge-

a closed

exhaustion

e v e r y Ms,

§3)

spaces

is w e a k l y

of M

We

weakly

(cf.

structure,

of

semialgebraic

structure

function

covering

family

§I).

set M B is c l o s e d

space M

space

(cf.

of a w e a k l y

subspace

category

that,for

semialge-

subspace

Def.

of M

following

7).

(cf.

holds.

of M s.

then X NM s 6H(M).

consist

lattice

ing p r o p e r t i e s

and will

B <s,

a faithful

of w e a k l y

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

structure

is an a d m i s s i b l e

PI.

The proof

the

M s in the

in §I,

Moreover Def.

let

whenever

E v e r y M s is, w i t h and

our

as w e s h a l l see.)

s p a c e M a. We e q u i p

as d e s c r i b e d

braic.

and

limits

"hard point"

the c o n d i t i o n s

in M a a n d

to t h e

of t h e

we abandon

to be w e a k l y

on.

spaces,

hard,

every M s carries R and

later

to be the

L e t M be a set,

semialgebraic respect

one)

assumed

useful

so v e r y

7.1.

space

spaces

on i n d u c t i v e

semialgebraic

(It is n o t

that

result

seems

Theorem

assume

of

of the n e x t

always

be v e r y

Its p r o o f

of w e a k l y

section.

are

limits

is said.

spaces

Theorem

inductive

and maps

following

braic

on

of t w o

exhaustions

hold

for e v e r y

steps.

In the

H s of the s 6 I.

first

spaces

s t e p we c o n s t r u c t

M s such that

the

follow-

55

QI.

If B < s ,

Q2.

If B < a

For

every

B <s.

and X 6Hs,

In o r d e r

lattice

H B c H s. then X NM B 6H8.

s 6 I we denote

on m ( a ) .

QI

then

For

b y m(a)

to c o n s t r u c t

the unique

exhaustion

the

index

the number exhaustions

of e l e m e n t s H a we

proceed

~ of w i t h m ( o ) = O we c h o o s e

Ho of M o. T h i s

is all

right,

B 6 I with

since

by

induction

an a r b i t r a r y

the c o n d i t i o n s

a n d Q2 a r e e m p t y .

L e t n o w m(a) MOa := M s

S B 6 HB,

U(MBIB
TM

we d e f i n e

= n >O.

We

choose

a lattice

of M a . (Of c o u r s e ,

H a as t h e

s e t of all

sets

exhaustion

L of the

if M ~ = ~ t h e n

X : S U

U(SB]B<s)

subspace

L : {9}-) with

Then

S 6L,

and

n [U(M~IB<s)]= u(sBl~
last

closed

condition

in M s.

means

{S d e n o t e s

If X , Y 6 H a t h e n c l e a r l y property Thus

E4~

QI

induction

We

E3,

E6

is e v i d e n t ,

hypothesis for the

start with

cribed T and

above, the

semialgebraic

closure

X U Y 6 H s.

of ~ ( M a ) . a n d Q2

a n d E4*.

family

Hs

subset

Using

the

induction

checked

It c o n t a i n s

is e a s i l y It r e m a i n s

(each set

empty

set.

checked,

again

by use

to v e r i f y

in H a b e i n g

indexed

and

let Y = T U

U (T~I B<s)

be a second

with M ° we obtain

T cS.

conditions.

Thus we have

The of the

the properties

be an e l e m e n t

above

and

also X NY 6H a .

the

U(S B ]B<s)

the

hypothesis

that

Let X = S U

fulfilling

X of M s is

of S in Ms.}

E3.

T B also

Intersecting

the

the

sets M B it is e a s i l y

H s is a s u b l a t t i c e

property

El,

for t h e

that

by

itself).

of H s , as d e s -

element Assume

only

of H s w i t h

that

Y cX.

finitely

many

s

possibilities obtain

for T = Y N M ~ .

Intersecting

with

M B for

Y N M B c X N M B. By Q2 b o t h X N M B a n d Y N M B a r e

some

~
elements

we

of H B.

56

Thus we have only

finitely

many

possibilities

for Y N M s .

that we have

finitely

many

possibilities

for Y, w h i c h

We

finally

(X n M ~ I X family

only

verify

£ H a)

that

is the

H a is an a d m i s s i b l e

s a m e as L u p to i t e r a t i o n

is an a d m i s s i b l e

(X N M B ] X

6 H a ) is the

same

as H B u p to

covering

ible covering

of M s. N o w El

family

H a . This

of M B. T h i s and,

finishes

H as t h e

set of all

for e v e r y

a 6 I and

S a %~

family that all

and

(Mala6I)

also the

Using

Y 6H

then

and property

X NY 6H.

lattices

QI

a n d Q2

one verifies

for t h e

that,

implies

lattice

It a l s o finite a 6J,

the

subset

this

3.]5,

indeed

X = U(S

finitely

Using

this

family family

is

evident

for

of M w i t h

S a 6H a

s E I. W e h a v e @ 6 H . property

families

H a one

of s u b s e t s

is g e n e r a t e d

Thus

proof.

]a6I)

many

family

H a is an a d m i s s -

a l s o E6 a r e

s t e p of o u r

for t h e

families

that

E3.

of M.

E4~

for t h e

checks

easily

It c o n t a i n s

by t h e m ) .

H a a n d E4~

for t h e

family

(Ma1~6I)

a 6 I,

c H a.

that

6H}

= {X 6 H [ X c M

statements

H trivially

fulfills

Thus

H is a l a t t i c e

for e v e r y

H a = {X N M a [ X

This

iterations.

X UY 6H.

Q2

(in f a c t

H a c H it f o l l o w s

(*)

Our

Thus

Ha

{X N M a l X 6 H }

Since

obviously

of e l e m e n t s . the

first

for o n l y

of M a. T h e

B
proves

subsets

proves

every

b y Th.

the

We define

If X 6 H

o of M a. F o r

covering

an a d m i s s i b l e

the

covering

We conclude

E3.

J of

PI

fulfills

Indeed,

I. A s s u m e

Y DM s c X NMs,

and

and both

P2

El,

if X £ H that Y nM

a

}

in t h e

E2, then

E4,

theorem.

E5

(E2 is t a u t o l o g i c a l ) .

X c U ( M a L a 6 J ) for s o m e

Y 6 H a n d Y c X. W e h a v e , , X N M s are e l e m e n t s

for e v e r y

of H a .

57

Thus

there

holds

are only

for e v e r y

possibilities

We

look

finitely

a 6J

set J c I w i t h

the

B <s

following

a 6J,

ii)

X c U ( M ala6J) .

a 6J

ringed

algebraic 1.6,

finitely

set X 6 H .

We choose

many

a finite

properties.

s e t X D M a is c l o s e d the

semialgebraic

structure space.

in Ma,

as a c l o s e d

If B < a

then

X with

the

space

family

(X D M a l a 6 J )

structure

This

X is a w e a k l y

family

of t h e

fulfills

semialgebraic

is a c l o s e d

subspace

EI-E3

space, of X.

subspace

a n d E4~.

limit

of s e m i -

Thus,by

X DMa,

J is f i n i t e ,

of

is a

inductive

and every

Since

since

XD M~

of X N M a. W e n o w e q u i p

spaces.

structure,

are o n l y

this

E3.

is a s e m i a l g e b r a i c

subspace

function

there

on a g i v e n

X D M a 6 H a. W e e q u i p X D M a w i t h

closed

that

for Y D M a. S i n c e

~ B £J

the

M a. T h e n X D M

possibilities

proves

structure

i)

For every

we conclude

for Y, w h i c h

for a s p a c e

many

Theorem

in its g i v e n the

space

X

is s e m i a l g e b r a i c .

If K is a n o t h e r above, that

then we

finite

find a third

L m J a n d L m K.

(X D M a l a £ L ) , structure

on the

finite

It is c l e a r

s p a c e X.

say

s p a c e Y is a c l o s e d ringed

Thus

f r o m II, subspace

space

By Theorem

exhaustion

I with

the properties

subset that the

the

L of

I with

the t h r e e same

space

i) a n d i),

families

inductive

structure

ii)

ii) such

(X D M a l a 6 J ) ,

limit

space

o n X is i n d e p e n d e n t

of J.

It is n o w c l e a r ,

spaces.

of

(X D M a J a £ K ) all y i e l d

of the c h o i c e

function

subset

Lemma

that

if Y 6 H ,

of X. W e e q u i p M w i t h

structure

of the

1.6 t h i s m a k e s

H, a n d e v e r y X £ H

1.2,

family

M a weakly

is a c l o s e d

the

Y cX,

then

inductive

the limit

H of s e m i a l g e b r a i c semialgebraic

subspace

space

with

of H in its g i v e n

58

space

structure.

It f o l l o w s

from the

a n d t h a t Ma, The

family

Thus,

proof

We

(M la6I)

limit

of the

state

Corollary subspace

7.1

and

7.2. of M.

7.1.

of s p a c e s

but

of M s i n c e

(Mala6I).

a 6 I

subspace

This

of M.

H is so.

on M coincides

nevertheless

with

the

finishes

the

Let H

o

semialgebraic

be a f a i t h f u l lattice

= {y N A I Y 6 H }

There

the

takes

first for

lattice

exhaustion

= H

by o u r p r o c e d u r e

One

interesting

consequences

its p r o o f .

and A a closed

exhaustion

of A.

Then

H of M w i t h

to c o n s t r u c t

I the

space

o

lattice

7.3.

set

L e t M be a set a n d

of M w i t h M I U ... U M r = M.

structure

of a

that M i NMj the

such that

(weakly

H in the p r o o f

exhaustion

H o could

{0,1}

the ordering

with

of

be chosen

subspace

e v e r y Mi,

of M.

weakly

structures

Then

let M I , . . . , M r be f i n i t e l y

Assume

semialgebraic)

is a c l o s e d

(I ~ i < j ~ r ) .

space

is a c l o s e d

covering

structure

for e v e r y

O
and

:= A, M I := M.

Corollary

that

weak,

a faithful

arbitrarily.

sets

space

family

that M a 6~(M)

structure,

Let M be a weakly

is e v i d e n t

puts M O

our

of the

rather

exista

Theorem

space

is an a d m i s s i b l e

3.15,

{X 6 H I X c A }

This

(*) a b o v e

theorem.

two

of T h e o r e m

there

in its g i v e n

by Theorem

inductive

relation

there with

space

the

(over R).

also

subset

in M i and Mj

on M a u n i q u e

its g i v e n

space

sub-

every M i carries

semialgebraic

on M i DMj

exists

that

many

Assume

of M i a n d Mj are e q u a l

structure

structure,

and

of a s p a c e

is a c l o s e d

sub-

59

Proof.

Let

partially

A denote ordered

iff J m K). closed

by the o p p o s i t e

For any J 6 A we put Mj

subspace

not d e p e n d

structure

on the choice

M i DMj

above.

weakly

semialgebraic

In the next

the set of all non e m p t y

Theorem

section

that the s y s t e m

7.1

inclusion

:= D(Mili£J).

in M i for some

i 6J.

applies

This

we will

to the o r d e r e d

and yields

use T h e o r e m

fulfills

EI-E5

of {I ..... r},

relation

(J J K

We equip Mj with

of i by the a s s u m p t i o n

spaces

(Msla6I)

of the

subsets

structure

on the

its

does

intersections

family

(MjIJ6A)

of

the result.

7.1 only instead

in the easier of EI-E3

case

and E4*.

§8 - S t r o n g

We

start

quotients;

with

Space(R)

(cf.

§I,

I. A m o r p h i s m

R is c a l l e d

of s p a c e s

generalities

and WSA(R)

Definition

IDI

some

gluing

identifying

on q u o t i e n t s

Def.

§2,

f : M ~ N between if the

following

Def.

6).

function

four

categories

ringed

properties

spaces

over

hold.

f is s u r j e c t i v e

.

ID2.

If U is a s u b s e t

ID3.

If

(UILI6A)

(UIII6A) If h

ID4.

of N w i t h

is a f a m i l y

If t h e s e

: U ~ R is a f u n c t i o n

properties

(via f).

This

terminology that

epimorphism,

hold

quotient

cf.

Def

[Gr,

on e v e r y

Since

Let

2b)

2.

that

N is a s t r o n ~

following

in the

proposition quotient

category

g,

there

which

Space(R).

as a m a p exists

quotient

(= s t r i c t

: M ~ L be m o r p h i s m s

and t h a t

f. T h e n

f is s u r j e c t i v e

(cf.

Definition

of

with

between

a unique

in S p a c e ( R ) . sets,

is

morphism

h of = g.

level.

Def.

say a l s o

f : M ~ N and g

fibre

set t h e o r e t i c §I,

(f-1(Ul) il6A) 6 C o y M t h e n

is a c a t e g o r i a l

2.2])

f is i d e n t i f y i n g

: N ~ L with

U 6 ~(N).

h 6ON(U).

by the

a strong

Assume

Proof.

we

is j u s t i f i e d

8.1.

constant

with

then

on a set U 6 ~(N)

then

then

Proposition that

6 ~(M)

6 Coy N.

of M

implies

f-1(U)

in ~(N)

ho(flf-1(U))60M(f-I(u))

h

2 and

in the

One n o w

using

the m a p h e x i s t s checks

properties

If M is a f u n c t i o n

directly

and that

is u n i q u e

on the

h is a m o r p h i s m

ID2-ID4.

ringed

space

over

R and

f : M ~ N

61

is a s u r j e c t i o n unique

from the set M to a set N, then there

structure

identifying. ture.}

of a f u n c t i o n

{The p r o p e r t i e s

We call N, with

In the f o l l o w i n g semialgebraic map b e t w e e n

spaces

Every p r o p e r proved [LSA,

179]

every Y 6 ~(N),

there

i) Let

The map

f is s t r o n g l y with

ible c o v e r i n g

f(M (cf.

iii)

of N by s e m i a l g e b r a i c

8.3.

strongly

the

struc-

of M by f.

a weakly

semialgebraic

is said.

This

can be

semialgebraic

spaces

to extend

this o b s e r v a t i o n

strongly with

(NBIBEJ) iff,

surjective

f(X)

that

for

D y.

be e x h a u s t i o n s

for every

if,

of M and N.

B 6 J, there

(f(Ma) laEI)

exists

is an admiss-

and

(XllIEA)

then sets.

is an a d m i s s i b l e

(f(Xl) il6A)

cover-

is an a d m i s s i b l e

In particular,

if M is of count-

type.

and s u r j e c t i v e

surjective

then

f is strongly

partially

p r o p e r map

surjective.

f : M ~ N

is identifying.

Proof.

Let

(MsIs6I)

(cf.

proper maps.

also N is of c o u n t a b l e

Every

else

f is

7) of N.

sets,

If f is s e m i a l g e b r a i c

Theorem

and

surjective

ing of M by s e m i a l g e b r a i c

able type then

We want

some X 6 ~(M)

surjective

Def.

a weakly

of locally

) D N B. This m e a n s

If f is s t r o n g l y

covering

ID4).

(MalaEI)

§3,

means

f : M ~ N is identifying.

exists

8.2.

ii)

map

quotient

a "space"

something

f : M ~ N is called

Remarks

some a E I

R, unless

of p a r t i a l l y

3. A map

the strong

in §5 and §6)

on N a

R, such that

tell us how to d e f i n e

structure,

as in the theory

class

over

R and a "map" m e a n s

for p r o p e r t y

to a suitable

Definition

over

surjective

similarly p.

this

over

space

ID2-ID4

(as a l r e a d y

space

ringed

exists

be an e x h a u s t i o n

of M.

Then

( f ( M ) La6I)

is an

62

a d m i s s i b l e c o v e r i n g of N by c l o s e d s e m i a l g e b r a i c f

: M a --~ f(M ) of f are p r o p e r maps b e t w e e n

h e n c e they are way,

identifying.

One now c o n c l u d e s

by use of T h e o r e m 3.15,

Definition

4. If a m a p

of M.

Given map

The restrictions

semialgebraic

spaces,

in a s t r a i g h t f o r w a r d

that f is i d e n t i f y i n g .

f : M ~ N is s u r j e c t i v e and p r o p e r then we say

that N is a p r o p e r q u o t i e n t of M and p a r t i a l l y proper,

sets.

(via f).

If f is s t r o n g l y s u r j e c t i v e

then we say that N is a p a r t i a l l y p r o p e r ~uotient

N o t i c e that in b o t h cases N is a s t r o n g q u o t i e n t of M.

spaces M,N,

a closed weakly

f : A ~ N we w a n t to

m e a n s to c o n s t r u c t

A

(8.4)

the i n c l u s i o n m a p

, MUfN

of w e a k l y

the c a t e g o r y LSA(R) R if f is proper.

semialgebraic

§10 that the a n a l o g o u s

of p a r a c o m p a c t

Our m a i n goal

in this

section

i

f l e x i b i l i t y of w e a k l y

semialgebraic

compact

locally semialgebraic

spaces

in

spaces over

is to c o n s t r u c t

for f p a r t i a l l y proper.

greater

semialgebraic

p u s h out e x i s t s

locally semialgebraic

in our b r o a d e r c a t e g o r y

reason why weakly

spaces over R, w i t h

f r o m A to M.

It has been p r o v e d in II,

(8.4)

(= "push out")

I g

in the c a t e g o r y WSA(R)

p u s h out

square

, N

I M

s u b s e t A of M and a

"glue M to N a l o n g A by the m a p f". This

a cocartesian

f

semialgebraic

a This

spaces c o m p a r e d w i t h para-

is p e r h a p s the m o s t

s p a c e s s e e m to be u s e f u l

important in s e m i a l g e -

b r a i c topology.

Our g l u i n g p r o b l e m can be r e g a r d e d as a q u e s t i o n a b o u t the e x i s t e n c e

63

of

suitable

quotients.

the d i s j o i n t

union

f(a) 6 N .

p

Let

Let

MUN

L

equip the

and

: MH N ~ L denote

L with

spaces

push out

The

that the

L is the u n i o n

strong

p(M ~A)

M~A

of M to the o p e n

We w i l l which

prove

that

is e v e n

an e a s y braic

8.5.

= g(M ~A)

that

images

gIM ~A

2 above).

: M ~ L,

and

and

of the d i r e c t

Def.

with

and g

g(M ~A)

under

p is the

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

subspace

L is w e a k l y weaker

p(M ~A)

j are

j(N).

We

sum M U N of

Then

(8.4)

is a

set M ~ A £ ~ ( M U N). from

the o p e n

subspace

of L.

semialgebraic than

of T h e o r e m

Then

is, v i a

p, a p r o p e r

phism

the

Assume

proper.

of N o n t o

Proof.

We

partial

II.I0.7

under

a condition

properness.

(needed

here

(Nala6J)

set.

By T h e o r e m

plM~ N

function

ringed

quotient

II.IO.7

space

of M U N.

subspace

an e x h a u s t i o n

is a g a i n

a 6 J, e q u i p p e d

t h a t M is s e m i a l g e b r a i c

a closed

choose

Then

sets

a £A

from

We

only

on

f

start with for

semialge-

spaces).

Proposition

B
(cf.

every

projection

Notice

structure

p

and p I M ~ A

slightly

consequence

set o b t a i n e d

Space(R).

of p ( M \ A )

Thus

natural

of the

quotient

in t h e c a t e g o r y

6 ~(L)

the

to M and N.

M and N by the m a p

preimage

the

of M and N by i d e n t i f y i n g

j : N ~ L its r e s t r i c t i o n s injective

:= M UfN d e n o t e

subset

the

structure

is a p r o p e r

map

from MU N

L~

is a c l o s e d

of L f u l f i l l s

inductive

limit

the

subspace

conditions

structure

The m a p

(Nala6I)

with

then

L is w e a k l y

f

: A ~ N is

semialgebraic

j = pin

and

is an i s o m o r -

of L.

an e x h a u s t i o n the

and

L

of N.

Let

of N. We

J :={a6ILN

regard

:= M U r N

:= M u f N a of L is,

of a s e m i a l g e b r a i c onto

L

of Ls. EI-E5.

on L of the

L

Df(A)}. as a

for e v e r y such

that

. If B 6 J is an i n d e x

with

The

system

Thus,

family

space

of

(Lala6J)

by T h e o r e m spaces

of sub-

1.6,

(Lala6J)

the makes

84

L a weakly subspace

-I

space

of L and (L~I~6J)

is p r o p e r p

semialgebraic

for t h i s

space

that our

Thus

space

that

every

is a n e x h a u s t i o n structure

(L a) ~ L a a r e p r o p e r .

This means

such

structure

L~

is a c l o s e d

o f L. T h e m a p

on L since

L is the

space

all

strong

the

same

: M~ N ~ L

restrictions

quotient

on L is the

p

of M U N b y p.

as t h e o n e

chosen

above.

We have

p-lp(N)

A U N ~ p(N) p(N) map

of L.

= A U N 6~(MU

N).

Thus

of p is a p r o p e r

map

f r o m A U N to t h e c l o s e d

Restricting

f r o m N t o p(N).

Definition the m a p

This

: A ~ N partially

restriction

of

f to t h e

in M is a p r o p e r

map

proper.}

Theorem

Assume

8.6.

strong

quotient

diagram

(8.4)

of N o n t o

in t h i s

Proof.

that

subspace

of L.

hence

subset

{N.B.

of M.

for e v e r y

of A w i t h This

proper

proper

Y 6 ~(X),

certainly

Then

Thus

The map

the of Y

holds

near M ~A.

WSA(R).

We call

the closure

semialgebraic.

in t h e c a t e g o r y

The restriction

proper,

a bijective

if f

the

the p is

j = p I N of p is an i s o m o r p h i s m

If f is p a r t i a l l y

L is a p a r t i a l l y

proper

proper

then

quotient

also

of M U N

case.

As

in t h e p r o o f

the setLwiththe space

to N.

L o f M U N b y p is w e a k l y

a closed

subspace

q.e.d.

X if,

A N~

f is p a r t i a l l y

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

p is p a r t i a l l y

near

intersection

that

surjective.

obtain

semialgebraic

proper

from ANY

is p a r t i a l l y

to N w e

The restriction

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

5. L e t X be a w e a k l y

f

strongly

p further

p(N) 6 ~ ( L ) .

structure this

space

strong

of the

preceding

quotient

structure

on L in a d i f f e r e n t structure

is t h e

proposition

w e do n o t

b y p. W e w i l l

equip

construct

way

and will

verify

strong

quotient

of M U N b y p.

a

a posteriori

65

If E is a w e a k l y E U N with g

g : A DE

city we d e n o t e

We choose

semialgebraic

this

our a s s u m p t i o n braic Pa

~ N obtained subset

an e x h a u s t i o n

the subset Xa U f N

space

(X la6I)

= p(Xa) Up(N)

such that

L

If B 6 I and B < a

Thus L B is,

family limit

(Lsls6I) space

structure

L is w e a k l y (Lal~£I)

in its given fulfills

of this

is an a d m i s s i b l e

ps = plX plA =

semialgebraic

We have

p

-I

quotient

U D L a 6 ~(L

We choose

a proper map

proposition

of a w e a k l y

of X

UN

and

semialge-

under

of X a U N. Thus L B 6 ~ ( L a ) . -I from Pa (L~) to the

a closed

subspace

family

of spaces.

L a is a closed of L. The

covering

Then,

family

inductive

by T h e o r e m

subspace

space

of L a. The

We equip the set L w i t h the

7.1,

of L, and

(Xala6I)

together

of M U N. The r e s t r i c t i o n s

semialgebraic

ID2-ID4

that our

maps.

Thus

We c o n c l u d e

also pIN and

that

p is w e a k l y

) for every

for p. Then we know that

space

structure

by p. Let U be a subset

-I N). We have p~ (U N L

an index

subset

semialgebraic.

the axioms

structure

The p r o o f s o f

a 6 I let L a d e n o t e

3.15).

which means

(U) 6 ~ ( M U

ID2.

(cf.

to v e r i f y

identifying, strong

are w e a k l y

simpli-

.

covering

U N of p are w e a k l y

(piN)of

set

then

every

w i t h A and N is an a d m i s s i b l e

For every

quotient

structure,

EI-ES.

semialgebraic,

For

structure

semialgebraic

The m a p ps gives by r e s t r i c t i o n L~.

f by restriction.

of L. By the p r e c e d i n g

is a proper

is a closed w e a k l y

is the

of M ~ A .

on L a the

-1 Pa (LB) = [XB U (A nXa)] U N

This

from

of M then p ( E U N)

of L by E UfN.

on f we have

:= pIX a u N.

subset

) =

p-1

ID4 are

is the

of L with

(U) N (X U N) 6 ~ ( X a U N).

s 6 I, and we c o n c l u d e

ID3 and

on L really

p is

that U 6 ~(L).

This

similar.

a 6 I. Then we know

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

8.5 that

Thus proves

66

p(N)

= ps(N)

L. We a l s o N onto

is a c l o s e d

know

p(N).

from M ~A joint)

union

is p r o p e r . proper.

8.5

Moreover,

onto

surjective.

from

the

open

every

is p a r t i a l l y

following

near M \A.

subspace

Remarks g

8.7.

i)

proper

near M~A of M t h e n

near

Z ~ (Z N A )

the M ~A

g

(M ~ A )

x T,

and

Z UhN

pLA

(dis-

p is s t r o n g l y Also

piN

is p a r t i a l l y

an a d m i s s i b l e

sets w e c o n c l u d e

cover-

that

p

f : A ~ N is p a r t i a l l y

obtained

p(M ~A)

of N w i t h

h

and N w i t h

p(N).

c Y then

range

of M U f N .

the m a p

space

is p a r t i a l -

If Z is a c l o s e d

~ N of f is p a r t i a l l y subspace

c y then

is p a r t i a l l y

and N is a c l o s e d

f(A)

of t h e

: Z NA

f(Z N A )

b y g l u i n g M t__oo N a l o n g A

subspace

with

proper

of M U f N .

the map h

proper

near

Thus

if

: Z NA

~ Y

Z ~ (Z h A )

and

of M U r N . then

the natural

fxid T map

: AxT

from

~ NxT

is p a r t i a l l y

(MxT) U f x i d (NxT)

to

proper (M U f N )

near xT

In s h o r t

(MxT) U f x i d (NxT)

type.

that

is p r o p e r .

clearly

is a c l o s e d

given with

space

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

iii)

M ~A

restriction and

p[Xs

Y is a s u b s p a c e

from f by restriction

If T is a n y

that

f by restriction

obtained

ii)

then

as an o p e n

Y 6 ~(N)

is a s u b s p a c e

L is t h e

from

q.e.d.

space

Z £~(M),

Z UhY

are

Since

of

isomorphism

case.

assume

and M U the

of L.

semialgebraic

If Y is a s u b s p a c e from

is an

with A and N form

identifying

: A ~ Y obtained

subspace

in t h i s

We call M UfN

of M U f N

ly p r o p e r

weakly

regard

pIM~A

restriction

proper

hence

that

it is n o w e v i d e n t

we always

by f. W e u s u a l l y

seen,

and

p(N)

of La,

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

p(M ~A)

the Xa together

proper

= psIN

subset

subspace

a 6 I the

ing of M U N b y c l o s e d

In t h e

that pin

If f is p a r t i a l l y

Since

semialgebraic

it is e a s i l y

of p ( M ~ A ) For

weakly

=

(M U f N )

If M a n d N a r e of c o u n t a b l e

x T.

type

then M UfN

is a g a i n

of c o u n t a b l e

B7

iv)

If M a n d N a r e w e a k

Notice

that,

v) A s s u m e

in t h i s

polytopes

case,

then M UfN

f is a u t o m a t i c a l l y

t h a t M a n d N are p o l y t o p i c

Then M UfN

is a g a i n

is a g a i n

polytopic

and

a weak

partially

that

polytope.

proper.

f is p a r t i a l l y

proper.

and

(M U f N ) + = M + U f + N +

Definition

6. L e t M b e a s p a c e

braic

subsets

plete

near

proper

X if t h e m a p

near

Example

of M with

X.

8.8.

A closed

L e t M be a s p a c e is p a r t i a l l y

weakly

is t h e d i r e c t of the

fies

the

exhaustion

If

(MsL~6I)

of M/A,

and

, ( M s ) as the

strong

Notice

that,

in o r d e r

space,

it s u f f i c e s

is n o t n e e d e d

Subexample M

(§6,

open

Def.

in P.

8.9.

projection

point

is an e x h a u s t i o n the r e s t r i c t i o n

quotient

Theorem

: M ~ M/A.

M ~A

1.6.

quotient

{~ is t h e m a p ~ identi-

isomorphically

: M s ~ ~(Ms)

The more

onto

(~(Ms) la6I)

as a w e a k l y

the

then M/A

The map

of M t h e n

sub-

of

is an

~ exhibits

A N M a %~.

semialgebraic difficult

Theorem

here.

Let M be polytopic

2), w i t h M r e g a r d e d {Such

M/A

We denote

strong

M /A N M s of M s , w h e n e v e r

to c o n s t r u c t

to a p p l y

no

is the

case.}

Y 6r(X).

to T h e o r e m

If A is e m p t y

M/A

special

~ and maps

{~}.

com-

is p a r t i a l l y

semialgebraic

According

space

~

{~}

for e v e r y

weakly

b y M/A.

Otherwise

in t h e p r e s e n t

set A i n t o o n e

(M/A) ~ { ~ } .

7.1

{~}.

space

near M\A.

one point

space

semialge-

s a y t h a t A is p a r t i a l l y

is a p o l y t o p e

complete

s p a c e M by t h e n a t u r a l (8.4)

We

and A a closed

semialgebraic

s u m of M a n d

in the d i a g r a m

in M.

that ANY

8.6 w e c a n g l u e M a l o n g A to t h e resulting

l e t X a n d A be t w o w e a k l y

f r o m A to the o n e p o i n t

This means

set of M w h i c h

and

completions

and

let M ~

as a s u b s p a c e

exist

P be

a completion

of P. A s s u m e

in a b u n d a n c e

if M

of

t h a t M is

is p a r a c o m p a c t

g

68

locally f(x)

semialgebraic,

= x for x 6 M

morphism

P/P ~M

We return proper

the

map

g

map

Let

glX a

8.10.

again

C a closed

the map

relative

assuming

a little

more

of X. W e

yields

(cf.

that

8.4)

(Xala6I)

about

spaces

the map ~VZ

is p a r t i a l l y

the

relation

and

let C b e a

for e v e r y

near M~A

relative

of M ~ A .

then

A.

B y Prop.

s 6 I. T h u s

pair

gET

turn

out

to be v e r y

8.11.

If

also

of

spaces

is a p a i r

is a c l o s e d point

say t h a t

pair

space

convenient

(M,A)

is p r o p e r

of

spaces

(X,C)

such

in t h e n e x t

is a r e l a t i v e

polytope.

is p a r t i a l l y

X is p a r t i a l l y

N is p a r t i a l l y

weak

(X,C)

of s p a c e s

{~}

look over-sophisticated

f : A ~ N into a space a relative

8.5 the

of X.

polytope

may

the

q.e.d.

f r o m X to the o n e

terminology

proper

f r o m Z to Y is p r o p e r .

proper

proper

be an e x h a u s t i o n

is p r o p e r

subspace

C. W e t h e n

Proposition

an iso-

f is p a r t i a l l y

s a y t h a t ~ is p a r t i a l l y

If f : A ~ N is p a r t i a l l y

weak

This

is a g a i n

(8.4)

Z £ ~(X ~ C ) ,

8. a) A c l o s e d

b) A r e l a t i v e that

6.6)

Y 6~(M ~A).

Definition with

f : P ~ M + with

M a n d N.

subset

: Xs ~ X a U f N

for every

map

(cf. Prop.

: X ~ Y be a m a p b e t w e e n

for e v e r y

: M ~ M UfN

Proof.

The natural

for x ~ M

to s a y

M UfN,

7. L e t ~

Proposition

= ~

II.}

situation

We want

spaces

C if,

f(x)

general

semialgebraic

relative

Chap.

-z~ M +.

to t h e

Definition weakly

and

near M ~A.

between

cf.

complete

at ]present.

proper

relative

C.

It w i l l

chapter.

weak

proper

polytope near M ~A,

In p a r t i c u l a r

then and

(N = { ~ } ) ,

any map (M U f N , N ) A is

69

partially

complete

: M/A ~ M UfN/N

The proof may

near M \A induced

be safely

We mention

that

properness

relative

e d by a p a t h

Proposition Then

completion

Let

Z 6 ~(M ~C) a closed

that

and dense triangle

obtained

with

~

h proper

and

inclusion

(cf.

We c h o o s e

a point

a

~ g with

:

[0,1]

stated,

subset

from

i

a

completed

b y h o a.

spaces

C iff t h e

following

such

a([O,1[)

a dense

~(1)

We

that

assume

and we have fiB w i t h

that

and

let C6T(M).

holds. c M ~C

f fulfills

to prove,

B

D of N which

a proper

for

If and

contains The

set

of h,

the

path

a given

:= Z is p r o p e r .

extension

embedding We w a n t

to

Since

Z is

x 6BxB.

: [O,1[

partial

f(B).

We

choose

Let h : B ~ D

Z is s e m i a l g e b r a i c i.e.

a co~utative

maps

II.12.10).

striction

- also

set c a n be c h a r a c t e r i z -

between

f by restriction.

of s ~ i a l g e b r a i c i

path

direction.

in B. W e c h o o s e

Br

properness

semialgebraic

relative

restriction

semialgebraic

be the m a p

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

then s can be completed.

just

the

The map

reader.

f : M ~ N be a m a p

~ M is an i n c o m p l e t e

condition

polytope.

criterion.

of t h e n o n t r i v i a l

lifting

to t h e

to p a r t i a l

proper

f.a c a n b e c o m p l e t e d

Proof

left

is a w e a k

: M ~ M UfN

to a w e a k l y

f is p a r t i a l l y

: [0,1[

by g

- similar

8.12.

and M/A

= x

and

~ Z of ~. By our

8

The

of

B,

prove

which that

dense

[O,1[)

c

in Z.

incomplete

ass~ption

we

regard

B = B.

Suppose

B there

exists

Now c o n s i d e r

path

hoa

as

the

an not. a path re-

in D c a n be

a can be completed

to a p a t h

70

B

: [O,1]

~ B.

diction

proves

Example

8.13.

tope

iff

every

We must that

have

~(I)

is

indeed

f]B

A closed

pair

incomplete

of

path

= B(1)

= x.

But

x (B.

This

contra-

proper.

spaces in M ~ A

(M,A) can

is a r e l a t i v e be

completed

weak in M.

poly-

§9 - T h e w e a k

We

continue

polytope

working

P(M)

in the

over

In t h e

following

M is a w e a k l y

semialgebraic

is an e x h a u s t i o n

of M. W e w a n t

to c o n s t r u c t

with

following

a weakly

universal

braic

map)

there

exists

of

Space(R)

spaces

together

R instead

category

semialgebraic For

f : Q ~ M from a weak

(9.0)

morphism

g~ 11~

ringed

just WSA(R).

property.

a unique

of f u n c t i o n

map

p

a weak

over

R and

polytope

: P ~ M which

every morphism polytope

g

space

Q

(over R,

: Q ~ P with

pog

P over

has

(= w e a k l y

(M Is6I) R

the

semialge-

as a l w a y s )

to N

= f.

IP

/ I I

M

f

Definition map p

I. We c a l l

: P ~ M,

Notice

that

morphisms

such a weak

a partially

it s u f f i c e s

f with

limit

that

any two partially

How

should

speak

P look

Q the one-point partially spaces of t h i s

of

hull the

In o t h e r

than

COVp(U), OM(U)

"the"

the

partially

same

of M a r e complete

words,

p must

P is t h e

polytope

is an i n d u c -

It is a l s o

hull

of M,

be b i j e c t i v e . we may

U 6 ~(M).}

c ~(P),

Since

(9.0)

that

set a n d p is the

~(M)

clear,

o v e r M.

Thus,

assume

for

if it e x i s t s .

property

set M equipped

{This means

for e v e r y

(9.0)

isomorphic

the u n i v e r s a l

underlying

the

property

Space(R).

: P ~ M exists,

s p a c e M.

C~M(P)

hulls

precisely

of M.

every weak

category

see t h a t

P, or m o r e

the u n i v e r s a l

Exploiting

p

core

since

complete

space we

complete

set.

in t h e

like?

P and M have

structure

to d e m a n d

of p o l y t o p e s

Thus we may

complete

Q a polytope,

tive

polytope

with

for

if the the

identity

map

a f i n e r space

and COVM(U)

c

id M is a m o r p h i s m

72

f r o m P to M, we a l s o

If K 6 ~ ( P )

then

have

~(P)

c ~(M).

K is a p o l y t o p e .

Thus

pIK

is a p r o p e r

K as a s u b s p a c e

of P to K as a s u b s p a c e

of M.

phism.

words,

subspace

M and

In o t h e r P are

the

Conversely, to the that

inclusion

the

structure limit gory

morphism

structure Space(R).

{Every

explicitly

ment

of ~(P)

~(P)

is an e l e m e n t

A function

flK N U 6 O K ( K

All hull

this

has

exists.

Definition of w e a k

structures

this

are

the

means

of K in

of C O V p

polytopes

~c(M)

can d e s c r i b e with

of p o l y t o p e s is i n d e x e d

following: every

the

~c(M)

by

A subset

K 6~c(M).

learn

the

inductive in the c a t e -

itself;

K 6 ~c(M),

of Coy K.

{This

means

union

U 6 ~(P)

of

U of M

A family

every

is the

the

the

that

finitely

the

many

is an e l e m e n t

is an ele(UIII6A)

in

family intersection sets

of O p ( U )

K NUI.} iff

K 6 ~c(M).

the

denotes

we n o w

for

under

We n o w m a k e

2. P(M)

and w e

(9.0)

inclusions.}

if,

for e v e r y

said

family

for

: U ~ R on some

been

property

K of M,

set M e q u i p p e d

of ~c(M)

the

iff K A U 6 ~(K)

DU)

the u n i v e r s a l

subspace

semialgebraic

element

:= U(UIII6A) f

M of the

P is the

is an e l e m e n t U

can a p p l y

of the d i r e c t e d

More

of K w i t h

the

be an i s o m o r -

= ~'c(M).

space

morphisms

(K N U I I I 6 A )

K~

P is w e a k l y

of P: The

transition

t h e n we

T h u s ~(P)

space

and

must

from

same.

if K 6 ~ c ( M )

K 6~(P).

Since

K £rc(M)

This

morphism

the

assumption

following

the

general

inductive

in the c a t e g o r y

that

limit

a partially

complete

definition.

of the d i r e c t e d

Space(R),

with

family

underlying

set M,

73

and PM

: P(M)

morphisms

Notice the

~ M is t h e m o r p h i s m

K ~

that,

set M.

M, w i t h

on the

By our

K running

discussion

is a l r e a d y

Proposition

9.1.

M.

Also

i)

polytope

~c(M)

ii) PM

If t h e : P(M)

Proof.

If M a d m i t s

= ~(P(M))

K is c l o s e d

and,

ringed

subset

s e t K in M a n d P(M)

not

need

P(M)

Let now a morphism is an e l e m e n t

f r o m Q to K a n d the

morphism hence

3ective verified

P(M)

is t h e o n l y

3. W e c a l l

half

of i n c l u s i o n

identity

of t h e

K 6 ~c(M),

m a p of

following

the

the s p a c e

c o r e of

structures

polytope

ii).

If K is a c o m p l e t e

it is e a s i l y

and that (cf.

then

the §I,

verified

subspace Def.

5;

that

structure

on

for t h i s w e d o

semialgebraic).

We equip

by

complete

of M.

claim

same

c o r e t h e n P(M)

same.

core

space M then

are

complete

is a w e a k

the

factorization morphism

j we have

f = PMOg

of

the real

K ~

of

property

As

g

f through (9.0)

field

the

inclusion

i = PMoj

Since PM'

struc-

has been

Denoting

:= jof.

K := f(Q)

g a morphism

just

factorization

f with

closed

M.

Then

its s u b s p a c e

f = i~g w i t h

of P(M).

the

factorization

universal

Q a polytope.

set K w i t h

K is a l s o a s u b s p a c e

the r e q u i r e d

Definition

first

the

second

inclusion

factorization

this

are

in P(M)

a unique

f r o m K to P(M)

the

P M is t h e

f : Q ~ M be g i v e n w i t h

i the

space

for e v e r y

the

is w e a k l y

of ~c(M).

in M. W e h a v e

the

complete

of t h e

semialgebraic

that

level,

~ M is a p a r t i a l l y

space

to p r o v e

the

proved,

: P(M)

~ M is a p a r t i a l l y

It r e m a i n s

family

~(M).

a partially

of M a n d P(M)

function

semialgebraic

ture

above

by the

clear.

and PM

on M as a s u b s p a c e

through

set t h e o r e t i c a l

proposition

is a w e a k

induced

of

i,

P M is bi-

and we have

of PM"

R sequential,

q.e.d.

if t h e r e

74

exists zero.

in R a s e q u e n c e Replacing

~i,~2,...,6k ~k'

even

Very many closure

of a field,

Our main

Theorem

9.2.

in t h i s

that

are

p.

~>O,

section

converging

~k by t h e m i n i m u m

~k ~

&k+1'

to

of

and omitting

some

k 6 ~ .

sequential.

is f i n i t e l y

[LSA,

that

every

elements

44].

For

generated

(These

such that

example, over

fields

(~nln6~)

are

~ or

every

real

]I~, is

even microbial,

is a n u l l

sequence.)

is to p r o v e

R is s e q u e n t i a l .

Then M admits

a partially

core.

proof

them

fields

which

[Du],

Assume

assume

for e v e r y

an e l e m e n t

goal

complete

The

cf.

contain

then

closed

of p o s i t i v e

a sequence

~k > ~k+1

real

sequential, i.e.

in s u c h

we may

that

(~klk6~)

of the

stated

theorem

will

in a s l i g h t l y

be b a s e d

more

general

on t h r e e form

easy

than

lemmas,

actually

t w o of

needed

for

that.

Lemma

9.3.

Assume

Let A £~(M). complete tope

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

c o r e of A.

Thus

it is e v i d e n t

fulfills

Lemma

a partially

P(A)

qA

complete

: q-1(A)

is a c l o s e d

core q

: Q ~ M.

~ A of q is a p a r t i a l l y

subspace

of the w e a k

poly-

P(M).

Indeed,

hull

Then

that M admits

from our

the u n i v e r s a l

subspace

property

theory

characterizing

(Prop.

3.2)

a partially

that qA complete

of A.

9.4.

polytope.

Assume Then

that,

P(M)

for e v e r y

is a w e a k

a 6 I, t h e

polytope.

space

P(Ms)

is a w e a k

75

Proof.

If S < a

Theorem Thus

7.1

there

then P(M B) is a closed

applies exists

set M such that, of Q, and

to the set M and the

a structure

( P ( M ) Is6I)

The

from Q to M w h i c h

a partially

complete

identity

fulfills

of spaces

semialgebraic

space

is an a d m i s s i b l e

of P(M a) by L e m m a

family

of a w e a k l y

for every a 6 I, the

is a weak polytope. map

subspace

P(M a)

covering

the u n i v e r s a l

cere of M. We c o n c l u d e

(P(Ma) Is6I).

space Q on the

is a closed of P(M).

map of M is a w e a k l y

subspace

This

space Q

semialgebraic

property

characterizing

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

9.1.i

that Q = P(M).

Lemma

9.5.

Tc(M),

contained

q.e.d.

Assume

with

that there

k 6 ~ , such that

in some Pk"

Then P(M)

is a weak

complete

Every

~c(Pk)

in the c a t e g o r y

P(M)

a countable

for every

Proof.

that

directed

exists

Pk c Pk+1

PM is a p a r t i a l l y

implies

space

Pk is the

is a weak

polytope

inductive

Space(R).

of spaces

family every

polytope.

(Pklk6~)

in

K 6rc(M)

is

{Thus,

by 9.1.ii,

family

of spaces

core of M.}

the space P(M)

family

9.3.

Thus

limit

of the

the h y p o t h e s i s

is the

inductive

limit

(Pklk6~)

. Theorem

7.1

(and that

(Pklk6~)

of the

lemma

in Space(R)

now

implies

is an a d m i s s i b l e

of the

that covering

of P(M)).

Remark.

In the proof

semialgebraic. the proof.

case

In that

But Th.

We n o w prove

of Th.

7.1

Theorem

9.2 this

case Th. is needed

9.2.

lemma will

1.6 instead in full

By Lemma

9.4

be only needed

of Th.

strength

7.1

suffices

to prove

it suffices

for M for

Lemma

to c o n s i d e r

9.4.

the

that M is semialgebraic.

Claim.

There

every k 6IN,

exists

a family

(Pklk6~)

such that every

K 6 rc(M)

in ~c(M),

with

is c o n t a i n e d

Pk c Pk+1

in some Pk"

for

76

Once

we h a v e

Choosing that

proved

for e v e r y

of

Pk

will

open

the

n £ ~o'

containing

claim

over

a family

for the

is c o m p l e t e

standard

t of o p e n

and we

o in ?(n) ~ M . a countable

for M.

R.

loss

of g e n e r a l i t y ,

It s u f f i c e s

(Pk(0) I k 6 ~ )

closure

Let

N

L 6 [c(N)

:= M U @

~ NM

is c o n t a i n e d

~kh6k+1

~O,lo(X)

some

null

for e v e r y

+

the c l o s e d

to find,

in ~ c ( O DM)

of ~ in M.

Then

Let

t > O. We

in q ( N )

the

. + In(X)

in M,

following

1

(x) > 6k ( i ~ J --

set Pk

in ?(n), h e n c e := Xk N Qk

and Pk c Pk+1"

holds,

since

that

a NM

Let

some

induction

open

Qk c Qk+1

to Hans

for

this

and

...

+

of p o s i t i v e for e v e r y

+ I

of a p o i n t n

(x)e

elements k 6 ~

the

n

=

x}

in R w i t h set ~)

k

li(x) >

}

complete.

We h a v e

be given•

clever

Xk D d = @

semialgebraic For

every

= # :

Delfs

exists

a numbering

*) I am i n d e b t e d

simplex

there

coordinates

is c o m p l e t e

K 6 ~c(M)

by

7(n)

o = ]eo,el,...,er[•

I, l o ( X ) e °

and we define,

semialgebraic

and X k C Xk+ I. The

choose

r

:= { x 6 ? ( n ) [ j=r+1

X k is c l o s e d

=

M is,

n-simplex

hypothesis

We

(~klk 6 ~ )

n

Xk

choose

that

such

that

If t = 0 t h e n M = ?(n)

such

Qk"

of V(n)

sequence

standard

in V(n) ~ M .

in some

to the c a s e

~(n). We p r o c e e d

the b a r y c e n t r i c

• .

k 6 ~,

retreat

n-simplex

(Qklk 6 ~)

Io(X),...,In(X ) denote

we

. By i n d u c t i o n

eo,e I ..... e n of the v e r t i c e s

We c h o o s e

of

simplices

are done.

family

x 6 ~(n){li(x)

Thus

a subcomplex

the o p e n

on the n u m b e r

tained

without

9.5.

:= U(Pk(°) Io6I(M))

for some

Let

complex

d 6 I(M),

the c l a i m

assume,

by L e m m a

sets

fulfill

every

we are d o n e

of M we

simplicial

simplex

fulfills

family

claim

a triangulation

M is a f i n i t e

which

this

definition.

and x £K

conthe

77

n

r

lj (x) = 0 =

T [ I i(x) i=O

j=r+1 By the

inequality

of L o j a s i e w i c z

closed

field,

[D,

k 6 ~

cf.

p.

43],

= 0

.

(generalized

[BCR,

Chap.

to an a r b i t r a r y

2, §6~)

there

real

exists

some

with n

r

k

j=r+1 for e v e r y

x 6K.

K c P s, and

our

Moreover, claim

there

is proved.

Corollary

9.6.

also

is of c o u n t a b l e

P(M)

Proof.

We

start

9.4.

exists every

with

n 6 ~ . Now by p o l y t o p e s

P(M)

is of c o u n t a b l e

9.7.

with

Assume

(cf.

as the

locally

We n o w a s s u m e not

assume

prove

this

about

that below

that

the

proof

of M.

is e v i d e n t proof

(Pn,klk6~)

Then

from

of T h e o r e m

type

9.2.

then

set

~ x ~[.

9.2,

of

there

by p o l y t o p e s ,

is an a d m i s s i b l e

index

(P(Mn) I n 6 ~ )

the p r o o f

of T h e o r e m

of P ( M n)

M is l o c a l l y

semialgebraic).

from

the

K c Q s. Thus

This

for

covering

of

proves

that

type.

complete

is e v i d e n t

as

in the

countable

locally

This

seen

covering

weakly

same

(Mnln6~)

of P(M),

just

always,

the

finishes

(Pn,kl (n,k) 6 ~ x ~ )

P(M)

s >_k w i t h

type.

covering

an a d m i s s i b l e

some

and M is of c o u n t a b l e

an e x h a u s t i o n

As we h a v e

Proposition

This

If R is s e q u e n t i a l

is an a d m i s s i b l e Lemma

exists

6.3).

Assume

Then

the

semialgebraic

also

that

function

semialgebraic

space

(and,

as

M is p o l y t o p i c ,

ringed

space

P(M)

Mlo c constructed

i.e. is

in I, §7.

1.7.8.

our

space

field

except

M that

P(M)

is a w e a k

R is s e q u e n t i a l .

in the

trivial

case

polytope.

On the c o n t r a r y , that

M is a w e a k

We do

we w i l l polytope.

78

We want the

to d e s c r i b e

s p a c e M.

have

Since

~(P(M))

c

subspace

9.8.

in s o m e

K1.

family

universal

thing

as a p a t h curve

for e v e r y

X 6 [(M),

this

every P(M)

such

selection

9.9.

set X t h e

E2-E5

lemma

same.

of

to M we D

I(M).

K 6 ~c(M)

The

follow-

3.16.

in

~c(M)

and every

it is e v i d e n t

([BCR],

T(P(M))

for e v e r y

are t h e

(KIII6A)

We conclude

the c l o s u r e s

~'(P(M))

P(M)

f r o m P(M)

and

and

exhaustions

is an e x h a u s t i o n

K £ ~c(M)

that

a path

f r o m Prop.

[DK 2,

of X in M and

§12],

is c o n t a i n e d

in M is t h e

3.6 a n d [DK4,

in P(M)

the

§2])

semi-

that,

are e q u a l .

We

by X.

is the

subspace

set of all X 6 ~(M)

structures

w i t h X £ ~c(M)"

on X in t h e

spaces

For

M and

are equal.

Proof.

If X 6 ~(M)

implies

a n d X 6 ~" (M) t h e n X 6 T(P(M)) c

t h a t X 6 ~(P(M)) . C o n v e r s e l y ,

X 6~(P(M)) and

of P M

set u n a m b i g u o u s l y

Proposition

= ~c(M),

family

in P(M).

and the

above)

by Corollary

fulfills

property

algebraic

denote

observed

on K in M a n d

An ordered

subsets

o f M is a m o r p h i s m

(as a l r e a d y

is n o w e v i d e n t

iff the

same

identity

structures

of P(M)

By the

semialgebraic

9.1.i we have ~(P(M))

ing p r o p o s i t i o n

Proposition

the

~(M)

By Proposition the

the

P(M)

= ~'c(M). both

The

coincide

subspace with

the

and X 6 ~(P(M)) . T h i s

if X £ ~'(P(M)) t h e n ,

structures subspace

on X w i t h

structure

b y Prop.

respect

3.6,

to M

in the p o l y t o p e

X.

q.e.d. We now

Theorem

state

a converse

9. 10. A s s u m e

polytope.

Then

to T h e o r e m

t h a t M is n o t

R is s e q u e n t i a l .

9.2.

a weak

polytope

but

P(M)

is a w e a k

79

Proof. y

Since

: [O,1[

c 6

]O,1[

M

is

cannot

such

yl [c,I[

that

p.

305f]).

since

y

is p r o p e r By

a weak

~ M which

[DK 4,

]O,1].

not

The

Lemma

set

(cf. 9.3,

polytope

be

completed

is

A

an

is

exists

(Cor.

embedding

:= y ( [ c , 1 [ )

II.9.9).

P(A)

there

The

is

5.8). (cf.

A

polytope.

incomplete

There

the

closed

subspace

a weak

an

path

exists

argument

some

in

semialgebraic of M

Thus

is

in M,

isomorphic

P(]O,I])

is

to

a weak

polytope.

Let

(Ksla61)

be

a faithful

complete

semialgebraic

(Kals£I)

all

for

every

sets

s 6 I.

interval

[~a,1],

some

a 6 I with

with

[c,I]

For

any

with set

We

by

there

exists

and

claim

K B mKy

we

tained =

two

some

Suppose

in

element thesis

contains

one

L We

than that

proved.

that

that

~s < & y '

Ls

see {~

that Im(s)

{6slm(s)

a with

n+1

with

L Y and and, the

the

set

~

n-l},

{~slm(s)

~n}contains

and

we

Let

s 6 I

_< n - 1 .

6~. of

n,

this

to

prove

since

Assume We

have

from

[~B,I]

Thus

~a

at m o s t the

elements.

that

and

6 B -< 6Y • is

con-

~y

or

one

more

induction Our

s

K s NK B =

Ks D K

From

Then

~y.

B 6 I

the

~ s -< 6¥ and

contains

n+1

elements

prove

= n.

hence

conclude

at m o s t

exists

We

n >O.

= m(B)

6 s _<

~n}

a closed

some

inteqer

nothing

m(y)

definition

of

elements.

L B mLy, say,

1 6K s

there

exists

number

= O.

y < B,

family

that

c 6 ]O,1[

a

L~.

is

m(s)

the

I in K s is

neaative

m(s)

y <s,

~B < 6Y'

, contradicting

non

K s is

assume

there

the

n = O there

indices

y 6 I such

c

before,

every

any

9.8,

from

may

L s of For

at m o s t

For

index

different

conclude

that

By.

only

for

Every

Omitting

[c,I]

as

P(]O,I]).

I, w e

Prop.

implies

that,

n.

contain

by

denote,

on

]O,1].

6 s 6 ]O,1].

this

~n}

of

component

Indeed,

m(s)

induction

B are

Ky w i t h

is

and

{~sls£I,m(s)

not

some

a £ I let

claim

£B

~

do

of

connected

with

c Ks,

B < a.

subset

which

The

exhaustion

claim

hypo-

80

Since

the

every

n 6

gn

{~alm(a)

_< n}

is f i n i t e

it m a k e s

sense

to d e f i n e ,

for

o'

:= m i n

We h a v e

~n > 0

a 6 I with proves

set

{~alm(a)

and

6a
that

~n h as

the

i n}

6n+I

.

• For

every

above • Then

shown

sequence

(~nln6~o)

c 6 ]O,1[

6m < c

there

for m

converges

exists

:= m(a).

to zero.

some

This

R is s e q u e n -

tial.

q.e.d.

In the

following

compare

the

M t o p and

strong

M and

spaces

in the

M t o p, w h i c h

P(M)

field

respectively

means

that

The

P(M)to p = Mto p

b)

Every

compact

c)

Every

converging

subset

PM

are

spaces with

sequential. M and

the

§3,

Def.

4).

P(M) top

These map

to

We d e n o t e

topologies

is f i n e r

by

of the

are

topological

from

P(M) top to

t h a n M t o p.

equivalent•

C of M t o p is c o n t a i n e d

sequence

We w a n t

P(M).

strong

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

topology

following

a)

(cf.

sense. the

R to be

of the

set M e q u i p p e d

classical

9.11.

the

topologies

P ( M ) t o p the

spaces

Theorem

we a s s u m e

(Xnln6~)

in some

K £ ~c(M).

in M t o p is c o n t a i n e d

in some

c ~c (M). d) M is p o l y t o p i c .

d) ~ a): W is a l s o every

Let W be an o p e n open

in Mtop,

a 6 I. Let

some

a set T 6 ~(M a) w i t h in

(Ma)to p.

The

in M t o p

with

subspace

the

Intersecting

with

which

a 6 I and

of P ( M ) t o p. We h a v e

means

that

a point

(Prop.

topology 3.5.ii).

topology

in P ( M ) t o p.

T we

that

see

Here

T is the with

T £ ~c(M), Thus

W N T is o p e n

in

be given.

of T c o i n c i d e s Since

to v e r i f y

W N M a is o p e n

x 6W n M

x 6 T and T c o m p l e t e .

strong

topology

subset

it also

W N T is o p e n in Ttop,

(Ma)to p for There

closure

the

that

exists of T

subspace coincides in

hence

(T) top" open

in

81

(Ma)to p.

Since

a) ~ b) : Let compactum Then

(KIII6A)

in M t o p = P ( M ) t o p w h i c h

exhaustion

indices

i 6A

element

xl

is c l o s e d

in M t o p.

The

in M t o p. This

Suppose

that

subspace

the

same

sequence

9.6).

§I.

Suppose

has

Since

that

the

set of p a t c h e s

set J of p r i m i t i v e I 6 J we c h o o s e

finite

subset

absurdity

proves

the c l a i m

an

intersection

P(M) top = M t o p,

for e v e r y

C is a

in any K 6 ~c(M).

For e v e r y

:= {xlll6J}

goes

The

the

M is not M

in N w h i c h

construct hold.

We c h o o s e (Recall

of A.

the

Thus

set A is c o m p a c t

with

set A A is

in M t o p,

b).

the

every

a point

is not

N of

n 6 ~

AN

from

the

has

Let

exists

some

a 6 I such

is i s o m o r p h i c

to

(K In6~) be an e x h a u s t i o n n

of

assumption

c)

and

9.8 t h a t

in some

sequence

K n.

for w h i c h

But

this

no c o m p l e t e We

neighbourhood further

in R c o n v e r g i n g

choose

to zero.

Let B ( x , ~ n)

center

x and

intersection

B ( x , ~ n) N N

is not

contained

x n ~ K n.

The

does

a sequence

in R p w i t h

would

now

in N.

ball

DN

every

contradiction.

complete.)

elements

B(X,~n)

contained

there

in N is c o n t a i n e d

locally

euclidean

with

Then

a converging

x 6 N which

the

R p.

be the d e s i r e d

of p o s i t i v e closed

some

converges

N is n o t

otherwise

x n 6B(X,~n)

will

polytopic.

of M is not l o c a l l y c o m p l e t e . M

"explicitly"

This

that

(6nln6~)

but

of

A c C,

It f o l l o w s

(cf.

since

end

in M t o p

contained

Since

P(N)

For

is not

A 6~(P(M)).

subset

denote

of P(M).

(K~II6A °) d e n o t e

cf.

set A

a semialgebraic

not

W is o p e n

is trivial.

c) = d):

we

that

@ is i n f i n i t e •

The

~ 6 A. T h u s

finite.

b) ~ c)

o C n K1%

6 C A K Io.

discrete

hence

I 6 A. Let

(KIII6A),

with

for e v e r y

also

proves

be an e x h a u s t i o n

C ~ K1 for e v e r y

of the

K~

x 6 W D T this

be c o m p l e t e . sequence

in K m for any m 6 ~ .

We

(Xnln6~)

choose

radius

6

n

in K n,

a point

converges

to x q.e.d.

82

Here

is a s o m e w h a t

of T h e o r e m

Example map y

example

9.1 2. L e t

is c o n t i n u o u s

~ M,

illustrates

the c o n t e n t s

R = ~, T(t)

and M : ~2 =

(t,e

_t-1

the a r c A u l t i m a t e l y

proof

is p o s s i b l e

of t h i s

P ( M ) t o p is the

inductive

see t h a t A is c l o s e d in M t o p b u t n o t

We n o w d i s c u s s we

over

). W i t h

respect

We consider

the

to M t o p, t h i s m a p

l i m y(t) : (0,O) . T h e set A := y ( ] O , 1 ] ) is n o t t~O B u t for e v e r y p o l y t o p e K in M t h e s e t K N A is c l o s e d

in K t o p s i n c e

As b e f o r e ,

~ (]O,1] × {0)).

and

in M t o p.

spaces

which

9.11.

: ]O,1]

closed

typical

limit

in a n y

functorial

assume

that

R induces

The

set C

K.

(A f o r m a l

of L o j a s i e w i c z . )

spaces

Kto p with

Since

K 6 ~(M)

:= A U { (O,O)]

we

is c o m p a c t

K 6 ~c(M).

behaviour

of p a r t i a l l y

R is s e q u e n t i a l .

a unique

the p o l y t o p e

inequality

of the

in P ( M ) t o p.

contained

the

by the

"leaves"

map

P(f)

A map

: P(M)

complete

cores.

f • M ~ N between

~ P(N)

such that

the

diagram

P(M)

P (f)

M

* P(N)

f

commutes.

This

, N

follows

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

have

a functor

P f r o m the

P(R)

of W S A ( R )

whose

versal of

P(R)

property

category

objects

of the m a p s

in W S A ( R ) .

are

WSA(R)

the weak

PM means

that

property to the

of PN"

full

polytopes

Thus we

subcategory

over

R. T h e

P is a r e f l e c t o r

uni-

[Mt, p. 129]

83

Remarks

9.13.

i) Let M f--~S and N g--~S be maps between spaces.

Let

M × s N denote the fibre product with respect to f and g, and let P(M) ×P(S) P(N) denote the fibre product with respect to P(f) The natural map from P(M ×S N) to P(M) ×P(S)P(N)

and P(g).

is an isomorphism.

In short,

P(M XsN)

= P(M) Xp(s) P(N).

This is a formal c o n s e q u e n c e of the fact that the functor P is a reflector.

In p a r t i c u l a r

p(MxN)

ii) Let

(S = o n e - p o i n t

= p(M) x p(N)

(MIII6A)

be a family of spaces.

direct sum of the family of

U (MIhI6A)

P(U

space), we have

(P(MI)II6A]

is an isomorphism.

Then the natural map from the

to the p a r t i a l l y c o m p l e t e hull

In short,

(MIII6A)) = U (P(M I) II6A)

Proposition

9.14.

If

(M,A)

Is a closed pair of spaces and f : A ~ N

is a map which is p a r t i a l l y proper near M ~ A from P(M) U p ( f ) P ( N )

to P(M UfN)

is the identity of M UfN}

P(M UfN)

then the natural map

{which on the set t h e o r e t i c a l

is an isomorphism.

In short,

= P(M) Up(f) P(N).

Proof. As in §8 let p denote the natural p r o j e c t i o n M U N to the space M UfN. P(M) U P(N)

level

to P(M UfN).

Then we are done,

Then P(p)

from the space

is a p a r t i a l l y proper map from

We will prove that P(p)

is strongly surjective.

since this implies that P(M UfN)

is the strong

q u o t i e n t of P ( M ) U P(N) by P(p), w h i c h is just P(M) U p ( f ) P ( N ) .

84

Given with

a set K 6 ~ c ( M U f N ) p ( C UD)

m K. W e c h o o s e

of K N (M ~ A ) tion.

we h a v e

in M.

We have

Then

D

f yields

p ( C U D) = K.

obtained

from p by restriction.

Example

9.15

ly c o m p l e t e

Final

(N = *).

9.16.

In the c a s e

if M

: X -~M

defined

the

core

with

follows

of M,

the partially of t h e

as

complete family

ordered

by

respect

to suitable

the

being

and

of M w h i c h

K is

is p a r t i a l -

to o b t a i n respect

~ M

core

P(M),

all

p.

can

take

as an

to g o o d

as w e l l

by

triangulations

is a h o m o t o p y

of M

135],

229].

The

complete

the

II.4.4.

is t h e in

inductive

[loc.cit.].

eouivalence.

We

[LSA,

partially

or w e a k

relation.

p.

to u s e

228f.].

Thus we may shall

Thus

limit with

it s u f f i c e s

of M

sub-

inductive

[LSA],

a refinement

limit

of the

(or all w e a k

theorem

now,

we

set of t h e s e

of t r i a n g u l a t i o n s

inductive

retracts

p.

~ ( c o X)

in M d e f i n e d

ordered

[LSA].

image

all p a r t i a l l y

as d e f i n e d

classes

[LSA,

(and

in

triangulations

set o f

cores

partially

P(M)

[LSA,

f r o m the t r i a n g u l a t i o n

One

deformation

t h a t P M : P(M)

in the

equivalence

of M,

X

semialgebraic

of M

to ~ as t h e

through

of all t h e

inclusion.

triangulations

strong

§10)

of M a l r e a d y

trianoulation

complex

~ running

spaces

are

(II,

is l o c a l l y

"cores"

respect

simplicial

is c o f i n a l

with

C U D with g

n : CO D ~ C UgD

subspace

that M

is a w e a k

triangulations)

cores

~ D by r e s t r i c -

q.e.d.

we defined

of M w i t h

of the

~(co X)

In o r d e r

identify

closure

then

Namely,

limit

: C AA

~ is p r o p e r

If A is a c l o s e d

as always)

cores

for C t h e

= P(M)/P(A) .

remark

co X

g

a n d D 6 ~c(N)

C O D is c o m p l e t e ,

paracompact,

core

choose

map

projection Now

C 6 ~c(M)

and we

a proper

the n a t u r a l

near M \A

p(M/A)

sets

By 8 . 7 . i w e m a y

K of M UfN,

Thus

find

:= K N N ,

subspace

complete.

to

confirm

the These

expect this

in

85

the next chapter for all weakly semialgebraic spaces by other methods, cf. Th. V.4.13.

§10 - T h e

In the

spaces

last

section

in a c a n o n i c a l canonical

way

weak

the

a short

field

which

space

N,

without

In this

a closed

changing

pair

the

if R is s e q u e n t i a l .

still work

over

(M,A)

subspace

again,

generalizations

into

section

into a partially

as t w o

time we

turn

a space

spaces

I. A c l o s e d

if the

Clearly

the

closure

(M,A)

say t h a t

a closed

some polytope

pair of

M ~A

relative

polytopes

proper

of the

a polytope

we

give

of

spaces

A and

two into

turn

a

map without

These

constructions

construction

an a r b i t r a r y

pair

real

M w--~ P(M).

closed

base

L in M,

(M,A)

i.e.

2. W e d e n o t e a relative

L UA

with

Notice

that

the u n i o n

family

of

L running

an e l e m e n t spaces,

precisely

is a r e l a t i v e

a relative

poly-

those

relative

weak

Alternatively

polytope

iff t h e r e

we

can

exists

L 6 ~ (M), w i t h M = L U A . c

closed

set of all

polytope

unions

is c a l l e d

is s e m i a l g e b r a i c .

be a f i x e d

the

(M,A)

in M is a p o l y t o p e .

are

M \A

(M,A)

Definition

is a g a i n

of M ~ A

for w h i c h

let

(K,A)

spaces

polytopes

Henceforth

and

h o w to t u r n

R.

Definition tope

seen

if R is s e q u e n t i a l .

polytope

m a y be r e g a r d e d

For

we have

f : M ~ N between

changing

and Pf(M)

constructions

a relative map

pA(M)

This

the transition

spaces.

K of M with

is t h e

K m A

set of all

~c(M).

intersection

of ~ c ( M , A ) .

of

subspaces

by ~c(M,A).

through

and t h e

pair

of a n y t w o

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

given

by

~Cc(M,A)

sets

in

~(M,A)

is a d i r e c t e d

inclusions.

87

Remark

10.1.

If

(X la6I)

an o r d e r e d

family

admissible

covering

limit

of this

~c(M,A). of

We n o w

map

assume

by PM

that

Theorem

(Q,B)

such

that

to the

3.15,

(Xa U A I ~ £ I )

M is the

: P(M)

the

(Xa U A i ~ 6 I ) It is a l s o

M is the

Space(R).

limit

is an

inductive If n o w

is a c o f i n a l

inductive

(M,A)

family

of the

in

family

A,

~ M and from

R is s e q u e n t i a l .

space

: P(A)

(pA(M),A)

strong

Then

there

p~

to

weak

by g l u i n g

:

P(M)

to A

(M) ~ M d e n o t e s

identity

of A.

Finally

the

P(M,A)

(M,A).

polytope.

The m a p

is

P(M,A)

sense.

f : (Q;B) ~ (M,A) b e a

(M,A).

obtained

and

the

is a r e l a t i v e

following

Let

field

denotes

A (PM, idA)

in the

tope

case,

E2-E5.

in the c a t e g o r y

then

the b a s e

(pA(M),A)

10.2.

by T h e o r e m

then

in S p a c e ( R ) .

that

the m a p

universal

spaces

polytope

of M ~ A

of M f u l f i l l i n g

Thus,

by the m a d P A

induced

Notice

of

3. pA(M)

P(A)

denotes

of M.

in this

%(M,A)

Definitions alona

weak

Thus,

spaces

of s u b s p a c e s

family

is a r e l a t i v e

is an e x h a u s t i o n

map

exists

from

a relative

a unique

map ~

weak

: (Q,B)

poly~

(pA(M),A)

triangle

(PA (M) ,A)

(Q,B)

f

• (M,A)

commutes.

Proof.

The pair

(Q,B)

((K,B) IK 6 ~c(K,B)) the

claim

for the

is the

by the

inductive

remark

restrictions

of

above.

limit Thus,

f to all

of the

family

it s u f f i c e s

the p a i r s

of

spaces

to p r o v e

(Q,B).

This

88

means

t h a t we m a y

assume

from

the b e g i n n i n g

that

(Q,B)

is a r e l a t i v e

polytope.

We

choose

and h

L 6 [c(Q)

of PM

PMO}

We have

(cf.

= gIL.

p~o¢

§9)

= g.

The m a p

p

:= P MA f r o m

ing the

theorem

p maps

p

-I

p(M,A)~

pA(M)

with

obtain

Corollary

Theorem

from

be a r e l a t i v e

also weak

10.4.

a unique

01~,h) is the

map

Let

By the ~

: Q ~ M

universal

: L ~ P(M)

to a m a p

unique

g

map

~

of

family

map

onto

implies

from

Let

: Q ~ pA(M).

from

(Q,B)

(K,A)

to

K. C o n v e r s e l y , Identifying

K 6 ~c(M,A). (M,A)

if

spaces 10.1

~c(M,A)

see

(L,A) 6 re(PA(M),A) set of

p we m a y

~c(PA(M),A)

say t h a t

are equal.

spaces

pA(M)

on the

set M is the

the

criterion

for a c l o s e d

pair

(M,A)

polytope.

~M

(= p u s h

inclusion out).

induc-

~c(M,A).

is a r e l a t i v e

weak

polytope

iff the d i a g r a m

,

PM with

that

above

structure

a new

and

we

* A

P (M)

mappings

to

Apply-

the u n d e r l y i n g

set of M by the b i j e c t i o n

of

such

q.e.d.

6 %(M,A).

space

(M,A)

P (A)

f to Q and B.

to M is b i j e c t i v e .

Remark

The

:=

inclusion

(p(L),A)

of the

10.2

Corollary

to the

10.3.

limit

pA(M)

families

exists

:= Q ~ B .

= f.

the u n d e r l y i n g

the d i r e c t e d We n o w

~

(K) i s o m o r p h i c a l l y

trivially

of

say L

~ and h fit t o g e t h e r

Clearly

with

tive

Q = L U B,

there

The m a p s

(pA(M),A)

then

with

: B ~ A be the r e s t r i c t i o n s

property that

some

as v e r t i c a l

arrows,

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

to

89

Indeed, P(M) map

We

that

Up(A)A of

that R.

come again

spaces

Let

f

second

be

real

any

means work

in

the

means

This

construction.

closed

field.

automatically the

happens

universal

larger

the

map

A PM

to b e

iff

the

that

property

For We

category

a short

also

a weakly

stated

time

abandon

of

identity

in T h e o r e m

the

the

base

10.2.

field

convention

semialgebraic

Space(R)

from

space

function

over ringed

R.

semialgebraic

a morphism

spaces.

in a u n i v e r s a l

from

isomorphism.

to o u r

: M ~ N be

Definition

an

fulfills

we

over

is

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

also

a "space"

Instead

diagram

to M

(M,A)

now

R may

the

4.

a weakly

(= w e a k l y

We want

semialgebraic

to

turn

f into

proper

core

of

map)

a partially

between proper

weakly map

way.

A partially semialgebraic

space

f is

P to M w i t h

a morphism the

~

: P ~ M

following

proper-

ties: a)

The map

b)

If a

M

such

phism

is p a r t i a l l y

: Q ~ M that

B

fo~

proper

is a m o r p h i s m

fos

is p a r t i a l l y

: Q ~ P such

that

Of

that

course,

then any

then

semialgebraic

there

exists

space

Q to

a unique

mor-

= ~.

, M

the

two

a weakly

proper,

~oB

Q ........ ~ P

Notice

from

maps

,, a, B a r e

partially

proper

automatically cores

of

f are

partially isomorphic

proper. over

M.

90

In the case that N is the o n e - p o i n t we have

found

Looking

for p a r t i a l l y

a set

~(f)

Definition

a partially

5. We call

K 6 ~(M) w h i c h

Remarks10.5.

flK

inclusion

This

implies

and the

field

R is s e q u e n t i a l

of f in §9, namely

in general

of M defined

PM : P(M)

it is n a t u r a l

~ M.

to study

as follows.

a semialgebraic

subset K of M proper

: K ~ N is a proper map.

are proper

We d e n o t e

over N,

if

the set of all

over N by Pc(f).

Indeed,

from K to M. Then that K = i(K)

If K 6 ~c(f)

core

cores

i) ~c(f) c ~ ( M ) .

the

ii)

proper

of subsets

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

proper

space

let K 6 ~c(f)

f.i is proper,

is closed

and L is a closed

and

hence

let i denote i is proper.

in M.

semialgebraic

subset

of K then

L 6 ~c (f)" iii)

The subset [c(f)

~c(f)

is a d i r e c t e d

(the t r a n s i t i o n iv) v)

maps

of ~(M)

family

f is p a r t i a l l y

Definitions

of closed

being

The u n i o n of all sets proper

6. a) Pf(M)

is closed

under

semialgebraic

unions.

Thus

subspaces

of M

inclusions). in

~ (f) is the w h o l e c

iff ~c(f)

denotes

with underlying

set M w h i c h

in the c a t e g o r y

Space(R).

b) The

identity

m a p of the

denote

this m o r p h i s m

set M.

= ~(M).

the f u n c t i o n

is the d i r e c t

ringed

limit

set M is a m o r p h i s m

space

over

of the family

from Pf(M)

R ~c(f)

to M. We

by pf.

c) We d e n o t e

the m o r p h i s m

f°pf

By a r g u m e n t s

very

(and including)

sition

finite

similar

9.1 one obtains

from Pf(M)

to N by f.

those w h i c h

gave us Propo-

gl

Proposition is w e a k l y ~c(f)

10.6.

pf

= ~(Pf(M))

If Pf(M)

and,

map

10.7.

generalize

Lemma

lemmas

Then

core

that the

space

of the w e a k l y

Lemma

10.9.

every

a 6 I, the m a p

core.

Then

Lemma

10.10.

in ~(M) the

f has

with

is c o n t a i n e d : Pf(M) partially

the

space

core

pf(M)

of f. A l s o

structures

f is p a r t i a l l y

9.4,

fiA

on K

proper

then

proper

Thus

~ N is p a r t i a l l y

proper.

f.}

core

q

same way.

: Q ~ M.

~ A of q is a p a r t i a l l y PfIA(A)

of M.

is a c l o s e d

Assume

that,

a partially

a countable

k 6 ~ , such

Pf(M)

in the

sub-

Pf(M).

f is p a r t i a l l y Then

which

for

proper

core.

exists

for e v e r y

core of

: q-1(A)

space

lemmas

proper

f r o m M s to N has

there

Pk ~ N of

of t h r e e

a partially qA

semialgebraic

can be p r o v e d

be an e x h a u s t i o n

fs = f l M

Pk"

and

f r o m A to N.

a partially

in some

9.5

weakly

core.

a chain

semialgebraic

that

every

proper

restriction

Pk c Pk+1

proper

proper

then

same.

and

then

down

f admits

(Mala£I)

Assume

restriction

write 9.3,

of the m a p

Let

the

core

core of f.

a partially

first

Assume

Let A £~(M). proper

we

the

10.8.

are

If R is s e q u e n t i a l

this

proper

to p r o v e

f : M ~ N admits

To p r o v e

K 6 ~c(f),

semialgebraic

proper

out

a partially

is a p a r t i a l l y

for e v e r y

is w e a k l y

start

Theorem

and pf

of M a n d of Pf(M)

is a p a r t i a l l y

We n o w

If f a d m i t s

semialgebraic

as a s u b s p a c e ii)

i)

{Thus,

that,

proper,

is w e a k l y

family

(Pklk£1N)

for e v e r y

and e v e r y

semialgebraic

b y Prop.

IO.6.ii,

k £ ~ ,

K 6 ~c(f) and pf

is a

92

By Lemma

10.9

it s u f f i c e s

s p a c e M is s e m i a l g e b r a i c . also

assume

K 6 ~c(f)

In o r d e r

M

Of c o u r s e ,

countable

is c o n t a i n e d

to f i n d 1

c

Theorem then

t h a t N is s e m i a l g e b r a i c .

to f i n d an i n c r e a s i n g every

to p r o v e

such

10.7

replacing

By Lemma

family

in t h e

10.10

(Pklk6~)

case

N by

that

f(M)

the

we may

it n o w

suffices

~c(f),

such that

in

in s o m e Pk"

a family

we c h o o s e

a commuting

triangle

, S

N

with

i an e m b e d d i n g

map

(cf.

are

semialgebraic).

mapping.

II.12.10;

We

:= S ~ M .

Lemma

things

choose

space

an embedding

x is an e l e m e n t

Since

g

is

proper equation

S N~T

lemma we

conclude

know

from

§9

Cor.

every

a countable P 6 ~c(T~A)

(M D P k L k 6 ~ ) every

X 6 ~c(f)

proof

of T h e o r e m

= X and

10.7.

in some

in s o m e

.

is p r o p e r .

in S. W e

can express

= ~.

q.e.d.

= {M N PIP 6 ~ c ( T ~ A ) } .

(Pklk6~) set Pk" family

set M n Pk"

reference)

This

implies

~c(f)

This

Now we

that

in ~ c ( T ~ A )

in

Let

of X in T b y ~T.

fiX = g l X

family

inclusion

as an i n c l u s i o n .

also by A N~T

increasing

i the

S into a complete

iff A Q XT - = ~

a formal

is c o n t a i n e d

is c o n t a i n e d

T of

to g i v e

increasing

since M and N

of S w i t h

X is c l o s e d

t h a t ~c(f)

9.6,

is a c o u n t a b l e

that

S and g a proper

the c l o s u r e

iff the m a p

this means

space

easy here

regard

X 6 ~ (f) c

of ~c(f)

this

(e.g.

we

S t

of M w e d e n o t e

From

exists

again

Let X 6~(M) . Then

Proof.

this by the

particularly

T which

For X a subset

10.11.

are

W e r e g a r d M as a s u b s p a c e

further

(semialgebraic) A

of M i n t o a s e m i a l g e b r a i c

there

such that that

such that

completes

the

93

Proposition morphism

10.12

between

semialgebraic locally

Proof.

(This g e n e r a l i z e s weakly

Given

semialgebraic

and polytopic.

semialgebraic.

some

K 6

Then

~c(f)

we

spaces.

the

shall

with

KcU

and U6

V 6 ~(M)

with

K cV

and a c o m m u t i n g

i

9.7).

Let

function

~cif).

prove

Then we

f : M ~ N be a

Assume

that

shall

t h a t M is l o c a l l y

ringes

(Here R is n o t n e c e s s a r i l y

U 6 ~(M)

V ~

Prop.

space

Pf(M)

is

sequential.)

there

exists

be done.

some

We choose

some

triangle

~ S

N

with

i a dense

embedding

map.

We

V as a s u b s p a c e

regard

locally choose

complete,and U as an o p e n

in S is c o n t a i n e d

We will write fw-~ Pf(M),

into

a semialgebraic

of S. T h e n V is o p e n

K is c l o s e d

in S s i n c e

semialgebraic

down

always

some

formal

assuming

properties

that

means

a weakly

semialgebraic

map.

the map data.

10.13.

N via

sometimes

f. T h i s

If f is p a r t i a l l y

Remark

10.14.

(cf.

Def.

think

PN(M)

We

V is finally

of K in V w h o s e

over

of a m a p

instead

then we

between

regarding

useful way

also

a "space" spaces

f : M ~ N as a " s p a c e M

of P f ( M ) ,

in a n a t u r a l

proper

Now again

R a n d a "map"

PN(M)

notation

starting

from

s a y t h a t M is

proper

map

then

semialgebraic ~(h°f)

=

map

~c(f),

as a

if s o m e h o w fixed

"partially

5 above).

If f : M ~ N is a n y w e a k l y

: N ~ S is a p a r t i a l l y

closure

of t h e c o n s t r u c t i o n

is a p a r t i c u l a r l y

f r o m M to N is g i v e n

o v e r N"

h

We

and then write

proper

is p r o p e r .

R is s e q u e n t i a l .

space

over

in S s i n c e

q.e.d.

semialgebraic

space

S and g a proper

in V.

a weakly

o v e r N",

f[K

neighborhood

means

Notations

space

and hence

94

Phof(M)

= Pf(M),

Proposition

10.15

Phof = Pf'

(h°f)~ = hof.

(Functoriality

of f~-~ Pf(M)).

i) Let

M

I

, M

(*)

N'

N

be a commuting

square of weakly

is partially proper. Pf(M)

semialgebraic

maps and assume that a

Then there exists a unique map y from P g (M') to

such that pfoy = 8Opg

ii) Assume

in addition

that the square

(~) is cartesian.

Then also

the square P (M') g

, P f (M)

Pf M'

is cartesian.

PN

Proof.

v

, M

In short,

(N' XNM)

if N' is partially

= N' xN PN(M)

proper over N, then

"

i) The map f.~opg = ~og from Pg(M')

to N is partially

Thus there exists a unique map ¥ from Pg(M')

to Pf(M)

ii) We look at the following diagram consisting squares

(solid arrows).

proper.

with pf-y = BOpg.

of two cartesian

95

rl / / /

Q .........

, Pf (M)

M' X M P f (M)

•

Pf

M

"~ M ~

,

N'

We have

to v e r i f y

that

is b i j e c t i v e

since

proper

s is a s s u m e d

sog°~

since

is p a r t i a l l y

It r e m a i n s proper proof

From

there

the

second

Corollary

that

exists

~ exists

10.16.

is b i j e c t i v e .

a n d then,

B and

proper.

that

foBo~

arrows

is p a r t i a l l y

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

If M

c o r e of g. T h e m a p

go~

is a n y

space

over

We

a : Q ~ M'

in the

conclude

with

with

that

proper.

go~

partially

~o% = ~. The

commutative

diagram

proper,

q.e.d.

and Remark

N a n d N'

n

6 are p a r t i a l l y

is p a r t i a l l y

< f r o m Q to M' X M P f(M)

by the dotted

part

The maps

for a g i v e n m a p

a map

since

proper

to be p a r t i a l l y

proper,

to p r o v e

is i n d i c a t e d

above.

pf

~ is a p a r t i a l l y

N

10.14 w e o b t a i n

is p a r t i a l l y

proper

over N then

PN(N'

Examples

XNM)

10.17.

= N' xN P N ( M ) .

i)

If L is a w e a k

polytope

and M

is a s p a c e

over

N

then

P N ( L xM)

L xM

being

=

L XPN(M) ,

regarded

as a s p a c e

over N by the obvious

map

L xM

~ M ~ N.

96

ii)

If M ~ N is a w e a k l y

plete

(hence closed)

semialgebraic

subspace

m a p and A is a p a r t i a l l y

com-

of N, then

~-I (A) = P(f-1 (A)) ,

as f o l l o w s

from Remark

10.14 and P r o p o s i t i o n

I0.15.ii.

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

for e v e r y y 6 N , ~-I (y) = p(f-1 (y))

Proposition Let

10.18

("Schachtelungssatz"

f : M ~ N and g : L ~ M be w e a k l y

for p a r t i a l l y semialgebraic

:= pf, ,' := pg. We f o r m the c a r t e s i a n

PN(M)

×MPM(L)

P2

mains

~

is a p a r t i a l l y

PN(M)

XMPM(L)

proper,

there

that, exists

given

Here

fo(goa)

core of f,g.

In short,

B exists

is p a r t i a l l y

and f ° g ° ~ ' ° P 2 a map s

is p a r t i a l l y

: Q ~ L with

is i n d i c a t e d

s i n c e gos proper.

by the f o l l o w i n g

is p a r t i a l l y

proper.

It re-

f. go~ p a r t i a l l y

a m a p 6 from Q to PN(M) xM P M ( L )

~ ' o p 2 o6 = a. The p r o o f gram.

proper

= PN(L).

is b i j e c t i v e

to p r o v e

for . and go~'.

, M

~'op2

~'°P2

and

L

PN(M)

Proof.

maps

cores).

J PM(L)

Pl

Then

square

proper

proper

such t h a t commuting

and y e x i s t s

diasince

97 B I

6

PN(M) XMPM(L)

Q .........

\

P2

' PM (L)

\ \

L

Pl "¥

h

\ \ x

PN(M)

M

N

Proposition

10.19

(Compatibility

with

fibre p r o d u c t s ) .

is a s p a c e o v e r T and M , N are s p a c e s o v e r S, h e n c e we h a v e the f o l l o w i n g

P T ( M XsN)

equalities

= PT(M)

H e r e the s e c o n d

XsPT(N)

equality

(Injectivity would

equality

to the reader.

characterizing

One v e r i f i e s

the p a r t i a l l y

10.20

generalizes

proper

= Pgf(M)

Ps(PN(M))

o v e r T:

(S) PT (N)

We leave

to M X s N

= Ps(M)"

of the first manner

10.16 above.

semialgebraic

a n d ~ := pf

In short,

that

property

c o r e of M X s N o v e r T.

Corollary

= pgf.

the p r o o f

has the u n i v e r s a l

(Idempotency) . G i v e n w e a k l y

and n O p g f ~

~ S is bi-

in a s t r a i g h t f o r w a r d

f : M ~ N and g : N ~ S, let L := Pf(M) Pgfn(L)

PT

that S

a l s o o v e r T. T h e n

s i n c e the m a p PT(S)

suffice.)

f r o m PT(M) X s P T ( N )

This p r o p o s i t i o n

Proposition

= PT(M) x

is o b v i o u s

jective.

the n a t u r a l m a p

of s p a c e s

Assume

maps

: L ~ M. T h e n

98

Proof. One verifies

in a s t r a i g h t f o r w a r d way that ~Opgf~

has the universal p r o p e r t y required tical level,

for pgf.

Since,

: Ps(L) ~ L ~ M

on the set theore-

~Opgf. is the identity of M the a s s e r t i o n follows.

§11

- The

quotient

Definition

I.

a)

semialgebraic the

i)

T contains The

hence iii)

onto)

and

onto)

T.

b)

T

phism

sense.

diagonal

the

is

an

third

equivalence of

x and

(x,y)

~

Pl ° TT"

Notice

also

section

x ~

The

T on

is

This

means

Diag M

of

p r 13

factor

(y,x)

c)

which

(x,y)

projection

~

~

equivalence

~

an

r e l a t i o n ~)

a space

M

equivalence

the

is

a weakly

relation

on M

following.

M x M. (y,x)

of

M x M maps

T

into

(and

T.

(x,y)

(x,y)

proper

relation

T of M xM

automorphism

natural

first

If

the

a partially

equivalence

theoretic

switch

The

An

subset

in

ii)

set

by

: M xM xM ~ M xM

M maps

(T x M)

relation

the

space

y of

that

on T by

T to M by

Pl

is

M

N

(M x T)

then

we

~T Pl

strongly

and and

into

(and

denote

the P2"

of M x M × M to

the

natural Notice

surjective

the

hence

automor-

projections that

s i n c e Pl

P2 has

= the

(x,x).

equivalence

relation

T

is

called

closed

if

the

set

T

is

closed

in M x M.

Example

11.1.

E(f)

is

a closed

:=

If

f

: M

{ (x,y)

a map

relation

=

between

spaces

then

f(y) }

on

M.

We

have

the

cartesian

square

, M Pl

Pl I

f

M

,

f

2.

(semialgebraic,

~)

is

£ M x M1f(x)

equivalence

E(f) I

Definition

~ N

We

N

call

proper)

This section may be used before Chapter third volume [SFC].

an if

equivalence the

skipped VII and

map

Pl

relation

T partially

: T ~ M - and

hence

proper also

at f i r s t r e a d i n g s i n c e it w i l l will have its main applications

P2

not be in t h e

100

is p a r t i a l l y seen that

Remark

proper

(semialgebraic,

a partially

11.2.

proper

If a m a p

(semialgebraic,

proper),

proper

square

in 11.1.

We ask

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

proper

quotient"

Definition and

the

tainly

3. A m a p

(cf.

§8).

(resp.

If t h e r e

exists

the d e f i n i t i o n

(Th.

of

braic

space

there

exists

R which

E(f)

is p a r -

from the cartesian

or even

above

function

cuotient

a "partially

relation

T. By

the

f is s t r o n g l y

that

and

strong

(resp.

surjective

cases

and

f is c e r -

from

case

a n d PT

is a s t r o n g

a partially

proper

(resp.

t h a t P T is p a r t i a l l y

to a s p a c e

E(g)

ringed

of M by P T

(§8, Def. 2) .

quotient

D T then

In G r o t h e n d i e c k ' s

proper

is a w e a k l y

exists

terminology

If

of M b y T

proper).

o f M b y T and g

there

of

semialge-

of M b y T.

quotient (resp.

sense

f is an i s o m o r -

quotient

proper}

space

bijection

§8 t h a t

R. T h u s M / T

of t h e

the natural

function

us a set t h e o r e t i c

It is c l e a r

map

classes

f : M ~ N of M b y T in t h e

f gives

If f : M ~ N is a s t r o n g

the

quotient

over

h o f = g.

in t h e s e

let PT d e n o t e

spaces

L with

proper

set of e q u i v a l e n c e

T,

quotient

then

ringed

in t h i s

of M b y T if T = E ( f )

a partially

We equip M/T with

ii)

such that

follows

eauivalence

and

Notice

relation

is the

a strong

then we conclude

relation

proper

8.3).

~ N w i t h f o PT = f"

phism

is c l o s e d .

is p a r t i a l l y

quotient"

f is c a l l e d

proper).

f r o m M to M/T. over

This

f : M ~ N is a s t r o n a

i) L e t M / T d e n o t e

structure

: M/T

proper).

of a " s t r o n g

set M b y t h e e q u i v a l e n c e projection

the eauivalence

o f M b y T if T = E ( f )

proper

11.3.

spaces

It is e a s i l y

following.

identifying

Remarks

respectively).

relation

of a s p a c e M by a g i v e n

quotient

partially

then

(semialgebraic,

f is i d e n t i f y i n g

proper)

equivalence

f : M ~ N between

tially

this we mean

proper

: M ~ L is a

a unique [Gr,

§2]

map

h :N ~ L

the map

f

101

is a s t r i c t

epimorphism

in the

category

WSA(R),

in

fact

even

in

Space(R). iii)

If t h e r e

know

from

Brumfiel

has

the p r o p e r

and

that

quotient

11.4.

M then

algebraic,

the

Proof.

We c h o o s e

such

that

regard there such

PT

defined

Assume

and p a r t i a l l y

shall

if M is

relation theorem also

proper.

on M t h e n

extends

call

(affine) indeed

readily

"Brumfiel's

proper

equivalence

M/T

is p a r t i a l l y

(cf.

relation

11.3)

proper.

on a

is w e a k l y

If T h a p p e n s

semi-

to be

is proper.

an e x h a u s t i o n

on M / T

(Mala 6 I) of M. F o r

T o := T N (M a × M a) on Ma the

of M / T

and

Pa

a,

of a

la 6 I)

and p a r t i a l l y space

s 6 I the

of a s e m i a l g e b r a i c

set M/T. (weakly

M a / T a - in its g i v e n

(Ma/T

every

is ~Droner-_ By B r u m f i e l ' s

: Ms ~ M a / T a

of the

structure

Pa of PT are p r o p e r

the p r e s e n t

structure

projections

the

for e v e r y

ly s u r 3 e c t i v e

This

space

sets M a / T a as s u b s e t s

that,

that

we

ringed

: M ~ M/T

the n a t u r a l

restrictions

know

function

set M a / T a c a r r i e s

exists

subspace

which

of M by T t h e n we

[B 2] that

equivalence

of M by T exists.

relation

the

be c l o s e d

theorem

If T is a p a r t i a l l y

then

the

important

statement

and PT

equivalence

T must

quotient

on.

proper

result

the

proper

and T is a p r o p e r

later

Theorem

11.2

proved

following

theorem"

space

a partially

11.1

semialgebraic

to the

exists

is proper. By T h e o r e m

structure

maps

that

conclude

(weakly

structure

coincides

space

Since

the

PT is a s t r o n g -

semialgebraic)

on M / T

1.6

- is a c l o s e d

of M/T.

proper

We

semialgebraic)

is an e x h a u s t i o n we

space

map.

with

the

We n o w one

above.

that

is a g a i n

Pl

proper.

: T ~ M is proper. It s u f f i c e s

We w a n t

to p r o v e

that

to p r o v e PT

that

PT : M ~ M / T

is s e m i a l g e b r a i c

(cf. 5.6).

102

L e t x £ ~(M/T) some

Y E r(M)

be given. with

a semialgebraic

Remark

11.5.

pT(Y)

set,

using

proper.

Brumfiel's

not be needed will

turn

Remark C.

result

recently

Scheiderer

makes

[LSA,

A]).

group

actions.

Definition braic)

space, group

We

over

SA(R)).

such

the map

space

the

map

f there

earlier

way.

obvious

object

11.4 w i l l

result.

space,

11.4

Indeed, to

and

it let T b e

strong quotient

semialgebraic

subset

D T is a p r o p e r

us

without

[SFC].

semialgebraic the

is

Theorem

"converse"

11.4 g i v e s

in the

K such that

map

semialgebraic

of M

o n t o M.

spaces

important

(cf.

case

of

definitions.

(locally

semialgebraic,

G in t h e c a t e g o r y

G is a s p a c e ( l o c a l l y

respectively)

x ~ x -I , b e t h

is a s p e c i a l

of T h e o r e m

possible

of a b s t r a c t

that

case

volume

that

exists

is i n d e e d

following

(M × K)

that the multiplication

i : G ~ G,

strength

is a b e s t

a closed

use

gluing

in an e s s e n t i a l

semialgebraic

This means

This

of s p a c e s

special

on M s u c h

R is a g r o u p

semialgebraic

full

complete

start with

there

q.e.d.

the

in the t h i r d

Theorem

4. A w e a k l y

group

LSA(R),

what

provided

volume

: T ~ M to

essential

We now explicate

the

relation

of Pl

= p2(p~Iy).

8.6 on g l u i n g

to d o t h i s

proved

Then M contains

the restriction

App.

11.4,

theorem has

p~1(X)

surjective

is s e m i a l g e b r a i c ,

important

Brumfiel's

is s t r o n g l y

Theorem

present

equivalence

by T exists.

Pl

since

L e t M be a l o c a l l y

a closed

PT Then

We decided

in t h e

Scheiderer

[Schd]:

since

theorem

o u t to be

11.6.

= X.

The previous

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

Since

map m

over

R and

: G x G ~ G,

are weakly

semialge-

WSA(R)

(resp.

semialgebraic also

an a b s t r a c t

(x,y)

semialgebraic

~ xy, maps.

and

103

Exampl e s

11.7.

C = R(V-~I)) ii)

i)

If G is an algebraic

then G(R)

The o r t h o g o n a l

U(n,R)

(rasp.

G(C))

groups O(n,R)

and s y m p l e c t i c

groups

over R as well

Sp(n,R)

over

by R) are s e m i a l g e b r a i c

use the

R in our n o t a t i o n

though

R(V-I)

scheme

R

over

R

(or over

group over

as the u n i t a r y

(cf.

groups

of unitary

and the q u a t e r n i o n s

over

is a s e m i a l g e b r a i c

there r e p l a c e d letter

group

[Ch, Chap.

over

R.

R are used

groups

I] w i t h

{We are

and s y m p l e c t i c

R.

just to

groups

al-

in the d e f i n i t i o n

of

these groups.} iii)

We embed O(n,R)

into O ( n + 1 , R ) a s

O(~,R)

:= lim O(n,R) n~

= U O(n,R) n

is a w e a k l y

semialgebraic

group w h i c h

by s e m i a l g e b r a i c

groups.

U(~,R),

All

Sp(~,R).

iv) A s s u m e over

that

R then the

is a p a r t i a l l y braic

group

in VII,

complete over

other

§4 and VII,

Definition operation

groups

space P(G)

complete

We shall meet

these

R is sequential.

scheme

a partially

Similarly

with

group R

examples

has

by A ~-~ ~

we have w e a k l y

semialgebraic abstract

In particular,

C) then

semialgebraic

of g e n u i n e l y

~(R)<jloc

groups

group

group

G

if ~ is any alge(rasp.

group over

weakly

(O(n,R) In6~)

complete.

the same u n d e r l y i n g R.

Then

semialgebraic

If G is a w e a k l y

over

~).

the e x h a u s t i o n

are p a r t i a l l y

(or over

locally

usual

~ ( C ) l o c)

is

R.

semialgebraic

groups

§9.

5. Let G be a w e a k l y

semialgebraic

group

of G on a space M is a left o p e r a t i o n

(over R). A left

of the abstract

group

G on the set M such that the map G x M ~ M,

(g,x) ~ gx is w e a k l y

semi-

algebraic.

over

We then call M a

If M is a G-space G gives

G-space

R.

and the group G is s e m i a l g e b r a i c

us an e q u i v a l e n c e

T(G)

(left)

relation

:= { (gx,x) Ix 6 M , g 6 G }

then the action

of

104

on M.

Indeed,

(g,x) ~

T(G)

(gx,x),

is t h e

and hence

equivalence relation M/T(G)

image

of t h e

a weakly

semialgebraic

is s e m i a l g e b r a i c

- if it e x i s t s

- will

is the

set

set of o r b i t s

Assume

now

t h a t G is c o m p l e t e .

fiel's

theorem

us the

(cf.

be denoted

underlying

gives

semialgebraic

Def.

more

m a p G × M ~ M x M,

subset

2).

The

briefly

of M x M. strong

b y G\M.

This

quotient Its

of G on M.

Then

P2

following

: T(G)

~ M is p r o p e r .

somewhat

special

Thus

result

Brum-

on o r b i t

spaces.

Corollary proper

11.8.

If G is a c o m p l e t e

quotient

G\M exists

U p to n o w w e d i d n o t the q u o t i e n t algebraic) gebraic.

care

Here

for e v e r y

G-space

for t h e q u e s t i o n

of a l o c a l l y

space

semialaebraic

semialgebraic

by an e q u i v a l e n c e

is a r a t h e r

special

then

the

which

conditions

M.

under

(and,

relation

group

as a l w a y s ,

is a g a i n

(and t r i v i a l )

weakly

locally

statement

semi-

semial-

in t h i s

direction.

Remark

11.9.

In the

semialgebraic,

Proof. braic

Let

subsets.

Then

Remark [Schd]:

(GU

is l o c a l l y

Is 6 I)

f r o m M to G\M.

semialgebraic

covering

also G\M

of C o r o l l a r y

(Usls E I) be an a d m i s s i b l e

the projection open

then

situation

in G \ M

for e v e r y

if M is l o c a l l y

semialgebraic.

covering

is a g a i n We have

11.8,

such

of M b y o p e n

a covering.

(pUsla 6 I)

Recently

q.e.d.

C.

Scheiderer

Let M be a locally

T a closed

p U s is

is an a d m i s s i b l e

o f G\M.

11.10.

Let p denote

p - 1 ( p U s) = GU n . T h u s

s, a n d

semialge-

semialgebraic

has proved

semialgebraic

equivalence

a much

partially

relation

better

complete

on M with

the

theorem

space

and

following

105

property. -I (*) If U 6 ~(M) then p2Pl U is open in M

(hence p2p~Iu 6 ~(M)).

Then the strong q u o t i e n t M/T exists and is again locally semialgebraic. {Scheiderer only deals w i t h the case that M is semialgebraic, s t a t e m e n t above follows t r i v i a l l y from this case. proves that

S c h e i d e r e r also

(*) just m e a n s that the map Pl is open.}

algebraic group acting space M and if T(G)

(from the left)

is closed then

but the

If G is a semi-

on a locally s e m i a l g e b r a i c

(*) is clearly fulfilled,

q u o t i e n t G\M exists and is locally semialgebraic.

hence the

In t h i s fixed

Chapter

V - Patch

complexes,

chapter

a "space"

means

real

closed

semialgebraic notions spaces

and

field

map,

and maps

explanation.

homotopy

classes

In t h e

is said.

the

III

All

over

the

from

of s p a c e s

spaces

spaces

retain

starting

category

a~ain

semialgebraic

between

in C h a p t e r

[f] of m a p s

f : M ~ N between

means

over

a

a weakly

homotopy

their

theoretic

sense

§2, w i t h o u t

over

R and

space

for t h e s e further

R is d e n o t e d

its m o r p h i s m s

by

are

the

spaces.

decompositions

following

Definition

are

else

be used,

The homotopy

Its o b j e c t s

§I - P a t c h

used

and will

HWSA(R).

a weakly

R a n d a "map"

if n o t h i n g

notations

and homotopies

M is a f i x e d

space.

I. A s e m i a l ~ e b r a i c

with

the

following

PDI.

If ~ a n d

PD2.

Every

partition

of M is a s u b s e t

I of

~(M)

properties.

T are

X 6 ~(M)

different

elements

is c o n t a i n e d

of Z t h e n

in t h e u n i o n

of

o D T = ~. finitely

many

elements

of Z. {The

For

letters

example,

patches

"PD"

if

(Msla6I)

{M oa l a 6 I °}

just means

that

Definition

2. L e t

i)

The

ii)

An element

to

"patch

decomposition",

is an e x h a u s t i o n

is a s e m i a l g e b r a i c

Z is an a d m i s s i b l e

boundary

We then write

refer

partition

covering

I be a s e m i a l g e b r a i c of ~ is t h e

set

3o

then of M.

of M

partition

the

set of

Notice

(IV,

§3,

of M,

that

Def.

and

face

of ~ if T D ~

PD2

7).

let ~ 6 Z .

:= ~ o .

T of Z is an i m m e d i a t e T < a.

of M,

see b e l o w . }

,@.

107

Every

~ 6Z

braic

set

has

PD3.

finitely

Do is c o v e r e d

Definition braic

only

by

3. a) A p a t c h

partition For every

b) A p a t c h X and

~ 61

letter

complex

X for of X.

Example

1.1.

wise

does

in I. T h e n finite

Example of M.

exist

Assume

Example of M.

1.3.

Assume

following

a number

is a s e m i a l g e -

condition.

n 6 ~ °

such

that

every

called

t h e h e i g h t h(o)

is a p a i r

(X,I(X))

of X. w e The

and

subsets

consisting

often write

elements

of Z(X)

the are

c o v e r M.

chain

of ~.

of a space single called

let Z be a f i n i t e

of M w h i c h

semialge-

the

s e t of p a i r -

Assume

that

chain

complex.

We

call

these

pairs

(M,Z)

the

complexes.

L e t M be a n y that

holds.

dimensional

space M

is t h e n

a cyclic

Thus

patch

of Z. T h i s m e a n s

and

let Z be

smaller (M,Z)

than

c,T

a semialgebraic

of Z w i t h

the dimension

is a p a t c h

complex.

~ < ~ the dimen-

o f o. T h e n

We call

partition

these

PD3

ob-

(M,Z)

the

complexes.

Again that

space

for any two elements

s i o n of • is s t r i c t l y viously

of ~.

r _< n.

complex.

is a p a t c h

elements

the

length

Z(X)

semialgebraic

(M,Z)

1.2.

(over R)

since

of t h e

the

Let M be semialgebraic

not

patch

many

faces

in ~ h a s

lengths

decomposition

disjoint

there

exists o

the patch

patches

finitely

decomposition

there

of t h e s e

a patch

immediate

I of M f u l f i l l i n g

% r < %r_ I ~ ... ~ T o The maximum

many

let M be any the boundary

that

every

space

and Z a semialgebraic

Da of e v e r y

immediate

face

~ 6Z

is a u n i o n

partition of e l e m e n t s

T of ~ is c o n t a i n e d

in

Do.

108

Then dim T < d i m o , and we see that We call these complexes

is a d i m e n s i o n a l patch complex.

(M,Z) normal patch complexes,

in topology,

normal patch complex

Of course,

(M,I)

cf.

[LW, p.9].

in analogy to normal cell

Every simplicial complex is a

(cf. IV.1.7 and IV.I.19).

a patch d e c o m p o s i t i o n ~ of our space M is called finite

(resp. dimensional,

resp. normal)

if the patch complex

(M,Z) is finite

(dimensional, normal).

We are interested in patch complexes as a very weak substitute of simplicial complexes.

Our spaces often cannot be t r i a n g u l a t e d

(cf. IV.4.8),

but they admit patch decompositions.

Proposition

1.4. Let

the set of patches

(Mala£I)

{M~La6I°},

be an e x h a u s t i o n of our space M. Then in the sense of IV, §I, is a patch de-

c o m p o s i t i o n of M.

Proof.

If M B° is an immediate face of M °a then M~ G M a ~ ,

IV.I.18.

Example

Certainly B %a,

hence

B
1.5. Consider the e x h a u s t i o n

d i m e n s i o n a l p r o j e c t i v e space ~ ( R )

and the axiom PD3 is evident.

(pn(R) I n 6 ~ o) (IV, Ex.

of the infinite

1.9). The c o r r e s p o n d i n g

patch d e c o m p o s i t i o n consists of the patches o n := ~ n ( R ) (n ~ 0 ,

read P

-I

(R)

hence B _
= ~). We have o k < o I iff k
~n-1

(R) & R n

This patch d e c o m p o -

sition is normal.

Example

1.6. We c o n s i d e r again the c o u n t a b l e comb from IV.4.8 w i t h

the e x h a u s t i o n

(Mnln£]N o ) indicated there.

O n := {~} x ]0,1] w i t h n £ ] N ,

and ~o

This time the patches are

:= [0,1] x {O}. The only immediate

face relations are o o ~ o n w i t h n 6 ]N . This patch d e c o m p o s i t i o n dimensional.

is not

109

Definition T 6Z

4. A s s u m e

is c a l l e d

that

a face

of a p a t c h

T = T r • Tr_ I < ... < ~ o a ~ro~er

face

It is e v i d e n t ordering

Remark

finitely

This

follows

only

finitely

1.4 t h a t

We n e e d

the

with

(each

we call

- -

face

T
a chain

If r > I then

- -

decomposition

relation

T

is a p a r t i a l

T ~o.

then

every

patch

~ 6Z

has

h(o)

from

then

decomposition

it is e v i d e n t

from

M ~ _< M °a i m p l i e s

extension

fact

that

~ has

of

the p r o o f

from

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

4 in I, §I.

subsets

semialgebraic

of M c o m i n g

B _<s-

of D e f i n i t i o n

(XIII6A)

for e v e r y

the

faces.

is the p a t c h

of M is c a l l e d subset

Y of M,

partially the

set of

~ ~ is finite.

to c h e c k

admissible

decomposition

semialgebraic

exists

T <~.

the

on the h e i g t

relation

Xl D Y

of any

patch

any p a t c h of M

(Mala£1)

it s u f f i c e s

the m e m b e r s

Any

{M~la6I °}

if,

that

just m e a n s

immediate

5. A f a m i l y

Of course,

of s o m e

many

face

(in M)

I 6A

T <~

if t h e r e

of M. A p a t c h

faces.

an o b v i o u s

Definition

all

PD3

many

If

an e x h a u s t i o n

finite

from

the a x i o m

decomposition

then write T <~.

by i n d u c t i o n

1.8.

~ 6~

of ~ and w r i t e

If Z is a p a t c h

only

Remark

~. We

=

of Z and t h a t

1.7.

Z is a p a t c h

set b e i n g

of M,

indexed

property

covering

for Y r u n n i n g

(IV,

§3)

of M,

(cf.

Def.

I),

e.g.

through the p a t c h e s

of M.

partition

decomposition

this

Z

of

M

is a p a r t i a l l y by

itself).

finite

and

in p a r t i c u l a r

family

of

subsets

110

Remark every the

1.9.

Let

A 1 is w e a k l y

set B

every

(AISA6J)

Definition patch

complex

in M.

intersection

is p a r t i a l l y

B NC

finite

Y of t h e

semialgebraic

in X,

of

the

subset

space

in M s i n c e ,

A,

for

The

B.

a subcomDlex

patches

subspace

that

J of

semialgebraic.

X is c a l l e d some

in M s u c h

for e v e r y

semialgebraic

in the

and

family

is c l e a r l y

space

X if Y is the u n i o n

finite Then,

is a g a i n w e a k l y

6. A s u b s e t

Y is w e a k l y

of X.

Then,

of the by

1.9,

Y of X t o g e t h e r

with

subset

E(Y) of

the

be a p a r t i a l l y

semialgebraic

:= U ( A I I I 6 J )

C 6~(M),

family

the

(AIII6A)

~(Y)

:= {(J¢E(X) IucY}

is a p a t c h

Definition subset

7. A s u b c o m p l e x

Y of the

Theorem

1.10.

o 6E(Y)

all

ii)

s e t H(X)

The

lattice

complex,

space

which

X is c l o s e d

(immediate)

faces

of all

(resp.

open)

space

X,

E(X)

in X if t h e

in X iff for e v e r y in Y.

subcomplexes

and

open)

in X.

of ~ are c o n t a i n e d closed

b y Y.

(resp.

Y of X is c l o s e d

finite

of the

denote

Y of X is c l o s e d

i) A s u b c o m p l e x

exhaustion

we a l s o

is the

of X is a f a i t h f u l set of p a t c h e s

of

H(X).

Proof.

a)

L e t Y be a c l o s e d

If ~ is an i m m e d i a t e this b)

implies

f a c e of d, t h e n

all

faces

L e t n o w Y be a f i n i t e

immediate 6 E(Y), c)

that

faces

of all

of X f u l f i l l s

of X a n d

• NY

subcomplex are

of X.

checked

that

i.e. the

the conditions

Then

Tcy.

Do c Y .

Of c o u r s e ,

in Y.

Then ~ = U(~I06E(Y)).

contained

t h a t Y = Y,

c £ E(Y).

~ ¢, h e n c e

o f o are c o n t a i n e d

o 6E(Y)

a n d we c o n c l u d e

It is n o w t r i v i a l l y

complexes

subcomplex

in Y,

then ~ cY

Y is c l o s e d

set H(X) E2',

E7,

for e v e r y

in X.

of f i n i t e

E3-E5,

If all

E8

closed in IV,

sub§I.

111

{Of c o u r s e , H(X) by

we equip

as a f a m i l y

itself.}

element

For

by

is an a d m i s s i b l e

of t h e d) W e

IV,

§I).

contains

hence

X(d)

index

from

IV,

c y by s t e p

I(X)

all

Y N Z has cular,

the

Notice

faces the

procedure

and

being

that

regard indexed

of o is an

o cX(o).

Since

Z(X)

is an a d m i s s i b l e H(X)

concludes

and

is an e x h a u s t i o n

H(X)

thus

than

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

elements

Thus

and

the X(o)

of H(X).

The

set

= X(O) ~ U ( X ( z ) l ~ < o )

is t h e has

set of p a t c h e s the property

for e v e r y

Z 6 H(X),

of H(X). that Y con-

the

subcomplex

we

conclude

that

a given

of the

Theorem

theorem.

1.10.ii

faithful

exhaustion

indicated

of T h e o r e m

r1(x) = X ( o ) .

o cy

is an e x h a u s t i o n ,

decomposition

the one

then

In p a r t i -

of all c l o s e d

spect to t h e p a t c h

x 6Y,

patch

s t e p b).

take

family

the unique

(cf.

by

lattice

for H t h e

exhaustion

Y n Z 6H(X)

1.4 a n d

how to enlarge

is t h e

Then,

the proof

Proposition

with

Thus

of X w h i c h

G 6Z(Y).

Since

of t h i s

let o d e n o t e of H(X)

I(X)

s p a c e M to a f a i t h f u l

structive

3.16

and

faces

a l s o H(X)

: X ~ H(X)

c X(o))

subcomplex

same property,

This

that

Cor.

X(o)

set o. T h u s

of a l l

Y N Z £~(Z).

Y 6~(X).

of all

are t h e p r i m i t i v e

to a g i v e n

Let now Y be any

tains

of H(X)

We have

X,

a) of t h e p r o o f .

through

is just

n

be g i v e n

X(o) ° = X(o) \ U ( X ( ~ ) I X ( ~ )

e)

space

If Y is an e l e m e n t

corresponding

and this

X(o)

of t h e p r o o f . of t h e

inclusion,

element

the u n i o n

function

Let x 6X x.

o running

patch

s t e p b)

each

by

X.

which

with

oEZ(X)

conclude

l o o k at t h e

e n d of

the ordering

in ~ ( X ) ,

covering

of X. W e

space

with

of s e t s every

of H(X)

covering

H(X)

finite

in t h e

1.10.i

exhaustion H

(Msol s 6 I ° ) .

together

(Prop.

(M Is6I)

IV.I.15).

subcomplexes This

proof

we obtain

g i v e us a n e w

Namely

of M w i t h

procedure

of Prop.

of a

is m o r e

IV.I.15.

we recon-

112

Corollary for any

1.11.

A subcomplex

~ 6I(Y)

every o 6Z(X)

and J 6[(X)

with

the union of all

subcomplex

of X w h i c h

Definition

8. This open

(or open

Y of a patch

contains

complex

T ~ ~ also

# 6 I(X)

o 6 I(Y).

with

subcomplex

of X is called

We now look for p a t c h d e c o m p o s i t i o n s

of a given

Definition every

the pair

(~,~)

of s e m i a l g e b r a i c

(N,V(n)) with N the union of the

R n+1

and

patch

the

open

star of ~ i_nn X

space M with

Z of M is c a l l e d

pair

some open

smallest

special

some more definitions.

9. A p a t c h d e c o m p o s i t i o n

~ 6Z,

for

G.

by Stx(O).

We need

in X iff

In particular,

~ < p is the

star of ~ i_nn X) and d e n o t e d

properties.

X is open

faces

spaces

standard

of ~(n).In this

special

(M,I)

for

is isomorphic

open n - s i m p l e x

case

if,

to a

~(n)

is called

in

a special

complex.

Notice

that,

forced

to be the w h o l e

Definition

if M is a weak

10. Let

decomposition

polytope

closed

(AILI£A)

of M and the

simplex

then,

is called

a simultaneous

and the

family

special

patch

complex

Theorem

1.12.

Let M be a space and

in ~(M).

Further

a simultaneous (AIII6A)

(AILI6A)

and

if,

A simultaneous

is a patch

of

special

in addition,

(M,Z).

patch

decomposition

Such

a p a t c h decom-

patch d e c o m p o s i t i o n

of M

M and every AI becomes

a

by this d e c o m p o s i t i o n .

let

(M is6I)

special

patch

(Mala6I).

in T(M).

(Alll6A)

of M such that e v e r y AI is a s u b c o m p l e x position

N is

V(n).

be a family family

in this definition,

(AIII6A)

a partially

be an e x h a u s t i o n decomposition

of M.

finite

Then there

of M and both

family exists

families

113

Proof.

This

angulation finite a 6 I°

(cf.

IV,

1.2.13.

§I,

Def.

I of a l l of M.

that

that

S of X a w i t h I n with

If a g a i n

either

I also

~ = Xs(S)

~ = x~(S),

B
to t h e p a i r

S and a l l

its

faces,

contained

in Xs.}

PD3,

(S n X

hence S NXs

semialgebraic

a 6 I let A(a) every

the

finite

finite

set of

of I s t h e n ,S).

{As u s u a l ,

denotes

of p a t c h e s ,

(M s n A l 1 1 6 A ( ~ ) )

images

xs(S)

that

T <

it is

decomposition

S denotes

the u n i o n

patch

partition

then

(~,~)

of M.

is i s o m o r the

of S a n d

decomposition

of all

the u n i o n

of S. T h u s

the p a i r

the

: X s --~ M s o f

of I are g i v e n

is a p a t c h

tri-

index

I ° is a s e m i a l g e b r a i c

~ = ×B(T)

hence

Xa

families

c M°'a It is e v i d e n t

through

denote

primitive

a triangulation

Z is a s p e c i a l

is a u n i o n

of the

B = s and T is a f a c e

is a n e l e m e n t

Xs,

AI

or

fulfills

Thus

% ~. F o r

the

Xs(S)

s running

via

M s and every

for a n y

triangulates

L e t Zs d e n o t e

If t w o e l e m e n t s

implies clear

sets

Indeed,

11) w e c h o o s e

simultaneously

simplices

consequence

I 6 A with M s hAl

(MBIB6I°,B<s).

open

phic,

theorem

set of all

M s which and

is a s t r a i g h t f o r w a r d

union

of

all

faces

of M.

Every q.e.d.

§2 - S o m e

We want braic

deformation

to extend

spaces.

Definitions of s p a c e s M ~A. b)

This

needs

(cf.

We then

§8,

Def.

call

the

triple

is c a l l e d

of o" a n d same

"normal".

We call

the

III,

(M,A,7)

§I to w e a k l y

derived

complex

semialge-

a closed

patch

Do d e n o t e s (in M).

from this

like

complex

i.e.

space

o 67,

set ~ ,

Notions

like

complex

"face

(M~A,7).

"dimensional"

(M,A,7)

(M,A)

complex.

the

patch

pair

7 of the

(M,A,7),

"absolute"

patch

of

a relative

patch

to t h e

relative

equivalences

decomposition

of t h e p a t c h

refer

homotopy

decomposition

of a in M a n d

the boundary

for n o t i o n s

of

is a p a t c h

of a r e l a t i v e

"height of o" holds

8)

related

preparations.

patch

IV,

the closure

and

results

various

I. a) A r e l a t i v e

then ~ denotes

The

s o m e of t h e

If ~ is a p a t c h

which

retractions,

finite

or

if 7 is

finite. c) A s u b c o m p l e x of M w i t h

A c L and

(L,A,7')

with

relative

patch

subcomplex

d)

~'

of

o 67',

In t h i s

the

complex.

T
has

The

(M,A,7)

Then,

~ £ 7 contained also

subspace

L a closed

product

(N,B,7')

(M,A,I)

this

is a s u b s p a c e

of c o u r s e ,

in L ~ A

triple

L is c l o s e d

the

with

sets

as the the

patch

decomposition

corresponding

patch

complex

of

the t r i p l e

is a g a i n

will

L

a

be c a l l e d

a

in M iff

(M,A).

of t w o closed

~ 67

relative pair

relative

o×T w i t h

e) A r e l a t i v e

(~,G)

subcomplex

× (N,B,I')

is d e f i n e d

equipped

as p a t c h e s

the pair

complex

of p a t c h e s .

Sometimes

o I ×T I ~ o 2 × T 2 iff o I ~ ~2 a n d

@ 67,

patch

~ T 67'

(M×N, (M×B) U (A×N)) which

a union

set of all

case we call

and

relative

L ~A

(M,A,Z).

The direct

(M,A,7)

of t h e

of

patch

and

patch

complexes

spaces

decomposition

T 6Z'.

Notice

that

TI ! T2 . 7 of

(M,A,7)

is i s o m o r p h i c

are

a closed called

to a p a i r

pair

(M,A)

special

iff,

and

the

for e v e r y

(N,V(n)) w i t h N t h e u n i o n

115 o

of the open

standard

n = dim a .) N o t i c e Def.

8) this

Remark

simplex V(n) and some

that

forces

2.1. We know

in M ~ A

a special

if Z'

of

relative

(M,A)

sitions

avoid b e l o w

then

a significant

of spaces.

role,

polytope

pair

(Th.

(M,A)

(IV,

1.12).

is a patch also I UI'

§8,

of

(M,A).

Converse-

p a t c h de-

of M.

If

is special.

with

notions

explicated

simultaneous

whenever

it is c o n c e p t u a l l y

a

Then the patches

decomposition

"relative"

instead But,

admits

of A and Z is a r e l a t i v e

the use of the

I, always w o r k i n g

of pairs

weak

patch d e c o m p o s i t i o n

then Z UZ'

both I' and I are special

Definition

closed

patch decomposition

is a patch d e c o m p o s i t i o n

composition

We could

is a r e l at i v e

from §I that every

special

ly,

(M,A)

of $(n).(Of course,

N = ?(n).

simultaneous give

if

faces

the patches

simpler

in

patch d e c o m p o in A do not play

to use these

relative

notions.

In the f o l l o w i n g

(M,A)

is a closed

a fixed

patch

decomposition

relative

patch c o m p l e x of I.

(This will

Notations height

2.2.

h(o)

w i t h h(o) M_I

(M,A,Z)

again by

be common

For any n 6 ~

= n by Z(n).

~ n by Mn.

course,

~ with

o running

the s u b s p a c e

(M,A).

practice

o

We denote through

also

later

structure

~a with

closed

subspace

We d e n o t e

of M(n).

by M(n).

a running

to any summand ~ of M(n)

by ~n

Z(M,A)

instead

~ 6 Z with

of A and all patches

sum

of

(IV,

(M,A). We put 1.10)

Here ~ carries, denote

through

: M(n)

(a 6Z(n))

the relative

on.)

in M. We further

sum of the spaces

we w r i t e

subcomplex

the direct

(M,A) with

denote

the set of patches

the union

I(n)

direct

striction

Sometimes

we denote

We denote

We equip

Z. We a b u s i v e l y

Every M n is a c l o s e d

= A for convenience.

spaces

pair of spaces.

of the of

by ~M(n)

Z(n).

This

the

is a

~ M n the m a p w h o s e

is the

inclusion

~

reM n.

116

We

finally

denote

restrictions

Both

maps

to

by

: 3M(n)

n

~ Mn_ I the m a p

obtained

f r o m Cn by

~M(n).

~n and ~n are p a r t i a l l y

~M (n)

, M

%0n

proper.

We h a v e

a commuting

square

n- 1

(*)

M(n) with tends

~n

inclusion ~n and

= M(n)

U

we o b t a i n

(cf.

IV,

relative

Mn_ I ~ M n w h i c h

and p a r t i a l l y

proper.

ex-

Thus,

by

n- I

M n by g l u i n g

2. We

patch

the n - b e l t ,

covering

call

M(n)

to M

n-1

and ~ n

Pn the

3. An

along

3M(n)

o

or

In the p r e s e n t filtration

spaces.

We

further

canonical

filtration

by the m a p

the n - b e l t ~M(n)

of

map

~n

~

of the

the b o u n d a r y

of

of the n-belt.

f r o m M(n) U Mn_ I to M n.

of a space

weakly

n 6 ~ . Instead o

call

projection

of X by c l o s e d

~

and M(n)

~ Mn_ I the a t t a c h i n @

admissible

(Xnln£~)

~

(M,A).

: 3M(n)

X n c Xn+ I for e v e r y like

M n the n - c h u n k

complex

we call

Definition

of

surjective

: M(n)U

§8).

Definitions

ible

Pn

M

~n

i.e.

sets

j. The m a p

(~),

n

Finally

i and

j is s t r o n g l y

the d i a g r a m

M

maps

' Nn

X is an a d m i s s i b l e

semialgebraic

subsets

with

we

shifted

index

also

allow

U {-I}.

situation

the

of M and t h u s

family

of c h u n k s

M is the

( M n l n 6 ~ o)

inductive

limit

is an a d m i s s -

of this

family

117

M c a n be other

" b u i l t up"

using

the maps

the

s p a c e A,

are

spaces

spaces. also

Remark.

of rather

on.

N n is o b t a i n e d procedure

Instead

of

M(n),

simple

(We t h e n

If M ~ A

would

from A by attaching

The whole

space

namely

be of c o u n t a b l e

semialgebraic

that

sums

be very

type

(Nnin ~ -I)

from Nn_ I by gluing general

the

of M is g i v e n

Notice

direct

after

of

the

by

belts

semialgebraic

useful

below

and

t a k e A = ~.)

filtration

in t h e

~n"

a space M will

often

one belt

structure

and the maps type,

to a n a l y z e

f i n d an a d m i s s i b l e

Our

~n"

the belts

This way

later

starting

case

spaces

(IV,

§4),

of M such

a semialgebraic is o n l y

we

slightly

use direct

sums

then we t h a t N_I

space more of

could = A and

to N n _ I. complicated.

semialgebraic

spaces.

In t h e

following

previous

chapters

We

the

study

Problem Find

2.3.

we denote (but n o t

following

We are

a set V 6 ~(D)

retract

Lemma

There

Let F

a set U 6 ~(A)

such that V NA

exists

and

I has

of R b y been

I, as

an i n d e x

a set D 6 ~(M)

= U a n d U is a s t r o n g

complex. loss

(M

a set X 6 ~(D n M o )

retract

:= A ~ U .

(M ~ F) n A = U a n d

without

IV w h e r e

[0,1]

in set).

problem.

given

a strong deformation

patch

Chapter

interval

with

D m U.

deformation

of V!

2.4.

Proof.

the unit

of X.

{N.B.

We have U c D ~F TM

F) ~ U = M ~ A.

Replacing

M,A,D

of g e n e r a l i t y

that

U = A.

that X QA

M o is t h e

and

Thus

by M ~F,

such

(M

(D ~ F ) TM

F,U)

U, D ~ F

= U a n d U is

O-chunk

NA

of M.}

= U. A l s o ,

is a g a i n

a relative

respectively

we assume

118

For

every

together

o 6E(O) t h e r e with

to ~a (cf.

III,

deformation ~M(o).

a

We

1.1

have from

implies

denote set

deformation

or

already G

M(o) U A

that

p

-I

and

map

H

~ W

from

Let

p

The

AxI

: XxI

~ X

The

the

A.

the

such

that

sum

p ~

X

in D N

xI * W ~

A ~ M o be

to W of the

A).

: wU

A ~ X of

factor.

the

diagram

every

so.

W

to

canonical

:= p ( W U

is

W0

strong

all

identifying

p

from a

is

since

first

Do

G O combine

direct

Since

proper

of

: W

p = po ) . Let

restriction

to

G

: M(o)U

= WU

partially

of

§2]).

(i.e.

(X)

X 6 ~(Mo).

projection

theoretic

: WxI

W o 6 ~(~)

retraction

[DK5,

to M °

Moreover

surjective the

a neighbourhood

-I W c ~ o (D N M o ) .

A c X c D D M o.

strongly

strong

retraction

projection

this

exists

Let

We

have

(IV.8.3) p

is

pr I :AxI *A

There

exists

a unique

ix,t)

6 SM(o)xI.

G U pr I (WxI) U (AxI)

~

W U A

~xid I

XxI

commutes. map

The

~ x i d I is

identifying.

Theorem exists F

: VxI

n 6

again

Vn_ I

that

partially the

map

retraction

Problem

2.3

a set

V 6 ~(D)

with

~ V

from

V

H

to

G(x,t)

= x

for

proper

and

strongly

is w e a k l y

from

2.5.

We

n ~

apply

-I

and

Lemma obtain

V n 6 ~ ( D n Mn) , w i t h with

is

surjective,

semialgebraic.

H

is

hence

a strong

X to A.

can

always

V NA

q.e.d.

be

= U and

U such

that

2.4

the

The

solved. a strong

F maps

More

precisely,

deformation

(V N M n ) XI

there

retraction

to V n M n

for

every

(M,M n)

for

o

Proof. all

reason

Thus

deformation

, X

H

strong (n h O ) .

relative

successively

V_I

deformation Let

to

= U and

for

the

complexes

n = -1,0,1,2,...

V n nMn_ I = Vn_ I for

retractions

V denote

patch

union

Hn of

all

: VnXI V n.

n ~ ~ Vn

Then

sets O,

together

from

Vn

V NM n = V n

to for

119

every

n >-I.

T h u s V £ ~(M),

V cD,

and V nA

= U. W e c h o o s e

an i n f i n i t e

sequence

s I = 1>So

> s I > s 2 > ...

_

of p o s i t i v e note

the r e t r a c t i o n

rn,m This

elements

in R

(say,

Hn(-,q)

sn = 2-n-I).

Let

Pn

: V n ~ Vn-1

f r o m V n to V n _ 1. If n > m ,

de-

let

:= Pm+1 ° Pm+2 ° "'" ° Pn

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

f r o m V n to V m.

We p u t

rn, n

:= id V

. We define

a

n map F n

: V n×I

~ V n as

If t 6 [ S k , S k _ I] a n d

follows.

L e t x 6 V n.

I ~ k ~ n,

If t 6 [O,s n]

then Fn(X,t)

:=x.

then -I

Fn(X,t)

F n is a s t r o n g

:= H k ( r n , k ( X ) , (Sk_1-s k)

deformation

retraction

(x,t) 6 V n×I w e h a v e

F n + I (x,t)

not m o v e

in V n. T h u s

map F

the points

: V×I

~ V.

an a d m i s s i b l e strong

This map

covering

deformation

Theorem

2.5

covers

of T h e o r e m

Corollary

2.6.

an o p e n w e a k l y a strong

the

same

semialgebraic subspaces

(U = A)

retract

for

pair

neighbourhood

to a set t h e o r e t i c

since

(Vnln6~)

(IV.3.15.c).

is

F is a q.e.d.

the

following

generalization

of

spaces

then

there

V of A in M such

exists

t h a t A is

of V.

a generalization as

the homotoDy~ H n + I d o e s

f r o m V to A.

is a c l o s e d

semialgebraic

state

every

III.1.1.

(M,A)

deformation

We now can exactly

If

since

the F n fit t o g e t h e r

of V b y c l o s e d

in p a r t i c u l a r

of a p a r t

f r o m V n to U. F o r

= Fn(X,t)

is w e a k l y

retraction

(t-Sk))

locally

of P r o p o s i t i o n semialgebraic

III.1.2.

spaces.

The

proof

is

120

Proposition as

2.7

in C o r o l l a r y

map

g

every

Also

a homotopy

Any map

f

G

: VxI

to a n e i g h b o u r h o o d ) .

: A ~ Z into

some

two extensions

of

~ Z with

Let M,A,V

space

Z extends

f to V t h e n

G o = g, G I = h a n d

be to a

there

G t [ A = f for

t 6 I.

Theorem

needs

2.8.

a strong

Proof.

(Def.

We

If

(M,A)

choose (I,{O})

We

to w e a k l y

retract

a relative

form the direct This

(N,B)

:=

with

Z(N,B)

of N ~ B .

is a c l o s e d

semialgebraic

spaces,

but

this

is t h e

the

decomposition

patch

product

closed

patch

then

(MxO) U (AxI)

is

complex

of t h e s e

of

with

two

(M,A). a unique

relative

We regard patch

patch

complexes

pair

decomposition

the patch

We have

=

Nn =

(MxO)

N(n)

U

o,~

ox]O,1]

(2.2)

(M(n)xO)

III.1.3

n > O,

spaces

(MxI, (MxO) U (AxI))

the notations

By T h e o r e m

patch

any two patches

iff T {o. T h u s

~N(n)

of

:= { o x ] O , 1 ] l o 6 E ( M , A ) }

For

(N,B).

pair

of Mxl.

as a r e l a t i v e

1.d).

together

We use

extends

deformation

the p a i r ]0,1].

III.1.3

a new proof.

Theorem

every

2.6.

of m a p s

: V ~ Z. If g a n d h are

exists

and

(Extension

of

(M,A) (N,B)

for b o t h

we have has

relative

the

Tx]O,1] { ox]O,1] same height

patch

complexes

as o.

(M,A)

= M(n) xI,

U

(3M(n)xI),

(MnxI)

(or a l r e a d y

a strong

of

[DK5,

deformation

Th.

5.1])

retraction

Gn

there

exists,

for

: N(n) xI ~ N(n)

from

121

N(n)

to

~N(n).

the proof

From

of Lemma

f r o m N n to N n _ I. 2.5,

we obtain

these 2.4,

G

we obtain,

n

strong

"Composing"

a strong

in e x a c t l y

deformation

all

t h e Hn,

deformation

the

retractions

as

same way Hn

in t h e p r o o f

retraction

F

: NxI

: NnXI

only

the

fact

as

that

in III,

(MxO) §I,

U

the

(AxI)

semialgebraic

spaces.

Corollary

2.9

extension

of a s p a c e M.

(Homotopy Given

~ Z with

a map

g

extension

theorem).

: M ~ Z into

F ° = g[A there

G o = g and GIAxI

is a r e t r a c t

following

to w e a k l y

: AxI

~ Nn

~ N from q.e.d.

immediately,

F

in

of T h e o r e m

N to B.

Using

as

exists

of M x I w e o b t a i n of C o r .

III.1.4

Let A be a closed

some

space

a homotopy

G

subspace

Z and a homotopy : MxI

~ Z with

= F.

It s e e m s m o r e

difficult

to g e n e r a l i z e

h o w to w e a k l y

semialgebraic

spaces.

the

second

We will

half

of

III,§1

say something

about

somethis

in §4.

We conclude

this

section

sequences

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

we do not

give

by writing extension

the proofs,

run exactly

as

Proposition

2.10

albeit

in t h e t o p o l o g i c a l

(cf.

down

[DKP,

2.9]).

several

theorem

some

2.9.

somewhat

formal

With

exceptions

of t h e m a r e

two

tricky,

since

con-

they

setting.

Let

A

M

f

be a triangle that

-

M'

of m a p s w h i c h

i is a c l o s e d

commutes

embedding,

i.e.

u p to h o m o t o p y

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

(fi = h ) .

of A onto

Assume a closed

122

subspace

o f M.

Then

there

exists

a map

g

: M ~ M'

such

that

g ~ f and

goi=h.

The

proof

is

almost

f~i

to

By

Cor.

h.

F(-,O)

= f and

trivial. 2.9

We

there

F o ( i x i d I)

choose

exists

= H.

a homotopy

a homotopy

The

map

g

H

F

: AxI

: MxI

:= F ( - , I )

~ M'

~ M'

has

the

from

such

that

required

properties.

Definition space from We

4

under ~ to

also

b)

is

call

category

: MxI

~ N

We

then

also

If

there

c)

The

of

are

the

is

say

M

are

maps

a map

usual

sense

whose

objects

classes

a ~omotopy

C is

under

M

over

over

R.

C,

if

there

is

no

consideration,

and

we

write

under

C

B to

with

C

C

is d e n o t e d a map

=

under

C and

from f

f to : M

spaces

is

equivalence

under

C

B is

B(z)

for

we

C-~N

is

we

if

[f]C

is

: a ~

that

foa =B.

which

f

: M

C-~N.

z 6C,t

6 I.

: MxI

C-~N.

H

write

f =

by

C

g.

[f]C

morphisms

A map

f under

isomorphism

in H W S A ( R ) c.

Theorem is

2.11

a map

closed

(Dold

under

C and

embeddings.

(forgetting

C).

Proposition

2.10

about

[Do

that

Assttme

Then

retractions,

, 3.6],

and

f is

the

[Do

2.18]).

structural

further

that

a homotopy

Theorem

cf.

[DKP,

Assume

maps f is

s

equivalence

2.11

imply

the

, 3.5,

3.7],

[DKP,

C

that

f

: M

: C ~ M,

B

: C ~ N

a homotopy under

following 2.27].

B

C

all

whose

an

f

a homotopy

denoted

H W S A ( R ) c.

A

doubt

write

C and

by

such

WSA(R)

g then

under

denoted

by

g :~ ~

H(s(z),t)

a map

the

: M ~ N

R.

A map

f

C of are

a space

a map

H under

under

C be

a space

a hpmotopy

a homotopy class

Let

into

f :~ ~

H is

a)

t__oo N u n d e r

under

and

that

homotopy

called

: C ~ M

from

a,B

the

use).

: C ~ N under

from

homotopy

category

B

s

spaces

H in

later

a map

maps"

exists

The

for

f a map

A homotopy

H

C

C

a space

"structural The

(also

equivalence C.

nice

results

bN

are

123

Corollary denote

2.12.

the

i) A s s u m e

there

of s p a c e s the

5. As

a map

system

closed

in M.

: M ~ Awith

is h o m o t o p i c

equivalence

we m e a n

decreasing

A map

of fo

2.13.

then

of s p a c e s

and

let

roi = i d A . Then

i : A ~ M

there

exists

to r.

A is a s t r o n g

from

fi

: Mi

deformation

of m a p s

re-

Corollary

decreasing

cf.

[BD,

5.13].

2.14

[BD,

5.13].

restrictions equivalences.

hence

closed

: Mi ~ Ni

subspaces

M i + I is a s u b s p a c e if e v e r y

such

Then

M i is

system that

of spaces.

is a h o m o t o p y

off

from

induction

one

Let

easily

[DKP,

p.

on r give

deduces

(No,N I , . . . , N r)

every

fi is a

(No,N I ..... N r) be Assume

that

equivalence.

f(M1)

M 2 ~ N2,

64].

c NI,

f(M2)

the

every Then

equivalence.

arguments in g e n e r a l .

following

between

subspaces

interesting

spaces.

Let

of N such

c N 2. A s s u m e

M I N M 2 ~ N I N N 2 of

f is a h o m o t o p y

The

the p r o o f

f : M ~ N be a m a p

of M and N I , N 2 c l o s e d

N = N I U N2,

M I ~ NI,

of M o. We

equivalence.

can be r e a d

r = I of 2.13

that M = M I U M2,

system

systems

this m a p

is a h o m o t o p y

M I , M 2 be c l o s e d

and

family

No .

and a s t r a i g h t f o r w a r d

corollary,

fi

a finite

M i is a s u b s p a c e

if M i + I c Mi, the

spaces

(Mo,M I,... ,M r ) to a s e c o n d

~ Ni of

r = I a proof

the c a s e

every

of

(fo,fl .... ,fr ) : (Mo,M I ..... M r ) ~

closed

( f o , f l , . . . , f r)

by a s y s t e m

that

We call

: Mo

Let

between

component

[LSA]

(fo'''''fr)

restriction

Theorem

in

I < i < r-1.

is a f a m i l y

there

r

(Mo,M I , . . . , M r ) such

for

From

exists

p : M ~ A which

of M i,

a map

pair

of M.

Definition

call

be a c l o s e d

mapping.

If i is a h o m o t o p y

tract

For

(M,A)

inclusion

a retraction ii)

Let

that

f are h o m o t o p y

the

124

Proposition

2.15

ible

subspace

closed

near M~A.

Then

equivalence. (M/A,A/A)

For

The

of the

one

needs

2.36]).

Assume

s p a c e M and projection

b y 2.13,

is a h o m o t o p y

2.12 w i l l skipped

spaces

modest

be needed

t h a t A is p a r t i a l l y p

: M ~ M/A

also the map

between

that

pxid

: MxI

generalization only much

and

2.16.

Let

~

pairs

p

: (M,A)

(M/A) xI is i d e n t i f y i n g ,

and easy

later

(Mo, .... Mr)

let C be a c l o s e d the

inclusion

valence.

the

system

retract

Then of

cf.

(VII,

consequence

§8)

and

of C o r o l l a r y

t h u s m a y n o w be

The

be a c l o s e d

subspace map

of M o. A s s u m e

the

case

(C,C n M 1 , . . . , C

r = O is t h e there

there

identity

by G(x,t)

exist

exists

Corollary

strong

N M r)

that,

is a s t r o n g

2.12

above.

deformation

a homotopy

:= E ( x , 2 t )

M I x I to M I and g i v e s

We

of

for e v e r y equi-

deformation

study

retractions homotopy

the

D k : MkXI ~M k

:= D o ( E ( x , 1 ) , 2 t - 1 )

deformation

retraction

f r o m M ° t o C.

o f M I to C D M I. F o r r ~ 2

restriction the p r o o f

starts

D I. N o w t h e m a p G : M o X I ~ M o, d e f i n e d

a n d G(x,t)

by

theorem

C which

I ~,

if O ~ t

case

extension

E : M o xI ~ M o r e l a t i v e

of M ° and e x t e n d s

~ t ~ I, is a s t r o n g

t i o n o n r.

system

f r o m C N M k to M k is a h o m o t o p y

f r o m M k to C N M k for k = O a n d k = I. By t h e 2.9)

decreasing

(Mo, .... Mr).

r = I. By 2.12

I

is a h o m o t o p y

equivalence.

k 6 {O,...,r},

with

complete

by the reader.

Proposition

(Cor.

t h a t A is a c o n t r a c t -

IV.8.7.ii.

following

Proof.

[DKP,

the natural

Thus,

the p r o o f

Remark

(cf. e.g.

a strong

runs

b y the

deformation same

if

It m a p s retraction

argument

and

inducq.e.d.

§3 - P a r t i a l l y

We will

prove

space M w h i c h

finite

open c o v e r i n g s

the e x i s t e n c e have

special

of p a r t i a l l y properties,

properties

of open c o v e r i n g s

properties

are i m p o r t a n t

turn out to be useful ing maps

If

but will

(UIlX6A)

finite

is a p a r t i a l l y

family

We have

in ~(M)

to live with

finite

sets,

We choose

a patch d e c o m p o s i t i o n

the patch

complex

3.1.

open c o v e r i n g

Proof.

Let

(M,Z)

in general,

p be a given patch

in general, M is not

of our

also

and coverchapter.

a partially

(UxII6A) 6 C O V M ( M ) .

the UX cannot

locally

be semi-

semialgebraic.

space M. A b u s i v e l y

(StM(a) [o6Z)

we denote

is a p a r t i a l l y

p has only

finitely many

expect

we cannot

of M. Then

then

faces

finite

this.

open w e a k l y

semialgebraic

sets of the c o v e r i n g

In our c o n s t r u c t i o n

Theorem

will,

the

in general,

are contractible.

we will

a partially

complex,

~ is a face q.e.d.

StM(o)

Nevertheless

to the case of a simplicial

that

intersection

(1.7).

the stars

from the p a t c h d e c o m p o s i t i o n ,

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

for any ~ 6 ~, the

iff p C S t M ( o ) , and this means

In general

ly many

certainly

i.e.

These

of the space M.

complex

trast

I

of M,

to common

also by M.

If M is a simplicial

starting

They will

of fibre bundles

covering

then

The family of stars

StM(O) D p is not empty of p. But

open

fact that,

algebraic

Proposition

since,

on M.

of a

spaces.

in the rest of the present

w i t h union M, the

topological

cohomology

in the theory

play no role

open c o v e r i n g s

some of them a n a l o g o u s

of p a r a c o m p a c t

for sheaf

later

finite

construct,

finite

subsets

covering

{but,

intersections

in conof

finite-

not be contractible}.

2.5 from the p r e c e d i n g

section

will

play

126

a crucial

role.

Theorem

3.2.

M.

there

Then

contractible

Proof. that

For

(DIII6A)

exists open

weakly

every

a finite

a £Z(M)

6 A(o)

finite

semialgebraic

we c h o o s e

in the u n i o n

covering

semialgebraic

of ~. This

be an a d m i s s i b l e

a partially

~ is c o n t a i n e d

choose ible

Let

sets

can be d o n e

simultaneously

(Uo,

covering (Uala6J)

sets w h i c h

refines

subset

of the D 1 w i t h

Is6K ~) of the

A(G)

as sets

(DIBI£A).

of A such Then we

~ by o p e n

the c o v e r i n g

o and

the

sets

Uo, ~ o p e n

space

of M by

I 6 A(c).

space

refines

by t r i a n g u l a t i n g taking

of the

covering

a finite

Uo, s w h i c h

and

open

contract-

(o D DIiI6A(o)) o DDI

stars

with

of this

triangulation.

We choose,

for e v e r y

Let J denote

the

Theorem

applied

2.5,

for e v e r y

tractible

the

covers

pairs

a set V

c V uta c StM(o)

finite

family

M since

6A

with

~ £Z(M)

and

(StM(C) ,~),

such

Uo, a c o D D a 6K

there

weakly

exists,

retract

3.1)

~ 6 Z(M)

of V

Since

and the

the

sets

is p a r t i a l l y

is c o v e r e d

~,a

. Since

family

U

~,a

(StM(o) lo6I(M))

K o are

finite

finite.

This

by the

is con-

we

conclude

family

subfamily q.e.d.

As a b y - p r o d u c t

Corollary

. By

that

(Vo,ala6K~).

Proposition

o

1 (o,a)

nDl(o,~)

(V ,am (o,s) 6 J) every

with

pairs

,s 6 ~(M)

deformation

(Prop.

l(J,a)

(c,a)

a l s o Vj, a is c o n t r a c t i b l e .

is p a r t i a l l y that

index

to the c l o s e d

is a s t r o n g

0,5

some

set of all

(0,s) 6 J ,

U@,~ and U

s 6K~,

we obtain

a second

proof

of a p a r t

of

IV.4.16.

3.3.

Every

semialgebraic

locally

semialgebraic

is p a r a c o m p a c t .

space M w h i c h

is also

127

Proof.

There exists,

spaces,

by the very d e f i n i t i o n of locally s e m i a l g e b r a i c

an a d m i s s i b l e covering

(DIlI6A)

of M by open s e m i a l g e b r a i c

sets. By the t h e o r e m this c o v e r i n g has a p a r t i a l l y finite r e f i n e m e n t (Uala6J) w i t h U a 6 ~(M). Since M is locally s e m i a l g e b r a i c the family (Uala6J)

is locally finite in the sense of

c o n t a i n e d in some DI , hence U

[LSA, p. 6]. Every U s is

6 ~(M). Thus we have found a locally

finite c o v e r i n g of M by open semialgebraics, paracompact

(I, §4, Def.

P r o p o s i t i o n 3.4 finite. ~(M)

which proves M to be

2).

q.e.d.

("Shrinking lemma").

Then there exist families

Let

(DIlI6A) 6COVM(M)

(Vlil6A)

in ~(M) and

such that M is the union of all VI and V I c A l

(VlII6A)

is again a p a r t i a l l y finite,

be p a r t i a l l y

(A~II£A)

cDl.

hence admissible,

in

{In particular, open c o v e r i n g

of M.}

This can be proved by use of Zorn's lemma, ing

(e.g.

Claim.

[Q, p. 79f]),

as in the t o p o l o g i c a l sett-

once we have v e r i f i e d the following

Let r be a subset of A and a an index in A ~r.

Assume we are given families

(V~I~6A)

in ~(M) and

Let r'

(A~IX6A)

:= r u{a}.

in ~(M)

such that Vl c A 1 c D 1 for every I 6 r, and

(*)

U(VxlX6F) U U(D

I~£A~F) = M .

Then there exist two sets V

6 ~(M)

and A

6~(M)

and

u(v~izer,) u U ( D I u 6 A ~ r ' )

=

M

.

In order to verify the claim we look at the set

X a := M ~ [ U ( V I I I 6 F )

U U(D

I~£A\F')]

.

such that V

cA

cD

128

This

set X a is c l o s e d

(Vlll6F)

and

and w e a k l y

(D i~6A~F')

are p a r t i a l l y

that X a c D a . By " U r y s o h n ' s algebraic

function ~

semialgebraic

lemma"

finite.

(IV.3.12)

: M ~ I with ~

-I

in M since

the f a m i l i e s

It f o l l o w s

from

there

exists

(O) m M ~ D a a n d ~

-I

(*)

a weakly

semi-

(I) m X a. We

define

V

These

:= ~

sets

Definition (~al~£J) the

]1,1]),

fulfill

I A a := %0-I([3,1] ).

the claim.

I. i) A p a r t i t i o n

of w e a k l y

family

x 6M

-I

semialgebraic

-I (~a (]O,I]) i~6J)

the s u m of the v a l u e s

ii) Let

(Dlil6A)

algebraic (resp.

be a p a r t i a l l y

finite covering

to this c o v e r i n g )

for e v e r y

c A l c D1 w i t h

the c l o s u r e

3.5.

cautious

-I of ~l (]O,1])

Given a partially semialgebraic

(BIII£A)

in w r i t i n g

is I. semi-

to this c o v e r i n g

is a p a r t i t i o n c D~

of u n i t y

(resp.

finite

covering

sets t h e r e

in ~(M)

d o w n the last c o n d i t i o n

is not n e c e s s a r i l y

subordinate

B l c V l c A l c D1 for e v e r y : M ~

J,

some set A1 6 ~ ( M ) ) .

By the last p r o p o s i t i o n

(AIJI6A),

through

for e v e r y

of M by w e a k l y

subordinate

-I i 6 A, ~i (]O,1])

such that

in M and,

a running

subordinate

such that,

: M ~ [O,1]

finite

strictly

(~lil£A) w h i c h is s t r i c t l y

and

with

~a

of u n i t y

by o p e n w e a k l y

Proof.

is p a r t i a l l y ~a(x),

We h a v e b e e n a l i t t l e

Theorem

functions

A partition

-I ~l (]O,1])

since

of a space M is a f a m i l y

sets.

(~lll6A)

N.B.

of u n i t y

exists

to

(Dlll£A)

with

~l

[O,=[ w i t h B l c ¢ ~

T h e n ~ :=

Z ~l is a w e l l d e f i n e d w e a k l y IEA

of u n i t y

(]0,~[)

(Vlil6A)

U(BILI6A)

We c h o o s e w e a k l y

functions

of a space M

a partition

families

respectively

-I

semialgebraic.

(Dlll6A).

there exist

I 6A.

weakly

c V1

in ~(M)

= M and

semialgebraic (cf.

semialgebraic

IV.3.12). function

on M

129

which have

is p o s i t i v e

everywhere.

the p r o p e r t i e s

In C o r o l l a r y

spaces.

One may

space M is locally

position

of M.

relati o n

m a y be very

a good

answer

Proposition

In general loose

if the p a t c h

3.6.

ly s e m i a l g e b r a i c finite.

in D e f i n i t i o n

3.3 we have obtained,

semialgebraic a give n

required

The functions ~l

ask

iff,

Of course,

patch

(cf.

complex.

o of M,

to check

criterion

since

view point.

M is normal

this

on locally that

from a p a t c h decom-

to be d i f f i c u l t

for every patch

it suffices

a result

starting

from a g e o m e t r i c

Let M be a normal

q.e.d.

for a "combinatorial"

seems

complex

and the sets AI

1.ii above,

as a by-product,

semialgebraic, this

:= ~i/~

the face

We can provide

1.3).

The

space M is local-

the complex

StM(d)

for the patches

is

of h e i g h t

zero.

Proof.

If all the stars

a cove r i n g

of M by open

(Prop.

Assume

now that M is locally

i.e.

with

This

implies

patches

hence

admissible.

o < p. If o ~ p then

since

sets.

contains

p nu ~.

U is semialgebraic.

This

(StM(O) 106Z(M)) covering

Thus

is

is locally

semialgebraic.

Let ~ be a p a t c h of M.

o. Let

o < Q, since

and then

then

Thus M is locally

semialgebraic.

some U 6 ~(M) w h i c h

~p N U % ~

complexes

semialgebraic

finite

There exists

3.1),

are finite

p be a patch

the patch

But U meets the c o m p l e x

complex

in StM(O), M is normal.

only f i n i t e l y m a n y StM(O)

is finite. q.e.d.

Sometimes "locally finite. open

it may be a d v i s a b l e finite",

as to be d e f i n e d

It also may be useful,

covering

struct

to work with

with

countable

such c o v e r i n g s

now,

instead

or at least

index

open

coverings

are

of just p a r t i a l l y

comfortable,

set at disposal.

out of a given p a r t i a l l y

which

to have

We now will

finite

an con-

open covering.

130 Definition finite

2. A f a m i l y

(in M)

M such that,

if t h e r e for e v e r y

(XlliEA) exists

of

subsets

of M

an a d m i s s i b l e

s £ I, the

set of all

is c a l l e d

open

covering

~ £A

with

locally (Uals6I)

of

X 1 n U s • ~ is

finite•

Notice

that

every

semialgebraic U a.

If t h e

finite", defined

Lemma

(Xlll£A)

is p a r t i a l l y

finite,

of M is c o n t a i n e d

in the u n i o n

of

s p a c e M is l o c a l l y

"locally

finite", p.

3.7.

Given

a partially

semialgebraic finite

6],

semialgebraic

as d e f i n e d

[LSA,

locally

all m e a n

sets w i t h

covering

We

choose

every

n 6 ~

we define

and

covering

countable

(Wnin6~)

a partition

now,

then

the

since

finitely

terms

"locally

any

many

"partially

finite",

as

the same.

finite

s u c h t h a t W n c U n for e v e r y

Proof• For

subset

family

in

weakly

sets

such

(Unin6~)

index

of M b y o p e n

set t h e r e

of M b y o p e n w e a k l y

exists

a

semialgebraic

n £ ~.

of u n i t y

a new weakly

(untn6~)

subordinate

semialgebraic

to

function

(Unin6~) w n :M

[0,1],

Wn(X)

-

~ ui(x)). i
-I := w n (]O,1]) • W e h a v e W n c U n . W e c l a i m

Let W n locally

Given

:= m a x ( O , U n ( X )

finite

a point

T h e n Wm(X)

covering

s e t s W n is t h e w h o l e

which

means

smallest

index with

t h a t x 6 W m.

Thus

s p a c e M.

o

We define

Nk

an i n c r e a s i n g

:= {x 6 M I

(Wnln61N)

is a

of M.

x in M let m be t h e

= Um(X) > 0 ,

that

family

I X ui(x ) >7} i
(Nklk61N)

in 3~(M) b y

Um(X)

% O.

the u n i o n

of the

•

131

Given

a semialgebraic

such that

subset'X

u I does

of M t h e r e

not vanish

exists

everywhere

a largest

on X. F o r

index

every

x 6X

have

~ u. (x) = I. T h u s X c N I. T h i s p r o v e s t h a t ( N k l k £ ~ ) i k t h e n W n N N k = 9. T h u s is i n d e e d

Theorem

a locally

3.8.

Given

finite

open

covering

an a d m i s s i b l e

a locally

finite

open

there

exists

open

every

connected

component

of every

(cf.

p.

By T h e o r e m

we

is an ad(Wnln6~)

of M.

q.e.d.

covering

covering

1 in

(Uala6I)

(Wnln6~)

of a s p a c e M

of M s u c h

set W n is c o n t a i n e d

that

in U a f o r

some

6 I.

Proof

[tD,

the covering unity that

397]).

(Usta6I)

(usls6I)

subordinate

I is i n f i n i t e .

of

I. F o r

qE

: M ~ R by

every

qE(x)

E 6 A we define

:= m i n u (x) a6E a

:= {x 6 M l q E ( x )

I with

the

assume

there

1 6E~F

x 6 V F,

Every

-

some

we have

the

choose

(Th.

set of f i n i t e

of

assume subsets

function

finite

IFI t O t h e n V E N V F = ~.

x in V E N V F.

Uk(X) < U l ( X )

We may

non empty

semialgebraic

that

u B(x)

IEI =

point

assume

a partition

3.5).

If E and F are d i f f e r e n t

cardinality

exists

We

covering

a weakly

max B6I~E

>O}.

finite.

already

since

Choosing

subsets

of

Indeed,

indices

k 6F ~E

x 6 V E, a n d Ul(X) < U k ( X )

since

a contradiction.

set V E is c o n t a i n e d

for a n y in ~(M).

Vn

is

same

to t h i s

Let A denote

Let V E

and

is p a r t i a l l y

3.2 we m a y

s 6E. This

Thus

-I in D(u a (]O r I ] ) I ~ 6 E ) .

(VEIE£A)

implies

that,

is c e r t a i n l y for every

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

a partially

n 6 ~ , the

finite

V E cU

family

set

:= U ( V E D E 6 A , IEI = n)

(open and)

weakly

semialgebraic,

and

that

the

family

(Vnin6~)

is

132

again

partially

covering the

of V n.

finite. Since

Also

the

sets

s p a c e V n is t h e d i r e c t

every

connected

Given

a point x 6M

finite

component

the

and non empty.

(Vnln6~)

is a p a r t i a l l y

(Wn[n6~)

of M w i t h

requirements

are p a i r w i s e

s u m of t h i s

family

of

of V n is c o n t a i n e d

finite

there

open

exists

of the t h e o r e m ,

in s o m e

s p a c e M,

a locally n.

This

open

disjoint

Thus

certainly

set U s .

x 6V n with

covering

W n c V n for e v e r y

spaces.

of all s 6 I w i t h

hence

s e t s V n is t h e w h o l e

lemma

is an a d m i s s i b l e

family

We have x 6VE,

of t h e

= n)

in t h i s

set E consisting

the union

By the preceding

(VEIE6A,IEI

us(x)

~O

is

IEq = n.

Thus

and we conclude

that

of M.

finite

open

covering

covering

fulfills

the q.e.d.

§4 - A p p r o x i m a t i o n

As b e f o r e

of

spaces

by weak

we consider

spaces

over

s p a c e M is l o c a l l y P in M w h i c h of M w i t h lation

This

is a s t r o n g

respect

of M w i l l

homotopy

result

and

in t h e

spaces

(Chapter

spaces

less

look

real

there

retract

barycentric

§I).

Thus

goal

Theorem

for

the

closed

exists of M.

field

a weak

map

R.

If a

polytope

Indeed,

subdivision

inclusion

III)

of

P a weak

4.1.

the

core

P

of a n y t r i a n g u P ~

M is a

be d o n e

A,B,C,D.

Definition

2. L e t

Z(f)

Z(f)

:=

Here we have

theory

of

to b e v e r y

locally

theory

are n o t

of t h e

proved

[D],

semialgebraic

[DK3],

available.

result

of a s p a c e M

[D2].

We neverthe-

above.

is a h o m o t o p y

equivalence

is t o p r o v e

space M has

by a string We will

a WP-approximation.

of

four

use mapping

lemmas

which

cylinders

we

label

and r e l a t e d

by capital more

compli-

("telescopes").

f : M ~ N be a p a r t i a l l y

of f is t h e

space

(cf.

IV,

p r o p e r map.

The m a p p i n g

§8)

(MxI) UM x l , f N .

identified

For

polytope.

section

Every

§I h a v e

of h o m o l o g y

now triangulations

constructions

cylinder

and also

some generalization

this

in III,

of h o m o t o p y

I. A W P - a p p r o x i m a t i o n

This will letters

refinements

development

: P ~ M with

cated

first

(III,

its

studied

Definition

The

then

deformation

to t h e do

a fixed

equivalence.

useful

the

semialgebraic,

poi/topes

Mxl

w i t h M in t h e

obvious

way

in o r d e r

to

134

glue

Mxl

Mapping

to N by

f.

cylinders

within

the

had

category

been

LSA(R).

mapping

cylinders

for

trouble

(in

sections)

one

in t o p o l o g y .

has

later

If

f is p a r t i a l l y

by

a closed

i

Although

partially

proper

embedding

M

constructed

i and

we

proper

that

then

in

we

we

now

not

§12

are

maps

do

have

II,

for

much

it w i l l have

a homotopy

better cause

them

a canonical

proper

for

maps off

us

having

sometimes

all

maps,

factorization

equivalence

of

as

f

p.

, Z (f)

(4.2) N

Here for

i and x 6M,

spaces

Lemma

of

A.

mations. ing

defined

by

t 6 [0,1],

y 6N.

Z(f),

IV,

Let

f

Given

cf.

As

§8.

: A ~ M be

i(x)

=

(x,O)

usual, We

any

we

=

M×[O,I[

M with

between

a WP-approximation

p(x,t)

regard

identify

map

and

the

spaces

: Q ~ A of

f(x), and

N as

subspace

which

A there

p(y)

have

=y,

sub-

MxO

of

Z(f).

WP-approxi-

exists

a commut-

square

with

Q¢

i

, p

A

f

, M

~ a WP-approximation

phism

of

Proof. have

p are

We

Q onto

start

a square

of M a n d

i a closed

a closed

subspace

with

WP-approximation

any

of

embedding

(i.e.

isomor-

P).

~I

: P

I

M of M.

Then

we

135

Q

g

--~ p I

A

which

~ M

f

commutes

g

up

to

homotopy.

: Q c~-~ Z(g)

of

g via

we

obtain

its

mapping

the

factorization

PI

cylinder

(which

exists

since

Q

is

a weak

polytope)

a square i

Qe

A

-~

Using

, p

--

M

~ f

with

P

:=

Moreover fo@.

We

Then

~

Lemma be

we

and

have

extend

4.3.

Let

which

is

: Z(f)

Proof. M UfN tion

is

a closed map.

~ N,

a homotopy

consider

= Mxl r

t 6 I,y 6N}.

UAxl,fN

the

the is

: L ~ M UfN We

: QxI G

~ M with

: pxI

a WP-approximation F(-,O)

~ M with of

subspace The

again

of

natural

= ~2oi

G(-,O)

M with

~oi

a space

M

and

= ~2 =

and

F(-,I)

(cf.

f-~.

let

of M.

2.9).

q.e.d.

f

: A ~ N

map

~ M UfN,

by

is

F

is

a WP-approximation

proper

UAZ(f)

which

a homotopy

induced

We

map

:= ~ 1 o P ,

a homotopy

A be

a partially

: M

~2

F to

:= G ( - , I )

q

p

Z(g)

now

identity

: M ~ M

and

the

natural

projection

is

evident

that

equivalence.

space

L

:= M x I

a strong being

prove

id M

that

UAxI,fN.

deformation

given also

by the

r[x,t]

It

retract = [x,1],

subspace

of

L,

the

retrac-

r(y)

= y

{x 6 M ,

M UAZ(f)

= MxO

UAxoZ(f)

=

136

of

L is a s t r o n g

q = roj

with

We

a strong

have

(M×O) topy

j

U (A×I), extends

G

from

: M UA Z ( f )

=

retract

~ L the

deformation

cf. to

: L×I

deformation

Theorem

a strong

(M×I×I)

of

L.

inclusion

retraction

2.8.

Then

Since

F

deformation

F

be d o n e

since

map.

: M×IxI

is

~ M×I

constant

of M × I

on A×I,

to

this

homo-

retraction

N×I

UA×Ixl,f×id

we will

~

L

q.e.d.

L to M U A Z ( f )

M / v

r

i~i!!i~!i~}}!i!~!~[~!~!~i~!~!~!~!~i~i~!!~}i~!~i!i~i~i~i{~!!!~!!~!~!~i{~i~i~i~i~i~}i~}i~i~i~!i!~i~!! i~~!~ ii!~!{i~i~i ~~!~!!!~!!!~!!~!~i!~i~!~

0

I

Lemma

B.

proper M UfN

Let

map. has

Proof.

a closed

Assume

that

subspace

M,A,

and

of M

N have

and

f

: A ~ N be

WP-approximations.

a partially Then

also

a WP-approximation.

Let

suffices

A be

f

1

to

i denote find

WP-approximation

the

inclusion

A~

a WP-approximation e

: A ~ A of

A.

M. of

By

the

By L e m m a

the

preceding

space A there

lemma

M UA Z ( f ) . exist

We

it

choose

commuting

squares

AC

with

i

, M

WP-approximations

A

@,X

and

closed

f

, N

embeddings

i,f.

(We w i l l

only

a

137

need

that

: Z(f)

i is a c l o s e d ~ Z(f).

P

,

z (f)

P

, N

natural

projections

P,P,X

are h o m o t o p y

Since

The m a p s

~ and ~ c o m b i n e

: M UiZ(f)

We

conclude

This

following

and u s e chunks

Lemma

Cor.

induce

a map

p commutes.

equivalences

also

¢ is a h o m o t o p y

equivalence.

to a m a p

that

~ is a g a i n

WP-approximation

we

choose 2.2.

a patch

a homotopy

equivalence.

of M U A Z ( f ) .

q.e.d.

decomposition

In p a r t i c u l a r

of a g i v e n

(Mnln£~)

is the

space

family

M of

of M.

C. E v e r y

spaces

this

and

by g l u i n g ~n

space

M(n)

has

a WP-approximation.

on n. M o is a d i r e c t

a WP-approximation.

to M n _ I a l o n g ~ Mn_ I

of s e m i a l g e b r a i c hypothesis

(cf.

the

§2).

spaces

and

a l s o M n _ I has

B t h a t M n has

The p r o o f

M n has

by i n d u c t i o n

thus

: ~M(n)

duction Lemma

p and

2.14

the n o t a t i o n s

We p r o v e

sums

X together

~ M UAZ(f)

is the d e s i r e d

In the

map

from

<0 and

diagram

z (3)

o;, with

The

embedding.)

of T h e o r e m

Let

subspace The

spaces

thus

have

sum of s e m i a l g e b r a i c

n >O.

~M(n) M(n)

T h e n M n is o b t a i n e d

by a p a r t i a l l y and

DM(n)

are d i r e c t

WP-approximations.

a WP-approximation.

We

proper

By in-

conclude

from

a WP-approximation.

4.1

will

be c o m p l e t e d

if we v e r i f y

the

following

138

Lemma

D. L e t ( M n l n 6 ~ )

every M n has

This

be an a d m i s s i b l e

a WP-approximation

lemma will

follow

filtration

then M has

from Lemma

A and

of a s p a c e M.

If

a WP-approximation.

some

elementary

facts

about

"telescopes" .

Definition

3. L e t ~

: AI

The

A 2

al

telescope

Tel(~)

(Teln(~) I n 6 ~ )

be an

which

a2

infinite

, A 3

is the

~

sequence

of p a r t i a l l y

proper

maps

...

inductive

is d e f i n e d

limit

of a f a m i l y

inductively

as f o l l o w s .

of

spaces

Tel I (0%) := Z(~I) T e l n(0f.) := T e l n _ I (O%) U A

Z(a n )

(n > I)

n Here we

regard

the o b v i o u s

Teln(~)

as a c l o s e d

subspace

of T e l m ( ~ )

for n ~ m

in

way.

A4 A3 A2

Sketch

This

inductive

limit

space

of Tel(0&)

space

T e l ((~) .

The ~in

telescope the

and

Tel(~)

category

exists

by

of Tel 3(~).

IV,

(Teln(0&) in6lN)

comes

HWSA(R).

close

§7.

Every

Teln(~)

is an ad/nissible

to an i n d u c t i v e

Indeed,

we have

is a c l o s e d filtration

limit

obvious

of t h e

inclusion

sub-

of t h e

sequence maps

139

Jn

: An ~ Tel(~).

~n

: An ~ N into

is e a s y But

to

find

a fixed a map ~

the h o m o t o p y

[~n ] since

the

Clearly

class

of the

Lemma

4.4.

space

[~]

~

Given

a family

that

uniquely

~Ojn ~n

determined

Thus

Tel(~)

is not

n 6 ~

it

for e v e r y by the

of ~ depends on the c h o i c e n.

of m a p s

~n = ~ n + 1 °an for e v e r y

~ N such

is not

for e v e r y

sequence

N with

: Tel(~)

construction

f r o m ~n to ~ n + i O S n limit

Jn ~: Jn+I°an"

n.

classes

of h o m o t o p i e s

a genuine

inductive

in HWSA(R).

Let ~2

AI

, A2

A3 <03

BI

BI

be a c o m m u t i n g phism

from

Assume

a i and

Tel(~)

that

This

can be p r o v e d

[Pu,

p.

N.B.

The p r e s e n t a t i o n

314,

function e

similar

to

as

4.4

of t e l e s c o p e s .

~n

~

We r e g a r d

to the

is a h o m o t o p y Then

the

this

lower

horizontal

equivalence

induced

as a m o r -

map

¢

and

that

: Tel(~)

equivalence.

topological

setting,

cf.

[BD,

p.

59ff]

or

7].

in

This

holds

We do not

purpose

spaces.

sequence

proper.

in the

= exp(-t)

-~]O,I].

(loc.cit.).

For our

Hrs.

e(t)

: [0,~[

every

a homotopy

B3

between

horizontal

Bj are p a r t i a l l y

is a g a i n

•

B2

ladder of m a p s

the u p p e r

sequence~. the

' B2

[BD] there does

is v e r y on p. not

in this

which

this m o r e

section

We h a v e

to r e p l a c e

59 by a s e m i a l g e b r a i c

destroy

for a l a d d e r need

lucid.

the only

general

it s u f f i c e s

argument. commutes fact

the

isomorphism

A statement up to h o m o t o p y

at p r e s e n t .

to c o n s i d e r

a special

kind

140

Definition (MnLn6~)

M]

with

Let

inclusion

M is d e f i n e d

the

maps

[P,

§7],

[L,

limit

in the

as

follows.

of Tel 3 a b o v e

The projection

of M x I

p~

Lemma

p ~ is a h o m o t o p y

follows

of M x I

procedure

We now to

: Tel(~)

f r o m the

, which

start

out

fact

Theorem

filtration.

tative

ladder

This

which

has

as u n d e r l y i n g

real valuation

ring

is a

set the

~of

R

(cf.

to p r o v e

Using

as a c l o s e d Teln(~%)

of t h e s e

can

as t h e

factor

of M x I

closed

.

subspace

U (Mn+IX{n})

subspaces.

n o w be d r a w n

first

subspace

(The u p p e r

lines

horizontally.)

restricts

to a n a t u r a l

equivalence.

that

2.8,

find a WP-approximation

ible

[O,n].

~ M.

c a n be p r o v e d

using

intervals

U ... U ( M n x [ n - l , n ] )

to the

projection

This

of the

Tel(m)

as the u n i o n

in the p i c t u r e

4.5.

sequence

We regard

U (M2x[1,2])

Tel(m)

of the

§5]).

This may be done

and

telescope

smallest

to v i s u a l i z e

of M x I

=

as t h e

polytope

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

(Mix[O,1])

filtration ~

in •

weak

elements

of an a d m i s s i b l e

.....

inductive

semialgebraic

non negative [KW],

Tel(k)

<-----~ M 2 • i2 , M 3 ~ ii

denote

locally

telescope

of a s p a c e

the

I

4. T h e

Tel(~)

by a r a t h e r

obvious

cf.

63f].

Lemma

[BD,

p.

D. By t h e

of the

Lemma

is a s t r o n g

lemma

telescope

A repeatedly

deformation

retract

inductive

just

Tel(~)

we obtain

proved

it s u f f i c e s

of the g i v e n an i n f i n i t e

admisscommu-

141 iI ~I

i2

•

' ~2 •

' ~3 ~

. . . . . .

t MI •

~ M2¢

iI

with WP-approximations

~n and closed

upper

be r e g a r d e d

weak

sequence. polytope

the ~n

M.

By Lemma

is a h o m o t o p y

Thus we have again

It m a y

Lemma

found 4.5

maps

~n

is a l s o

This

finishes

Looking

a homotopy

at L e m m a

that

: Tel(~)

Clearly

the

inductive

A we draw

hence

Let ~

denote

filtration

~ Tel(m)

Tel(~)

of T e l ~ )

equivalence,

of L e m m a

in"

as an a d m i s s i b l e

a WP-approximation

it is c l e a r

.....

embeddings

the m a p ¢

equivalence.

the proof

again

4.4

,

~ M3 •

i2

the

of a

induced

by

is a w e a k

polytope.

as d e s i r e d .

(Using

limit ~

: M ~ M of t h e

a WP-approximation

D a n d of T h e o r e m

4.1.

from Theorem

the

4.1

of M.)

following

con-

sequence

Corollary tion ~

4.6.

: P * M there i

P(

M

with

Given

a map exists

,

f

4.7.

space M over

and

a WP-approxima-

square

N

Assume R,

that

and

the

the partially

is a h o m o t o p y

equivalence.

Thus,

case,

in t h i s

a commuting

spaces

~ O

~ a WP-approximation

Theorem

f : M * N between

every

i a closed

field

R is s e q u e n t i a l .

complete

space

has

embedding.

core PM

a canonical

Then,

: P(M)

~ M

for e v e r y (cf.

IV, §9)

WP-approximation.

142

In o r d e r f,g

to p r o v e

: X ~ Y are

this

maps

topy

from

P(F)

is a h o m o t o p y

is

f to g,

a homotopy

we

start

between

then

we

from

spaces

have

P(f)

equivalence,

with

a very over

P(X×I)

R,

and

F

= P(X)xI

to P(g).

then

general

P(h)

We

: XxI

(cf.

conclude

: P(X)

observation. ~ Y

is a h o m o -

(IV.9.13.i),

that,

~ P(Y)

If

and

if h : X ~ Y

is a g a i n

a homotopy

equivalence.

Let

n o w M be

there Then

exists P(~)

Since

(cf.

is a h o m o t o p y

equivalence

from

maps

in

gluing

and

proof

between

closed

that

A'

be

Let

f

f,

of

are

field

R.

: Q ~ M with

homotopy

Q to

By Theorem Q a weak

P(M)

and

equivalences

of

Lemma

B yields

spaces

which

will

of

polytope.

PMOP(~)

also

4.1

PM

= ~. is a

also

turn

out

a very to

general

be u s e f u l

homotopy

later

on

4.11).

(~0,~)

pairs

: (M,A)

of

~

spaces.

and

¢

(M',A') (N.B.

By

be

a homotopy

2.13

: A ~ A'

are

with

f and

partially

Then

the map

equivalence

it s u f f i c e s

homotopy

to k n o w

equivalences.)

for

Let

J N

, N'

square

equivalence.

:M UfN

induced

P(~)

proof

<0 : M ~ M'

a commuting

homotopy

~ and

the

4.8.

A

sequential ~

Theorem

this

the

equivalence.

on

§7

over

equivalence

method

result

space

a homotopy

both

homotopy

The

any

~ M'

f'

proper

Uf, N'

b y <0,~, X is a g a i n

a homotopy

equivalence.

and

(~ and)

X a

143

In o r d e r

to p r o v e

a n d M' UA, Z(f') then

applies

Corollary

this

one replaces

respectively.

Corollary

4.9.

Let

M Of N a n d M' U f , N '

This

is j u s t i f i e d

b y M 0A z ( f )

by Lemma

4.3.

One

2.14.

(M,A)

be a c l o s e d

pair

of

spaces

into

a space

(cf.

§2,

with

A a weak

polytope. i)

If f,g

homotopy ii) then M

: A ~ N are h o m o t o p i c equivalent

to M U N u n d e r g

If f : A ~ N is a n y m a p the m a p

u

The

second of i)

is a w e a k

polytope,

let F

with

arrows

the

space

same

M×I

are

the

the

equivalence

identity

map

of

case

of T h e o r e m from

4.8.

f to g.

In the

Since

f,g,F

are

partially

proper.

(x,O)

and

j

The

A is

Consider

, N

id N

injection

homotopy

Thus,

equivalent

is h o m o t o p y

:= iLA.

by Theorem to M U f N

eauivalent

under

to M U g N

N.

under

2.11,

For N.

verti-

the

This q.e.d.

Def.

will

turn out

three

4.8 a n d

claim

WP-approximations

These

x ~

equivalences.

is h o m o t o p y

(MxI) UF N first

N

F

We now discuss 5).

by h and

a special

three maps

f

the

(MxI) U F N

reason

proves

all

AxI

i : M ~

induced

is

4).

: N ~ L is a h o m o t o p y

~ N be a homotopy

F

<

cal

is just

: AxI

A

M×I

Def.

then M UfN

diagram

M *

F

N

N,

equivalence.

claim

situation

the commuting

and h

: M U f N ~ M Uho f L

is aaain a h o m o t o p y

Proof.

maps

of d e c r e a s i n g

to b e

important

systems

of

in t h e n e x t

spaces chapter.

(§2,

144

Definition a finite Po for

5.

i) A s y s t e m

family

(Po,PI,...,Pr)

I < i < r. W P - s y s t e m s

spaces.

In the c a s e s

of w e a k

polytopes,

ii)

of w e a k

polytopes,

of w e a k

form

polytopes

a special

r = I, r = 2,...

or W P - p a i r s ,

A WP-approximation

or W P - s y s t e m

we

class speak

WP-triples,

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

with

of c l o s e d

...,

for

of

short,

Pi a s u b s p a c e

of pairs,

system

for

systems

is of

of

triples,

...

short.

spaces

( M o , M I , . . . , M r)

is a m a p

: (Po,Pl .... ,Pr ) ~ with

(Po,...,Pr)

of ~ a h o m o t o p y

If the that

system

(Mo,M I ..... M r )

a decreasing

WP-system

and

every

component

~i : Pi ~ M.1

equivalence.

( M o , . . . , M r)

is c l o s e d

any W P - a p p r o x i m a t i o n

of

then

( M o , . . . , M r)

it f o l l o w s

from

is a h o m o t o p y

Theorem

2.13

equivalence

of

systems.

Theorem

4.10.

i) E v e r y

WP-approximation then ii)

: (Po,...,Pr)

Pr can be c h o s e n If ~

exists

topy,

such

first

applied

a map

that

part

several

~of

of the times.

theorem The

We w i l l

to say m o r e

of §6.

WP-approximations".

( M o , . . . , M r)

(Mo,...,Mr).

map

about

has

a

space. WP-approximation

(Qo ..... Qr ) , u n i q u e

f is a h o m o t o p y

follows

second

theorem

~

of s p a c e s

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

(M ° ..... M r ) is a s e c o n d

~ ~. T h i s

extension

at the end

~

f : (Po .... ,Pr)

the h o m o t o p y

have

system

as a o n e - p o i n t

: (Qo ..... Qr ) ~

there

The

~

decreasing

part

2.9 and

4.1

can be p r o v e d

in the n e x t

section

These

are d e f i n e d

as

and Cor.

easily

4.6,

by use

of

2.13.

WP-approximations

Already

up to h o m o -

equivalence.

from Theorem

Theorem

then

of d e c r e a s i n g

we w i l l

follows.

need

systems

"relative

145

Definition

6. L e t

approximation (P,j (A))

of

(M,A)

be a c l o s e d

(M,A)

a relative

consists

weak

pair

of s p a c e s .

of a c l o s e d

polytope

and

A relative

embedding

a commuting

j : At

WPp with

triangle

A

P

~ M

~0

with

i

the

that

then,

Identifying

inclusion

from

by

theorem

Dold's

A with

j (A) w e

approximation

of

between

such that

pairs

identity

Theorem

(M,A)

2.33,

~0 a h o m o t o p y

~

is

alternatively

as a h o m o t o p y (P,A)

4.11.

equivalence.

a homotopy

may

is a r e l a t i v e

equivalence

regard

equivalence weak

Notice

~

a relative : (P,A)

polytope

~ and

under

A.

WP(M,A) ~ is the

Let

Q r

k

~ P

Ar

i

• M

square with

of M a n d A r e s p e c t i v e l y . (M,A),

el.

4.6.}

: (P U x A , A )

induced

M and

on A.

be a c o m m u t i n g

pair

A to

by the

P "

k

~

k a closed

{N.B.

Then

Such

=

(M,A)

diagram

A

~Q

id A M -

i

~A

, A

id A

a square

the map

(M U A A , A )

commuting

embedding

and

exists

X,~ W P - a p p r o x i m a t i o n s for e v e r y

closed

146

is a r e l a t i v e

Proof.

WP-approximation

It is o b v i o u s

it f o l l o w s

that

from Theorem

of

(M,A).

(P U X A , A )

4.8

is a r e l a t i v e

that ~

: P U

X

weak

polytope,

A ~ M is a h o m o t o p y

and

equi-

valence.

Example pair

4.12.

(M,A)

Assume

that

the

of s p a c e s

over

R the m a p

P(M,A)

(cf.

IV,

=

§10)

(p~,id)

theorem

is e s p e c i a l l y

nice

dimension

Also

the

as

Theorem

4.13.

: Pf(M)

Proof. We know

Let

will

semialgebraic

:= Pf(M).

that

, Pf(M)

closed

(M,A),

relative

and

as

is c l e a r

by

WP-approximation

(pA(M) ,A) has

the

same

that map

pf

By

4.7

the

field

over

IV.10.20 that

in IV,

§10 gives

us a h o m o t o p y

later.

f is a h o m o t o p y

from Theorem

and conclude

f~

be u s e f u l

Assunte a g a i n

~ M of

L

of

This

it is c a n o n i c a l

construction

which

be a n y w e a k l y pf

4.7.

every

(M,A)

WP-approximation

and T h e o r e m

since

~

For

(M,A).

other

equivalence

R is s e q u e n t i a l .

: (pA(M),A)

is a r e l a t i v e

the p r e c e d i n g

field

The

partially

Let

proper

f :M ~ N

core

equivalence.

we have

P M and

is a h o m o t o p y

R.

R is s e q u e n t i a l .

PL a r e

P(L)

= P(M)

homotopy

equivalence,

and Pf"PL =PM-

equivalences, q.e.d.

§5 - T h e t w o m a i n

If M is a space, subset

(n ~ I )

also the

(before

If

Theorem with

~o(M,x)

groups

III.6.3)

their

a n d A is a w e a k l y

homotopy

and

and

the

in f o r c e

homotopy

,n(M,A,x)

in t h e

semialgebraic stated

semialgebraic

absolute

groups

~I(M,A,x)

sets

remain

there

(n ~ 2 )

same way

spaces.

groups

as h a s

The more

on pp.

in the p r e s e n t

and

formal

265-270 setting,

to-

proofs.

is an e x h a u s t i o n

x then

sets

then we define

§6 for l o c a l l y

of t h e s e

(M la6I)

point

sets

in III,

properties

x,

of M,

and the relative

pointed

been done

on homotopy

x is a p o i n t

of M c o n t a i n i n g

~n(M,x)

gether

theorems

of M s u c h

that

every M s contains

the

clearly

~n(M,x) = lira ~n(M~,x) ~n(M,A,x)

From this rem

fact

= lira ~ n ( M

it is e v i d e n t

More

also

the

groups

(III.6.3,

6.4)

remain

true

for w e a k l y

generally,

the m a i n

theorems

on homotopy

sets

III.4.2

to w e a k l y

this,

later

and also

"composition"

Lemma

used

5.1.

for

Let

use,

ofinfinitelymany

in t h e

of a s p a c e M. such that

first

and

second main

theo-

semi-

spaces.

c a n be g e n e r a l i z e d

been

,x) .

that

for h o m o t o p y

algebraic

,A N M

proof

Gn(-,1)

we make

spaces.

explicit

homotopies.

In o r d e r

an e a s y

(This

to p r o v e

lemma

lemma has

III.5.1

on t h e

already

of I I I . 4 . 2 . )

(Cnbn£~) Assume

semialgebraic

and

that

be an a d m i s s i b l e (G n

= Gn+1(-,O)

s o = O < s I < s 2 < ...

: MxI

filtration

~ N1n6~)

(cf.

is a f a m i l y

a n d G n is c o n s t a n t

o n C n.

§2,

Def.

3)

of h o m o t o p i e s

Let

148

be an i n f i n i t e a homotopy

sequence

= Gk+ I (x, (Sk+1-s k)

-I

in

[0,1[.

Then there

and

= Gn(X,O)

(x,t) £ C n X [ S n _ 1 , 1 ] .

Proof.

Using

these

formulas

we o b t a i n

a well defined

homotopy

F n : C nxI ~ N f r o m G I (-,0)JC n to G n ( - , O ) L C n. We h a v e F n L C n _ i X I since Gn_ I is c o n s t a n t homotopy

on Cn_ I. Thus the F n fit t o g e t h e r

I. We call F the c o m p o s i t e

(GnJn£~) , alon~

the s e q u e n c e

( M , A I , . . . , A r) and

with every A i closed

to the d e s i r e d

about relative

5.2.

containing

i)

of the

f a m i l y of h o m o t o p i e s

( S n l n 6 ~ o)

(N,BI,...,Br)

be two s y s t e m s

of s p a c e s o v e r R

in M. We fix a m a p h : C ~ N on a c l o s e d

s p a c e C of M w i t h h(C NAi) c B i for

Theorem

= Fn-1

F.

Definition

Let

exists

(t-Sk))

(x,t) 6 C n X [ S k , S k + 1 ] , 0 < k < n - 2 ,

F(x,t)

if

increasing

F : MxI ~ N such that

F(x,t) if

strictly

homotopy

I < i
sets e s t a b l i s h e d

(First m a i n

theorem).

sub-

We use the n o t a t i o n s

in III,

§4 and III,

Let S be a real c l o s e d

§5.

field

R. T h e n the n a t u r a l m a p

< : [ (M,A I ..... Ar ) , (N,B I ..... Br ) ]h ~ [ (M,A I ..... Ar ) , (N,B I ..... Br ) ]h(s ) is b i j e c t i v e . ii)

(Second m a i n t h e o r e m ) .

Assume

that R = ]!R. T h e n the n a t u r a l m a p

h I : [(M,AI, .... Ar), (N,BI, .... Br)] h ~ [ (M,AI, .... Ar), ( N , B 1 , . . . , B r ) ] t o p

149

is b i j e c t i v e .

The two claims Theorem

As

can be proved

III.4.2

in III,

and

§4 w e

III.5.1

see t h a t

a n d as t h e r e

we retreat

theorem

We

and we

2.9. look

We choose dard

a relative

notations

2.2

course) • We want every

n ~-I,

n ~0,

such that

ed t h i s w e

are done.

Composing

the

the

sequence

indices

(the l e t t e r

are

Hn(-'O)

We have

shifted

C(S)

In o r d e r

to construct

with

h_1

= h and

for

i
f-1

a map

in

b y -I)

q0n

M (n)

n-1

~ M

~n

n

fn

~ N(S)

= fn"

[O,I[ R w i t h we obtain

the

replaced

s_1

~ N(S),

: M(S) relative

stan-

b y C, of

~ N(S)

for

Mn_1(S)

hnlMn_ I = hn-1' Once we have

g : M ~ N with

proceed

= f' of c o u r s e .

(*)

: M n ~ N,

Hn(-'1)

h S : C(S)

and we use

being

of h o m o t o p i e s

hn,fn,H n we

~ M

(M,C)

= h,

extension

gs = f rel. C(S).

A there

h_1

of <,

f-1

for = f'

accomplish-

g l M n = h n for e v e r y

along = 0

a homotopy

some

strictly

(cf. L e m m a G

in-

5.1,

: M(S)XI(S)

~ N(S)

as d e s i r e d .

the pushout

~ M (n)

holds:

(Hnin ~ O )

f r o m f t o gS'

hn

i).

surjectivity

extending

h with

from

of

the homotopy

~ N(S)

of

starting

the proof

the

using

: M(S)xI(S)

= fn-1'

(snln ~ - I )

relative

We use

Hn

following

family

r =0

f : M(S)

maps

give

to prove

decomposition

§2

same way

We

: M ~ N extending

and a homotopy the

case

a map

to construct

(hn)s'

creasing

it s u f f i c e s

patch

from

f n l M n (S) =

n.

g

the

respectively.

t o the

are given

for a m a p

in e x a c t l y

diagram

by

induction

on n. W e

start

Assume

that

the hi,fi,H i are

over

(cf.

2.2)

R

given

150

By base S. We

field

extension,

introduce

:= hn_1O%0n

un

:=

that

: 3M(n)

extending

k n together

By T h e o r e m with

: M(n)(S) x I(S)

relative

(~) the m a p s

: M(n) (S) ~ N(S)

interval

By this

I(S)

diagram

Mn_I(S)xI(S) relative

to N(S)

(Vn) S. By the

Hn(-,O)

Hn to a h o m o t o p y

We put

fn

We are But

:= H n ( - ' 1 ) "

with

Remark

5.3.

Hn

This

closed

above

pushout

to a m a p the

a pushout

the m a p

vn

: M(n)

= fn_11Mn(S)

Hn

hn

~ N

diagram

(*)S w i t h

(cf.

and Hn(-,1)

induction

from ~ N(S)

=

(hn) S. We

Hn(-,O) step

the

IV.8.7.ii).

(hn_1)s(X)

with

of the

: M n ~ N with

: Mn(S)xI(S)

~ N(S)

the

property

diagram

(x,t) ~

to a h o m o t o p y

finishes

(as a l r e a d y

and

= fn-1 the w h o l e

5.2 m a i n l y in III,

in the

§4-§5)

"absolute

case"

it is n e c e s s a r y

C = ~.

to w o r k

homotopies.

Theorem

5.2

remains

(M, (AIlI6A)), (N, (BILI6A)) A

again

: M(S)xI(S)

in T h e o r e m

proof

relative

a map

sum of semi-

5.2.i.

interested

for the

exists

"Multiplying"

F n and

combine

with

of T h e o r e m

there

is a d i r e c t

--~ N(S)

we o b t a i n

extend

proof

M(n)

v n and hn_ I c o m b i n e

the m a p

Mn_I(S)

space

III.4.2

hnO% n = v n and h n l M n _ I = hn_ I. unit

(*)S o v e r

diagram

a homotopy

~M(n) (S) f r o m u n to

diagram

(*) a p u s h o u t

,

(kn) S. The

u n extends

spaces.

n

~ N

(fn_11Mn(S))O(~n) S

algebraic

F

from

the m a p s

kn

Notice

we o b t a i n

of

true

spaces

in M,

of course.

This

with more

notational

effort.

for

locally

instead

of

can be p r o v e d

finite

finite

systems

ones,

in a s i m i l a r

with way

as

all

151

It will

be m o r e d i f f i c u l t

fer p r i n c i p l e s

here

given by the main

do not know whether a given isomorphic

to

Recall field

the h o m o t o p y

theory

in WSA(R)

is the

in a unique way "prime

field"

since we R is

system of spaces

algebraic more

sets,

(M,AI,...,A r) over

for some

R o of real

III to apply the trans-

on h o m o t o p y

s y s t e m of spaces

field

that R ° embeds and thus

in C h a p t e r

theorems

(N(R),BI(R) .... ,Br(R))

(N,BI,...,B r) over the will make

than

numbers. ~) This

laborious

into any o t h e r

in real

algebraic

than

in LSA(R).

real closed geometry.

§6 - C o m p r e s s i o n s

In t h i s

section known

Whitehead.

For

the

main

theorem

Let

say t h a t map g F

and

on homotopy

I. L e t

further

steps,

pA(M)

f

theorems

"compression which

(steps b)-d)

spaces

Definition

some

classical

some

of g e n e r a l i t y

n-equivalences

we present

ly b y w e l l

need

and

are

in the Pf(M),

sets

: (M,A)

L be a weakly

which

arguments"

important

proof

~

(N,B)

semialgebraic

subset

g(M)

: (MxI,AxI)

~

(N,B)

from

f to g a c o m p r e s s i o n

f : (Ek,s k-l) usual,

Def.

polytope sion at

This

(M,A)

6.1.

a n d that, l e a s t m.

simpler

complexes. to c o m p r e s s with

of

spaces

then

that

right

and

level

we will

the

first

(M,A)

ball

of

spaces.

of N c o n t a i n i n g

call

relative

B. W e A to a

a homotopy

of f t o L. W e

n-connected

say t h a t

S k-1

relative

number

if e v e r y m a p

is c o m p r e s s i b l e .

in R k and

be a s p e c i a l

patch

Here,

as

its b o u n d a r y .

complex

(cf.

of

spaces

(M,A)

is a r e l a t i v e

weak

for a f i x e d m 6 ~ ,

every

patch

~ 6Z(M,A)

has

dimen-

argument,

which

(M,A)

pair

pairs

n = 0 o r n is a n a t u r a l

and k ~ n

unit

the

the

to J . H . C .

to B.

either

k 6 N °

(M,A)

Then

is

by a very

than

in t h e

W e are g i v e n f to A.

i.e.

closed

Let

I). A s s u m e

U~,

with

the

c a n be p r o v e d

is e v e n

(M,A)

a pair

E k denotes

Proposition §2,

~

c L. W e

if f c a n be c o m p r e s s e d

We call

§10,

if f is h o m o t o p i c

with

o r n = ~.

due

6.8 b e l o w ) ,

be a m a p b e t w e e n

(N,B)

o

essential-

5.2.i.

~

2. L e t n 6 ~

to a t t a i n

in IV,

: (M,A)

Definition

mostly

of T h e o r e m

introduced

f c a n be c__ompressed to L,

f is c o m p r e s s i b l e

c a n be p r o v e d

f(E k) c M'.

basic

"cell

topological

a map

There

(m-1)-connected.

f

exists

Replacing

by cell" setting,

: (Ek,s k-l) a finite M b y M'

~

say,

(M,A)

closed we may

for

with

simplicial k <m.

subcomplex assume

here

that

We have

(M',A)

of

the r e l a t i v e

153

complex

(M,A)

(~,~)

is finite.

~ (Ed,sd-1),

a maximal patch.

patch

Then

image

Thus

(M ~ o , A )

of

M ~

Proposition

The n a t u r a l

b)

~k(M,A,x)

Of c o u r s e ,

now

by

choose

of a n o t h e r

Since

is not

contained

retract

of M ~ { p } ,

follows

A. We

a face

(M,A).

deformation

retract

proposition

is not

of

then

relative

p 6 ~ which

is a s t r o n g

d,

and

of ~ ~ { p } .

f can

induction

in

be com-

on the n u m b e r

A pair

can be e x p r e s s e d

(resp.

of

sets

spaces

in t e r m s

of r e l a t i v e

if k = I).

(M,A)

is n - c o n n e c t e d

iff the

follow-

hold.

map

no(A)

~ No(M)

= O for e v e r y

of A s i n c e

x 6A

is s u r j e c t i v e . and e v e r y

to d e m a n d

the

natural

(b) for one

fundamental

number

point

groupoid

k ~n.

x in e a c h

7-[(A)

operates

connected on the

(~k(M,A,x) Ix6A).

6.2

is an

immediate

consequence

of the

following

more

gene-

lemma.

Lemma

6.3

be a m a p Assume

This

o which

a point

deformation

it s u f f i c e s

Proposition

for

~

~k(M,A,x)

6.2.

a)

ral

exists

complete

subcomplex

n-connectedness

two p r o p e r t i e s

system

a patch

of d i m e n s i o n

in M ~ A .

groups

component

i.e.

set

The

generally,

homotopy

ing

there

f. The

to M ~ .

of p a t c h e s

M is p a r t i a l l y

is a c l o s e d

is a s t r o n g

pressed

Quite

since

in M \ A ,

d i m f(E k) < d i m ~ the

If ~ is a p a t c h

(Compression from

that

some

a closed

lemma, pair

f is h o m o t o p i c

L 6 ~(N)

containing

c a n be p r o v e d ,

as

of

cf.

[W, p.

spaces

to a m a p B. T h e n

in the

g

7Of]).

(M,A)

Let

f : (M,A)

to a p a i r

: (M,A)

~

(N,B)

of s p a c e s with

f can be c o m p r e s s e d

topological

theory,

by

~

g(M)

(N,B) (N,B). c L

to L.

applying

the

154

homotopy cf.

extension

[W, p.

Having

theorem

the

preliminaries

we h e a d

for our m a i n

ploited

afterwards.

result

Definition

3. The

dimension

dimensions

dim X a

of the

of the

of X. We h a v e

chosen

dim(M,A)

Notice

and

(MxI, (AxI) U (Mx~I)),

on c o m p r e s s i o n

dim X

of a space

semialgebraic

of X.

of maps,

If

X~

Of course,

(M,A)

which

X is the

spaces

U~.

and n - c o n n e c t e d n e s s

is a p a i r

will

be ex-

supremum

of the

for some dim X

exhaustion

is i n d e p e n d e n t

of s p a c e s

t h e n we

set

:= d i m ( M ~ A ) .

that,

Theorem

6.4

a relative of

Example polytope retract

We

(M,A)

on c o m p r e s s i b i l i t y

dim X 6 ~o

exhaustion

if

semialgebraic

pair

to the p a i r s

71].

finished

(X la6J)

2.9

start

(Ylil6A)

sets

I n6~N o)

IV,

(Compression weak

spaces.

6.5

Then

every

is a l s o

§3,

Def.

7),

map

If

covering

then

Let d 6 ~

of d i m e n s i o n

(f = i d ( M , A ) ) .

which

admissible

theorem).

polytope

d,

and

f : (M,A)

(M,A)

d-connected,

o

~

dim X

of X by w e a k l y = sup(dim ylll6A).

U~.

Assume

that

(N,B)

(N,B)

that

(M,A)

is

is a d - c o n n e c t e d

is c o m p r e s s i b l e .

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

relative

weak

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

deformation

of M.

out

to p r o v e

decGmposition of the We w a n t

(cf.

is any

pair

to c o n s t r u c t with

the

ii)

Hn(X,t)

iii)

Hn(MnXl)

Here

(for n = O)

Theorem (M,A),

6.4.

= Hn_1(x,1)

choose

and we use

inductively

following

We

a family

properties:

for x 6 M n _ 1 ,

the

a special standard

of h o m o t o p i e s

i) Hn(-,O)

relative

patch

notations (H n

: MxI

2.2. ~ [4

= Hn_1(-,1).

t 6 I.

c B. H_I

: MxI

~ N means

the c o n s t a n t

homotopy

H_I (x,t)

=

155

f(x). and

Once

thus

this

is d o n e w e c a n

can compose

obviously

compresses

apply

Lemma

5.1

with

t h e H n to a h o m o t o p y

F

f to B.

(s n n £ ~ o)

The

sequence

: MxI

C n = M n _ I, G n = H n _ I ,

~ N relative

A which

c a n be c h o s e n

here

in a n y way.

Suppose above un the gn

that

for

some

if n = O).

We want

: (Mn,Mn_ I) ~ standard

(N,B)

map

~n

: (M(n),~M(n))

pairs

which

natural

are

numbers

n ~0

the h o m o t o p y

to c o n s t r u c t

defined

(N,B).

isomorphic k
Now

~

• We

(N,B)

consider

= Hn_1(x,1).

(Mn,~Mn_1)

(M(n),~M(n))

to p a i r s

Since

n

b y Un(X)

: (M(n) ,~M(n))

~

H

H n _ I is a l r e a d y

is d - c o n n e c t e d

as

the m a p

we obtain

with

(H_I

Composing

is j u s t

(Ek,s k-l)

given

a map

a direct

s u m of

- perhaps

there

u n with

varying

exists

a

compression

G

n

: (M(n)×I,~M(n)×I)

~

(N,B)

of g n to B. T h e d i a g r a m q0nX id I ~M(n) xI

, Mn_1 ×I

ix id I

[j × id I

M(n) xI

, M ~nxidi

obtained

f r o m the

I is a g a i n homotopy

Un

: MnXI

(x,t) 6 M n _ I X I . homotopy

Hn

(cf.

finishes

if,

Thus

A map x £M,

(2.2)

by multiplying

there

exists

= G n and Un(X,t) theorem

= Hn_1(-,1).

the p r o o f

for e v e r y

in

extension

Hn(-,O)

5. L e t d 6 ]lq U ~ . o

a d-equivalence

(*)

UnO(~nxid)

the h o m o t o p y

~ N with

This

square

IV.8.7.ii).

~ N with

Using

: MxI

U n ( M n X l ) c B.

Definition

cocartesian

cocartesian

×I n

of T h e o r e m

induced

= Un(X)

for

U n to a

Hn(Mn×1)

=

6.4.

f : M ~ N between the

a unique

we extend

Clearly

with

spaces

homomorphism

is c a l l e d

-

156

f.

: nq(M,x)

surjective topy

~ ~q(N,f(x))

for q = d.

equivalence

Example

6.6.

If

homotopy

Remark

6.7.

the c l o s e d

(M,A)

since

Starting

this

of

spaces

then

is d - c o n n e c t e d .

proper

(Z(f),M), This

say t h a t

in case

d %~,

f is a w e a k

homo-

the

This

inclusion follows

map A ~

from

the

M

long

III.6.1.

follows p

remark

6.4

into

a very

6.8

(cf.

e.g.

map

with

the p r o j e c t i o n

from

theorem

(M,A)

A partially

is d - c o n n e c t e d . Z(f)

is a p a i r

iff

also

and,

holds.

sequence

pair

for 0 ~ q < d

If d = ~ then we

if this

is a d - e q u i v a l e n c e exact

is b i j e c t i v e

f : M ~ N is a d - e q u i v a l e n c e

Z(f)

from

the

: Z(f)

we are

general

the m a p p i n g

cylinder

factorization

(4.2)

~ N is a h o m o t o p y

able

to t r a n s f o r m

theorem

about

of f of

iff (cf.

§4)

f via

equivalence.

the

lifting

compression of m a p s

up to

homotopy.

Theorem for some at m o s t f,

between

d 6 ~o d and

U~.

[Spa,

Let

(L,C)

let h be a m a p

: [L,M] h ~

relative

p.

404]).

Let

be a c l o s e d from

C to M.

f : M ~ N be a d - e q u i v a l e n c e pair

of s p a c e s

Then

the m a p

of d i m e n s i o n

[L,N] f°h

homotopy

sets

(cf. n o t a t i o n s

in III,

§4)

induced

by

f is s u r j e c t i v e .

Remark. pair ing

More

explicitly

of s p a c e s square

this m e a n s

of d i m e n s i o n

of m a p s

(solid

at m o s t

arrows)

the

following.

d. A s s u m e

Let

there

(L,C)

be a c l o s e d

is a g i v e n

commut-

157

h

C

(*)

, M

v 11 ~ C

f

/ I

L I

,

w

N

with

f a d-equivalence

into

L.

vlC

Then

= h and

Proof.

a)

polytope

there fov

We

classical

exists

deal

argument

for

the

Consider

the

square

the

p.

case

(solid

arrow

(dotted

relative

404f] of

vertical

: L ~ M

proper.

convenience

h

v

to w

with

[Spa,

left

a map

f is p a r t i a l l y

argument

C

the

is h o m o t o p i c

first

and

and

(L,C)

In t h i s

the

inclusion

arrow)

such

of

C

that

C.

that

remains

the

case

valid.

is a r e l a t i v e Z(f)

exists

We

reproduce

This

square

weak

and

the

this

reader.

arrows)

,M

I i

/ /"

U 3oW

with

j

: N ~

co~ute,

but

there

exists

Z(f)

which

extends

u

: L ~

square

commutes

Theorem map

v

Z(f)

6.4

we

: L ~ M

nical

after can

of

the

inclusion

p

a homotopy the

compress

u

:

(L,C)

we

Z(f).

: LxI

u.

Z(f)

~

(Z(f),M)

obtain

from

from

According

i o v = u rel. C.

have

~

homotopy

jow by

~ N we

HICxI

H

obvious

h with

: Z(f)

homotopy

of N i n t o

replacing

extending

projection

nature

the

rel. C.

to

i°h. 6.7

Thus

we

with

the

By

rel. C.

not

to a m a p

to Remark

to M.

w = pojow =nou

jow

j°wlC

Composing

fov =pou

does

The and

have

the

a

canospecial

Thus

f o v = w rel. C. b)

We

field also

now R is

prove

the

sequential

C is a w e a k

theorem and

polytope.

under

the

L is a w e a k But

we

additional polytope,

do n o t

assume

assumptions which that

implies

that that

f is p a r t i a l l y

the

158

proper.

We w i l l

in IV,

§10.

we o b t a i n

use

the p a r t i a l l y

Starting

from

a commuting

the

proper

given

diagram

of

core

pf

commuting

solid

: Pf(M)

square

arrows

with

~ M introduced

(~) of s o l i d

pfoh

arrows

= h,

Pf C

~ Pf (M)

[

L

since

the m a p w~i the

By step

has

a)

of the p r o o f

with

voi

now p r o v e

the

the

(~) of

(L,C)

under

f is a g a i n

v from

The m a p v

the w e a k e r

is a r e l a t i v e

solid

(Of c o u r s e , proper

and

a d-equivalence.

L to Pf(M)

(dotted

:: p f o v

from

L to M

assumption

that

R is se-

arrows

weak

we

look

polytope.

Starting

at the c o m m u t i n g

again

diagram

from of

arrows

P(L)

(L,C)

tesian

f r o m P(L) relative

dimension to M P(C).

f o v o p L : fov.

~d.

(dotted

fov = w rel. C,

, N

complete

weak

The

arrow

since

b)

of the

above)

the

the

v

square

left h a n d and

proof

extending

fov.PL ~woPL

also

of L and C

(P(L),P(C))

from 9 a map

that

cores

polytope

pairs

By step

We o b t a i n We k n o w

f

W

is a r e l a t i v e IV.10.4).

. M

v/

PC the p a r t i a l l y

(Cot.

relative

~

* L

PL

PL and

Since

h

~ C

.

that

proper.

f is p a r t i a l l y

Thus

a map

fov = w rel. C.

theorem

P(C)

with

The m a p

4.13).

exists

partially

properties.

and

square

(Th.

there

= h and

sequential

solid

f r o m C to L).

equivalence

the r e q u i r e d

c) We

f r o m C to N is c e r t a i n l y

inclusion

is a h o m o t o p y

arrow)

~ N

W

i denotes pf

,M

(cf. square

(L,C)

there hoPc

have

exists

such

and

§9).

is c o c a r the

same

a map

that

: L ~ M extending rel. P(C)

IV,

fo~ = w O P L

h such

conclude

9

that

from

this

159

P(C)

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

pair PLC

still

PLXi d

~ LxI

(IV.8.7.ii).

assume

of d i m e n s i o n

: pC(L)

(P,C)

C×I I

,

P(L) xI

d) We

PcXid

xI

I

that

at m o s t

~ L described

is a r e l a t i v e

homotopy

R is s e q u e n t i a l ,

weak

equivalence

of ~ u n d e r

C. We

d.

polytope

of d i m e n s i o n

C.

Let

~

By the p r e c e d i n g under

C such

under

C and

f o v

=

step

that

, L

fov

of the p r o o f

C

uo~.

C u-~0o@ -~C u

Then

be any

closed

WP-approximation

at m o s t

diagram

: P ~ L. T h u s

d and ~ is a

a homotopy

(solid

inverse

arrows)

, M

,

u

c)

may

t h i s m a p by ~

: L ~ P denote

at the c o m m u t i n g

, P

(L,C)

relative

We d e n o t e

h

L

the

now

in 4.12.

under

look

We n o w use

but

N

there

v

exists

a map 9

: P ~ M

:= voW, : L ~ M is a g a i n

a map

,

as d e s i r e d . e) We

finally

prove

the t h e o r e m

T h e n we c h o o s e

a real

example

take

field

R(t)

of R(t) ly we

over

R

in w h i c h

can

unique null

we can

take

for S the

(one

real

t is p o s i t i v e

and

a commuting

rational

small

of R((t)) w i t h In b o t h square

sequential.

is s e q u e n t i a l .

respect

infinitely

closure

R is not

of the

t) w i t h

t is p o s i t i v e .

in S. We have

that

S of R w h i c h

closure

indeterminate

in w h i c h

case

overfield

for S the r e a l

ordering

sequence

closed

in the

For

function

to the o r d e r i n g over

R. A l t e r n a t i v e -

respect

cases

to the

(tnln6~)

is a

160

[L,M] h

f*

, [L,N] f°h

K

~

[L,

]h(s)

' [L,N]f°h(s) (fs) ,

where

the vertical

t o S".

They

(fs)~

arrows

are b i j e c t i v e

is b i j e c t i v e

Applying instead

Theorem of

Corollary

we obtain,

In the

dim(L,C)

~ d-1.

Then

The most

interesting

means

a bijection that

Theorem

6.10

of s p a c e s

M and N.

Indeed,

f.

Let

[L,N]

theorem").

extension

from R

lower horizontal

Thus

map

f, is b i j e c t i v e ,

=

q.e.d.

(LxI, (Lx~I) U (CxI))

way,

of T h e o r e m

6.8

assume

that

even

is b i j e c t i v e .

is t h a t ~

The

(L,C) x (I,~I)

C = ~ a n d d = ~. for e v e r y

equivalence.

(Mnln6~) Let

that

the

equivalences.

f clearly

a genuine

: [L,M]

"base

s~ace

Thus we have

Every weak

Then

L over

f

: M ~ N

R. T h i s

proved

homotopy

equivalence

be a d m i s s i b l e

filtrations

equivalence.

6.11.

homotopy

the p r o o f .

f. a b o v e

case here

maps

5.2.i.

in the u s u a l

the map

("Whitehead's

Corollary

Assume

of

situation

f is a h o m o t o p y

is a h o m o t o p y

n 6 ~ .

by Theorem

to the p r o d u c t

6.9.

yields

the n a t u r a l

b y s t e p d)

6.8

(L,C)

are

homotopy

and

(Nnln6IN)

f : M ~ N be a m a p w i t h restrictions Then

is a w e a k

fn

: M n ~ Nn of

f is a h o m o t o p y

homotopy

equivalence.

{The

f ( M n) c N n for e v e r y are

eauivalence.

equivalence, result

f all

6.11

hence could

by the also

theorem

have

been

161

g a i n e d by the t e l e s c o p i c

methods

of §4.}

We n o w look for a g e n e r a l i z a t i o n of s p a c e s

(cf. §2,

Definition between

Def.

equivalence

U~.

systems

if d = ~)

6.8 to d e c r e a s i n g

systems

5).

7. Let d 6 ~ o

decreasing

of T h e o r e m

We call a m a p

of s p a c e s

f : (Mo . . . . M r ) ~

a d-equivalenc e

if all the c o m p o n e n t s

fi

(N o ..... N r)

(or w e a k homotop~y

: Mi ~ Ni of f are

d-equivalences.

Remarks

6.12.

implies

t h a t the i n d u c e d h o m o m o r p h i s m

is b i j e c t i v e

i) If r = 2 and M r , N r are one p o i n t

if I ~ q < d

ii) By T h e o r e m s

systems

In the p r e s e n t

s y s t e m of s p a c e s equivalence

~

surjective

of s p a c e s

terminology

if d 6 ~

closed decreasing

If t h e s y s t e m We c o n c l u d e

s y s t e m of s p a c e s

{y} this

~ ~q(No,NI,Y)

equivalence

between

equivalence. of a d e c r e a s i n g

5) is just a w e a k h o m o t o p y

(Mo ..... M r ) w i t h

ing s y s t e m of w e a k p o l y t o p e s . equivalence.

is a h o m o t o p y

§4, Def.

{x},

and q = d.

a WP-approximation

(Mo, .... M r ) (cf.

: (Po ..... Pr ) ~

is a h o m o t o p y

: ~q(Mo,M1,x)

6 . 1 0 and 2.13 e v e r y w e a k h o m o t o p y

closed decreasing iii)

and

f.

spaces

(Po ..... Pr ) a d e c r e a s -

( M o , . . . , M r)

from Theorem

is h o m o t o p y

is c l o s e d t h e n

4.10.i

equivalent

that every

to a d e c r e a s -

ing W P - s y s t e m .

Theorem

6.13.

between

decreasing

be a c l o s e d

f*

systems

decreasing

and h be a m a p i) If d i m P

Let f : (Mo, .... Mr)

from

~

(N o ..... N r) be a d - e q u i v a l e n c e

of s p a c e s

system.

(d 6 ~ o

Let f i n a l l y

(C,C N P I , . . . , C

n p r ) to

U~).

Let f u r t h e r

C be a c l o s e d

(Po,...,Pr)

subspace

of Po

(Mo,...,Mr).

< d t h e n the i n d u c e d m a p o --

: [ (Po ..... pr ), (M ° ..... Mr ) ]h ~

[ (Po .... ,pr), (N o ..... Nr ) ]foh

162

between ii)

relative

If d i m P

Proof. claim

ii)

< d-1 --

o

Once

we

by

homotopy then

i)

to

(PoXI .... ,PrXI)

by

inductions

We

prove

claim

u

the

~

first

of

by

we

that

have

claim

i) we

on

be

a map

to

find

and

obtain

second

(CxI)U(Po×~I)

Let

r ~

extending a map

v

instead

We

I, a n d foh.

let

In o r d e r

to

: (Po' .... Pr ) ~

denote

(0 < i < r ) .

the

r.

6.8.

fov ~ u rel. C. W e

i respectively

solid

first

Theorem

( N o , . . . , N r)

claim

h such

ui,vi,fi,h

square

the

r = 0 is c o v e r e d

extending by

first

: (Po,...,Pr)

prove

is b i j e c t i v e .

applying C.

case

surjective.

proved

(Po' .... Pr ) a n d

The

f~

is

have

of

the

sets

the

first

components

consider

(Mo'''''Mr) of

the

u,v,f,h

commuting

arrows h I

(C N P I , . . . , C

NPr )

, (M I .... ,M r )

~

v'

(PI' .... Pr ) with

h'

left

vertical

exists

=

v'

=

extending

F'

with

u'

(hl, .... hr),

a map

=

arrow

and

means

we

inclusion

map.

By

: (PI ..... Pr ) ~

=

(Ul,...,Ur),

induction

and

hypothesis

the there

(MI'''''Mr)

: (PlxI ..... PrxI)

, F' (-,I)

(c,t) 6 (C N P I ) X I .

This

(fl ..... fr ) , u'

a homotopy

(F I ..... F r)

= u O and

f' =

(NI ..... Nr)

arrow)

(v I .... ,v r)

F' (-,0) = u '

Fo(-,O)

an

(dotted

h'

f'

We

extend

Fo(c,t)

have

= f'ov', F I to

= foho(C)

a homotopy

and

~

F1(c,t)

a homotopy for

(N I ..... N r)

every

Fo

= foh1(c) : PoXI

(c,t) 6 C x I

for

every

~ N O with (cf.

2.9).

163

F =

( F o , . . . , F r)

constant Then

o n C.

w'

: (PoXI ..... P r X I ) Let

= f'oV'.

w

:=

Now

~

( W o , . . . , w r)

consider

the

(N O ..... N r) w i t h

F(-,O)

:= F ( - , I )

:=

commuting

a n d w'

square

of

= u and

F

(w I ..... W r ) .

solid

arrows

eo CUP

I

I

P

with

j~ M o

// . "" v ° / / /

o

hypothesis)

by

there

= eo

(Mo,...,Mr). w

, N

w0

e ° obtained

Vo[C UPI

"[

= u rel.

and Then

C

.

fo

o

gluing

exists

h o and

a map

v°

v I. B y

Theorem

: Po

Mo

fo o v ° = w ° rel. C U PI"

Let

f o v = w rel. C U P1"

On

Thus

f o v ~ u rel.

C,

6.8

(dotted v

:=

the

(or b y arrow)

(Vo'V')

other

induction such

that

: (Po .... 'Pr ) ~

hand

as d e s i r e d . q.e.d.

Remark. that

The

As

dim(Pr,C

special

tion

nPr ) ~d

and

6.14.

a decreasing

that

shows

case

a decreasing

a map

proof

d = ~,

g from

Let

~

it s u f f i c e s

by

of

the

a universal

: (Po .... 'Pr ) ~

system

of

WP-system

spaces

to

~I

I

' (M O . . . . . M r)

claim

for

implies

i)

to assume

O < i
a characteriza-

property.

Let

(Po .... ,pr ) , u n i q u e

(Po,...,Pr)

f

theorem

(M o, .... M r ) .

i

(Qo ..... Q r )

first

( M o , . . . , M r)

~ o g = f.

g11

the

(M ° ..... M r ) .

(Qo' .... Q r ) to

(Qo,...,Qr)

for

d i m ( P i , (C ~ P i ) U P i + 1 ) ~ d

C empty,

of WP-approximations

Corollary of

the

up

be

a WP-approximation

f be Then

any map

from

there

exists

to homotopy,

such

164 This

universal

Corollary

property

6.15.

(No' .... Nr)

For every map

a m a p g from

functoriality

Let ~ : (Po .... 'Pr ) ~

: (Qo .... 'Qr ) ~ of spaces.

implies

(Mo' .... Mr)

and

be W P - a p p r o x i m a t i o n s

f from

(Po ..... Pr ) to

of W P - a p p r o x i m a t i o n s .

(Mo,...,M r) to

(Qo ..... Qr ) , u n i q u e

of d e c r e a s i n g

(No,...,N r) t h e r e up to homotopy,

systems exists such

that ~og = fo~.

(Po'''''Pr)

, (M o . . . . M r )

g

f

(No,..

(Qo .... 'Qr ) If f is a w e a k

The last

homotopy

statement

here

equivalence

is a l r e a d y

,N r )

then g is a h o m o t o p y

clear

by T h e o r e m

equivalence.

4.10.ii.

§7 - C W - c o m p l e x e s

Definition complex CWI.

I. a) A r e l a t i v e

(M,A)

with

The p a t c h

the

For

every

closed and b)

If

CW-complex

or r e l a t i v e )

call

Examples plex

7.1.

(§2,

(M,A)

over

If

I) and

where

(They

should

be c a l l e d

theory

[LW,

Every

iii)

iv)

The d i r e c t

again

I.c)

defined to the

over

~n

: En

~

from

the

isomorphically

onto

the

weak

underlying

call

M a

pair

of

spaces

of

polytope.

call,

here

t h e n we

R.

if n e c e s s a r y ,

more

classical

precisely

the

(absolute

semialgebraic

CW-complexes

if

is a d i m e n s i o n a l (M,A)

the m a p s

is also These

special

a relative are

Xo in CW2

regular

simplicial

If L is a c l o s e d Def.

then

which

we w i l l

relative weak

precisely

can be c h o s e n

CW-complexes,

patch

polytope

those as

com-

then

relative

isomorphisms.

as in the

topological

78].)

closed

§2,

maps

and A is e m p t y

of CW2,

CW-complex.

CW-complexes

ii)

~ which

R = ]lR we w i l l

(M,A)

is a r e l a t i v e

p.

if T i o

cw-complexes.

i)

Def.

a m a o_ Xo

exists

CW-complex

is a r e l a t i v e

in c o n t r a s t

topological

onto

CW-complex)

CW-complexes

CW-complexes,

(i.e.

~o.

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

are w o r k i n g

patch

properties.

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

there

(n = d i m o)

(or a b s o l u t e

a relative

of R is a r e l a t i v e

additional

of M ~ A

o 6Z(M,A)

onto

two

(M,A)

§~).

is a r e l a t i v e

that,

If we

patch

S n-1

CW-complex

Notice

cf.

n-ball

(M,A)

following

decomposition

d i m T < d i m o, CW2.

CW-complex

then

a relative

subcomplex

(L,A)

product

complex

is a g a i n ({12, Def.

CW-complex.

(cf.

IV.I.7)

of a r e l a t i v e a relative 1.d)

is a C W - c o m p l e x .

CW-complex

(cf.

CW-complex.

of two r e l a t i v e

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

(M,A)

if

(M,A)

CW-complexes is a r e l a t i v e

is

166

CW-complex,

then

also

(M,A) x (I,~I)

is a r e l a t i v e ox]O,1[ v)

If

with (M,A)

o running

proper

with

are those

of

We

the

to the

(M,A). map

For

Some

into the

tion.

(M,A)

more

Notice

with

M n of

precisely

just

by

the chunks.

Notations

7.2.

I n , denotes

(M,A) the

We

of

way.

{As u s u a l

the

CW-complexes

if

(M,A)

(open)

is a

cells

of

a characteristic of t h e

characteristic

w e do n o t

follow

this

of a s p e c i a l

denote

often and

conven-

type.

of A a n d a l l the

this

cells

n-skeleton

of

is not n e c e s s a r y .

that

(Mn]n>1)

with

the

is an a d m i s s -

M.

to w o r k

a modification

set of c e l l s

of M / A

choice

should

be a r e l a t i v e

iden-

In p a r t i c u l a r The c e l l s

Thus,

as t h e u n i o n

space

we

a

c e l l A/A.

Xj as a b o v e

complexes

(M,A)

natural

Thus we need

sets

is a g a i n

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

a fixed

3) of the

it is m o r e

Let

a map

patch

= A.

(M,A).

~ E Z(M,A)

(M,A) n, b u t m o s t

(§2, Def.

CW-complexes

the

is d e f i n e d

-I

(M U r N , N )

CW-complex.

is c o m m o n

of a C W - c o m p l e x .

(M,A)

the

f : A ~ N to a s p a c e

zero-dimensional

incorporate

are

are

of M U f N . }

in the o b v i o u s

t h a t M n is a s u b c o m p l e x

filtration

briefly

further

R, w e c a l l

We put M

as

(M UfN) ~ N

~ we call

authors

Our CW-complexes

of d i m e n s i o n

For

cell

any map

the pair

is an a b s o l u t e

which

over

then

same patches

setting

structure

The n-skeleton

ible

M/A

terminology

a given

and

subspace

and one

CW-complex

for ~.

maps

(M,A)

the

(M,A) x(I,~I)

~(M,A).

near M ~A,

space),

of

CW-complex,

with

semialgebraic

relative

through

the o p e n

(N = o n e p o i n t

extend

The p a t c h e s

is a r e l a t i v e

CW-complex

tify M ~A

(MxI, (AxI) U (M×~I))

CW-complex.

N is p a r t i a l l y relative

=

skeletons

of the notations

CW-complex.

of d i m e n s i o n

In(M,A), n of

than 2.2.

or m o r e

(M,A).

We

choose

167

for e v e r y with ~o

the

map

Xo

: E n ~ ~.

is d e n o t e d

by

~o" A b u s i v e l y

d 6Z n a characteristic inclusion

~

~n

: En ~ M n a c h a r a c t e r i s t i c

M n-1

is c a l l e d

partially

an a t t a c h i n g

proper

map

for ~

Its

•

composite we a l s o

restriction

of q. We h a v e

~o

a com~uting

of Xa

call

: S n-1

square

of

maps ~on

l xS n-1 n

(*)

map

The

M n-1

i

xE n

,

n

defined

as

follows.

The

I nxE n is the d i r e c t I nxS n-1

is the

of t h e s e

n,

attachin~

i and

j are

proper

Thus

the

is d e f i n e d

Mn =

~n'

in d i m e n s i o n

(~)

(ZnxEn)

n,

The m a p s

: (InxEn) U M n-1

of E n i n d e x e d

called

by ~n(a,x)

surjective

by the

the big

:= Co(x).

is o b t a i n e d Cn and

space.

set Zn,

j combine

and

of the b o u n d a r i e s characteristic

The m a p ~n'

from

Thus

map

called

the

~n by r e s t r i c t i o n • into

a partially

map

~ M n.

is c o c a r t e s i a n , U

as a d i s c r e t e

of In xEn c o n s i s t i n g

The m a p

inclusions.

square

I n is r e g a r d e d

subspace

of E n.

map

strongly

Pn

set

sum of c o p i e s

closed

copies

in d i m e n s i o n big

Mn

~n

which

means

M n-1 n

We

spell

out

the

standard

There

is no d i f f e r e n c e

(M,A)

and

Proposition has

(N,B)

7.3.

dimension

at

are

facts

least

"cellular

to the t o p o l o g i c a l

relative

Assume

about

m.

t h e o r y here.

CW-complexes

that, Then

for

over

a fixed

(M,A)

is

approximation" In the

of maps. following

R.

m 6 ]I~, e v e r y

(m-1)-connected.

cell

of

(M,A)

168

The

proof

is v e r y

Definition (N,B) b)

similar

2. a) A m a p

is c a l l e d

every

f from

cellular,

Consequently,

if H m a p s

This

the pair

H

of

spaces

: (M×I,AxI)

(Mnx~I)

forces

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

~

the m a p s

H(-,O)

of

and

to the p a i r

n >0.

(N,B)

U (Mn-lxI)

6.1.

(M,A)

if f(M n) c N n for e v e r y

a homotopy

the n - s k e l e t o n

n ~O.

to the p r o o f

is c a l l e d

(MxI,AxI)

H(-,I)

cellular

into

from

N n for

(M,A)

to

(N,B)

to be c e l l u l a r .

Applying to all Lemma

Proposition

the

relative

5.1,

7.4.

to a c e l l u l a r

such

are

which

we have

Applying of the

a very

there

Cellularity

7.5.

rough

7.4

destroying

and

following

cell

(N,Nn),

successively

and u s i n g

is h o m o t o p i c

approximation of

(M,A)

the

maps

map

finally

relative

A

CW-complex

approxima-

out of bounds.

(MxI, (AxI) U (Mx~I)) we o b t a i n

: (M,A)

from

Cellular

approximations

n o w are

~

(N,B)

are

homotopic

f to g.

is a n e c e s s a r y

obvious

f.

simplicial

(I,3I)

f,g

structure.

rather

of

product

and

homotopy

to a n o t h e r

the

6.4

theory,

(N,B)

to the d i r e c t

a cellular

without

~

III but w h i c h

If two c e l l u l a r

a CW-complex

theorem

(N,B).

substitute

of the a t t a c h i n g

by the

~

in C h a p t e r

to g l u e

stated

f : (M,A)

CW-complexes

exists

(Mn,M n-l)

g a cellular

used

relative

map

: (M,A)

Proposition

Corollary then

g

a map

tions

the c o m p r e s s i o n

as in the t o p o l o g i c a l

Every

map

and

CW-complexes

w e obtain,

Proposition

We call

7.3

assumption

along

a closed

It is also proposition.

if one w a n t s subcomplex

sufficient,

as is

169

Proposition

7.6.

f : (L,A)

(N,B)

~

Let L be a closed be a c e l l u l a r

subcomplex

map.

Then

of

(M,A)

the pair

and let

(M UfN,B)

together

w i t h the p a t c h d e c o m p o s i t i o n

I(M UfN,B)

is a r e l a t i v e since

(M,L)

(M UfN,B).

= I(M,L)

UI(N,B)

CW-complex.

{Notice

is a r e l a t i v e The natural

that

f is p a r t i a l l y

weak polytope.}

map

from

(M,A)

to

proper

N is a closed (M UfN,B)

near M ~ L

subcomplex

extending

of

f is again

cellular.

We now start out to prove equivalent

to a C W - c o m p l e x

real a l g e b r a i c topologically ly we will

numbers,

patch

complexes many

we will

establish

subcomplexes.

(M,A)

can be d e f i n e d that every

refined

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

3. Let

polytope

to a s e m i a l g e b r a i c

prove much more

theorems

Definition

which

and also,

equivalent

these

between

that every weak

which

We need

and

(N,B)

(f,a)

of a map

f : (M,A)

(N,B)

a : Z(M,A)

~ Z(N,B)

axioms

PWI-PW3

hold.

For any s u b c o m p l e x

aL

:= B U

~

For any ~ £1(M,A)

L of

(M,A)

7.11).

In relative

equivalences this precise.

patch complexes

over

R.

to

(N,B)

is a pair

between

pairs

of spaces

such that

the f o l l o w i n g

we w r i t e

a

is

IR. A c t u a l -

between

to make

(M,A)

R o of

CW-complex

over

7.10,

homotopy

be r e l a t i v e from

and a b i j e c t i o n

(7.8,

a definition

equivalence

field

topological

induce

homotopy

R is h o m o t o p y

the

equivalences

A patch w a t c h i n ~ consisting

over

CW-complex

theorems

homotopy

over

instead

three of a(o).

let

U(ao{o6X(L,A)).

This

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

of

PWI.

If o and T are patches a face of o. Thus,

(N,B).

of

(M,A)

and aT is a face of ao then T is

if L is a closed

subcomplex

of

(M,A)

then

aL

170

is a closed PW2.

f(L)

subcomplex

c aL for every

it suffices a patch PW3.

For every (aL,B)

closed

obtained

cates

lemma

Lemma

7.7.

f itself

(M,A)

the m a p b e t w e e n

7.8.

complex

of

{Notice

that

(M,A)

of A and the

the m a p

faces

of

from

(L,A)

to

equivalence.

equivalence.

a fixed 6.11

real

closed

and T h e o r e m

equivalences

(M,C)

be r e l a t i v e

(automatically

from

there

(Mn,A)

field R. The 2.13.

It indi-

can be c o n s t r u c t e d

f : (M,A)

by

closed

triple

is a r e l a t i v e

weak

from

a relative

(with the p r e g i v e n

patch

of spaces polytope.

(A,C)

CW-complex

complex

fn-1 pairs

~ Z(N,B)

(f,a)

be

is a p a t c h

and A a closed

p at c h c o m p l e x

(M,A,C),

cf.

(g,8)

(M,A)

2.1.}

Assume

on any

that

be a p a t c h w a t c h i n g (B,D).

B as a closed

(B,D))

sub-

is special.

can be c h o s e n

CW-complex

containing of

homotopy

(N,B).

decomposition

cell d e c o m p o s i t i o n

of M

respectively.

fn extends

a n . Then

to

to a r e l a t i v e (N,D)

(N,B)

be the m a p b e t w e e n

patch

Let

and

let a : Z(M,A)

(M,A)

such that the r e l a t i v e

filtrations

such that

all the maps from

and let

a patch w a t c h i n g

(N,B)

fn' and

be a r e l a t i v e

such a r e l a t i v e

(M,A)

(Nn,B) ~

complexes

admissible)

is given

to

all the maps

(M,C)

patch

M n and N n of

equivalence

Let

topy e q u i v a l e n c e exists

(N,B)

sets e x t e n d i n g

homotopy

decreasing

be

an_ 1. Let

extending

Theorem

over

homotopy

subcomplexes

(fn,an)

and a n e x t e n d s

watching

(Of course,

f to L is a h o m o t o p y

by C o r o l l a r y

for every n 6 ~

equivalence

of spaces

and

(NnLn6~)

that

(M,A).

procedure.

and N by closed Assume

(M,A)

is a h o m o t o p y

still work

is obvious

Let

(Mnln6~),

L of

by r e s t r i c t i n g

that patch w a t c h i n g

an i n d u c t i v e

L of

for L the union

subcomplex

some time we will

following

this

subcomplex

(M,A).)

In particular,

For

(N,B).

closed

to check

~ of

of

Then

homothere

subcomplex

and a p a t c h w a t c h i n g

171

homotopy

equivalence

and a e x t e n d homotopy

Proof.

(f,a)

from

B. In p a r t i c u l a r

equivalent

inductively

homotopy

for e v e r y n ~ - I ,

s t a r t i n g w i t h N_I

Nn_ I is a c l o s e d

subcomplex

subspace Th.

7.1).

relative complex

of

(f,a)

from

the u n i o n

(fn_1,an_1)

to

(Mn_I,C)

U

from

(Mn,C)

(Nn,D) to

and

(Nn,D)

= g' s-1 = B, such t h a t

structure

fn-1'

and a n e x t e n d s

of the f a m i l y of s p a c e s

semialgebraic.

is an a d m i s s i b l e

E v e r y N n is a c l o s e d

filtration

of (N,D) w h i c h m a k e s subcomplexes.

to a p a t c h w a t c h i n g

(N,B),

T h e n we c o n s t r u c t

M n = M(n)

is

patch complex

CW-complex

E of all the sets of c e l l s

of N

(IV,

E(Nn,D)

(N,D)

is a

a relative

By L e m m a

CW-

7.7 the

homotopy

equivalence

homotopy

equivalence

as d e s i r e d .

for some n ~ O , from

(fn,Sn)

= B, f-1

the N n as c l o s e d

fit t o g e t h e r

(M,A)

that,

given.

space

patch decomposition

(fn,an)

g

We e q u i p the u n i o n N of all the sets N n

(Nnln~-1)

containing

relative

(Nn,B) , fn e x t e n d s

s p a c e N is w e a k l y

Clearly

pairs

Assume

limit

of N, and

f extends

every weak polytope

a relative

equivalence

an_ I. T h e n w e w i l l be done:

This

such that

of the s p e c i a l

a patch watching

(Nnln~-1).

(N,D)

(A = @, B = ~),

the n - c h u n k

construct

w i t h ~.e i n d u c t i v e

to

to a C W - c o m p l e x .

Let M n d e n o t e

(M,A). We w i l l

(M,C)

the p a t c h w a t c h i n g

to a r e l a t i v e (Nn,D)

and

CW-complex

(fn,an)

(Nn_I,D)

as f o l l o w s .

is a l r e a d y

We h a v e

Mn_ 1 n

w i t h M(n)

the n - b e l t

of the n - b e l t

of

(cf. 2.2).

(M,A)

and ~n

Let u

: 3M(n)

f n _ i O ~ n . By C o r o l l a r y

4.9.ii

extends

equivalence

to a h o m o t o p y

h : M n ~ M(n) U u N n _ I in a c a n o n i c a l

way.

: 3M(n)

~ M n _ I the a t t a c h i n g m a p

~ Nn_ I d e n o t e

the h o m o t o p y

equivalence

the c o m p o s i t e fn-1 : Mn-1 ~ N n - 1

172

Now

(M(n),DM(n))

running

is the d i r e c t

through

the set X(n)

(M,A). We fix some ~ 6 X ( n ) . of ~n to Da. This L(o)

denote

We w r i t e mapped (~,3~)

AL(a)

= fn_1O~a

is i s o m o r p h i c

with

image

All the v a w i t h homotopic

= X ( M n , M n _ I) of p a t c h e s

Let ~a

closed

I := an_ I. S i n c e

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

: Da ~ Mn_ I d e n o t e

u~

subcomplex

(fn_1,l)

(Ed,s d-l)

: Do ~ AL(o)

in the

with d

~ 6 X(n)

together

X(Nn_I,D)

:=

(M(n)

This m a k e s

Combining

relative

(Nn,D)

the s p a c e (Nn_I,D).

D0.

Do is

The p a i r

:= d i m o . By P r o p o s i t i o n

7.3

to a m a p v a : Do

kL(~) N N d-1

of

: DM(n)

(IL(a),D).

~ Nn_ I w h i c h

patch decomposition

patch decomposition

a relative

bijection

T w i t h the b i j e c t i o n

a bijection the a x i o m PWI

:= a n in o r d e r

We fix a g a i n of

equivalence h

of

containing

is

CW-complex

of

g i v e n by (M(n),DM(n))

containing

Nn_ I as

subcomplex.

We h a v e a t a u t o l o g i c a l

complex

n of

U v Nn_I,D)

and the o b v i o u s

(cf. 2.1).

obeys

IL(o)

form a m a p v

and e q u i p this p a i r w i t h the r e l a t i v e

obtain

of h e i g h t

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

(Mn_I,C)

of u is h o m o t o p i c

(d-1)-skeleton

with

f r o m D~ to M n _ I. Let

is p a t c h w a t c h i n g

into the s u b c o m p l e x to

of

(~,D~)

to u. We d e f i n e

(Nn,D)

a closed

of s p a c e s

is just the i n c l u s i o n m a p p i n g

the s m a l l e s t

by u

sum of the p a i r s

an_ I = I f r o m X ( M n _ I , C )

a n f r o m X(Mn,C) and e x t e n d s

to X ( N n , D ) .

to X ( N n , N n _ I ) .

to X ( N n _ I , D )

By c o n s t r u c t i o n

an

an_ I . In the f o l l o w i n g we w i l l w r i t e

to get rid of indices.

some ~ £Z(n).

Let Q(~)

(M,C) c o n t a i n i n g h from above

: Q(a)

y f r o m X ( M n , M n _ I) = X(n)

g. We h a v e Q(O)

restricts

~ ~ U u kL(o)

denote

the s m a l l e s t = ~ UL(O).

to a h o m o t o p y

closed

sub-

The h o m o t o p y

equivalence

we

173

w h i c h extends the h o m o t o p y e q u i v a l e n c e

fn_11L(~)

from L(~)

to XL(~).

By C o r o l l a r y 4.9.i there exists a h o m o t o p y e q u i v a l e n c e

k

: ~ U 0

IL(o)

~ ~ U

U0

under lL(o).

~L(~)

= ~Q(~)

V0

The c o m p o s i t e fa

:= ko°h ~ is a h o m o t o p y e q u i v a l e n c e

Q(o)

to ~Q(o) w h i c h extends the h o m o t o p y e q u i v a l e n c e

L(o)

to IL(o).

fn_11L(G)

from

from

All the maps fa fit together to a map fn : Mn ~ Nn w h i c h extends fn-1" By c o n s t r u c t i o n the pair closed s u b c o m p l e x of

(fn,~) obeys the axiom PW2.

If L is a finite

(Mn,C) we obtain from C o r o l l a r y 2.14 by induction

on the number of patches in L ~ (L n M n _ I) that the r e s t r i c t i o n fnlL is a h o m o t o p y e q u i v a l e n c e from L to ~L. By T h e o r e m 6.10 this also holds for L infinite.

The map fnlL extends the h o m o t o p y e q u i v a l e n c e g b e t w e e n

the closed subspaces C and D of L and ~L respectively. fnlL is a h o m o t o p y e q u i v a l e n c e v e r i f i e d that

from the pair

(L,C) to

Thus

(Th. 2.13)

(~L,D). We have

(fn,~) obeys PW3 which finishes the proof.

We discuss the b e h a v i o u r of relative C W - c o m p l e x e s under base field extension.

Let K denote a real closed o v e r f i e l d of R. {The letter

is now needed for spheres.}

Let

(M,A)

We choose c h a r a c t e r i s t i c maps ~ ing maps ~o

: sn-1 (R) ~ Mn-1

more p r e c i s e l y En(R),Sn-I(R)

"S"

be a relative C W - c o m p l e x over R.

: E n (R) ~ M n and c o r r e s p o n d i n g

for all the cells ~ of instead of En,sn-1.}

(M,A).

Then

attach-

{We write

(M(K),A(K))

is

a r e l a t i v e C W - c o m p l e x over K with the sets of cells

Z(MCK) ,A(K)) -m- {o(K)Ia6Z(M,A)}.

The n - s k e l e t o n M(K) n of of M(K).

(M(K) ,A(K))

c o i n c i d e s with the subspace Mn(K)

The cell a(K) has the c h a r a c t e r i s t i c map

attaching map

(~o)K" We have ~(K)

if T and a are cells of

= a(K)

(T~o)K

and ~(o(K))

=

and the (Do) (K). Thus,

(M,A), then T ~ o iff T (K) ~ o(K) . It is now

174

evident

that

extensions

the

L(K)

We n o w p r o v e spirit

consequences

a relative

of t h e

then

I(L,A).

Theorem tive

(M,A)

the

CW-complex

from (N,D)

(with t h e p r e g i v e n equivalence a extends

(f,a)

6.

R ° of r e a l

Proof.

We denote

the p r o o f

For

~o

: Ed(K)

~ Md

keep

the

the

In f a c t

algebraic

that

to

base

same

they

K and

watching

we

regard

to Z ( N ( K ) , B ( K ) ) .

all c e l l s

no w i t h

be a r e l a t i v e

are

(N,B)

homotopy a as a If ~ is

of I(M,A)

~ running

(g,B)

on

(B,D))

(N(K),D(K))

to N(K)

(B,D)

through

there

exists

be a r e l a -

over

be a c e l l w a t c h i n g

Then

B = ~,

Let

CW-complex

B is a c l o s e d

(A = ~,

K, a n d

homotopy

a relative

subcomplex

of

(N,D)

and a cell watching

homotopy

such

g and

that

R = Ro),

for s o m e

f extends

every

CW-complex

CW-complex

N over

M

the

numbers.

the n-skeleton

every

over

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

Let

structure (M,C)

complex

usually

than

(B(K),D(K)).

equivalent

of T h e o r e m

(N,B)}.

sets.

on C W - c o m p l e x e s ) .

(M,C)

(M,C).

to

cell

"skeleton

of

just

(N,B).

theorem

In p a r t i c u l a r

field

(f,s)

of

R such

from

K is h o m o t o p y

which

is a p a t c h

then

rather

of B a n d

of

over

(f,a)

to I(N,B)

R. L e t

(A,C)

If

patch

(N(K),B(K))

union

over

R.

to Z(N,B)

subcomplex

equivalence

N and

to

(First m a i n

CW-complex

A a closed

over

over

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

7.10

are

(M,A).

5.2 on h o m o t o p y

be a r e l a t i v e

f r o m I(M,A)

aL d e n o t e s

L of

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

theorems

(M,A)

f r o m I(M,A)

any bijection

(M(K),A(K))

subcomplexes

theorems"

complex

from

of

latter.

Let

patch

equivalence bijection

"main

two m a i n

7.9.

subcomplexes

of the c l o s e d

two

as t h e

Notations

closed

of

by skeleton"

(M,A)

by Mn(n~-1).

proceeding

7.8 a b o v e

{but t h e r e

~ 6I(M,A)

we

(d = d i m ~)

choose

and denote

We will

in a s i m i l a r

the N n were

not the

a characteristic the

way

corresponding

build as

up

in

skeletons

map attaching

175

: sd-1 (K)

m a p by ~o

We w i l l

construct

a family from

to

f-1 = g' a-1

that

(fn_1,an_1).

and

is the

((Nn,D) In>-1)

homotopy

inductively,

= B, such t h a t N n-1

(Nn-I,D)

CW-complexes

of p a t c h w a t c h i n g

(Nn(K),D(K))

n (f ,a n ) e x t e n d s

n >O.

a f a m i l y of r e l a t i v e

((fn,an) In~-I)

(Mn,C)

Assume

Md-1

equivalences

and

(fn,an)

s t a r t i n g w i t h N -I = B, (n-1)-skeleton

of

(Nn,B)

and

T h e n we w i l l be d o n e by use of L e m m a

(fn_1,an_1)

are a l r e a d y

constructed

7.7.

for some

We h a v e Mn

using

=

(I n

xEn(K))

the n o t a t i o n s

h e r e as a d i s c r e t e

7.2.

from I nxsn-1(K)

denote

the s m a l l e s t

denote

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

s p a c e o v e r K.} Let

a map

respectively

M n-1

U~n

n

:= I

closed

the m a p o b t a i n e d

subcomplexes

sets

of

(M,A)

(M,C)

(or a l r e a d y

{Abusively

we r e g a r d

and L(a)

containing

~ and

Do

: S n-1 (K) ~ IL (0)

u to { a } x s n - 1 ( K ) .

the vo into a m a p v f r o m I n xsn-1(R)

t o p i c to v K.

:= f n _ i O ( ~ n ) K ,

let Q(o)

III.3.1)

a m a p v a : S n-1 (R) ~ 1 L (o) such t h a t uo is h o m o t o p i c combine

considered

'

Let u

7.8) . Let u~

by r e s t r i c t i n g

5.2.i on h o m o t o p y

n

~ := Sn-1"

to Nn-1 (K). For a n y a 6 Z n

(as in the p r o o f of Th.

main theorem

z

to N n-1

to

By the f i r s t there

exists

( v ) K. We

T h e n u is h o m o -

Z n a l s o as a d i s c r e t e

space o v e r R.}

We d e f i n e N n := This

(ZnxEn(R))

space contains

patch decomposition cal b i j e c t i o n sition

N n-1

as a c l o s e d

Z ( N n , N n-l)

of

(Nn,N n-l)

and the p a t c h d e c o m p o s i t i o n

(Nn,D).

subspace,

y f r o m X n = Z ( M n , M n-l)

w i t h the s t r u c t u r e of

U v Nn-1

Moreover

of a C W - c o m p l e x I and Y c o m b i n e

a n d we h a v e an o b v i o u s

together

to Z ( N n , N n - I ) .

of

(Nn-I,D)

with a tautologiThis p a t c h d e c o m p o -

together

such that N n-1

is the

into a bijection

equip

(Nn,D)

(n-1)-skeleton

a n f r o m /(Mn,C)

176

to E(Nn,D)

w h i c h obeys the axiom PWI

(modified in the obvious way,

since now the image of ~n consists of cells over R instead of K).

Using C o r o l l a r y 4.9 we obtain a h o m o t o p y e q u i v a l e n c e fo

: Q(o)

= En(K) u

L(J)

~ [En(R)

U v ~L(~) ] (K)

for every a 6 1 n which extends the h o m o t o p y e q u i v a l e n c e L(o)

to iL(o) (K). These maps fo and fn-1

fn : Mn ~ Nn(K) w h i c h extends fn-1" fulfills PW2. Theorems

2.13 that

topy e q u i v a l e n c e

into a map

By c o n s t r u c t i o n the pair

from

(fn,an)

(fn,an)

(Mn,C)

to

also fulfills PW3, hence is a homo(Nn(K),D(K)).

q.e.d.

(Second main theorem on CW-complexes).

r e l a t i v e s e m i a l g e b r a i c C W - c o m p l e x over JR. Let

Let

(B,D)

(M,C). Let

be a cell w a t c h i n g topological h o m o t o p y e q u i v a l e n c e

from

Then there exists a relative

be a

(M,C) be a relative

t o p o l o g i c a l C W - c o m p l e x and A a closed subcomplex of

(Btop,Dtop).

from

It is now easily v e r i f i e d by use of C o r o l l a r y 2.14 and

6.10,

Theorem 7.11

fit together

fn_iLL(o)

(g,B)

(A,C) to

semialgebraic CW-complex

(N,D) over ]R such that B is a closed s u b c o m p l e x of

(N,D)

(with the

p r e g i v e n cell structure on

(B,D)) and a cell w a t c h i n g t o p o l o g i c a l

homotopy equivalence

from

tends g and a extends

(f,s)

(M,C)

6. In p a r t i c u l a r

to

(Ntop,Dto p) such that f ex-

(A = ~, B = ~), every topologi-

cal C W - c o m p l e x M is h o m o t o p y e q u i v a l e n t to the u n d e r l y i n g t o p o l o g i c a l space Nto p of a s e m i a l g e b r a i c C W - c o m p l e x N over IR.

The proof follows the same pattern as the p r e c e d i n g one.

Instead of

the first m a i n t h e o r e m on h o m o t o p y sets one uses the second one. Further one uses some well known facts from the theory of topological CW-complexes

(in p a r t i c u l a r the t o p o l o g i c a l r e l a t i v e W h i t e h e a d theorem

p. 222]).

[W,

177

Theorem theory

7.10 of

ing a l s o sults

is a u s e f u l

topological into account

further

tool

Theorem

we

go o n to a r b i t r a r y

IV,

§9).

We give

bers

n ~I

j~

induced

and m~O.

Proof. are

and

a) W e

By base

first

valence

b)

real

closed

Assume

in a s i m i l a r

complete

cases

cores

closed

such

we

(cf.

The claim

that M

now

we obtain

the claim

(M,B)

for

is b i j e c t i v e

for R = ~

see t h a t field

by that

some

num-

A'

holds

e.g.

is a l s o

theorem

f(A)

cf.

there

of M',

c A' (R),

(M,A,B,x) with

polytope.

Theorem

7.8

t h a t M is an a r b i t r a r y (M,A,B,x)

space

f r o m the

but

R = R o-

claim

with

by

a CW-complex

c B' (R) a n d

equif yields

(A' n B') (R)

x running

y running

instead

for

the

exists

6.21].

and a homotopy

f(B)

Then we

and A,B

[Sw,

holds

R we now obtain

a n d B'

for

for

by the homotopy

to A'(R), B to B'(R), and A N B to

is a w e a k

for

is m - c o n n e c t e d

CW-complexes,

(M',A',B',y)

way by using

c) A s s u m e

sub-

in A N B. A s s u m e

t h a t M is a C W - c o m p l e x

holds

base

that

fromA

~

case

R o to IR w e

for

L e t A a n d B be c l o s e d

(B,A n B )

claim

subcomplexes

it h o l d s now

the

Indeed,

equivalences

since

favourable

re-

~ nq(M,B,x)

The

from

7.10.

by restriction.

and

topological

f : M ~ M' (R)

homotopy

A NB

for

extension

Ro,

the

Tak-

the homomorphism

consider

of M.

theorem

of T h e o r e m

In v e r y

L e t x be a p o i n t

for q = n+m.

use

over

= M.

surjective

an a r b i t r a r y

CW-complexes.

can transfer

partially

theorem).

j : (A,A N B )

For

M'

using

inclusion

subcomplexes

excision

A UB

Then

: ~q(A,A NB,x)

I ~q
spaces

is n - c o n n e c t e d

by the

polytopes.

excision

of a s p a c e M w i t h

(A,A N B )

sometimes

homotopy

an e x a m p l e .

(Homotopy

that

7.8,

weak

even

spaces

from the

to s e m i a l g e b r a i c

to a r b i t r a r y

7.12

results

CW-complexes

can

Theorem

to t r a n s f e r

through

can verify

through A' N B'. the

claim

of 7.10. R is s e q u e n t i a l .

claim

for

Then

(P(M),P(A),P(B),x).

178

{Notice

that

d) A s s u m e real

finally

closed

that

field

We conclude ready

P(A) N P(B)

that

the c l a i m

7.13.

and

n-connected

partially

P~

induced point

this of

complete

b y the

spaces

obtain

the

claim

7.10,

7.11 power

short,

we

for

for sets

a sequential

(M(K),A(K),B(K),x). 5.2.i

(or a l -

(M,A,B,x).

pair

of

m >O

IV,

the

p

§8).

q.e.d.

spaces

with A m-connected

a n d n > I. A s s u m e Then

: (M,A)

if 2 < r < m + n

polytope claim

then

~

t h a t A is

the homomorphism

for

if r =

7.12

Proposition

to the

is,

CA = I×A/I×A

Theorem

partially

similar

(M/A,.)

surjective

the c o n e

using

t h a t A is o n l y steps

and

by a p p l y i n g

and then

by two

any base

m +n +I.

exists.

to the

2.15,

cf.

triple

[Sw,

complete

near M ~A

steps

a n d d)

c)

call

write

down

some consequences

which

will

be widely

4.

theorems

i) A s y s t e m

is a f i n i t e

every

family

Pi a c l o s e d

it a C W - p a i r ,

Pi of a C W - s y s t e m (Po,...,Pr)

In

p. 84]. we now

in the q.e.d.

of the

Definition

and

holds

for h o m o t o p y

some numbers (cf.

claim

choose

above,

finally

full

case

holds

projection

(M U C A , M , C A )

general

theorem

Then we

~ ~r(M/A,*)

natural

obtain

R. T h e

be a c l o s e d

near M ~A

bijective

In t h e

We

for

If A is a w e a k

case we

proof

(M,A)

: ~r(M,A,x)

x in A,

Proof.

Let

sequential.

K of

first main

Corollary (M,A)

R is n o t

extension

by the

III.6.3)

= P(A NB).}

central

in t h e n e x t

results

chapter.

7.8,

Here

the

R, or C W - s y s t e m

for

is n o t n e e d e d .

of C W - c o m p l e x e s (Po,PI,...,Pr)

subcomplex

of Po"

a CW-triple,

(Po,...,Pr)

decreasin~

used

of our

etc.

over with

Po a C W - c o m p l e x

In the c a s e s Notice

is a C W - c o m p l e x .

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

that We

R

r = I, r = 2, every

call

system

over

of

component

the C W - s y s t e m spaces,

i.e.

...

179

Pi+1

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

of Pi for

ii) A C W - a p p r o x i m a t i o n

I
of a d e c r e a s i n g

s y s t e m of s p a c e s

(Mo,M I , . . . , M r )

o v e r R is a m a p

~0 : with

(po,P1,

(Po'''''Pr)

.... Pr)

~

a CW-system

of ~ a h o m o t o p y

equivalence.

WP-approximation

(§4, Def.

Remark.

In t o p o l o g y

are w i d e l y used,

spaces,

topological

Theorem

o v e r R and e v e r y Notice

~i

: Pi ~ Mi is a

5).

of s y s t e m s

but t h e r e one o n l y d e m a n d s

means

component

that e v e r y C W - a p p r o x i m a t i o n

CW-approximations

m a p are w e a k h o m o t o p y gical

(Mo,M I, .... Mr)

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

spaces

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

of the

e q u i v a l e n c e s , cf. [W,Chap. 5, §3], which,

less t h a n h o m o t o p y

equivalence.

for t o p o l o -

We call

t h e m here

CW-approximations.

7.14.

Every decreasing

s y s t e m of s p a c e s

( M o , . . . , M r) has a

CW-approximation

: (Po,...,Pr) If M r is c o n t r a c t i b l e M i is c o m p l e t e ,

Proof.

that

with

properties

CW-complex.

to c o n s i d e r

WP-system.

the case

Applying

(Po,PI ..... Pr )

a decreasing

s t a t e d above,

Since b o t h s y s t e m s

If

s y s t e m of s p a c e s has a

it s u f f i c e s

a decreasing

space.

a map

(Mo,M I ..... M r ) ~

(Po,...,pr)

Thus

as a o n e - p o i n t

as a f i n i t e

that every decreasing

is a l r e a d y

7.8 we o b t a i n

i~ :

then Pr can be c h o s e n

(Th. 4.10.i).

(Mo,MI,...,Mr)

Theorem

(M o .... ,M r )

t h e n Pi can be c h o s e n

We k n o w a l r e a d y

WP-approximation

~

CW-system,

at w i l l w i t h

and all c o m p o n e n t s

of s p a c e s

the a d d i t i o n a l

~i of ~ h o m o t o p y

equivalences.

are c l o s e d we k n o w by 2.13 that ~ is a

180

homotopy of

equivalence.

A homotopy

inverse

~ of % is a C W - a p p r o x i m a t i o n

(M o ..... Mr).

Theorem

7.15.

q.e.d.

i) L e t

K be a r e a l

( M o , M I , . . . , M r) b e a d e c r e a s i n g decreasing

CW-system

closed

overfield

WP-system

(Po,P1,...,Pr)

over

over

o r R. L e t

K.

Then

R together

there

with

exist

a

a homotopy

equivalence

<0 : ( P o ( K ) , P I ( K ) ..... Pr(K))

If M r is c o n t r a c t i b l e M i is s e m i a l g e b r a i c ii)

If

(Qo,...,Qr)

then

then

~

(Mo,M 1 ..... M r )

Pr c a n be c h o s e n

Pi c a n be c h o s e n

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

g

(Po .... ,Pr ) , u n i q u e

~ ° g K ~ f"

(M° ..... M r ) t h e n

If f is a h o m o t o p y

as a f i n i t e

WP-system

(Qo(K),...,Qr(K)) : (Qo ..... Qr ) ~

as a o n e - p o i n t

there

over exists

then

also

If

CW-complex.

R and

f a map

from

a map

up to h o m o t o p y ,

equivalence

space.

such that

g is a h o m o t o p y

equivalence.

Proof.

In o r d e r to p r o v e p a r t i) w e m a y a s s u m e , by t h e p r e c e d i n g t h e o r e m , t h a t

( M o , . . . , M r) is a l r e a d y a C W - s y s t e m .

: (Mo ..... M r ) ~

such

that

(Po,...,Pr)

Using

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

additional

¢i

~ Pi (K) of ~ is a h o m o t o p y

is a h o m o t o p y as d e s i r e d . and the

Using ing

last

first main

instead

properties

equivalence

The

second main

claimed

b y 2.13.

claLms in the

theorem

of t h e

7.11

CW-system above,

A homotopy theorem

theorems

and

now

sets

5.2.ii

over

a map

R, at w i l l

and e v e r y

equivalence.

on homotopy

first main

theorems

7.10 we obtain

(mo(K) ..... Pr(K))

ing t h e : Mi

Theorem

hav-

component

This map

of

systems

inverse

~ of ~ is a m a p

follow

from Theorem

5.2.i.

7.10 and we o b t a i n

6.14

q.e.d.

5.2.i

the

correspond-

181

Theorem there with

7.16.

exists

i) Let

( M o , . . . , M r) be a t o p o l o g i c a l

a semialgebraic

a topological

homotopy

: (Po' .... Pr ) ~ If M r is c o n t r a c t i b l e If M i is a f i n i t e

CW-system

(Po,...,Pr)

CW-system. over

~

Then together

equivalence

(Mo' .... Mr)

then

Pr can be c h o s e n

CW-complex

as a o n e - p o i n t

space.

then

also

Pi can

be c h o s e n

as a f i n i t e

is a W P - s y s t e m

over

~R and

f is a c o n t i n u o u s

CW-complex. ii) from

If

(Qo,...,Qr)

(Qo ..... Q r ) t o p

algebraic

map

such

~ - g = f in the

that

homotopy

g from

to

equivalence

(M ° ..... M r ) then

(Qo ..... Qr ) to topological

then

there

exists

a weakly

(Po ..... Pr ) , u n i q u e sense.

g is a h o m o t o p y

map semi-

up to h o m o t o p y ,

If f is a t o p o l o g i c a l equivalence.

weak

ChapterVI

We now have

- Homology

enough h o m o t o p y

lized h o m o l o g y an a r b i t r a r y

theory

and c o h o m o l o g y

real closed

for

field.

and c o h o m o l o g y

at our d i s p o s a l (weakly

to build

semialgebraie)

This will

be done

up genera-

spaces

over

in the present

chapter.

In §7 we shall p r e s e n t Brown's

representation

([Bn],

[Sw,

Chapter

V.

chapter,

Chap.

This

up to some easy right

now.

representation

and c o h o m o l o g y could b r i n g

theories

spectra

give a l t e r n a t i v e homology, logy

9]).

theorem

for c o n t r a v a r i a n t

of the p r e c e d i n g

results

(We will

theorem

in §I and give

by spectra

into play

long way

of reduced

We can t r a n s f e r

logy of w e a k l y

semialgebraic

topological

and s e m i a l g e b r a i c

Already

stage.

spaces

arguments

references.)

of h o m o l o g y chapter

approach

we

This w o u l d in co-

algebraic

representation by spectra

would

and thus

of some results

in c l a s s i c a l

theorem

topoto a

[Sw, Chap.

14].

setting,

but,

to g e n e r a l i z e d

be too much

to

in the p r e s e n t

cross

to the s e m i a l g e b r a i c this

functors

some notations,

proofs

theories

and taste,

sections

earlier

of

as an a d d e n d u m

In the p r e s e n t

from Brown's

this d e s c r i p t i o n opinion

(§8).

easier

homology

to the authors

homotopy

to a d e s c r i p t i o n

at a much

but not so in homology.

description

of two v a r i a n t s

the n e c e s s a r y

leads

and sometimes

it is a rather

analogues

section m a y be r e g a r d e d

It is i n d e p e n d e n t

can be read

Brown's

the s e m i a l g e b r a i c

a mixture

homoof

and w o u l d use too m u c h machinery.

§I - The basic

In this

section

homology tions.

categories,

suspensions

we set the stage

and cohomology.

and cofibers

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

We first

compile

of g e n e r a l i z e d

the basic

(Some of them have been used before.)

categorial

nota-

Let R be any real

closed

field.

Notations

1.1.

and ~(2,R)

denotes

F u r t h e r ~*{R) This

a) ~(R)

space ~(R)

(M,x)

the c a t e g o r y

the c a t e g o r y

denotes

is a full

denotes

of pairs

the c a t e g o r y

subcategory

as the c a t e g o r y

(cf. V,

§2, Def.

the pair

(M,~).

of w e a k

polytopes

polytopes

(M,{x}).

Alternatively

under

every weak

In this way also ~(R)

becomes

a full

R ~) ,

R.

over R.

a pointed we may view

the one point

polytope

over

over

weak p o l y t o p e s

since we m a y regard

polytopes

4). We identify

of weak

of p o i n t e d

of 2(2,R)

as the pair of spaces

of weak

M over

space

~

R with

subcategory

of

(2,R). b)

Similarly

category

we regard

WSA*(R)

the c a t e g o r y

the c a t e g o r y

of p o i n t e d

WSA(2,R)

spaces

of pairs

WSA(R) over

of spaces

of spaces

R as full over

over R and the

subcategories

of

R.

c) InW S A ( R ) we further have the full s u b c a t e g o r i e s of p a r a c o m p a c t locally s e m i a l g e b r a i c spaces LSA(R) and of s e m i a l ~ e b r a i c spaces

SA(R),

finally

the c a t e g o r y

LSAc(R)

SAc(R)

= SA(R) n ~(R)

of p o l y t o p e s

over

spaces

LSA~(R),

R. These

SA*(R) ....

... All c a t e g o r i e s d) For any of these homotopy

category.

H ~ are the h o m o t o p y

~) This

= LSA(R) Q~(R)

categories

lead to c a t e g o r i e s

and of pairs

mentioned

so far are

categories

of spaces full

~ we denote

H ~ has the same objects classes

is a s u b s t i t u t e

WSAc(R).

and the c a t e g o r y

of maps

of the m o r e

between

of p o i n t e d

LSA(2,R),

subcategories

SA(2,R), of WSA(2,R).

by H ~ the c o r r e s p o n d i n g as ~

but the m o r p h i s m s

objects

systematic

in ~. All

but c l u m s i e r

of

these

notation

184

c a t e g o r i e s H ~ are full s u b c a t e g o r i e s of HWSA(2,R). e) We denote by ~

the category of t o p o l o g i c a l CW-complexes.

More pre-

cisely this means the full s u b c a t e g o r y of the c a t e g o r y TOP of H a u s d o r f f topological logical) pairs

spaces which has as objects the spaces which admit a

CW-decomposition.

(M,A) of t o p o l o g i c a l

We denote by

~(2)

the c a t e g o r y of closed

spaces such that M admits a C W - d e c o m p o s i t i o n

w h i c h m a k e s A a s u b c o m p l e x of M. S i m i l a r l y we denote by ~ gory of pointed CW-complexes. a one point space.

(topo-

These are the pairs

the cate-

(M,A) 6 ~9(2) w i t h A

~2 and ~/2. are full s u b c a t e g o r i e s of ~ ( 2 ) .

We de-

note the c a t e g o r y of finite t o p o l o g i c a l C W - c o m p l e x e s by #/2F and the c o r r e s p o n d i n g full s u b c a t e g o r i e s of ~/2. and ~/2(2) by ~ If ~

and

~f/2F(2).

is one of these categories then again H~ means the c o r r e s p o n d i n g

(topological)

h o m o t o p y category.

All c a t e g o r i e s m e n t i o n e d finite direct sums.

so far admit finite direct products and

Some of them,

in fact the most i m p o r t a n t ones,

admit a r b i t r a r y direct sums. We spell this out in a special case.

Remark 1.2. i) Let pointed space M inclusions

(MIII6A)

:= V(MIBI6A)

iI : MI~

c a t e g o r y WSA*(R).

be a family of pointed spaces over R. The (cf. IV,

1.8) together with the natural

M is the direct sum of the family

If we replace the i I by their

(MIEI6A)

(pointed)

in the

homotopy

classes

[il] we obtain the direct sum of this family in HWSA~(R).

ii) Let

(MIiI6A) be a finite family of pointed spaces over R w i t h base

points x I. We equip M

:= ~-[(MIiI6A ) with the base point x

Then M, t o g e t h e r w i t h the natural p r o j e c t i o n s Pl direct p r o d u c t of the family their h o m o t o p y classes family in HWSA*(R).

(Mlil6i)

in WSA*(R).

: M ~ MI,

:= (xlil6i). is the

R e p l a c i n g the Pl by

[pl ] we obtain the direct product of this

185

If A is finite then the pointed space V(MIBI6A)

can and will be identi-

fied with the closed subspace of q-[(MILI6A) c o n s i s t i n g of the points (yll~£A) w i t h Yl = x~ for all indices

~ except at most one.

As a c o n s e q u e n c e of some central results

in Chapters

III and V we can

write down various natural e q u i v a l e n c e s between the h o m o t o p y c a t e g o r i e s listed above.

Remarks

1.3. i) The inclusions HP(R) ~

H~(2,R) ~

HWSA(2,R)

HWSA(R),

H~*(R)~

are e q u i v a l e n c e s of c a t e g o r i e s

(Th. V.4.10).

same holds for the inclusions HSAc(R) ~

HSA(R),

HSAc(2,R) ~

inclusions w i t h

by "LSA" ii)

HSA(2,R),

and the analogous

HWSA*(R),

HSA~(R) ~

The

HSA*(R), "SA" r e p l a c e d

(III.§1, V.2.13).

If K is a real closed field e x t e n s i o n of R then the functor

extension"

(M,A) ~

(M(K),A(K))

from 2(2,R)

to 2(2,K)

"base

yields an equi-

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

K : H~(R,2)

~ , H~(K,2).

The r e s t r i c t i o n s of < to H~(R) to H~(K)

and from H ~ ( R )

to HP*(K)

Theorems V.5.2.i and V.7.15. LSAc,

SAc, LSA$,

and HP*(R)

If ~

are e q u i v a l e n c e s

respectively.

from H2(R)

This follows from

is one of the " m e t a - c a t e g o r i e s "

SA$, LSAc(2,-) , SAc(2,- ) then it is already clear

from C h a p t e r III that < yields by r e s t r i c t i o n an e q u i v a l e n c e ~(R)

from

to ~(K).

iii) The functor from ~(2,IR) weak p o l y t o p e s topological

(M,A)

to TOP(2) w h i c h sends every pair of

over IR to the pair

(Mtop,Ato P) of u n d e r l y i n g

spaces yields a full e m b e d d i n g

: H~(2,]]~) ~

HTOP(2),

i.e. an i s o m o r p h i s m of H~(2,IR) M o r e o v e r every object

in ~(2)

onto a full s u b c a t e g o r y

~(2) of HTOP(2).

is isomorphic to an object in I&~(2), and

186

every object in the image categories ~, ~*, ~ F ' ~ '

~ F (2) of H~(IR) ,

H9*(IR) , HSAc(IR),

HSA~(~R) , HSAc(2,]I~)

is isomorphic to an object in

I@2, ~I)*, ~/2F, ~/2~,

4~F(2 ) respectively.

All this follows from Theorems

V . 5 . 2 . i i and V.7.16.

As a c o n s e q u e n c e we can m o d i f y I to an e q u i v a l e n c e

of c a t e g o r i e s

X'

: H#(2,IR)

, ~9(2)

which r e s t r i c t s to e q u i v a l e n c e s HSAc(2,IR)

to

~2, ~W)~, ~ F '

Notice that in remarks

from HP(IR) , H2*(IR) , HSAc(IR) , HSA~(IR),

~J/)* F' ~ F

(2 ) respectively.

ii) and iii) we may replace the letter

"WSA", and similarly SA c , LSA c, by SA, LSA,

'~" by

as a c o n s e q u e n c e of i).

But in the following p a r t i a l l y complete spaces will play a dominant role.

We now focus a t t e n t i o n to the category?*(R)

of p o i n t e d weak p o l y t o p e s

over a fixed real closed field R. Since we have e q u i v a l e n c e s H~*(R o)

~ ~ H2*(R),

H~*(Ro)

- ~ H~*(IR) , H~*(IR)

~ H~9*,

the reader

may b e l i e v e that the h o m o t o p y theory of pointed weak p o l y t o p e s over R is somewhat identical to the h o m o t o p y theory of t o p o l o g i c a l CW-complexes, and that this in p a r t i c u l a r holds theories, theory

w h i c h in t o p o l o g y

(cf. e.g. the book

for reduced h o m o l o g y and c o h o m o l o g y

are known to be a part of stable h o m o t o p y

[Sw]). We shall v e r i f y this to a large extent

in the present chapter.

U n f o r t u n a t e l y the remarks this,

1.3 do not yet imply a m e t a t h e o r e m like

since the e q u i v a l e n c e between two c a t e g o r i e s

is not such a strong

statement as one might wish. We shall resort to the more e x p l i c i t form of our main theorems

in V,§7 and other h o m o t o p y results.

In the following we most often denote a pointed space

(M,x o) briefly

by M and the set of h o m o t o p y classes of maps from M to a pointed space

187 N by

[M,N]

(instead

of

[M,N]*

which

would

fit b e t t e r

with

Def.

4 in

v, §2).

Definition M,N

over

I. a) The

R is the

if M and N c a r r y is a p o i n t e d the

product

cannot

The

the

define

every

Using

of p o i n t e d

Notice

sphere

of two p o i n t e d

base

in a n a t u r a l

products

that,

MvN

pointed

polytope

polytopes

Notice then

also MAN of

to t o p o l o g y ,

we

spaces.

by its n o r t h

M is the

S I = I/{0,I}

that

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

in c o n t r a s t

as p o i n t e d

weak

identification

MvN/MvN.

since

of a r b i t r a r y

S n = Sn(R)

point

weak

CW-complexes

way,

also

SM of a p o i n t e d

a fixed

M AN

with

structure

MxN.

smash

product

MxN/MvN

CW-complex

suspension

SIAM.

space

complex

b) We r e g a r d

smash

smash

p o l e ~. product

we m a y w r i t e

SM = I x M / ( I x x o) U (OxM) U (I×M).

c)

Any

induce

two maps a map

f : M ~ M',

fag

: MAN

g

: N ~ N' b e t w e e n

~ M'AN'

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

pointed

weak

f induces

polytopes

a map

Sf f r o m

SM to SM'. d)

For

skM. skM

We h a v e

k £ ~

sphere

we d e n o t e

o

S ° M = M and

is the k - f o l d

Every ~k

every

the

sk(sIM)

iteration

smash

= sk+IM.

of the

S k, k > I, is e q u i p p e d

with

in H2*(R),

X 6WSA*(R)),

gives

object

is a b e l i a n

of the

smash

~kAidM

turns

skM

which

for

the

group

every

for k > 2. N o w

product

with

such

respect

point

preserving

to d i r e c t

=

also

[sk,x].

from

sums

then

to M.

map

(S k ~ ,[~k ]) is a c o g r o u p

(and t h e n

on ~k(X)

by

if k > O ,

applied

that

it is e v i d e n t

briefly

functor

a base

X 69*(R)

structure

skAM more

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

suspension

: Sk ~ skvs k, unique_ up to h o m o t o p y ,

object

product

every This

cogroup

the d i s t r i b u t i v i t y

(= wedges)

that

also

: skM ~ s k M v s k M

into

a cogroup

in H~*(R)

for e v e r y

pointed

weak

polytope

M,

188

which

is a b e l i a n

k >I,

the

abelian

The

set

for k > 2. Thus,

[skM,x]

for e v e r y

is a g r o u p

in a n a t u r a l

~I can be d e f i n e d

([2t],[O]) Ul ([t])

Here

[t] d e n o t e s

take

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

skvs k =

the

image

(SIrs I) A S k-l,

plication

in ~k(X).

equally we h a v e

Lemma

way,

X and

and this

every

group

is

1.4.

well the

S(Pk)

Definition

of t in SI

Pk the m a p cf.

the

PiAidsk_1

formal

If k > 2

from

in III,

p.

argument

any o t h e r of the

following

--Pk+1

follows

= I/{0,I}.

formula

By a p u r e l y employ

as

0
= ([O],[2t-I])

Thus

space

if k > 2 .

comultiplication

could

pointed

t h e n we

S k = SI^s k-1 265

can to

for the m u l t i -

(e.g.

[Sw,

2.24])

one

k factors

in S k = S I ^ ... A S

polytopes

then

fact.

for e v e r y

k_> I.

2. If M and N are p o i n t e d

weak

the

suspension

homomorphism

SM, N

: [M,N]

~

is d e f i n e d

by SM,N[f]

In g e n e r a l

this

[SM,SN].

But

it f o l l o w s More

H~(R).

:= [Sf].

is just

a map

if M is a l r e a d y

from

generally

group

[SM,SN]

it can be s h o w n

[SM,SN].]

lemma

that

it does

not m a t t e r M,

resp.

the

[M,N]

formal

way

that

factor

N to d e f i n e

L then

homomorphism.

or N is a g r o u p

we use the

group

to the g r o u p

polytope

SM, N is a g r o u p

object

whether

set

SL of a w e a k

in a p u r e l y

if M is a c o g r o u p

SM = S I ^ M or the c o g r o u p on

the p o i n t e d

a suspension

the p r e c e d i n g

homomorphism {Again

from

SM, N is a object

in

S I in

multiplication

189

We will

often

denote

long as no c o n f u s i o n is used

the s u s p e n s i o n is possible.

for the s u s p e n s i o n

homomorphism

(In m u c h

homomorphism,

SM, N b r i e f l y

of the l i t e r a t u r e but we need

by S, as the

letter

"Z" to denote

sets of patches.)

Theorem

1.5

polyto p e

(Freudenthal's

which

suspension

is n - c o n n e c t e d

theorem).

for some

n >O.

Let N be a p o i n t e d

Then the

suspension

weak

homo-

morphism

SSq,N

: ~q(N)

is b i j e c t i v e

for

I < q < 2n and

This can be d e d u c e d Of course

~ ~q+1 (SN)

from Cor.

we can obtain

surjective

V.7.13

the t h e o r e m

topological

Freudenthal

suspension

the t h e o r e m

to the s u s p e n s i o n

for q = 2n+I.

as in topology, also by direct

theorem.

homomorphisms

cf.

[Sw, p. 85].

transfer

In §7 we will

from the

generalize

S SqM,N

Definition

3. a) Let f : M ~ N be a map

space N. The cofiber ed" m a p p i n g subspace us to use with

OxM.

{It will

subspace

of Z(f).}

the base p o i n t of M. This that the natural

projection

C(f) ~

more

C(f)

N ~

convenient

which

weak p o l y t o p e

subspace

C(f)

C(f)

preserves

of C(f)

is a h o m o t o p y

for space as a

is the compoZ' (f) ~

via

C(f).

M to a p o i n t e d

C ( f ) / I x x O with

space has the virtue, j : N~

"switch-

complete

as a pointed

C{f)

is the q u o t i e n t

pointed

closed

Z' (f)/OxM of the

Z' (f) and the p r o j e c t i o n

C(f)

embedding

We regard N as a p o i n t e d the natural

N~

from a p o i n t e d

cofiber

M to a

The space N will be r e g a r d e d

by the e m b e d d i n g

embedding

f : M ~ N be a m a p

We regard

polytope

by the p a r t i a l l y

turn out to be s l i gh t l y

of C(f)

space N. The r e d u c e d

C(f),

(IxM) UIxM, f N

base p o i n t OxM/OxM.

site of the natural b) Let

of f is the q u o t i e n t

Z' (f) =

Z' (f) instead

natural

(closed)

cylinder

C(f)

from a w e a k

xo

not shared by base points.

j. Notice

equivalence

that

of p o i n t e d

190

spaces

(cf. V . 2 . 1 5 ) .

If N is a w e a k In c o n t r a s t arbitrary

polytope

to t o p o l o g y

spaces

Examples

1.6.

over

i) T h e

t o p e M is t h e c o n e has a base reduced

x

pointed

cofiber

then

o

smash

of

the

the r e d u c e d

is a w e a k

cofibers

of m a p s

polytope.

between

identity

map

its v e r t e x cofiber

id M of a w e a k as b a s e

poly-

point.

of this m a p

If M

is the

point

O (XoXI)

product

0.

the c o f i b e r by gluing

tion M UCA

(We w i l l

C(i) the

is j u s t i f i e d

and their

p

C(f)

IAM,

the u n i t

always

interval

adopt

this

I being

convention

equipped about

with

I as a

space.)

obtained

C(i)

resp.

define

CM = IxM/OxM with

If A is a p a r t i a l l y

M then

we cannot

C(f)

R.

:= I x M / ( O x M )

is t h e

the base

ii)

point

also

cone

CM This

then

union

: M UCA

is a h o m o t o p y

complete of the space since

is C(i).

~ M U CA/CA

equivalence

- hence

closed

- subspace

inclusion

i : A~

M is the

C A to M a l o n g

A by the map

b o t h M a n d C A are The

closed

natural

projection

pointed

spaces

of a s p a c e space M UCA

i. The n o t a subspaces

of

= M/A

between

since

C A is c o n t r a c t -

ible. iii)

If A is a p o i n t e d

M then

the

reduced

M UCA/IX{Xo},

which

partially

cofiber we

C(i)

justly

complete of the

denote

subspace

inclusion

by M U~A.

is a h o m o t o p y

~ S UdA/CA

equivalence

= M/A

between

pointed

A ~

Again

projection

: M UdA

of a p o i n t e d

spaces.

M is the

space

space

the natural

191

iv)

Let

point

f : M ~ N be a map

x O to a p o i n t e d

of t h e

switched

Z' ( f ) / I x x

space

mapping

the reduced

o

natural

closed

points,

which

of p o i n t e d

from a pointed

cylinder switched

embedding we regard

N. W e m a y

regard

polytope

mapping Z(f),

the quotient

cylinder x ~

inclusion.

M with

Ixx O as a c l o s e d

Z' (f). W e c a l l

i : M~ as an

weak

base subspace

Z(f)

:=

of

f. W e h a v e

a

[ (O,x)],

preserving

base

We have

a commuting

diagram

spaces

M(

i

~ ~(f)

.

i)

B M

with

canonical

8 : C(i)

Both

, N

f

a and

projections

~ and

= Z(f) U C M

~ Z(f)/M

B are homotopy

a canonical

homotopy

ed as t h e r e d u c e d unreduced

Lemma

1.7.

polytope.

~ C f)

equivalences

equivalence,

cofiber

= C(f).

of an

of p o i n t e d

every

reduced

inclusion.

spaces.

cofiber

{A s i m i l a r

Thus

may

u p to

be r e g a r d -

remark

holds

for

cofibers.}

Let

For

(M,A)

every

be a c l o s e d

pointed

pair

space

of p o i n t e d

X the

natural

spaces

with

A a weak

sequence

of

pointed

We

could

sets

[M U CA,X]

~

[M,X]

i__~

[A,X]

is e x a c t .

This place

is o b v i o u s the

Let now

f

first

from the definitions term

in t h e

: M ~ N be a m a p

(cf.

exact

sequence

between

pointed

[Pu, by

p.

305]).

[M/A,X].

weak

polytopes.

Clearly

re-

192

C(f)/N

Lemma

= SM.

1.8.

We denote

We have

the

canonical

an i n f i n i t e

projection

commutative ,

M

f

, N

J

~ C(f)

- j

N

j

~ t(f)

q

M ~

Here

j,j',j",..,

the p r e c e d i n g obvious

natural

pointed

weak

the

right

All

this

of

their

projections.

polytopes.

if w e

apply

4. W e

sequence

The

the

c a n be c h e c k e d

Definition cofiber

into

call

as

of f a n d

all

lower

suspension

the

lower

Sj

, SCf

target

spaces

The vertical

sequence

maps

are

three

,

of

the

equivalences

moves

,

of

steps

to

functor.

in t o p o l o g y

long

of the

are homotopy

long

, ~(j")

, SN

injections

They

'"

~ d(j')

Sf

cofibers.

the upper

j

J

~ SM

S M b y q.

diagram ,,

, t(j)

are the natural

maps

C(f) ~

(cf.

[Pu,

sequence long

p.

3OSff]).

in t h e d i a g r a m

sequence

the Puppe

the Io~ sequence

f.

As a c o n s e q u e n c e 1.7,

of o u r d i s c u s s i o n

1.8 w e o b t a i n

Theorem

1.9

pointed

weak

Applying exact

the

following

(Barratt[Ba],

the

polytopes functor

sequence

and

[-,X]

of p o i n t e d

Puppe

of c o f i b e r s

theorem

[Pu]).

of b a s i c

Let

sets

Puppe

and,

and the

lemmas

importance.

f : M ~ N be a m a p

let X be a f u r t h e r to the

above

pointed

sequence

starting

weak

between

polytope.

of f w e o b t a i n

from the

fourth

a long

term,

groups

[M,X] ~

Starting

from

[N,X]

the

~

[C(f),X]

seventh

term

-

[SM,X] ~

the

groups

[SN,X]

are

~

[SC(f),X]

abelian.

-

[S2M,X]

~...

193

In the

special

Corollary another

1.10.

pointed

case

If

of an i n c l u s i o n

(M,A)

weak

is a p a i r

polytope

map we obtain

of p o i n t e d

then we have

weak

polytopes

a natural

a n d X is

long exact

sequence

[A,X] ~

Definition

[M,X] ~

[M/A,X]

5. W e c a l l

the

-

[SA,X] ~

sequence

in t h i s

corollary

the s u s p e n s i o n

If f : M ~ N is a n y m a p sequence

of

sequence

of t h e p a i r

between

f c a n be i d e n t i f i e d (Z(f),M),

[S2A,X] . . . .

....

sequence

pointed

weak

in a n a t u r a l cf.

[S(M/A),x] ~

of m a p s

A ~ M ~ M / A ~ S A ~ S M ~ S(M/A)

used

[SM,X] *

1.6.iv.

of the

polytopes way with

closed

then the

pair

the

(M,A).

Puppe

suspension

§2 - R e d u c e d

cohomolo@y

The category

~*(R)

category the

Hg*(R)

of p o i n t e d

both

suspension

of w e a k

are

functor

polytopes

weak

equipped

S. U s i n g

polytopes with

S we

of r e d u c e d

homology

category ~*(R),

as

in t o p o l o g y

CW-complexes

[Sw,

is d o n e Chap.

first.

In the

we w i l l

F

: H~*(R)

we w i l l f* as

~ Ab.

often

long

a family with

abusively

(onln62)

all

Exactness

axiom.

and e v e r y

n 6 Z the

For

P*

Here

theories ~*

work-

on the of p o i n t e d

every

pair

of a b e l i a n

contravariant between

groups.

functors

pointed

weak

homomorphisms

polytopes

F([f])

by

feared.

(semialgebraic)

of n a t u r a l

category

the g r o u p

is to be

: kn+1 oS -L*k n such

on

k n (M/A)

the

many

of c o n t r a v a r i a n t

functors)

is exact.

consider

denote

I. A r e d u c e d

a family

and w e l l

on the c a t e g o r y

If f : M ~ N is any m a p

(knln62)

endomorphism,

natural

and c o h o m o l o g y

Let Ab d e n o t e

as no c o n f u s i o n

Definition

give

its h o m o t o p y

7].

We do c o h o m o l o g y following

R and

a distinguished

can

ing d e f i n i t i o n s

over

c~homology

functors

kn

: HP*(R)~

equivalences

that

the

(M,A)

theory

k* o v e r Ab

together

(= i s o m o r p h i s m s

following

of p o i n t e d

two

weak

R is

axioms

between hold.

polytopes

over

R

sequence

i* ~ k n (A)

k n (M)

i denotes

the

inclusion

A~

M and

p denotes

the p r o j e c -

t i o n M - ~ M/A. Wedge

axiom.

every

n 6 Z the m a p

(i~)

For

every

family

: kn(V(Mlli6A))

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

Here

(MItI6A)

of p o i n t e d

weak

polytopes

and

~ ]-F(kn(MX) [ICA)

i I denotes

the n a t u r a l

embedding

of M l into M.

195

Actually some ral

it

bound

suffices n o E Z.

equivalences

Then on

S(V(XlII6A))

Moreover, for

if t h e

n ~n o such

tend

the

to d e m a n d they

since

follow

S(M/A)

of

these

for

the

= SM/SA

axioms

other

for

n by

n ~n o with

use

of

the

natu-

and

= V(SX~II6A).

functors

that

the

of

these

family

each

k n and

axioms

the

above

functors

equivalences hold

k n to

for

on are

these

a reduced

n,

defined

then

we

cohomology

only

can

ex-

theory

by defining

k no-r(x)

for

"= k n ° ( s r X )

r > O.

Notice

also

exactness then

that

axiom.

the

wedge

Indeed,

axiom

for

A finite

if M I a n d M 2 a r e

M I V M 2 / M 2 = M I a n d M I v M 2 / M I = M 2.

By

two the

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

weak

exactness

of

the

polytopes,

axiom

the

diagram ii M I,

of n a t u r a l

injections

abelian

groups

We

call

the

of

the

§4 w e

theories

We

draw

k•.

and

the

theory" will

will

some

be

need

projections

on

theory.

index

- M2

becomes

a direct

sum

diagram

of

k n.

equivalences

"cohomology from

under

cohomology

o omitting

P2 i2

~ M I v M2.

Pl

n.

We

: k n+1 o S C ~ k n t h e usually

In t h e

instead

of

to be m o r e

denote

following "reduced careful

we

suspension by

isomerphisms

them

all

the

also

say more

letter

briefly

cohomology

theory".

since

also

then

same

(Starting

unreduced

studied.)

consequences

from

the

axioms

of

a given

cohomology

theory

196

If

(M,A)

every

is any p a i r

by composing (n*) -I

jection

polytopes

then we define,

for

~ kn+1(M/A)

: kn(A) -t~kn+1(SA)

exactness

projection 1.6.iii).

with

~ k n+1 (M/A).

f r o m M U C A to M U C A / M = SA,

(Ex.

Here

as

: k n+1 (SA) ~ k n+1 (M U C A )

q~

q denotes

in L e m m a

f r o m M U C A to M / A w h i c h

As a consequence

of L e m m a

the natural

1.8,

and

pro-

~ denotes

is a h o m o t o p y

equiva-

1.8 w e d e d u c e

f r o m the

axiom

Proposition abelian

o -I

: kn(A)

: k n+1 (M U CA)

the natural lence

weak

n 6 Z, a h o m o m o r p h i s m

6 = 6n(M,A)

and

of p o i n t e d

2.1.

groups

For

every

pointed

(going to i n f i n i t y

6_~ k n (M/A)

P~

k n (M)

WP-pair at b o t h

(M,A)

the

long

sequence

of

sides)

6 J kn+1 (M/A)

i * ~ k n (A)

is exact.

Corollary

2.2.

For

we have

a natural

at b o t h

sides)

every map exact

-~ kn(~(f))

This

follows

the Puppe same

(cf.

Let

(M,A,B)

since,

pointed

groups

weak

polytopes

(going to i n f i n i t y

~ k n+1 (~(f))

u p to c a n o n i c a l suspension

homotopy

sequence

of

equivalences, (Z(f),M)

are

§1).

be triple

n 6 2 we define

of a b e l i a n

f* ~ kn(M)

of f and t h e

e n d of

A = ~n(M,A,B)

sequence

, kn(N

2.1

sequence

the

every

from

~

f : M ~ N between

of p o i n t e d

weak

polytopes

a homomorphism : k n (A N B )

~ kn +I (M) ,

with M = A UB.

For

197

as the

composite

k n ( A N B)

with

8 = 6n(A,A DB),

by the n a t u r a l natural

space

2.3.

"Mayer-Vietoris A.

isomorphism

=

inclusion

This

follows

[Sw,

p.

a

of

the g r o u p

A / A N B -t~M/B,

WP-triple

(M,A,B)

isomorphism and p

with

maps

B are d e f i n e d

: A~

induced

: M ~ M/B

the

M = A UB

the

infinite

A , kn+1 (M)

by

: i~u - i~v

M,

Proposition

J2

2.1

: B ~

M,

in a w e l l

i I : A DB ~

known

way,

A,

cf.

i 2 : A DB~B.

[ES, p. 39ff] ,

that

= -An(M,A,B),

38],

[Sw,

2.4.

Let

3) of a p o i n t e d

q 6 ~, a n a t u r a l

~(u,v)

Jl

Notice

An(M,B,A)

--~ k n (A NB)

kn(A) @ k n ( B )

and

from

105].

Proposition

every

(j~Z,j~Z),

with

p.

For

~

Here

~(Z)

Def.

inverse

kn+1 (M)

sequence"

.kn(M) . .

is exact.

[ES,

y the

~ ~ kn+1 (M/B)

projection.

Proposition

cf.

kn+1 (A/A n B)

~

exact

p.

(Mnln6~) weak

the h o m o m o r p h i s m

M n~

M.

This

follows

be an a d m i s s i b l e

polytope

M.

Then

filtration

there

exists,

(cf. V,

for e v e r y

sequence

O ~ l i m I k q-1 (M n) n Here

106].

a , kq(M )

B is induced,

The h o m o m o r p h i s m

by a p p l y i n g

B , lira k q ( M n) ~ 0 n of c o u r s e ,

a is e x p l a i n e d

the p r e c e d i n g

in

by the

[Mi],

proposition

[Sw,

inclusions p.

to the

128].

reduced

§2,

198

telescope suitable [Sw,

NB.

Tel(~) closed

p.

128],

Tel(R)

closed

had been

It is c o n t a i n e d

goal

cohomology with

the

Definition families

natural

in V,

following over

~

:=

(Mnln6~)

A U B = T61(~) of Thm.

§4 a n d h a d b e e n x o denotes

any

terms

real

and

(cf.

[Mi],

6.6 b e l o w ) .

identified

the base

1 • be t w o

isomorphisms

transformations

T

with

point

Tn+I(sx)

(cf.

a

of M.

terms,

[Sw,

Chap.

uniquely

7])

In o r d e r

the

o n the

to e x p r e s s

terminology.

cohomology (~nln62)

theories and

over

R with

(Tnin6~).

f r o m k ~ to 1 • is a f a m i l y

functors

a n d n 6 2, the

in v a g u e

R correspond

CW-complexes.

: k • ~ i*

between

that,

field

theories

we n e e d m o r e

transformation

kn+1(SX)

closed

topological

2. L e t k • a n d

X 6 2*(R)

is to p r o v e

cohomology

of s u s p e n s i o n

for e v e r y

with

in the p r o o f

. Of c o u r s e ,

of p o i n t e d

a) A n a t u r a l

family

in e v e r y M n.

in t h e

in p r e c i s e

of Tel(~)

s t e p b)

of M x I

theories

~W*

A,B

defined

topological

category this

subspaces

see a l s o

subspace

Our main

:= T e l ( ~ ) / X o × I ~ of t h e

(T n

: k n ~ inln6~)

such

of that,

square

, in+1(SX)

_-- 7 n ( x )

on(x)

kn(X)

i n (X)

Tn(x) commutes. b) W e c a l l if,

T a natural

in a d d i t i o n

Tn(X)

equivalence, is an

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

isomorphism

for e v e r y

f r o m k ~ to i*

X 6P~(R)

and

every

n 6 2.

Proposition between

2.5.

Let T,U

cohomology

: k* ~

theories

over

1 • be t w o R.

natural

transformations

199

a) A s s u m e

that

Tn(s °)

= u n ( s °)

for e v e r y

b) A s s u m e

that

Tn(s °)

is b i j e c t i v e

n 6 2. T h e n

for e v e r y

T = U.

n 6 ~. T h e n

T is a n a t u r a l

equivalence.

The

proof

axiom

runs

and

similar

Propositions

Proposition

2.5

ly d e t e r m i n e d , of a b e l i a n groups

later

Let

groups

K be a real

K "restricts"

(2.6)

{Notice

us t h a t

theory

closed

123f]

cohomology

made

These

precise

groups

theories

over

using

the w e d g e

are

theory there,

called

k*

is u n i q u e -

by the the

sequence

coefficient

over

IR w i t h

of R.

theory

different

topological

Every

base

cohomology

cohomology

(k*) R o v e r

fields

theories.

theory

R in the

and

k * over

following

for n 6 • and X 6 2" (R),

(kn)R(x)

:= kn(X(K)

(<~n)R(x)

:= <~n(X(K)

(SX)(K)

p.

k*.

to a c o h o m o l o g y

that

sense

overfield

We d e f i n e ,

[Sw,

a reduced

cohomology

theories

in

2.4.

(kn(S°) In62).

to c o m p a r e

way.

proof

in a r e s t r i c t e d

cohomology

obvious

2.1,

tells

of c o h o m o l o g y

We w a n t

to the

= S(X(K)

.} We

further

define,

for

any m a p

f:X~Y

(kn)R([f])

Every theory to

natural i* o v e r

:= kn([fK])

transformation K restricts

(i*) R, d e f i n e d (Tn)R(x)

T

: k* ~ I* f r o m k* to a n o t h e r

to a n a t u r a l

transformation

cohomology

T R from

(k~) R

by

= Tn(X(K))

Let H o m ( k * , l ~) d e n o t e

the

set of n a t u r a l

transformations

f r o m k*

to l*.

200

Proposition

2.7.

The

H o m ( ( k ~ ) R, (i~) R)

restriction

is b i j e c t i v e .

a natural

equivalence

The p r o o f

is an e a s y

that

the base

Let Coho(R) theories

which

denote

over

between

them.

dealing

with

field real

which

the

maps

an object

to T R. W e c a l l

Theorem

2.8

the cohomology

theories

to c o n s t r u c t , theory

L e t M 6 P*(K) consisting

are

over

2.7 m e a n s

be given.

of a p o i n t e d

to

One

uses

the

fact

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

the

are

the

natural

eohomology

transformations

is a r e m i n d e r For

every

every

that we

real

embedding

are

closed

R~

K into

a

functor

restrictions

over

T of C o h o ( K )

T to R.

theories),

to C o h o ( R ) .

theories

K

.

res R is an

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

K correspond

up

uniquely

to

R.

that

with

res~

is f u l l y

theory

a natural

We d e n o t e

equivalence

of k ~ a n d

for c o h o m o l o g y

cohomology

weak

(k*) R a n d a m o r p h i s m

Coho(K)

cohomology

K together

[~] o f a h o m o t o p y

objects

a restriction

theorem

for a g i v e n

k* o v e r

~ Hg~(K)

(more p r e c i s e l y ,

category

the

Proposition

theory.

notation

k* of C o h o ( K )

to isomorphism

Proof.

equivalence.

theories.}

(k~) R and T R t h e

from the

K is

suspensions.

whose

in t h i s

T over

C o h o (R)

(First m a i n

equivalence

with

morphisms

K) w e h a v e

: C o h o (K)

HP~(R)

cohomology

K of R

res K R

transformation

in c a t e g o r y

category

{The t i l d a

field

A natural

functor

R and whose

extension closed

exercise

commutes

reduced

T ~ T R f r o m H o m ( k ~ , l ~) to

iff T R is a n a t u r a l

extension

of c a t e g o r i e s

map

by

~

: X(K)

i* o v e r

equivalence

I(M,R)

polytope

faithful.

the

X over ~ M.

It r e m a i n s

R a

cohomology

T

: i* -t~ (k*)R

set of all p a i r s

R and

the homotopy

Such pairs

exist

by

(X,[~]) class

201

*) Theorem then, X

V.7.15.

again

If

by V . 7 . 1 5 ,

: X ~ Y such

we have

that

i as above, transition

for any maps

are

kn(M)

Roughly

with

(Y,[~])

exists,

system

transition

pairs

(X,[~]

isomorphisms

direct

system.)

choice

about

the

X (K)

6 I(M,R)

abusively

we denote neither

system

a unique

could

also

the

natural

specifying

g

(X,[~])

"approximating

(Since

the

be r e g a r d e d

the

are

: X ~ Y over

6 I(M,R)

pair"

space

given,

R such

f r o m kn(M)

X nor

weak

the n u m b e r

polytope then

that

without

(X,[~]).

projection

pointed

(Y,[~]) 6 I(N,R) map

in I(M,R).

for any

f r o m M to a n o t h e r

6 I(M,R),

~

groups

6I(M,R))

a definite

up to h o m o t o p y ,

map

We d e f i n e

= in(x)

(X,[~])

a unique

maps

, (Y,[¢])

this

kn(M)

if

I(M,R)

r

two

If f : M ~ N is a m a p

of

equivalence.

of a b e l i a n

we put

by ~*,

two e l e m e n t s

and X is a h o m o t o p y

:= lim(in(x) [ (X,[~])

(X,[~])

are

up to h o m o t o p y ,

speaking,

any

K and

and

inverse

EI(M,R))

as a g e n e r a l i z e d

in(x)

~OXK =~,

: in( Y ) ~ i n ( X )

×*

For

there

a generalized

(in(x) I (X,[~])

making

(X,[~])

to n.

N over

there

exists,

the d i a g r a m

, M

gK Y (K)

commutes dotted

*) We

~

, N

up to h o m o t o p y .

arrow

ignore

a suitable

which

makes

the p r o b l e m

We d e f i n e the

: kn(N)

~ kn(M)

as the

square

whether

interpretation

kn([f]

of

I(M,R) "all".

as a set.

It can

be s e t t l e d

by

202

k n (M)

i n (X)

in[g]

kn[f]

kn(N)

commutative.

~ 4"

, in(y)

The h o m o m o r p h i s m

(X,[~])

and

(Y,[~]).

variant

functors

kn[f]

In this w a y we o b t a i n

f r o m HP*(K)

o

some

arrow

Then

such that

(S~) *

k n+1 (SM)

We c h e c k

M

Clearly

on(M)

(SX)

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

of the c h o i c e

axiom

For e v e r y

:= V ( M I I I 6 A ) ,

is an e l e m e n t natural

the s q u a r e

• I n (X)

the w e d g e

in 9 * ( K ) .

We d e f i n e

I n (X)

k n (M)

independent

(SX,[S<0]) 6 I(SM,R).

, i n+1

(M)

commutes.

of c o n t r a -

isomorphism

(X,[~]) 6 I ( M , R ) .

(M) as the d o t t e d

n

(knln6Z)

of

: kn+1 (SM) c ~ k n ( M )

on(M)

n

a family

on the c h o i c e

to Ab.

We n o w look for a s u s p e n s i o n

We c h o o s e

does n o t d e p e n d

X

T T k n (MI)

(X,[~]).

for the

It is n a t u r a l

functors

we c h o o s e

a pair

k n. Let

Let jl

: X1 ~

We h a v e a c o m m u t i n g
, I n (X)

~ T T I n (Xl)

groups which

(MIII6A)

~ M. T h e n

X and i I : M I ~ square

is

in M.

be a f a m i l y

(Xl,[%01]) 6 I ( M I , R ) .

:= V ( X I I I 6 A ) and <0 = V ~ l : X(K)

of I (M,R).

embeddings. k n (M)

I 6A

of

of a b e l i a n

Let (X,[<0])

M denote

the

2O3

By the wedge also

the

axiom

topes

the

(B,[~]) £ I(A,R) exactness

of t h e

k n(A)

from

the e x a c t n e s s

Thus

We denote pair

a natural

2.8

of p o i n t e d

V.7.15.

(~,~)

{ (X,B)

a homotopy

(X/B) (K) . U s i n g

poly-

: (X(K),B(K))

can even

equivalence

the pairs

weak

be c h o s e n

X :X(K)/B(K)

_t~

(X, [~]) 6 I(M,R) ,

it is n o w e a s y

to deduce

the

~ In(X/B)

a cohomology

the pair

projection

Tn

equivalence

theory

(X,[idx(K)])

It d e p e n d s

equivalence

to a n a t u r a l

(X,B)

of p o i n t -

~ k n(M/A)

the n a t u r a l

by Tn(x).

be a pair

equivalence

(X/B,x) 6 I ( M / A , R )

is i n d e e d

any X 69~(R)

(M,A)

Thus

of

~ in(x)

(knln6Z)

Let

a pair

a homotopy

~ induces =

is b i j e c t i v e .

sequence

~ k n(M)

in(B)

For

and

for k n.

by T h e o r e m

X(K)/B(K)

arrow

is b i j e c t i v e .

axiom

with

CW-pair.}

Moreover

vertical

K. W e c h o o s e

is p o s s i b l e

as a p o i n t e d M/A.

over

R together

This

arrow

exactness

polytopes

over

(M,A).

i n the r i g h t

left vertical

We now check ed w e a k

for

-t~in(x)

of

I(X(K),R).

associated

on X in H 9 ~ ( R ) .

: (kn) R ~ i n of T

K.

is an e l e m e n t

kn(X(K))

functorially

over

functors.

The

: (k*) R ~ 1 ~ o f c o h o m o l o g y

to t h i s

Thus we obtain T n fit t o g e t h e r theories.

Theorem

is p r o v e d .

We call

the c o h o m o l o g y

tension

of

isomorphism More

(2.9)

theory

k • over

K constructed

1 • t_~o K, a n d w r i t e

k ~ = i~

. We

T

: (i~) R ~ 1 ~ a n d

explicitely, n IK(X(K))

for e v e r y

= in(x)

thus

X 6P~(R)

feel and

have

the b a s e

established

justified every

here

a canonical

to say t h a t

n 6 ~,

ex-

(i~) R

1• .

204

Conversely

let k* be a c o h o m o l o g y

theory

over

K.

Let M E 9 * ( K ) ,

n 6 ~.

Then

((kn)R)K(M)

=

lira ((kn)R(x) L (X,[~])

= lim (2.6) " We h a v e

a canonical

component

at any

6 I(M,R))

(kn(X(K)) I (X,[%0]) E I(M,R))

group

isomorphism

(X,[~])

U n(M)

is kn([<0]).

This

: k n(M) _m. ((k n)R)K(M)

gives

us a c a n o n i c a l

whose

natural

equivalence

U

By use

: k* -7~ ((k*)R) K

of U we

(2.10)

We n o w

identify

((k*)R)K

look

for a t h e o r e m

semialgebraic theories. (knln6Z)

= k~ .

cohomology

similar theories

A topological of c o n t r a v a r i a n t

topological

with

suspension

isomorphisms

ness

axiom

Let ~

and the w e d a e topological

denote

the

morphisms

are

to d e f i n e

a restriction

more

the n a t u r a l

complicated

the u n d e r l y i n g necessarily

than

(cf.

(cf.

[Sw,

p :~

together

fulfills here

cohomology defined

~ Coho(]I~) of the

the

S means

easy

We w i l l details

theories. as above.}

. This

functors

M t o p of a w e a k

CW-complex. the

7],

1.1)

cate-

exactof

functor).

the d e f i n i t i o n space

cohomology

k • is a s e q u e n c e

notations

Chap.

transformations,

p leaving

theor~

the

: H~9 * ~ Ab on the h o m o t o p y

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

topological

of

kn

relates

topological

: k n + 1 o S C-~k n w h i c h

functor

a topological

the d e f i n i t i o n

on

suspension

category

]I~ to the

CW-complexes

axiom

2.8 w h i c h

cohomology

functors

of p o i n t e d

the

over

(reduced)

gory

course

to T h e o r e m

will

res~

polytope

indicate to the

{The We w a n t

be s l i g h t l y

above, over

the

reader.

IR

steps

since is not of

205

Let

k ~ be a t o p o l o g i c a l

denote

the

set of all p a i r s

~0 : M ~ X a h o m o t o p y by T h e o r e m

V.7.8.

can be r e g a r d e d and

canonical

family.

For

theory.

(X,[~])

equivalence

with

transition

For

system

spaces.

family

let J(M)

CW-complex Such

pairs

(kn(Xtop) l (X,[%0])

of a b e l i a n

isomorphism

any M 62~(IR)

X a pointed

of p o i n t e d

any n E Z the

as a d i r e c t

between

groups

with

and exist

E J(M))

an o b v i o u s

any two m e m b e r s

of the

We d e f i n e

(kn)sa(M)

One

cohomology

extends

:= limp

this

(kn)sa : H ~ * ( I R )

(kn(Xtop) I (X,[%0]) 6 J ( M ) ) .

definition -~ Ab

to a d e f i n i t i o n

in the o b v i o u s

way.

of a c o n t r a v a r i a n t

One

then

defines

functor

suspension

i somorphisms on

~ (k n ) s a : (kn+1 )Sa°S -~

in a n a t u r a l

straightforward

way, and v e r i f i e s for the f u n c t o r s

(k n ) s a the

e x a c t n e s s a x i o m and the w e d g e axiom. For e x a c t n e s s one n e e d s C W - a p p r o x i m a t i o n s of p a i r s of p o i n t e d w e a k p o l y t o p e s ,

i.e. C W - t r i p l e s w i t h a

one p o i n t

s p a c e as last c o m p o n e n t . T h e y e x i s t by T h e o r e m V . 7.14 (or a l r e a d y V.7.8) .

In this

way

a cohomology

we

the

semial~ebraic

call

algebraic

CW-complex

(2.11)

via

(kn)sa(M)

the

Every

injection

natural

theories

leads

t h e n we m a y

square

IR

of k*.

is e s t a b l i s h e d

If M

which

is a p o i n t e d

semi-

identify

= kn(Mtop)

(M,[idM])

to a n a t u r a l way

(k*) sa o v e r

restriction

transformation

straightforward n 6 ~, the

at

theory

such

T

E J(M)

: k* ~ 1 • b e t w e e n

transformation

that,

for e v e r y

Tsa

topological : (k ~)sa

M E2*(IR)

~

cohomology

(l~)sa

in a

, (X,[~]) 6 J(M), and

206

kn

(Xto p

, (k n) sa (M)

)

I

(T n) sa (M)

Tn(Xtop) in

(i n) sa (M)

) (Xto P

commutes. the

Here

the h o r i z o n t a l

"coordinate"

We n o w h a v e

from ~

Theorem

2.12

striction cular,

the d e s i r e d

restriction

(Second m a i n p :~

topological

theorem

equivalence.

cohomology

similar

to the

proof

points

Theorem

V.7.16

is needed.

but m e n t i o n

with

a natural

(namely

can

such

theories

theories

runs

]19, we

that,

(k~) sa,

and

This

is d o n e

as f o l l o w s .

[~] Such

(X,[~])

the

homotopy

pairs

jective

exist

limit

T is one

with

of the

For

at

(k~) sa,

of the

system

inverse with

1~ ~

semialgebraic

from

the

theorem

semi-

2.8.

the d e t a i l s

cohomology

At

to the

theory

1~

I(M)

denote

the

CW-complex

the g r o u p

of a b e l i a n

to

set of

(over

equivalence i ntop(M)

]19) and ~

: X ~M.

as the p r o -

groups

isomorphisms Theorem

adjoint

way

morphisms).

homotopy

We d e f i n e

emanating

In p a r t i -

isomorphism•

i~o p r i g h t

any M 6 ~ 0 • let

transition

re-

to 1 ~ in a c a n o n i c a l

adjunction

system

with

main

leave

The

cohomology t h e o r y i~o p t o g e t h e r

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

by V.7 • 16.i.

(in(x) I (X,[~]) 6 I(M)) members

of the

up to

first

We

(l[op)Sa

a functor

X a pointed

class

of the

a semialgebraic

T from

we o b t a i n

correspond

IR u n i q u e l y

a topological

equivalence

that

over

given

construct

k~ ~

pairs

p :k* ~

is a n a t u r a l

~ Coho(IR)

The p r o o f

over

functor

theories).

cohomology

reader

injections

for c o h o m o l o g y

algebraic

crucial

the n a t u r a l

to Coho(]l~)

functor

the

denote

(X, [~]).

established

T ~ T sa,

arrows

between

V.7.16.ii.

any

two

207

If X is a p o i n t e d pair

semialgebraic

(X,[idx]) • The p r o j e c t i o n

(X,[idx])

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

(2.13)

CW-complex i ntop(Xtop)

which

then

I(Xto p) c o n t a i n s

~ i n (X) at the

is n a t u r a l

the

"coordinate"

in X. Thus we m a y

identify

i t o p ( X t o p) = in(x)

If M 6~ ~ (]i~) then n (itop)Sa(M)

= lira (i t op(Xtop) P (X, [~]) E J ( M ) ) = lim

We d e f i n e

n sa (M) --~ ~ i n (M) as the i s o m o r p h i s m : (itop)

Tn(M)

at the c o o r d i n a t e

We call theory

(X, [~0]) is the i s o m o r p h i s m

i[o p the t o p o l o g i c a l i s. Since we have

(l~op)Sa

to i s we feel

(2.14)

Given

(ln(X) I (X,[t0]) 6J(M))

extension

a canonical

justified

whose

in[k0].

of the s e m i a l g e b r a i c

natural

to identify,

component

equivalence using

cohomology

T from

T,

~ sa lS (ito p) = a topological

cohomology

theory

k * we have,

for any M 6 ~ s and

n6~, ((k n) sa) top (M) =

.lim

=

lira

(2.117 Thus we have U n(M) whose

a natural

I (X, [~])

6 I(M))

(Xt°p)

isomorphism

at the c o o r d i n a t e

to a n a t u r a l

A g a i n we identify, (2.15)

(k n

~ ((k n) sa) top (M) : k n (M) --~

component

fit t o g e t h e r

((k n)sa(x) [ (X, [t0]) 6 I(M))

using

((ks)sa)top

(X,[t0])

transformation U,

= ks

is kn[~].

These

U from k s to

isomorphisms

((k~)sa)to p.

208

As a c o n s e q u e n c e discussions

we

of the m a i n learn

duces,

for e v e r y

theory

R k~ o v e r

(2.16)

with

R k*

R ° the

over

R as

:=

K then,

field

and

2.13

and

the

cohomology

R, a s e m i a l g e b r a i c

subsequent

theory

k*

in-

cohomology

follows:

of real

,

algebraic

theories

way.

(k~) sa

correspond

(iRkS)top

respect

numbers.

Moreover,

with

the

in this

way

cohomology

the

theories

We h a v e

K is an e m b e d d i n g

with

2.8

topological

(((k~)Sa)R°) R

R in a u n i q u e

if R ~

every

closed

cohomology

IR k~ = and

real

field

topological

that

theorems

= k•

of a real

to this

closed

field

into

a real

closed

embedding,

R

(K k*) By t h e s e use

the

= R k~,

wonderful following

Notations

2.17.

a pointed

weak

the c o h o m o l o g y

In p a r t i c u l a r

correspondences simplified

polytope group

(Rkn) [f]

we n o w

real

simply

theories

and

symplectic

K-theory, between

by kn[f]

all

(singular

over

or,

over more

the w e l l

theories

any

field

by kn(M).

cohomology;

cobordism them

we

cohomology

closed

polytopes

can e v a l u a t e

cohomology

and m a p s

any

weak

simply

equalities

topological

over

pointed

and

feel

justified

to

notations.

(Rkn) (M)

cal

topes

= K k~

If k • is any

f : M ~ N between homomorphism

(Rk~)K

real

theory R then

Similarly

known

for

...)

closed

by

group

f~.

topologi-

unitary,

on p o i n t e d field.

any m a p

the

reduced

orthogonal,

if M is

we d e n o t e

R we d e n o t e briefly,

and

weak

poly-

§3 - C e l l u l a r

Having the

settled

question

closed use

homology

(to the

which

field

reduced

R we n o w m a y

an o b v i o u s

Definition

same

"dual"

extent

cohomology be b r i e f

notation

I. A r e d u c e d

(knln6~)

of c o v a r i a n t

family

(Onln62)

of n a t u r a l

two a x i o m s

Exactness

axiom.

and

n 6 ~ the

every

k n(A)

theories

about

kn

equivalences

in §2,

theory

topology)

a aiven

We

I.

k~ o v e r

R is a

~ Ab t o g e t h e r

: kn --~kn+1OS

real

theories.

Definition

: HP~(R) on

over

homology

homology

functors

algebraic

exist

reduced

to the one

(semialgebraic)

family

following

as in c l a s s i c a l

such

with

that

a

the

hold.

For e v e r y

pair

(M,A)

of p o i n t e d

weak

polytopes

over

R

sequence

-. 1 . ~ k n(M)

p , , k n(M/A)

is exact.

Wedge

axiom.

every

n 6 2 the m a p

ix,)

For

:

every

family

(MIIX6A)

of p o i n t e d

weak

polytopes

and

~ k (M X) ~ k ( V MX) XEA n i~ X6A

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

Dually

to D e f i n i t i o n

(reduced) homology logical

homology theories

homology

[Sw.

Chap.

With

the o n l y

and

2 in §2 we d e f i n e

theories over

over

which

transformations

R and e s t a b l i s h

R. A n a l o g o u s l y

theories

natural

live

we d e f i n e on H ~

the the

instead

category category of H ~ ( R ) ,

between Ho(R)

of

~ of topocf.

7].

exception

the c o r o l l a r y

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

in §2 have

obvious

2.4

all the

"duals"

which

properties,

theorems,

can be p r o v e d

in an

210

analogous

way.

The h o m o l o g y

then that p r o p o s i t i o n

counterpart

and reads

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

as follows

(cf.

[Mi],

2.4 is simpler [Sw, p.

121f]

for a proof).

Proposition weak

3.1.

polytope

R and every

Let

(MnLn6~)

M over

R. Then,

be an a d m i s s i b l e for any reduced

q £ Z, the natural

filtration

homology

of a p o i n t e d

theory

k, over

map

l i m kq(M n) ~ kq(M) n is an isomorphism.

The t o p o l o g i c a l homology phism,

reduced

theories

and we are

we can e v a l u a t e weak

polytopes

Definition reduced

over

homology any real

justified

closed

field

topological

and maps b e t w e e n

topological

correspond

homology

homology

theory

theory

analogous

homology

them over

with

R uniquely

to use a n o t a t i o n

any reduced

2. A reduced

theories

theory

up to isomorto 2.17.

Thus

on pointed

any real closed

k, over

the reduced

field

R.

R - and s i m i l a r l y

a

- is called

ordinary,

if kq(S °) = 0

from T h e o r e m

3.3 b e l o w

(and is well

for q % O.

Remark known

3.2.

It will

become

in the t o p o l o g i c a l

abelian

group G, there

reduced

homology

coming

theory

obvious

theory,

exists, k

cf.

[Sw, Chap.

up to isomorphism,

over

R with

ko(S°)

from r e d u c e d

singular

homology

H~(-,G)

Let k~ be a reduced

ordinary

homology

theory

Going

to m o r e c o n c r e t e

torial

"cellular"

M over

R, w h i c h

terms we want

description

is c o m p l e t e l y

10])

analogous

for a given

a unique

& G. This

ordinary

is the theory

on the c a t e g o r y

over R and G

to e x p l i c a t e

of the groups

that,

kn(M)

~0 ~.

:= ko(S°).

an almost

combina-

of a given C W - c o m p l e x

to the well

known

topological

211

case,

cf.

Recall

[Sw,

that

Chap.

10].

the d e ~ r e e

with

[f]

= m[idsn]

also

if f d o e s

on ~ n ( S n ) . }

deg(f)

of a m a p

in nn(Sn) , p r o v i d e d

not preserve

base

the

two p o i n t s

of

n >O.

points,

If n = O we put deg(f)

interchanges

f : S n ~ S n is the

= +I

{N.B.

since

This

n1(S n)

if f = idso,

S ° , and deg(f)

integer

makes

acts deg(f)

m

sense

trivially = -I

if f

= O in the r e m a i n i n g

cases.

For every n

cell

:= d i m o

o of M w e

(cf. V,

§7).

fix a c h a r a c t e r i s t i c

attaching

map

obtained

from

dimension

n-1

we h a v e

a well

~T

~ S n-1

: sn-1

S n-1

such

that

~°

: sn-1

Let ~

~ M n-1

defined

the

, M n-1

induced

[~:T]

Notice

Cn(M)

set In(M)

not

admitted

given

associated

every

cell

preserving

diagram

T of

map

commutes.

- M n - I / M n-1 ~ T

~T"

are

canonical

projections,

and ~T

is an iso-

We put

= O if T is not

cellular

If n ~ O

the

relative

arrows

by

[o:T]

the

= O.

the

= deg(~).

that

We d e f i n e

: E n ~ M n with

En-I/sn-2

the u n a d o r n e d

morphism

For

base-point

following

~j

denote

¢~ by r e s t r i c t i o n .

sn-1

Here

map

then

of c e l l s

Cn(M)

formula

complex is the

of d i m e n s i o n

as an e l e m e n t

CW-complex

by the

chain

an i m m e d i a t e

of Zo(M),

(M,{Xo}). (n >O)

C. (M) as free n.

The b o u n d a r y

M has

If n < O

group

the b a s e

map

of o.

follows.

abelian

Here

since

face

generated

point

by

X o O f M is

to be r e g a r d e d

~ : Cn(M)

then

as a

~ Cn_I(M)

is

212

2(0)

with

:= ~

T running

second

through

pointed

There

3.3.

exist

aM such

Zn_1(M).

CW-complex

in the o b v i o u s

Theorem

[o:t]t

way.

Let

H

N induces

It w i l l

soon

canonical

for e v e r y

a chain

f

: M ~ N to a

C.(f)

obvious

that

theory

f : M ~ N,

the

: C.(M) indeed

and

G

~ C.(N)

~

= O.

:= ko(S°).

~ kn(M)

cellular

map

, k

aM

n

square

(M)

f,

(C. (N) ®G)

n

map

homology

Hn(C. (f) ® G ) [

H

map

isomorphisms

(C. (M) ® G )

n

cellular

become

k, be an o r d i n a r y

: Hn(C. (M) ~ z G )

that,

Every

, k

°N

n

(N)

commutes.

This

can

theory,

be d e r i v e d cf.

is given. wedge

[Sw,

The

gives

indeed

8~ = O

In the

special

canonical the

~) cf.

10] w h e r e point

also

S n,

= O for q # n

an e a s y

the

axioms the

as

in the

explicit

is the o b s e r v a t i o n

sphere

case

that

proof

one

copy

topological

description

that

for e a c h

and k n ( M n / M n-l)

that

of a M

M n / M n-1

is the

o 6Zn(M).

This

= Cn(M) ® z G .

for the b o u n d a r y

M is a g e o m e t r i c

K the a b s t r a c t i o n simplicial

characteristic chain

also

from

maps

The

case

in C. (M)

[loc.cit.].

IKI R w i t h

the a b s t r a c t

with

of the

k q ( M n / M n-l)

G = Z also

M =

Chap.

starting

of c o p i e s

implies

directly

p.

complex 314 below.

of M

complex maps

simplicial

(III,

K in one

§4),

we

of m a n y

at our d i s p o s a l .

C. (K) of the a b s t r a c t

can

ways

Now

complex, order and

hence [ES,

then

have

C. (M) c o i n c i d e s

pointed

p. 67] ~)

and o r d e r e d

213

simplieial complex K, and we obtain the following truly c o m b i n a t o r i a l d e s c r i p t i o n of kn(M).

Corollary

3.4. For every n 6 2 and every abstract pointed ordered simpli-

cial complex K there exists a canonical

isomorphism

a K : Hn(K,G) -m+kn(IKl R) such that,

for any simplicial map f : K ~ L, the following square

commutes:

Hn(K,G)

kn(IKT R) aK

(IflR),

Hn(f) H

n

(L,G)

(x

, kn (ILl R)

L

If the reduced h o m o l o g y theory k, is not o r d i n a r y then again a cellular d e s c r i p t i o n of the groups k~(M)

for M a pointed C W - c o m p l e x is possible,

a l t h o u g h this can be e n o r m o u s l y complicated. theory

[Sw, Chap.

As in the t o p o l o g i c a l

15] one proves

T h e o r e m 3.5

(Atiyah-Hirzebruch,

homological

spectral sequence

G.W. Whitehead).

There exists a natural

r ,d r ) with E 2 = H (C. (M) ® k sO)) (Ep,q P,q P q

c o n v e r g i n g to k,(M).

In c o h o m o l o g y duals of these theorems 3.3 and 3.5 can be proved In an analogous way.

~4 - H o m o l o g y

of p a i r s

of w e a k

We n e e d

some

formalities

without

base

points.

Notations direct weak

sum MU

a)

with

spaces

we have

[M,N °] =

of course,

classes

of maps

of b a s e

point

M+

(MIII6A)

over {+},

with

and

R then M + denotes

the

regarded

{We h e s i t a t e conflict

spaces

to u s e

with

this

Def.

over

R then N ° denotes

hand

side denotes

as a p o i n t e d notation

3 in IV,

the

space

for

§6.}

N forgetting

[M+,N]

the

left

polytope

A

X+ =

homology

category

pairs

spaces theory.

classes

hand

of m a p s

side

(free) denotes

f r o m M + to N.

homotopy the

set

If X is

of w e a k

polytopes

then

( U MI) + 16A

trary

from the

right

set o f

then

is a f a m i l y

reduced

homotopy

the

the

(M ×X) +

in t h i s

extended

+.

space

between

bijection

Our objective

of

point

this would

space

relations

polytope

f r o m M to N ° w h i l e

+ v M1 = X6A

(4.4)

point

preserving

a second weak

If

since

a natural

(4.3)

the

point.

(4.2)

where,

a one

base

If N is a p o i n t e d

the b a s e

about

If M is a w e a k

{+} w i t h

polytope

arbitrary b)

4.1.

polytopes

section

the n e x t

or c o h o m o l o g y

HP~(R)

over

and

theory

to the h o m o t o p y

R and

one

to u n d e r s t a n d

is to e x t e n d

over

category the

an a r b i -

R in the c o r r e c t HWSA(2,R)

formal

way

of a l l

properties

of the

215

In this

section

polytopes). HP(2,R)

have

at W P - p a i r s

of t h e s e

pairs

endomorphism

(= p a i r s

and the

E which

maps

of w e a k

homotopy a pair

category (M,A)

to

(A,~).

we w r i t e

as a full

or c o n t r a v a r i a n t

F(M)

instead

subcategory

map

between

by

f~ in the

Analogous

In the

attention

2(2,R)

a natural

If F is a c o v a r i a n t usually

focus

The c a t e g o r y

both

the p a i r

we w i l l

WP-pairs

then most

conventions

last

cohomology

sections than

of F(M,~).

of H P ( 2 , R ) ,

covariant

case will

functor

cf.

often

and by

1.1.}

For

t h a t we r e g a r d ~

contravariant

of the

space

is a F[f]

case.

categories.

time m o r e

we n o w

HP(R)

(N,B)

the h o m o m o r p h i s m

for o t h e r

justice

to Ab t h e n

If f : (M,A)

we d e n o t e

been most

on h o m o l o g y .

{Recall

f~ in the

be o b e y e d

we h a v e

f r o m HP(2,R)

give

explicit

preference

on to

homology.

Definition braic

I. A h o m o l o @ y

homology

functors

hn

: HP(2,R)

transformations

Exactness.

For

--~ hn(A) is exact. from

Here

(M,~)

to

the W P - p a i r

Excision. M = A UB

i~

theory)

~n

theory

h~ o v e r

(more p r e c i s e l y , R is a f a m i l y

~ Ab t o g e t h e r

with

: hn ~ hn-1 °E w h i c h

every

WP-pair

±~"~ hn(M) i and (M,A).

(M,A)

the

sequence

(hnln62)

a family

satisfy

long

J~ ~ hn(M'A)

j denote The

the

an unreduced

of c o v a r i a n t

(3nln6~)

the

is c a l l e d

of n a t u r a l

following

axioms.

sequence

Dn(M'A) ....hn-1(A)

inclusions

semialge-

from

(A,~)

the h o m o l o @ y

.... to

(M,@)

sequence

and of

(M,A).

If A a n d B are c l o s e d then

the

inclusion

: h n ( A , A nB)

subspaces

i : (A,A OB)

-%~hn(M,B)

of a w e a k ~

(M,B)

polytope

induces

M with

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

216

for e v e r y

n 6 ~.

Additivity. weak

If M is the

polytopes

(il~) for e v e r y

These

: ~ 16A

additive)

are

CW-complexes

from

For

theory,

some

7].

Let

more

{x}.

that

with

7]. We call

precisely

from not

Thus

C~hn(~)

(MIII£A)

of

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

such

the

~9(2)

over

M over

the h o m o l o g y

[ES,

a topological homology

very

Chap.

well I],

theory.

known

[Sw,

R.

R there

exact

a canonical

topological

cf.

(strongly

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

here

by a r g u m e n t s

here,

theory

of a

of p a i r s

a homology

axioms

repeated

axioms

is a u n i q u e

sequence

retrac-

of the p a i r

isomorphism

~hn(M,{x})

then

hn(M,{x])

A a strong

of the

an u n r e d u c e d

polytope

and we have

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

pair

of a f a m i l y

: M 1 ~ M induce

on the c a t e g o r y

x of a w e a k

hn(M)

conclude

iI

analogues

h~ be a h o m o l o g y

splits

(4.5)

Chap.

and m o s t l y

f r o m M to

(M,{x})

theory

consequences

any p o i n t

tion

inclusions

the p r e c i s e

[Sw,

topology

Chap.

(IV,I.IO)

h n ( M I) -L~hn(M)

homology

homology

the

sum

n 6 ~.

axioms

We d r a w

then

direct

{x} ~

= O.

M is a h o m o t o p y

(More

deformation

generally,

retract

equivalence

whenever

of M t h e n

h

n

and we

(M,A) (M,A)

is a

= O for

all n.)

Proposition

4.6

(Homology

sequence

decreasing

WP-triple,

and

denote

associated

inclusion

the

inclusion

(A,~)

~

(A,C).

let

of a t r i p l e ) .

i : (A,C) maps

Finally,

for

~

(M,C),

Let

(M,A,C)

j : (M,C)

of W P - p a i r s . any n 6 ~, we

Let

~

be a (M,A)

i I denote

introduce

the

the

217

the composite

homomorphism

3n(M,A,C)

: hn(M,A)

~n(M,A ) , hn_ I (A)

(il) *

J

hn_ I (A,C)

The long sequence

~n+I(M,A,C) is

hn(A'C)

i,

hn(M'C)

J*

hn(M'A)

~n (M,A,C)

hn_ I (A,C)

exact.

Proposition

4.7

(Mayer-Vietoris

spaces of a weak polytope An(M,A,B,C)

denote

hn (M'C) with inclusion

maps

a(x)

=

Let A,B,C

be closed

sub-

M with A UB = M and A NB m C. Let A n =

the composite

Ii~' hn (M'B) ~ maps

Jl~~ hn (A'ANB)

J1' II" The followina

An+l , hn(AnB,C ) s where

sequence).

hn(A,C ) ~hn(B,C)

(i I~ (x) ,i2~ (x)) and B(u,v)

~n (A,ANB,C)

~ hn_ I (ADB,C)

long seauence B

is exact.

hn(M,C ) A n hn_I(ANB,C )

= i3~ (u) -i4~ (v) with

inclusion

ii, i2, i3, i 4. Moreover, A

n

(M,B,A,C)

= -A

(M,A,B,C)

n

We now study the restrictions gory HP~(R) x then,

of HP(2,R).

hn of the functors

If M is a pointed

h n to the full subcate-

weak polytope

with base point

by definition, hn(M)

= h n(M O,{x})

For every WP-pair

(M,A) with A not empty

space with base point A/A.

Proposition projection

If A = ~ then we put M/A = M +.

4.8. For every WP-pair p : (M,A) ~

we regard M/A as a pointed

(M/A,A/A)

(M,A)

induces

and every n 6 ~ the natural an isomorphism

218

p,

: hn(M,A)c~

n (M/A)

Proof. This is evident if A = ~ .

Assume now that A i s not empty. L e t M U CA

denote t h e c o f i b e r of the i n c l u s i o n A ~ M

(cf. 1.6.ii). W e h a v e a c o m m u t i n g

triangle of WP-pairs (M,A) •

i

~ (MUCA, CA)

(M/A, A/a) with n a homotopy

equivalence.

spaces without base point. axiom hn[i]

{Illegally

We omitted

is an isomorphism.

here M UCA,

CA, M/A,

A/A mean

the upper index o.] By the excision

Thus also hn[P]

is an isomorphism,

q.e.d.

Given a pointed weak polytope M = (N,x) we obtain an isomorphism On(M) from hn(M)

to hn+1(SM)

with CN the reduced

for every n 6 ~ as follows.

cone of N and x its natural base point, which is the

image of the point x 6 N c CN under the natural {Again illegally,

sequence

the boundary maps

projection

p : CN ~ CN/N.

we regard CN and CN/N as spaces without base point

omitting the index o.} Since CN is contractible homology

Recall that SM = (CN/N,x)

of the triple

(CN,N,{x})

Dn := ~n (~N'N'{x})

h

n

(CN,{x})

(cf. Prop.

are isomorphisms.

4.6)

= O. The shows that

We define

On(M)

as the composite

hn(N,{x})

2- I n+l

, hn+ I (CN,N)

P~

hn+ I (CN/N,{x})

The second arrow is an isomorphism by Proposition is an isomorphism.

Proposition equivalences

4.9.

It is natural

4.8 above.

in M.

(~n In6~) together with the family

o n : hn

hn+1OS

Thus On(M)

is a reduced homology

(OnEn6~)

of natural

theory over R.

219

Proof. We obtain the exactness axiom WP-pair

(§3, Def.

I) for a given pointed

(M,A) from 4.8 and the h o m o l o g y sequence 4.6 of the triple

(M°,A°,{x}), with x the base point of M and A. It remains to v e r i f y the wedge axiom for a given family

(M~II6A)

of pointed weak polytopes.

We c o n s i d e r the direct sum N := U I M1o and its closed subspace B := HI {x~}, w i t h xl the base point of M I. We have N/B = V M I. We denote this pointed weak p o l y t o p e by M. Since B is a retract of N the h o m o l o g y sequence of the pair

(N,B)

(cf. Def.

I) splits into short exact sequences.

Using

(4.8) and the a d d i t i v i t y axiom for h, we obtain short exact sequences

O ~ @ hn({Xx})

~ @ hn(M~)

~ hn(M)

~ O

The cokernel of the first map in the sequence Thus we obtain an i s o m o r p h i s m from • hn(Ml)

is • hn(Ml)

to hn(M)"

(cf. 4.5).

It is easily

checked that this i s o m o r p h i s m is induced by the inclusions

il : MI ~

M.

q.e.d. We now start with a reduced h o m o l o g y theory k, over R and want to "extend"

it to an u n r e d u c e d h o m o l o g y theory k,. For any pair

(M,A)

of

weak p o l y t o p e s over R we define

kn(M'A)

:= k n(M/A)

,

and thus indeed obtain extensions k k

n : HP*(R) kn(M)

Let

(M,A)

: HP(2,R)

Ab of the functors

~ Ab. Notice that for any w e a k polytope M,

:= kn(M'~)

= k n(M +)

be a pair of weak p o l y t o p e s over R. Then

of pointed weak polytopes.

to kn(M/A)

The h o m o m o r p h i s m kn[ p] from k n ( M +)

coincides with the h o m o m o r p h i s m kn[J]

clusion map j : (M,~) ~

(M+,A +) is a pair

We identify M + / A + = M/A. Let p denote the

natural p r o j e c t i o n from M + to M/A.

the pair

n

induced by the in-

(M,A). A p p l y i n g the dual of P r o p o s i t i o n 2.1 to

(M+,A +) we obtain a long exact sequence

220

, k

~n+1

n

(A)

with homomorphisms

4.10.

theories

over R with these natural

Unred

there

(~nln6Z)

of natu-

theory over R.

Excision

is evi-

(4.4).

q.e.d.

from

theories

the category Ho(R)

transformations

(cf.

[Sw,

of homology

theories

as morphisms.

to a functor Red :Ho(R)

k, ~ k, extends naturally

~ Ho(R).

to a functor

~ Ho(R).

and Ho(R),

the homology

reduced homology

natural equivalences

i.e. there exist natural equivalences

idHo(R ) - ~ U n r e d o Red and id~o(R ) ~ R e d o U n r e d . morphism,

depend on the pair

follows

Theorem 4.11. These functors are q u a s i - i n v e r s e between Ho(R)

' kn-1 (A)

~n

is an obvious notion of a natu-

h, ~ h, extends naturally

the assignment

: Ho(R)

(M,A)

T : h~ ~ g, between homology

7]), and we can introduce

Similarly

n

together with this family

of kn" A d d i t i v i t y

As for reduced homology

The assignment

, k

axiom has just been established.

dent by the definition

Chap.

j,

~n : kn ~ kn-1 °E is a homology

The exactness

ral transformation

(M)

(M,A).

(knln62)

ral transformations

Proof.

n

3n = ~n (M'A) which naturally

(M+,A +) and hence on

Proposition

, k

i~

theories

In particular,

over R correspond

up to iso-

uniquely with the

theories over R.

Since we have been very explicit

on similar matters

in §2 we now can

safely leave the proof of this theorem to the reader.

Analogous

natural

equivalences

gories ~ and ~ of unreduced

h~ ~ h,, k~ ~ k~ exist between

and reduced

topological

homology

the catetheories,

221

as is well known and proved in the same way.

Henceforth

we u s u a l l y

semialgebraic

If K D R

will

x k,

identify

and the topological

= k,,

~,

= h,,

both

in

setting.

is a real closed base field extension

then it is clear

Conversely,

of h, t_oo R, such that

(h~)~ = (~,)R.

of g, t__ooK, such that

In this way the homology theories theories

theory h,

every homology theory g, over R yields a homology

(g*)K over K, called the extension

theory

((g*)K)~ = (g*)K"

over R correspond with the homology

over K uniquely up to natural

For any WP-pair

from §2 R

that for every homology theory h, over K we have a homology over R, called the restriction

the

equivalence.

(M,A) over R we have formulas,

in analogy to

(2.6

and

(2.9), (4.11)

hR(M,A)

= hn(M(K) ,A(E))

(gn)K(M(K) ,A(K)) and similar

formulas

Every topological homology

= gn(M,A)

for the boundary maps

homology theory h, "restricts"

theory h~ a over IR such that

every semialgebraic

homology

(,hsa,~ , } =

(g*)top such that

way the semialgebraic

homology

with the topological

ones,

(M,A) of semialgebraic hSa(M,A) n

to a semialgebraic (h,) sa . Conversely,

theory g, over IR "extends"

gical homology theory

(4.12)

3n

((g,)top) ~ = (g*)top"

theories over IR correspond

up to natural equivalence.

CW-complexes

over IR,

= hn(Mtop'Atop)

(gn)top(Mtop,Atop)

to a topolo-

= gn(M,A)

In this

uniquely

For every pair

222

Consequently, over

each

real

Rh~

which

given

=

closed

cients

group

unreduced

2. A

h~we

to T h e o r e m

can d e s c r i b e

cohomology

(semialgebraic

unredueed)

gether

a family

(6nln£~)

the

Exactness.

exists

following

For every

h n(M,A) - 5 ~

H~(-,G)

with

on a C W - p a i r 3.3.

For

Rhn(M,A)

coeffi(M,A)

over

an e x t r a o r d i n a r y in a c e l l u l a r

theories.

of c o n t r a v a r i a n t

that

there

3.5.

(hnln£~)

such

homology

G c a n be e v a l u a t e d

is a f a m i l y with

h,,

theory

singular

analogous

theory

to T h e o r e m

We n o w d e f i n e

Definition

theory

to h,.

R of

procedure

homology

analogous

over

abelian

R by a c e l l u l a r

way

R a homology

canonically

RH~(-,G)

in some

topological

field

homology

((h~a)R°)R

corresponds

The p e n d a n t

a topological

three

pair

h n(M)

cohomolo~y

functors

of n a t u r a l axioms

(M,A)

-~

(notations

polytopes

h* o v e r

: HP(2,R)

transformations

hold

of w e a k

hn

theory

the

~ Ab to-

6n : h n o E

as

in Def.

long

R

~ h n+1 I).

sequence

, hn+1(M,A)

h n(A) 6n(M,A)

is exact.

Excision. A UB

If A and B are c l o s e d

= M then

i*

for

: hn(M,B)

every

subspaces

n 6 Z, the n a t u r a l

of a w e a k

polytope

M with

map

~ h n ( A , A NBI

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

Additivity.

If M is the d i r e c t

topes

for e v e r y

then,

sum of a f a m i l y

n £ ~, the n a t u r a l

map

(MIII£A)

of w e a k

poly-

223

(i~)

:

h n

(M)

-,

L .. I 16A

h n

(M~)

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

All our c o n s i d e r a t i o n s word

for c o h o m o l o g y

analogous nation.

on h o m o l o g y

theories.

to the ones above,

They which

theories

can be r e p e a t e d

lead to n o t a t i o n s we will

word

by

and results

use w i t h o u t

further

expla-

224

§5 - H o m o l o @ [

of p a i r s

of s p a c e s

We w a n t

to e x t e n d

H2(2,R)

to the h o m o t o p y

R. This

will

similar

to the p r o c e d u r e

We

first F

(M,A)

(X,B)

mation

~

triple

X

~

We d e f i n e

of all p a i r s

of W P - a p p r o x i m a t i o n s

problem

I(M,A)

over

R and

(M,A).

instead

an a b e l i a n

groups

category

of s p a c e s

(cf. V,

h o w to e x t e n d

§4)

over

in a w a y

denote

{It t u r n s

ignore

~

(Y,C)

We thus

way.

class

[~],

although

~X

have

~,

a pair

(X,B,[~])

of a W P - a p p r o x i -

this

to w r i t e

exists,

Given

any

up to h o m o t o p y ,

and X is a h o m o t o p y

a canonical

a

is r e d u n d a n t

theoretic d i f f i c u l t i e s . } there

covariant

Given

of t r i p l e s

to be c o n v e n i e n t

in I(M,A)

with

set

homotopy

out

set

(Y,C,[~])

the

the

of the c l a s s

we

a single

in a c a n o n i c a l

[~]

group

F(M,A)

(F(X,B) I (X,B,[~]) maps

(X,B, [~]) 6 I(M,A)

~0~ : F(X,B)

the

R f r o m the

equi-

isomorphism

~F(Y,C).

as t r a n s i t i o n

denote

HWSA(2,R)

R let

: (X,B)

h~ o v e r

in §2.

(Th. V . 4 . 1 0 . i i ) .

: F(X,B)

theory

~ A b to H W S A ( 2 , R )

(X,B,[~]),

map

valence

category

simpler

As e l s e w h e r e

triples

abelian

the

WP-pair

(X,B,[~])

a unique

X~

a

homology

by use

over

: (X,B)

notation. two

with

: H~(2,R)

of s p a c e s with

be d o n e

deal

functor

a given

between

any

as the d i r e c t 6 I(M,A))

with

two m e m b e r s

limit these

of the

of the

family

isomorphisms

family.

For

X~

every

let

~ F(M,A)

canonical

map

emanating

from

the d e f i n i t i o n

of lim

. It is

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

Let

now

[f] be a h o m o t o p y

of s p a c e s F(M,A)

(M',A').

to ~(M',A')

of

class

of m a p s

T h e n we o b t a i n as

follows.

from

a natural

We c h o o s e

(M,A)

to a s e c o n d

homomorphism

WP-approximations

F[f]

pair from

225

: (X,B) exists

~

(M,A)

a n d ~'

u p to h o m o t o p y

~'. g ~ fo~.

We define

: (X',B')

a unique F[f]

~

map

(M',A').

g : (X,B)

as t h e d o t t e d

F[g]

F (M,A)

F[f] ...............

F(M',A')

commutative.

It is e a s i l y

s e e n t h a t ~[f]

o f ~ a n d ~'.

In t h i s w a y w e o b t a i n

It is c l e a r homotopy

Given

equivalence

(M,A)

(M,A,[id(M,A)])

last

functor

F to the

does

not

a covariant

sentence

f : (M,A)

over

~

in V . 6 . 1 5 (M',A')

R there

in I ( M , A ) .

isomorphisms

(X',B')

such

makes

there that

the

square

, F(X',B')

(id(M,A)) ~ : F(M,A)

These

V.6.15

depend

on t h e

choice

functor

~ Ab.

from the

a WP-pair

~

arrow which

F (X,B)

: HWSA(2,R)

B y Cor.

every weak

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

exists

Thus we have

that F turns

a distinguished

a canonical

F[f].

element

isomorphism

C~F(M,A)

fit t o g e t h e r

restriction

to a n a t u r a l

FIH2(2,R)

equivalence

of F. W e

feel

from the

justified

to

identify

FIHP(2,R)

and

= F,

thus we write

over

R and

If ~

: (X,B)

then

the homomorphism

free

to a b b r e v i a t e

R.

any map

F(M,A)

~

= F(M,A)

f between

(M,A)

a n d F[f] = F [ f ]

F[f]

from above by

(M,A)

such pairs.

is a W P - a p p r o x i m a t i o n ~

for a n y W P - p a i r

o f a p a i r of

coincides

with

f~ for a n y m a p b e t w e e n

F[~]. pairs

spaces We now

over

R

feel

of s p a c e s

over

226

We call tion.

~ the e x t e n s i o n

It is e a s i l y

seen

f r o m F to a s e c o n d to a n a t u r a l

of t h e that

functor

every

G

transformation

functor

F to HWSA(2,R)

natural

: H2(2,R) ~

transformation

~ Ab extends

: F ~ G,

by W P - a p p r o x i m a -

called

T

: F ~ G

in a u n i q u e

the extension

way

of T b y

WP-approximation.

By an a n a l o g o u s we

can

extend

category

procedure,

any covariant

of pairs

: HTOP(2)

~ Ab which

isomorphisms.

More

generally

is t h a t

for d i a g r a m s

If h~

direct

h n on H W S A ( 2 , R ) . (6nln62)

of t h e s e

of e x c i s i o n to i g n o r e

question

or H~O(2) S e t of

with

sets

The whole

is a d i s t i n g u i s h e d

R then we obtain

to u n d e r s t a n d

holds

equivalences

functors.

there

to a f u n c t o r

instead point way

limits.

over

functors.

theorem

this

theory

We want

category

for c o n t r a v a r i a n t

inverse

homotopy

on H P ( 2 , R )

or the

[W, p. 224]

~ A b on t h e h o m o t o p y canonically

weak

functors

in A b or Gr or S e t

and

is a h o m o l o g y

: H MO(2)

topological

Gr of groups also

F

CW-approximation

CW-complexes

all t h i s w o r k s for

of c o u r s e ,

to p r o d u c e

functor

turns

in the c a t e g o r y

of A b and,

classical

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

into

values

using

Here

for s o m e

the properties

the difficult

for the 6 n. time

for e v e r y

It t u r n s first

of t h e

question out

hn a functor family

is w h a t

sort

to be c o n v e n i e n t

dealing

with

easier

matters.

Definition Important of

I. L e t Of. be a s p a c e cases

subspace.}

spaces {We u s e

the

H0~(2) d e n o t e let E denote

letter

a) A p r e h o m o l o g y

{We are

a r e OC = ~ ( R ) , WSA(R) , ~0, T O P w i t h

Let

in 0~, a n d

category.

E uniformly

theory

the the

homotopy

a bit vague the

category

endomorphism

usual

here. notion

of p a i r s

(M,A) ~

(A,@)

of o f H0~(2).

for all 0~. }

h, on OZ is a s e q u e n c e

(hnin62)

of c o v a r i a n t

227

functors

hn

: H~(2)

transformations the

~n+1 (M,A)

from

~ hn(A)

i and

to

(M,A).

b) A n a t u r a l

i,

j denote

transformation

is a s e q u e n c e t h a t for e v e r y

(Tnln6~)

T

hn(M)

J*

the i n c l u s i o n s

the f o l l o w i n g

(M,A) 6~,

~n (M,A)

from

to

(A,~)

square

(M,~)

prehomology Tn : h

hn_ I (A)

n

and

theories

on

~ h' such n

commutes.

(M,A) n

, h'n(M,A)

~'(M,A) n

hn_ I (A)

natural

of n a t u r a l

hn (M'A)

transformations

~n(M,A)

{As in p r e v i o u s

(Dnln62)

for e v e r y pair

T : h~ ~ h~ b e t w e e n

of n a t u r a l

(M,A) 6 ~(2)

hn(M,A )

a natural

a sequence

sequence"

Here

(M,@)

with

Dn : hn ~ hn-1 °E such that,

"long h o m o l o g y

is exact.

~ Ab t o g e t h e r

, h'

Tn_1 (A,~)

sections

equivalence

equivalence

(A)

n-1

we w r i t e

hq(A)

(or i s o m o r p h i s m ) (i.e. e v e r y

Let h, be a p r e h o m o l o g y ~n : hn ~ hn-1 °E e x t e n d s

of h q ( A , ~ ) . }

We call T

from h~ to h~ if e v e r y T n is a

Tn(M,A)

t h e o r y on ~(R). to a n a t u r a l

instead

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

Every natural

transformation

t r a n s f o r m a t i o n ~n

: hn ~

(hn-1"E)~

v

and it is p r e t t y

obvious

Proposition

The s e q u e n c e

sequence ~

5.1.

of n a t u r a l

that

(hn_1OE) ~

= hn_ I .E .

of f u n c t o r s

transformations

(~nln62)

(~nln£~)

together

with

is a p r e h o m o l o g y

the

theory

on W S A ( R ) .

Proof.

Given

: (X,B) ~

a pair

(M,A) of s p a c e s o v e r R we c h o o s e

(M,A). We then can c o m p a r e

(X,B) w i t h the 6 ~ - h o m o l o g y • ~, ~ ,

X~ i n d u c e d

by ~ and

sequence

of

a WP-approximation

the h ~ - h o m o l o g y (M,A)

its r e s t r i c t i o n s

using

sequence

of

the i s o m o r p h i s m s

~ : (X,#)

~

(M,~),

228 X : (B,~) ~

(A,~) which are again W P - a p p r o x i m a t i o n s .

logy sequence of

(X,B)

Since the h,-homo-

is exact the same holds for the h , - h o m o l o g y

sequence of

(M,A).

q.e.d.

Proposition

5.2. For every natural t r a n s f o r m a t i o n T : h, ~ g, between

p r e h o m o l o g y theories on ~(R)

the sequence

(Tnln67)

is a natural trans-

formation T : ~, ~ g, between p r e h o m o l o g y theories on WSA(R).

If T is

an i s o m o r p h i s m then T is an isomorphism.

We leave the easy proof to the reader. tension of h,

Similarly,

(resp. T,) to HWSA(R)

We call ~,

(resp. T,) the e~x-

by W P - a p p r o x i m a t i o n s .

by c l a s s i c a l C W - a p p r o x i m a t i o n every p r e h o m o l o g y theory h~

on ~/) extends to a p r e h o m o l o g y theory h, on TOP and every natural transformation

(resp.

isomorphism)

T : h, ~ g, between such theories

extends to a natural t r a n s f o r m a t i o n

(resp. isomorphism)

T : ~, ~ g,.

Given any h o m o l o g y theory g~ over R we want to u n d e r s t a n d the formal p r o p e r t i e s of the p r e h o m o l o g y theory g,. For all q u e s t i o n s we have in m i n d it does not matter w h e t h e r we look at ~, or at a p r e h o m o l o g y theory isomorphic to 5,. We know from §4 that there exists a topological h o m o t o p y theory h,, unique up to isomorphism, morphic to Rh,. By P r o p o s i t i o n phic to

such that g~ is iso-

5.2 the p r e h o m o l o g y theory ~

(Rh,) v . Thus it suffices to study the theory

is isomor-

(Rh,) v . This allows

us to simplify n o t a t i o n s drastically.

Notations

5.3. For the rest of this section,

a fixed t o p o l o g i c a l h o m o l o g y theory i) If

(M,A)

~n(M,A) ii)

If

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

instead of ~n(M,A)

and in §6 as well,

h, is

(living on H f~9(2)). spaces then we write hn(M,A)

and ~n(M,A).

(M,A) is a pair of spaces over R then we write hn(M,A)

and

and

229

~n(M,A)

instead

of

stead of 3n(M,A) iii)

more briefly,

If M = (N,x)

(RDn )* (M,A)

(Rhn)" (M,A) and

but abusively,

is a pointed

and rather ~n or even

space over R or a pointed

often in-

2. topological

space we define hn(M) iv)

:= hn(N,{x})

If M is a pointed weak polytope

isomorphism On(M).

hn(M)

~ hn+1(SM)

Similarly,

topological

hn(M) /-~hn(SM)

(cf.

[Sw, Chap.

[Sw, Chap.

often write more briefly

Theorem

5.4

homology

topological

On, or even o, instead

field extension

space,

7]) again by On(M).

h

n

: hn(M'A) 2-*hn(M(K)'A(K)) n

(M,A)

~n(M,A )

isomorphism

In both cases we

i) Let K be

field R. For every pair

such that

a natural

all

isomor-

the squares

(M,A) , hn_1(A)

~-

hn (M (K) ,A (K))

its

of On(M).

(M,A) over R and every n E ~ there exists

phism
Rh~ by

we denote

for homology),

of a real closed

the suspension

theory

2] by SM and the natural

(First and second main theorem

a real closed of spaces

of the reduced

if M is a pointed

suspension

over R then we denote

I
n (M(K) ,A(K))

, hn_1(A(K))

commute. ii) For every pair of weakly exists

a natural

In(M,A)

semialgebraic

isomorphism

: hn(M,A)

such that all the squares

C ~ h n ( M t o p , A t o p)

spaces

(M,A) over IR there

230

n

h n (M ,A)

(M,A) , hn_ I (A) I

1

(M,A)

n

I ~i ~n-1 (A,¢)

~-

h n (Mto p , Ato P)

n (Mtop' A t o p

)

' hn_ I (Ato p)

commute.

Indication ~n(M,A), tified these

of proof.

In(M,A)

hn(M,A)

for p a i r s

pair of s p a c e s

i) Let

(M,A) o v e r R

(M,A)

Thus the d i r e c t regarded

of

(M,A)

system

We d e f i n e

of the f i r s t

(resp.

isomorphisms

and then h a v e respectively

isomorphisms

]19) l e a v i n g

such

idenby

for an a r b i t r a r y

the v e r i f i c a t i o n

of the

to the reader.

If ~ : (X,B)

t h e n ~K is a W P - a p p r o x i m a t i o n

(hn(X,B) I (X,B,[~]) 6 I(M,A))

as a s u b s y s t e m of

n i c a l way.

these

be a pair of s p a c e s o v e r R.

WP-approximation

(M,A),

and h n ( M t o p , A t o p )

We n o w d e f i n e

commutativities

established

of w e a k p o l y t o p e s

with hn(M(K),A(K))

isomorphisms.

various

In §4 we have a l r e a d y

~

(M,A) of

(M(K),A(K)).

from above

(hn(Y,C) I (Y,C,[¢]) 6 I ( M ( K ) , A ( K ) )

~n(M,A)

as the n a t u r a l

s y s t e m to the d i r e c t

map

is a

can be

in a c a n o -

f r o m the d i r e c t

limit of the s e c o n d one.

limit

It is an

isomorphism. ii) Let

(M,A) be a p a i r of w e a k l y

I' (M,A) d e n o t e (X,B,[~]) Further

the s u b s e t of I(M,A)

with ~

(Mtop,Atop)

d e n t m a p an(M,A) system

: (X,B) ~

of

a topological

l i m i t of the s e c o n d

system,

isomorphisms.

can be r e g a r d e d

with

~ hn(M,A).

an eviThe

canonically

We d e f i n e

is h n ( M t o p , A t o p ) .

We d e f i n e

(Y,C,[~])

C W - a p p r o x i m a t i o n . We h a v e

limit of the f i r s t

which

]19. Let

CW-approximation.

: lira (hn(X,B) I (X,B,[~]) 6 I' (M,A))

f r o m the d i r e c t

over

of all t r i p l e s

the set of t r i p l e s

(hn(Y,C) I (Y,C,[%,]) 6 J ( M t o p , A t o p ) ) .

the i n d u c e d m a p

are

consisting

spaces

a semialgebraic

(hn(X,B) I (X,B,[~])) 6 I' (M,A)

subsystem

Bn(M,A)

(M,A)

let J ( M t o p , A t o p) d e n o t e

: (Y,C) ~

semialgebraic

as a

Bn(M,A)

as

s y s t e m to the d i r e c t B o t h Sn(M,A)

and

231

%n(M,A)

= Bn(M,A)Oan(M,A)

-I

(End of our e x p l a n a t i o n s c o n c e r n i n g Th.

5.4).

We now are justified to identify hn(M,A) hn(Mtop,Atop))

with hn(M(K),A(K))

by the i s o m o r p h i s m
(resp. %n(M,A)).

(resp. This means

that most often we write

hn(M,A)

= hn(M(K),A(K)),

resp.

hn(M,A)

= h n ( M t o p , A t o p)

= 3n(M(K),A(K)) , resp.

3n(M,A)

= ~n(Mtop,Atop)

and, of course,

~n(M,A)

also

Example 5.5. Let h~ be singular h o m o l o g y H~(-,G) some abelian group G. For

(M,A)

over R the groups Hq(M,A,G) morphisms, [DK 3]

,

a pair of locally s e m i a l g e b r a i c

defined now coincide,

with the groups Hq(M,A,G)

and d e s c r i b e d

in III,

with c o e f f i c i e n t s

in

spaces

up to natural iso-

inventend by Delfs

([D],

[DI],

§7.

This is evident for M and A p a r t i a l l y c o m p l e t e from the simplicial description

3.4 of these groups.

since there exist "enough" pairs K c M and L c A

It is then also clear in general (K,L)

(with L c K, of course)

retracts of M and A r e s p e c t i v e l y

of p a r t i a l l y complete subspaces w h i c h are strong d e f o r m a t i o n

(cf. formula 3.6 in [DK3]).

In the following we shall not make use of this fact. On the contrary, the c o n s i d e r a t i o n s in III,

of the present chapter will give the results,

§7 anew and in a somewhat easier way than Delfs'

But one should keep in mind that Delfs'

say,

theory does.

theory has a deeper content

than the present one by its inherent c o n n e c t i o n with sheaf c o h o m o l o g y w h i c h is absent here. For example, groups Hq(M,G)

by his theory it is clear the the

of a locally s e m i a l g e b r a i c

space M {defined as

(Rhq)~ (M)

232

for h* = H*(-,G)

= singular

cide w i t h the c o h o m o l o g y M with

groups

with coefficients

in G} c o i n -

H q ( M , G M) of the c o n s t a n t

sheaf G M on

s t a l k s G.

Open Question that Hq(M,G)

B.

We return

In w h i c h

= Hq(M,GM)?

of the r e l a t i v e

generality How about

cohomology

to an a r b i t r a r y

Proposition

5.6.

groups

Proposition

5.7.

Let

the d i r e c t

inclusion maps

il

(il,)

homology

sequence

cf.

t h e o r y h,.

of any t r i p l e

(M,A,C)

of the long h o m o l o g y

[ES, p. 25ff.].

be a f a m i l y of s p a c e s o v e r R. Let

sums of the M1 and the A1 r e s p e c t i v e l y .

: (MI,AI)~-~

: @9 hn(Mi,A l)

of

is exact.

of the e x a c t n e s s

((MI,AI) II6A)

M and A d e n o t e

interpretation

Hq(M,A,G)?

as in 4.6,

of a pair of spaces,

for a space M o v e r R

a sheaf theoretic

The long h o m o l o g y

is a f o r m a l c o n s e q u e n c e

sequence

is it true

topological

s p a c e s w i t h A ~ C, d e f i n e d

This

cohomology

(M,A)

induce

The

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

C~hn(M,A)

16A for e v e r y n 6 Z.

Proof.

It s u f f i c e s

empty.

T h e n we o b t a i n

sequences choose

of

this

the r e s u l t

(M,A) and the

a WP-approximation

:= ~ I

in the s p e c i a l in g e n e r a l

for the f a m i l y

case that all A 1 are

by use of the long h o m o l o g y

(MI,A I) and the f i v e - l e m m a . ~I

: XI ~ MI"

(Xlll6l).

we see t h a t the c l a i m h o l d s

for

For any

Let X := ~ X I ] I £ A ) .

f r o m X to M is a W P - a p p r o x i m a t i o n

claim holds hn[~]

to p r o v e

Using

The m a p

of M. We k n o w t h a t the i s o m o r p h i s m s

(MI[16A).

I 6 A we

the

h n [ ~ I] and q.e.d.

§6 - E x c i s i o n

and

The conventions cular,

h~

on p a i r s

Definition call

a triad

i,

map

6.1.

then

in 4.7,

which

In p a r t i will

be

5.3).

is c a l l e d

a triad

induces

an

theory

if M = A U B . h~)

if t h e

isomorphism

If

(M,A,B)

is an e x c i s i v e

triad

a n d C is a s u b s p a c e

of the q u a d r u p l e

(M,A,B,C),

defined

is exact.

is a g a i n

logy

sequences

a formal

to the c r u c i a l

cf.

6.2.

the

triad

closed

in M is e x c i s i v e .

Proof.

We know

If the

field

tially

complete

from

§4 t h a t

R is s e q u e n t i a l core

a n d P(A) N P(B)

PX

of the p.

Which

triads

triads

then,

of

We have

are

long homo-

37ff].

excisive?

The

follow-

semialgebraic

spaces.

spaces

over

R with A and B

polytopes

for e v e r y

~ X exists

of the

p.

of w e a k l y

of w e a k

Now(P(M),P(A),P(B))

= P ( A N B).

[ES,

topological

(M,A,B)

: P(X)

exactness

IO5f],

"tameness"

to a r b i t r a r y

Every

(ef. V . 4 . 7 ) .

[Sw,

question:

witnesses

in c o n t r a s t

Proposition

consequence

of p a i r s ,

ing p r o p o s i t i o n

of X

theory

(for the h o m o l o g y

(M,B)

the M a y e r - V i e t o r i s

This

spaces

~

(cf.

(M,A,B)

excisive

i : (A,ANB)

homology

in force.

n 6 Z.

Proposition

We come

R

remain

: hn(A'ADB)-L~hn(M'B)

for e v e r y

as

over

of s p a c e s

(M,A,B)

section

topological

of s p a c e s

I. A t r i p l e

inclusion

of A D B

from the preceding

is an a r b i t r a r y

evaluated

We

limits

and

space

X over

excisive. R, t h e

par-

is a W P - a p p r o x i m a t i o n

is a t r i a d

a commuting

are

of w e a k

square

polytopes

234

i)

(P(A),P(A)nP(B))

,(P(M),P(B))

(PM,PB

(PA,PADB) (A,A~B)

a n d P(i) cals

is just

,

the

inclusion

are WP-appreximations.

clude

that

(M,A,B)

Now consider closed

the

overfield

a triad with excisive. theorem

Remark

Since

case

that

the

pairs

(P(M),P(A),P(B))

R is n o t

K of R w h i c h

A(K)

and

for h o m o l o g y

If

between

above.

The

is e x c i s i v e

vertiwe

con-

is e x c i s i v e .

B(K)

=

5.4.i

(M,A,B)

sequential.

is s e q u e n t i a l .

closed

S i n c e A(K) n B ( K )

6.3.

map

(M,B)

in M ( K ) .

(A DB) (K) w e

that

(M,A,B)

is an e x c i s i v e

Then we choose

(M(K),A(K),B(K))

Thus

is a g a i n

(M(K),A(K),B(K))

conclude

by the

then

is

first main

is e x c i s i v e ,

triad

a real

q.e.d.

also

(M,B,A)

is

excisive.

This

c a n be s e e n

in m u c h

position

6.2

subspace

(AxO) U (CxI) U (Bxl)

closed

subspaces

Theorem

6.4.

be a m a p map"

g

Let

over

for e v e r y

triads

~

same way

(M',A',B') of MxI,

(AxO) U (C×[O,

(M,A)

R which

: (M,A)

g,

Proof.

to t h e

the

is p a r t i a l l y

: hn(M,A)-L~hn(M

(cf.

in t o p o l o g y

and with

pair

C

IV.8.4)

M'

U (Bxl)

of M',

over

near M ~A.

induces

cf.

R and

Then

Pro-

is t h e

:= A n B, a n d A' ,B'

of s p a c e s

proper

by applying

(M',B',A'),where

]), (Cx[~,1])

be a c l o s e d

(M U f N , N )

as

are

the

[Sw, p. IO3f~.

f : A ~ N

the

"push out

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

UfN,N)

n 6 Z.

i) W e

Now we have

first

consider

a commuting

the

diagram

case

t h a t ~< a n d N a r e w e a k

polytopes.

235

(M,A)

, (M UfN,N)

(M/A,*)

, (M UfN/N,*)

with g an i s o m o r p h i s m and canonical p r o j e c t i o n s p and q. We know from §4 that hn[ p] and hn[ q] are isomorphisms. isomorphism.

Thus hn[g]

Of course also hn[g]

is an

is an isomorphism.

ii) We now prove the theorem in the case that the field R is sequential. Then we have a c o m m u t i n g square

(P(M),P(A))

P(q)

, (P(M UfN),P(N))

P (M,A)

, (M UfN,N)

with W P - a p p r o x i m a t i o n s p,q. M o r e o v e r P(M UfN) and P(g)

(IV.9.14),

is the pushout map between the pairs above. We know from step

(i) that hn[P(g)] iii)

= P(M) Up(f)P(N)

is b i j e c t i v e and conclude that hn[g]

is bijective.

If finally R is not sequential then we choose a sequential real

closed field K m R. We know from step

(ii) that hn[g K] is b i j e c t i v e and

conclude by the m a i n t h e o r e m 5.4.i that hn[g]

In the special case that N is the o n e - p o i n t

is bijective,

q.e.d.

space the theorem reads

as follows.

C o r o l l a r y 6.5. For every closed pair of spaces c o m p l e t e near M ~ A

(M,A) with A p a r t i a l l y

and every n £ • we have a natural i s o m o r p h i s m

hn (M,A) -t~~n (M/A)

We now start out to prove a very general theorem about the h o m o l o g y of direct limits of spaces.

236

Theorem

6.6.

Let

(M,A)

s y s t e m of subspaces Def.

7 in IV,

§3).

be a pair

of M w h i c h For every

16Alim hq(X I,A NXI)

of spaces

and

(XAII6A)

is an a d m i s s i b l e

q 6 Z the natural

a directed

covering

of M

(cf.

map

~ hq(M,A)

is an isomorphism.

Proof.

a)

It suffices

the c l a i m

follows

five-lemma. pair

in general

Thus h e n c e f o r t h

this

in the case that A is empty.

by using

long h o m o l o g y

we c o n s i d e r

a single

sequences

Then

and the

space M instead

of a

(M,A).

b) We first prove M

to prove

(cf. V,

Milnor's

§2,

Def.

trick

(Xnln6IN),

the claim

[Mi].

cf. V,

equivalence

3).

for an a d m i s s i b l e

Then the t h e o r e m

We c o n s i d e r

In T we have

A

:= U ( X n x [n-l,n]l

n even)

B

:= U ( X n x [n-l,n]i

n odd)

with A U B = T. By P r o p o s i t i o n close

inspection

(T,A,B,ADB)

c) We now prove (XIiI6A) assume without

the claim,

the t h e o r e m

is an e x h a u s t i o n

that

the closed

subspaces

the e x h a u s t i o n

exhaustion

(Prop.

such that

(XIII6A)

IV.I.15).

of

by use of of the

family

is excisive.

A

of the q u a d r u p l e

[Mi].

Omitting

our d i r e c t e d

is cofinal

with respect

sequence

is faithful

The

(Xnln6~)

: T ~ M is a h o m o t o p y

(T,A,B)

in the case that

of M.

loss of generality,

complexes

p

6.2 the t r i p l e

cf.

T cMxI~

projection

of the M a y e r - V i e t o r i s

gives

can be v e r i f i e d

the t e l e s c o p e

§4. The natural

(V.4.5).

filtration

the d i r e c t e d

superfluous

(Prop.

space M carries

in the

family

a patch

§I.

lattice

decomposition

finite

cf. V,

we may

We enlarge,

to a faithful

of all

to this d e c o m p o s i t i o n ,

indices

IV.I.14).

family

system

closed

sub-

237

H e n c e f o r t h we assume that M is a patch complex and

(XlII6A)

is the

family of finite closed s u b c o m p l e x e s of M. We look at the chunks M n and belts M(n)

of M and also at the chunks XI, n and belts Xl(n)

of

the s u b c o m p l e x X I. Since X 1 is closed in M, we have Xl, n = M n n X I. M o r e o v e r Xl(n)

is just the direct sum of all closed patches d of M of

height n w i t h ~ c X I. In particular, k e e p i n g I fixed. We have ~Xl(n)

Xl(n)

is empty for n large,

= Xl(n) n ~M(n).

By T h e o r e m 6.4 the pushout map g : (M(n),~M(n))

~

(Mn,Mn_ I ) yields

isomorphisms

g~

: hq(M(n),~M(n)) -t~hq(M n,Mn_ I)

For every I 6 A the map g restricts to the pushout map gl from (XI(n),~Xl (n)) to

gl~

(Xl,n,Xl,n_ I) w h i c h yields

isomorphisms

: hq(Xl(n)'~Xl(n)) -t~hq(Xl,n'Xl,n-1)

We obtain a c o m m u t i n g square w i t h b i j e c t i v e vertical

lim h (X. (n) 3X 1 (n)) i , q A '

, h

l i m hq (X)tAMn , X)~ANn_ ] )

, hq(Mn,Mn_ 1)

(M(n),3M(n))

q

(M(n) ,~M(n))

is the direct sum of the pairs

through all patches of height n and every sum of some of these pairs some space Xl(n).

arrows

(~,~o) w i t h o running

(Xl(n),~Xl(n))

(~,~d). Moreover,

is the direct

every ~ is c o n t a i n e d in

Thus, by the a d d i t i v i t y of hq, the upper h o r i z o n t a l

arrow is an isomorphism.

This implies that a is an isomorphism.

Using

long h o m o l o g y sequences and the f i v e - l e m m a we see by induction on n that the natural map

lim hq(X 1 A M n) ~ hq(M n)

238

is an i s o m o r p h i s m for every n. The d i r e c t e d system

(X 1 N M n L (l,n) 6A x ~ )

of subspaces of M gives us a natural c o m m u t i n g square of h o m o m o r p h i s m s

lim (l,-~

lim h

hq(X 1 N M n)

q

, lira hq(M n)

n [

, h q (M)

(XA)

The upper h o r i z o n t a l arrow is b i j e c t i v e by what has just been proved. The v e r t i c a l arrows are b i j e c t i v e by step b) of the proof. lower h o r i z o n t a l arrow is bijective,

Thus the

as claimed.

d) We obtain the theorem in general by somewhat r e p e a t i n g the last argument. family

We choose an e x h a u s t i o n

(X A nMals6I)

(Msls6I)

of M. For every A 6A

the

is an e x h a u s t i o n of X A. The d i r e c t e d system

(XA nMal (~,a) 6 ixi) of subspaces of M gives us a natural c o m m u t i n g square

(llim,~ hq(X 1 N M

)

lim hq(X I)

* lims,hq(M a)

~ hq(M)

The v e r t i c a l arrows are b i j e c t i v e by step c) of the proof.

The upper

h o r i z o n t a l arrow is t r i v i a l l y b i j e c t i v e since every M s is c o n t a i n e d some X I. Thus the lower h o r i z o n t a l arrow is bijective,

We now can prove that the elements polytopes

in our h o m o l o g y groups

in the given space. For any pair of spaces

denote the set of pairs

as claimed,

(M,A)

"live"

let ~c(M,A) sub-

L c K. This set ~c(M,A)

a d i r e c t e d system of pairs of polytopes under the partial o r d e r i n g given by inclusion.

q.e.d.

in

(K,L) with K and L complete s e m i a l g e b r a i c

sets of M and A r e s p e c t i v e l y and, of course,

in

is

23g

Theorem

6.7.

For e v e r y q 6 2 the n a t u r a l

lira (hq(K,L) I (K,L) 6 ~c(M,A))

map

~ hq(M,A)

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

Proof.

i) W e c h o o s e

surjectivity

an e x h a u s t i o n

of the map.

Let ~ 6 h

(M Ls6I) q

(M,A) be given.

s p a c e s of M a n d A w i t h X m Y then we call preimage the

of 5 if ~ m a p s o n t o

inclusion

(X,Y) ~

By the p r e c e d i n g 6hq(Ma,ANM of

Theorem

cf.

III,

sequences

retracts

to

(Ms,ANMa)

image

in hq(K,L)

ii) We n o w p r o v e ~26hq(K2,L2)

: hq(X,Y)

sub-

~ of hq(X,Y)

~ hq(M,A)

with

a i

some a 6 I and some

of ~. We c h o o s e

§2. Let K and L d e n o t e Then

(K,L) 6 ~ c ( M , A ) .

gives

we c o n c l u d e

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

and this

Moreover

for some p a i r s

51 and 52 h a v e the same

K and L are

(KI,L I) and

from

Thus n has a pre-

of 5.

elements

~16hq(K1,L1)

(K2,L 2) in ~ c ( M , A ) ,

in h I ( M , A ) .

strong

long h o m o l o g y

in h o m o l o g y .

We are g i v e n

image

Using

of M~ and A N M a

t h a t the i n c l u s i o n m a p

is also a p r e i m a g e

injectivity.

a good t r i a n g u l a t i o n

the c o r e s

of M and A r e s p e c t i v e l y .

a n d the f i v e - l e m m a

(K,L)

i,

If X and Y are

an e l e m e n t

6.6 t h e r e e x i s t s

is a p r e i m a g e

in this t r i a n g u l a t i o n . deformation

< under

the

(M,A).

) which

(Ma,ANMa),

of M. We f i r s t p r o v e

We h a v e

and

such that

to find a pair

(K,L) 6 ~(M,A) w i t h K~ U K 2 c K and L I U L 2 c L such t h a t ~I and ~2 h a v e the same

image

By T h e o r e m

in h q ( K , L ) .

6.6 t h e r e

and 52 h a v e the same

exists image

some a £ I w i t h K I U K 2 c M a such that ~I in h q ( M

t i o n of the s y s t e m of s p a c e s

,AAMa).

We c h o o s e

(M , A N M a , K I , K 2 , L I , L 2 ) .

a good

triangula-

Let K and L d e n o t e

the c o r e s of M a and A N M a. T h e n K I U K 2 c K and L I U L 2 c L. The sion m a p Thus

from

~I and

(K,L)

52 h a v e

to

(M ,ANMa)

the same

gives

image

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

in h q ( K , L ) ,

inclu-

in h o m o l o g y . q.e.d.

240

We return

to e x c i s i o n

Lemma

6.8.

which

is an a d m i s s i b l e

for e v e r y

Let

problems.

(XIII£A)

be a d i r e c t e d covering

I 6 A, t h e t r i a d

system

of M.

Let

(XI,A N X I , B

of

subspaces

(M,A,B)

N X I)

of a s p a c e M

be a t r i a d

is e x c i s i v e °

such that,

Then

(M,A,B)

is e x c i s i v e .

This

is an e a s y

rather

general

consequence excision

of T h e o r e m

result.

6.6.

In o r d e r

We are

to

now next

state

door

it w e n e e d

to a

two defi-

nitions.

Definition {i.e.

2. A s u b s e t

L closed

in t h e

and

semialgebraic

semialgebraic

closure

X NL

X of M is c a l l e d

space

in o t h e r

terms,

N.B.

It s u f f i c e s

to c h e c k

this

sets

of a g i v e n

closed

in L for e v e r y

If t h e

s p a c e M is l o c a l l y

M is an i n t e r s e c t i o n A = X. W e a l s o

with

can write,

> O,

with

g(x)

The word

"basic"

alludes

M and,

at

3. A t r i a d

for every

with

is l o c a l l y is o p e n

U 6 ~(L)

closed

in its and A £~(L).

L running

through

By t h e way,

then X NL

is l o c a l l y

then

(M,X,Y)

every

basic

and A 6~(M),

the

subset

X of

and we can choose

case,

> O}

functions

f and g on M

to the p o s s i b i l i t y

least

L 6~(M),

U 6 }(M)

in t h i s

semialgebraic

Definition

that X NL

L £~(M)

with

semial~ebraic

locally

description"

for e v e r y

set X N L

= U NA

property

of M.

two

"basic

X AL

if,

L 6 ~(M).

U NA

X = {x 6 M l f ( x )

in M } , t h e

L. T h i s m e a n s

or,

exhaustion

basic

for t h e

sets X nL

is c a l l e d

there

exist

in g e n e r a l

basic

closed

(cf.

1.4.15).

to g i v e

such a

in L.

if X a n d Y are b a s i c semialgebraic

subsets

in A

241

a n d B of L w i t h check

this

last property

exhaustion

Examples triad") ii)

A c X NL,

i) E v e r y

triad

in t h i s

(M,U,V) case

L running

(M,A,B)

iii)

More

with

X and Y denoting

generally,

Then

riors

U a n d V of L n x

1.4])

A c U,

B cV

Theorem

there

6.10.

Every

Applying

exist

even

the

to

sets of a given

A and B closed

and L DY

closed

in M

("closed

basic

preceding

place

our

system

since

this

core

III.2.1).

the

finite

A and B are also

B 6~(M)

is b a s i c .

with

of M s u c h

Y in t h e

A cU,

(in its

that M =X UY,

strong

let L 6 ~ ( M ) . L NX

semialgebraic

(M,X,Y)

Lem~a the

semialgebraic

without

(finite

subsets

Indeed,

lemma

triad

loss

of

triad")

Tihen t h e

and L DY

inte-

respectively.

semialgebraic subsets

topology

version

A and B of L w i t h

= L.

and

assume

("open

sets A £ ~ ( M ) ,

in L c o n t a i n

shrinking

a good

then

in M

of X a n d

triad.

a n d A U B = M. W e c h o o s e

that

the

it s u f f i c e s

lemma V.3.4.

interiors

to consider

closed

exist

shrinking

the

exist

and A UB

it s u f f i c e s

(Prop.

with

U and V open

is a b a s i c

= L. B y t h e

[DK5,

system

= L. A g a i n

through

let X a n d Y be b a s i c

of M.

U UV

(M,X,Y)

with

there

a n d A U B = M b y the

there

with

£riad

B cV

that

and A UB

is b a s i c .

Indeed,

Proof.

QL,

of M.

6.9.

Every

Thus

Bcy

closed.

subsets

by

A and

(cf.

III,

(M,X,Y,XNY,A,B)

system

§2).

Bcy,

is a t a m e

We now may

re-

(co M, X Q c o M, Y N c o M, ...)

retract

henceforth,

this

system

see

Now

B of M w i t h A c X ,

of t h e

that

its c o r e

deformation

complex

of M w e

t h a t M is s e m i a l g e b r a i c .

complexes

assume

simplicial

case

triangulation

(M,X,Y,XNY,A,B)

Thus we

6.8 to an e x h a u s t i o n

of g e n e r a l i t y

simplicial

is a s t r o n g

is e x c i s i v e .

of the

without

M is c l o s e d .

Then

given

loss the

system

of g e n e r a l i t y , subcomplexes

242

Let C a n d D d e n o t e

the c o r e s

We know

from

X,Y,XNY

respectively.

the

III,§1

com~uting

that

of X and Y.

C,D,CND

Moreover,

Then

are

strong

A cC,

BcD,

C ND

is t h e

deformation hence

core

of

retracts

CUD

= M.

We

X NY.

of

l o o k at

square

h q (C,CnD)

~ h q (M, D)

L h q (X,XNY)

with

all

arrows

, h q (M,Y)

induced

by

isomorphism

b y Prop.

we

the v e r t i c a l

see t h a t

tal map

is an

6.2.

inclusions. Using

maps

isomorphism,

long

are

which

The

exact

upper

sequences

isomorphisms. means

horizontal

that,

Thus

indeed,

map

and

the

the

lower

(M,X,Y)

is a n

five-lemma horizon-

is

excisive. {N.B.

The whole

argument

works

more

generally

if M is l o c a l l y

semi-

algebraic.}

Let

n o w h • be a t o p o l o g i c a l

of t r i a d s results cult may

q.e.d.

with

6.2

since come

theorem,

Theorem

to h* as

- 6.5 r e m a i n the

higher

i n t o play. which

6.11.

admissible exact

respect

true

sequence

derivatives

lim (n) of

(M,A)

is a s p a c e

(Mnln6~)

be a p a i r

of M.

of a b e l i a n

For

effort

I. T h e of the

over

q 6 Z there

excisiveness

analogues

limits

state

trick

of o u r

is m o r e

inverse

we can

telescope

of s p a c e s

every

We define

diffi-

limit

functor

the

following

[Mi].

R and

(Xnin6~)

exists

an

a natural

groups

of c o u n t a b l e

. In t h i s

any extra

by M i l n o r ' s

O ~ lim(1)hq-1(Xn,AnXn

If M

in D e f i n i t i o n the t h e o r y

Without

filtration

theory.

but

is e v i d e n t

Let

cohomology

) ~ hq(M,A)

type

case we obtain

(IV, the

~ °lim h q ( X n , A N X n)

§4)

then M has

following

~ O

an e x h a u s t i o n

excision

result

by the

243

above methods.

Theorem

6.12.

is exc i s i v e

This

6.10).

for M locally

then every basic

semialgebraic

of M is of c o u n t a b l e

special

type

triad

(M,X,Y)

for h ~.

also holds

component this

If M is of c o u n t a b l e

case w i t h o u t

using

type.

since

then every

One can give a direct

a limit

ar g u m e n t

(cf. Proof

connected

proof

in

of T h e o r e m

§7

- Representation

Some notations. category weak

Set*

by

polytope element

study

then,

f*.

pseudo-mapping

contravariant sets.

with

i

: A ~

functors

w e shall o f t e n

If A is a p o i n t e d

i~(x),

spaces

F f r o m H?*(R)

If f : M ~ N is a m a p

as b e f o r e ,

M and x is an e l e m e n t

restriction

An

We

of p o i n t e d

polytopes

F([f])

theorems,

closed

M the

denote

subspace

of F(M)

between the

of a p o i n t e d

b y xIA.

pointed

induced

t h e n w e shall o f t e n

inclusion,

to the

weak

denote

We call

map

the

x i A the

of x to A.

isomorphism

by our usual equivalence

in H P * ( R ) ,

sign

" ~"

M ~N,

i.e.

for

which

a homotopy

isomorphisms.

we do not care

equivalence, If t h e r e

to m a k e

will

exists

be denoted

a homotopy

explicit,

then we write

M ~N.

If F is the

functor

then

it f o l l o w s

ment

(cf.

(W) W e d g e

[Sw,

[-,L]

on H?*(R)

from Remark p.

axiom.

153])

If

1.2.i

for

and

is any

pointed

an e a s y

that F fulfills

(Mlil6A)

some

the

family

space

homotopy

following

in P * ( R )

L over

extension

R argu-

two axioms.

then

the natural

map F(V(MII~6A)) x ~

~ ]-[(F(M I) II6A),

((xIMl)ll6A),

is b i j e c t i v e .

(MV) M a y e r - V i e t o r i s

are pointed x I 6F(AI), there

W e are version

closed

to

If X is a p o i n t e d

subspaces

x 2 6 F ( A 2) are

exists

about

axiom.

aiven

s o m e y £ F(X)

state

of B r o w n ' s

of X w i t h

a strong

with

with

polytope

X = A I U A 2 and

and AI,A 2 if e l e m e n t s

x 1 1 A ] N A 2 = x 2 J A I N A 2, t h e n

y i A 1 = x I a n d y L A 2 = x 2.

converse

representation

weak

of this

theorem

[Bn].

fact,

a refined

245

Definition be

1. L e t

a pointed

space

weak

• to L.

sending from to

such

In t h i s sal

element

Theorem fills

7.1.

the

following exists such

for

We

refer

be

Remark

(W)

holds.

Given

u yields

that

a natural

f~(u) the

6F(X).

natural

to t b ~ s e e l e m e n t s exists

In t h i s

of

F(L).

a space

(= n a t u r a l

a classifying

as

transformations The

point

L and

and

of

[-,L]

F is

an e l e m e n t

equivalence)

space

follows

from

functor

u £F(L)

functors.

u is c a l l e d

and

(MY)

functor

is

a weak

F

: HP*(R)

representable.

polytope

C and

~ Set*

More

a univer-

an e l e m e n t

space

and

u is a u n i v e r s a l

a pointed C as

reader

to

word

for

[Sw,

by word

p.

155ff]

where

is g i v e n .

to

the

of T h e o r e m

be

chosen

with

ulC

element

as

there = v

for

a pointed

F.

CW-

a detailed

[?his - v e r y

semia]~gebraic

7.1

giw~s

the

map

us

proof

beautiful

of

the

- proof

setting.

for

free

the

following

theorem. some

for X = S q w i t h CW-complex

L can

the

v 6F(C)

u 6F(L)

ful-

a subcomplex.

theorem

proof that

CW-complex

element

which

precisely

L is a c l a s s i f y i n g

that,

one

L

is t h e

way,

an

of

Let

transformation

and

The

the

~(u)

(L,C)

7.2.

relative

such

from

CW-complex

adapted

jective

97],

contravariant

topological

i) A s s u m e

p.

if t h e r e

axioms

the

refinements

F(L)

the map

functor.

F.

containing

analogous can

[Mt,

to

isomorphism

Every

If C is a l r e a d y complex

of

Then

[X,L]

L is c a l l e d

a relative

that

6

uniquely

T u is a n

case

a contravariant

let ~ d e n o t e

of F ( ~ ) .

representable,

that

and

an e l e m e n t

[f]

lemma

F correspond

~ Set ~ be

~ F

an e l e m e n t

Yoneda's

called

u be

element

: [-,L]

U

: H~(R)

polytope

Let

distinguished

T

F

n 6 Do, 0 ~q

(L,C)

can

and

Tv(X)

surjective

chosen

without

: [X,C] for

~ F(X)

X = S n.

cells

is b i -

Then

the

of d i m e n s i o n

246

ii)

If Tv(X)

is b i j e c t i v e

a classifying

Remark

7.3.

of g r o u p s there

space

morphism groups

of F w i t h

universal

then

it f o l l o w s

a

of g r o u p s

map

u

is a g r o u p

for e v e r y

X £~*(R).

Theorem

7.1

applies

to t h e

over

Indeed

Here we 7.1.

present

complex Taking

other

space

L over into

over

again

anew

with

spaces Theorem

: L ~ N which

as d e f i n e d

(V.4.1)

in V,

§7.

[X^M,N]

Iv] 6 F ( C ) a second

space

systems

7.1

clearly there

F.

spaces.

(N,B)

of p o i n t e d applying

functor

V.7.14

CW-

Thus we have

: C ~ B ~

of T h e o r e m

N.

F =

[-,N]

Starting

on C W - a p p r o x i -

of s p a c e s .

and M a p o i n t e d

on H~(R)

By Theorem

of t h e m a p v

proof

decreasing

to the

N

this means

for p o i n t e d

argument

with

represents

C ~ B and then

class

holds

theorem

a pointed

a CW-approximation

Let now N be a pointed

Vietoris-axiom.

exists

choosing

of a r b i t r a r y

X ~

there

a WP-approximation

extension

theory

(MV)

o n HP*(R)

of a p a i r

from this one obtains

functor

[-,N]

a CW-approximation

and the homotopy

mations

7.1

F =

and

representation

of C W - a p p r o x i m a t i o n s

and a h o m o t o p y

abelian

in §8.

of t h e

functor

of N,

u p to

cohomology

by d e f i n i t i o n

application

u

that

is an iso-

of F a r e

we obtain

by first 7.1

holds

a map

that N has

the existence

generally

the

way

in H ~ ( R ) .

k n of a r e d u c e d

R. By T h e o r e m

R together

account

this

formal

a n d Tu(X)

If t h e v a l u e s

applications

u is a C W - a p p r o x i m a t i o n

proved More

some

(W) t h e r e

to the c a t e g o r y

: LxL ~ L, u n i q u e

in HP*(R)

group

functors

exploit

First we consider

any pointed

that

We will

f r o m HP~(R)

preserving)

is a n a b e l i a n

2.3.

v.

in a p u r e l y

(L,[~])

b y Prop.

functor

(L,[~])

the axiom

element

7.1

(base p o ± n t

such that

X = S q t h e n C is a l r e a d y

from Theorem

then

R.

sphere

If F is a c o n t r a v a r i a n t

exists

homotopy,

for e v e r y

weak

fulfills exists

polytope

the wedge

a pointed

over

and

weak

R. T h e

the Mayer-

polytope,

247

which we denote by Map(M,N),

6M

and a map

: Map(M,N) A M ~ N

,N

such that,

for every X 6P*(R),

[X,Map(M,N)] [f] ~

~ [X AM,N],

[~M,N°(f ^idM)]

is bijective.

We call Map(M,N)

e v a l u a t i o n map. M AMap(M,N)

Composing

a pseudo-mapping

space and ~M,N an

~M,N w i t h the switching i s o m o r p h i s m from

to Map(M,N) A M we obtain a map

~/I,N : 51 AMap(M,N) such that,

the map

~ N

for every X 6 ~ * ( R ) ,

the map

[X,Map(M,N) ] ~ [M AX,N] , [f] ~

[£M,N ° (idM ^f)]

is bijective.

The p s e u d o - m a p p i n g equivalence.

space is d e t e r m i n e d by M and N only up to h o m o t o p y

To obtain straight n o t a t i o n s

once and for all,

let us agree that we choose,

for every pair M,N as above a p s e u d o - m a p p i n g space

and an e v a l u a t i o n m a p and denote these by Map(M,N)

and 6M, N. Then we

obtain a b i f u n c t o r

Map(-,-)

: H~*(R)×HWSA*(R)

w h i c h is c o n t r a v a r i a n t

~ HP*(R)

in the first and c o v a r i a n t

in the second argu-

ment. W i t h o u t spelling out all the details, we remark that there exist natural h o m o t o p y e q u i v a l e n c e s

(7.4)

M a p ( M , N IxN 2) -L~Map(M,N I) × M a p ( M , N 2)

(7.5)

M a p ( M IAM 2,N) -Z-~Map(M 1,Map(M 2,N))

248

for M , M I,M 2 E ~ * (R) , N I , N 2 , N E W S A * (R) . T h e s e natural

~

string

=

[ (X A M 7) ^ M 2 , N ] .

of e q u a t i o n s

no(Map(M,N))

us t h a t

the

as the h o m o t o p y

=

[S°,Map(M,N) ] =

connected

classes

of m a p s

better

maps

f r o m M to N in a n a t u r a l

will

see,

We

to h a v e

already

[M A S ° , N ]

components

much

space

o f the

Ix A M , N I .) x Ix A ~ , N 2 ] ,

[X ^ (M I ^ M 2 ) , N ]

tells

consequences

bijections

[X ^ M , N I x N2]

The

are

a space

points

way,

space

c a n be

Of c o u r s e ,

correspond

as o n e

a pseudo-mapping

[M,N]

of M a p ( M , N )

f r o m M to N.

whose

=

has

interpreted

it w o u l d

uniquely

in t o p o l o g y .

instead

be

with

But,

the

as w e

of a true mapping

c a n be u s e f u l .

turn

to a s p e c i a l

type

of p s e u d o - m a p p i n g

spaces,

the pseudo-loop

spaces.

Definitions

2.

i) For

space Map(SI,N) the

switched

N which [X,~N] ii)

evaluation

Since

(~N,[~])

and

~

: ~N×QN

is a g r o u p

such

~l,N

b y qN"

X E?*(R),

every map

for all,

in HP*(R) ~ 2N,

Once

and

a multiplication

~N a r e r e g a r d e d

For

N we denote

it a p s e u d o - l o o p Thus

the p s e u d o - m a p p i n g

space

of N. W e d e n o t e

qN is a m a p

a bijection

a map

as

g

the

If] ~

w e have,

from S~N to

[qN-(Sf) ] f r o m

by R e m a r k

u p to h o m o t o p y ,

above

bijections choose,

~ and denote

between

: ~N ~ ~N'

§I)

for all w e

ingredients

f : N ~ N'

(cf.

unique

in H?* (R) a n d

isomorphisms.

N 6WSA*(R),

iii)

for e v e r y

S I is a c o g r o u p

group

qN and

map

space

[SX,N].

a multiplication

are

by ~N and c a l l

induces, to

any pointed

such that,

[X,~N] -:~ [SX,N]

it b y PN"

spaces

for e v e r y

that

for e v e r y

of t h e p s e u d o - l o o p pointed

such

7.3,

Both

maps

space

we choose, X E~*(R),

~N. once the

249

diagram

[SX,N]

f~

[X,~N]

~ [SX,N']

[X,dN']

,

g,

commutes.

The v e r t i c a l

jections.

We d e n o t e

morphisms

we h a v e

arrows

this

map

here

mean,

g by ~f.

of course,

Since

the

the m a p s

canonical

bi-

f~ are

group

homo-

have

obtained

a diagram UN

~N×~N

~ aN

isf

ef~efI ~N'x~N'

, aN' ~N'

which

commutes

up to h o m o t o p y .

functor

N ~

objects

in H ~ ( R ) .

iv)

pseudo-loop

The

aN,

[f] ~

(= a d j o i n t

in

S

~ HP~(R)

~N

: Hg~(R)

have,

choosen

[Mt])

to the with

mapped

for e v e r y

once to

and

[idsM]

from

functor

: S~N ~ N are the

also

[~f]

All

~

M 62"(R),

under

~ H ~(R)

suspension

functor

adjunction

homotopy

the n a t u r a l

adjoint

class

bijection

~ HWSA*(R).

maps

in

[Mt,

map

6M

[M,QSM]

[M,~SM]

p.

The m a p s 118].

which

-7~[SM,SM].

is Thus,

idsM =nSM°(S~M ) ,

v)

characterizes

For

functor

any

r 6 ~

by ~ r .

equivalence

~M up to h o m o t o p y .

we d e n o t e By

(7.5)

the

r-fold

we have,

iteration

for e v e r y

of the p s e u d o - l o o p

N 6WSA~(R),

We

: M ~ QSM,

by d e f i n i t i o n ,

which

a

of g r o u p

is r i g h t

j : HP*(R)

adjunction

a right

in the

category

j°S of the

inclusion left

t h a t we

to the

: HWSA*(R)

composite

the

means

HWSA~(R)

associated

for all

this

a homotopy

25O

~r N = M a p ( s r , N ) .

From

the d e f i n i t i o n s

nq(~ r N)

(7.6)

We

illustrate

it is o b v i o u s

that,

for e v e r y

q hO

and

r hO,

= nq+ r (N)

the

usefulness

of the p s e u d o - l o o p

functor

by an

example.

Theorem

7.7

polytopes.

(General Assume

suspension

theorem).

t h a t N is n - c o n n e c t e d

L e t M a n d N be p o i n t e d

for

some

n 6~

. The

weak

suspen-

O

sion homomorphism

SM, N

(§I, Def.

: [M,N]

is b i j e c t i v e

~

Freudenthal's

map

~N

: N ~ ~SN

The

claim

We

digress

"free"

[SM,SN]

if d i m M

Proof.

now

is a

short

pseudo-mapping

< 2n and

surjective

suspension

follows

for

2)

theorem

1.5 m e a n s

(2n+1)-equivalence from Theorem

from our spaces

if d i m M

(cf. V,

= 2n+I.

that §6,

the a d j u n c t i o n

Def.

5 a n d Def.

V.6.13.

general

7).

q.eod.

theme

c a n be o b t a i n e d

in o r d e r for

to i n d i c a t e

spaces

without

how

base

points.

Theorem

7.8.

variant

functor

presentable. ~M

such

,L

L e t M be a w e a k X ~

Thus

[XxM,L]

there

polytope f r o m H2(R)

exists

a weak

: M a p ( M , L ) xM ~ L

that,

for e v e r y

[X,Map(M,L)] [f] ~

~

X 6P(R),

[XxM,L]

[ ~ M , L ° (fxids)]

the map

and

L a space

to the polytope

over

category Map(M,L)

R. T h e

of

sets

contrais r e -

and a map

251

is b i j e c t i v e .

Proof.

We use

pointed

the n o t a t i o n s

space

any X 6 9 ( R ) ,

(L,y)

in s l i g h t l y

[XxM,L]

(cf.

by N.

=

4.2,

4.3)

Corollary

7.9.

4.1.

We c h o o s e

Then

N ° = L. Let

sloppy

notation,

[ (XxM)+,N]

=

[X+AM+,N]

some T

y 6L

and d e n o t e

:= M a p ( M + , N ) °.

= [X+,Map(M+,N) ] =

the

Then

for

[X,T]

q.e.d.

If M 6 2 ( R )

and N 6

WSA~(R)

then

M a p ( M , N °) ~ M a p ( M + , N ) °

We

return

to p o i n t e d

spaces

and

It is an a n a l o g u e

of a famous

Adams,

cf.

[Sw,

in the

same w a y

ed

[Adl],

Th.

as there.

state

another

representation

9.21].

It can

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

representation theorem

be p r o v e d

due

here

no t r a n s f e r

theorem.

to B r o w n word

and

by w o r d

principle

is n e e d -

for the proof.

Theorem gory

7.10.

HSA~(R)

Assume axiom

that

Then

of p o i n t e d F fulfills

(W) for

section.

Let F be a c o n t r a v a r i a n t

finite

Of c o u r s e

there

exists

[-,L]

on HSA~(R)

Again

we

not

be n e e d e d

right law)

call

track.

families

now

all

a group

(MIII6A)

the

sequel,

It i n d i c a t e s and v e r y

semialgebraic

spaces

object

is n a t u r a l l y

in the

over

{cf.

that

weak

useful spaces.

(MV)

have

such

of groups.

of the

to be p o l y t o p e s . } .

F.

us a h o l d

even

cate-

and the w e d g e

that

functor

polytopes

objects

category

(= i s o m o r p h i c )

of the

it g i v e s

the h o m o t o p y

the b e g i n n i n g

involved

equivalent

but

axiom

L in HP~(R)

space

from

R to the

the M a y e r - V i e t o r i s

L a classifying

are n a t u r a l

in c o m p l e t e

polytopes

functor

the

functor

to F.

Theorem

that we

7.10 will

are

(with a h o m o t o p y if one

is o n l y

on the group

interested

§8 - ~ - S p e c t r a

We n o w h a v e cohomology theory

theories

of s p e c t r a

spectra

serve

polytopes. here

the p r e r e q u i s i t e s for d r a w i n g

setting

We w i l l

not

delve

be c o n t e n t

with

to r e p r e s e n t

cohomology

theories

us

with

will

allow

"homotopy

already

called

spectra.

connection

but w i l l

This

called

and

the

maps"

appears

to w o r k

- between

spectra,

in G. W h i t e h e a d ' s

reduced

deeply

into

the

point

that

suitable

the v i e w

on the

a naive

between

level

notion

which

of w e a k

of m a p s

in the

fundamental

-

topological

paper

[W 2]

(there

"maps").

Definitions pointed

I. a) A

weak

(base p o i n t

(semialgebraic)

polytopes

(Enrn£~)

preserving,

b) A s p e c t r u m

E over

for the d e f i n i t i o n E n is an a b e l i a n

as always)

family

(fnln6~)

with E 6n

R is a f a m i l y

a family

(g~In6~)

object

(cf.

f : E ~ F from

of m a p s

fn

functor

in H~*(R)

of

of

: SEn ~ En+1" E ~n : En

an 2 - s p e c t r u m if the m a p s

of the p s e u d o - l o o p

group

map

maps

E over

E E above,are h o m o t o p y e o u i v a l e n c e s n

~(~n+1 ) ° ~n f r o m E n to ~2 En+ 2 c) A h o m o t o p y

together

R is c a l l e d

w h i c h are a d j o i n t to the

spectrum

via

~).

In this

the h o m o t o p y

(cf.

~En+1

§7

case

every

equivalence

§7) • a spectrum

E to a s p e c t r u m

F n between

: En

spaces

such

F is a

that

the

diagrams Sf

n

SE n

~ SF n

sE

~F

n

n

En+1

fn+1

~ Fn+1

commute

up to h o m o t o p y .

spectra

see

[Ad]

d) A h e m o t o p y f : E ~ F such

or

{For

[Sw,

equivalence that

every

a more

useful

notion

of m a p s

between

Chap. 8].} between map

fn

spectra : En ~ Fn

E,F

is a h o m o t o p y

is a h o m o t o p y

map

equivalence.

253

e) A n a l o g o u s l y instead

we d e f i n e

of ~ ( R ) ,

(topological) valences

N.B.

and

loop

between

topological

functor

~,

topological

E

spectra,

~-spectra,

further

using using

homotopy

the the

maps

category

/60*

genuine

and h o m o t o p y

equi-

spectra.

E

gn and

The m a p s

topological

qn h e r e

have

nothing

to do w i t h

the

a's and

q's

in §7.

Definition

2. Let

f,g

: E ~ E'be

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

spectra.

of

preserving)

(base p o i n t

and Fn(-1)

= gn"

and we h a v e with

F(-,O)

a spectrum

pointed

weak

(8.1)

limit

Fn

t £ I this

of h o m o t o p y

polytope

F from

h o m o t o p •i e s

every

E over

Hn(X,E)

the

A homotop~

= f, F(-,I)

the d e f i n i t i o n

Given

Here

For

two h o m o t o p y

:= l i m k

is t a k e n

way.

For

§I t h a t

every

8.2.

H~*(R)

to Ab

a "weak"

only

The

=

,

family

with

respect

a map

F(-,t)

of h o m o t o p y

fits

= fn : E ~ E~ well

above.

to the

group

Hn(X,E)for every

limits

transition

maps

)

(~;n+k * [sk+lX,En+k+l] are

indeed

for

[sk+IX,En+l+k ]

=

of c o n t r a w ~ r i a n t with

cohomology

I in §2 w i t h

granted

Fn(-,O)

[ s k X , E n + k ].

these

lim k

together

reduced

Definition

notion

(Fnln6Z)

abelian

groups

in a n a t u r a l

X 6])* (R) we h a v e

~n+1 (SX,E)

Remark

semialgebraic

X and n 6 Z as follows.

E

from

F gives

an a b e l i a n

[skX,En+k ] S [sk+lx,SEn+k] We k n o w

: E n xI ~ E n/ w i t h

e
R we d e f i n e

between

f to g is a f a m i l y

family

= g. This

maps

finite

the

families).

maps

H* (-,E)

the e x c e p t i o n

~

[skX,En+k ]

functors

identity

theory

lim k

= ~n

(Hn(-,E) In62)

~n+1 (-,E)oS

over

(X,E)

R,

of the w e d g e

i.e. axiom

from

~ Hn(-,E) it f u l f i l l s (which

is

is

254

This is obvious write En(x)

(recall 1.2.i and 1.7).

instead of Hn(X,E)

In the following we briefly

and E* instead of H*(-,E).

D e f i n i t i o n 3. We have an obvious notion of natural t r a n s f o r m a t i o n s b e t w e e n weak reduced c o h o m o l o g y theories over R, cf. §2, Def.

2. Every

h o m o t o p y map f : E ~ F between spectra over R induces a natural transformation from E* to F* in the ev:.dent way. We denote this natural t r a n s f o r m a t i o n by Uf.

P r o p o s i t i o n 8.3.

If E is an ~ - s p e c t r u m then,

for every X 6P*(R)

and

n 6 Z, the evident map [X,E n] ~ En(x)

is an i s o m o r p h i s m of abelian groups.

Proof.

E* is a reduced c o h o m o l o g y theory.

We have c o m m u t a t i v e diagrams

[sk+Ix,SEn+k .]

[sk+Ix ,En+k+ I ]

[skX,En+k ]

( ~ ~ which tell

[skx,~En+k+1 ]

us that the t r a n s i t i o n maps for the inductive limit

are isomorphisms.

(The u n a d o r n e d arrow is the a d j u n c t i o n i s o m o r p h i s m

made e x p l i c i t in §7). Thus the functor E n is isomorphic to this implies that E n obeys the full wedge axiom

(cf.

[-,En],

1.2.i).

Remark 8.4. M o r e o v e r it is clear that the h o m o t o p y classes topy maps

(8.1)

and

q.e.d.

[f] of homo-

f : E ~ F between a - s p e c t r a c o r r e s p o n d u n i q u e l y to the natural

t r a n s f o r m a t i o n s T : E* ~ F* via T = Uf, and T is a natural e q u i v a l e n c e iff f is a h o m o t o p y equivalence.

255

Theorem

8.5.

For e v e r y

an ~ - s p e c t r u m

Proof. that,

reduced

cohomology

E over R together

We k n o w f r o m the w e d g e

with a natural

axiom

for e v e r y n £ Z, the f u n c t o r

(W) and exists

(MV) r e q u i r e d

T n : [-,E n]

a(X)

~ Ab f u l f i l l s

theorem

7.1.

E n over R together with we h a v e

exists

T : E~ -?~k *.

I) and P r o p o s i t i o n

k n : HP~(R)

~~ k n. For e v e r y X 6 P * ( R )

: [X,E n] ~ [ X , ~ E n +

k* o v e r R t h e r e

equivalence

(§2, Def.

in the r e p r e s e n t a t i o n

a pointed weak polytope

valence

theory

2.3

the a x i o m s

Thus t h e r e

a natural

equi-

a bijection

1]

such t h a t the d i a g r a m

[X,E n]

, kn(X)

a(x) I

Tn(X)

[X,~En+I] commutes.

The a(X)

to

[-,~En+1].

~n

: En ~ ~En+1'

families family

together

By the Y o n e d a unique

(Enln62)

(Tnln£~)

and

form a natural lemma t h e r e

(~nln6Z)

8.6.

natural

equivalence

homotopy map

is d e t e r m i n e d

equivalence.

by k* u n i q u e l y

We n o w c o n s i d e r

In short,

a from

a homotopy

and V

[-,E n]

equivalence

(~n)..

an ~ - s p e c t r u m ,

f r o m E • to k ~.

The and the q.e.d.

: F • -L~k~ is a g a i n

a

f r o m 8.4 that t h e r e e x i s t s

a

up to h o m o t o p y ,

such t h a t V o U f = T, and

the ~ - s p e c t r u m

up to h o m o t o p y

s p e c t r a over d i f f e r e n t

spectra.

define

~-spectrum

f : E ~ F, u n i q u e

f is a h o m o t o p y

together

then it is e v i d e n t

equivalence

such that a =

equivalence

If F is a s e c o n d

~ kn+1(SX)

exists

up to h o m o t o p y ,

is a n a t u r a l

Remarks

topological

~ Tn+I(sx)

~ ~ [SX,En+I]

E in the t h e o r e m

equivalence.

real c l o s e d

f i e l d s and a l s o

256

Proposition

8.7.

Every

spectrum

E over R w i t h

all m a p s

E ~n b e i n g

This

follows

Definitions

map

all

4. a)

Let

over

obtained

homotopy

map

R.

b)

L e t E be a s p e c t r u m

above).

Replacing

we obtain

the

E.

Every

as a h o m o t o p y

Theorem

8.8

i)

CW-complexes

and

over

K which

extension over

IR w i t h

all

Etop,

map

theorem

of R,

from

and

let

and every

we

call

the

R to K. E v e r y

R yields

a homotopy

map

way. spaces

En pointed

called

topological

for

spectra).

CW-complexes.

by t h e p r o p o s i t i o n

topological the

f : E ~ F between

map between

spaces

topolo@ical spectra

spectra

(En)to p

spectrum

over

]R c a n

ftop : Etop ~Ftop"

L e t K be a r e a l

closed

of R. spectra

g

: E(K)

~ F(K)

If F is a s p e c t r u m with

follows

Analogously,

E over

uniquely

over

the h o m o t o p y

classes

of h o m o t o p y

to the h o m o t o p y

classes

of h o m o t o p y

the

K then

[g] = iff

there

equivalence

~

first main

theorems

[fK ]. A h o m o t o p y

fK is a h o m o t o p y exists

: E(K)

R is an ~ - s p e c t r u m

from the

using

R then

equivalence

a homotopy

A spectrum

over

b y the r e l a t i o n

f : E ~ F is a h o m o t o p y

this

field

spectrum

maps

All

to a

E n b y En(K)

underlying

f : E ~ F correspond

iii)

space

E n by t h e i r

maps

together

equivalent

extension

of g e n e r a l i t y

(First m a i n

If E a n d F are

ii)

field

restriction

homotopy

be read

overfield

over

a topological

underlyin@

pointed

E(K)

spectra

in t h e o b v i o u s

is no e s s e n t i a l

every

a spectrum

from E by base

: E(K)

closed

Replacing

f : E ~ F between

~ F(K)

E n being

K be a r e a l

fK

(This

spaces

R is h o m o t o p y

and V . 7 . 4 .

E (£n)K w e o b t a i n

spectrum

over

cellular.

from V.7.14

E be a s p e c t r u m E 6 n by

spectrum

V.5.2.ii

equivalence.

a spectrum

E over

R

~ F.

iff E(K)

theorem

map

is an ~ - s p e c t r u m .

V.5.2.i

and

and V.7.16.i,

T h e o r e m V.7.15.i,

we obtain

257

Theorem

8.9

algebraic

(Second m a i n

spectra

homotopy

maps

homotopy

classes

[ftop]

[g].

=

equivalence ii)

Given

E over

iii)

of h o m o t o p y

A homotopy

iff

ftop

every

homotopy

over

space

g

with

homotopy the

classes

exists

of

relation

(semialgebraic)

homotopy

semi-

(topological)

: E t o p ~ Fto p by the

En a semialgebraic

spectrum

cohomology

of R we h a v e

K.

uniquely

F there

~

If E and F are

(semialgebraic)

(topological)

spectrum

i)

homotopy

equivalence. a semialgebraic

CW-complex,

spectrum

and

a

(topo-

: Eto p ~ F.

E over

IR

is an a - s p e c t r u m

iff E t o p is

Q-spectrum.

k • a reduced

k K over

spectra),

f : E ~ F is a

equivalence

A semialgebraic

extension

the

maps

map

is a

a topological

a topological

For

~R t h e n

for

f : E ~ F correspond

IR, w i t h

logical)

over

theorem

Similarly

IR w e h a v e

over

constructed

for

a reduced

consequence

R and

K a real

in §2 a r e d u c e d

1 • a reduced

constructed

1~ . As an i m m e d i a t e top

theory

field

cohomology

semialgebraic topological

closed

theory

cohomology cohomology

of the d e f i n i t i o n s

(cf.

§2)

theory

theory one

ob-

tains

Proposition

8.10.

evident

canonical

and

i) For

every

~-spectrum

E over

R there

exists

an

isomorphism

E ( K ) * -?-~ (E~)K

of r e d u c e d ii)

cohomology

theories

If E is a s e m i a l g e b r a i c

algebraic

CW-comples

then

over

K.

~-spectrum there

exists

over

an e v i d e n t

phism

(Etop)* -L~ (E*)to p of t o p o l o g i c a l

reduced

cohomology

IR w i t h

theories.

every

and

E n a semi-

canonical

isomor-

258

By the r e s u l t s the real

relations closed

8.5 - 8.10 between

base

rather

obvious.

Chapter

V and

Thus

we h a v e

Notice

theories

and

of R, t h e n i~,--~ for

remark

logy

m-spectra

trum).

natural

to

be r a t h e r

For

of a b e l i a n

"sa"

brief

Let

groups

~n+k(Ek^X)

cohomology have

new

access

if K is a real

the e x t e n s i o n

E(K)

out

of

labour.

to §2.

between

cohomology

field

cohomology process

is the n a t u r a l

different becomes

serious

closed

for r e d u c e d

over

obtained

comfortable

E~--~

extension

theories

i*~,--~ I~ w h i l e

one.An

analogous

"top".

about

polytope

the

connection

X and

over

R

~n+k+1(SEkAX)

between

spectra

(not n e c e s s a r i l y

any n E 2 we h a v e

(~n+k(EkAX) Ik m O) w i t h S --~

been

of the r e l a t i o n

of §2 on

theories

without

than

half

theories

7.1

E be a s p e c t r u m

any weak

cohomology

results

process

and

second

theorem

is that,

the p r o c e s s

theories.

these

feature

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

pertains

We w i l l

that

pleasant

of the

topological

a rather

1 *R is m o r e

spectra

and

representation

gained

A particularly

contents

semialgebraic

fields

the

the

obvious

and h o m o -

an ~ - s p e c -

a direct

transition

system maps

(6 ~Aid x) , ' ~ n + k + 1 ( E k + I A X )

We d e f i n e

Hn(X,E) In this

:= lira ~ n + k ( E k ^ X ) k

w a y we o b t a i n

in a b e l i a n

groups.

up to a sign)

covariant

We d e f i n e

obvious

way:

functors

suspension

On(X)

H

n

(-,E)

on H ~ ( R )

isomorphisms

is i n d u c e d

by the

with

in the family

(perhaps

of h o m o m o r -

phisms S

~n+k(EkAX) with

~ the

-4

switching

n n + k + l (SEkAX) isomorphism

~ , ' nn+l+k(EkASX)

from

S I AE k AX

values

,

to E k A S I A X

.

259

T h e o r e m 8.11.

i) The families

(Hn(-,E) In6~) and

form a reduced h o m o l o g y theory H,(-,E)

(On[n6~)

together

over R.

ii) Given a reduced h o m o l o g y theory k, over R there exists a s p e c t r u m E over R together with a natural e q u i v a l e n c e H,(-,E) ~ k ,

Proof. [W2], sets

.

The t h e o r e m is well known to be true in the t o p o l o g i c a l [Sw, Chap.

(or groups),

8 and Chap.

setting

14]. By our m a i n theorems on h o m o t o p y

h o m o l o g y theories and spectra we know w i t h o u t further

labour that the theorem holds in the s e m i a l g e b r a i c setting.

Of course,

it would be more s a t i s f a c t o r y to prove the theorem d i r e c t l y

in the s e m i a l g e b r a i c ment

(cf.

setting.

[W2, p. 249f.];

This is rather easy for the first state-

in contrast to c o h o m o l o g y the wedge a x i o m

does not cause any troubles).

In order to prove the second statement

d i r e c t l y it seems a d v i s a b l e to use the m o d e r n s o p h i s t i c a t e d

language

of C W - s p e c t r a and maps between them as d e s i g n e d by B o a r d m a n and Adams (cf.

[Ad, Part III] and some chapters

in [Sw]), w h i c h

a n y w a y is indis-

p e n s a b l e for u n d e r s t a n d i n g the deeper aspects of stable h o m o t o p y the