Topological Stability of Smooth Mappings

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

552 C. G. Gibson K.WirthmQller A.A. du Plessis E.J.N. Looijenga

Topological Stability of Smooth Mappings

~

~..

Springer-Verlag Berlin.Heidelberg 9New York 1976

Authors Christopher G. Gibson Department of Pure Mathematics University of Liverpool P. O. Box 147 Liverpool, L69 3BX/Great Britain Klaus WirthmLiller Fachbereich Mathematik d. Universit~t Postfach 397 8400 Regensburg/BRD Andrew A. du Plessis School of Mathematics and Computer Science University College of North Wales Bangor Gwynedd, LL57 2 UW, Wales/Great Britain Eduard .1. N. Looijenga Mathematisch Instituut der Katholieke Universiteit Toernooiveld Nijmegen/The Netherlands

AMS Subject Classifications (1970): 57D45, 58C25 ISBN 3-540-07997-1 ISBN 0-387-07997-1

Springer-Verlag Berlin 9 Heidelberg 9 New York Springer-Verlag New York 9 Heidelberg 9 Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin 9 Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.

Preface

During the academic year 1 974-75 the Depar~nent of Pure Mathematics in the University of Liverpool held a seminar on the Topological Stability of Smooth Mappings:

the main objective was to piece together a complete proof of the

Topological Stability Theorem (conjectured already by Them in 1 960, and proved by Mather around ~ 970) for which no published account existed.

This volume comprises

a write-up of the seminar by four of its participants. There are several acknowledgements which should be made.

Any mathematician

working in this area is conscious of his debt to the inventiveness of Them, and to the technical work of Mather which has placed much that was conjecture on a firm mathematical foundation.

As far as the seminar is concerned I would like to

single out the special contribution of Eduard Looijenga, who showed us how to fill in gaps which otherwise might have remained open.

Also, I would like to

acknowledge the considerable help offered by Terry Wall, from the inception of the seminar to the production of the typescript.

Further acknowledgements are due

to the seminar audience (who frequently had good reason to appear confused) for their patience arKl in particular to Tim Ward who helped out with the talks;

to

the British Scientific Research Council who provided financial support for Eduard Looijenga, and the University of Wales whose financial assistance enabled Andrew du Plessis to participate in the seminar;

to Los Lander who was inveigled

into drawing the diagrams, and did an excellent job;

to Dirk Siersma and

Klaus Lamotke whose careful reading of parts of the manuscript removed many errors; and finally to Evelyn Quayle, Jean Owen and Margaret Walker who produced a first-class typescript.

Liverpool, July 1976o

C. @. Gibson.

Contents

Introduction Chapter

I

: Construction of Canonical Stratifications

w

Whitney Stratified Sets

w

Semialgebraic Sets

17

w

Thom Stratified Mappings

22

Chapter

II

:

9

Stratifications and Flows

w

Tubes

36

w

Tube Systems

41

w

Vector Fields

50

Flows

53

Applications

58

w Chapter III

: Unfoldin~s of Smooth Map-Germs

w

Introduction

65

w

Pre] ~m~ uaries, mostly algebraic

73

w

Infinitesimal Stability

80

w

Versality

86

Contact- equivalence

94

w

w Chapter

IV

Determ~ nacy

1o5

Jet-spaces and a transversality theorem

111

@enerioity

t20

: Proof of the Topological Stability Theorem

w

Multi - transversality

125

w

A stratification of the jet space

131

w

Properties of the stratification

w

Topological stability as a transversality property

145

References

15o

Index

152

Symbols

155

Introduction Motivation and some historical remarks. The

C~176

f

and

f'

from a manifold

to be

C l -equivalent

(l = 0, l, 2, ..., oo)

h

N

P

of

and

h'

of

equivalenoe relation: 'C g -behaviour'. of

f

We call a

to a manifold

if there exist

f' = h' o f o h -1.

it simply says that

~ith respect to

function space

such that

N

Coo-mapping

f

and

f'

~

C e -stable

if the e ~ i v a l e n c e

This presupposes a topology on

increases.

Coo-mapping can be approximated by a C~

mappings,

f

class

in the

C~176 P):

we choose

Obviously, this property

There is an important question, related to

this notion, which has also some physical interest.

the set of proper

This is clearly an have the same

the Whitney topology (see Ch. IV for the definition). becomes stronger as

are said

C~-automorphisms

C t- equivalence) forms a neighbourhood of

Coo(N, P).

P

Namely, whether any proper

C l -mapping or more precisely, whether among

Cpr(N , P),

the

C t-stable

ones are dense.

The Morse lemma and the Whitney embedding theorem imply that the answer is yes if

dim P = 1

or greater than

2 dim N (and

e

arbitrary).

H. ~%itney seems to

have been the first to investigate this question in its own right. he showed that the answer is affirmative in case included a fairly simple characterisation of

In 1955 [W~]

dim N = dim P = 2.

His proof

Coo-stable mappings between surfaces.

A few years later, in 1959, Thom [TL] gave examples of proper smooth mappings from I~n

to

I~n

(n ~ 9)

which cannot be approximated by a

A complete answer to this question for 1967.

I = oo

In his fundamental series 'Stability of

C 2 -stable mapping.

was given by J.N. Mather around Coo-mappings'

[I-VII

obtains among other things a (mult~ltransversality criterion for For certain pairs

(n, p):= (dim N, dim P),

he

Coo-stability.

this transversality criterion

involves unccuntably many tramsversality conditions.

As one may expect, in such

a case these transversality conditions cannot be simultaneously satisfied by a dense subset of

Coo (N, P). pr

Thus Mather was able to determine the pairs

for which the answer to our question proper mappings are dense in of the following conditions

(with

Cpr(N , P)

~ = oo)

is yes:

if and only if

the

(n, p)

(n, p)

~~ satisfies one

n


dim X .

obtained from another, Also,

submanifold of X - ZX,

Y -

The set

~P

f : X I -~Y

- f(zx)

dim Y ,

dim X ,

is a semialgebraic smooth

since it is obtained from another,

with

In this way we obtain a surjective

dim YI > dim X I .

A point in

YI

f : XI -~Y1 "

We conclude that indeed

is a regular

f(X I) .

has at least one regular value - by Sardis Theorem, for instance -

surjectivity of

is a

since it is

if and only if it does not lie in the image

I

is

~Y, b y deleting a closed semialgebraic set of lower

of dimension

f : XI -*YI

YI = Y - z Y

of dimension

by deleting a proper closed subset.

Value of

f(ZX)

Using the induction hypothesis

X I = X - f-1 (ry U f ( Z X ) )

I~n

smooth mapping

f

(2.3) .

is semialgebraic, of

aim ( Z X ) < dim X .

semialgebraic smooth submanifold of

dimension.

ZX

But

contradicting

dim Y ~ dim X . Q. E. D.

(3.3)

Let

f :

be ~@mial~eb~aic, with PToof on

A~ain,

dim Y .

~n flX

Y = f(X)

When

-~

Rp

be a pol.ynomial mapping, and la~

finite-to-one:then

dim f(X) = dim X .

is semialgebraic.

dim Y = 0

This time we proceed by induction

the result is clear.

and that the result holds for lower dimensions. ZY

X _. R n

is semialgebraic, of dimension < dim Y .

Suppose

dim Y

is positive,

By (2.5) the singular set Using the induction hypothesis,

and (3.2) , we have dim (flX)-I(zY) = dim (Zy) Now

XI = X - ZX

dim X ,

- (fiX) -I (ZY)

< dim Y ~ dim X .

is a smooth submanlfold of

m n

of dimension

since it is obtained from another, X - ZX, by deleting a closed

semialgebraic subset of lower dimension . submanifold of

~P

one smooth mapping

of dimension f : Xl -*Ys "

dim Y .

And

YI = Y - z Y

is a smooth

In this way we obtain a finite-to-

By shrinking

XI

we can suppose

f : XI -*Ys

30

has constant rank.

Assume

dim X 9 dim Y .

construct a non-trivial smooth vector field

It will then be possible to ~

on

xI

with

Such a vector field has a non-trivial flow line, on which contradicting the finite-to-one condition.

Thus

f

6(XI) _C ker T(flXl), is constant,

dim X = dim Y . Q.E.D.

(3,4)

Let

f : 1~n -~ m p

~e a Dolvnomi@l mapping, and let

8 semial~ebraic smooth submanifold, fiX

The @et

fails to be i=~ersive is semial~ebraic.

one then

Sg(flX)

Proof

First, we show that

X

A

TX

X

subset of

=

subset of with

x,y ~ X

~n x ~n x ~n

x, y E X

and

will correspond to

Now we show that suppose

f

and

precisely

T ( ~ n) :

Sg(flX) ,

Finally, assume

comprising triples

Let

f

necessary.

(x, y, v)

of points

A be the diagonal in T I~n

with

is semialgebraic. (Factor

9

A x I~n

~n x ~n :

the semialgebraic

f

Observe first that we can

as the composite of the natural Let

8(~n)

denote the

By the above the set ker Tf ~ TX ~ s(~n)

its image under the projection

is

T(1~n) -~ ~ n

is

ar~l semialgebraic, by the Tarski-Seidenberg Theorem. flX

finite-to-one.

so there is an open set" U C X suppose

of

TX .

Sg(fIX )

T(~n)

(x, y, v)

x $ y 9

embeddi~ in its graph, and a linear projection.) umit sphere bundle in

~n

x = y .

a linear projection,

semialgebraic in

is finite-to-

For hhis we use the fact

comprising triples

then, under the natural identification of A ~ B

where

We put

Clearly, both A, B are semialgebraic,

set

fl x

X

is the limiting position of lines in

~ n • l~n • ~ n

cellinear points with B

Moreover, if

is semialgebraic.

at two distinct points. =

of points in

be

has dimension < dim X .

that a tangent line at a point on cutting

Sg(flX)

X C ~n

Suppose that

at no point of which

has constant rank on

U ,

replacing

U

f

dim Sg(flx) = dim X ,

is immersive.

We can

by a smaller open set if

It will now be possible to construct a non-trivial vector field

31

on

U

on which

with f

~(U) ~ ker Tf .

Such a vector fiel~ has a non-trivial flow line,

is constant - contradicting the hypothesis that

It follows that indeed

f

is finite-to-one.

dim Sg(flX ) < dim X . q. E. D .

The preliminary work is now complete and we are in a position to establish the existence of canonical partial stratifications for generic polynomial mappings.

(3.5) an~

Let

f : A ~ B

A, B

be semial~ebraic open sets in

a pol.ynomial mappin~ for which

admits a canonical ~artial stratification

~

flZ(f)

~n,

~p

respectively ,

is finite-to-one,

f

having only finitely ma~v

semial~ebraic strata. Proof

Observe that the set

that the set

C

of critical points of

f

is semialgebraic, so

of critical values is likewise semialgebriac, by the Tarski-

Seidenberg Theorem.

Put

C = Cc D - Cc-1 -D 00''" . dim Cj ~ j ,

Z

c = dim C .

of

C

We shall construct a filtration

by semialgebraic sets, closed in

C , with

such that the following conditions are satisfied for each

j

with

Mj : C. - C j ~ (PS0) which each

M. ~

(PSi)

is empty, or a smooth suhmanifold of

is Whitney regular for f-IM. ~ Z

f-1~ - Z

is a smooth submanifold of

is Whitney regular over

Suppose inductively that to this prescription. dim C. = j .

If

N ,

for

k > j .

f-lMj ~ ~

for

k ~ j .

over

we take

have been constructed according

Cj_ 1 = C j

We put

:

j

closure { u w %

which is semialgebraic, of dimension < J .

k>j

,

Now p u t

c~ = f-l(cj_~t)n~.

and the restriction

f-IM. ~ Z

Cc, Cc_ I ,..., Cj

dim Cj9 < j

of dimension

k > j .

f : f-IM. ~ Z ~ M . is a local diffeomorphism, J J (PS2) f ' I M k ~ Z is Whitney regular over (PS3)

P

ar~l are done.

Suppose

32 which is semialgebraic of dimension R2

j , by

We define

(3.3) 9

R 2' U R2"

=

where R 2'

=

closure

w(cj, ,

n

closure {kYj W(0j ,

-

and R2" Clearly

R 2' , R2"

(and hence

R2)

are semialgebraic, of

dimension < j

.

Next,

put R3 which is semialgebraic, of

=

clos~e of

dimension < J, Ca"

=

=

closure of

J ,

(3.~) 9

Also write

and

W(Cj", f'IfCj" - P)

which is semialgebraic, of dimension < J .

c j-1

by

- R 2)

Cj' - R 2 - R 3

which is semialgebraic, of dimension

R$

s~(flcj'

Finally, define

= ~1 uf(R 2 u ~ 3 U ~ )

which is semialgebraic, of dimension ~< (J - I) , by (3.2) . That completes the induction step of the construction. reader the task of formally checking that the partition of the required canonical partial stratification of

C

We leave to the by

the

M

J

is

f . Q.E.D.

Now let f : A -~ B (3.5)

f

A, B be semialgebraic open sets in Euclidean spaces, and

a polynomial mapping which satisfies the Generieity Condition. admits a canonical partial stratification

a Thom stratification

(~, l~),

And by (3.1)

f

admits

namely the Thom stratification associated to

this we call a canonical Thorn stratification for main result of Chapter I .

6 .

By

f .

@ :

Sunning up, we obtain the

33

(3.6) f : A ~ B then

Let

A, B

be semialgebraie open sets in Euclidean spaces, and let

be a polynomial mapping which satisifes the Generioi~y Condition :

f admits a canonical Thom stratification

(I, i~ ) .

It is this result which enables us to construct the crucial stratification of the jet space in Chapter IV .

We conclude the present chapter by listing

the basic properties of canonical partial stratifications require~ for Chapter IV. First, it is invariant under smooth equivalence of smooth mappings; precisely

(3.7) g, h

with

Le__~t fl' f2 f

I stratification @2

~e smooth m~ppings ~or which there exist diffeomoruhisms

o g = h o f2 " ~

9

and let

Suppose that @2

fi

admits a canonical partial

be the stratification induced ~7

is ~ canonical partial stratification for

h :

then

f2 "

The proof consists of no more than a (tedious) formal checking - which is also the case for

(3.8) stratification then

@'

Let

f : N * P

@ .

Let

be a smooth mappin~which admits a canonical partial

P' ~ P

be open, let

N' = f-Ip, ,

is a canonical partial stratification for

an~ let

f : N' ~ P '

@' = C]P' :

.

Another result for which it does not seem worthwhile writing down a proof is

(3.9)

Le__~t f : N -~ P

stratification

@ .

of

@

f .

Then

And let

be a smooth man~in~ which admits a canonical uartial U C N

be an open set containing the critical set

also provides a canonical partial stratification for

flU .

The final fact which we shall need to know later is

(3.1o)

Foxr I ~ J ~ s

le_tt fj9 : N.J -*p

domains having the same dimension, and let

be smooth mappings~ with

f : N -~ P

be their disjoint sum.

34 Suppose each

fj

admits a Them stratification

are in ~enera I position, ~' = ~ ~j' .

(~, ~)

.

f

(

,

admits a Thorn stratification

Moreover, if the

(~,i~)

with

(Ej, lj') are all canonical then so too is

CHAPTER

Stratifications

II

and Flows

Klaus WirtmOller

$6

w

Tubes

This chapter is concerned with the construction of continuous flows on stratified sets.

Since strata are smooth manifolds we can clearly obtain a smooth

flow on each stratum of a stratification by integrating a smooth vector field.

But

in general we cannot expect to obtain a continuous flow on the whole set by just putting the parts together. special nature:

Therefore the vector fields we consider will be of a

in a sense to be made precise they will be controlled along the

boundary of each stratum in such a way that we do obtain continuous flows.

Let us

illustrate the idea by means of an example. (i. i )

Example.

field on

Stratify the plane

~ 2 by

IO, I~2 - O~

and let

~

be the vector

2 2 defined by

~(0) = O, (in standard coordinates).

~(x, y) = ~

~/~ IF(~, y -ei

o~J

for

(x, y) i o

Clearly, the origin is a fixed point and all other flow

lines are circles centred at the origin.

Note that

jjl 9

/

outside the origin all flow lines have constant speed velocities

II~(x, y)IIAI~x, y~I

have constructed a flow on

tend to infinity as

II~ll : 1 (x, y)

whence their angular

approaches

0 .

2 2 which is not differentiable at the origin.

Thus we On the

other hand distance to the origin is constant along any flow line, and this implies at once that the flow is continuous. This simple tYpe of control by the distance to the smaller stratum will not be sufficient to ensure continuity of the flow unless this stratum is a discrete set.

37

The next example indicates another type of control over a stratum of positive dimension. (1.2)

Example.

Split

it3-~t~ it2

and stratify it by

[it x 0,

~ x(m2

- 0)1 .

Then w(t, x, y) = a/at + ~(x, y) defines a vector field

W

on

it • it2 , with

~

as in (1.1).

linear flbw on the t - axis whereas the other flow

U

induces a

lines spiral round the t - axis.

This time the proof that we have a continuous flow

on

~3

is basea on two

observations : (1)

the flow preserves distance to the t - axis,

(2)

under the canonical retraction (t, x, y)

(say)

and

it x it2 ~ ~ x 0 the flow line through

is mapped to the flow line through

(t, 0, 0)

on the t-axis.

Our strategy in the general case will be to look for vector fields having properties

(I) and (2) locally, i.e. in some neighbourhood of the "smaller" stratum.

We will give a meaning to "distance" and "retraction" in this context by taking this neighbourhood to be a tube.

But first of all let us introduce the following

notion which will be convenient later on. (I. 3 )

Definition.

that two maps

Let

gl' g2

X, Y, Z

defined on

if there exist neighbourhoods coincide on

Z ~ U

(on

U

be subsets of a topological space Z

of

Z N U ~ V,

define the same X

and

V

of

respectively).

~erm at Y

in

N

X

(at

such that

N ~

We say

(X, Y)) gl

and

g2

38

Note that we do not require that clear that the composition

X

or

Y

be contained in

Z.

It should be

of two suitable germs, or one germ and a map, is a well-

defined germ. Now we give the precise definition of "tube". (1.4) (X

Definition.

Let

X _c N

need not be closed in

where

~ : E~

X

N).

diffeomorphism,

p

T

(at

X)

= p o e

-1

retraction

A tube at

E,

and

e : E-* N

of the inclusion : N~

9

X _c N.

go : ~

(2) means that

retracting

that

~3

T = (E, ~, p, e)

T

~(X) so that

g,

x go

SR2-. ~

be the projection

~

~3

g o ~

(I. 2 ) once

to the first factor then

to hhe linear flow on the real

S/at.

Conversely,

given any submer-

suppose we wish to construct a flow on ~

I~3 ,

under

g.

For

(2) and the reason for this is that

tube rather than a tube which is compatible with

g

in

sense.

Definition.

and

and

By abuse of language we will

of tubes let us look at Example

maps the flow on

onto

Let

T

(or a germ of such a map at g

is the

T.

(2) refers to the standard

(1.5)

e o ~

X ; we call them the germs of the T .

no such flow will have the property

the following

of a local

= ~ o e -1 : N ~ X

preserving the strata and mapping to the standard linear flow on general

N

the quadratic function

~ : X-~ E

Observe

llne generated by the constant vector field g

~

is the germ at

and the distance function defined by

If we let

sion

p : E~

are well defined germs at

Before we come to the construction

property

is a quadruple

commuting with the zero section

often omit the superscript

more.

X

is a (smooth) vector bundle,

of a Riemannian metric on

germ

be a submanifold of ~he smooth manifold

T

be a tube as in ( 1 . 4 ) X).

T

and let

is compatible with

g

g : N ~ P if the germs

be a map (at

X )

are equal.

Thus each fibre of

T

should be contained in a fibre of

to find compatible tubes only for very special

g,

g .

We can expect

and the remainder of hhis

section is devoted to the proof of the following existence result.

39

(1.6)

Theorem.

Let

N, X C N

be a smooth map germ at

X

such that

(relatively) owen subsets of compatible with such that

g,

and

X

with

P

be smooth manifolds, and let

glX

is a submersion.

XI ~ X _c X u .

then there exists a tube

T

-If -

Suppose

TO

a_~t X ,

XI c X~

is a tube at

are

Xo,

also compatible with

g,

TIX I = ToIX I.

Proof.

Our proof depends on a standard method of constructing tubes, see [Lang]

or [BrScker-J~nich]

:

choose any spray

~

on

N

and let

e~

be the germ at

of the associated exponential map, restricted to a normal bundle in

g : N-. P

N.

Then

T = (E, ~, 99 , e~ )

is a tube at

E.

Let

X

T

from a careful choice of

Let us deal with the latter first. Tg : TN-* TP

the kernel rank of

g

denote the differential of

g.

Since

is constant in a neighbourhood of

generality to assume that this neighbourhood is all of suhbu_ndle of Let

of

X

We shall obtain the required extra properties of and

~ : E ~ X

X

TN, and furthermore we have

To = (Eo' ~o' Po' eo)"

is submersive

X, and it is no loss of N.

ker Tg + TX = TN

The differential ~

gIX

e~

Then

ker Tg

over

X.

is a

along the fibres of

~o

is a monomorphism Tfibrer ~e o) : E o -* TNIX ~ of vector bundles, by means of which we~dentify we must have

E ~ _c (ker Tg)IX ~

since

TO

E~

with its image in

is compatible with

Now if we pick a Riemannian metric on the bundle we also get a metric

~o

on

(ker Tg)IX ~

orthogonal sum of the Riemannian bundles Po

Then

g.

TX ~ ~ (ker Tg)IX ~ = kerT(glXo)

splitting the latter bundle into the

ker T(glXo)

and

E~

(here we think of

as a Riemannian metric rather than a Quadratic function).By a standard extension

process we find a metric E

TNIX o.

~

on

(ker Tg)IX

be the orthogonal complement of

and let and leave

p

be

~IE.

EIX I = EolXI, X

Then

E

ker(TglX)

such that in

~ = ~o

(ker Tg)IX

is a normal bundle of

i.e. for any choice of the spray

in the right direction.

X ~

over

X I.

We let

with respect to

such that

E _c (ker Tg)IX

its solution curves will

40

T = (E,T, p, %)

Thus

will h~ve all required properties if the spray

satisfies

(1)

~(ker Tg) ~ T(ker Tg)

(2)

e~I(EIX l) = eol (EIX I) . But finding such a spray is a local problem in the following sense:

construct of

X

locally near any point

by open subsets of

N

x e X

X,

we

can

then we find a locally finite covering

on which the local parts of

~

A

are defined.

partition of unity subordinate to this covering gives us a spray neighbourhood of

if

~

On some

satisfying (1) and (2), and this is clearly enough.

We deal with the local problem of defining

~

near a point

x e X

by

linearising the situation, as follows. (a)

let

x E X o.

for

P

Since

near

gx

gIX e

is submersive we find charts for

with respect to which

glX ~

linear coordinates in the normal space x :

via

e

we obtain an

O

Note that since

T

N-chart

EolX near

is compatible with

g

and a trivialisation of x

g,

N,

if

x E X - X

Choose E~

extending the chart for g o e

and

near

X . O

is constant along the

N-

and

P - charts.

Now let

be the standard geodesic spray on Euclidean space.

leave it to the reader to check that (b)

x

O

is automatically linear in the

in the chart on

near

is a linear projection.

O

fibres, so

X~

we define

~

~

~,

We

satisfies (1) and (2).

as in case (a)

but use any chart on

N

which

O

linearises

g .

Then

neighbourhood of

x

~

clearly has property (1), and if we shrink to a

staying well away from

X1

we may ignore (2).

Note that Theorem (1. ~ ) still holds if we allow the components of

KS P

to have

different dimensions. (1. 7 )

Corollary.

Theorem (1. 6 ) remains true if

0nly at an open subset Proof.

Xg

o_~f X

Choose an open subset

Theorem (1. 6 )

to

with

X 1 ~ Xg,

TO

over

N

g

containing N

g

in place of

of N .

hence a tube at

N

g

is not a germ at

X

but

X - X 1.

such that

X

g

= X ~ N g

We obtain a tube at X 1 U Xg = X . []

X

g

and apply which coincides

41

w We return to the problem of means of tubes.

Tube Systems controlling vector fields on a stratified set by

Let us first study the cross-~atio example mentioned in the

introduction in more detail. (2.1)

Example.

Let

Y : JR-*(0, co)

be a smooth function and let

A c SR3 be

defined by the equation (.)

xy(x + y)(x - y(t). y) = 0 .

x

A

is stratified by the

t - axis

canonical Whitney stratification). Whitney stratification of

I~3 .

X

and its complement

Adding

IR3 - A

(in fact this is a

as a third stratum, we obtain a

Suppose now we are looking for flows on

(i)

preserve the strata ,

(2)

project to hhe standard linear flow on the t - a x i s (t, x, y)

Y

and under the retraction

-* t.

First notice that this problem cannot have a smooth solution. every

point

tangent spaces

2 3 which

(t, O, 0)

on the

t-axis

TxA (x E A, x -* (t, 0, 0))

containing the t - axis.

,

For at

the limiting positions of consist of four planes in

The cross-ratio defined by the corresponding

SR3 ,

concurrent

lines in the projective plane is, after suitable ordering of the lines, just A smooth flow satisfying

(1) and (2) must preserve the cross-ratios,

each

which is

~(t).

42

impossible unless 7

happens to be constant.

any strata-preserving

C1

Note that in fact we have shown that

flow must be trivial on the t - a x i s .

Next let us try to piece together a continuous solution, smooth on each stratum. Trivially condition (2) determines our flow on

X .

If we take

~

then we clearly find a unique solution on

A

canonical tube at

X

controlled by

(i.e. commutes with the retraction and preserves

function of

~ ~).

But for an arbitrary tube

TY

at

Y

impossible to extend this solution further to a flow on both

~

and

TY.

retract neither along the fibres of

which is the distance

it will generally be I~3 which is controlled by

The obstruction comes from the fact that X ~

to be the

~Y

nor along the surfaces

will in general X p = constant.

The conditions we have to impose on the tubes in order to make such extensions possible are most conveniently (2.2)

Definition.

and let

~

and

Let

TY

phrased in terms of germs (see (I. 3 )). X

and

be tubes at

Y

X

be submanifolds of the smooth manifold

and

Y.

We write

X

instead of

~ TX

N, etc.

Let us introduce the commutation relations (CRvr)

~rX O r Y

=

T/X

(c~p)

pXo Y

=

px ,

both being equations between germs at submanifold of another smooth manifold map

Y

into

y'.

(CRf)

(X, Y). N',

Furthermore, let

with a tube

Y'

TY' , and let

be a f : N -* N'

Then we will also have to consider y, f o ~Y = ~ o f (as germs at Y) .

The reason for introducing this last relation will become clear shortly.

Note

that the three commutation relations, put in a different way, require that TY be y, compatible with X , pX and ~ o s respectively. In view of the submersiveness condition in the existence theorem for tubes

( 1 . 6 ) the following lemma is the key

to further progress. (2.3) manifold

Lemma.

Let

N, and let

the map germ

(at

~

X

and

Y

be disjoint smooth submanifolds of the smooth

be a tube at

X.

X)

(x, px~l Y

: Y~Xx

If

Y

is Whitney regular over

X

then

4S

is submersive. Proof. x e X

If it is not, we find a sequence such that

( X, pX)iy

isation of the tube X = A px 0 at

Yi

in

and

kernel of APx

( X, pX)iy

~Yi

we may assume L

in

in An .

(TyiY) ~ T

a line in

near

An

x

Since

converging to some point

Yi"

Choosing a local trivial-

TX

is the standard tube at

( X, pX) : A n ~ A p x A

is not we conclude that Yi"

Y

we may assume that

T iY

is not transverse to the

By compactness of the Grassmannians of subspaces of and

(xy i-* ) -* L

where

T

so

Y

+

L in

~n.

An

is a p - dimensional subspace and

Using a simple convergence lemma we see that

+ L ~ T,

is a submersion

This kernel is the orthogonal complement of

to the orthogonal complement of l~Px 0 TxX + L = A p x 0

in

is not submersive at

N = A n = A p x A n-p.

( X, pX)

0 +

TX

(yi)

T

is net transverse

But this implies

is not Whitney regular over

X

at

x . KS

The strategy outlined so far will enable us to lift a smooth flow on a manifold to a controlled flow on a stratified set. necessary)

to look also at a more general problem:

f : A -* A' (say)

and a controlled flow on

A

which is mapped to the given one under

X

and

and

Y

X'

and

on

A'

Y

in

A

A' f ?

A'.

But if

does there exist a controlled flow on If we restrict attention to two strata

f(X) ~ X'

and

f

f(Y) ~ Y'

are different strata then the requirement that the flow on

: ~ x ~2...4,, ~ 2 .

maps

X

(say) where A

will interfere with the control exercised by the distance function

simple example is provided by (I. 2) and (I. 1 )

f

given a stratified map

then there is nothing new about the situation provided

into a single stratum in Y'

It is natural (and will in fact become

lift that pX.

together with the projection

A

44

The example suggests that the control by

pX

should be disposed of in these instances

since a similar control is already effected by the map the target:

f

the t - a x i s .

f

together with the flow in

keeps the spiralling flow line automatically at constant distance to X ~ ,

On the other hand we do need control by the retraction

is why (CRf) was introduced. but involving

7

X

and

f

and this

We are heading now for a result analogous to ( 2 . 3 ) X ~

rather than

pX.

and

Before we can state it we

need a bit of technical preparation. Let N

f : N~N'

be a smooth map sending the smooth submanifolds

submersively into

submanifolds of such that

X'

N'.

and

Y'

respectively, where

X'

and

Furthermore, suppose we are given tubes at

(CRy) holds for the pair

Replacing the germ (at

X')

~

X'

(X', Y' )

: Y' ~ X'

and

(CRf)

Y'

X

and

Y

in

are smooth X, X'

holds for

X

and and

Y' X'.

by a smooth representative we form the

pullback

proj X xx, Y'

- -@

Y'

II proj

~

X'

I

X

)

X Xx, Y' = I(x, y') E X x Y' : fx = X ' y , ~ ) .

(i.e.

submersion

X •

the germ of

Y'

X'

Since

f : X ~ X'

is a

is a smooth manifold, and the commutation relations imply that

(X, Y) X, (

actually maps into

y!

X Xx,

~

Y'

o f) : N ~

X x Y'

.

Clearly, this property does not depend on the choice of a representative above, and we are now ready to state the following. (2. ~ )

Lemma.

with respect to

If, under the assumptions above, f

Y

is Thom regular over

X

then

(x,

f)[y

: y ~ x •

Y'

y,

is a submersive ~erm at Proof. such that

X

(notice that

If not, we find a sequence (~X, f)iy

f = ~

(yi)

is not submersive at

in

o f Y

Yi "

o_~n Y).

converging to some point

x E X

If we put x i = ~Xyi, y~ = fYi

and

45

x i' = fx i

then the tangent space to

Using the fact that

Tf

sends

T

X • Y

onto

Y'

at

yj) -

(xi'

Ty~g',

is

T

X • T , Y'. x.~ ~x! X' Yi i

a simple calculation shows that

Yi ( x, f)iy

T

X

fails to be submersive at

ker T

Yi

(~Y) ~ k e r

T

Yi

(fiX)

Yi

if and only if the linear map

is not onto : the relevant maps are displayed in

xi

the diagram

0 -~ ker Ty(flY)

) T Y

TxX XTx,X , Ty, Y'

y'

L\ 0

) ker Tx(fIX )

~

)

TxX

> Tx, X' .

Passing to a suhsequence we may assume that all spaces same dimension and converge to a proper subspace (ker Tyi(flY))

itself converges to

converges to ker Tx(flX), X T x (T) = S c ker Tx(flX ). ker Tx(flX) at

x.

we cannot have

T CTxN ,

S

Tyi~X[ker Tyi!flY)]

of

say.

TxX,

havel the

~hile the sequence

Necessarily

(ker Tx.(fIX)) I

and we obtain the proper inclusion X Since T x ~ is a retraction whose image contains ker Tx(flX ) _C T,

so

Y

is not Thom regular over

X

D

The main result of the section will be that we can (under suitable conditions) assign a tube to each stratum so that all relevant commutation relations are satisfied.

Let us first fix some notation to which we will stick throughout this

and the next two sections. Let

(A, A)

be a Whitney stratified subset of the smooth manifold

a = dim A (i.e. the maximal dimension of a stratum in we denote the union of all i-dimensional Mi

~tr~t~ in

is a smooth submanifold of N of dimension

i.

A). ~

by

For each Mj .

N ,

and let

i=0,1,..~, a

P y II.l.l)

Notice al~o ~n~t ~he

~i

each at8

46

locally finite unions of strata, so a commutation relation holds for a pair (M i, M j) X ~ M i,

if and only if it holds for all pairs of strata Y ~M j .

manifolds for

Mi

with

It will in fact turn out to be more convenient to work with the rather than with single strata whenever possible.

Next we put,

-I ~ a < ~ .< a , A ~a =

Note for later use that each A~

(X, Y)

are locally closed in

U i=~+1

A~

and

1

is a relatively closed subset of

A , again by (l.l.1).

Notice that the

filtration by dimension already used in Chapter I.

f : N -~ N'

A~

provide the

If, more generally,

is a Whitney stratified Subset of another smooth manifold stratifies a smooth map

and that the

A

N'

(A', A' )

such that

(A, A' )

then let us agree to use the same notation in

source and target but distinguish by a prime all symbols referring to the latter. (2. 5 )

Definition.

manifold each

N.

~,

Let

(A, A)

A tube system for

for

be a Whitney stratified subset of the smooth A

consists of one tube

i = 0, l, ..., a.

(CRy) hold;

furthermore all (CRp) are satisfied 9

i a (T)i=0

tube system

relations of type strata in

A

Let

f : A ~ A' 9

for

A

If

we call it controlled if

f : N ~ N' ( ,k~a' ,T "k=O

is controlled over

be smooth and let

is a t u b e system for

(T 'k)

if the

9

Ti

fX U fY C M 'k

for some

Note that in general "controlled over (unless, e.g. all strata of

A'

(A, A' ) A'

then a

satisfy all

(CRy) and (CRf) and if (CRp) holds for those pairs

with

(X, Y)

of

k. (T'k)"

does not imply "controlled"

have the same dimension).

We can now state the result announced above. (2. 6 ) map.

Theorem. Let

f : A-+ A'.

Le__~t N

and

N'

be smooth manifolds,

A ~ N , A' _c N', and supwose

(A, A' )

By putting

(Ti)

for

A

f : N -~ N'

(T 'k)

which is controlled over

N' = point we obtain

a smooth

is a Thorn stratification for

Then for each weakly controlled tube system

exists a tube system

at

We say that the tube system is weakly controlled

if all commutation relations of type

be a stratification of

T i = (E i , w i , p i , e% )

for

(T'k).

A'

there

47

(2.7) Proof

Corollary. of (2. G ).

Every Whitney stratification admits a controlled tube system.KS We proceed by double induction as follows:

already constructed a tube system (there is nothing to show for Ma

(Ti)

a = 0).

for

A a-I

which is controlled over

By Theorem (i. 6 ) we find a tube

satisfying condition (CRf) (with respect to the tubes

various commutation relations involving strata in We do this by downward induction on

a < a :

Aa

a = a - 1

that

(T i)

(T 'k)

restricted to

be complete, for

a = 0

(for

A aa-1

A

ei

we have

A a = Ma).

is also controlled over

a .

(T 'k)

If we can arrange

then the proof will

gives the result.

neighbourhood of i

~

in

Aa

is controlled over i e ,

and

N , and we denote the latter by

pi

as maps defined on

for

~ ~ i < a:

ITil.

ITil

Note

Let for each

i

Ei

with

a , Qi =

We p a r t i t i o n mapped by points

In order

is a diffeomorphiem of some neighbo~rhood of the zero section in

that we may consider u.
e , for a n X 'q s A'

IT'qlx'I

i

(in view of (I.I.i)

are strata such that (by continuity of

Qi(k, ~), f o i

= ~,k o f

fX cX'

then

f

maps

f) for

~ ~ i < a

and

0 ~< k ~ I t,

(5)

f o i

(6)

(~, 9 pi)

(7)

( i, f) : Qi(k ' g) ~ ~ 0 (N',x;)

integration

be the of

uke~e~i,t (where

" Uk~+t'61k

~ i , t ( y v) = ~ i ( Y V , t ) , ~ i , t ( x ' )

Now d e f i n e

The u s e o f Im(i)

!

!

r : (N ,Xo) § (N,Xo)

= ~i,t ~F

- ~i(x',t)

, s : (P',y~)

).

-~ ( P , y o )

by

r(x') = i -I o ~k,_Vk~ . . . o ~i _Vl(X' )

(where

v i = u i o~ ~ F(x') )

s(y')

(where

w.l = u.1 o~(y')

i-l,j

j-lo ~k,_Wk ~ ... ~ l , _ w l ( Y ' )

-1

makes s e n s e :

- (~ o F ) - l ( u o ) .

retractions

' F~

to

i,j

recall

that

And i t i s a l s o c l e a r

)

I m ( j ) = ~ - l c u O) , that

r,s

are respectively

.

Now for(x') .'I

= J

= f o i-I o ~k,_v k o ... o # i _v I (x') = j-I o F o ~k _Vk o ... o ~i _Vl

~

o...o~i vk

oFCx') = soF(x') ,-v I

(since wiCFCx')) = vi(x') ) and so

f o r = s o F , as required:

thus

(F,i,j)

is trivial.

Cx')

73

w (I.i)

Preliminaries, mostly algebraic

Definition The local ring

~,Xo

is the ring of functlon-germs

(N,Xo) § 9

(with

addition and multiplication defined pointwise). This ring has a maximal ideal (N,x o) § (m,o)

consisting of all germs

.

We shall write (1.2)

mN,xo

4 ' mN

for

% , X o ,mN,xo

when no confusion can arise.

Lemma Let

Then

(Xl,...,Xn)

x I ..... xn

be a system ~f local co-ordinates for

represent a (N-basis for

(N,xO) .

mN .

Proof Let

#

represent an element of ~(x) = r

mN .

Then

- r n

=

~ (r

.... O,x i ..... x n) - r

.... O,Xi+l,...,x n)}

i=l -

~ II ~ i=l 0

{%(O ..... O,txi,xi+ I ..... Xn)}.dt

i!ixill ~-~~xi (O,. " " '0'txi'xi+l" "" 'xn) }.dr O Thus the germ of X. WS 1

~

is indeed a Z-linear combination of the germs of the

,

Further, by taking differentials (giving a CN-linear homomorphism mN + T*Nxo ) we see at once that the dxi's (1.3)

xi's

must be linearly independent (since the

are); and so they do indeed form a basis. Definition Let

f : (N,xo) § (P,yo)

f* : ~ § CN

is defined by

Note that

f*mp c mN .

be a map-germ.

f*(~) = ~ o f .

Then a ring homomorphism

74

Now let us consider a situation which has already arisen in our study of unfoldings: (1.4)

Lemna Let

(u,u0) be a sequence of germs such that -I

_i., i

(V,v0)

~,

(W,w0) a submersion with

is an immersion and

(wO) = Ira(i) . Then the sequence

o § ~*(mw).C v = c v .i*, c U + o is exact and (non-canonically) split. Proof Let

(Wl,.,.,We)

(Vl,...,Vk,W 1

g,

o

be a system of local co-ordinates for

...,We o ~)

be a system of local co-ordinates for

a system exists because

is a submersion). -

mV NOW

i* : % +

%

(W,wO)

.C V

acts as follows:

i*(1) = I , i*(vj) = vj o i , i*(wj o w) . wj o w o i = 0 . Thus it is clear that for

(U,uO) , so, by (1.2), Hence

i*

(v I 9 i,...,VkO i) v l o i ..... v k o i

is a system of local co-ordinates form a basis for

mU .

is onto.

Also, clearly

Ker i* - <w I o ~,...,w e o w>.C v = w*(mw).C v , so that the

sequence is indeed exact. And of course the sequence is split via our choice of co-ordinates.

(1.5)

Now observe that

(Xl,...,xn) ~x---i..... ~

8N

is a free CN-mOdule of rank

is a system of local co-ordlnates for

= dim.N

(if

(N,Xo) , the vector fields

are a basis for it).

Now we define the CN-mOdule

%f , for any map-germ

to be the module of germs of vector fields along

f

f : (N,xO) § (P,y0) ,

(Chat is, germs

75

$ ; (N,Xo) + TP

such that the following diagram commutes; TP ~//~ (N,xo)/-

the CN-mOdule action is defined by free CN-module of rank for

= dlm, P

It is worth noting that

(1.6)

(if

A

is an R-module,

$ : A § B

~,8

all

8N

(1.7)

B

E

R, a,b

r

tf : 8N -> 8f over

an S-module, and $

if

8f

is a

is a system of local co-ordlnates f

along

f

81N .

defined by f*

are a basis for it).

tf(~) = Tf o ~ ; and

defined by

$ : R § S

wf(n) = ~ o f .

a ring homomorphism,

@(ca+ 8b) = $(e).$(a) +$(8).$(5)

then

for

A ),

We may interpret

sense~ elements of

It is clear that

is the same object as

wf : 8p + 8f

is a homomorphlsm over

);

(yl,...,yp)

~-~--~ylf'" "" , % o

There is a %-homomorphism

there is also a homomorphism

( ,Yo )

f

(~'$)x = Sx'$x '

(if

(P,yo) , the vector fields

~rojection

Bf

8f

as 'infinitesimal perturbations v of

are 'tangent vectors' at

specifying as they do for each tangent vector there.

x E N

how

f(x)

f

f : in some

in the space of germs,

is to be deformed by giving a

We should thus expect to he able to derive elements of

ef

from paths in the germ-space, which we do as follows:if

g : (N•

B E TUuo

Oxu0)

§ (p,yo)

is such that

there is a well-defined element

c : (~,O) § (U,Uo)

represents

f(x) = g(X,Uo)

8.g e 8f

B , define

defined as follows:

~,g(x) = d

is irmmedlate (by Taylor's theorem and differentiation

, then for each if

(g(x,c(t)))it= 0 ,

of products)

that

~.g

It is

well-defined. Such parametrized families of germs give rise to unfoldings F : (NxU,x OxuO)

§ (p xU,y O x u O )

of

f

by

F(x,u) - (g(x,u)~u)

it is worth noting that %,g = {iF(O,8 opN ) - wF(O,8o pe)}IN• (where we identify

8Nx U = ep N @ 8p u , 8p• U = 8pp 9 8p~ ).

, and vice versa;

76 Conversely, given an element of

8f , can we construct 'a parametrized family

of germs (and hence an unfolding) realising it?

(I. 8)

Lemma If

# l ' " ' ' # k ( Of , then there exists a germ g : (N•

such

As to this, we have:

that

g(x,O) = f(x)

and

-~ (P,yo)

8--~." g " ~i 1

(i = I ..... k).

Proof Let can write

(yl,o..,yp)

be a system of local co-ordinates for

r

. ( ~~

= j=l ~ r

g : (Nx~k,xoxO)

§ (P)yo)

o f)

(i=l ..... k) , where

r

(P,yo) . ~ ~

"

Then we

Now define

by k

Yi o g(x)u I ..... Uk) = Yi of(x) + Then clearly

(1.9)

g(x,O) = f(x) , and

~-~-." g = ~i i

~ u.~j,i(x) . j=l J

(i=l,...,k)

, as required.

Having introduced soma of the modules with which we will be concerned) as

well as indicating some of their relationship with unfoldings, we proceed to the theorems of module algebra which we will need.

(I.I0)

Lemma Let

such that Let

R

(Nakayama) be a commutative ring with identity) and let

1 + m c R* E

Then if

( R*

is the group of units of

m c R

be an ideal

R ).

be a flnitely-generated R-module. m.E = E , E - O .

Proof Suppose that

E + 0 ) so that, since

minimal spanning set {el,... ,ek} k el = j-l~x.j . ej , where x~ ~ m .

for Thus

E

E

is finitely-generated,

over

R .

Since mE = E ) we can write k x.e. and so, since (I - Xl).e I -j=2Jj

i - xI

is invertible)

there is a

k e I = [ (I - Xl)-ixj.e. j=2 3

P

'

77

So

eI

minimal.

is dependent on

Thus

(i. II)

e2,,..pe k

in contradiction to

{el,.,,,e k}

being

E = 0 ,

Corollary Let

A

SeA

be a finitely-generated R-module (R as in (i.i0)), and let

be a finite subset, Then spans

S

A/mA

generates over

A

over

R

if and only if the image of

S

in

A/mA

R/m .

Proof VOnly if':

If

Thus spans 'If':

S

generates

A

over

R , we have

R.S = A ,

(R/m).S ~ R.(S/mS) = A/mA , so that indeed the image of

A/mA

over

We have

A

in

A/mA

R/m . (R/m).S = A/mA , whence

A/R.S = m.(A/R.S) generates

S

.

over

Thus, by (I.i0)

A = m . A + R.S ,

A/R.S - 0 , or

so that

A ffi R.S .

Thus

S

R .

Remark Our use for (I.I0), (i. II) is precisely that conditions of (I.i0) (the inverse of (l+f)-l(x)

(1.12)

I + f , for

R ffi ~

f 9 mN

, m = mN

satisfy

the

is of course given by

= l/(l+f(x))).

Corollary Let

B

be a sub-module of a finitely-generated ~ - m o d u l e

A , and suppose

that

d i m ~ { A / ~ + I A + B} < r . Then

~ A c B .

Proof The n o n - d e c r e a s i n g

dimlR{A/A+B} consists of hypothesis.

r+2

sequence < ... < d l m ~ { A / ~

A+B}

non-negative integers, and is bounded above by

Hence there exists an integer

dim]R{A/~ A+B}

< ,.. _< d i m ~ { A / ~ + l A + B }

ffi d i m ~ { A / ~ + l A + B }

, whence

r , by

i , O -< i < r , such that 4

A+B

" 4+IA+B

9

78

"

Thus

"

i

(m~ A + B ) / B = (mN+IA+B)/B = mN . (mN A + B ) / B

(m.l~ A + B ) / B

= 0 .

Hence

~

a ~-module, if

then

a E A, ~ ~ ~

A

.

AcB

N

Now let us observe that if

, and s o , by ( 1 , 1 0 ) ,

f : (N,Xo) + (P,yo)

is a map-germ and if

may also be considered as a ~-module

, we define

~. a = f*(#) . a ).

us, in this situation, to find generators for

via

f*

A

is

(that is,

It will often be important for

A as a Cp-module.

As to this, we

have, firstly:

(1.13)

Theorem Let

(Malgrange)

f : (N,x O) § (P,yo)

~-module.

Then

d i m • {A/f* (rap)A}

A

be a map-germ, and let

is finitely-generated

A

as a ~-module

be a finltely-generated if and only if

is finite.

This may look like a theorem of algebra; in fact, however, it is a very deep result in analysis, first proved by Malgrange

(at the instigation of Thom).

Another, slightly easier, proof is due to Mather [MaIV].

We will not give a proof

here.

(1.14)

Corollary Let

subset.

A

be a finitely-generated

Then

A/(f*mp)A

S

spans

generates A/(f*mp)A

A

~-module,

and let

as a ~ - m o d u l e

over

S c A

be a finite

if and only if the image of

m(~/mp)

.

A/(f*mp).A

over

S

in

Proof W0nly if' : clear. 'If' :

If the image of

S

spans

is finite, so by (1.13)

A

is finitely-generated

by (i. Ii) that

(1.15)

S

generates

A

over

~

~

, then

dim]R{A/(f*mp)A}

as a Cp-module.

It then follows

.

Corollary Let

E

be a finitely-generated

generated sub-~-module

(via

f* ).

CN-mOdule , and let

EI c E

be a finitelyT

79

If E =E' f o r some

r > dim]R{E'/mpE'}

then

+ (f*~+~

r+l._ >~

E = E'

Proof The inclusion

Ev c E

induces a vector-space homomorphism

E'/~,E' which is onto by our hypothesis,

§ E/Cf*~ + ~+1)~ so that

di~/(f*mp So, ~ince

E

,

is a finitely-generated

+ ~I)E~

~ r .

CN-mOdule , it follows from (1.12) that

E c (f*.p).E . Thus the hypothesis may now be written E = E ' + (f*~)E So the vector-space homomorphism onto, and thus by (i.13)

E

EV/mpE v ~ E/(f*mp)E

is finitely-generated

But the reqrltten hypothesis also imPlies (1.10),

E/E' = 0 , or

.

E - E' .

induced by inclusion is

as a ~-module.

E/E v = (f*mp).(g/E v) .

So, by

80

w

Infinitesimal

Stabilit Z

In this section we state and prove an algebraic condition equivalent to stability, and derive various consequences.

(2.1)

The condition is:

Definition A map-germ

f : (N,x O) + (P,yo)

is infinitesimally

stable if

ef = tf(SN) + wf(Sp) (for notation see (1,5), (1.6)).

The main result of the section is:

(2.2)

Theorem A map-germ is stable if and only if it is infinitesimally

stable.

Proof 'Only if' : Let

Suppose

# ~ 8f , and let

g(x,O) = f(x)

and

( F , ~ x O,ip x O) be trivial: for

let

- soF Let

f : (N,x O) -~ (P,yo)

~

of r,s

g : (Nx~,Xo•

9 g = ~ f

by

is a stable map-germ. + (P,yo)

(as in (1.8)).

Then

f(x,u) = (glx,u),u)

be retractions to

be such that g

determines an unfolding

, which, since

~ • O, IF x 0

f

is stable, must

respectively such that

. ~ {s(y,u) }In--O ~ 8p . ~ = ~u (r(x,u)}Iu=O c e~, ~ - ~fu

the identity

f o r(x,u) = s(g(x,u),u)

tf(~) = # + wf(~) , so that

with respect to

~ E tf(SN) + wf(Sp)

Since this is true for any

~ ~ 8f ,

f

Differentiating

u , we obtain

.

is infinitesimally

stable.

Before proving the 'If v part of the theorem, it will be convenient for us to derive a consequence of infinitesimal stability relating to vector fields along unfoldings. Let

f : (N,x O) § (P,yo)

: (P',y~) -~ (U,u O) Let (where

d~

be a map-germ,

a submersion such that

8F, ~ - Ker{d~ : 8F + 8 oF} is defined by

d~(~) = T~o# ).

(F,i,j) ~l(uo)

an unfolding of

= Im(j) .

f , and

81

Then there is a ~,-module morpbism as a ~,-module via

i* ) defined by

u : 8F, ~ § 8f

~ § Tj -I o ~ o i

(where

8f

is considered

(this makes sense because

~(i(x))c Ker TwF(i(x) ) - Im(TJf(x)) ). Further, O § (~ o F)*m u . eF, ~ c eF, ~ ~

ef + O

is an exact sequence of %,-modules. (If

~ ~ 8f , then

dj o ~

Tj o ~ o i-I : (Im(i),x~) § TP' is a submanifold of germ

~'

Thus

a

Nv

is a vector field along is a vector field along

and

j o f = Fo i , so that FIIm(i) .

Since

Im(i)

Im(~) c Im Tj = Ker T~ , this may be extended to a

of vector field along

F

with

Im(~') c Ker T~ ; and then

a(~ ~) = ~ .

is onto.

Also,

Ker a = { ~

8F,~Idj-lo ~ o i = O} = oi

= {+~F,~I~oi = (Ker i*)eF, ~ But, by (1.4),

(Ear i * ) % , = (w o F)*mu. % , ; so the sequence is indeed exact).

Thus there is an isomorphism of %,-modules (a)

8f ~ 8F,~/(~ o F)*mu . 8F, W

Similarly (replacing

f

by

IN,I P

respectively), we have isomorphisma

(b) 8N ~ 8N, ~ o F / ( Z o F ) * m u. 8N, ~oF

(where

8N,,~oF = Ker t(~oF)

(c) 8p = 8p,,~/(Z*mu) . ep,,~

(where

ep,

(2.3)

)

= Ker t~ ).

Lemma If

f

is infinitesimally stable, then 8F, = tF(eN,,~oF) + wF(Sp,,~)

Proof It is easy to see that the morphism tF(SN, ~oF)

onto

tf(eN)

and

(F*mp,)eF, ~

~ : eF, ~ § Of

defined above maps

onto (f*mp)ef .

(~ o F)*mu " F*(~*mu) = F*mp, , we can deduce from

(a)

Hence, since

an isomorphism of

~,-modules (*)

8F,~/tF(eN,,woF) + (F*mpv)eF, ~ ~ 8f/tf(8N) + (f*mp)Sf

82

Since

f

is infinitesimally stable,

finite ~ - b a s i s some finite

for the

RHS

of

(*)

, so that a

is given by the projection of

wf(S)

, for

S c 8p .

By (c), isomorphism

S

is the projection of a finite set

(*) ,

wF(S')

must generate this % , - m o d u l e

@F, = tF(SN, ~ F )

S v c Spy ~ ; and then, by the

must project to an ~ - b a s i s

Thus, by (1.14) applied to the ~ , - m o d u l e wF(S')

8f = if(8 N) + wf(@p)

for the

8F, /tF(SN,,~oF)

as a ~ , - m o d u l e .

LHS

of

(*) ,

, the pzojeetion of

Certainly,

then,

+ wF(Sp, ~) .

Now we can give: Proof

of 'If' in (2.2) Let

f : (N,x O) + (P,yo)

any unfolding of

f .

be infinitesimally

To show that

f

stable, and let

(F,i,j)

is stable, we must show that

a trivial unfolding; by (0.6) it will suffice to construct,

for any

vector fields

and

(where

E 9 8N, , q 9 @p,

~ : (P',y~) § (U,u O)

such that

tF(E) = wF(n)

is a submersion such that

be

(F,i,j) T E 8U ,

t~(q) - w~(T)

~-l(u O) = Im(j)

, as

usual). So - let and

~

~ 9 8u , and let

respectively,

be lifts for

T

over

that is, T(~oF)

(Such lifts exist because Then

E 1 ~ 8NV,q I 9 8p~

o E 1 = ~. (~ oF) ,T~ o n I = T o

~ ~ F,~

are submersions).

T~ o nlO F - Tx o dFo E 1 = O , so wF(n I) - tF(E I) ( Ker dr = 8F, ~

Thus, by (2.3), there exist

E 2 9 8Nt,xoF , n 2 9 8p,,~

such that

wF(ql) - tF(~l) = wF(n2) + tF(E 2) 9 Define Then

E = E 1 + E2 , n = n I - n 2 .

tF($) = tF(EI) + t F ( E 2 ) =

wF(~l) - wF(N2) = wF(q)

t~(q) m t~(nl) - t~(q2) = r . ~ = w~(T) Thus

E,q

and

.

are the required vector fields, and the proof is complete.

z oF

is

83

(2.4)'

Let

space

f : (N,x O) § (P,yo)

be a map-germ, and let ~ f

be the JR-vector

8f/tf(8 N) + (f*mp)Of . Let

pf : 8p/mpOp §

be the ]R-homomorphism induced by

wf

(this is

wf(mpep) c (f*mp)Of),

well-defined since

Then: Corollary f

is stable if and only if

pf

is surjective,

Proof 'Only if' : clear. 'If' :

If

pf

is surjective, then

of - if(0 N ) + wf(Oe) + (f*mp) 0f . Then by (1.15) (with

E = Of , E' - tf(SN) + wf(Op) ) we have

8f = tf(SN) + wf(Sp) , so that

f

is infinitesimally stable, and hence, by

(2.2), s t a b l e . We want to go on from this to derive a condition under which a map-germ will possess a stable unfolding if

PF

(F,i,j).

Of course

F

is stable if and only

is surjective; we want to express this in terms of the module

8f .

As

a first step, we have

(2.5)

Lemma There is an ~ - l i n e a r isomorphism

q F , f : ~ f +~/F J

defined as follows: if

~ r Of , let

Then

~ ( OF

be such that

~o i = dj o ~ .

qF,f([~]) " E~] 9

Proof Let

~ : (P',y~) ~ (U,uO)

be a submersion such that

~'l(u O)

Ira(j)

Consider the ~,-module morphism

%F,~/tF(8N, ~oF) § 8F/tF(SN,) induced by the inclusion

8F, ~ c 8F .

f

This morphism i s i n j e c t i v e ,

because

o

84

8F, ~ n tF(SN,) = Ker d~ n Im tF = tF(ker t(~ oF)) = tF(0N, ~ F ) . On the other hand, we have

8F = 8F, ~ + tF(SN, )

of local co-ordinates at co-ordinates at

(for if

(Ul,...,u k)

(U,u O) , then we may choose a system of local

(P',y~) of form (yl~.,.pyp,Ul,~p,..,ukP~)

such that { ~ }

. L

8F .

is a system

spans

J

But d~(wF(~)

- tF(~(uj~oF )) = O

(j

so tha= also

{wF(~S--~)dyi, tF(~(Uj=S ~F))}

spans

8F ).

Thus the CN,-morphism is actually an isomorphism. an isomorphism of Cp,-modules (via

It is, therefore, also

F* ), and so gives rise to an ~-linear

isomorphism 8F,~/tF(eN,,~oF) + (F*mp,)SF, ~ % 8F/tF(eN,) + (F*mp,)8 F 9 Composing the inverse of this isomorphism with the isomorphism in (2.3), we obtain

(*)

a : 8F, ~ § 8f (2.6)

derived

qF,f "

The characterization of the action of the isomorphism

(*)

qF,f

follows from the definition of

of (2.31, this isomorphism being induced by the morphism

defined by

a(~) - dj -10 ~@ i .

We define the R-homomorphism PF,f : TP~y~ §

as the composite

-i Tp ~

, ~ 8p~/rvSpT~Fr~ YO

We have Le=~a

(F,i,j)

is a stable unfolding of

Proof Immediate from (2.4) and (2.5). This

leads

us

to

f

if and only if

0F, f

is surjective.

8S

(2.7)

Definition Let

f : (N,xO) § (P,yo)

Define

Xf = d i m ~ f

We say that (2.8)

f

be a map-germ,

e {O,I,2 .... ,|

.

is of finite singularity type (abbreviated FST) if

Xf < ~

Theorem f

has a stable unfolding if and only if

f

is of FST.

Proof 'Only if' :

If

(F,i,j)

so that certainly 'If' :

If

is a stable unfolding of

f , then

0F, f

is surjective,

Xf < dim PV .

k - Xf < ~ ~ let

g(x,O) = f(x) , and

{ ~

a

g : (N • ~ k ~ Xo • O) § (P~yo)

k " g}i=l

be such that

maps to an ]R-basis in ~ f

(such

g

exists

i

by (1.877. by

Then

g

determines an unfolding

(F,1 N • O,ip • O)

of

f p defined

F(x,u) = (g(x,u),u).

ha~e PF,f(~a----lyo 7 o u i = q~l.[wF(~a-~-Y]

We

~I

Also,

a

a-~7.9 g =

{ t F ( a ~.) - wF(~,~--)}IN.~.• O

1

1

au i {[au~" g]}

(see (1.7)), and so

1

[a_s So, since

@

~U i

a g2--pF,f(~---,yo7 .oui,

spans ~/f , it follows that

1

( F , ~ • O,Ip x O)

is a stable unfolding of

f .

PF,f

is surjective, so that

86

w

Versality

In this section w~ show the equivalence of versality and stability for unfoldings. In fact, the key result of this section is a uniqueness theorem: (3.1)

Theorem Any two stable unfoldings of the same dimension of a map-germ

f

are

isomorphic as unfoldings.

The main step in proving this is establishing a Vcontinuoust version of it, which is the following: (3.2)

Lemma

Suppose that the commutative diagram F

/ ix ii~

(JR ,I)

j • fIR

is a one-parameter family of stable unfoldings of a map-germ a e I Fa

(

I

is the unit interval [0,i]) the unfolding

is defined by

x0

(Fa,i,j)

(~,I)

of

f , where

(that is, there exist retractions

respectively such that the whole diagram

(N ' ,x~)

(N' : < ~ , x ~ •

Fo

~

__~F

(P' ,y~)

(p~ • 2 1 5

(*) i ~ l]R

(N • IR,x O• I) conanutes).

(that is, for each

F(x',a) = (Fa(x'),a) , is stable).

Then the family is trivial over to

f

(JR ,I~

.i • i N

' (p x IR,yo • I)

r,s

87

Proof

We will make the natural identifications eN'x~ ,x~Xa = epN ' 9 e .n..p,. o F

Opwx~, ,y6Xa (where, as usual

epp, @ e # ~

PN' : (N' x IR ,x6 x a) § (N' ,x6)

and similarly for

is the natural projection;

Pp, ).

We claim that to construct

aE

=

r,s

it will he sufficient to find, for each

I , {a = (~a'~u = ~:S~~

F) ( eN'•

,

rla = (na,~-~-uo ~]R ) ~ ep,x]R,y~X a

such that

~a

is a lift for

(p' x R 'Yo x a)

~a

over

Fa

is the germ-restriction

(where of

F a : (N t x ~ , x 6 x a) §

F ) , where

~ ,~ Ker(i x l~,)*OpN '

~a' E Ker(j x l]R)*Bpp @ ( na

is, of course, automatically a lift for

~

over

~

)

To see this, we proceed as follows:let

be flows for

~ba : ((N' x:]R) x :JR, (x6Xa)

x O) § (N' x ~ , x O'xa)

~a : ( (P' xIR) x m ,

x O) -+ (P' x m , Y o X a )

~a,na

representatives

for

x O'

[-~,r

x

[a,a+c]

Hence, since

x

I

0 = a 0 < ... < a

(YoXa)

respectively.

We may in fact suppose (via choice of

F,~a,~ a ) that

~a,~ a

YO' x [a,a+E]

x [-~,r

is compact and connected, such that

respectively,

for some

of

~ > O .

there exists an increasing sequence

Sa.,~a. are germs defined on neighhourhoods of 1 1 x O' x [ai,ai+l] x [ai-ai+l'ai+l-al] ' YO' x [ai,ai+l] x [ai-ai+l,ai+l-ai] respectively. e

= I

are germs defined on neighbourhoods

88

Then

r,s

may be defined by

(r(x',a i +t),O) = ~ao,ao_a I o ~al,al_a 2 o ... = ~a

a -a o ~al,ai~t (x~ ,ai + t) i-l' i~l i

(s(y''ai+t)'O) =

a -a i-l' i-i i

~ ~ao'ao-al

for

~ "'' ~ ao'ao~al

0 < t (P,yo) for

To see this, recall that if

is defined by

~ ~ 8U .

F(x,u) = (g(x,u),u)

Then (defining

g"

from

, then (as in (2.8)I F"

in the same way

90

as

g,g'

were defined from

F,F ~ ) we have

PF",f(80) = [-~'g"] = ~-h(8),g'~ = PF',f(h(~o)) = PF,f (~) and the claim is proved. Now let define

yl,...,y p

ga : ( N •

be a system of local co-ordinates for

O•

§ (P,yo)

(a ~ I)

by

Yi o ga(x,u ) = (I ~ a) 'Yi o g(x,u) + a,y i o g"(x,u) Then, defining clearly have

Fa : (NxU,xo•

§ (p•215

by

P a = (I + = . F ,f a)PF,f aPF",f PFpf

surjective, and hence so also is Now define the map-germ F(x,u,a) = (Fa(x,u),a) iNx U • O,ip• U x O

.

P a " F ,f

(P~yo) , and

Thus

(i=l,.,.,p)

.

Fa(x,u) = (ga(x,u),u) So in particular

Fa

P a F ,f

is stable for each

F : (N•

§ (PxUx~,YO

Then, by (3.2), there exist retractions

is

a ~ I .

• r,s

, we

by to

respectively such that the following diagram co=mutes :

(N• U,Xo• O )

~F

:

(p •

)

rT

Define

rI : (N•215

and define

sI : ( P x U , Y o •

Then clearly unfoldings, (3.3)

§ (NxU,xoXO)

r l(x,u) = r(x,u,l)

by

§ (PxU,Yo•

(rl,s I) : ( F " , ! N X O , I p •

by

,

sl(y,u) = s(y,u,l)

§ (F,INXO,Ip•

.

is an isomorphism of

and so the proof is complete.

Corollary Let

(F,i,j),(F',i',j')

dim.(F',i',j') Then

- dim.(F,i,j)

(F',i',j')

and

be stable unfoldings of

f

such that

= e Z O . (FxIRe,i•

•

are isomorphic unfoldings.

Proof By (3.1), we need only show that of

f .

(F x l~e ,i x O,j x O)

is a stable unfolding

91

Let

g : ~'x~e,x~•

-~ (P',y6)

(g(x',t),t) = ( F •

Then

be defined by

.

PF•

,F(~-t. I

) ffi [- 8-~ ~

1

" 0

(where

tI . . . . .

te

are the usual

1

YO xO co-ordinates

in

~e)

: (P'x~e,y6xO)

so that

PFxll~e ,F " PF o T~ , _ where YoXU '

§ (P ' ,yo) ,

is the natural projection.

Thus PFxlRe,f Hence, since (Fxl~e,ix0,j

ffi (qF,f)-I~

PF,f ' T~yo•

x0)

e ,F

are surjective,

= PF,f o T#YoX O . so is

is indeed a stable unfolding of

PFxl]R e,f

; and thus

f .

Now we can prove

(3.4)

Theorem An unfolding of

f

is versal if and only if it is stable.

Proof 'If' :

Suppose that

(F,i,j)

any other unfolding of since

X F, ffi Xf

(F",I,J)

.

(via

Then

Then, if

f . qF, f) ,

is a stable unfolding of Since

f

has a stable unfolding,

XF , < |

(F",loi',Joj')

f ; let

and so

F'

Xf < ~

be So,

has a stable unfolding,

is a stable unfolding of

e = Idim.(F",loi',Joj')

(FV,iV,j v)

- dim (F,i,j)I

say

f .

, we have, hy (3.3),

either an isomorphism (~,~) : ( F x l l ~ e , i x O , j

x0 ) ~ ( F " , l o i ' , j o j ' )

or an isomorphism (~,~) In the first case, (where

: (F,i,j) = ( F " X l ] R e , l o i ' x O , J o j ' x O )

((~oF) o ~ o l , # o ~ o J )

~ : (p' x R e ,y~ x O) + (p',y~)

second case, So

see that

is a morphism

(F'.i',j')§ (F,i,j).

is versal.

Suppose

(F,i,j)

(F',i',j') + (F,i,j)

is the natural projectlon); while in the

(~ o (IN," xO) o I , ~o (ip,,, xO) o j)

(F,i,j)

'Only if' :

is a morphism

.

(F,i,j)

is a versal unfolding of

f .

The nearest way to

is then a stable unfolding is via the naturality of

PF,f '

92

which we make explicit (3.5)

as follows

:

Sublemma Let x~)

(N

F

~,Xo)

+

f~. (p,yo)

*

-

be a m o r p h i s m Then

:

(r

~

(F,i,j) § (F',i',j')

PF,f " 9F',f

(p,y~)

(F",y~)

of unfoldings

of

f .

OT~y0

Proof Define an ~ - h o m o m o r p h i s m if

A' e 8 F,

A ~ e F ~ let

K([A])

= [A']

-I = qF,f

and let

A' ~ eF,

qF,,f[a]

= [A']

(for if

"

But

.

V r 0p,

as follows

§

A' o i' = Tr o A o

A' oi'

A ~ 8F

ffi V' o , o F o i

K ( [ V o F]) ffi [V' o F']

i .

Then define

be such that

- Tj' = a .

A' o i' = Tj' o a - T r , and let

:

for it is not hard to see that

a ( 8f , let

be such that

(V' oF') oi' so that

be such that

This well-defined,

.

-I qF',f o K

Now let

K :~F

V' ~ 0p,,

Then

oa=

ffi V' o j' o f

qF,f[a]

T~oAo

be such that

A o i = Tj o a - [A] ,

i , so

K([A])

V' o j ffi T r

ffi T ~ b o V o j o f

- T~o

,

=[A']).

j .

(VoF) oi

Then

,

, and thus

Now we return to the proof of :'Only if ~ in (3.4). Suppose not stable,

that

(F,i,j)

so that

PF,f

[el ~ Im(PF, f) , and let and ~uu" g, = a

(such

(F',iNXO,ipXO)

of

is a versal unfolding is not surjective.

g' : ( N x ~ , x o •

g' by

there exists a morphism

(r

F'(x,u)

f , and suppose that

Then let -~ (P,yo)

exists by (1.8)).

f

of

is

be such that

be such that

g'(x,O) f f ( x )

Then define an unfolding

= (g'(x,u),u)

: (F',INXO,ipxO)

a r

F

.

Since

-> (F,i,j)

(F,i,j)

is versal,

, and, by (3.5), we have

93

PF',f = PF,f =T~yOxO ' But

~F,,~(~I

) = E-

~. ~

~ [-~

YO xO Thus we have arrived at a contradiction;

so

, ~hi~h is not contained in

F

must be stable,

Im(PF, f)

94

w

Con=act-equivalence

Definition (4.1) Map-germs

fi : (Ni'xi) § (Pi'Yi)

germs of diffeomorphism

(i = 0,I)

are equivalent if there exist

h : (No,Xo) § (Nl,Xl) , k : (Po,Yo) § (Pl,Yl)

such that

fl o h = k o fo " Definition (4.2)

a)

A pair

(h,H)

consisting of germs of diffeomorphism

h : (No,X O) § (Nl,Xl) , H : (NO •

+ (N I x P l , x l x y l )

such that the following diagram commutes: INo• (No,x O) ~ - -

h

PN O ~ (NoXP0,xO•

~ (No,X O)

IN; Yl

(N l,x I)

PN1 ~ (N I • 2 1 5

)

(Nl,x I)

is a contact eRuivalence.

If

NO = NI'Po

=

P1 '

(h,H)

is a~-equivalence; i f also

h

=

~0

,

(h,H)

is a~ -equivalence. b)

Map-germs

f'l : (Ni'xi) ~ (Pi,Yi)

(respectively~-equivalent, (respectively~-equivalence,

(i = 0,I)

~-equivalent) C-equivalence)

are contact-equivalent

if there exists a contact-equivalence (h,H)

such that

(l,fl) o h = Ho (l,fo) . We have, of course, already used the notion of equivalence extensively - in particular, in w map-germ

f

that of

f .

we said heuristically that a versal parametrized unfolding of a

'contains' representatives of all 'nearby' equivalence classes to

Now, as we pointed out in w to a parametrized unfolding. w

any unfolding of

f

gives rise (non-uniquely)

In particular, then, a stable (equlvalentlyp from

versal) unfolding does so, but of course the parametrlzed unfolding so obtained

need not be versal in the category of parametrized unfoldings.

@5

However, what is true is that such a parametrized unfolding 'contains' representatives of all 'nearby' contact-equivalence classes to that of

f

- more

precisely, we should set up a category of parametrized unfoldings and Vcontactmorphisms' and show that in this category an unfolding is versal if and only if it is stable as a map-germ. That this is true seems as non-obvious as the equivalence of stability and versality for (non-parametrized) unfoldings, and it is equally difficult to offer any suggestion as to why it should be true other than a detailed proof of the type of those given in w 'contact-versality'

We shall not give such a proof here, since the notion of in this form is not very useful for the purposes of this book.

Instead, we shall obtain, in w parametrized unfolding of

f

a different version of the 'fact' that a

which is stable as a map-germ 'contains w all 'nearby'

contact-equivalence classes to that of

f , via a transversality theorem.

Instead, let us use the discussion above as motivation:

the close

relationship outlined here between equivalence and contact-equivalence in the presence of stability suggests the following : Theorem

(4.3)

Let where

FO,F I

fO fl The

be stable unfoldings of the same dimension of germs

fO,fl ,

are contact-equivalent.

FO,F I

are equivalent.

which has the important consequence~ Corollary (4.4) If

fO,fl

are contact-equivalent stable map-germs, then

fO fl

are

equivalent. Proof fO,fl

are zero-dimensional stable unfoldings of themselves; so the result

follows at once from (4.3). In order to prove the theorem, we will need to relate contact-equivalence to the algebra of vector fields which we have been using to study unfoldings.

The

96

connecting link is provided by " Lemm~

(4.5) f,f~ : (N~x-O) § (P,y0)

Map-germs

are ~-equivalent

if and only if

f*mp.C N = f'*mp.C N . Proof '0nly if' : (IN,H)

Suppose

such that Now

f,fv

are~-equivalent,

so that there exists a~-equivalence

(l,f') ~ IN - Ho (l,f) o

H(x,y 0) = (x,Y0)

for all

x e (N,x0)

(from the diagram of (4.2) a)),

H*(p~mp, ~ x e ) c p~mp~ CNx P .

so that But

H

is a diffeomorphism,

H*(p~mp, CNx P) = p~mp, % x p

so

H*

is an isomorphism,

and thus

"

So f'*mp.C N - (1 ,f ' )* (p~mp . CNxP) .CN = (1 , f ) * (H*p~mp . CNxP) ) .CN = (l,f)*(p~, Vlf':

Suppose Let

f*mp.C N - f V * m p . ~

Xlp...,x n

N

= f*mp.C N .

,

be a system of local co-ordinates at

system of I bcal co-ordinates at

(I

CN I

Suppnse

a I ..... a n

co-ordinates at

Thus, by (4.5),

(h-l) *

(l,fo) , so that provides the required

. ~ : CNO § C N L

is an isomorphism carrying

be local co-ordinates at

(Nl,X I) .

(N0,x0) , blp...~b n

Then we may write

~-l(b i) =

n ~ h..a. j~! ~J J

( i = I ..... n) .

fo*mp.CNo

local

to

98

Since

~

(and #-I)

are isomorphlsms, the matrix

invertible for

b = x0

and hence for

diffeomorphism

h : (No,Xo) + (Nl,Xl)

b

near

H b = [hij (b)]

is

x 0 , and so we may define a

by

n

b i ~ h = j~lhij.a.j Then

~-I = h* , so that

Thus by (4.5) So

fo,fl

fl 9 h,fo

(fl

(i = l,...,n)

9 h)*mpCNo

= f~mp.CNo ,

are~-equivalent.

are indeed contact-equivalent,

This leads to the geometrical result| (4.7)

Lemma Let

(Fk,ik,Jk)

(k= 0,I)

be unfoldings of the same dimension of map-germs

fk " Then

Fo,F I

are contact-equivalent

germs if and only if

fo,fl

are,

Proof First note that, via composing with a germ of diffeomorphism

' ' + (P{'Jl (~l),y~) , we may assu~e k : (P~,Jo(P#ty 6) (= P' say),

Po

m

~i (= F say), r~ = F 1 !

=

Jo = Jl (= j say).

Now let us observe that, in general, if

(F,i,j)

is an unfolding of

f ,

then

i*(F*~p,.%,) = (i*F*~,).%

(since i*%, =

= (f*j*mp,).C N = f*mp.C N

(since

%)

j*mp, = mp) .

Now for the proof : 'If' :

Let

SO certainly Let

h

(h,H)

be a contact-equivalence

(fl o h)*mp.CNo

=

(l,fl) o h = Ho (l,fo) ;

f~mp.CN 1

be a germ of diffeomorphism

o iO = i I o h .

such that

(NO',x0') § (Nl,Xl); '

such that

Then

i~E (FlO ~)*mp, .%~I = (fl o h)*mp.CNo

But Ker i~ c F ~ , .C~6

(for

m

f~mp.CNo

=

i~[F~mp; .CN~3 .

Ker i~ ~ F~(Ker j~)), and

so

99

h)*mp,.CN~= F~mp,.CN~

(FI" Thus

Fo,F 1 o h

'Only if' :

are~-equivalentp

Suppose

isomorphism

Fo,F I

: CNo § CNI

and so

.

Fo,F 1

are c o n t a c t - e q u i v a l e n t .

are contact-equivalent; so that there is an carrying

F~mp,.CN~

onto

F~mp,.eNl .

Let us first prove the result in the case where rank

(dfo)xo (= rank

(dfl)Xl) = O . In this case, Let

Ul,...,u k

9

2

Ker 15 - mN~

=

2

F~mp~,CN~ ~ mN~ ,

be local co-ordinates in N~

such that

Zm i 0 =

and such that they extend to a system of local co-ordinates at

w

{u l = . . .

=Uk=O}

u I ..... Uk,X l .... ,xn

,

(No,xO) 9 We may write (non-uniquely) 9

#(u i) = v i + a i , where

2

V i E Ker i~ , a i s F~mp,.CN~ n mNl. Then define a homomorphism

~v : CN~ + CN~

2'(ui

(i = I, ....k)

= vi

'(~j)

by

~(xj)

(j = i ..... n)

.

is the same as for

This is an isomorphism (for the induced map

0 "'0 "'I "'I into, hence onto, F~m~.CN! (for

) and it carries F~mp,.eN~ F~mp,CN{ ).

Moreover, it carries

Ker i~

onto

Ker i~ ; and thus induces an

isomorphism i~ (F~mp ,CN~) -+ i~ (F~mp,CNI) i.e. an isomorphism Thus

fo,fl

f~mp.CNo

f~mp.CNl

are contact-equivalent.

Now let us consider the case where rank let

(Po,Yo)

(dfo)xo

be a submanlfold-germ of

such that

fo

70 : (No,Xo) + (Po,Yo)

is transverse to

O

at

xO ; also,

~0 : NO + NO'30 : PO § P

PO "

he the restriction of N

rank

(P,yo)

(dfo)xo (= rank

(df l)xl) > 0 :

of codimension equal to rank Let fo "

NO = fol(Po) ; and let Clearly, then,

TO

is of

N

(fo,io,Jo)

is an unfolding of 70

are the inclusions), as also is

(where (Fo,io o io,Jo o ~O ) .

100

Similarly, we can construct fl

and

fl : (Nl'Xl) § (PI'Yl)

which is unfolded by

FI .

Then, since analysis,

Fo,F I

are contact-equivalent, so are

Hence, by the Vlf" part of the lenmm,

fo,fl , by the foregoing are contact-equlvalent.

fo,fl

Now, to connect with our algebra of vector fields, we have:

(4.s)

L~,,ma Let

(h,H)

be a~-equivalence between map-germs

e(h,H ) : 8f -~ 8f~ dHo (O,~) o h -I Then

f,f~ ; and let

be the CN-mOdule isomorphism defined by (9 ~ 8f)

e(h,H )

(where we identify

(O,e(h,H)~) =

8(l,f ) - 8N 9 8f) .

induces an ~ - v e c t o r space isomorphism E(h,H ) : ~ f §

(where ~/f = 0fTtf(S N) + f*mp.0f ) . Proof We may write H' = (h- l • r

e(h,H )

as the composite

e(h,h•

) o e(IN,H,) , where

o H , and so it is sufficient to show that each of

~(h,hx~)

is well-defined. First, then, the case We have Thus

H = h • Ip .

f' = f o h "I , e(h,h• ) _

e(h,hXlp)(f*mp,Sf) = (h

Also, if

= wh "I . )*f*mp.Sf, = fV*mp. Sf, .

~ c ON , then wh'l(tf(~)) = wh " I o T f ' o dho ~ = Tf' o dho ~ o h -I = tf'(Tho ~ o h "l) .

Thus

e(h,hXlp)(tf(SN)) = tf'(SN) . So

E (h,h•

is indeed well-deflned.

Now let us consider the case By (4.5),

h = 1N .

f*mp.C N = f'*mp.C N , so that, since

restriction of the homomorphlsm

dH : 8(l,f ) § S(l,f,)

e(1,H ) (f*mp.ef) = f'*mp.ef,

e(1,H) over

is just a H* , we have

'

101

Also, if

~ 9 8N , then

(O,e(l,H)(tf(6))

= dH(O,Tf o 6) = THo T(l,f)o 6 - THo (6,0).0 (l,f) = T(l,f')

Now

o 6 - T}{o (6,0 ) o (l,f)

THo (6,0) o (l,f) = (~,~) , where

H(x,y O) = (X,Yo)

for all

Thus we have

.

~ 9 f'*mp.Sf.,

, because

x ( (N,xO) .

e(l,H)(tf(6))

= tf'(~) + ~ , and so we have shown

e(l,H)(tf(SN) ) c tf'(SN) + f'*mp. Sf . Thus, also,

e(l,H)

is well-defined,

We can now give Proof of Theorem (4.3) First of all, let us observe that it is sufficient to consider the case rk(df O

~

germs

ffi rk(df

) = O (for as in Le.~na (4.7), fO,fl are unfoldings of rank I f~O,~l which, by that le~mm are contact-equivalent since fO fl are~

and of course

FO,F I

are stable unfoldings of

~O,~I ),

Next, let us remark that it is sufficient to consider the case when are~-equivalent, all

y 9

(for if

with the~-equivalence

between them such that

fO,fl

H(xo~Y ) = y

for

(P,yo)

(l,f I) oh = Ho (l,f O) , then

(I,

(HxO).l

x o fl o h) = (h-l,h -I• (H O)'l)oHo(l,f) xO

(where

H xO : (Po,Yo) -> (Pl,Yl)

replace

fl

is defined by

by the equivalent germ

and also replace

FI

(y) = H(xo,Y))

, so that we can

(HXo) -I o fl o h , which is ~-equivalent

to

fO ,

~-I k o F I = h , where

by an equivalent germ

: (No,Xo) ' I) ' , k : (Po,Yo) ' ' § (Pl,Yl) ' ' § (Nl,X ' ,

that

H

are germs of diffeomorphism such

ho i0 = i I o h , k o J0 = Jl = HxO ) " Let us also observe that, by Theorem (3.1), it will be enough to show that

any stable unfolding of

f0

(of the relevant dimension,

any stable unfolding of

fl

of dimension

Now, in general, if (pF,f)-l~f (For

q 9 8F

(F,i,j)

k

say) is equivalent to

k .

is any unfolding of a map-germ

f , we have

= Im TFx~ is such that

qy6 9 Im TFx~$r--~there exists

6 9 8N,

such that

102

wF(q) - tF(~) 9 mN,.OF~=~DF(qy~)

9 mN,~F

~===>pF,f(ny~) = (qF,f)-ipF(DYo) 9 Also, we have d f Sop since

F

= mN~ f + [wf(Te)] , and

is a stable unfolding

surjective (by (2.6)), it follows that of

F

of

mN.)~F

).

[wf(TP)] = pF,f(Im(Tjyo) ) .

f

if and only if

PF,f

is

is stable if and only if the restriction

PF,f PF,f ; Im TFx~ + m N f

is surjective. Now let us consider our ~equivalent 0

at

fO,fl , which are of rank

map-germs

x0 . Let

U

be a neighbourhood of

be a map-germ

s.t.

gO(x,O) = fO(x)

0

in

~k

, and let gO ; (NxU,x O x O ) +

and such that

{[~.o gO~}

span

(P,yo)

m N ~f

1

(Such fO

gO

exists by (1.9); it is possible to span

does have a stable unfolding of dimension

Then the germ

F 0 : (NxU,x0xO)

is a stable unfolding of Now let (where

fO

-> (PxU,YoXO)

~ 'g i e(l,H)(-~-" gO) = ~-~-. 1

F I : (NxU,xoXO)

§ (PxU,YoXO)

stable unfolding of germs.

fO

fl .

k

elements~ since

(and hence rank

k

x 0f

at

).

FO(x,u) = (gO (x,u) ,u)

(9

~

o

PFo,fo ~Uily xO ) = -[~--~i" g ] )"

be defined by

is the C-equivalence between

gl(x,O) = fl(x) , and

with

defined by

(for, as in (2.8),

gl : (NXU,x OxO) + (P,yo)

(IN,H)

k

~f

and

(x,g (x,u))

H(x,g (x,u))

fl ) ; then clearly Thus, if we define

1

by

FI(x,u) = (gl(x,u),u)

Thus it remains to show that

, then

FO,F I

FI

is a

are equivalent

This we will do by showing that they are part of an (~,I)-family of

stable germs trivial over Let

{Xl, .... x n}

{Yl .... 'YP} Define

(IR,I) .

be a system of local co-ordinates at

be a system of local co-ordinates at F a : (NxU,xoxO)

-> (PxU,Yo•

(P,yo) . (a E (1%,1))

I Yi ~ Fa = (l-a).Yi o F 0 + a.y i o F I u i o F a = u.l

(N,xO) , and let

by

103

so

that

Fa

is an unfolding of Yi

Indeed

9

fa : (N,Xo) § (p,yo)

=

fa

a fO

(l-a) "Yi

(l,f a) = H ao (l,f O) , where

defined by ~

+ a.Yi

Ha : ( N x P , x o X Y o )

fl

'

+ (Nxp,x OxyO)

is defined

by X. oHa = X. 1 1 Yi~ H a = (l-a)y io INx P + ay i , H so

that

(IN,Ha)

Of course

is a~-equivalenee. Fa

is of the form

Fa(x,u) = (ga(x,u),u) , where

a

0

yiog and clearly we have

= (l-a)Yiog

i

+ a.Yiog

@ B a e(l,H a)(8-~." gO) = 8~. " g 1

1

Hence

PFa,fa(.~_il~

YoXO

) = _[ Bu.' 3..~_ ga] = _[e(1,Ha )(~u3. . gO)] z

z

= E(l,Ha)~FO,fO(8~ilYoX O) " Thus

~Fa,f a

Now let

is surjective, so that

G : (NxU•

G(x,u,a) = (Fa(x,u),a) .

OxOxl)

Fa

§ (PxU•

We aim to show that

is, that there exist retractions

r,s

is a stable unfolding of

to

•215 G

fa .

be defined by

is trivial over

iNx U • O , Ipx U x O

(]l,l) ; that

respectively such

that the diagram F0 (NxU,xoXO)

--

~ (PxU,YoXO)

rl

Ts G

(NxUx ~ , x 0 x

~ ~ (PxUx~,Yo

xOxl)

F,

(~ ,I) commutes. germs

rI

If we show this, then of

(N•

OxO)

, sI

sl(y,u) = s(y,u,l) , so that

F I = s I o F O o (rl) -I , where the diffeomorphismof

FO,F I

(PxU,y O•

are given by

will indeed be eqdivalent.

As in (3.2), to construct the desired retractions to

find, for each

a E (~ ,I)

'

rl(x,u) =r(x,u,l),

elements

r,s

it will he sufficient

~a' ~ .* ,n 8" * " ~NxlT'NxU rNxU , ~'a s PpxUmNxU 8pp• U

104 such that

~a = (~" ~ = ~ ,

(Here

G a)

Ga : (N•215

restriction of

is a lift for

O•215

over

~a = (qa'~u , 2 =~

§ (P•215

•215

Ca

is the germ-

G ; and we have identified

eNxUxB=

e

9 e

PNxU

Ga

~o

,

ePxUx~=

e

PpxU

e ev

),

To do this, we proceed as follows: Ca = tGa(0'~" ~R~ Ga) " WGa(0'~~ ~

let

) "

P

Then

~a = i~IP~• Now

Fa

Ha x l U x l U )

~ FI - Yi ~ F0)'wGa(~-~-o~yiPP•

is ~-equivalent to

F0

O) .

(the~-equivalence being (essentially)

, so that = F0*mpxu.CNxu

Fa*mp• Thus

0

Ga*P~Ump•

= P ~ x U F *mpxu'CN•215

Yi o F I s FI*-

Now

C

~PxU" NxU

= F 0.-

C

mpxu" NxU "

So

* u(Yi ~ F I - Yi ~ FO) ~ P~ PN~

UFO*~PxU" CNxUx~ = Ga*P~ x l~p•215215

and thus Ca ~ Ga*P~ x umpxu'SGa,~ ' Now, by (2.3), we have eGa,~ ~

= tGa(eNxUx~,~

oG a) + WGa(epxuxl~, ~

)

so that Ga*P~,~ ump•

m c tGa(P~ •215

~R~ Ga ) + WGa(P~ x Dmpxu'Sp•

W]R )

Hence, via the natural identifications OPNxu we can f i n d

~1~ ~ Ga

PP•

~av E p ~ x UmNxU. ePNx U , qa, ~ p~•

such that So, since

ONxU•

U

Ca = tGa(~"O) - WGa(na,O) . Ca = tGa(O'~u~ w~OGa) - wG a ( O , ~ o ~R) , we have

wGaCnl,~o~> ~a = (~a'~u ~ ~ )

so that over

a ' !~u ~a ~ ~t~' o

~

o

%)

tG a(~a,~o w~1~

is indeed a lift ~or

G a , as required, and the proof is complete.

=

105

w

Determinacy

We now turn our attention to the question of whether a map germ is equivalent (or contact-equivalent) to a polynomial. (5.1)

Definition Map-germs

f,f! : (N,Xo) ~ (P,yo)

are r-jet equivalent if

r+l

(f* - f * ' l m p c m N

This is clearly an equivalence relation; we call the equivalence class of j rf .

A more usual definition of r-jet, in terms of partial derivatives, is equivalent to that given above, as is shown in the following: (5.2)

Le~mm Let Then

f,f' : (N,xO) + (P,y0) jrf ffi jrf,

be map-germs.

if and only if the partial derivatives of

f

and

f'

(with

respect to some, and hence any, systems of local co-ordinates) agree for all orders _< r .

To prove this, we will require, first

(5.3)

Sublen~a Let

~ E CN,xo .

Then

~ ~ ~+I

if and only if all partial derivatives of

# , with respect to any system of local co-ordinates for

(N,xO) , of orders

~ r

vanish. Proof Let If

Xl,...,x n

be a system of local co-ordinates for

(N,xO) ,

r = 0 , the result holds (by definition!).

Otherwise let us assume inductively that the result holds for jets of order < r ~ If

r+l # ~ mN , then

generate

m N , by (1.2)).

I ~i'xi , with i=l

~i E ~

(since

Xl, .... x n

By our inductive hypothesis, then, the partial

#i

vanish for all orders

xi.#i , and so

, vanish for orders

derivatives of of

n

=

-< r - I -< r

, so that the partial derivatives (differentiation of products').

106 Conversely, let

if the partial derivatives of

be the function defined by

~i

~i (x) =

~ =

~(0,... ,Xi+l,... ,x ) n

~ ~i.xi . i=l

r ~i e m N ;

~i

Jl ~~x (0,.

"''txi'xi+l"' "

Now

_< r ,

,,Xn).dt

"

xi.~i(x) = ~(0, .... x i .... ,Xn) -

so that, since the partial derivatices of

vanish, so do those of derivatives of

vanish for all orders

0

n

Then (as in (1.2))

#

xi.~i(x)

.

~

for orders

Then it is easy to see that the partial

must vanish for orders

_< r - I ; so by our inductive hypothesis

r+l ~ E mN

whence

So the inductive step, and hence the proof, is complete.

Now we have Proof of (5.2) All partial derivatives of same is true for

vo f,vo f'

f,f'

for any

of orders

~ r

agree if and only if the

v E mp ; which, by (5.1), is true if and

r+l vo f - v o f, = (f* - f,*)(v) ~ m N

only if

for any

v ~ mp .

We can derive further consequences of jet-equivalence as follows: yl,...,y P

8f

and

(5.4)

(P, yo) .

be a system of local co-ordlnates at

wf(~),...,wf(~y~) 8fv

is a free basis for

8f

let

Then

as a CN-mOdule.

We will identify

wf(~y~) ~-+ wf'(~y~.) .

in the following l~r,~ by

Lemma Let

f,fv : (N,Xo) + (p,yo)

be such that

jrf = jrf, .

Then

a)

tf'(ON) c tf(SN) + ~ S f

f'*mpOf,

b)

r§

c (f*mp + m N

)Of .

Proof

a)

s

x I,.. 9 ,xn

Then, since

be a system of local co-ordinates at

8Xl"'''~-~n

(N,x O) .

is a free basis for the CN-mOdule

n

write

Thus

= i[=l i'~x i tf(

(~i E C N) )

-

n1 % t i-i

< r

for any

f(~

~ ( ON 9

" I 1

1,3

~x~(f* (yj)).wf (~y~) i

8 N , we can

107

and so Since

tf'(~) = tf(~) + z,j'~'~i~x~-~.(fV*(yj ) I f,f'

have the same r-jet, all partial derivatives of

orders

~ r

orders

s r-I

vanish, and so all partial derivatives of

So, by ( 5 . 1 ) , b)

Since

- f*(yj)).wf(~y~.) . fV*(yj) - f*(yj)

~x~-~.(f'*(yj) - f*(yj)) 1

of of

vanish,

~--~-(f*(yj) 1 f,f'

- f'*(yj))

E ml~ , and thus

tf'(~)

- tf(~)

~ m~T8f .

r+l f'*mp c f*mp + m N , and the result

have the same r-jet,

follows at once.

45.5)

Definition A map-germ

f : (N,Xo) § (P,yo)

isF~-determined by its r-~et (respectively

determined by its r-jet) if every map-germ jrf = jrf,

f' : (N,Xo) + (P,yo)

is~Mu-equlvalent (respectively equivalent) to

Thus, if

f

such that

f .

is determined by its r-jet, it is equivalent to a rather

particular polynomial - its Taylor series (with respect to any local co-ordinates) truncated after order

r .

However, determinacy is rather hard to calculate; but

for stable map-germs can clearly be arrived at via~i~-determinacy (using (4.4)), for which we now give a sufficient condition: (5.6)

Theorem Let

f : (N,xO) § (P,yo)

be a map-germ such that

~ Sf r Then

f

tf(6 N) + f*mpSf .

is~-determined by its (r+l)-jet.

Proof Let

fv : (N,xO) § (P,yo)

Let

(yl,...,yp)

define

be a map-germ such that

be a system of local co-ordinates at

fa : (N,Xo) + (p,yo)

y : (N•

g : (Nx~,XoXl) Xo•

+ fN•

(P,yo) .

by

Yi~ fa = (l-a).y io f + a,y io f' Then define

jr+If = jr+If,

+ (P,yo) Xo•215

by by

(i=l .... ,p)

g(x,a) = fa(x) , and y(x,a) = (x,g(x,a),a) .

For

a ~ ~,

108

We aim to find retractions

the

r,$

1NX O,INx Px

to

0

respectively such that

following diagram commutes:

(N,xo)

(1,f)

,l

(Nx~,XoXI)..~

Y

,

(Nx P'Xo Y O ) sx ]

and

Let

(rl,s I)

between

ga : (N•215

"

O) , defined by

f

and

gp7

(for

rl(x) = r(x,l)

sl(xpy) = S(x,y,l) , give a

fv , and the proof will be complete,

"~ (P'Yo) 'Ta ~ ( N • 2 1 5

be the germ-restrictions of

(~ixO)

) (Nx~,XoXT)

r I ; (Npx O) § (NpXo) , defined by

sl : ( N x P , X o X Y O) § ( N x P , x o •

~-equivalence

PN• ....

) (Nxpx~,XoXYoXl)

'~ (1% 1) If we can do so, then

~

-> ( N • 2 1 5 2 1 5

a ~ I ) .

We shall identify, in the natural way, 0

-O

Ya

@O

PN

8NxPx]~.,X o X Y o X a

@0

ga

=

~

OpN @ OPP 8 0~]R

ONx~, XoXa - Op~l @ Or~ (where

pN,p P

(N,Xo),(P,y O)

(N •

are the natural projections of respectively, and

p~

(N x p x R, x 0 x YO x a)

to

is the natural projection

XoXa) § (N,Xo)). By similar arguments to those of (3.2), for the construction of

possible it will be sufficient to find, for each ' ~ mNmpSpe =

PN•

such that over

~a

to he

a r I , elements

(E~,~8~o ~ )

is a lift for

Ya

With respect to the system of local co-ordinates P

(yl,...,yp)

+a = i=ll (#a).Wgla (~-~'~i'--)oy (~a)i c CNxR, XOXa and it is easy to calculate that

r,s

, we can write

109

(@a) i = Y i d fv . Y i ~

r+2 c Nx~t

~ mN

~a E _r+2 mN 8ga "

so that Now We claim that

~ mNrega ~ tga(SNx~' %_.

+ g*mp'Sga .

To see this, we argue as follows : from the hypothesis of the theorem, we have ~ S f c tf(SN) + f*mp.Sf . Since

jr+ira = jr+If ,

(*)

it follows, by applying (5.4), that mNr8 fa c tfa(SN ) + (fa*mp + mNr+l)Sfa

Now recall (from (2.2)) that there are natural isomorphisms 8ga/m~ , aSg a ~ 8fa (ga gives rise in the obvious way to an unfolding identify

8ga with

8F

w ' where

~

Fa

of

is the projection

(px~,YoXa)

a,

0NxIR' ~ / m R ,

aeNxi~., ~ . ~

fa , and we can § (~,a))

ON .

So from (*) we can deduce mNr 8ga c tga(SNx~, w~) + g*amp.Sg a +ranr+l 8ga ' Thus it follows from (I.i0) that the CN-mOdule (via { ~ 8 ga + tga(eN•

' w~) + g*~.8 ga }/{tga(eNx~, ,~) + ga*mp.Sga} = O '

and the claim follows. It certainly follows, then, that

~a ~ tga(mNONxP., ~ R ) + mN~ so that there exist such that

~'a ~ mNSNx~, ~ , , n"a ~ mN'g*mp'eg a ~a = tga(~av) + ~'' " a

We will, as usual, identify

~.'~ mNeNxm, ~,~

a

8Nx~, ~

with (~',o> ~ ep~ |

%~.

with

8p~ , and so identify

110 7a

Now

qa' o Ya = q''a "

such that

na

1lit for

is an immersion, so, since

(Ea

=

ga = PP o Ya ' there exists

It remains to show, then, that

, 8 ~ P N x R 'qa'~u = wl~ ) over

n~ c mNmp8pp

~a = (~a'~u ~

7a :

We have TU~

=

~ PNx

tga( O

'

= ( C o PN~m'

tga(C)

~a' ~ ~

+

= (~a" PNxl~' qa '~u

~m )

~m ) = na =7a

and so the proof is complete,

(5.7)

Corollary If

f

is of

FST , then

f

is ~-determined by its

(Xf + l)~jet.

Proof By hypothesis,

~ Sf

Xf = dim~{ef/tf(SN)

+ f*mp. Sf} = r < ~ .

c tf(ON) + f*mp.8f , and so, by (5.6),

(5.8)

f

Thus, by (i.12),

is~-determined by

jr+If .

Corollary If

f

is determined by its

(p + 1)-jet

is stable, then

f

is stable, then

pf : TPy 0 + 8f/tf(eN) + f*mp,0f

(where

p = dim. P ). Proof If that

f

dim~{Sf/tf(SN) Now let

f'

Thus, by (1.12),

+ f*mp. Sf} ~ p .

jp+If, = jp+If .

be such that

is surjective, so

~Sf

c tf(8N) +f*mp.8f .

Then, by (5.4), we can conclude

that tf'(SN) + f'*mp.ef, ~ tf(O N) + f*mp.8f so that A/f,~, But then

can be identified. pf,pf,

can be identified (since these are induced respectively by

wf,wf' , and our identifications are given by

wf(~--~.) +-+ wf'(~-~) ), so that vJ i.

is also surjective, and hence

f'

But, by (5.6), it follows that (4.4),

f'

~

is stable.

is actually equivalent to

f'

is~-equivalent to

f .

f , so that, by

P f,

111

w

Jet-spaces and a transver~ali~y

theorem

In this section we will obtain the 'contact-transversality'

theorem

advertised in w First of all, however, we will require some material on jet-spaces. (6.1) germs

The set

jr(N,P)

(for any integer

(N,x) + (P,y) , for any

x E N , y ~ P.

There is an obvious projection function assigning for

(x,y)

r ~ I) consists of all r-jets of

pr,O : jr(N,p ) § N • P

to any map-germ

(N,x) ~ (P,y) .

induced by the We write

jr(N,P)xo

(pr'O)-l{xo} x {pO} , and

jr(N,P) x PY (or jr) for (pr,O)'l{xo} x {yo} . 0 There are also projections pr,S : J~(N,P) ~ jS(N,P) (IN s N r ) given by

pr,S(jrf) = jsf . jr(N,P) pr'S(o ~ s ~ r)

is topologized as a smooth manifold~

in such a way that the

are projections of smooth fibre-bundles,

by giving the following

local charts: if

(Xl,... ,Xn)

neighbourhood

U

of

is a system of local co-ordinates defined on a xO

in

N ~ and if

co-ordinates defined on a neighhourhood where

i=l,...,n

, j m l , .... p

I -< c I + ... + on ~ r 0 E jr(N,P)xo,Y 0

and

(where

(pr'O)-iu x V , where, if f : ( N, x) §

and

(YI'''" 'YP) V

of

YO

k--l,...,p,o

is a system of local

in P , then = (al,...,a n)

(Xi,Y j ,Zk, O) , with

o i >- 0 , is a system of local co-ordinates at 0

is the r-jet of the constant germ) defined on z E (pr'O)-Iu • V

is the r-jet of a germ

(P,y) , then Xi(z) - xi(x) Y'3(z) = yj(y) 01§ ,, '+~ Zk,o (z) "

k o f)

Ol ~x I

on 9 9 .~x

n

(That these functions are well-defined follows from (5.21 ; that they form a system of local co-ordinates becomes clear by considering germs of polynomial mappings).

112

(6.2)

In fact~ we can be rather more exact about the fibre-bundle structure

involved: let us write

jr(n,p)

the group of r-jets at

0

jr (n,p)

jr(~n

Lr(n) c jr(n,n)

~P)o,o ' and let

(~n ,0)

of germs of diffeomorphlsm of

multiplication defined by on

for

jrh.jrh' = jr(h - h v) ).

(with

L r(n) x L r(p)

Then

be

acts

..rh ,3.rk.) = Jr( k ~ f o h -1) . jrf.(3

by

(To see that this action, and the multiplications in

Lr(n),Lr(p)

are well-defined,

we observe the general fact that the r-jet of a composite depends only on the r-jets of its constituents; for suppose that

f~f~ : (X,Xo) § (Y,yo)

same r-jet, and that

also have the same r-jet.

g,g' : (Y,yo) + (Z,zO)

(gof)*

SO that, since

- (g, of,)* = f*o (g*-g'*)

(g*-g'*)C z c ~ + I

r+l c mx , so that

g ~ f , g' o fv

+ (f*-f'*)

, (f*- f'*)Cy c ~ + I

, (

(go f),. (g, = f,),Cz

have the same r-jet). jr~N,P)x0pY 0

jr(n,p) ; clearly different choices correspond to the action of

Lr(n) • Lr(p)

on

jr(n,p) , and so

fibre bundle with fibre

(6.3)

Then

og'*

Now a choice of local co-ordinates provides an isomorphism of with

have the

If

jr(n,p)

and structure group

f : (N,xO) + (P,yo)

jrf : (N,Xo) + (jr(N,p), jrf) is a representative of

f

pr,O : jr(N,p ) + N • P

Lr(n) x Lr(p) .

is a map~germ, then a germ

is defined by the assignment

and

may be viewed as a

f

is its germ at

x ,

.i-2. x --+ j fx ' where Clearly

jrf

is a germ

X

of section for the projection

(6.4)

PN o pr,O : jr(N,p ) § N .

L er~ma Let

f : (N,xO) § (P,yo)

be a map-germ, and let

x = jrf

i

There are natural identifications (as ~ - v e c t o r spaces) a)

T(jr(N,P)xo) z ~- 8 flm " Nr+l 8f .

b)

T(jr(N,P)xo,Yo) z ~ ~N -- 8 flmN " r+l 8f .

Proof a)

Let

that

g(x,O) = f(x)

~ ~ 8f , and let

g : (N x ~ ,

and ~ u " g = ~

x 0 x O) + (P,yo)

(as in (1.8)).

be a map-germ such

Then the germ of path

113 u -~ jr(g~ element

(where

gU : (N,Xo) § (p,yo)

~ ~ T(jr(N,P)xo) z

gU(x) = f ( x )

9 -linear;

; which is determined by

+ u ~u 3g (x,O) + O(u 2)

Thus we can define and

r(r

where {yl,...,yp}

is defined by

r

= 0

gU(x) = (x,u)) gives an

r , since

(with respect to a n y local co-ordinates).

: 8f § T(jr(N,P)xo) z if and only if 3.rr

are local co-ordinates in

by

= O

r § ~ . (where

This is clearly P r = [ r i=l

P ) i.e. by (5.3), if and only if

r+l

, ~ m N 0f. So

r

induces an injection in

a); but

N Of T(jr(N,P)xo) z , 8f/m " r+l

the same l~-dimension, so this induced map is surjective also.

have

It gives our

identification.

b) This follows at once, by restricting r to T(jr(N,P)xo,Yo) z c T(jr(N,P)xo) z .

(6.5) a)

Corollary

Let

(yl,...,yp)

he a system of local co-ordinates for

This induces a local trlvialisation of induces a splitting

jr(N,P)xo

over

(P,yo) (P,yo) , and hence

T(jr(N,P)xo) z ~ T(jr(N,P)xo,Yo) z @ TPy 0 , which, with respect

to the identification of (6.4), is given by ~r(sf)- ~r(mNSf) @ Ig .

b)

Let

(Xl,... ,xn)

be a system of local co-ordinates at

This induces a local trivialisation of induces a splitting Let

over

(N,Xo) , and hence

T(jr(N,P)) z = T(jr(N,P)xo) z @ TNxo

f : (N,xO) + (P,yo)

above,

jr(N,P)

(N,Xo) -

be a map-germ.

rctfc ))l '

3

With respect to the splitting

~

,

Proof a)

The splitting via local trivialisation is given by translations in

while the map-germs

g : (N•

-> (P,yo)

g(x,u) = f(x) + (O, .. . ,u,.. . ,O) translations on

f ,

Since

(i=l,. ,. ,p)

~u" g = wf(

.

defined by

give the action of these , the result follows.

P ;

114

b)

This is a straightforward calculation. Let

$i(x,u)

#i : (N • ~ ,x0 • O) § (N,Xo) = x + (O,...,u,...,O)

by the translation flow given by

(i=l,...,n)

.

T(JrOxo ( lxi O) = r(jrf)xo d

.r

d

.r

Then

(xo'u lug)

~u(Jf~(x0,u))[u=0 -I

=~-ju(J (z~

d ..r(f

-I

,u))~(Xo,U))lu=O-Tu u

)

~

,u))xo ]u=O

(by the chain rule) The second term in the last expression is

-~r(tf(- ~x~.)) . Now, with 1 (,u) has the same Taylor series f o ~-I

respect to the chosen local co-ordlnates, at

~(Xo,U )

as

f

has at

x 0 ; so that

jrfx 0 § jr(f o ~(i u))~(Xo,U )

translation with respect to the local trlvialization by

is just

(0,...,u,...,0) .

Thus

indeed, with respect to the local trivialization splitting, T(jrf)xo(~[x0)

(6.6)

= (r(tf(~),~,Xo)

"

Now let us see how the notion of ~requivalence gives rise to a group-

action on

jr .

Let ~

be the group of germs of diffeomorphism of

multiplication given by composition).

/~ acts on germs

(N,xO)

(with

f s (N,Xo) § (P,y0)

by

f.h = f o h "I Let

~

( N x P , x o X Y O)

be the subgroup of the group of germs of diffeomorphism of whose elements

diffeomorphism-germ terminology on w by

H

are of the form

(P,yo) § (P,y0) H

for each

is a ~-equlvalence).

H(x,y) = (X,~x(y))

x ~ (N,xO) .

with

~

~ acts on germs

= ~ , . ~ (semi-direct product).

f : (N,xO) § (P,yo) H o (iN,f o h -1) .

by

f : (N,xO) -~ (P,y0)

Thus, in the terminology of w

-equivalence between ~-equivalence

between

This acts on germs

f.(h.H) = (f.h).H , that is, by

(~,f.(h.H)) =

(h,Ho (hx Ip))

f

and

f. (h.H) ; and similarly if

f

and

f' , then

a

(thus, in the

(1N,f.H) = H o (IN,f) . Now let

Hx

is a

(h,}{) is a

f' = f.(h,Ho (h-lxlp)).

115

Now let I~ r c jr(N,N)x0,Xo multlplicatlon defined by

be the group of r-jets of elements of ~,

jrh.jrh' = jr(h o h') ).

~r

acts on

jr

(with

by

jrf.jr h = jr(f .h) . Also, let ~ r c Jr(N x P,N x P)xoXYo~XoXY 0 of

~

.

~r

acts on

jr

by

be the group of r-jets of elements

jrf.jrH = jr(f.H) .

Finally, let 1- n ,

c f*mp

component

:

r+l +...+Xnrp§

by polynomials

r W c , each of which,

dim. (prc'r)'Ic , that is, of greater I~RK r' = c r each irreducible component c '

. r+l r+l ~x I ,...,x n ,0,...,0)

f(xl' .... Xn) =

that

if

of

than some irreducible

z' ~ (pr''r)-Iz

I(

generated

than

components

it will clearly be enough to find

We proceed as follows let

Thus,

codimension

w~'

codim.

Finally, z ~ Wr

c

.

j = 1 .

122

Let us write

Y~" = Yi - Yi-i

for

i=2,,.. ,p .

With respect to the

co-ordinates

(yl,yl, ..,,yp) , f has the form , r+l r+l r+l r+l r+l - x~+I ) f(x I ..... x n) = ~xI +.., +Xk+ I ..... Xk+i+ l - x i ,...,x n (where we write

First consider the case Let

q

be such that

k = n - p ).

i ~ i ~ k + i .

(q-l)(k+l)

+ i i).

It follows that

has regular intersections relative

((JIF)-IAt(N • T, P • T)) F

are transverse to

N x Itl

This proves (ii) as well as the last statement.

(ii) implies (i) uses (1.7, i => li)

F

and for

The proof that

and is left to the reader.

A little more than the hypotheses of (3.4) is needed to ensure that the family

Ift~tE T

(3.5)

is topologically trivial.

Corollary (to (3.3), (3.4) and (I, 5.8))

Keep the notations of (3,4) a n d

assume that one of the conditions (3.4 - i, ii) ~hat

F

Suppose moreover

is proper and that there exists a proper smooth function

that the restriction of the composite ((Jtft)-IAe(N'-- P))ft

~ oft

is transverse to

and homeomorphisms

: N ~ ~

~ c I~

trivial family of mappings, i.e. for any t E T

is satisfied,

Then

@

t E T,

h : N x U-~ N x U,

$ : P ~ I~

such

to any stratum of F

defines a locall2

there exist a neighbourhood

h' : p x U-* P x U

U

of

such that the

diagram below commutes F

N•

NxU

)

P•

~

PxU

ft x idU

In particular I a~v

ft' (t' E U)

Observe that if ~(P) = 189 9 F

P

is topologically equivalent to

is compact such a

The existence of ,

at infinity in case

P

~

always exists;

ft" we simply let

is postulated in order to have some control on

is not compact.

145

Proof F

Let

(At, A~)

respectively,

N • It'l

AIN x It'l

innocent refinement of

any intersection if we let

@

(A, A' )

X ~ (N • It'J ;

and

19-~Z ,

position and hence so are B = A ~ (~ o N T o F)-l@ F

HT

then and

~T o F

(~ o ~

X E A, t' E T . X E A, of

A

I~

and

o F)-l~ Hence

is transverse

HT o F

maps

T .

Now,

whose strata are the connected ($ o ~

o F)-I@

.

The pair ~ (B,

maps the strata of B'.

are in regular

We put

B

B') Thom

submersively 9o

Moreover each stratum of

Since

~-l[n, n + l]

to the closure of any stratum of F

is transverse to

submersively to

(~ o NT)-l@ .

$-l[n, n + I] • T.

the restriction of

A

Our aim is to let an

B' = 4' ~ (* o N ~ - l @

B'

T, HT~ B'

is

is compact, the

is proper.

to the closure of any stratum of

B

As

F

is

is also

Hence the hypotheses of ~I, 5.8) are satisfied for the diagram (N x T, B)

The result follows from (11,5.8)

w

F

(p • T, B,) ~T and

f : N ~ P

summarised by (3.5).

.

(11,5.9).

~

sufficiently large) any proper smooth

which is multi-transverse

topologically stable. stability theorem.

T

Topological stability as a transversality property

We now aim to prove that (for mapping

At .

imply that

does the same with the strata of

restriction of

proper.

A' and

and since

contained in a subset

proper,

~

X F] (~ o Np o F) -1 2Z ,

~

Then by (3.4)

of ft' and

satisfy the hypotheses of the Second Isotopy

denote the stratification

components of

stratifies

denote the Thom stratifications

corresponds with

The assumptions regarding

to each intersection

$ o NT

(A, A')

obtained by applying (3.3).

and

Lemma.

and

with respect to

AI(N, P)

is

In view of (1, l) this will imply the topological The progress we have made so far in this direction is best With an application of this last result in mind the

following proposition must be crucial.

146

(4.1)

Proposition.

transverse with respect to an open subset

f : N-+ P

The proper smooth mappings

O~(N,

P)

AI(N, P) of

which are multi-

J~f(N) n W l(N, P) = r

and satisfy

form

C~(N, P).

The proof requires a bit of preparation in the form of lemmas (4.2) and (4.3) below. (4.2)

Le__~t f ~ Coo(N, P)

Lemma.

cO

and let

[fj e Coo(N, P)l j= 1

converge to

f.

Then a subse~uence of

Ifj ~ embeds in a one-parameter family, more precisel.z ~

there exists .a family

IFt : N-* P~tem

IF1/k]k--I Oo

is smooth and Proof.

of mappings such that

respect to this metric and do the same for metric

p~

on

Jr(N, P)

for which

as a submanifold of

first factor. Vp

P • P,

N

In particular~

such that

N

P .

Jl(N, P)

We choose a tube in the sense of P • P

Ifj l~__l.

is a subsequence of

Choose a Riemannian metric for

(x, t) b~ Ft(x )

-- f.

F~

is complete with

These determine for every

t

is complete.

(II, 1.4, 1.5) for the diagonal of

compatible with the projection onto the

This exists by (II, 1.6).

The tube furnishes us a neighbourhood

of the diagonal together with a smooth mapping

F : Vp • [0, 1] * P

such

that Y(YI' Y2' 0) --Yl and ~(Yl' Y2' I) = y2 Ifj (k )1oo k=l

We now select a subsequence

of

Ifjloo j=l

such that

supI~t(Jfj(k)(X), Jbf(x)) : x ~ N1 < k -k2 and (fj(~+1)(x), fj(k)(X)) E V~ for all

x e N.

(-~o, 0]

Let

o : I~ -* [0, l]

and equals I on [i, co).

be a smooth function which vanishes on

Then define

g(x, t) = w(fj(k+l)(X), fj(k)(X), a(t + 1 - k)) g(x, t) = f(x)

if

t < 0.

with the property that

Then

g

is a smooth

suplp~(Jgt(x),

g : N x I~ if

(x, 0) . (t / O)

extends smoothly over

It follows that the family and

Fo(X) = f(x)

P

Coo-mapping on

J~f(x)) : x e N I < k -k2

N x m

We use (4.2) to prove the following

and N x (I~\[01)

if

Itl > k .

(x, t) ~ g(x, t -1)

by giving it the value

IFt}tel~ defined by

is as required.

by

t ~ [k - l, k]

We leave it to the reader to deduce that the mapping N x (SR k 101)

a

f(x)

on in

Ft(x) = g(x, t-1)

147 (4.3)

Lemma.

the set

Let

O K of

A~(N, ?)

f E

.,

~t

(N, P)

f' E ~(N, P)

with

is a neighbourhood of

Proof.

Suppose not.

As

and let

f

K

f'If-lK

f

[fj E Coo(N, P)Ij= I

always refers to a neighbourhood of

a

subsequenoe of

an

multi-transverse with respect to

converging to

A~(H, P).

f-IK

in

there exists a smooth one-parameter family Ifj 1~176

-

Then

has a countable neighbourhood basis there

is not multi-transverse with respect to

{Fl/klk_ I

P.

i_.nn C~176 P).

oo

then exists a sequence

be a compact subse t of

f

such that

fjlf-IK

(As usual, such a condition

N.)

By the previous lemma (L~.2)

IF t : N -* ]~ItEl~

with

F~ = f

and

It follows from (3.2) that there exists

j=l"

E > o

such that

Following (3.2),

JIF

is transverse to

N x IO~

is transverse to

intersection coincides with

~ ,

f ,

at

K • [-~, ~].

Since

and their

(Jtf)-lA~(N, P)

it follows from (1.7) that for a possibly

(J~F)-IA~(N • 3,

intersections relative

P x 3)

(Jg)-IAt(N • I~, P x 2)

(Jlf)-lA~(N, P) x IOl 9

has regular intersections relative smaller positive

A~(N • 2,

P • 2) I F-I(K • [-e, ~])

has regular

F , and that the strata of

AK, e = (JeF)-IA~(N x I~, P • 2) I F-I(K x [-e, ~])F are transverse to

N • I0J .

we may then choose t E [-e, r to

AK, ~

so small that

for all

t e [-e, r

is a Whitney stratification by (3.3),

N • ItJ

Then (1.7) implies that

At(N, P)

I t = ~

~

Since

FtlF;IK

.

is transverse to

AK, r

for all

is multi-transverse with respect

In particular this is so for

-I k > ~ , which contradicts our assumption.

if

Proof of (4. i).

Let

f E O~(N, P).

Choose a pair

{Vi cWil

of iEl

locally finite coverings of ~i c W.

for all

i E I.

P

by relatively compact open subsets for which

The set

D

of

f'E Coo(N, P)

with

Jf'(N)~Wt(N,P)=~

i

and

f, (f-I ~i ) c W i

any

f' E 0

AlE I O~i

is proper.

(O~i

intersection

for all

i E I

is clearly open.

On the other hand it follows from (4.3) that

as there defined) is a neighbourhood of

0 ~ NiE I ~ i

By the last condition

is a neighbourhood of

f.

f .

Hence their

It is clear that this

148

intersection is contained in

Of(N, p).

(4.4)

Theorem.

Any

f E Or(N, P)

is topolg~ically stable.

(4.5)

Corollary.

(The Topological Stabili~ Theorem)

The topologically

C_oo_(N, P)

stable mappings intersect the set of proper smooth mappings

in a

9r

dense subset. Proof.

In view of (4.4) it suffices to show that

Coo (N, P) pr

for some

e

codim We(n~ p) > n.

By (III, 7o 2)

The density of

(1.1) once we have show that codimension

> n .

Of(N, P)

W e (N, P)

Je(n, p)

canonical stratification. structural group

there exists an in

Lt(n) x L~(p)

Proof of (4.4).

of

Y(YI' Y2' I) = Y2 " P)

(III, 7.1).

Recall that

of the Jet-bundle Wt(N, P)

Jl(N, P) -~ N x P

into strata of codimension

F : Vp x [0,I] -~ P Now let

a stratification

~ c ~.

N. = ($ o f)-iIj ~ and if f'j : ~.'-*jPj

Vp

close to

f ,

then

(4, 4')

For any

for

f .

j E Z~ we put

f' E C~176 P)

neighbourhood of

Nj

enables us to compare

onto

is close to Nj,

then

Nj , ~l~j'

$

Note that

so if

> n .

P x P

~(YI' Y2' 0) = Yl

Let

$ : P-* I~

be

to any stratum of

4'

P. = $-IIj~ , 0 = ($ o f')-IIj I and let

we set f'.

and so

Then (3.3) constructs out

It follows from (3.4) that ~.' ~

is a

of the diagonal in

such that

f E ~e(N, P).

denote the restriction of

compact submanifolds.

Wt(n, p)

Hence it admits by (I, 2.7) a

a proper smooth function such that the restriction of is transverse to

will follow from

By choosing a tube for the diagonal in

and a smooth mapping

(Jls

such that

admits a stratification into strata of

we obtain (as in the proof of (4.2)) a neighbourhood

and

is dense in

This stratification is then invariant under the

determines a stratification of

P x P

l

~pr(N, P)

This is easily done as follows.

semi-algebraic subset of

D ~(N, P)

N.0 and

P.j are

fj E D~(Nj, Pj). ~

If

f'

is

is a smooth retraction of a

will be a diffeomorphism.

f' with f . Then (4.1) applied to f J 0 yields a neighbourhood Of of f such that f' E Of implies

and

This f. (j ~ 2Z), J

149

(a)

For any t ~ [0, IS , the mappi~

f~ E C~(N, P),

defined by

f~(x) ~ ~(f(x), f'(x), t), belongs to ~(N, P). (b) and

The composite ~ oft' is transverse to ~ c 9 for any t E [0, IS f, f~,j E n2(Mj t, Pj) for any j E ~ . We prove that any f' E Qf is

topologically equivalent to (Jef~)-IA~(N, P)

f .

According to (3.3) property (a) implies that

determines a stratification

(At, A~)

and property (b) above it follows that the strata of A~ manifolds stratum of f'

Pj, At

J s 2~,

of

f~.

are transverse to the

in particular, that the restriction of

is transverse to

are topologically equivalent.

~.

From (3.4)

$

ft

I~ then follows from (3.5) that

to any f

and

Referenees

[Bo ]

Borel~ A.

Linear Algebraic Groups, W.A. Benjamin, Inc. New York, 1 969.

[B=]

~,

,a

.

Brocker I T. & Jamch~ K.

Einf~hrung in die Differential topologie.

Springer - Verlag, 1 973.

[Gi ]

Gibson. C.G,

Regularity of the Segre Stratification,

Math. ~roo.

Camb. Phil. Soc. (To appear)

[Go ]

@olubits~1 M. & Guillemin I V.

Stable Mappings and their Singularities,

Springer - Verlag, 1 973.

[Hi ]

Hironaka, H t

"Number Theory, algebraic geometry and commutative algebra".

Volume in honour of Y. Akizuki.

Published

by Kinokuniya, Tokyo, 1973.

[~o]

Lan~. S.

Differential Manifolds,

~o~asiewicz I S~

Ensembles Sumi-Analytiques,

Looi.~enga, E.J.N.

Structural Stability of Smooth Families of functions.

[Ma]

Addison-Wesley,

1 972.

IHES Lecture Notes, 1965. C~ -

Thesis, University of Amsterdam, 1974.

Morlet, C~

Seminaire H. Caftan, Expos~ 4, 1961-62.

Mather t J. N~

I. Notes on Topological Stability, Lecture Notes, Harvard University, 1 970~ II. Stratifications and Mappings, Proceedings of the Dynamical Systems Conference, Salvador, Brazil, July, 1 971, Academic Press. III. Finitely-determined map germs.

Publ. Math.

IHES 35 (i968) p p . i 2 7 - i 5 6 . IV. Classification of Stable Germs by

9

algebras.

Publ. Math. IHES 37 (1969) pp.223-248. V. Transversality.

Advances in Mathematics 4 (1970)

pp.301 - 336. VI. The Nice Dimensions.

Proc. Liverpool

Singularities Symposium I, Springer Lecture Notes in Maths. 192 (1971).

151

[Th]

Thorn,R.

Propri~tes Diff~rentielles Locales aes Ensembles Analytiques, Seminaire Bourbaki, 1964/5. exp. 28i.

[Wa]

Wall, C.T.C.

Regular Stratifications. Warwick 1974".

'~ynamical Systems -

Springer Lecture Notes in Math.

No. ~68, P.332-344.

[Wa]

Wass erman, C-.

Stability of Unfol~ings, Springer Lecture Notes in Mathematics, 393 (i974).

[~]

vait~,, ~.

I. Local Properties of Analytic Varieties, pp.205244 in Differential an~ Combinatorial Topolo~, Princeton, i 965. II. Tangents to an Analytic Variety, Annals. of Maths.

8~ (~ 965), ~-% - ~ 9 .

Index

associated Thom stratification

28

bad set

~9

canonical partial stratification

26

canonical Thom stratification

32

canonical Whitney stratification

16

cl-equivalent mappings cl-stable mapping

I I

codimension

134

commutation relations

42

compatible tube

38

contact equivalence

94

control conditions

50

controlled vector field

36

controlled vector field

50

cross-ratio example

41

determined map-germ

307

diagonal stratification

127

dimension

18,68

distance function

38

equivalent map-germs

94

filtration by dimension finite singularity type

9,15,46 (FST)

Frontier Condition general position Genericity Condition generic subset

85 17 12,13,125 25 127

germ

37

induced stratification

13

infinitesimally stable map-germ

8O

ISS

Isotopy Lemma, First

4,59

Isotopy Lemma, Second

4,62

integrable vector field r-jet equivalent germs

54 105 5O

lift Local Finiteness Condition

9

local ring

73

local trivialisation

58

locally trivial stratification

59

locally trivial stratified set

58

minimal partial stratification

26

minimal Whitney stratification

15

multi-transverse mapping

5,127

Nakayama's Lemma

77

partial stratification

26

pinch map

24

Preparation Theorem of Malgrange

78

product stratification

12

refinement

14

retraction

38

regular point

18

regular intersection

14

restriction semialgebraic

126

set

17

singular point

19

singular set

19

special representative

134

spray

39

stable map-germ

69

154

9,22

stratification stratified vector field

9

stratum Tarski-Seidenberg

50

Theorem

17

Thom mapping

23

Thom Regularity Condition

23

Thom regular strata

23

Thom stratification

4,23

Thom

(Whitney stratification)

jet space topological equivalence Topological Stability Theorem topologically trivial family

5,131 2 148 3

transverse mapping

13

trivial stratified set

58

trivial unfolding

68

tube

38

tube system

46

tube system, controlled

46

unfolding

66,67

versal unfolding

67,69

Whitney regular strata

10,11

Whitney regularity Whitney Regularity Condition

3 11

Whitney stratification

4,11

Whitney stratified set

11

Whitney's Theorem

19

Whitney topology

127

Whitney umbrella

9

Symbols A, A, A' w A' a, a' B(X,Y) (CR.), (CRp), (CRf) CN,x O' mN, Xor C N, m N f, f X jr(N,P) jr(n,p) jrf jrf L r (n) Mi,M ,i pr,S Qi, Qi(k,l). Qpi

46 45,46 19 42 73 112 111 112 112 105 112 45,46 111 47,48

Sg(f)

3O

ITil

47 75 50,51 2O

tf VF., VFp, VFf, VFpf W(A,B)

W r (n,p) wf xf A1 (N,P) 8f C Dr K Nf R

120 75 85 134 126 94,114 127 94,114 83 114

Xc

17

(U.V)

128

R I(N,P)

146 70

eN,x o, e N ef

74