Stratified Mappings - Structure and Triangulability (Lecture Notes in Mathematics, 1102) 3540138986, 9783540138983

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Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann Su bseries: Mathematisches Institut der Universitat und Max-Planck-Institut fur Mathematik, Bonn - vol. 4 F. Hirzebruch Adviser:

1102 Andrei Verona

Stratified Mappings Structure and Triangulability

Spri nger-Verlag Berlin Heidelberg New York Tokyo 1984

Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann Su bseries: Mathematisches Institut der Universitat und Max-Planck-Institut fur Mathematik, Bonn - vol. 4 F. Hirzebruch Adviser:

1102 Andrei Verona

Stratified Mappings Structure and Triangulability

Spri nger-Verlag Berlin Heidelberg New York Tokyo 198

Author Andrei Verona Department of Mathematics and Computer Science California State University Los Angeles, CA 90032, USA

AMS Subject Classification (1980): 57 R05, 58A35; 32B25, 32C42, 57R35, 57R45, 57S15, 58C25, 58C27 ISBN 3-540-13898-6 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13898-6 Springer-Verlag New York Heidelberg Berlin Tokyo

This work IS subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.

© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

INTRODUCTION

For several reasons, most of them stemming from algebraic topology, it is important to know whether a topological space, or more generally a continuous map, is triangulable or noto

Cairns [Cal proved the triangulability of

smooth manifolds; another proof, also providing a uniqueness result, is due to J. H. C. Whitehead [Wh],

First attempts to prove the triangulability of al-

gebraic sets are due van der Waerden [W], Lefschetz [Le], Koopman and Brown [K-B]

and Lefschetz and Whitehead [L-W]o

Rigorous proofs, in the more general

case of semianalytic sets, were given by Lojasiewicz [Lo] and Giesecke [Gi], Later, Hironaka [Hi tic sets.

1)

and Hardt [HaZl proved the triangulability of subanaly-

The most general spaces known to be triangulable are the stratified

sets introduced by Thom [T

Z]

and the abstract stratifications introduced by

Mather [Mal] (Mather's notion is slightly different from Thom's one, but it is more or less clear that the two classes of spaces coincide, at least in the compact case); they include all the spaces mentioned above and also the orbit spaces of smooth actions of compact Lie groups,

Their triangulability was pro-

ved by several authors (Goresky [GIl, Johnson [J to mention only the published proofs.

Z]'

Kato [Ka] and Verona [Ve

A more detailed discussion of these

proofs and of others can be found in the introduction of Johnson's paper or at the end of Section 7 of the present work).

The more difficult problem of

the triangulability of mappings was considered by much fewer authors: Putz [P] proved the triangulability of smooth submersions, Hardt [Ha

proved the Z] triangulability of some, very special, subanalytic maps and I proved in

[Ve

3]

the triangulability of certain stratified maps.

In [T

Thom con-

I]

sidered the problem of the triangulability of smooth maps and (implicitely) conjectured that "almost all" smooth mappings are triangulable. aim of this paper to prove this conjecture. Theorem. dary.

Let

M and

To be precise, we shall prove

N be smooth manifolds without boun-

Then any proper, topologically stable smooth map from

triangulable.

It is the

M to

N is

3]

IV

Since the set of proper and maps from M to

N

M to

stable smooth

N is dense in the set of all proper smooth maps from

(Thom-Mather theorem*) we obtain a positive answer to the

above mentioned conjecture. As a matter of fact, we prove a more general result concern-

ing the triangulability of certain stratified mappings (Theorem 8.9) which implies the theorem stated above and also the following result (first proved by Hardt [Ha

2]):

any proper light subanalytic map is trian-

gulable (light means that the preimage of a discrete set is discrete). Our Theorem 8.9 is not as general as one would expect it.

It applies only

to proper and nice abstract Thom mappings (nice means that the mapping is finite to one when restricted to a certain subspace).

It is natural to con-

jecture that the theorem is true for any proper abstract Thom mapping. positive answer would solve another conjecture of Thom [T Thom mapping (in Thom's triangulable.

1]

A

: any proper

terminology "application sans eclatement") is

The main difficulty in proving this more general version of

Theorem 8.9 is explained in Section 8.15.3. Since we are dealing with stratified spaces as introduced by Mather in [Mal] and since these lecture notes have never been published, I thought it would be useful to collect in a first part of the present work (Chapters 1, 2, and 3) the main results of the theory.

Some of the

proofs presented here are new and simpler than the original ones.

For tech-

nical reasons, we are obliged to work with certain manifolds with corners, called here manifolds with faces. presented in Chapter 4.

The necessary facts concerning them are

In Chapter 5, we extend the theory of abstract strati-

*) This result was conjectured by Thom [T a possible proof.

Later Mather ([Ma

2]

1].

and [Ma

In [T 3])

3]

and [T

2]

he outlined

filled in the details, made

it rigorous and, by slightly changing some of Thorn's concepts, cleaned up many technical points. in [Gib]

0

Another proof (only slightly different) is presented

v fications and abstract Thom mappings to the case when the strata are allowed to be manifolds with faces; most of the proofs are copies of the proofs presented in the first three chapters and so they are omitted.

In Chapter 6,

we prove some theorems concerning the structure of abstract stratifications and of abstract Thom mappings.

In some sense, they can be viewed as a kind

of "resolution of the singularities" in the

ceo -case.

For example, Theorem

6.5 can be interpreted as saying that any abstract stratification of finite depth can be obtained from a manifold with faces by making certain identifications on the faces.

Chapter 8 contains the main results of the paper

(they were mentioned above).

In an appendix, I have collected some facts

from PL-topology which are needed in Chapters 7 and 8.

Acknowledgement. the preparation of this

For their support and/or hospitality during

work, I wish to express my gratitude to the Institut

des Hautes Etudes Scientifiques in Bures-sur-Yvette, the Alexander von Humboldt Foundation, the Sonderforschungsbereich Theoretische Mathematik of the University of Bonn, the National Science Foundation (Grant MCS-8I088I4 (AOI), the Institute for Advanced Study in Princeton and, last but not least, the Max-Planck-Institut fur Mathematik in Bonn.

Especially, I wish

to thank Professor F. Hirzebruch, whose confidence and understanding during the preparation of this paper were of great helpo

TABLE OF CONTENTS

O.

NOTATION AND CONVENTIONS

1.

ABSTRACT STRATIFICATIONS 1.2.

4

1.3.

Existence of partitions of unity

8

1.4. The sheaf Notes

C; -

is fine

CONTROLLED VECTOR FIELDS

9 9 9

2.1.

Definition and elementary properties

2.2.

Flows

11

9

2.3.

Existence of flows

12

2.4.

Lifting of vector fields

14

2.5.

Locally trivial controlled maps

16

2.6. 2.7.

First isotopy lemma The local structure of abstract stratifications

16

2.8.

Saturated subsets

2.9-2.10.

3.

2

Tubes Definition and elementary properties of abstract stratifications

1.1.

2.

2

Preimage of saturated subsets

17 17

17

2.11. Fibre products

19

2.12. The abstract stratification Notes

19

ABSTRACT TI!().1 MAPPINGS

20

3.1.

Regular squares

20

3.2. 3.3. 3.4.

Definition and elementary properties of abstract Thom mappings Examples The sheaf X f

21

B

=

20

22 23

3.5-3.6. Lifting of controlled vector fields

24

3.7.

27

Notes

The second isotopy lemma

28

VIII

4.

5.

MANIFOLDS WITH FACES

29

4.1.

Definition and elementary properties

29

4.2.

Maps compatible with the faces

31

4.3.

Examples and remarks

32

4.4-4.5. Vector fields parallel to the faces

36

ABSTRACT STRATIFICATIONS (WITH FACES)

37

5.1.

Definition and elementary properties

37

5.2.

Maps compatible with the faces

39

5.3.

Examples, remarks and constructions

40

5.4.

Controlled vector fields

48

5.5.

Abstract Thorn mappings

49

5.6.

Lifting of controlled vector fields

50

5.7.

Second isotopy lemma

51

5.8.

First isotopy lemma

51

5.9.

More examples and remarks

51

5.10-5.11. Applications of the isotopy lemmas 5.12. 6.

On the controlability of the

p

functions

54 56

THE STRUCTURE OF ABSTRACT THOM MAPPINGS

57

6.1.

Decompositions of abstract stratifications

57

6.2.

Existence of decompositions of abstract stratifications

65

6.3.

Total decompositions of abstract stratifications

69

6.4.

Restriction of decompositions

70

6.5.

Existence of total decompositions of abstract stratifications

71

6.6.

The core of an abstract stratification

71

6.7.

Admissible squares

72

6.8.

Decompositions of abstract Thorn mappings

83

6.9-6.10. Existence of decompositions of abstract Thorn mappings 6.11.

Total decompositions of abstract Thorn mappings

6.12. Existence of total decompositions of abstract Thorn mappings 6.13-6.15. More cores

96 111 117 119

IX

7.

TRIANGULATION OF ABSTRACT STRATIFICATIONS

120

7.1.

120

Triangulation of relative manifolds

7.2-7.5. Good triangulations of cores

8.

121

7.6.

Smooth triangulations of

stratifications (definition)

7.7.

Extending a good triangulation of the core to a smooth triangulation of the corresponding abstract stratification

126

7.8.

Existence of smooth triangulations of abstract stratifications

127

7.9.

Triangulation of subanalytic sets

127

7.10. Triangulation of orbit spaces Notes

128

TRIANGULATION OF NICE ABSTRACT THOM MAPPINGS 8.1. Notation

129 129

8.2.

129

Remarks on good triangulations of

8.3.

Regular triangulations of

8.4.

Nice abstract Thorn mappings

128

c(f)

c(f)

8.5-8.7. Properties of regular triangulations of 8.8. 8.9.

9.

126

130 131

c(f)

The canonical extension of a regular triangulation of c(f) Triangulation of proper nice abstract Thorn mappings

133 135 138

8.10. Topologically stable mappings

138

8.11. Triangulation of proper topologically stable mappings 8.12. Triangulable smooth mappings are generic

140

139

8.13. Triangulation of proper light subanalytic mappings 8.14. A remark of Hironaka

140

8.15. A possible generalisation

140

APPENDIX

141

9.1. 9.2.

143

Simplicial complexes and triangulations Product of triangulations

140

143

9.3.

Fibre product of triangulations

145

9.4. 9.5.

Mapping cylinders Two lemmas

146

9.6.

Piecewise linear maps

152

151

REFERENCES

153

SUBJECT INDEX

156

SYMBOL INDEX

158

O. NOTATION AND CONVENTION

0.1.

A topological space is called nice if it is Hausdorff, locally compact,

paracompact and with a countable basis for its topology. 0.2.

is a topological space and

X

denotes the closure (resp. interior) of

If

A

X

0.3. If A and joint union. 0.4.

A , then

C

in

Bare topo logical spaces, then

For any set

A, I

A

(or

clA(X) (resp. intA(X))

A AU B

denotes thei r dis-

idA) denotes the identity map of

A.

00

0.5.

Smooth means always differentiable of class

0.6.

The connected components of a smooth manifold may have different dimen-

sions. Given a smooth manifold

C

M we denote by

its tarrr,ent bundle. If

x E: M , TM denotes the tangent space of M at x manifold and f : M N is smooth, df TM TN x E: M

f ; if

then

The smooth map 0.7. Let

Let f

and

equals U

of

A

g Z

f

df x

TMx

is called submersive if

g be maps defined on Z

(denoted

X

f = g

and

other similar situations (for example a neighborhood 0.8.

R

df

11

of

Z

such that

X, Y

and

Z

x E: M.

be subsets of

respectively. We say that

A.

f

The same terminology is also used in

n

Y U

denotes the field of real numbers

R: = {r E: R ; r > O}

df.

Z) if there exists a neighborhood

X X

is another smooth

is surjective for any

x

Y

near

fl X n u = g] Y n U

such that

N

denotes the differential of

denotes the restriction of

TNf(x)

be a topological space and let

near

x . If

near Y

n

Z

means that there exists

U ).

r

O}

and

1.

ABSTRACT STRATIFICA TION S

1.1.

Let

A

be a nice topological space and let

closed subset. TI

T

X

T

Px 10,

X

---+

X X

Let

---+

dx) < 6(x)

be an open neighborhood of

be a continuous retraction (i .e . ,

R+

X ---+ R

6

TX

X c A

be continuous and such that

-1

x EX;

if

10 < 6

in

A, and

TIXIT X = IX)

"x (0)

we shall use the following notation

for any

X

be a locally

X.

10 < 6

Given if

set

X x (10, 6) ={(x, t)E X xR; dx) < t < 6(x)} X x{s}={(x,t)EX x R; t = dx) } X x [10, 6)

(X x (10, 6» U (X x i c J)

X x (10, 61

(X x (10, 6) ) U (X

x {6})

X x [10, 6J

(X x [10, 0)) U (X

x {6})

TE: X SE: X

{a E T

< E:(TIX(a))} X; PX(a)

{a E T

X; PX(a)

dTIX(a»}

In such an expression a real number will be considered as the corresponding constant function on 10

(or

X.

10 , E: ' 1 Z

values in

* R+,

From now on, unless the contrary is specified, 6 , ... ) 1

6,

will be a continuous function taking

its domain being determined by the context; if in addition

the domain is a smooth manifold, then

10

will be assumed to be smooth.

When no confusion can arise we shall denote the restrictions of to and

X

and

by the same symbols; otherwise we shall use the notation respectively. and

after shrinking

( 1. 1. 1) if

TI

T X'

X cUe A

be as above.

One can verify that, possibly

the following assertions are true: with

U

open, then

for some

10;

3

(1.1.2)

:

X x [0, c)

--i>-

is proper for some

c,

and a neighborhood

U

As an immediate consequence (1.1.3) given a compact subset in

A,

X

such that

Let

T

The triples (T

X'

hood and

1T

X

X' X (T

of

PXIT

= X

X

o

1T

X'

(T

S

-1 (V) X Px

and

1T

(T

U.

equivalence class of triples

x)

X' X'

near

such that

= PXtTX n

and a neighborhood

of

V

K

of K

in

n T XS c u.

X' X' PX) A

*

E R+

have the same properties as

' P X)

X

in

U

1T

and

1T

' PX)

U

there exists

Kc X

P

1T

X,

X

X

that is, there exists a neighbor-

= T X ()

T X () U

1T

and

1T

are called equivalent if

U,

1T

X

By definition, a tube of

(T X'

T X'

' P X),

IT X X

() U in

= 1TXIT X n A

U

is an

In order to simplify the notation,

we shall not distinguish between a tube and a triple which represents it; however we shall consider only triples which verify (1.1.1) and (1.1.2) (and therefore (1.1. 3) too). Let now and

X

nY

means that Ty

= {T y'

yeA

be another locally closed subset such that

(in this situation we shall write X < Y 1Ty

or

' py }

X

= Y).

Let

be tubes of

X

T

X

X < Y;

= (T X'

and

Y

1T

X

Xc c'J

as usual

' P X)

A

X < Y

and

respectively.

We shall consider

"control conditions" of the form (1. 1. 4) there exist y (a ) E TX

1T

(1.1.5) there exist y (a) E TX

1T

N ext let

S

and S

B

such that

(a ) ) = X(1Ty 6

1T

X

a E T

S

X

n T y6

implies

6 Ty

implies

(a ) ;

such that

a

T

S

X

n

P ( 1Ty(a)) = PX(a). X

f : B ---+ A

f(y) c X.

6

1T

and

and

nice topological space. such that

and

Let

be a continuous map, where Yc

Given tubes

B

and

X c A

and

B

is another

be locally closed subsets of

X

(Y)

in

A

and

respectively, we shall also consider control conditions of the form

Y

in

4

(1.1.6) there exists

(1.1. 7) there exists py(b) 1. 2.1.

8

f(T y) c

T

and

X

8

PX(f(b)), bE T y.

:=

6

A weak abstract stratification (w. a. s.)

A;

topological space of

bE T y:

such that

8

and

X

8

= 'ITX(f(b)),

f('ITy(b))

T

such that

8

(ii ) a locally finite family

(called strata) such that

A

consists of (i) a nice

A

of locally closed subsets

is the disjoint union of the strata;

A

(iii) a family of tubes of the strata,

{eX; X E A}.

The strata and their

tubes must satisfy the following four axioms: (1. 2.1.1) if

X. YEA

and

n

X

d

A

(Y)

'*

-

t = t'

is proper).

too.

It follows

s > 0

Now, if

is small

and = 1T

x(At;(At;(a,

t'), s )

= At;(1T

X(At;(a,

t'», s )

t'), s ) = At;(1T , t'+s) = At;(At;(1T X(a), X(a) similarly

We deduce that (Z.3.1), (Z.3.Z) and (Z.3.3) are valid for

o -
0, and

Y E V. n

By

(Z.3.4) and (Z.3.3)

o < PX(Y n)

PX(At;(Y n, s n + p -

S

PX(At;(At;(Y, s n ), s n+p - s n »

n»

= PX(At;(Y, sn+J) = PX(Y n + p) thus the sequence

(PX(Yn»

cannot converge to zero.

this contradicts the continuity of To prove (ii), let

part of the proof and choose

t

1

and

PX(x)

0,

Thus (i) is true.

P X'

(x , t) E Dt;'

Since

let the notation be as in the first t

z

with the additional property

14

that

"z:

t1 < t
- Y

shows that 3.4.

Let

g

"x

Let also by

(6c!

n

fl: =

g (b)

= (IT Y (b),

and define

f(b)).

A direct verification

is an a. T .rn ,

f : fll-----;>-/i

V c B

be an a.T.m. and

be either a locally

closed subset which is a union of strata or an open subset. by

(V)

the subset of

Xi (V)

A, nlf-l(X)

(3.4.1) for any X

consisting of those

n V is a controlled vector field on

f

It is obvious that the collection

sub sheaf of the sheaf

Let

n

f:

r;

be an a.T.m.,

(B)

f

X fl

is a

f

nO

such that

X;(A), B

and

c B

be

df vr; 0 = r;.

Suppose that niB 0

O

Then

df'n = r;.

Because of the existence of controlled partitbns of unity, the

assertion is local. depth(fl)

is a

B}

In fact

we shall denote it

a closed union of strata and

Proof.

open in

V

and also a sub sheaf of Lie algebras of

Ci-submodule of

there exists

which verify

n v.

alf-l(X)

3.5. LEMMA.

n

We shall denote

= 0,

Thus we may assume that

If

is finite.

then there are no incidence relations between the strata of

and the assertion is trivial.

Assume that

depth(fl) > 0

and that the

assertion is true for any a. T . m , whose domain has depth less than Let X E A

B

B, Y

be the stratum which contains

be the stratum which contains

"x

g : also

b

=

controlled submersion.

f(Y).

band

Let

be the a.T.m. constructed in 3.3.4. and

f

X

= flB X

: B

X

----+

X.

By (:\.2.4)

Let f

X

As a consequence of Lemma 2.4. there exists

is a

24 I; E X B

(B

such that

X)

(3.5.1)

df-I;

and (3.5.2) Let

B1 = B0 U B

of (3.5. 2) , df'n

and

X

it is a closed union of strata and, as a consequence

;

nO

= 1:;.

l

Let nO I:;

such that

determine a vector field

"x

cO=cn B l, g' =glc:

nIl Co

= (I; IY) x (I:; I

and

E XXx/1 (Y x

X "x

is a controlled vector field on

(60 I

Notice that actually and

dg'

-n 0

= 1:;.

we may apply the inductive hypothesis to _ g' f such that Thus there exists n E (C) c (C)

g',


y},

then

B,

be the

Choose now a compact neighborhood

6(Z) : Z

and

R+

x.

t > O.

real numbers


-

(G, F) 6

0

x

Bn

Set

=

is obviously an isomorphism from to

f:

I-

----i>-

6,

the proof is Q.E.D.

The notions introduced in this chapter are again due to

Thorn [TIL while the presentation follows closely that of Mather [M However the proof of Lemma 3.5 is much simpler here.

I].

A

29 4.

MANIFOLDS WITH FACES

4.1.

Let

M

boundary of

be a smooth manifold with corners (see [C I) , M

!VI

and

= M

-6,

and

be an a s . of depth zero.

f' I B'o : =0 B I f-- -----* A'

is an a.T.m.

fiB I = f '

Q

verifies (5.3.7.1).

verifies (5.3.7.1).

shows that

g

X

to

f" ) -vertical face of

(resp.

be an isomorphism of such that

f"

(r esp .

f'

!i ' UljJ !i"

5.9.8.

5.9.9.

n f-l(TE)

8 8 = Ty E

be the corresponding face of

AD)

a

B' = B"

X

is an a. T . m ,

be an isomorphism of

(j.l )

v

Q

6, x

-----+

-----'>- 6,'

the w a s ,

that

over

is an a. T .rn , from

g

be an a.T.m. and

such that v

g : B

and define

Then

f : !i1------*6,

f x l

and

E X)

and

•

Then

B' 0

X

of

Y 1 = y 2)

then

1

0

f :

is

an a cTv m , (Indeed, (5.5.1) is obvious, (5.5.2) and (5.5.3) are easy consequences of our additional assumption, and (5.5.4) follows from 3.1.2).

E X

54 5.10. PROPOSITION. subset of t

R

A

g : W _

and

l'?1- -be

f :

R

g -l( C-00, t])

= A tl

At)

B[t

= ClCA[t)

ft]

: Bt l _

Then

B[t

([a, t])

([a,O])

O a E A

lim A£\(a,-t), tJ'l

F+o(a)

=

+ a EU 0

(A£\(a, 1-

A

A

*

A

A

Given an open subset

of

U

A

and

[, E X A (U)

set

{a E U;

Then

The above relations imply that

£\

is determined by

[,

and

E.

More precisely 6.1.4. only if

LEMMA. E = E,

Two decompositions and

£\

and

6 near

of

A are equal if and

61

6.1.5.

Given a decomposition

p, E, ¢}

=

A such that

n

A.

A*

A:

=

A+ n A.

n A*

A.

induces in a canonical way a Pi' E i , ¢i}

=

are faces of

A

and

of

where

A+ respectively,

+

A.

and

E.

of

p,

E and

6.1.6.

¢

respectively.

A containing

=

¢.

n A.

A

=

and

A-IA., =

are the r as t r ic tions

Note that

p, E, ¢}

Let

open subset of

A and a face

(this is equivalent to

or to

decomposition

of

A.

be a decomposition of

A and

Then

n

=

D,

U be an

p,

E,

¢}

is

a decomposition of 6.1.7.

Let

*

:

p, E, ¢}

=

P : A

+

A

Set

- +

O A = {a E A-;

*

A,

*

+

A+ = A ...... (i..-'AO).

and

*

R+ and

¢ : C(p)

A

+

by

Then one can endow a v s , structures

decomposition of u

A-

and

such that

A

A and

,'\

.

=

A =

-1

«_00,1]),

u

Proposition 5.10 gives us

A

,'\

u

u

= (Indeed,

A and let

{a E A-;

=

= E: A

be a decomposition of

0

=

E

and

A

-,

=

= (u

P

Define

,

-

=

and

A+ with unique

p, E, ij)}

is a and

TI,'\)E:,'\

. r. = d/dt u the rest being a simple matter of

AO

and

u

and

d

verification. ) 6.1.8.

=

and i

A

2 2 A A = lA UB A , B being a face of both lA and = i i i = {i A- A+ Let p, E, i-

A+

depth( 2

>

* A* = 1 A,

Then

1 l A+ U 2 A A = , A+ B 1 E = E and W = lw.

Take

A* = A* ,

Case II:

depth(

2

> depth(

1

This case is similar to

Case I.

= depth(

Case III: 1 A

n

l(IB

2A

=

0,

A*

U

n A+

l AO U2 AO.

W: C(p)

->-

B

Define

EI i A * = i E, we define

* A,

= depth(

A-

by

i

2

-

1 -

P : AO

= 1,2.

WIC(i p)

->-

2-

A*

Then =

iljJ,

and S(p)E

i

Then

>

is face of both

Take

=

=

AO = A-

1 A*

=

2

1

=

1,2.

l A+

+A E

and l A+ U 2A+ B

= :

A

*

->- R

1

*

+

by

then 2

S(l p) EUS(2 p) E and

63

Case IV:

=

hypothesis

26 1B 1A*

n

A-

CClp)

B

n 1A*

n

2A*

B*

n

2 A

B

2A-

B

n 1A-

B

lA+

n 2A+

B

n lA+

2 B n A+

lAO

n 2AO

B n lAO = B

E,

n CC 2p)

Then, by

Notice that

B

n

1 A-

i

q, 6,

B+

1 A

U B

and we can define E: Ii A*

2 * A

=

lA* U 2A* ; = B*

A*

Take

=

i

= =

2 A

and

°

p : A 1,2.

CCq).

->

lA+

A+ A*

n 2AO =

and

Notice that

B+ BO

=

U +

2

A+

then

B

*

E : A

cCp)

Thus we can define

->

cCp)

=

A-

n A+

p!iAO= i p,

R* by +

CC\) U CC ¢

AO

->

2p) A-

and by setting

lAO U 2AO

64

ICCip) that

i.

As in the other cases, a straightforward verification shows

16 VB 26

6.1.9. 6

=

of

Let A

p,

=

}

E,

is a well defined decomposition of

be an a.s. of depth zero and let

q

A-

+

M.

A

A decomposition

is called q-compatible if near

A

or, equivalently dq • If (if .

near

A

is q-compatible then

qlA+

is a weak morphism from

is not q-compatible it may happen that

the face

6.1.10.

0

=

A0

0

f

+)

•

Consider a diagram

qiA+

to

is not compatible

65 in which

= 0 =

and

be the fibre product of

=

q-compatible decomposition of pg, Eg, (resp. qlA+ :

of

.

'\,-

A

and thus we can cons i.de r =

°

A xM\i)

A.

Then

!* ..

"'-1"'0 A A = '\,+1'\,0 A A.

-

h

+

,

be a

, p, E,

We shall define a decomposition h

° °-

=

qlA :

g

A xM!:!

(re sp ,

=

-

M

+

and g are transverse

"'+

=

+

A

",*

x M

",*

*,

'K 0

=

. g '\,0 Deflne p : A

A- n 'K+

"

*

A x ..

-+

is a face of both

",* g ",* A , E : A

-+

*

M

A.O

N

='

= A*

x N; M

g

and g

R+ and

"'0 AOx N' p g (a,x ) = ( p(a),x), (a,x) E A M '

by s e t t i. ng : (a,x) E A

with respect to q and g (see

an a.s. Let

*-

'"

are transverse. Let

and identify its underlying topological space wi th a closed

subset of and

g

as follows. Notice first that q!A-:

qlA * :

+-

and

and

Ais

= 0,

5.9.3); since

q

g : C(p )

Eg(a,x)

([(a,x),t]) = ([( a,t]),x), [(a,x),t]

E() a ,

e

g

C(p).

A direct verification shows that

so definp.d is a decomposition of

If Uh is small enough, we can take

U

=

(U

(a,x)

t::,

e

xN)nA '" U

"'A

-+

and then

•

t::,g'

(a,x) E U

.

t::,g'

THEOREM.

6.2.

and let

e

i

Let

q : A-

+

M.

(b) 1

=1

(c)

then and

0

Suppose that there exists

< 00, Ie I

A

=

depth(i;) ; there is given a (qIAi)-compatible decomposition A;, p. , =1 1 if

6,[A. n A. 1 1 J A,) • =J

i'

e

I,

then

°,

is a decompositi-

is a total decomposition of

A

be a total decomposition of

}

= n-k VIA

is a decomposition;

easily by induction on a face of

1

} where

such that

decomposition of of which

n+l

If

°,

=

i

I

>

°. Then

= n-j+l

j). Since

=

IA.

1

n-k

and

1

be the a.s. (this follows

it follows that

Moreover

A.

induces a total

as follows. Let

then

and we can consider

V

=

A.

1

n

A

j

A.

1

is a

is

70 j

non empty face of Define

viA.

A

for all

> k+1

=

1

by

1

PI Ai

""k+2'A n Ak+2 , ... , ""n+l'I·A.1 n An+ 1 • I i

{""k+ l lAi'

=

An inductive verification shows that

of

j

j

depth(A IA. n A )

and

viA.

is indeed a total decomposition

1

A. =1

6.3.3.

Let

f:

-

= {""I, ",,2, ... , tl

n+ l

,

=

of

}

is called

tible decomposition of

=

6.3.4.

A total decomposition

{",,2,•.• , ""n+]}

n

=

of

° or

til

n > 0,

V =

is an f-compa-

the total decomposi-

is f!A+-compatible. V

° and then

n >

V+ = {",,2, ••• , ""n+l}

decomposition

f-compatible if

and, inductively if

tion V+

regular if either

°. A total decomposition

of

p-compatible total decomposition of

+

(inductively)

the total

V+ lAO

is regular and

(recall that

is a

AO

a face of 6.3.5.

Let N

A

q

An + 1}

of

tively if

'"

=

=

1

2

,

, ••• ,

,n+ 1 }

L1

be a q-compatible total

We define a total decomposition x

n>O,

g

---+ M

V = {t:.

be as in 6.1.10 and let decomposition of

I

M

as follows.

{t;2, ...

Set

g V

... ,

=

(see 6.1.IO)

and, induc-

=

The following lemma is an immediate consequence of the definitions introduced above.

6.4. LEMMA. position of

Let

f: and let

-

with Ai

= 0,

let

be a non empty face of

V

be a total decomIf

V

is regular

71

(resp. f-compatible) f

Ai

i:

M i

V!A i

is regular (resp. fi-compatible)

is the restriction of

6.5. THEOREM.

f:

Let

i E I

A.

Assume that

let

f).

with

depth(!:!)

O. Let

=

I

c

I

and for

V.IA. n A. 1

1

=

J

V.IA. n A. J

A. n A.

if

J

1

J " I/)

1

(i.j E I) •

Then there exists a regular and f-compatible total decomposition

viA.1

such that

Vi

Proof.

there is nothing to prove. Assume that

required properties.

n Ai

1

since

III

AD n Ai "

if

lI

1

-,

=

+

,

>

p,

o.

with the

a regular and

VO of

0. i E I. Notice that VO

is f-compatible and thus

f IA0

=

is also f-compatible

' * )op. Next construct, again (fiA

V+

by induction. a regular and (fIA+)-compatible total decomposition

... ,

lIn+ I}

+

of

II

I

=

Let -

•

+

V=

{lI

• P. E.

{lI

l• }

inductively

=

n 1} 2 lI ••••• lI +

If

n

* A U

=

+

Q.E.D.

be a total decomposition of A.

D set

n > 0

If

with define

+

.V i.

It is

obvious from the definition that

is a manifold with faces; it is

called the V-core of

6.6.2.

V and

Given two total decompositions

seen that

and

0'

are diffeomorphic

of

it is easily

(we shall not use this

assertion) • 6.6.3.

Let

V

be a total decomposition of

A and =

Then an easy inductive argument shows that A. n 1 that

c(A=1.• vIA.) 1

2•

such that

when this makes sense. Finally set

6.6.1.

=

Construct

of

}

E,

Next construct by induction on

p-compatible total decomposition V. iAO

A

i E I.

for any

first an f-compatible de compo s i t i on

=

of

V

The proof is simple and we shall only sketch it. If

O.

=

(here

V.1 be a regular and fi-compatible total decomposition of

any =1

then

is a face of

•

A. 1

=

be a face of

A

c(A .• vIA.)

and

=1

1

=

72

6.7.

From now on

will be an a.s. of depth zero and p:

-

will

be a proper submersive weak morphism sending strata onto strata, all the faces of

being p-vertical (recall that

0


0

set

Os

is proper;

YE

B

are q(Y)

N defined

is the obvious total decomp osition

Now it is easy to check by inducti on that

for any

[ O,IJ .......

total decomp osition of

is regular .

(10)

x

f.

of the a.s. of depth zero S

"

f(b,t) = feb); then there exists a weak isomorp hism such that 8 ( b , 0 ) = b , 8 (Bh x {1}) = BO S S (thus q(B x) c N-) and (8 , I is an i somor-: 5 A)

inducti vely as follows :

°

giN

being

B

h = BX1B = I

=

be the a.T.m. given by

*

M

P

the q -horizo ntal face of

is basic,

g

q-verti cal; is closed in

N

75 (II) Z

C

for any strata

q(Y)

there exists

w

X E A,

flw

ZEN

with

fey) c X and

such that the diagram

(l

Y

Y E Band

n

g

-I

(n zog)1

(l

(T ) Z

I

w

)z

I

1

glZ

t X

p(X) pix

is regular. let

(12)

g A

p

A ..,. l1

(with respect to by

a.T.m. from

feb)

B to

p

g)

and

(fCb) , g(g(b» )

A

p

g-compatible,

S

and

p(a,x)

=

g

gOg

f

B ..,. A

x.

g

t A

S

,

8

S

=

B+ IIB+

and Let

and is an

I

1 A

P

8

s

f\.../

Os is

of depth

n

on

N

I_

-f T t:.

Then

A are p-vertical and, if

induces a structure of admissible B

and

and define

11

--+

is a proper, submersive, weak morphism sending

strata onto strata, all the faces of

as follows:

g: N -

be submersive weak morphisms such that

l1

x l1

M be an a.s. of depth zero and let

t

-

g

l1

and the squares q

+

>

_-----7

p

N+

I

1 l1

g!N+

s+

and

S

are respectively

76 q

*

N

B

I

_T f E

t

1

A

giN

*

M

p

Later it will be useful to consider the squares

B

fX

(sx)

q

x

x N

T A

T - - - -----';>

g

giN

-

M

P

and B

B

=e (S ) e

f

e

U h B

X

qe

N

B

T

I

1

J A

g

M

P

whose maps are the restrictions of the corresponding maps in

S.

From now on the notation introduced in this section will be used without any other mention.

6.7.2.

Consider the commutative diagram q

r f

f

A

I

!

-----

p

77

and let

S

S)

(resp.

denote the exterior

(resp.

interior) square.

quadruple

is called an isomorphism from

is an isomorphism from

f

Assume now that n = 0,

isomorphism.

(l)

and

Sand

S

are admissible squares of depth

S

to

(3)

s+

to

S

are isomorphisms.

is also called an admissible

S

6 -x

-+

-x B ,

If!

B

= B

...

B

W,

n-l.

An isomorphism

'r h

,

6',

and

(B

and

B

h : B ... B ,

let

: N

If!

be the restrictions of

(If!+, ¢+,

and

-+

+

N

¢

If!+

,

B+ N

B

...

,

N

respectively;

is an admissible isomorphism from

W,

(If!-,

(5)

0

is an isomorphism from

S+

to

A,

to

N

S

S

S

equal, denoted (lB' IN' lA' 1

to

S

G

Two admissible squares of depth

S

If

(this makes sense by induction); (4)

A

n.

is called admissible if

= B ,

* N ... N

*

and

S

is compatible with

(2 )

...

if

Assume we have defined inductively the notion of admissible

from

: B

S

f

isomorphism between admissible squares of depth

X

to

to

an isomorphism from

and

S

The

= S,

S

B = B,

Sand

A = A,

S,

N = N,

M

M and

(i.e.,

f,

called equal, denoted

(lB' IN' lA' 1 q=q,

g

g

are called admissibly

M

is an admissible isomorphism from

M)

N,

if

n,

S

M and to

S.

If

B

=

B,

is only an isomorphism from

M)

p = p)

and

then

Sand

S

are

S = S.

Consider again the diagram from the beginning of this subsection and aSSume that (If!,

S

is admissible of depth

is an isomorphism from

structure of admissible square of depth

S

to n

S

n, S.

is arbitrary and Then

such that

S

has a unique

(If!, 1J,

is an

admissible isomorphism (the construction is obvious and left to the reader).

78 6.7.3.

Let B

(S)

q

'\;

--;;-

A

P

be an admissible square of depth be the corresponding face of

N.

N

I

T

f

if and only if

)

n

N .

and

1

g

M

B.

Notice that

1

Consider the square

=1

T g.

1

J

1

A.

1

M.

p.

=1

=1

1

S

(if

and

M.

whose maps are the restrictions of the corresponding maps in

faces of Ai

=

on

p

-1

n)

A

and

»·

(M i

!:1 i

A. = A and =l

is f-horizontal, then

=

n

i

then

Suppose

n

= 0

i

and

SIB

n. < n •

Then

1

structure and therefore Case II: 1

1

are

1

g -1 (M.), 1

1

S!B

(by induction

i

as follows.

n.

1

If

is obviously admissible of depth zero.

i

N.

n

X

B

respectively.

n.

1

and Take

n.

=

B-:1

=

+

B.

=1

too.

Take

6 S IB.

liS INi

=

1

n

=

+1 + Bi

1

B

,

B.

is a face of

1

can be endowed with the required

slBi

B.

+

a face of

1S

1

By induction

B.

B.

n > O.

Case I:

=

1

N. We shall endow

=

A.

with a structure of admissible square of depth

n = 0,

1

!:1 ;

otherwise

M respectively and Let

1

N.

T

f.

N.

q.

B.

=1

(siB.) 1

Let

is f-vertical (horizontal)

B.

is g-vertical (horizontal).

1

B .

be a face of

1

Notice that

are faces of x B.

=1

=

=

1

+ B , and

X B

and

B

=

=

i

+ B. 1

1

n

B+ ,

B

B-IB-:-

=

B.

1

Since

79

+ q.

+ B.

=1

1

=

S+IB: 1

n N.

=1

1

+

g.

1

!ji

1

it is admissible of depth

1

B-:-1 n

+ N.

t p--:--7

A. is just

)

I

T

f+

1

1'\./

=1

n

B.

=

1

Bh

8 s i B. :

(by induc t ion) . 1

1

Gs

restriction of

, we can take

n-l

x

[0,1]

-+

x

B.

1

Since

to be the

direct verificati.on shows that all the conditions

A

involved in the definition of an admissible square are satisfied.

6.7.4.

Consider now two admissible squares 'B

(S')

T

rv

q'

')

I g'

t

t

f '

'N

A

M

P

and

"B

(S")

f"

r-v

of

'B

n'

and

and "B

l

n"

)-

"N

I

t

T

1 A

of depth

q"

g"

M

P

respectively.

Let

'B

be an f"-horizontal face of

be an f'-horizontal face

l

"5

admissible squares q' 'B

=1

(S'I •B l)

f'

1

T

I

1 A

and

1

t"lJ

t P

!1

g' 1

Suppose that the

80

"B (S"I"B )

T

f" 1

l

0.r

q" 1

"N

I

J

t A

g" 1

M

P

are admissibly equal and that we can

'B U,S

B

, q

q

"B

N l and

q"

U'B

"N

U,

'N

N

1

Let f' U,S

U,

g'

g

S' U,

S

square

S"

B

g"

N l

1

B >-

f"

A

-+

,

1 N

Consider the

M

given by

l

B

rv

q

N

I

T

(s)

1

--:>;t P

A

Let

n

max{n' ,n"}

1 g

and

m

!1

n' + n".

Using induction on

shall introduce a structure of admissible square of depth follows.

If

m

0,

Assume next that Case 1 : 'N

l

n 'N

face of

n

0.

S

on

we

S

as

is admissible of depth zero.

m > O. n' > nil.

Hence

'S+

it is clear that

n

m

'B

Then depth ('N )

n

l

0

'B

'B

l

n

depth ("N ) < n" 'B

and thus

-

x

'B

and therefore

is a

l

Thus

'S+ [' B l and, by induction, the square Take

D S

D,S U'N D"S

S+= 'S+ U'B

"S.

Thus

'B

B S+

'S+

i

U'B

"s

1

X

B

is admissible of depth

'B

x

B+

s admissible of depth

'B+ n-l.

U'B

"B

n-l. and

1

The other conditions

1

in the definition of an admissible square are easily checked.

Case 2:

n

n " > n'.

Case 3:

n

n'

This case is similar to Case 1.

n" > depth(

Then

'N

n

':-I

81

=

-

,

"N n liN l

IB

hence 'B

and therefore

I

n 'B

IB

'B+

is a face of

l

n 'B

I

x

=

r/J = "B

"B

and

I

(=

l

x n "B

=

'B

is a face of

1)

"B

I

n "B "B

+

Thus

's+ I ' B 1

, +

and, by induction, the square Take

6

=

6,S U'N 6"S 1 "B+ and U'B 1

S 'B+

depth

n-1.

S

U'B

is admissible of depth

n-1.

1

B

, +

S

U'B

"5+

Thus

is admissible of

1

The other conditions in the definition of an admissible square

are easily checked. Case 4: 'B

1

n 'B-

n

'f-

set by

1(

'N

n'

n"

=

depth(

=

1

n

'B

construct

'B+ = liB

and

"B

B

'B

In

=

t

h is case

n 'N*) = "f- 1C"N n "N*) = "B n "B-

1

1

= 'B n 'B

Similarly, let 'B

both

=

l

n "B+

1

Then

x

= "B

1

n

Cre s p ,

'B-

I

'B-

x

"B

is a face of

"B+)

and we can

+ B

"B

and

'B+) 1

(resp. U

Denote this

1

'B

1

+

"B+ U + 'B

Notice that

and thus, by indue t ion

'S+ U

, +

"S+

i

s an admissible square of depth

n-l.

B l

'S+ U

, +

"S+.

Then

S+

is admissible of

B 1

depth

n-l.

The other conditions

the definition of an admissible square

are easily checked. 6.7.6.

(S)

LEMMA.

Given an admissible square of depth

f

B

N

T

Ig

! A

-

l

--"7 M P

n

82 and f

S

E: M Ceq)

:

->-

R*

C(p)

+

consider

=+

Seq)

and

f

S

[g(x)

.o l ,

is an a.T.m. from Seq)

to

Proof. an a.s.

(see 5.3.6).

Define

by [f(b),tJ,

Then

S(p)E

°
-

f

i

q

(N)

is an

q

Z ' e i (N). q

with

Seq +)

also consider the a.T.m. 's

zC

with

and of 5.9.4 and

i (N),

be an admissible square of depth

defined in the same way as

Seq)

5.3.6 (ii) ard of the fact that

Z < Z'

5.3.6(v) for strata 6.7.6.

z'

Finally (5.5.4) is a

n.

S(p)E

Q.E.D.

Besides and

f

S

e

f

S

we can

C () = qe

1--+

o

and we can consider

°

g_(p)

is a

g_(q+) U C(q) • C(q ) = e

In view of 5.9.8, we can also consider the a.T.m. g_(q +) U C(q ) C(qO) = e G

S(q+) U S(qe)-C(qO)

5.3.8

(notice that

(G, IC(p»

q

-+

Seq)

C(p)E.

=

Let also

be the weak isomorphism constructed in

+ qUo q). B e

is an isomorphism from

I-- --+

A direct verification shows that

f

to

f ' S

In particular, the diagram

E

83

is commutative. 6.7.7.

Later on certain constructions will lead us to consider squares q

rv

B

N

')

I

T

f

t

!

A

in which

Nand

M

P

B, or A and

B, or

B

are the empty sets. Such a square

will also be called admissible of depth definition).

f : B

finite depth and

A*

to

we can consider

*

*

->

* * E:A----+R

=+

=

T

E

[x

X'

Y E

strata of

f.

Since

o.

Let B*

f-l(A * )

=

B*

A

and

by

-1

*

*

f*

f

* ' x E A* By (5.1.9), TO!Y n TOlY' = 0 if A* and Y Y B*

E)

(U*

YE B*

and choose

I

TE X X

0:

*-

n T E IX' = 0 if

Y';' Y'

X'

< co,

->

Tol y

y

if so let

*

and define

*

and

and

X,;, X'

are strata of

ToI Y) n f-l(A d ) Y

exists;

* = 0;

such that all the conditions involved hold on

B O) =

be


-

is an a.T.m.

Since

v

denoted

being of

B

is a closed union of strata,

is an a s .

By 5.9.4,

and

f* : B*

From Remark (2) in 5.5, it follows that

denote it

f** :

=

(depth(0)

A will be an a.T.m. ,

f- ->-

>

the restriction of

from

=

n

All the constructions performed until now are still valid.

From now on

6.8.l.

g

are Let

84 B8) = BIB 8» =E:) = d

* A

to

* f*

with respect to

*

(resp. and

Tf

c)

*

Consider the fibre product

•

we can define a map

by

*

A

Ad =

/OTf8) = Tfd o(fIB 8» * S) c)

Since

(see 5.9.3) •

x

8)

g(b)

(TfE:)(b),

Locally this map is just the map constructed in 5.9.5 and therefore Remark (1) in 5.5)

(c L,

6.8.2.

A decomposition of (L)

L',

(Li )

B

(resp.

B

+

face of

B

(resp •. B ) =+ f

B = B O

n

is an a.T.m.,

B

f

B

s

A

f

f

A

+

('JI,¢)

if

(resp. f+-vertical) B

U BO

A determine an

°f

to the

U A

IBO

and

exists and

is an a.T.m.); n =

J

1*

f*

A

P

is an isomorphism from the a.T.m.

r-

-+

A-

namely

I

T O A

'JI([b,l]) = b

f+

B

(as a consequence

fO

f

C

where

A

one can construct

is an admissible square of depth

to the a.T.m.

S, 'JI},

B

(resp.

and

Ad

x * A

,

is an f -vertical

A c A

(S)

(Lv)

-+

=

= (iii)

-

B+ = f-l(AO)

c B,

l - -+

B

is a decomposition of

fIB_:

isomorphism from the a.T.m.

{L'"

\!

*

to

is a w.a.s. structure on

(resp.

the inclusions

a.T.m.

is a qu i n t up l-e

f

A+ p, E, ¢}

f-l(A+»,

B8) =E)

is an a.T.m. from

g

such that

'JI([b,O)) = b

f

t - ->-

S: if

b E B*

and

b E B • O

Remarks. (I)

If

= 0, giving a decomposition

V =

of f is equivalent to giving the regular and f-compatible total decomposition

Os of

of f, then B_

=

= 0 = BO'

If B.

=

o and

V

=

is a decomposition

and V is completely determined by

85

(2) Let V be a decomposition of f. Since f

+

is "simpler" than f. On the other hand, since f

:

ly determined by S and since

and f

into f + S

6.8.3.

Let

notice that f

v

B

= {I'I,

,

called compatible with

(G

(i)

F

(ii)

if

, F- )

G

: B

-+

f

v

to

A

and

1'1

to

G and

F

G,

then

G,

then

(lB' lA)

o'

f

and

v

of

f

is compatible with

f : Bo--

(G,F)

and

->-

A and

(G,F)

f: B

S

to

= tv, B=-,

Let

B. # 0

be a fac.e of

( recall that

A.

A

S, 'I'}

B

and

if

B.

f. i

v

are called equal, denoted v

and

l-

v.

A be a.T.m. 's,

-+

and

I - -+

f

to

f,

v

be a

f.

denoted

It is G*v,

G*v.

be a decomposition of B.

S;

a < t < 1.

be an isomorphism from

is compatible with

6.8.4.

v

(G+, F+)

respectively; then

b E B v

is

f

easy to see that there exists a unique decomposition of such that

(G,F)

f

(see 6.1.2);

1'1

is an admissible isomorphism from

Two decompositions

decomposition of

f

and

be the restrictions of

Let again

•

+

G('I'([b,t]»

if

S

if

f

f

to

f

denotes the restriction of

B +

f

be decompositions of

denotes the restriction of

B

+

S, iji}

let

(v)

v,

-+

G : B + +

(GO' G*, Fa, F*)

v

and

v

0,

can be decomposed

be an isomorphism from

is compatible with

if

(iv) -+

S

=

A be a.T .m, 's,

1-4

{6, B ,

=

(G,F)

is an isomorphism from

* A

v

B

f

is an isomorphism from

(iii)

* F

and

and

S, 'I'}

respectively and

f

*)

s e (see 6.7.6) which in turn are "simpler" than : B t-- ->- A

f

is complete-

s

- I and

f S is "simpler" than f. If

and

- I,

=

A.

is f-horizontal).

f.

Let also

be the restriction of Suppose that

f

86 We shall define a decomposition

) = follows.

Notice first that we can consider

* (B i)* = f-:-l(A. n A ) = B.i. n B* face of face of

(B. )

B B

=+

B n B.

x

x

(B. )

is a face of

(B. )

(Bi)O = (B. ) + n (B. ) - = BO n B.

and

is the quasimorphism of the square

q : and thus

SI(Bi)O

f.

as

Next note that

lIlA i

is a face of

of

I,7I Bi

-

B n B.

is a

B n B.

is a

i.

+

+

i.

is a face of S,

then

q

-1

If

«B i)*)

(Bi)O

is the square

I (B i) 0

fO,i

1

p.

the mappings being restrictions of the corresponding mappings in

S.

It is

Finally, a straightforward verification shows that

a.T.m.

f.

6.8.5.

To a decomposition

A.

1,7 =

{lI,

of

two weakly controlled vector fields role in what follows.

and

sl,7

f

one can associate

which will play an important

Before defining them, let us fix (and recall) some

more notation. S

being the square

T fO

I

t

f*

O - - - 4 A* A p

+

p*,

a,

w*}

be the associated decomposition of

let

87 +

B+ =0

I

T

f+

(s+)

£'"

O A

1'" '" A

P

be the admissible square of depth

f+

n-l

associated to

S

(n

and let

'"

q

-.-,

I

'" T

(S-)

J

fO

O A

6

S

:

h

x [ 0 ,11

-

-+

°

x

B

O

the other data associated to + F B 0

Let next + Tl B 0

P

+

u+

B O

+

'"

A

P

be the basic square associated to

,

A

the face

AO

of

r

+

(I

A

A+

...

U 0

A

Bh = B n B 0 O O' O B+ O B + B n B; x

Let also

S.

X B n B+ and 0 O

q

0

-

'"

0

S. +

B x R O +

U

PB

0

and

+ R +

+

B O

B of O

be the data associated to the face + O U 0'+ A

'"

f",

+ B

O

B =+

+ Tl 0

,

and

+

BO

+

n 0

A

and the diagram

>B

O

x R

+

1

f

+

F 0 A

O A x R +

be

'"

r

+ B 0 F

+

u+ + R B +

and

O

0

A

u+

O A

+

O A x R+

be the data associated to

A

We can (and shall) assume that

df . Tl

x

O

x lR

+

88 is commutative.

If in addition

f

is proper, we can (and shall) assume that

+

DB U U+

B

B O

f

o

+

bE DB

q

:

C(q)

B* and

and

j

+

(n

A

and define

bE DB

0

+

(D 0)'

[b t ] E C(q),

t,

d('¥

-1

o

'

o

'

4J ) • (0 x d/dt), q

B x [O,lJ O

4J q

+

C(q)

are defined in 5.3.6).

It

is clear that

{ (b , t ) E

V

'¥([b,tJ)

(b,t-l),

=

V

lim tl'l

(b,-t) V

= q(b),

bE B ' O

t.::. O}

0 < t < 1

bE B O

F; (b)

o

(b,t) E

V

(b j t ) E

V

In a certain sense

Sv

determines the "horizontal" structure of

B

However, in general, this is not enough to determine the structure of

B near

B* :

has also a certain "vertical" structure.

Part

89 which we

of this structure will be determined by the vector field proceed now to construct. Let

of

h B 0

and

in

and

O B in 0

B+ 0

the associated vector fields. U- U BX U u" h BO o B 0 O f O and define 1;0 E X (W B O) =0

W o

C

olu+o B O

R

ox

Let

+

T1

h B 0

be the collars

+ T1 0 be B O

and

Set

by

,

+ = T1 0 B O

Next let

W

+

+ {b E U B 0

p; (b) E W O} 0

determined by

and

Since

O B

B O

des· (0 x d/dt)

1;

respectively.

->

q

is proper, we may assume that

f

and let

1;+ E X B+(W) =+

be

90

and the diagram qlu+

o

B O

U+

O B

>

0

+

F 0

B O

1 0

B O

x R

+

q

0

) x

lR

+

+ U 0

B*

1

0

B*

x

R

+

+

F 0

B*

91 +

A

lT

7(PB

(b))

o

if

b E W+

(the right sides are

already defined;

o

< t

1 (the right sides are already defined);

WV(b)

'4J

=

6S

Notice that

* and

to

(b),

TIv(b)

= IT 6

S

if

(b)

is controlled,

WIl

dldt,

dW Il·

is a submersive weak morphism from

lTv

0,

=

0,

For later use set B

6.8.6.

Let

Il

{ll,

=

decompositions of B

B

l,v

l,ry

'I'(C(q

S, 'I'}

f.

U = Ui? v near

x

x

».

and

B

{6, B ,

fj

It is obvious that if B

near

,

=

near

V

V B

v

S, Iji}

be

B

B

then W = Wry v

near

and

B

B

The converse is not true because the above equalities do not imply

s

=S

(however, they imply that

thus, if in addition

6.8.7. Il

=

of

Let

6

=

containing

6.8.8.

{6[U,

then

V

p, S, '¥}

and contained in Ilv,u

= S,

=

{A, A

S

6,

6

A

=

f-l(U).

n

=+

be a decomposition of f.

Let

U

V be an open subset of

Set

fV,U

V, S, 'I'}

Given a decomposition

=+

V

=

A

=

ijI;

L'

and

be an open subset B

flV :

containing Then

is a decomposition of {6,

'¥

v).

be a decomposition of and let

Band

B

B

S, 'I'}

of

fV,u f,

set

B

92

+ -1 + B+ ; (PB) (B O)' +

B+

'¥(C(q+»,

and

'¥+ ; '¥IC(q+) : C(q+)

°

B+

->-

+ B

Consider also

q[B

O

BO

->-

UL\

°O

BO

->-

B*) .

°and

B+ n BO V+IBO form

of

fO.

respect to

°

BO

to

'¥(C(q »,

and

'¥o ;

°

UL\

B+ with

+

+

{L\luL\'

+

+

S • '¥}

°

-e- B_

BO

°

,¥O ; '¥+ IC(q).

f+

is a

*

x * A

f** '

VO;

SO;

BO

->-

L\

of

and

BO ; B+ n BO, + + O V is of the

A

be given by

fO

is an a.T.m.

A determines a decomposition In view of (12) in 6.7.1

as in 6. I • 10.

q

°

'Lr-f

O 0

_ .

I

T

1

1 __

f

O

*

;*

p

determines an admissible square 0

O B ;0

BO

;*

T

\

1 ;,0

A

P where the identification of

A

BO -'

.

A direct verification shows that

A

U

the fibre product being taken with

'

and let

of

}

E,

+ B

O +

that

it is obvious that

where

The decomposition

A

is a face of

B

We can therefore consider the decompositior

SO, ,¥O},

and

(B

the square

(SO)

->-

respectively such that

+ B+; (PB)

Notice that

A;

, p,

B+ and +

-1 °O) ' BO ; ° ( recall '¥IC(q °) : C(q 0°)

BO

We shall denote it

TIL\

B:,

B+

'/

and

(fO(b), TIV(b». from

B+ ;+ and

to

VO; {L\luL\'

Let now

f+ ; f IB+ : + B

u B+

f+.

decomposition of

flB

B+ +

One can endow

+

canonical w.a.s. structures is an a.T.m. from

B+

A

j,

x * A

*

Pj,

\

; B*

;*

with

*

is given by

93 (a,b)....- b of

and

is the weak morphism involved in the decomposition

(in order to apply (12) of 6.7.1 we have to use (7) of the same

subsection).

6.8.9.

be a decomposition of (0,1).

-+

df'

(see 6.1. 7) .

f

as follows.

verifies

f,

+ A

let

Since

6

f

is proper. -

+

, f; ,

=

is proper, we may assume that

(resp.

0

B) +

E

U

v

p,

=

¢}

B

B*) ;

f-l(I\+);

+

then

df . B

thus =

(u 0 116 ) 6

(resp.

G: B + B and O O

F:

AO

by setting G(b)

(b,

- 1) ,

V

and notice that in the commutative diagram

.:

q

1 O

fO

T

/V

I

J

.:

O A

AO

»

q

p

P

t*

of

S,

with a w.a.s. structure

Next define

6)

of

fi,

f*

')

A

=

which

6.8.2(ii) and such that the vector field associated to its face

B is a restriction of O

f

E,

l f- ( U

=

V u

and

f

S,

and let

p, E, }

Ii = f-l(A-)

By definition

Ii

is a

V = {f:"

Let

We shall define a decomposition

Set He can endow

f: B

Consider the decomposition

=

A

s°,

{E,

fO

Assume now that

: A* -

VO =

Now it is easy to check that

decomposition of

and

f:,s

+

AO

•

94

(G,F)

is an isomorphism from the a.T.m.

6.7.2, we can endow admissible square.

S,

fa

to the

a.T.m.

fa.

By

the interior square, with a structure of an

Finally

Sv

will be determined by the condition

S. II

U

It is useful to observe that we can take

v11

O/ll

l()v

01]6 0

f) l()V'

l,V II

11

6.8.10.

Assume again that

decomposition of

let

be a decomposition of

f.

f: B

A

*

ll:

(0,1)

of

f

6

V = ll

such that U

(in contrast with 6.8.9, we must now "enlarge" U

be a

V

and let

done exactly as above (see 6.8.9) if we can take

Clearly, we can take

6

let

Later on, we shall be interested in finding a

V = {6,

decomposition

is proper,

l,v'

v = f-

l

(U

6

to equal

V.

This can be and

)

II -1

f

(U

U/';

».

= U6

)J

6 Then using Proposition 5.6 and a partition

U/';' )J

of unity argument, we can construct a vector field such that of

B

O

S

in

condition:

=

S

near

B

df . S = S6

and

V)J

B

=+

Now we can take the collar

)J

f- I ( U

+

to be defined on

U-

"its associated vector field is

this means that

' A- ) UB

6

B O

slu::B ". O

O

and determined by the

From the construction

U

v)J

As a matter of fact, we shall need a slightly more general version of this construction.

Namely assume that in addition to our previous

hypotheses there is given

v.

f.

Ie I

B

B.t---+ A.

notice that, since take

V

6.8.11.

6!A.i.

Consider the a.T.m. and

respectively, such that

f: of

exists,

->

B,

i EI

a decomposition (here

such that

with the additional property that

closed subsets 2B

and for any

VIB.

*

A.

*

=A.n A).

V.

Then we can

i E I

f; and assume that there exist

endowed with w.a.s. structures

and

95 iB

(2) (3)

-, A ·

T m, , an a ••

1,2;

i

is an if-horizontal face of one can construct

lB

U

lB

and the inclusions

2B =

OB

IBI B = 2 B [ B = 0 = 0

and

C

Band

determine an isomorphism from B

f

J- ->-

A ove rIA •

Let also

1, 2,

i VI U

v

OB

i.

such that {lI, B

2

i

V. = {lI,

lB =*

face of both

B

i

,

,

s,

IB

B* 2

and

over

B =*

as follows. 2

U

and the inclusions If

*

2

U

OB*

f

*

IB

= lB

n B*

OB" = OB

B*,

*

C

B*

n 2B

*

2

and

2 B f-lB U * OB* *

->-

is a

*

B

* A

*

C

B*

to

1 *

A

Next notice that lB [ B = 2 B I B =- 0 =- 0 -

We shall construct a decomposition f

*

if,

be a decomposition of

'l' . }

i,

of

'l'}

determine an isomorphism from B J---> A* =*

s. ,

B

=+'

vlloB = v210B

First notice that

f*

to

lB

OB_

n 2B

and we can construct

argument, we can construct

is a face of

=+

B

(i = 1,2) , By a similar

B Let

B

i

B n B +

f-l(AO);

Denote this w.a.S. = O structure on B be the restriction of f. by Let f : Br-.-+ A =0 O O 2 and 2 B C B Note that B = lB U B and the inclusions lB C B 0 O O 0 0 0 O 2 2 B f---> AO to lB U determine an isomorphism from If U f =0 OBO =0 o OBO 0 it is a face of both

f

O

: B

both

O

=0

over

J- ->- A

lB=0

=

an d

and

B

1 0

B =+

and

(here

oB0

A

2

'

etc.).

- IB n 2 B 0 0

Consider the diagram

is a face of

96

in which the exterior square is

Sl U B

o

S2

the weak isomorphisms mentioned above and the diagram commutative. Sl U B S2 o0

Let

S

(see 6.7.4), q

Sl U B S2 o0

to

and

denote the interior square.

S,

are

\i

is the unique map which makes

is an admissible square (c f • 6.7.4) and

isomorphism from

11

0

Since

(11, \i, 1, 1)

we can endow

S

is an

with a canonical

structure of an admissible square (see 6.7.2). Finally, let

C(q)

B

be the unique map which makes commutative

the diagram

'1

1

(i

=

1,2)

decomposition of 6.9.

THEOREM.

ijiIC(q.) l

l

is the homeomorphism defined in 5.3.8.

vI

verify easily that

B

is the quasimorphism of the square

is the restriction of and

2

OB_

B

'J'

where

1,2)

U

1

C(q)

(i

lB

)B

C(ql) UC(qo) C(q2)

U B

o

v2

=

{6,

:

C(q.) l

i

B

-

Now one can

so defined is a

f. Let

of finite depth and

f: B* f

I - ->-

f:; be a proper a.Lm.,

0. Let 6

=

{f:;-, f:;+, p,

A and

B being

be a decomposition

97 of

let

For any

A.

(recall that i EI

A

f. : B.t-

i E I

if

h,

f).

A.

------+

Let

be the restriction of

I c I

B

and assume that for any

the following conditions are satisfied. (a)

;

=

of

there is given a decomposition

(b)

f. : B -+A. ,where =i

the a.T.m.

if

(c)

j EI

and

6.

i.

=

6!A.

depth(A.!A. nA.) =1

J

i.

then

=

v. [s, n B • J

1

n B.

V.[B. J

J

Then there exists a decomposition such that

V\B

i

Proof.

= Vi'

V

{6,

S, 'l'}

If

=

0,

0

> 0 •

> 0

f' : B'

I - -+

The case

is "easy" and left to the reader (the arguments are similar

to those which follow, but much simpler). =

f

in view of Remark (I) in 6.8.2, the theo-

We shall proceed by induction on =

of

i E I.

rem follows from Theorem 6.5. We assume therefore that

m

f

A'

Thus we

assume that

and that the theorem is true for any proper a.T.m. with

< m,

of the proof, we shall also assume that

In order to point out the main steps I

=

0 (however, at a certain point,

we shall use the inductive hypothesis in its all generality).

At the end

we shall indicate the necessary changes for handling the general case. Consider the a s ,

and the weak morphism

v

f*

=

flB* :

1--+

A*

By Theorem 6.2 there exists an f*-compatible decomposition +

/),*

For any

p*, 8, *

b E

A*

* f*

A with respect to =

*

= 0,

p =

is an a.s. (see

determines a decomposition

(0 x

=

x

since

of

}

*

P =

l

=

Notice that we can take

•

(use the fact that

and (6.4.3»

f

is a weak morphism,

we can define

v.

g: V

X E A*

and

P

with

by f(Z)

C

x.

Consider the commutative diagram

* B*

x

*

A

A

in which - the fibre product

n

(Z

ul'> )

*

Xx

T X

- the horizontal map is given by

is taken with respect to

b

(TIz(b), f(b»;

in view of 5.9.5

and Remark (2) in 5.5 it is an a.T.m.; - the right oblique map is given by

(b,a)

(IT(b), a);

it is easily

seen to be an a.T.m.; - the left oblique map is the restriction of

g

in view of 5.9.9

it is an a.T.m. Thus

glW

--+!: a.T.m.

from

is an a.T.m. from

is locally an a.T.m. is an a.T.m. to

V to

The decomposition possibly after shrinking

P

and Remark (1) in 5.5 implies that

Since for any

(cf. 5.9.5),

!:' 1'>* Ul'>

where of

*

'

* Y E B*,

glT y

it follows that

V

g

is an

itself

=

being f*-compatible we can assume, that

df . S 1'>*

it follows that

=

O.

dg' SI'>*

Since =

O.

By Proposition 5.6

101

applied to

*

I1!U 'B* = tI

* such that

on Z

g!W,

conditions

Wn TZSCZ)

/

/

--'

\,

,/

\

/

-

A

\

l' +

A

103

1/

°

°

{6,

=

SO, 'jI0};

clearly

hypothesis to

f+:

and find a decomposition

=

g

are proper, we may assume that

+ S , 'jI+}

B+ {6, B+ =-' =+' -1 g (U p) 1/

=

*

*

f+

of

A : D

W,

I/0

U+)-1(U ) 6

U 1/+

Since

v(b)

B

O

There exists also a controlled

(-1,0]

= A(BO x (-v,a))

B+ and

n.

Since

and

D

c

V

x (-v,O) c D

BO

a

A

R* +

C

such that

D

= AlBa x (-v,a) :

BO x (-v,a)

A direct verification (by now it is standard) shows that

°

isomorphism from (0 x d Zdt

)

=

x (-v,a)

(notice that

to

W'

W'.

is a weak is open in

v' : BO

B)

n,

Using arguments similar to those in Lemma 6.10 we can construct continuous

and

1,

O B x (-v,a)

and

f

it follows that

By Lemma 6.10 there exists a continuous

W'

=

the flow associated to

x

Let

I/+IB O

such that

O B

Consider now

* = n[U , B* 6

fO: BO f-

we can apply the inductive

+ 1/

9°

is a decomposition of

and

Since

U

I/0

R* +

such that v'(b) = 1, is controlled, V'

(b) < v(b),

a

-->

A

104

Set

S

v/v' •

=

isomorphism

One can now construct (the construction is standard) an

F : BO x (-S, a)

(i)

F(b,t)

(ii)

F(b,t)

Let

l; =

del,

B

O

x

a)

BO x (-v, a)

with the following properties:

(b,t), (b,t) E BO x (-v', a).

0

F) • (0 x d/dt) E

and the definition of

Notice also that

+

c

From (Li )

above

A,

S(b)

S(b) > 1

for

1

c D l;

of

g

°:

for

0

-+

and recall

be its first associated vector

that field.

X

Define

E,' E X f (W')

by

B

E,'

d(A

0

(CO x 0).

F)

v

From the construction, it is evident that

and d n • E,'

Consider the a.S.

P

=

*

By Proposition 5.6, there exists

0.

x * A -

and the vector field

E," E Xg_(B-,B:) B

or equivalently

such that

dg

'E,"

tD

..,

D"

lb

:lo

t

r

*

* I

++-

-

+-

+-

+-

+-

+-

+-

....

+-

+-

+-

....

+-

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

....

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

+-

+-

+-

....

....

....

.... +-

01

1- t-

t

+-

t t t t

x

t: II

t

+-

I:>l

t f 1

t

tD

I:>l

t

t

t t t t t t t t t

tD

'0

t t t t

\ttttt

otD I

t t t t t t t t

tD

* *

....

\

t:l:' aD"

++++-

....

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

+-

.... ....

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

-e-

+-

I:>l

....

Ox

.... +-

.... ....

+-

....

-e- +-

....

+-

.... ....

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

.... ....

....

-e-

.... .... .... .... +-

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

+-

tD

....

10

+-

+-

l

....

.... .... .... .... +- .... .... .... ....

++jt::;j .... .... 0 0

. . +/

+-

/

+-+t:l:'o

.... ....

+-

+-

+ +-

a

CJ1

106 and

dT[ •

Let

W = (\ of)(-IS, a).

S'

patch together

and

S"

S"

0.

Using a partition of unity argument, we can

sEX

and obtain

(B ,B;)

such that

B

dn •

Since

and

B

SO,

+ B ,

U V+

we may define

by

(the definition is correct because

S+

and

and

V

parallel to the faces). endow

B

f-l(A-)

and

Clearly

B

+

df'

S

= f-l(A+)

=

s6

S

are

By Proposition 5.10, we can

with w.a.s. structures

Band

respectively which verify (ii) of the definition of a decomposition of an a.T.m. (see 6.8.2).

Moreover, these w.a.S. structures can be chosen such

that the vector field associated to the face (resp.

B ) =+

Define

l+t,

It is clear that

d\jJ • 1;

Set now

of

= B

by the relation

\jJ(!I(F(b,t»)

and

O

S.

is a restriction of \lJ E C;(W)

B

=

d Zdt

(b

j

t )

E BO x

(-IS,

a) .

B

107

B n 1j; -1([0,1]) O +

=

F)(Bg x [0,1]) ,

0

h

x

(B (B U B » U B O' O O O

from the constructiDn of

Since

f

df . C;

it fDllows that

X O(W'

and therefDre

= 0

can apply Proposition 5.11 (to

B

and endow

and

+

with w.a.S. structures

and

and

o

We

+

and

BO'

respectively which verify

condition (3) of the definition of an admissible square (with replaced by

f

0

t - ->

O:

see 6.3.1).

,

f: B

-+ A

Moreover we can choose

these w.a.S. structures such that the vectDr field associated tD the face O

B

0

0

Define

e q

e

h

: B

+

of

B

x [0,1] ->-

O

X

B

0

by

0

Bh =0

x

e (b, t )

Ar;(b, c) X

B

to

[0,1]

is a restrictiDn of and notice that

We can define

=0

by setting

->-

(here

: S(p*)O and

decomposition

q

+

v+

is the isomorphism given by the decDmposition

->-

+

+

to

S+

is the quasimorphism Df the square

6*

Df the

Df

Observe nDW that

-

BOcB

-

=

-1 lC «(-oo,lJ)CV

be the restriction of

submersive weak morphism from

*

V ->- B* .

11 :

to

*

is the unique q--horizDntal face Df and

x

x

(resp. X

is a weak i somo rph i sm from

x

of

h

h BO' BO' B0 )

(resp.

is closed in

Clearly

q

q:

is a

sending strata onto strata; MoreDver, since

one checks easily that

B O'

and we can define

q

f

is proper

is prDper.

Consider

the square q

f

*

)

-1

I

OJ A

O

-----')-

P

A

* f*

*

It obviously verifies cDnditions (i),(iii),(iv) of the definition of a basic square (see 6.7.1).

One checks directly that condition (ii) of that

definition is also satisfied (or one can use Propositions 5.10 and 5.11 and

C;.

108

the fact that

g:

YI--+

P

Define

is an a.T.m.).

Thus the above square is basic.

by setting q

Summing up the above remarks it follows that the square

tV fO

)

B*

I

T

t A

is admissible of depth

q

O

f*

*

A

m.

v

In order to complete the construction of the decomposition it remains to construct the weak isomorphism

'V : Seq)

f-

of

f,

To this

-+ B

end we set

Let

b E B O'

'V([b,c])

\(b,t-l),

'V([b,O])

q(b),

'V([b,O])

b,

Case I:

Case II:

b E B O bE B* •

q(b).

t-l)

bE B+ 0

Then

and

(b,t-I)

t-I)

is true.

(* )

t/

X

Then

bE B 0

for some

e(bO's)

b

t-l), s ) ,

t-l) =

Since

0 < t < 1,

Let us check that lim tliO

(* )

b E BO'

h b E B 0 O

and

0 < s < 1.

this case can be reduced to

Case I. Case III: to

-1

Let

b E B O

(sn)

and such that the sequence Since

df

x H ,OJ)

>

be a sequence in sn»

(-I, 0),

converges to

it follows that

b* E B* •

is an open neighborhood of

converging Set

b*

Notice that

B*'B**

in

Band

109

any

Assume that

t > -l. b

n

is of the form

y E N'E*

E N'B*

b

the other hand

b

It follows that

n

with

= 71;:;(b, s )

n

* • b* E B*

with

71;:;(b n,O' s )

n

and

YO E

n,

Then, for a sufficiently large

b* E B*-B*

and therefore

with

71;:; (yo' t)

y 1,

Since

,

b E B;

;:;IB-'B*

Since on

b n, 0 E

we get a contradiction.

= ;:;

and

dn

. ;:;

= 0 , we

deduce that n(b)

n(b ) n

q(b)

and therefore lim n(b )

q(b).

n

Now an easy tcpological argument (based on the fact that

f

is

proper) shows that this implies (*). There are no more difficulties in verifying that isomorphism and that

V

=

{6,

is a decomposition of

Return now to the general case when the same lines as above.

is a weak

In order that

0. The proof follows exactly

I VIB

f.

i

=

Vi

i E I,

for any

the

data constructed above must satisfy the following additional conditions: (a)

6* I (B i)* = 6 S.

for any

E

i

J =

{j E I ;

i.

{j E I ;

(6 .

=

(B.)x'B:

-

(B.)x'B: c W, W 2 ,•

Choose open subsets (V ) c Un

n

sup{t E R

a

n Vn _ l

(n >

then

V

n

2)

{a}

+

[O,t) c D}.

is open in

U n

(n > 1)

of

A = n O

and

x

A,

AO C Un

A such that Let also

Vn

verify

n E

(i)

1,

til (a) n

a E V

n

(iii) Choose A x [O,f

It

l)

f l E C:(A)

c D.

a EV n f Ia )

and

A

n > 2

For

is obvious that x

= lim fn(a). n->-oo

such that

n)

c

°

define inductively

for any a E A, ° D. Finally, let f

f (a) > n

[O,f

flea) >

for any f

n

a E A and

E

by

f (a) = 1 - lin n

*+

: A->R

be given by

A straightforward verification shows that

required properties.

for any

f

has the Q.E.D.

111

6.11.

As before

depth.

Let

6.11.1.

if

n =

I

n > 0

D+

V

+ BI- .... A

being of finite

is a sequence

f

1

D

=

+

D = {VI} ,

then

the definition makes sense by inductiB

+ = I'l = A ,

+

thus in this case a total decomposition of f-compatible total decomposition of A total decomposition as follows:

= {til,

fj,

2

6.11.2. face of

D of ncO

if

where

1 n+1 D = {V , •••• V }

and

= n-k.

.

}

, ••• ,

Let

f i:

--+-

.

B

j

A

Let

=

f

determines a total decomposition

} V = D ... , tl n + l } = DD

set

vk +j

6.11.3.

,vn + I}

=

Ai

is a face of

vk +j [a, n Bk+ j , 1.

Two total decompositions of

f

V D

set

+

be the restriction of

f, B i

be a

f

Let Nn+1 •••. ,V } of

DIB = {V i

fi

j B

have the same meaning as in 6.3.2 and set

is a decomposition of fj

arguments,

,

n > 0

if

be a total decomposition of

is an a.T.m. from

sj. o/j}

(see 6.7.1 (7) );

reduces to a regular and

has a canonical w.a.s.

Then

structure

DS

A and

B

B

and

Take

f

One defines a total decomposition

as follows.

},

+

) = n-r l ,

is a regular and f-compatible total decomposition of

DD of

n+J

{V , ••• , V

S, o/} is a decomposition of f and, inductively 2 n+1 {V , ••• ,V } is a total decomposition of f

=0 ,

n

If

and

0

(since

=

=+

on).

is an a.T.m.,

A total decomposition of 1

where

!P- --+-

f:

to

and

if

and therefore J, ••• ,n-k+1

n • By depth

is a face of

B.

1.

Notice that

D= {VI, ... ,vn + l }

and

are called equal • denoted D = D' , if

VI

V DIB.1.

• VDIA i.

=

= {'Vi, ... ,

D'

, 1 'l

nn+1

,""

v

'V + 1 n

6.11.4. = 'B

Assume now that

n "B

and

f = 'f Uc"f

, where

is an 'f (resp. "f)-horizontal face of

'f :

and

_ " I 11 n+l - { V •••• , 'V }

such that

B = 'B U "B = = C =

'Die

"f:

Let

'D = {'V

be total decomposi tions of 'f

= "Die.

c

=

and I,

•••

and

,'v n + 1} and "f

We shall define inductively (on

"D

respectively a total

112

decomposition

'D U "D = {\,II, ••. ,\,In+l}

(a)

\,II = '\,II U "\,II

(b)

if

'D+IC+ = "D+IC+

U

and

,where

c+

C+ = 'B+

Let

U

'D

U

+

c "D)+

=

C +

omposition

D = {II , ... ,II

n+1

"D

then

+

c+"D+ •

feb) • Clearly 1

C = "B+ n C

= 'D+ U

be a compact a.s. of depth zero and define

fM(b,x)

n

by hypothesis (and by definition)

and thus, by induction, we can consider

('D

by

as follows:

is determined by (a) and the relation

c

6. II. 5.

=

n > 0 , then

is a face of both

f

(see 6.8. I I) ;

C

'D U "D

of

c

}

f

M

of

is a proper a.T.m.

f

n+l = {\,IM, .. ·,II as follows. } of f M M I {t; B SM' 'I'M} , where =+

f

:BxM--+

=

M

=

Given a total dec-

we define a total decomposition

1

If

S

"l

III = L'I I

,

set

is the square

x

(fO)Ml

=

('l'([b,t]),x) •

It is obvious that

structure of admissible square on (SM)+ = (S+)M)' by

III

n

=

SM

6.11.6. f

Let

f'

is the obvious one; Otherwise

to

f'

be another a.T.m., and let

D

M

let

for example is determined

(G,F)

D be a total decomposition of

G* (D) -- {'III , ••• , , \,I n+l}

6.11.7.

is a decomposition of

0 , we are done.

exists an obvious total decomposition then

I M

defined above and the relation

M

phism

If

11

with

Consider the regular square

G*(D)

of I

f'

(if

'III = G*(II ) , etc.)

f D

be an isomorThen there {III, ••• , \,In+l}

113

_0--.. Ii f

(s)

I

T

1

g

--.. !1

IT and o(b»

let

fS :

A=S = A xMz::; N =

be the a.T.m. given by

fS(b) = (f(b),

(see the Remark following the definition of a regular square). A

V

decomposition (i) (ii)

fi

of

2

f

is called s-compatible

if

is IT-compatible;

the total decomposition

of

D S

(see 6.7.1 (7»

is

(OIB*)-compatible; (iii)

B

From (iii) it follows that

(o[B*)olT

V

°

near

B

and from (ii)

we deduce that (iv)

do- s V = 0

Assume that composition

fis

z

X

near

B U B

is s-compatible. Then by (i) we can consider the de-

V

fig

of

induction on

(see 6.1,10).

) shows that the square

(AS)O _ =

p

s

is admissible (the notation is obvious). B =+ ,

is a decomposition of B+

+

Nand

IT+

fS. Z

0+

B =+

f+

-----+ N

T

t

A

I i -

+

It foJLlows that

--->- M

=

g

S

V = {fis,

From (i.) and (ii) we also deduce

lTIA+ : A+

and the square

(s+)

A direct verification (using

+

M are weak morphisms

114

is regular. A total decomposition s-compatible if decomposition

S

D

D+

{V

2,

n 1

v +

••• ,

VD

n > 0 , (D

6.11.8.

Let now

I

:t

s)+

2

"n+l}

IJ , ••• ,

v

f+

f

of

is called

n > 0 , the total

is (s+)-compatible.

If

D

and we can define a total decomof

fS

(D+)s+

=

of

}

is

{VI, ••• , Vn + l }

vely if

"V

is s-compatible and, inductively if

V

is s-compatible then position

D = {V

as follows:

VI

S

and, inducti-

= V

Notice that

a

rv----(s)

f

T

r

i

A

-

a

m

and, if

+

t:!

T f+ I

(s+)

g

---+ M

be an admissible square of depth +

I

+

I

g

t

A

m > 0 , let

+

M n

X B - a

(sx)

fX

x --->

T

N

I --+

n

g

M

and

f

-T

t

be the associated squares (s+ sic, hence regular). ble if either

m

=0

a

*

I g*

is admissible of depth

A total decomposition and

D of

m-I f

and

s

is ba-

is called s-compati-

D is s-compatible (in this case

s

is regular)

115

m > O an d t h ere ex i. s t t ota 1 de compos i.t.i.ons

or

D+ , DX

and

D

of

f+

respectively such that (I)

D

is s -compatible (s

(2)

D+

is s+-compatible (this makes sense by induction);

(3)

(4)

x

is regular I);

h

-

D = (8 s) *«D I B ) [0, I ]) (see 6.7.5 and 6.7.6); X D IBO = D+ iBO

(5)

U h D B

6.11.9.

n l} V+

D = {VI,. '"

Let

be a total decomposition of D

or

n > 0

(i) (ii)

f:

is called regular if ei ther

and

V

is a regular total decomposition of

D+

is a regular total decompositioll of

makes sense by induction on

Bf--+ =+

f+

(this

n );

(iii)

.... x

DO = D+IB O is an S-compatible total decomposition of x + and the total decompositions DO of f+ : + .--+ AO 0 = ' DO

B ..... =0

n = 0

and

D

of

f

O

: B

=0

associated with

t--

f

O

of

:

f

x

O

(see 6.11.8)

DO

are regular. 6. I I. 10.

face of

Let

D be a regular total decomposition of

and let

total decomposition 6.1 I. 11.

f . : B.t--+A. 1

=1

DIB

of

i

f

i

f , let

be the restriction of

=1

f

B. 1

.

be a

Then the

is regular.

Consider a regular square

B

(s)

f

_ _0 - . . N

T 1

Ig ---+

11

and let

D be a regular and s-compatible total decomposition of

direct verification shows that the total decomposition -+

(see 6.11.7)

is regular.

S D

:

of

f. fS: B

A

116

6. II. 12.

The notation being as in 6.11.4,

'D

sitions

OlD

and

are regular.

assume that the total decompo-

'D U OlD

Then the total decomposition

c

is also regular. 6.11.13. Consider the submersive weak morphisms and

being

rr-\ertical. Let

0

rr:

-

and

f depth zero and all the faces

=

and let

f: B -

0

f

a:

and

g : being

-

be the

canonical projections. The square

B

_a--"' N

Ig

T

(s)

f

!

A =

-M tr

is clearly admissible of depth zero. Given a decomposi t i on V decomposi t i on

0

there exists a unique s-r compa t i b Le regular total

f

D

off D

To construct De fine

+

x

such that

we proceed as fo l l.ows , Let

l = {AI, nV U

11_, _

B S , til} =+' T

N

B =0

A0 x N = M

Then

M=

rr-compatible, regular, total

B = A0 x = M =0 f

x

by setting B = A* x N M =* = q

and

* x

1

,

AO

2,

N and M=

S

••• , lIn+l} B

=+

is the square

M

I

T O

V = {lIl, lI

1

f*

A*

P

the mappings being the obvious ones. '!' C(q)

on

is the obvious homeomorphism of

B

By induction on

n , we can define now

V+

D+ = D

2

n+1

= {V , ••• , V

}.

117

6.12.

Let

THEOREM.

B r--+

f

---T 0

B

(s)

be proper, let

f

N

I

g

J t!

--.........---> 7T

be an admissible square of depth

m and let

tible total decomposition of

Let

V

I c

be a regular and rr-compa-

be a regular and (sIB.)-compatible total decomposition of

f.

1

V D.1

Assume that

VIAl'

=

D.IB. n B.

and

11

D.IB. n B.

=

J

J1

J

let

i E I

and for any

for any

V =V

Proof. n.+1 17 i 1

},

Let

where

n.

,

=

depth(\,j,)

=

the a. T .rn,

=

I {ll , .. -, lln+1 }

V=

Then

0

Let

{17 , ••• , 1

is regular and we can consider

s

siB'

D.

of

1

1

D of

f

of (As). ,

B. t-

=1

=

1

such that

such that

D. , i E I . Now, using induction on

=

=

i E I • Then there exists a unique regular and

s-compatible total decomposition 1

Di

be a regular total decomposition of

and

DIB.

and

the regular total decomposition

and the regular total decompositions i E I

f

We begin with the following remark.

< n

=

1

m

Assume

n

D of

i E I .

for any

and

D

=1

i,j E I

Then there exists a regular and s-compatible total decomposition such that

1

A.

=1

1

D.

m

1

D=

S

D

,

V D

=

V

and

(as in Steps II, III and IV

below), we can see that the Theorem follows from the following weaker assertion :

(*)

"Let

of

,

be proper, let

let

si tion of =

f I c f.

1

D·IB. n B. J

si tion

1

J

D of

and for any B.I--+- A. =1

:

=1

for any f

We shall prove

(*)

i E I let

Assume that

V

VD. 1

be a regular total decompo-

D.1 =

V!A

i

and

D.IB. n B. 1

1

=

J

Then there exists a regular total de compo-r

i,j E I

such that

be a regular total decomposition

V

D

= V

and

by induction on

DIB.

1

=

D. 1

n =

is an a.s. and a regular total decomposition of

for any • If

f

i E I

"

n = 0 ,

reduces to a regular

B

=

118

and f-compatible total decomposition Theorem 6.5.

n > 0

Assume now that

too) is true for any a.T.m.

of

(*)

and that

r

f':

the assertion follows from

--*

(and hence the Theorem < n

with

By Theorem 6.5 there exists a decomposition

s,

of

'l'}

f

such that

1

VI lB.

Vi

1

for any

i E I

Vi with

=

Step II.

Let B* =*

T

-

(S )

I*

1 f

1

*

*

- - ------7

P

be the basic square associated to regular and

v_

S -compatible total decomposition

V+IAO•

DO

n B = (D O i)

DO

x

total decomposition of 1

.

=

DO

of

f O such that

i E I

- h (6 S ) * « DOIBO) [ O. I J )

x

Step III.

n B.

S. Then. by induction. there exists a

f O such that

is a regular and

VDx

O V+IA •

=

0

i E I.

SX -compatible

x h - h DOIB O = DolBo

and

Step IV. By induction there exists a regular and S+-compatible to+ DO

tal decomposition

+ I B+ n B Do i O

and

+

Step V.

D+IB+

n B1.

Step VII. of

f

such that

0

--*

(D)+ i O. -

such that 1·

C

c

I

is a regular and S-compatible total

B O

--*

such that

DD

and

o

By induction there exists a regular total decomposition

n l {V ..... V + }

and

=

X

B O

2

D+

+

DO U 0 DO U h DO

decomposition of

Step VI.

+ f O:

of

=

of

f+:

(D.) IB 1

D

=

V D

+

+

such that

n B.1 •

{VI. V2 .....

=V

and

VD

+

V+.

D+IBO = DO

i E I

vn + 1 }

is a regular total decomposition

DIB. = D. • i. E I. 1

1

Q.E.D.

119

6.13.

n l} D = {Vi , •.. , v +

Let

=

be a total decomposition of

Os

and let n = 0

If

of

f . Let

vI =

be the corresponding total decomposition if

set

n > 0

define inductively

It is obvious from the definition that

.

is a manifold with faces and that

c(f,D) = f

:

is a submersion compatible with the faces.

is called the

of If

B i

c(B.,DIB.)

is a face of and

i.

c(B.,DIB.)

"f : "s

and

=

-+ "f

'f

D = 'D U "D C

(see 6.\1.0\).

and

and

f = 'f U "f, c 'D

and

respec ti vely such that Then

c("B__ ,"D)

'DIC)

B n i

c(B=,D) .

are a.T.m. "s • Let

posi tions of

= c('!!_, tD) U

is a face of

i.

Assume next that t--+A

then it is easily seen that

where

' f : 'B

"D be total dec omand let

'Die = "Dlc and

=

=

(this follows directly from the definiti-

ons). 6 • 14 •

Le t now

a B =

(s)

T

f

N

rv-

I

t

t

g

M 1T

be

an

admissible square of depth

tal decomposition of

f ; let also

associated data (see 6.11.8). If

m and let

D be an s-compatible to-

(s+), (sx), (s-), D+ m

° ,set

and

D

=

m > 0 , define inductively

be the and if

•

is

s

is called the (s,D)-core of

B

it is clearly a manifold with faces. If

m > 0, notice that -

x

,D) and

x

x

,D )

is diffeomorphic through

es

f, g, a and

1T

taking the restrictions of of manifolds with faces

x

,D) to

,0+IBO)

+

+

,D)

x [O,IJ •

we obtain the regular square

By

120

cs(O,D) •

1

1

c (f,D) s

c Ctr , V

If

B.

is a face of

1

c(g,V

s)

'M D)

B, we can consider siB. 1

and

DIB. •

It

1

1S

obvious tha t ,DIB.) = c (B,D) n B. c s IB . (B. =1 1 S = 1 1

and this manifold with faces is a face of 6.15.

7.

c (B,D). s =

The assumptions and notation are as in 6.11.13. Then

TRIANGULATION OF ABSTRACT STRATIFICATIONS

In this chapter we shall prove that any a.s. of finite depth can be triangulated. All notions concerning simplicial complexes and triangulations of topological spaces can be found in the Appendix. 7.1.1. ces

A relative manifold

(V,oV)

oV

such that

(with corners)

is a pair of topological spa-

is a closed subset of

V and

V' oV

is a

manifold with corners. Examples.

(I)

If

X is a manifold with corners, then

(X,0)

is

a relative manifold. (2) clA (X)

and

7.1.2.

Let

Let

be a w.a.s. and

x = X'OX. (V,6V)

(ii)

101 '16KI

(X,

oX)

Set

X

=

is a relative manifold.

be a relative manifold. A triangulation

is called smooth if (i)

Then

X be a stratum of

K contains a subcomplex =

6K

of

such that

6V

for any closed simplex

°


M +

+

, c(f ) : M , L

A.

:

and

--+ M.

c(f.)

be the restriction of 7.2.2.

Let

(K, 0

f

(L

with

and that the assertion is true