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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