Singularities of Smooth Functions and Maps 0521233984, 9780521233989

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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor.I.M. James, Mathematical Institute, 24-29 St Giles,Oxford 1

1. 4. 5. 8. 9. 10. 11. 12. 13. 15. 16. 17. 18. 20. 21. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.

General cohomology theory and K-theory, P.HILTON Algebraic topology, J.F.ADAMS Commutative algebra, J.T.KNIGHT Integration and harmonic analysis on compact groups, R.E.EDWARDS Elliptic functions and elliptic curves, P.DU VAL Numerical ranges II, F.F.BONSALL & J.DUNCAN New developments in topology, G.SEGAL (ed.) ·symposium on complex analysis, Canterbury, 1973, J.CLUNIE & W.K.HAYMAN (eds.) Combinatorics: Proceedings of the British Combinatorial Conference 1973, T.P.McDONOUGH & V.C.MAVRON (eds.) An introduction to topological groups, P.J.HIGGINS Topics in finite groups, T.M.GAGEN Differential germs and catastrophes, Th.BROCKER & L.LANDER A geometric approach to homology theory, S.BUONCRISTIANO, C.P. BOURKE & B.J.SANDERSON Sheaf theory, B.R.TENNISON Automatic continuity of linear operators, A.M.SINCLAIR Parallelisms of complete designs, P.J.CAMERON The topology of Stiefel manifolds, I.M.JAMES Lie groups and compact groups, J.F.PRICE Transformation groups: Proceedings of the conference in the University of Newcastle-upon-Tyne, August 1976, C.KOSNIOWSKI Skew field constructions, P.M.COHN Brownian motion, Hardy spaces and bounded mean oscillations, K.E.PETERSEN Pontryagin duality and the structure of locally compact Abelian groups, S.A.MORRIS Interaction models, N.L.BIGGS Continuous crossed products and type III von Neumann algebras, A.VAN DAELE Uniform algebras and Jensen measures, T.W.GAMELIN Permutation groups and combinatorial structures, N.L.BIGGS & A.T.WHITE Representation theory of Lie groups, M.F. ATIYAH et al. Trace ideals and their applications, B.SIMON Homological group theory, C.T.C.WALL (ed.) Partially ordered rings and semi-algebraic geometry, G.W.BRUMFIEL Surveys in combinatorics, B.BOLLOBAS (ed.) Affine sets and affine groups, D.G.NORTHCOTT Introduction to Hp spaces, P.J.KOOSIS Theory and applications of Hopf bifurcation, B.D.HASSARD, N.D.KAZARINOFF & Y-H.WAN Topics in the theory of group presentations, D.L.JOHNSON Graphs, codes and designs, P.J.CAMERON & J.H.VAN LINT Z/2-homotopy theory, M.C.CRABB Recursion theory: its generalisations and applications, F.R.DRAKE & S.S.WAINER (eds.) p-adic analysis: a short course on recent work, N.KOBLITZ Coding the Universe, A.BELLER, R.JENSEN & P.WELCH Low-dimensional topology, R.BROWN & T.L.THICKSTUN (eds.)

49. Finite geometries and designs, P.CAMERON, J.W.P.HIRSCHFELD & D.R.HUGHES (eds.) 50. Commutator calculus and groups of homotopy classes, H.J.BAUES 51. Synthetic differential geometry, A. KOCK 52. Combinatorics, H.N.V.TEMPERLEY (ed.) 53. Singularity theory, V.I.ARNOLD 54. Markov processes and related problems of analysis, E.B.DYNKIN 55. Ordered permutation groups, A.M.W.GLASS 56. Journees arithmetiques 1980, J.V.ARMiTAGE {ed.) 57. Techniques of geometric topology, R.A.FENN 58. Singularities of smooth functions and maps, J.MARTINET 59. Applicable differential geometry, F.A.E.PIRANI & M.CRAMPIN 60. Integrable systems, S.P.NOVIKOV et al. 61. The core model, A.DODD -----62. Economics for mathematicians, J.w.S.CASSELS 63. Continuous semigroups -in Banach algebras, A.M.SINCLAIR 64. Basic concepts of enriched category theory, G.M.KELLY 65. Several complex variables and complex manifolds I, M.J.FIELD 66. Several complex variables and complex manifolds II, M.J.FIELD 67. Classification problems in ergodic theory, w.PARRY & S.TUNCEL

London Mathematical Society Lecture Note Series.

58

Singularities of Smooth Functions and Maps

JEAN MARTINET Professor of Mathematics University of Louis Pasteur, Strasbourg

Translated by CARL P. SIMON Associate Professor of Mathematics and Economics University of Michigan

CAMBRIDGE UNIVERSITY PRESS Cambridge London

New York

Melbourne

Sydney

New Rochelle

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 32 East 57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia

© Cambridge University Press 1982 First published in 1982 Printed in Great Britain at the University Press, Cambridge

Library of Congress catalogue card number 81-18034 British Library cataloguing in publication data Martinet, Jean Singularities of smooth functions and maps.(London Mathematical Society lecture note series, ISSN 0076-0552; 58) 1. Mathematics-1961I. Title II. Series 510 QA37.2 ISBN 0 521 23398 4

TABLE OF CONTENTS

xi

Introduction List of Symbols Part One: Chapter I.

xiv

General Information. The ring of germs of differentiable functions of n real variables. 1. The ring

E

n

of genns of functions of

n variables

1

3. The algebra of jets

3 3

4. Ideals of finite codimension in E

n 5. The study of some other ideals of E n 6. Remarks and examples CHAPTER II.

1

2. The maximal ideal and its powers

The 9:rou12 of local diffeomorEhisms of 1. The group

origin of 2. The group

5 7

nt

L of local diffeomorphisms at the n JRn

9

Lk

9

of k-jets of local diffeomorphisms n 3. The action of L and Lk on E and Jk n n n n 4. Infinitesimal generators of L and Lk n n

11

12

CHAPTER III. Elements of the classifications of germs of functions of n variables. 1.

Introduction

17

2.

Morse's Lemma (first proof)

18 21

3.

Morse's Lemma (second proof)

4.

Generalization

24

5.

Orbits of germs of finite codimension

26

6.

Comments and examples

29

CHAPTER IV.

Introduction to the study of deformations 1.

Introduction and definitions

2.

Tangent space and codimension of a germ of a scalar function

3.

The notion of a universal deformation of a germ of finite codimension

4.

The fundamental geometric lemma in the theory of deformations

5.

The universal deformation of a Morse germ and the Decomposition Lemma. 3 Universal deformation of x

6. CHAPTER

v.

Generic singularities of mappings of the plane to the plane 1.

Introduction

2.

Folds and cusps for mappings of the plane to the plane.

3.

Whitney's Theorem

4.

Proof

5.

Generic .singularities of mappings from :IRs

CHAPTER VI.

into

:~~ 2

(s ~ 21

The division theorem of order two. 1.

Introduction. Statement of the general division theorem

2.

Reduction to a particular case: the canonical division theorem

3.

Reduction of the division theorem of order two to a lemma of Whitney

4.

Preliminaries for the proof of Whitney's lemma

5.

Differentiable functions on real subsets of ~n

6.

Proof of Whitney's Lemma

7.

Proof of Lemma 5.5 Appendix

CHAPTER VII.

Thorn's transversality theorem 1.

Introduction and review

2.

Sard's Theorem

3.

Proof of Sard's Theorem

4.

The fundamental lemma of transversality

5.

First application: the generic behavior of the rank of differentiable mappings

6.

Thorn's Transversality Theorem

7.

Application: mappings from

Part Two:

generic singularities of

nl

to

m?

The differentiable preparation theorem.

CHAPTER VIII. The importance of flat functions

CHAPTER IX.

~!

I I

,.~ I' I

CHAPTER

x.

1.

Flat functions on a real vector subspace

2.

The extension theorem of Lojasiewicz

3.

A division lemma

The division theorem 1.

Introduction

2.

The analytic case

3.

The differentiable theorem of Newton

4.

Proof of the canonical division theorem of order k .

The Malgrange-Mather preparation theorem 1.

Introduction

2.

Statement of the Malgrange-Mather Preparation Theorem

3.

The division theorem as a special case of the preparation theorem

4.

A first generalization

5.

A second generalization

6.

The preparation

theor~m

in general

Universal deformations of germs of real-valued functions

Part Three: CHAPTER XI.

Universal deformations of real-valued functions 1.

The fundamental theorem of universal deformations

2.

Universal deformations and transversality

3.

Universal deformations of potentials

4.

A remark concerning the Weierstrass Preparation Theorem

CHAPTER XII.

Classification of germs of real-valued functions of dimension less than six; the elementary catastrophes of R. Thorn 1.

Preliminary remarks on ideals of finite codimension in E

2.

Introduction to the classification of germs of codimension less than or equal to 5

3.

Classification of germs of corank l

4.

Classification of germs of corank 2 (and codirnension ~ 5)

5.

Geometric description of germs of codimension < 5

6.

A transversality theorem

7.

The elementary catastrophes of R. Thorn

n

Part Four:

Singularities of differentiable mappings

CHAPTER XIII.

Introduction to the local study of differential mappings. Tangent space.

1.

Introduction.

2.

Rank and unfoldings

Definitions

3.

Jets of mappings.

4.

Tangent space

Orbits

5.

Tangent space to the orbit of a map germ

6.

Algebraic description of the tangent space

7.

Tangent spaces: examples of computations

CHAPTER XIV.

Universal unfoldings 1.

Introduction

2.

The geometric lemma of the unfolding theory

3.

The algebraic· lemma of the unfolding theory

4.

Proof of the universal unfolding theorem Application: the singularities El' ••• ,1

s.

of CHAPTER XV.

lRn+l

to

IRn+l Cn .::._ 1)

Classification of stable map germs 1.

Introduction

2.

Stable unfoldings of germs of finite type

3.

Characterization of map germs of finite type

4.

The contact group; contact orbits

5.

Stable map germs: the main geometric characterization

6.

Description of the contact orbits

CHAPTER XVI.

Classification of stable germs 1.

The fundamental theorem and its consequences

2.

Study of germs embedded in an unfolding

3.

A fundamental lemma

4.

Proof of the fundamental lemma

5.

Orbits of stable germs

CHAPTER XVII.

Generic singularities:

examples

1.

Introduction. singularities

2.

The study of

3.

Generic singularities of rank p from lR.p+l to lRp+l

4.

K(p,l,2)

5.

Generic singularities of rank p for mappings from lRp+l to 1Rp+ 2

Stability of generic K(p,l,l)

(p ..::._ ll

6.

K(p,2,1)

7.

Stability is not generic in

00

C (lR

9

,m.8 )

8. 9.

The study of some Singularities of

K(s,s,2) germs, s .

rnapp~ngs

from JR

2s

~

243

2.

to IR

s+2

247

(s ~ 2)

Bibliography

249

Index

253

xi INTRODUCTION

These notes arose from a seminar held at the University of Michigan during the winter 1973-74, and a course given at the Pontificia Universidade Catolica do Rio de Janeiro (P.U.C.) from March to June 1974.

My aim was to present a detailed study of the most important features of the singularity theory for differentiable mappings, as it had developed in the fundamental work of H. Whitney, R. Thorn, J.N. Mather and some other mathematicians. This subject is so rich that one has to make careful choices to give a coherent and significant idea of the theory.

In this

text, I have made the following choices£ 1)

Only the local theory is developed, that is, the study of

germs of functions or mappings; but the transversality theorem, which is actually a local result, is included. 2)

A prominent role is given to the notion of an unfolding,

which I consider one of the most important concepts in this theory, technically as well as conceptually. 3)

I

highlight the

differen~es

and analogies between the

singularity theory of functions (lRn ~ IR; only the group of local diffeornorphisrns of

lRn

is involved), and of mappings

both groups of local diffeomorphisms of 4)

1Rp

and

IRq

(lRP

-+

lRq;

are involved).

A proof of the division theorem for differentiable functions

is included; it is the only part of the theory where analysis is needed, but it is fundamental.

I choose Lojasiewicz's beautiful

xii

proof, which is a pleasure to develop; moreover, it represents an introduction to some important techniques in differential analysis. These notes should be, it seems to me, easily understood (at least until the beginning of Part 4) by.a good graduate student. The prerequisites are: a)

a very good understanding of Taylor's formula, the

implicit function theorem, and the existence and uniqueness theorem for solutions of differential equations, ab~ut

b)

some basic ideas

c)

a reasonable familiarity with the structure of a module

Lie group actions, and

over a commutative ring. The progression is as follows: Part one, which represents (with Part three) the lectures given at P.U.C , is self contained.

It introduces the main ideas by

means of a detailed exposition of examples:

classification of

function germs (III), universal deformations of some functions (IV), m.Z to m 2 (V) , and the division theorem

folds and cusps from "of order 2" (VI).

This part contains only two general results:

Theorem III.5.2 which is the main tool for classifying function germs, and Thorn's transversality theorem (VII), which is stated here in its simplest version. Part two is devoted to the proof of the division theorem (VIII and IX; this may be omitted by the reader mostly interested in singularities), and the Malgrange-Mather preparation theorem, an algebraic version of the division theorem (X) .

I draw some conse-

quences relative to unfoldings (Theorems X.5.4 and X.6.3), which are the forms under which the preparation theorem is used in the theory. In Part three, the preparation theorem is applied to the theory of universal deformations of functions germs (XI), from which we draw the classification of Thorn's "elementary catastrophes" (XII). Actually, I recommend that the reader look at the beginning of

xiii Part 4 is a kind of cross-section of J. Mather's papers on singularity theory; most of Mather's results are stated, in their local version.

I did not include the result on the finite determinacy for

map germs of finite codimension, but instead I prove a theorem about universal unfoldings of such mappings.

I do not compute the "nice

dimensions" (i.e., those for which stability is a generic property of mappings) , but I describe the method which allows one to solve this problem, and work it out completely on some examples. The point of

v~ew

presented in this part leads naturally to the

study of topological stability, but

this problem lies beyond the

scope of this book. The bibliography is reduced to the most important references. A more extensive one can be found in the excellent book by M. Golubitsky and V. Guillemin {Stable Mappings and their Singularities), and in the "bible" of the subject, The Proceedings of the Liverpool Symposium. A last important remark:

this book

i~

concerned· about the

singularity theory for differentiable mappings only; but the theory holds (without any major modifications) for analytic mappings in the complex domain.

xiv

LIST OF SYMBOLS

SYMBOL

PAGE

E

l

E(lRn, 0)

l

Mk

l

Mk

2

n

Jk n

3

Jk(lRn ,0)

3

/t

3

Ik,I

k

n

lR [ [x , •• ,xn]J 1

M"" L

n

PAGE

77

sf

83

ck

92

k. This algebra Jk is also called the algebra of k-jets of C~ functions n

.

quot~ent

Similarly, the

space

Mk /Mk+l

can be canonically identi-

fied with the real vector space of homogeneous polynomials

in n

variable of degree k. If f is in En' its projection into Taylor polynomial of order k at 0.

J~ can be considered as its

One denotes this canonical pro-

jection by j

k

k : En+ 3 n'

Ideals of finite cod :!mens ion in E • nLet I be an ideal in r: • Designate by 4.

rc.En 1 ~1 ~

n

Jk n

the projection_, of I into Jk. n

Proposition.

The following conditions are equivalent:

1)

I-;, /)r-

2)

I:::llhMk+l, i.e., I+

Proof.

1) => 2)

iJr-+1 ::JMk,

is evident,

The proof of 2)

= 1)

is an algebraic

pearl which will be generalized later (see Nakayama's Lemma in Chapter X).

Let 1 s_prove this implication. k

Let {g , ••• ,gr} be a system of generators of M • For example, 1 {g , ... ,gr} can be the set· of canonical monomials of degree kin 1

x , ... ,xn. 1

For each i • l, ••• ,r, there exists by hypothesis a

germ fi in I such that: k+l (l)gi-fieM.

4

k+l Every element in M j

M.

where 11 's are in (2) gi • f i

k

= 1'l•M

Thus, one can write (1) in the form:

j

T

~ 11 gj,

is, by definition, of the form

j•l

r j j jil\ligj, where the 11 1 e: M.

+

Alternatively, r

(3) gi-

I

.

11igj • fi' j=l

i•l, ••• ,r.

The equations in (3) can be summarized by a matrix-vector equation (4)

M.s.= !. '

where M is the rxr square matrix whose entries are oji - \lji Here,

o{

is the Kronecker symbol, whose value is 1 when i

0 otherwise.

In (4),

~is

E

= j

En' and

the column vector of the gi's and f is

the column vector of the fi's. The key point now is the fact that M is invertible as a matrix with entries in E . n

In effect, the determinant of M has as its

value at 0 the determinant of M(O). is the identity matrix and detM(O)

Since the

= 1.

~~~ are in M, M(O) ]._

Thus, detM as an element

The classic cofactor algorithm provides of E is invertible in En n an inverse matrix Mrl whose entries are in E • One then has: n -1

(5) A .. M .f· This equation means that the g 's are linear combinations of the 1

fj 's with coefficients in En. the gi's are in J.

Hence,

Since the fj's are in the ideal I,

M~I.

Q.E.D.

k k f ~ in M generate M if and only if k+I1 p the reductions modulo M generate the vector space of homogeneous

Corollary 1.

The germs

polynomials of degree k. Corollary 2.

Let ICE

11

be an ideal.

The following conditions are

equivalent:

1)

I has finite codimension in En as a vector space over IR,

2)

There is an integer k such that I :>Mk.

Proof. in E . n

2) => 1) is clear since Mk itself has finite codimension

5

1)

=

2):

Consider the following nested sequence of subspaces

(ideals):

+

I

M :>I

+

2 M :> .•. ::>I

+

Mk:H

+

Mk+l::> •••

Since I has finite codfmension in IR, there exists a k such that

I+ Mk =I+ Mk+l. But then, I + Mk+l ::> M\ whence I ::>Mk by the preceding Proposition.

5.

A study of some other ideals in

Consider lR.n = mP x IRq, p

E~

+ q .= n, p and q

coordinates by t 1 , ... ,tp,x , ... ,xq. 1 tion homomorphisms

> 0.

Denote its

There are two natural restric-

E + E n P

*

f(t,x)

f(t,O)

and

E + E

q

n

f(t,x)

*

f(O,x).

In the following discussion, we will often denote

Ep

and

Eq

by Et

and Ex respectively, to keep track of the name of the variables under consideration. Let I be the kernel of the homomorphism E

n

+

E , i.e., the ideal p

of germs in E which are identically zero on mP x fO} ("the tn

axis").

Similarly, let J be the kernel of the restriction homoi!Dr-

phism E + E • n

q

Similarly, let Ik+l be the ideal of f in En such that f and all its partial derivatives of order less than or equal to k belong to I, i.e., the ideal of functions which are k-flat on mP x { 0}. 5.1 Proposition.

a)

I is generated by the coordinate functions

xl' •.. ,xq; b)

c) Proof.

k

Ik = I ; J(\Ik= J • Ik. The proof is easy and is based on the same principle as

6

tl~t

of Proposition 2.

For a) and b), one writes

Jl a

:12 xi af (t,;l.x)d). if f{t,O) • 0. i=l 0 xi

f(t,Y) ..

One easily checks that the functions hi(t,x) •

1 af

J0 axi (t,).x)d).

belong to Ik-l w'!lere f is in Ik. For

c), one writes

= ~2,

fl at

3f (Xt,x)d). t 1 i•l 0 i since f is in J, and one easily verifies that the functions gi(t,x) = f(t,x)

J0l ;rti (At ,x)d). belong

to Ik when f is in Ik.

(Note that f e:l k implies

(lf at

k k k EI for each i = l, ... ,p.) This proves that Jni CJ • I . i k The inclusion J • I CJnrk is straightforward,

that

5. 2 Consider an ideal L in E . n

Denote by LCE

q

the set of restric-

tions to {O} x lRq of elements of L; one thus obtains an ideal of We remark that if LC:: I k (the ideal of functions which are (k-1)

E • q

-

k

flat along m.P x { 0}), then L CMx, where Mx denotes the maximal ideal of E

q

Propos i t ion.

= EX •

The following result will be useful later:

Let LCI

k

be an ideal of En.-- Then, L

=

I

k

i f and only

= Mk. ='----'-xif 'L

Proof.

The proof is essentially the same as that of Proposition 4

(Nakayama's Lemma).

'L =

By the above remark, it is enough to show that

Mk

k

implies that L =I . For this, let {g , ... ,g} be a system 1 x k r of generators of I (for example, the set of canonical monomials k of degree kin x , ••• ,xq). Since Mx = L by hypothesis for each 1 i ~ l, ••• ,r, there exists an £ in L such that• 1 gi(x)- f~(O,x), i.e., g1. - f

1

is .~:ero on I~) .,.. IRq.

But since gi and fi are in Ik too, one has gi - fi e: JJ1Ik, where J is the ideal of function zero on {0} x IRq as defined above. 5.1 c), one can write:

By

7 r

j

j~lAigj'

gi- fi •

j

where Ai & J.

Alternatively, r

gi +

.

L Aigj

.. fi. j•l One concludes, just as in section 4, that the gi's are linear comk

binations of the fi's and thus that I c;L.

k

by hypo-

Q.E.D.

thesis, L = Ik.

6,

Since LCI

Remarks and Examples

6.1 A germ f

£

E is called Flat if all of its partial derivatives n

--

are 0 at the origin, that is to say, f

£

"" k n M • k=l

The flat germs

clearly form an ideal which is naturally denoted as M"".

This ideal

does not reduce to just the 0 element. For example, for n • 1, the 1 function f(x) = exp(-~) (f(O) = 0) defines a non-zero element in

If'.

The quotient E /M "'=

if f

£

n

J'"'n

is the ring of jets of infinite order;

E , the projection j""f £ J"" is the jet of infinite order of nn f at 0. It can be identified with the Taylor series of f at the origin, i.e., an element of the ring of formal power series inn variables, IR[ [x , ••• ,xn]], Recall that every formal power series 1 is the Taylor series of a germ of a c"' function; in other wcrds,

This is a famous result of Emile Borel, a proof of which will be sketched in chapter VI, §7,4, One final remark:

the ideal M"" does not admit a finite system

For, suppose that {f , ••• ,fr} is a finite system 1 of generators of ~!"" noting that M • M"' = M"", one can write for

of generators.

each i = 1,. ••• ,r: f

i

..

One could then deduce that each fi Proposition 4, i.e., that M"'

= {0},

=0

by the same argument as in

a contradiction.

6,2 Some examples of ideals of finite codimension To study an ideal IC:En of finite codimension, it is natural

8

to consider the sequence of nested ideals:

(See §4.) The numbers c , c , c , ••• represent the codimensions of the corres2 3 1 k-1 pending inclusions. If ck = 0, then I + M = I + Mk and I ::>Mk-1 • The codimension of I is then equal to c 1 + c 2 + ... + ck-l' The finite sequence {c ,c , ••• Jck_ } clearly contains important informa1 1 2 tion concerning the ideal I. In E (germs of functions of one variable), there is 1 k exactly one ideal of codimension k, namely M .

Examples.

In E (germs of functions of two variables: x andy), let us 2 study the ideals of codimension 3. A priori, we have either

First case.

That c

2

=1

(i.e., c 2 ; 2) means that I contains a

function whose first partial derivatives are not all zero at 0. Up to choosing convenient local coordinates in the plane, we can suppose that the coordinate function x belongs to I.

Let [x] denote the

Pass to the quotient E2/[x], which is the ring f of germs of functions of y. One then sees that I is the 1 3 ideal generated by x and y • ideal generated by x.

Second case.

Stnce c 3 .. 0, I::JM

2

and one has

E ;:) M ?I::>M 2 • 1 1 2

But the codimension of I •

2

M,

M2

in M is equal to 2.

the ideal of generated by

~

')

2

, xy, y •

It follows that

9 CHAPTER II THE GROUP OF LOCAL DIFFEOMORPHISMS OF IRn

1. The group Ln of germs of local diffeamorphisms at the origin of lRn. By definition, a germ of a local diffeomorphism at the origin of IRn is an equivalence class (for the equivalence relation defined in Section I.l) of C~ mappings ~:

U +

JRn

(_U

an open set in

JRn containing 0)

such that

0

1)

ojl (0)

2)

ojl defines a diffeomorphism of an open subset U C U contain-

a

I

ing 0 onto ojl(U'); this is equivalent, by the Inverse Function Theorem, to the statement that the Jacobian matrix Doll(O)

of .P at 0 is invertible.

Such a germ will be denoted by

It is clear that the set of all germs of local diffeomorphisms at 0 forms a group under composition, which we will write as L(IRn,O)

or L • n

2.

The group Lk of k-jets of local diffeomorphisms. n

Let k be an

integer~

1.

Let ojl

E

Ln.

The Taylor Series of ojl

at 0 of order k can be written as

(1) ojl(x)

= P1 (x) + P2 (x) + ... + Pk(x) + £(x),

where P

"' D¢(0) is a linear automorphism of IRn,

1

10 P

n

is a polynomial mapping lRn + IR , homogeneous of degree 2,

2

Pk is a polynomial mapping lRn + IRn homogeneous of degree k. Finally, the components of

belong to /.!

£

k+l

•

A germ

cf>

will be

called k-flat with respect to the identity if, in (1) P

1

=I

identity on lRn; P 2 • ...

D

This means that the mapping

cf>

-

= Pk • 0.

I has all its components in /.(k+l.

The germs which are k-flat with respect to the identity form a subgroup of Ln•

(Exercise:

prove this last statement,)

The quotient group of Ln by the subgroup of germs which are kflat with respect to the identity is called the group of k-jets of local diffeomorphisms at 0 and is denoted by Lk. n

One writes.

k

for the canonical projection.

By Taylor's Formula, the group Ln

can be canonically identified with the set of polynomial mappings: :&"

+ P (x) = P (x) + • • • + P k (x)

1

n

n

into m. , where P1 is homogeneous of degree one (linear), ••• , Pk is homogeneous of degree k, such that P is an automorphism of 1 IRn. The group operation P • Q on Lk can easily be seen to be the of lR

following operation:

a polynomial mapping P o Q of degree to remove all terms of degree Proposition. Proof.

k

n

one composes the mappings P, Q >

~

£

L

2

k , one then truncates P o Q

k.

The group Lk admits a natural Lie group structure. n

First note that Lk can be considered as an open subset of k

n

the vector space Ln of all polynomial mappings of degree P(~)

in

to obtain

n

• P (x) + P (:K') + 1 2 k

~

k

+ Pk(x): k

effect, Ln is the subset of those elements of L for which P

1 k is invertible, i.e., determinant P1 ; 0. This gives Ln a smooth manifold structure and furnishes it with a system of global con

ordinate functions defined by the coefficients of the polynomials Pr, 1 < r < k.

11 k

k

k'

The group operation Ln x Ln + Ln is an algebraic mapping; in effect, it results from the composition of polynomials, followed by a truncation. We need only show that the mapping ~ + ~-l of Lk to Lk is c"' . kn n for any fixed $ e Ln' consider the

We will show this as follows: k

k

m. -linear mapping of Ln to Ln defined by

W+ Wo

$, truncated to degree k. k

This is an automorphism of the finite-dimensional vector space Ln; one shows that it is surjective by applying the Inverse Function Theorem to

This shows that, if one considers the mapping

~.

LkxLk+Lk n n n

its partial derivative with respect to ~ is an automorphism of Lk. n

One then applies the Implicit Function Theorem to the equation

W• ~=·I

to demonstrate that~+ ~-lis

c"'.

The Proposition is thus proved. Remarks.

1.

1

The group Ln can be identified with the group Gt(n,IR)

of automorphisms of IRn. For k' ~ k, there is a canonical projection Lk'+ Lk defined n n' ~ k. This mapping is naturally

2.

by truncating off the terms of degree a Lie group homomorphism. 3.

.:::Ac::c..:;t.=:i;:;;on::.s::....;o:..:f:._;:Lon-_::ac::n:.::d-=oL k on E and Jk. n---n---n

3.1 The group Ln operate in a natural way on the ring En of germs of

c"'

o.

functions at

If f

En and

E

~ E

definition, f o $.-- lf ~*:

I ..;

I

Ln' the transformation of f by ~

will be, by

E + E n

f + f

n 0

"'

is an automorphism of the ring En' k

~

e Ln is fixed, the mapping

M for every k.

Two germs f, g

E

Note right away that ~*(Mk) En will be called isomorphic,

12 if there exists a

~

~

~

Ln such that g = f o

in other words, if f

and g belong to the same orbit of Ln on En.

3.2 Similarly, the group of k-jets of diffeomorphisms L nk (k -> 1) operates on the algebra Jk of k-jets of functions; if f E Jk and ~ t

k

n

n

Ln' one composes the polynomial mappings f and ~

cates f o

to degree k.

~

and then trun-

This represents an algebraic action of

the Lie group Lk on the algebra Jk which is a linear representan n' tion of Lk.

--

n

3.3 By construction, the actions of L on E and of Lk on Jk are nk n k n n compatible with the projections L + L and E + Jn. This just means that g • f o Remark.

+ implies

k

n

k n

j g • j f o j

k

+.

n

The action of L on Mk/Mk+l (space of homogeneous polyn

nomials of degree k) is compatible with the canonical action of GR. (n,lR)

4.

on this space.

Infinitesimal Generators of L

and of Lk.

4.1 The following construction will be fundamental in the remainder of this book. Consider the space lR x lR n with coordinates denoted by (t ,x , x 2 , .•• ,xn).

Let a vector field X be defined and C~ on an open

1

neighborhood of 1R x { 0} in lR x IR n and have the form (1) X(t,x)

il

=at+

a

n

~ X (t x ) -

i=l i

•

ilxi

where the functions Xi all satisfy Xi(t,O)

=0

for all t.

The orbits of a vector field X on 1R x lRn are the integral curves (i.e., the graphs of solutions) of the first order system of differential equations: dxi (?) d't"~ Xi(t,x) i • l, ... ,n. By hypothesis, 1R x {O} is an orbit of X.

Designate by t +

(t,~

(t,x)) the solutions of (2) corresponding to the initial condition ~(O,x)

• x.

13 By the fundamental theorems of first order differential equations, there is an open set 1)

U

c

JR

lRn

x

containing [0, 1]

x {

0}

such that:

U is the union of orbits of X passing through the points (O,x) & U;

2)

Each orbit in U is defined for t on a neighborhood of the interval [0,1];

3)

For each t

u0

[0,1], the mapping

&

ut

-+

x-+ Ht,x) whe:re Ut .. {x ~t(O)

and

lR n/(t,x) e: U} is a diffeomorphism,

&

= 0.

(Note that

~

0

is the identity map.)

Thus, one has a mapping X+ ~1

from the space of germs of vector fields along [0, 1] x 0 c:. JR x JRn , which takes its values in Ln • . This mapping is clearly not surjective; for the determinant of D~t(O)

is positive for all

4.2 Theorem.

Every

~t

germ~ ~

as constructed above. Ln such that

D~(O)

has a positive

determinant can be obtained by the preceeding construction. Proof.

Let's write

~(x) -

Ax + w(x)

where A stands for D~(O) and t is a mapping whose components are 2 in M • We assume that det (A) > 0. 00

Let t + A(t) be a C I and A(l)

= A.

path (t

~

lR) in GHn,IR)

such that A(O)

Such a path exists since the group of matrices

with positive determinant is connected. Set

~(t,x)

c

A(t)x

For each t e: IR, x +

+0

+ tt(x).

~(t,x)

is a germ of a diffeomorphism at 0 and

is the identity. Set X(t,x)

a + ~1.. 3 ~1 _a_ - -at i=l - GR.(n,lR) such that

= I,

1)

H(Q

2)

Q= Q

0

) 0

the identity matrix,

H(Q) for each Q in U.

•

The proof of this fact is left as an exercise; it is a consequence of the impl·icit function theorem. B,)

Given the above, the expression (1) of f can be written

f(x) = Qx(x)

(2}

where

~stands

for the quadratic form whose coefficients are qi.(x). "'

J

Set Hx • H(~). The map x ->-Hx is defined and C on a neighborhood of 0; furthermore, H • H(Q ) • I and one has 0

0

~

=Q

0

•H,

0

X

It follows that (3}

f(x) • .

Let q,(x) = Hxx. lRn ,. Since H

0

0

~

(x)

= Q0 (HX x),

One thus defines a germ at 0 of a mapping from

4>(x) • x +terms with components in· Thus, D4>(0)

=I

nt to

= I; one has

M2 •

and 4> is a local diffeomorphism,

One then has Q.E.D.

f

20

2.2.

Remarks on the preceeding proof. 2.2.1.

L

In fact, Morse's Lemma describes a family of orbits of Consider the canonical projection

n

M2 ~ M2/M3

= P2. n·

1P

The action of L on n

is compatible with the action of Gl(n,m.) on

p2n (see II.3 .3). 2 The inverse image under j of every open orbit of the 2 • Gt(n,lR) action on Pn is an orbit of the L action on E •

Proposition.

n

n

This is a trivial consequence of Morse's Lemma. The preceeding remark suggests the following. Let r k k be an orbit of the group Ln on the space Jn of k-jets of functions, 2.2.2.

having the following property: For every f in g

E

r,

Jk, jk-l(g-f) k

n

there exists an£> 0 such that

= 0 and

II g-f II




g £ r.

Then, the inverse image of r in

En by the mapping

is an orbit of L in E • n n This conjecture is motivated by the following false proof, which is exactly modeled on 2.1.

(Exercise:

find the error.)

It is clr.arly enough to show that if f it

a

F0

•

Q for each t e: [0,1].

In particular, F 1

= f = Q 0 cj>~l

Q.E.D.

The fact that the components Xi of Y. are 1-flat on [0,1] implies that Dcj>t(O) = identity on :IRn for each t. Remarks.

1.

X

{O}

(See II.4.2,)

2 We have not used the fact that Q has in Pn an open

orbit under the action of Gl(n,lR).

In fact, as we will soon see,

24

this will be a consequence of the argument used,

z.

One can avoid using a partition of unity and can prove Morse's

Lemma using a local version of Lemma 3.2.

This local version uses

the argument of 3.3 to show that for each T

e;

IR, there exists an

c > 0 such that F t is isomorphic to F T if It-T I
M• J(f).

I f A £ J(f), one has A ..

n af 'I' [ . ".. i ax-.•

i=l ~i E

En.

A"'

Write ~i(x) • ~i(O) + vi(x), where vi£

r

i .. l

11

i

M.

~

It follows that

co)~+ ~l. Ex

has infinite codimension over JR.. 3.

n

The notion of a universal deformation of a germ of finite codimension.

3.1.

Let F be a p-parameter deformation off

h (0) =

c, be a germ

r.:

(lRq

o:f

€

En. Lets-> t=h(s).

a c"" mapping

,Ol ... (mP ,Ol.

One can now define a q-parameter deformation of f, called the pullback of

f'

n,

by

=

h*F(s,xl

by F(h{s),x).

For any deformation F , denote by with respect to its various

F.l. (x)

def

Then, if we set G

~~.

Fi e En

(O,x)

l.

h*F, one quickly finds via the Chain Rule that

p (

}. j=:

the initial speed of F

p~rameters:

.t .. ] 1 •

• •

1~

39

In particular, this means that the initial speeds of G belong to the real vector space generated by the functions

Fj

in En'

More generally, if G is a deformation of f isomorphic to h*F for some mapping h: (lRP, 0) + (lRq, 0) ,

then one has the same conclusion

modulo J(f), that is to say that the projections of the Gi's into the quotient space E /J(f) belong to the real vector space generated n

by the projections of the

Fj's.

(Exercise).

Two p-parameter deformations F and G of f are called eguivalent if there exists a local diffeomorphism h: (lRP, 0) + (lRP, 0)

such that G

is isomorphic to h*F. We will see that it is with respect to this notion, a little larger than that of isomorphism, that it will be convenient to classify the deformations of a given germ f in En.

3.2.

Definition,

One says that a deformation F of f

£

E~

universal deformation of f if any other deformation G of f is isomorphic to the

pull back

of

F

by a suitable Coo mapping between the two

parameter spaces. Proposition.

If f

£

En admits a universal p-parameter deformation F,

then f is of finite codimension and the initial speeds Fi generate E/ J(f) as a vector space over lR,

Proof.

i.e.,

Let g be an arbitrary germ in En'

deformation off:

G(s,x)

=

f(x) + sg(x).

Consider the one-parameter By hypothesis, G is iso-

morphic to h*F for some germ of a mapping h: (lR, 0) G

=g

+

(lRP, 0). By 3 .1,

belongs, modulo J(f), to the real vector space generated by the Q.E.D.

3,3,

Proposition,

Let F and G be two p-parameter universal deforma-

tions of the same germ f in En' where p

= codim

f,

Then, F and G are

equivalent. Proof.

Since F is universal, there is a germ of a C~ map

40 h: (lRP, O) -+ (lRP, O) v + u

=

h(v)

such that G is isomorphic to H

h*F.

Sihce G is universal, so is H.

Furthermore,

•

H

:-

i

t:! L

ah~ ---oL (

j=l avi

0) • F

j"

But the systems {H , ••• ,Hp) and {F , ••• ,Fp} are both linearly inde1 1 pendent over lR by Proposition 3,2 since codim f = p. Therefore, the matrix Dh(O) is invertible and h is a local diffeomorphism,

q;E,D. One shows in the same way that, if F is a p-parameter universal deformation of f, where p = codim f, and if G is a q-parameter universal deformation of f, with q > p, then G is isomorphic to h*F where h: (lRq,O)

+

(lRP.o) is a submersion, i.e., Dh(O) is surjective.

Thus G

is equivalent to a constant deformation of F.

3.4.

One of the goals of this book is the proof of the

Theorem,

Let f

£

En be a germ of finite codimension.

Then,

2_E::

parameter deformation F of f is universal if and only if its initial speeds are such that J(f) +JR. {Fl' ... ,Fp} ~ En. In other words, they generate En/J(f) as a real vector space. In this chapter, we are going to indicate with a couple of examples some of the mechanisms in the proof of the above fundamental theorem. We will present the complete proof in Chapter XI.

3.5. 1,

Exercises: Let f

E

En be a germ of finite codimension and let k be an integer

such that M • J(f) ::> Mk.

Show that there exists a number

£

that: i f g is

rorm on

< e: (for some fixed

> 0 such

41

2,

Let f.

En be a germ of finite codimension and let F be a deforma-

E

tion off such that J(f)

R{F1 , ••• ,Fp } mE. n

Let k be an arbitrary

Show that, for each n > O, there exists a number

integer.

that, whenever g

E

a (t,x) in lRP x lRn

j~t(x),

morphic to 3.

+

E satisfies JJj kg - j kf JJ n

( IJt II


pbg defined bl•

f;

t

and by h: (lRP,O)-+

43

it is clear that F is isomorphic to h*F , the isomorphism being de1 fined by integrating the vector field X, Remark,

5.

The converse of the preceeding lemma is true,

The universal deformation of a Morse germ

5,1.

Theorem,

Let f be a Morse germ in E,

formation F(t,x)

5.2.

= f(x) +

n

The one-parameter de-

t is a universal deformation of f.

Proof.

5,2.1. u

(Exercise)

Let us first consider G(u,x) an arbitrary deformation of f, Denote by E +n the ring of germs of C"' functions of (u ,x)

t: lRq,

at the origin of :R

q q n x

1R •

Denote by Eu the ring of germs of c""

functions of u at the origin of lRq.

We will identify E with the u

subring of Eq+n composed of functions independent of x. Denote by J (G) c Eq+n the ideal generated by the n partial derivatives

()G ax i

t:

E +n' i q

=

1, ... ,n,

Note that the restriction of J(G)

to the "fiber" jO} " lRn is the ideal J(f) in E • n

Proposition.

If f is a Morse germ, and G is an deformation of f,

~

with the above notation

(In fact, this equality is equivalent to the statement that f is Morse.) This proposition implies that every function of (u,x) can be decomposed as a sum of an element of J(G) and a function of u, Proof.

Consider the mapping:

'dG

(u,x 1 , ••• ,x) + (u., - (u,x), ... , 3xl n

3G axn

(u,x)).

This mapping is a local diffeomorphism at the origin:

j !'

I:

vertible since the Hessian of f is nondegenerate. In the target space, clearly

Eq+n · • I+ Eu'

D~(O)

is in-

44 where I denotes the ideal generated by the coordinates x , ••• ,xn' 1 which we know can be identified with the ideal of functions that are (Proposition 5.1 of C~pter I.)

zero on JRq x {Q}. ~*E

q+n

.P*Ic:

But,

$k~~

~*E

s

q+n

+

~ ~*I

~*E u

•

J(G), by the definitions of J(G) and of ~

fu• since D

Eq+n'

It follows that

.P,

is an unfolding

since 2

Folds and Cusps for Mappings of the Plane to the Plane

2.

Let

be a germ of a

F:

the coordinates on the source space by on the target space by (1)

with

I=

(X,Y)

(x,y)

• Then, F

.

C

mapping.

Denote

and the coordinates

is defined by:

X= X(x,y)

Y

Y(x,y)

X(O,O)

= Y(O,O) = 0

•

Denote the groups of local diffeomorphisms at the source and at the target by

L

s

and

respectively

(s for "source" and t for

"target"). The derivative, or Jacobian of

F , will be denoted by

DF •

All the properties and conditions which we will state and work with should be understood to be local ones, that is, true only on a sufficiently small neighborhood of

0 •

The text would become

unreadable if we had to make this statement explicitly every time we use it.

52 2.1.

If

then

F

DF(£)

is invertible, i.e., if

F

has maximal rank (two),

is a local diffeomorphism (by the inverse function theorem)

and is isomorphic to the mapping

(Caution:

do not call this the "identity map"; such a statement

would not make sense since we are considering the source and target as distinct planes.) 2.2.

Suppose now that

Lerrana.

In this case,

I~

(2)

F

X

f(x,y)

has rank one.

DF(~)

where

is isomorphic to a mapping of the form:

~

(0)

3y

-

= 0

We can find linear changes of coordinates

Proof.

source and target respectively under which Let

F

ax

=B

ax IR (s ~ 2)

plane, it is true in general that a global map has only folds and cusps as its singularities.

The points where

has rank 1 form a closed submanifold of dimension one. curve, the cusp points are isolated.

On this

F

65

CHAPTER VI THE DIVISION THEOREM OF ORDER TWO

Introduction.

1.

The statement of the general Division Theorem

El+n = Ex,y

Consider the ring

1.1.

functions

f(x,y)

where

(x,y) •

of germs at

(x,y , ... ,yn) 1

0

of

c

DO

belongs to

1Rl x 1Rn

Definition. k

P in E has the division property of order x,y , there exist Q x , if for any f in E x,y in E (germs which depend only on y) y

A germ

with respect to

in E and x,y such that

k

f(x,y)

(l)

P(x,y) • Q(x,y) +

~

k-i ri (y)x

·•

i=l Example.

xk

P(x,y) k f(x,y) -

~

i=l

has this property: ak-if

1

(k-") 1

=0

•

~ (O,y) • 1

1 •

is, by construction, (k-1) x

The function. k-i X

ax

flat at every point of the hyperplane

Consequently, it belongs to the ideal generated by

(Chapter I; Proposition 5.1). k f(x,y) .. x. • Q(_x,y) +

~ l.

i=l

1 ilk-if • xk-i (k-1.)! ilxk-i (O,y)

and the division property is established for Remark.

xk

Thus,

xk

It easily follows that the division property of order

is also true for

k

66 P(x,y) = xk • h(x,y),

where

t-

h(O)

0 •

(Exercise. ) Definition.

1.2.

order

k

P(O)

in

~

ax

x (0)

A

germ

in

P

Ex,y

is called regular of

if

=

k-1

=E...____!: (0")

0

axk-1

This means that the restriction of

where

h(O)

P

t-

and to the x-axis can be written

0

and also that where

The Division Theorem of Order

k •

(B. Malgrange, 1962).

Every germ which is regular of order division property of order 2.

k

x

with respect to

satisfies the

~

Reduction to a particular case; the canonical division theorem.

2.1.

Consider the polynomial:

in the real variables x, o , o , •.. ,ok • 1 2 canonical polynomial of degree k. Consider the ring of

in

k

c"'

functions of

as an element of

Ex,y,o x,y, and

Ex,y,a

of germs at

a

We will call

0

in

lR

"

It is clear that

is regular of order

jc

in

Pk

Jk.-, pk

~

the

"

JJI.i j

(5)

ar. aa.

~ {0,0)

-Q(O,O,O)

J

With all this in mind, consider the eguations r (y,cr) 1

0

(6)

ar. [ dO'~

The matrix

in the unknowns

J

(0,0)

is in-

J

vertible; in fact, by (5) it is triangular and all the diagonal entries are equal to

- Q(O,O,O), which is non-zero by (3)

There-

fore, by the implicit function theorem, the system (6) has one and only one solution Set

a(y) , which is

a = a(y) = (a (y), ••• ,ak (y)) 1

C"" ,

in

with

(1)

•

a(O)

= 0

(by (3)).

One obtains

k k-1 . P(x,y) = (x + a (y)x + ••• +ak(y))Q(x,y,a(y)) 1

(7)

We have thus established the following fact: if

Pk

P(x,y)

has the division property of order which is regular of order

k

in

x

k , then any function can be written in the

form (81

where

P(x,y)

Q(O,O)

= [xk F0

+ o (y)xk-1 + .•• + ok(y)]Q(x,y) 1 and

a , ••• ,ak 1

belong to

Ey ,

in fact, to

MY

69 Second Step.

We now show (still under the hypothesis that

the division property) that any germ order

k

Let

in f

x

P(x,y)

Pk

has

which is regular of

has the division property.

Ex,y

be in

and divide it by

Pk •

Consider this· germ as a germ in

Ex,y,cr

We find

f(x,y)

Now replace the independent variable function

cr(y)

f(x,y)

X

(9)

k

in this equation by the

I

One finds

k-1 + a (y)x + ••• +ok(y)JQ (x,y,cr(y)) + 1 1

k +

a

obtained in the first step.

si(y,a(y))

• X

k-i

i=l Equations (8) above can be written as (don't forget that we are working with germs of functions and that

~ = Q-l x

k

is

c"'. on a neighborhood of

Q(O)

is not zero; so

0):

+ cr1 (y)xk-1 + ••• + ok(y) • P(x,y) •Q-1 (x,y) •

Substitute this equation into (9) to obtain the division of

f

by

Q.E.D.

P •

2.3.

Comments. We have now reduced the

order

k

proof of the division theorem of

to the proof of:

The canonical division theorem of order

satisfies in the ring

Ex,y,a

k •

The canonical polynomial

the division property of order

k .

70 On the other hand, we have shown that a consequence of this theorem is the existence of the form (8) for a function which is regular of order

k

in

x •

This result, which is very important

in its own right, is called the Preparation Theorem.

It was first

proved for analytic functions by Weierstrass. Let's see why it is important (and why it is called the "Preparation" theorem). germ of a function

f

Suppose that one is given a at the origin in

one wants to study the se·t Suppose that

f

l:

Rn+l

(~n+ll

{f=O} , assuming that

C"' (analytic) and that f(O)

=0

is not flat at the origin, i.e., that its partial

derivatives· at

0

are not all

0

(in the analytic case, that

f J. 0)

(x,y , ••• ,yn) in 1 x of some finite

We can always choose linear coordinates lRn+l

(resp.,

order

k

f

0 .)

ai:

~n+l)

such that

f

is regular in

(The smallest such integer Equation (8) shows that

k l:

is called the order of is also given by the

equation

since

~

Q(O,O)

(x,y)

0 •

~

Consider the mapping

y

The above equations show that and less than

k

points in

the above polynomial in reduces to

around

k 0

-1

(y)

contains at most

k

points

~n

if and only if the discriminant of 1 is zero at y • (In particular, n- (0)

0.)

E appears then, locally, as a branched covering, with

The set at most

x

1f

branches on a neighborhood of in

~n) •

0

in

lRn (k branches

This illustrates that the preparation theorem

is an essential tool in the local study of the level sets of a function.

71 3.

The reduction of the division theorem of order 2 to a lemma of Whitney.

3.1.

Preliminary remarks. The proof of canonical division theorem of order 1 is simple.

One has

E

x,y,o r(y,cr) X

+ 0

P (x,o) = x + o , for x,o 1 Set r(y,a) f(-cr,y,cr)

in

Let

consider

By its construction, the germ

=0

IR. g

f

be in

g(x,y,cr)

=

f(x,y,o)-

is zero on the hyperplane

Thus, by Proposition 5.1 in Chapter I , it belongs to

the ideal generated by

(x+cr)

in

E ; and this gives us our x,y,

quotient. On the other hand, the proof of the canonical division theorem of order two is certainly non-trivial.

We will carry out the

proof in this chapter by reducing it to an old (1939) result of Whitney on even functions. 1)

There are two reasons for this strategy:

The lemma about even functions (see below) was the "tool Whitney

used to obtain the first non-trivial canonical forms for singularities of differentiable mappings (1940) , and we want to emphasize the fact that it is all one needs to prove the results in Chapters IV and V 2)

We will prove Whitney's Lemma by using some ingenious ideas

that Lojasiewicz used in his proof of the general canonical division theorem (1970).

Thus we will have the essential ideas of his

proof but in a form that is technically simplified. the general division theorem in Chapter IX • 3.2.

Whitney's Lemma. Consider the mapping

We will prove

72 Denote by E (resp. E ) the ring of germs of c"" x,y u,y at the origin of the source (resp,, the target) and by

Eu,y

4>*:

+

Ex,y

the homomorphism

= g(x 2 ,y)

(cf>*g) (x,y)

Whitney's Lemma.

= cf>*g

For each

f

Ex,y

in

which is even in

f(-x,y)) , there exists a

(i.e., f(x,y) f

functions

g

x

in

Eu,y

such that

r

and

in

•

In fact, this lemma is equivalent to For each

Lenuna. E u,y

f

Ex,y ,

in

there exist

1

such that

f

f(x,y)

i.e-.,

= xr 1 (x2 ,y)

2 + r (x ,y) 2

We need only show that the first lemma implies the second. let

E x,y

be in

f

f(x,y) + f(-x,y)

and Then,

2

By the first lemma, other hand, in

y ,

Chapter

since I.)

write

u(x,y)

v

1

v(x,y)

= +*r

=

2

• f (x, yl + f (-x,y)

Then, w

=

cf>*r

in 1

x

is even io

2

for some

2

f(x,y) - f (-x,y) +

r}.

iP

= 2 [f(x,y)-f(-x,y)) = xw(x,y)

is odd

n

f(x,y)

So

Eu,y where

X

on the w is even

(Once again use Prop. 5.1 of

for some

r

1

in

Eu,y

and it follows

that f = x • cf>*r

3.3.

1

Proposition.

+ cf>*r 2 • The canonical division theorem of order two is

equivalent to Whitney's Lemma.

.

73 Proof.

1)

form) •

Let

degree 2

Set

The division theorem implies Whitney's lemma (second 2

.

r (u,y) 1

s

(u,y)

s

r

2

+ cr 2 , the canonical polynomial of and divide it by P in E . : 2 x,y,o

P (x,cr) = X 2 Let f be in

1 2

(y,O,-u) (y,O,-u).

We need only replace

-x

0

2

in (1) tc

verify that

The converse.

2)

Consider the (global) diffeomorphism:

lfi:

]R

X

JR2

01

(x,y,o ,o l. 1

]Rn

X

(x

2

012

+2,y' 0 1'- 0 2 +4)

It is clear that P

2

o

ljJ

-1--

denoting by Let

f

-2

(x,o)

=X

x,y,o be in

.- (x ,y)

is surjective.

Suppose

a (unique) function

f(x,y)

g(u,y)

is even in

x •

There exists

such that

f(x,y) u ;J. O,

For

is analytic on a neighborhood of

g

(u,yf; in fact,

is a local (analytic) diffeomorphism at each poing

~

2)

The function

g

tinuous and the mapping

is continuous; in effect, ~is

proper (exercise).

(x,y), x"" 0 . f

is con-

Therefore, by a

classical theorem about analytic functions, the function which is analytic off the hyperplane

u

=0

g ,

and continuous every-

where, is analytic. 4.2.

In the differentiable case, we are going to establish Whitney's

Lemma by mimicking the preceeding argument. 11

a: :.;

I'l

lRn, consider the set

E :: i(x,y)lxec: So, $: E

'

is real or pure imaginary}

E

=E

E

.... lR

X

~

i

surjective map.

U F'

0 ,

if

u < 0

is obviously a non-zero

.,

c

element in

Differentiable functions on real subsets of A real form in O:n

5.1.

a)

dim

b)

E

E =

n i.E

n,

= {0}

o:n

is any real vector subspace

such that

•

Eu

E

of

a:n

76 Conditions a) and bl are equivalent to the condition

and also to the condition that every basis a real vector space is a basis (over real form

E

~n

in

~)

{e , ••• ,en} of E as 1 of ~n , Thus, for any

there is a complex automorphism

h

~n

of

such that h(E) = m.n c: a:n Examples. In since

a:

(the "canonical real forro").

In the complex plane 2

iE

, the (real) plane

=E

•

E

a: , every line is a real form.

a:

x {O}

is not a real form

m. x lR c: a: x a:·

But, E

is a real form,

In the Grassman manifold of real n-planes in

Exercise.

the subset of real forms is open and dense. Let

5.2.

E

defined on

be a real form in

~n

E (or on an open subset of

and let

f

(!r. ,

.

be a

C

E) with values in

function ~

•

Let

{e , .•• ,en} be an arbitrary basis of E, which is also a basis 1 of !I'r.' a:-'d le ... x , ••• ,xn be coordinates ill a:n relative 1 to this basis. Then,

f

is a

..

C

function of the real variables

Its Taylor series of order

k

at a point

a

in

(x , •.. ,xn). 1 is defined

E

by the identity:

whereo

( = , .. J.' •••

•\,>

-

(xJ-a 1 , ... ,xn-an)

is in

m.n, p

homogeneous polynomial of degree i (l -< i < k) and Mk+l The identity (l) only makes sense for { .in {

l.

r

is a

is in

lRn' but the

Taylor polynomial

is defined on the ambient space

a:n .

One easily shows that it

does not depend on the choice of basis (e , ••• ,en) of 1

E

77 This shows that i f in

f

is a

c

function on a real form

~n , it~ Taylor polynomial of order

k

identified with a polynomial of degree write of

j~f

k

can be canonically on

~n •

We will

for the latter and call it the complex Taylor series

f The above definition of the complex Taylor series is the

easiest definition to work with; the following equivalent definition is more "intrinsic". derivative of. f

At each point

is an

a

of

E , the kth order

lR-multilinear, symmetric mapping:

(Here, we are only using the structure of

~

as a· two-dimensional

real vector .space.) Now since

E

is a real form of

canonically extend

Dkf(a)

~n (E + iE

=

a:"l , one can

to a mapping that is a:-multilinear and

symmetric:

For example, for

k

=

1:

Dcf(a)l; = Df(aln + iDf(a)n', for where of the

n,n'

in

D~f(a)

E for

are determined by k > 1

l; l;

in

a:n ,

n + in'

The definitions

are similar.

Important Remark. If

f

is an analytic (holomorphic) function on

clearly for each integer (2)

where

a:n

1

then

k (a e E)

fE

denotes the restriction of

f

to any real form

a:" and the right side of (2) represents the complex Taylor polynomial

f

at

a .

E

in

78 The complex Taylor polynomials of

c "'

functions on a real form

obviously have all the formal properties which the usual Taylor polynomials have, e.g., the polynomial of a product is the truncated product of the polynomial factors. Definition.

5,3,

A subset

~n

of

E

will be called real if it

is a finite union of real forms:

U

E

Ei

i=l, ••• ,r

. c

A mapping

U+

f:

, where

~

is an open subset of

U

is called

E

if: ll

for each· i = 1, ••• ,r is

Ei n. u 2)

c

the restriction

r

for each

i,j (i

j~fi (x)

j~fj (x)

and

j)

~

l.

If

f

is

Let

n

Ej

k > 0

jcfi (x)

=

of

nu

f

r

to

one has

.

jcfj(x); where

f •

.,

E (see the above paragraph) is

C

F

E = lR U ilR

defined by

l.

an analytic function on an open subset of

~n , then its restriction to

for any real subset

Ei

for any integer

is the· complex Taylor series of

Examples.

2.

iT'

X

Condition 2) can also be written as jcf(x)

f.

, let:

c~

f:

E + IR

be the function

1

-2 f(x) =e f(x) =0

X

for

x e lR

for

x

e

i:lR.

Condition 1) is obvious, so is condition 2) since the restrictions of

f

to each of the axis are

not thP. restriction to

E of

c"'-flat at

~

C •

Note that

holomorphic function.

f

is

79

3. Let

Let

f

x

(where

So,

E

be the same set as in the preceeding example.

be the restriction of the function

ocfl (O) ~

I;; , where

DCf

-~;;

2 (0)~

..

satisfied. Given any

C

of a real set

E

function

k

independent of

£

I

x

f:

E

U

to

£

to f

lR,

to~

iiR.

since condition 2) is not

~ ~

, where

U

is an open subset

~n , we can set

where

X e Ei

nu

•

This definition is

i ; by 2).

Taking into account 5.2 and the above definition, the usual properties of differentiable functions also hold for functions 00

which are

C

on a real subset of

functions on· a real set c(E) •

The germs of

E

Remark.

+: are

E~

E C. ~n

Let ~p

c"" E •

Thus,

••:

E(F) f

f +

~

00

C

form a ring, which we will denote by 0

in

E

also form a ring,

F c~k

be real subsets.

mapping, i.e., its· components

functions on

be clear that if on

In particular, the

c(E,O). and

c""

be a

~n

functions at

Coo

which we will write as

E

.

is in

Suppose that

f(F)

,

then

op(E)

CF

•+p

It should

is a

f • +

Let

.1 .... .

c"" function

defines a ring homomorphism:

E(E)

t-o. f • • •

A similar remark applies to germs. 5,4.

Theorem (Lojasiewicz, 1969).

subsets of

~n

with

E ~ E •

a:

One has:

is the restriction of

2

a:

from

is the restriction of

1

function on

in

for any integer

f

, where

is not a C~

f

x).

stands for the conjugate of

~

x

Let

E

and

E

be two real

The restriction homomorphism

80 E(E,O) + E(E,Ol

is surjective. This is a theorem about extensions. a

c"'

function at the origin of

a

cao

-function

can be extended to a germ of

E

at the origin of

f

It says that any germ of

E

The proof is not easy; it will be carried out in Chapter VIII. In this chapter (§7)

we will prove a particular case, sufficient

to establish Whitney's lemma.

At the same time, this particular

case will be an essential lemma in the proof of the general case. Lemma.

5.5.

Let

lR " lRn C 0: " ctn

E

(c • lR) x lRn c ct x a:n , where

=

E

Le~

E U E' •

c

and let

E' =

is a non-real complex number.

The restriction homomorphism

ECE,O) + E(E,O)

is surjective. We first assume this lemma and show how this lemma implies Whitney's lemma.

6.

Proof of Whitney's lemma. Let

6.1.

$:

E .... 2

(x,y) >-+-

i

where

a:

x

a:n.

6.2.

,y)

I

(x,y)

e

E.

cc "

a:n , be the map

1

= E U E", E ., 1R x lRn c. a: x a:n , and

So,

1R x lRn

(X

a: )( a:n

;p (E)

1R

x

lRn

and

q,

is a c"'

E' ., (i • lR) x lRn c:::

mapping of

E into

, since it is the restriction of an analytic mapping.

Let

f e E(E,O)

be a germ of a function, which is even in

By the preceeding lemma, there exists an

an extension of ffx,y)

f •

f

Set

f(x,y) +f(-x,yl 2

(x,y)

e

B •

in

E(E,O)

which is

x.

81 Then,

f

is a new extension of

f(x,y) 6.3.

f(-x,y) , for

f , which is itself even in

x

x:

real or pure imaginary.

There exists a unique germ of a function

g:

(lR x lRn ,0) +a:

such that f = 9., ~

The function since

~

where

x

~

9

is

C00

'1- 0

on the open set

(C 00 )

is a local

u

I 0

( (u,y) € lR x lRn)

E

diffeomorphism at all points of

Note also that

g

f

is continuous, since

is

and

is a proper mapping.

6.4.

To show that

g

is in

to show that the derivatives of admit continuous extensions.

E(lR x lRn ,0), it is sufficient g , all well-defined for

u

t

0 ,

This implication follows from a

classical lemma, which is demonstrated in the appendix of this chapter. We will now show that these extensions of the derivatives of g

exist by induction on the order

k

of the derivative.

the notation simple, we will only carry out the first induction argument.

To keep

step in this

The rest of the argument will be clear.

Notice first that if

(x,y)

is in

E

and

x

t 0 , then we

have

(1) Let

Dcf = [ (Dcg) • ~) • DC$

that the matrix for

DC~

everywhere that it makes sense. is

e: ~) where

I

equation

!'

represents the identity on

a:n .

Consider now the

Note

82 (2)

A(x,y) • Dc(x,y)

where

(E,O) +

A:

l(~ ~ ~n.~l

is unknown.

This is a linear equation whose coefficients are on and

I

E C"'

claim that (2) has a solution

and which is even in

X

A

,

c"' functions

which is unique

0

To see this, multiply (2) term by term by the adjoint matrix •.

ljl(x,y)

(1

=

0

0

)

2x·I

It follows·that: Dcf(x,y) • ljl(x,yl =

(3:

The left side of (3) i$ It is odd in

2x. A(x,y) C"'

on

E , with values in

So each component function of the left side of

x

(3) belongs to the ideal generated in one need only apply to

E

and

I.5.1

E(E,O)

by

x; to see this,

to the restrictions of each component

E' •

This proves the existence of uniqueness and parity in even and continuous on

x

A , a

c"'

solution of (2)

should be evident.

Now, since

E , there exists a unique continuous

mapping (by the inductive hypothesis)

such that (4)

A = Al o

.

If we compare (1), (2), and (4), it is clear that

and we have shown that

has a continuous extension.

Its A

is

83

by

Now, carryout the same analysis, but replace f and g 2 A and A1 , to show that o g has a continuous extension;

and so on. 7.

Proof of Lemma 5.5.

7.1.

Actually we are going to prove the global version of this

extension lemma. Proposition.

Let

a: x a:n , and let number.

lR x lRn c.a: x a:n, let

E

E • E

U E' , where c

E'

=

(c·R) x lRnc

is a non-real complex

Th.en, the restriction homomorphism.

is surjective. 7.2. a(x)

be a

C

on a neighborhood of

Let

0

Proof.

=1

For any integer

a(x)

n

~

function on and .a(x)

=0

lR

such that lxl ~ 1

for

0 , consider the linear mapping

E' -+- E

h : n

(x,y) !-+" (2n • ~ , y) • c Finally, let

An ·be a sequence of complex numbers,

n

~

0 ,

chosen so that the series

I

ll (z)

A

n~O

n

z

n

converges in the whole complex plane and defines an entire function

11

Let (1)

sf

f

be in

.E(E)

L A • (fl n>O n

0

Set

•

f (x,y) 1

o:(x) • f(x,y)

h ) n

I claim that the series (1) converges at every point of defines a

C~

function

and

sf

on

E' •

E'

and

It suffices to verify that

this series converges locally uniformly, as well as·all the

84 formally differentiated series.

~(y)

Denote by

the upper bound

of the absolute values of the partial derivatives of order ~

k

of

£

1

= Qf

IR x {y} _c E •

on the set

One shows easily

that

where

ck- is a constant which depends only on

k •

The local uniform convergence of the series for the k-th derivative of (l) follows from the absolute convergence of the series for

~

z = 2k •

at the point

We have constructed an extension f on f

t

of

f

to

E

E ,

= sf onE' ,

if we have chosen the coefficients

A

Definition 5.3 is satisfied for

at those points

E

n

E' ,

i.e. , where

x

=

f

n

so that condition 2) in (x,y) in

0 •

So let jf

= L

a

p,q~O

p,q

t;p · 11q

be the Taylor series of n

11" (., 1 , ••• ,11nl €: lR; q ql

~

111 , .• ·•11n that the variables

f

at such a point; where (q , ••• ,~), qi 1

The complex series

t;,n are now allowed

The complex series

jc(sf)

~ 0,

jcf

i~

~

e IR,

and the same, except

to be complex.

can be obtained from the above

series by simple substitution, since the mappings

E' + E

are restrictions of complex automorphisms of

It follows

that

r

n,p,q These two series are identical if and only if

85

I

for every

~ Q

p

I

n~O

~(2P)

i.e., if and only if

= cp

for every

p ~ 0 •

However; we know that there exists an entire function which satisfies these conditions:

sequence

{6 }

in

p

~

The mapping

Remark.

which goes to infinity and another

, there always exists an entire function

ap

such that

by a theorem of Mittag-Leffler, ~

{zp } in

given a sequence

for all

p

This finishes the proof.

E(E) -+ E(E')

f I-+ sf is continuous if these two rings are given the topology of uniform convergence on compact sets for the functions and all their partial derivatives.

In this case, the above mapping

1 is called an extension operator.

7.3.

The local Lemma 5.5 follows trivially from the above

Proposition. function let

g

=

If

f (l •

is a germ in

f

which is

,

f

where

c (l

DO

E(E,O),

on an open set is a

c"'

equal to 1 on a neighborhood of

0

centered at

u

0 and contained in

Proposition to

g

in

E(E) ,

f

be the germ of

and let

E(E)

represent

f

by a

u containing

function on

m

and equal to

0

Then,

0

mn

)(

which is

outside a ball

Now, apply the preceeding

to obtain an extension g

in

E(E,O)

g

of

g

in

•

Note that in the local lemma one loses the algorithm described in 7.2. Important Remark •

Proof 1:2, that of Whitney's Lemma, and the proof of 3.3 show that the canonical division theorem of order two is true globally, i.e., on the ring of m x lRn x m2

c""

functions on the space

(x,y,a), and that in this case there is a division

operator which furnishes for each

f

a quotient and a remainder

86 depending continuously on

compact sets for

f

f

(topology of uniform convergence

on

and all its partial derivatives) • This fact

has important consequences in the global theory of singularities. 7.4.

As an exercise, a more elementary (but less sharp) proof

of Lemma 5.5: E. Borel's Lemma.

1)

there exists a at

Given any sequence of complex numbers f(x),

fun~tion

x

suggestion. hood. of

Let and to

c

be a

for

00

function equal to ~

1

and show that there is a sequence

t

0

I

f (x)

0

enough so that

n

~

is

f

aPf

Consider

n

in

(O,y) = pia

lR

which grows fast

•

Given a sequence of

JRn , there exists a

axP

.

on a neighbor-

n

Generalization.

-

lxl

1

a cp(t x)xn

n >0

3)

n

·axn n

I n>O

2)

a

whose Taylor series

is

0

y in

e IR,

c"'

Coo

function

functions

f(x,y)

ap(y) ,

such that

(y)

P

The preceeding result implies Lemma 5.5.

Appendix. Let equations

E c JRn

by an algebraic set:

E

is defined by some

87 ~ere

•et

the f:

Pi

functions are polynomials

lRn + lR

·pen set

be a continuous function which is

lRn

C"' on the

lRn \ l: • If the derivatives of

,f

(not identically zero) •

then

I

f

...

is

f

admit continuous extensions to all

on

C

It is sufficient to prove the lemma in the

'roof.

If

case •

ne has f (a+b) - f (a)

1)

or every

a 1b

apping of

in

lRn

to

( lRn

G(a+tb)bdt 1

where

L ( lRn 1 lR)

G:

x

then

t-+

f

is

G (x)

c

1

is a continuous and

G

= Of

(funda-

1ental theorem of calculus). ,f

With this fact in mind, suppose that G is a continuous extension Df, i.e., G: lRn + L(JRn ,JR) is continuous and G(x) = Df(x)

or all x

=

X

a+tbjt

in

e

ertainly has

IRn \l: [0,1]} (1).

Then, if the segment meets

l:

If not, since

{a ,b } such that p p .he segment [ap ,ap +bp l meets sequence

Exercise). f

f

and of

ap l:

l:

...

is algebraic, one can construct a, b

Q.E.D.

p

+b

and for each

p

,

in a finite number of points.

Formula (1) is then true for G .

[a,a+bj :=

at a finite number of points, one

[a,b]

by the continuity

88

CHAPTER VII THOM'S TRANSVERSALITY THEOREM

1.

Introduction and Review

1.1. a

We assume that the reader is familiar with the notions of suhmanifold of

Cr»

manifold.

1Rn

00

and of a

function

C

such a sub-

OJ"

In order to set the notation used later on, we remind

tPe reader that an embedded submanifold of codimension of dimension

n-p) in

following property: borhood

of

1J

x

lRn

io; a subset

for every ir

Jlln

~

and a

.

of

!'

in

S

r

mapping

lRn

p (or

having the

there is an open neigh-

4•

U + mP

such

that

11

?

is a submersion (its derivative

surjective for all 2)


p

as

I

measure zero if

If

is by (1) with side of length 21< • (t/n)k+l

is covered by at most

np

(~/n)k+l ; the sum of their vclurnes in

of length

~/n •

f

This quantity tends to Thus,

i.e.,

k > £ - 1

0

f(Ck

n

cubes each lRq

as P)

is


0 , there exists an f + A:

X 1-+

f(x) +Ax

at

x

Df(x), the derivative of

transverse to all the submanifolds Theorem.

operates in a

(Cw) mapping

~-+

Df:

x Gt(q,lR)

where

,

Clearly, the manifolds

5.2.

G.l.(p,lR)

by setting

fi C L

if

Of

is

p,q

be any differentiable mapping. A e L p,q such that is nice. (Here,

IIAII

II II

For

< c

is· any norm

on

L , for example, the maximum of the absolute values of the p,q coefficients of A .)

Proof.

Consider the mappings lRP x L

F:

lRP x L

G·

G(x,J) The mapping

set of

p,q

....

G

L

+ A.

X

p,q o~

FJ

a+

7

•

is clearly a submersion; so it is transverse to all Ei

A's for which

j ; e>

lRq

Df(x) + ;1. , the derivative

the submanifolds

ii ...

....

t---+ f(x)

(x,A.) and

p,q

of G

L By the fundamental Lemma 4.4, the p,q is not transverse to at least one of the

se+- cf measure zero-

Q.r.D.'

99 5.3.

Interpretation of the preceeding theorem in some special cases.

A

"Df(x)

{O}

= , (1Rp) w =

Lp,l

function

CCXJ

q = 1 (scalar-valued functions on

Case where

1)

Here, one has

=

0"

in

L

f:

m.P + lR

implies

"Df

I:p-l U I:p

I

lRq).

where clearly

is therefore nice if and only if

is transverse at

x

to

p,l "

This means that at every singular point

has maximal rank

p

x

of

f , the germ

But the Jacobian of

Df

at

x

is nothing

else but the matrix

of the Hessian'quadratic form of 3.)

f

at

x .

(See Chapter III.2 and

The transversality condition means therefore that at every

singular point

x

of

f , the germ of

f

at

x

is Morse.

This

proves the following proposition. Proposition.

Let

f:

JFt

-+ 1R

e: > 0 , there exists a linear map

and

f +A

em function.

A on 1RP

For every

such that

JIAJJ


p

q

= 2p

Here, one has

Lp,2p By the formula of Proposition 5.1, the for .all

i > 1

for all

i > 1

means therefore that Df(JRP) 1'\ l:i

=

'

The condition "f nice''

E

100 in other words,

f

immersion from

mP Let

Proposition. an

e: > 0 , there exists

For every

f + A

and

is an immersion.

This proposition is the key to the proof of the famous

Remark.

theorem of Whitney: dimension

p

every abstract differentiable manifold of 2

m

can be immel!'sed into

The case where

3)

2 mP -+ m P.

f:

such that

in

A

has maximal rank p everywhere; it is an 2 into m P. So, we have the following.

P •

p = q = 2 (mappings of the plane to the plane).

Here, one has L

= EO U 1:1 ll 1:2 '

codim 1: 0

with

m2

2,2

m2

. to

=0

and

codim 1:

1

=1

•

So, a nice mapping

- Its rank is greater than or equal to since

f

from

has the following properties:

codim 1:

2

=4

- At each singular point of function determinant condition for

f

at every point,

l

> 2 •

f , the differential of the

(Df)

is not zero; this is the

to be transversal to

1:

1

•

This proves the Proposition. an

A

Let

in

f:

€.

-+

m2

•

For each

(rank(f+A)

E

> 0

there exists

satisfies at each

such that

of its singular points of

m2

= 1)

the hypothesis

(H ) 1

V.2.3. Thorn's Transversality Theorem. We are now going to generalize all of the above in a rather

natural fashion.

101

6.1.

Jk

Denote by

p,q

(k

mappings of degree .::_ k term.

~

1)

from

the vector space of polynomial 1Rp

to

1Rq ,

without a constant

Give this space any norm, for example, the maximum of the

absolute values of the coefficients of the components. m

c

be a

k ~ 1 ~ there is a

mapping.

For every integer

em mapping

defined by jkf(x)

= the

degree

k

Taylor polynomial of

This polynomial is also called the k-jet of Jk p,q

is called the space of k-jets from

..

S

Let

6.2.

c

set of

be a subrnanifold of

mappings

is transverse to

~ + 1Rq

f:

at to

x , and 1Rq.

Jk Denote by T(S) the p,q such that jkf: mP + Jk p.q

S •

Thorn's Transversality Theorem. differentiable mapping. Jk p,q

f

1Rp

f - f(x)

Let

For every

f:

mP + 1Rq

be an arbitrar·r

c > 0 , there exists a

P

such that

IIPII

< E ,

f + P

Proof.

e T

It uses the same idea as that of Theorem 5.2.

Consider

F:

,t

F(X,A)

'r I·

G:

=

f(x) + A(X) Jk

p,q

rl

n

['11:.

t

lf 0

is a continuous function that is zero on

where

is a simple converse to this statement: Let

f

be a function which is defined and

...

c

on

and which satisfies the condition: (1)

for every compact set of integers that

k,r

K in

m.n

and for any pair

there exists a constant

C such

108 Proposition. on

mP,

Such a function, extended so that it equals

c"'

is

on

m.n

0

mP

and flat on

The proposition is obvious; by extending

f

and its derivatives

to be zero on

IRP ,

one obtains by (1) functions which are continu-

ous on all of

mn •

The Proposition is then a consequence of the

lemma proved in the Appendix

1.2.

Chapter VI •

to

Rough multipliers (G. Glaeser) We will continue to use•the notation of 1.1.

rough multiplier for and

c""

on

lRn -

lRP C lRn

Jif

any function

and which satisfies the condition:

for every compact subset

( 2)

We call a A which is defined

positive integer

of

K

mn

and every

k , there exists a positive

integer

r

and a constant

IJokA(x,y)

II


0

such that

if (x,yl € K -

JRP.

IJyllr Condition

implies that

(2)

A and its derivatives do not grow

too quickly as one approaches the subspace

lRP.

This definition

is motivated by the Proposition. is

nf ,

then

It suffices to note that

) • f

is

on

lRn

m.2

A is a rough multiplier for and flat on

c"'

A• f

is

and if on

1Rn

-~

and

c"'

f

nf

and flat o::

that

If

Dk(A • f)

f,Df, .•• ,Dkf

c...

Oll

is a "universal .. polynomial in It follows then that

A• f

:mn

;>.., DA, ••• ,oi\,

satisfies cor.dition (l)

and the preceeding proposition applies. Example.

The function

which is non-zero outside function

f

A = (y

Jif

can be written as

2 2 -1 + ••• + y ql 1

is a rough 1.1uh:iplier

It follows then that any f:at

109

(y~ + ••• +y~)

f where

2.

g

c"" on

is

• g

IRn

and flat on

The extension theorem of Lojasiewicz (cf VI.5,4)

2.1.

Preliminaries.

2 .1. l.

Let

Lemma.

and

F

1 exists a rough multiplier a)

A

b)

(A-1)

be two subs paces of

2

A for

F

is flat for

F

(F

-

1

is flat for

F

F

-

2

led to the case where

,

K.]._ "' F.]._ (\ s

i

and radius l

0

n

(F

c""

s

.1.

The function

A

c""

is

K2

n F 2 , we are 1 In- this case, set

lRn

>. 0 K 1

by

F

on the manifold and

Xe

for all on

2

is the sphere with center at

function

(-X-) llxll

0

such that

are disjoint compact subsets of

on a neighborhood of

.l.(x)

lRn - { 0}

1

is flat on

0

for

2

- {0}.

M = sup

which

IRn - {0} and flat on

Fina~ly,

since by the homogeneity of

II Dk.l. (x) II where

F

S

s .

on a neighborhood of

- { 0} 1 For the same reason,

since it is zero on a neighborhood of this set.

(A-1)

There

(\ F l

1

{O}

F2 1 where

and

Kl

So, there exists a equals

F

1,2,

n F2

1

lRn .

F l 2

~

1

By taking the quotient of

Proof.

0

IRP •

.1.

F

is a rough .multiplier

.1. , one has

< ~ , x e IRn - { O} ,

- llxll

II DkA (x) Jl

xes F , ... ,Fp (p

2.1.2.

Lemma.

Let

I

Let

and

be two

il

h

\

I

~

l I I

f

flat on on

g

1

c""

(F 1J ••• U Fp-l) 1 lRn such that:

~

l)

functions on ('I

Fp

be vector sub-spaces of lRn

such that

Then, there exists a

(£-g)

c""

lRn. is

function

110 a)

f- h

b)

9' - h

is flat on

F U , •• V Fp-l , 1

is flat on

This means that i f

f

F

p

g

and

have the same Taylor's series at

...

UF ) (F V then there exists an h 1 p- 1 U Fp I having the same Taylor series as f on (FlU and IJ Fp-l) the same Taylor series as 9' on F p each point of

Proof. p

=

...

We will use induction on

p (the result

is obvious for

By the inductive hypothesis, there is a

l)

f - h

1

is flat on

F V 1

is flat on

F

Use lenuna 2.1.1

1

such that

l)p

p-2

p

to construct a rough multiplier

which is flat on

h

with

P-ll

flat on

F

A for

F

p-1

U F

p

Then

p

set

One easily verifies that 2.2.

h

satisfies the conclusion of the lemma.

Proof of the theorem of Lojasiewicz Let us review the problem (cf. VI.S.l. Consider in the a:n two real subsets E and )!. with E~ E

2.2.1.

complex space

and also the restriction homomorphism

Er any fnnction 7 - f • •· •

:·(x,y,a), let

T'"'is •; < 11 motivated by

N)

121

P(x,z)

where

is the canonical polynomial.

?

2.3.2.

Let

f(x,y,z).

f(x,y,a)

be any holomorphic function and consider

are going to divide

'He

f

by

!' ,

a simple task because

of the following algorithm: Consi

E

as follows: \

I

1\

E=E\J(U

!,1.

r

E.) Tr~

Tti

where for each involution ET,i

is the set of zj -

zT (j)

in

{l, ••• ,k} II: x

i

and

ll:n x rek

~

l, •.. ,k

such that

j = l, •.. ,k

= 0

cx-z. >

-

{

T of

(x,y,z)

l.

i.e., x- z. em

0,

l.

y e lR.

Clearly, each under

E

.

T,l.

Tk , just as Given an

4.2.2.

and then extend

f f

is a real form, and the set E

f

in

Eove equation to F and then projects it intc lR ,.. l!ln ,.. JR.]~. u~in~ Propositio< 3.2 (with parameters) ·::.o complete the 4.4.

existence of the division.

If one compares the above ?roof with that of the division

theorem ot order 2, one sees that the essential simplification in thP. order 2 case arose because we could easily and directly reduce that case to the lemma about even functions, analo9ous lemma here is theorem of Newton.

~1e

The

differentiable version of the

129

CHAPTER X THE MATHER-MALGRANGE PREPARATION THEOREM

1.

Introduction We will assume that the reader is familiar with the notion

of a module over a commutative ring with unit. 1.1.

Let

M be a module over a commutative ring

words, an A-module.

Let

c

I

A

be an ideal.

A , in other

Set

I· M

=

r

I i=l I •M

a.m.: 1

ai

1

is a submodule of

e

I, mi

e

Clearly,

M, r arbitrary} •

M • M j·I • M ; its structure as an A-mdoule can be

Form the quotient

identified with a module structure over the quotient ring In effect, in

I

Cl • m

=

0

Let

A

= En

variables, and let

M induces on is an Let

M

M

I

in

the ring a·£

M/I • M

I

=M

whenever

A/I

a

is

n

real

The

1R C En).

En).

Then,

Mn/M = 1R.

En-module structure of the quotient

An A-module

a finite family

functions of

E -module structure n structure of the a vector space over lR

(the maximal ideal of

identified with its

of elements of

c"'

En -module.

be an

lR-algebra and

Therefore, the

1.2.

m

.

Example.

of

for every

structu~~

as a vector space over

M/I • M

can be

lR.

M is said to be of finite type if there exists

{m , ••• ,~} , called a system of generators, 1 M such that each m in M can be written as

.130 k

e

A

Note that, in this case, if

I

}; aimi

Ill ..

a

I

i=l

1

is an ideal of

A

,

then

k

f:

I· M • {me Mjm =

aimi' a 1 e I} •

1

A nontrivial example 00

Consider the

~·

(JR

C

mapping

" lRn ,0) +

(x2,y)

(x,y) Denote by

Es

x JRn ,0)

(lR

(respectively

Et) the ring of germs of

at the source (resp. target) of

~

•

Then,

00

C

functions

~

induces a ring

Et

1

homomorphism

One can thus consider for each ;\ ' f

f

E

in

5

.)

•

as a module over

Es and

~

in

by defining

Et :

f

We have shown (Whitney's Lemma, Chapter VI) that each

f in

E

s

can be written as: f(JC.y)

This means that

2 2 x • r (x ,y) + r Cx ,y), i.e., 2 1

{l,x}

is a system of generators for

E

s

as an

Et-module. The goal of this chapter is to generalize this fundamental example.

131

1.3.

Nakayama's Lemma. We have worked frequently in the first part with particular

forms of this lemma. Let

We now present the general version.

be a commutative ring with unit.

1
formation of the

function (cf. III.6.2.c) Let

c E

I

x,y,u

be the ideal generated by

and

af ily

Show that

f x,y It follows that E x,y,u 6.

~he

preparation theorem in general.

We will now prove the general preparation theorem announced

6.1.

in 2.3

Decompose

f

into an injection followed by a surjection

by the classical trick which uses the graph of

(x,f(x))

X

Let

E

s.t origil"' ""

f :

f(x) .

denote the ring of germs of functions of JRS

X JRt

•

(x,y)

at the

143 The homomorphism i

is an immersion.

i

•

E

s, t

-+

E

is clearly surjective, since

s

Es -module M

So, the

Es,t-module. M/f*Mt • M = M/1r*Mt • M

also of finite type as an clear that

of finite type, is

On the other hand, it is

The general theorem now reduces to Theorem 5.1. 6.2.

Q.E.D.

Examples

1)

Let

f:

(,IR2 ,0) -+ (m2 ,0)

be

2 I-+ (u =x ,v =y2)

(x,y)

This mapping is called "the handerkerchief folded into quarters" (exercise: explain why).

Let's study the ring

of x,y) as a module over Et 2 2 Then, f*Mt [x ,y J CEs

Therefore,

{l,x,y,xy}

(the ring of functions of So,

f

in

Es

over

= c;(x 2 ,y 2 J +xfl(x 2 ,y 2 )

2

2

+ yy(x ,y ) a, fl, y, o e Et .

This is a "two-fold Whitney lemma."

Prove it using the

Whitney lemma established in Chapter VI. Let

2)

f:

(IR

2

,0) -> (lR_2 ,0)

(x,y)

t--> (u = x

This mapping represents, in Z

from

a:

=X

+

to

a:

One has

iy

I->

z

2

=

~eal

(X+

Et .

Es can be written in the form

2 2 + xyo(x ,y ), where Exercise.

u,v) .

f*Mt:::l M\ and more precisely,

is a system of generators for

This means that every f(x,y)

Es (of functions

iy)

2

be the mapping 2 - y , v = 2xy) •

coord~nates,

the mapping

2

Let us again study Es as an Et-module. 2 2 f*Mt = [x - y ,xy]; and one can easily check that

2 2 f*Mt + m{l,x,y,x +y } = Es

144 It follows that Es is of finite type over 2 2 {l,x,y,x +y } as a system of generators. Exercise.

Show from this that any

= f(x,y)

f(-x,-y)

C""

and admits

f(x,y)

which satisfies

, i.e., is invariant with respect to symmetries

about the origin, can be written as

where 3)

~

..

c

itself is

Let

N:

( lRn

, 0) + ( lRn , 0)

(See the previous Chapter.) finite type over Exercise.

be • the real newton mapping.

One easily shows that

E s

is of

Et

Use this fact to discover another proof of the differen-

tiable theorem of Newton.

(See Malgrange:

Ideals of differentiable

functions. ) 6.3.

We are now going to establish an important corollary of the

preparation theorem, which will be used constantly in the fourth part. Consider a germ of a mapping

(u,x) Defino;:

fJ:

(u,f(u,x))

~

(ll!-s,O) + (lRt,O) x

•~

Recall that such a germ

f (x) 0 F

by

= f(O,xl

is called an unfolding of

f

0

•

(See

Chapter V, and for more details Chapters XIII and on.) Denote by E (resp. E ) the ring of germs of functions u,x u,y on the source (resp. target) of F , and by E (resp. E ) the X

y

ring of germs of functions on the source (resp. target) of

f

0

145 Then,. we have the following corriim.itative diagram of homo-

morphisms: F*

E u,y

!

E u,x

!

*

fo

E y

E

X

where the vertical arrows represent restrictions to their respective kernels Now, let

M be an

M (see 5.4 above).

E -module, via F* u,y Ey-module, via f~ Theorem. l.

M

similarly,

M

with

=

0

Let

M

0

=

of the elements

can be considered as an can be considP.red as an

M 0

is of finite type over

only i f

0

is of finite type over

More ,erecisel:t:,

Mo

M is 9:enerated bJ::

is generated over

E y

Ey m , ••. ,mk over E if and u,y 1 by the projections 1ni,O _..£!.

rn.

Proof.

1

M/F*M •M Note that, by the * . u,y F M c E is generated by u,y u,x and the components of f . One easily deduces

Consider the quotient

definition of {u , .•. ,up} 1

Eu,x-module of finite type. The module

u

M ·E u u,x

The following conditions are equivalent:

2.

the

and

EX -module of restrictions to u

M/u ·M, the of

Mu • Eu,y

F , the ideal

(exercise) that:

The theorem is then a trivial consequence of the preparation theorem.

146 THIRD PART:

UNIVERSAL DEPORMA'l'IONS OF

REAL-VALUED FUNCTIONS CHAPTER XI UNIVERSAL DEFORMATIONS OF REAL-VALUED FUNCTIONS

1.

The fundamental Theorem of Universal Deformations.

1.1.

We begin by recalling some notation and by restating the

fundamental ·theorem.

Let

function at the origin of

f:

(nt, 0)

-+

lR

be a germ of a

e En

lRn , L e., f

't

Suppose J(f) c E

finite codimension, i.e., the Jacobian ideal

n

c"' is of

has, as

a real vector subspace, a finite dimensional complement. A p-parameter deformation

Theorem.

and only if its initial speeds

We

lR{F , ••• ,:F l 1 P

+

J(fl h~ve

Fi , i

F

of

f

= l, ••• ,p

is universal if , are such that:

= En

already established the necessity of this condition

in Chapter IV (3.2).

We need only prove the sufficiency.

Practically speaking, we will construct universal de-

Remark.

formations of

f

as follows:

given

f , we will choose functions

g , ••• ,gr j~ [P such that their canoninal projections into 1 En/J(f) form a basis of this space as a real vector space. This means that and ~¥

r

1R{g , ... ,gp} c En is a complementary space to J(f) 1 • Then, we will form the deformation (universal

= codim f

the above theorem)

147 Any other q-parameter deformation of F

f

will then be equivalent to

q = p , or equivalent to a constant deformation of

if

3.1

q > p , by Proposition

F

if

of Chapter IV.

Proof of the Fundamental Theorem

1.2.

The idea is the same as that of the proof of Theorems 5.1 and

6.1 in Chapter IV.

cmP

F:

X

Let

F

be a deformation of

f:

lRn ,0) + 1R

(u,x) ~ f(x) + f(u,x), f(O,x)

0,

such that J(f) +

F.l. (x)

where Let

G:

m {i-1 , .... ,i-p l 3F au:-

E

n

(O,x)

l.

(lRq x lRn ,0) + 1R

f(x) + g(v,x), g(O,x) = 0 ,

(v,x) 1-+

be an arbitrary deformation of

f

,

Form the direct sum

(v,u,x) r+ Denote by

v1 = 0 , v1 H

q

q

and

parameters G:

=F

f(x) + g(v,x) + f(u,x).

H , ••• ,Hq the restrictions of H 1 v = 0 , ..• ,v = v =... vq 2 1 2

and the restriction of

Denote by .E

V,UrX

be the ideal of

E

to the subspaces

=

0

respectively.

H to

u

0

is

G

the ring of germs of C"' lRp X lRn (resp. lRq X lRp).

Ev,u )

(resp.

functions at the origin' of J(H)

F

(lRq x lRP x 1Rn,O) + 1R

H:

So,

with the

H of

lRq

v,.u,x

X

Let

generated by the partial derivatives

By Theorem 5.3 of Chapter X (the corollary of 3H/3x1 , ..• ,3H/3xn the first form of the preparation theorem), we have

148

J(Hl + E {aaH v,u u

(1)

~l .. E

I • • • I

au -

1

In effect, for

v,u,x

p

u = v = 0 , we have (in the notation of

..!!.._) J(Hlo = J(f); (.~Clu. 0

X.5.3)

i = 1, ••• ,p

F.

J.

J.

J(f) +

and

Now consider ilH

(2)

avl

E , by hypothesis.

lR{:F , ... ,:FP} 1

oH

avl

n

~ = i=l

X

E

in

Xi(v,u,x)

By (1), we can write

v,u,x

ilH

+

a;-J.

1

~i (v,u)

i=l

ClH

1hi":"

I

J.

which can be formulated as follows: There exists a germ of a vector field il

X

avl -

r

n

~i(v,u)

i~l

x·H

such that

a L 1hi":" - i=l J.

0

X.

(v,u,x)

l.

a

k

J.

.

By the geometric Lemma 4 of Chapter IV, there exists a germ of a submersion

is isomorphic to

such tha'::

By an obvious induction argument, we can conclude that there exists a germ of a submersion h : q

tb'lt

i'UCh

t-: v

t tRq

(lRq " ~ ,0) ...

H

,o; _.

~

h

t'l

1:: ~ Ql

q

If

G be a p-parameter deformation of

(lRP x lRn ,0) ->- Jk n

G

M • J (f)

is equivalent to

is transverse to Fp

~f

the fact that the deformation

F

p

+ t

(t

,i!t the

as deformation of a potential

The proof is easy and is left as an exercise; one need

is, by the definition of

f

e

o~ly

use

lR an additional parameter)

Fp , a universal deformation of

f .

Deformations which satisfy the above transversality conditiQn are called universal p-parameter deformations of the potential

3.5.

f

Examples. 1.

If

f € En

is a Morse germ, then

J(f) =

deformation represents a universal defotmation of which should be intuitively obvious. 3 3 2. Let f = x + y in E Then, 2 J(f) + lR{l,x,y, xy} = E J(f) + lR{x,y,xy}

=

M •

2

and

M•

The constant

f

a fact

152 So, a universal deformation of the potential F(u,v,w,x,y)

x

3

+ y

3

f

is:

+ ux + vy + wxy •

We will see some more examples in the next chapter. 4.

A remark concerning the Weierstrass preparation theorem. Let

f(O) = 0 .

that of

C"'

.be a germ of a

function such

Suppose that we want to study the level surfaces

f l:

If

(:Rn+l ,0) -+ :R

f:

{f=d , for

E

f

E

is not flat at

small in

lR.

0 , we have shown

appropriate choice of coordinates

(VI.2.3) that by an

(x,y , ... ,yH) 1

in

:Rn+l, we

can write

=

f(x,y)

(ll

k

[x

+ a (y)x 1

k-1

+ ••• +ak(y)]Q(x,y),

Q(O~)

'0.

(Weierstrass· Preparation Theorem) Thisallows us to study geometrically because

(1)

does not tell us anything about

f(x,O) = xkQ(x,O) (1)).

So,

f

(by construction, all the

r 0 , but not Q •

ai

are zero at

is ann-parameter deformation of the

which in turn is isomorphic to + xk a 1miversal deformation of

xk

ge~

By

~ee

lRn

0 in

f(x,O),

By the fundamental Theorem 1.1

is

So, there actually exists a local system of coordinates at the origin in

rF,£ # 0,

But now note that

(x,v , ... ,yn) 1

ln which

of this form. the level surfacev

t1

can all be

studied at the same time as branched covers of the space

lRn (y) •

remark shows how the division theorem is more powerful than the Weierstrass preparation theorem.

This

153

CHAPTER XII THE CLASSIFICATION OF GERMS OF REAL-VALUED FUNCTIONS OF CODIMENSION LESS THAN SIX; THE ELEMENTARY CATASTROPHES OF R. THOM

En

Preliminary remarks on ideals of finite codimension in 1.1.

Let

I c En

be any ideal.

Consider the nested sequence of

ideals:

ck (k ~ O)

Denote by

the codimension of

considered as real vector spaces over

Mk+l + I :ffi.;

Mk + I/Mk+l + I

of the quotient space

Mk + I

in

'

it is also the dimension

The number

can be

ck

interpreted as follows: Let

Ik

jk(Mk+l)

=

= 0

jk(I) c: ~;note that

Ik = jk(Mk+l+I)

since

Consider now the canonical projection (truncating degree

11:

k

to degree (k-1)) and

Clearly, we have rr(Ik) = Ik-l" So, Ik c: -1 in rr (Ik-l) , transverse to the kernel

Ik

is,

(the space of

homogeneous polynomials of degree k). It is clear that

ck

is the codimension of

so it is also the codimension of

1.2.

~ Pkn

in

-

Keeping the same notation, suppose that

integer I::> Mk

Ik

.

k

This means that It follows that

0

ck Mk + I =' Mk+l + I

k+r

=

0

for all

in

Ik

pkn .

.

r > 0

0

for some

so, by Chapter I,

154 Proposition.

Let

the codirnension tient space

of

I

c

be an ideal of finite codirnension.

n

(i.e., the dimension of the quo-

is equal to:

J.

In this case, the integers

finite range.

ci

zer~

are all

in

Ik

outside some

L

In all these cases, the partial sum

equal to the codimension of 1.3.

En

in

Then,

c.

i=O Remark.

E

En/I)

L

c =

I c

i=O

J~ •

c.1

is

An application to the Jacobian ideal of a function. Let

be in

f

En

and suppose that

shown (Chapter IV, §2.4) that if codim M • J(f)

f

f

M3

is in

•

We have

is of finite codimension, then

n + codim J(f)

Consider the sequence of ideals of 1.1 with

I

M • J(f) •

Since

3

M • J(fl c. M , it can be writteP as:

E ::sM ""M2 ::s MJ ::s M4

t

11

So,

,

cu =' ~codim

c:

1

= n,

M- J (f)

and

c2

> 1 + D

M • J(fl

::I

= n(n+l) 2

+ n(n+l) 2

This proves the Proposition.

If

f €

M3

has finite codimension, then

1 .. r + !'(n+l) < codim II • J(f' 2 -

Remark.

The sum

codimer~ie~ 2. 2.1. l11

c

0

+ •.• +ck

of thP orbit

rkf

11 ~·

relative to

codfm o!'f'

M • J(fl

represents the

ltl

Introduction to the classification of germs of codimension We propose to classify, up to isomorphism, the elements f_,(for ar-y

rl

!!'UCh

that

codim f • codim J(fl < 5 ,


codim J(f) Hence, p

~

codim J(g)

~

1 +

~

2 •

In the rest of this chapter, we are going to present the classification of germs of codimension < 5

in a manner analogous to

Thorn's presentation in "Structural Stability and Morphogenesis." We will work successively with the cases of

corank l and corank 2.

In each case, we'll make up'a list of "canonical forms" for all the germs of the given codimension, using the form (1). 3.

Classification of germs of corank 1. This case is very easy.

3.1.

[

in

f

n

is of codimension

codimension

1

if and only i f

r

to

,

g

in

But this is obviously equivalent to

g(xl = xr+l •h(x), h(O) ~ 0

and [

r •

Using the form (1) and Proposition 2.2,

Thus,

g

xr•l (reven) or .!:xr+l (>:odd).

g

e

is isomorphic, in

So,wehavethe

In E every germ of co rank l and of codimension n ' is isomor12hic to one (and only one) of the followins serms:

Theorem. r .::..

2

n-1

I

i=l

2

e:ixi +

n-1 2 Y. e:.x 1 l. i=l Remark. 3.2.

X

~ X

r+l n

1

r+l n

±1

,

if

r

is even

=±1

,

if

r

is odd

M :)M 2 ";:) M3 :::~ M4 + M. J(gl-:;, ••• 2

1

2

3

c3

(where the symbols under the inclusion signs represent the codimensions), it follows that

c

3

=

0 (use 1.2).

M• J(g) ::::oM 3 , and in fact, M • J(g)

(2)

This inclusion implies that

g

= M3

So,

•

should be isomorphic to its Taylor

polynomial of order 3 and thus to its (cubic) hessian at 0 , by Theorem 5.2 of Chapter III • We need only find among the homogeneous polynomials of degee 3 in two variables

(x,y) , those that satisfy the above equality (2).

First, note that these polynomials can easily be classified up to isomorphism (i.e., under the action of the linear group

Gi(2,IR))

every cubic form is isomorphic to one (and only one) of the following: a)

x

3

- xy

2

= x(x-y)

(x+yl1 this characterizes the forms

P(x,y) for which the. equation P(x,y)

=0

defines

three real distinct lines. b)

x

3

+ xy

2

2 x(x2 +y ); this characterizes the forms

P(x,y) for which

P

0

defines one real line and two

complex (conjugate) lines.

158 2 x y , which represents the forms

c)

P

P = 0

such that

defines one simple line and one double line

(i.e., of

multiplicity two). 3

d)

x , which represents the case where

P

0

defines one

triple line. It is easy to verify that only cases a) and b) (the nondegenerate cases) satisfy equation (2) · : so we have proven the

3 E2 , every germ g in M of codimension 4 is isomorphic to one (and only one) of the germs:

Theorem.

In

From this result, it is easy to deduce the Theorem.

In

En

(n~2)

, every germ

f

of corank 2 and codimension 4

is isomorphic to one (and only one) of the germs: Q- 2

L

i=l

F

2 3 2 x ~ x ± xn-lxn , i i n-1

Vocabulary:

For the case

x

3

Fl.,

+ xy

2

~ ±1 • (respectively, x

3

2 - xy ) , we say

that we are working with an elliptic umbilic (resp. hyperbolic umbilic). 4.3.

Codimension 5.

Propositiov 2.2 shows that

codim M • J(g)

7 •

By considering

the sequence, M2 .;:) M3 ::1 M4 + M • J(g) ::1 M5 + AI • J 3 c c3 4 ii: f'>llows that CJ 1 and c = 0 • (see l-2). 4

r

~

M

::>

I. 1

a)

"!.eaning of

(q~

:"'

...

,

c3

By 1.1, this means that the codimension of

P~ ~ j 3 (M • J(g)) in

P~ (the space of cubic forms of two variables) equals one. intersection depends only on the cubic hessian of

g

at

This 0

It

159 easily follows tl1 M , by the preceeding results. We are 6 now going to work in the space J of jets of order 6 , n

First consider in this vector space the orbits fi

range over the set of models

the germs of codimension < 5 6 manifolds of J and

where the

which we found in

and

§4 for

This gives us a finite set of sub-

n '

Codim rGf.

l.

1

if codim f.

0

n+l

i f codim f.

1

n+r

if co dim f. = r

l.

l.

l.

Now consider the submanifolds

Codim rEif. = l.

r 6 f.l.

t'f.

l.

0

if co dim f.

n

if codim f. = 1

' 2 < r + JR.

< ~

It follows that

0

l.

l.

n+r-1 i f co dim f.

l.

I

rl 2 ~ r < 5

164 On the other hand, if a germ not belong to the union of the

E

in

f

is such that

n

does

fGf. , then the codimension of

f

l.

is certainly greater than 5 • With all this in mind, the above theorem is a consequence of Thorn's transversality theorem on the one hand and of Proposition 3,4 of Chapter XI on the other, once we have proven the

.

-6 In J6 the complement E of \) r f. (where f. ranges ~n l. l. l. over the set of models described in §5) is a finite union of submanifolds of codirnension > n + 5 The proof of this lemma is fairly straightforward; so we will only sketch it here. a)

Let

sp

f0ld of codirnension is in the complement

n+£(p+l)

V~f. i

(Exercise.)

2

It is a subrnani-

corank p

be the set of jete of

If

p

~

3 , Sp

by 2.3 , and it is a submanifold of

l.

codirnension greater than or equal to

n + 6 •

s _n r and 52., r Using the de1 composition lemma, we can easily show that sl (\ E is a submanifold b)

We need only study

s

o:C

of codimension 4

of codirnension To handle

So,

sl 1\

r

is a submanifold of

J6

n

n + 5 • s

of codimensions

2

n l:

, decompose it into a union of three subrnanifolds

n + 5 , n + 5, and n + 7

respectiv~ly,

depending

upon the nature of the cubic hessian in the g-term of the decomp-

o~ition lemma (~espective)y ~ 2 y,x 3 ,or

01.

The elementary catastrophies of R. Thorn 7.1.

In his book Structural Stability and Morphogenesis, R. Them

proposed a qualitative description of the evolution in space-time (mon>hology) of a "physical system", modeled as follows. One is given

,.

~

fibration

165 (where

00

M is a differentiable manifold) and a

C

function

M -+- lR.

E:

4

11-l (u) , u e lR , represents the manifold of possible

The fiber

states of a system considered at a point

in space-time

u

For local situations, one can suppose that

4

M = lR.

lR.

4

•

x lR" (n an

arbitrary integer, representing the number of parameters necessary to describe the state of the system at each point) and that 'II:

(u,x)

t--+- u

is the canonical projection. The function

E .represents a "potential" to which the system

under consideration is supposed to be subjected. each given

u . in

4

lR.

are represented by the points

u

minima of the restriction

E

u

= {(u,x) I x {(u,x)

x

of

We are thus led to study in ~

This means that for

, the only "stable" states of the system at

in E

11-l(u)

to

M the subset

is a critical point of

aE I~ = 0

, i

which are local

~~-l(u)

=

Eu}

l, ••• ,n} ,

~

which clearly contains all the locally stable states.

~ ax.

= 0,

i

= l, ••• ,n,

The equations

can be called the state equations.

l.

Example.

n = 1, E

function represents a universal unfolding of the potential

(This 4 .) The

x

state equation is

It defines a hypersurface

I:

in

4 M ,. lR. x lR.

whose equation is

166

Consider the projection 11:

(x,u)

t-+ u ,

4 4 the restriction of the canonical projection of m. K lR into m • 4 2 In fact, let's replace lR by m. (u ,u l for the rest of this 1 2 4 discussion since these are. the only two coordinates of m which matter here. The singular points of 02

E

12x

ax 2

2

+ 2u

1

=

11

are defined by

0

and they project into the subset

s ul u2

- 6x

ax

3

r

of

JR

2

defined by

2

.

u2

5

If

u

lies in the set

C 1 S , then

O\ttside the cusp w- 1 (u} c ~ consists of

exactly one point, which is a minimum of

E

other hand, if set 11

-1

(u)

C

E

consists of three points

0

2

insidt•,

On the

u

is in the

u ~;;

, then

two minima and one local

n1 , the observed state of the system is completely determined by the state equation; further-

maximum (an unstable state) .

If

u

wore, it depends continuously on

~

in

n

is in

•

On the contrary, if

, there are two possible states for

u

2 tion is needed to determine the observed state.

u

is

and additional informaFor example,

"Maxwell's convention" asserts that one should always choose the

167 smallest of the local minima. states that as

u

On the other hand, the "delay rule"

moves across

S

from

n1

to

0

2

additional local minimum appears for the corresponding

and an E

u

, the

observed state will be the one determined by the local minimum over

n1 -When it does disappear, there will

that evolves continuously from the unique minimum over until that minimum disappears.

be a discontinuity in the state of the system as the state is now determined by the other minimum. of

u

Since the curve

for which the"number of minima of

Eu

S

is the locus

changes, it is the

set of points around which one will observe a discontinuity in the state of the system.

There will be a jump from one sheet to another 2 n: E + IR • The curve S is called

of the branched covering the catastrophe set,

and it represents, by definition, the morpho-

l£9y of the system. Let us return to the general situation and make the hypothesis:

7.2. (H)

The function

E

satisfies the (generic) properties of

Theorem 6 .1. This is equivalent to a "stability" hypothesis for the physical laws. In this case, we know beforehand all the possible local structures of

E , by using successively Theorems 6.1, 3.1, 4.2, and 4.3 of this

Chapter and Proposition XI.3.4 on universal deformations of potentials. In a precise way,

is isomorphic (as a deformation of a potential)

E

at each point of the set

E to one (and only one) of the following

germs (all considered at the origin) : n

2

0)

}: c.x. , i=l l. l.

1)

Fold:

2)

Cusp:

n-1

2

iil cixi + x

3

+ u 1x

168

3)

Swallow-tail:

4 l B.utterfl y •

5)

n-.:

2

3

Elliptic umbili~ (hair): iil Eixi + x + xy

2

2

2

+ u1 (x -y) +

+ u x +u y 2 3 6)

Hyperbolic umbilic (crest of a wave)

n~ 2

2 i~l Eixi

7)

+ x

3

- xy

2

2

2

+ u 1 (x ty ) + u 2x + u 3y

Parabolic umbilic (mushroom) n-2

Eix~ + x 2 y + y 4 + u1 y 3 + u 2 y 2 + u 3x + u 4y i=l ~

of

By virtue of these models, we are able to describe at each point 4 n 4 ~ c lR x lR , the local structure of the projection 11: ~ + lR

and to identify the stable states of the system as a function of the 4 4 u in lR • As in the example treated in 7 .1 , the set S c lR 4 (the singular set in the target of 11: ~ + lR or the "apparent

point

contour" of this projection) is called the catastrophe set of the 4 system: it is the set of points in JR in a neighborhood of which one can observe the discontinuities of the state of the system under consideration.

The models 1 through 7 above (model 0 corresponds

to a non-singular point of the projection 11) allow us to describe 4 in JR , up to diffeomorphism, the local structure of S at each ~-

This is why these seven models are called the elementary

catastrophies. The names which have been given to them arise from 4 the geometric study of the projection 11: ~ -+ JR in each of these cases.

For details of such a study and for some physical

examples, see the article by A.N. Godwin and Thom's book. Remarks. 1)

The theorem we have been discussing is qualitative in the

followirtg sense; to carry out the classification of elementary

169 catastrophies, we have used, on the fiber bundle

n:

local states, the group of local diffeomorphisms of

M+ m

4

of

M which

respect the fibration (cf. the notion of equivalence of deforma-

in particular, the group of local diffeomorphisms of

tions) m

4

and that of the fiber of

Tr

•

In reality, space-time and the

n will often carry richer structures and a "quantitative"

fiber of

theory will require some strong restrictions on the group used (for 4 example, in m , one may have to restrict oneself to the Galilean group or the Minkowski group) . 2)

One remarkable aspect of the classification of elementary

catastrophies is the following: structure of

S

only uses

~

the description of the local (in Cases 1,2, 3, 4) or two (in

Cases 5, 6, 7)

of the state variables

independent of

n

(x ,x , ... ,xn) , and it is 1

2

(which measures the "complexity" of the system) .

The model described in 7.1 corresponds to a description of

3)

the "morphology" of a system. But it can be used for other obJeCt1ves. For exampl~ consider the projection:

(u,x) and a function Suppose that

1--->-

u

E:

mP x mn + m.

(u,x)

represents the state of a physical system represents some control variables, which

and that

an experimenter can change as he wishes: a

E

represents as usual

potential to which the system is subjected.

will call the catastrophe set mP

S

c:

mP

As in 7.2" we

the sets of points

u

in

in a neighborhood of which the state of the system undergoes

some discontinuities (or catastrophies). If

p

~

4

and if one accepts the hypothesis (H) on

E , the above

theory provides all the possible elementary ca tastrophies. if

p

= 2,

only. Cases 1 and 2 of 7.2 (codimension

~

For example,

2) can arise.

170 PART FOUR:

SINGULARITIES OF DIFFERENTIABLE MAPPINGS CHAPTER XIII

INTRODUCTION TO THE LOCAL STUDY OF DIFFERENTIABLE MAPPINGS:

1.

TANGENT SPACE

Introduction;Definitions.

t be the space of germs at o e lRs of c"" mappings s, t from IRs to lR • For f in E we will use the notation s,t f: (lRs ,0) + IRt. In what follows, we shall always deal with germs 1.1.

f

E

Let

such that

(lRt ,0). germs

f(O)

= 0

, represented by the notation

f:

Es,t Two are isomorphic (A-isomorphic in Mather's notation)

The space of such germs will be denoted by

f,g

(IRs ,0)4

0

e E0s,t

if there exist germs of diffeomorphisms

~

e Ls ,

~

e Lt

in the

source and the target spaces, such that:

g=~·f·~-1 1.2. A

Let

f., v

0 e Es,t

be a map germ.

p-parameter unfolding of F;

(mP

X

lRS ,0) +

F ~

'If

,

fo

(mt'

is a X

c"'

map germ

]Rt 1 0)

such that 1)

T •

where

,..,

Jil?

Y

lRs +

mP

and

are the canonical projections; this means that

,.,

mP

x lRt + JRP

171 1.3.

Two unfoldings

meters) of

f € 0 diffeomorphisms

F

FfJs,t

+€

t

and

(with the same number

G

are isomorphic if there exist germs of such that:

s,p

+€

Recall that (Chapter 4, §1.3)

Ls,p

unfolding of the identity mapping of

(u,x) The same holds for

~

r+ (u,$(u,x))

.

Let h:

F

Define

is a p-parameter

~(O,x)

where

=x

.

(u,x)

(u,f (x)).

~

f

0

0

e Es,t

0

, and let

(1Rq ,0) + (1Rp ,O)

,_.. u = h(v)

c"" map germ. a q-parameter unfolding of

G

(v,x) The unfolding unfoldings

•

, i.e. ,

be a p-parameter unfolding of

v

be a

means that

lRs

unfolding is called trivial if it is

An

isomorphic to the constant unfolding: 1.4.

of para-

p

F

isomorphic to

0

, by

(v,f(h(v) ,x))

1--+-

G is called the pull-back of and

f

G of

the parameter spaces of F

by

h •

are called eguivalent if

fo h

h*F , where

F

Two G

is

is a germ of diffeomrophism between

and

G

This is clearly an equivalence

relation. An unfolding of

f

0

F

of

is isomorphic to

f

0 h*F

e Eo is universal if every unfolding s,t for some mapping h

172 1.5.

Our aim will be to study the classification of map

their unfoldings.

germs and

In many ways, this theory will be similar to the

theory we have developed for numerical functions.

Nevertheless,

there are important differences, some of which are the following: (lRs -+- ~) , the classifica-

In the case of scalar functions

1)

tion problem was solved (Chapters III and XII) by using only Nakayama's Lemma.

In·the case of mappings, we shall have to

use the Preparation Theorem (this was hinted by the study of mappings from the plane to the plane, in Chapter

Vl.

As was also shown in Chapter V, the classification problem

2)

is clearly related to the study of unfoldings. A new concept will play a fundamental role:

3)

a germ

f

trivial.

0 e Es,t

0

stable map germs;

is called stable if all of its unfolding are

In the case of functions this notion is not very inter-

ing: the only stable germs are the non-singular ones (and the Morse functions when one considers potential functions).

In

this more general case, there will be many stable germs; and it is a difficult problem to classify them, even in low dimensions. 2.

Rank and unfoldings. Let

2.1.

at Let

f:

(~s ,0)

->

(lRp+t ,0)

be a map germ having rank

0 , i.e., the rank of the Jacobian matrix i:

t

and transverse at that f 1 f- (J:t)

0

to the subspace

is transverse at =

0

tangent space at

0

to

Es

(E 5 ,0)

-+

(l:t'O)

c Bp+t , this means

to the subrnanifold

is a subrnanifold of codimension

i:s

is

is of rank

Therefore lRp+s , the

p

Ker Df(O) (in fact, 0

0

e

lRp+s).

at the origin 0

for each choice of local coordinates in

Es

is equal to p.

p , containing 0 ,

Im Df(O)

considered as a germ of submanifold at f 0:

Df(O)

c IRp+t be a submanifold of codimension

p

and

i:s

must be

The restriction and defines,

Et , a germ of

173 a mapping from

lRs

to

lRt •

define in this way are said to be 2.2.

imb~dded

,o>

~to f

that one can

Let

c:mP, o>

+

-1

~t

be a germ of submersion, such that

~s

s,t f •

in

We continue to use the above notation. p q-p

parameters, is parameters) of

a minimal universal unfolding. The proof is very easy, and analogous to the one given in Chapter IV, §3.3.

This theorem will be an important tool for classifying

some singularities, as shown in Section 5 below. To construct a universal unfolding of a mapping may proceed as follows: spar-e supplementary to

select a basis in

E s,t

{g , ... ,gp} 1

f .s,t ; so that:

f

0

, one

of a sub-

191

Then, set: F(u,x)

= (u,£ 0 (x)

!

+

i:l

ui • gi (x)) , u = (u , ••• ,up) 1

This is a universal unfolding of

f

JR.p.

F.

, since

0

e

~

Examples. 1)

Consider

where

n

£ :

0

"*

(lR ,0)

is an integer~ 3 •

(lR ,0)

One has

Tf

0 Therefore, a minimal universal unfolding of F:

(lRn- 2 x lR,O) + (lRn- 2 n

n-2

1

Consider

f : 0

f

is:

0

., lR,O)

(u,x) t-+ (u,x +u x 2)

2 + 1R{x,x , ••• ,xn- 2 }

(lR ,0) + (JR

2

+ ••• +un_

2

• x)

,0)

(ordinary cusp).

X

We have shown in the previous chapter (6.2) that

Thus, a minimal universal unfolding of F:

(lR

X

lR ,0) -* (lR

(u,x) 1.3.

Theorem.

I-+-

X

f

0

is

2 1R ,Ol

2

(u,x ,x 3 +ux).

An infinitesimally stable germ is stable.

Indeed, infinitesimal stability means that fore

f

stable.

Tf

Es,t

is its own universal unfolding, which means that

Theref

is

192 2.

The geometric lemma of the unfolding theory. This lemma is analogous to the one proved in Chapter IV, §4 • Let

(u,x) be an unfolding.

1--+ (u,f(u,xl)

Let

be the restriction of Lemma.

F

to the space

nf-l

defined by

The following conditions are esuivalent:

a)

The unfolding

is isomorphic to

(lRP ,0) + (lRp-l ,0)

h: b)

F

h*Fl

.

u

1

where

is a submersion.

There exist germs of vector fields X lRP X lRS and nt X IRt

and

origins of

y

at the

respectivell'• such

that: (1)

X=

y

_a_+ aul

= _a_+ aul

1 i=2

f

~-

~

s

a

(u) --+ au. 1.

L

X; (u,x)

l=l

_a_ ax.

l.

t

{i

i=2

a (u) _a_+ L Yi(u,y) ayi au :.. L=l

OF • X ~ Y oF

(:')

The proof, similar

to

the one given in

~hapter

IV.4, is left as

an exercise.

Condition (l.) means that vector field spacP

m.P

.

~

=

a -aul

+

I

i~2

~

~'

0 •

y

and

a

(u) au.

l.

are "liftings" of a defined in the parameter

193 The fibers of the submersion ~

integral curves of

h:

JRP + JRp-l ~

(the expression of

curves are transversal.to the subspace

u

are the

ensures that these

Ol ,

1

Condition (2) means (see Chapter XIII, Section 3.1) that a trivial unfolding of

F

1

Let us interpret condition (2) analytically. p s

DF

....-"'---.

~

P{ (

au

0

(lf axl

af

df

t{

au-p

1

F

df

ax-s

l

We have:

(I

identity matrix)

il I~~

So that (2)

r,i·

(3)

f i II

where

•,.

of aul Y:

is equivalent to: p +

L F,;. (u)

i=2 ~

(JRp·

X

s (lf (lf X. (u,x) ~= YoF + ~ ~ i=l ~ l.

L

JRt ,0)

components are

I

the vector field 3.

I

... JRt

represents the mapping whose (i.e., y is the "vertical'' part of

II

{I

Yl, ' ' ' 'yt

'Y>

The algebraic lemma of the unfolding theory. Let us still consider:

(u, x)

1--" (u, y=f (u, x) )

a p-pararneter unfolding of some germ

f

0

.

With the notation

of Chapter XIII, §6, we have:

+ c

Es,t

Now, set: TF

J(F)

+

T(F)

is

•

c:: (E

u,x

)t

194

{ !L E ax1 u,x def

J(F)

where

f

and def

u,y

a£

•···•-axs

{e , ••• ,e} t 1

Then, we have the following The following condi.tions are equivalent:

Lenuna. a)

b)

gi (O,x) • Proof.

=-

al

=

b) b):

generated

a)

Set

is obvious by restriction to M

;

Note that

M 0

Now, condition a) implies that

M 0

E

By Theorem

y

Eu,y

over

X.6.3

Remark.

thus,

M

= Mfu•M

=0

is a finitely .is exactly

is finitely generated over

, we km;w that

M is finitely generated

Finally, it is enough to apply Theorem x.5.4

and the projection of

wit~

= {Eu,x ) t/J(F)

f u,x -JOOdule.

u

t{F) into

to

M

M to obtain the result.

Using the quotient space by

finitely generated modules over

J(F)

f

u,y

allows us to deal only or

The above proof uses the Preparation Theorem two times through its successive applications of Theorem X.6.3 and X.5.4. 4.

Proof of the universal unfolding theorem. We have already proved the necessity.

The sufficiency part

uses the same idea that we used in Chapter II for scalar functions Let

J'

t>e an unfolding of

f

0

, such that:

195

Let

G be an arbitrary unfolding of

write

f

0

, with

q

parameters;

G as G(v,x) = (v,f (x) + g(v,x))

where

0

Consider the unfolding

Denote by

H ,H , .•• Hq 1 2

subspaces

v

=.0,

1

v

e

lRq

g(O,x)

and

H which is the "direct sum" of

= (v,u,f(u,x)

H(v,u,x)

v

+ g(v,x)

= h(v,u,x))

= v

= 0, •.. , v = 0 .

2

Using the algebraic Lemma 3, and Condition

and

•

the successive restrictions of 1

F

= 0.

Thus

H

H

q

to the F .

(1) above, we

obtain: (2)

TH + Ev u{ ~~ ,

1

Consider

, ••• , ~~ } p

in

(f

v,u,x

(f )t v,u,x

lt

From (2) , we may write: (3)

5 ah P ~h ilh xi (v,u,x) -il- + y. H + ·~;i (v,u) ~ ilvl = i=l xi i=l ui

L

L

Comparing (3) with the analytic expression in geometric Lemma 2 above, we find that h : 1

H is isomorphi~ to

(lRq x ~ ,0) ... (lRq-l x lRP ,O)

h~Hl

where

is a submersion.

5.

Application.

lRn+l

(n .::_ 1)

5.1.

Consider a

I

One

Q.E.D.

finishes as in XI.l.2. The singularities

•

map germ:

!: 1 '" •. ,l

from

JRn+l

to

G:

196 F:

(lRn+l,O) -+ (lRn+l,O)

with rank

n

at the origin

0 •

This germ may be written, using suitaole coordinate systems

F

X n

where

= Xn

f(O) = 0

.££ ay

and

1 E F

The singular set of

OF

(0)

'" 0

•

(the set of points where the kernel

has dimension one) is defined by the equation: ilf ay =

0 •

The origin is called a

Definition.

El,l, ••• ,l,O- singularity of

F (the symbol contains r ones; it will be abbreviated as l ,0 E r ) , with l < r < n + l , if the following conditions are fulfilled:

(l)

(2)

~f

a

ily

(Ol

a2 £ :ol =- •••

= Cly.2

af

The functions -

~

arf

--

ily ' ••• , Clyr

()

,

are independent

~t

Q ,

(1}

This definition, which generalizes the definition of folds and cusps for tion.

n = l

(See Chapter V), has a very simple geometric interpreta-

197 1)

~ 1 F is a codimension 1 submanifold through 0 , since by

Condition (2), the differential

2)

1

= ••• = dxn

0

is contained in the tangent space to This means that the restriction The set

3f ay

Fj

1 E F

since has rank (n-1)

1 1 1 E ' F c E F , consisting of those points of

' restriction of

F 2

0,

a

3y

0 •

DF(O) , which is the "vertical" line:

The kernel of dx

does not vanish at

0.

at 1 EF

where the

has rank (n-1) , is defined by the equations

f

=

2

0

~

I

In view of (2),

~l,lF

is again a submanifold, of codimension 2. 1 1 i One considers the restriction of F to r • F to define in an t: 1 1 1 - analogous manner the subset l: ' ' F ; and so on. 3)

1 4)

Conditions (1) and (2) mean that

" a codimension restriction of

'15.2. i

r

submanifold. F

to

l: rF

!which has the origin a

l: r

and

ElrF

is

Any map germ

F:

lRn+l

+

,o -singularity (1 < r < n+l)

, isomorphic to the following mapping:

X n

rF

Condition (3) means that the

is an immersion.

Theorem (B. Morin, 19~5). 1

0 €E

X

n

Y = yr+l + xlyr-1+ .•• +xr-1 • y •

is

JRn+l

198 Proof.

We may interpret

above) of

f :

F

as an n-pararneter unfolding (see

(lR ,OJ + (lR ,0)

0

y t-- Y = f(O,y) l E r

Conditions (l) and (3) in the definition of is isomorphic to the mapping

Y

= yr+l

,o show that

f

0

Looking back to 1.2.

Example l, and using the universal unfolding theorem we see that is isomorphic (as an unfolding, which is more precise than as a general mapping from

xl

xl

X

X

lRn+l

to

lRn+l)

to:

F n

y

n

= y r+l

r-1 + u (x) y + •.. + ur-1 (x)y 1

We have not yet used Condition (2); an easy computation shows that it is equivalent to the following condition: du , .•• ,dur-l 1

are independent at =

and

tn~t

F

0.

to

This implies in fact that

Ey

is a universal upfolding of

we use Theorem 1.2

the differentials

finish the proof.

f

0

y

r+J

Finally,

F

199 CHAPTER XV CHARACTERIZATION OF STABLE MAP GERMS

l,

Introduction

1.1.

The aim of this chapter is to present an important and very

useful description of stable map germs. We already know (Chapter 0 XIV, Theorem 1.3) that f in E is stable if and only if it is s,t infinitesimally stable, i.e., Tf

= Es,t

We are going to look for some effective ways of checking this condition. 1.2.

Our starting point is a simple remark:

restrictions on the rank at

0

e lRs

there are strong

of a stable map germ.

More

precisely, one has the: 'Proposition. ir

Let

f: (lRs ,0) ~ (lRt ,0)

be a stable map germ, and

Then the mapping

Df: (lRs ,0) + Jl f at 0 s,t Es-r c Jl at 0 • ·is transverse to s,t Here Es-r c Jl is the submanifold of all (s x t) matrices s,t of rank (These manifolds were studied in Chapter VII). r ,, be the rank of

I

I I!

Using the defintion of transversality (cf Chapter VII, §4), this

11

[i

II

.

The proof is easy and left to the reader.

1.

result implies that the rank

If:

s

t

(lR ,0) + (lR ,0)

~,

at

is such that:

0

of a stable map germ

200 codim r..s-r < s

I

i.e.

(see Chapter VII, §5,1)

(s-r) (t-r) < s

Chapter VII (§5.3), Examples.

0

r

In particular

f

implies

The above inequality implies the following:

If

s "' t

2,

then

r > 1

If

s = ...

3,

then

T >

If

s = t

4,

then

r > 2

If

s = 2, t.= 3,

~

(lRt ,0)

then

0

2 r

with rank

r

> 1

has rank

0

at

(§2)

that a germ

f:

(:Rs ,0) +

.Q. , may be considered (in many ways) as

at

an r-parameter unfolding of an f

in this case, from

must be a Morse function.

We have seen in Chapter XII

1. 3.

=1

t

f : 0

(lRs-r ,0) + (lR t-r ,0), where

0 ,

It is then very natural to consider the following problem: given a map germ

F

which is an unfolding of a germ

possible to characterize the stability of

F

and of the "initial speeds" of the unfolding

f

, is it 0 as a property of f

F ?

This problem has a fairly simple answer which will give rather effective criterion for stability. 2.

Stable unfoldings of germs of finite type.

2.1.

Theorem.

Let

F·

mponel"= in

respectively, while

lRt.

X and

a similar way, wP

where

s

Y

in the parameter space

The components

{'

and

n

w~lte

any element

is the horizontal component

Now, cnnciitic;n

(~)

equation:

ha=

DF • :X. t Yo ~elutions

r •

Y and

Z

Y

IRP

and

in the

X and

Y

Y are called the "vertical components".

z

of

of

Z

component.

(31

and

as

parameter space are called the "horizontal components" of

r~

lRP

means that for apy

(F

and

u,x

Z

)p+t

as

the vertical

203

This equation can be written in the equivalent form: I

(

(4)

0

of

of

au-1

a£

au-p

of

ax;

ax-s

)(:)

+

(:.:) (:) or ~

(5)

\

I

+ n • F

~

s

of

i=l

a£ L X·--+ i ax.

;.l. -a-+ _ui

i=l

z

YoF

l.

that is:

= l; (6)

\'

!

n•

F

(ni oF) •

i=l

C,.

s

of

au.

+

L

X

i=l

l.

I;

= (1;1, ••• , l;p)

e

CE

n

Cn 1 , •••• npl

e

4>

of the base

{0} •

¥s,t

diffeomorphisms of

is a subgroup of

Ls+t (the group of local

1Rs+t); it is called the contact group, for a

reason which will soon be made clear. We shall denote by by the r-jets at

Kr CLr s,t s+t

e 1Rs

0

x

1Rt

obvious!·,· a Lje-subgroup of For example,

1

K s,t

(r ~ 1)

the subgroup defined

of the elements of

K s,t

it is

(exerc-ise).

is the group consisting of the linear

transformations: s

t

1 K s,t Notice that there is a canonical injection:

where

A

and

4.2,

I~finitesimal

B

are invertible; thus

x

Gl(t,lR).

viewpoint:

C:OP""ider a germ of of the form:

Gl(s,1R)

.:1

c."'

V"Ctor f.'eld a.-

0

io

lRl1 x mt,

209

s

a

2

(2)

xi (x)aX," + ~

i=l with here

X. eM

a -a-

t

L

":i. (x,y) j=l J

yj

Y.(x,O) = o, i.e., Y. eM · E (We write J J y x,y s t x= (xl'"''Xs) eJR, y= (yi''"'yt) eJR I Ex' Ey and Ex,y ~

and

x

denote the corresponding rings of germs of functions.)

Then,

one proves easily (exercise) that:

~u = exp

uX

€ Ks,t

for

u

e

1R

(see II.4.3).

Ks,t

Let

be the vector space of vector fields of the form (2)

above; this space may be considered as the Lie Algebra of

K s,t

Set

Kr

(r .::_ 1) The space ~ may be s,t s,t identified with the space of ,eol:inomial vector fields of the form (2)' ~

with coefficients of degree Proposition.

The space

Kr s,t

r is the Lie algebra of the Lie group

Kr s,t The proof is straightforward and is left to the reader. 4.3.

Action of

Let

f

Consider the lRs x JRt ~

on ~

and ~aph

be elements of of

f:

0

E and respectively. s,t it is an s-dimensional submanifold of

, transverse at the origin. to

{0} x 1Rt.

The diffeomorphism

transforms this submanifold into another submanifold having the

same properties; therefore, it is the graph of a new map germ that we shall denote as

Analytically, if (3)

0

~.f € Es,t • ~(x,y)

(h(x) .w(x,y))

and

g

~.f

, one has:

goh(x) =w(x,f(xll.

We have in this way defined an action of

Ks,t

on the space

E~,t

210 It follows from (3) and from 4.l.c) that if set

{f

=O}

c lRs

is transformed into the set

the diffeomorphism

this means that

h

have the same "contact" with

lRs

= +. f

=0}

graphs of

th~

, the

c 1Rs f

by

and

g

(this will be made more pre-

x {0}

cise later on); that is why the orbit be called the contact-orbit of

g {g

0

s,t • f = Es,t of f will Ks,t itself is called the

K

f. , and

con tact-group .. Similarly, for any integer of

jrf

r

~

~

under the action of

1 ,

We now consider the infinitesimal action of

4.4.

Let map germ from lRS Pro~sition.

at

-

X

the orbit

we denote by

Kr s,t

be a vector field in K s,t exp ux. f to lRt. If f u

Ks,t

E's,t l

on

and let

f

be a

then:

u

X·f=-~

(4)

def

au

u=O

The proof is left as an exercise. The subspace: TKf

= def

{X· fjx e K5

t) C '

0

E5

t '

will be called the tangent space at of

f

to the contact-orbit

f

Kf

.

TKf = M

Proposition. Proof.

s

+

• J(f)

Note that, in (4),

"1:

f*M

t

•

E

s,

t

•

is of the form (see 4.2)

t

"1:"

= I

j=l

since

Y(x,O)

y. ·Yj(x,y) J

=0

.

Thus:

and one obtains all the elements of arbitrary.

Q.E.D.

f*M

t

· f

s,t

since the

Yj

are

211 Passing to jet-spaces, we have the easy jr !M • J(f) + f*M • E I c:: Jr . s t s,t s,t

Proposition. def is the tangent space at Important remark.

jrf

is clear that a map germ is a subspace of

5.

Stable map germs:

(u,x)

(lR.p

X

lRS

(u,xl

fu

jr f (x)

(fu(x)

is of finite type if and only if

the main geometric characterization.

(u,f(u,x))

---+

3.1, it

of finite codimension (as a real subspace),

§3.2, we consider

Define, for any integer

u

f

Es,t

be a map germ, with the rank of

F

the notation of

where

Kr ·f.

Combining the previous results with

TKf

Let

to the orbit

F

u € lRP

at

·p

p •

0

Using

as an unfolding I

X

e

lRS

,f(u,x)

e

lRt •

r > 1: ,0)

-+

Jr s,t

~-->- jrf (x) u

represents the r-jet at

x e lRs

of the

.,

C

mapping

= f(u,x))

The map germ F is stable if and onlJ::: if the ma,EEin9: Theorem. .p+lf (lRP X lRS ,0) -+J p+l is transverse at 0 to the contact Jl : s,t Kp+l CJp+l orbit s,t s,t fo Proof.

Use condition (4) in Proposition 3.3, which may be written:

Project this into the jet space above one obtains:

Jp+l s,t

Using the results of 4.4

212

+

TKp+lf

s,t 0

lR {

I • • • I

This last equality is equivalent, by definition, to the transversality condition.

Q.E.D.

We shall give applications of this theorem in Chapter XVII.

6.

Description of contact Qrbits.

6.1.

Proposition.

map germs

local diffeomorphism

he L

f,g

0

E belong to the same s,t K-orbit (i.e., are contact-equivalent) if and only if there exists a Two

in

such that:

s

This last condition means that the ideals Es

are isomorphic.

f*Mt

and

g*Mt

in

These ideals are generated respectively by

f , ... ,ft and g , ... ,gt: they define (in the sense of algebraic 1 1 geometry) the subsets {£ "'0} and {g = 0} in (~ ,0) • Thus, contact equivalence of of equations

f

Ii

{fi = 0

and

g

simply means that the two sets {gi = 0

= l, ... ,t} and

equivalent, up to a local diffeo100rphism of

Ii

• 1,.; .,t}

are

m.s •

Proof of the Proposition a)

Assume that

is an element of Since

~

(x, 0)

Ks,t

t ·

(x,y)

1

+=

where

f

Then

j ojli. (x,y) • yJ.

lj;

of

i

(x,y)

~

~(x, f(~))

go h(x)

0, each component t.

~i

g "'

~

(h(x),~(x,y))

(see (3) in 4.3).

can be written

1 ••••• t

.

j"'l

Then, the components of t

o

h

can be written as:

r

·~ (11:, f(x))

txt-matrix

[ljl~l. (0,0))

9 ... o " ( K) =

Now, the

g

diffeomorphism.

j•l

This shows that

f*M t

• ! J (x)

is invertible since

t

is a local

213 b).

Suppose that

is enough to show that write

g

for

g •h • t

g

o

h

Now

$~ (x) f,

I j=l

=

h* (g*Mtl

J

l.

and

f*Mt f

f*Mt

for some

e

h

Ls

Then it

are contact equivalent; we shall

g*Mt

so that we have

(x)

i

l, ... ,t

i

l, ... ,t

and t

= L ljl~l. (x)g.J (x)

fi(x)

j=l

for some functions

~~l.

~~l.

and

Write these equalities in matrix

notation: g = ~- ·f

(1)

'I'. g

f

It is possible to choose

Claim: (and

and

'I' (0))

is an invertible

proof of the proposition:

(x,~(x) • y)

for ~·

~

[!J>

+

matrix

= I - 'I' •

!J>

that

(x,y.) g

1--+

=~ ·f

C (I- 'I' • !J>)] • f

I

Note that

denotes the identity

(0) + C • ' (0)

txt-matrix.

Ker ~· (0) n Ker ~ (0) = 0. C

Set

This implies

(with constant coefficients)

is invertible.

Ker (A • ~' (0)) is supplementary to

such that

$:

From (1) we have

C , where •

~(0)

This will complete the

indeed the mapping

that there exists a txt-matrix such that

'I' ) such that

is then a diffeomorphism, and obviously

Proof of the claim. g =

(and

~

txt-matrix.

(Hint:

Choose

A

Ker !J> (0) ; then choose

Im(B • A!J>' (0)) is supplementary to

such B

Im !J>(O); C =/B ·A

answers the question) . Examples. 1)

f: (lR ,0) .... (JR

2

,0)

X

=

2 [x ] =

One has

f*Mt

only i f

g' (0) = 0

M~ .

and

Then it is clear that

g" (0)

t-

0 •

g

e

Kf

if and

214

2 2 (lR ,0) + (lR ,0)

f•

2)

(x,y) t--+

(X

2

2 ,y )

In particular, Kf contains all the germs 3 to M • E since in this case: s s,t

g

=f +

t

where

c

belongs

(and use Nakayama's lemma). Thus

Kf

c: E0

2

f·

(JR

j 2 , of the orbit

is the inverse image, by

s,t in 2-jet space.

A similar remark holds for the mapping•

,o)

(m? ,OJ

-+

2 2 (x,y) r-+ (x -y ,2xy)

6.2.

Finite determinacy of K-orbits. We.have defined in 4.4 the

~ Ms •· J

TKf

(f)

sub~pace

0

+ f*M t • Es,t c:: Es,t

which represents the results of the action of the Lie Algebra on the germ Trf)

an

f .

Ks,t

This space, as was already pointed out, is (unlike

E -submodule of s

E0s,t

This means that contact orbits may be studied with the same technique used for orbits of the group

L

n

acting on

En (Chapter

III), i.e., mainly Nakayama's lemma. In particular, one has the following important Theorem.

be a map germ.

conditions are equivalent:

.., '

T:Kf::;,

2'

Tiff -+ Mktl

IJk • s

s

r...t .E

~.t

::;)

1l · r::>,t p

I

The following

215 Kk f c Jk is k-open. Moreover, under s,t s,t ~~~~~~~~~~~-k~~=-each of these conditions, Kf is the inverse image of K f through 3)

.k

The orbit

Eo

... Jk In this case, one says that Kf is k-determined. s,t s,t The proof is completely analogous to be the proof of Theorem 111.5.2; J :

it is left to the reader. type if and only if

i f and only if Remark.

TKf

TKf ::l

Note that

a map germ

f

is of finite

E0s,t

is of finite codimension in

Mk • E s s,t

for some

, therefore

k (by Nakayama's Lemma) •

We shall give another proof of this theorem in the next

chapter as part of the classification theorem of stable map germs. Exercise.· Ms • J(f) 6.3.

t

= 1,

compare the spaces

TKf

and

Local algebras. Let

that

For the case

on the examples of Chapter XII.

f

e

0

E s,t

be a germ of finite type, and let

TKf ::> Mk • E s s,t

k

be such

Using 6 .1 and Theorem 6. 2, one -obtains the

following Theorem. the ideals jets

/-

= 3.~.

f

lRt ,0)

0

af , ... ,-~-} af {-ax1

X

lu,f(O,u,xl)

is a stable unfolding of [u,x]

(:mi'

lRS ,0) +

a~

af

af

• [u,y] {e1 , ••• ,eeau-• ... •au-} 1 p

(u,x] { 1!_ , ... , ~= axl s

}

+ F* [u,y] • {e , ... ,et} • 1

Explanation of the notation. As usual, we denote by

fA,u,x

(re~p.

fA,u,y

the ring of

germs of functions at t:»e origin of IP > IR"'J .~ 1R 5 (resp. lR " :11 2 ·· :Rt' We denote b:r [u,x• 0: f lh~ ideal A,U,X

225 generated by the coordinate functions similarly

[u,y] c

E>..,u,y

u , ••• ,up,x , ••. ,xs; 1 1 is the ideal generated by

u , ... ,u ,y , ... ,y; F*[u,y] c f, 1 p 1 z ",u,x u , ... ,up,f , ... ,ft (where f , ... ,ft 1 1 1 Note that F* [u,y] c [u,x].

is the ideal generated by are the components of

f).

With this notation, the above formula sits in the module (E

) t , which one can also consider as an E -module via >.,u,x >.,u,y F*. The derivatives 3f/3xi are elements of this module, and

[u,x]{3f/ax , ... , f/axs} represents the E, -submodule of 1\,u,x 1 linear combinations of the 3f/3xi with coefficients in [u,x). Similarly,

[u,y]{e , ... ,et, af/au , ... ,af/aup} 1

1

t,1\,u,y-submodule of

(E,

",u,x

It

e , • .. ,et , (the canonical basis of :at) and 1 coefficients in

[u,y] c

The last term

represents the

made up of linear combinations of af at aul•···•aup , with

E>,.,u,y .

F*[u,y){e , ••• ,e} represents the E 1 t t >.,u,x (E).,u,xl whose components are in

submodule of elements of F* [u,y]. 3.3.

where

Proof of the Lemma.

F0 ,l.. (x)

Qi_ (O,O,x). aui

E).,u,x-module

and to the M 0

to obtain

Ex-module

= M/[).,u)

• M

Now apply Theorem X.6.3

to the

226 Multiply through by

[u,y], considering both sides as

modules, and note that

*

F* [u,y]

af

df

F [u,y] ' { ax ' ' ' ' 'ax 1 5 F*[u,y] IE,

A,u,x

}

=

[u,y] • E,

",u,x

..

at

a£

[u,x) {-~-· ••• ,-a-}

x5

oXl

[u,x] ::> F* [u,y].

This Lemma shows, in particular, that the left side is in

fact an

Note that its "restriction"

EA,u,x-submodule of = 0} is precisely

the tangent space

0, u

{A

to the orbit

3.4.

af

at

)t = F*[u,yl{e , •• ,e} 1 ~

to both sides of this equation, since

to

-

+ [u,y]{el, ' .. ,et'au ' ' " 'llu/ • 1

The statement of the Lemma follows·when one adds

Remark.

E

t.,u,y It follows that

•

Kf . 0

The geometric significance of this Lemma. Return to the hypotheses and notation of the Lemma. df

a>.

that the speed

at e IT

(1)

~i

In particular,

>.

FA:

satisfies

satisfies:

0

for all

A •

...

,et

Then,

} •

£:~,0,0)

=0

•

lR , the mapping

(u,x) ,..._. (u,f(A,u,x))

=

and therefore can be considered as

(0,0)

0

E

an element of

for all

=

(A,O,O)

in

FA (0,0)

Corollary.

F

f axl, at af £ ... , axs.1 + f* [u,y) e 1 ,

[u,x)

So, for each

of the unfolding

Suppose

p+s,p+t

•

This remark implies the

Under the hypothesis (1), )

FA

is isomorphic to

sufficiently small.

The notion of isomorphism here is the usual one, via the group L

p+s

X

Proof.

L

p+t By XIII.4

and

5 , all that we need do to establish this

result is to find germs of vector fields and target of

F

such that:

X and

Y at the source

227 YoF

DF.X

(a)

(b)

X ..

a

TI +

I 1

il

y

ai+ ~i,Xj

where

I

ilu.

i=l (c)

s

_a_+

f;i P.1u1x)

1). 1

i•l

au.

j=l

Y. J

I

_a_

~~~u,y)

e [u,y]

ni'~

and

l:

+

1

e [u,x]

ax. J

t

il

p.,u,x)

_a_

xj (>. 1u 1x)

j=l

1

()yj

I

trivial calculation,

A

similar to the one in XV.2.3, shows that one can solve equation

~~ e

(a) subject to the conditions (b) if and only if

af ... ·-a af-} {-a-,

ilf

[u,x] •

ilf

Hypothesis (1) + [u,y]. {ell••••et 1 aul'"""'aup} xl xs and Lemma 3 .1 guarantees that ()f satisfies this condition. Q.E.D. a>. 4.

Proof of the Fundamental Theorem.

4.1.

Proposition.

f

Let

p-parameter unfoldings of

0 t

be in

K(p,s,t).

Then, any two stable

are isomorphic to each other.

0

Keep in mind that we are not speaking about isomorphisms of unfoldings 1 rather isomorphisms with respect to the group Lp+sx

Lp+t

Proof.

a)

Let

=

F(u,xl

be a stable unfolding of g(u,x) =

!

(u,f (x) + g(u,xl) 1 where g(O,x) 0 , u e RP • We can write 0

u.h. (u,x)

i=l

1

1

f uihi (O,x) i=l r. (u,x)

where

1

0,

£

h. e IEu,x )t 1

1

p

I

+

uiri (u,x)

I

i=1

h. (u,x) - h. (O,x) 1

1

Now set p F(>.,u,x)

f 0 t~l

+

l: i=l

p uihi (O,x) +

=

().,u,f().~,xl)

I

i=l

and consider the mapping F(A 1 u,x)

~

1

u.r. (u,x), 1

1

>. e

JR;

228 defined on a neighborhood of with values in

x {O} x {O}

JR

lR x JRP x JRt .

in

x :Rp x :R5

JR

By construction,

f(A,O,O)

,

=0

•

;.late that FA:

(u,x)

is, for any

(u,f(A,u,xl)

1-4

A in

:R, a stable unfolding of

same initial speeds as

~~

Now we have

F

e

u • (E

whenever

F),

A-

:>t

0

In particular,

F

= F

1

1

in

0

lR , FA

is isornor-

is sufficiently small.

0

It follows trivially that I .

c:F*[u,y] • {e , ••• ,et}.

)t

u,x i=l ~ ~ By the Fundamental Lemma (3.1), for each >. phic to

0

= F1 .

1 ~.r.

=

Ql\

f : for, it has the

FA

is isomorphic to

F

0

for all

is isomorphic to the linear unfolding

Fo. b)

All we need prove now is that the linear unfoldinqs are isomor-

phic.

Let p

F(u,x)

I

u 1 hi (xl)

I

uiki (x))

(u,f (x) + 0

(u,f

G(\l,X)

0

i=l (x)

+

and

i=l

be two stable linear unfoldings.

Using standard arguments (exercise), C~

one can easily show that there exist

functions

g. (A,X) ,>. € lR, ~

such that p

F (u,x)

=

(u,f (x) + 0

1\

L

uigi

().,x\ =

£:>.,u,xl)

i=l

ia a stable unfolding of

r0

for all

Sin·ce Of ll'

3gi

0

! i=rl

u

•

(!\

o.,x)

€' \1 •

(f ~

'J,X

)t c F*lu,ylfe , .• ,et} 1

one can now use the argument in a) to show that rnorphic- to

·a "'

Q.E.').

Fl

G

is iso-

'

229 4.2. f ,g 0

\~e

Let 0

e

F

and

K(p,s,t).

G

e

S(p,s,t)

Suppose that

be stable unfoldings of f

and

0

g

are

0

Ks,t-isomorphic.

know, by XV.6.1 that there is a local diffeomorphism

in

h

Ls

such that

So, consider the composite mapping

G'

It is by definition isomorphic to the diffeomorphism g

0

o

h

I x h

G (even as an unfolding) via

e Lp+s ., and it is an unfolding of

= g~ Since g *Mt = f;Mt , we know by XV.6.1 that:

0

go(x> where

M(x) • f

M is a

first that

0

(x)

,

t x t-matrix with

det M(O) < 0 •

M(O)

invertible.

Ixk

where

m.t .,. m.t

k:

is negative.

:ay its construction,

0

= k o M(x) o f

det k

o

X

JR

G":

t 1

is a linear automorphism whose determinant

is a stable unfolding of g

Suppose

Consider the composite mapping

0

(x)

0=

G

,

k o

G" g

0

is isomorphic to

G'

and

where

and

M(x) > 0 •

Taking into account 4.1, all the above remarks reduce the proof of the Fundamental theorem to that of the following lemma. 4.3.

Lemma.

Let

(1)

g (x) 0

= M(x)

where

M(x)

is a

positive.

t ,g e K(p,s,t) 0 0

such that

• f (x),. 0 t x t

matrix with the determinant of

Then, there exist unfoldings

which represent isomorphic germs.

F,G

e S(p,s,t)

M(O) ~

f

0

,g

0

,

230 Proof.

Let

S (p,s,t):

in

F

p (u,f (x) + u.h. (x)) 0 i=l ~ ~ ' Let be a stable, linear unfolding of fo

(u,x)

F:

L

I-+

(the connected group of

em

be a

t x t

path such that

where

M(~,x)

M().,O)

2

1

=

g

(x)

0

A(l) = M(O), }

e

lR.

Let

• f !x) 0

+ M(x) - M(O)

(Note that

and for all

0 t

). t

A(~~

and

By definition, we have

f* M = f*M

(3)

M(~,xl

= M~ (x)

A().).)

f (x)

matrices with positive determinant)

A(O) .,. I

fA (x) = f().,x)

(2)

I

ll

in

lR •

Now, define F).:

(u,x)

(m,P

"m.s ,0)

-+

!:nf "

JRt ,0)

(u,f(~,u,x) = M(>-,x)

I->

• (f (x)+ 0

I

i=l

u.h.(x)ll). ~ ~

f*[u,y] = [u,f J in the notation of Lemma 3.1, where 0 stands for the ideal generated in f' A,u,x by

Note that:

and the components of

a£ aM ~=""'· CA

!

~

0

.

Thus,

(f + l. u.h.) eF*[u,y]·(f, ) 0 i=l ~ ~ AtUtX

CA

t

•

Consequently, using Lemma 3.1, we will have demonstrated that P

is isomorphic to

is isomorphic to that

F0

F

0

F

for every

0

in

lR

and thus that

A A is a stable unfolding of fA for all But this last step is fairly simple. By the stability of F

and

JCV.3.2, we have

..

0 t

cs,t

It follows from (3)

that

(.0)

)

J lfol

f*ll • [ A t

f*M •

s,t

F

1

(i.e., the goal of this Lemma), once we show

= f*M

o

t

Eo

• lR fh , ••• ,h } 0 1

•F

~.r

for all

s,t

~

•

231 On the other hand,

~=aM>. axi

M>. (0)

Since

at 0

• • fo + MA

axi

axi

is invertible, we can easily conclude that:

J(f>.) + f~Mt • Es,t = MA(J(f 0 l + f Mt • E5 ,t) •

0

Then, we can multiply (4) through by

+ f{Mt • E6 ,t +

J(fA)

:R {MA •

M>.

h 1 , ... ,M>. • hp}

By

XV.3.2, this implies the stability of

5.

Orbits of Stable Germs Let

F € S(p,s,tl

to obtain:

FA •

be a stable unfolding of

t

e K(p,s,t).

0

Ne saw in XV.3.3 that this implies that

+ f*M • E

J (f )

0 t

0

s,t

:;:,

E

Mp+l •

s

s,t

Therefore, TKf

(1)

= Ms • J

0

~I~n~t~h~e~a~b~o~v~e~s~~~·t~u~a~t~io~n~·~t~h~e~o~r~b~i~t

Proposition.

b r ;:>+2F EY

image of

.p+2 J 2f p+ K

. . f ~nverse ~mage o

Proof.

If l + f*M · E :l MP+ 2 • E 0 0 t s,t · s s,t

0

Similarly, the orbit

E.z:.

is the inverse

fF

Kf

jp+2

is the

0

By 1.2.2), we need only prove the second part of this

proposition relative to the contact orbit.

This is essentially

Theorem XV.6.2. But we can also obtain this result by using Lemma 3.1. suffices to show that if g

0

= f

+

0

£

,

where

£

e

satisfies condition (1) and if

f~

MP

It

2

s

•

E

s,t

, then

g

0

e

Kf

0

•

To show

this, consider F:

where of

t

F 0

,

0

(A,u,s)l-+- (A,u,f(u,x) =

F:

(u,x)>-+ (u,f(u,x))

For any

( cf. XV. 3. 2 . )

+ AE(x) = f(A,u,xl),

A

in

:R ,

FA

is the given stable unfolding is a stable unfolding of

fA .

232 On the other hand, it is clear that:

af" ar=

af"

3t

£8 [u,x] {-- , ••• , --} 3x 3xs 1

by hypothesis (1).

+ -. F [u,y]{e , ••• ,e} 1 t

So,

F). is isomorphic to Fo for any A In particular, Fl' an unfolding of g = fo + £ I is isomor0 phic to · F d hence 9o is embedded in Fo and is K-isomorphic to f

0

by 2.3.

Remark. f

0

e

Q.E.D.

We will see in the' examples of the next Chapter that for

K(p,s,t) TKf

0

:::.

we will often have

Mr • E

for an integer orbit

fF and

s

s,t

r

much smaller than

and the orbit Kf 0 under J.r

p + 2 •

In this case, the

will be inverse images of

233 CHAPTER XVII GENERIC SINGULARITIES: EXAMPLES

1. 1.1.

Introduction.

Stability of

generi~

singularities.

In this chapter we intend to solve, in some particular

cases, the following two problems: (1)

Given

s

and

germs from (2)

t

lRs

, list and classify the stable map to

, up to isomorphism;

Determine whether the generic singularities of mappings from

lRs

To be more precise, let (global) mappings the germ

lRt

f

xo

f:

to

are stable.

S(s,t) ~ C (s,t)

IRs + IRt

is stable.

IRt

(Here

00

be the set of

such that for any f

xo

C

x

0

in

stands for the germ of

IRs f

0 at x considered as an element of E by setting f (~) 0 s,t x 0 f(x +~) - f(x ).) The second problem above is to decide whether 0 0 or not S(s,t) is dense (more precisely, residual) in C (s,t). 00

00

For example, we have shown in VII.7 that generically in C (2,2) 2 2 a map from 1R to 1R has oniy cusps and folds as singular points and in XV.2.2 that folds and cusps are stable germs.

Since

mapping germs at a regular point (where the rank is two) are always 00 2 2 stable, this shows that 5(2,2) is residual in C (IR ;IR ). 1.2.

Before describing the method we will use to solve problems

(1) and (2), we introduce s6me notation. ( 1.3), denote by

As in the previous chapter

S(p,s,t) the set of map. germs from

(lRp+s ,0)

to

234

(lR p+t, 0)

which are stable and which have rank

p

at

0

Recall

that the transversality condition on the rank of a stable germ (XV.l.2) implies that 1p

(l)

~

s (t -1)

00

Now let mappings

J.

S(p,s,t) p + s . (Recall that R(p,s,t) ia t!>.e in·.rerse image of it;s projection into Jp+2.) s,t

'

•I

R(p,s,t)

To solve problem (2), the main tool is the following result of J. Mather:

I I'

I

235 1.3.

Theorem.

The following conditions are equivalent:

1)

S(p,s,t)

is a residual subset of

2)

H(p,s,t)

is tru.e.

C~(p+s,p+t);

Condition 1) implies that the generic singularities of rank p of

C~ maps from

lRp+s to lRp+t

are all stable.

Theorem 1.3 has the following immediate consequence. 1.4.

Corollar:i·

Let

s'

t'

and

be positive inte9:ers.

Then,

the followin9: conditions are eguivalent: S{s',t ')

is residual in

2)

H{p,s,t)

is satisfied for all triJ2lets

that 1.5.

c"'(s' ,t'J

1)

p + s

= s'

I

p + t

= t',

and

(p,s,t)

such

p > s(t-1)

Comments on Theorem 1.3. This is the only important result which we will not prove.

Its proof can be found in Mather {1970a). However, note that

"2) implies 1)" is essentially proved

using Theorem XV.S (the characterization of stable germs of rank p) and the Transversality Theorem..

A complete proof presents no

essential difficulties. The converse ("2) implies 1)")

is a bit more delicate.

We

will give an idea of how its proof goes through some examples later in this chapter (especially in section seven). The rest of this chapter is devoted to the solution of problems 1) and 2) for some values of

p,s,t •

In each case,

we will follow the strategy suggested by Theorem 1.3: a)

We study. K{p,s,t) Th~s

and compile a list of normal forms.

provides us (via stabilization) of a list of normal

forms for elements of b)

S{p,s,t);

We compute the "codimension" of Problem 2) ·'

R(p,s,t) to solve

2.

The study of

236

K(p,l,l). f:

Here we want to study map germs with

f

(JR,O)-+ (:ffi,O)

M! , up to contact equivalence.

in

(1) in 1.2 tells us that

p

(s=t=l)

Note that inequality

can be any non-negative integer.

It

is obviously true that

..

f*Mt

[f] .. Mn+l s f(x)

if and only if

xn+lh(x), where

=

dir~t

XV.6.1 .(or by a simple

h(O) 'I 0

there is exactly one K-orbit of eodimension K.xn+l

K

n

= Mn+l

Therefore, by

argument) ·for each integer

n > 1 ,

n , namely:

_ Mn+2

s

s

The following Proposition then is a simple consequence: Proposition.

3.

3.1.

a)

K(p,1,1l

b)

R(p,l,l)

M~+J

has codimension

Generic· singularities of rank p from

lRp+l to

p + 2

in

Ms •

1Rp+l.

Proposition 2 above and theorem 1.3 prove the Generically, a

THEOREM.

c""

mapping

at every point where its rank is

f,

lRp+l .... lRp+1

is stable

p •

Moreover, one can compute normal forms for all the stable germs of rank

p

by constructing stable p-parameter unfoldings of the 2 • In this way, one obtains (easy exercise)

functions- x 2 ,x3 , ••• ,xp+ \

the singularities cribed in 3.2.

L

1 ,0 r

(r .::._ p

+

2)

that have already been des-

xrv.s.

Corollary

l) The generic singularities of mappings from the

plane to the plane are stable; the only such singularities are the fold and the cusp.

237 3 3 The generic singularities of mappings from IR to IR 1 0 1 1 0 stable; they are the germs E ' (fold), E ' ' (cusp), and 2)

El,l,l,O (swallow-tail). To prove this, it suffices (by Corollary 1.4 above) to notice that: 1)

p + s = 2, p + 5 = 2, and

p

2 (s = t

or

0)

p ::_ s (t-1)

P. = 1 (s = t

generic singularities from by 2)

p

IR2

is equivalent to 1). Thus, IR2 are classified

to

K(l,l,l)

+ s = 3 =

p

3

+

p

(s = t = 0)

and

t

or

singularities from

IR

p.::_ s(t-l) 2 (s

p

3

to

=t JR

3

is equivalent to

= 1); thus, the generic are classified by

K(2,1,1). 4.

K(p,l,2).

Because of the inequality p ::_ s(t-1), we must have p ::_ 1 2 in this case.• We need to study map germs f: (IR ,0) + (IR ,0) 2 (s=l,t=2) with f in M E Writing f=(f ,f J , we have s s,t 1 2 f*Mt

= [f1 ,f2 J

M:+l

if and only if n+l

df(O) dx

0 , and

~(0) n+l dx

0

.#

r

•

Then, using Proposition xv.6.l again, we have Kf = K. (xn+l, 0) In this way, we obtain a unique K-orbit of codimension integer

2n

for each

This proyes the

n ::_ 1

Proposition. a)

K(p,l,2) = K.(x2 ,o) v K.(x 3 ,0) V ••• V K(xn+l,O), where n

b)

is the greatest integer such that

R(p,l,2) = Mn+a • E s

s,t

2n

p + 1 . 0 has codimension::_p+2 in E ~

s,t

=M

s

•E

s,t

•

238 5.

IRp+l to

Generic sin2ularities'of rank !: for mappin2s from IRp+2 (p ~ 1). The above Proposition and Theorem 1.3 prove the

5.1.

Theorem.

Generically, a mapping

every point where its rank is

IRp+l + Rp+ 2

f:

p

We now look for normal forms. f : n

defined

by

cm,ol x

+

1->-

is stable at

Consider the map

em? ,ol

(xn, 0)

•

Using the notation of XV.3.2, we have: · Cf) + f*Mt • Es, t + lR qxo) n

J

n-1

' (o) x ' ... ' (xo ),( o n- 1) ' X

(

on ) } = Eo s,t

X

Therefore, a "minimal" stable unfolding (i.e., one involving the smallest number of parameters) of F

defined

n

:

(IR 2n-l x m ,0) +

(lR 2n-l

f

n

is:

" m2, 0)

by

ul

ul

un

un

\"

v

1

F

~

n

X

vl

v

n-1 Jl n-1. x +v x ••• + v 1'\-l.x 1 n .,. u x n-1 u x • .•• + u X 1 2 n

These examples of stable germs of normal forms.

...

F

n

represent the desired list

239'

5.2.

Corollary.

2 3 (lR ,0) -+ (JR ,0)

(u,x)

F : 1

(H. Whitney, 1944).

The map germ from

defined by 1-+-

(u,ux,x 2 )

is a normal form for the only generic singularity that occurs for mappings from Proof. 2

m

m

m3

to

One checks easily that the stable singularities from

to

IR

3

are

cla~sified by K(l,l,2); then one applies the

above theorem. X

-----------1----r------------U

This singularity is called a "cross-cap" or "Whitney's umbrella."

It sends the vertical lines (u =constant) to a family 2 of parabolas (X= x , Y = ux) • 3

4

Exercise. Classify the stable si_ngularities from IR to m 4 5 6 5 from m to m , and from IR to JR. Show in each case that the generic singularities are stable. 6.

K(p,2,1) This is the first non-trivial case.

K-orbits of codimension f:

2

(JR

~P

We need to study the

+ 2 of maps

,0) -+ (lR,O)

(s = 2,t = 1) such that

f

is in

/is .

Note that

p ~ 1

by

L2 (1).

We will limit ourselves to some partial· results which have already been stated in a different form. denote the coordinates in the source

We will use (x,y) to 2

:JR •

240 6.1.

Proposition

(p=4).

a)

K(4,2,1)

consists of twelve K-orbits

represented by the following normal forms (the codimension of each

x

K-orbit is given in the parentheses): 2

X

=

b)

4

y (4); X

2

R(4,2,1)

5

+ y (5);

X

2

6

~

y (6); X

has codimension

~

7

3

2

·~

2 2 3 + -y (2); x + y (3); -2 2 4 xy (5); X y ~ y (6).

Ms

in

The proof follows easily from the results of Chapter XII, and Moreover, one can easily verify that

we leave it to the reader. these K-orbits are actually

L -orbits. s As a consequence, for any p (1 ~p ~ 4), the singularities of rank p of a generic map from 1Rp+ 2 to 1Rp+l are all stable. 6.2.

The case

p = 7 •

The interesting phenomenon is that Condition satisfied.

H(7,2,l) is not

More precisely, one has the

Proposition.

R(7,2,1) = M~'\K(7,2,1)

The set

therefore has codimension

9

i

in

4 M , the ideal s

Now,

therefore it is enough to show that TKf has codimension at least 4 4 M4 Consider TK f = j (TKf) I the tangent space at j4f s 4 to the orbit K f in J4 ; the 4-jet of f may be identified with

one in

2

the degree-four "Hessian" of

f

at

0 , which is a ho100geneous

polynomial of degree four. 4 Then, TK f is the subvector space of

(the space of

homogeneous polynomials of degree four in 2 variables) generated by X

ClP ax

1

However, since while

P~

C)P y ax 4P

()P

t

X

ay

= X C)P ax

t

y

i!P

ay

+ y aP

has dimension 5.

ay

1

I

p •

4 TK f

has dimension at IOOSt 4

This completes the proof.

241 7.

Stability is not generic in

9 8 C"' (:IR , lR ) •

7.1.

Proposition 6.2 shows that Condition

fied.

Using Theorem.l.3, this proves the

H(7,2,1)

is not satis-

Proposition. Actually we are going to prove this directly without using Theorem l. 3.

We will present an example of a mapping

with the following property:

8 vcc"'oR 9 ,m )

ofF

words, every map

G

m?

->-

m8

there exists a neighborhood

suchthat

S(7,2,l)C\V

sufficiently close to F

topology defined i.n VII. 6. 3)

F:

isempty; inother

(with respect to the

has a non-stable germ at some point

(with rank= 7 at this point). 7.2.

(u , .•• ,u ,x,y) and (v , ..• ,v ,z) denote the coo:dinat:s 1 7 7 1 8 m respectively. Consider the mapping F: lR -+ lR

and

in

defined by

+

z = f(u,x,yl.

f (x,y) = xy (x-y) (x+y) . 0

where

The germ of F at 0 is a 7-parameter unfolding of the function M4 (in the notation of §6) and is ·not stable in view of 6. 2. s

actually has codimension 10 by III.6.2.d)

and the transversa-

lity condition XV.S cannot be satisfied.) 3 Now consider the subset E c J · defined as follows: 9,8

f a)

is in

E if and only if

the linear part

A of .f

has rank 7, and

212 the map

b)

Df

f:

IR

"'Ker A C.lR

~

2

9

1R

.!,.

1

defined by the composition:

8

:IR

~ coker A "'

IR

is zero. It is easy to show (exercise) that 9 3 1R + J

9,

=9

r

codimension

.

is a submanifold of

It is also clear that the mapping 3 l: at 0 (j F (0) € l:)

is transverse to

a

a

9

Thus, if G: lR -> lR is sufficiently close to E' in the 4 Whitney c -topology (see VII.6.3), then there exists a point p 9 3 in :IR (close to 0) such that j G(p) is in E • This means that: a) at

has rank 7 at p ; using convenient local coordinates

G

and

p

9 :IR

G (p) in

8 IR , the ,germ of

and

G

at

p

can be

written in the form (u , ..• ,u ,x,y) 1

The function

b)

G is not stable at 7.3.

Remark.

1

7

g(O,x,y) p

has a zero 3-jet.

Therefore,

by the same argument we used for

at

F

0 •

Consider once mere the mapping

f (x,y)

xy(x-y) (x+y)

0

from (1R2 ,C)

(u , ••• ,u ,g(u,x,y)j.

M

7

to

The K-orbit of

fo is equal to th'e L;;-orbit of f (see IIL6.2d) and has codimension 10 in M 0 s Therefore, f has a stable unfolding (which is in fact a universal (lR ,0)

0

unfolding) with 8 parameters: F:

tiR 8

m.2

x

,o> ~

Similarly, the germs fo'a(x,y)

r

fO,a

not isomorphic if

B

(lR

2

,0)->

xy(x-ay) (x+y),

admit stable unfoldings 4-tuplcs

tm.8 , :m ,o)

a 1

F

a

:m

(a

10

-+

S , because

_ (lR.,O),

'I

0,· 1) 9

IR •

1

F- (0) a

Now, F and

a

and 1

F~ (0)

of straight lines in a plane, with cross-ratios

respectively.

FB

are·

consist of

a

and

Therefore, there exists a continuous (infinite) 10 9 :IR to :IR .

family of distinct stable germs from

243

B.

The study of some

B.l.

K(s,s,2) germs,

Notice first that

K(p,s,2)

s > 2

is defined only i f

consider the simplest case:· p = s

~

p

s ,

ll'e

Thus, we are interested in

map germs

2 (f € M • E ) and for which s s,t 2 TKf M • J (f) + f*M • E c M • E . has codimension < 2s in s t s,t s s,t: 0 2 Es,t = Ms • Es,t • Notice that Ms • Es,t c: Ms • Es,t has codimension 1 2s: indeed the quotient space M • E 1M 2 • r J is the space s s,t s Ls,t s,2 of linear maps from lRs to m2 • which have rank 0 at the origin

Therefore, (1)

2 TKf = M • s

K(s,s,2)

is exactly the set of germs

f

such that:

Es,t

In view of Theorem XV.6.2, Condition (1) is equivalent to: (2)

is 2-open in

Property (2) is a property of the 2-)et of f at 0 , which is a 2 lRs to lR , i.e., its components are

quadratic mapping from

homogeneous polynomials of degree 2 • Denote by p2 the vector space of quadratic mappings from s,2 2 ]RS to lR. We have just shown that the study of K(s,s,2) reduces to the study of the open orbits of the Lie group K2 s.,2 2 acting on p2 But the action of K2 on Ps,2 reduces to ·the s,2 s,2

8.2.

action of the linear group

1

K , because we are dealing with s,2 2-jets with zero linear part. Recall from XV.4.1 that Kl s,2

= Gi.(s,lR)

x Gi.(2,IR).

summarizing our results so far, we can state the Proposition: .2 2 J : Ms • Es, Gt 12

.m>

The K-orbits in K(s,s,2) are the inverse images by 2 + Ps, ·of the open orbits of the action of G£ (s,lR) x 2 2

£!!.

r;,

2

244 In other words, we are led to the problem of classifying 2 quadratic mappings from JRs to JR relative to the most natural notion of isomorphism:

"'0 f 0