M
"know" a p r i o r i
the abstract
choice
of Delfs
{We t a k e
defined
t h e groups
a remarkable
an isomorphism we
problem
way.
reaches
polytope
a n d H ( M , A ; G)
/
{We d e s c r i b e d
3
Here
n
H (M A;G)
WSA(R).
of ordinary
which
of
affine
i s more
{But notice cohomology
comthat
with
sheaf
How
cohomology.}
about
an
singular means
a
interpretation
s i m p l i c e s , as
V(n)
i n R ^
over
R
we
t o M.
can
logy.
The
to
ordinary
an
noting from
this
Delfs
point
I have
for the
way.
chain open
As
semialgebraic
tries
of
the
C.(M,A;G) a s
=
a
G,
natural
here simplex
spaces in
H^(C.(M,A;G))
Q
in vain
prove
theory
one
to
a
i n the
given
to
case
like
finite
topo-
f i t together with
*
de-
isomorphism
open
iterated
the
trouble
reason
to
proof
prove
an
the
for a
t o make
some o t h e r
a
that
some
But
no
such
e x c i s i o n even
applying
s i m p l i c e s have
find
always
would
chain.
to
was
b a r y c e n t r i c s u b d i v i s i o n or
subdivision,
standard
semialgebraic
H (C.(*,0;G)) imply
of
s i n g u l a r simplex closed
weakly
chains
+
not
i n the
the
by
/
H (-,G).
n
by
n
a
groups
would
difficulty
respect
H (M A;G)
complex
that
H (C.(M,A;G))
sets)
from
the
and
This
to
could
simplices
that
theory
The
in classical
"small" with
singular
prove
of
course,
(M,A)
for years
groups
Of
singular chain
theory
We
pair
space.
tried
archimedean.
topes.
the
homology
geometric
theorem
one
one
elements
(= m o r p h i s m )
any
i s to
homology
and
direct
not
problem
the
map
For
define
the
i n topology?
semialgebraic n +
of
a
in
excision
field
triad given
covering
a
R
i s
of
poly-
singular
(with
two
subdivision to i s t h a t , as
s o r t of
become
long
finite
small
the as
linear
i f R
i s
not
archimedean.
The
last
problem
Chapter V I I -
convincing algebraic
We set
proceed =
along
very
single spaces
of
the
different
i s s u e , up are
roughly
as
present
really
to
book
lines. now,
contains
This
to
a
solution
demonstrate
solution
of
i s perhaps that weakly
the the
most
semi-
useful.
f o l l o w s . Every
s e m i s i m p l i c i a l complex,
simplicial
i n other
set
K
terminologies)
(= s e m i s i m p l i c i a l can
be
"realized"
as
a
weak p o l y t o p e
in
topology
complex.
If
simplicial of H
space
(K,L)
pair
0
0
K
[May])
If
M
i s a of
can
K
"abstract"
i s a weakly
simplicial
set
-* M.
iSinMl
only -
the
non
[LW]
H
q
from
Since
group
we
=
geometry.
wide Thus, sets,
Simplicial
sets
branches this
of
of
i s a
H
and
i s proved
groups
subspace
of
the
open
for
finally, only
have
can
we
use
can
to
of
form
an
M,
then
be
J
or
=
H
on
M
The
the
J
( S i n M,
methods
fact that
a
§7) i s
M
homology
gives .
map
(VI,
simplicial
(M,A)
q
M.
{in topology
ordinary
j
of
semialgebraic
using
to
M
are
our
the
theory.}
homotopy
equi-
Thus
S i n A ; G) ,
i n the
sets
[DK^,
enormously
in
homotopy
previous
complexes"
be
used
groups
singular
equivalence
simplicial
abolish
in particular now
a
definition.
maps
simplicial
proved
texts
already
D
the
L
[La]
the
simplices
Weingram.
R
canonical
CW-
description
(cf.
form
weakly
|SinA| )
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can
homotopy
most
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n
topology,
material
i s a
m
In
R
a
homology
known
singular
( I S i n M l , I S i n A I ; G) i\ K
q
R we
canonical
Lundell
(I S i n Ml ,
pair
of
with
cellular
ordinary
well
over
the
equivalence. this
course,
known
n
a J
the
i s
structure
(of the
this
H (K,L;G).
space
that
from
the
as
natural
sets
that
with
groups
i s Hg(C.(M,A;G))
simplicial
of
[LW]
-
know t h a t
i s now
above
identified
prove
i f A
the
(M, A; G)
this
i t follows
s i n g u l a r homology
generally,
valence
door
We
same way
simplicial
comes w i t h
R
book
i n
topological
More
of
the
carries a
R
c o n s i s t i n g of
a weak homotopy
but
and
be
Sin M
J
following
l K l
semialgebraic
I S i n Ml
:
i n much
then
homology
realization M
K),
R
homology mentioned
(iKl ,I LI ;G)
n
over
D
[ M i ^ ] . The
subset
ordinary
IKI
p.
useful
theory
semialgebraic
of
equivalences in
the
semialgebraic
verdict
"no
124].
in
many
fibrations.
geometry.
Some
Much
applications in
t o the theory of semialgebraic
the next
But
volume
in
One n e e d s
simplicial
sets.
By a s i m p l i c i a l
space
WSA(R),
i . e . a sequence
(X ln€N )
over
R with
between
simplicial
Roughly the
various
them
half
f o rfuture
able
to construct X. B y t h i s
of weakly
Q
sets
instead
R we m e a n
semialgebraic
be g i v e n
of just
a simplicial
semialgebraic
be regarded
as
object spaces
face and degeneracy may
simpli-
maps discrete
R.
Chapter
of simplicial
ties
t i a l l y
over
of our last
fundamentals
space
n
weakly
spaces
X over
(VII, §1). Simplicial
spaces
w i l l
[SFC].
one needs more.
cial
fibrations
application
V I I i s devoted
spaces
w i l l
we m e a n
a
proper. Fortunately
and t h e i r
arise
the realization
t o an e x p l i c a t i o n
from
realizations.
the fact
that
IX| of a partially R
simplicial discrete
space
a l lwhose
simplicial
spaces
we
of
Difficulare only
proper
simplicial
f a c e maps
are par-
are partially
proper.
A
reader having worked
spaces
a n d maps
similar
stuff
fort
indicate
Let
we
i n Chapter
about
meant
of
M
We
now
feel
spaces.
semialgebraic
from
by a p r i n c i p a l
\exactly
I V may
simplicial
some o r t h o g o n a l g r o u p i ti s clear
t h e fundamentals
now b y a n e x a m p l e
G be a complete
;then
through
0(n,R).}
pose
t o meet
To g i v e
such
that
this
stuff
group
over
R.
I f M
i s an a f f i n e
G-fibre
bundle
tp : E -> M
a reader
semialgebraic
the following
some
i s really
think
semialgebraic geometry
com-
useful.
{For instance
o v e r M.
a finite
semialgebraic
i n Chapter V I I
the beginnings of semialgebraic
as i n topology, o f course w i t h
by open
bored
of weakly
space, what i s
The d e f i n i t i o n
trivializing
of
i s
covering
subsets.
problem.
L e t S be a r e a l
closed
overfield
of
R a n d l e t \p : F
M(S) be a p r i n c i p a l
exist
G - b u n d l e cp : E -> M o v e r
ip
s
It
a principal
: E ( S ) -> M ( S )
seems h a r d
solve of
i s isomorphic to
to solve
G.
This
Example VII.1.2.v (partially M^Gl.
One
elements
finds
the
t h e base
conclude
over
that
geometry
the
like
existing
define
one
realization
classes
main
of
G-princi-
with the
theorem
on
clear
from
homo-
III.3.1).
o f N
similar
become
amply
the notion
equivalence.
shall
R
The
space"
into
a
fibra-
plan,
maps
from M t o
section of
the unit
Chapter
interval
i n the preface
IV and w i t h only weakly
i tpays
that
(=
of
this
would of
f o r the categories other
as e a r l y as p o s s i b l e .
ways
[LSA],
Uberlagerun[SFC].
spaces
i n the theory
i n many
of
volume
semialgebraic
proofs
i n t h e same way
coverings
i n t h e next
I realized
s i n c e many
spaces
[0,1],
announced
do t h i s
how w e l l
semialgebraic
In the last
sufficiently
i s sequential.
I n t h e meantime
realized
semialgebraic
f o rM = N =
to introduce
c a n be done
s t r u c t u r e on a
of weakly
i n Chapter
o f arguments,
coverings
WSA(R).
this
the field
fibrations
i n Chapter
later
as w i l l
one has t o work w i t h
a given
b i g f o r some p u r p o s e s ) .
that
from
with
There
and
we
i n point.
something
i n t h e case
then
s u b s t i t u t e and
i n s t e a d o f homotopy
o f t h e s e t Map(M,N)
(sufficiently we
But this
con-
substitute of the
f o r a l l purposes,
turns
This
V I I we a r e
spaces
s u b s t i t u t e o f t h e t o p o l o g i c a l "path
strategy
subset
VII
I
equivalence
Chapter
as a canonical
of fibrations.
[Bn].
1 7 ] M a p ( S i n M, S i n N)
space Map(M,N).
are not sufficient
Using
semialgebraic
s e t [May, p. R
i s a case
Another
N
up t o homotopy.
7 . 2 ] , [DKP, 5.3]) w h i c h
tion,
r e p r e s e n t a t i o n theorem
i M a p ( S i n M, S i n N) I
f o r a
s u b s t i t u t e s . Our c o n s t r u c t i o n i s
a r e any weakly
mapping
homotopy
question ([W,
and N
i n the theory
fibre
only
not existing
constructions
such
o f Brown's
i s canonical
better
from
at a
cause
a
fibrations LSA(R) to
and
intro-
I
thank
Rainer I
Professors, Ronnie Vogt
for useful advice
f u r t h e r thank
for help
here.
In
particular,
these
persons
previous
to
my
with
for proof
versions
of
without
Regensburg,
March
losing
Fritsch,
i n tackling
details
Appendix
secretary Marina
versions
Rudolf
Hans D e l f s , Roland
Schwartz
in
Brown,
of C
reading
Huber, the
and
volume.
Franke
for a
a
Peter
proofs too due
numerous to
very
special
efficient
in c r i t i c a l
Manfred
Knebusch
I
for
thanks
typing of
situations.
1988
and
to
Huber.
and
homotopy.
Scheiderer
successful search Finally
May,
simplicial
Claus
i s entirely
this
patience
with
J.
be
Niels listed
also
thank
mistakes are
due
a l l these
TABLE
OF
CONTENTS
page CHAPTER
§1
-
IV - Basic
Definition
theory
o f weakly
semialgebraic
and c o n s t r u c t i o n of weakly
spaces
semialgebraic
spaces
1
§2
- Morphisms
§3
-
Subspaces
§4
-
Spaces
of countable
§5
-
Proper
maps
§6
-
P o l y t o p i c spaces?
§7
-
A
§8
-
Strong
§9
-
The weak p o l y t o p e
theorem
15 and products
23
type
36
and p a r t i a l l y
proper
the one-point
on i n d u c t i v e l i m i t s
quotients; gluing
A
P (M)
maps
42
completion
49
o f spaces
54
o f spaces
60
P(M)
71
§10 -
The spaces
§11
The q u o t i e n t by a p a r t i a l l y
-
and P (M)
86
f
proper
equivalence
relation
CHAPTER
V
1
99
- Patch
complexes,
§1
-
Patch
§2
-
Some d e f o r m a t i o n
and homotopies
again
decompositions
106
106
retractions,
and r e l a t e d
equivalences
homotopy 11 4
§3
-
Partially
finite
open
coverings
§4
-
Approximation
§5
-
The two main
theorems
§6
-
Compressions
and n-equivalences
§7
-
CW-complexes
o f spaces
125
by weak p o l y t o p e s
on homotopy
sets
133 147 152 165
page
CHAPTER V I - Homology
and cohomology
categories;
182
§1
- The b a s i c
§2
- Reduced
§3
- C e l l u l a r homology
§4
- Homology
of pairs
o f weak p o l y t o p e s
214
§5
- Homology
of pairs
of spaces
224
§6
- Excision
and l i m i t s
§7
- Representation
§8
-
cohomology
suspensions
and cofibers
183
o f weak p o l y t o p e s
194 209
233
theorems, pseudo-mapping
spaces
244
ft-spectra
252
CHAPTER V I I - S i m p l i c i a l s p a c e s
2 60
§1
- The b a s i c
§2
- Realization
§3
- Subspaces
280
§4
- Fibre
2
9
2
§5
- Quotients
3
0
3
§6
- Semialgebraic
§7
- The s p a c e I S i n Ml a n d s i n g u l a r
§8
- S i m p l i c i a l homotopy,
§9
- A group o f automorphisms o f [0,1]
APPENDIX open
C
sets
definitions o f some
260
simplicial
spaces
268
products
realizations of simplicial
(to Chapter of M
t
Q
p
?
i s f(M)
homology
3
11 3
homology
and s i n g u l a r
I V ) : When
sets
again
2
0
331 341
a basis
of 352
References
355
Symbols
359
Glossary
363
Contents
of Chapters
I - I I I
3
75
Chapter
§1
R
IV - Basic
- Definition
i s a
fixed
topological missible)
ly
M =
Definition
sets
o f weakly
As
semialgebraic
i n I , §1
we
and Cov^ t h e s e t o f
and examples
a
subset M
Definition
K of M
i s a locally
subset
of M
open
M we
i n M
the union subset
space
over
a
below).
i f , f o r every
i s already
semialgebraic
give
o f a weak-
( D e f i n i t i o n s 6,7
small
U € f(M) of the
o f A.
R then
every
i n M.
2. a) A
function
ringed
space
space
M equipped
with
a sheaf
topological
the s e t of (ad-
a space
a suitable finite
i s small
generalized
to the definition
the set UflK
\ running through
a
(admissible)
such
a n d a weak p o l y t o p e
(U^UGA) G C o v ( U ) ,
semialgebraic
leading
spaces
consider
M
call
1.1. I f M
spaces
(M,3*(M) , C o v ) . H e r e T ( M ) m e a n s
space
1. We
(1K w i t h
Example
semialgebraic
c f . I , § 1 , D e f . 1. S t a r t i n g w i t h
of definitions
every
field.
subsets of M
semialgebraic
and
closed
space
coverings, chain
and c o n s t r u c t i o n
real
open
theory of weakly
M
over £>
M
R
i s a
generalized
of rings
of
R-valued
M,N
R i s a
functions. b)
A morphism
map
f
that h*f c)
between
: M -* N
( i nt h e sense
f o r every 1
: f ~ (V) We
denote
Example ringed
V € f(N)
semialgebraic
ringed
and h € &
N
(V)
over
R.
maps.
over
continuous
topological spaces),
the composite
such
function
1
of © (f" (V)). M
the category of function
locally
spaces
of generalized
R i s an element
1.2. E v e r y space
function
ringed
semialgebraic
The morphisms between
spaces
space such
over
over
R by
R i s a
spaces
Space(R).
function
are the
locally
Henceforth a
small
ringed UOK
l e t M
subset space
be a
of M
over
then
M
induces
I f (V^IAEA)
(V, | A € A ) G Cov__ i f a n d o n l y A
space
(always
over
on K t h e s t r u c t u r e
R) . I f K i s
of a
function
i s the set of a l l intersections
i s a family
i fthere
i n f(K)
exists
then
a finite
subset
A'
of A
K
such
that
A £ A'. in
ringed
R a s f o l l o w s . T(K)
uef(M).
with
function
the set V
{As usual,
I , §1
:=
we
U(V^IAGA)
then
are clearly
i s already
write
the union
(V^IXGA) £ C o v ( V ) . K
fulfilled.}
^
i s the sheaf
K
of a l l
with
The axioms
associated
i - v i i i
to the
p r e s h e a f 1 9 ° d e f i n e d a s f o l l o w s . A f u n c t i o n h : V -* R o n s o m e V € T ( K ) o • i s a n e l e m e n t o f ° ( V ) i f f t h e r e e x i s t s some U G T(M) a n d some g G(0 (U) K
M
with
U (IK D V
that
U n K = V.}Thus
of
a n d h = g l V . {We
^ ( V ) i f f there
for
g
۩M^)
±
1
means
0
just
(F).
about assume
f i n i t e
i n
We
close this
given
weakly
have
some
section
some
semialgebraic
redundancy
Definition
with
easy
space
which
observations
M.
on exhaustions
An e x h a u s t i o n
(M la£I)
of M
a
of a may
c a n be e l i m i n a t e d .
8. A n e x h a u s t i o n
(M^laEI)
of M
i s called
faith*)
ful
i f , i n s t e a d o f E2, the f o l l o w i n g 0,3 £ I,
E2' ) F o r a n y two i n d i c e s
I P r o p o s i t i o n 1.14. I f ( M l a € I ) M
then
? (MglaCI')
: Proof. :with
there
i s a
Throw
y M c:Mp.
a
N
i n the
Y
as
weakly
semialgebraic the
closure
semimap.
f : X -* Y
map.
a
semialgebraic
the
restriction
space flM
N
:
i s weak-> N
i s
semialgebraic.
f In particular, I R
are
2.3
precisely
i s already
the weakly the
semialgebraic
elements
evident
from
of
(
^ (M). M
maps This
the definition
from
M
special of
an
to the case
real
of
exhaustion.
line
Theorem
Definition
3.
From
semialgebraic the
braic
Remark
of 0 (U)
map
every
DM^IaEI).
the map.
of
f
a map
i n f ( e e J I f (M )
J
3
space
K
c a l l ,
i s again
2.3,
N
since M
4. We
call
semialgebraic t o N.
conflict
This
with
i s a morphism
i s a morphism,
finite).
i f f , f o r every
a €I ,
i . e . a semialgebraic
map
9 ] ) .
i s the inductive limit
i ti n a special
Definition
from
: M -> N
fIM^ :
(cf.
observed
f
(UjlJ N ( S ) i f both
semialgebraic f
(resp. M
and
: M N(S)
N
space
-* N
over
(resp.
M(S))
are locally
g
over
R we
obtain,
: N -* M) S.
This
over
i n a R a
similar morphism
i s t h e same map
semialgebraic.
as
§3
-
Subspaces
As
before,
fixed
M
1.
every
space
This
of
subset the
X
set
semialgebraic
set
T(M)
semialgebraic
of
M
X flM
i s called
notation of
subsets
of
not
from
Proposition
i s evident
In
particular,
M
If
X
depend
G 7(M)
a
weakly
M
on
[LSA],
X nM
i s denoted
the
2.7.
choice
f o r every
are
weakly
semialgebraic
subsets
X^Y
are
again
weakly
semialgebraic
i n M.
that
the
preimage
: N
Weakly sets
the
of
M
a
follows.
3.1.
for
X nM
By
E3
this
=
that
hence
X^
i n X^.
every
a
can
(MglaGI).
subset
of Let
any
X G T(M)
semialgebraic
sets
i ) Assume
contained
of
Recall
( c f . 1.17),
algebraic
that,
(X)
i s weakly
exhaustion
Remarks
is
f
semialgebraic
as
of
-* M
and
(M laGI)
a
a
Every
X
the c
T(M)
of
(M^laCI),
. Also
of
?(M)
cJ(M).
that, the
then
a
X UY, easily
weakly
by
i s an
"collecting" the
(open)
family
verified
semialgebraic of
patches
semialgebraic
subset
space.
aGI,
there
i s given
whenever
3 A
Remark g
Given
X
o f U,
of Cov (U) M
i n t h e sense
are just
of this
definition,
€A.
[0,1] denotes
the closed
unit
interval
the
i n R.
Admissible
coverings
Proposition (X^|X€A)
behave w e l l under
3.14. I f f
: N -* M
i s an a d m i s s i b l e
admissible
covering
L e t B € r(N) . Then
A with
f(B)
closed
N
i s a
i s an exhaustion
every o f M.
B G r(M) (Again
then
€ Cov
(U fl A ) .
A
A
semialgebraic i f f
i s weakly maps).
semialgebraic.
A map
R i s a morphism
be an o r d e r e d
that
A
semialgebraic N over
o f U,
(X^flA^GJ)
: U -> R i s w e a k l y
: A
A
3.16. L e t (A IXGA) E2-E5.
f
f o rweakly
the restriction
Corollary
o f T(M)
t h e f u n c t i o n fIU nA
a function ringed
X GA,
o r even
do t h e sames
covering
i s a family of subsets
a function
(Gluing principle into
sets,
A
(X^I36J)
I f U € 3*(M) , t h e n
Then
sometimes
q.e.d.
A
7(A )).
x
I f U C X(M) a n d
for
they
t h e i n t e r s e c t i o n X fl
(X I3CJ) € C o v ( U ) c)
(X ) I X G J ) .
J of
subsets.
T (A ) , r e s p .
&
1
subset
manner.
semialgebraic
(resp. b)
and nevertheless
3 . 1 5 . L e t (A-^IXGA) b e a n a d m i s s i b l e
subset
a finite
c a n be u s e f u l . They
This
weakly
i s an
A
exists
exhaustions,
Theorem
(f (X )|X6A)
f (B) C 2T(M) . T h e r e
semialgebraic
straightforward
map a n d i f
1
of M then
than
a
preimages.
o f N.
Proof.
Admissible
taking
i f f ,
f
: M ~> N f o r every
morphism.
family
i n ?(M) w i t h
i s contained i tsuffices
the
i n some
A . A
to l e t B run
through
the sets
Mg.)
Indeed,
Theorem
3.15.d)
space
i sthe inductive
M
Definition val x
i nM
limit
i nM
[ 0 , 1 ] i n R t o M.
of Y
8. A p a t h
(aswell
of the family
i s t h e s e t o f a l l y€M
open weakly
s e m i a l g e b r a i c i n M.
of the different
Proof. union
o f path
It
a
C
i s easily
disjoint
tion
path
path
f o r every
that
the unit
component
there exists
c o m p o n e n t C(x,M) The space
i s closed
a
inter-
C(x,M)
path
and also
M i sthe direct
c o m p o n e n t s o f M,
sum ( c f .
considered, of course,
C(x,M) n
a €1, the intersection
components o f t h e s e m i a l g e b r a i c space r ( M ) fl?(M
) . The c l a i m
a
seen
that
non empty
Definition
such
(A^IAGA).
o f M.
Clearly,
C(x,M) n M
from
us that t h e
= x a n d y ( 1 ) = y.
3.17. E v e r y
subspaces
o f spaces
i s a s e m i a l g e b r a i c map
Proposition
as
tells
F o r any point x o f M t h e path
: [ 0 , 1 ] -* M w i t h y ( 0 )
1.10)
as Th. 3.15.a)-c)}
9. We
open weakly
call
3.17 we c a l l
a path
such
the path
M^.
i s a
Thus
follows.
component X i s n o t t h e u n i o n
o f two
semialgebraic subsets.
a space
X connected.
Justified
components o f M a l s o
by
Proposi-
t h e connected
compo-
n e n t s o f M.
Let
N be a second
exhaustion spaces with
o f N.
M a n d N.
weakly We w a n t
s e m i a l g e b r a i c space
to construct the direct
We
equip
the cartesian
the inductive
limit
space
semialgebraic
spaces
(M
over
xN
product
structure
I(a,3) GIxj),
R a n d (N^I3GJ) product
MxN
where
ofthe
of the sets
of the ordered
an
M,N
family of
Ixj i s t h e d i r e c t
product tions
of the ordered
i n Theorem
the
s p a c e MxN
(M
x I
Q
with
braic
spaces
product spaces
the given M
semialgebraic the subspace
Using
topology
XxY
i s closed
All
this
I f X (resp.
i s obvious
N
i
n
MxN
of the
1.6,
WSA(R) o f w e a k l y
coin-
semialge-
checked
that
are weakly
projections, i s the
3.19.
a n d N.
semi-
direct
semialgebraic
open)in
from
and
Then
to
(resp.
the definitions.
p r o j e c t i o n pr^|T(f)
-» L a n d g
on M
sets
than
the
N.
t h e s e t XxY
i s weakly
s t r u c t u r e on product
open)
i n M
XxY
of the sub-
then
XxY
and N
i s
then
MxN.
Using
Theorem
2.3
also
the
verified.
weakly
construct
have more open
I f X and Y a r e s e m i a l g e b r a i c
and Y a r e c l o s e d
Let f
r ( f ) o f MxN
may
t h e s t r u c t u r e as t h e d i r e c t
r ( f ) of f i s a closed
finally
product
and t h e subspace
N be
subspace
x
: MxN
2
a n d Y € X(N).
: M
natural
Theorem
i t i s easily
p r
topologies
proposition i s easily
Proposition
Thus,
by
assump-
exhaustion
2.3
these
o n MxN
i n t h e s p a c e MxN,
semialgebraic.
: M
with
L e t X € T(M)
X and Y of M
following
MxN,
the
structure of
-» M,
i n the category
coincides with
spaces
with
Theorem
: MxN
of the strong
3.18.
semialgebraic
graph
that
strong
product
MxN
a n d E5.
f u l f i l l s
R.
The
Proposition
and
and N
over
direct
f
i n a d d i t i o n E4
family
s t r u c t u r e as t h e d i r e c t
a n d N^.
Q
maps,
of M
Caution.
We
and
I and J . This
natural projections pr^
algebraic
in
and
i s weakly
(a,3) E I x j )
cides
the
1.6
sets
from
a weakly
semialgebraic
semialgebraic
r(f) to M
i s an
subset
map.
The
o f MxN.
isomorphism
The
of the
M.
fibre
products
: N -» L b e w e a k l y
i n the category semialgebraic
WSA(R).
maps o v e r
R.
Let Then
f xg
MxN
:
-* LxL
MxN
is
i s again a weakly
semialgebraic
preimage
weakly
semialgebraic
of the diagonal
A
=
T
subset
r(id )
t h e subspace
Theorem
and
2
Caution. gory
M x^N
We
do n o t c l a i m
3.21. I f M
ringed
and N
i s again l o c a l l y
locally
semialgebraic
product
MxN Li
This
A l l
i n
i s rather
objects
tension
equip
MxN LI
The f o l l o w i n g c a n now
be
verified
diagram
projections,
t o M X^N, i s a c a r t e s i a n
of function
Remark
f x g . We
i sthe
N
and q the n a t u r a l
p r
i n MxN.
3.20. The c o m m u t a t i v e
L
p
under
M x^N
manner.
MxN
with
since
Li
structure
a straightforward
o f MxN,
o f LxL
T
L
in
and
:= { ( x , y ) €MxN| f ( x ) = g ( y ) }
a closed
with
map
that spaces
square
this
then
i s cartesian
i n the cate-
R.
locally
semialgebraic.
of pr^
i n WSA(R).
diagram
over
are also
i.e. the restrictions
semialgebraic
I f a l l three
our space MxN Li
spaces,
spaces
i s t h e same
M,N,L
as t h e
then are fibre
[LSA].
evident
defined
from
i n this
the definitions
and P r o p o s i t i o n
s e c t i o n behave w e l l
( c f . D e f . 7 a n d D e f . 8 i n §2)
t o some
real
under
base
closed
2.12.
field
field
S
exR.
Remarks
3.22.
set
a
X fl M
(X D M sets The
a) L e t X € T(M) . F o r e v e r y
of M
yields
a
) (S) o f M ( S )
by base
[DK ,
a
3
field
base
field
notion X(S)
extension
a semialgebraic
p . 142]. L e t X(S) d e n o t e
i n M(S) . We h a v e X(S) PIM subspace
a G I , t h esemialgebraic
(S) =
t h eunion
subsubset
o f these
(X nM ) ( S ) , h e n c e X ( S ) € T(M(S) ) . fl
s t r u c t u r e o n X(S) i n t h e s p a c e M ( S ) i s t h e s a m e a s t h e
extension
o f t h esubspace
X(S) h a s no a m b i g u i t y .
i sc l o s e d
(resp.
open)
s t r u c t u r e o n X i n M. T h u s t h e
I f X i sc l o s e d
( r e s p . open)
i n M ( S ) . I f X € JT(M)
then
i n M,
then
X(S) €JT(M(S)).
We h a v e X ( S ) flM = X. b)
I f (X^IXGA)
subsets, c)
then
The space
weakly
covering
o f M by weakly
(X (S)|X€A) i san a d m i s s i b l e
covering
A
I f (C^IXGA)
(C^(S)IA€A) d)
i sa n a d m i s s i b l e
i st h e f a m i l y o f connected
i st h ef a m i l y o f connected
(MxN) (S) i s t h e s a m e a s M ( S ) * N ( S ) .
semialgebraic
map, t h e n
o f M(S) .
components
components
t h esubsets
r ( f
g
semialgebraic
o f M,
then
of M(S).
I ff : M
N i sa
) and T(f)(S) o f
M(S) x N(S) a r e e q u a l . e)
I f two weakly
then
semialgebraic
t h etwo subsets
maps
M(S) x^^NtS)
(M x N)(S) o f (MxN)(S) a r e e q u a l . sion
The be
f u n c t o r WSA(R)
easy left
C-^ (S)
If
The be
proofs
M i sconnected
then
c l a i m c a n be proved Then x £ M (S) Q
means
t h e maps
fg'9 ^
that
t h e base
field
cartesian
easy
a
n
d
s
exten-
squares.
statements
o n c e we know t h a t
may
safely
t h e spaces
thefollowing claim:
M(S) i s c o n n e c t e d .
a s f o l l o w s : We f i x a p o i n t p € M . L e t x € M ( S ) f o r some
qGM^. (This
t o a simplicial
: [0,1] -+ M w i t h
c) i sa l s o
This
f r o m x t o some p o i n t
6
We c o n c l u d e
a r e given
from
o f a ) , b ) , d ) , e) a n d o f s i m i l a r
a r econnected.
morphic
(coming
WSA(S) p r e s e r v e s
t o t h ereader,
given.
f : M -> L , g : N - + L
complex
a E I . There
exists
a path
i sevident
since,
s a y , M^ i s i s o -
over
R.) T h e r e
6(0) = q a n d 6(1) = p . T h e n
exists 6
q
a
y i n ^ ( S )
path
i sa path
i n M(S)
from
q
to
p.
The
composite
path
y*6
connects
x
to
p.
§4
- Spaces
In
this
small
introduce able
o f weakly
which
behaves
admit
well
ly
i t does
In
the following
notsuffice
R. A l s o ,
then
we a l w a y s
with
i t s natural total
able
index
Remarks
mean
that
1. A s p a c e
t r i v i a l
spaces,
simple
always
section
t h e "spaces
exhaustions.
constructions i nthis
f o r a l l purposes
i f we w r i t e d o w n
(X^|A£A)
covering
most
a "space"
almost
semialgebraic
particularly
under
over
Definition
type
a n d , u p t o some e x a m p l e s ,
a class
type",
spaces
o f countable
i n semialgebraic
means
a weakly
an ordered
of
This
paper.
we
class of
Unfortunate-
topology.
semialgebraic
family o f sets
t h e s e t IN o f n a t u r a l n u m b e r s
count-
space
(X |n6 3N)
i s equipped
ordering.
M i so f countable
by semialgebraic
sets
type
i fM has an a d m i s s i b l e
( c f . § 3 , D e f . 7) w i t h
count-
s e t A.
4.1. i ) Of c o u r s e ,
every
semialgebraic
space
i so f countable
type. ii)
I f M i so f c o u n t a b l e countable
i i i ) The d i r e c t again
type,
then
also every
subspace
type. product
o f countable
MxN
o f two spaces
type.
Indeed,
M,N
o f countable
K of iv)
countable, MxN
then
o f countable
sets with
type
i so f countable
closed overfield
countable
then
M(S)
countable
of a family
I f S i sa real type
with
( X ^ x y I(A,K) £AXK) i s a n a d m i s s i b l e
U(M^|A£A)
sum
K
o f M and N by semialgebraic sets,
by semialgebraic
The d i r e c t spaces
v)
coverings
type i s
a n d ( Y IK£K) a r e
i f (X,|A£A) A
admissible
of M i s of
(M^IAEA) type
o f R and M
i sagain
index
covering
s e t A xK. o f non empty
i f f A i s countable.
i s a space
of countable
A and
type
over
(cf.
R of
3.22.b).
Proposition an
4.2.
exhaustion
with
a(n)
B ^
is
a map
f
proper
map
Now
i s a complete
This b)
M
0
a
subset
by an easy that
We
Definition A€^(M),
3. A map
complete
f
: M
-» N
weakly
the
a
map
restriction
: M
-> N
T h i s means
Of
course,
i tsuffices
case want
c f . [ L S A , p.
that
M
( I , §5)
i s t h e one p o i n t
t o prove T
N
that E
S
E
M
i s semi-
T
Q
a n d t h e same
discrete
59], that
A
space.
We
i s a finite set.
partially
i s proper.
M
of f are semialge-
i s semialgebraic.
proper
The
space
q.e.d.
i f , f o r every M
t o t h e one p o i n t space
A6!f(M)
these
o f o u r e x h a u s t i o n o f M.
N
a l l fibres
i s called
t o check
i f f f
i s indeed semialgebraic.
know t h a t
every
that
a point X
f | A : A -» N
that
sense
s e m i a l g e b r a i c spaces
i s a complete
and M
semialge-
i n the present
s e m i a l g e b r a i c i n M,
A
from
locally
i ssemialgebraic.
a n d we
choose
5.2
i f t h e map
proper.
M
f
Thus
now
by Prop.
ly
sets
we
the restriction
partially
the
o f A.
1° i s f i n i t e ,
conclude
of locally
and weakly
argument,
In the general case
braic.
i s proper
space,
a £ 1 ° we
{x loc€I } i s closed
means
X €A,
consider the special
For every
f o r every
-» N
i n the category
space.
conclude
of semialgebraic spaces, c f .
s e m i a l g e b r a i c and a l s o
: M
f i r s t
holds
i f f f (M)
( e . g . an e x h a u s t i o n o f N ) . Then
are weakly
a ) We
:=
i s proper
-> f (M) b e t w e e n s e m i a l g e b r a i c
i f f , f o r every
Proof.
A
-+ N
of f i s proper.
5.4. E v e r y
algebraic.
" f ": M
: M
be an a d m i s s i b l e c o v e r i n g o f N by c l o s e d
semialgebraic : M
f
§9].
L e t (B^IACA)
f
a n d t h e map
a map
i s called i s partial
i s complete.
properties
Notice that
f o r A running the partially
through complete
spaces
(resp.
topes
(resp.
Partially describe is
spaces)
polytopes)
proper their
easier
Remarks
complete
than
5.5.
be
i n the center of our interest.
properties
i n I , §5
Let f
: M
since
-* N
we
and
g
: N
ii)
I f g«f
i s partially
proper
then
i i i )
I f g«f
i s partially
proper
and
then
i s partially
proper,
and
A l l
this
ties
surjective"
-> L b e
proper
then
here
maps.
g°f
i s partially
f i s surjective
{ i tsuffices 3
theory
we
3.6.}
f i s partially
( c f .Def.
{The
Thus
proper. and s e m i a l g e b r a i c
t o assume
i n §8)
proper.
instead
that
of
f i s
surjective
semialgebraic.}
follows
of proper
Remark
explicitly.
can use Prop.
I f f and g
"strongly
are partially
rather
i)
g
poly-
d e f i n e d i n §1.
maps w i l l
formal
a r e t h e same o b j e c t s a s t h e w e a k
5.6.
immediately
from
the definitions
and
formal
proper-
maps.
A map
f
: M
-» N
i s proper
i f ff
i s partially
proper
and
semialgebraic.
This
i s evident
Definition y
4.
: [ 0 , 1 [ -> M
We
An
be
the definitions,
incomplete
from
are interested
completed, can
from
the half
whether
i . e .extended
a t most
one
Remark
5.3.iii
path
i n M
i s a
open
unit
interval
a given t o a path
completion y
•
semialgebraic
incomplete y
and
i n R to
path
• [ 0 , 1 ] -» M.
Y
Theorem
5.4.
map
M.
i n M
Notice
can that
be there
Theorem the
5.7
(Relative
following
f i s partially
b)
I f y
criterion).
F o r a map
f
: M
-» N
proper.
i s an incomplete
i n N,
completion
are equivalent.
a)
ed
path
then
y
path
i n M,
such
c a n be completed
that
6
:=
f«Y c a n b e
complet-
i n M.
M
[0,1[
[0,1]
Proof.
a ) => b ) : T h e c l o s u r e A
(Prop.
3 . 6 ) . Thus
A by r e s t r i c t i o n follows
from
(1.6.8,
[DK ,
b)
f ( A ) C IT (N) i s proper.
of Y([0,1[) a n d t h e map
The p a t h
i s semialgebraic i n M
: A -» f ( A ) o b t a i n e d
from
i n f ( A ) . The c l a i m
path
completion
b)
now
criterion
2.3]).
=> a ) : T h e s e m i a l g e b r a i c r e l a t i v e
that
h
6 runs
the semialgebraic relative 4
i n M
f l Ai s proper
f o r every
path
A eF{M).
completion
This
means
criterion
that
implies
f i s partially
proper.
Corollary weak
5.8
polytope
Proposition map
f
This
i f f every
completion
incomplete
5.9. The p u l l
back
: M -> N b y a n a r b i t r a r y
c a n be p r o v e d
criterion. nitions
We
(Absolute path
path
f' : M map
g
i n M
N
: N'
a s i n I , §5 b y u s i n g
I n c o n t r a s t t o [LSA] a l s o
criterion).
1
can be
-> N
-> N
The
1
M
i s partially
directly
i s a
completed.
of a partially
the relative
a proof
space
proper
proper.
path
completion
from
the
defi-
i s possible.
indicate
this
second
proof.
Let M
1
:= M x
N
1
and l e tg
f
:M '
-* M
denote the By
the
pull
closure of
back
of
g
by
f.
1
in M
by
B
g (A)
P r o p o s i t i o n 3.6 B x^c
and
we
€ ^(M').
have
We
Let and
B€?(M)
have
a
some A 6 ^ ( M ' ) the
closure of
and
ceJ(N').
cartesian
square
be
given.
f'(A) Then
of
D
We
denote
i n N' :=
by
C.
f(B)
semialgebraic
€f(N)
maps
f ' B
with
X C D
f«j g , j , f
,g^j o b t a i n e d
f
hence f Jj I A
f^j i s p r o p e r . i s proper.
Proposition
Since
Since
5.10.
from
For
i s partially
A
f,g,f',g'
i s closed
C€jf(N)
this
a
: M
map
f
by
restriction,
semialgebraic
implies that
-* N
the
a)
f
b)
If P
i s a
weak p o l y t o p e
i n N
then
c)
If Q
i s a
polytope
i n N
with
dim Q
tope
i n
a)
b)
in B
f^ X D
i s
C,
proper, also
f ' l A i s proper,
following
are
q.e.d.
equivalent.
proper. -1
Proof. from
The
is
Theorem
a
5.7
5.11.
algebraic.
A
map
defined
sense
of
in
[LSA]
i n N
f
here
=> c )
i t s Corollary
of
Assume
sense
Indeed,
and
criterion),
polytope
Remark
(P)
a )
follows
absolute the
set
path 6([0,1])
subspace
i n R°° a n d ,
Example
i f
(x) > 0}
C
set
K
a
G- ( x ) > 0, . . . , G
generally,
for only finitely
G (x)
is
(x) >0,
we
semialgebraic
A.
have
the
i . e . an
the
Does
of
every
(even
space
i n the
M
i s a
isomorphism
inclusion
constructed a
space
space
map
M'
have
a
M
onto
embedding a
dense
tp : M
->P
subspace
P.
completion?
completion category
of
dense
f o r any LSA(R)).
paracompact Also,
i n I,
locally §7,
we
have
constructed
semialgebraic struct M
space
N.
Starting
a c o m p l e t i o n M } i s a n e l e m e n t
+
inclusion
language.
description
i fM
the
M.
polytope then
i s semialgebraic
description
Proposition
c)
M
polytope, which
and t h e o n e - p o i n t
an e x p l i c i t
is
=
subspace
space
subspace
i s a weak
a weak
gave
b) A
M
t h e space
I , §7 w e
either
have
i s n o t y e t a weak
i s already
the definition
similar
:
the subset
i s a closed
i s a closed
In
a
then
the one-point completion of
Remark
case,
of the set M
M .
Definition M
equip
union
+
i n
: M
a € I we
I f 3 . F o r e v e r y
the structure
described
set
the s e t which
i f f (Ih \{oo}|X€A) M
A
o f Cov».. M
I f U € f(M) , t h e n 0
(u)
=0
, (u)
. I f UET(M ) +
and » £ U then
a
func-
tion and
f
: U -> R i s a n e l e m e n t
f (°°)
(i.e., M
i s the limit
f o r every
c u and
Proposition and Q
l e tf
t o M.
with
map
K
to
f extends
0 0
to a
f o r every
choose
restrictions
with
g^(x) =
0 0
6.7.
extends
Conversely,
exists
Every
i fg
: N
+
-> M
r
e
Thus
+
such
that
from
a weak
an open
polytope
subspace
g
V of
: Q -* M
+
There
f o r every
R
proper
maps
extend
exists 3 € J
between
t o maps
c f . 1.7.6. These
a
monotonic
( c f . 2.6) . T h e locally
g^
maps
:
com-
-+
M
+ K
^ j
f i ttogether
+
M .
: N
+
(M)
L e t Q be
map
. «
they
(V 0 0 ^ ) ,
f
c
x G U ^ {«>}
s e m i a l g e b r a i c ) map
) c: M
partially
t o a map
K € T
( U M « } )
x GM ^ K } .
( Q ^ I 3 G J ) o f Q.
-*
: Q ->
g
some
M
.
a
f o rx E Q ^ map
proper
(weakly
f (V n Q
that
f o r x -* ,
i s polytopic.
an e x h a u s t i o n
f^ : V D
the desired
spaces
M
x G Q ^ V
semialgebraic spaces.
Corollary
there
€ ®
0 0
f (x)
I f ( x ) - f (») I < e f o r e v e r y
: J -> I s u c h
plete
i n R
6.6. A s s u m e t h a t
Then
We
of the values
e >0
i f ff l u M « }
M
: V -> M b e a p a r t i a l l y
g(x) =
Proof.
of ^ +(U)
proper +
-* M
i s a map
+
map
f
with
f
with
g
: N -» M +
—1
between
polytopic
(») = °°.
(«>) =
{
goal
complete
The
m
of a field,
contain
Theorem
e
real
sequential,
Our
w
that
many
i.e.
(c^lkGlN)
stated
w i l l
i n a slightly
be based
on three easy
more
general form
admits
a partially
than
lemmas, two o f
actually
needed f o r
that.
Lemma 9.3. A s s u m e Let
that
M
A € T (M) . T h e n t h e r e s t r i c t i o n
complete tope
c o r e o f A. T h u s P ( A ) i s a c l o s e d
i ti s e v i d e n t f r o m
f u l f i l l s
Lemma
core
q
: Q -> M.
( A ) -* A o f q i s a subspace
partially
o f t h e weak
poly-
P (M) .
Indeed,
hull
1
q^ : q
complete
the universal
our subspace
theory
property characterizing
( P r o p . 3.2) t h a t a partially
q
complete
o f A.
9.4. A s s u m e
polytope.
that,
f o revery
a € i , t h e space
Then P(M) i s a weak p o l y t o p e .
A
P (M ) i s a
weak
Proof.
I f 3
case
standard
V(n). We
induction
in in
x € V(n){A (x) >0,A (x) + ... i
t
. By
v e r t i c e s of
A (x),...,A (x)
closed
i n V(n)
Let
k
e
the
simplices
( Q l k 6 1N)
n
Q
of
retreat
n-simplex
i s contained f
T h u s we
standard open
o
M.
subcomplex
of
(N)
c
M
t
e
We
we
done
< >)
claim
family
t
L€
)
open
\M.
' 1»-«-/e
Let
M
are
o € I (M) , a
claim
, a
Q
number
a
o
the
the
on
0
the
k
some n € 3N
containing
e
P
f u l f i l l
every
we
sets
: =
k
claim
simplicial
simplex
f u l f i l l s
family
this
t r i a n g u l a t i o n of
every
w i l l
proved
r
of
a
+ A (x)e R
p o s i t i v e elements
point R
= in
x}. R
with
*)
e
k
> e
k
+
for
1
every
k e i ,
, X,
:=
n I
(x€V(n)
k
i s closed
and
X eX k
tained
k
am
/
J
i n M,
following
I
+
The and
holds,
indebted
set P
k
k
k
:=
X
n Q
k
V
a DM
=
Delfs
k
for
/ r TT
e,
i=0
i n V(n), h e n c e
k +
Hans
define,
~
j . L e t
since
to
P
>
3
semialgebraic .
we
A . (x)
'j=r+1 X
and
every \
A
the
1
1
1
J
complete.
be
We
have
semialgebraic
given.
For
every
0 :
for
this
set
(x)
i s complete
K E X* (M)
k
k£]N
clever
definition.
no
=
and x €K
0 conthe
n
r
I
X . (x) = 0 3
j=r+1 By
field,
k € IN
of Lojasiewicz
X
j=r+1 every ,
g
also
£
(x) >
/ r
Proof.
v
3
We
start
with
exists
an admissible
i s proved.
n € IN . N o w by polytopes
P(M)
i sof countable
Proposition
weakly
locally
complete
i s evident
now assume
this
some
finishes
below
K Q. •
Thus
s
the proof
i s of countable
o f T h e o r e m 9.2.
type
then
(M |n€3N)
o f M.
o f P(M), as i s evident seen
i n the proof
Then
from
(P(M )|n€IN) n
t h e proof
o f Theorem
9.2,
of
there
covering
with
countable
index
that
M i s locally
( c f . 6 . 3 ) . Then
from
s e t INxiN.
This
proves
that
type.
s e m i a l g e b r a i c ) . Assume
about
that
c
s >k with
type.
just
same a s t h e l o c a l l y
prove
This
an exhaustion
9.7. A s s u m e
always,
assume
exists
(P ,|kGIN) o f P ( M ) b y p o l y t o p e s , f o r n, K n ( P , I ( n , k ) €3Nx3N) i s a n a d m i s s i b l e c o v e r i n g o f n , jc
P(M)
not
there
covering
9 . 4 . A s we h a v e
We
some
/
9.6. I f R i s s e q u e n t i a l a n d M
Lemma
This
1
i=0
x £ K . Moreover,
an admissible
the
exists
X (x)
P(M) i s o f c o u n t a b l e
every
2, §6]) t h e r e
real
xk
TT
k
and o u r c l a i m
Corollary
is
( g e n e r a l i z e d t o an a r b i t r a r y
with
I
K c P
.
c f . [D, p. 4 3 ] , [BCR, Chap.
n
for
A . (x) = 0
i=0
the inequality
closed
=> TT
also
semialgebraic that
M i s p o l y t o p i c ,i . e .
the function ringed
semialgebraic
space M
1
q
(and, as
c
space
P(M) i s
constructed
i n I ,§7.
1.7.8.
our space M that
the field except
P(M) i s a weak
polytope.
We
R i s s e q u e n t i a l . O n t h e c o n t r a r y , we
i n the t r i v i a l
case
that M
i s a weak
do
w i l l
polytope.
We
want
the
space
have By
to
describe M.
Since
)T(P(M))
c
subspace
ing
proposition
P(M)
in
some
By
the
same
9.8.
An
universal as
a
path
in
for
X € }f(M) , t h e
every
this
set
Proposition
9.9.
such are
in M
and
by
the
morphism above)
^(M), P(M)
are
exhaustions
from
and
and
Corollary
family
and
to
T(P(M))
for
the
P(M) =>
every
same.
M
we
T(M)
K € T
The
of
. (M)
c
follow-
3.16.
(K^IX€A)
in
E2-E5
and
i t is
evident
X* (M)
every
i s
an
K E ^(M)
exhaustion
i s
contained
set
p
P(M).
selection
the
conclude
lemma
([BCR],
closures
8*(P (M) ) X
M
We
unambiguously
X £ y(M)
that
X€JT(P(M)) P(M)
of
of
by
i s the
subspace
X
that
from
[DK # 2
in M
and
a
path
Prop. §12], in
3.6
in M and
[DK ,
the
§2])
4
P (M)
i s
are
the semi-
that,
equal.
We
X.
set
of
a l l X € r(M)
structures
on
X
in
with
the
X € JT (M) .
spaces
M
For
and
equal.
If
implies
and
a
observed
f u l f i l l s
property
curve
Proof.
K
ordered
algebraic
P(M)
on
evident
family
is
h a v e f (P (M) ) =
we
i s now
M
of
already
subsets
.
thing
every
identity
structures
i f f the
denote
semialgebraic
(as
9.1.i
the
Proposition
the
JT(M)
Proposition
of
the
=
and
X € JT (M)
then
X € T ( P (M) )
x e n P ( M ) ) . Conversely, ^ (M). c
both
The
coincide
subspace with
the
and
i f X £ JT(P(M))
structures subspace
on
X
X € f (P (M)) then,
with
structure
in
by
.
Prop.
respect the
This
to
3.6, M
polytope
X.
q.e.d.
We
now
state
Theorem polytope.
a
9.10. Then
converse
Assume R
i s
to
that
Theorem
M
i s not
sequential.
9.2.
a
weak
polytope
but
P(M)
i s
a
weak
Proof.
Since
M
i s n o t a weak
T
:
[ 0 , 1 [ -» M
c
€
]0,1[ such
[DK ,
p.
since
y
4
]0,1].
which that
305f]).
Lemma
be
yl[c,1[
The
i s proper By
cannot
polytope
completed
:=
y([c,1[)
(cf.II.9.9). P(A)
exists
an
incomplete
(Cor. 5.8). There
i s an embedding
set A
9.3,
there
path
exists
some
( c f . the argument i n
i s closed
The
subspace
i s a weak
polytope.
A
semialgebraic
of M
i n M,
i s isomorphic
Thus
P(]0,1])
i s a
Every
i s a
to weak
polytope.
Let
(K^laEI)
complete (K
[e ,1J,
a E I with
with
any
a E I
3
P (M)) f
Let
(*)
be is P
f
a commuting partially (M) s u c h
ii)
Assume
the
square
proper.
Then
3°P
i n addition
that
p °y f
semialgebraic
there exists
=
that
maps
a unique
P
g
M
map
y
from
that
a
1
P
(M ) t o
•
g
the square
(*)
i s cartesian.
Then
also
Proof, Thus
(M)
f
1
cartesian.
P
. (N'
x
In short,
M)
i ) T h e map
there exists
i fN
= N' x
f 3°p
P
-
g
1
i s partially
proper
We
squares
look
- a*g
a unique
map
at the following
(solid
arrows).
over
N,
then
(M)
from y
P^(M')
from
P
to N
diagram
i s partially
(M') t o P ( M ) f
g ii)
and assume
square
P (M*)
is
of weakly
consisting
.L
o f two
with
proper.
p «y r f
cartesian
= 3 °
—
M' x P M
P
(M)
f
f
(M)
N'
We
have
is
bijective
proper
to verify
that
since p^ i s b i j e c t i v e .
since a
i s assumed
aog T
then
Grothendieck's
M
M
by
§8
proper
there
T
by
natural
i n the
f
i s a
T
and
exists
terminology
a
space
Def.
2) .
sense
of
i s an
isomor-
weakly of
quotient
(resp.
(§8,
T
the
bijection
quotient
proper)
M
p
that
T h u s M/T
of
by
theoretic
from
the
of
function ringed
of
of
classes
denote
T
the
quotient
spaces
that
T
of
and
strong
gives
f.
set
T,
equip
quotient
ringed
In
strong
i f T=E(f)
i s the
T
f
relation
then
f ° p
g.
i s a
denote
strong
L with =
a
quotient"
8.3).
partially
-> N
by
proper).
t o M/T.
in this
ii)
"strong
-> N
T
(Th.
above
with
a
a
( c f . §8).
R which
conclude
h»
: M
(resp.
function
that
of
equivalence
definition
we
spaces
equivalence
proper).
space M
f
i) Let
over
of
such
a
map
from M
phism
then
A
exists
: M/T
relation
between
the
respectively). It i s
following.
proper
projection
the
of
the
11.3.
by
-> N
then
existence
identifying
Remarks
: M
equivalence
(semialgebraic,
quotient
partially
If
f
i s identifying
proper)
set
map
proper
11.1.
Definition and
proper
proper),
quotient"
we
(semialgebraic,
M of
semialge-
by
T.
If
M
by
T
proper). g
: M
-» L
u n i q u e map [Gr,
§2]
the
i s h
a :N
map
-* f
L
is
a strict
epimorphism
i n the category
WSA(R),
i n fact
even i n
Space(R). iii)
I f there
know
from
Brumfiel
exists
11.1
to
proper
later
then
then
Proof.
We
p
such
such
relation
that,
p
proper.
i s
on M
(affine)
then
extends
indeed
readily 1
also call
"Brumfiel s
equivalence
M/T
relation
on
( c f .11.3) i s w e a k l y
proper.
I f T happens
a
a,
M a
/
*M )
Q
and
(M /T la
of p
T
a
Q
a
~
space
5
M
of a
~*
a
i s proper.
M
a semi-
t o be
a
/
i
of a
i
n
i t s given
s
a
(weakly
proper
we
structure
proper.
structure
-
1.6
i s a
o f M/T. that
space
We
space closed
Since the
p,p i s a
semialgebraic)
o n M/T
Brumfiel's
semialgebraic)
conclude
(weakly
By
By Theorem
€ I) i s an exhaustion maps
a G I the
semialgebraic
T
o f t h e s e t M/T.
are proper
and p a r t i a l l y
on M
Q
the structure
T
For every
Q
the structure
the present
map.
coincides with
strongWe
now
t h e one
above.
Assume t h a t p^ again
theorem
( K I a € I ) o f M.
as subsets
a
f o r every
o f M/T
that
proper space
:= T fl ( M
Q
carries
a
o n M/T
surjective
defined
is
T
T
a
restrictions
know
shall
t h e n a t u r a l p r o j e c t i o n s p^
exists
relation
This
i s partially
an exhaustion
t h e s e t s M- /T
subspace
ly
: M -> M/T
T
a
regard there
i s a partially
t h e s e t ^- /
that
we
i fM
2
we
i s proper.
T
choose
equivalence
[B ] that
equivalence
the function ringed
and p
proper
theorem
which
by T then
on.
space
algebraic,
quotient of M
be c l o s e d and p a r t i a l l y
by T e x i s t s .
statement
11.4. I f T
result
i s a proper
quotient of M
Theorem M
the important
and T
the following
theorem"
proper
a n d 11.2 t h a t T m u s t
has proved
semialgebraic the
a partially
: T -» M
proper.
i s proper.
I t suffices
We
t o prove
want
t o prove
that p
T
that
p
T
: M ->
i s semialgebraic
M/T ( c f . 5.6)
X € ?r(M/T)
Let
Yejr(M)
some a
with
semialgebraic
Remark case
not
Brumfiel's
turn
Remark
Let M
closed
the
a c t i o n s . We
braic)
i s a best
i s a
case
earlier
o f Theorem
volume
way.
such
a closed
that
n T
use of abstract
w i l l
Theorem
11.4
[SFC].
space,
the strong
semialgebraic
(M x K)
to
i s
without
11.4
possible result.
semialgebraic
special
f there
Indeed,
t h e f o l l o w i n g "converse"
on M
space,
over
SA(R)).
toi t
and l e tT quotient
subset
i s a proper
K
map
semialgebraic
start
with
11.4 g i v e s obvious
semialgebraic
R i s a group
us
of M
such onto
be
that M.
spaces ( c f .
i n the important
(locally
object
G i s a space ( l o c a l l y
that
semialgebraic
space
r e s p e c t i v e l y ) over
i
the multiplication
: G -* G , x
x
, both
semialgebraic,
G i n the category
means
that
case
of
definitions.
This
such
map
Theorem
4. A w e a k l y
group
LSA(R),
the
essential
strength
indeed
q.e.d.
i n an e s s e n t i a l
complete
: T -> M
e x p l i c a t e what
Definition
group
makes
special
exists
i s
t h e g l u i n g map
i n the third
relation
o f p^
this
volume
theorem
contains
This
on g l u i n g o f s p a c e s
the full
important
Then M
2
8.6
t o do
since
locally
1
(X) = P ( p ^ Y ) .
i s semialgebraic.
r e c e n t l y has proved be a
1
there
App. A ] ) .
now
group
p~
11.4, p r o v i d e d
i n the present
restriction
Scheiderer
We
result
i s strongly surjective
T
Theorem
decided
equivalence
T exists.
[LSA,
We
11.6. 3 r u m f i e l ' s
[Send]:
p^
theorem
o u t t o be
Scheiderer
by
set,since
p
Then
T
proper.
be needed
will
C.
p ( Y ) = X.
of the present
using
Since
11.5. The p r e v i o u s
partially
a
be g i v e n .
map
m
are weakly
WSA(R)
(resp.
semialgebraic
R and a l s o
: G x G -> G ,
semialge-
an
(x,y)
semialgebraic
abstract xy, and
maps.
Examples C
11.7.
R(V-T))
=
ii)
The
use
and
the
though
i i i )
then
G(R)
by
letter R(\/-1)
embed
semialgebraic
Assume
that
R
is
partially
then
group
We
shall
in
VII,
§4
operation the
R
scheme
R
as
well R
as
(cf.
groups
R
are
R
(or
group
the
[Ch,
over
u n i t a r y and
over
over
semialgebraic
over
=
U n
space
scheme
unitary
{We
i n
the
R. groups
I]
are
symplectic
used
over
Chap.
R.
over
with just
]R to
groups a l -
definition
of
other
a
space
M
such then
G-space
us
T(G)
an
a
we are
the
A
has
the
°y.
•—>
Then
have
weakly
partially
semialgebraic
i s a weakly
semialgebraic
same u n d e r l y i n g In
(or over
C)
then
i
semialgebraic
group
abstract
weakly
group
9(C)
(resp.
oc
group
i f Of i s a n y
particular,
genuinely
groups
complete.
R.
of
(0(n ,R)In€lN)
exhaustion
over
weakly
that call
and
M
the
equivalence
:=
by
over
G
alge-
1 q c
)
is
R.
semialgebraic
groups
§9.
be
set
with
examples
G
on
i s a
R
Let G
We
groups
locally
VII,
5.
which
Similarly
group
usual
0(n,R)
group
P(G)
over
complete
and
as
i s sequential. If G
of
algebraic.
gives
groups.
complete
meet
Definition
G
the
partially
M
of
0(n+1,R)
Sp(«>,R). A l l t h e s e
over
If
into
weakly
U(°°,R),
on
Sp(n,R)
quaternions
l i m 0(n,R) n->°°
semialgebraic
G
the
over
notation
:=
by
a
0(n,R)
i s a
semialgebraic
0(«,R)
a
braic
are
0(n,R)
is
a
G(C))
groups
i n our
and
a l g e b r a i c group
groups.} We
iv)
groups
R)
R
i s an (resp.
symplectic
replaced
these
If G
orthogonal
U(n,R) there
i)
semialgebraic
group
M
i s a
left
operation
the
map
G * M
-> M,
a
(left)
group
G
€
over
G)
the
»-> g x
R).
A
left
abstract
group
i s weakly
semi-
R.
i s semialgebraic
relation
{ ( g x , x ) l x €M,g
G-space
(g,x)
of
(over
then
the
action
of
on
M.
Indeed,
(g,x)
T (G)
(gx,x),
i s the
and
hence
equivalence r e l a t i o n M/T(G)
-
i s
i f i t exists
underlying
set
Assume
now
that
i s the
f i e l 's
theorem
G
image
the
a weakly
- w i l l
be
set
orbits
of
us
the
semialgebraic
semialgebraic
semialgebraic
i s complete.
gives
of
( c f . Def.
denoted
Then
more G
p
: T(G)
following
on
GxM
subset
2).
The
briefly
of
2
map
of
-> M * M ,
M * M.
strong
by
GXM.
This
quotient Its
M.
-» M
somewhat
i s proper.
special
Thus
result
on
Brumorbit
spaces.
Corollary proper
Up
to
the
11.8.
I f G
quotient
now
we
did
quotient
algebraic) gebraic.
GXM
of
Here
complete
exists
not a
space
i s a
care
i s a
an
every
for the
locally by
for
semialgebraic G-space
question
semialgebraic
equivalence
rather
special
(and
then
the
which
conditions
M.
under
(and,
relation
group
as
always,
i s again
trivial)
weakly
locally
statement
semi-
semial-
in
this
direction.
Remark
11.9.
In
semialgebraic,
Proof. braic the open
Let
projection
Remark [Send]: a
then
Then
closed
be
GXM
an
of
i s locally
admissible
(G0* la€I) a
to
GYM.
i n GXM
Corollary
covering
have
for every
such 1
p
a,
i f M
i s
M
(P
and
u a
of
a )
M
by
open
covering. =
G
U a
(pU
•
T
n
l a £ I)
u
s
semialge-
Let P
p
u
i s an
denote i
s
a
admissible
q.e.d.
Recently be
locally
semialgebraic.
i s again We
11.8,
GXM.
11.10. Let
also
from M
semialgebraic of
situation
(Uglot 6 I )
subsets.
covering
T
the
a
C.
Scheiderer
locally
semialgebraic
has
semialgebraic
equivalence
proved
a
much
partially
relation
on
M
better
complete with
the
theorem
space
and
following
property. (*)
I f U € £(M) t h e n
Then
the strong
{Scheiderer statement proves
space
above
that
algebraic M
quotient
only
(*)
group
1
P P^ U 2
quotient deals follows just
i s open
M/T
with
exists
t r i v i a l l y
(from
and i f T(G) i s c l o s e d G\M
exists
that
from
t h e map
the left) then
and i s l o c a l l y
(hence
1
p p~ UG 2
and i s again
the case
means t h a t
acting
i n M
(*)
M
this p^
locally
) . semialgebraic.
i s semialgebraic, case.
Scheiderer
i s open.}
on a l o c a l l y i s clearly
semialgebraic.
I f G
but the also
i s a
semi-
semialgebraic
f u l f i l l e d ,
hence t h e
In
this
fixed
Chapter
V
- Patch
complexes,
chapter
a
"space"
means
real
closed
semialgebraic notions spaces
field
map,
R and a
i fnothing
and n o t a t i o n s used a n d maps
explanation.
and w i l l
a weakly "map"
else
I t so b j e c t s a r e the spaces
homotopy
classes
§1
- Patch
In
the following
Definition with
1. A
M
i s a
of {The
For
"PD"
fixed
means
that
I
refer
sense
§2, w i t h o u t
over
f o r these further
R i s denoted
R and i t s morphisms
-» N b e t w e e n
of M
by
are the
spaces.
i s a subset
elements
a fix
o f I then
i n the union
=
I o f tf'(M)
0.
of finitely
The b o u n d a r y
ii)
An element write
t o "patch
decomposition",
i s a n e x h a u s t i o n o f M,
i s a semialgebraic partition
many
elements
of a
o f M.
x of I x < a.
semialgebraic partition
i s t h e s e t 9a i s an immediate
:=
see below.}
then
i s an a d m i s s i b l e c o v e r i n g o f M
2. L e t I b e a
i)
then
from
their
theoretic
space.
i s contained
i f (M^laGI)
{M°la€l°}
Definition
We
weakly
X.
example,
just
a
properties.
X 6IT(M)
letters
patches
: M
semialgebraic partition
the following
Every
I I Ir e t a i n
over
means
a
decompositions
PD1 . I f a a n d T a r e d i f f e r e n t PD2.
f
spaces
over
A l l t h e homotopy
o f spaces
HWSA(R).
[ f ] o f maps
between
starting
category
again
s e m i a l g e b r a i c space
i s said.
i n Chapter
be u s e d ,
The homotopy
and homotopies
the set of Notice that
( I V , §3, D e f . 7 ) .
o f M,
and l e to E I .
a ^ a . face
of a
PD2
i f x n 9 a * 0.
Every
a G I
has only
braic
s e t 3a
Definition braic PD3.
X
3.
R
I
of M
a G I
< T _ < | < ... * < T R
patch
complex
and a patch
letter
X
Example
1.1. L e t M
wise
does
of
M.
sion
i s a pair
The
We
i s a
such
Q
m,
n
->
n-1
de-
l e t
n
V
t o V . We p u t r := i d . . . We n m n, n V : V x i -> v as f o l l o w s . L e t x e v . I f t 6 [0,s ] then n n n n
define
a
n
map
F
c
If
n
t € [ s ^ ^
8
^ . ^
F (x,t) n
F
i s a
n
strong
(x,t) €V xi n
not
move
map
F
an
2.5
a part
We
now
exactly
have F
1
n
'
j
k
n
( x ) , ( s _ - s ) - ( t - s ) ) k
This
of V
the F
since
t o U.
V
For
every
t h e homotopy
f i t together
semialgebraic
by c l o s e d from
n
n
subspaces
to a
since
H
set theoretic (V
ln€IN)
(IV.3.15.c).
F
t o A.
(U = A)
does
n + 1
i s
i s a q.e.d.
the following generalization
III.1.1.
i s a closed
semialgebraic
t h e same
f r o m V"
n
i n particular
I f (M,A)
can state
( x , t ) :=x
.
k
i s weakly
retraction
deformation
k
= F (x,t)
. Thus
map
covering
1
retraction
^(x,t)
+
n
then
1
n
i n V
o f Theorem
2.6.
k
1
k
covers
open w e a k l y strong
1
H ( r
deformation
Corollary
a
d
deformation
-> V.
admissible
Theorem
an
:=
n
the points
: Vxi
strong
of
we
a
F
retract
pair
of spaces
neighbourhood of
V
of A
then i n M
there such
exists
that
A i s
V.
a g e n e r a l i z a t i o n o f P r o p o s i t i o n I I I . 1 . 2 . The p r o o f as f o r l o c a l l y
semialgebraic
spaces.
i s
Proposition as
g
: V
exists every
Also
Z.
2 . 6 . A n y map
Theorem
: V x i -» z w i t h
G
a new
III.1.3
extends
Proof.
2 . 8 . I f (M,A)
We
choose
pair
]0,1].
(I,{0})
We
form
together
We
N^B.
retract
a relative
Z extends
= h and G
T
to a
there
IA = f f o r
:=
We
3N(n) N
Theorem
semialgebraic spaces,
but
this
o f spaces
then
(MxO)
U (Axi) i s
Mxi.
patch
decomposition
patch
product
complex
of these
of
with
(M,A). a unique
two r e l a t i v e
We
regard
patch
patch
complexes
pair
decomposition
{ a x ] 0 / 1 ] I a£I(M,A)}
F o r any two patches
(N,B).
of
i s the closed
the patch
use the notations
every
= g , G^
q
t o weakly
as a r e l a t i v e
with the patch
T < a. Thus
and
space
:= (Mxl,(MxO) U (Axi))
Z(N,B)
iff
G
i s a closed pair
the direct
1.d). This
(N,B)
of
some
be
proof.
strong deformation
(Def.
: A -* Z i n t o
L e t M,A,V
I f g and h a r e two e x t e n s i o n s o f f t o V t h e n
a homotopy
Theorem
the
f
t o a neighbourhood).
t €I.
needs
By
( E x t e n s i o n o f maps
i n Corollary
map
a
2.7
n
n > O,
of
ax]o,1] o f
(2.2) f o r both
have N(n) =
=
(M(n)xO) U
=
(MxO)
III.1.3
a,T
U
(M,A)
we
have
Tx]o,1] z
1.4
subspace
homotopy with
= g a n d GlAxl = F.
seems
how
more
d i f f i c u l t
to weakly
in
§4.
We
conclude
sequences we
as
that
semialgebraic
space
: Axi
n
Theorem
t o B.
Using
G
these
o f Lemma 2.4, s t r o n g d e f o r m a t i o n r e t r a c t i o n s ' =»
r
from
From
do
run
semialgebraic
this
of
Proposition
section
as
spaces.
by w r i t i n g
t h e homotopy
not give
exactly
to generalize We
w i l l
down
albeit
i n the topological
( c f . [DKP,
half
o f III,§1
say something
several
e x t e n s i o n theorem
the proofs,
2.10
the second
2.9.
some o f t h e m
somewhat With
some-
about
this
formal
con-
two e x c e p t i o n s
are tricky,
since
they
setting.
2.9]). Let
A
M
be
1
a triangle
that
i i s a
• M'
o f maps w h i c h
closed
commutes
embedding,
i . e . an
up
t o homotopy
isomorphism
(fi - h ) .
of A
onto
a
Assume closed
subspace
o f M.
Then
there
e x i s t s a map
g*i
= h.
The
proof i s almost
f * i
t o h . B y C o r . 2.9 t h e r e
F(-,0)
t r i v i a l .
= f a n d F^ixid-j.)
We
g
choose
: M -* M'
a homotopy
e x i s t s a homotopy
= H.
such
T h e map
g
F
H
that
: A x i -» M'
: M*I
:= F ( - , 1 )
g - f and
M
1
from
such
that
has t h e required
properties.
Definition space from We
4
under
(also
C i s a map
a t o a space
also
f o rlater
call
3
a
u s e ) . a) L e t C be a s p a c e
: C -* M
into
: C -» N u n d e r
f a map
from M
a space
C i s a map
t o N under
C,
M over f
R.
R. A m a p
: M
i fthere
over
N
such
A f
that
i s no doubt
3
: a
f«a = 3
which Q
"structural
The b) H
We
maps"
category
o f spaces
A homotopy : Mxi
-> N
then
also
a,3 a r e under
H from
a n d maps u n d e r
a map
i n the usual
say that
consideration,
C i s denoted
f : a -» 3 t o a m a p
sense
with
a n d we w r i t e
H i s a homotopy
under
b y WSA(R)
g : a -• 3 i s a
H(a(z),t)
f p
—>N.
: M
.
homotopy
= 3(z) f o ra l l z € C , t £ I . p
C a n d we w r i t e
H
—>N
: Mxi P
If
there
e x i s t s a homotopy
H under
C from
f t o g t h e n we w r i t e C
f *
g. C
c ) T h e h o m o t o p y c l a s s u n d e r C o f a m a p f : M —>N i s denoted by [ f ] . The c a t e g o r y whose o b j e c t s a r e t h e s p a c e s under C and whose morphisms p a r e t h e h o m o t o p y c l a s s e s u n d e r C i s d e n o t e d b y HWSA(R) . A map f u n d e r P
C in
i s c a l l e d a homotopy
under
C i f [f]
i s an
isomorphism
HWSA(R) .
Theorem is
equivalence
C
2.11
a map
closed
under
[Do , 3 . 6 ] , [DKP, 2 . 1 8 ] ) . Assume
C and that
embeddings. Assume
(forgetting
Proposition about
(Dold
C ) . Then
t h e s t r u c t u r a l maps further
that
f i s a homotopy
2.10 a n d Theorem
2.11
a
: C-»M,
f i s a homotopy
equivalence
imply
that
under
the following
r e t r a c t i o n s , c f . [Do , 3.5, 3 . 7 ] , [DKP, 2 . 2 7 ] .
f
: M
-^->N
3
: C-»N
equivalence C.
nice
results
are
Corollary denote i) a
2 . 1 2 . L e t (M,A)
the inclusion
Assume
ii)
p
of
of
of
: M -» A w h i c h
closed is
Q
i n
M.
a family
A map
of f
1
Then
there
exists
deformationr e -
:
closed
system that
family
M.
of
Q
We
subspace
i fevery
-* ISL s u c h
(M M Q
map
i s
(N,N-j,...,N) Q
every
R
f^ i s a
F
, . . . ,M)
1
-> ( N ,N M
-» N . o
o
f ^ :
0
M,j ,M
-
and hence
o f maps f ^ :
; M
A
Q
( f , f ^ , . . . , f
From
cz M^,
2.13. L e t (f , f , . . ., f )
component
there
A
i s a strong
every
Q
r
map b e t w e e n c l o s e d
r =
and l e t i
t o r.
A
(M ,...,M )
from
o
For
that
decreasing i f
(f ,...,f )
restriction
Theorem
i s nomotopic
such
R
f o r 1 < i < r - 1 . We
I
: M -* A w i t h r » i - i d
equivalence then
(M ,...,M )
the system
M,
r
5. A s i n [ L S A ] we m e a n b y a s y s t e m
spaces
call
o f spaces
M.
Definition
a
a map
I f i i s a homotopy
tract
pair
mapping.
there exists
retraction
be a c l o s e d
2.14
[BD, 5.13].
ke closed M = M
1
UM,
restrictions equivalences.
2
subspaces N = N
Then
of M
UN,
1
-*
Let f
, M
2
and
f (M)
2
1
-> N , 2
: M -> N b e a m a p b e t w e e n ,N
closed
2
c N
1
nM
2
, f(M )
M/A
t h e map b e t w e e n
that
pxid
: Mxi
i s a homotopy
pairs p
(M/A)xi
g e n e r a l i z a t i o n and easy only
much
retract
later
( V I I , §8) a n d t h u s
t h ei n c l u s i o n
Then t h e system
subspace
map f r o m
of M. Q
Q
The case
Assume t h a t ,
C 0 M^. t o My i s a h o m o t o p y
(C ,C D M«j , . . . ,C n M )
i sa s t r o n g
r
r = 0 i s t h eC o r o l l a r y 2.12 a b o v e . exist
exists
the identity J
of M
:= E ( x , 2 t )
strong
deformation
M^
tion
equi-
deformation
We s t u d y
o
a homotopy
E :M
and extends
i f0 < t < ^ ,
xi-»M
and gives
t o C D M^. on r .
extension
relative
C which
theorem starts
D. . N o w t h e m a p G :M xi->M , d e f i n e d 1 o o r
and G(x,t)
deformation
the case
r e t r a c t i o n s D^ : M^xi -»M^
:= D
retraction
^
—
M^ x I t o M^ of
f o r every
r
2.9) t h e r e
G(x,t)
—
system o f
o f (M ,...,M ).
1 - < t < 1, i s a s t r o n g z
of Corollary
may now b e
decreasing
f r o m My t o C n M y f o r k = 0 a n d k = 1. B y t h e h o m o t o p y
by
-*
i si d e n t i f y i n g , c f .
consequence
r
= 1. B y 2 . 1 2 t h e r e
with
: (M,A)
reader.
and l e t C be a closed
valence.
(Cor.
complete
equivalence.
2 . 1 6 . L e t (M ,...,M ) b e a c l o s e d
k€{0,...,r),
Proof.
A i s a contract-
IV.8.7.ii.
2.12 w i l l
r
o f t h espace M and t h a t
the natural projection p
Thus,
t h ep r o o f
Assume t h a t
from
(E(x,1),2t-1) i f M
t o C. I t m a p s
O
by r e s t r i c t i o n
F o rr > 2 t h e p r o o f
a strong
runs
deformation
retraction
b y t h e same a r g u m e n t
and inducq.e.d.
§3
- Partially
We
w i l l
space
prove
M which
f i n i t e
open
the existence have
special
properties
o f open
properties
are important
turn ing
but w i l l
(U^IX€A)
If
finite We
i n J'(M)
to live
with
algebraic
sets,
We
choose
a patch
patch
complex
the
open
covering
the fact
o f t h e space
n p i s n o t empty
If
M
In
general
But p has only
i s a
starting of
M
trast
given
simplicial we
from
cannot
t o the case sets
of stars
patch
€Cov (M). M
cannot
be
semi-
semialgebraic.
Abusively
we
denote
(St (o)la€I) i s a partially
f i n i t e
M
o f M.
Then
f o r a n y a [ 0 , 1 ] s u c h
t o this
1. semi-
covering
i sa p a r t i t i o n
o f unity
-1 (cp^|X€A) cp^
1
such
( ] 0 , 1 ])
a A
been
the closure
Theorem by
c
x
N . B . We h a v e since
that,
3.5. G i v e n
open weakly
D
x
with
-1 o f cp^ ( ] 0 , 1 ] )
a partially
B^
Then
\p :=
\p
x
I XGA
finite
sets
i n T (M)
down
the last
there
exists
(D^IXeA)
B^ c
condition
semialgebraic.
o f a space
a partition
M
o f unity
t o (D^IXeA).
exist
families
respectively with
iK i s a w e l l defined A
covering
X € A . We c h o o s e
: M -* [ 0 , » [ w i t h
(resp.
i s n o tn e c e s s a r i l y w e a k l y
subordinate
c= V ^ c A ^ c D^ f o r e v e r y
functions
i n writing
proposition there
(B^|X€A)
c D^
s e t A ^ € 5*(M) ) .
cautious
semialgebraic
By t h e l a s t
(A^IXGA),
and
some
a l i t t l e
(tp^lXEA) w h i c h i s s t r i c t l y
Proof.
X e A , cp^ ( ] 0 , 1 ] )
f o revery
U(B^|AeA)
weakly
(] 0,»[) c V^ weakly
(V^|X€A)
i n T(M)
= M and
semialgebraic (cf.
semialaebraic
IV.3.12). function on M
which have
In
i s positive
everywhere.
the properties
Corollary
required
3 . 3 we
have
semialgebraic
spaces.
a
M
given
space
position
o f M.
relation
may
a
good
answer
Proposition ly
may
In general loose
seems
be a n o r m a l
i f f , f o r every
Of course,
i tsuffices
1 . i i above.
M
view
i s normal
patch
patch
a o f M,
t o check
point.
complex.
this
locally
a patch
since
(cf.
on
criterion
from
t o be d i f f i c u l t
A^
q.e.d.
a result
starting
a geometric
complex
and t h e sets
"combinatorial"
semialgebraic,
from
:= ip^/ip
as a by-product,
ask f o r a
this
i fthe patch
semialgebraic
finite.
One
3.6. Let M
i n Definition
obtained,
i s locally
be v e r y
T h e f u n c t i o n s tp^
We
that decom-
the
face
can
provide
1.3).
The
space M
t h e complex
i s local-
St (a)
i s
M
f o r the patches
of
height
zero.
Proof. a
I f a l l the stars
covering
finite
(Prop.
Assume There
of M
now
i.e.
with
This
implies
patches
M
p.
i s locally
I f a *p
3p (1U *0
since
U
sets.
admissible.
then
contains
o 0 } . I f E a n d F a r e d i f f e r e n t h,
with
and
i spartially
a
(u la€I)
By Theorem
£
since
V
E
f]V
HVp. Choosing x€V" , E
and
finite
p
subsets
= 0.
of
Indeed,
indices k€ F ^ E (x) < u ^ ( x )
since
a contradiction.
-1 i n n (u ( ] 0,1 ] ) I a € E ) . I n p a r t i c u l a r , cx
s e t V„ i s c o n t a i n e d h>
any a E E . Thus
(V^IEEA) i s c e r t a i n l y hi
a partially
finite
V c U hi ot D
family
*
in
. This
V
is
n
(open
:
=
U
(Q , . . - / Q ) *
unique
r
part
from
Theorem
4.1
c a n be proved
t o s a y more
WP-approximations
We
will
at
t h e e n d o f §6. A l r e a d y i n t h e n e x t These
then
u p t o homo-
f i s a homotopy e q u i v a l e n c e .
2.9 a n d T h e o r e m 2 . 1 3 .
WP-approximations".
i s contractible
r
WP-approximation
e x t e n s i o n theorem
about
has a
space.
follows
t i m e s . The second
o f spaces
r
( P , . . . , P ) -> ( M , . . . ,M ) . I f M
c a n be chosen
there
The
tp :
d e c r e a s i n g system
are defined
section as
easily
by use o f
of decreasing
we w i l l
follows.
a n d C o r . 4.6,
need
systems
"relative
Definition
6.
approximation (P,j(A))
L e t (M,A) of
(M,A)
a relative
be a c l o s e d
pair
consists of a
weak
polytope
o f spaces.
closed
A
embedding
and a commuting
relative j
: A
WPP
with
triangle
A
with
i the inclusion
from
that
then,
theorem
by Dold's
Identifying
A with
approximation
of
between
such
identity
Theorem
pairs on
be
a commuting
of
M
pair
induced
:
(M,A)
a n d tp a h o m o t o p y
that
equivalence.
2 . 1 1 , tp i s a h o m o t o p y
alternatively
as a homotopy
may
regard
equivalence
(P,A) i s a r e l a t i v e
equivalence
weak
tp :
a relative ( P , A ) -»
polytope
Notice under
A.
WP(M,A)
a n d tp i s t h e
Let
square
with
respectively.
(M,A),
tp
j ( A ) we
to M
A.
4.11.
and A
A
c f . 4.6.}
(P U
A,A)
k a closed
{N.B.
Then
Such
square
t h e map
-> (M U A , A ) A
by t h e commuting
a
embedding
=
(M,A)
diagram
Q id
7
a n d x,ty
exists
WP-approximations
f o r every
closed
is
a relative
Proof.
WP-approximation
I t i s obvious
that
of
(P U
(M,A).
A,A)
i s a relative
weak
polytope,
and
A.
it
follows
from
Theorem
4.8
that
t p : P U A - + M i s a
homotopy
equi-
X
valence.
Example pair
4.12.
(M,A)
P
(cf. the is
Assume
that
the field
of spaces
over
R the
(M,A)
I V , §10)
4.13.
any weakly
Pf
: P f (M)
and
know
from
conclude
(
and
since
w i l l
Assume
p A
For every
closed
map
M
< >'
A )
( M
'
A )
WP-approximation
Theorem
4.7.
This
i t i s canonical
be
again
semialgebraic
-* M
Let L
:
construction
which
be
Proof.
i d )
i s sequential.
of
(M,A),
relative
and
(P
as
i s clear
by
WP-approximation
(M),A)
has
the
same
(M,A).
the other
Theorem
We
nice
as
equivalence
?M'
theorem
especially
Also
(
i s a relative
preceding
dimension
=
R
:=
of
P
fn*
> P^(M)
useful
later.
that
the f i e l d
map
over
f i s a homotopy
f
(M) . B y
I V . 1 0 . 2 0 we
4.7
that
i s a homotopy
f
R.
that
p^
R
The
gives
us
i s sequential. partially
a
homotopy
L e t f :M
proper
-» N
core
equivalence.
Theorem p
i n I V , §10
and
have p
L
P(L) =
P(M)
a r e homotopy
equivalence.
and
p "P =PMf
L
equivalences, q.e.d.
If
M
i s
a
subset
space,
of
Ti (M,x)
M
also
the
been
done
in
properties
(M
point
la€l)
i s an
then
=
n
for
fact
More can
this,
Lemma of
a
such
and
also
used
in
5.1.
groups
remain
in
stated force
in
homotopy
n (M,A,x) n
the
the
on
groups
(n>2)
same way
spaces.
there
in
semialgebraic
absolute
semialgebraic
sets
of
TT^ (M„
and
as
The
has
more
pp.
265-270
present
setting,
formal
to-
M
such
that
every
M^
contains
the
,x)
>
n
a
a
that
(III.6.3,
the
main
that
M.
for
the
weakly
later
proof
(C
also
the
6.4)
remain
true
for
homotopy
sets
III.4.2
n
f i r s t
and
second
main
weakly
theo-
semi-
0 < s.
i
< s
n
0
2.
+
1
we
make
spaces.
explicit
homotopies.
an
(This
In
and
order
easy lemma
III.5.1
to
prove
lemma on has
the
already
III.4.2.)
be
that
G
=
on
semialgebraic
use,
of
|n€3N)
Assume
G (-,1)
=
to
theorems
of i n f i n i t e l y many
Let
space
o
the
weakly
l i m n ( M , A 0M ,x) .
generalized
s
a
spaces.
"composition" been
i s
TT^ ( M , A , x )
and
exhaustion
groups
generally, be
homotopy
locally
i t i s evident
homotopy
algebraic
define
and
Q
A
proofs.
=
n
rem
n (M,x)
groups
lim
TT (M,A,x)
this
we
and
clearly
n (M,x)
From
M,
relative
III.6.3)
their
of
then
for
these
with
x
the
I I I , §6
of
point x,
sets
Theorem
gether
If
and
pointed
(before
i s a
containing
( n > 1)
n
x
an (G
( ~ ' 0 )
< .. .
admissible : Mxi and
G
f i l t r a t i o n
-* NI n E I N )
R
i s a
i s constant
on
( c f . §2,
family C n
of
Let
Def.
3)
homotopies
be
an i n f i n i t e
a homotopy
F
e c
x
(x,t)
= G
t s
n
F(x,t)
if
F
=
k
k
+
1
, s
k
+
n
Using
since
G
n
r
s
k
1
r
( t - s
formulas
every
on C
„ . Thus n-1
defined
homotopy
t h e F_ f i t t o g e t h e r n ^
F t h e composite
t h e sequence
A^ c l o s e d
C of M with
about
relative
i n M.
of thefamily
t o the desired
n
Q
We
be two systems
f i x a map h
o f spaces
over
R
: C -» N o n a c l o s e d s u b -
h ( C fiA^) c i B ^ f o r 1 < i < r . We u s e t h e n o t a t i o n s
homotopy
R. T h e n
sets
established
thenatural
r
i n I I I , §4 a n d I I I , § 5 .
L e t S be a r e a l
closed
field
map
: [(M,A ,...,A ),(N,B ,...,B )] 1
o f homotopies
(s lnG3N )
5,2. i ) ( F i r s t m a i n t h e o r e m ) .
containing
is
a well
G- (-,0) I C t o G (-,0) I C . We h a v e F I C xi = F , \ n n n n n-i n-1
r
Theorem
) )
k
we o b t a i n
(M,A^,...,A ) a n d ( N , , . . . , )
space
K
that
,1] .
1. We c a l l
n
1
h
h
[(M,A^,...,A ),(N,B ,...,B )] (S)
r
r
1
r
bijective.
ii)
X
1
these
(G ln€lN) , along
with
exists
F.
Definition
Let
+
there
], 0 < k < n - 2 , and
1
„ i sconstant n-1
homotopy
k
Then
n
: C x i -> N f r o m n
n
( x , ( s
sequence i n [0,1[.
G (x,0)
(x,t) ^ C x [ s _
Proof.
increasing
: Mx I -* N s u c h
F(x,t)
if
strictly
(Second main theorem).
Assume t h a t
: [ (M,A.j , . . . , A ) , ( N , B , . . . , B ) ] r
1
r
h
R = IR . T h e n
thenatural
map
-» [ (M,A , . . . , A ) , ( N , B , . . . , B ) ]£ 1
r
1
r
is
bijective.
The
two
claims
Theorem
As
in
and
III.4.2
I I I , §4
as
We
we
and
we
see
are
for
a
a
2.2
want
every
n>-1,
and
n > 0,
such
that
=
(h
we
ed
this
n.
Composing
creasing the
)
o
to
- 1
i N ,
=
h, =
f
-> N
a
h IM n
n
s_^
we
proceed
by
course.
Assume
that
diagram
over
(cf.
pushout M
n-1
Mn
=
1
glM
0
the
N(S),
by
h^^,
=
h
some
for =
f,
accomplishfor
n
of
for
f_^
every
strictly
( c f . Lemma G
C,
.j ( S )
M
have
n
stan-
-* N ( S )
i n -
5.1,
: M(S)*I(S)
i n d u c t i o n on
of
the
-*
"*
N(S)
desired.
f,
use
-
homotopy
n
=
n
along =
:C(S)
g
replaced
. O n c e we
with
K,
extension
use
relative
n
f ^
we
: M(S)
n
-> N ( S )
with
obtain
and
f
h
of
- f r e l . C(S) .
g
being
homotopies R
g
n and
n
-» N ( S )
h_^
[0,1[ we
as
g
h
homotopy
(-,1)
, H n
n
shifted
M(n)
i
(H ln>0)
R
n
1
a map
(s ln>-1)
f
n
of i ) .
the
: M(S)*I(S)
R
proof
using
: M
n
the
from
surjectivity
: M(S)
letter
give
starting
the
extending
(the
have
family
same way
prove
r =0
following holds:
construct
h
§2
We
to
decomposition
homotopy
done.
from
f
-» N
construct
, H n
sequence
In
h
a
case
a map
: M
from to
the
C(S)
for
n
are
relative
with
g
the
patch
the
i n d i c e s are
order
map
relative
We
(S)
given
the
r e s p e c t i v e l y . We
i t suffices
We
notations
IM n n
that
2.9. look
in exactly
III.5.1
retreat to
course).
f
proved
we
choose
dard
be
there
theorem and
can
R
the 2.2)
n.
h ,f ,H i
i
We
i
are
start
given
By
base
field
extension,
S. We i n t r o d u c e
k
u
Notice
n
:
=
n
:
=
h
n-1 ( f
that
o c p
l
algebraic
spaces.
extending
k
F
M
(
n
S
9
M
(
)
)
o
n
(
)
N
"* }
relative
^n s
(
k n
with
M
(
T
) »
9M(n)(S)
from
(*) t h e m a p s v
interval
By
this
M _
1
n
n
relative H
We p u t f proof
u
with
M
n - 1
n
I
(S) w i t h
s
over
N
S
-* < ) •
space
M(n)
there
i sa d i r e c t
exists
sum o f
a map v
n
:= H ( - , 1 ) . n
relative
By t h e pushout
g
n
semi-
: M ( n ) -> N
n
n
n
This
, IJM^ ( S ) n— i n
: M(S)xi(s) finishes
(cf.
IV.8.7.ii). from
n
H
n
: M (S)xi(s) n
a n d H„(-,1) n
-» N ( S ) w i t h
=
-> N ( S )
( h ) . n o c
H (-,0) n
t h ei n d u c t i o n step
with
(*)_ w i t h t h e S
3
diagram
o f the
-* N
R
( x , t ) »->• ( h _ ^ ) g ( x )
t o a homotopy = f
: M
n
t h ed i a g r a m
3
a pushout
a n d t h e map
M
H
J
property
t o a map h
"Multiplying" -
again
H (-,0) n
= f
We ^ .
and t h e whole
5.2.i.
i nTheorem
5.2 m a i n l y
( a sa l r e a d y
i nt h e "absolute
i n I I I , §4-§5)
case"
i t i s necessary
C = 0. t o work
homotopies.
5.3. Theorem
5.2 r e m a i n s
(M, ( A | X € A ) ) t ( N , ( B | X G A ) ) A
)
a n d h _-| c o m b i n e
n
t o a homotopy
n
f o r t h e proof
A
(*)
homotopy
n
t o N(S) combine
o f Theorem
Remark
e
s
t o (v ) .
n
t h e map F
We a r e i n t e r e s t e d But
) (
I I I . 4.2
a
I ( S ) we o b t a i n
diagram
(S) x i ( s )
extend
n
s
h ° i|> = v a n d h IM . = h n n n n n-1 n-1 unit
diagram
' ;
By Theorem
together
n
(*) a p u s h o u t
:M(n)(S) x I ( S ) — • N ( S )
R
diagram
:
n
extends
R
from
t h e maps
n-1
u
we o b t a i n
true
o f spaces
f o r locally
finite
instead of finite
systems
ones,
with a l l
A
c l o s e d i n M, o f c o u r s e . T h i s c a n b e p r o v e d A above w i t h more n o t a t i o n a l e f f o r t .
i na similar
way a s
It
w i l l
fer do
be more
principles
difficult
here
than
g i v e n by t h e main
theorems
n o t know whether a g i v e n system
isomorphic
to
(N (R) ,B-j (R) ,. . . , B
r
i n Chapter
I I Ito apply the
on homotopy
o f spaces
sets,
(M,A^,...,A )
(R) ) f o r some
system
of
since
over
r
transwe
R i s
spaces *)
(N,B,j , . . . , B ) r
w i l l
make
Recall field
over
the field
t h e homotopy
that
R
Q
and thus
R
Q
of real
t h e o r y i n WSA(R)
embeds
i n a unique
i s the "prime
algebraic more
way
field"
numbers.
laborious
into
i n real
This
than
any other r e a l algebraic
i n LSA(R).
closed
geometry.
§6
- Compressions
In
this
ly
by w e l l
section
Whitehead. of
(steps
t h e spaces
main
theorem
further
say
that
map g
classical
"compression
F o r some s t e p s ,
need
Let
we p r e s e n t s o m e t h e o r e m s
known
generality
Definition
and n-equivalences
P
which
b)-d)
(M)
1. L e t f
arguments"
semialgebraic
t o L, i ff i s homotopic
F
: (Mxi,Axi)
(N,B) f r o m
f
i s c o m p r e s s i b l e i ff c a n be compressed
or
n = °°. We
call
Q
a pair
g (M) c L . We
§2,
polytope
and that,
sion
at least
This
c a n be proved
is
even
m.
simpler
to
compress
(M,A)
with
We
and k < n ball
i n R
be a s p e c i a l
the pair
o f spaces
m C IN,
i s
a r e g i v e n a map f
1
f ( E) c M.
o f f t o L . We
say that
exists
i fe v e r y
number
map
i s compressible. Here, as k k— 1 and S i t s boundary.
relative (M,A)
p a t c h complex
(cf.
i sa relative
e v e r y p a t c h a £I(M,A)
weak
has
dimen-
(m-1)-connected.
"cell
i n the topological
f t o A. T h e r e
homotopy
t o B.
(M,A)
k complexes.
a
We
A to a
n-connected
by a very b a s i c
than
call
relative
B.
o f spaces
(M,A)
will
o f spaces.
n = 0 or n i sa natural
f o ra f i x e d
Then
pairs
U (M,A) w i t h k € 3N k usual, E denotes t h e closed unit
Proposition
level
i n I V , §10, and t h e f i r s t
subset of N containing
-* ( N , B ) w i t h
2. L e t n € I N
6.8 b e l o w ) , we
(N,B) b e a map b e t w e e n
: (M,A)
Definition
the right
5.2.i.
: (M,A)
f c a n be compressed
essential-
m o s t l y d u e t o J.H.C.
are important t o attain
(M), i n t r o d u c e d
sets
L be a weakly
c a n be proved
i n t h e p r o o f o f Theorem
and
on homotopy
which
:
by c e l l " setting,
argument,
which
here
say,f o r simplicial
k-1
( E ,S
a finite
R e p l a c i n g M b y M'
) -> (M,A) closed
with
k - 1 )
relative
n
n
space
a
( f '
of (N ,B),
be done:
N
f extends
relative
a relative
with
space
B = 0) , e v e r y
of the special
starting
7 . 1 ) .C l e a r l y
Assume
t h e n-chunk
limit
that
CW-complex.
equivalence
the i n d u c t i v e
(N ln>-1).
( A = 0,
homotopy
„. T h e n w e w i l l
with
(M,C) t o (N,D) s u c h
construct inductively
i s a closed
n
a
denote
n
w i l l
every
N _^
In particular
equivalent to a
LetM
(M,A). a
3-
(f,a) from
: 3 M ( n ) -> N
t h e homotopy
equivalence
u
N
1
n-1
n -
^
denote
equivalence
map
t h e composite f - :M - ->N . n - i n-1 n-1
Now a
(M(n),3M(n))
running
M _ ^
a
the inclusion
^. S i n c e
restriction
and
i s just
i s isomorphic to
f
A l l
a € I (n) . L e t t p
the smallest
by u ^ =
(o 3a)
X
X
write
the
f i x some
t o 3a. This
n
sum o f t h e p a i r s
through the s e t I ( n ) = I(M ,M
(M,A) . We of
i s the direct
denote
have
Q(o)
the smallest
closed
= a U L ( a ) . The
t o a homotopy
sub-
homotopy
equivalence
we
which By
extends
the
Corollary
k
4.9.i there
: a U
X
a
X
under
to
L(o)
to
A l l
X
maps
f
Q
the
pair
on
number
homotopy
for
L
the
closed
n
l L
i s a
is
now
We
choose
ing more a
The
of
that
(f
the
Let
K
i
s
a
homotopy
homotopy
equivalence
from
equivalence f _^lL(o)
from
n
n
f
M(K).
and
y)
behaviour denote
a
n
tp
: S ~
1
n
n
are
of
L
Let
and
=
n
the
PW2.
Corollary the
Theorem
6.10
homotopy
n
instead
of
the
i s a
2.14
by
this
the
a
of
n
n
n
a
induction f
' L i s
n
also
Thus
holds between
(Th.
(^L,D).
under R.
2.13)
We
have
base
{The
field
letter
CW-complex
of
1
(M,A) .
Then
ex-
"S" over
corresponding
E ,S ~ .} of
finite
proof.
and
cells
sets
to
relative
-* M
n
equivalence g
(L,C)
overfield be
I f L
respectively. pair
f _«j-
extends
restriction
CW-complexes
: E (R)
with
which
n
axiom
finishes
f o r a l l the
K
N
{We
R.
attachwrite
(M(K),A(K))
i s
cells
{a(K)IaGI(M,A)}.
of
(
a u
L(o)
The
^Q(a)
the
By
equivalence f _^lL(a)
a
L(a).
Q(a)
homotopy
(M(K),A(K)) has
the
have
characteristic
a(K)
(M,A),
coincides
=
then
with map
a (K)
and
T-1).
proceeding
{but we
by
there
choose denote
the a
the
i n a N
We
w i l l
similar not
the
characteristic
map
R
were
way
corresponding
build as
up
i n
skeletons
attaching
map
by
We
w i l l
a
cp
d
: S ~
Q
n
n
a
9'
n
(f ,a )
Assume
to
_ i
(N
=
that
n >0.
We
M
a
n
of
a
n
\D)
n
N
n
1
T
n
watching
i s the e
n
w
e
homotopy starting
be
are
n
((N ,D)In>-1) equivalences
with
(n-1)-skeleton
w i l l
(f,® _ ( M , B )
i s bijective
claim
t o IR
that
holds
such
we
base by
that
M
i s a CW-complex
for R =
see that field
that A'
R we
f(A) c
B'
by
now
o f M',
f o r (M,A,B,x)
i s a weak
i s an
with
polytope.
Theorem
7.8
y
f o r (M,A,B,x)
from
[Sw,
B'(R)
x
running
to
we
can v e r i f y
instead
of
7.10.
but R
the claim
and
by
f
yields 1
) (R)
through A' fl B' . the
i s sequential. f o r
.
equi-
( A ' fl B
running through
R
CW-complex
a homotopy
f(B) c
with
6.21].
the claim
exists a
and
A,B
f o r R =
Then
a r b i t r a r y space
and
homotopy
holds
obtain
there
A'(R),
the
c f . e.g.
i s also
theorem
and
IR
f r o m A t o A' (R) , B t o B' (R) , a n d A R B holds
for
n+m.
the case
closed
using M
closed
(M,B,x)
f
M
and B be
homomorphism
f o r (M',A ,B',y)
by
that
the r e -
favourable
partially
Tak-
q
claim
i t holds now
and
subcomplexes
equivalences
since
-> TT
Indeed,
M' (R)
restriction.
CW-complexes.
In very
f o r t o p o l o g i c a l CW-complexes,
extension
R ,
using
= M.
the
consider
o f M.
theorem
base
over
A UB
Then
f i r s t
homotopy
sometimes can t r a n s f e r
theorem).
surjective f o r q =
By
by
with
the inclusion j
subcomplexes
excision
M'
M
i s n-connected
and
we
polytopes.
excision
: TT ( A , A 0 B , x ) q by
7.8,
the
example.
and m>0.
+
induced
are
an
r e s u l t s from
to semialgebraic
t o a r b i t r a r y spaces
(Homotopy
( A , A D B)
Proof.
Theorem
t o a r b i t r a r y weak
give
that
1 0
IV,
§8).
p
(M,A)
q.e.d.
spaces
and
with
n>1.
Then
the
A
m-connected
Assume
that
claim and
two
A
:
then by
then
and
down
some
which
w i l l
be
power
of
the
Definition
4.
i)
i s a
every
P^
the
using
->
cone
(M/A,*)
CA
=
Theorem
Proposition
partially
similar
to
finite
family
closed
c a l l
i t a
P
of
CW-system
(P ,...,P ) r
theorems
system
a
CW-pair, (P
decreasing
consequences
widely
A
we
a
i s
homomorphism
surjective
applying
i s only
steps
write
f u l l
short,
A
the
i s , for
i f r =
any
m+n+1.
I*A/1xA
exists.
7.12
the
to
2.15,
cf.
complete
near
steps
and
c)
base
triple
[Sw, M^A
d)
In
p.
84].
we
i n
now
the q.e.d.
7.11
Q
for
a
above.
7.10,
i
(cf.
choose
holds
homotopy
pair
some n u m b e r s
polytope
that
by
for
i f 2 (M , . . . ,M )
r
r
at w i l l
and a l l components
o f spaces
are closed
we
with
the additional
^ o f ^ homotopy e q u i v a l e n c e s know b y 2.13 t h a t
^ i s a
homotopy of
equivalence.
A homotopy
inverse
cp o f
i sa CW-approximation
(M ,...,M ). Q
Theorem
7.15. i ) L e t K be a r e a l
(M ,M ,...,M ) Q
q.e.d.
R
1
be a d e c r e a s i n g
R
decreasing
CW-system
closed
overfield
WP-system
over
(P ,P^,...,P ) over Q
o r R. L e t
K. T h e n
R together
r
there
with
a
exist
a
homotopy
equivalence
cp
If
M
( P
( K )
Q
f
P
semialgebraic
I f (Q , . . . , Q ) r
(Q (K),...,Q (K)) Q
g
(K) ,. . . , P
1
i scontractible
R
is ii)
:
R
then
then
P
R
(K) )
r
t o (M ,...,M )
/
M
1
,...,M )
.
R
Q
r
WP-system
then
R
- f . I f f i sa homotopy
K
Q
P^ c a n b e c h o s e n a s a f i n i t e
i sa decreasing
r
( M
c a n be c h o s e n a s a o n e - p o i n t
: (Q , . . . , Q ) -* ( P , . . . , P ) ,
cp«g
-*
there
unique
over
CW-complex.
R a n d f a map
exists
then
also
from
a map
up t o homotopy,
equivalence
space. I f
such
g i sa
that
homotopy
equivalence.
Proof. (M
I n o r d e r t o p r o v e p a r t i ) we may a s s u m e , b y t h e p r e c e d i n g t h e o r e m , t h a t
,...,M ) i s a l r e a d y a CW-system.
I\J :
such ing ^i
that
(M ,...,M ) Q
Q
M
i ~* P - L ^ ) K
a homotopy
as
desired.
ing
( K ) ,... , P
o f ^ i sa homotopy equivalence
R
main
main
Theorem
7 . 1 0 we o b t a i n
CW-system
over
equivalence.
theorem on homotopy
R, a t w i l l
above, and every
b y 2.13. A homotopy
main
a map
( K ) )
This
sets
5.2.i.
hav-
component
map o f s y s t e m s
inverse
c l a i m s i n t h e t h e o r e m now f o l l o w
instead of thefirst
second
Q
properties claimed
The l a s t
thefirst
Using
( P
r
theadditional :
-
(P ,...,P ) i sa decreasing
is
and
R
Using
cp o f ip i s a m a p from
Theorem
6.14
q.e.d.
t h e o r e m s 7.10 a n d 5 . 2 . i t h e c o r r e s p o n d -
t h e o r e m s 7.11 a n d 5 . 2 . i i
we
obtain
Theorem there with
7.16. i ) L e t (M ,...,M ) Q
exists
a s e m i a l g e b r a i c CW-system
a topological
homotopy
cp : ( P , . . . , P ) Q
If
M
If
r
be a t o p o l o g i c a l
r
i scontractible i sa f i n i t e
Q
then
p 0
'*-*'P ) over r
Then
IR t o g e t h e r
equivalence
(M ,...,M )
r
(
CW-system.
r
P
CW-complex
r
.
c a n be chosen
as a one-point
space.
then
also
P^ c a n b e chosen
as a
finite
I f (Q , . . . , Q ) i s a W P - s y s t e m
over
IR a n d f i s a c o n t i n u o u s
CW-complex. ii) from
r
(Q '•••rQ )top Q
t
r
o
(
M 0
'**-'
M
) then
algebraic
map g f r o m
such
cp^g - f i n t h e t o p o l o g i c a l
that
homotopy
(Q ,...,Q )
r
Q
equivalence then
r
there
exists
t o (P , . . . , P ) / r
sense.
g i s a homotopy
a weakly
unique
semi-
up t o homotopy,
I f f i sa topological equivalence.
map
weak
We
now
lized an
have
enough
homology
homotopy
theory
and cohomology
arbitrary real
closed
at our disposal
f o r (weakly
field.
This
w i l l
to build
semialgebraic) be done
up
genera-
spaces
i n the
over
present
chapter.
In
§ 7 we s h a l l p r e s e n t
Brown's ([Bn],
representation [Sw, Chap.
Chapter
V.
chapter, can
and
give
right
bring
now.
section
spectra
r e s u l t s i n §1
theorem
give
by s p e c t r a
into play
of reduced
We
can transfer this
to
the authors of weakly
topological
long
way
as an addendum t o
sections
cross
(§8). I n t h e p r e s e n t earlier
proofs
semialgebraic
stage.
and semialgebraic
thus
references.)
o f some
This
Brown's
representation by spectra
this
approach
would
arguments
would
algebraic theorem
topoto a
[Sw, C h a p . 1 4 ] . setting, but,
to generalized
be t o o much
and would
we
results i n co-
i n classical
spaces
and
present
chapter
Already
theories
and t a s t e ,
i n the
a n d some n o t a t i o n s ,
description to the semialgebraic
opinion
functors
t o a d e s c r i p t i o n o f homology
easier
homology
homotopy
be r e g a r d e d
a t a much
from
o f two v a r i a n t s of
the necessary
leads
b u t n o t so i n homology.
i ti s a rather
may
of the preceding
(We w i l l
theories
analogues
f o rcontravariant
a l t e r n a t i v e and sometimes
description
logy
easy
representation
homology, logy
9 ] ) .T h i s
u p t o some
cohomology
could
theorem
I t i s independent
be read
Brown's
the semialgebraic
a mixture
u s e t o o much
homoof machinery
§1
- The b a s i c
In
this
categories,
section
homology tions.
we
suspensions
s e t the stage
and cohomology.
(Some o f t h e m
We
have
cofibers
f o r a discussion
first
been
and
compile
used
of
the basic
before.)
generalized categorial
L e t R be
nota-
any r e a l
closed
field.
*) Notations and
1.1.
P(2,R)
a)
3>(R) d e n o t e s
denotes
the category
F u r t h e r >>* ( R ) d e n o t e s This
i s a
space
f u l l
subcategory
9 *(R) as t h e c a t e g o r y V,
the
pair
o f spaces
In this
o f weak
since
(M,{x}).
we
every
also
polytopes over
polytopes over
may
polytopes regard
Alternatively
polytopes under
identify way
o f weak
o f p o i n t e d weak
of ?(2,R)
o f weak
§ 2 , D e f . 4 ) . We (M,0).
of pairs
the category
(M,x) a s t h e p a i r
(cf.
the category
weak
(R) b e c o m e s a
f u l l
,
R.
over
R.
a pointed we
may
view
t h e one p o i n t space
polytope M
R
over
R
*
with
subcategory
of
? (2,R) . b)
Similarly
category the
we
regard
WSA*(R)
the category
of pointed spaces
c a t e g o r y WSA(2,R)
of pairs
WSA(R) over
of spaces
of spaces
R as f u l l over
over
R and t h e
subcategories of
R.
c ) I n WSA (R) we f u r t h e r h a v e t h e f u l l s u b c a t e g o r i e s o f p a r a c o m p a c t s e m i a l g e b r a i c s p a c e s L S A (R) a n d o f s e m i a l g e b r a i c s p a c e s the of
category polytopes
spaces ... d)
LSA (R)
= L S A (R) np(R)
over
These
Q
LSA*(R),
R.
SA*(R),...
categories lead and o f p a i r s
A l l c a t e g o r i e s mentioned F o r any o f these
homotopy
category.
HOi a r e t h e h o m o t o p y
This c
H(X h a s t h e s a m e
i s a substitute
WSA (R).
of spaces
o f maps
o f t h e more
denote
SA
c
(R)
finally = SA(R) n P(R)
to categories of pointed
so f a r a r e f u l l
c a t e g o r i e s 01 w e
classes
and t h e c a t e g o r y
SA(R),
locally
LSA(2,R),
SA(2,R),
s u b c a t e g o r i e s o f WSA(2,R).
b y HOC t h e c o r r e s p o n d i n g
o b j e c t s a s Ot b u t t h e m o r p h i s m s between
o b j e c t s i n &.
of
A l l these
systematic but clumsier
notation
categories e ) We
HOC a r e f u l l
denote
cisely
this
b y *0 means
topological logical)
(M, A )
which
makes
a
the full
spaces
which
the category
admit
direct
sums.
space
inclusions category classes
points
i
M
family
such
nO(2) that
Similarly
M admits
[ i ^ ] we
sums.
M
we d e n o t e
then
again
We
together
M
with
spell
i n HWSA*(R).
CW-decomposition
by
W*
the catewith
A
o f * f e ? ( 2 ) . We d e by ^
the
finite
direct
and t h e
F
^ , ( 2 ) .
corresponding
t h e most
this
products important
out i n a special
of pointed
spaces
^by t h e i r
sum o f t h i s
family
of pointed
:= T T ( M ^ | X € A ) w i t h
family spaces
t h e base
(M^|X€A) obtain
i n WSA*(R). the direct
ones, case.
R.
The
the natural
(M^|X€A)
i nt h e
homotopy
i n HWSA*(R). over
point : M
and
over
(pointed)
t h e n a t u r a l p r o j e c t i o n s p^
[ p ^ ] we
with
sum o f t h e f a m i l y
the i
the direct
of the family
homotopy c l a s s e s
HOC m e a n s
(topo-
of closed
b y #??* a n d
( c f . I V , 1.8) t o g e t h e r
replace
be a f i n i t e
equip
a
( M , A ) € W{2)
o f # ? * a n d %0(2)
i s the direct
I f we
obtain
a
Hausdorff
admit
the category
t o p o l o g i c a l CW-complexes
s o f a radmit
: = V(M^|XeA)
WSA*(R).
product
which
subcategories
Some o f t h e m , i n f a c t
^ :
x ^ . We
T h e n M,
their
by
TOP o f
pre-
category.
direct
L e t (M-jJXeA)
direct
t h e spaces
1 . 2 . i ) L e t (M^IXGA) b e a f a m i l y
pointed
ii)
o f M.
categories
mentioned
arbitrary
Remark
denote
subcategories
homotopy
categories
finite
We
of finite
f u l l
(topological)
of the category
a n d 2fe?* a r e f u l l
I f OL i s o n e o f t h e s e
All
subcategory
CW-complexes. These a r e t h e p a i r s
space.
corresponding
HWSA(2,R).
has as objects
A a subcomplex
one p o i n t
of
o f t o p o l o g i c a l CW-complexes. More
o f t o p o l o g i c a l spaces
of pointed
note
the category
CW-decomposition.
pairs
gory
subcategories
x
R with :=
M^,
Replacing product
of
base
(x^lXGA).
i s the the this
by
If
A
i s finite
fied
with
(y^|X6A)
As
a
then
the
closed
with
=
consequence
write
the
of
1.3.
HP(2,R)^
for a l lindices
some c e n t r a l natural
i ) The
f o r the
are
and
c
"LSA" If K
(III.§1, i s a
real
closed
(M,A)
valence
categories
K
of
H?(K)
: HP(R,2)
and
— ^
of
from
. LSA*,
from
Chapter
Ot(R)
t o OC(K) .
i i i )
The
functor
topological
X
an
Moreover
HP*(R)
the
H W S A ( R ) ,
points
one.
I I I and
V
homotopy
(R)
C
H P * ( R ) ^—
categories
we
can
categories
HWSA* (R) ,
(Th. V . 4 . 1 0 ) .
HSA(R) , H S A * ( R ) ^
analogous
field
inclusions
extension of
LSA
(M, A )
P(2,R)
R
to
yields
H P * (R)
with
then
the
?(2,K)
(2,-),
IR a
The
H S A * (R) , "SA"
replaced
functor
yields
"base
an
equi-
equivalences from
the
then
which
pair
This
follows
H?(R)
from
the "meta-categories"
restriction
t o TOP(2)
f u l l
of
SA(2,-)
by
to
are
respectively.
I f OC i s o n e
P(2,IR) over
: HP(2,IR)
and
K yields
from
isomorphism
from
t o HP*(K)
V.7.15.
SA*
spaces
every
HSA
K t o HP(R)
I I I that
weak p o l y t o p e s
i.e.
the
the
identi-
HP(K,2) .
Theorems V . 5 . 2 . i and , SA
HP(R)
(M(K),A(K))
restrictions
LSA
of
be
V.2.13).
extension"
to
w i l l
a t most
i n Chapters
equivalences of
inclusions
H S A ( 2 , R ) 0,
then
lr
S M
i s the k-fold
iteration
of the suspension
functor
applied
to
M.
lr Every U
k
: S
object
sphere k
S
k
-* S v s
, k >1, k
i s equipped
, unique
i n HP*(R), w h i c h
up
t o homotopy,
f o r every
X GWSA* (R) ) , g i v e s t h e g r o u p object of
i s abelian
u Aid k
turns
S M
k
M
: S M
into
-
Now
a base such
X€P*(R)
structure
f o r k>2.
t h e smash p r o d u c t
with
on
k
a cogroup
that
(S
(and then TT^ (X) =
[S
i t i s evident from
with respect to direct
S M
point preserving
sums
k
,ly J) k
also
i s a
cogroup
every
,X] . T h i s
the
map
cogroup
distributivity
(= w e d g e s )
that
also
k
v S M
i n HP*(R)
f o r every
p o i n t e d weak
polytope
M,
which
i s abelian
k>1,
the
abelian
The
set
for
k>2.
[ S M,X]
i s a
r
L
tt] denotes
take k
S vS
as k
Thus
1
=
Lemma
1
in
in
every a
pointed
space
n a t u r a l way,
the
image
1.4.
k
well
By
a
as
0
spaces
maps
are
of
the
equivalences three
p.
of
steps
to
the
long
308ff]).
i n the
sequence
diagram
the
of
Let
basic
f
: M
Puppe
and
sequence
starting
from
sequence
the
lemmas
importance.
-> N
further pointed
Puppe
and,
the
c o f i b e r s above
[Pu]). be
1
• SCf
§ 3
( c f . [Pu,
f o l l o w i n g theorem
and
of
sequence
discussion of
( B a r r a t t [ Ba] , P u p p e
» C(j")
functor.
long
lower
q.
J
sequence moves
i n topology
upper
we
We
as
suspension
the
1.8
4.
the
call
cofiber of
apply
lower
— i ^ -
, SN
natural injections
The
by
,
SM
their
SM
diagram
I
C(f)
into
p r o j e c t i o n C(f)
commutative
II
j ,j ,j " , . . . a r e preceding
the
infinite
—3—»
1
Here the
SM.
of
be
weak
f we
the
a map
between
polytope.
obtain
fourth
a
long
term,
groups
[M,X]
Starting
-
[N,X]
from
the
-
[C(f),X] -
seventh
term
[SM,X] -
the
groups
[SN,X] -
are
[SC(f),X]
abelian.
-
2
[S M,X]
•
In
the special
o f an
1 . 1 0 . I f (M,A)
Corollary another
case
p o i n t e d weak
inclusion
i s a pair
polytope
then
map
we
obtain
o f p o i n t e d weak we
have
polytopes
a natural
long
and X i s
exact
sequence
[ A , X ] +-
Definition
A
[M,X]
5.
-> M
We
[ M / A , X ] «* [ S A , X ]
call
-> M/A
t h e sequence
S A -> SM
used
i n this
corollary
If
: M->N
i s a n y map
f
*(R) o f
H?*(R)
the
suspension
ing
definitions
category
P*
do
In
the
F we
will
f*
as
[Sw,
-> A b .
Chap.
7].
first.
Let
1.
A
If
f
n
of
(a |nGZ)
a
: k
Exactness
axiom.
and
n € Z
n
+
For
n
n
tion
M -»
n € Z
(i*)
is
an
natural
on
the
the
map
a l l the be
between group
endomorphism, and
well
theories f(0*
category
category
i t s homotopy
of
work-
on
the
of
pointed
abelian
groups.
functors
pointed
weak
homomorphisms
polytopes
F([f])
by
feared.
cohomology
functors
k
n
that
the
pair
(M,A)
of
k*
over
: HP*(R)-> Ab
equivalences
such
theory
(=
pointed
i s
together
isomorphisms
f o l l o w i n g two
R
axioms
between hold.
weak p o l y t o p e s
over
R
-i^->
denotes
the
n
k (A)
i n c l u s i o n A «-» M
and
p
denotes
the
projec-
M/A.
Wedge a x i o m . every
k (M)
i
i s any
natural
every
and
many c o n t r a v a r i a n t
i s to
1 1
give
cohomology
(semialgebraic)
of
R
sequence
k (M/A)
Here
-+ N
over
distinguished
can
denote
denote
^ S -^k
the
Ab
a
and
contravariant
n
family
exact.
: M
reduced
functors)
is
S we
consider
confusion
(k ln€Z)
n
w i l l
abusively
Definition
every
Using
i n topology
no
a
S.
with
i s done
as
with
equipped
polytopes
(R) , a s
often
family
weak
homology
long
a
are
polytopes
reduced
f o l l o w i n g we (R)
pointed
functor
cohomology
: HP*
weak
of
CW-complexes
We
both
of
For
the
every
family
(M^IAGA)
of
pointed
weak
polytopes
and
map
n
: k ( V ( M l XGA)
isomorphism.
A
Here
) -> T T ( k ( M ) I n
A
i , denotes
the
X€A)
natural
embedding
of
M,
into
M.
Actually some ral
i tsuffices
bound
n
t o demand
€ Z. T h e n
Q
equivalences
a
n
S(V(X IA€A))
since
=
X
they
S(M/A)
such
tend by
f o r n >n
with
Q
n by use o f t h e natu-
= SM/SA a n d
X
for
that
axioms
V(SX IXGA).
i fthefunctors k
Q
o f these
follow f o r theother
Moreover, n >n
each
n
and t h eequivalences
t h e axioms
t h e family o f these
above
functors k
hold
n
o
n
f o r these
t o a reduced
aredefined n, then
only
we c a n e x -
cohomology
theory
defining
k o" (x) n
for
n
r
:= k ° ( S X )
r
r > 0.
Notice
also
exactness then
that
t h e wedge
axiom.
M^vM /M 2
Indeed,
= M^
2
axiom
i f M^
a n d M^vM /M 2
f o r A finite
and M 1
2
= M . 2
i sa consequence
a r e two pointed
weak
By t h e exactness
of the
polytopes,
axiom t h e
diagram
M • 1 1
of
p
- M, v M « 1 2
natural injections under
We
call
theequivalences
of
t h e cohomology
"cohomology from
§4 we w i l l
theories
We k*.
draw
w i l l
some
need
be
sum d i a g r a m o f
k .
o
theory.
theory"
0
n
groups
omitting t h e index
- M 2
2
a n d p r o j e c t i o n s becomes a d i r e c t
abelian
o
> i
0
1
n
: k
n +
^»S ^ k
1
1
t h e suspension
We u s u a l l y d e n o t e
n. I n t h e f o l l o w i n g i n s t e a d o f "reduced t o be more
careful
them
we a l s o
isomorphisms
a l l b y t h e same s a y more
cohomology
theory".
since
also
then
letter
briefly (Starting
unreduced
studied.)
consequences
from
t h e axioms
o f a given
cohomology
theory
If
(M,A) i s a n y p a i r
every
n
1
(TT*)"
jection the
1
a"
: k
from
n
+
: k (A) -
n
: k 1
(M U C A ) -+ k
polytopes
then
we d e f i n e , f o r
n + 1
n + 1
(M/A)
(SA) w i t h
(M/A).
q*
Here
: k
n
As a consequence
1
(SA) - k
q denotes
S A , a s i n Lemma
M U C A t o M/A
+
which
o f Lemma
n
+
1
(M U C A )
the natural
pro-
1 . 8 , a n d TT d e n o t e s i s a homotopy
equiva-
1.8 we d e d u c e
from t h e
axiom
Proposition abelian
n + 1
M U £ A t o M U CA/M =
(Ex. 1 . 6 . i i i ) .
exactness
k
(A) ^ - * k
n a t u r a l p r o j e c t i o n from
lence
2.1. F o r every
groups
(going
N
L,
is
n
= 6 (M,A)
composing
and
weak
n £ Z, a h o m o m o r p h i s m
6
by
of pointed
k ( /A) M
pointed
t o infinity
N
- E l .k
WP-pair a t both
(M,A) t h e l o n g sides)
n
( M ) - i l *
sequence o f
k (A)
k
n + 1
(M/A)
-
exact.
Corollary
2.2. F o r e v e r y
we h a v e
a natural
at
sides)
both
n ~ —>k (C(f)) n
This
follows
from
the
Puppe sequence
the
same
Let
(M,A,B)
every
f
: M -> N b e t w e e n
sequence
of abelian
pointed
groups
(going
"i * n f * n n+1 ~ - J — * k ( N ) -±—» k ( M ) - k (C(f)) n
2.1
n
s i n c e , up t o c a n o n i c a l
o f f and t h e suspension
weak
infinity
-
homotopy
sequence
to
polytopes
of
equivalences, (Z(f),M)
are
( c f .end o f§ 1 ) .
be t r i p l e
n € Z we
A
exact
map
n
define
= A (M,A,B)
of pointed
weak
polytopes
a homomorphism
: k
n
(A f)B)
-> k
n
_
M
(M) ,
with
M = A UB. F o r
as
the
composite
n
k (A
n B)
-r-* k
n + 1
(A/A
P.B)
k
o with by
6
=
the
natural
n
space
2.3.
=
inclusion
This
follows
[Sw,
p.
a
from
105].
p.
Proposition 3)
the
group
flB ^M/B,
isomorphism and
p
: M
induced
-» M/B
the
-
1
follows
p.
exact
q
: A ^
1
A UB
the
infinite
=
M,
-
(M)
->
i*v
: B ^
2
2.1
n + 1
by
i*u
J
k
M,
i
well
: A fl B
induced,
lim •* n
of
i s explained
the
preceding
k
q
(M
course, in
n
) -
by
[Mi],
0
the [Sw,
proposition
to
.
inclusions p.
the
128],
reduced
§2,
telescope
T e l (fa)
suitable [Sw,
p.
NB.
128],
been
step
Mxi^.
Of
i s contained
in
every
M
main
goal
with
the
theories
^ *
of
Definition families A
2.
of
terms
Let
k*
n
course,
+
1
T
x
n
+
real
1*
=
been
ln€IN)
T e l (fa)
Thm.
denotes
Q
closed
theories
more
be
6.6
and
(cf.
below).
identified
the
[Mi],
base
with
point
of
a M.
that,
i n vague
field
R
(cf.
[Sw,
cohomology
: k*
between
n
(SX)
(a |nGZ) -* 1 *
from
functors
the
terms,
correspond Chap. In
uniquely
7])
order
the
on
to
the express
terminology.
two
T
n €Z,
1
had
prove
isomorphisms
and
(SX)
and
of
(M
.
need
transformation
X € 9*(R)
every
§4
proof
AUB
t o p o l o g i c a l CW-complexes.
and
transformations
k
i n V,
any
we
suspension
natural
natural
over
f a m i l y fa :=
T e l (fa) w i t h
following i s to
pointed
in precise
the
i n the
t o p o l o g i c a l cohomology
category this
i n the
of
b)
defined
of
cohomology
for
also
X o f
Q
A, B
subspace
Our
a)
T e l (fa) / X
subspaces
see
T e l (fa) h a d
closed It
closed
:=
(T
n
theories
: k
R
with
n
and k*
over
(x |nGZ). to n
1*
i s a n
-> l | n € Z )
family such
of that,
square
i
n
n
0 (X)
+
1
(SX)
n
T (X)
n
i (x)
n
k (X) n
T (X) commutes. b)
We
call
T
a
natural n
i f ,
i n a d d i t i o n T (X)
n G
2.
Proposition between
2.5.
Let
cohomology
equivalence, i s an
T,U
an
isomorphism
: k*
theories
or
^
over
1*
be R.
isomorphism,
for
two
every
natural
from
XGP*(R)
k* and
to
1*
every
transformations
n
= U (S°)
n
i s bijective
a)
Assume
that
T (S°)
b)
Assume
that
T (S°)
n
f o r every
n € Z. T h e n
f o revery
T =
n € Z. T h e n
U. T i s a
natural
equivalence.
The
proof
axiom
runs
2.5 t e l l s
ly
determined,
of
abelian
groups
later
a reduced
i n a restricted (k (S°)In€Z).
t o compare
cohomology
K be a r e a l
sense
n
groups
i n [Sw, p .
123f]using
t h e wedge
2 . 1 , 2.4.
us that
o f cohomology
want
Let
t o the proof
and Propositions
Proposition
We
similar
theory
closed
made
These
precise
groups
theory
k*
i s unique-
t h e r e , by t h e sequence
are called
the coefficient
k*.
cohomology
theories
cohomology
theories
over
over
3R w i t h
overfield
o f R.
different
topological
Every
base
fields
cohomology
cohomology
theory
and
theories.
k*
over
ID
K
"restricts"
obvious
t o a cohomology
w a y . We n
(2.6)
define,
R
:= k ( X ( K ) )
R
that
f
Y
: X
n
R
Every
natural
R i n the following
,
:= a ( X ( K ) ) .
(SX)(K)
(k ) ([f])
over
n
(a ) (X)
{Notice
(k*)
f o r n € Z a n d X € ?* (R) , n
(k ) (X) n
theory
= S(X(K)).}
n
:= k ( [ f ] ) R
We
further
define,
f o r a n y map
.
transformation T
: k* -* 1* f r o m
k*
t o another
cohomology
R theory to
1* o v e r R
(1*) ,
to a natural
transformation T
from
(k*)
d e f i n e d by n
R
(T ) (X)
Let
K restricts
R
Hom(k*,l*)
n
= T (X(K))
denote
.
the set of natural
t r a n s f o r m a t i o n s from
k* t o 1*.
Proposition
2.7.
R Hom((k*) a
The
restriction
,(1*)
) i s bijective.
natural equivalence
The
proof
that of
i s an
the base
Coho(R)
theories
dealing
with
closed
res^
to
R
T .
We
K
HP*(R)
Hom(k*,l*)
to
K)
we
-+
object
k*
( k * )
(First from
(more
R
have
T
R
One
uses
i s an
the
fact
equivalence
are the
are the natural i s a
theories.}
every
restriction
cohomology
transformations
reminder
For every
t h a t we
real
embedding
are
closed
R
K
into
a
functor
to
(k*)
R
and
the restrictions
Coho(K)
t h e o r i e s over
means
a morphism
o f k*
theorem f o r cohomology
the category
P r o p o s i t i o n 2.7
objects
notation
~ o f Coho(K)
the cohomology
cohomology
theory.
Coho(R)
and
main
K i s
equivalence.
-> H P * ( K )
precisely, a
T over
suspensions.
whose
i n this
cohomology
of R
field
isomorphism
Proof.
functor
tilda
reduced
call
2.8
i n category
commutes w i t h
: Coho(K)
equivalence
the
from
natural transformation
and whose morphisms
{The
maps an
Theorem
to
R
extension
which
T -> T
i s a natural
exercise
extension
over them.
real
A
denote the category
between
field
i f f T
easy
categories which
Let
map
R
K
T
theories).
t o Coho(R).
theories over
and
T
of
to
R.
r e s
R
~ Coho(K)
i s an
In particular,
correspond
up
uniquely
to
R.
that
res
i s fully
D
faithful.
I t
remains
K to
construct,
theory
Let
k*
for a
over
M e P * (K) b e
consisting [cp] o f a
K
given
together
given.
of a pointed
homotopy
We
cohomology with
equivalence
1*
over
R
a natural equivalence
d e n o t e by
weak
theory
I(M,R)
polytope tp : X ( K )
X
T
cohomology ~ R : 1 * —•* ( k * ) .
the set of a l l pairs
over -> M.
a
R and Such
t h e homotopy
pairs
exist
by
(X,[tp]) class
Theorem then, X
V.7.15.
a g a i n by
: X -* Y
We ( l
have n
there
^°X
that
: l
as
n
( Y )^ l
f o r any maps
n
k (M)
making
For n
l (X)
f
( X )
two
t o homotopy, homotopy
of abelian
transition
of
I(M,R)
a unique
map
equivalence.
groups
maps
(X,[cp]),
system.)
this We
(Y,[^]) system
i n I(M,R).
could
also
(Since
be
the
regarded
define
n
put k
choice
cp*, a b u s i v e l y
-* N
i s a map
n
(M)
=
about
we
l
from M
n
( X ) f o r any
the natural
specifying
to another
g
: X
-* Y
the space
X
weak
are given,
over
R
without
(X,[cp]).
projection
pointed
(Y, [^]) € I ( N , R ) map
(X,[cp]) € l ( M , R )
the "approximating pair"
denote
neither
a unique
X(K)
from
k
n
(M)
n o r t h e number
to n.
polytope N
over
then
exists,
such
that
n
-
there
the
diagram
M
f
gK N
Y(K)
*)
a
elements
:= ^ l i m ( l ( X ) I ( X , [cp]) e I ( M , R ) ) .
t o homotopy,
dotted
i s
system
pairs
i f (X,[cp]) € I ( M , R ) ,
commutes
X
d
up
a r e two
r
(X,[cp]) £I(M,R)
: M
and
with
direct
a definite
by
n
inverse
s p e a k i n g , we
any
a
are isomorphisms
a generalized
Roughly
n
(Y,[iJ>])
exists,
V
( X ) I ( X , [cp]) € I ( M , R ) )
transition
up
(X,[cp]) a n d
V.7.15,
such
X as above,
K
If
a generalized
X*
If
*)
up
t o homotopy.
arrow
We
ignore
a
suitable
which
makes
the problem
We the
define
k (N)
n
k (M)
as
the
square
whether
interpretation
n
k ([f])
of
I(M,R) " a l l " .
i s a
s e t . I t c a n be
settled
by
n
k
i (x)
(M)
n
n
k [f]
n
l [g]
n
k (N)
commutative. (X,[cp])
We
now
The
and
variant
functors
look
choose
n
as
: k
some the
k
n + 1
HP*(K)
k
[ f ] does
way
we
to
Ab.
suspension
obtain
€I(M,R).
arrow
such
Then that
check
M is
:= an
(Sep) •
n
i
of
contra-
+
1
We
define
square
(SX)
Clearly a of
the
element
(M)
of
n
V(X IX€A)
and
X
We
have
j a
x
: X
a
«-> X
commuting
> l (X)
(jj)
TTk (M,) n
A
= 7TTTl (x,) n
A
k .
.
x
and
i
square
x
be
e i ( M ^ R ) .
- M.
: M
which
x
i s
M.
(M^IXGA)
Let
: X(K)
groups i n
(X^fcp^])
cp = Vcp x
abelian
I t i s natural
pair
n
^
of
functors
choose
I(M,R) . L e t
embeddings.
k (M)
(X,[cp]).
f o r the
X € A we
:=
isomorphism
of
axiom
every X
i s an
choice
t h e wedge For
i (x)
cp*
x
A£A
(k |n€Z)
n
(M)
V(M IXeA),
natural
n
family
of
n
n
i n ? * ( K ) .
choice
a (X)
independent
We
the
(SX,[Scp]) € I ( S M , R ) . the
n
commutes.
on
isomorphism
a (M)
k
a
depend
^*k (M)
(X,[cp])
(SM)
not
n
(SM)
dotted
n + 1
In this
from
for a
n
a (M)
homomorphism
(Y,[^l).
a (M)
We
n
l (Y)
Then M
a
family
Let (X,[cp])
denote
the
By
t h e wedge
also
axiom
the left
vertical
We
now
ed
weak p o l y t o p e s
topes
check
over
(M,A). as
(B
K.
of the
n
from
n
For We
(k ln£Z)
natural
to 2.8
point-
can even x
equivalence
the pairs
i t i s now
poly-
(cp,ij;) : ( X ( K ) ,B ( K ) ) ->
{(X,B)
a homotopy
chosen
:X(K)/B(K)
(X,[cp])
easy
be
€I(M,R),
t o deduce t h e
.
a cohomology
the pair
(X,[id
I t depends T
n
equivalence
x
(
R
)
theory
])
k (X(K))
functorially :
( k
T
:
n
)
R
(k*)
^ l
on X n
l
over
K.
i s an element
n
projection
equivalence
a natural
of
(X,B) o f p o i n t e d weak
equivalence
€I(M/A,R)
be a p a i r
of
the natural
by T ( X ) .
Thus
k (M/A)
i s indeed
n
pair
a pair
( X / B ) (K) . U s i n g
n
any X e * * ( R ) denote
cp i n d u c e s
l ( X ) -> l ( X / B )
n
Thus
choose
L e t (M,A)
by Theorem V.7.15.
n
-
i s bijective.
n
k (M)
the exactness
l (B)
n
f o r k .
a homotopy
=
arrow
sequence
n
-
We
(X/B,x)
and
vertical
i s bijective.
axiom
with
X(K)/B(K)
[ip]) 6 I ( A , R )
r
k (A)
We
over
i s possible
Moreover
exactness
a
the exactness
a pointed CW-pair.}
M/A.
the right
arrow
R together
This
n
f o r l
n
( X )
of I(X(K),R).
associated to
i n HP* ( R ) .
of functors.
T h u s we
The T
-» 1 * o f c o h o m o l o g y
n
this obtain
f i t together
theories.
Theorem
i s proved.
call
tension
the cohomology o f 1*
isomorphism More
(2.9)
T
t o K, :
theory
k*
over
K constructed here
and w r i t e
k*
=
. We
(1*)
explicitely,
l£(X(K))
R
-» 1 * a n d t h u s
f o r every
=
1*
n
l (X)
X GP*(R)
.
feel
have
established
justified
and every
t h e base
n
a canonical
to say that
GZ,
ex-
R
( 1 * ) = 1*
Conversely
l e t k*
be
a
cohomology
=
lim
theory
over
K.
Let
M€P*(K),
2.
n €
Then
n
R
((k ) ) (M) K
n
= lim (2.6) * We
have
a
canonical
component
at
any
R
( ( k ) ( X ) I (X,[ip]) (k
group
(X,[cp])
n
(X(K))l(X,[tp])
n
k ([tp]).
6 I(M,R) )
n
isomorphism is
€I(M,R)) .
n
U (M)
n
: k (M)
This
gives
R
((k ) )
us
a
(M)
canonical
whose
natural
equivalence
U
By
use
: k*
of
U
(( k * ) )
we
((k*) )
now
look
for
semialgebraic theories. n
(k |n€Z)
A of
with
suspension
ness
axiom
and
are
to
a
the
the
theories
the
wedge
the
underlying a
definition
of
n
the
n
: k
n
(cf.
P
+
the
the
topological
theory
relates
k*
on
i s
the
^ S
-^k
1 1
which
Chap.
7],
cohomology
a
sequence
homotopy 1.1)
S
cate-
together
f u l f i l l s here
the
the
exact-
means
of
functor).
: £
t Q
p
CW-complex.
leaving
which
(cf. notations
[Sw,
definition space ^
2.8
: H # 7 * -» A b
topological
functor
topological p
c
to
cohomology
transformations,
topological
of
k
suspension
restriction
IR
CW-complexes
axiom
natural
Theorem
cohomology
functors
category
than
to
over
(reduced)
isomorphisms
complicated
necessarily
similar
topological
the
morphisms
more
.
theorem
topological
denote
define
k*
contravariant
pointed
the
=
topological
of
Let
a
K
cohomology
gory
course
.
K
identify
R
(2.10)
We
R
^
easy
defined
theories. as
a b o v e . ) We
-> C o h o ( I R ) . T h i s of
the
functors
of
a
weak
We
w i l l
details
will
K res_ R
polytope
indicate to
the
{The
be
want slightly
above, over
the
reader.
since
IR
i s
steps
of
not
Let
k* b e a t o p o l o g i c a l c o h o m o l o g y
denote
t h eset
by
Theorem V.7.8.
can
be regarded
and
canonical
family.
We
n
(k )
One 1 1
S
a
equivalence
with
F o rany M£P*(IR)
X a pointed
o f pointed
F o rany n G Z t h efamily
as a direct
transition
system
spaces. n
(k
(X
of abelian
isomorphism
between
l e tJ(M)
CW-complex a n d Such
pairs
exist
) I ( X , [ < p ] ) € J (M))
f c
groups
with
an obvious
a n y t w o members
of the
define
S a
(M)
extends
(k )
(X,[tp])
of a l l pairs
cp : M -» X a h o m o t o p y
theory.
this
: H3>* ( I R )
n
:= l i r n ^
(k (X
definition
-
t
o
)
p
I ( X , [cp])
G J(M) ) .
t o a definition
Ab i nt h eo b v i o u s
of a contravariant
way. One t h e n
defines
functor
suspension
isomorphisms
n
o
.
( k
n+1
) S
a.
s
^
(
n
k
)
a
S
i n a n a t u r a l s t r a i g h t f o r w a r d way, exactness
a x i o m and
and v e r i f i e s f o r t h e f u n c t o r s ( k
the wedge axiom. F o r e x a c t n e s s
t i o n s of p a i r s o f p o i n t e d weak p o l y t o p e s ,
one
n
)
s
a
the
needs CW-approxima-
i . e . C W - t r i p l e s w i t h a one
point
s p a c e a s l a s t c o m p o n e n t . T h e y e x i s t b y T h e o r e m V . 7 .1 4 ( o r a l r e a d y V . 7 . 8 ) .
In
this
way a cohomology
we
call
t h esemialgebraic
algebraic
n
(2.11)
via
CW-complex
(k )
theinjection
Every
(M)
then
=
leads
straightforward
we may
n
k (M
Q
p
M
t o a natural
t h e square
t
a t (M,[id ])
way such
(k*)
s
a
restriction
natural transformation
theories
nGZ,
S a
theory
T
over
i se s t a b l i s h e d
o f k*. I f M
which
i sa pointed
semi-
identify
)
GJ(M)
.
: k * -* 1 * b e t w e e n
transformation
that,
TR
for
every
T
S
a
t o p o l o g i c a l cohomology
: (k*)
s
a
-+ ( l * )
M G ? * ( I R ) , (X,[cp])
S
a
i na
GJ(M),and
k
T
T
)
(X,. ) top
1
the
(k
w
main
leave
a semialgebraic
theorem
2.8. A t
the details
cohomology
a t o p o l o g i c a l cohomology
theory
tothe
theory l*.
1* together
o p
s a with
a natural equivalence
(namely k*
such
>-> ( k * )
This
S
a
,
i s done
pairs
(X,[cp])
that
with
Such
pairs exist
n
(l (X)l members
a functor
o
p
)
t o 1* i n a c a n o n i c a l
1*
as f o l l o w s . F o r any M€#9*
t h e homotopy
limit
we o b t a i n
(l*
l*_
and T i s one o f t h e a d j u n c t i o n
[cp]
jective
T from
X a pointed
right
o p
of the inverse
( X , [ c p ] ) € I (M)) o f t h e system
with
We
define
system
emanating
from
t h e s e to f (over
equivalence
t h e group
of abelian
transition
denote
CW-complex
c l a s s o f a t o p o l o g i c a l homotopy by V.7.16.i.
adjoint to
morphisms).
l e t I (M)
semialgebraic
way
l£
Q p
(M)
IR) a n d cp : X
as t h e pro-
groups
isomorphisms
between
Theorem V . 7 . 1 6 . i i .
^M.
any two
If
X
i s a pointed
pair
(X,[id l). i s an
x
2
1 3
< If
The
x
(X,[id ])
>
semialgebraic
(l£
o p
S a
)
n
(M)
at
the coordinate
We
call
p
)
1*. S
a
n
(
x
)
i n
i s natural
x
a
^ ^
t
t
i n X.
) contains h
e
the
"coordinate"
T h u s we
may
identify
•
:
=
li5U(l2
=
lim, ( l
o p
n
(X
n
( i t o p ^ ^ ^ (X,[cp])
Since
we
t o 1 * we
^ t o p )
3
=
*
a topological
1
€J(M))
-~*l (M)
as t h e i s o m o r p h i s m whose
extension
l [cp] .
of the semialgebraic
a canonical natural
justified
*
component
R
i s the isomorphism
have
feel
)l(Xjcp])
t o p
(X) I ( X , [cp]) £ J (M) ) .
the topological
Q
(2.14)
Given
(M)
l£ p
theory
i
~*
I (X^.
then
define
o
T
=
then
i^op^top^
isomorphism which
We
( l *
projection
^ o p ' W
M G P * (IR)
CW-complex
equivalence
to identify,
using
T
cohomology from
T,
*
cohomology theory
k*
we
have,
f o r a n y M €W*
and
n 6 2,
n
S
((k )
a
)
t
Q
p
(M)
=
^Lim
n
((k )
S
a
(X) I ( X , [cp] ) € I (M) )
n
= l i m (k (X. (2.1lT Thus
we
have a n a t u r a l
n
whose f i t
together
Again
(2.15)
we
n
: k (M)
component
(
((k )
to a natural
(
k
*
)
S
a
)
t
Q
p
at the coordinate
identify,
S
a
)
using
t o p
.
isomorphism
n
U (M)
) I ( X , [cp]) € I ( M ) ) P
=
*
'
n
(X,[cp])
transformation U,
k
(M)
i s k [cp] . These U
from
k*
to
isomorphisms k
(( *)
S
a
)
t
these
use
k
a
,
R
=
R
k*,
wonderful
f f i
< k*) R
weak
polytope
-» N
homomorphism
In
group
between
k*>
=
K
K
theory
semialgebraic
k* i n -
cohomology
k*
a
real
cohomology
theories
and
symplectic
K-theory,
and maps b e t w e e n
with
i n this
way
the cohomology
the
theories
,
closed
and
real
simply
equalities
closed by
into
we
by
k
n
field
n
k (M).
polytopes
evaluate
over
a
feel
real
closed
justified
over
theories
any
real
R
R we
then
known
closed
on
to
the
by
i s
denote map
group
f*.
reduced
topologi-
unitary,
pointed
field.
i f M
f o r any
orthogonal,
...)
and we
denote
briefly,
a l l the well
cobordism
theory
Similarly
[ f ] o r , more
( s i n g u l a r cohomology;
them
field
.
any
weak
simply
can
Moreover,
t o p o l o g i c a l cohomology
n
R
numbers.
=
k*
R
cal
topes
subsequent
notations.
over
pointed
now
the
embedding,
( k )(M)
n
we
t o p
of
i s any
( k )[f]
particular
a
correspond
correspondences
a
: M
and
have
to this
I f k*
cohomology
R,
algebraic
We
(
2.17.
the
2.13
,
R
theories
Notations
f
R
) °)
the following simplified
pointed
and
t o p o l o g i c a l cohomology
field
embedding
respect
K
By
* )
s
ffik
then,
S a
(((k*)
i n a
2.8
follows:
topological R
every
closed
:=
R
q
that
theorems
weak
poly-
Having the
settled
( t o t h e same
question which
closed use
field
R we
an obvious
Definition family
now may
"dual"
n
(a ln€Z) n
be b r i e f
of natural
axiom.
and
n € Z t h e sequence
is
-TT
n
(
axiom.
every
n E Z t h e map
homology
n
theory
: HP*(R) n
: k
k
+
We
1.
over
R i s a
-* A b t o g e t h e r w i t h
~*
n
real
theories.
t o t h e one i n §2, D e f i n i t i o n
K
)
pair
k
TT
For every
family
X€A an
k n
+i°
s
such
a
that the
n
(M,A) o f p o i n t e d w e a k
(
M
M
polytopes over
R
)
(M^IXGA)
o f p o i n t e d weak
polytopes and
XGA
isomorphism.
Dually
to Definition
(reduced) homology logical
homology theories
homology
[Sw.
Chap. 7 ] .
With
the only
and
a given
exact.
Wedge
is
over
topology)
hold.
F o re v e r y
k
algebraic
exist
reduced
equivalences a
Exactness
A )
about
of covariant functors k
two axioms
V
theories
( s e m i a l g e b r a i c ) homology
following
every
as i n c l a s s i c a l
cohomology
notation
1. A r e d u c e d
(k ln€Z)
family
reduced
extent
2 i n § 2 we
theories over
over
define natural
R and e s t a b l i s h
R. A n a l o g o u s l y w e
theories
which
live
i n §2 h a v e
obvious
the category
Ho(R) o f
define the category ^ of
o n EKO* i n s t e a d
exception of Proposition
the corollary
t r a n s f o r m a t i o n s between
2.4
o f HP*(R), c f .
a l lthe properties,
"duals" which
topo-
c a n be proved
theorems, i n an
analogous
way.
then
proposition
for
that a
R
3.1.
polytope M
and
every
an
q € Z,
topological
phism,
-
R.
Then,
follows
i s simpler
( c f . [ M i ] , [Sw,
p.
121f]
an
admissible filtration
f o r any
reduced
homology
of a pointed
theory k
over
+
map
k (M) Q
reduced
theories
a n d we
over
reduced
2.
homology any
real
are j u s t i f i e d
p o l y t o p e s and
Definition
for
as
be
n
the natural
can e v a l u a t e any
weak
reads
2.4
isomorphism.
homology
we
counterpart of Proposition
(M ln€IN)
over
n
The
and
Let
l i r n ^ k (M ) n ^ is
homology
proof).
Proposition weak
The
A
closed
t o use
reduced
topological
homology
R
a notation
them
homology
correspond
field
topological
maps b e t w e e n
reduced
theories
theory k
theory
-
analogous
any
+
the
reduced
to
isomor-
u n i q u e l y up
homology
over
with
t h e o r y on
real
over
i s called
t o 2.17.
R
closed
-
and
Thus
pointed field
R.
similarly
a
ordinary,
i f k (S°) =
3.3
(and i s
a
0
q * 0.
Remark known
3.2.
I t w i l l
become
i n the topological
obvious
from
Theorem
t h e o r y , c f . [Sw,
abelian
g r o u p G,
there exists,
reduced
homology
theory k
up
over
Chap.
10])
t o isomorphism,
R with
k (S°) = Q
from
reduced
singular
homology
H*(-,G)
Let
be
reduced
ordinary
homology
theory over
+
Going
a
t o more
c o n c r e t e t e r m s we
torial
"cellular"
M
R,
over
which
description
want
on
to explicate
of the groups k (M)
i s completely analogous
n
that,
for a
a unique
G.
coming
k
below
This
an
and
i s the
:=
almost
of a
to the well
G
given
known
given
ordinary theory
t h e c a t e g o r y HO
R
well
*.
k (S°) . Q
combinaCW-complex
topological
case,
c f . [Sw,
Recall with
Chap. 1 0 ] .
t h a t the degree
[f] = m[id s
also on
n
i f f does n
n
(S ).}
n
deg(f)
1 i n n
f
n
n
(S ), provided
not preserve
I f n
o f a map
= 0 we
base
: S
=
-* S
n>0.
n
{N.B.
This
makes
s i n c e TT^ ( S ) a c t s
+1
i f f =
i d s
interchanges
i s the integer
n
points,
put deg(f)
n
t h e two p o i n t s o f S°, and d e g ( f )
=
o
0
sense
t r i v i a l l y
, deg(f)
i n the
m
=
-1
i f f
remaining
cases.
For n
every
cell
:= d i m a
a o f M we
( c f . V, map
obtained
dimension
n-1
we
.n-1
characteristic S
§ 7 ) . L e t 0
the
set ^
not
admitted
n
i f T
with
cell
T of
map
commutes.
T
2
a n d ip^ i s a n
iso-
put
(
M
) of cells
C (M) n
given
the formula
complex
of I (M),
(M,{x }). Q
(n >0)
immediate
C.(M)
i s the free
of dimension
as an element
CW-complex
i s n o t an
chain
then
relative by
n
n
= deg(cp^) .
that
=0.
1
-* M
the associated
are canonical projections,
b y ty^. We
define the cellular
C (M)
diagram
E - /S -
n
preserving
..n-1 /..n-1 M /M
.n-1
: E
a
For every
defined base-point
.n-1
ip
^ denote
restriction.
that the following
.n-1
n
map
Q
The
n.
as
of
follows.
abelian
Here
3
I f n N t o a
C. ( f )
obvious
homology
f
that
theory
: C. (M) -> C. (N) 93
indeed
and G
:= k
Q
= 0 .
(S )
isomorphisms
: H ( C . (M) ® G ) - k ( M )
M
n
that,
Z
f o revery
H (C.
cellular
map
f
k
(M) ® G )
n
H (C.
n
n
: M -» N , t h e s q u a r e
(M)
(f) ®G)
n
H (C.
(N)
n
k (N)
®G)
n
commutes.
This
c a n be d e r i v e d
theory, is
c f . [Sw, Chap.
given.
The s t a r t i n g
wedge o f c o p i e s implies
k
n
In
gives
39 = 0
n _ 1
)
|K|
the
abstract
R
canonical with
10] where point
the explicit
n
S ,
one copy
that
n
1
n
that
f o reach n
n
description of
(M /M ~ )
M /M a £I
= C
n
f o rt h e boundary
n
(M). This
(M) ® maps
^ i s the
z
G.
The
case
i n C.(M)
[loc.cit.].
with
that
M
i s a geometric
K the abstraction simplicial
complex
characteristic
the chain
also
also
as i n t h e t o p o l o g i c a l
i s the observation
proof
complex
of M
simplicial
K i n o n e o f many ways
C.(K) o f t h e a b s t r a c t
p. 314 b e l o w .
complex,
( I I I , § 4 ) , we c a n o r d e r
maps a t o u r d i s p o s a l .
*) cf.
t h e axioms
= 0 f o rq * n and k
an easy
t h e s p e c i a l case
M =
from
o f t h e sphere
(M /M
q
G = Z also indeed
directly
Now
[ E S , p . 67]'
and then
C.(M)
pointed
hence
and
have
coincides ordered
simplicial
complex
description
Corollary cial
3.4.
complex
= V'
that,
and
we
obtain
the
following
truly
combinatorial
k (M). n
For K
K
°K such
of
K,
every
there
G)
^
f o r any
n € Z
exists
k
n
simplicial
f
: K
the
following
square
commutes:
VK-G)
^
»
k
n
(
l
K
l
R>
K H (f)
( l f l
n
L
V'
If
the
reduced
description although theory
Theorem
5T L
G)
of
this
[Sw,
3.5
homological
homology the can
Chap.
>
n
(
I
theory
groups be
k
L
k*
k (M)
one
R
)
i s not
for M
+
enormously
15]
I
a
ordinary pointed
complicated.
In
to
cohomology
analogous
way.
again
CW-complex i n the
sequence
(E
r
Whitehead). r
,d )
with
P191
converging
As
then
a
cellular
i s possible,
topological
proves
( A t i y a h - H i r z e b r u c h , G.W. spectral
) *
R
E
2
There =
H
P19I
exists
a
natural
( C . ( M ) «>k P
(S°)) 9i
k (M). +
duals
of
these
theorems
3.3
and
3.5
can
be
proved
in
an
§4
- Homology
We
need
some
formalities
without
base
points.
Notations direct weak
We
o f weak
I f M
{+} w i t h
spaces
have
the relations
polytope
a one p o i n t space
since
p o i n t +. this
{We
would
[M,N°]
hand
side
the left
p o i n t p r e s e r v i n g homotopy
If
weak
M
+
(M^|X€A)
(4.4)
A X
from pairs
v
reduced
=
+
Mt
t o N° w h i l e
as a pointed
t o use t h i s
with
notation for
Def. 3 i n IV,
N° denotes
the
t h e space
N
§6.} forgetting
=
X )
theory.
classes
(free)
hand
side
denotes
o f maps
from
M
+
t o N.
homotopy the set I f X i s
+
polytopes
then
.
A
section
and t h e next
o r cohomology
HP*(R)
over
the right
the set of
.
M,)
X€A
homology
of spaces
+
o f weak
( U
i n this
denotes
then
(M x
A
the category
extended
M
i s a family
objective
trary
from
polytope
A6A
Our
denotes
[M ,N]
of
(4.3)
+
M
w i t h anc
+
=
o f maps
second
conflict
R then
R then
{+}, r e g a r d e d
hesitate
over
classes
a
over
spaces
bijection
of course,
base
between
point.
a natural
(4.2)
where,
about
i s a p o i n t e d space
base
polytopes
i s a weak
polytope w i t h base
I f N
the
4.1. a)
sum MU
arbitrary b)
of pairs
theory
t o t h e homotopy
R and t o understand
one
i s t o extend
over
category
R
an
arbi-
i n the correct HWSA(2,R)
way
ofa l l
the formal properties
of the
In
this
s e c t i o n we
polytopes). HP(2,R) the
both
pair
If
F
i s
a
map
we
i n
the
a
attention
?(2,R)
natural
or
of
at
these
WP-pairs
pairs
endomorphism
instead of
WP-pairs
the
covariant
cohomology
most
case w i l l
sections
of
E
and
(=
pairs
the
which
we
have
than
on
homology.
1.
homology
cf.
often
and be
functor
F(M,0).
HP(2,R),
then
conventions
last
contravariant
F(M)
subcategory
Analogous
In
have
write
f u l l
f*
category
covariant
between
by
focus
of
weak
homotopy
maps
a
pair
category
(M,A)
to
(A,0).
a
usually as
The
w i l l
by
f*
been For
I f
the
for
we
(M,A)
to
we
then HP(R)
-»
i s
(N,B)
homomorphism
space
time
now
Ab
regard
contravariant
the
justice
:
the
other
of
that
f
denote
in
most
HP(2,R)
{Recall
1.1.} we
obeyed
from
a
F[f]
case.
categories.
more
give
explicit
preference
on to
homology.
Definition braic
A
homology
functors
h^
theory)
: HP(2,R)
transformations
Exactness.
is
exact.
from the
=
A )
to
WP-pair
AUB
i
N
every
i
and
n
j
i s
-*• h ^ ^ E
n
(
R
M
a
together
WP-pair
h
(more p r e c i s e l y , an
over
+
Ab
: h
(M,A).
)
IT
denote
family
with
which
(M,A)
h
n
(
the
the
M
'
A
)
a
(h ln€Z)
satisfy
sequence
are
closed
semialge-
covariant
(3 ln€3) n
the
of
following
natural axioms.
sequence
3 (M,A)'
h
n
i n c l u s i o n s from
The
of
n
family
long
unreduced
i s called
the
n - 1
(
A
)
(A,0)
homology
to
(M,0)
and
sequence
of
(M,A).
If A then
+
3
"17*
Here
(M,0)
Excision. M
For
V
h
theory
and the
B
inclusion i
: h ( A A fl B ) n
f
:
-^h (M,B) n
subspaces (A,A
D B)
of ->
a
(M,B)
weak
polytope
induces
an
M
with
isomorphism
for
every
n G Z.
Additivity. weak
I f M
polytopes
for
every
These
A
additive)
draw
from
homology
For
from
pair a l l
7 ] . We
and mostly
M
call
family
induce
+
from
be a homology
a n d we
h (M) ^ h n
n
(M^|A€A)
of
an
isomorphism
of a
(strongly
i s contractible that
with A
h (M,{x}) n
theory
©h (M,{x}) n
{x}
M
of
topological
a
topological
homology
very
well
c f . [ E S , Chap.
theory.
known
1 ] , [Sw,
R.
R there
exact
i s a unique
sequence
of the
retracpair
isomorphism
.
i s a homotopy
(More
a strong deformation
by arguments
over
over
here
topological
here,
a canonical
- 0.
a homology
the axioms
t h e homology
then
of pairs
an unreduced
polytope M
have
( * )
o f the axioms
such
not repeated
t o { x } . Thus
splits
conclude
-> M
o n t h e c a t e g o r y W(2)
consequences
7]. Let h
(4.5)
M
^ :
analogues
t h e o r y , more p r e c i s e l y
some
(M,{x})
If
theory
any p o i n t x o f a weak
tion
i
of a
^ h ( M ) n
[Sw, Chap.
topology
Chap.
h (M, ) n X
A
(IV,1.10)
the inclusions
are the precise
CW-complexes
We
e
sum
n € Z.
axioms
homology
then
: ©
(JL*) X
i s the direct
generally,
retract
of M
equivalence and
whenever then
(M,A)
h^(M,A)
we
i s a
= 0 f o r
n.)
Proposition decreasing denote
4.6
(Homology
sequence
W P - t r i p l e , and l e ti
the associated inclusion
inclusion
(A,0)
-+ ( A , C ) .
:
of a triple).
( A , C ) -» (M,C) , j
maps
Finally,
L e t (M,A,C)
of WP-pairs.
f o r any n GZ,
we
:
(M,C)
be
a
(M,A)
L e t i ^denote
the
introduce the
the
composite homomorphism
3 (M,A,C)
: h (M,A) - J ^ x y -
n
The
long
3 is
n + 1
(M,A,c)'
4.7
h
(
n
A
C
'
)
^ 7
h
n
(
M
(Mayer-Vietoris
o f a weak
A (M,A,B,C)
M
V
polytope
denote
n
'
C
)
h
— 7 7
n
(
C
'
)
sequence).
M with
M
A
'
)
h
3 (M,A,c) ' n - 1
(
A
'
n
C
)
"
with
h
— T T
inclusion
M
B
n( ' )*-T7T
where
a(x) =
( i ^ ( x ) , i +
^3'
^-4*
n
gory
study
by
L e t A^
=
A
3 (A,An ,c)
2
( x ) )
+
h
'
B
n-1
M triangle of
( c f . 1 . 6 . i i ) . We h a v e a c o m m u t i n g
WP-pairs c
(M,A)
>
(MUCA,CA)
(M/A,A/A)
with
TT a h o m o t o p y
spaces
w i t h o u t base n
axiom
n
Given
C i ]
a
h (M)
with
CN
{Again
the
of
n
+
the
i l l e g a l l y ,
homology
sequence
the
maps
=
3
:=
x
MUCA,
upper
n n
t p l
we
n € Z as
the
i s an
obtain
and
CN/N
as
(CN,N,{x})
3 (CN,N,{x})
are
n
By
the
that
point,
projection
spaces
A/A
q.e.d. ]
SM =
: CN
without base =
n
( c f . Prop.
4.6)
isomorphisms.
We
a^(M)
i s the j
-* C N / N . point
0.
;
(CN/N,*
which
p
h (CN,{x})
mean excision)
isomorphism
Recall
base
i s contractible
triple
M/A,
isomorphism,
an
follows.
natural
CA,
i n d e x °.}
i t s natural
under
S i n c e CN
the
also
(N,x)
and
CN
of n
N
r e g a r d CN
i n d e x °.}
boundary
of
x €N we
Thus
f o r every
cone
here
omitted the
polytope M
(SM)
1
point
the
We
isomorphism.
reduced
omitting
as
point.
weak
to h
n
image
i s an
pointed
from
the
equivalence, {illegally
j \
The
shows define
that a
n
(M)
composite
h
n
(
N
,
{
x
}
_ ^
)
. h
(CN,N)
n + 1
h
n + 1
(CN/N.{x})
.
n+1 The is
second an
arrow
i s an
isomorphism.
Proposition equivalences
4.9. a
I t i s natural
(H |n€Z) n
: n
isomorphism
n
n n
n
+ 1 °
by
i n
Proposition
i
s
a
r
e
d
u
above.
Thus
a
(M)
M.
together with s
4.8
c
e
d
the
family
homology
(o ln£Z) n
theory over
of R.
natural
j
Proofj_
We o b t a i n
WP-?air
t h eexactness
(M,A) f r o m
(M°,A°,{x}), w i t h the
wedge
axiom
We c o n s i d e r
axiom
( § 3 , D e f . 1) f o r a g i v e n
4.8 a n d t h e h o m o l o g y x t h ebase
f o r a given
t h ed i r e c t
point
sequence
4.6 o f t h e t r i p l e
o f M a n d A. I t r e m a i n s
family
pointed
(M^|X€A) o f p o i n t e d
t o verify
weak
polytopes.
sum N
:= U M ° a n d i t s c l o s e d s u b s p a c e B := U { x , } X X o f M^. We h a v e N / B = V M^. We d e n o t e t h i s p o i n t X X
with
x^ t h ebase
ed
weak
of
t h epair
(4.8)
point
polytope
b y M.
B i sa retract
( N , B ) ( c f . D e f . 1) s p l i t s
and t h ea d d i t i v i t y 0 - © h ({x >) X n
The
Since
x
cokernel
axiom
-> © h ( M ) X
A
n
of thef i r s t
T h u s we o b t a i n
f o r h
A
o f N t h ehomology
into
short
we o b t a i n
+
-> h ( M ) - 0
sequences.
short
exact
Using
sequences
.
n
map i n t h e s e q u e n c e
an isomorphism
exact
sequence
i s© h X
(M, )
(cf.
4.5).
from
© R (M,) t o (M).I t i s e a s i l y X i s i n d u c e d b y t h e i n c l u s i o n s i ^ : M^ «-+ M. n
checked
that
this
isomorphism
q.e.d.
We n o w s t a r t tend" weak
with
i tt o an unreduced polytopes
over
k (M,A)
k
n
thus
: HP*(R)
:= k ( M , 0 )
n
of
pointed
n
pair
weak
extensions that
+
n
M
+
coincides with
(M
+
+
r
A )
R and want t o
k . F o rany pair
(M,A)
+
n
: HP(2,R)
"ex-
of
We
Ab o f t h e f u n c t o r s
polytope
M,
.
polytopes
over
identify
t o M/A.
+
M /A
R. T h e n +
+
The homomorphism k t p ] n
n
Applying
a long
+
(M ,A )
exact
i sa
pair
= M/A. L e t p d e n o t e t h e
t h ehomomorphism k t j ]
(M,A).
we o b t a i n
k
f o r a n y weak
= k (M )
polytopes.
m a p j : (M,0)
theory
k* o v e r
,
o f weak
p r o j e c t i o n from
k (M/A)
clusion the
n
(M,A) b e a p a i r
natural to
obtain
theory
define
Ab. N o t i c e
k (M)
Let
R we
n
indeed
homology
homology
:= k ( M / A )
n
and
a reduced
t h edual sequence
induced
from
+
k (M ) n
by t h e i n -
o f P r o p o s i t i o n 2.1 t o
k
with
(A)
homomorphisms
+
+
(M ,A )
> k
:
n
9 n+1
3
n
i
*
=
3
hence
on
Proposition
4.10.
(k ln€2)
Proof. dent
As
The
by
ral
over
The
7 ] ) , and R with
Similarly Unred
h
T
These
between
Ho(R)
and
reduced
Since safely
have
leave
Analogous gories $
+
with
«E
1
on
this
i s a homology -*
just
been
the
pair
+
k
+
i-> k
between
of
n
theory
over
natu-
R.
J
follows from
i s an
obvious
homology
the category
naturally
as
(4.4).
notion
theories
Ho(R)
evi-
q.e.d.
of a
natu-
( c f . [Sw,
o f homology
theories
morphisms.
to a
extends
+
(3 ln€Z)
established. Excision i s
. Additivity
-* g
family
3
n-1
extends
Ho(R), and
t h e homology
homology
we
depend
(A)
1
n
functor
naturally
Red
to a
functors are quasi-inverse natural
u /r>\ ~~* U n r e d • R e d Ho(R)
morphism,
> k ~
5
n
:Ho(R)
->
Ho(R)
functor
-> H o ( R ) .
4.11.
d
naturally
theories there
: h*
«-». h
+
— 3
natural transformations
Theorem
i
of k
the assignment
: Ho(R)
-» k n
can introduce
these
assignment
: k
homology
we
which
together
axiom has
the definition
transformation
Chap.
3_ n
exactness
f o r reduced
(M,A)
n
*
(M,A).
n
transformations
• k j
(M,A)
and
ral
(M)
n
i . e . there i d ~ Ho(R)
very
the proof
natural equivalences and £
of unreduced
and
R
equivalences
In oarticular, -
correspond
uniquely
up c
with
toi s o the
R.
explicit of this
natural
Red*Unred.
theories over
t h e o r i e s over
been
exist
equivalences
on
similar
matters
theorem to the
h
+
»-* h * ,
reduced
k
+
i n §2
we
now
can
reader.
k
+
exist
topological
between
homology
the
cate-
theories,
as
i s well
known and p r o v e d
Henceforth
we
usually
semialgebraic
If
K => R
w i l l
i n t h e same
identify
and t h e t o p o l o g i c a l
i s a real
closed
base
f o r every
homology
over
R,
the restriction
Conversely, (g*)
over
K
In
this
theories
For
K, way
every
theory h
homology
called
ft
+
= h^, both
+
extension then
over
+
of h
theory
K we
+
t o R,
+
over
g
have such
theories
a homology
such
that
K u n i q u e l y up t o n a t u r a l
equivalence.
R we
have
formulas,
§2 h
+
*~ R (h*) .
a homology
R correspond
(M,A) o v e r
from
theory
R (h^)"=
that
over
any WP-pair
i n the
i ti s clear
R yields
t h e e x t e n s i o n o f g ^ t o K,
t h e homology
over
= k ,
+
setting.
field
that
called
k
way.
theory
((g*) )~
= (g*)
K
w i t h t h e homology
i n analogy
to
(2.6) and
(2.9) ,
(4.11)
h*(M,A)
=
h (M(K),A(K)) n
(g ) (M(K),A(K)) n
and
similar
Every
every
with
theory
homology
h^
a
homology
g (M,A) n
f o r t h e b o u n d a r y maps
over
theory
IR
theory
( g * )
t
o
g
a
h* (M,A)
t
to a semialgebraic ( h ) +
S
a
.
Conversely,
"extends" o
p
)~
=
to a
^*^top*
IR c o r r e s p o n d
topoloI
n
3R ,
•
t
h
i
s
uniquely
equivalence. For every
)
( M
.
n
(h^ )"
that
(M/A) o f s e m i a l g e b r a i c C W - c o m p l e x e s
(4.12)
a
that
such
p
8
"restricts"
+
theory
t h e s e m i a l g e b r a i c homology the topological
h
such
s e m i a l g e b r a i c homology
gical way
formulas
topological
homology
=
K
pair
Consequently, over
each
given
real
a
t o p o l o g i c a l homology
closed
field
R
a
homology
theory
h , +
there
exists
theory
R
R
which
The
h„
, s a . Ov ((h„ ) )
=
corresponds
pendant
+
cients
i n
R
cellular
by
a
some
topological way
We
define
Definition is
a
with
that
every
N
h^we
evaluated
on
Theorem
3.3.
to
can
cohomology
of
be
describe
R
a
with
coeffi-
CW-pair For
h (M,A)
an in
n
(M,A)
over
extraordinary a
cellular
theories.
unreduced)
contravariant n
(6 |nGZ)
pair
of
(M,A)
N
of
h (A)
n
polytopes
> h
h*
: H?(2,R)
(notations
—
1
h
theory
transformations
hold
weak
n
- r * - h (M)
functors
natural
axioms
cohomology
as
the
n
+
1
6
N
-> A b n
: h »E
i n Def.
long
(M,A)
over
R
to-»
h
1).
sequence
-
N
6 (M,A)
exact.
Excision. UB
=
M
i *
is
can
H*(-,G)
3.5.
3
A
G
following three
For
+
s i n g u l a r homology
(semialgebraic
family
h .
analogous
theory
unreduced
-> h ( M , A )
is
of
group
Theorem
(h |n€Z)
the
Exactness.
to
A
a
R
procedure
n
family
gether such
2.
over
abelian
homology
analogous
now
canonically to
H (-,G)
R
R
an
If A then
and for
B
are
every
N
: h (M,B)
closed n € Z,
subspaces
the
of
natural
a
weak
polytope
M
with
map
n
-* h ( A , A D B)
isomorphism.
Additivity.
If M
topes
for
then,
i s the every
direct
n 6 Z,
the
sum
of
a
natural
family map
(M^IXGA)
of
weak
poly-
n
+
1
(i*)
is
an
A l l word
n
: h (M)
n
- T T
h (M,)
isomorphism
our
considerations
f o r cohomology
analogous nation.
on
homology
theories.
t o t h e ones
They
above, which
theories lead we
c a n be
repeated word
to notations
w i l l
and
by
results
use w i t h o u t f u r t h e r
expL
§5
- Homology
We
want
t o extend
H?(2,R) R.
similar
functor
F
tp :
triple
a
unique
+
We
-» A b
define
t o HWSA(2,R)
over
As elsewhere
: F(X,B)
problem
R l e t I(M,A)
(X,B,[cp]), x
:
R
from
of a l lpairs
R and
ignore
the category o f spaces
( c f . V,
§4)
over
i n a way
t o extend
i n a canonical
a single
the set of triples
[cp] t h e h o m o t o p y
class
thus
a
of a
[cp] , a l t h o u g h t h i s
there
i p » x -Wr
have
a n
( X ' , B ' )
F [ f ] as t h e dotted
F[g]
F(X,B)
!
(X ,B )
arrow
which
such
makes
there that
the
square
F ( X * ,B' )
F[f] F (M' ,A')
F (M, A )
commutative. of
tp a n d tp' . I n t h i s
£
It
I t i s easily
homotopy
from
the last
equivalence f
a WP-pair
(M,A,[id,
( i d
These
we
that
£[f]
obtain
does
n o t depend
a covariant
on t h e c h o i c e
functor
: H W S A ( 2 , R ) -* A b .
i s clear
Given
way
seen
(M,A)
)
1
(M,A) *
F
(M,A)
over
. v ] )i n I ( M , A ) .
isomorphisms
functor
:
sentence
F
M
< '
A
)
i n V.6.15 f
1
-* ( M , A )
into
R there exists T h u s we
~* ^(M,A)
have
an
F
turns
every
weak
isomorphism F [ f ] .
a distinguished
a canonical
element
isomorphism
.
f i tt o g e t h e r t o a n a t u r a l
to the restriction
that
FlHP(2,R)
o f F.
equivalence from the We
feel
justified
to
identify
FlHP(2,R)
and
thus
over
If
we
=
write
R a n d a n y map
tp :
(X,B)
F,
F(M,A)
= F(M,A)
f between
(M,A)
t h e homomorphism tp
free
t o a b b r e v i a t e F [ f ] by
+
from f
+
(M,A)
pairs.
i s a WP-approximation
then
R.
such
and F [ f ] = F [ f ] f o r any WP-pair
above
of a pair
coincides
f o r a n y map
with
between
o f spaces
F [ t p ] . we pairs
now
over
R
feel
of spaces
over
We
f
call
tion.
It
the
i s easily
from
F
to
natural
a
extension
to
a
of
seen
second
the
that
functor
functor
every
G
: F
to
natural
: H ( 2 , R) i
transformation
F
HWSA(2,R) by transformation
-» A b
-* G,
WP-approxima-
extends
called
the
in a
T
: F
-»
unique
extension
of
G
way T
by
WP-approximation.
By
an
analogous
we
can
extend
category F
of
any
-> A b
of
generally
values
turns
F
: Hfc2(2)
-> A b
on
the
homotopy
[W,
p.
224]
homotopy
canonically
t o p o l o g i c a l weak
i n the and,
a l l t h i s works f o r category
Ab
is
that
to
produce
direct
If
h
homology
h
on n
functor
CW-approximation
t o p o l o g i c a l CW-complexes
which
of
(n
classical
to
a
functor
equivalences
isomorphisms.
More
n
using
covariant
pairs
: HT0P(2)
into
procedure,
of
for diagrams
is a
+
course,
HWSA(2,R).
|n€Z)
of
of
also
or
want
functors.
of
e x c i s i o n theorem
to
ignore
Gr
holds
question
or
R
Set
then
category
or
is a
H«0(2)
Set
functors.
there
we
understand Here
the
for the
for
the
HP(2,R)
of The
with
sets whole
instead point
distinguished
way
limits.
over
to
or
on
contravariant
inverse
theory
We
groups
for
i n Ab
and
these
this
Gr
functors
obtain
the
n
some t i m e
I t turns first
every
properties
d i f f i c u l t
n -
for
of
question out
to
dealing
h
a
n
the
family
i s what
be
functor
sort
convenient
with
easier
matters.
Definition Important of
cases
subspace.}
spaces {We a)
1 . L e t c9t b e
use A
i n OL, the
are
Let and
prehomology
space
(X = P{R)
H0t(2) l e t E
letter
a
E
category.
{We
are
, W S A ( R ) , HO,
TOP
with
denote denote
the the
uniformly
theory
h
+
on
homotopy
«
i s a
b i t vague the
category
endomorphism
for
a
(M,A)
usual of
here. notion
pairs (A,0)
of of
HCX(2).
a l l # . } sequence
(h
|n€2)
of
covariant
functors
h
n
: H0l(2) 3
transformations the
n
"long homology
e x a c t . Here
from b)
(M,0)
h
n
(T l n € Z ) n
(M)
n
T
a
natural
Let 3
: h
n-1
and
)
(
A
)
"
M
(A,0) t o (M,0) a n d
prehomology
transformations T
the following
square
n
: h
theories n
-> h
1
n
on
such
commutes.
h (M,A)
H
T _ (A,0) n
i ti spretty
sequence
1
n
we w r i t e
g
(
A
( A )
) instead
( o risomorphism)
( i . e .every
obvious
from
T (M,A)
h
i s an
n
theory on ?(R). Every to a natural that
5.1. The sequence
of natural
A-1
+
o f h ^ ( A , 0 ) . } We c a l l t o h^ i f every
T
R
T
i sa
isomorphism).
natural transformation
t r a n s f o r m a t i o n 3„ : & -> ( h ^ E ) * n n n - i '
(h °E) n-1
V
= n" - * E . n-1
of functors
(h lnGZ) n
transformations (3 ln€Z) n
together with the
i s a prehomology
theory
o n WSA(R).
Proof. cp : (X,B) h
from
5 ( ,A) ' n - 1
)
3 • (M, A ) n
equivalence
n
A
n
h* be a prehomology
n
'
(M, A )
n
i nprevious sections
natural
M
: h * -> h ^ b e t w e e n
n
{As
(
n
the inclusions
9 (M,A)
h
( M , A ) € (X,
h
h
i r
of natural
( M , A ) € (X{2)
(M, A )
R
pair
sequence"
transformation T
f o r every
h
f o r every
of natural
(M,A).
OC i s a s e q u e n c e that
(3^1n£Z)
a sequence that,
n
i and j denote
t o
A natural
: h ^ -> h _^E s u c h
h (A)
3 ^ (M,A) n+1 is
-* A b t o g e t h e r w i t h
Given
a pair
(M,A) o f s p a c e s
( X , B ) -> ( M , A ) . We t h e n with
t h e h -homology +
over
c a n compare sequence
R we c h o o s e
a
t h e h^-homology
WP-approximation sequence
of
o f (M,A) u s i n g t h e i s o m o r p h i s m s
X * i n d u c e d b y cp a n d i t s r e s t r i c t i o n s
: (X,0) -> (M,0) ,
X
(B,0)
:
logy
-> ( A , 0 )
sequence
sequence
of
of
which
are again
(X B) i s exact
WP-approximations.
t h e same h o l d s
f
Since
t h e h -hcmo+
f o r the h -homology +
(M,A).
q.e.d.
Proposition
5.2. F o r e v e r y
prehomology
t h e o r i e s on ?(R) t h e sequence
(T lnGZ)
formation
: n ^ -* g
t h e o r i e s on WSA(R).
T
an
isomorphism
We
leave
tension
the easy o f h*
Similarly, o n HO
isomorphic
homotopy
phic
( h*)
R
v
R
to simplify
Notations fixed
i) n
ii)
g
over
+
T
t o g * . We
know
topological
I f (M,A) i s a p a i r
I f T i s
(resp.
+
T )
the ex-
t
every
prehomology
theory
and every
natural
h * o n TOP
such
isomorphism)
want
f
t o understand
§4
that
there
formal
we
have i n
t o study
prehomology
exists
such
the prehomology
a
that
theory
the theory
theories
the
+
a t g* o r a t a
+
: fi^ -> g ^ .
g . For a l l questions
look
h
g
( h ) R
topologig* i s i s o -
+
v
i s
isomor-
. This
+
allows
drastically.
homology
instead of H
h
up t o i s o m o r p h i s m ,
i tsuffices
i s a pair
call
(resp.
from
B y P r o p o s i t i o n 5.2
notations
i s a natural trans-
: h * -» g * b e t w e e n
theory
h*, unique
. Thus
between
+
WP-approximations.
R we
w h e t h e r we
5.3. F o r t h e r e s t
I f (M,A)
3 (M,A)
by
theory
isomorphism)
theory
theory
t o h*»
a
(resp.
We
CW-approximation
not matter
morphic
us
t o HWSA(R)
+
o f t h e prehomology
i tdoes
to
to the reader.
T )
: h * -> g
isomorphism.
t o a prehomology
any homology
theory cal
proof
T
n
prehomology
to a natural transformation
properties mind
f i s an
by c l a s s i c a l
transformation
Given
between
+
(resp.
extends
extends
then
natural transformation
of this
section,
theory
of topological (M,A) a n d
(living spaces
a n d i n §6 a s w e l l , on
h
then
we
w r i t e h (M,A) and n
n
over
R then
we
i s
HM0(2)).
§ (M,A).
o f spaces
+
write h
(M,A) a n d
3
n
(M,A) i n s t e a d o f 3 (M,A)
stead
I f M
space
=
we
I f M
:= h ( N , { x } )
n
Similarly,
(M) - ^ h
often
a
real
of
(SM)
phism
V
( J J K n
(M,A) , a n d r a t h e r 3
but abusively, space
over
i fM
often i n -
3.
o r even
n
R or a pointed
i s a pointed
f
[Sw, Chap.
field
: h
a
R then
of the reduced
topological
main
a,
closed
field
n € 2 there exists that
n
R
h
+
by
denote i t s isomorphism cases
we
o f o (M).
f o rhomology).
: L
n
we
(M) . I n b o t h
(M,A) • > h ( M ( K ) , A ( K ) ) s u c h 3
space,
instead
theorem
the suspension
theory
2 ] b y SM a n d t h e n a t u r a l
extension of a real
h (M,A)
denote
homology
7]) again by
R and every
we
topological
, o r even
and second
(M,A) o v e r
K (M A)
(SM)
briefly
(First
closed
n
n + 1
polytope over
( c f . [Sw, Chap.
more
5.4
spaces
weak
suspension
write
Theorem
and
.
n
h ( M ) -> h
topological R
briefly,
i s a pointed
isomorphism
h
(M, A )
define
n
n
v
0
(N,x) i s a p o i n t e d
h (M)
a (M).
( h ) K n
more
R
i i i )
iv)
of
R.
i ) Let K
For every
a natural
be pair
isomor-
a l l the squares
(M, A ) '
R
h
n-1
< (M,A)
(
A
V - 1
n
h ( M ( K ) ,A(K)) n
h
3 ( M ( K ) ,A(K))
n
-
1
)
A
< '0)
(A(K))
n
commute. ii)
For every
exists
a natural
A (M,A) n
such
pair
that
of weakly
semialgebraic
isomorphism
: h (M,A) n
a l lthe squares
h
n
(M
t < } p
, A
f c o p
)
spaces
(M,A) o v e r
3R
there
3
(M,A)
h (M,A)
> h
n
n
_
(A)
1
= j A __ ( A , 0 )
A (M,A) n '
n
1
I V ^ o p ' W
V M
t
o
, A
p
t
Q
)
p
h
'
(
n-1
A
t0P
}
commute.
Indication K (M,A),
\ (M,A)
n
tified these pair
n
and
We
these
now d e f i n e
(M,A) o v e r
established
R
(resp.
M
(
n
(M,A),
t
o
A
*
p
t
o
isomorphisms
and then
have
) respectively
p
isomorphisms
IR) l e a v i n g
such
f o r an
iden-
by
arbitrary
the verification
ofthe
commutativities t o the reader.
WP-approximation
regarded
o f spaces
o f (M,A) t h e n
Thus t h e d i r e c t
of
n
h (M(K),A(K))
L e t (M,A) b e a p a i r
nical
already
o f weak p o l y t o p e s
n
isomorphisms. o f spaces
I n §4 w e h a v e
f o rpairs
n
h (M,A) w i t h
various
i)
of proof.
system
the first
I f tp : ( X , B ) -> ( M , A ) i s a
cp i s a W P - a p p r o x i m a t i o n
o f (h (Y,C)I (Y,C, (M, A)
o f (M(K),A(K)).
K
from
n
define system
R.
( h ( X , B ) l (X,B,[cp]) 6 I ( M , A ) )
as a subsystem
w a y . We
over
limit
c a n be
) G I (M ( K ) ,A (K) ) i n a
as t h e natural
t o thedirect
above
map
from
cano-
the direct
o f t h e second
limit
one. I t i san
isomorphism. ii)
L e t (M, A )
I'(M,A)
denote
(X,B,[tp]) Further
J
map a
system
n
M t
o
p
A
'
t
A
(M,A)
o
p
) denote a
t o
spaces
0
of (h
n
c a
l
1
(Y,C) I ( Y , C ,
o f t h e second
the direct system,
a r e isomorphisms.
We
(Y,C,[^1)
M t
o
limit
which
i sh
define
p
A
'
t
p
0
(
M t
o
p
i
have
an
n
)
}
*
W
e
d
A
t
Q
e
f
i
n
3 (M,A)
e
p ^ *
system B
o
t
n
evi-
The
canonically n
of the first n
with
(M, A ) ) - » h ( M , A ) .
(M,A) c a n b e r e g a r d e d ) €J
^ t o p ' t o p ^
dent
n
with
l e t (
o f weakly
the subset
M
\\) :
the
be a p a i r
as a as
t o the direct
a (M,A) and n
X (M,A)
=
n
(End
We h
n
n
f
.
n
of our explanations
now
1
3 (M A)«a (M,A)
are justified
concerning
to identify
most
often
h (M A) n
and,
we
n
of course,
3 (M,A)
Example
3 (M(K),A(K)), n
5.5. L e t h
some
abelian
over
R t h e groups
morphisms, [DK D
This
group
with
i s evident
since
there
exist
K c= M
and L c A of M
But than
groups
anew
i s absent g
H (M,G)
3
t o prove
obtain
this
the result
i n the special
X
to M
that
and t h e five-lemma. ->
. Let X
i s a WP-approximation
f o r the family
see that
cp^ : X ^
case
a l l A^ a r e
i n g e n e r a l by use o f t h e long
(M,A) a n d t h e (M^,A^)
from
holds
[cp] we n
W
i s i ttrue
topological
(M^,A^)
coefficients
of the constant sheaf
a sheaf
5.7. L e t ((M^,A^)IAGA)
and A denote
inclusion
M
about
i s a f o r m a l consequence
sequence
H (M,G )
generality
(M, G ^ ) ? How
Proposition
This
with
g
groups
In which
H (M,G) =
with
cohomology
G.
O p e n Q u e s t i o n B. that
singular
:=
o f M.
F o r any X G A
U(X^IAGA). We
homology
The
know t h a t
(X, | A G A ) . U s i n g t h e i s o m o r p h i s m s A t h e c l a i m h o l d s f o r (M | AG A) . A
we
map
the [cp
h n
] and A
q.e.d.
§6
- Excision
The
conventions
cular,
h
1. A
i *
map
i
of
A fl B
as
i n 4.7,
6.1.
then
I f (M,A,B)
a
of pairs,
to the crucial
proposition
a triad
an
i fM
t h e o r y h^)
be
= A U B.
i ft h e
isomorphism
triad
and C
i s a
subspace
(M,A,B,C) , d e f i n e d
i n contrast
i s excisive.
Proof.
We
the field
Every
from
triad
§4
that
R i s sequential
complete
core
( c f . V.4.7). P(A) flP(B)
p
105f],
x
triads then,
Now ( P ( M ) , P ( A ) , P ( B ) )
= P(ADB).
We
have
follow-
semialgebraic
over
R with A
polytopes
f o r every
: P ( X ) -> X e x i s t s
The
spaces.
o f spaces
o f weak
homo-
37ff].
are excisive?
of weakly
topological
(M,A,B)
of the long
[ E S , p.
triads
t h e "tameness"
to arbitrary
i n M
know
of the exactness
c f . [Sw, p.
witnesses
closed
and
induces
q u e s t i o n : Which
6.2.
X
i s called
i s an e x c i s i v e
formal consequence
Proposition
of
w i l l
i s exact.
sequences
t i a l l y
parti-
(cf. 5.3).
( f o r t h e homology
-* ( M , B )
In
theory which
the Mayer-Vietoris of the quadruple
logy
If
i n force.
n € Z.
i s again
spaces
homology
(M, A , B )
excisive
(A,ARB)
R
remain
n
This
ing
over
o f spaces
n
every
come
:
topological
o f spaces
(M,A,B)
section
: h (A,AHB) -^h (M,B)
Proposition
We
the preceding
triple
a triad
inclusion
for
from
on p a i r s
Definition call
limits
i s an a r b i t r a r y
+
evaluated
We
and
space
X
and i s a
i s a triad
a commuting
are over
excisive. R,
the par-
WP-approximation
o f weak
square
and B
polytopes
P(i)
(P(A) ,P(A)np(B) )
(A, AflB) and
P(i) i s just
cals
Now
that
consider
closed a
the inclusion
(M,A,B)
triad
with
excisive. theorem
Remark
Since
that
the pairs
(P(M),P(A),P(B))
above.
The
i s excisive
vertiwe
con-
a n d B(K)
i s sequential.
closed
S i n c e A(K) D B ( K ) =
f o rhomology
T h e n we c h o o s e
(M,A,B)
i s an e x c i s i v e
a real
(M(K),A(K),B(K))
i s excisive.
then
also
(M,B,A)
i s
j
q.e.d.
j
i s
\
excisive.
This
•]
c a n be seen
position
6.2
subspace
(AxO)
closed
be
g
1
to the triads
over
(AxO)
R which -> (M U
f
and with
1
(M ,B',A'), where M C
:= A R B ,
U (Cx [ 0 , - 1 ] ) , (Cx [-1,1 ] ) U ( B * 1 )
i s partially f
1
(M',A ,B )
U (Cxi) U ( B x 1 ) o f M x i
: (M, A)
:
for
t h e same way a s i n t o p o l o g y b y a p p l y i n g
6.4. L e t (M,A) b e a c l o s e d
a map
map"
i n much
subspaces
Theorem
N,N)
pair
o f spaces
proper
( c f .IV.8.4)
near
induces an
a n d A',B'
o f M',
over
M^A.
1
i s the
n
i ) We
Now we h a v e
first
: A -* N
t h e "push o u t
isomorphism
f
consider
a commuting
the case
diagram
that
I
arethe
h (M,A)^h (MU N,N) n
1
c f . [Sw, p . 1 0 3 f
R and f
Then
Pro-
e v e r y n € Z.
Proof.
].
main
conclude by t h e f i r s t
triad
|
(M (K) , A (K) ,B (K) ) i s a g a i n
i n M(K). Thus
( A f l B ) ( K ) we
5.4.ithat
6 . 3 . I f (M,A,B)
j;
R i snot sequential.
K of R which
A(K)
map b e t w e e n
i s excisive.
the case
overfield
(M,B)
i
are WP-approximations.
clude
(P(M),P(B))
M a n d N a r e weak p o l y t o p e s .
(M U
(M,A)
(M/A, *)
with §4
g
ri^Ep]
isomorphism. ii)
We
Then
(M U N/N,*) f
an isomorphism
that
now
we
and canonical
Thus
n n
t g l
i s an
t h e theorem
a commuting
i n t h e case
and
P(g) i s the pushout
(i)
that
h [P(g)]
I f finally field
conclude
K
R.
We
by t h e main
t h e s p e c i a l case
as
follows.
complete
M \A
n
now
of
direct
know
that
N
6.5. F o r e v e r y near
n
start
of
that
also
In^£«g3
from
i s an
the f i e l d
R
i s sequential.
N,N)
P(M U N) the pairs
t h e n we
step
( i i )
5 . 4 . it h a t
n n
and every
n € £ we
above.
u P
(
We
f
)
p
( N )
know
(IV.9.14)
from
step
h [g] i s bijective. n
choose
a
sequential
real
that
h [g ] i s b i j e c t i v e and n •* K i s bijective. q.e.d.
space
o f spaces have
P(M)
that
t g ]
i s the one-point
pair
=
f
and conclude
from
closed
t h e theorem
(M,A) w i t h
a natural
A
reads
partially
isomorphism
.
out t o prove
limits
between
theorem
h (M,A) ^ R ( M / A )
We
map
R i s not sequential
In
Corollary
p,q. Moreover
i s bijective
n
course
know
(P (M U.N) ,P (N))
(M U
WP-approximations
a n d q . We
square
P(cr)
with
p
isomorphism.
(M, A )
closed
Of
n
(P(M) ,P(A) )
i i i )
projections
and h [ q ] a r e isomorphisms.
prove
have
N,N)
f
a very general theorem
spaces.
about
t h e homology
Theorem
6.6. L e t (M,A) b e a p a i r
system Def.
o f subspaces
7 i n IV, §3). For every
h„(X,,Anxj
lim
q
ATT is
an
follows
five-lemma.
b)
f i r s t
( c f . V,
Milnor's
t o prove
i n general henceforth
this
of M
(cf.
map
i n t h e case
by u s i n g we
long
consider
the claim
that
homology
a single
f o ran a d m i s s i b l e
[ M i ] . We
t h e theorem
consider
A
i s empty.
sequences
space
M
Then
and the
instead
I n T we
have
:= U ( X
n
x [n-1,n]I
n
B
:= U ( X
n
x [n-1,n]I
n odd)
the closed
of a
now p r o v e
(X^|X€A) assume without
the claim,
t h e theorem
i s an exhaustion
that
the exhaustion
that
complexes
(Prop. (X^IAGA) with
IV.1.15).
by use o f
: T -> M
i sa
homotopy
(T,A,B)
sequence
i sexcisive.
A
of the quadruple
c f .[Mi].
i n t h e case
o f M.
Omitting
i s faithful
The s p a c e
i scofinal
respect
of
T cMxi^ o f t h e f a m i l y
that
the directed
superfluous (Prop.
t o this
M carries
i n the family decomposition,
system
i n d i c e s we
IV.1.14).
loss of generality, our directed family
exhaustion
n
subspaces
= T. B y P r o p o s i t i o n 6.2 t h e t r i p l e
gives
(X ln€iN)
even)
inspection of the Mayer-Vietoris
We
c a n be v e r i f i e d
§4. The n a t u r a l p r o j e c t i o n p
(V.4.5).
(T,A,B,ADB)
f i l t r a t i o n
the telescope
A
A UB
close
prove
, c f . V,
equivalence
such
Thus
trick
n
c)
q £ Z the natural
covering
q
§2, Def. 3 ) . Then
(X ln€!IN)
with
i s an a d m i s s i b l e
a directed
(M,A).
We
M
(X-^IAEA)
h,(M,A)
A
a) I t s u f f i c e s
claim
pair
A
and
isomorphism.
Proof. the
of M which
o f spaces
We
enlarge,
to a faithful a patch
of a l l finite c f . V, §1.
may
lattice
decomposition closed sub-
Henceforth family and
we a s s u m e
of finite
belts
that
closed
M i sa p a t c h complex
subcomplexes
M(n) o f M and a l s o
o f M . We
a t t h echunks
subcomplex
Moreover height
X^ (n)
i sjust
n with
keeping
By
X^ . S i n c e
a ( n ) o f
,n
A
X^
sum o f a l l c l o s e d X^ (n)
i s the
a t t h e chunks
i n M, w e h a v e
. In particular,
We h a v e
look
X, A
the
a n d (X-^IXEA)
= M
n
patches
i s empty
f o r n
n
n
.
a of M of
large,
fl3M(n).
: (M(n),3M(n))
-> (
M n
'
M n
_-|)
yields
isomorphisms
: h
For
every
(M(n) ,3M(n) ) ^ h ( M , M _ ) q
n
X € A t h e map g r e s t r i c t s
(X, ( n ) , 3 X , ( n ) ) t o (X. ^ , X , A A A ,n A , n— J
We o b t a i n
a commuting
square
yields
bijective
map g ^
vertical
U B , \ ^ X
> q
n '
m
X
X
n - ^ —
m
i st h ed i r e c t
a l lpatches
space
X^(n).
Thus,
homology
that
thenatural
lim
h
q
(
sequences
x
x
n
This
n and every
implies
that
and t h efive-lemma
h
q (
M
n ' n - l '
M ) n
arrows
'
(o,3o)
with
a running
(X^(n),3x^(n)) i st h e d i r e c t
(a, 3 a ) . Moreover,
map
nM ) -
M
by t h ea d d i t i v i t y
i san isomorphism.
long
(
sum o f t h e p a i r s
o f height
sum o f some o f t h e s e p a i r s
h
from
isomorphisms
• h„(M(n) ,3M(n)) q
through
arrow
t o t h epushout
which
with
.
1
l i m h (X, ( n ) , 3 X , ( n ) ) ——> q A A
(M(n),3M(n))
some
n
every a i scontained i n
of h
g
, t h eupper
horizontal
a i san isomorphism. we s e e b y i n d u c t i o n
Using on n
is
an
isomorphism
of
subspaces
lim,
of M
h ( X , q
lim
h
gives
flM
X
us
upper
The
vertical
)
lirn^h n
system
( X ^ fl
square
I (X,n)
of
CA
x IN) I
homomorphisms
(M ) n
h (M) Q
arrows
arrow are
horizontal
arrow
d)
obtain
theorem
the
We
choose
(X^ f l M l a G l ) a
(X^ D M I (X,a) 6 A x i ) a
i s bijective
bijective
lower
family
directed
n a t u r a l commuting
n
horizontal
argument.
a
The
A
The
We
n.
(X.)
q
IT
f o r every
by
step
i s bijective,
an
i n general
exhaustion
i s an of
by
as
by
what b)
somewhat
of
of
M
lirn^h
(M
)
the
proof.
been
repeating
o f M.
a
subspaces
just
proved.
Thus
the
claimed.
(M la€i)
exhaustion
of
has
X^.
The
gives
us
For
the
every
directed a
last
X €A
the
system
natural
commuting
square
lim (X,a)
lim
The
X,.
now
polytopes denote sets
the
of M
by
arrows
the
prove
A
lower
that
bijective
pairs
elements
space. (K,L)
respectively
system
of
inclusion.
pairs
of
by
step
bijective
horizontal
the
given
set of and
are
i s t r i v i a l l y
i n the
directed
given
r
arrow
Thus
can
h M)
(X,)
vertical
some
a
( X . n M j . *
h
horizontal
We
h
For with and,
c)
since
arrow
K of
L
of
M
spaces
under
L
c the
K.
The
upper
i s contained
a
as
claimed,
groups
complete
course,
polytopes
proof.
every
homology
pair and
the
i s bijective,
i n our any
of
"live"
(M,A)
partial
q.e.d
i n
l e t 2T (M,A) c
semialgebraic This
i n
set
sub-
2T (M,A) c
ordering
i s
Theorem
6.7. For every q € 2 the natural
lim,
is
an
( h ( K , L ) I ( K , L ) € 2T (M,A) ) -
i ) We
choose
surjectivity spaces
o f t h e map.
( M ,AflM )
of
(M ,AflM ),
in
this
a
Theorem
sequences
image
i n h (K,L)
a
L
f
^2^ q^ 2' 2^ n
F ^ and £ p
a
v
e
o
r
t
n
s
o
e
c
By
image
Theorem
and tion the
£
2
m
K
i n h
map
Thus
^
of M from
and £
a
P
a
exists
we
(M,A) w i t h
i
some a € I a n d some choose
r
s
We
are given
c
and
U L
map
from
Thus n has a
pre-
of
2
C-j € h ^ (K^ , L ^ ) a n d
i n ^ (M,A), c
have
to find
cz L s u c h
2
homology
the inclusion
(K ,L )
i n h ^ ( M , A ) . We
K and
long
elements 2
and A 0 M
of
exists image
some a € 1 w i t h i n h (M ,AnM ). a
q
o f spaces
that
Q
2
U K
(M ,AnM ) a
t h e same
We
2
C
c: M
choose
(M ,ADM^,K ,L^/ L ) .
Then
a
a
U K
a
2
gives
image
c
a 1
such
that
pair
and £
have
2
q
such
that
^
a good t r i a n g u l a -
2
an isomorphism
i n h (K,L).
Q
L e t K and L
2
K a n d L^ U L
Q
K and L a r e s t r o n g
i n homology.
a preimage
(K-j/L^)
image 2
a good t r i a n g u l a t i o n
the cores
conclude that
i s also
a n d AflM .
have
a
Q
( X , Y ) -» h
o f £ • We
an isomorphism
i
U K
(K,L) t o 2
n of h (X,Y)
(K,L).
t h e same
o f the system
sion
1
6.6 there
have
cores
e
same
( K , L ) € # (M,A) w i t h same
an element
* : h
prove t h e
I f X and Y a r e sub-
( K , L ) £ if (M,A) . M o r e o v e r
injectivity.
2
the
i
first
and A r e s p e c t i v e l y . Using
and t h i s
now p r o v e K
gives
a
c j
h
Then
and t h e five-lemma
t o (M ,AnM )
We
6.6 there
retracts of M
(K,L)
ii)
? under
i s a preimage
triangulation.
deformation
call
c f . I l l , §2. L e t K a n d L d e n o t e
a
We
(M,A) .
which
a
a
onto
o f M.
(M,A) b e g i v e n .
X r> Y t h e n w e
(X,Y)
the preceding q
q
(M l a € I )
Let £ €h
o f £ i f n maps
inclusion
D Gh
an exhaustion
o f M and A with
preimage
By
h (M,A)
C
isomorphism.
Proof.
the
map
c
L. The
denote
inclu-
i n homology. q.e.d.
We
return
to excision
Lemma 6.8. which for
L e t (X^IXEA) be a
i s an a d m i s s i b l e
every
is
problems.
d i r e c t e d system
covering
o f M.
of subspaces
L e t (M,A,B)
be
a triad
(X^,A nX^,B nX^) i s e x c i s i v e .
A £ A , the triad
of a
space
such
Then
M
that,
(M,A,B)
excisive.
This
i s an easy
rather
general
consequence
o f Theorem
excision result.
6.6.
In order
We
a r e now
to state
next
i t we
door
need
to a
two
defi-
nitions .
Definition {i.e. in
2. A
L closed
subset
I t suffices
sets
of a given
If
i n L
M
A
= X . We
=
X
The
also
two l o c a l l y word
"basic
"basic"
M
3. A
and, f o r every
with
L
By t h e way,
semialgebraic
> 0,
U £ T(M)
i n this
g(x) >
semialgebraic alludes
triad
X fl L
that
X n L = U fl A w i t h
o f M.
L
£T(M)
i s locally i s open
U € r{L)
running
then
(M,X,Y)
L£}f(M),
then
X fl L
closed
i n i t s and A €?(L) .
through i s
the
locally
basic
subset can
X of choose
case,
0}
functions
f and g on M
f o r the sets
i s called
there
every
a n d A £ T (M) , a n d we
to the possibility
description" at least
Definition
means
property
U fl A w i t h
can write,
{ x € M | f (x)
i f , f o r every
L € 2T(M) .
i s locally
i s an i n t e r s e c t i o n
This
this
exhaustion
M
L.
basic
i n M}, t h e s e t X n L
terms,
t o check
f o r every
the space
with
space
X D L or, i n other
N.B.
closed
i s called
and s e m i a l g e b r a i c
the semialgebraic
closure
X of M
exist
i n general
(cf.
1.4.15).
to give
such
a
X f l L i n L.
basic
closed
i fX
and Y
are basic i n
semialgebraic
subsets
A
and
B
of
check
L
with
A c X D L ,
Bey
flL,
this
last
property
with
L
exhaustion
Examples triad") ii)
of
Every
Indeed,
i i i )
More
with
X
M.
AUB
Thus
U UV 1.4])
A HP* ( R )
which
i scontravariant
i n the first
ment.
Without
out a l lthedetails,
natural
up t o homotopy
spelling
homotopy
Map(M,N xN ) ^ M a p ( M , N )
(7.5)
Map(M AM ,N) ^
1
2
2
1
xMap(M,N ) 2
M a p ( M ,Map ( M , N ) ) 1
2
i n t h e second
we r e m a r k
equivalences
(7.4)
1
and covariant
that
there
arguexist
for
M,M.| ,M
natural
€P*(R),
2
[ X A ( M
string
X N
1
A M
1
2
2
]
us that
=
t o have
maps
from M
to N
w i l l
see, already c a n be
turn
X [ X A M , N
[ ( X A M ^
A M
2
2
] ,
, N ] .
[S°,Map(M,N)] =
classes
much b e t t e r
We
=
the connected
t h e homotopy
space
of the
of equations
Q
as
[ X A M , ^ ]
=
) , N ]
n (Map(M,N))
tells
a r e consequences
2
bijections
[X A M , N
The
, N , N € WSA* (R) . T h e s e
[MAS°,N]
components
o f maps
a space
i n a natural
[M,N]
o f Map(M,N)
from M
whose
=
t o N.
points
c a n be
interpreted
Of c o u r s e , i tw o u l d
correspond uniquely
be
with the
way, as one h a s i n t o p o l o g y . B u t , as
a pseudo-mapping
space
instead
of a true
we
mapping
useful.
to a special
t y p e o f pseudo-mapping
spaces,
the pseudo-loop
spaces.
Definitions
N
i ) For any pointed
1
space the
2.
Map(S ,N)
b y ftN a n d c a l l
switched evaluation which
[X,«N]
map
space
N we
denote
i ta pseudo-loop
e^i ^
b y n_ « N
Thus
n
space N
i n d u c e s , f o r e v e r y X € 9 * (R) , a b i j e c t i o n to
the
pseudo-mapping
o f N . We
i s a map
denote
from
SftN t o
[ f ] >-* [ n ^ ' ( S f ) ]
from
[SX,N].
1 ii) a
Since S
multiplication
(ftN,[y]) are
N
group
and y
i i i ) and
y
:
i s a group
N
ftNxQN
such
,
->
Once
F o r e v e r y map a map
: N -* N
a n d f o r a l l we
g
:
1
1
-> ftN
such
by Remark
bijections choose,
u and denote
between
have,
up t o homotopy,
and t h e above
as ingredients
f
( c f . §1) we
unique
a multiplication
are regarded
f o ra l l ,
i n HP*(R)
i n HP*(R)
isomorphisms.
N €WSA*(R), r)
i s a cogroup
such
that,
[SX,N]
f o r every
i tby y
spaces
that
[X,fiN] ^
. Both
of the pseudo-loop pointed
7.3,
we
maps
s p a c e fiN. choose,
once
f o revery X eP*(R), the
diagram
[SX,N]
» [SX,N']
[X,flN]
[X,fiN»]
commutes.
The v e r t i c a l
jections.
We d e n o t e
this
morphisms
we h a v e
diagram
a
arrows
here
mean, o f c o u r s e , t h e c a n o n i c a l b i -
m a p g b y flf. S i n c e t h e m a p s
f * a r e group
homo-
N
QfxSlf
V which
commutes
up t o homotopy.
functor
N
objects
i n HP*(R).
iv)
QN,
A l l this
[ f ] •+ [ f t f ] f r o m
The p s e u d o - l o o p
HWSA* (R)
f u n c t o r ft : H W S A * ( R )
(= a d j o i n t
i n [Mt]) t o t h ecomposite
S
-» H P * ( R )
n
: HP*(R) N
: SW
also
choosen mapped by
the inclusion
t o f
left
j
d S
M
^
under
t h enatural
obtained
-* H * ( R ) i s r i g h t
adjoint
: H P * (R)
class
bijection
HWSA* (R) . T h e m a p s
maps
adjunction
functor
[ M t , p . 1 1 8 ] . We
map C
M
i n [M,ftSM]
[M,ftSM]
: M -» ftSM, which i s
-^[SMjSM].
Thus,
definition,
which
SM
n
SM*
(
S
5
M
characterizes
Forany r 6 1
functor
by £
equivalence
r
)
c
M
up t o homotopy.
we d e n o t e
. By
ther-fold
( 7 . 5 ) we h a v e ,
a
t o t h e category o f group
adjunction
and f o r a l l i n t h e homotopy i
we h a v e
j*So f the suspension
f o r e v e r y M € P * (R) , a r i g h t
once
id
v)
with
-* N a r e t h e a s s o c i a t e d
have,
means t h a t
iteration
o f t h e pseudo-loop
f o revery N€WSA*(R),
a
homotopy
r
r
QN
From
thedefinitions
(7.6)
We
^Map(S ,N) .
r
n (^ N)
=
q
illustrate
n
i t i sobvious
g
+
( N )
r
that,
f o re v e r y
q > 0 a n d r >O,
.
t h eusefulness o f t h epseudo-loop
functor
by an
example.
Theorem
7.7
polytopes. sion
Assume
homomorphism
S
is
(General suspension
:
M,N
bijective
[
M
'
N
that
]
[
1
Freudenthal s
map
C
: N
The
claim
We
digress
"free"
S
i f dimM
ftSN
N i sn-connected
L e t M a n d N b e p o i n t e d weak
f o r s o m e n G 3N
. The
suspen-
(§1, D e f . 2)
Proof. N
theorem).
M
'
S
N
]
< 2n and s u r j e c t i v e
suspension
i sa
now f o l l o w s
theorem
(2n+1)-equivalence from
f o rs h o r t from
Theorem
i f dimM
1.5 m e a n s
that
the adjunction
( c f . V, §6, D e f . 5 a n d Def.
V.6.13.
o u r g e n e r a l theme
pseudo-mapping spaces
= 2n+1.
c a n be o b t a i n e d
7 ) .
q.e.d.
i norder
t o i n d i c a t e how
f o rspaces
without
base
points.
Theorem
7.8. L e t M b e a weak p o l y t o p e
variant
functor
presentable.
e
such
M
L
that,
X *-» [XxM,L] f r o m
Thus
there
: M a p (M, L ) xM
f o r every
->
exists
[ X , M a p ( M , L ) ] -* [XxM,L] [f]
~ [e
M,L
over
HJ>(R) t o t h e c a t e g o r y
a weak
L
XGP(R),
and L a space
t h e map
p o l y t o p e Map(M,L)
R.
The contra-
o f sets
i s re-
a n d a map
is
bijective.
Proof.
We u s e t h e n o t a t i o n s
pointed any
space
(L,y)
X GP(R),
(cf.
some y £ L a n d d e n o t e t h e +
b y N. T h e n N ° = L . L e t T
i nslightly
[XxM,L] =
4 . 1 . We c h o o s e
sloppy
Then
+
+
+
+
= [ X , M a p ( M , N ) ] = [X,T]
4 . 2 , 4.3)
Corollary
for
notation,
[X AM ,N]
+
[(XxM) ,N] =
:= M a p ( M , N ) ° .
q.e.d.
7.9. I f M C P ( R )
and NG
WSA* (R)
then
+
Map(M,N°) -Map(M ,N)° .
We r e t u r n It
t o pointed
i san analogue
Adams,
cf.
spaces
o f a famous
[Ad^],
t h e same w a y a s t h e r e .
ed
for
that
o f pointed F f u l f i l l s
(w) f o r
section. Then
finite
Of course
there
exists
[-,L]
o n HSA*(R)
Again
we c a l l
not
be needed
right law) in
theorem
I t can be proved
In particular,
7.10. L e tF be a c o n t r a v a r i a n t
HSA*(R)
axiom
representation
theorem.
due t o Brown a n d
here
word
by word
no t r a n s f e r p r i n c i p l e
i s need-
t h e proof.
Theorem
Assume
another
representation
[Sw, T h . 9.21].
in
gory
and state
track.
over
families
(M^lXGA)
now a l l t h e s p a c e s a group
object
i sn a t u r a l l y
i nt h esequel,
and very
semialgebraic
{cf.
useful spaces.
objects
o f groups.
have
such
that
us a hold
even
(with
of the
t o be
polytopes.}.
the functor
(= i s o m o r p h i c )
polytopes
cate-
(MV) a n d t h e w e d g e
of the functor
but i t gives weak
t h e homotopy
t h ebeginning
involved
equivalent
I tindicates that
axiom
L i n HP*(R)
space
from
R t o t h ecategory
t h eMayer-Vietoris
L a classifying
arenatural
complete
polytopes
functor
t o F.
F. Theorem that
7.10 w i l l
we a r e o n t h e
a homotopy
i f one i sonly
group
interested
§8
- ^-Spectra
We
now
have
cohomology theory
theories
of spectra
spectra
setting
w i l l
not delve deeply the view
us t o work
maps"
appears
- between
i n G.
with
into
point
on t h e l e v e l
which
suitable
o f weak
o f maps
i n the
Whitehead's fundamental
the
that
a naive notion
spectra,
reduced
paper
-
topological [W 1
(there
2
"maps").
Definitions pointed
A
We
be c o n t e n t w i t h
allow
"homotopy
already
called
b)
but w i l l
This w i l l
called
(base
and spectra.
t h e c o n n e c t i o n between
serve t o represent cohomology theories
polytopes. here
the prerequisites f o r drawing
1. a ) A
(semialgebraic)
weak p o l y t o p e s
point
(E |n£Z)
spectrum
E over
E over
together with
n
preserving,
spectrum
as always)
R i s called
maps
R i s a
a family
family
(e
Inez)
of
of
e
an ^-spectrum
i f t h e maps
n
n
:E
n
-•fiE^^^
E which areadjoint t othe for E ^
the definition
n (
c)
n
above,are homotopy e q u i v a l e n c e s
o f the pseudo-loop
functor
i s a n a b e l i a n group o b j e c t i n HP*(R) n+1 n n ^ n+2 * ' )
o
n
f
r
o
A homotopy
family J
m
E
t
map
(f ln€Z) n
o
f
E
( c f
: E -* F
o f maps c
f n
§
from : E n
7
Q).
In this
v i a t h e homotopy
( c f . §7 case
every
equivalence
)
a spectrum -> F n
E t o a spectrum
between
spaces
such
F
that
i s a the
diagrams
"•n+1 commute
up t o homotopy.
spectra
see [Ad] o r
d) f
A homotopy : E -> F
such
[Sw,
{ F o r a more u s e f u l
o f maps
between
Chap.8].}
e q u i v a l e n c e between that
notion
e v e r y map
f
R
spectra : E
n
-» F
E,F n
i s a homotopy
map
i s a homotopy e q u i v a l e n c e .
e)
Analogously
instead
of
we
(R) , a n d
(topological)
loop
valences
between
N.B.
maps
The
in
topological
functor
ft,
using
fi-spectra,
using
the
category
the
f u r t h e r homotopy maps
KO*
genuine
and
homotopy
equi-
topological spectra.
E n
and
E here n
n
2.
Let
f,g
: E
topological spectra.
of
(base F
and
n
and
point =
(~1)
we
with
have
nothing
the
F(-,0)
spectrum
pointed
weak
H (X,E)
the
know
limit
way.
from
For
H
n + 1
H?*(R)
to
"weak"
n + k
]
§1
1
granted
:=
JL.
=
i n
X
homotopy
to
do
with
the
e's
and
for
t GI
F(-,1)
R we
with
these we
to
as
n
+
k
+
F
-* E ^
gives
notion
n's
family with
a
of
semialgebraic
map
(F lneZ) n
F (-,0)
=
n
F(-,t)
homotopy
:E
fits
f
-*• E *
well
abelian
n
group
H (X,E)for
every
follows.
].
to
the
transition
k
[ S
are
indeed
^ X , E
abelian
n
+
maps
k
+
1
]
.
groups
in
a
natural
have
k
+
1
X,E
n
+
1
+
k
]
=
contravariant
cohomology the
the
identity
theory
k
lim,[S X,E k
maps H
of
n + 1
over
the
J
N | ]
=
n
H (X,E)
n
functors
H*(-,E)
exception
families).
n
above.
]
k
i s a
n
equivalence
an
g
: E x i
This
g.
limits
with
finite
n
n
=
respect
E
F
f
family
n G Z
[ S - V s
with
from
this
k
of
F
define
and
lim,[S k
homotopy maps b e t w e e n
homotopies
lim,[S X,E
family
§2
two
homotopy
over
together
reduced
Definition
of
X G J>* (R)
The Ab
f,
that
(SX,E)
8.2.
only
=
i s taken
every
Remark
A
every
polytope
[S*X,E
We
E
n
(8.1)
Here
Eor
n
definition
a
it E'be
preserving)
9f «
have
Given
a
e
topological spectra,
§7.
Definition or
define
(H (-,E)InGZ) (-,E)°S R,
wedge
n
-* H ( - , E )
i.e. i t axiom
from
f u l f i l l s
(which
i s
i s
This
i s obvious N
write
E (X)
Definition between
(recall
instead
3 . We
map
formation
f
from
transformation
Proposition
is
t o F*
by
U^.
I f E
-» E
We
which
t e l l
are
have
this
implies
Remark topy
f
i s an
that
R,
R
c f . § 2 , D e f . 2.
induces
w a y . We
then,
transformations
a natural
denote
this
f o revery
Every trans-
natural
X €P*(R)
and
E
n
obeys
E*
maps
arrow
cohomology
the functor the f u l l
i ti s clear
E
equivalence.
n
i s isomorphic
wedge axiom
that
ft-spectra
f o rthe inductive i s the adjunction
(cf.
t h e homotopy correspond
: E * -> F * v i a T = U ^ ,
f i s a homotopy
i s a reduced
theory.
diagrams
(The u n a d o r n e d
: E -> F b e t w e e n T
groups.
the transition
i n §7). Thus
transformations
over
over
ft-spectrum
of abelian
8.4. M o r e o v e r
maps
spectra
H*(-,E).
of natural
theories
i n the evident
commutative
isomorphisms. explicit
notion
of
briefly
( X )
us that
made
i f f
N
instead
we
map
an isomorphism
Proof.
cohomology
E*
n 6 Z, t h e e v i d e n t
r
a n d E*
: E -» F b e t w e e n
8.3.
[X,E ]
fi (X,E)
an obvious
weak r e d u c e d
homotopy
N
of
have
1 . 2 . i and 1.7). I n t h e f o l l o w i n g
and T
limit
(8.1)
isomorphism to t ~ ^
1.2.i).
classes
uniquely
i s a natural
E n
] '
a n
&
q.e.d.
[ f ]o f
homo-
to the natural equivalence
Theorem an
8.5.
For every
ft-spectrum
Proof. that, (W)
We
know
over
from
f o r every
and
exists
(MV)
T
n
t h e wedge
weak
[~'
E
l
n
axiom
over
polytope
-^k
1 1
E
n
Def.
1)
: HP*(R)
over
. For every
[X,flE
n + 1
R there
a natural equivalence
(§2, n
k*
and
-> A b
R
together
XGP*(R)
we
exists
T : E*
^ k * .
Proposition
f u l f i l l s
i n the r e p r e s e n t a t i o n theorem
n
the
with
the functor k
: [X,E ] ^
that
cohomology theory
together
required
:
a(X)
R
n €Z,
a pointed
valence
such
E
reduced
7.1.
with
have
the axioms
Thus
there
a natural
a
2.3
equi-
bijection
l
diagram
(SX)
commutes. to
[-,ftE
n
: E
n
The
By -> ^
n
a(X)
E n +
together
the Yoneda
-| ' unique
families
(E lnez)
family
n
n
(T |n€Z)
and
8.6.
natural
equivalence
f
map
I f F
f
determined
We
now
by
consider
topological
lemma t h e r e
exists
t o homotopy,
(n ln€Z)
i s a
such
together
n
second
then
: E -* F ,
i s a homotopy
is
up
a natural equivalence
i t i s evident unique
equivalence. k*
uniquely
spectra
spectra.
ft-spectrum
over
up
and V from
the
t o homotopy
different
real
a =
an E*
8.4
t o homotopy,
In short, up
that
from
a
from
a homotopy
define
i s a natural equivalence
Remarks
homotopy
form
[-,E
equivalence
(n )*» n
ft-spectrum,
The and
t o k*.
such
i s again
there
exists
t h a t V«U^
fi-spectrum
E
= T,
i n the
fields
and
a a and
theorem
equivalence.
closed
the
q.e.d.
: F* ^ k * that
]
also
Proposition
8.7. E v e r y
spectrum
E over
a l l
e
maps
This
being
follows
Definitions E
R with
map
e
by
n
(e ) , w e n K T
homotopy
map
f
from
spectrum
E by base
: E -* F b e t w e e n
over
read
E. E v e r y
8.8
overfield
IR w i t h
E
and l e t
by E ( K ) and
n
every
n
(First
map
main
t
f
between
theorem
o
E
^ r called
call
the
R t o K.
Every
a homotopy
R
pointed
map
CW-complexes.
by t h e p r o p o s i t i o n spaces
(
the topological
t
0
n
)
t
Q
p
IR c a n
E
p
E
spectrum
spectra over
spectra f
f o r spectra).
F
• j- p ~* top" 0
L e t K be a r e a l
closed
o f R. R then
t h e homotopy
classes
of
homotopy
u n i q u e l y t o t h e homotopy
classes
of
homotopy
g
: E ( K ) -> F ( K ) b y t h e r e l a t i o n i s a homotopy
I f F i s a spectrum with
follows
Analogously,
equivalence over
a homotopy
spectrum
this
R yields
topological
maps
together
we
e x t e n s i o n from
: E -* F b e t w e e n
: E -> F c o r r e s p o n d
: E -* F
K which
underlying topological
f
A l l
space
a l l spaces
maps
A
e x t e n s i o n o f R,
of generality
^
map
I f E and F a r e s p e c t r a over
i i i )
and
way.
spectrum
homotopy
as a homotopy
Theorem
ii)
field
s p e c t r a over
R e p l a c i n g t h e E^ by t h e i r
underlying
f
p o i n t e d CW-complexes
E(K) over
field
restriction
obtain a topological
i)
closed
Replacing every
obtain a
i s no e s s e n t i a l
above).
be
R.
L e t E be a spectrum
(This
being
n
equivalent to a
V.7.4.
: E ( K ) -> F ( K ) i n t h e o b v i o u s
R
we
V.7.14 and
over
obtained
b)
E
R i s homotopy
E
spectrum
f
a l l spaces
a) L e t K b e a r e a l
be a spectrum E
over
cellular.
from
4.
spectrum
E over
from
R
K then
[g] =
[ f
A
K
3 -
homotopy
i f f f ^ i s a homotopy there
exists
map
equivalence.
a spectrum
E over
R
e q u i v a l e n c e cp : E ( K ) -* F . i s an
the f i r s t
ft-spectrum
main
i f f E(K) i s an
theorem
using the theorems V . 5 . 2 . i i
V.5.2.i
ft-spectrum.
a n d Theorem V.7.15.i,
and V.7.16.i,
we
obtain
Theorem
8.9
algebraic
(Second
spectra
over
homotopy
classes
[f
[ g ] . A homotopy
p]
=
equivalence ii) E
Given
over
a
i f f f
3R , w i t h
Q
p
homotopy
extension
over
3R
l* p.
uniquely
with
: ^^ ^
"* t o p
g
: E -> F
i s a
space
E
spectrum
q> :
i e
r
e
i
a
t
exists
n
i °
homotopy
equivalence. a semialgebraic
CW-complex,
spectrum
and a
(topo-
-> F .
E over
IR
i s an
ft-spectrum
cohomology theory
o f R we
have
K.
Similarly
we
have
constructed
over
i n §2 a r e d u c e d
f o r 1* a r e d u c e d
constructed
R and K a r e a l
i f f E ^ ^ i s
a reduced
closed
field
cohomology
semialgebraic
theory
cohomology
t o p o l o g i c a l cohomology
As an immediate consequence o f t h e d e f i n i t i o n s
Q
^
(semialgebraic)
a semialgebraic
n
t
^
( t o p o l o g i c a l ) homotopy F there
classes of
the (topological)
F
Q
homotopy
ft-spectrum.
k* a r e d u c e d
over
f
equivalence
A semialgebraic
For
map
i s a
every
topological
k*
t
maps
a t o p o l o g i c a l spectrum
logical) i i i )
o f homotopy
i ) I f E and F a r e semi-
the (semialgebraic)
: E -* F c o r r e s p o n d
maps
0
theorem f o r spectra).
IR t h e n
homotopy
t
f
main
theory
theory
( c f . §2) o n e o b -
tains
Proposition evident
8.10. i ) F o r e v e r y
and canonical
E(K)*
of ii)
reduced I f E
algebraic
( E * )
ft-spectrum
E over
an
K
i s a semialgebraic CW-comples
then
K.
ft-spectrum
there
exists
over
(E*) top
t o p o l o g i c a l reduced
3R
with
an evident
phism
of
exists
isomorphism
cohomology t h e o r i e s over
(E. )* ^ top
R there
cohomology t h e o r i e s .
every
E
n
a
and c a n o n i c a l
semiisomor-
By
the
the
relations
real
closed
rather
obvious. V
T h u s we
have
of
R,
for
remark
logy
of
For
W V
X
of
i f K
process than
to
and
weak
groups
a
T T
)
N + K
n
the
X
(
S
E
+
+
the
k
A
X
any 0)
>
R
over
out
field
labour.
extension theories
1 * «vw—* 1*,
process
of
cohomology
cohomology
one.An
with
)
between
(not
while
analogous
have
obvious
' "
E A l d
k
spectra
necessarily
n € 2 we
( E
We
becomes
§2.
to
between
natural
different
serious
closed
reduced
connection
and
( E ^ A X ) |k
n k 1
access
extension i s
spectrum
polytope
(
new
on
"top".
about
be
without
real
over
obtained
7.1
i s a
§2
of
theories
been
relation
for
the
E n*—> E ( K )
E
the
half
theories
have
theorem
i s that,
brief
second
cohomology
comfortable
feature
natural
Let
any
abelian
rather
"sa"
the
results
process
rather
theories.
trum).
these
restriction
the
be
a
of
t o p o l o g i c a l cohomology
that
ft-spectra
pertains
w i l l
and
pleasant
i s more
spectra
contents
representation
gained
the
the
semialgebraic
fields
the
and
1*
l**v—>
We
and
then
8 . 1 0
Notice
particularly
theories
-
between
base
Chapter
A
8.5
results
n
+
a
an
direct
transition
k
+
1
(
E
k
+
1
A
and
X
homoft-spec-
system maps
•
)
x'*
define
H (X,E)
:=
n
In
this
in
abelian
up
to
a
way
we
lim^
obtain
groups.
sign)
n
+
k
(E AX)
.
k
covariant
We
define
obvious
way:
a
functors
suspension
n
(
S
E
x
)
i s
H (-,E)
HP*(R)
on
n
isomorphisms
induced
by
the
in
with
the
family
of
(perhaps homomor-
phisms
n
n + k
(
E
k
A
X
)
n
n+k+1
(
k
A
X
)
n
n + 1 + k 1
with
cp t h e
switching
isomorphism
from
S
A E.
(
E
A X
k
A
S
to
X
)
•
E,
A S
1
A X
values
.
Theorem form
a
ii) E
8.11. reduced
Given
over
a
R
Proof.
[Sw,
2
(or
labour
in
the
(cf.
does
not
the
theory
theorem
be
p.
2
any
and
Part
Chap.
holds
more
249f.];
to so.
the
to
do
or
also with
natural.
He
By
R
This
to
maps b e t w e e n
them
have
choice
weak
polytopes,
the
to
theorems we
know w i t h o u t
easy
for
the
the
aspects
by
latter
first
wedge
second
Boardman
which
of
stateaxiom
statement
only
perhaps
with
i s
i s
homotopy
language
reader
and
anyway
stable
transfer this
work
the
directly
s o p h i s t i c a t e d language
[Sw]),
the
further
setting.
theorem
prove
setting
homotopy
the
designed
and
on
prove
modern
in
spectrum
topological
cohomology
to
as
deeper
setting,
will
to
to
the
possible to
semialgebraic
i n the
a .
+
spectra
order
some c h a p t e r s the
H (-,E) -^k^
i s rather
use
exists
semialgebraic
in contrast
together
n
there
main
satisfactory
advisable
the
true
our
i n the
t r o u b l e s ) . In
i s perfectly
be
(o ln€Z)
R.
over
+
t h e o r i e s and
for understanding I t
k
to
14].
setting.
I I I ] and
over
natural equivalence
homology
i t would
CW-spectra
theory.
theory
and
8
i t seems
pensable
a
H^^E)
Chap.
[W ,
[Ad,
theory
i s w e l l known
cause
directly
(cf.
with
and
n
homology
semialgebraic
ment
(H (-,E)In€Z)
theorem
that
course,
of
reduced
groups),
Of
families
homology
together
The
[W ], sets
i ) The
and
invited
CW-complexes
being
more
Adams indis-
Chapter
As
already
eye
to
[SFC].
said
From
i.e.
we
(cf.
1.2.ix
could
The have
jects,
For
2.5
theory of
the
the
below).
this
of
fibrations
sets
occuring
This
chapter
preceding
simplicial
assume t h a t
basic to
would
i s written i n the
chapters
instead of
with
third
volume
i t would
be
simplicial
simplicial
spaces
t r i v i a l i z e
the
are
major
an
spaces, discrete
part
cf.
every
non
the
monotonic
N.B.
In
a l l our
Definition
a
by
. I f a
below be
Let
simplicial
i.e. X
to
1.
more
C
be
object
functor X :
i f i £ J
nonempty
[n]
denote
simplicial
sets
[n]
totally
ordered
could
and
ob-
in C
[n]
the
set Ord
denote
whose morphisms
ordered
sets,
{a
:
are
[n]
->
[m]
C. -» [m]
we
implies a(i)
totally we
the
these
unique
o
i t s natural total
o b j e c t s are
maps b e t w e e n
study
i t seems
A
with
a
on
[Cu].
equipped
exists
a l l finite
standard terminology
i n t e g e r n € 3Sf
whose
of
the
negative
monotonic
There
not
some o f
[ L a ] , [May],
category
called
definitions
recall
the
a)
i n the
deal with
spaces
introduction
viewpoint
and
{ 0 , 1 , 2 , .. . ,n}
is
Simplicial
§1-§5.
§1. We
the to
-
i n the
applications
sufficient
of
VII
with
the
category
small
Ord
Ord
category
Ord.
but more o f t e n
than
than
with
Ord.
category. is a The
c o n t r a v a r i a n t f u n c t o r from value
i s a monotonic
X([n]) map
w i l l
then
the
usually
Ord be
morphism
to
C,
denoted
X(a)
:
3
:
if
-+ [m]
->
morphisms b) f
Let : X
a*
X
and
-> Y
functor c)
w i l l [p]
i s a
the
transition
Y
be
i n C
denoted
by
d)
i s a
I f X
i s a
denoted
monotonic morphisms
natural
objects
briefly
(3a)*
map, of
by
a*.
Notice
= a*3*.
We
that, c a l l
these
X.
i n C.
A
transformation
simplicial
from
the
morphism
functor
X
to
the
simplicial
:
= T
the
simplicial
X
X
n
s
=
:
?
X
n
-
object
n-1
degeneracy
i
objects
and
simplicial
morphisms
i n C
i s
sC.
d
i
s
second
simplicial
category of
d
be
Y.
The
and
usually
i n C
then
we
define
the
face
morphisms
< O l i < n )
morphisms
X
n 1
«> [ n ]
[ n + 1 ] -» [ n ]
:
1
1
=
1
the the
monotonic monotonic
injection
which
omits
the
value
i
surjection
which
takes the
value
i
has
unique
twice.
Notice
that
every monotonic
map
a
:
[q]
-* [ n ]
a
decomposi-
tion a = 6
with
n > i ^
q+s
=
a
T3
=
This
^1
n+t
>
...
[ L a , p.
with
^s
...6
3
>
a
J1
. . . a
j+ 1 •
s i m p l i c i a l morphism : X
n
=
f
n
d
n
i
f
-» Y „ i n C n
which
(0 Y b e t w e e n f
:
x
Y
h
a
s
property
Q i f f e v e r y map
We
use a l lthe vocabulary which
shall
agreement w i t h o u t much
n
n
naturally
explanation.
p o l y t o p e X means, o f c o u r s e , a s i m p l i c i a l
that
every X
a
Such
1.2. i )L e t X b e a s i m p l i c i a l
singular [Frd]). to
X(R).
varieties
with
maps
ofX
i
)
:
R
namely jects
space
x
(
n
R
) ~*
n
space
1
= Mx
monotonic
space
R gives
then
space
homotopy
(
R
R,
over
) ->
x
i . e .
R.
([AM],
Z(R) from
R which
C
we d e n o t e b y
( X ( R ) I n € 3N ) R
R
n
theory
Z
-i ( )
variety
X over
Q
a
n
d
t
h
e
dege-
X(C) over
C := R ( \ / - 1 )
us a constant s i m p l i c i a l
from
Ord° t oWSA(R), w h i c h maps
gives
R.
a t o identity.
X over
spaces
over
space
maps
over
R,
a l l ob-
We d e n o t e
R as follows.
R. S t a r t i n g X
R
this
simplicial
to
f ( x ) = ... =
v)
L e t G be a weakly
map f r o m
from
i s the fibre
f we product
S . I f a : [ n ] -* [ m ]
a* : X -> X i s t h e m a p ( x . . . . ,x ) m n o m
an obvious
with
n
g
have
space
x
over
i n t h e Hodge t h e o r y o f
over
x M o f n+1 c o p i e s o f M o v e r
We
G
such
by M .
a simplicial n + 1
R
(R) •
L e t f : M -> S b e a m a p b e t w e e n
(M/S)
space
:
R
algebraic
simplicial
M over
+
(dj_)
[n] t oM a n da l lmonotonic
obtain
this
simplicial
schemes
X(R) i s t h e sequence
x
the constant functor
simplicial
variety
f o r example
simplicial
maps
a simplicial
a semialgebraic Every
a
X over
: O r d ° -> C a n d t h e f u n c t o r
the boundary
( s
algebraic
[De] a n d i n e t a l e
i s a semialgebraic
Similarly
i i i )
iv)
an important role
I nmore c o n c r e t e terms,
neracy
space
i n the category C o f algebraic
The composite
together
us
play
algebraic
WSA(R)
ii)
object
objects
from
i s a weak p o l y t o p e .
n
simplicial
emanates
F o r example,
weak
Examples
spaces has
p r o p e r t y Q.
n
further
simplicial
X t o S, w h i c h
(x , a (o;
. . . ,x
sends
(
w
X 0
i s
, ,) . a (n;
'"'
, / X
m^
f ( x ) .
a monoid
m
semialgebraic structure
monoid,
(associative,
i . e . a weakly with
unit
semialgebraic
element
e) such
that
t h em u l t i p l i c a t i o n
braic. the
m a p G x G -> G ,
T h e n we c a n d e f i n e a s i m p l i c i a l
n-fold
product
G
N
(h..,...,h ) w i t h i m
oc(i-1) < k < a ( i )
{empty
1
product
-> x y ,
space
= G x ... x G . I f a
a*(g ,...,g ) = i n
if
(x,y)
i sw e a k l y
semialge-
WG a s f o l l o w s :
(^G) i -
: [ m j -> [ n ] i s m o n o t o n i c
1
h. t h eordered
= e } . T h u s we
product
s
N
then
o f a l l g, w i t h .K
have
1 < i < n-1,
d
d
and,
( g
o
V
1
(
n 9i
9 >
( 3
(
n
next vi)
NG t h e n e r v e
volume
" - - '
g
n
XxY x
[n]
x n
[SFC]
obvious the
over
Y
=
(g,
o f t h emonoid
' 9 i + l " - " 9
product
vii)
F o rany family
rect
sum X
of this over
:= U ( X J A € A ) A
)
play
*
a role
R then
Clearly
maps p r ^
XxY
we o b t a i n a
i n t h eobvious
i n the
a
simplicial
functor
together with t h e
: X x y -* X, p r
of simplicial
only
book).
o f X and Y i nt h ecategory (X^IXCA)
- 1
t h efunctors X and Y into
Ord° t o WSA(R). projection
n
G. I tw i l l
spaces
R by combining
simplicial
direct
e
9 i '
(cf. introduction
from
n
'
9„_1>'
I fX and Y a r es i m p l i c i a l
space
)
i € [n],
i 9l'---'9 -.,>
We c a l l
2
=
n
f o r every
S
=
2
: X x Y -» Y i s
sWSA(R).
spaces
way, X n
we may f o r m
:= U(X, IAEA) An
the d i (cf.
IV.1.10). viii)
I fR i sa r e a l
space
X over
functor WSA(R)
X
ix) of
R yields
sets,
a simplicial
Every
map f ^ : X ( R )
Every
field
: O r d ° -» W S A ( R ) w i t h
-> W S A ( R ) .
p l i c i a l
closed
simplicial
extension o f R then space
t h ebase
simplicial
map f
X(R) o v e r field
every
simplicial
R by composing t h e
extension
: X -> Y o v e r
functor
R yields
a
sim-
Y(R).
s e t K, i . e . s i m p l i c i a l
gives us a s i m p l i c i a l
space
K
R
object i nt h ecategorySet
by regarding every
set K
as
a
discrete
space
importance x)
over
f o r us,
cf.
Conversely i f X
space
X
6
by
simplicial space
In
X^
the
t o X.
simplex
3.
x
The
a
:
x
i s called by
DX^
Proposition : Xg
ii)
n>q
and N
=
1.3.
and
the
i)
a*
i s a
closed
If a
:
X
X
simplicial 6
-» X
i s a
primary
space
are
R
over
called
a
map
n-simplices
from
of
e x i s t s a monotonic that
The
degenerate
nondegenerate
discrete the
X.
such
set of
as
R.
the
i f there
X
simplicial
x
=
X.
n-
surjection
a* (y) .
Otherwise
n-simplices
n-simplices
An
by
of
NX^
X
i s
{D
=
:
[n]-»
injection
[q]
which
embedding
and
i s a monotonic
has
a weakly
a*(X )
i s a
g
i s a monotonic
has
a weakly
semialgebraic
r e t r a c t of
injection
semialgebraic
surjection
cosection.
X . n
t h e n a* : X ^ q
section.
then
X
i s a n
In particular
a*
strongly surjective.
A l l a
of
a
simplicial
of
J
is
be
obtain
set of
some y € X
set of
I f a
[ n ] «-» [ q ]
which
w i l l
R
"new"}.
i s an
surjection
K
t h e n we
discretization
degenerate,
-> X ^
Thus
of
nondegenerate.
"degenerate",
a*
the
map
simplicial
points
[n]—**[q] w i t h
denoted
X^
i s a
i s called
spaces
space,
underlying
identity
call
following X
simplicial
simplicial
the
The
We
These
§6-§8.
i s a
regarding space.
Definition
R.
this
left
The
part
subspace
Remark let
inverse
first
open
i s evident
X
1.4.
n, a
a
has
a
i n the
second
of
proposition
the
of
Keep
denote
since
right
inverse
i n the
first
case
and
case.
implies
that
NX
=
n
X \ n
DX
R
i s
an
X . R
n
fixed.
the
open
For
every monotonic
subspace
a*(NX
q
) of
s u r j e c t i o n a : [ n ] -» a*(X
q
). This
i s a
[q]
locally
closed x £ X
subspace
has a unique
n
surjection subsets
X
and y
Y
of X
n
p,q € 3N
q
N
i f
such
notion
c i a l
space
space
as
Y
I f f
and i f f
I f k < n
n
n
(
f =
We
we
: NX^ —>X q n, a
surjections
from
obtain
a
z n
c
)
n
(Y ln£]N n
define,
locally
closed)
locally
closed)
i n X .
expectations.
map
g a simplicial
Y
i s a
i
: Y ^
a
simplicial
from
f o r every k £ 3N I f k > n
q
map
from
simpli-
have
X i s
a
Z t o Y.
, a subspace
then
i n X
n
Q
Y^ o f X^
i s the union of the
through the f i n i t e l y
many
mono-
[n].
1.5. I f 3
: [p]
i s a closed
o f X and w r i t e
reasons
: [ p ] -> [ q ] ,
f o r e v e r y n € ] N , t h e n we
R
a running
[k] t o
a
t h e n we
[q] i s monotonic
subspace
n
Y = X .
shall
write
X n
subspaces
(open,
i s a simplicial
= X^.
with
resp.
) of
O
map
o f t h e space
way, and t h e i n c l u s i o n
i°g w i t h
(Y, | k € l N ) K. O
f o rother
Y
[ n ] t o [ q ] . We
as a s t r a t i f i c a t i o n
the usual
X
t h e n Y^
the n-skeleton
sk (X).
a*
closed
open,
Y meets
and D e f i n i t i o n . Thus
p
i s called
: Z
from
) c Y
monotonic
union of the
i s a sequence
i n the evident
surjections
needed =
map.
of X
(resp.
a*(X )
subspace be
R
Y
Y
subspaces
q
[q] a
i s the disjoint
a* (Y^) c Y ^ f o r every monotonic
be g i v e n .
q
Proposition 3*(Y
that
over
follows.
closed
o f a l lX n, a
subspace
factorization
n £ 3N
tonic
through the monotonic
o f subspace
Z t o X,
unique
n
: [n]
every
X
i s closed
simplicial
Let
-
. The subspace N
X
a
7f ] that
4
4. A
every Y
Thus
t h e isomorphisms
u
This
a
nondegenerate.
of the family
Definition
[ L a , p.
bijection
a running think
u
known
d e s c r i p t i o n x = a*y w i t h
3
U U NX q=0 a
may
. I t i s well
. Combining
n, a
semialgebraic
with
of X
o f X.
We
call
I f superscripts more
then this
n
elaborately
w i l l
Proof.
I t s u f f i c e s t o study
The
assertion
i s evident
and
q>n
the assertion
q n then of
3*x
n
N(X )
k
and
above,
i s empty. I f
m
X .
i n the category
limit
of the
sWSA(R) , w i t h
family the
inclu-
§2.
R e a l i z a t i o n o f some s i m p l i c i a l
Starting
with
IXI
R by r e p l a c i n g
over
gluing
these
Definition (i.e.
simplices
1. a ) We
n-simplex
e
e
T
o' 1''"' n^ ' t
h
e
o
i
n
V(a) i s t h e a f f i n e e
stead
of V (a).
b)
a
X denotes
over c)
such
a covariant
^
Q
o
+1
over
R)
functor
( i nt h e c l a s s i c a l f
v
([n])
are
+ ... + t
1
n
t
=
we w r i t e
n
e
from
n-simplex
and
tuples
from
O r d t o WSA(R)
V([n])
sense,
1. I f a
(= l i n e a r ) m a p
Usually
V
as f o l l o w s .
i s the closed
with
the vertices
(t ,...,t ) Q
€
n
R
n + 1
: [ n ] -* [m] i s m o n o t o n i c
V([n])
V(n) instead
on t h e s e t X t h e c o a r s e s t
~
t o V([m])
o f V([n])
which
a n d a*i n -
X
xv(n) n
equivalence
relation
~
(x,a*t) t€V(n)
equivalence
(s x,t) i
x € X
n
,
x € X
n
,
relation
1
~
(x, ( o )
~
and n
a value
n
, n>0,
sections
(x, t)
d i f f e r e n t base
stead
of V(n).
that
and
n>1.
(starting
more b r i e f l y
classes
the natural
by
A If
one such
that
L
: X -» |X| d e n o t e s
v
: [ n ] -> [m] . N o t i c e
the coarsest
the s e t of equivalence
x
a
t)
0 < i < n , t 6 V ( n - 1 ) ,
t o I X I .I n l a t e r
note
1 t
i s also
map
{6 )*t)
(x,
|X| d e n o t e s
lation,
and monotonic
0 < i < n , t G V (n+1)
(d.x,t)
X
space
that
this
d)
geometric
a
t h e d i r e c t sum |J(X x V ( n ) | n e n N ) o f t h e s p a c e s n o
a n y x € X^,
for
n
to build
1 )
introduce
(a*x,t)
for
x € X
X we w a n t
R.
We
for
and t
(j_)«
by a t r u e
s
then
e^ t o
n +
t
every
sends
each
space"
i n R P
> 0
space
define
with
i
simplicial
together.
a "cosimplicial
standard e
the given
spaces
from
of this
equivalence
projection
3.6) we
l x , t l ( x € x
from
shall
r e -
theset
u s u a l l y de-
, t € V (n) ) . n
fields
a r e under
consideration
we
write
v
( n )
R
i n -
e) r
X denotes
: X -> I X I d e n o t e s
x
means
In
the interior
We
want
semialgebraic We
shall
then
We
space
succeed
shall
shall
Lemma
call
need
2.1
two w e l l
this
under
known
appropriate
£
i nX contains
a unique
the proof reference.
o f X then
o f t h e second
be
Write
a
We
obtain
x =
we
conclude
again
2.2
also
two
points i n
3*oc*x =
3 * z . Thus
p = n, a =
Z i s a subspace Z which and s h a l l
Z w i l l
: [n]
a monotonic
Z be a subspace
T h u s we may rarely
z = a*x w i t h
exists
that
Let
§8)
on X and
of the simplicial
space
X
facts.
I n other
point
(x,t)
words,
with
since
every
xGNX , R
f o revery
Z n
3
i n X
surjection
then
and
ao3=idj-^
: [ p ] [ n ] w i t h z i s nondegenerate
i n Z
z €NX . n
Z i s a subspace from
i n X are already
I Z las a subset equivalence
an
flNX„ c N Z . L e t n o w z € N Z n n n
[p] a monotonic
hence
n € IN . o
I d i dn o t f i n d
z € Z^ i s n o n d e g e n e r a t e
I t i s now c l e a r
are equivalent
consist of f u l l
fl NX_ = N Z n n
x € Z^. S i n c e
o f X. T h e n
regard
n
injection
i d ^ - j ,
o f X.
Z
lemma,
I f an n-simplex
3
There
hypothesis
(IV,
Q
i n Z. T h u s
xGNXp.
weakly
n € ]N .
z i s nondegenerate
given.
i n some
i s identifying
x
i sbijective.
x
certainly
J
n
an a d d i t i o n a l
combinatorial
p . 3 6 ] .T h e map
role
the structure of a
IXI t h e r e a l i z a t i o n
c
give
an a u x i l i a r y
t h e map
2.2. I f Z i s a subspace
shall
play
R such
that
as usual, V(n)
V(n).
with
i n doing
class
t e V (n) , some
over
simplex
only
o f X and
t o X. H e r e ,
x
t h e s e t IXI
t h e space
[La,
equivalence
Lemma
t o equip
Q
of n
o f t h e geometric
X w i l l
x V(n) In6 3N )
n
the restriction
t h e f o l l o w i n g t h e space
proofs.
I
subspace LI(NX
t h e open
o f X. B y Lemma
2.1
that any
equivalent
o f IXI { a l t h o u g h
classes
Lemma
of X}.
i n Z.
only
In
particular
tice
For
that
1
We
n
regard
IX I
IXI i s t h e u n i o n
convenience
IX" I
we p u t X
as a subset
o f these
of
IXI f o re v e r y
n G 3N
. No-
Q
subsets.
^ = 0, t h e e m p t y
simplicial
space,
hence
= 0.
shall
need
2.3. n
Lemma
Proof.
v
A
two more
(
x
n
£ = n (x,t)
write
(Lemma
shall
2.1).
n o t need
=
n
maps
v
A
with
x
£
easy
v(n))
x
n
Of course,
:
by
we
We
v
A
x
( ))
v
x € N ( X
n
)
k
Choose
into
,
k < n .
=
n
n
X^ x v ( n ) n
have
this.)
combinatorial
lemmas.
I* I
for every
n
n
n
IX I.
t 6 V ( k ) , (Moreover
n € 3N
L e t £ £ IX I
both
be given.
uniquely
surjection
We
determined
x € NX^ b y Lemma
some m o n o t o n i c
. o
2 . 2 , b u t we : [ n ] -» [ k ] .
a
o
There
exists
€
A
Let
s €V(n)
= H (x,a s) x
+
denote
n
A
some
a (s)
q.e.d.
t h e complement
n
U
o f NX
t h e boundary
that
A
subspace
2.4. n
Proof. If
We
x 6X
then
i sa closed
n
have
Y
choose
DX
n
n
onto
n
=
v
A 1 some m o n o t o n i c
n
n
lies
Using
Lemma
V(n),
as usual.
Notice
x V(n).
n
. Thus
c e r t a i n lJ y 1
(6 )
+
n
(s) 1
i n lX ~ |.
with Thus
n (DX A n
xV(n))
Y
v
n
1
|X ~ |.
i e [ n ] , s € V (n-1) ,
some n
p+q
i nt h e c l a s s i c a l
sequence the
t o h
us a homological
E as
(h ( X ) ) q
p
+
q
(
from
I
X
P
|
P
1
a natural
p
X
"
P
1
|
)
V (|xP|/|xP- |).
=
t h esubspace
isomorphism P
IX I/|X
P
1
g
( c f . 2.7) t h a t
along
(X /DK ) ^ p
I
above
t o |X " I
A
'
-
1
I,
IX^I
c a nbe obtained
( X x 3V ( p ) ) U P
by gluing
(DX x V ( p ) ) . P
Thus
E
V V V =W V D
s
P , q
The
theorem
;f.
[ S e , p.
now
D
f o l l o w s by
a c a r e f u l study
of the differentials
d P /4
If
h*
109f.J
i s a cohomology
cohomological
E
but a
=
H
P(h (X))
i n general,
quotient h
P + q
of h
(|X|)
over
s p e c t r a l sequence q
P,q
theory
:
=
+
q
h
we
obtain
i n t h e same way
a
with
,
as a consequence p
R then
( I XI), P + q
o f V I , 6.11, t h i s
w i l l
converge
to
fact
i n [Se, §5]).
namely
(|X|) / l i m
(
1
)
h
P
+
q
1
P
~ ( I X | )
P (cf.
If
[W,
h*
13, §3], Segal
i s ordinary
group it
Chap.
G
cohomology
( i . e . h°(S°)
turns
out that i n [w,
argument
= G,
seems t o i g n o r e
H*(-,G)
q
h (S°)
= 0
with
631]).
Thus
coefficients
f o r q * 0,
the lim^-subgroup
p.
this
of h
f o r a n y G,
P
+
q
i n some
c f . V I , § 3 , D e f . 2)
( l X l ) i s zero
we
abelian
have
then
(cf. the
a converging
spectral
sequence P
q
H (H (X,G) ) = * H P
Example c i a l
P
+
q
(IXl
2.16. Assume t h a t
set (cf. 1.2.ix).
,G)
X
.
(2.15)
i s discrete, i.e. X = K
I f h*
i s ordinary
homology
R
with
H*(-,G)
K a then
simplih
(X) = 0 q
for
q*0.
Thus
t h e homology
an
isomorphism
from
of
the s i m p l i c i a l
s p e c t r a l sequence
the c l a s s i c a l
set K
collapses.
" a b s t r a c t " homology
( c f . §7 b e l o w )
to H
(|K_.I,G).
p,
sequence
an i s o m o r p h i s m
f o r H*(-,G) from
P
H (K,G)
also
collapses
and gives
P
.
to H (lK l,G) R
group The
us
Hp(K,G)
cohomology
K
P spectral
I t gives
us, f o r every
§3.
Subspaces
In
t h e whole
If
Z i s a subspace
cal
generally
subsets
we w i l l
look
(after
simplicial
then Lemma
at simplicial
space IZI
we r e g a r d
over
R.
as a
2.2).
subsets
Z o f X, f o r t e c h n i -
Z
n
subset
of X
a*(Z ) c q
such
n
that
Z o f X i s a sequence z
P
f o revery
(Z^lnenN^)
monotonic
of
map
: [ p ] -» [ q ] .
we m a y the
simplicial
s a y more
subspaces
sets
formally
with discrete
that
level
:= U ( Z
Z
:= U ( N Z x v ( n ) |n€3N ) n o
have
x V ( n ) |n€lN
as
Z
n
Z = z n x
X
)
IZI
subsets
over
of X are just
(Ex. 1.2.x).
I x l .
o f I X I . We
Thus
every
simplicial
define subsets
Z and
.
2.2),
and n ( Z ) = x
ly
closed,
s e m i a l g e b r a i c , ...) i n X
in
x
In
particular,
subset
Z of X
C (Z) = X
i scalled
i fe v e r y
IZI.
closed
(open,
local-
Z^ i s c l o s e d
(open,
...)
.
nothing
else
a weakly than
R
follows.
2.
n
=
spaces
,
( c f . Lemma
simplicial
of X
6
Definition
A
6
|X I
we h a v e
Z o f X gives us a subset
Z o f X and X r e s p e c t i v e l y
simplicial
the simplicial
of the discretization
the settheoretic
subset
We
proper
(cf. §1, Def. 4 ) ,
of X
1. A s i m p l i c i a l
Identifying
On
i s a partially
reasons.
Definition
a
X
o f I X I , a s e x p l a i n e d i n §2
subset
More
section
semialgebraic simplicial
a subspace
o f X.
subset
Z of X i s
Proposition
3.1.
L e t Z be
i)
IZI i s a w e a k l y
ii)
IZ | =
n
a
subspace
semialgebraic n
I Z I fl l X
| f o r every
of
X.
subset
n e UN
of IXI.
. {Recall
that
X
n
means
the
o n-skeleton iii)
o f X,
I f Z i s closed with
map
£
clude ii)
that
We
lX |
=
c
lz l
=
C
n
x
NZ
We
look
X
K
spaces. =
proper.
of the
i n X.
Since
( T h . 2 . 6 ) , we
i n IXI
the con-
(cf. IV.5.1.ii)
^ x
The
xV(k)) I
,
s
P
a s s e r t i o n now
square
by
theb i -
of set theoretic
maps
x
a
r
maps.
Then
t
a
l
l
equip
a l l t h e maps
that i
We
y
Z i s closed proper.
i t c a n be Thus
obtained n
z
i
s
by
of
i n X.
Then
implies c
: z
Z.
i t s subspace
are morphisms
Z i s closed
that
Z -> | Z |
restricting
strongly
IZI i s t h e r e a l i z a t i o n
t h e s e t |Z| w i t h
i n the square
This
set theoretic bijection
subspaces.
follows
>|X|
assume i
i
,
a t the commuting
i n |X|.
o
x V(k))
e
j inclusion
Now
since
space
semialgebraic
semialgebraic
(Lemma 2 . 2 ) . T h e
j
i and
z
k
n
structure
to
NZ.
zl
with
map
( LI k=0
V
C
of
IZI*
j°n
NX
Z^ n N X ^
=
k
jectivity
n
|Z|
Z.
i s semialgebraic
i s weakly
X
(Jj
Y X
i i i )
C (Z)
IZI =
space
i s weakly
between spaces
the set
have n
and
Z of X
i n I X I , and
i n IXI i s t h e r e a l i z a t i o n
simplicial
subset
: X -> |X|
x
IZI i s c l o s e d
structure
proper
i ) The
1.5.}
i n X then
i t ssubspace
partially
Proof.
cf.
n
z
i s a
i s
i n X.
We
Thus
partially
semialgebraic
the semialgebraic
surjective.
between
conclude
that
map our
c
x
Since j
j
,
l
f
i s partially
z
i s partially
proper.
subspace
inclusion).
Caution.
I f a
is
not open
Remark s t i l l
speak
n
Z
lz l
=
lx l
of
H
Z and W
Definition Z
n
c= W
be
3.
that
two
We
We
3.3.
have
C~
cz NW
for
every
n.
The
p r o p o s i t i o n s 3.1
pond We
1
take
a
union
V
:=
of
of
Z.
i s a
this
subspace
subsets
of
happen
o f X, Z as
of X
X
6
q.e.d.
that
n
we
can
then
x
a discrete
and
the
(IZI)
1
simpli-
formula
i s contained
of
i t i s a
IXI.
i n
(Regard
Formally
(since
map
c l o s e d embedding,
subset
subset
the
i n part i i )
.
X.
subset
o f W,
and
write
hence
Z c
w.
1.4
that
Zcw,
i f
.
t
i s a morphism
i n p a r t i c u l a r continuous,
we
between
obtain
from
this
lemma t h e
following
Proposition
3.7. L e t Z be a s i m p l i c i a l
i)
I f |Z| i s a s u b s p a c e
ii)
I f |Z| i s c l o s e d
We
head
f o ra n answer
o f |X| t h e n
(open)
subset
o f X.
Z i s a subspace
i n |X| t h e n
t o the following
Z i s closed
problem:
o f X. (resp.
F o r which
open)
i n X.
subspaces
Z
of
X i s t h e s e t IZI s e m i a l g e b r a i c
Lemma
3.8. F o r e v e r y
closed
embedding.
Proof.
We
X
n
with
lx,tl
jections u€X of
verify
=
that
ly,tl.
have
\p
3
The
map
braic.
lu,a tl
Since that
t h e map every
C
x
sets
to
that
iJ^nCL
x be a n e l e m e n t
monotonic s u r -
2.1)
(Lemma
a = 3.
Thus
since
a t and 3*t are points
that
p = q, u = v, a * t = 3 * t .
+
x = y.
n
i spartially
x
o f X . We
now p r o v e
be done:
ip
be proper,
must
t
subset
n (L x V ( k ) ) x V(k))
preimage.
We
proper
and
i|> i s
semialge-
t
hence
a closed
bijection
L€^(NX^),
i s semialgebraic
of this
that
o f IXI i s c o n t a i n e d
with
simplices
Since
subspace
semialgebraic
many
Let
lv,3*tl.
: X -* I X I i s a s e m i a l g e b r a i c
finitely prove
L e t x and y be p o i n t s i n
x = ot*u, y = 3 * v w i t h
=
+
i s p a r t i a l l y proper
T h e n we w i l l
( c f . 3.6) i s a
: [ n ] -» [ g ] a n d n o n d e g e n e r a t e
equality implies
x {t} i s a closed
: X ^ -> |X|
i s i njective.
t
We w r i t e
: [ n ] -» [ p ] , . We
t h e map
V ( p ) a n d V ( q ) we c o n c l u d e last
n
a
, v € X
The
X
first
tGV(n)
i n IXI?
i t i s evident
i n t h e union
s o m e k. T h u s i n X
have
lx,tl
=
of
i t suffices
f o rsuch
n
embedding.
a s e t L.
iy,sl
with
some
0
y€L, tion
sGV(k).
We w r i t e
and u €NX . m
We
x = a*u w i t h
have
lu,a*tl
=
a
: [ n ] -» [m] a m o n o t o n i c
ly,sl
and conclude
surjec-
m = k, u = y ,
-1 a t = s. Thus +
algebraic
from
^
sets
t
n (L xV(k))
a*L w i t h
[n] t o [k] .
i s t h e union
a running
( I np a r t i c u l a r ,
of the finitely
through
the monotonic
—1
°
n(LxV(k))
i s empty
many
semi-
surjections
i fk > n . ) q.e.d.
Lemma 3.9. I f Z i s a s i m p l i c i a l for
every
Proof.
subset
o f X then
Z
n
= Z n X
n
n.
B y 3.2 w e h a v e
n
IZ I
=
n
IZI n I X I
=
n
IZ n X I .
We
conclude
b y 3.3
that
Z
= Z DX .
n
n
(Of c o u r s e ,
one could
prove
t h e a s s e r t i o n i n a more
c o m b i n a t o r i a l way.)
3.10. L e tZ be a subspace
Proposition
i n IXI
algebraic
Proof. Then Z
n
Assume
Z = Z
n
Assume
3.9, a n d IZI
t o be s e m i a l g e b r a i c
o f I X l we h a v e
(cf.
3 . 3 ) . Now l o o k
This
map i s s e m i a l g e b r a i c
by
Lemma
3.6. Thus
n
f o r some n .
Izl c|x l n
f o r some n .
IZI t o b e
implies
Since
n
b y Lemma
n
this
t h e s e t |Z| i s s e m i -
and ZcX
= n(Z *V(n))
Izl i ssemialgebraic.
filtration
and Z c X
Z i ssemialgebraic
that
b y Lemma
now t h a t
missible
i f f Z i ssemialgebraic
f i r s t
i sassumed
o f X . Then
( IX I n
I n € 3N
2.3. Since semialgebraic.
) i sana d -
f o r some n , h e n c e
Z I X I f o r s o m e m a n d t € V (m) . t m p r o p e r ) b y Lemma 3 . 8 , a n d ^ ( l Z l ) = Z
a t t h e map (even
t
1
m
Z i s semialgebraic. m 3
Given host
a nonnegative
o f semialgebraic
every
k E 3N
A* K
with
i n t e g e r n we d e s c r i b e subspaces
we d e f i n e
Q
a subset
a running
through
N A * c= A S b e
and l e tF denote
Definition
i s an isomorphism
to study cartesian
their
f
this
fibre
simplicial
t h e map
f xg
maps
from
between
simplicial
X *Y to S KS,
product o f X and Y w i t h
respect
t o f and g i s
subspace y
g
:
=
F ~ (Diag S) 1
x y.
of
X
In
more
concrete terms,
Xx
Y i s t h e s u b s p a c e (X_ x Y I n € IN ) o f XxY, S n S n o t h e u s u a l f i b r e p r o d u c t o f t h e s p a c e s X^ a n d Y n n c
n
where X with are
x
n
S
n
respect c
denotes
n
to f
n
: X
p a r t i a l l y proper
complete)
We
Y
have
9
and g^ ^n
n
: Y
->
n
n
. Notice
proper, semialgebraic,
h a s t h e same
a commuting
g
->
(resp.
t h e n X x^Y
X x Y
n
t h a t , i f X and Y '
complete,
partially
property.
square
Y
>
I g
of
simplicial
canonical
spaces
with
projections
straightforward
way
(4.10)
p and q t h e r e s t r i c t i o n s
p r ^ : X x y -* X ,
that
this
diagram
p r
2
to X ^
: X x Y -* Y.
i s cartesian
One
y of the checks
i n sWSA(R).
in a
Assume now
that
X,Y
and S are p a r t i a l l y
Lemma 4 . 1 1 . T h e i s o m o r p h i s m
h
:
v
proper.
I X x y | ->
\ x I x |y | maps
|X x
X , x the
fibre
|X| x
product
(
g
|y|
|
Y|
onto
o
of
IXI a n d
lYl with
r e s p e c t t o IfI and
Igl.
Proof.
The isomorphism
and
Thus
Y.
IX
h
Y
a
p
Theorem the
|Y|
of 1
If
I D i a g SI
( D i a g S) I
S
I
X
IS I
IX x
=
Finally, X
S
Y
l
°
n
t
°
4.12. Assume
diagram IX x
s
g
X
l
X
Y|
s,s
isi
I f l x |g| i s IXI
|Y|,w h i l e t h e
Y|,
g
g
maps
I D i a g SI
onto
D i a g ISI . Thus q.e.d.
I X I
lSl
again that
( c f . 4.10
respect to X
IFI i s
b y 4.9, h l
x
under
under
with
x SI
ISI
I x | g |
of Diag
3.20). m
IS
functorially
square
h
x
IF"
X
a commuting
X,Y
preimage
h
behaves
v
IFI
preimage
(cf.
have
x Yl
IXI
The
we
h
X,Y
and S a r e p a r t i a l l y proper.
Then
above) |Y| igl
IXI
is
ISI
cartesian
Proof.
We
i n t h e category o f spaces
compare
this
diagram
with
WSA(R)
the canonical
cartesian
square
IXI x
|Y|
l s l
I Yl
ISI
{Of
course,
the
theorem h
with It
=
T T
T T
1
/ 2
a
r
e
commutes
f ,n g
TT^h =
t
h
e
n
: IX x Y l s
r
a
projections.}
us a unique
ISI
Since the diagram i n
map
|Y|,
have
to verify
that
h i s an
isomorphism.
the diagram
IXI x
g
l
IXI x
that
h
IX x Y|
u
| q | . We
2
checked
t
i tg i v e s
| p | , TT " h =
i s easily
a
|Y|
| s |
3 I X x y|
IXI x ( y l % Y
with
i and j i n c l u s i o n
mappings,
commutes.
{Recall
that
h
= A ,I
( I p r ^ I , I p r I ) .} We 2
learn
from
Lemma
4.11
above
that
h i s indeed
isomorphism.
In
q.e.d.
t h e course o f t h i s
Corollary is
an
4.13.
p r o o f we
The n a t u r a l
a restriction
have
seen
isomorphism
of t h e isomorphism
h
h
from
f
IX x ^ y l t o I x l x ^
|y|
. A ,Y
We
now a r e amply
shall have
also
do s o i n l a t e r
g
t o identify sections.
x
IX g Y | w i t h
Under
this
IXI x ^
|Y| a n d
identification
we
shall
the equation I(x,y),tl
for
j u s t i f i e d
=
(lx,t|,ly,tl)
a n y t € V (n) , x e X , R
y € Y
: Y -* Y' a r e s i m p l i c i a l
spaces
over
a common
n
(4.14) with
f (x) = g (y) . I f f
maps between p a r t i a l l y
partially
proper
space
: X -> X' a n d
proper
S then
l f
x q
simplicial g l
=
I f lx
|g|
We
present
an
application of
Definition
4.
A
group cial and
object space
G
the
G
a*
be
a
unit
proper.
use
We
(m (x,y)
a map
on
unit
4.1.}
I G l . The
f o r y.
1
that
Proposition m.
a*e
4.15.
Assume t h a t
IGxGI
=
sition
Iml
Q
the
=
e
=
R
:
IGl
|e ,t| = n
a
above.
gives
us
a discrete simplicial
group
| r
p l i c i a l group
sets.
object
R.
For ZK
Such
group
example,
i n sSet
object
with
simpli§11)
-» [ n ] m o n o t o n i c ) .
group"
instead
of
denote
y € G
n
. )
by
y
(simplicialI) denote
n
G
I m I
X
n
n.
discussion
partially a
us
G-space
h
on
n
defines
space
over
famous
abelian
give
simplicial
straightforward
spaces
IGI
H*(-,TT)
the
are
for a l ln
(left)
(left)
group
86ff.]
A
a
x |X|.
example,
5.
is
IGI
p.
R
cohomology
Definition
p l i c i a l
[EM,
lK(TT,n) |
realizations
together
another
of
one
obtains.
Assume identify
the
weakly
that
the
IGxXI
sim=
semialgebraic
We
want
space
to analyze the realization
by a s i m p l i c i a l
We
first
to
another
not
equivalence relation
a very
special
one a l o n g a c l o s e d
y e t need
Assume f
discuss
that
Brumfiel's
A
i s a closed
Z as A
n
:= X U
follows.
Z
In this
The
n
transition
call
A
map
Z = X U
i
-
favorable of a
subspace.
conditions.
simplicial
In this
of a simplicial
simplicial
situation
we
X U,- Y_ n f n
map
define
space
from
A
case
space we
do
X and
t o a
a simplicial
obtained by gluing J
3
3
n
have
maps a*
f
p r o p e r map
a*
f
commuting
: X^
( c f . I V . §8) . I f a
second space
n
X^
t o Y_ n
along 3
: [ k ] -» [ 1 ] i s
squares
• X^ a n d a*
: Y^
> Y^. c o m b i n e
into
the
: Z^ -> Z^ .
Y the simplicial space
space
Y may
obtained by g l u i n g
and w i l l
o f Z i n t h e o b v i o u s w a y . We f
simplicial
IV.11.4.
proper
r
A by_ f . T h e s i m p l i c i a l subspace
the gluing
simplicial
subspace
i s t h espace
t h e n we
transition
under
of a
Y
by t h e p a r t i a l l y
monotonic
We
space.
case:
theorem
: A -» Y i s a p a r t i a l l y
simplicial
of the quotient
have
be regarded
a commuting
X to Y
as a
closed
square
• Y | j
(5.1)
with
i and j i n c l u s i o n
maps a n d g t h e o b v i o u s
from
X t o Z e x t e n d i n g f . We
know f r o m
simplicial
I V , §8 t h a t
every
along
map component
g
n
is
:
X
Z
n
"* n
°
strongly
§1),
that
jective that
f
g
g
a
P
r
t
i
a
l
i s partially
(and p a r t i a l l y
the diagram spaces.
Theorem
5.2.
:=
s
I f X
IAI
—
means,
according
(g ,j ) n
proper).
One
checks
i n a
proper and
:\
|J Y
f
Yl
Let a
we
|X| U
may
| f
a
n
(cf.
straightforward
then
sWSA(R)
>ay ofsim-
the simplicialspace
the diagram
( c f . 5.1)
(*)
*
U
a
*
Recalling and
from
shall
Proposition
4.6
that
If i s
identify
j IYI .
: [ k ] -> [1]
XiU*! —
Z
> IZI
i n WSA(R).
=
-
> lY!
proper,
IX U
R
-» Z i s s t r o n g l y s u r -
i n the category
are partially proper
n
to our terminology
( g , j ) : X|JY
partially
^
partially
be m o n o t o n i c .
'
x k
U p
Pi •I
We
have a commuting
square
Y k
k
1
with
strongly
tion
about
This
implies
We
that
I j l
cocartesian
X
surjective partially and Y that
reasoning
T h e map
and
and
and Y
i s again
IXI
the
Proper
(5.1) i s c o c a r t e s i a n
XU^Y
Proof.
y
proper
lil
is
l
s u r j e c t i v e . This
p l i c i a l
Z
i
the upper the lower
i n the proof
proper
horizontal horizontal
conclude
by
(Igl,Ijl)
Proposition :
IXLJYI
=
arrow arrow
o f 4.6). Thus
( g , j ) : X | j Y -»• Z i s s t r o n g l y 4.6
and p^.
a*|Ja* a*
By
IZI
our assump-
i s p a r t i a l !
i s partially
Z i s partially
proer
proper (cf.
proper.
s u r j e c t i v e and p a r t i a l l y
that
IXI U IYI -
maps p ^
prper.
is
again
We
a r e done
sian
strongly i f we
know
that
on t h e s e t t h e o r e t i c
jective
Ij|
and
IXI ^ IA I
maps
Let
s u r j e c t i v e and p a r t i a l l y
level.
i s injective.
injectively
Write
=
lg (x),tl.
9 (x) =
a*(z) with
=
that
Igl
a
only
Suppose
that
need
£ =
lx,tl
with
I g l (£) € l Y l .
g (x)
= a*(z)
€ Y
conclude
that
indeed
return
conclude
Write and
This
I X l ^ l A l
t o the point
from
Ix
lg (x),tl
1
contradicts
x € NX ,
t € V(n).
n
Then
s u r j e c t i o n and
that
this
= t
our assumption
have
z = g
R
x
= x', hence
a =
point
x' € NX , m
,
Since
g
id^-j,
n
(x ) with '
that
x
1
and then
x
^ A
We
= a*x^ . S i n c e
x
*d
q
R
t ' € V (m) . T h e n
|g|(£')
=
g 'x) eNZ n
b y L e m m a 2.1
i s injective
i s the one-point
the following.
Thus
.
on X
n
^ A
n
,
Igl(£). g (*') e NZ
R /
that
£ , as desired.
Y
£ £A.
€ X
1
•
1
£ =
this
i . e . g ( x ) € NZ^.
i n I X I ^ |A| w i t h
conclude
m
n
that
n
implies
, g (x) = g (x').
the case
g ( x ) = g («*x^),
| g ( x ) , t ' | . We
n
theory
i Z l ^ l Y l .
I g l (£) . We
,t ' I w i t h =
n
means
that.Igl
that
into
i s a second
t
In
above
that £' =
f
to verify
i s sur-
.
n
xCA^.
nondegenerate
Assume
(lgl,ljl)
By o u r subspace
q
is
that
: [ n ] -» [ q ] a m o n o t o n i c
and i m p l i e s
maps
We
already
i s cocarte-
I Z I ^ IYI .
Write
z € NY^,
n
We
we
i n the theorem
Iz,a*tI,
a* (t) € V ( q ) .
means
know
identifying.
Then
Igl(?) and
(*)
hence
n
n
z € NZg.
We
Thus
into
£ € I X I ^ IA I b e g i v e n .
Igl(5)
the diagram
proper,
m
m
we
=
n,
obtain
q.e.d.
simplicial
space
{*}
the theorem
m
Example t i a l l y
complete
{with tion p
5.3. L e t X be a p a r t i a l l y
(X/A) I pi
=
n
:
We
now
way.
=
every
the definitions
that
call
proper,
...)
proper,
proper,
Example
5.4.
E(f)
spaces.
with
respect
equivalence
T
...)
1X1/1AI
This
relation
f o r every
: X -* Y
denote
equivalence
a*
: X /T
We
denote
n
relation
simplicial
i n short,
on
equivalence automati
simplicial
T
on t h e space
o n X^
n
subspace
(partially i s closed
X
n
f o r
proper, (partially
n.
the first
o n X . We
map
then
have
and t h e second E(f)
n
T
T
i
s
t
n
e
s
n
e
t
the fibre
n
factor
product
i s a closed
) f o r every
o n X . We
T t o X and by T a r e now
n.
denote
simplicial
as f o l l o w s :
theoretic
map
induced
(X/T)
from
by a*
b y p^
the switch
t
maps.
:
the simplicial
x n
We
i s theset
n
: [ p ] -> [ n ] i s m o n o t o n i c ,
the natural projection
s e t X/T.
( f
relation
set defined I f a
E
=
(as i n IV, §11). These
classes X /T .
-* p / p by p
i s a
T closed
i s a simplicial
the simplicial
X
n
map
i n a somewhat
relation
relation
T i s an equivalence
of T
of
realiza-
y
relation
b y X/T
o f I V , §11
T on X
i s an equivalence
n
t o f i n both
automorphism
projection t o IX/AI,
c a n be done
t h e two n a t u r a l p r o j e c t i o n s from
2
and the
X/A
x X
the following p
proper
space
par-
space.
i fthe equivalence
:= X
from
and A a
the simplicial
i s partially
and r e s u l t s
the equivalence
I f f
Then
space
9
1. A n e q u i v a l e n c e
n . We
n}
an isomorphism
IXI / I A I
such
o f X.
simplicial
of the natural simplicial
L e t X be a s i m p l i c i a l
T of X xx
and
subspace
f o r every
n
to simplicial
Definition
In
/A
induces
extend
relations
n
|X| -> I X / A I
: X -> X / A
IX/AI
(closed) X
proper
then ~*
x p
«
set X to the
Definition (a
2.
s i m p l i c i a l map
p a r t i a l l y proper
and
every
map
partially
It
i s clear
strong
I t
X
by
space,
Brumfiel's
Theorem and
i s
of
§8
the
T.
In
We
want
to
this of
We
assume closed
again
We
that
i s a
cT
we
; the
switch
We
conclude
I (TxX) and
there set of
X
i s a
X/T
X
by_
quotient
T,
i f E(f)
surjective
=
T
and
T
of
the
space
fl ( X x T )
I =
ITXXI
again
by
X/T
of
a
a
sim-
strong
quo-
this
simpli-
set.
as
follows.
T
on
X
i s
closed
p a r t i a l l y proper
X.
Then
i s a
i s the
fl I X x T l 3.19,
switch
= IXI =
space
subspace
cf.
itself
under
simplicial
relation
X xX
IXXXXXI
the
closed
IXI c IT I,
into
behaves
simplicial
Diag
i s mapped
the
relation
equivalence
that
by
relation
Then
i s a
T
strong
quotient
exists.
on
ITI
mean
immediately
proper).
p
i s a exists
structure
that
we
quotient i f there
unique
simplicial
equivalence
by
that,
such
course,
extends
the
in
then,
of
strong
(is strongly
§11)
IV,
previous
and
i s an
|T|
in
p a r t i a l l y proper
automorphism that
(as
relation
conclude
Finally,
3.4,
|T|
quotient)
a
p a r t i a l l y proper
equivalence
p a r t i a l l y proper,
Diag X
• cf.
of
an
equivalence
verify
; 3.19.
X
every
case,
(resp.
quotient)
that
i s called
identifying
then
the
that
how
-+ Y
proper
simplicial
Assume
know
T,
IV.11.4
Theorem
proper
that
by
p a r t i a l l y proper
(the
i s
n
evident
X
on
a
: X
proper).
IV,
instead
5.5.
-> Y
i s also
space
of
cial
from
f
quotient,
: X^
n
quotient
p l i c i a l tient
f
proper,
quotient.
a
A
by
x [XI
on
IXI.
4.9.
The
of
realization.
and
that
space IX x XI
Indeed,
T
T i s
=
of
IXI
this
automorphism,
x |x|
we
have
( I T l x | X l ) fl ( I X I x | T l ) ,
IXI
x
|XI.
from
realization
automorphism
i s
of x cf.
|X|.
l p r [ ( T x X ) n (XxT)]l
=
13
with
factor. f i r s t pr
t
p r ^
n
natural
I p r ^ l
and
(TxX)
Proposition partially
Proof. and
third
proper
This
E(lfl)
factor.
l [ (lT|x|xi )
: X
space
Y
from
IXI x
fl(|X|x|Tl)],
X x x x x
projection
to
from
the
f i r s t
I X I x |xi
and
x |xi
(ITlx|xI) n (|X|X|T|)
third
to
into
the ITI
since
T.
-> Y
i s a
then
§4
|x|,
from
I t maps
into
I f f
follows =
natural
n (XxT)
5.6.
1 3
projection
i s the
the
maps
1 3
e
lpr
simplicial
E(|f|)
(Th.
the
=
4.12
fibre
map
from
X
to
another
|E(f)|.
and
Cor.
products
4.13),
using
since
the
maps
E(f) =
Xx
f
If I
and
y
x
respectively.
Example De X
Let
-* X
Rel X
relation
De X (cf.
by
4.7).
by
denote
We
of
define the
by
a
direct
proper
We
ready
on
and
proper
that
n
and
we
i
s
our
i s a
know
partially
deployment
E ( x
x
) .
We
partially
this
quotient n
already
from
:
x
could
2.6
that
on be n
x
of
X I X I
quotient
relation
also
Rel X
s u r j e c t i v e by
proper
equivalence
(Prop.
call
proper
strongly
space
proper
p r e v i o u s map
and
course,
the
simplicial
partially
x
the
and
a
proper x
proper
relation
s
Then
X
Proof used
verified i s
par-
surjective.}
main
T
^
x
x
of
I X I . (Of
the
partially
closed
x
map
i s just
strongly
Assume the
surjective
that
I R e l XI
state
partially
partially
conclude
proper.
equivalence
The
realization
to
5.8.
relation
X.
computation,
t i a l l y
Theorem
the
a
realization
. Thus
to
t i a l l y
i s strongly
space
4.6.
i s partially
i s again
I t i s again
I R e l XI
are
that X 1)
R e l X . The
position X
Def.
s °e X
x
4.7). the
Assume
( c f . §4, x
and
5.7.
result
i s a proper
equivalence
of
partially
this
proper
simplicial relation
section.
on
space
closed X.
IXI . The
equivalence
Then
ITI
i s a
simplicial
par-
space
X/T
( c f . Th.
of
the
IXI
i s again
simplicial
by
|T|.
Proof. (P
5.5)
I f
:=
In
map
p
short,
: X
T
IX/TI
[q] -*[n]
a
upper
maps
horizontal
horizontal
map
partially
from
the
G
are map
a*
then
be
a
we
i.e.
every
the
we
have
of
a
I
quotient
commuting
i s a
of
square
partially
proper.
Case
the
T(G)
the
image
closed X.
simplicial This
t i a l l y
by the
t i a l l y
proper
case
and
G
X
i n the Of
natural
and
X
The
partially
g i v e s us
an
action
of
X
map
theorem
cases.
G*X
a
The
the
lower
space
now
X/T
follow
group
We
1:
complete,
We n
denote =
G \ X R
We
the
G
learn
IXI
IXI
and
both •-»
first
X
assume
i s
these
cases
(gx,x) , i s
case
a
relation
and
par-
q u o t i e n t X/T(G)
more
i n the from
i s
equivalence
f o r every
R
quotient i n the on
In
closed
actions.
G-space.
(g,x)
i n the
quotient of
IGI
Case
-* XxX,
I t i s proper
case.
of
i s discrete.
i s proper
(G\X)
case
simplicial
hence
case.
proper
proper
that
simplicial
semialgebraic) space,
X*X,
projection. second
the
i n the
two
space
of
second
conclude
i n the
left
(hence
simplicial
i s a
a
following
course,
i n the
IG\X| a
group
subspace
G\X.
denote
|X|
theory
equivalence relation
proper
briefly
:
of
our
2:
Thus
We
surjective.
4.6.
of
complete
strongly
proper.
assertions
and
the
and
proper.
other
5.6
outcome
i n one G^
i s partially
The
simplicial
are
proper
i s partially
Propositions
that
on
proper
|p
IXI/ITI.
partially a*
proper.
explicate
Ipl
partially
realization
/T_ q
q
vertical
of
the
—
x
Let
and
T
T
We
=
i s a
i s monotonic
V n
is
X/T
proper
P ) X„
The
partially
n.
first
Theorem
by
p :X
case 5.8
one.
The
-*
and
G\X
par-
that
I T (G) I i n t h e
second
( c f . 4.17)
Let
first
action
w i t h T(!G|)
=
of
IT(G)I
(cf.
3.19).
proper usually well
Thus
quotient) |G|
beyond
will
the p a r t i a l l y IGIMXI n o t be
IV.11.8.
proper quotient
exists, a
and
IGIMXI
semialgebraic
group.
( i n the f i r s t =
case
IG\XI. N o t i c e
Thus
this
result
even
that lies
§6.
Semialgebraic
Let
K be
a
discrete
simplicial
simplicial
briefly with
by
and
R
ively.
l f l
that
and
l
f
l
o
and
p
top
There [Ca]
=
=
f t
t
In
principle
parts
formulated
o
p
field
from R
-> L
extension
(IfI ) ^
complexes
by
polytope
|K|
associated
or
R
stresses
R
K to a R
by
|K |
over
analogy
( c f .I I ,
o r by
R
more
the
simplicial
I f I
realization
§3).
set L
then
| f | . We
R o f K and f
call
respect-
(2.8.v).
of R then
clearly
IKI
R
, c f . 2.13.
R
[Mi^] of K
respectively.
We
have
realizations which
return
literature
introduction,
and t h e books
and f w i l l IKI
t
Q
p
be d e n o t e d
= ( I K I
m
)
t
Q
by
and
p
we
have
every
simplicial
the articles
[La] and
within
whether remain
known
theorem
the category
or not suitable
true
sets.
[Cu] and
We
mention
[Gu] f o r a
[May] f o r t h o r o u g h
involve
other
of a normal
a characteristic
at our disposal of simplicial
results
treatments
topological s e t K.
(cf.
map
n
We
V.1.3) : V(n)
which
sets,
involving
f o r our semialgebraic
t o our s i m p l i c i a l
structure
on
of the theory.
entirely
t o check
define
: K
R
i s a weak
an e x t e n s i v e
survey,
basic
results
R
realizations
f o r a pleasant
have
map
of the
^ ' m ' t o p .
of
the
If|
l f l
exists
concise
We
closed
topological t
IK|
of
simplicial
the semialgebraic
R
Notice
R
of f
a n d 2.5)
instead
R
of abstract
the realization
(|K| )(ft)
| K l
|K|
sets
the realization
( c f .1.2.ix
R
i f f i s a simplicial
I f ft i s a r e a l
The
K
denote
notation
the realization
denote
|KI
of simplicial
s e t . We
space
I K I . The
Similarly, we
realization
c a n be
but
we
topological
realizations,
l e t alone
spaces. start
out to establish
CW-complex. |K|
by
n
For every ( t ) :=
on
K
x € K
|x,tl.
n
We
we
denote
t h e image o f n
Notice
that
by
x
lxl° i s a
n (V(n))
I x l and t h e s u b s e t
semialgebraic
subset
of
x
of
IKl and
I x l by
lxl°.
Ixl i s a
poly-
tope.
If
a
:
[ p ] -* [ n ] i s a m o n o t o n i c
map
then
clearly
the
triangle
(6.1)
commutes.
From
the diagram
la*(x)l
=
( 6 . 1 ) we
I x l and
la*(x)l° =
V(n)
and V(p) onto V ( n ) .
If
i s nondegenerate
x
Moreover,
conclude
then
lxl°,
n
maps
x
V(n) i s t h e preimage
phism
from
V(n) t o
lxl°.
{|xl°
I x €NK^, n €3N } q
since
i n this
(6.1),
x € N K
t h e s e t I x l ^ lxl° i s a u n i o n
y € NK
As
,
, f o r some
usual n
(K )
R
we
we
studied
have
X =
we
n
of
v(p) onto
lxl°
(cf.
i s an
2.1).
isomor-
IKl. Using
see that,
of finitely
x
maps
that
many
the n-skeleton of the simplicial
of the family
of closed
the relation
simplicial case
2.1
partition
injective,
+
onto
. Thus
from
i s the n-skeleton of the simplicial limit
special
clear
a
then
again
f o r every
"cells"
lyl°,
p I K
n
i
with
n
the discrete
o f NK
a n d DK
n
the diagram
n
space
. Thus
i n Lemma
we
Z .
The
R
obtain
the
2.7
I
j 4
NK
xV(n)
>
T
l K
n
|
^n for
every
n>0.
and
^
the
diagram
there,
n
H e r e (x,t) n
=
n
maps tp
j a r e i n c l u s i o n mappings, { i n the case isomorphism
from
K
o Notice
of the natural
x
xV(o)
=
K
o
( t ) f o r a n y x € NK^,
to
n
=
R
0
lK°l.}
o
t € V (n) . T h e
following i s
evident.
Theorem
6.2.
The
partition
{lxl°
lx€NK ,
of
IKl which
gives
IKl the structure
decomposition n
complex. x €NK , n
lK l
i s the n-skeleton n
t h e map
x
n
of this
n € JN }
of
Q
IKl i s a
patch
o f a normal
CW-complex
: V ( n ) -> I K l i s a c h a r a c t e r i s t i c
CW-
and, f o r every map
f o r the
c e l l
!x I .
In
t h e f o l l o w i n g we
merely
a weak
Proposition
shall
polytope -
6.3.
L e t B be
Then
there
This
i s easily
verified.
of
such
|x| c B .
K
from A
:=
exists a
that
the diagram (A |n>0) n
always i n this
(unique)
We
I f a
I K l as a CW-complex - n o t
way.
a closed
(6.1) t h a t
i s a
regard
subcomplex
simplicial
define
A
R
o f t h e CW-complex
subset A
of K
such
that
|K|. B
= l A l .
as the s e t of a l ln-simplices
x
: [ p ] -» [ n ] i s m o n o t o n i c ,
then
i t follows
|a*x| c
that
the
simplicial
|x|. This
subset
o f K.
implies
Clearly
IAI =
B.
family
If
f i s a simplicial
x €
map f r o m
f(x)
= n
x
Ifl(lxl) instead
of
IL I . T h i s
As
i s well
cial
f
(
=
x
every
(6.4)
)
lf(x)l,
o ff (x).}
known
sets - there
that
somewhat
a tthe
i s a close relation
( c f . I I , §3) and
(cf.
§1]).
[Ca,
Convention.
I nt h i s
classical)
o fP b y E(P)
Definition
together with
t
i s a face
a total
restriction
marginally
from
o fthe the
b)
A simplicial
is
a map f : E(P) o f E(Q)
6.5.
complex
i n the
the
set
given
map f f r o m -* E ( Q )
We r e g a r d
o fthe
empty
one
which
subsets
maps
o fL equipped
L. In p a r t i c u l a r
with
simplicial
we r e c a l l now
a closed
(=
the
P i s a simplicial s o fP such
{This
ordered
set
complex
that,
i f
ordering o ft i s
definition
order
differs
complex
s o fP onto
preserving)
set
o fthe the
simplicial
simplex
vertices
L. Thes i m p l i c e s
o f simpli-
textbooks.}
ordered
The
cell
by S(P).
o fs) the
every
(= w e a k l y
way:
P means
simplex
subset
P t oa n o t h e r
a totally
following set
complex
i n most
a
A s i n I I , §3 we d e n o t e
o r d e r i n g o f s t ot .
i n a monotonic
Example
elements
empty
theory
sets which
complex
o r d e r i n g o feach
o f s ( i . e . a non
onto
closed abstract
o fsimplices
simplicial
write
i s a c e l l u l a r map.
simplicial
set
o f IKl
cell
between
complex.
the
1. a ) A n o r d e r e d
P
f(s)
and
{We b r i e f l y
roots o fthe
a simplicial
abstract simplicial
vertices
the
suitable
chapter
l f ( x ) l ° .
If!
o fcourse,
- and
=
If I m a p s e v e r y
Thus
R
implies,
a n d If I ( I x l ° )
complexes
of
for
n
hence
non
set L then,
NK ,
lfl-n
of
K t oa s i m p l i c i a l
complex
complex
L are
restrictions
we o b t a i n a n o r d e r e d
simplex
way.
L as anordered
o fthe
a
Q
simplicial
L are the
o fthe
simplicial
the
finite ordering
complex
[n]
f o r every
n € JN . Q
Definition
2. a ) G i v e n
simplicial
s e t P as follows.
maps is
from
[ n ] t o P.
the composite
ordered b)
Q we
I f a
complex
a simplicial
nonsense
of
simplicial
to
Q.
We
shall
Notice
maps
often
that
we
map
denote
then
^
V 0
P we
define a
The n - s i m p l i c e s o f P a r e t h e and x € P
a i s a simplicial
map
simplicial then
n
a*(x)
from t h e
[p] t o [n].} f from
map
f
P t o another
: P -> Q b y f
obtain i n this
from
complex
: [ p ] -> [ n ] i s m o n o t o n i c
define a simplicial
abstract
simplicial
x ^ a . {Notice that
simplicial
Given
an ordered
way
n
ordered (x)
simplicial
:= f ° x . B y
a bijection
an n-simplex
/*««/V }
x € P
by
R
Q
i s a n m.-simplex
N
categorial
f »-> f f r o m
P to Q to the set of simplicial
n
complex
maps
theset
from
with
of P with m o
Then
every
of P
face of x
i s
N
> with
In this
v 0
''*«'
w a y we
v n
ordered
nondegene-
i s again non-
s o f P g i v e s us a nondegenerate
f - « » / V
< v ^ < ... < v »
(closed!) s i m p l i c i a l
"restriction" Conversely
(unique) NP
at the simplicial
n-simplex
the vertices
obtain a bijection
(=
elements)
s
s
from
R
Q
the
2.10).
t o NP .
n
If
look
(cf.
i f fthe v^ are a l ldifferent.
degenerate. s
[ n ] ~ = A(n)
we
i fA
S (Q) = n
o f P, o f c o u r s e
o f t h e o r d e r i n g o f P, t h e n
simplicial
have
subcomplex
i s a simplicial
subcomplex NA . R
subset
equipped
Q i s a simplicial of P then
Q o f P w i t h Q = A.
subset
there exists
Identifying
with
a
S (P) = n
Definition to
P
In
this
cal
3.
We
call
a
f o r some o r d e r e d
case
way
as
denote from
can
follows.
the
[0]
we
simplicial
simplicial
choose
P
We
E(P)
put
i - t hvertex
map,
[n] w h i c h
sends
to
{ v ( x ) , ( x ) , . . . , v ( x ) } Q
ing
o r d e r e d by
hedral,
every x 6 K
n-simplex v
(x)
1
IPI
R
I Pi of
P,
l e t e
denote
that
we
y .
simplicial
simplicial
{e ,e^,...,e }. Q
R
We
(Notice
n
x
by
of
that,
the
from
unique
P
x a
: K
i
are
->
n
the
the
such
since
K
map
sets
set K
Q
be-
i s
poly-
sequence
to f
P
each
i s nondegenerate map
l e t v
canoni-
i n d u c e d by
n-simplices
£v (x).
one
complex
P we
forgets
the
i f fthe v^(x)
are
K which
an
with
l
compare
ordering
of
set. Let
the vertices
sends
y ,
V (X) =
Q
Q
of
the P)
s be
s with
e o
the
a
simplex of
vertices
complex
define
now
simplicial
corresponding closed
identified
map
i n a
n
.e.,...,e be i n
the
have
geometric
=
n
i € [n]
n
P to the
i t s associated
o |s|
v (x)
( c f . I I , §3,
zation
any
r u n n i n g t h r o u g h NK ,
simplicial of
ordered
over
x
i s isomorphic
cp : P ^ * K
transition
i . The
that
,. . . , y > n
Q /
1
an
and
and
cp i s t h e
u ,.../
=
Given
i d s e t K,
g
S
e
sSet
from
t
a weakly
of
D t o the
semialgebraic
s e t K.
I DK I t o |K| , w h i c h
I K l . { i n other words,
t
R
maps
i s a
every
simultaneous
K
by a homotopy F ( | x l * [0,1])
such
i n the topological ([We],
and e r r o r s .
The p r o o f
They
that
setting
c f . also have
been
g i v e s Theorem
closed
i n the
small-
.
of Weingram-Fritsch
also
f o r each
i s contained
contains t_.(|xl)
of Barratt
and thus
F
6.8
by
Weingram
[LW]).
Weingram's
bridged
and
correc-
i s completely of over
any
real
add t h e f o l l o w i n g
L e t A be a s i m p l i c i a l
obvious
subset
of a
simplicial
s e t K.
In the
-1 situation of
|DK|,
o f Theorem hence
t
\
precise-
R.
u s e we
6.9.
simplicial
IKI w h i c h
p i o n e e r i n g work
by F r i t s c h
closed
l y l of
contained
triangulated.
[ F r t ^ ] .}
properties.
t h e space
I DK I t h e i m a g e
cell
by F r i t s c h
j
|K| a n d a l l i t s c e l l s . }
I A (K) I i s h o m o t o p i c
T3)
of
{A t h o r o u g h
D of the category
simplicial
the following
from
about
theorem.
transformation A
and, f o r every IKI w i t h
n o t be e x p l i c i t
given
s e t can be
an endomorphism
a natural
functor
triangulation
For
exists
i s an isomorphism of
has been
important
i s polyhedral f o r every
v
cell
There
but shall
t o an e x t e n s i v e l i t e r a t u r e .
o f any s i m p l i c i a l
: IDKI
T1 ) D K
way
of subdivisions
the following
simplicial
ted
the reader
realization we
i n an e s s e n t i a l
1
6.8
(|At) =
the preimage IB I w i t h
t
some
(|A|)
i s a closed
subcomplex
(polyhedral) simplicial
subset
B
o f DK.
A (K) of
We
maps B
spaces
simplicial F
conclude into
A.
(|DK|,|BI) map
from
from We
may
read
t o the pair (DK,B)
i n T3) a s a h o m o t o p y
IX(K) I .
T3) t h a t
from
to
t
IX ( K ) I m a p s K
as an isomorphism
o f spaces
(|KI,|A|)
(K,A) . M o r e o v e r
t h e map
IBI i n t o
of pairs
we may t
R
|A|,
from
hence
the pair
a n d X (K) a s a read
t o t h e map
t h e homotopy of
pairs
§7.
The space
For
any space
n-simplex dard
gular
R we
i s a
define
simplicial
I f a xooc*
map
from M
Sin
: Sin M
o f M.
I f x
n-simplex
o f M.
of x
and a
maps
space
have
by a
set Sin M
x from
of M V (0)
N gives
functor
the geometric then
: V ( p ) -> V ( n ) . We
+
(Sin f)
as f o l l o w s .
i s monotonic
and t h e elements
which
defined
T h u s we
map
i s a point
t o a second
-> S i n N
simplicial
: [ p ] -> i n ]
set of M
singular O-simplex,
Every
a
homology
(semialgebraic)
V ( n ) t o M.
n-simplices
f
and s i n g u l a r
as t h e composite
ponding
of
t h e n we
stan-
a*(x)i s call
(SinM)^ denote
An
Sin M
the
sin-
the corres-
t o x, by x.
us a s i m p l i c i a l
(x) =
f°x f o r x
S i n from
a
map singular
the category
WSA(R)
sSet.
There
i s a close
functor. i
that
n
other map
hand, M
:
we
M
In
order
by
J (x,t)
defined
by
have, M
=
J ( *Y/t)
n
=
j
: t n ] -* [ p ] m o n o t o n i c ,
One now
verifies
setting
[ L a ,Chap.
Theorem
7.1. F o r e v e r y
a natural
f o r any x € K
t »-» l x , t l
space
realization
simplicial
. {Recall
map
from
§6
n
M,
from
a natural
V ( n ) t o I K l . } On t h e (weakly
semialgebraic)
by
(x € ( S i n M )
map
have
S i n and t h e
x
f i r s t
= x(t) f o r (x,t) € (SinM) = y a ^ ( t )
functor
s e t K we
map
defined
x ( t ) ,
this
i„(x) = is,
f o revery
to establish this
M
a
simplicial
I S i n Ml
J ( l x , t l )
M
between
i s the characteristic
x
j
relation
For every
: K -> S i n l K l
v
a
over
of Sin M
singular
to
M
n-simplex
defined the
I S i n Ml
n
, tG V(n) ) .
define n
a map
x v ( n ) and then
(y,0c* ( t ) ) f o r y
:
(SinM )
observe
A
-»
M
that
a s i n g u l a r p - s i m p l e x o f M,
a n d t € V (n) .
t h e f o l l o w i n g theorem
p r e c i s e l y as i n the t o p o l o g i c a l
I I ,§6].
simplicial
s e t K we
have
^lK| and
0
|
i
K'
f o r every
(
S
i
n
)
3
0
i
the functor
tion
functor
More
explicitly,
I
M we
=
i
d
g
S i n M
•
-* s S e t
space
M
and
between
-> S i n M , w h i c h
i s right
adjoint
-
1
a
)
(
7
-
1
b
)
to the realiza-
v i a t h e a d j u n c t i o n maps J
-» W S A ( R )
given a
: K
7
have
S i n : WSA(R)
I : sSet
(
a
simplicial
t h e maps
f
:
s e t K,
I K l -+ M
c a n be; c h a r a c t e r i z e d
and i
m
there
.
i s a
and t h e
by e i t h e r
R
simpli-
one o f
two e q u a t i o n s
f
Here Let
=
i °igi/
g =
M
i s a first (K^IAGA)
be any diagram
a small
(IK^I
IAEA)
limit
(= c o l i m i t
category
ip^
-* K d e n o t e
:
Corollary (IKJI
This
:=
l i m (K. ) > A n
7.2.
IACA)
o f Theorem
the realization
:=
I tp^ I
:
consequence
functor.
sets,
gives
Lirt^
realizations.
i . e . a functor
us a diagram
of
^
spaces
there exists
of the first
A
the direct
diagram.
n € JN . } F o r any A €A, l e t o
f o revery
IKl i s the direct
i s an immediate
to the theory of
I n the category sSet
the canonical
b y t h e maps
7.1
of simplJLcial
[ M t , I I , §2]) K
K
.
K
A to sSet. This
by r e a l i z a t i o n .
{Define
n
(sinf)-i
application
from
of
'
correspondence
maps
the
| K l
S i n M
Thus
cial
d
space
M
one-to-one
i
=
simplicial
l i m i t IK
x
map
from
of the diagram
of
spaces
I -> I K l .
of t h e existence
Indeed,
t o K.
f o r any space,
of a right i n short
adjoint hand
notation,
H o m ( I K I ,M)
= Horn ( K , S i n M) =
l i m H o r n ( K , S i n M) = * ~ A f
A
Let
us recall,
f o r later
use, the notion
l i m Horn ( I K, I ,M) . ** A
of simplicial
X
homotopy.
Definition a)
1 .
F o r any s i m p l i c i a l
short, of
s e t K a n d i 6 { 0 , 1 } we
the simplicial
map
K t o the n-simplex
composite the
realization
x* of K*A(1).
map
Ic
±
i d
maps
simplicial G
I i s t h e map
from
homotopy
simplicial
homotopy
The
relative
algebraic
g
G -e
homotopy,
one-to-one
x «
1
simplicial
ICI f r o m
stated
set L with
Ig
Q
I to
=
IKI x [ 0 , 1 ]
g IC
sim-
= g^lC. A
Q
map
i s a constant projection
I G I : I K l x I —• I L I i s a
Ig-jl.
between s i m p l i c i a l
i n Theorem
0 t o i . The
C i s a simplicial
then
i sthe
: K ^ L be two
of the natural
IC. N o t i c e t h a t
this
x
K t o K xA(0) and
= g ^ a n d G I C x A (1)
Q
f o r
an n - s i m p l e x
t o lKxA(1)l Qf
i . e .the composite g
from
o f K a n d l e t 9 g^
= g , G-
Q
sends
: [ 0 ] -> [ 1 ] s e n d i n g
t o g^ r e l a t i v e
Q
correspondence
maps
isomorphism 6
by e^(K), o r
Notice that
( x , i )f r o m l K l
subset
from
: C x A (1) -* C w i t h
1
with
K t o another
: K x A (1 ) -* L s u c h
pr
1
xA(6 )
R
L e t C be a s i m p l i c i a l
p l i c i a l
K t o K * A (1) w h i c h
of the evident simplicial
simplicial
b)
from
denote
7.1
behaves w e l l
maps
and weakly
with
semi-
r e s p e c t t o homo-
topy.
Proposition and
M a space.
same map g
g
o' 1
F
:
K
Let f
from ->
s
i
n
a homotopy
Q
a n d f ^ be maps
I C I t o M. M
a
respectively,
is f
7.3. L e t K be a s i m p l i c i a l
n
d
G
:
set, C a simplicial
from
L e t F b e a map
K xA(1)
-* S i n M
from
g
Q
to g
relative
1
7.1. Then
C i f fF
o f K,
restrict
to the
I K l x [ 0 , 1 ] t o M. L e t
be t h e l e f t
as e x p l a i n e d i n Theorem
from
IKl t o M which
subset
adjoints
of
f f^, Qf
g I C = g ^ l C , and G Q
i s a homotopy
from
f
Q
to
v
This
i s a straightforward
Theorem
7.1
Definition
(cf.
[ L a , p.
and Remark
consequence
of the uniqueness
statement i n
47f]).
7.4. A
(finite)
system
of simplicial
sets
i s a
tuple
(K,A^,...,A ),
subsets other with a
A«| , . . . , A
system f(A^)
c: B^.
Analogously
homotopy
(L,B,| , . . . , B )
and
Proposition
7.3
and systems
come
Theorem
s e t K and
f from
of course,
a
to a simplicial
) t o an-
/
simplicial have
maps
subset
immediately
simplicial
(K,A^,... A
1 we
two s i m p l i c i a l
generalize
of
map
to Definition
between
relative
R
now
simplicial ) means,
F
to
We
o f K. A
r
(L B^,...,B
simplicial
sets
consisting of a simplicial
r
map
f :K
the notion
from
of
(K,A^,...,A ) r
C o f K.
t o systems
-* L
Theorem
of
7.1
simplicial
spaces.
t o t h e main
result
7.5. F o r any space M
of this
t h e map
J
section.
:
I S i n Ml
M
M
i s a
homotopy
equivalence.
In
order
t o prove
to
verify
f o r every :
(J >* M
is
ISinMl contains
We
shall
that
J
reader
We
M
[LW, p .
we
shall
identify
give
Q
that
theorem"
V.6.10,
t h e map
the vertex
. {Notice
that
x of Sin M
every
(see above)
connected
component
x.}
t h e arguments
i n t h e book
of Lundell
i n the topological setting
equivalence).
only
For the convenience
and
prove
of the
a l l details.
the pointed
with
the realization
cial
s e t (L,«>) a r i s i n g
°° h a s t o b e a v e r t e x ,
n € 3N
identified
(which
i s a weak homotopy
"Whitehead's
n
reproduce
102ff.]
by
Ti (M,x)
ISinMl
a point
essentially
Weingram
have
lx,1| of
of
and every
,5) -
H e r e we
the point
i tsuffices,
x EM
V l S i n M l
bijective.
with
this
n-sphere
(S ,«>) , w h e r e oo d e n o t e s n
(|L|,«>) o f a s u i t a b l e from i.e. a
pointed
some t r i a n g u l a t i o n O-simplex}.
of
the north
polyhedral n
(S , M
exists
f = jjjJ °lg|. T h e map
a
be point-
(j^)*
M
( j ^ ) * we
S
simplcial
a pointed
nnaeed
n
s e t annnd
pdyhedral
a lemma w h i c h
preservino
f
: S
n
simnuiplicial
( i . e .a t r i a g u l a t i o n
-* I K l a
w i l l
n
offf
pointed
s e t T and an
(S ,°°) ) s u c h
t o the reaaalization
that
iso-
f*cp
|g| o f a
simpli-
: T -» K .
lemma t h e i n j e c t i v i t y o j ( J ) # M
point We
preserving have
np
f
homotopic.
t h e constant map w h i c h
simplicial sends
T h e map
conclude
from
relative
{t}*. This
constant
map
map f r o m
M
poiiint
i s t h constant
implies
thatj
( t ) , i n otier
such
i s null
o f T,
irm.nap
that
We J °f M
homotopic. J °lgl i s M
t € T
Q
.
Let k
i . e . the unique
subset
{x}*of Sin M
from
the pointed
relaaative ( t )t o J * l k l . M
i s simplicc^ially
o f couce,
as f o l l o w s .
ISinMl
T t o S i i i n M,
M
7.3
seen
:= S i n I £ M . T h e m a p
J °lgl i s henotopic
I k lr e l a t i v e
-» •
T to thesimplicial
J ©|k|
Proposition
n
f itseealf
L e t t d e n o t e :he b a s e
by { x } . Then
|T| t o M.
can tboe
: S
t o prove t h a t
T,tp,g a s i n t h e l e m m a , w i n K
simplicial
space
exists
homotopic.
generated
and
n (M) as the s e t o f
Indddeed,
m a p . *y T h e o r e m sua that
a l l maps
regarrcd maps
our notation. A l l
[ I g l ] t oi f ] .
(base p o i n t
a base
null
denote
anddl
sujiective.
the injectivity c
cp : I T I ^
map
Using are
there
homotopic
cial
s
:f r o m
afterwards.
Then
morphism
We
presrving
: L -» S i n M
Lemma 7 . 6 . L e t K b e a p o i n t e d map.
be m i n t e d , poins.
point
preserving g
t h e :>ase p o i n t s
t h e base
o f base
seen
omit
sets
t o preserve
i s easily
ed
shall
that words,
Iglli
homotopic
i s homotopic
I hi g I i s n u l l
We
to k
to the
homotopic.
The
If I
map
i s homotopic
lgl°tx i m a t i o n i s r a t u r a l
a map
( c f .again
t h e decreasing system
l ) t c c o 7K w h o s e
t h e map
respecting
barycsntric
well.
( c f .6.9). f = t ^ o hi s
a decreasing system
f roomm
as
theorem
of the pointed
[Spa, p.
^i s a CW-aappproximation i n M
n
i s a simplicial
f o r t h e seconndd
My
is
ibbe
immediately obtaiinn
V.2.13
Remark
r
S
are essentially
set S i n t h e map
points
u
apprcoDximations
(M ,...,M^)
i ,...,| S i n M
preserving
i n Chapter
{T i s a n i t t c e r a t e d
i s proved.
k (cf.
which
of simplicial setting.
classsses
homotopic
l e th denote
i s homotopic map
i s null
the triangulation
: ITI
l u lo f a s i m p l l i i c i a l
7.5
j ^
| T l -* I DK I I
III.5.6)
appl^
resppoecting base
on c o n t i g u i t y
Remark
We
a triangulatiicon
h°tp
realization
and
|X(K)l°h
exists
such
t h e leercmma.
, htnce
It
S
t o prove
ttco
tthen
i ti s easily
ISin N |
i n 3Bfc. I n d e e d ,
checked
that
i ft h e systems
i f f
: M -*
the square
N
commutes. a
map
This
between
Corollary
implies
decreasing
7.9. E v e r y
equivalent
t h e commutativity systems
closed
t o a decreasing
of
square f o r
spaces.
decreasing system
of t h e analogous
system
of closed
o f spaces
geometric
i s homotopy simplicial
complexes.
This
follows
(cf.
also
The
case
logy
r =
recall
One
denote
s e t K gives
simply
from
For
map
n
are,
simplicial
group
from
K
associated
n
, i fn > 0
group
:= C. (K) a
we
z
simplicial
abelian
n
group
and " s i n g u l a r
algebra.
Z[K] such
by t h e s e t
: O r d -> S e t w i t h Ab o f A b e l i a n
with
homo-
now.
Z[K ] generated
n
n
the cochain
"singular chains"
notions
C ( K ) t o C _^ (K) b e i n g
G any a b e l i a n
by
explain
Z[K ] f o rn>0
t o 2r[K]
homology
shall
the functor
complex
R
C*(K,G)
The
6.8
us a d e s c r i p t i o n o f o r d i n a r y
Set t o the category
by C (K) =
from
Z[K]
abelian
from
the chain
C. ( K , G ) and
and t h e t r i a n g u l a t i o n Theorem
gives
known
us a
composes
group"
defined
ary
7.7
some w e l l
i s the free
n
abelian
is
1 o f Theorem
r e s p e c t i v e l y , a s we
simplicial
n.
7.7
( V I , § 3 , D e f . 2) a n d c o h o m o l o g y
f i r s t
Z[K]
o u r Theorem
Remark 6.9).
cochains"
We
from
f o r every
groups.
"free
L e t C. ( K ) 2[K].
group
f o rns of
homology
technical
off.
of
singular
contiimuous
that
field
singular
of
the
settir.mg.
of
but
introduction
the
;
call
the
i t complettiely
§3,
t
we
G.
IV,
g r o u p s are
a
( S i n M , S i n A ;G)i)
in Chapter
better
cohomology
n
cohomology
(cf.
to
( S i n M , S i n A A. ; G)
and
theory
strict
C
(M,A)\)) , a n d
cohomolog^jy/ g r o u p
homology
Indeed,
way
pair
topological
t o the t o p o l o g i c a l
homology
gives
n t
i
the
of
is justified
from V(n)
impossible
we
elements
coefficients in
already
for
of
singular
elements
singular
elementary
then
the
maps
contrast not
the
n-th
terminology
algebraic
group
the
terminology
standard
and
honologY
(M,A)
.
, « eevery
isomorphiissms
abelian
group
G
n
n
H ( S i n M , S i n A ;G)
Proof.
We
prove
analogous. and
2.16
assume
isomorphic H
n
( ISin M
VI,
§4.
5.3. H
n
f i r s t
( S i n M,G)
now
f o r hio.O)mology.
this
considle.e:r
together give
H
We
We
• i H (M,A;G) .
-—•
that
to
H
us
H
A
Thus
i s n o t t empty.
obtain
( S i n M , S i n A ;G)
applying
i s jjuuist
the
from
to
A
H
the
may
2 2.. 16
a
be
naturality
o f f : the
(M,G)
A ;Q)
n
slated
essentially
sane a s
H
the
simplicial
from
H (K,G)
with
7.8
to
n
icentified
gives
H us
f rornn a
(IKI
natuial
isomorphism
n
; G) . { N o t i c e t h a t
isomojpaism
from
H
K
to
,G) ..}.} O n
a naturaill
A(0)
the
ve
otler
isomorphism
obtain
(K,G)
hand, (j^)*
a
Theorem
I t follows groups
instead
7.10,
simplicial
naturally and
A I ;G) , c f .
i n 2.16, to H
by
(lKl,G) n
natural
Theorem
7.7
isomorphism together
( | 3 i n M l , I S i n A I; G)
from
q.e.d.
cohomology
HWSA(2,R)
7.5
from
n
7.11.
be
ISin M /Sin A I , c f .
with
H (M,A;G).
Remark and
to
map
Theorem
above,
( I S i n M | / |3 i
n
i s
n
to
w i l l
.
Then H ( S i n M , S i n
( I S£i.n M I , I S i n A
R
i s empty. Then
» H
( S i n M/Sin.P.Av,G) , a s
n
that
f o r cohomology
i i.somorphisns
I S i n M I / I S i n A'i I I
we
case
arguments
( I S S i i . n M |,G)
I , I S i n A I ,G)
But
the
The
of
fircom
can
1 b>e
subset
of
read
that
as f i n c t o r s
the on
singular
the
homology
homotopy
category
WSA(22„R).
i t stands3,,
as
T h e o r e m 7.10
a
l e a v e s something
sirmpiplicial
se: K then
t o be the
desired.
I f L
obvi.oas s h o r t
i s
a
exact
sequence
0 gives
-» C. ( L , G ) us
a
long
3 (K,L) n
C. (K,G>) ) exact
seqmuence
: H (K,L;G) n
-» C. ( K , L ; S )
- +
H _ R
1
-
0
i n honology (L,G)
with
connecting homomorphisms
Similarly ing
we
n
question
connecting morphism the
a canonical
e x a c t sequence
i n cohomology
with
connec
homomorphisms 6 (K,L)
The
have
n
: H (L,G)
arises
-+
n
6 (M,A)
n
isomorphisms
from
We
p r e s e n t i n §8 a
other
natural
2.16
i t may
(K,L;G).
i n the case
(K,L) =
correspond, perhaps of ordinary
constructed
Starting shall
n + 1
whether
homomorphisms
3 (M,A),
H
up t o s i g n ,
homology
these
t o t h e homo-
and cohomology
under
above.
be l a b o r i o u s second
isomorphisms)
( S i nM , S i n A)
t o check
whether
p r o o f o f Theorem
where
this
problem
7.10
this
(with
disappears.
i s true. perhaps
§8.
Simplicial
One
obtains
8.1. The
We
want
to exploit
on
the level
Recall
some o f w h i c h
then
the relation
maps
from
K t o L may
transitive,
the
generally
creasing • t h e K, K I£
:=
system
first
on t h e l e v e l
start
deserve
with
independent
[Cu]) that
we
f i xthe following
but i fL
sets.
subsets of K
concerning the functors o f spaces
\t h e d e c r e a s i n g s y s t e m simplicial
map
i s a
(IK
o
then
simplicial
maps
of
simplicial
simplicial
from
6J :=
case
we
denote
K t o L by
[K,L].
(K ,...,K
) i s a de-
By t h i s
we
with
r> K. => ... z> K . M o r e o v e r i r
K o
I I.
S i n and
( S i nIK
(i„ , . . . , i „ ) f r o m *o r
mean, o f c o u r s e ,
o f Kan s e t s .
I , . . ., IK
o f Kan s e t s
on
and
i s a Kan s e t then i t
In this
setting.
Q
system
M
13f.])
interest.
of simplicial
of simplicial
o f homotopy
on t h e s e t Map(K,L)
(L ,...,L ) i s a decreasing system
-decreasing
o f any space
i f K and L a r e a r b i t r a r y
"homotopic"
classes
( e . g . [ L a , p.
general results
n o t be t r a n s i t i v e ,
are simplicial
;n o t a t i o n s
[the
set SinM
hence an e q u i v a l e n c e r e l a t i o n .
s e t o f homotopy
More
fact,
( c f .[ L a ] , [May],
again
extension condition.
o f h o m o l o g y . We
sets
is
simplicial
Kan's
this
homology
argument as i n t o p o l o g y
singular
set, i . e .f u l f i l l s
homotopy
and s i n g u l a r
b y t h e same e a s y
Proposition Kan
homotopy,
r
Thus
We
that
use obvious
\
o
Proof.
\£>\ , C = 0) , a n d u t h e
e q u i v a l e n c e , and g i v e s us t h e c l a i m
{Of c o u r s e , Ji(S) means
means
If
)
/ t=
conclude
extension o f R then
Sin A
o f Kan sets. Q
C = 0). We
(with
be a d e c r e a s i n g system
R
then
8.2
t o Sin\£\ , 3 i s t h e a d j u n c -
q.e.d.
Q
R = 3R
i ^ from £
^i s a homotopy
(M ,...,M )
closed
sist
i n Proposition
i n 8.4
h.
Let
by t h e inclusion
of Sin M
cp m a p s V ( n )
o f t h e second
i sa strong deformation i s a strong deformation
l e m m a a n d 8.6
. Assume
o
into
M . k
that
b
This
one i s even more
retract retract
i ts u f f i c e s
trivial.
o f S i n Jt(S) . of Sin ^
t o prove
t
o
p«
that, f o r
any
s p a c e M,
lence a
and,
i n case
homotopy
R =
a
: S i n M S i n M ( S )
IR , a l s o
i s a homotopy 3
the inclusion
equiva-
: S i n M S i n M ( S ) . I t i s e a s i l y
commutes,
lal ISinMl
^M*S
1. | S i n M ( S ) l
-
s
\
/
s
^M(S)
M(S)
We
know
We
conclude
8.4,
(from
that
the
fact
Let
us
that
Theorem
that
a
I a I
i
J
and
M
J ^ j M
a
r
e
homotopy e q u i v a l e i c e s .
s
equivalence
equivalence.
{N.B.
We
and
then,
by
constantly
CorolLary
exploit
8.1.}
now
s
that
i s a homotopy
g
i s a homotopy
look
at the second
the following
,
7.5)
n
M
|
t
o
triangle
' ' &
p
t
0
P
3.
inclusion
I t i s again
easily
checked
commutes.
•
ISinM
t
o
p
l
t
o
p
^M^ t o p M, top It
i s known
from
topology
that
j
([Mi^]; I 31
t
Q
p
this
c a n be
i s a weak
proved
homotopy
homotopy
equivalence,
lary
this
8.4
Remark space M
8.11. over
equivalence
as
hence
Alternatively
o u r Theorem
instead of
just
and
a homotopy
i s a homotopy
we
can conclude map
a weak
j
7.5
[LW]).
We
then
that
I 3 I
equivalence
3
the continuous
homotopy
equivalence
top
equivalence
implies that
IR
i s a weak
M M
conclude i s a
(V.6.10).
t.iat weaic
By
equivalence.
q.e.d.
directly
for
that,
Corol-
every
i s a ( t o p o l o g i c a l ) homotooy ^top homotopy e q u i v a l e n c e . Indeed, t h i s M
follows
from
homotopy
Theorem logy.
the
topological
equivalent
8.10
We
to
gives
us
f i x some
a
abelian
1.
homotopic,
i f ,f o r e v e r y
homotopic
decreasing are
to
Lemma 8.12.
to
Let
maps.
f*,g* in
g^.
the
+
simplicial
I t i s well
C.(K )
to
and by
C.(L )
C.{q^)
use
Lemma
from of
the
8.13.
(M,A)
to
i n
singular
maps
{0,...,r}, we
call
f
and
the
known
that
induced
Q
are
a pair
two of
two
f,g
maps
ft,
:
g
from &
f ,g
Proof.
=
g*
From
: H
the
n
i s
1
cohomo-
to £
pseudo-
£,
:
-*
components
of
f
g.
are
pseudohomotopic
sim-
; G)
the
i n d u c e d maps C . ( f
homotopic C.(L^)
and
are
also
chain
the
o
) and
C.(g
J
o
)
i n d u c e d maps
homotopic.
The
from C.Cf^)
claims
follow
q.e.d.
pseudohomotopic spaces
and
: H* ( L , L
chain
C.CK^)
by
component
pseudohomotopic,
Assume
1
are
Q
k
K ;G)
homology
Proof.
q
1.
since,
G.
simplicial
spaces
the
Q f
to
c o r r e s p o n d i n g components
=
Then
: H (K
group
two
of
r
approach
Similarly
systems
homotopic
p l i c i a l
call
theorem
CW-complex.
new
Definition'
is
We
a
Whitehead
( S i n M , S i n A ;G)
commutativity of
-> H
the
n
( S i n N , S i n B ; G)
square
.
pair same
of
spaces
homomor-
ISin fI
( I S i n Ml , I S i n A l )
(iSin Nl , i S i n Bl )
3
(M,A) (M,A)
o f t h e analogous
the
maps
are
pseudohomotopic.
and
C = 0)
to
IS i n f I
that
(SinN,Sin
(N, B)
f
and
and
square
f o r g we from
Then
conclude
the simplicial
B)
s e e , by u s e o f Theorem
IS i n g I we
(N,B)
(I S i n Ml ,I S i n A | )
maps
from
S i nf
a r e pseudohomotopic.
This
gives
that
(I S i n N I , I S i n B I )
to
C o r o l l a r y 8.4 and S i n g
7.7,
(with
from
r =
0
( S i n M , S i n A)
the desired
result.
q.e.d.
Definition
2. C l e a r l y t h e f u n c t o r s t o Ab, t o g e t h e r
described
a t t h e e n d o f §7 c o n s t i t u t e a p r e h o m o l o g y WSA(R),
homology
over
Theorem
as defined
R with
8.14. S i n g u l a r
ordinary
homology
with
coefficient
N.B.
Recall that,
homology gives
theory
i n V I , § 5 . We
group
call
( c f . Chapter
o n WSA(R)
R with
The analogous
be t r u e .
I f R =
n
theory
on t h e space
theory
singular
coefficients
VI, starting
with
there
coefficient
exists group
us an i n t e r p r e t a t i o n by s i n g u l a r chains
Proof.
this
9 (M,A)
from
i n G
i s an
V I , § 3 , D e f . 2)
G.
up t o i s o m o r p h i s m ,
constructed
homomorphisms
i n G.
homology over
theory
theory
to
the connecting
coefficients
from
r
HWSA(2,R)
category
with
( M , A ) -* H ( S i n M , S i n A ; G )
only G.
one
Thus
ordinary
Theorem
of the ordinary
homology
i n Chapter V I .
result
IR
i n algebraic
t h e n we
obtain
from
topology
i s very
well
known
Theorem
8.10.ii
canonical
morphisms
H
which
n
8.4
( S i n M , S i n A ;G)
- ^ H
are compatible with
n
( S i n M^. ,SinA^_ ;G) top top
the connecting
homomorphisms.
Thus t h e
iso-
theorem
holds
R =
R
Let
finally
o
, S =
f o r R =
TR)
IR . We
now
i n t h e same way J
R be an a r b i t r a r y R
that
real
homology
theory
over
with
homology
theory
t o a homology
Q
obtain
from
Theorem
t h e theorem
closed
coefficients theory
8.10.i
holds
field.
We
f o r R = R . o
denote
i n G b y h . We R a s we
the
singular
extend
+
h£ over
(with
have
this
learned i n •p
Chapter again If VI
V I . Since
- we K
i s any p a i r have
:=
Sin M n
values
R
of n a l l these
with
the connecting
to singular latter
group
now
( S i n IKI _ o
n
(K,L;G)
=
i s again
A
We
R with
t h e same way
:
)
R
we
sufficient
(M,A)
write
;G)
.
field
R
, this
group i s
come
from
Thus
found
different
t h e same
they
are
homo-
compatible
an isomorphism
coefficients
i n G and thus theory
with
from know
that
coefficient
t o work
theory
have
notion
that
singular
cohomology
i s an o r d i n a r y
theory.
been
t o prove
Theorem
i n the category
o f WSA(2,R),
prehomology would
-*
R
may D
G.
Chapter
(IK| ,|L| )
Thus
an o r d i n a r y homology
one v e r i f i e s
Remarks. I n o r d e r
instead
group
R
sets.
have
h£
i n contrast to
(Sin M,Sin A;G). For
of simplicial
over
M
coefficient
, S i n IL I o
R
homomorphisms.
homology
theory
j (
canonical isomorphisms
of pairs
-
know t h a t
q.e.d.
cohomology
The
t h e same
we
G.
just
Final
to
theory
at our disposal.
) = H
R
isomorphic
equivalence
or
R then
:= S i n A
( I K L J L L o o
topy
In
over
P r o p o s i t i o n 8.8, a p p l i e d t o t h e g r o u n d
canonically
the
with
o f spaces
and L
= h
n
h^
theory
the canonical CW-approximation
h*(M,A)
By
i s an o r d i n a r y homology
+
i s an o r d i n a r y homology
(M,A)
with
h
o n c e we
know
8.14
P(2,R) that
i twould
of pairs
o f WSA(R).
-
t o use Theorem
o f pseudohomotopy
enables
us
o f weak
been polytopes
(M,A) >-> H* ( S i n M , S i n A ; G)
on t h e whole sufficient
have
Notice
also that
8.10
to avoid
i s a
i t was
i n t h e case
r =
a serious use of
0.
homotopy about
theory
systems
up
for pairs to
8.10
of
spaces
deserve
or
Kan
interest
sets. on
But
their
our
own
results
for
r>0.
§9.
A
In
group
this
section,
volume, want
o f automorphisms
we
deviate
R.
constructing
Definition tonic
of thought
a sufficiently
large
weakly
w i l l
g
In
_
, t
1
]
i
this
0 < s
g
(
Remark. avoid
=
/
B
1, with
+
i - i
(
s u c h map
inverse
these
t
i - i
l
maps
-
l
< s
1
V
t
g
the semialgebraic
PL
Aut ([0,1]), reflects
)
i - i
" '
stupid i
(
that
R, w h i c h
i n a
acts
of natural
simplicial
space
the principles f o r
exists
closed {t }
=
s
_
-
]
r
i - i
we
again
< t^
)
allow
t__^ =
^ =
t
} .
±
a sequence
and, f o rt € [
i
t
f
V
i
mono-
subinterval
i ft
±
form i
a sequence
i
_
1
. T h e r e a s o n w h y we
^
i
s_^ ,
'
= t ^ . We
t
(
9
= ^
1
,
}
could
do n o t do
form
a monotonic a subgroup
space
[0,1].
We
of the semialgebraic PL-automorphism of t h e group denote
i n the present section that
V
group
We
this
soon.
i s again
together
+
i f ft
i
g i s an automorphism
of
+
-
i
b y t h r o w i n g o u t some t
become a p p a r e n t
Every
sign
t
s
At present i tlooks
this
w i l l
The
)
=
semialgebraic
proper
on each
:= q ( t ^ )
s^
chapter.
of [0,1] i s a b i j e c t i v e
put [ t ^ , ^ ]
the points
i n this
gained i n Chapter IV.
there
q linear ^
of [0,1] { 0 < i < n ;
n
that
one o f t h e p r e s e n t
t o apply
PL-automorphism
i n R with
situation,
t
spaces
: [ 0 , 1 ] -> [ 0 , 1 ] s u c h
< .. . < s
Q
u s a new o c c a s i o n
semialgebraic
0 < t 1
t h e s t r u c t u r e o f a weak p o l y t o p e .
of
M
constant
f i x a constant
tive
Let
t h e s e t o f a l l maps
subsets M(c).
We
with
denote
i
denote
n
n-simplex
V(n)
)
< . , . < t =1. i— — n
coordinates
t of the standard
coordinates",
, . . . , t
3
a point
Here
of t . { i f t =
I ?
t . i s t h e sum o f t h e f i r s t I
u =
0
i
e
i '
then
t h e s e t o f a l l (s,t) € V ( n ) *V(n) such
t
that
i
=
l^
M
(cd)
n
m+n P ,
x
Then
vertical
Let
points €M
:
y
=
t
:=
n
(d) —
i tw i l l
arrows
u
, (c) , m+n
(
m,c t n
n
u
, d
. m,n,c,d
> M
be e v i d e n t
i n the diagram
(u,v)G M ^ c )
n
m+n cd /
(c) M
m
the
X
knew
map
T) xn . m, c n, a
(x,v*s)
shall
the diagram
k
(*)
T h e n we
* L ( c ) xft ( d ) -> M , ( c d ) m n m+n
f
such
map
semialgebraic.
C a m,n c,d
C =
this
and
(v*s,y)
'
v
)
l
(
S
f
t
(
)
]
"
v
1
*
s
(
V
that
u
^ ^ , i s semialgebraic, since m,n,c,d a r e i d e n t i f y i n g s e m i a l g e b r a i c maps.
(s,t)GM (d) n
€M
)
(cd) m+n
, (d) a s m+n
be g i v e n .
We
define
new
pairs
follows:
' S
)
( u
'
*
"
Then
I
Wn,c
Vn,d We
define
gram proof
(*)
(
( x
v
' * v
* ' s
s )
y
)
=
n
-
m,c
, 1
the desired commutes.
that
n,d map
( s
v )
'
t )
£ by
I t follows
£ i s indeed
' • £((u,v),(s,t)) from
the previous
semialgebraic.
:=
(x,y). steps
Then
the dia-
a) a n d c ) o f t h e q.e.d.
The
orbits
Indeed,
i f s
example, takes
of
+
We
the
—•
t
are
9.5.
|XI
As
of
following
to
to
If X
= M
A2)
If X
=
A3)
If f
: X
the
the
prove
a
the
open
given
a
:
[p]
by
the
-> [ n ]
faces
face
of
of
V(n).
V(n)
{for
formula
following i s easily
map
assume
(9.1)
checked.
the
semialgebraic
map
i s constant then
that
R
proper
i s s e q u e n t i a l . There G
=
exists
+
PL
Aut ([0,1])
simplicial
space
X
on
such
the that
a realizathe
hold.
then
the
result.
a c t i o n of
partially
properties
-> Y
be
same o p e n
gGG
beautiful
semialgebraic
A(n)
i n the
element
Also
to
G-equivariant.
before
three
seen
points
monotonic
i s
every
A1)
easily
two
s.
every
ready
are
then
V(n)
unique weakly tion
V(n)
t
For
are
Theorem
and
point
: V (p)
now
on
i n V(n)},
L e m m a 9.4. ot
G
the
a c t i o n of
action of
i s a
simplicial
on
i s given
G
on
map
G
V(n)
on
i s as
If I
then
|X|
:
=
M
just
IXI
i s
t r i v i a l .
described.
IY|
i s
G-equi-
variant . This
a c t i o n of glx,tl
If
X
X
Proof. have
already
(gx,gy),
formula
(x€X,tGV(n)) (*)
the
any
two
orbits
defined
the since
partially
the
of
G
on
a c t i o n of the
G
on
G
IXI
are
the
IX x Y l
for
and
simplicial
any set
=
we
|X|
|X|
open
cells
of
the
X
and
Y,
we
and
we
are
x |Y|
spaces
|Y| by
to
IXI
M x A(n)
of
a
have
to
then
the
x |Y|
IXI
product A(n)
simplicial
on
p r o j e c t i o n s from
Thus,
standard
proper
a c t i o n of
G-equivariant. a
the
IXI.
If, for
define
by
lx,gtl.
i s d i s c r e t e then
CW-complex
to
=
G
formula
and
IYI
constant
define
the
forced
g(x,y)
must space
G-action
be M on
=
x A (n) I = M x V ( n )
IM
Let
now
X
be
simplicial this
map
(cf.
This
X
E(n )
on
x
action
of
IXI
by
to
have
of
any
formula
the
§6
of
If
X
use
and
that
9.6.
set
proper
G
of
i s not =
N
of
subset
set. local
of
M.
{This
means
intrinsic
be
s t a b l e under
of
open
cells
of
every IKl.
M,
n
we
x
for which
are
forced
A3.
be
A1
A2.
and
(*)
and
we by
can
be
s t i l l
of
M,
9.4
and
we
G-equivariant. on
the
The
our
realization
from
verified
the
in
last
a
assertion
description in
have
an
semialgebraic on
l e t M a
:
the be
action of
cells a
|K|
for points
a
1
(N)
|X|,
over
with
of
i s a weakly
hence
of
space M
i s intrinsic,
the
automorphisms.
triangulated, cf.
holds, P
Lemma
q.e.d.
formulated
property
to.
relation
It i s clear
I t can
IXl
the
holds,
G-action
X.
of
semialgebraic
become
to
example,
P
of
->
X
has
isomorphism
property
: X
sum
a weakly
i s transitive
For
automorphism
nx
theorem
space
|X|
that M
s i n c e our
map
discrete.
on
canonical
realization
i n the
formula
some
as
a
the
direct
semialgebraic
action
exists
Indeed,
(*)
the
useful.
and
have
equivalence
obtain
sequential then
this
the
the
f u l f i l l
Aut [0,1]
of
we
since
for X
there
a l l points
with
also
+
PL
i s sometimes
assume t h a t
IXl
We
a c t i o n . I t f o l l o w s from
f u l f i l l
from
X,
(x,gt),
simplicial
they
.
space.
to
on
=
formula
actions
that
cells
group
some
that
formula,
these way
If R
which
be
g(x,t)
semialgebraic
i s evident
open
simplicial P
this
De X
G-action
formula
such
i s d i s c r e t e then
fact
the
(x,gy)
strongly surjective
i s identifying
x
partially
the
proper
=
simplicial
deployment
established a weakly
(*)
abstract
n
IXI
theorem
Remark
the
g (x,y)
action i s compatible
straightforward in
proper
define
a weakly
on
forced now
We
Since
G
formula
the
partially
this
X.
from
x
4.7).
i s indeed that
are
x
xV(n)
fi
the
partially
i s the
above
We
map
Prop.
spaces
any
by
K
6.8}. M.
a R
a Let
Then
the
semialgebraic the
must
set be
a
N
must
union
Epilogue.
i s sequential, although passing
we
have
seems
to a
used
Anyway,
axiom
our patch
version not
spaces
seems
which
of
real
that
that
i s s t i l l
gation
I V , §1 E 3 . On
constructions
a theory
feasible
i n Chapter
admit
theorem
more
d i d i n this
+
[oral
extension
spaces
results,
our base be
field
surmounted
o f R,
a
trick
occasions.
group
of weakly
The
trouble
axiom
seems
and thus
V,
§6.
On of
Aut ([0,1])
semialgebraic
t o be
seems
+
PL
comes
from
the
the other
the
crucial
t o be
as f o r example
inductive limits
i s an h o n e s t
N i e l s Schwartz semialgebraic
( c f . [Sch1]
of weakly
V
a
largely strong hand,
why
semialgebraic
book?
PL A u t ( [ 0 , 1 ] ) reasonable.
that
semialgebraic
i n Chapter
general
"abstract" weakly
closed
field
this
to establish
trouble might
f o r some p u r p o s e s . t h e one hand,
o f Whitehead's
t h a n we
had t o assume
and VI a t v a r i o u s
f o r some o f o u r b e s t
sometimes
i n order
a deficiency of our d e f i n i t i o n
i n Chapter
responsible
of
i n ChaptersV
closed
our c o n s t r u c t i o n of the weakly
exhaustion
that,
i n practice this
sequential real
to reveal
spaces
for
a r t i f i c i a l
s t r u c t u r e o n P L A u t ( [ 0 , 1 ] ) , we
by
It
somewhat +
space R
I t looks
and
semialgebraic
communication].
space
i n some g e n e r a l
r e c e n t l y s t a r t e d an spaces
[LSA, App. spaces
based
investi-
on h i s theory
A ] ) . He
without
sense
gained
axiom
E3
evidence
i s
s t i l l
Appendix
C
We
discuss
shall
(toChapter
this
I V ) : When
question
i s T(M) a b a s i s
mostly
denotes
t h es e t o f p o s i t i v e elements
Example
C.1. L e t M b e a c o u n t a b l e
IV.4.9. Then M i sn o t a l o c a l l y
o f open
by examples.
sets
of
M t
?
In thefollowing
o f R.
o r uncountable
semialgebraic
comb, c f . IV.4.8 a n d
space.
Nevertheless i s
o
T(M)
a basis
Example
o f open
sets
o f M, top
c.2. L e t R = IR a n d l e t M b e t h e s u b s e t
HR x I R x ] R +
+
u
+
3 ]R xIR x{0} u +
be
{(0,0,0)} o f IR . F o r a n y f i n i t e
+
t h esemialgebraic
3R^ . U s i n g
I V , 1.6
semialgebraic closed
subspace
we e q u i p
space
such
semialgebraic
Jx I R x I R +
M with
that
M , T
J o f JR
u 1R XIR X{0} +
t h eunique
every
subspace
+
subset
u
+
l e t Mj
{(0,0,0)} o f
structure of a
i ni t s given
+
weakly
structure, i sa
o f M a n d (M_|JcIR, ,J f i n i t e ) - i
i sanex-
J
haustion of
M. T h e r e
basis not
o f M. U = { ( x , y , z ) exists
o f thestrong
€M|z C o h o ( I R )
204
a*
260
n n
x
DX
265
n
261
i
i
KM,R)
200
J(M)
205
d.,s.
261
KM)
206
sC
261
209,229
f
262
194
sk (X),
215
M
6 (M,A)
222
x
3 (K,L)
329
IXI
268
6 (K,L)
330
If 1
276,
n 0 3 (M,A) n
n
n
n
261
n n
X
n
266 263
6
265
311
Ix,tl
268
X
268
X
269
De X
294f
Rel X
308
C
x
269
X
X
295
A*
286
A(n)
276
A(a)
277
K
264f
R
IKl,
IK|
Ifl,
l f l
l
K
n
x
l
t o p '
[LSA,104],311
R
f
l
t o p
lxl°
n
S(P)
f
3
Sin f
BG
[LSA,99],314
320
322
Q
WG
312
320
M
e , PL
1
315
S i n M,
V
1
312
E(P), P,
3
311
Ixl, K
l
[LSA,104],311
R
+
Aut [0,1]
341 xiv,264 x i v
Glossary absolute
path
completion
admissible
criterion
covering
29
f i l t r a t i o n attaching base
map
field
116,
117,
116
[LSA,
extension of
subset triad
belt,
n-belt
cohomology
a
homology
of
Brumfiel's cell cell =
21,
theory [LSA,
a
simplicial
a
space
a
simplicial
a
spectrum
of
a
space
theory
34,
221
19f]
map
21,
203
264 [LSA,
space
19]
264
256 240
24Cf
116
big
boundary
273]
a
a map
basic
46
a t t a c h i n g map
167
characteristic
map
a
belt
a
patch
theorem
167
116 106
101
166 watching patch
homotopy
watching
cellular
htp.equ.
chunk,
n-chunk
classifying
f o r CW-complexes,
approximation
168
chain
211
complex
homotopy
characteristic
equivalence:
map
16 8
map
166,
168
311
116
space
245,
251
cf.
169
closed
equivalence relation pair
of spaces
semialgebraic
coefficient cofiber
26
simplicial
subset
subcomplex
110
of a
6,
[LSA 2 1 , 99]
280
simplicial
space
system
of spaces
weakly
semialgebraic subset
123
199
189 theory, reduced
1 9 4 , 204
unreduced comb
39,
complete
space
40
of an incomplete path of
composite
222
43
completion
a
space
of a family
compression
152
to
a map
compress
compressible
of homotopies
t o a subset
lemma
152
153
theorem
154
190
connected
31
connected
component
n-connected
31
152
constant
homotopy
[LSA, 249]
simplicial [ L S A , 2 2 9 ] , 7 1 , 89
countable
type
36
45
50, [ L S A 94, 175]
152
compression
core
3,
complex
groups
cohomology
cone
68
simplicial
subspace
9 9 , 306
space
263
148
27
266
CW-approx i m a t i on CW-complex
179
165
CW-system
178
decreasing
system
of
CW-complexes
of
spaces
degeneracy
morphism
degenerate
simplex
degree
of
a map
deployment dimension
direct
261 265
211
154,
[DK ,
§8]
2
patch complex
product of
sum
discrete
spaces
equivalence
simplicial
of
relative
of
spaces
of
simplicial
relation
264
patch complexes
114
7
set
spaces
264
42 space
264f,
279
on
a
space
99
on
a
simplicial
space
306
155
map
N
axiom
194,
excision
axiom
215,
excisive
triad
233
exhaustion
spaces
265
n-equivalence
exactness
114
31
simplicial discretization
107,
of
space,
evaluation
123
2 94
dimensional direct
178
247 209,
215,
222
222
4
extension
of
a
(co)homology
extension
by
WP-approximation
face
109,
[LSA,
face
morphism
100]
261
theory to a 226,228
real
closed
overfield
203,
221
faithful fibre
exhaustion
product
finite
11
of
spaces
of
simplicial
patch
33
complex
relative 1
Freudenthal s function
fundamental G-space
space
groupoid
along
a
height
107,
114
homology
homology
theorem
114
189
1 [LSA,
268]
closed
sequence
theory,
subspace
of
a
pair
of
a
triple
reduced
homotopic
[LSA,
122,
253,
C
C
incomplete
[LSA,
equivalence
between
theorem
321f
spectra
v i i i , 121
265ff],
147
between
spectra
252
subset
106 45 a
122
[LSA,
simplicial
of
C
theorem
60
path
249],
under
map
face
216f
215
equivalence
group
immediate
spaces
122
extension
a
of
215
321f.
excision
of
spaces
185ff]
232]
under
identifying
[LSA,
209
relative homotopy
66,
of
unreduced
index
complex
103
gluing
image
298
107
patch
suspension
ringed
spaces
point
function
13
13
289
177
252
intersection lattice
of
a
family
exhaution
Lipschitz
complete
locally
finite
[LSA,
family
15, of
semialgebraic
group
main
theorems
sequence
[LSA,
map
under
mapping
C
102], 4
19
4]
174,
76
theories
200,
homology
theories
homology
groups
229f,
homotopy
groups
147,
homotopy
sets
i i ,
206
221 [LSA, [LSA,
148f,
cylinder
243,
249f,
257]
133
axiom
244
sequence
197
maps
260
between
18,
function
equivalence
of
ringed
[0,1]
341
spaces 1
(= i s o m o r p h i s m ) b e t w e e n
transformation
between between
x i v , 264
nondegenerate
270f]
[LSA,
between
nerve
280]
256f
PL-automorphism
natural
102]
122
monotonic
natural
22,
22,
...
Mayer-Vietoris
morphism
6,
192
f o r CW-complexes
49,
[LSA
[LSA, 7 ] ,
spectra 42,
[LSA,
102
cohomology
"map"
130,
complex
space cofiber
283
75]
sets
map
long
subsets
342
simplicial locally
simplicial
11
constant
locally
of
simplex
265
cohomology
cohomology
theories
198
(pre) homology t h e o r i e s
theories
198
(pre)homology theories
209,
227
227
normal
patch
one-point
complex
108,
completion
open
51,
simplex star
114 [LSA,
[LSA,
subspace
110
2,
266
simplicial family
subset
homology
^-spectrum
complex
theory
[LSA,
210
43]
complete
44 near
X
67
relative core
C
68
71
partially
finite
109,
partially
proper
i v , 44,
283 47
near
X
64
over
N
93
relative core
C
68
89
equivalence quotient
patch
13,
of
unity
relation
62,
simplicial partition
314
252
paracompact partially
280
3
simplicial ordinary
20]
112
subcomplex
ordered
78]
100,
space
99f,
306
307 x i i i ,
271
128
107 complex
v,
107
decomposition watching
107
homotopy
equivalence
169
path
31 component
polyhedral polytope
simplicial
i i i ,
polytopic preimage
proper
set
316
3
49 of
prehomology primitive
31
a
simplicial
theory
index
i v , 43,
subset
226
13 47 over
N
90
equivalence face
[LSA,
simplicial pseudohomotopic
Puppe
sequence
quotient
xivf,
178],
space
62,
100,
307
271
space
2 48
xivf,
247
192
of
a
simplicial
complex
a
simplicial
map
276,
a
simplicial
set
311
a
simplicial
space
cofiber
theory
(co)homology cone
[LSA,
103]
311,
[LSA,
104]
269
189
cohomology
194,
groups
204
217,
327
190
homology
theory
switched
mapping
telescope regular
306
100
realization
reduced
99f,
337
space
pseudo-mapping
relation
109
quotient
pseudo-loop
289
locally
209 cylinder
197f
semialgebraic
CW-complex
191
165
space
[LSA,
42]
relation
space
relative
308 CW-xomplex homotopy
165
group
[ L S A , 2 6 5 ] , 147
patch
complex
patch
decomposition
path
114
completion
polytope weak
criterion
restriction
polytope
68
(co)homology
(co)homology CW-complex
function group map
2 0 0 , 2 0 5 , 2 1 5 , 221
194, 209
relation
99f
1 8 4 ] , [ L S A , 6]
2
102
16, 42,
[DK , 2
184]
106
realization
311
restriction
of a topological
space
always
(here
spectrum subset
affine)
25
closed
subset
field
280
73f
265, [LSA, 20, 99]
simplicial
approximation complex group
theorem
332
[ L S A 2 1 , 9 9 ] , 6,
314
301
(co)homology homotopy
321f
2 1 2 f , 279,
(co)homology
theory
[ D K , § 7 ] , [ L S A , 4,
252
simplicial
simplex
theory
[DK ,
partition
real
theory
165
equivalence
sequential
145
245
of a
semialgebraic
46
86
WP-approximation representable
114
326f
2
205,215
42]
simplicial
map
262,
314
morphism object set
262
302
subset
280,
subset
of
282
X
generated
decomposition
triangulation singular
chain x i , cochain
[LSA
skeleton
166,
266,
small
"space" space
42,
under
49, C
special
star
122,
strictly
106]
327f,
338
set 47]
2
187 ... 122 patch
complex
patch
decomposition patch
112 112
complex
114
252 [LSA
22,
138]
locally
finite
subordinate strong
26,
1
relative spectrum
112
312
subspace product
groups
[LSA,
subset
smash
286
x i
simplicial 127,
A
328
simplex
lemma
by
328
(co)homology
shrinking
323
264
x i i i ,
patch
99],
260
G-space
simultaneous
24,
261
x i ,
space
[LSA,
q u o t i e n t 60, topology
25f
simplicial
partition 307
of
complex unity
128
[LSA
22,
102]
strongly
surjective
subcomplex
110,
61
[LSA, 22,
subordinate
partition
subspace
24,
2,
of unity
by
A
sequence theorem of
195,
spaces weak telescope
138, extension
topological
178
sets
323f
polytopes
144
140 theorem
28
CW-approximation
179
CW-complex
165
cohomology
theory
theory
extension
of a
realization spectrum morphism
204
216 semialgebraic
(co)homology
311
253
260
233
triangulation
sided
union
209
123
homology
two
of a
286
250
simplicial
triad
space)
193
CW-complexes
transition
simplicial
188
isomorphism
Tietze's
(ina
187 homomorphism
system
128
266 generated
suspension
100]
of a
space
of
system
a
Lipschitz family
universal
element
Urysohn's
lemma
29
i , 4,
of spaces
constant
26,
106]
[LSA, 233]
342
of simplicial 245
[LSA,,
subsets
282
theory
207,221
vertex
of a simplicial
complex
of
s e t 316
a simplicial
map weak
316
homotopy
equivalence
polytope
i i i ,
semialgebraic
map
[LSA 135, 233]
18
102
i v , 17,
monoid
263
space
i v , 3
19
simplicial
group
simplicial
subset
subset wedge
4,
function group
156
4
triangulation weakly
[ L S A , 99]
301 280
23
6f axiom
Whitehead's
theorem
WP-approximation
144
160
of a of
WP-system
194, 209
space
133
a decreasing
system
of spaces
144
LOCALLY SEMIALGEBRAIC SPACES H. D e l f s , M. Knebusch (Lecture Notes i n Mathematics V o l . 1173) TABLE
OF
CONTENTS page
Chapter I - The b a s i c d e f i n i t i o n s
1
§1 - L o c a l l y s e m i a l g e b r a i c spaces and maps
1
§2 - I n d u c t i v e l i m i t s ,
some examples o f l o c a l l y
semialge-
b r a i c spaces
11
§3 - L o c a l l y s e m i a l g e b r a i c s u b s e t s
27
§4 - Regular and paracompact spaces
42
§5 - S e m i a l g e b r a i c maps and p r o p e r maps
54
§6 - P a r t i a l l y p r o p e r maps
6
3
§7 - L o c a l l y complete spaces
75
Chapter I I - C o m p l e t i o n s and t r i a n g u l a t i o n s
8
§1 - G l u i n g paracompact spaces
87
§2 - E x i s t e n c e o f c o m p l e t i o n s
94
§3 - A b s t r a c t s i m p l i c i a l complexes
99
§4 - T r i a n g u l a t i o n o f r e g u l a r paracompact spaces
7
106
§5 - T r i a n g u l a t i o n o f weakly s i m p l i c i a l maps, maximal complexes 56 - T r i a n g u l a t i o n o f amenable p a r t i a l l y
11 f i n i t e maps
3
124
§7 - S t a r s and s h e l l s
1
§8 - Pure h u l l s o f dense p a i r s
146
§9 - Ends o f s p a c e s , t h e L C - s t r a t i f i c a t i o n
156
§10 -Some p r o p e r q u o t i e n t s
178
§11 - M o d i f i c a t i o n o f pure ends
189
§12 -The S t e i n f a c t o r i z a t i o n o f a s e m i a l g e b r a i c map
198
§13 - S e m i a l g e b r a i c spreads
211
§14 -Huber's
219
theorem on open mappings
3
8
C h a p t e r I I I - Homotopies
226
§1 - Some s t r o n g d e f o r m a t i o n r e t r a c t s
226
§2 - S i m p l i c i a l
232
approximations
§3 - The f i r s t main theorem on homotopy s e t s ; mapping
spaces
24 3
§4 - R e l a t i v e homotopy s e t s
249
§5 - The second main theorem; c o n t i g u i t y c l a s s e s
257
§6 - Homotopy groups
265
§7 - Homology; t h e Hurewicz theorems
278
§8 - Homotopy groups o f ends
286
Appendix A - A b s t r a c t l o c a l l y s e m i a l g e b r a i c spaces
295
Appendix B - C o n s e r v a t i o n o f some p r o p e r t i e s o f spaces and maps under base f i e l d e x t e n s i o n
309
References
315
List
319
o f symbols
Glossary
322
L E C T U R E
N O T E S
XJLST
E d i t e d b y A. D o l d
M A T H B M A T T G
3
a n d B. Eckmann
Some g e n e r a l r e m a r k s o n t h e p u b l i c a t i o n o f monographs and s e m i n a r s
I n what f o l l o w s a l l r e f e r e n c e s t o monographs, a r e a p p l i c a b l e a l s o t o m u l t i a u t h o r s h i p volumes such as seminar n o t e s .
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Manuscripts or plans f o r Lecture Notes volumes should be s u b m i t t e d e i t h e r t o one o f t h e s e r i e s e d i t o r s or to SpringerV e r l a g , H e i d e l b e r g . These p r o p o s a l s a r e then r e f e r e e d . A final d e c i s i o n c o n c e r n i n g p u b l i c a t i o n c a n o n l y be made o n t h e b a s i s o f the complete m a n u s c r i p t s , b u t a p r e l i m i n a r y d e c i s i o n c a n u s u a l l y be b a s e d on p a r t i a l information: a fairly detailed outline d e s c r i b i n g t h e p l a n n e d c o n t e n t s o f each c h a p t e r , and an i n d i c a tion o f t h e e s t i m a t e d l e n g t h , a b i b l i o g r a p h y , a n d one o r two sample c h a p t e r s - o r a f i r s t d r a f t o f t h e m a n u s c r i p t . The editors w i l l t r y t o make t h e p r e l i m i n a r y d e c i s i o n a s d e f i n i t e as t h e y c a n on t h e b a s i s o f t h e a v a i l a b l e i n f o r m a t i o n .
§3.
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Vol. 1205: B.Z. Moroz, Analytic Arithmetic in Algebraic Number Fields. VII, 177 pages. 1986. Vol. 1206: Probability and Analysis, Varenna (Como) 1985. Seminar. Edited by G. Letta and M. Pratelli. VIII, 280 pages. 1986. Vol. 1207: P.H. Berard, Spectral Geometry: Direct and Inverse Problems. With an Appendix by G. Besson. XIII, 272 pages. 1986. Vol. 1208: S. Kaijser, J.W. Pelletier, Interpolation Functors and Duality. IV, 167 pages. 1986. Vol. 1209: Differential Geometry, Pehiscola 1985. Proceedings. Edited by A.M. Naveira, A. Fernandez and F. Mascaro. VIII, 306 pages. 1986. Vol. 1210: Probability Measures on Groups VIII. Proceedings, 1985. Edited by H. Heyer. X, 386 pages. 1986.
Vol. 1236: Stochastic Partial Differential Equations and Applications. Proceedings, 1985. Edited by G. Da Prato and L. Tubaro. V, 257 pages. 1987. Vol. 1237: Rational Approximation and its Applications in Mathematics and Physics. Proceedings, 1985. Edited by J. Gilewicz, M. Pindor and W. Siemaszko. XII, 350 pages. 1987. Vol. 1238: M. Holz, K.-P. Podewski and K. Steffens, Injective Choice Functions. VI, 183 pages. 1987. Vol. 1239: P. Vojta, Diophantine Approximations and Value Distribution Theory. X, 132 pages. 1987.
Vol. 1211: M.B. Sevryuk, Reversible Systems. V, 319 pages. 1986.
Vol. 1240: Number Theory, New York 1984-85. Seminar. Edited by D.V. Chudnovsky, G.V. Chudnovsky, H. Cohn and M.B. Nathanson. V, 324 pages. 1987.
Vol. 1212: Stochastic Spatial Processes. Proceedings, 1984. Edited by P. Tautu. VIII, 311 pages. 1986.
Vol. 1241: L. Garding, Singularities in Linear Wave Propagation. Ill, 125 pages. 1987.
Vol. 1213: L.G. Lewis, Jr., J.R May, M. Steinberger, Equivariant Stable Homotopy Theory. IX* 538 pages. 1986.
Vol. 1242: Functional Analysis II, with Contributions by J. HoffmannJorgensen et al. Edited by S. Kurepa, H. Kraljevic and D. Butkovid. VII, 432 pages. 1987.
Vol. 1214: Global Analysis - Studies and Applications II. Edited by Yu.G. Borisovich and Yu.E. Gliklikh. V, 275 pages. 1986. Vol. 1215: Lectures in Probability and Statistics. Edited by G. del Pino and R. Rebolledo. V, 491 pages. 1986. Vol. 1216: J. Kogan, Bifurcation of Extremals in Optimal Control. VIII, 106 pages. 1986. Vol. 1217: Transformation Groups. Proceedings, 1985. Edited by S. Jackowski and K. Pawalowski. X, 396 pages. 1986. Vol. 1218: Schrodinger Operators, Aarhus 1985. Seminar. Edited by E. Balslev. V, 222 pages. 1986.
Vol. 1243: Non Commutative Harmonic Analysis and Lie Groups. Proceedings, 1985. Edited by J. Carmona, P. Delorme and M. Vergne. V, 309 pages. 1987. Vol. 1244: W. Muller, Manifolds with Cusps of Rank One. XI, 158 pages. 1987. Vol. 1245: S. Rallis, L-Functions and the Oscillator Representation. XVI, 239 pages. 1987. Vol. 1246: Hodge Theory. Proceedings, 1985. Edited by E. Cattani, F. Guillen, A. Kaplan and F. Puerta. VII, 175 pages. 1987.
Vol. 1219: R. Weissauer, Stabile Modulformen und Eisensteinreihen. Ill, 147 Seiten. 1986.
Vol. 1247: Seminaire de Probabilites XXI. Proceedings. Edite par J. Azema, P.A. Meyer et M. Yor. IV, 579 pages. 1987.
Vol. 1220: Seminaire d'Algebre Paul Dubreil et Marie-Paule Malliavin. Proceedings, 1985. Edite par M.-P. Malliavin. IV, 200 pages. 1986.
Vol. 1248: Nonlinear Semigroups, Partial Differential Equations and Attractors. Proceedings, 1985. Edited by T.L. Gill and W W . Zachary. IX, 185 pages. 1987.
Vol. 1221: Probability and Banach Spaces. Proceedings, 1985. Edited by J. Bastero and M. San Miguel. XI, 222 pages. 1986. Vol. 1222: A. Katok, J.-M. Strelcyn, with the collaboration of F. Ledrappier and F. Przytycki, Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities. VIII, 283 pages. 1986. Vol. 1223: Differential Equations in Banach Spaces. Proceedings, 1985. Edited by A. Favini and E. Obrecht. VIII, 299 pages. 1986. Vol. 1224: Nonlinear Diffusion Problems, Montecatini Terme 1985. Seminar. Edited by A. Fasano and M. Primicerio. VIII, 188 pages. 1986. Vol. 1225: Inverse Problems, Montecatini Terme 1986. Seminar. Edited by G. Talenti. VIII, 204 pages. 1986. Vol. 1226: A. Buium, Differential Function Fields and Moduli of Algebraic Varieties. IX, 146 pages. 1986. Vol. 1227: H. Helson, The Spectral Theorem. VI, 104 pages. 1986.
Vol. 1249: I. van den Berg, Nonstandard Asymptotic Analysis. IX, 187 pages. 1987. Vol. 1250: Stochastic Processes - Mathematics and Physics II. Proceedings 1985. Edited by S. Albeverio, Ph. Blanchard and L. Streit. VI, 359 pages. 1987. Vol. 1251: Differential Geometric Methods in Mathematical Physics. Proceedings, 1985. Edited by P.L. Garcia and A. Perez-Rend6n. VII, 300 pages. 1987. Vol. 1252: T. Kaise, Representations de Weil et GL2 Algebres de division et GL . VII, 203 pages. 1987. n
Vol. 1253: J. Fischer, An Approach to the Selberg Trace Formula via the Selberg Zeta-Function. Ill, 184 pages. 1987. Vol. 1254: S. Gelbart, I. Piatetski-Shapiro, S. Rallis. Explicit Constructions of Automorphic L-Functions. VI, 152 pages. 1987.
Vol. 1228: Multigrid Methods II. Proceedings, 1985. Edited by W. Hackbusch and U. Trottenberg. VI, 336 pages. 1986.
Vol. 1255: Differential Geometry and Differential Equations. Proceedings, 1985. Edited by C. Gu, M. Berger and R.L. Bryant. XII, 243 pages. 1987.
Vol. 1229: O. Bratteli, Derivations, Dissipations and Group Actions on C*-aigebras. IV, 277 pages. 1986.
Vol. 1256: Pseudo-Differential Operators. Proceedings, 1986. Edited by H.O. Cordes, B. Gramsch and H. Widom. X, 479 pages. 1987.
Vol. 1230: Numerical Analysis. Proceedings, 1984. Edited by J.-P. Hennart. X, 234 pages. 1986.
Vol. 1257: X. Wang, On the C*-Algebras of Foliations in the Plane. V, 166 pages. 1987.
Vol. 1231: E.-U. Gekeler, Drinfeld Modular Curves. XIV, 107 pages. 1986.
Vol. 1258: J. Weidmann, Spectral Theory of Ordinary Differential Operators. VI, 303 pages. 1987.
Vol. 1259: F Cano Torres, Desingularization Strategies for ThreeDimensional ector Fields. IX, 189 pages. 1987.
Vol. 1290: G. Wustholz (Ed.), Diophantine Approximaioron scendence Theory. Seminar. 1985. V, 243 pages. 1987
Vol. 1260: l.H. Pavel, Nonlinear Evolution Operators and Semigroups. VI, 26 pages. 1987.
Vol. 1291: C. Mceglin. M.-F. Vigneras, J.-L. Waldspurger,