Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
552 C. G. Gibson K.WirthmQller A.A. du Plessis E.J.N. Looijenga
Topological Stability of Smooth Mappings
~
~..
Springer-Verlag Berlin.Heidelberg 9New York 1976
Authors Christopher G. Gibson Department of Pure Mathematics University of Liverpool P. O. Box 147 Liverpool, L69 3BX/Great Britain Klaus WirthmLiller Fachbereich Mathematik d. Universit~t Postfach 397 8400 Regensburg/BRD Andrew A. du Plessis School of Mathematics and Computer Science University College of North Wales Bangor Gwynedd, LL57 2 UW, Wales/Great Britain Eduard .1. N. Looijenga Mathematisch Instituut der Katholieke Universiteit Toernooiveld Nijmegen/The Netherlands
AMS Subject Classifications (1970): 57D45, 58C25 ISBN 3-540-07997-1 ISBN 0-387-07997-1
Springer-Verlag Berlin 9 Heidelberg 9 New York Springer-Verlag New York 9 Heidelberg 9 Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin 9 Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
Preface
During the academic year 1 974-75 the Depar~nent of Pure Mathematics in the University of Liverpool held a seminar on the Topological Stability of Smooth Mappings:
the main objective was to piece together a complete proof of the
Topological Stability Theorem (conjectured already by Them in 1 960, and proved by Mather around ~ 970) for which no published account existed.
This volume comprises
a write-up of the seminar by four of its participants. There are several acknowledgements which should be made.
Any mathematician
working in this area is conscious of his debt to the inventiveness of Them, and to the technical work of Mather which has placed much that was conjecture on a firm mathematical foundation.
As far as the seminar is concerned I would like to
single out the special contribution of Eduard Looijenga, who showed us how to fill in gaps which otherwise might have remained open.
Also, I would like to
acknowledge the considerable help offered by Terry Wall, from the inception of the seminar to the production of the typescript.
Further acknowledgements are due
to the seminar audience (who frequently had good reason to appear confused) for their patience arKl in particular to Tim Ward who helped out with the talks;
to
the British Scientific Research Council who provided financial support for Eduard Looijenga, and the University of Wales whose financial assistance enabled Andrew du Plessis to participate in the seminar;
to Los Lander who was inveigled
into drawing the diagrams, and did an excellent job;
to Dirk Siersma and
Klaus Lamotke whose careful reading of parts of the manuscript removed many errors; and finally to Evelyn Quayle, Jean Owen and Margaret Walker who produced a first-class typescript.
Liverpool, July 1976o
C. @. Gibson.
Contents
Introduction Chapter
I
: Construction of Canonical Stratifications
w
Whitney Stratified Sets
w
Semialgebraic Sets
17
w
Thom Stratified Mappings
22
Chapter
II
:
9
Stratifications and Flows
w
Tubes
36
w
Tube Systems
41
w
Vector Fields
50
Flows
53
Applications
58
w Chapter III
: Unfoldin~s of Smooth Map-Germs
w
Introduction
65
w
Pre] ~m~ uaries, mostly algebraic
73
w
Infinitesimal Stability
80
w
Versality
86
Contact- equivalence
94
w
w Chapter
IV
Determ~ nacy
1o5
Jet-spaces and a transversality theorem
111
@enerioity
t20
: Proof of the Topological Stability Theorem
w
Multi - transversality
125
w
A stratification of the jet space
131
w
Properties of the stratification
w
Topological stability as a transversality property
145
References
15o
Index
152
Symbols
155
Introduction Motivation and some historical remarks. The
C~176
f
and
f'
from a manifold
to be
C l -equivalent
(l = 0, l, 2, ..., oo)
h
N
P
of
and
h'
of
equivalenoe relation: 'C g -behaviour'. of
f
We call a
to a manifold
if there exist
f' = h' o f o h -1.
it simply says that
~ith respect to
function space
such that
N
Coo-mapping
f
and
f'
~
C e -stable
if the e ~ i v a l e n c e
This presupposes a topology on
increases.
Coo-mapping can be approximated by a C~
mappings,
f
class
in the
C~176 P):
we choose
Obviously, this property
There is an important question, related to
this notion, which has also some physical interest.
the set of proper
This is clearly an have the same
the Whitney topology (see Ch. IV for the definition). becomes stronger as
are said
C~-automorphisms
C t- equivalence) forms a neighbourhood of
Coo(N, P).
P
Namely, whether any proper
C l -mapping or more precisely, whether among
Cpr(N , P),
the
C t-stable
ones are dense.
The Morse lemma and the Whitney embedding theorem imply that the answer is yes if
dim P = 1
or greater than
2 dim N (and
e
arbitrary).
H. ~%itney seems to
have been the first to investigate this question in its own right. he showed that the answer is affirmative in case included a fairly simple characterisation of
In 1955 [W~]
dim N = dim P = 2.
His proof
Coo-stable mappings between surfaces.
A few years later, in 1959, Thom [TL] gave examples of proper smooth mappings from I~n
to
I~n
(n ~ 9)
which cannot be approximated by a
A complete answer to this question for 1967.
I = oo
In his fundamental series 'Stability of
C 2 -stable mapping.
was given by J.N. Mather around Coo-mappings'
[I-VII
obtains among other things a (mult~ltransversality criterion for For certain pairs
(n, p):= (dim N, dim P),
he
Coo-stability.
this transversality criterion
involves unccuntably many tramsversality conditions.
As one may expect, in such
a case these transversality conditions cannot be simultaneously satisfied by a dense subset of
Coo (N, P). pr
Thus Mather was able to determine the pairs
for which the answer to our question proper mappings are dense in of the following conditions
(with
Cpr(N , P)
~ = oo)
is yes:
if and only if
the
(n, p)
(n, p)
~~ satisfies one
n
<
n
<
V6 p §
8
6
p - n ~ 4 ,
and
3 ~P-n
~0
p<
8
and
p - n = -i,
p<
6
and
p - n =-2,
and
p - n = -3.
p<7 Mather also showed that for negative.
and
= 1
the answer to our question is in general
So we are lead to consider the case
In the previously mentioned article [TL], would be yes in that case. to how this might be proved.
stratified mappings.
(topological equivalence).
Thorn conjectured that the answer he gave some indications as
There he introduced an intermediate between the the category of stratified spaces aud
Part of its interest comes from Whitney's regularity lemma
which asserts that any complex-analytic
Thom's sense.
t = 0
In later papers [T1, 2]
smooth and piecewise- linear categories :
[W2]
,
S. Lojasiewicz ILl
set admits a stratification in
generalised this later to semi-algebraic sets
and used this extension to prove that any closed semi-algebraic set is triangulable. Thom's work on stratified sets culminated in his famous isotopy lemma's IT1, T3, HI] to which we shall return below. on
C~176
and Thom's philosophy, Mather obtained around 1970 a definite
proof of Thorn's conjecture. J
.
By fully exploiting his own work
An outline of this proof has been given in
.
SemLualre Bourbaki by Cheneiner [C] (see also [M2]).
A complete version will
appear in a book Mather is writing. The proof presented in these notes is somewhat different from Mather's (as it does not use the existence of global universal unfoldings) and follows Thom's indications more closely.
As far as prerequisites are concerned, we
require of the reader no more than some familiarity with differential topology and commutative algebra, both on an elementary level. we refer to other work, however. sets
(I-(2.1),
the erem.
For a few not-so-standard facts
These are various properties of seml-algebraic
(2.2), (2.4), (2.5)
and
(2.6))
and the Malgrange preparation
Structure of the proof. Actually the proof will yield a stronger result. us call U
f e C~(N, P)
such that any
stron61y topologically stable
f' e U
parameter family
can be joined in
{ftlte I
topologically trivial.
(where
I
f0 = h~ o f t
o
deserves its name:
U
if it has a neighbourhood
with
stands
f
for the unit interval) which is
h~1
for all
it implies
lhtltE I
t e I.
of
N
N
and
lh~ItE I
of
P
such
Strong topological stability clearly
C ~ -stability.
Our aim is to prove that the
strongly topologically stable mappings are dense in we take
by means of a smooth one-
By the latter we mean that there exist continuous
families of topological automorphisms that
In order to state it, let
C~r(N , P).
For simplicity
compact here.
At some stage we have to construct one-parameter families of homeomorphisms. 0nly one geometric technique is known that can furnish these, namely integration of vector-fields. manifold
M
As is well-known, a
C 1 -vector-field
generates a one-parameter family of
~
on a compact
C 1 - diffeomorphisms.
In certain
situations this is mimicked by possibly discontinuous vector-fields which satisfy certain control conditions. homeomorphisms.
These will then generate a one-parameter family of
A typical and very illuminating example of such a vector-field
is described in (II, 1.1). going further.
We advise the reader to have a look at it before
Roughly speaking in Thom's stratification theory one considers the
case where the vector-field
~
on the manifold
up into smooth submanifolds
IMi~
such that
M
is such that
~IM i
M
can be broken
is smooth andtangent to
In order to he more precise, we need the notion of ~/~itney regularity. and
Y
be disjoint submanifolds of a euclidean space.
Whitney regular over
X
both converging to some and the tangent planes
if for any pair of sequences x~ e X TYiY
such that the lines
converge to a plane
Then
Y
M
instead of euclidean space.
explained b y the following property.
If
Y
X
[Yi e Yli=I
converge to a line
p , we have
~p.
notion is coordinate invariant and has therefore still a meaning if in a manifold
Let
is called
Ix i E X}i=l, xiY i
M..
This X
and
Y
The interest of this notion is is Whitney regular over
X,
then
are
X
admits a (tubular)neighbourhood
such that
TX
~X : (Tx' TX A ~' X) -* X
in
M
with retraction
is a topologically locally trivial fibre
bundle triple with 'cone-like' fibres.
The proof of this (which is not hard)
consists in showing that any ordinary vector-field trolled ~cssibly TX
such that
as follows. partition
~
on
X
extends to a
con-
discontinuous, but at any rate integrable) vector-field
WIY Let
A
~X : TX-~ X
on
is tangent along
Y
and
be a subset of
M
which is endowed with a locally finite
A
into submanifolds of
Whitney regular over
X .
M
d~x o W = ~ o ~X"
W
such that for any pair
This generalises
X, Y E A,
Y
is
Such a partition is usually called a Whitney
stratification and its members strata. Now suppose that of
p
to any
asserts that
X E 4
p : M-* M'
is a proper submersion such that the restriction
is also a submersion.
p : (M, S, IX E At) ~ M'
is a locally trivial fibre bundle.
Especially interesting is the case where ordinary projection; are isotopic in
N
Then Thom's first isotopy lemma
M = N x ~
then it follows that
and
p : N • ~ ~
I~
A t = Ix E N : (x, t) E A1
(hence the name isctopy lemma).
and
A~
We will rabher be concerned
with the second isctopy lemma which deals with families of mappings (instead of families of spaces). stratification for M'
such that
F
Let F
F : M ~ M'
be a smooth mapping.
is a pair of Whitney stratifications
maps a:~ stratum of
A
A Them A
of
M
and
submersively into a stratum of
4'
of
4',
and satisfies another condition (Them regularity) which we do not describe here but which is 'almost always' fulfilled. M = N • I~, M' = P • I~ (F(x, t) = ft(x), t). if
F
and
F : N • I~ ~
maps the strata of
topologically trivial f~mily. transverse to
N x It~
intersections
A
and
P • I~
is of the form
In this situation the second isotopy lemma tells us that
admits a Thcm stratification
P x i~ ~ I~
The case that interests us is when
and A'
4'
(4, A' )
such that the ordinary projection
submersively to
I~,
then
Note that then the strata of P • Itl.
A
IftltEl~ and
4'
is a must be
It is not hard to show that b y taking
determine a Them stratification
(At, A~)
for
ft"
We therefore may paraphrase the second isotcpy lemma by saying that the family Iftl
is topologically trivial if it admits a smooth family of Them stratifications
I(A t, A[)I . In view of this, it seems to be important to construct Thom stratifications for an open-dense set of mappings.
This is best done by means of a certain
Whitney stratification of the jet space partition of extension A
o
Jl(N, P)
into manifolds and
J~f : N ~ J~(N, P)
= (Jf)-lA
Jl(N, P). f : N~
P
A
is a locally finite
such that its. ~ - jet
is transverse to the members of
is a finite manifold partition of
transverse with respect to
If
A
if moreover
f
N.
then
A ,
We call
f
multiA
maps the members o9
in a O
regular way to
P ,
more precisely if for any ~P
(with
X
E A X
containing
~
|
the canonical homomorphism
(TyP/~(TxXx))
x ~ f-l(y)
x)
y E P
is a surjection.
It can be shown (in much the
O
same way as the Thorn transversality lemma is proved) that the mappings which are multi-transverse with respect to transverse with respect to
A,
A then
are dense in A
C~176 P).
If
f
is multi-
can be refined in a natural way to a O
manifold partition
Af
algebra generated by
(namely, the set of indecomposables of the Boolean A~
and
have either disjoint or equal
If-lf(x) : X E Aol) f-images.
We then obtain a partition of
(not necessarily into manifolds, of course) . partition of
P
which we denote by
By adding
f ~ CC~
P)
(Af, A~)
At(N, P)
C~176 P).
of
J~(N, P)
Now let
f
f
with respect to (N, Aft),
A ~ (N, P).
(P, Aft )
we obtain a
with the preperty that for any
AZ(N, P)
A~(N, P)
the corresponding pair
f .
Moreover the mappings
are not only dense, but also open in
be multi-transverse with respect to
by a smooth family
f(N)
sufficiently large there exists
actually defines a Thom stratification of
arcwise connectedness it has a neighbourhood joined with
t
multi-transverse with respect to
multi-transverse with respect to
P\f(N)
Af
f Af.
In the last chapter it is shown that for a Whitney stratification
such that two members of
Iftl
U
A~(N, P).
such that any
f' E U
By local can be
of mappings which are multltransverse
Then the corresponding Thom stratifications
fit together to form a Them stratification
(A, A' )
of the
associated
F : N • I -~ P x I
submersively to
I.
with the property that strata of
By the second isotopy lemma
f
and
f'
A'
map
are then
topologically equivalent. We shall not discuss the construction of
A~(N, P)
here, but only remark
that for its definition a very basic r~le is played by the concept of a canonical stratification.
In chapter I this notion is investigated in its relation ~o the
semi-algebraic category. of Thom.
The next chapter states and proves the isotopy lenlnas
Chapter III is an exposition of the Thom-~ather unfolding theory and
thus contains somewhat more than we need. two previous chapters. stratification
A~(N, P)
It is completely independent of the
As we have said, in the last chapter the Whitney is constructed and by means of this stratification the
proof of the stability theorem is completed. Related problems. Finally, we mention two related problems, which have already been discussed by Thom in [T2] and [T4] . Problem I (t' ~ t)
Has any jet
which is
z E Jl(N, P)
C~
an extension
z' E Jt'(N, P)
(i.e. such that any two representatives of
z'
are topologically equivalent) ? This problem has been solved affirmatively by A.N. VarchenkO IV], obtained in fact a much stronger result. Problem 2 C~176 P)
Apparently still open is
Does there exist a Whitney stratification
U
a parameter manifold) is
associated map the case
Nijmegen,
A
of
(neglecting a subset of infinite codimension) into submanifolds of
finite codimension with the fo,llowing property: (with
who
U * Coo(N, P)
g~
is transverse to
a smooth family
If t E Coo(N, P)l
(in an obvious sense) if the A ?
This is known to be so in
dim F = 1 .
May 1976
E.J.N. Looijenga
References cited in the introduction. [C]
A. Chenciner,
Travaux de Thom et Mather sur la stabilit~ topologigue.
I
Sem. Bourbaki, F{vrier 1973, no. 424. [~]
S. ~ojasiewicz,
[MI-VI]
Ensembles
Stability of
89 p.254-291, p.223-248,
(Svrtnger Lecture Note 383)
semi-analytiques,
available at IHES (1965).
CO
C -mappings I-VI, Ann. of Math. 87 p.89-104, Ann. of Math.
Publ. Math. IHES
35
p.127-156, Publ. Math. IITES 37
Adv. in Math. 4 p.301-366,
Proc. Liverpool Syrup. I (Springer
Lecture Note 192 ) p.207-253. IN1] Harvard notes on topological stability (1970). [M2] 'Stratifications and Mappings' in Dynamical Systems, M.M. Peixoto (ed.) p.195-223,
Academic Press (1973).
[TI] R. Thom,
La stabllzte topologique des applications polynomlales,
L'Enseignement Math. 8 p.24-33 (1962). [T2] 'Local topological properties of differentiable mappings' Analysis,
p.191-202,
in Differential
Oxford U.F. (196&).
[T3] Ensembles et Morphismes 8tratzfzes,
Bull. Amer. Math. Soc. 75 , p.240-284
(1969). [TL~] 'The bifurcation subset of a space of maps' in Man~folds-Amsterdam
1970
(Springer Lecture Note 197) p.202-208. [TL] R. Thorn and H. Levine,
'Singularitiez o~' d lf't'er~nti~l cJ~L~t>iri~r reT>~ir,te~
in Proc. Liverpool Syrup. I (Springe] Lecture Note ]9[_~) p.l-~9. Iv]
A.N. Varchenko,
Algebro-Geometrica]
E~uic.~n~u!ar!ty ~,nd Loc~! Topologies•
Classification of Smooth Mappings, Proc. of the Inlt. Congress of Math. Vancouver, 1974, p.&27-431. [W'l] H. Whitney,
On singularities of mappings of Eudlidean spaces I,
of the plane into the plane,
Mappings
inn. of Math. 6__22, ~.374-410 (1955).
[W2] Tangents to an analytic variety,
Ann. of Math. 8_~ p.469-549 (1965).
CHAPTER
I
Construction of Canonical Stratifications
Christopher Gibson
w
Whitney Stratified Sets
For the purposes of this volume a stratification of a subset manifold
M
is a partition
X
of
V
V
into smooth submanifolds of
M
of a smooth (called the
@trata) which satisfies the
Local Finiteness Condition Every point in
V
has a neighbourhood in
M
which meets only finitely
mar~v strata.
A good example to keep in mind is that of an algebraic set (i.e.
a set defined by the vanishing of finitely many polynomials).
the~set
ZV
of all singular points of
lower dimension) and that
V - ZV
Vd_~Vd_ ~ D .......
of
defining
Vi_ 1
to be
V
ZV i if
V
by taking dim V i = i ,
Recall that
is another algebraic set (of strictly
is a smooth manifold.
There are finitely many differences i
~ _C lqm
V d = V,
where
an~ to be
V i - Vi_ I ,
(or empty) yielding a finite stratification of
One obtains a filtration d = dim V,
V.z if
and
dim V.l < i .
each a smooth m ~ 4 f o l d of dimension V .
A simple illustration of this naive construction is provided by the algebraic set
V _C ~ 3
defined by
x
2
= zy
2
(the Whitney umbrella).
The resulting
stratification (into a surface with two connected components, and a line) is i~icated in Fig. I .
LZ
Z
/
J
/
Y
4
Fig.1
Fig. 2
10
Ideally (returning tO the general situation) one seeks stratifications for which the local topological type of the pair
(M, V)
is constant on each stratum.
From this point of view the stratification of Fig. 1 is unsatisfactory.
Indeed,
if we take a point on the z-axis and sketch the intersection of a small ball centred at that point with
z
we obtain
z=O Clearly,
z: O
something rather special happens at the origin, where the local
topological type changes. z < 0
V
If we agree to split up the z-axis into
z > 0 , z = 0,
we obtain a second stratification of the Whitney umbrella (indicated in
Fig. 2) in which the ~rawbaek of our first stratification is no longer present. The question of just what happens at a point on a stratum where the local topological type changes was analyzed by [Whitney I ] .
To avoid such situations
as that in Fig. I he introduced the following condition for smooth submanifolds X p, Yq when
of a smooth manifold M C
~m
M m,
is an open set.
and a point One says that
x E X . Y
Take first the case
is Whitney regular over
X
m
~t
x
when the following holds.
respectively (with spaces
(TyiY)
Is
(xi) , (yi) are sequences in
x i ~ yi ), both converging to
converges to a subspace
q-dimensional subsRaces of
~
m
),
T C
I~m
x,
X, Y
if the sequence of tangent
(in the Grassmannian of
and if the sequence
---'~ (xiY i)
of lines containing
11
x. - y. i i
converges to a line
subspaces of
I%m) ,
then
L C I~m
L C T .
(in the Grassmannian of I -dimensional (On several occasions we shall tacitly use
the fact that then automatically we have following. mapping at
x
Given a diffeomorphism of
X, Y, x
to
X', Y', x'
if and only if
Y'
TxX _C T .) M C
respectively,
arbitrary smooth manifold. it is so at every point in
Y
of
V
Y
M' C ~ m
is Whitney regular over X' x
at
x' .
X
It follows that
over
X
at
in the case when
M
is an
And we say that
Y
is Whitney regular over
X
when
X .
A Whitney stratification of a subset /
onto another open set
is Whitney regular over
we can define Whitney regularity of
stratification
I~m
One checks easily the
V
~f a smooth manifold
M
is a
which satisfies the
Whitney Regularity Condition Any stratum
Y E /
is Whitney regular over any other stratum
X ~ / .
The rea~er will easily convince himself that the stratification of the Whitney umbrella in Fig. I is not a Whitney stratification (the surface fails to be Whitney regular over the z - axis at the origin).
On the other hand the
stratification of Fig. 2 is a Whitney stratification.
In the course of Chapter II
it will be verified that Whitney regularity is sufficient for our purposes; indeed that on a connected component of a stratum of a Whitney stratification the local topological type remains constant. For the purposes of this volume it will be convenient to agree that a Whitney stratified set is a pair M,
and
I
(V, I)
a V~hitney stratification of
with
V
a subset of a smooth manifold
V .
Before proceeding any further we shall establish the few facts we shall require concerning Whitney stratifications. of ~ather
7 ] 9
The first is the following lemma
12
(I .1)
Let
X,Y
x E X ~ (Y - X),
be smooth submanifolds
and let
Y
of a smooth manifold
be Whitney regular over
X
a_~t x :
M,
let
then
aim X < dim Y . Proof
We can suppose
converging to (TyiY)
x .
M = ~m
Compactness
i
T _C m m
And, again using compactness
lines
xiY i
we have
that
converge to a line L _ T .
perpendicular
to
x
L .
But the line
T X.
Thus
in
Y - X
allows us to assnme that
E X
Since
Y
For to
we can suppose that the
is Whitney regular over
TxX _. T,
xiY i
Y).
which minimizes the distance
of the Grassmannian,
Also, we shall have
TxX + L C T .
(yi)
(of the same dimension as
there is a point
Yi "
x
Choose a sequence
of the Grassmannian
converges to a subspace
sufficiently large
9
X
at
as was observed above,
is perpendicular
to
Tx.X , z
so
so L
is
dim X = dim T X < dim T = dim Y .
X
X
Q.E.D. The next three results are all concerned with the subject of gaining new Whitney stratifications /1 ,..., X
from old.
be stratifications
First of all we can take products.
of subsets
V I ,..., V
n
Let
of smooth manifolds. n
Clearly, we obtain a stratification sets of the form
X 1 • ... x X
n
of
with
wroduct stratification , and denote it
--
(I .2)
Let
~I ' ' " '
of smooth manifolds~
Xn
the product
V I x ... x V n X. E I. i
1
for
~I x ... • I
by taking its strata to be I ~ i ~ n :
n
.
be Whitney stratifications stratification
we call this the
of subsets
X I x ... x I
V I ,..., V n
is a Whitney n
stratification
V I • ... x V n "
The proof is sufficiently
obvious to justify its omission
than compactness
of the Grassmannian).
is as follows.
Recall first that smooth submanifolds
manifold
M
,...,
In
Another way of gaining new stratifications X 1 ,..., X n
of a smooth
are in general position when the natural map
TxM -* * TxM/TxX i /I
(using no more
is surjective for each point
be stratifications
of subsets
x
in the intersection.
V I ,..., V n
Now let
of a smooth manifold
M .
IS
We say that
~
,..., 9
are in 6eneral position when
X
X
n
I
general position for all ~ ... ~ I n to be sets
the stratification of
X I ~ ... ~ X
(1.3)
X I E /I ,..., Xn E ~n "
Let
~
n
with
X
,..., I
~ ... ~ In
Proof
M .
V 1 ~ ... ~ V
obtained by taking the strata
n
E I ,..., X E I . I n n
I
be Whitney stratifications of subsets
Y = Y1 ~ "'" ~ Yn
If
/I ' " ' '
~n
M = ~m 9
Let
are in general position then V 1 ~ ... ~ V n
to
x E X, x
and let
with
for which the lines
Xs, Y
to a subspace
T ~ lqm
X, Y
converge to a line
~ i ~i
and E ~
s
be sequences in
(xi) , (yi)
"
X = X I ~ ... ~ Xn
~ ... ~ I
be strata in
n
n
Let
V 1 ,..., V
-
is a Whitney. stratification of
We can suppose
are in n
And in that case we denote by
n
9f ~ smooth manifold
'''''
(of the same ~imension as
for
I ~ s ~ n .
s
respectively converging
L,
and
Y).
(Ty Y)
converges
We have to show
L ~ T .
Using compactness of the Grassmannian we can suppose that the sequence
(~i
converges to a subspace
YS)
I ~ s ~ n .
Since each
Y
Ts
(of the same dimension as
is Whitney regular over S
L _sTC
for
I ~ s ~ n ,
and hence
L_C n T s
,
"
x
we have
We will show that
By hypothesis the subspaces
are in general position (with inter~ection --
at
for
S
which will clinch the r e s u l t .
T C n T
X
Ys )
TyiY)
for all
T = nTs
TyiY1 , . . . , i :
'
TyiY
it follows that
using an obvioue lemma concerning convergence in the ~rassmeamian.
S
By previous remarks we automatically have
T
hypothesis hhe subspaces
are in general position, and hence
TJI
,..., T X
D T X S
--
X
for
1 ~ s ~
m.
But by
S
xn
T I ,..., T
are as well.
dim T = dim n Ts,
That implies
and it follows that
n
T=nT
S
9 Q. E. D.
One final method of producing new stratifications is as follows. be a stratification of a subset
V'
be a smooth mapping transverse to
I'
Clearly, we obtain a stratification f'Ix'
with
X' E X' :
we call
of a smooth manifold
I
M'
and let
Let
f : M ~M'
(i.e. transverse to all the strata of 2
of
f-Iv'
I'
I') .
by taking the strata to be sets
the induced stratification on
f-Ix' .
14
(I .4)
Le__~t f : M -~M'
of a subset f-IV'
:
if
Proof and
V'
~'
of
- -
and let
$
9
f
as the composite
the restriction to graph f
suffices to show that (I .2)
..
M Fgraph
~"
f ~M'
of the projection
diffeomorphically to a stratification
by
be the induced stratification on
is a Whitney stratification then so too is
Factorize ~
M'
be a smooth mapping transverse to a stratification
~"
with
But this latter stratification is
M • V'
F
maps
and it
To this end observe that
is a Whitney stratification, It follows from
(M • V') ~ graph f ~"
.
of a subset of graph f ,
which by hypothesis is transverse to graph f . naturally induced stratification on
F(x) = (x, f(x))
M • M' ~ M '
is a Whitney stratification.
the product stratification on
9 .
(I .3)
that the
is a Whitney stratification.
and we are done. Q. E. D.
A particular case of the above is provided by the following situation. Let
~
U C_ M
be a stratification of a subset be open.
The inclusion
U ~M
there is an induced stratification on restriction of
X
to
then the restriction
U . ~IU
And by
V
of a smooth manifold
M,
and let
is automatically transverse to U ~ V (4.4)
which we denote by if
X
~IU
~,
so
and c a r the
is a Whitney stratification
is likewise a Whitney stratification.
The usefulness of Whitney stratifications depends on being able to find a sufficlently large class of sets for which they exist.
In fact it is not merely
existence which we seek; we require also uniqueness in the sense of a definition to be given below. First however a word of explanation may be welcome, since the definition we intend to give is not the obvious one.
Consider all Whitney stratifications
of a given subset
M .
refinement
(if
every stratum of
V
of a smooth manifold
These are partially ordered by
~, ~
are stratifications of
V
then
~
is contained in a stratum of
~').
~
re_fines
~'
when
One might hope to obtain
a greatest element (i.e. one which is refined by any other) under this partial order, but this is crying for the moon, as the following example shows. V C
~2
be the set comprising the origin and the set defined by
y2 K x
Let 2
.
~5
These yield a Whitney stratification the Whitney stratifications of must be it.
V
9
of
V ,
and it is easy to see that if
have a greatest element under refinement then
But the x - a x i s and its complement in
stratification
~
of
V
which does not refine
V
~ .
also yield a Whitney
We conclude that there is
no greatest element.
With the above in mind we follow L a t h e r definition. of
V
Given a subset
V
of a smooth manifold
we define the associated filtration of
filtration
(Vi)
of
V
8 S in introducing the next
obtained by taking
Vi
V
M,
and a stratification
b y dimensiqn to be the
to be the union of the strata in
of dimension ~ i . We can now introduce a rather different partial order on the stratifications of
V .
Let
~, ~
be two such, and let
the associated filtrations by dimension. an integer
i
for which
V. C V.' but 1
1
We write V. = V ' S
2 ~ ~
for all
(Vi), (V~)
be
when there exists j ~ i .
In this way
S
we induce a partial order ~ on the Whitney stratifications of
V .
If this
~artial order has a least element we call it a minimal Whitney stratification of V . Notice incidentally the following application of Whitney stratification of a subset
V
(I .I).
of a smooth manifold
M,
Let
9
be a
and let
(V i)
be the associated filtration by dimension; then (i)
V i - Vi_ I
is empty, or a smooth submanifold of
M
of dimension
i ,
16
and (ii)
V i - Vi_ I
does not meet the closure of
Vj - V j_I
unless
i ~< j ~
For the sets which arise naturally in this area of mathematics there is little doubt that the "minimal Whitney stratification" is the correct notion to aim for.
But for arbitrary sets the notion has a drawback.
is a subset of a smooth manifold I ,
and that
restriction
UC IIU
M
M
Suppose that
admitting a minimal Whitney stratification
is an open set :
there is no reason to suppose that the
is the minimal Whitney stratification of
V N U .
therefore introduce a stronger notion, which is obviously local. Whitney stratification of dimension.
We call
I
the largest subset of dimension j > i .
i ,
V. l
V ,
V
and let
(Vi)
We shall
Let
be a
be the associated filtration by
canonical when for each index
i
the set
V i - Vi_ I
which firstly is a smooth submanifold of
and secondly over which the
I
Vj - Vj_ I
M
is
of
are Whitney regular for
Clearly a canonical Whitney stratification of
V
is necessarily minimal.
And for the sets we have in mind it will follow from the very construction of the canonical Whitney stratification that the two notions coincide.
Notice
incidentally that these are notions of differential topology in the sense that a diffeomorphism of
M
onto a smooth manifold
M'
,
mapping
V
onto
map the minimal (respectively canonical) Whitney stratification of
V,
V'
9
will
if such
exists, onto the minimal (respectively canonical) Whitney stratification of
V' .
Before proceeding to these constructions we should say something about the definition of the term '~hitney stratification" we have adopted, since it is not the accepted one.
It is usual, pursuing the analogy with cellular subdivision
in topology, to insist that Whitney stratificatiorsshould in addition satisfy the
Frontier Condition Let
X, Y
be strata with
X~Y
$ 0 :
then
XCY
(i.e. the frontier of
a stratum is a union of strata).
The Frontier Condition plays no role in the proofs of the results stated in this volume.
Moreover in practice it is something of an embarrassment, since
17
it is not preserved under natural operations on stratifications (e.g. taking intersections of stratifications in general position).
For these reasons we feel
it better to dispose with the Frontier Condition altogether.
In this connexion
it is perhaps of interest to mention the following fact, which will be proved in Chapter II. subset
V
Suppose that
9
is a Whitney stratification of a locally closed
of a smooth manifold
M.
Det
connected components of the strata of
~c
~ .
be the partition of It turns out that
Local Finiteness Condition, hence is a stratification of stratification.
Moreover
~c
V,
~
~2
~c
into the
satisfies the
indeed a Whitney
satisfies the Frontier Condition.
is minimal (canonical) if and only if
V
Clearly,
X
is.
Semial~ebraic Sets
We seek a useful class of sets admitting canonical Whitney stratifications. One hopes for a class large enough to include algebraic sets, and (for practical reasons) closed under as many set theoretic and topological operations as is possible.
For the purposes of this volume semialgebraic sets (defined below) will
suffice - though larger classes with more or less identical properties exist. As was pointed out in the introduction to this volume we shall merely quote the basic results about semialgebraic sets here, referring the interested reader to the publications of [Lojasiewicz] for proofs. The class of semialgebraic subsets of Boolean algebra of subsets of
3~m
Rm
is defined to be the smallest
which contains all sets of the form
Ix~ m m:f(x)>01 with
f : I~m -~ ~
a polynomial function.
Let us first dispose of the trivial facts. the definitions that the semialgebraic subsets of
It follows immediately from Rm
unions, finite intersections, complements and products. image of a semialgebraic subset of f : I~n ~ ]Rp
is semialgebraic.
RP
are closed under finite Moreover the inverse
un~er a polynomial mapping
The first substantial fact is the Tarski -
18
Seidenberg Theorem.
(2.1) ma~oin~
The image of a semlal~ebraic subset of
f : ~n ~ ~p
under a polynomial
is semialgebraic.
[Lojasiewicz'sS proof of
(2.1)
the Finiteness Theorem of [Whitney
(2.2)
~n
yields another basic result, generalizing
2 S .
A semial~ebraic set has only finitely many connected components,
each of which is semialgebraic. A delightful little application of the Tarski- Seidenberg Theorem is provided by
(2.3)
The closure (hence also the interior and the frontier) of a
semialgebraic set
A _C I~m
Proof
B _C ~ m x ~ m x I~
Clearly
x E A
and
Ix - Y l
projections that
< ~
is semialgebraic. comprising all triples
is semialgebraic.
- p(B)) ,
with
Now consider the polwnomial
I~m x (I~m x ~) P+ ~ m x I~ ~ ~ m .
A = I~m - q( I~m x m
(x, y, r
A minor computation verifies
which is semialgebraic by
(2.1) ~ Q.E.D.
We can introduce a notion of "dimension" for semialgebralc sets as follows. Let
V
be a subset of a smooth manifold
dimension
d)
when
smooth submanifold of
x
M.
has a neighbourhood M
(of dimension
A point U
in
d) .
the two definitions are equivalent.
Assume that
point; then the dimension
V
of a regular point.
of
is re~ula r
for which
U ~ V
is a
The reader is warned that this is
not quite the definition used in [Lcjasiewicz S :
dim V
M
x E V
however it can be proved that V
has at least one regular
is defined to be the maximal dimension
(Of course one has to adopt some convention concerning the
dimension of the empty set,
e.g. that it is
-I .)
semialgebraic set is always defined follows from
That the dimension of a
19
(2.4)
Every non-void semial~ebraic set has at least one regular point;
in fact the regular points lie dense. It is not hard to show that for semialgebraic subsets
A, B
of
~m
one has (i)
If
(ii)
A ~ B
then
dimA
~ dim B .
dim ( A U B) = max (dim A, dim B) .
(iii)
dim A = dim A .
The points in a semialgebraic set dimension, are called singular. singular.) of
V .
ZV
which fail to be regular, of maximal
(Thus a point in
And the singular set Notice that
V
ZV
of
is closed in
V
V
can be both regular and
is the set of all singular points
V .
The next result lies rather
deeper than the preceeding ones.
(2.5) of dimension
The sin~u!ar set
ZV
of a semialgebraic set
is
semial~ebraic,
( dim V .
Now we come closer to the matter in hand. of a smooth manifold
M .
points
Y
x E X
V
whore
We define
Let
X, Y
be smooth submanifolds
B(X, Y) - the bad set - to be the set of
fails to be Whitney regular over
X
at
x .
The basic
fact about the bad set is Whitney's Theorem, which lies at least as deep as any of the preceeding propositions.
(2.6) B(X, Y)
Let
X, Y
be semial~ebraic smooth submanifolds of
is semial~ebraic~ of dimension
~m
:
then
< dim X .
A corresponding result in the complex case appeared in a long and difficult paper of [Whitney ~ ] . illuminating account. by [Wall
] .
Shortly afterwards
[Thorn I ] published a more
A recent account along the same basic lines is provided
The reader versed in algebraic geometry is referred to the elegant
proof given by [Hironaka] using his resolution of singularities. In or~er to reduce the symbolism it will be convenient at this point to introduce, for ar~ semialgebraic sets
A, B
in
Rm
the set
2O
W(A, B) : 2A U B ( A -
7A, B -
It follows from the preceeding results that dimension
W(A, B)
.
is semialgebraie, of
< dim A .
(2.7)
Anv semialgebraic set
stratification Proo___~f w
ZB)
I
V ~
~m
admits a canonical Whitne~
having finitely ma~7 semialgebraic strata.
The stratification of an algebraic set described at the beginning of
works equally well for a semialgebraic set.
What we do is to modify the
construction by deleting from any proposed stratum the (closure of the) set of points where previously defined strata fail to be Whitney regular over it. Formally, we proceed as follows. V = Vd D
Vd_ 1 D
dim V i ~ i
....
of
V
V i - Vi_ I
Suppose inductively that
according to this prescription. If
dim V i = i
If
closure Vi_ 1
remarks.
V.
closed in
V
with
a smooth manifold of dimension
Vd, Vd_ I ,..., V i dim V. < i l
in
V
of
{
we put
i
have been constructed
Vi_ 1 = V.1 and a r e d o n e .
V i - Vi_ I
smooth manifold
V i - 2V i
W(Vi V
j= +I
is semialgebraie, of dimension And
We construct a filtration
we put
Vi'1 That
d = dim V .
by semialgebraic sets
and each difference
(or empty).
Let
~ (i - 1),
"
follows from the preceeding
is a smooth manifold since it is obtained from the by deleting a closed set.
stratification of
V
whose strata are the
construction that
9
is a Whitney stratification.
many semialgebraic strata.
) a - VJ -1
We take
V i - Vi_ S .
It remains to show that
9
to be the
It follows from the
And of course it has finitely i
is canonical, which fact
again follows immediately from the construction. Q. E. D. Since the literature on Whitney stratified sets is rather devoid of examples this may be an appropriate point to digress from the main line of thought and describe a few examples culled from the folk- lore of the subject.
In the few
known situations where the Whitney Regularity Condition has been verified the
21
Whitney Theorem on the bad set has been the main tool.
A particularly nice set-
up is provided by a semialgebraic stratification (i.e. one with semialgebraic strata) of a set
V C ~m
which enjoys the following homogeneity property; that
given any two points on the same stratum there exists a diffeomorphism of a neighbourhood in
I~m
of the one point onto a neighbourhood in
point which preserves strata.
~m
of the other
It follows isw,ediately from the Whitney Theorem
that such a stratification must be a Whitney stratification. Take for instance a Lie group action on a smooth manifold finitely many semialgebraic orbits.
V _C ~ m
having
Here the o~bits enjoy an obvious homogeneity
property as a result of which they are non- singular semialgebraic sets, using (2.~), hence smooth submanifolds of
V
providing a stratification.
Ar~ the same
homogeneity property ensures that this is a Whitney stratification. explicit illustration, let let
~
be the subset of
kernel rank
i :
the
V, W
be finite - dimensional real vector spaces, and
Hem(V, W)
~i
By way of
comprising all linear maps
provide a Whitney stratification of
they are the orbits under the natural action of the Lie group
V -~W
having
Hem(V, W)
since
@L(V) • GL(W) ,
and clearly semialgebraic. A similar situation yielding examples of Whitney stratifications is provide~ by a smooth action of a compact Lie group
G
on a smooth manifold
M.
Recall
that locally there are only finitely many orbit types, and that the union of all orbits having a given type is a smooth submanifold of provide a stratification of
M.
M .
Thus these unions
And using slices it is not difficult to
establish the Whitney Regularity Condition. Here are two further examples where positive work is required to verify the statements.
CoDsi~er the partition of
a = (a 0, a I ,..., an)
obtained by separating points
according to the number and the multiplicities of the
complex zeros of the polynomial
a0zn + alzn'1 + ... + an
semialgebraic Whitney stratification. algebra :
I~n+1
this provides a finite
Finally, an example drawn from linear
the partition of the set of endomorphisms of a finite dimensional real
vector space by Segre symbol provides another finite semialgebraic Whitney
22
stratification.
A proof can be found in [GibsonJ.
w Let f(A) c B
f : N ~ P
.
Thorn Stratified Mappings
be a smooth mapping, and let
A stratification of
stratifications of
A, B
f : A-~ B
A cN,
is a pair
B _cP
(X, ~')
be sets with with
~, ~'
Whitney
respectively for which the following conditions are
satisfied. (i) (ii) then
f
maps strata into strata (but not necessarily onto).
Let
f : X-~ X'
X
be a stratum of
~,
mapped by
f
into a stratum
X'
of
~' :
is a submersion.
A simple example of a stratified mapping is provided by the Whitney cusp mapping of the plane, i.e. the m~pping u = x, v = y3 - xy. image under
f
f : ~2,
i~2
given by
(x, y) * (u, v)
The set of critical points is the parabola
the cuspidal cubic
4u 3 - 27v 2 = 0.
with
And the inverse image of this
curve is the union of the parabola with another - see below.
V U
x = 3y 2,
with
23
There are certainly circumstances where the mere existence of a stratification for a smooth mapping provides useful topological information. However, for the purposes of the Topological Stability Theorem we require a little more - precisely, we require stratifications of smooth mappings to which Thom's Secon~ Isotopy Lemma applies.
To this end we introduce a condition on
stratifications of smooth mappings analogous to the Whitney Regularity Condition. Let N
f : N *P
for which the restrictions
We say that holds. (ker T then to
be a smooth mapping, let
Y
Let
flX
is Thorn regular over
and X
fly at
X, T
be smooth submanifolds of
have constant rank, and let x
(yi) be a sequence of points in
relative to y
f
x E X .
when the following
converging to
x
such that
(flY)) converges, in the appropriate Grassmann bundle, to a plane T ; Yi ker Tx(flX) _C T . An~ we say that Y is Thom regular over X relative
f
when it is so at every point Now let
f(A) _C B, (~, ~ )
f : N ~P
and let
x E X .
be a smooth mapping, let
(2, ~')
be a stratification of
AC
N, B _
P
f : A -~B .
be sets with We call
a Thom stratification when it satisfies the
Thom Re&~Alarity Condition For any strata
X, Y E 9
we have
Y
Thom regular over
X
relative to
f .
For the purposes of this volume it will be convenient to agree that a Thom stratifie& mapping is a triple (~, ~ )
a Thorn stratification for
(f, ~, ~') f .
with
f
a smooth mapping, aud
It is customary to reseve the term
Thom m a ~ i n g for a smooth mapping which admits a Thom stratification. An example where the Thom Regularity Condition has a simple geometric meaning is provided by the smooth function f
-* ~
given by
admits an obvious stratification, indicated in the figure below.
take a sequence of points u
f :
on the positive
x-axis.
the tangent lines at the x - axis.
(vi)
vi
f(x , y) = xy. Suppose we
in the open first quadrant converging to a point The Thom Regularity Condition merely asserts that
to the fibres through
vi
must converge to the
Clearly then we have a Thom stratification of
f .
24
f
b
\\ In this last example we have a natural stratification for the mapping which automatically satisfies the Them Regularity Condition. always the case is illustrated by the pinch map (x, y)
-*(u, v)
that the y - a x i s the
where
u = x, v = xy .
collapses to a point :
a - process).
f : ~ 2 -~ ~ 2
That this is not given by
(It "pinches" the plane in such a way algebraic geometers will reeognise it as
Here again there is a natural stratification o~
f,
indicated
in the figure below.
J,"
w
X u f
P
U
X
/ The strata in the domain are complement Clearly
Y .
Take
x s X ,
ker Ty1(flY). = IOl ,
l
X = critical set of
and a sequence
(yi)
in
f = y - axis, and its Y
converging to
so the sequence of kernels converges to
Were the Thorn Regularity Condition to hold we should have
x .
IOl 9
ker Tv(flX) _C IOl ,
25
hence = 101 .
But thisis f~Ise since the kernel is
X .
In fact we can improve upon this argument to show that the pinch map does not admit a Them stratification. I claim that for any stratum Certainly
dim U ~ 2 .
U E ~
The case
suffices to show that
U, X
is a smooth submanifold of x E U ~ X . to
x .
the intersection
dim
does not arise since otherwise if submersively into a stratum in
For suppose it did admit one
U = 0
the pinch map
~
.
is countable.
f
dim U = 2
could not map
We are left with the case
dim U = I .
U It
intersect transversally (since then the intersection ~2
of dimension
0,
We can certainly find a sequence
Since
(2, i~)
is trivial; and the case
U ~ X $ ~
~' .
U ~ X
f
hence countable)
(yi)
in
~2
.
So choose
. X - U
converging
is locally finite there will be a stratum
subsequence - which we also denote
(yi) - converging to
V
x .
containing a All the
ker Ty (flV) are trivial, so the Them Regularity Condition implies that i And that entails that U, X intersect transversally at ker Tx(flU) is trivial. x .
Finally, observe that we must have
X
=
U U ~ X , UsE
wBich provides the
required contradiction since the left-hand side is uncountable, but the righthand side is countable (being the countable union of countable sets) . For the purposes of the Topological Stability Theorem one wishes to associate Them stratifications with infinitesimally stable mappings, and since these locally look like polynomial mappings (a statement which will be made precise in Chapter III) the natural starting point is to try and associate Them stratifications with polynomial mappings.
The example of the pinch map makes it clear
that we must impose some kind of restriction, preferably one which is "generic" . We therefore impose on our mappings the
@enericity Condition The restriction
fie
of
f : N -~P
to its critical set
Z
is both proper
and finite-to-one.
In Chapter III it will be shown that this condition is indeed satisfied b y some representative of any infinitesimally stable map germ, so is "generic".
26
We come now to the explicit construction of Thom stratifications for generic polynomial mappings. of Leoijenga.
Our constructions will be simplified b y the following i~ea
Let
f : N ~ P
set of critical points of Notice that if
f
f,
be a smooth mapping. and
C = C(f)
f
f : f-Iu ~ E ~ U
C,
U, V
is a smooth submanifold of
N,
~
of
is closed 9
of ~
C
f-Iv ~ Z
is Whitney regular over
f-Iu ~ Z .
(PS3)
f-Iv - ~
is Whitney regular over
f-Iu ~ ~ .
that there is a partial order ~ on the stratifications
induced partial order has a least element we call it stratification of
f .
G
the following conditions, for each
If the
minimal partia~ there is no
We introduce therefore a stronger
be a partial stratification for
associated filtration by dimension.
M
a
f .
As in the simpler situation discussed in w
reason to suppose that this is a local notion.
(P$0)
which
(possibly equal) .
inducing a partial order on the partial stratifications of
Let
A
and the restriction
(PS2)
notion.
for the
is a local diffeomorphism.
Now recall from w of
C(f)
is a Whitney stratification
satisfies the following conditions for any strata f-Iu ~ Z
Z = Z(f)
for the set of critical values .
satisfies the Genericity Condition then
partial stratification for
(PSS)
We write
f ,
and let
Certainly then the
(Ci)
be the
M i = C i - Ci. I
satisfy
j .
is empty, or a smooth submanifcld of
P
of dimension
j
over
0 which each
Mk
is ~ i t n e y
regular for
k > J .
(PSI)
f-IM. ~ Z is a smooth submanifold of N , and the restriction J f : f-IM. ~ Z ~ M . is a local diffeomorphism. 0 0 (PS2) f-IM k N Z is Whitney regular over f-IMj ~ Z for k > j ~PS3)
f-1~
_ Z
We shall call the largest subset of
is Whitney regular over
g
f-lM. ~ ~ 0
for
a canonical partial stratification of
C
k ~ j . f
when
M. 0
is
for which the above conditions hold, for each index 0
Clearly a canonical partial stratification of
f
is necessarily minimal 9
And
for the mappings we have in mind it will follow from the very construction of canonical partial stratifications that the two notions coincide.
Without more
j
.
27
ado we shall come to the point of partial stratifications.
(3.1)
Le_~t f : N - + P
values is closed ~
if
f
be a smooth manpin~ whose set
admits a partial stratification
C @
of critical then
f
admits a
Thom stratification. Proof
Fist, note that we obtain a Whitney stratification
augmenting
@
P :
(P - C)
since
by
any stratum of
(P - C),
which is open in
is open in
@ .
P
~'
Z
f-1(p _ C) ,
N .
in
~'
with
~
f(X) C U , f(Y) C V .
or disjolut from (i)
~ ,
f
by
of
N
by taking with
I claim first that
To this end let
From the very construction of
X
f-Iw ~ P , f'dW - $
the critical set of
a Whitney stratification of
P
hence a smooth submanifold of
And clearly we obtain a stratification
and
of
it will automatically be Whitney regular over
its strata to be sets of the form a stratum of
P ,
~'
X, Y
be strata in
~
~
is
~ .
it follows that there are strata
Since any stratum in
W
U, V
is contained in
z
there are four possible cases to consider.
X • Z : ~
and
Y _C Z
Since
Z
is closed
have
we
X O Y : ~
and
Whitney Regularity follows trivially. (ii)
X ~ Z = ~
t~ausverse to over
f-1 U _ r , (iii)
by
U, V . so
and
Y ~ Z = ~
is a submersion so certainly
It follows from (I .~) that Y
is Whitney regular over
X C P
ar~l Y _C ~.
X C Z
an~
In this case
f-Iv - Z
is Whitney regular
X . Y
is Whitney regular over
X
(Ps2) . (iv)
which case
Y
Y N Z = ~
(PS3) .
Next, I claim that
(~, ~')
maps strata into strata.
strata in ~' ~
~
Y = f-1(p _ C)
of the form
f'Iw ~ Z
is a stratification of
X :
f .
f-1(p . C)
or
f I N - r
f-1 W _ r ,
with
is a submersion 9
this follows from (PSI) o
W
in
the remaining
By construction
We have to show that it ~oes so submersively.
this is immediate since
of the form
One possibility is that
(being open) is certainly Whitney regular over
possibility is covered by
f
fiN - Z
a stratum of
And for strata in
For
28 It remains to show that the stratification Thom Regularity Condition. Cyi)
Take strata
be a sequence of points in
into strata (i)
U, V
of
X ~ Z .
~' .
ker Tx(f{X) = IOl ,
to
f.
of the point
X n Z = x
.
converging to
x .
Necessarily
Z
for Let
f
satisfies the
x ~ X ,
As before
and let
X, Y
map
And again we consider cases.
and trivially
Y
and
f : X * U
is a local diffeomorphism,
is Thom regular over
Y ~ Z = ~
X = f-lf(x)
we have
submersive outside
in
By (PSI) the restriction
so
(ii)
Y
X, Y
(~, ~')
as well .
Y = f-lf(y) 9
X
at
x
relative
On a neighbourhood And since
f
is
we have ker Tyi(f{Y)
=
Tyi(f-lf(yi))
ker Tx (f{X) = Tx (f-lf(x)) so the former converge to the latter, again showing that X
at
x
relative to
Y
is Thom regular over
f . Q. E. D.
The Them stratification
(X, i~) constructed in the above proof will be
called the Thom stratification associated to the partial stratification
g .
Thus
the problem of constructing a Thom stratification for a generic polynomial mapping reduces to that of constructing a partial stratification.
Since this is
merely a special type of Whitney stratification of the set of critical values it is to be expected that the construction will be no more than a complicated version of that described in (2.7) .
And indeed this will be the case.
In
order to make the construction work we need two preliminary results.
(3.2)
Let
be semial~ebraiq : Proof
Put
f :
~n
then
Y = f(X) ,
Seidenberg Theorem.
*
~p
be a nolvnomial manning I and let
X ~
~n
dim f(X) ~ dim X . which is of course semialgebraic by the Tarski -
We proceed by induction on
dim X .
The case
dim X = 0
is clear, since a semialgebraic set has zero dimension if and only if it is finite.
29 Suppose
dim X
is positive, and that the result holds for semialgebraic sets of
dimension < dim X .
By (2.5) the singular set
&imension < dim X .
The Tarski- Seidenberg Theorem tells us that
semialgebraic, as is its closure, by dim ~ ) =
dim f ( ~ X ) ~
Let us suppose
dim Y > dim X .
obtained from another, Also,
submanifold of X - ZX,
Y -
The set
~P
f : X I -~Y
- f(zx)
dim Y ,
dim X ,
is a semialgebraic smooth
since it is obtained from another,
with
In this way we obtain a surjective
dim YI > dim X I .
A point in
YI
f : XI -~Y1 "
We conclude that indeed
is a regular
f(X I) .
has at least one regular value - by Sardis Theorem, for instance -
surjectivity of
is a
since it is
if and only if it does not lie in the image
I
is
~Y, b y deleting a closed semialgebraic set of lower
of dimension
f : XI -*YI
YI = Y - z Y
of dimension
by deleting a proper closed subset.
Value of
f(ZX)
Using the induction hypothesis
X I = X - f-1 (ry U f ( Z X ) )
I~n
smooth mapping
f
(2.3) .
is semialgebraic, of
aim ( Z X ) < dim X .
semialgebraic smooth submanifold of
dimension.
ZX
But
contradicting
dim Y ~ dim X . Q. E. D.
(3.3)
Let
f :
be ~@mial~eb~aic, with PToof on
A~ain,
dim Y .
~n flX
Y = f(X)
When
-~
Rp
be a pol.ynomial mapping, and la~
finite-to-one:then
dim f(X) = dim X .
is semialgebraic.
dim Y = 0
This time we proceed by induction
the result is clear.
and that the result holds for lower dimensions. ZY
X _. R n
is semialgebraic, of dimension < dim Y .
Suppose
dim Y
is positive,
By (2.5) the singular set Using the induction hypothesis,
and (3.2) , we have dim (flX)-I(zY) = dim (Zy) Now
XI = X - ZX
dim X ,
- (fiX) -I (ZY)
< dim Y ~ dim X .
is a smooth submanlfold of
m n
of dimension
since it is obtained from another, X - ZX, by deleting a closed
semialgebraic subset of lower dimension . submanifold of
~P
one smooth mapping
of dimension f : Xl -*Ys "
dim Y .
And
YI = Y - z Y
is a smooth
In this way we obtain a finite-to-
By shrinking
XI
we can suppose
f : XI -*Ys
30
has constant rank.
Assume
dim X 9 dim Y .
construct a non-trivial smooth vector field
It will then be possible to ~
on
xI
with
Such a vector field has a non-trivial flow line, on which contradicting the finite-to-one condition.
Thus
f
6(XI) _C ker T(flXl), is constant,
dim X = dim Y . Q.E.D.
(3,4)
Let
f : 1~n -~ m p
~e a Dolvnomi@l mapping, and let
8 semial~ebraic smooth submanifold, fiX
The @et
fails to be i=~ersive is semial~ebraic.
one then
Sg(flX)
Proof
First, we show that
X
A
TX
X
subset of
=
subset of with
x,y ~ X
~n x ~n x ~n
x, y E X
and
will correspond to
Now we show that suppose
f
and
precisely
T ( ~ n) :
Sg(flX) ,
Finally, assume
comprising triples
Let
f
necessary.
(x, y, v)
of points
A be the diagonal in T I~n
with
is semialgebraic. (Factor
9
A x I~n
~n x ~n :
the semialgebraic
f
Observe first that we can
as the composite of the natural Let
8(~n)
denote the
By the above the set ker Tf ~ TX ~ s(~n)
its image under the projection
is
T(1~n) -~ ~ n
is
ar~l semialgebraic, by the Tarski-Seidenberg Theorem. flX
finite-to-one.
so there is an open set" U C X suppose
of
TX .
Sg(fIX )
T(~n)
(x, y, v)
x $ y 9
embeddi~ in its graph, and a linear projection.) umit sphere bundle in
~n
x = y .
a linear projection,
semialgebraic in
is finite-to-
For hhis we use the fact
comprising triples
then, under the natural identification of A ~ B
where
We put
Clearly, both A, B are semialgebraic,
set
fl x
X
is the limiting position of lines in
~ n • l~n • ~ n
cellinear points with B
Moreover, if
is semialgebraic.
at two distinct points. =
of points in
be
has dimension < dim X .
that a tangent line at a point on cutting
Sg(flX)
X C ~n
Suppose that
at no point of which
has constant rank on
U ,
replacing
U
f
dim Sg(flx) = dim X ,
is immersive.
We can
by a smaller open set if
It will now be possible to construct a non-trivial vector field
31
on
U
on which
with f
~(U) ~ ker Tf .
Such a vector fiel~ has a non-trivial flow line,
is constant - contradicting the hypothesis that
It follows that indeed
f
is finite-to-one.
dim Sg(flX ) < dim X . q. E. D .
The preliminary work is now complete and we are in a position to establish the existence of canonical partial stratifications for generic polynomial mappings.
(3.5) an~
Let
f : A ~ B
A, B
be semial~ebraic open sets in
a pol.ynomial mappin~ for which
admits a canonical ~artial stratification
~
flZ(f)
~n,
~p
respectively ,
is finite-to-one,
f
having only finitely ma~v
semial~ebraic strata. Proof
Observe that the set
that the set
C
of critical points of
f
is semialgebraic, so
of critical values is likewise semialgebriac, by the Tarski-
Seidenberg Theorem.
Put
C = Cc D - Cc-1 -D 00''" . dim Cj ~ j ,
Z
c = dim C .
of
C
We shall construct a filtration
by semialgebraic sets, closed in
C , with
such that the following conditions are satisfied for each
j
with
Mj : C. - C j ~ (PS0) which each
M. ~
(PSi)
is empty, or a smooth suhmanifold of
is Whitney regular for f-IM. ~ Z
f-1~ - Z
is a smooth submanifold of
is Whitney regular over
Suppose inductively that to this prescription. dim C. = j .
If
N ,
for
k > j .
f-lMj ~ ~
for
k ~ j .
over
we take
have been constructed according
Cj_ 1 = C j
We put
:
j
closure { u w %
which is semialgebraic, of dimension < J .
k>j
,
Now p u t
c~ = f-l(cj_~t)n~.
and the restriction
f-IM. ~ Z
Cc, Cc_ I ,..., Cj
dim Cj9 < j
of dimension
k > j .
f : f-IM. ~ Z ~ M . is a local diffeomorphism, J J (PS2) f ' I M k ~ Z is Whitney regular over (PS3)
P
ar~l are done.
Suppose
32 which is semialgebraic of dimension R2
j , by
We define
(3.3) 9
R 2' U R2"
=
where R 2'
=
closure
w(cj, ,
n
closure {kYj W(0j ,
-
and R2" Clearly
R 2' , R2"
(and hence
R2)
are semialgebraic, of
dimension < j
.
Next,
put R3 which is semialgebraic, of
=
clos~e of
dimension < J, Ca"
=
=
closure of
J ,
(3.~) 9
Also write
and
W(Cj", f'IfCj" - P)
which is semialgebraic, of dimension < J .
c j-1
by
- R 2)
Cj' - R 2 - R 3
which is semialgebraic, of dimension
R$
s~(flcj'
Finally, define
= ~1 uf(R 2 u ~ 3 U ~ )
which is semialgebraic, of dimension ~< (J - I) , by (3.2) . That completes the induction step of the construction. reader the task of formally checking that the partition of the required canonical partial stratification of
C
We leave to the by
the
M
J
is
f . Q.E.D.
Now let f : A -~ B (3.5)
f
A, B be semialgebraic open sets in Euclidean spaces, and
a polynomial mapping which satisfies the Generieity Condition. admits a canonical partial stratification
a Thom stratification
(~, l~),
And by (3.1)
f
admits
namely the Thom stratification associated to
this we call a canonical Thorn stratification for main result of Chapter I .
6 .
By
f .
@ :
Sunning up, we obtain the
33
(3.6) f : A ~ B then
Let
A, B
be semialgebraie open sets in Euclidean spaces, and let
be a polynomial mapping which satisifes the Generioi~y Condition :
f admits a canonical Thom stratification
(I, i~ ) .
It is this result which enables us to construct the crucial stratification of the jet space in Chapter IV .
We conclude the present chapter by listing
the basic properties of canonical partial stratifications require~ for Chapter IV. First, it is invariant under smooth equivalence of smooth mappings; precisely
(3.7) g, h
with
Le__~t fl' f2 f
I stratification @2
~e smooth m~ppings ~or which there exist diffeomoruhisms
o g = h o f2 " ~
9
and let
Suppose that @2
fi
admits a canonical partial
be the stratification induced ~7
is ~ canonical partial stratification for
h :
then
f2 "
The proof consists of no more than a (tedious) formal checking - which is also the case for
(3.8) stratification then
@'
Let
f : N * P
@ .
Let
be a smooth mappin~which admits a canonical partial
P' ~ P
be open, let
N' = f-Ip, ,
is a canonical partial stratification for
an~ let
f : N' ~ P '
@' = C]P' :
.
Another result for which it does not seem worthwhile writing down a proof is
(3.9)
Le__~t f : N -~ P
stratification
@ .
of
@
f .
Then
And let
be a smooth man~in~ which admits a canonical uartial U C N
be an open set containing the critical set
also provides a canonical partial stratification for
flU .
The final fact which we shall need to know later is
(3.1o)
Foxr I ~ J ~ s
le_tt fj9 : N.J -*p
domains having the same dimension, and let
be smooth mappings~ with
f : N -~ P
be their disjoint sum.
34 Suppose each
fj
admits a Them stratification
are in ~enera I position, ~' = ~ ~j' .
(~, ~)
.
f
(
,
admits a Thorn stratification
Moreover, if the
(~,i~)
with
(Ej, lj') are all canonical then so too is
CHAPTER
Stratifications
II
and Flows
Klaus WirtmOller
$6
w
Tubes
This chapter is concerned with the construction of continuous flows on stratified sets.
Since strata are smooth manifolds we can clearly obtain a smooth
flow on each stratum of a stratification by integrating a smooth vector field.
But
in general we cannot expect to obtain a continuous flow on the whole set by just putting the parts together. special nature:
Therefore the vector fields we consider will be of a
in a sense to be made precise they will be controlled along the
boundary of each stratum in such a way that we do obtain continuous flows.
Let us
illustrate the idea by means of an example. (i. i )
Example.
field on
Stratify the plane
~ 2 by
IO, I~2 - O~
and let
~
be the vector
2 2 defined by
~(0) = O, (in standard coordinates).
~(x, y) = ~
~/~ IF(~, y -ei
o~J
for
(x, y) i o
Clearly, the origin is a fixed point and all other flow
lines are circles centred at the origin.
Note that
jjl 9
/
outside the origin all flow lines have constant speed velocities
II~(x, y)IIAI~x, y~I
have constructed a flow on
tend to infinity as
II~ll : 1 (x, y)
whence their angular
approaches
0 .
2 2 which is not differentiable at the origin.
Thus we On the
other hand distance to the origin is constant along any flow line, and this implies at once that the flow is continuous. This simple tYpe of control by the distance to the smaller stratum will not be sufficient to ensure continuity of the flow unless this stratum is a discrete set.
37
The next example indicates another type of control over a stratum of positive dimension. (1.2)
Example.
Split
it3-~t~ it2
and stratify it by
[it x 0,
~ x(m2
- 0)1 .
Then w(t, x, y) = a/at + ~(x, y) defines a vector field
W
on
it • it2 , with
~
as in (1.1).
linear flbw on the t - axis whereas the other flow
U
induces a
lines spiral round the t - axis.
This time the proof that we have a continuous flow
on
~3
is basea on two
observations : (1)
the flow preserves distance to the t - axis,
(2)
under the canonical retraction (t, x, y)
(say)
and
it x it2 ~ ~ x 0 the flow line through
is mapped to the flow line through
(t, 0, 0)
on the t-axis.
Our strategy in the general case will be to look for vector fields having properties
(I) and (2) locally, i.e. in some neighbourhood of the "smaller" stratum.
We will give a meaning to "distance" and "retraction" in this context by taking this neighbourhood to be a tube.
But first of all let us introduce the following
notion which will be convenient later on. (I. 3 )
Definition.
that two maps
Let
gl' g2
X, Y, Z
defined on
if there exist neighbourhoods coincide on
Z ~ U
(on
U
be subsets of a topological space Z
of
Z N U ~ V,
define the same X
and
V
of
respectively).
~erm at Y
in
N
X
(at
such that
N ~
We say
(X, Y)) gl
and
g2
38
Note that we do not require that clear that the composition
X
or
Y
be contained in
Z.
It should be
of two suitable germs, or one germ and a map, is a well-
defined germ. Now we give the precise definition of "tube". (1.4) (X
Definition.
Let
X _c N
need not be closed in
where
~ : E~
X
N).
diffeomorphism,
p
T
(at
X)
= p o e
-1
retraction
A tube at
E,
and
e : E-* N
of the inclusion : N~
9
X _c N.
go : ~
(2) means that
retracting
that
~3
T = (E, ~, p, e)
T
~(X) so that
g,
x go
SR2-. ~
be the projection
~
~3
g o ~
(I. 2 ) once
to the first factor then
to hhe linear flow on the real
S/at.
Conversely,
given any submer-
suppose we wish to construct a flow on ~
I~3 ,
under
g.
For
(2) and the reason for this is that
tube rather than a tube which is compatible with
g
in
sense.
Definition.
and
and
By abuse of language we will
of tubes let us look at Example
maps the flow on
onto
Let
T
(or a germ of such a map at g
is the
T.
(2) refers to the standard
(1.5)
e o ~
X ; we call them the germs of the T .
no such flow will have the property
the following
of a local
= ~ o e -1 : N ~ X
preserving the strata and mapping to the standard linear flow on general
N
the quadratic function
~ : X-~ E
Observe
llne generated by the constant vector field g
~
is the germ at
and the distance function defined by
If we let
sion
p : E~
are well defined germs at
Before we come to the construction
property
is a quadruple
commuting with the zero section
often omit the superscript
more.
X
is a (smooth) vector bundle,
of a Riemannian metric on
germ
be a submanifold of ~he smooth manifold
T
be a tube as in ( 1 . 4 ) X).
T
and let
is compatible with
g
g : N ~ P if the germs
be a map (at
X )
are equal.
Thus each fibre of
T
should be contained in a fibre of
to find compatible tubes only for very special
g,
g .
We can expect
and the remainder of hhis
section is devoted to the proof of the following existence result.
39
(1.6)
Theorem.
Let
N, X C N
be a smooth map germ at
X
such that
(relatively) owen subsets of compatible with such that
g,
and
X
with
P
be smooth manifolds, and let
glX
is a submersion.
XI ~ X _c X u .
then there exists a tube
T
-If -
Suppose
TO
a_~t X ,
XI c X~
is a tube at
are
Xo,
also compatible with
g,
TIX I = ToIX I.
Proof.
Our proof depends on a standard method of constructing tubes, see [Lang]
or [BrScker-J~nich]
:
choose any spray
~
on
N
and let
e~
be the germ at
of the associated exponential map, restricted to a normal bundle in
g : N-. P
N.
Then
T = (E, ~, 99 , e~ )
is a tube at
E.
Let
X
T
from a careful choice of
Let us deal with the latter first. Tg : TN-* TP
the kernel rank of
g
denote the differential of
g.
Since
is constant in a neighbourhood of
generality to assume that this neighbourhood is all of suhbu_ndle of Let
of
X
We shall obtain the required extra properties of and
~ : E ~ X
X
TN, and furthermore we have
To = (Eo' ~o' Po' eo)"
is submersive
X, and it is no loss of N.
ker Tg + TX = TN
The differential ~
gIX
e~
Then
ker Tg
over
X.
is a
along the fibres of
~o
is a monomorphism Tfibrer ~e o) : E o -* TNIX ~ of vector bundles, by means of which we~dentify we must have
E ~ _c (ker Tg)IX ~
since
TO
E~
with its image in
is compatible with
Now if we pick a Riemannian metric on the bundle we also get a metric
~o
on
(ker Tg)IX ~
orthogonal sum of the Riemannian bundles Po
Then
g.
TX ~ ~ (ker Tg)IX ~ = kerT(glXo)
splitting the latter bundle into the
ker T(glXo)
and
E~
(here we think of
as a Riemannian metric rather than a Quadratic function).By a standard extension
process we find a metric E
TNIX o.
~
on
(ker Tg)IX
be the orthogonal complement of
and let and leave
p
be
~IE.
EIX I = EolXI, X
Then
E
ker(TglX)
such that in
~ = ~o
(ker Tg)IX
is a normal bundle of
i.e. for any choice of the spray
in the right direction.
X ~
over
X I.
We let
with respect to
such that
E _c (ker Tg)IX
its solution curves will
40
T = (E,T, p, %)
Thus
will h~ve all required properties if the spray
satisfies
(1)
~(ker Tg) ~ T(ker Tg)
(2)
e~I(EIX l) = eol (EIX I) . But finding such a spray is a local problem in the following sense:
construct of
X
locally near any point
by open subsets of
N
x e X
X,
we
can
then we find a locally finite covering
on which the local parts of
~
A
are defined.
partition of unity subordinate to this covering gives us a spray neighbourhood of
if
~
On some
satisfying (1) and (2), and this is clearly enough.
We deal with the local problem of defining
~
near a point
x e X
by
linearising the situation, as follows. (a)
let
x E X o.
for
P
Since
near
gx
gIX e
is submersive we find charts for
with respect to which
glX ~
linear coordinates in the normal space x :
via
e
we obtain an
O
Note that since
T
N-chart
EolX near
is compatible with
g
and a trivialisation of x
g,
N,
if
x E X - X
Choose E~
extending the chart for g o e
and
near
X . O
is constant along the
N-
and
P - charts.
Now let
be the standard geodesic spray on Euclidean space.
leave it to the reader to check that (b)
x
O
is automatically linear in the
in the chart on
near
is a linear projection.
O
fibres, so
X~
we define
~
~
~,
We
satisfies (1) and (2).
as in case (a)
but use any chart on
N
which
O
linearises
g .
Then
neighbourhood of
x
~
clearly has property (1), and if we shrink to a
staying well away from
X1
we may ignore (2).
Note that Theorem (1. ~ ) still holds if we allow the components of
KS P
to have
different dimensions. (1. 7 )
Corollary.
Theorem (1. 6 ) remains true if
0nly at an open subset Proof.
Xg
o_~f X
Choose an open subset
Theorem (1. 6 )
to
with
X 1 ~ Xg,
TO
over
N
g
containing N
g
in place of
of N .
hence a tube at
N
g
is not a germ at
X
but
X - X 1.
such that
X
g
= X ~ N g
We obtain a tube at X 1 U Xg = X . []
X
g
and apply which coincides
41
w We return to the problem of means of tubes.
Tube Systems controlling vector fields on a stratified set by
Let us first study the cross-~atio example mentioned in the
introduction in more detail. (2.1)
Example.
Let
Y : JR-*(0, co)
be a smooth function and let
A c SR3 be
defined by the equation (.)
xy(x + y)(x - y(t). y) = 0 .
x
A
is stratified by the
t - axis
canonical Whitney stratification). Whitney stratification of
I~3 .
X
and its complement
Adding
IR3 - A
(in fact this is a
as a third stratum, we obtain a
Suppose now we are looking for flows on
(i)
preserve the strata ,
(2)
project to hhe standard linear flow on the t - a x i s (t, x, y)
Y
and under the retraction
-* t.
First notice that this problem cannot have a smooth solution. every
point
tangent spaces
2 3 which
(t, O, 0)
on the
t-axis
TxA (x E A, x -* (t, 0, 0))
containing the t - axis.
,
For at
the limiting positions of consist of four planes in
The cross-ratio defined by the corresponding
SR3 ,
concurrent
lines in the projective plane is, after suitable ordering of the lines, just A smooth flow satisfying
(1) and (2) must preserve the cross-ratios,
each
which is
~(t).
42
impossible unless 7
happens to be constant.
any strata-preserving
C1
Note that in fact we have shown that
flow must be trivial on the t - a x i s .
Next let us try to piece together a continuous solution, smooth on each stratum. Trivially condition (2) determines our flow on
X .
If we take
~
then we clearly find a unique solution on
A
canonical tube at
X
controlled by
(i.e. commutes with the retraction and preserves
function of
~ ~).
But for an arbitrary tube
TY
at
Y
impossible to extend this solution further to a flow on both
~
and
TY.
retract neither along the fibres of
which is the distance
it will generally be I~3 which is controlled by
The obstruction comes from the fact that X ~
to be the
~Y
nor along the surfaces
will in general X p = constant.
The conditions we have to impose on the tubes in order to make such extensions possible are most conveniently (2.2)
Definition.
and let
~
and
Let
TY
phrased in terms of germs (see (I. 3 )). X
and
be tubes at
Y
X
be submanifolds of the smooth manifold
and
Y.
We write
X
instead of
~ TX
N, etc.
Let us introduce the commutation relations (CRvr)
~rX O r Y
=
T/X
(c~p)
pXo Y
=
px ,
both being equations between germs at submanifold of another smooth manifold map
Y
into
y'.
(CRf)
(X, Y). N',
Furthermore, let
with a tube
Y'
TY' , and let
be a f : N -* N'
Then we will also have to consider y, f o ~Y = ~ o f (as germs at Y) .
The reason for introducing this last relation will become clear shortly.
Note
that the three commutation relations, put in a different way, require that TY be y, compatible with X , pX and ~ o s respectively. In view of the submersiveness condition in the existence theorem for tubes
( 1 . 6 ) the following lemma is the key
to further progress. (2.3) manifold
Lemma.
Let
N, and let
the map germ
(at
~
X
and
Y
be disjoint smooth submanifolds of the smooth
be a tube at
X.
X)
(x, px~l Y
: Y~Xx
If
Y
is Whitney regular over
X
then
4S
is submersive. Proof. x e X
If it is not, we find a sequence such that
( X, pX)iy
isation of the tube X = A px 0 at
Yi
in
and
kernel of APx
( X, pX)iy
~Yi
we may assume L
in
in An .
(TyiY) ~ T
a line in
near
An
x
Since
converging to some point
Yi"
Choosing a local trivial-
TX
is the standard tube at
( X, pX) : A n ~ A p x A
is not we conclude that Yi"
Y
we may assume that
T iY
is not transverse to the
By compactness of the Grassmannians of subspaces of and
(xy i-* ) -* L
where
T
so
Y
+
L in
~n.
An
is a p - dimensional subspace and
Using a simple convergence lemma we see that
+ L ~ T,
is a submersion
This kernel is the orthogonal complement of
to the orthogonal complement of l~Px 0 TxX + L = A p x 0
in
is not submersive at
N = A n = A p x A n-p.
( X, pX)
0 +
TX
(yi)
T
is net transverse
But this implies
is not Whitney regular over
X
at
x . KS
The strategy outlined so far will enable us to lift a smooth flow on a manifold to a controlled flow on a stratified set. necessary)
to look also at a more general problem:
f : A -* A' (say)
and a controlled flow on
A
which is mapped to the given one under
X
and
and
Y
X'
and
on
A'
Y
in
A
A' f ?
A'.
But if
does there exist a controlled flow on If we restrict attention to two strata
f(X) ~ X'
and
f
f(Y) ~ Y'
are different strata then the requirement that the flow on
: ~ x ~2...4,, ~ 2 .
maps
X
(say) where A
will interfere with the control exercised by the distance function
simple example is provided by (I. 2) and (I. 1 )
f
given a stratified map
then there is nothing new about the situation provided
into a single stratum in Y'
It is natural (and will in fact become
lift that pX.
together with the projection
A
44
The example suggests that the control by
pX
should be disposed of in these instances
since a similar control is already effected by the map the target:
f
the t - a x i s .
f
together with the flow in
keeps the spiralling flow line automatically at constant distance to X ~ ,
On the other hand we do need control by the retraction
is why (CRf) was introduced. but involving
7
X
and
f
and this
We are heading now for a result analogous to ( 2 . 3 ) X ~
rather than
pX.
and
Before we can state it we
need a bit of technical preparation. Let N
f : N~N'
be a smooth map sending the smooth submanifolds
submersively into
submanifolds of such that
X'
N'.
and
Y'
respectively, where
X'
and
Furthermore, suppose we are given tubes at
(CRy) holds for the pair
Replacing the germ (at
X')
~
X'
(X', Y' )
: Y' ~ X'
and
(CRf)
Y'
X
and
Y
in
are smooth X, X'
holds for
X
and and
Y' X'.
by a smooth representative we form the
pullback
proj X xx, Y'
- -@
Y'
II proj
~
X'
I
X
)
X Xx, Y' = I(x, y') E X x Y' : fx = X ' y , ~ ) .
(i.e.
submersion
X •
the germ of
Y'
X'
Since
f : X ~ X'
is a
is a smooth manifold, and the commutation relations imply that
(X, Y) X, (
actually maps into
y!
X Xx,
~
Y'
o f) : N ~
X x Y'
.
Clearly, this property does not depend on the choice of a representative above, and we are now ready to state the following. (2. ~ )
Lemma.
with respect to
If, under the assumptions above, f
Y
is Thom regular over
X
then
(x,
f)[y
: y ~ x •
Y'
y,
is a submersive ~erm at Proof. such that
X
(notice that
If not, we find a sequence (~X, f)iy
f = ~
(yi)
is not submersive at
in
o f Y
Yi "
o_~n Y).
converging to some point
x E X
If we put x i = ~Xyi, y~ = fYi
and
45
x i' = fx i
then the tangent space to
Using the fact that
Tf
sends
T
X • Y
onto
Y'
at
yj) -
(xi'
Ty~g',
is
T
X • T , Y'. x.~ ~x! X' Yi i
a simple calculation shows that
Yi ( x, f)iy
T
X
fails to be submersive at
ker T
Yi
(~Y) ~ k e r
T
Yi
(fiX)
Yi
if and only if the linear map
is not onto : the relevant maps are displayed in
xi
the diagram
0 -~ ker Ty(flY)
) T Y
TxX XTx,X , Ty, Y'
y'
L\ 0
) ker Tx(fIX )
~
)
TxX
> Tx, X' .
Passing to a suhsequence we may assume that all spaces same dimension and converge to a proper subspace (ker Tyi(flY))
itself converges to
converges to ker Tx(flX), X T x (T) = S c ker Tx(flX ). ker Tx(flX) at
x.
we cannot have
T CTxN ,
S
Tyi~X[ker Tyi!flY)]
of
say.
TxX,
havel the
~hile the sequence
Necessarily
(ker Tx.(fIX)) I
and we obtain the proper inclusion X Since T x ~ is a retraction whose image contains ker Tx(flX ) _C T,
so
Y
is not Thom regular over
X
D
The main result of the section will be that we can (under suitable conditions) assign a tube to each stratum so that all relevant commutation relations are satisfied.
Let us first fix some notation to which we will stick throughout this
and the next two sections. Let
(A, A)
be a Whitney stratified subset of the smooth manifold
a = dim A (i.e. the maximal dimension of a stratum in we denote the union of all i-dimensional Mi
~tr~t~ in
is a smooth submanifold of N of dimension
i.
A). ~
by
For each Mj .
N ,
and let
i=0,1,..~, a
P y II.l.l)
Notice al~o ~n~t ~he
~i
each at8
46
locally finite unions of strata, so a commutation relation holds for a pair (M i, M j) X ~ M i,
if and only if it holds for all pairs of strata Y ~M j .
manifolds for
Mi
with
It will in fact turn out to be more convenient to work with the rather than with single strata whenever possible.
Next we put,
-I ~ a < ~ .< a , A ~a =
Note for later use that each A~
(X, Y)
are locally closed in
U i=~+1
A~
and
1
is a relatively closed subset of
A , again by (l.l.1).
Notice that the
filtration by dimension already used in Chapter I.
f : N -~ N'
A~
provide the
If, more generally,
is a Whitney stratified Subset of another smooth manifold stratifies a smooth map
and that the
A
N'
(A', A' )
such that
(A, A' )
then let us agree to use the same notation in
source and target but distinguish by a prime all symbols referring to the latter. (2. 5 )
Definition.
manifold each
N.
~,
Let
(A, A)
A tube system for
for
be a Whitney stratified subset of the smooth A
consists of one tube
i = 0, l, ..., a.
(CRy) hold;
furthermore all (CRp) are satisfied 9
i a (T)i=0
tube system
relations of type strata in
A
Let
f : A ~ A' 9
for
A
If
we call it controlled if
f : N ~ N' ( ,k~a' ,T "k=O
is controlled over
be smooth and let
is a t u b e system for
(T 'k)
if the
9
Ti
fX U fY C M 'k
for some
Note that in general "controlled over (unless, e.g. all strata of
A'
(A, A' ) A'
then a
satisfy all
(CRy) and (CRf) and if (CRp) holds for those pairs
with
(X, Y)
of
k. (T'k)"
does not imply "controlled"
have the same dimension).
We can now state the result announced above. (2. 6 ) map.
Theorem. Let
f : A-+ A'.
Le__~t N
and
N'
be smooth manifolds,
A ~ N , A' _c N', and supwose
(A, A' )
By putting
(Ti)
for
A
f : N -~ N'
(T 'k)
which is controlled over
N' = point we obtain
a smooth
is a Thorn stratification for
Then for each weakly controlled tube system
exists a tube system
at
We say that the tube system is weakly controlled
if all commutation relations of type
be a stratification of
T i = (E i , w i , p i , e% )
for
(T'k).
A'
there
47
(2.7) Proof
Corollary. of (2. G ).
Every Whitney stratification admits a controlled tube system.KS We proceed by double induction as follows:
already constructed a tube system (there is nothing to show for Ma
(Ti)
a = 0).
for
A a-I
which is controlled over
By Theorem (i. 6 ) we find a tube
satisfying condition (CRf) (with respect to the tubes
various commutation relations involving strata in We do this by downward induction on
a < a :
Aa
a = a - 1
that
(T i)
(T 'k)
restricted to
be complete, for
a = 0
(for
A aa-1
A
ei
we have
A a = Ma).
is also controlled over
a .
(T 'k)
If we can arrange
then the proof will
gives the result.
neighbourhood of i
~
in
Aa
is controlled over i e ,
and
N , and we denote the latter by
pi
as maps defined on
for
~ ~ i < a:
ITil.
ITil
Note
Let for each
i
Ei
with
a , Qi =
We p a r t i t i o n mapped by points
In order
is a diffeomorphiem of some neighbo~rhood of the zero section in
that we may consider u.
N').
at
assume the tube system induced on
First of all, we choose representatives for The germs
thus each 6nto a
in
Ta
of dimension less than
Therefore let us assume the tube system induced on (T'k).
T 'k
(T 'k)
Aa-I so as to satisfy the
to complete the proof we have to modify this tube near
is controlled over
assume we have
f .
y E Qi
n
9
Qi according to ~ e r e points in So let, for all such that
f
0 ~ k,t ~ a',
maps
y
into
Qi
Qi(k,~)
M '~
turn out that (CRy) and (CRf) are relevant over
and
be the subset of all iy
Qi(k, t)
/
Q'(1,D
O'(O,1)
and t h e i r projections to
into for
M 'k. k < t
It will whereas
are
48
over
Qi(k, k)
we have 59 look at
(CRy) and (CRp).
We therefore set
a I
i QP
=
U qi(k, k) . k =0
We obtain a partition Qi
i = Qp U
U
Qi(k, t)
k~t of
Qi
into a finite number of open subsets : openness follows at once from the
fact that
f
maps strata into strata.
Now we choose representatives for the tubes ei
domains of all diffeomorphisms
e,k
and
T 'k
too.
We can shrink the
so that the following statements become
true. (I)
as germs at
Qi,
i
o a
=~ i
for
a<
(possible by downward induction
i
hypothesis) (2) (3)
Qa ~ Qi,
as germs at
a ~
as germs at
Qp
Qi,
a
o i
pa
o e
= a i
=
for p
a
a
for
a < i < a ((2) and (3) by
upward induction hypothesis) ($)
M'~ N
as germs at
[T'kl
,
=,k o =,t
= ~,k
for
0 ~ k < t ~ a'
(since
(T 'k)
is weakly controlled) (5)
Qi(k, e ) =
(6)
if
~
Xp E A
if and
ITPIX[ into (7)
as germs at
k > e , for a n X 'q s A'
IT'qlx'I
i
(in view of (I.I.i)
are strata such that (by continuity of
Qi(k, ~), f o i
= ~,k o f
fX cX'
then
f
maps
f) for
~ ~ i < a
and
0 ~< k ~ I <. a '
(again by induction hypothesis) a : Qp a (8) (~, pa)IQp
Ma • I~ is a submersion (by Whitney regularity and Lemma
(2.3))
(9)
(a
f)lQ~(k, t) : Qa(k, t)*
0 ~ k < t ~ a'
M~
xM, k
(M,t n IT'kl)
~ a submersion f o r
(by Thom regularity, (7) and Lemma (2.4)).
Now we apply Corollary (1.7)
with
49
the following
data:
X = r
Qi,
Thus we have to specify
g
Qi,
as a germ at
pa)
g = (a,
r
Xo =
:
Qa = Qpa U kU
Xg = Q~, Q~(k,
x
N-~k s
g= (.~,.,eof)= N ~
To = TalXo'
•
(M, t n l T ' k l )
l):
we set
near
a qp
near
Qa(k, t).
M 'k
Then
glXg
over each
is a submersion Qi,
a < i < a,
by ( 8 )
a
(9),
and
TolXg is
i
a
u
0 /r
a
a
= /r
(2) a
= p
a
O?r
o /f
i
0 7/
0 ~
Ta
is compatible
with
to which
(1.7)
further,
obtain open subsets
involved
8re normal and
X - Xg
in
X
f
can be applied
X
o
we can arrange
= ?r
(1)
a
(3) and
compatible
with
g ; for
we have
7r 0 ~
p
and
= p
0 /r
we shrink in
Qi
(2)
i
(1)
b y assumption.
i QI
a
c~ = '/T
i
0 ?T
= p
J
(3) Finally,
to get a subset
XI ~ X
the tubes
Ti
(~ < i < a)
a bit
and set
XI =
i U QI " u< i
Since all spaces
is a neighbourhood
of the relatively
X - Xg c X 1 _CXl ~ X _CX ~ .
closed subset
50
Applying (i. 7 ) we obtain a new tube extends
Ta~xI
relations:
and is compatible with
T
at
g.
T
X
(with retraction
satisfies all required commutation
this follows from the very construction of
where we have to verify (CRf).
g
everywhere but over
(x Qp = U Qa(k, k), and over
But
~ , say) which
Q~(k, k)
Q;,
we
compute ~, k
f o ~=
o f o ~;=f (7)
~
o
o ~=f
o~
~
= (7)
We complete the induction step by (shrinking and) extending
~, k
T
of.
over all
is done by another application of Theorem (1. 6 ), this time with the resulting tube at
Ma
be the new
w
T a.
M a.
g = f.
This
We let
D
Vector Fields
We shall now give precise definitions fo~ the class of vector fields we propose to integrate. (3. I)
Definition.
Let
(A, A)
be a stratified subset of tRe smooth mmnifold
A stratified vector field
on
M i , for
By abuse of language, we also denote by
i = O, i, ..., a.
general discontinuous) which equals
~i
conditions for
on ~
A
vector field on M i.
If
(T i)
at a stratum
(VFp)
Tp Y o ~ = 0
f : A ~A'
c N'
A
lifts
~'
(VFf) holds on
A.
if
the (in N ,
then the control
; We call
~
Y E A , and controlled" if furthermore
Next let
A
on
are
T~ Y o ~ = ~ o ~Y
be a stratified vector field on on
is a tube system for
Y.
~
~i
with values in the tangent space of
(VF~)
holds for all
~'
A
Y s A
both are equations between germs at
Y s A.
is a collection of ~nooth vector fields
weakly controlled if (VFp)
holds for all
be a smooth map, stratified by A'.
(VF~)
(A, A'), and let
We say that the stratified vector field
~he condition
Tfo~
N.
=~' o f
Instead of (VFp) we also consider the following weaker condition on
51
at a stratum
Y E A :
if
f
(VFpf) We say that
~
maps
Y
into
Y' E A
then, as a germ at
Y,
Tp Y o ~ I f-lY ' = 0 .
is controlled over
~'
if
~
satisfies all conditions of type
(VFf), (VF~) and (VFpf). The reader may verify himself that the vector fields in ( l . 1 ) controlled in the appropriate
sense.
and (i. 2 ) are
The following existence result shows that the
notions just defined are wide enough to be useful. (3. 2 ) and
Theorem.
A' c N'
Le__~t (T i)
Let
f : N -~ N'
be a smooth map between manifolds,
be subsets I and let
and
(T 'k)
controlled over
(~, A')
A
and
A',
~
o__nn A
A _c N f : A,
and suppose that
Then any weakly controlled vector field
a lift (i.e. a stratified vector field over
be a Thom stratification for
be tube systems for
(T'k).
let
~'
(Ti)
on
A'
A'. i_~s
admits
liftin~ it) which is controlled
~' .
Proof.
We proceed by induction on the dimension a of
already defined
~
on
A a-1.
A.
We choose representatives
So let us assume we have for all tubes
Ti
and
as in the proof of (2. 6 ) such that the following hold (using the same notation): (I) ~ i o ~ J 9
= i on "
(2) p l o ~ J =pi
Qi ~ Qj, i
on
(3) Qi(k, ~)=~
Qp ~ QJ,
0 ~ i < j < a
for
0 ~ i < j < a
if k>t,
(5)
f o i
(6)
(~, 9 pi)
(7)
( i, f) : Qi(k ' g) ~ ~ 0<~k<
for
= ~,k o f
: qpi
on ~
0~
Qi(k, Z),
for
0 .< i < a,
x ~{ is a submersion, for x ,k(M,t f] iT,kl)
0 .< k .< t .< a'
0 .< i < a
is a submersion,
for
0~i
~.< a'
(8)
T~ i o ~ = ~ o ~ i
(9)
Tp i o ~
on
(y) = 0
if
(10) T~ 'k o ~' = ~' o ~,k (ii) Tf o ~ = ~' o f
on
An-1 F3
ITil, for
y E A a-1 ~ ITil on
0 .< i ~< a - 2 and
fy = f iy
A' f3 IT'kl
A a-I
By slightly shrinking the tubes
Ti
we also obtain subsets
F i C Qi
which are
T 'k
52
closed in
Ma;
of course (I) to (Ii) still hold with
Fi(k, t) = F i ~ Qi(k, t)
Fi
and
in place of
qi
F iP = F i ~ Q~).
will be complete if we can construct a smooth vector field
Ca
on
Ma
(we put
The proof which
satisfies
for all
(V!~)
T~ i o ~a = ~i o i
on
Fi
(VFpf)
Tp I o C a : 0
on
Fi P
(~f)
~
on
Ma
pi
are now maps rather than map
i = 0, ..., a - i
germs ).
o f
o~a=~'
(notice that
i
and
Clearly these conditions are local and define a convex subset of the
linear space of all smooth vector fields on
M a, so we may define
glue the pieces together by means of a partition of unity on
~a
M a.
locally and
Let
y E ~a:
there are three cases to consider.
(a)
y r
U
Fi
since
:
f
maps strats into strata it sends an open neighbourhood
o~i
y
in
near
y,
Ma
into some
M 'e ,
fIM a
is,
smoothly equivalent to a linear projection of vector spaces.
we can find a vector field Ma ,
and since it does so submersively
Ca
satisfying (VFf) on a neighbourhood of
U
and we choose this neighbourhood disjoint from
Fi-
Thus y
in
Then (VF~)
O~i
y 9 Fj
for some
Then either (b I) y e F j# : near
y
o .< j < a.
j ,
y e Fj P
or
We choose
y e FJ(~, m)
by (6), (~J, pJ) : QpJ-~ M j x ~
Then
(VF~)
and
(VFpf)
maximal with this property.
for some
as a lift of the vector field
case (a).
j
o~l
<m~
a'.
is a submersion.
We choose
~J x 0
under
(~J, pJ),
arguing as in
hold for
i = j
by construction, and all
other conditions follow by computation : Tf o C a = Tf o T~J o C a = Tf o
(5) and for
~J
o
~J
=
~, o
f o
(ll)
~J
= ~, o f
(5)
i < j T~ i o C a = T~ i o T ~ j o C a = T~ i o ~ j o~J = ~J o i o ~ J
(l)
(8) (8)
= ~io~J
(i)
Tp i o C a = Tp i o T~ j o ~a = Tpi o ~J o ~J
(2)
Ca
0
(9)
53
(b2) y E FJ(t, m), vector field
I < m. ~J x
Here (I0) and (ii) imply that there is a well-defined t @,m
on
Mj x
M'
lift to
QJ(t, m),
using (7).
construction, and (VF~)
(3. 3 )
Corolla~.
Then
follows for
QJ(t,m).
(VFpf) is empty on
Le__~t f : N-* P
P.
smooth vector field on Proof.
Le__~t (Ti) P
which we may (locally)
(VFf) and (VF~), i < j
for
as in (a).
i = j,
hold by
Finally condition
F]
(A, A) a Whitney stratified subset of submersively into
t (M'm ~ IT'll)' M'
be a smooth map between manifolds I and let N
such that
f
maps each stratum of
be a controlled tube system for
A .
Then each
can be lifted to a controlled vector field on
This is nothing but the special case
A' = N' = P,
f : A-* P .
w
A.
A' = IPI of ( 3 . 2 ) .
The reader may check for himself that under the assumptions made indeed a Thorn stratification for
A
(A, IPI)
is
D
Flows
The final step in our discussion will be the proof that controlled vector fields do define continuous flows.
But let us first look at the ways in which an
arbitrary stratified vector field may fail to do so. (4. i)
~
.
~(x, o) : a/ax
Let
~
be the vector
and ~(x,y) : 0
vector field in the obvious way.
field on the plane
for y l O ,
(x, y) E m 2.
(x,
Example.
Let
y) : @/@X +
a/ay
f
be defined on
for
y ~ 0.
obtained from t # 0 .
~ x I~2
~
This is a stratified
~2 as
A less trivial example is
f(x, 0)--a/ax
and
Recall t h a t we expect a flow on
defined at least on a neighbourhood of all maximal flow lines of
given by
We can integrate it and obtain a flow
]R x ~ 2 -~ i{2 which is discontinuous along the x - axis.
(a. 2)
~2
0 • I~2 in
we obtain a mapping
~ x I~2 . D-+ ~ 2
by deleting all points of the plane
~2 to be
But if we collect
where the set It = Yl
D
is
for which
Clearly the difficulty is that the flow lines through points just below
54
T"
the x-axis are too short-lived as they approach the x - a x i s
(in the picture we have
marked time intervals of equal lengths on the flow lines). We return to the general discussion: Integrating each vector field where
Di
~R x M i
~i
on
containing
0 • M i.
x E A,
Di
Of course
(sending
e : D -~ A
t
to
We let
e(t, x)).
~i ;
ei d
Di
is given b y e i (t, x) : ~(x).
Ox : (tx' t+) ~ A D =
Ua i=0
~
(t;, t+) x x,
with
denote the flow line through Di
we obtain a map
A
though in general it
be open).
The stratified vector field is locally integrable if
contains a neighbourhood of then
D
e i : Di-*
is an open subset of
which has the algebraic properties of a flow on
Definition.
D = I~ x A
x,
be a stratified vector field on
we obtain smooth flows
Now setting
will not be continuous (nor need (4.3)
=
~
intersects I~ • Ix] in an open interval
-co ~< tx < 0 < t+x <~ co x
~
is the maximal domain belonging to
ei(0, x) For any
let
0 x A on which
e
is continuous.
is called 81obally integrable.
The following lemma characterises the maximality of
D .
D
If furthermore
A.
(4.4)
Lemma.
Let
~
be locally inteErable.
is continuous ever.ywhere on
D
.Furthermore, if 9
e x : [0, t;) -~ A
Then
D
is open in
x E A
and
'
t+
8
t + < co then
- -
is a ~roper map, and
R x A and
X
is unique with respect, to this property
X
(of course a similar statement is true for Proof.
t;).
The first part of the lemma is, for smooth flows, a standard result in
differential topology, and the proof easily generalises J2nich]).
(see [Lang] or ~rScker-
It is also well-known ~hat under the assumption
t + < co the inverse X
image of a compact set subinterval of
C c A
[0, t;),
under
8x![0 , tx+)
is actually contained in a compact
and so is itself compact.
Conversely, if
positive number then the set
8xI[0 , t) -1 (ex[0 , t]) - [0, t)
exI[0 , t)
O
cannot be proper.
t < t+x
is not compact, so
Note that the lemma applies in particular to the smooth vector fields i
A ,
is a
~i
on
so we have
(4.5)
Corollary.
X E ~
l_~f ~
~uch that
is any stratified vector field on
t+x < Go
the map
0x : [0, tx) -* ~
A
then for
any
is proper.
We are now ready to prove the main result. (4.6)
Theorem.
Let
A _c N
Let
~
Le__~t N
an__~d A' _c N'
and
~'
Proof.
A
N'
be smooth manifolds and
be subsets, and
(A, A')
be stratified vector fields on
is controlled over that the set
and
~'
By downward induction on
is locally integrable.
N.
Then
A
respectively
~
u = a, a - i, ..., 0
contains a neighbourhood of
(2)
8
is continuous at all points of a neighhourhood of [0, oo)
0 x Mu
Di
is open in
in
for
4 .
Suppose
we may assume that
~' ~IA a
[0, co) x A,
in the first statement by
the induction hypothesis and (4.4) imply that Ua i=~+l
such that
The inductive step consists in proving
D
since
(T i)
A'
f : A-~ A'.
is locally integrable if
(I)
For we may replace
a smooth map.
a stratification for
with respect to some tube system
is locally closed in
f : N ~ N'
D
~
eI ~ i=~+l
the result will follow.
0 x Mu (look at Di
in
I~ • M u .
-~), and then
is continuous,
and
is.
56
For the proof of (I) it will be convenient to shorten J x C C
is a subset of
D .
Now let
a compact neighbourhood of
we find a number
t > 0
containing
IT I
fX).
E
and
r
A a~-l N ~-l([0, t ] C ) ~
Tp o ~ = 0 ~
and
p-l[0, ~], N,
E(.,
8) = A a=-i
s(.,
8) = E ( . ,
0 < 8 ~ r
[0, t] x f(C) ~ D',
8)
such that
e
D .
x,
and X
We choose a
mapping the total space of a
onto an open set
f-lx' ~ ITI
IT I c N
(X' E A'
as maps defined on
, such that
is the stratum
ITI.
For a sufficiently
E([0, t]C , ~) , defined as
[0, t] x f(E(C, r
Aa
the set
~
D'
ITI
for
Aa
~ore generally, we define
lifts
[0, tic .
is contained in ~'
is open in
we have necessarily m x A'.
Since the compact set that for some
[-s, 0] x S([0, tic, ~) is contained in
cannot meet
[0, s] x E(C, 8)
C , and we find a positive number
is disjoint from is contained in
Making
D' (the maximal
the inductive assumption guarantees
e([-s, O] x S([0, tic, r
We show that
is contained in with
if
n p-l(8)
~'), for since
e([-s, 0] x S([0, t]C, ~))
JC
n p-l[o, 8]
and by (4.4),
s < t
to
is a smooth vector field on
is a compact subset of
o -l(.)
is contained in
positive number
p
~IX
and the dot stands for an arbitrary subset of
domain associated with
~)
on
C)
the stratum containing
is a locally compact space.
smaller, we may assume that
S([0, t ~ ,
[0, t] x C
the "disc bundle"
locally closed subset of
where
Since
diffeomorphically
We regard
small positive number
X.
T ~ = (E, ~, p, e)
small open disc bundle in on
in
such that
representative of the tube
T~ o ~ = ~ o ~
x
x E M a, X E A
x
e(J
D .
But
8 <
E(C, 8). D .
So let
y E E(C, 8).
There are three cases to consider. (a)
y E C :
this case is trivial.
(b)
fy ~ X' : we want to show
s < ty, + so let us assume
J = (eyI[0, t~))-IIE([0, J
is an open subset of
siC, ~)r~ f-l(e~y[0,
[0, ty)
t+ < s Y
and look at the set
s])l .
for the control conditions imply
~ o e
'
and
f o ey = e'fy, and by construction
the other hand, as
e' ~y
is a compact subset of
does not meet A aa'
=e y
eyI[0 , t~) X'
avoids
E([0, siC, r
S([0, s]C, r ~ f-l(e'fy[O,
~Y
On s])
so the inductive hypothesis implies via (4.4) that
57 §
J
is compact.
Since
0 E J
we conclude that
ty) + is compact, which is absurd.
[0,
Thus
J = [0, ty),
hence that
tY§ as desired.
s <
We are left
with (c)
fy E X' Aa
F~ ITI
u-I
f
but
by
p
y ~ C, (A'
'
and
hence
A' )
'
~'
by
py > 0:
we argue as in (b), replacing
[0, co), stratified by the origin and its complement,
by the zero vector field.
Now the proof of (1) is complete.
In order to prove assertion (2) we use the
same construction in a slightly different way, as follows. First of all, we choose contained in in
D .
C, t
and
~
This is possible
so small that since
D
I~ • A by what we have already seen.
where
r
we make
is any number in C
interval
and
~
(-t, t).
containing
e(l-t, t] x s([-t, t]c, ~)) into
I-t, t] x E(C, 8)
E([-t, t]C, ~).
K
E(C, 8), ~
of
e(r, x) V
in
A
for some
such that and we show that
e
maps
it suffices to prove that
9
sends
So let
as in (I) above.
V
(r, x)
y E E(C, 8) : we may restrict Look at
tic, ~) ~ f-l(8~y[-t,
is open and closed in
t])l
[-t, t],
and since
0 E K
we
N' = point deserves a separate statement.
Corollar~J.
A controlled vector field on a locall,y compact stratified set
is locally inte~rable.
[]
The hypothesis that N) cannot be dropped: x-axis
8 > 0
(0, x)
is continuous in
is contained in
We find a
into
fy ~ X
e
is
K = [-t, t] . []
The case (4.7)
E(RC, r
By control through
It follows easily that
contains a neighbourhood of
is disjoint from
K = I-t, t] F3 8ylIE([-t,
obtain
r .
x E M ~.
I-t, t] x E([-t, t]C, ~)
We prove that
V.
ourselves to the case
Again let
Given a neighbourhood
even smaller so that
R ._c (-t, t)
R • E(C, 8)
Aaa_l by
A
be locally compact (or, eguivalently, locally closed in
stratify the plane
and its complement.
22
(with coordinates
x
By deleting all points of the y - a x i s
we obtain a Whitney stratified set on ~hich the vector field but not locally integrable. We conclude the section with the following observation:
a/ax
and
y)
but the
by the origin
is controlled
58
(4.8)
Lem~a.
Let
stratification for ion
f : N-~ N' f : A -~ A'.
f : X ~ A -~ A'
vector fields on
is
A
Further let
such that
Then if
Corollar;/.
f : A ~ A'
A'
A _c N'
~'
~
lifts
~
~'
and
Proof of (4.8)
~'
x
the restrict-
be stratified
~
x e A
I
By (4.4), 8x [0, t+)
then
f
such that is proper.
actually maps into
ex
(tx, tx +) / R ,
X,
and
If
X e A
By (4.4) again,
t fx + = tx+ < co'
cannot be globally integrable.
~'
so
w Let
A
A , and
a smooth map from
We say that the stratified set are
a
stratified set
f = prp o h,
sending
projection to y e P
A
P).
(A, A)
(F, ~-)
V.
The homeomorphisms
is proper. [3
V
N ,
A
is (topolo6ically) trivial over h : A ~ P • F P • ~
locally trivial over such that
h : A ~ f-~
(A ~ f - b ,
~ V • F
a
into ~mother smooth manifold
and a homeomorphism
(A, A)
has an open neighbourhood
over
N
to the product stratification
We call
is the
Applications
be a subset of the smooth manifold f
and
e x : [0, tx+) -~X ~ A
8' = f o 8 : [0, t+) ~ A' fx x
stratification of
is
such that
hence the composition
Definition.
~
is~ too.
is also proper,
(5.1)
P
(prp
P
Theorem.
smooth manifold X E A ,
fiX
Let N,
with denotes the
if each point
A ~ f-b)
is trivial
are called local trivialisations.
and let
P.
be a locally closed Whitney stratified subset of the f : N-~ P
is a submersion and
locally trivial over Proof.
(A, A)
P.
if there
The following result gives a sufficient condition for local triviality over (5.2)
a
[]
< co.
stratum containing
(A, A')
and assume that
is globally inte6rable then
If not, we find a point
t+x
and
X e A
The conclusion holds in particular for all
is proper.
we may assume
A _.cN,
Assume that for each stratum
a proper map.
and
locally inte6rable. (~.9)
be a smooth map,
be a smooth map such that for each stratum
flX N A
is a proper map.
Then
(A, A)
is
P.
The assertion bein~ local with respect to
We apply (2.6), (3.5), (4.6) and (4.8):
P,
we may assume that
P = I~p .
each globally integrable smooth vector field
59
on
~P
admits a globally integrable stratified vector field on
In particular, let
~I' "'" ~p
a/~tl, ..., ~/~tp
on
If we let
be the fibre
(F, F0
~P,
then a trivialisation of
lifting it.
be lifts of the constant coordinate vector fields
and denote the flows generated on A n f-l(0)
(A, A)
~P
A
over
x
~P
*
by
el, ..., ep.
with its induced Whitney stratification is given by
( A n f-l(0))
(tI ..... tp, x)
A
~
A
el(tl, e2(t 2 ..... ep(tp, x))).
D
Theorem ~.2) is known as Thom's First Isotopy Lemma, for in the special case A = N, P = I~ the proof we have given consists in constructing an isotopy of moving each stratum of
A ~ f-l(0)
to one in
A ~ f-l(1).
N
Note that we cannot
expect to find smooth local trivialisations as we have already seen in Example (2.1). The next result gives us a great deal of information on the local structure of Whitney stratifications. (5.3) Let
Definition. x
Let
be a point in
A
(A, A) and
be a stratified subset of the smooth manifold
X e A
the stratum containing
is (topologically) locally trivial at and, for some
y E F,
product stratification
IYl
X x ~.
h : A ~ X • F
is a stratum of A
We say that
if there exist a stratified set
a local homeomorphism
such that the one-polnt set
in
x
x.
~
sending
and
h
sends
x
A
(F, ~') to
A
N.
(x, y)
to the
is ipcally trivial if it is so at every point
A .
(5.4)
Theorem.
Let
smooth manifold
N,
(A, A)
an~ let
be a locally closed Whitney stratified subset of the X
be a stratum in
A .
Then there exist a smooth
sphere bundle
@ : M-* X , a closed Whitney stratified subset
neighbourhood
Z
of
X
in
N
(S, ~ )
is locally trivial over
(2)
there exists a homeomorphism from the pair (Zr
obtained from qby E X).
Z$IS)
X
(with respect to
M
and a
+ x
4)
(Z, A ~ Z)
which is the identity on
M x K0, I]
of
such that
(I)
cylinders
(S, ~)
by identifying
X
onto the pair of mapping
(recall that
Zr
(y, 0) E M x [0, i]
is with
60
Proof.
Choose a tube
mapping the closed a neighbourhood on
X).
~ -disc bundle
ITI
of
By making ITI
~
(I)
A ~
(2)
for any stratum
X
in
p
N
at
X
IY E E : py .< r (here
~
ITI (for
Y E A, Y ~ X ,
I 2
..cE diffeemorphically
e onto
is a suitable smooth positive function
A
is locally closed in
the map
(~, p) : Y ~
(by(2.4) and local finiteness of
by
and pick a representative for
small enough we may suppose that
is closed in
submersion Replacing
T = (E, ~, p, e)
~p ~o~
normalises
is proper.
I~I ~ - X
is ~ i t n e y
of the form
Y n Iwl ~ ,
ITI ~ ~ X
is a
and then the map
x (o, 2)
stratifie~ by the open set
Y ~ A, ~ d
x I~
A).
~ = 2,
(~, ~) : ITI ~ - x - , x
N)
ITI ~ - A
we may a~ply Theorem (5.2) to
a,,1 strata
ITI ~ - ~
~d
But, recalling the proof of (5.2), we restrict the choice af local
(~, P).
coordinates in : Rdim
X x (0, 2)
X x R*X
for this application:
x (0, 2)
we allow only charts
of the form (x, t)
~ (~(x),
2e__L_ t ) i +e t
Where
~ : ~dimX_, X
M ~ ITI
by
(S, ~ ) = ( A n
p ~ 1
is a local ch~rt f ~ and
M, A n M)
p = I
X .
respectively,
(notice that
M
We define
and let
$ = ~IM
z_c l~l
&d
sna
intersects each stratum in
A
transversely).
On application of (5.2), or rather its proof, we obtain (I)
(S, ~ )
is locally trivial over
vector field (2)
a/at,
X
(with respect to
$) :
just ignore the
and
there is a homeomorphism
h : M x (0, l] -+ Z - X,
making
h M x (0, i]
~
Z - X
(~
D)
x • (0, i]
commutative and mapping a/at
only, obtain a flow
S • (0, I] 8
on
to
(A ~ Z) - X :
JTI ~ - X,
and let
lift and integrate
61 r 2--U~ , y ) .
hCy, r) = @Clog From hhe diagram in (2) it follows at once that Z$ ~ Z
having the required properties.
Notice incidentally that (x, t)
~ t . e -I x ;
[X~ U
~
and
Z$
Z~
is embedded in the vector bundle
(good) topological
spaces
by
and is Whitney stratified by
(5.5) Corollary.
Let
Z
From this observation and the fact that, for
U, V, W
W x Zf,
and a map
f : U -~ V ,
Zid W • f
is naturally
we obtain at once the following. N
be a smooth manifold and
Then any Whitney stratification of
A
A _c N
a locall.y closed subset.
is topologically locally trivial.
Again, we cannot expect to find smooth local trivialisations: counterexample.
E
and that we have in fact constructed a homeomorphism between
preserving stratifications.
homeomorphic to
extends to a homeomorphism
KS
(x, t) E M • [0, I],
• (0, 1]
h
KS
(2.1) provides a
On the other hand, topological local triviality is sufficient for
the following application: (5.6) Theorem.
Let
smooth manifold
N .
of all strata in Proof.
(A, A)
A
be a locally closed Whitney stratified subset of the
Then the partition
Ac
of
is a Whitney stratification
of
into the connected components A.
We only have to check the Local Finiteness Condition,
induction on the dimension of the ambient space notation) of Theorem (5.4) as follows : subset
A
S _cM
(for
meets only a finite number of strata in ~c .
is homeomorphic to
different
meeting
strata in
A ~ Z N ~-lc
we may apply the inductive hypothesis to the
in
But since
Ac
Ac
X
then
8 A $-lc Z$1S
A ~ Z ~ w-lc intersect
is a neighbourhood of
x E X
must also intersect
S ~ $-lc x
in
Corollar~r.
Ac
and
C
is a compact
by a homeomorphism preserving strata each S ~ $-lc,
in different strata of A
and of course ~c.
Thus
meeting only finitely many strata of
A c. KS (5.7)
(and the
Thus if
x
stratum of
using the conclusion
dim M = dim N - I).
neighbo~rhood of A ~ Z
N ,
and we do this by
satisfies the Frontier Condition.
62
Proof.
Let
X
and
Y
be strata in
local triviality it is also open in
xn~=~
or
X~.
A c.
X .
X ~ Y Since
X
is clearly closed in
X .
By
is connected we have either
D
The result needed in Chapter IV is an analogue of (5.2) for stratified mappings It is well-known as Thom% Second Isotopy Lemma.
rather than stratified sets. (5.8)
Theorem:
Let
N
F
f
N'
let
be a diagram of smooth manifolds and maps t closed subsets~ and let that
f
(A, A')
maps each stratum of
for all strata f : X' N A '
X E A
~ P
and
A'
suhmersively into
P.
the restrictions
are proper maps.
A' ~ N'
of
y
P, in
such
F : X ~ A-* A'
and
(F : A - ~ A ' , A, y E P
4')
there
and hcmeomorphisms
h :V
•
F-lf-l(y) n A
. ~-If-l(v) n A
h':V
•
f-l(y)
.
hA'
F : A-* A'
i.e. for every point P
be local~y
Suppose furthermore that
Then the stratified map
is (topologically) locally trivial over V
and
A_cN
be a Thom stratification for
X' E A'
exist an open nei~hbourhcod
p
f-l(v) n A' ,
preserving the natural stratifications, such that Pr V
id x F V • F-lf-l(y) ~ A
;
F -I f-l(v) ~ A
~
commutes
.
Proof.
As in (5.2), w e may assume
vector fields on and (3.2).
~P,
lift them to
V • f-l(y) ~ A'
f-l(v) ~ A'
P = ~ P (=V). A',
We define the trivialisatio~
h
and
h'
;
V
Again, consider the coordinate
and lift them further to
By (4.6) and (4.8) we obtain a flow on
V
A
as in (5.2).
A,
covering those on O
Now a standard property of fibre bundles implies the following.
using (2.6) A'
and
P.
SS
Ifj under the assumvtions of ~5.8),
(5.9) Corollary.
P
is connected then the
topological type of the map F
:
F-if-l(y) ~ A
f-l(y) ~ A'
Y Riven by restricting
to the fibres over
F
y
is independent of
y E P .
t
Proof.
Partition
P
this is a partition of
according to hhe topological type of P
Fy, y ~ P.
By (5.8)
into open subsets, hence it is the trivial partition.
[]
The Second Isotopy Lemma can easily be generalised to hhe case of a sequence of smooth maps
Nt
Ft ~
~ N/_ I
Details are left to the reader.
..- ~
NI
FI ~
NO
~
P .
CHAPTER III
Unfoldings of smooth map-germs by Andrew du Plessis (U.C.N.W., Bangor).
85
SO.
Introduction
Throughout this chapter, all manifolds, mappings etc. will be taken to be smooth. The mathematics described here arises from the study of smooth stability of mappings, and in particular the associated local theory. For mappings,
smooth stability is easily defined:
a mapping is smoothly
stable if all mappings sufficiently close to it (with respect to the Whitney topology) are equivalent to it under the natural action of diffeomorphisms
of the
source and target manifolds, For map-germs, however,
the immediate
Tlocal' version of this (which, to
make sense~ would have to use as equivalence relation the action of diffeomorphisms preserving source and target points) is not the fright' one (in the sense that one would hope that all germs defined by a stable map should be stable germs) - for, with respect to this definition, only germs of maximal rank would be stable (for clearly such germs form an open equivalence class whose complement is not a neighbourhood of any of its points, and so a f o r t i o r i classes which are neighbourhoods
contains no equivalence
of any of their points).
Moreover, the topology
for the space of map-germs implicitly being used here (via comparison of all derivatives at the source point) is apparently too coarse - it is, of course, completely appropriate example, the germs
for analytic map-germs, but it does not separate, for 2 ( ~ , O) § (~, O) defined by e "x and O , any of whose
representatives will always be separated by the Whitney topology. The difficulties,
of course~ arise from our dancing too close attendance on
the source and target points.
Thus to match more closely the stability theory for
mappings, we seem forced to consider representatives 'nearby' germs, we need to consider representatives
of germs:
essentially,
for
defined on Vnearby' open sets.
This looks a very difficult theory to work with .,. The concept of unfolding of these difficulties.
(originally due to Thom) permits a very neat finesse
In the simplest version, one considers germs of finite-
66
dimensional par~metrized families of germs as a germ
F : (Nx~, X o • o) ~ (PxU, yo• which preserves u-levels (i.e. representatives F(N x {u}) c P x {u}
for
u
a family an unfoldin~ of that
F
of
in some neighbourhood of f , where
(f(x),u O) = F(x,u O) .
)
F u0
f : (N,Xo) + (P,yo)
If the germ
f
have the property that in
U ).
We call such
is the map-germ such
is smoothly stable, we expect any
such unfolding to be trivial up to equivalence (that is, there should exist levelpreserving germs of diffeomorphism such that
K o Fo H -I = f Xlu)
H
of
(N x U, x O X Uo) ,
K
of
(p x U, YO X Uo)
.
Of course it is not quite clear that this notion of stability is equivalent to stability as it might be introduced via 'uniform representatives' formalized that notion properly!).
(if we had
However, it does stand in the 'right v
relation to stability of mappings (see, for example, Guillemin and Golubitsky [go ]), and we will take this as justification for considering stahility of mapgerms in this way). It turns out that stability in this sense has a very neat algebraic equivalent ('infinitesimal stability') leading to a very elegant classification of stable mapgerms (which was first obtained by Mather;
in fact, in this chapter we will meet
a large proportion of the beautiful mathematics he introduced, in ~ l l l ] p
[Mal'V]
[MaV], to do this). However, it is not entirely satisfactory to restrict attention to stable mappings, since they are not always generic (that is, there exist (compact) manifolds
N,P
such that the set of stable mappings
space of all mappings
N + P
N § P
is not dense in the
(with respect to the Whitney topology);
see ~MaV]).
It will, therefore, be the aim of this chapter to describe an extension to the notion of stability for map-germs which is almost as easily handled algebraically and whose 'globalization' is generic. Let
f : (N, x0) + (P, yo )
germs 'nearby'
f
be a map-germ.
If
f
is not stable, so that
are not all in the same equivalence class, we might nevertheless
hope that all 'nearby' germs are 'contained'
(up to equivalence) in some unfolding
87
F : (N•
x O•
+ (P xU, Y0 •
unfolding
G : (N•
'
More precisely we hope that for any
x Oxu~) + (P•
Y0XU~)
of
f
there exist germs
~,~
such that the following diagram commutes~
(N•
x 0xuO) ~ (PxU, Y0 •
(NxU', Xo• where
~,?
~ (e •
Y0 •
have the form
~(x,u') = (~u,(X), ~(u')) , ~(y,u') = (~u,(y), ~(u')) with
#uS = iN '~u~ = iF
and
I
a map-germ
(U',u~) ~ (U,uO) ; that is,
versal in a category of 'parametrized unfoldings' of
f
F
is
(whose morphisms (the pair
(#,~)) we have described above). Sadly, the condition that
f
should have a versal parametrized unfolding,
while still being fairly easily computable, does not always have generic 'globalization' either (counter-examples are described by Mather in [~eSS3; but see also, for example, the work of Wasserman [ Wa] where, following unpublished notes of Mather, the case where
f
is a function is extensively studied.
In this case,
since stability itself is generic for functions, the condition is quite satisfactory; and indeed very much stronger genericity statements can be made (see Looijenga [ Lo3)). Thus we n e e d t o t h i n k d e p e n d e n c e on t h e p a r a m e t e r
a little
harder.
What we s h a l l
space.
With this
in view,
let
do i s
to remove the
us b e g i n
the formal
mathematics of this chapter: (0.i)
Definition Let
f : (N,xO) + (P,yo)
An unfolding of immersions and
j
f
be a map-germ.
is a triple
is transverse to
(F,i,j)
of map-germs, where i,j
F , such that the diagram
(N',x~)~ (P',y~) +i
§
(N,xO) ~ (P,yo)
are
88
is cartesian
(that is, the diagram is coTsnutative, and the mapping
into the submanifold
{(x',y) c N' xPIF(x')
The dimension of
(F,i,j)
given by
u I = ... = u k = 0 .
co-ordinate
system for
{UlOF,...,UkOF} for
(N',x~)
system for
Of course, then,
(P,yo)
.
Moreover,
extends to a co-ordinate
,
F
is
(P',y~)
let
such that
(Yl o j,..,,yp o j)
since
j
system
Im(j)
is
is a
is transverse to
F ,
(Xl,...,Xn,UlOF,,..,UkOF)
Then, since the diagram is cartesian,
Ul o F = ... = uk o F = 0 , so that
N
d i m . N ' - dlm.N.
for this situation as follows:
be a co-ordinate
of
is a diffeomorphism onto).
as an unfolding of
We may choose co-ordinates (Yl ..... yp'Ul' .... Uk)
= j(y)}
(i,f)
(x I o i,...,x n o i)
Im(i)
is defined by
is a co-ordinate
system for
(N,xo) Clearly, with respect to these co-ordinates,
F
'preserves u-levels';
that this notion of unfolding is just a 'co-ordinate-free' introduced previously.
version of that
From this point of view it is worth observing that
essentially what we are doing here is choosing a submersion such that
so
~ ~ (P',y~) + (U,u o)
~-l(u O) = Im(j) - one choice amongst many possibilities,
whereas no
choice is given us in the parametrized version, (0.2)
Definition An unfolding
i
and
j
(F,i,j)
respectively
(Recall that
is trivial if there exist retractions
such that
(N• and
P' = P x U XoXUo)
,
s
to
is a retraction to
i
if
r o i = IN ;
s o j = ip )
With respect to co-ordinate N' = N •
and
f or = s oF
r : (N',x~) § (N,Xo)
similarly, of course,
r
.
Then
S = (S,ip)
(f x IN) o R = S o F .
systems as chosen previously,
R = (r,~)
is a level-preserving
is a level-preserving
we may identify diffeomorphism of
diffeomorphism of
(PxU,Yo•
So this notion of triviality is precisely the
'co-ordinate-free' version of that introduced earlier.
60
(0.3)
Definition A map-germ
is stable if any unfolding of
F : (N,Xo) ~ (P,yo)
F
is
trivial.
Thus far, then, there is no difference between this co-ordinate~free approach and the theory suggested earlier.
However, in the case that
asking that all 'nearby ~ germs to a germ some unfolding of
f
f
f
is not stable,
are ~contained ~ (up to equivalence) in
is again presumably to seek a 'versal' unfolding, but this
time in a different category, in which the objects (unfoldings of
f ) are
essentially the same as before, but in which there are many more morphisms. (0.4) (a)
Definition A marphism (~,~) : (F,i,j) § (F',i',j')
of unfoldings of
f
is a commutative diagram of map-germs:
N'
(b) of
An unfolding
(F',i',j')
~
of
f
P'
is ve=sal if for every unfolding
(F,i,j)
there exists a morphism (F,i,j) + (F',i',j') .
Let us observe, then, that a map-germ versal unfolding of itself (the retractions (F,i,j) § (f,iN,l P) ). unfolding
(F,i,j)
stable (when we call
f
is stable if and only if it is a
(r,s)
of (0.2) giving the morphism
In fact this statement generalizes as follows:
of a map-germ F
f
an
is versal if and only if the map-germ
a stable unfolding).
F
is
This seems to be a very deep fact
(certainly, I am not aware of any straightforward geometric approach to it), more or less the key to the classification of stable map-germs, and the proof of it therefore yields many other interesting results.
In fact, it is these subsidiary
70
results which are the useful ones for the purposes of this book (indeed, we could have avoided the concept of versality altogether, though with consequent more or less complete loss of (I)
motivation);
these useful facts are:-
A stable (equivalently, versal) unfolding is 'essentially ~ unique (for details, see w
(2)
A stable unfolding is, with respect to suitable co-ordinates, a germ of polynomial mapping (see w
(3)
The equivalence class, as a germ, of a stable unfolding is determined by the contact class of the germ it unfolds (we will introduce the notion of contact-equlvalence in w
moreover, a stable unfolding satisfies an
important transversality condition relative to the action of the contact group in the jet-space. (4)
(see w
The property of having a stable unfolding is 'generic v (in a sense which will be explained in w Proofs of these facts (and of the equivalence of versality and stability for
unfoldings) are based upon some rather subtle manipulations of modules of vector fields over rings of function-germs. presented in w
The preliminaries to the algebra are
For the moment, some of the style of argument we will be using
can be understood from the lemma which follows, which will be useful in the succeeding sections. However, just before stating and proving this lena,
it seems worthwhile to
state explicitly that this chapter is mostly expository rather than original. Many of the concepts are due originally to Thom, while most of the deep methods were developed by Mather. their profound ideas.
I am happy to have this opportunity of expounding
I am also grateful to my co-authors, in particular E,J.N.
Looijenga, and also to C.T.C. Wall, for much help with the material presented here,
(0.5)
Definition Let
8N
8N,xo
be the set of germs of vector fields on
for this where no confusion will arise).
(N,xo)
(we shall write
71
(0.6)
Lemma Let
let
(F,i,j)
be an unfolding
~ : (P',y;) § (U,u O) Then
n ~ 8p,
(F,i,j)
f : (N,x O) §
be a (germ of) submersion
is trivial
, ~ r ONv
for the map-germ
such that
if and only if for any
(P,yo)
~'l(uo)
T r OU
, and
= Im(j)
.
there exist
such that Ii
is a lift for
T
over
~
(i.e. T~ o n = T o z)
is a lift for
u
over
F
(i.e. T F = $ = D =F)
Proof 'Only if': (0.2).
If
Suppose
(F,i,j)
(Here
pp
(germ of) vector
x 0'
field on n
PxU
be the retractions
(r,,oF)
YoXUo ) ~ e 8p,
projection,
as in
T
a unique
~
such that
splitting
over
: (N',x~) + ( N x U ,
T(s,~) o ~ = T o pp ~ o pp
T(PxU)
as a
= TP $ TU)
.
~ .
XoXUo)
8N,
is a germ of
and we identify
via the natural
is a lift for
so that there exists
,
r,s
: (P',y;) + ( P x U ,
is the natural
then,
Similarly, at
(s,~)
so that there exists a unique
: P •U § U
In particular,
let
T~8 u
It is clear that diffeomorphism,
is trivial,
is a germ of diffeomorphism
such that
T(r,~ Q F) o ~ = T a PN
Thus T ( f o r,w oF) o ~ = T ( f • ~ ) But
(f o r,~ o F) = (s,~) o F
o (T o pN ) = (T Opp) o F = T(s,~) o n ~ F
(from the definition
T(s,~) o T F ~ Since
(s,~)
so that 'If':
is a lift for
Let
(Ul,...,Uk)
n
over
r,s ), and so
~ = T(s,~) o n ~
is a germ of diffeomorphism,
~
of
we have,
. therefore,
that
TF o ~ = q o F ,
F .
be a system of local co-ordinates
at
(U,u O) , and let
a ni
be a lift for
~
over
~ '
gi
a lift for
ni
over
F , for each
l i = l,...,k Let
. ~i : (p' x]R,
germs of local flow for vector
fields,
YO' xO) § ni,$i
see Chapter
ll).
(P',y~)
, #i : ( N ' x ~ ,
respectively
x;xO)
(for facts about
Then we have
-> (N',x;)
integration
be the of
uke~e~i,t (where
" Uk~+t'61k
~ i , t ( y v) = ~ i ( Y V , t ) , ~ i , t ( x ' )
Now d e f i n e
The u s e o f Im(i)
!
!
r : (N ,Xo) § (N,Xo)
= ~i,t ~F
- ~i(x',t)
, s : (P',y~)
).
-~ ( P , y o )
by
r(x') = i -I o ~k,_Vk~ . . . o ~i _Vl(X' )
(where
v i = u i o~ ~ F(x') )
s(y')
(where
w.l = u.1 o~(y')
i-l,j
j-lo ~k,_Wk ~ ... ~ l , _ w l ( Y ' )
-1
makes s e n s e :
- (~ o F ) - l ( u o ) .
retractions
' F~
to
i,j
recall
that
And i t i s a l s o c l e a r
)
I m ( j ) = ~ - l c u O) , that
r,s
are respectively
.
Now for(x') .'I
= J
= f o i-I o ~k,_v k o ... o # i _v I (x') = j-I o F o ~k _Vk o ... o ~i _Vl
~
o...o~i vk
oFCx') = soF(x') ,-v I
(since wiCFCx')) = vi(x') ) and so
f o r = s o F , as required:
thus
(F,i,j)
is trivial.
Cx')
73
w (I.i)
Preliminaries, mostly algebraic
Definition The local ring
~,Xo
is the ring of functlon-germs
(N,Xo) § 9
(with
addition and multiplication defined pointwise). This ring has a maximal ideal (N,x o) § (m,o)
consisting of all germs
.
We shall write (1.2)
mN,xo
4 ' mN
for
% , X o ,mN,xo
when no confusion can arise.
Lemma Let
Then
(Xl,...,Xn)
x I ..... xn
be a system ~f local co-ordinates for
represent a (N-basis for
(N,xO) .
mN .
Proof Let
#
represent an element of ~(x) = r
mN .
Then
- r n
=
~ (r
.... O,x i ..... x n) - r
.... O,Xi+l,...,x n)}
i=l -
~ II ~ i=l 0
{%(O ..... O,txi,xi+ I ..... Xn)}.dt
i!ixill ~-~~xi (O,. " " '0'txi'xi+l" "" 'xn) }.dr O Thus the germ of X. WS 1
~
is indeed a Z-linear combination of the germs of the
,
Further, by taking differentials (giving a CN-linear homomorphism mN + T*Nxo ) we see at once that the dxi's (1.3)
xi's
must be linearly independent (since the
are); and so they do indeed form a basis. Definition Let
f : (N,xo) § (P,yo)
f* : ~ § CN
is defined by
Note that
f*mp c mN .
be a map-germ.
f*(~) = ~ o f .
Then a ring homomorphism
74
Now let us consider a situation which has already arisen in our study of unfoldings: (1.4)
Lemna Let
(u,u0) be a sequence of germs such that -I
_i., i
(V,v0)
~,
(W,w0) a submersion with
is an immersion and
(wO) = Ira(i) . Then the sequence
o § ~*(mw).C v = c v .i*, c U + o is exact and (non-canonically) split. Proof Let
(Wl,.,.,We)
(Vl,...,Vk,W 1
g,
o
be a system of local co-ordinates for
...,We o ~)
be a system of local co-ordinates for
a system exists because
is a submersion). -
mV NOW
i* : % +
%
(W,wO)
o
and let
(V,vO)
(such
Then, by (I.2)
~,.,.~w e
o~>.C V
acts as follows:
i*(1) = I , i*(vj) = vj o i , i*(wj o w) . wj o w o i = 0 . Thus it is clear that for
(U,uO) , so, by (1.2), Hence
i*
(v I 9 i,...,VkO i) v l o i ..... v k o i
is a system of local co-ordinates form a basis for
mU .
is onto.
Also, clearly
Ker i* - <w I o ~,...,w e o w>.C v = w*(mw).C v , so that the
sequence is indeed exact. And of course the sequence is split via our choice of co-ordinates.
(1.5)
Now observe that
(Xl,...,xn) ~x---i..... ~
8N
is a free CN-mOdule of rank
is a system of local co-ordlnates for
= dim.N
(if
(N,Xo) , the vector fields
are a basis for it).
Now we define the CN-mOdule
%f , for any map-germ
to be the module of germs of vector fields along
f
f : (N,xO) § (P,y0) ,
(Chat is, germs
75
$ ; (N,Xo) + TP
such that the following diagram commutes; TP ~//~ (N,xo)/-
the CN-mOdule action is defined by free CN-module of rank for
= dlm, P
It is worth noting that
(1.6)
(if
A
is an R-module,
$ : A § B
~,8
all
8N
(1.7)
B
E
R, a,b
r
tf : 8N -> 8f over
an S-module, and $
if
8f
is a
is a system of local co-ordlnates f
along
f
81N .
defined by f*
are a basis for it).
tf(~) = Tf o ~ ; and
defined by
$ : R § S
wf(n) = ~ o f .
a ring homomorphism,
@(ca+ 8b) = $(e).$(a) +$(8).$(5)
then
for
A ),
We may interpret
sense~ elements of
It is clear that
is the same object as
wf : 8p + 8f
is a homomorphlsm over
);
(yl,...,yp)
~-~--~ylf'" "" , % o
There is a %-homomorphism
there is also a homomorphism
( ,Yo )
f
(~'$)x = Sx'$x '
(if
(P,yo) , the vector fields
~rojection
Bf
8f
as 'infinitesimal perturbations v of
are 'tangent vectors' at
specifying as they do for each tangent vector there.
x E N
how
f(x)
f
f : in some
in the space of germs,
is to be deformed by giving a
We should thus expect to he able to derive elements of
ef
from paths in the germ-space, which we do as follows:if
g : (N•
B E TUuo
Oxu0)
§ (p,yo)
is such that
there is a well-defined element
c : (~,O) § (U,Uo)
represents
f(x) = g(X,Uo)
8.g e 8f
B , define
defined as follows:
~,g(x) = d
is irmmedlate (by Taylor's theorem and differentiation
, then for each if
(g(x,c(t)))it= 0 ,
of products)
that
~.g
It is
well-defined. Such parametrized families of germs give rise to unfoldings F : (NxU,x OxuO)
§ (p xU,y O x u O )
of
f
by
F(x,u) - (g(x,u)~u)
it is worth noting that %,g = {iF(O,8 opN ) - wF(O,8o pe)}IN• (where we identify
8Nx U = ep N @ 8p u , 8p• U = 8pp 9 8p~ ).
, and vice versa;
76 Conversely, given an element of
8f , can we construct 'a parametrized family
of germs (and hence an unfolding) realising it?
(I. 8)
Lemma If
# l ' " ' ' # k ( Of , then there exists a germ g : (N•
such
As to this, we have:
that
g(x,O) = f(x)
and
-~ (P,yo)
8--~." g " ~i 1
(i = I ..... k).
Proof Let can write
(yl,o..,yp)
be a system of local co-ordinates for
r
. ( ~~
= j=l ~ r
g : (Nx~k,xoxO)
§ (P)yo)
o f)
(i=l ..... k) , where
r
(P,yo) . ~ ~
"
Then we
Now define
by k
Yi o g(x)u I ..... Uk) = Yi of(x) + Then clearly
(1.9)
g(x,O) = f(x) , and
~-~-." g = ~i i
~ u.~j,i(x) . j=l J
(i=l,...,k)
, as required.
Having introduced soma of the modules with which we will be concerned) as
well as indicating some of their relationship with unfoldings, we proceed to the theorems of module algebra which we will need.
(I.I0)
Lemma Let
such that Let
R
(Nakayama) be a commutative ring with identity) and let
1 + m c R* E
Then if
( R*
is the group of units of
m c R
be an ideal
R ).
be a flnitely-generated R-module. m.E = E , E - O .
Proof Suppose that
E + 0 ) so that, since
minimal spanning set {el,... ,ek} k el = j-l~x.j . ej , where x~ ~ m .
for Thus
E
E
is finitely-generated,
over
R .
Since mE = E ) we can write k x.e. and so, since (I - Xl).e I -j=2Jj
i - xI
is invertible)
there is a
k e I = [ (I - Xl)-ixj.e. j=2 3
P
'
77
So
eI
minimal.
is dependent on
Thus
(i. II)
e2,,..pe k
in contradiction to
{el,.,,,e k}
being
E = 0 ,
Corollary Let
A
SeA
be a finitely-generated R-module (R as in (i.i0)), and let
be a finite subset, Then spans
S
A/mA
generates over
A
over
R
if and only if the image of
S
in
A/mA
R/m .
Proof VOnly if':
If
Thus spans 'If':
S
generates
A
over
R , we have
R.S = A ,
(R/m).S ~ R.(S/mS) = A/mA , so that indeed the image of
A/mA
over
We have
A
in
A/mA
R/m . (R/m).S = A/mA , whence
A/R.S = m.(A/R.S) generates
S
.
over
Thus, by (I.i0)
A = m . A + R.S ,
A/R.S - 0 , or
so that
A ffi R.S .
Thus
S
R .
Remark Our use for (I.I0), (i. II) is precisely that conditions of (I.i0) (the inverse of (l+f)-l(x)
(1.12)
I + f , for
R ffi ~
f 9 mN
, m = mN
satisfy
the
is of course given by
= l/(l+f(x))).
Corollary Let
B
be a sub-module of a finitely-generated ~ - m o d u l e
A , and suppose
that
d i m ~ { A / ~ + I A + B} < r . Then
~ A c B .
Proof The n o n - d e c r e a s i n g
dimlR{A/A+B} consists of hypothesis.
r+2
sequence < ... < d l m ~ { A / ~
A+B}
non-negative integers, and is bounded above by
Hence there exists an integer
dim]R{A/~ A+B}
< ,.. _< d i m ~ { A / ~ + l A + B }
ffi d i m ~ { A / ~ + l A + B }
, whence
r , by
i , O -< i < r , such that 4
A+B
" 4+IA+B
9
78
"
Thus
"
i
(m~ A + B ) / B = (mN+IA+B)/B = mN . (mN A + B ) / B
(m.l~ A + B ) / B
= 0 .
Hence
~
a ~-module, if
then
a E A, ~ ~ ~
A
.
AcB
N
Now let us observe that if
, and s o , by ( 1 , 1 0 ) ,
f : (N,Xo) + (P,yo)
is a map-germ and if
may also be considered as a ~-module
, we define
~. a = f*(#) . a ).
us, in this situation, to find generators for
via
f*
A
is
(that is,
It will often be important for
A as a Cp-module.
As to this, we
have, firstly:
(1.13)
Theorem Let
(Malgrange)
f : (N,x O) § (P,yo)
~-module.
Then
d i m • {A/f* (rap)A}
A
be a map-germ, and let
is finitely-generated
A
as a ~-module
be a finltely-generated if and only if
is finite.
This may look like a theorem of algebra; in fact, however, it is a very deep result in analysis, first proved by Malgrange
(at the instigation of Thom).
Another, slightly easier, proof is due to Mather [MaIV].
We will not give a proof
here.
(1.14)
Corollary Let
subset.
A
be a finitely-generated
Then
A/(f*mp)A
S
spans
generates A/(f*mp)A
A
~-module,
and let
as a ~ - m o d u l e
over
S c A
be a finite
if and only if the image of
m(~/mp)
.
A/(f*mp).A
over
S
in
Proof W0nly if' : clear. 'If' :
If the image of
S
spans
is finite, so by (1.13)
A
is finitely-generated
by (i. Ii) that
(1.15)
S
generates
A
over
~
~
, then
dim]R{A/(f*mp)A}
as a Cp-module.
It then follows
.
Corollary Let
E
be a finitely-generated
generated sub-~-module
(via
f* ).
CN-mOdule , and let
EI c E
be a finitelyT
79
If E =E' f o r some
r > dim]R{E'/mpE'}
then
+ (f*~+~
r+l._ >~
E = E'
Proof The inclusion
Ev c E
induces a vector-space homomorphism
E'/~,E' which is onto by our hypothesis,
§ E/Cf*~ + ~+1)~ so that
di~/(f*mp So, ~ince
E
,
is a finitely-generated
+ ~I)E~
~ r .
CN-mOdule , it follows from (1.12) that
E c (f*.p).E . Thus the hypothesis may now be written E = E ' + (f*~)E So the vector-space homomorphism onto, and thus by (i.13)
E
EV/mpE v ~ E/(f*mp)E
is finitely-generated
But the reqrltten hypothesis also imPlies (1.10),
E/E' = 0 , or
.
E - E' .
induced by inclusion is
as a ~-module.
E/E v = (f*mp).(g/E v) .
So, by
80
w
Infinitesimal
Stabilit Z
In this section we state and prove an algebraic condition equivalent to stability, and derive various consequences.
(2.1)
The condition is:
Definition A map-germ
f : (N,x O) + (P,yo)
is infinitesimally
stable if
ef = tf(SN) + wf(Sp) (for notation see (1,5), (1.6)).
The main result of the section is:
(2.2)
Theorem A map-germ is stable if and only if it is infinitesimally
stable.
Proof 'Only if' : Let
Suppose
# ~ 8f , and let
g(x,O) = f(x)
and
( F , ~ x O,ip x O) be trivial: for
let
- soF Let
f : (N,x O) -~ (P,yo)
~
of r,s
g : (Nx~,Xo•
9 g = ~ f
by
is a stable map-germ. + (P,yo)
(as in (1.8)).
Then
f(x,u) = (glx,u),u)
be retractions to
be such that g
determines an unfolding
, which, since
~ • O, IF x 0
f
is stable, must
respectively such that
. ~ {s(y,u) }In--O ~ 8p . ~ = ~u (r(x,u)}Iu=O c e~, ~ - ~fu
the identity
f o r(x,u) = s(g(x,u),u)
tf(~) = # + wf(~) , so that
with respect to
~ E tf(SN) + wf(Sp)
Since this is true for any
~ ~ 8f ,
f
Differentiating
u , we obtain
.
is infinitesimally
stable.
Before proving the 'If v part of the theorem, it will be convenient for us to derive a consequence of infinitesimal stability relating to vector fields along unfoldings. Let
f : (N,x O) § (P,yo)
: (P',y~) -~ (U,u O) Let (where
d~
be a map-germ,
a submersion such that
8F, ~ - Ker{d~ : 8F + 8 oF} is defined by
d~(~) = T~o# ).
(F,i,j) ~l(uo)
an unfolding of
= Im(j) .
f , and
81
Then there is a ~,-module morpbism as a ~,-module via
i* ) defined by
u : 8F, ~ § 8f
~ § Tj -I o ~ o i
(where
8f
is considered
(this makes sense because
~(i(x))c Ker TwF(i(x) ) - Im(TJf(x)) ). Further, O § (~ o F)*m u . eF, ~ c eF, ~ ~
ef + O
is an exact sequence of %,-modules. (If
~ ~ 8f , then
dj o ~
Tj o ~ o i-I : (Im(i),x~) § TP' is a submanifold of germ
~'
Thus
a
Nv
is a vector field along is a vector field along
and
j o f = Fo i , so that FIIm(i) .
Since
Im(i)
Im(~) c Im Tj = Ker T~ , this may be extended to a
of vector field along
F
with
Im(~') c Ker T~ ; and then
a(~ ~) = ~ .
is onto.
Also,
Ker a = { ~
8F,~Idj-lo ~ o i = O} = oi
= {+~F,~I~oi = (Ker i*)eF, ~ But, by (1.4),
(Ear i * ) % , = (w o F)*mu. % , ; so the sequence is indeed exact).
Thus there is an isomorphism of %,-modules (a)
8f ~ 8F,~/(~ o F)*mu . 8F, W
Similarly (replacing
f
by
IN,I P
respectively), we have isomorphisma
(b) 8N ~ 8N, ~ o F / ( Z o F ) * m u. 8N, ~oF
(where
8N,,~oF = Ker t(~oF)
(c) 8p = 8p,,~/(Z*mu) . ep,,~
(where
ep,
(2.3)
)
= Ker t~ ).
Lemma If
f
is infinitesimally stable, then 8F, = tF(eN,,~oF) + wF(Sp,,~)
Proof It is easy to see that the morphism tF(SN, ~oF)
onto
tf(eN)
and
(F*mp,)eF, ~
~ : eF, ~ § Of
defined above maps
onto (f*mp)ef .
(~ o F)*mu " F*(~*mu) = F*mp, , we can deduce from
(a)
Hence, since
an isomorphism of
~,-modules (*)
8F,~/tF(eN,,woF) + (F*mpv)eF, ~ ~ 8f/tf(8N) + (f*mp)Sf
82
Since
f
is infinitesimally stable,
finite ~ - b a s i s some finite
for the
RHS
of
(*)
, so that a
is given by the projection of
wf(S)
, for
S c 8p .
By (c), isomorphism
S
is the projection of a finite set
(*) ,
wF(S')
must generate this % , - m o d u l e
@F, = tF(SN, ~ F )
S v c Spy ~ ; and then, by the
must project to an ~ - b a s i s
Thus, by (1.14) applied to the ~ , - m o d u l e wF(S')
8f = if(8 N) + wf(@p)
for the
8F, /tF(SN,,~oF)
as a ~ , - m o d u l e .
LHS
of
(*) ,
, the pzojeetion of
Certainly,
then,
+ wF(Sp, ~) .
Now we can give: Proof
of 'If' in (2.2) Let
f : (N,x O) + (P,yo)
any unfolding of
f .
be infinitesimally
To show that
f
stable, and let
(F,i,j)
is stable, we must show that
a trivial unfolding; by (0.6) it will suffice to construct,
for any
vector fields
and
(where
E 9 8N, , q 9 @p,
~ : (P',y~) § (U,u O)
such that
tF(E) = wF(n)
is a submersion such that
be
(F,i,j) T E 8U ,
t~(q) - w~(T)
~-l(u O) = Im(j)
, as
usual). So - let and
~
~ 9 8u , and let
respectively,
be lifts for
T
over
that is, T(~oF)
(Such lifts exist because Then
E 1 ~ 8NV,q I 9 8p~
o E 1 = ~. (~ oF) ,T~ o n I = T o
~ ~ F,~
are submersions).
T~ o nlO F - Tx o dFo E 1 = O , so wF(n I) - tF(E I) ( Ker dr = 8F, ~
Thus, by (2.3), there exist
E 2 9 8Nt,xoF , n 2 9 8p,,~
such that
wF(ql) - tF(~l) = wF(n2) + tF(E 2) 9 Define Then
E = E 1 + E2 , n = n I - n 2 .
tF($) = tF(EI) + t F ( E 2 ) =
wF(~l) - wF(N2) = wF(q)
t~(q) m t~(nl) - t~(q2) = r . ~ = w~(T) Thus
E,q
and
.
are the required vector fields, and the proof is complete.
z oF
is
83
(2.4)'
Let
space
f : (N,x O) § (P,yo)
be a map-germ, and let ~ f
be the JR-vector
8f/tf(8 N) + (f*mp)Of . Let
pf : 8p/mpOp §
be the ]R-homomorphism induced by
wf
(this is
wf(mpep) c (f*mp)Of),
well-defined since
Then: Corollary f
is stable if and only if
pf
is surjective,
Proof 'Only if' : clear. 'If' :
If
pf
is surjective, then
of - if(0 N ) + wf(Oe) + (f*mp) 0f . Then by (1.15) (with
E = Of , E' - tf(SN) + wf(Op) ) we have
8f = tf(SN) + wf(Sp) , so that
f
is infinitesimally stable, and hence, by
(2.2), s t a b l e . We want to go on from this to derive a condition under which a map-germ will possess a stable unfolding if
PF
(F,i,j).
Of course
F
is stable if and only
is surjective; we want to express this in terms of the module
8f .
As
a first step, we have
(2.5)
Lemma There is an ~ - l i n e a r isomorphism
q F , f : ~ f +~/F J
defined as follows: if
~ r Of , let
Then
~ ( OF
be such that
~o i = dj o ~ .
qF,f([~]) " E~] 9
Proof Let
~ : (P',y~) ~ (U,uO)
be a submersion such that
~'l(u O)
Ira(j)
Consider the ~,-module morphism
%F,~/tF(8N, ~oF) § 8F/tF(SN,) induced by the inclusion
8F, ~ c 8F .
f
This morphism i s i n j e c t i v e ,
because
o
84
8F, ~ n tF(SN,) = Ker d~ n Im tF = tF(ker t(~ oF)) = tF(0N, ~ F ) . On the other hand, we have
8F = 8F, ~ + tF(SN, )
of local co-ordinates at co-ordinates at
(for if
(Ul,...,u k)
(U,u O) , then we may choose a system of local
(P',y~) of form (yl~.,.pyp,Ul,~p,..,ukP~)
such that { ~ }
. L
8F .
is a system
spans
J
But d~(wF(~)
- tF(~(uj~oF )) = O
(j
so tha= also
{wF(~S--~)dyi, tF(~(Uj=S ~F))}
spans
8F ).
Thus the CN,-morphism is actually an isomorphism. an isomorphism of Cp,-modules (via
It is, therefore, also
F* ), and so gives rise to an ~-linear
isomorphism 8F,~/tF(eN,,~oF) + (F*mp,)SF, ~ % 8F/tF(eN,) + (F*mp,)8 F 9 Composing the inverse of this isomorphism with the isomorphism in (2.3), we obtain
(*)
a : 8F, ~ § 8f (2.6)
derived
qF,f "
The characterization of the action of the isomorphism
(*)
qF,f
follows from the definition of
of (2.31, this isomorphism being induced by the morphism
defined by
a(~) - dj -10 ~@ i .
We define the R-homomorphism PF,f : TP~y~ §
as the composite
-i Tp ~
, ~ 8p~/rvSpT~Fr~ YO
We have Le=~a
(F,i,j)
is a stable unfolding of
Proof Immediate from (2.4) and (2.5). This
leads
us
to
f
if and only if
0F, f
is surjective.
8S
(2.7)
Definition Let
f : (N,xO) § (P,yo)
Define
Xf = d i m ~ f
We say that (2.8)
f
be a map-germ,
e {O,I,2 .... ,|
.
is of finite singularity type (abbreviated FST) if
Xf < ~
Theorem f
has a stable unfolding if and only if
f
is of FST.
Proof 'Only if' :
If
(F,i,j)
so that certainly 'If' :
If
is a stable unfolding of
f , then
0F, f
is surjective,
Xf < dim PV .
k - Xf < ~ ~ let
g(x,O) = f(x) , and
{ ~
a
g : (N • ~ k ~ Xo • O) § (P~yo)
k " g}i=l
be such that
maps to an ]R-basis in ~ f
(such
g
exists
i
by (1.877. by
Then
g
determines an unfolding
(F,1 N • O,ip • O)
of
f p defined
F(x,u) = (g(x,u),u).
ha~e PF,f(~a----lyo 7 o u i = q~l.[wF(~a-~-Y]
We
~I
Also,
a
a-~7.9 g =
{ t F ( a ~.) - wF(~,~--)}IN.~.• O
1
1
au i {[au~" g]}
(see (1.7)), and so
1
[a_s So, since
@
~U i
a g2--pF,f(~---,yo7 .oui,
spans ~/f , it follows that
1
( F , ~ • O,Ip x O)
is a stable unfolding of
f .
PF,f
is surjective, so that
86
w
Versality
In this section w~ show the equivalence of versality and stability for unfoldings. In fact, the key result of this section is a uniqueness theorem: (3.1)
Theorem Any two stable unfoldings of the same dimension of a map-germ
f
are
isomorphic as unfoldings.
The main step in proving this is establishing a Vcontinuoust version of it, which is the following: (3.2)
Lemma
Suppose that the commutative diagram F
/ ix ii~
(JR ,I)
j • fIR
is a one-parameter family of stable unfoldings of a map-germ a e I Fa
(
I
is the unit interval [0,i]) the unfolding
is defined by
x0
(Fa,i,j)
(~,I)
of
f , where
(that is, there exist retractions
respectively such that the whole diagram
(N ' ,x~)
(N' : < ~ , x ~ •
Fo
~
__~F
(P' ,y~)
(p~ • 2 1 5
(*) i ~ l]R
(N • IR,x O• I) conanutes).
(that is, for each
F(x',a) = (Fa(x'),a) , is stable).
Then the family is trivial over to
f
(JR ,I~
.i • i N
' (p x IR,yo • I)
r,s
87
Proof
We will make the natural identifications eN'x~ ,x~Xa = epN ' 9 e .n..p,. o F
Opwx~, ,y6Xa (where, as usual
epp, @ e # ~
PN' : (N' x IR ,x6 x a) § (N' ,x6)
and similarly for
is the natural projection;
Pp, ).
We claim that to construct
aE
=
r,s
it will he sufficient to find, for each
I , {a = (~a'~u = ~:S~~
F) ( eN'•
,
rla = (na,~-~-uo ~]R ) ~ ep,x]R,y~X a
such that
~a
is a lift for
(p' x R 'Yo x a)
~a
over
Fa
is the germ-restriction
(where of
F a : (N t x ~ , x 6 x a) §
F ) , where
~ ,~ Ker(i x l~,)*OpN '
~a' E Ker(j x l]R)*Bpp @ ( na
is, of course, automatically a lift for
~
over
~
)
To see this, we proceed as follows:let
be flows for
~ba : ((N' x:]R) x :JR, (x6Xa)
x O) § (N' x ~ , x O'xa)
~a : ( (P' xIR) x m ,
x O) -+ (P' x m , Y o X a )
~a,na
representatives
for
x O'
[-~,r
x
[a,a+c]
Hence, since
x
I
0 = a 0 < ... < a
(YoXa)
respectively.
We may in fact suppose (via choice of
F,~a,~ a ) that
~a,~ a
YO' x [a,a+E]
x [-~,r
is compact and connected, such that
respectively,
for some
of
~ > O .
there exists an increasing sequence
Sa.,~a. are germs defined on neighhourhoods of 1 1 x O' x [ai,ai+l] x [ai-ai+l'ai+l-al] ' YO' x [ai,ai+l] x [ai-ai+l,ai+l-ai] respectively. e
= I
are germs defined on neighbourhoods
88
Then
r,s
may be defined by
(r(x',a i +t),O) = ~ao,ao_a I o ~al,al_a 2 o ... = ~a
a -a o ~al,ai~t (x~ ,ai + t) i-l' i~l i
(s(y''ai+t)'O) =
a -a i-l' i-i i
~ ~ao'ao-al
for
~ "'' ~ ao'ao~al
0 < t <- ai+l-a i , and (with
a0 - O~a e
ffi i
~at(~a+t)
respectively) for
t
near
Ker(i x 1 ~ ) * c p~,mN, , Ker(j x I R ) * c p~,mp, , so that PN''
n' 9 (p~,mp,).e a
(x;a§247
=
r(x~ • I) - ~ ,
(*)
so
that
Then arguments similar to those of (0,6) show
co,mutes.
More strongly, the facts that will ensure that
Thus we w i l l have
Ppv
,at(Y~a*t) - (y~a§247
s(y~ • ~) - YO' .
that the top pentagon of
i (N) ,j (P)
O,e
.
Of course
~a' e ( P ~ ' m N ' ) ' 8
i replaced by
Sal,ai_t (y',ai + t)
~av ~ Ker (i K l ~ ) * S p N v~ n'a 9 Ker(j x13%)*epp,
r[i(N) x 3%, slj(p ) x 3%
are the natural projections to
respectively, and hence that the whole of
Thus it remains to construct
~a,na
for each
(*)
commutes.
a .
Let
~a " tFa(O'~~ ~3%~Fa) Then it is easy to see that that
~a 9 Ker(i • 13%)*8Fa,~3% Now, since
is an unfolding of
Fa
T~
"
WFa(O'~~ ~3%)
o ~a = 0 ~ and that
~a ~ (i • 13%) -- 0 , so
.
is stable for any
a ~ I , and since
(Fa,INv x O,IF, • O)
F a , we have by (2.3) that 8Fa,~3% = tFa(eNT•
o Fa) + W F a ( S p , •
)
and thus Ker(i x 13% )*OFa,~3% c tF (Ker(i • 13%)*eN,•% ,w3%o F a) + WFa(Ker(j x l ~ ) * e p i x 3% '~'~. )
Hence there exist that
~" 9 Ker(i x 13%)*eN,x3 % '~3% oF ,ha 9 Ker(j x l ~ ) * e p , x 3 % , ~ 3% a
~a = -tFa(~a) + WFa(qa) " Now observe that the identification
eNVx3 % = 8pN , @ e i%o F
identifies a
epN v with
8N'x]R ,73%0 F a ' and similarly
8pp w
is identified with
epTx3 % '~3%
such
89
With respect to these identifications,
~a= (~' , ~'u ~ ~ ~ ~ Pa )
define
~a = ( n ' , ~ o ~ )
,
Then tFa(~a) " WFa(~a) = ~a - (-tFa(~a) + WFa(qa)) = O so that Thus
Sa
is a lift for
~a,qa
qa
over
Fa
have the required form, and the proof is complete.
Now we can give Proof of (3.1) Let dimension
be two stable unfoldings of
(F,i,j) , (F',i',j') k .
By finding submersions
-I (O) = Im(j) , ~' : (P",y~) § (U,O) neighbourhood of (NxU,xo•
0
in
I~k
F,F'
such that
such that
remains to show that Since
~ : (P' ,y6) -> (U,O)
, we may identify
§ (P•
(F,iNxO,ipxO)
f
z"l(o) F,F'
of the same such that
= Im(j')
, where
U
is a
as level-preserving map-germs
F(x,O) = (f(x),O) = F'(x,O) , (FV,INXO,IpXO)
, so that it
are isomorphic unfoldings.
are stable, the ~-homomorphisms PF,f' PF',f : T(P XU)YoXO -~ 8f/tf(eN) + (f*mp)Sf
are both surjective, and hence so also are the ~-homomorphisms PF,f ' PF',f : TUo § 8f/tf(ON) + wf(Sp) + (f*mp)Sf induced by restriction and projection
(recall that
Thus there exists an isomorphism PF,f = PF' ,f Let then define
oh
of
TU O = ~ k
is induced by
wF ).
such that
.
g' : ( N x U , x o X O )
-> (P,yo)
F" : ( N x U , x o X O )
is an unfolding of
f , and
(F", 1N x O, ip x O)
be defined by
§ (PxU,YoXO) (IN•215
is an isomorphism of unfoldings,
We claim
h
PF,f
by
(x,g'(x,u)) = F'(x,u)
;
F"(x,u) = (g'(x,h(u)),u)
.
-I) : ( F ' , i N X O , i p •
F"
§ (F",IN•
so that it remains to show that (F,IN• O,Ip xO)
,
are isomorphic. PF",f = PF,f "
g : (NxU,xoXO) PF,f(80) = [-~'g3
-> (P,yo) for
To see this, recall that if
is defined by
~ ~ 8U .
F(x,u) = (g(x,u),u)
Then (defining
g"
from
, then (as in (2.8)I F"
in the same way
90
as
g,g'
were defined from
F,F ~ ) we have
PF",f(80) = [-~'g"] = ~-h(8),g'~ = PF',f(h(~o)) = PF,f (~) and the claim is proved. Now let define
yl,...,y p
ga : ( N •
be a system of local co-ordinates for
O•
§ (P,yo)
(a ~ I)
by
Yi o ga(x,u ) = (I ~ a) 'Yi o g(x,u) + a,y i o g"(x,u) Then, defining clearly have
Fa : (NxU,xo•
§ (p•215
by
P a = (I + = . F ,f a)PF,f aPF",f PFpf
surjective, and hence so also is Now define the map-germ F(x,u,a) = (Fa(x,u),a) iNx U • O,ip• U x O
.
P a " F ,f
(P~yo) , and
Thus
(i=l,.,.,p)
.
Fa(x,u) = (ga(x,u),u) So in particular
Fa
P a F ,f
is stable for each
F : (N•
§ (PxUx~,YO
Then, by (3.2), there exist retractions
is
a ~ I .
• r,s
, we
by to
respectively such that the following diagram co=mutes :
(N• U,Xo• O )
~F
:
(p •
)
rT
Define
rI : (N•215
and define
sI : ( P x U , Y o •
Then clearly unfoldings, (3.3)
§ (NxU,xoXO)
r l(x,u) = r(x,u,l)
by
§ (PxU,Yo•
(rl,s I) : ( F " , ! N X O , I p •
by
,
sl(y,u) = s(y,u,l)
§ (F,INXO,Ip•
.
is an isomorphism of
and so the proof is complete.
Corollary Let
(F,i,j),(F',i',j')
dim.(F',i',j') Then
- dim.(F,i,j)
(F',i',j')
and
be stable unfoldings of
f
such that
= e Z O . (FxIRe,i•
•
are isomorphic unfoldings.
Proof By (3.1), we need only show that of
f .
(F x l~e ,i x O,j x O)
is a stable unfolding
91
Let
g : ~'x~e,x~•
-~ (P',y6)
(g(x',t),t) = ( F •
Then
be defined by
.
PF•
,F(~-t. I
) ffi [- 8-~ ~
1
" 0
(where
tI . . . . .
te
are the usual
1
YO xO co-ordinates
in
~e)
: (P'x~e,y6xO)
so that
PFxll~e ,F " PF o T~ , _ where YoXU '
§ (P ' ,yo) ,
is the natural projection.
Thus PFxlRe,f Hence, since (Fxl~e,ix0,j
ffi (qF,f)-I~
PF,f ' T~yo•
x0)
e ,F
are surjective,
= PF,f o T#YoX O . so is
is indeed a stable unfolding of
PFxl]R e,f
; and thus
f .
Now we can prove
(3.4)
Theorem An unfolding of
f
is versal if and only if it is stable.
Proof 'If' :
Suppose that
(F,i,j)
any other unfolding of since
X F, ffi Xf
(F",I,J)
.
(via
Then
Then, if
f . qF, f) ,
is a stable unfolding of Since
f
has a stable unfolding,
XF , < |
(F",loi',Joj')
f ; let
and so
F'
Xf < ~
be So,
has a stable unfolding,
is a stable unfolding of
e = Idim.(F",loi',Joj')
(FV,iV,j v)
- dim (F,i,j)I
say
f .
, we have, hy (3.3),
either an isomorphism (~,~) : ( F x l l ~ e , i x O , j
x0 ) ~ ( F " , l o i ' , j o j ' )
or an isomorphism (~,~) In the first case, (where
: (F,i,j) = ( F " X l ] R e , l o i ' x O , J o j ' x O )
((~oF) o ~ o l , # o ~ o J )
~ : (p' x R e ,y~ x O) + (p',y~)
second case, So
see that
is a morphism
(F'.i',j')§ (F,i,j).
is versal.
Suppose
(F,i,j)
(F',i',j') + (F,i,j)
is the natural projectlon); while in the
(~ o (IN," xO) o I , ~o (ip,,, xO) o j)
(F,i,j)
'Only if' :
is a morphism
.
(F,i,j)
is a versal unfolding of
f .
The nearest way to
is then a stable unfolding is via the naturality of
PF,f '
92
which we make explicit (3.5)
as follows
:
Sublemma Let x~)
(N
F
~,Xo)
+
f~. (p,yo)
*
-
be a m o r p h i s m Then
:
(r
~
(F,i,j) § (F',i',j')
PF,f " 9F',f
(p,y~)
(F",y~)
of unfoldings
of
f .
OT~y0
Proof Define an ~ - h o m o m o r p h i s m if
A' e 8 F,
A ~ e F ~ let
K([A])
= [A']
-I = qF,f
and let
A' ~ eF,
qF,,f[a]
= [A']
(for if
"
But
.
V r 0p,
as follows
§
A' o i' = Tr o A o
A' oi'
A ~ 8F
ffi V' o , o F o i
K ( [ V o F]) ffi [V' o F']
i .
Then define
be such that
- Tj' = a .
A' o i' = Tj' o a - T r , and let
:
for it is not hard to see that
a ( 8f , let
be such that
(V' oF') oi' so that
be such that
This well-defined,
.
-I qF',f o K
Now let
K :~F
V' ~ 0p,,
Then
oa=
ffi V' o j' o f
qF,f[a]
T~oAo
be such that
A o i = Tj o a - [A] ,
i , so
K([A])
V' o j ffi T r
ffi T ~ b o V o j o f
- T~o
,
=[A']).
j .
(VoF) oi
Then
,
, and thus
Now we return to the proof of :'Only if ~ in (3.4). Suppose not stable,
that
(F,i,j)
so that
PF,f
[el ~ Im(PF, f) , and let and ~uu" g, = a
(such
(F',iNXO,ipXO)
of
is a versal unfolding is not surjective.
g' : ( N x ~ , x o •
g' by
there exists a morphism
(r
F'(x,u)
f , and suppose that
Then let -~ (P,yo)
exists by (1.8)).
f
of
is
be such that
be such that
g'(x,O) f f ( x )
Then define an unfolding
= (g'(x,u),u)
: (F',INXO,ipxO)
a r
F
.
Since
-> (F,i,j)
(F,i,j)
is versal,
, and, by (3.5), we have
93
PF',f = PF,f =T~yOxO ' But
~F,,~(~I
) = E-
~. ~
~ [-~
YO xO Thus we have arrived at a contradiction;
so
, ~hi~h is not contained in
F
must be stable,
Im(PF, f)
94
w
Con=act-equivalence
Definition (4.1) Map-germs
fi : (Ni'xi) § (Pi'Yi)
germs of diffeomorphism
(i = 0,I)
are equivalent if there exist
h : (No,Xo) § (Nl,Xl) , k : (Po,Yo) § (Pl,Yl)
such that
fl o h = k o fo " Definition (4.2)
a)
A pair
(h,H)
consisting of germs of diffeomorphism
h : (No,X O) § (Nl,Xl) , H : (NO •
+ (N I x P l , x l x y l )
such that the following diagram commutes: INo• (No,x O) ~ - -
h
PN O ~ (NoXP0,xO•
~ (No,X O)
IN; Yl
(N l,x I)
PN1 ~ (N I • 2 1 5
)
(Nl,x I)
is a contact eRuivalence.
If
NO = NI'Po
=
P1 '
(h,H)
is a~-equivalence; i f also
h
=
~0
,
(h,H)
is a~ -equivalence. b)
Map-germs
f'l : (Ni'xi) ~ (Pi,Yi)
(respectively~-equivalent, (respectively~-equivalence,
(i = 0,I)
~-equivalent) C-equivalence)
are contact-equivalent
if there exists a contact-equivalence (h,H)
such that
(l,fl) o h = Ho (l,fo) . We have, of course, already used the notion of equivalence extensively - in particular, in w map-germ
f
that of
f .
we said heuristically that a versal parametrized unfolding of a
'contains' representatives of all 'nearby' equivalence classes to
Now, as we pointed out in w to a parametrized unfolding. w
any unfolding of
f
gives rise (non-uniquely)
In particular, then, a stable (equlvalentlyp from
versal) unfolding does so, but of course the parametrlzed unfolding so obtained
need not be versal in the category of parametrized unfoldings.
@5
However, what is true is that such a parametrized unfolding 'contains' representatives of all 'nearby' contact-equivalence classes to that of
f
- more
precisely, we should set up a category of parametrized unfoldings and Vcontactmorphisms' and show that in this category an unfolding is versal if and only if it is stable as a map-germ. That this is true seems as non-obvious as the equivalence of stability and versality for (non-parametrized) unfoldings, and it is equally difficult to offer any suggestion as to why it should be true other than a detailed proof of the type of those given in w 'contact-versality'
We shall not give such a proof here, since the notion of in this form is not very useful for the purposes of this book.
Instead, we shall obtain, in w parametrized unfolding of
f
a different version of the 'fact' that a
which is stable as a map-germ 'contains w all 'nearby'
contact-equivalence classes to that of
f , via a transversality theorem.
Instead, let us use the discussion above as motivation:
the close
relationship outlined here between equivalence and contact-equivalence in the presence of stability suggests the following : Theorem
(4.3)
Let where
FO,F I
fO fl The
be stable unfoldings of the same dimension of germs
fO,fl ,
are contact-equivalent.
FO,F I
are equivalent.
which has the important consequence~ Corollary (4.4) If
fO,fl
are contact-equivalent stable map-germs, then
fO fl
are
equivalent. Proof fO,fl
are zero-dimensional stable unfoldings of themselves; so the result
follows at once from (4.3). In order to prove the theorem, we will need to relate contact-equivalence to the algebra of vector fields which we have been using to study unfoldings.
The
96
connecting link is provided by " Lemm~
(4.5) f,f~ : (N~x-O) § (P,y0)
Map-germs
are ~-equivalent
if and only if
f*mp.C N = f'*mp.C N . Proof '0nly if' : (IN,H)
Suppose
such that Now
f,fv
are~-equivalent,
so that there exists a~-equivalence
(l,f') ~ IN - Ho (l,f) o
H(x,y 0) = (x,Y0)
for all
x e (N,x0)
(from the diagram of (4.2) a)),
H*(p~mp, ~ x e ) c p~mp~ CNx P .
so that But
H
is a diffeomorphism,
H*(p~mp, CNx P) = p~mp, % x p
so
H*
is an isomorphism,
and thus
"
So f'*mp.C N - (1 ,f ' )* (p~mp . CNxP) .CN = (1 , f ) * (H*p~mp . CNxP) ) .CN = (l,f)*(p~, Vlf':
Suppose Let
f*mp.C N - f V * m p . ~
Xlp...,x n
N
= f*mp.C N .
,
be a system of local co-ordinates at
system of I bcal co-ordinates at
(I <- i,j < p)
CN•
(P,yo) .
(N,x0) ,
Then there exist
yl~...,y P
a
vij 'Wij ~
such t h a t
Yi ~ f " .!lVlj ' (Yj ~ f') 3 Yi~ f'=
P ~lWij,(yj o f)
J For each w
x
=
x E (N,x0) , we define real
[w..(x)]
i3
Let (such
C
C
that if
V x = [vij (x)] ,
.
be a real (pxp)-matrix
exists:
Ker a n Ker b = ~
line = c(a(Ker b)) AoW u = 0 , then
such that
C(I-VxoWx0)
for, in general, given endomorphisms
then there exists an endomorphism only if)
(pxp)-matrices
c
such that
- just choose
and such that
.(I-VxoW) s 4 0
c
ca § b
a,b
+ Wx0
is invertible
of a vector space
is invertible if (and
to be any endomorphism of
Imc % ~nb = V .
V
such
In our particular case,
= a , so the condition of non-intersecting
V,
97
kernels is satisfied). P Now let
w'j ffik~l cik(6kj - s
Then the matrix near
x0
P
W' ffi [w'. (x)] x ij
vk~w~j) + wij r CN " is invertible for
9
P y:• o f, ffi ~ w~.(y, o f) ; and so jffil 1j j
Also,
i
is such that
Corollary
(~=I, .... n)
P ]=i w [ [j (xl' ., .,Xn).y j
yi o 11
defined by
(IN,H) ~
xi 0 H = xi
(i ffii ..... p)
(l,f') ffiHo (l,f) .
(4.6)
Map-germs only if
x - x 0 , and hence for
fi : (Ni'xi) § (Pi'Yi)
dim PO = dim PI
(i=O,l)
are contact-equivalent if and
and there exists an isomorphism
CNo § CNI
carrying
f =eo.CNo to f I.CNI . Proof
Of course the condition well assume
dim PO = dim PI
PO = PI (=e' say), since if
diffeomorphism, then 'Only if' :
is trivial; and indeed we may as
k : (Po,Yo) § (Pl,Yl)
is a germ of
k*mPl = n~O . Let
(h,H)
be a contact-equivalence such that
(l,f I) = h = Ho (l,f O) .
Then
(l,f I = h) ffi ( ( h ' l •
fl ~ h'fo
are~-equlvalant.
isomorphism
CN o
'If' :
Let
->
CN I
Suppnse
a I ..... a n
co-ordinates at
Thus, by (4.5),
(h-l) *
(l,fo) , so that provides the required
. ~ : CNO § C N L
is an isomorphism carrying
be local co-ordinates at
(Nl,X I) .
(N0,x0) , blp...~b n
Then we may write
~-l(b i) =
n ~ h..a. j~! ~J J
( i = I ..... n) .
fo*mp.CNo
local
to
98
Since
~
(and #-I)
are isomorphlsms, the matrix
invertible for
b = x0
and hence for
diffeomorphism
h : (No,Xo) + (Nl,Xl)
b
near
H b = [hij (b)]
is
x 0 , and so we may define a
by
n
b i ~ h = j~lhij.a.j Then
~-I = h* , so that
Thus by (4.5) So
fo,fl
fl 9 h,fo
(fl
(i = l,...,n)
9 h)*mpCNo
= f~mp.CNo ,
are~-equivalent.
are indeed contact-equivalent,
This leads to the geometrical result| (4.7)
Lemma Let
(Fk,ik,Jk)
(k= 0,I)
be unfoldings of the same dimension of map-germs
fk " Then
Fo,F I
are contact-equivalent
germs if and only if
fo,fl
are,
Proof First note that, via composing with a germ of diffeomorphism
' ' + (P{'Jl (~l),y~) , we may assu~e k : (P~,Jo(P#ty 6) (= P' say),
Po
m
~i (= F say), r~ = F 1 !
=
Jo = Jl (= j say).
Now let us observe that, in general, if
(F,i,j)
is an unfolding of
f ,
then
i*(F*~p,.%,) = (i*F*~,).%
(since i*%, =
= (f*j*mp,).C N = f*mp.C N
(since
%)
j*mp, = mp) .
Now for the proof : 'If' :
Let
SO certainly Let
h
(h,H)
be a contact-equivalence
(fl o h)*mp.CNo
=
(l,fl) o h = Ho (l,fo) ;
f~mp.CN 1
be a germ of diffeomorphism
o iO = i I o h .
such that
(NO',x0') § (Nl,Xl); '
such that
Then
i~E (FlO ~)*mp, .%~I = (fl o h)*mp.CNo
But Ker i~ c F ~ , .C~6
(for
m
f~mp.CNo
=
i~[F~mp; .CN~3 .
Ker i~ ~ F~(Ker j~)), and
so
99
h)*mp,.CN~= F~mp,.CN~
(FI" Thus
Fo,F 1 o h
'Only if' :
are~-equivalentp
Suppose
isomorphism
Fo,F I
: CNo § CNI
and so
.
Fo,F 1
are c o n t a c t - e q u i v a l e n t .
are contact-equivalent; so that there is an carrying
F~mp,.CN~
onto
F~mp,.eNl .
Let us first prove the result in the case where rank
(dfo)xo (= rank
(dfl)Xl) = O . In this case, Let
Ul,...,u k
9
2
Ker 15 - mN~
=
2
F~mp~,CN~ ~ mN~ ,
be local co-ordinates in N~
such that
Zm i 0 =
and such that they extend to a system of local co-ordinates at
w
{u l = . . .
=Uk=O}
u I ..... Uk,X l .... ,xn
,
(No,xO) 9 We may write (non-uniquely) 9
#(u i) = v i + a i , where
2
V i E Ker i~ , a i s F~mp,.CN~ n mNl. Then define a homomorphism
~v : CN~ + CN~
2'(ui
(i = I, ....k)
= vi
'(~j)
by
~(xj)
(j = i ..... n)
.
is the same as for
This is an isomorphism (for the induced map
0 "'0 "'I "'I into, hence onto, F~m~.CN! (for
) and it carries F~mp,.eN~ F~mp,CN{ ).
Moreover, it carries
Ker i~
onto
Ker i~ ; and thus induces an
isomorphism i~ (F~mp ,CN~) -+ i~ (F~mp,CNI) i.e. an isomorphism Thus
fo,fl
f~mp.CNo
f~mp.CNl
are contact-equivalent.
Now let us consider the case where rank let
(Po,Yo)
(dfo)xo
be a submanlfold-germ of
such that
fo
70 : (No,Xo) + (Po,Yo)
is transverse to
O
at
xO ; also,
~0 : NO + NO'30 : PO § P
PO "
he the restriction of N
rank
(P,yo)
(dfo)xo (= rank
(df l)xl) > 0 :
of codimension equal to rank Let fo "
NO = fol(Po) ; and let Clearly, then,
TO
is of
N
(fo,io,Jo)
is an unfolding of 70
are the inclusions), as also is
(where (Fo,io o io,Jo o ~O ) .
100
Similarly, we can construct fl
and
fl : (Nl'Xl) § (PI'Yl)
which is unfolded by
FI .
Then, since analysis,
Fo,F I
are contact-equivalent, so are
Hence, by the Vlf" part of the lenmm,
fo,fl , by the foregoing are contact-equlvalent.
fo,fl
Now, to connect with our algebra of vector fields, we have:
(4.s)
L~,,ma Let
(h,H)
be a~-equivalence between map-germs
e(h,H ) : 8f -~ 8f~ dHo (O,~) o h -I Then
f,f~ ; and let
be the CN-mOdule isomorphism defined by (9 ~ 8f)
e(h,H )
(where we identify
(O,e(h,H)~) =
8(l,f ) - 8N 9 8f) .
induces an ~ - v e c t o r space isomorphism E(h,H ) : ~ f §
(where ~/f = 0fTtf(S N) + f*mp.0f ) . Proof We may write H' = (h- l • r
e(h,H )
as the composite
e(h,h•
) o e(IN,H,) , where
o H , and so it is sufficient to show that each of
~(h,hx~)
is well-defined. First, then, the case We have Thus
H = h • Ip .
f' = f o h "I , e(h,h• ) _
e(h,hXlp)(f*mp,Sf) = (h
Also, if
= wh "I . )*f*mp.Sf, = fV*mp. Sf, .
~ c ON , then wh'l(tf(~)) = wh " I o T f ' o dho ~ = Tf' o dho ~ o h -I = tf'(Tho ~ o h "l) .
Thus
e(h,hXlp)(tf(SN)) = tf'(SN) . So
E (h,h•
is indeed well-deflned.
Now let us consider the case By (4.5),
h = 1N .
f*mp.C N = f'*mp.C N , so that, since
restriction of the homomorphlsm
dH : 8(l,f ) § S(l,f,)
e(1,H ) (f*mp.ef) = f'*mp.ef,
e(1,H) over
is just a H* , we have
'
101
Also, if
~ 9 8N , then
(O,e(l,H)(tf(6))
= dH(O,Tf o 6) = THo T(l,f)o 6 - THo (6,0).0 (l,f) = T(l,f')
Now
o 6 - T}{o (6,0 ) o (l,f)
THo (6,0) o (l,f) = (~,~) , where
H(x,y O) = (X,Yo)
for all
Thus we have
.
~ 9 f'*mp.Sf.,
, because
x ( (N,xO) .
e(l,H)(tf(6))
= tf'(~) + ~ , and so we have shown
e(l,H)(tf(SN) ) c tf'(SN) + f'*mp. Sf . Thus, also,
e(l,H)
is well-defined,
We can now give Proof of Theorem (4.3) First of all, let us observe that it is sufficient to consider the case rk(df O
~
germs
ffi rk(df
) = O (for as in Le.~na (4.7), fO,fl are unfoldings of rank I f~O,~l which, by that le~mm are contact-equivalent since fO fl are~
and of course
FO,F I
are stable unfoldings of
~O,~I ),
Next, let us remark that it is sufficient to consider the case when are~-equivalent, all
y 9
(for if
with the~-equivalence
between them such that
fO,fl
H(xo~Y ) = y
for
(P,yo)
(l,f I) oh = Ho (l,f O) , then
(I,
(HxO).l
x o fl o h) = (h-l,h -I• (H O)'l)oHo(l,f) xO
(where
H xO : (Po,Yo) -> (Pl,Yl)
replace
fl
is defined by
by the equivalent germ
and also replace
FI
(y) = H(xo,Y))
, so that we can
(HXo) -I o fl o h , which is ~-equivalent
to
fO ,
~-I k o F I = h , where
by an equivalent germ
: (No,Xo) ' I) ' , k : (Po,Yo) ' ' § (Pl,Yl) ' ' § (Nl,X ' ,
that
H
are germs of diffeomorphism such
ho i0 = i I o h , k o J0 = Jl = HxO ) " Let us also observe that, by Theorem (3.1), it will be enough to show that
any stable unfolding of
f0
(of the relevant dimension,
any stable unfolding of
fl
of dimension
Now, in general, if (pF,f)-l~f (For
q 9 8F
(F,i,j)
k
say) is equivalent to
k .
is any unfolding of a map-germ
f , we have
= Im TFx~ is such that
qy6 9 Im TFx~$r--~there exists
6 9 8N,
such that
102
wF(q) - tF(~) 9 mN,.OF~=~DF(qy~)
9 mN,~F
~===>pF,f(ny~) = (qF,f)-ipF(DYo) 9 Also, we have d f Sop since
F
= mN~ f + [wf(Te)] , and
is a stable unfolding
surjective (by (2.6)), it follows that of
F
of
mN.)~F
).
[wf(TP)] = pF,f(Im(Tjyo) ) .
f
if and only if
PF,f
is
is stable if and only if the restriction
PF,f PF,f ; Im TFx~ + m N f
is surjective. Now let us consider our ~equivalent 0
at
fO,fl , which are of rank
map-germs
x0 . Let
U
be a neighbourhood of
be a map-germ
s.t.
gO(x,O) = fO(x)
0
in
~k
, and let gO ; (NxU,x O x O ) +
and such that
{[~.o gO~}
span
(P,yo)
m N ~f
1
(Such fO
gO
exists by (1.9); it is possible to span
does have a stable unfolding of dimension
Then the germ
F 0 : (NxU,x0xO)
is a stable unfolding of Now let (where
fO
-> (PxU,YoXO)
~ 'g i e(l,H)(-~-" gO) = ~-~-. 1
F I : (NxU,xoXO)
§ (PxU,YoXO)
stable unfolding of germs.
fO
fl .
k
elements~ since
(and hence rank
k
x 0f
at
).
FO(x,u) = (gO (x,u) ,u)
(9
~
o
PFo,fo ~Uily xO ) = -[~--~i" g ] )"
be defined by
is the C-equivalence between
gl(x,O) = fl(x) , and
with
defined by
(for, as in (2.8),
gl : (NXU,x OxO) + (P,yo)
(IN,H)
k
~f
and
(x,g (x,u))
H(x,g (x,u))
fl ) ; then clearly Thus, if we define
1
by
FI(x,u) = (gl(x,u),u)
Thus it remains to show that
, then
FO,F I
FI
is a
are equivalent
This we will do by showing that they are part of an (~,I)-family of
stable germs trivial over Let
{Xl, .... x n}
{Yl .... 'YP} Define
(IR,I) .
be a system of local co-ordinates at
be a system of local co-ordinates at F a : (NxU,xoxO)
-> (PxU,Yo•
(P,yo) . (a E (1%,1))
I Yi ~ Fa = (l-a).Yi o F 0 + a.y i o F I u i o F a = u.l
(N,xO) , and let
by
103
so
that
Fa
is an unfolding of Yi
Indeed
9
fa : (N,Xo) § (p,yo)
=
fa
a fO
(l-a) "Yi
(l,f a) = H ao (l,f O) , where
defined by ~
+ a.Yi
Ha : ( N x P , x o X Y o )
fl
'
+ (Nxp,x OxyO)
is defined
by X. oHa = X. 1 1 Yi~ H a = (l-a)y io INx P + ay i , H so
that
(IN,Ha)
Of course
is a~-equivalenee. Fa
is of the form
Fa(x,u) = (ga(x,u),u) , where
a
0
yiog and clearly we have
= (l-a)Yiog
i
+ a.Yiog
@ B a e(l,H a)(8-~." gO) = 8~. " g 1
1
Hence
PFa,fa(.~_il~
YoXO
) = _[ Bu.' 3..~_ ga] = _[e(1,Ha )(~u3. . gO)] z
z
= E(l,Ha)~FO,fO(8~ilYoX O) " Thus
~Fa,f a
Now let
is surjective, so that
G : (NxU•
G(x,u,a) = (Fa(x,u),a) .
OxOxl)
Fa
§ (PxU•
We aim to show that
is, that there exist retractions
r,s
is a stable unfolding of
to
•215 G
fa .
be defined by
is trivial over
iNx U • O , Ipx U x O
(]l,l) ; that
respectively such
that the diagram F0 (NxU,xoXO)
--
~ (PxU,YoXO)
rl
Ts G
(NxUx ~ , x 0 x
~ ~ (PxUx~,Yo
xOxl)
F,
(~ ,I) commutes. germs
rI
If we show this, then of
(N•
OxO)
, sI
sl(y,u) = s(y,u,l) , so that
F I = s I o F O o (rl) -I , where the diffeomorphismof
FO,F I
(PxU,y O•
are given by
will indeed be eqdivalent.
As in (3.2), to construct the desired retractions to
find, for each
a E (~ ,I)
'
rl(x,u) =r(x,u,l),
elements
r,s
it will he sufficient
~a' ~ .* ,n 8" * " ~NxlT'NxU rNxU , ~'a s PpxUmNxU 8pp• U
104 such that
~a = (~" ~ = ~ ,
(Here
G a)
Ga : (N•215
restriction of
is a lift for
O•215
over
~a = (qa'~u , 2 =~
§ (P•215
•215
Ca
is the germ-
G ; and we have identified
eNxUxB=
e
9 e
PNxU
Ga
~o
,
ePxUx~=
e
PpxU
e ev
),
To do this, we proceed as follows: Ca = tGa(0'~" ~R~ Ga) " WGa(0'~~ ~
let
) "
P
Then
~a = i~IP~• Now
Fa
Ha x l U x l U )
~ FI - Yi ~ F0)'wGa(~-~-o~yiPP•
is ~-equivalent to
F0
O) .
(the~-equivalence being (essentially)
, so that = F0*mpxu.CNxu
Fa*mp• Thus
0
Ga*P~Ump•
= P ~ x U F *mpxu'CN•215
Yi o F I s FI*-
Now
C
~PxU" NxU
= F 0.-
C
mpxu" NxU "
So
* u(Yi ~ F I - Yi ~ FO) ~ P~ PN~
UFO*~PxU" CNxUx~ = Ga*P~ x l~p•215215
and thus Ca ~ Ga*P~ x umpxu'SGa,~ ' Now, by (2.3), we have eGa,~ ~
= tGa(eNxUx~,~
oG a) + WGa(epxuxl~, ~
)
so that Ga*P~,~ ump•
m c tGa(P~ •215
~R~ Ga ) + WGa(P~ x Dmpxu'Sp•
W]R )
Hence, via the natural identifications OPNxu we can f i n d
~1~ ~ Ga
PP•
~av E p ~ x UmNxU. ePNx U , qa, ~ p~•
such that So, since
ONxU•
U
Ca = tGa(~"O) - WGa(na,O) . Ca = tGa(O'~u~ w~OGa) - wG a ( O , ~ o ~R) , we have
wGaCnl,~o~> ~a = (~a'~u ~ ~ )
so that over
a ' !~u ~a ~ ~t~' o
~
o
%)
tG a(~a,~o w~1~
is indeed a lift ~or
G a , as required, and the proof is complete.
=
105
w
Determinacy
We now turn our attention to the question of whether a map germ is equivalent (or contact-equivalent) to a polynomial. (5.1)
Definition Map-germs
f,f! : (N,Xo) ~ (P,yo)
are r-jet equivalent if
r+l
(f* - f * ' l m p c m N
This is clearly an equivalence relation; we call the equivalence class of j rf .
A more usual definition of r-jet, in terms of partial derivatives, is equivalent to that given above, as is shown in the following: (5.2)
Le~mm Let Then
f,f' : (N,xO) + (P,y0) jrf ffi jrf,
be map-germs.
if and only if the partial derivatives of
f
and
f'
(with
respect to some, and hence any, systems of local co-ordinates) agree for all orders _< r .
To prove this, we will require, first
(5.3)
Sublen~a Let
~ E CN,xo .
Then
~ ~ ~+I
if and only if all partial derivatives of
# , with respect to any system of local co-ordinates for
(N,xO) , of orders
~ r
vanish. Proof Let If
Xl,...,x n
be a system of local co-ordinates for
(N,xO) ,
r = 0 , the result holds (by definition!).
Otherwise let us assume inductively that the result holds for jets of order < r ~ If
r+l # ~ mN , then
generate
m N , by (1.2)).
I ~i'xi , with i=l
~i E ~
(since
Xl, .... x n
By our inductive hypothesis, then, the partial
#i
vanish for all orders
xi.#i , and so
, vanish for orders
derivatives of of
n
=
-< r - I -< r
, so that the partial derivatives (differentiation of products').
106 Conversely, let
if the partial derivatives of
be the function defined by
~i
~i (x) =
~ =
~(0,... ,Xi+l,... ,x ) n
~ ~i.xi . i=l
r ~i e m N ;
~i
Jl ~~x (0,.
"''txi'xi+l"' "
Now
_< r ,
,,Xn).dt
"
xi.~i(x) = ~(0, .... x i .... ,Xn) -
so that, since the partial derivatices of
vanish, so do those of derivatives of
vanish for all orders
0
n
Then (as in (1.2))
#
xi.~i(x)
.
~
for orders
Then it is easy to see that the partial
must vanish for orders
_< r - I ; so by our inductive hypothesis
r+l ~ E mN
whence
So the inductive step, and hence the proof, is complete.
Now we have Proof of (5.2) All partial derivatives of same is true for
vo f,vo f'
f,f'
for any
of orders
~ r
agree if and only if the
v E mp ; which, by (5.1), is true if and
r+l vo f - v o f, = (f* - f,*)(v) ~ m N
only if
for any
v ~ mp .
We can derive further consequences of jet-equivalence as follows: yl,...,y P
8f
and
(5.4)
(P, yo) .
be a system of local co-ordlnates at
wf(~),...,wf(~y~) 8fv
is a free basis for
8f
let
Then
as a CN-mOdule.
We will identify
wf(~y~) ~-+ wf'(~y~.) .
in the following l~r,~ by
Lemma Let
f,fv : (N,Xo) + (p,yo)
be such that
jrf = jrf, .
Then
a)
tf'(ON) c tf(SN) + ~ S f
f'*mpOf,
b)
r§
c (f*mp + m N
)Of .
Proof
a)
s
x I,.. 9 ,xn
Then, since
be a system of local co-ordinates at
8Xl"'''~-~n
(N,x O) .
is a free basis for the CN-mOdule
n
write
Thus
= i[=l i'~x i tf(
(~i E C N) )
-
n1 % t i-i
< r
for any
f(~
~ ( ON 9
" I 1
1,3
~x~(f* (yj)).wf (~y~) i
8 N , we can
107
and so Since
tf'(~) = tf(~) + z,j'~'~i~x~-~.(fV*(yj ) I f,f'
have the same r-jet, all partial derivatives of
orders
~ r
orders
s r-I
vanish, and so all partial derivatives of
So, by ( 5 . 1 ) , b)
Since
- f*(yj)).wf(~y~.) . fV*(yj) - f*(yj)
~x~-~.(f'*(yj) - f*(yj)) 1
of of
vanish,
~--~-(f*(yj) 1 f,f'
- f'*(yj))
E ml~ , and thus
tf'(~)
- tf(~)
~ m~T8f .
r+l f'*mp c f*mp + m N , and the result
have the same r-jet,
follows at once.
45.5)
Definition A map-germ
f : (N,Xo) § (P,yo)
isF~-determined by its r-~et (respectively
determined by its r-jet) if every map-germ jrf = jrf,
f' : (N,Xo) + (P,yo)
is~Mu-equlvalent (respectively equivalent) to
Thus, if
f
such that
f .
is determined by its r-jet, it is equivalent to a rather
particular polynomial - its Taylor series (with respect to any local co-ordinates) truncated after order
r .
However, determinacy is rather hard to calculate; but
for stable map-germs can clearly be arrived at via~i~-determinacy (using (4.4)), for which we now give a sufficient condition: (5.6)
Theorem Let
f : (N,xO) § (P,yo)
be a map-germ such that
~ Sf r Then
f
tf(6 N) + f*mpSf .
is~-determined by its (r+l)-jet.
Proof Let
fv : (N,xO) § (P,yo)
Let
(yl,...,yp)
define
be a map-germ such that
be a system of local co-ordinates at
fa : (N,Xo) + (p,yo)
y : (N•
g : (Nx~,XoXl) Xo•
+ fN•
(P,yo) .
by
Yi~ fa = (l-a).y io f + a,y io f' Then define
jr+If = jr+If,
+ (P,yo) Xo•215
by by
(i=l .... ,p)
g(x,a) = fa(x) , and y(x,a) = (x,g(x,a),a) .
For
a ~ ~,
108
We aim to find retractions
the
r,$
1NX O,INx Px
to
0
respectively such that
following diagram commutes:
(N,xo)
(1,f)
,l
(Nx~,XoXI)..~
Y
,
(Nx P'Xo Y O ) sx ]
and
Let
(rl,s I)
between
ga : (N•215
"
O) , defined by
f
and
gp7
(for
rl(x) = r(x,l)
sl(xpy) = S(x,y,l) , give a
fv , and the proof will be complete,
"~ (P'Yo) 'Ta ~ ( N • 2 1 5
be the germ-restrictions of
(~ixO)
) (Nx~,XoXT)
r I ; (Npx O) § (NpXo) , defined by
sl : ( N x P , X o X Y O) § ( N x P , x o •
~-equivalence
PN• ....
) (Nxpx~,XoXYoXl)
'~ (1% 1) If we can do so, then
~
-> ( N • 2 1 5 2 1 5
a ~ I ) .
We shall identify, in the natural way, 0
-O
Ya
@O
PN
8NxPx]~.,X o X Y o X a
@0
ga
=
~
OpN @ OPP 8 0~]R
ONx~, XoXa - Op~l @ Or~ (where
pN,p P
(N,Xo),(P,y O)
(N •
are the natural projections of respectively, and
p~
(N x p x R, x 0 x YO x a)
to
is the natural projection
XoXa) § (N,Xo)). By similar arguments to those of (3.2), for the construction of
possible it will be sufficient to find, for each ' ~ mNmpSpe =
PN•
such that over
~a
to he
a r I , elements
(E~,~8~o ~ )
is a lift for
Ya
With respect to the system of local co-ordinates P
(yl,...,yp)
+a = i=ll (#a).Wgla (~-~'~i'--)oy (~a)i c CNxR, XOXa and it is easy to calculate that
r,s
, we can write
109
(@a) i = Y i d fv . Y i ~
r+2 c Nx~t
~ mN
~a E _r+2 mN 8ga "
so that Now We claim that
~ mNrega ~ tga(SNx~' %_.
+ g*mp'Sga .
To see this, we argue as follows : from the hypothesis of the theorem, we have ~ S f c tf(SN) + f*mp.Sf . Since
jr+ira = jr+If ,
(*)
it follows, by applying (5.4), that mNr8 fa c tfa(SN ) + (fa*mp + mNr+l)Sfa
Now recall (from (2.2)) that there are natural isomorphisms 8ga/m~ , aSg a ~ 8fa (ga gives rise in the obvious way to an unfolding identify
8ga with
8F
w ' where
~
Fa
of
is the projection
(px~,YoXa)
a,
0NxIR' ~ / m R ,
aeNxi~., ~ . ~
fa , and we can § (~,a))
ON .
So from (*) we can deduce mNr 8ga c tga(SNx~, w~) + g*amp.Sg a +ranr+l 8ga ' Thus it follows from (I.i0) that the CN-mOdule (via { ~ 8 ga + tga(eN•
' w~) + g*~.8 ga }/{tga(eNx~, ,~) + ga*mp.Sga} = O '
and the claim follows. It certainly follows, then, that
~a ~ tga(mNONxP., ~ R ) + mN~ so that there exist such that
~'a ~ mNSNx~, ~ , , n"a ~ mN'g*mp'eg a ~a = tga(~av) + ~'' " a
We will, as usual, identify
~.'~ mNeNxm, ~,~
a
8Nx~, ~
with (~',o> ~ ep~ |
%~.
with
8p~ , and so identify
110 7a
Now
qa' o Ya = q''a "
such that
na
1lit for
is an immersion, so, since
(Ea
=
ga = PP o Ya ' there exists
It remains to show, then, that
, 8 ~ P N x R 'qa'~u = wl~ ) over
n~ c mNmp8pp
~a = (~a'~u ~
7a :
We have TU~
=
~ PNx
tga( O
'
= ( C o PN~m'
tga(C)
~a' ~ ~
+
= (~a" PNxl~' qa '~u
~m )
~m ) = na =7a
and so the proof is complete,
(5.7)
Corollary If
f
is of
FST , then
f
is ~-determined by its
(Xf + l)~jet.
Proof By hypothesis,
~ Sf
Xf = dim~{ef/tf(SN)
+ f*mp. Sf} = r < ~ .
c tf(ON) + f*mp.8f , and so, by (5.6),
(5.8)
f
Thus, by (i.12),
is~-determined by
jr+If .
Corollary If
f
is determined by its
(p + 1)-jet
is stable, then
f
is stable, then
pf : TPy 0 + 8f/tf(eN) + f*mp,0f
(where
p = dim. P ). Proof If that
f
dim~{Sf/tf(SN) Now let
f'
Thus, by (1.12),
+ f*mp. Sf} ~ p .
jp+If, = jp+If .
be such that
is surjective, so
~Sf
c tf(8N) +f*mp.8f .
Then, by (5.4), we can conclude
that tf'(SN) + f'*mp.ef, ~ tf(O N) + f*mp.8f so that A/f,~, But then
can be identified. pf,pf,
can be identified (since these are induced respectively by
wf,wf' , and our identifications are given by
wf(~--~.) +-+ wf'(~-~) ), so that vJ i.
is also surjective, and hence
f'
But, by (5.6), it follows that (4.4),
f'
~
is stable.
is actually equivalent to
f'
is~-equivalent to
f .
f , so that, by
P f,
111
w
Jet-spaces and a transver~ali~y
theorem
In this section we will obtain the 'contact-transversality'
theorem
advertised in w First of all, however, we will require some material on jet-spaces. (6.1) germs
The set
jr(N,P)
(for any integer
(N,x) + (P,y) , for any
x E N , y ~ P.
There is an obvious projection function assigning for
(x,y)
r ~ I) consists of all r-jets of
pr,O : jr(N,p ) § N • P
to any map-germ
(N,x) ~ (P,y) .
induced by the We write
jr(N,P)xo
(pr'O)-l{xo} x {pO} , and
jr(N,P) x PY (or jr) for (pr,O)'l{xo} x {yo} . 0 There are also projections pr,S : J~(N,P) ~ jS(N,P) (IN s N r ) given by
pr,S(jrf) = jsf . jr(N,P) pr'S(o ~ s ~ r)
is topologized as a smooth manifold~
in such a way that the
are projections of smooth fibre-bundles,
by giving the following
local charts: if
(Xl,... ,Xn)
neighbourhood
U
of
is a system of local co-ordinates defined on a xO
in
N ~ and if
co-ordinates defined on a neighhourhood where
i=l,...,n
, j m l , .... p
I -< c I + ... + on ~ r 0 E jr(N,P)xo,Y 0
and
(where
(pr'O)-iu x V , where, if f : ( N, x) §
and
(YI'''" 'YP) V
of
YO
k--l,...,p,o
is a system of local
in P , then = (al,...,a n)
(Xi,Y j ,Zk, O) , with
o i >- 0 , is a system of local co-ordinates at 0
is the r-jet of the constant germ) defined on z E (pr'O)-Iu • V
is the r-jet of a germ
(P,y) , then Xi(z) - xi(x) Y'3(z) = yj(y) 01§ ,, '+~ Zk,o (z) "
k o f)
Ol ~x I
on 9 9 .~x
n
(That these functions are well-defined follows from (5.21 ; that they form a system of local co-ordinates becomes clear by considering germs of polynomial mappings).
112
(6.2)
In fact~ we can be rather more exact about the fibre-bundle structure
involved: let us write
jr(n,p)
the group of r-jets at
0
jr (n,p)
jr(~n
Lr(n) c jr(n,n)
~P)o,o ' and let
(~n ,0)
of germs of diffeomorphlsm of
multiplication defined by on
for
jrh.jrh' = jr(h - h v) ).
(with
L r(n) x L r(p)
Then
be
acts
..rh ,3.rk.) = Jr( k ~ f o h -1) . jrf.(3
by
(To see that this action, and the multiplications in
Lr(n),Lr(p)
are well-defined,
we observe the general fact that the r-jet of a composite depends only on the r-jets of its constituents; for suppose that
f~f~ : (X,Xo) § (Y,yo)
same r-jet, and that
also have the same r-jet.
g,g' : (Y,yo) + (Z,zO)
(gof)*
SO that, since
- (g, of,)* = f*o (g*-g'*)
(g*-g'*)C z c ~ + I
r+l c mx , so that
g ~ f , g' o fv
+ (f*-f'*)
, (f*- f'*)Cy c ~ + I
, (
(go f),. (g, = f,),Cz
have the same r-jet). jr~N,P)x0pY 0
jr(n,p) ; clearly different choices correspond to the action of
Lr(n) • Lr(p)
on
jr(n,p) , and so
fibre bundle with fibre
(6.3)
Then
og'*
Now a choice of local co-ordinates provides an isomorphism of with
have the
If
jr(n,p)
and structure group
f : (N,xO) + (P,yo)
jrf : (N,Xo) + (jr(N,p), jrf) is a representative of
f
pr,O : jr(N,p ) + N • P
Lr(n) x Lr(p) .
is a map~germ, then a germ
is defined by the assignment
and
may be viewed as a
f
is its germ at
x ,
.i-2. x --+ j fx ' where Clearly
jrf
is a germ
X
of section for the projection
(6.4)
PN o pr,O : jr(N,p ) § N .
L er~ma Let
f : (N,xO) § (P,yo)
be a map-germ, and let
x = jrf
i
There are natural identifications (as ~ - v e c t o r spaces) a)
T(jr(N,P)xo) z ~- 8 flm " Nr+l 8f .
b)
T(jr(N,P)xo,Yo) z ~ ~N -- 8 flmN " r+l 8f .
Proof a)
Let
that
g(x,O) = f(x)
~ ~ 8f , and let
g : (N x ~ ,
and ~ u " g = ~
x 0 x O) + (P,yo)
(as in (1.8)).
be a map-germ such
Then the germ of path
113 u -~ jr(g~ element
(where
gU : (N,Xo) § (p,yo)
~ ~ T(jr(N,P)xo) z
gU(x) = f ( x )
9 -linear;
; which is determined by
+ u ~u 3g (x,O) + O(u 2)
Thus we can define and
r(r
where {yl,...,yp}
is defined by
r
= 0
gU(x) = (x,u)) gives an
r , since
(with respect to a n y local co-ordinates).
: 8f § T(jr(N,P)xo) z if and only if 3.rr
are local co-ordinates in
by
= O
r § ~ . (where
This is clearly P r = [ r i=l
P ) i.e. by (5.3), if and only if
r+l
, ~ m N 0f. So
r
induces an injection in
a); but
N Of T(jr(N,P)xo) z , 8f/m " r+l
the same l~-dimension, so this induced map is surjective also.
have
It gives our
identification.
b) This follows at once, by restricting r to T(jr(N,P)xo,Yo) z c T(jr(N,P)xo) z .
(6.5) a)
Corollary
Let
(yl,...,yp)
he a system of local co-ordinates for
This induces a local trlvialisation of induces a splitting
jr(N,P)xo
over
(P,yo) (P,yo) , and hence
T(jr(N,P)xo) z ~ T(jr(N,P)xo,Yo) z @ TPy 0 , which, with respect
to the identification of (6.4), is given by ~r(sf)- ~r(mNSf) @ Ig .
b)
Let
(Xl,... ,xn)
be a system of local co-ordinates at
This induces a local trivialisation of induces a splitting Let
over
(N,Xo) , and hence
T(jr(N,P)) z = T(jr(N,P)xo) z @ TNxo
f : (N,xO) + (P,yo)
above,
jr(N,P)
(N,Xo) -
be a map-germ.
rctfc ))l '
3
With respect to the splitting
~
,
Proof a)
The splitting via local trivialisation is given by translations in
while the map-germs
g : (N•
-> (P,yo)
g(x,u) = f(x) + (O, .. . ,u,.. . ,O) translations on
f ,
Since
(i=l,. ,. ,p)
~u" g = wf(
.
defined by
give the action of these , the result follows.
P ;
114
b)
This is a straightforward calculation. Let
$i(x,u)
#i : (N • ~ ,x0 • O) § (N,Xo) = x + (O,...,u,...,O)
by the translation flow given by
(i=l,...,n)
.
T(JrOxo ( lxi O) = r(jrf)xo d
.r
d
.r
Then
(xo'u lug)
~u(Jf~(x0,u))[u=0 -I
=~-ju(J (z~
d ..r(f
-I
,u))~(Xo,U))lu=O-Tu u
)
~
,u))xo ]u=O
(by the chain rule) The second term in the last expression is
-~r(tf(- ~x~.)) . Now, with 1 (,u) has the same Taylor series f o ~-I
respect to the chosen local co-ordlnates, at
~(Xo,U )
as
f
has at
x 0 ; so that
jrfx 0 § jr(f o ~(i u))~(Xo,U )
translation with respect to the local trlvialization by
is just
(0,...,u,...,0) .
Thus
indeed, with respect to the local trivialization splitting, T(jrf)xo(~[x0)
(6.6)
= (r(tf(~),~,Xo)
"
Now let us see how the notion of ~requivalence gives rise to a group-
action on
jr .
Let ~
be the group of germs of diffeomorphism of
multiplication given by composition).
/~ acts on germs
(N,xO)
(with
f s (N,Xo) § (P,y0)
by
f.h = f o h "I Let
~
( N x P , x o X Y O)
be the subgroup of the group of germs of diffeomorphism of whose elements
diffeomorphism-germ terminology on w by
H
are of the form
(P,yo) § (P,y0) H
for each
is a ~-equlvalence).
H(x,y) = (X,~x(y))
x ~ (N,xO) .
with
~
~ acts on germs
= ~ , . ~ (semi-direct product).
f : (N,xO) § (P,yo) H o (iN,f o h -1) .
by
f : (N,xO) -~ (P,y0)
Thus, in the terminology of w
-equivalence between ~-equivalence
between
This acts on germs
f.(h.H) = (f.h).H , that is, by
(~,f.(h.H)) =
(h,Ho (hx Ip))
f
and
f. (h.H) ; and similarly if
f
and
f' , then
a
(thus, in the
(1N,f.H) = H o (IN,f) . Now let
Hx
is a
(h,}{) is a
f' = f.(h,Ho (h-lxlp)).
115
Now let I~ r c jr(N,N)x0,Xo multlplicatlon defined by
be the group of r-jets of elements of ~,
jrh.jrh' = jr(h o h') ).
~r
acts on
jr
(with
by
jrf.jr h = jr(f .h) . Also, let ~ r c Jr(N x P,N x P)xoXYo~XoXY 0 of
~
.
~r
acts on
jr
by
be the group of r-jets of elements
jrf.jrH = jr(f.H) .
Finally, let 1<.r =~,r4~ r , which acts on (The multiplications in . ~ r , ~
jr
by
jrf.(jrh.jrH) = jr(f.(h.H)).
, and their actions on
jr , are well-defined
by the argument of (6.2)). It is clear that (with respect to any systems of local co-ordinates for (N,Xo) , (p,y0)) . ~ r , ~ r ~ We shall write Borel [ Bo ~
are algebraic groups acting algebraically on
z~,z~r,z~
for their orbits through
z ~ jr .
these orbits are actually submanifolds,
jr .
By a theorem of
We calculate their tangent
spaces as follows:
(6,7)
Lemma If
f : (N,xO) + (P,yo)
is~:a map-germ, and
r Z = j f , then
.r(tf(mNeN))
a)
T(z~) z =
b)
T(z~r) z = ~r(f*(mp)ef)
c)
T ( z ~ ) z " .r(tf(mNON) + f*(mp)ef)
Proof We use the following: T(xG) x = TaG(TGI ) a)
Since~.~ r
for any
if
G
is a Lie group acting on a manifold
x ( M , where
is an open subset of
jr(N
aG;Gx ~ -> M ,N)xo,X 0
is the mapping ' ~N
Now the following diagram is clearly com~nutative: tf
I
wr
T~ N and so
T(Z~)z
= ~r(tf(mNeN)) .
1,7r - - ~
Tjr z
=
M , then
g § x.g
~r(mNeN) "
116
Any element
b) where
HU ~
of
, H0 = ~ •
defined by
T~-r NxP .
Let
H(x,y,u) = HU(x,y).
PN : (N•
+ (N'Xo)
Thus
may be represented by a path H : (~•215215215
Then
a
~u' H 9 mpSNxp,pN
u § -jr(Hu) - ,
+ (N•215
0)
be
(where
is the natural projection) p and
r a ~ (~u.H) ~. ~ .
T~r~x P = ~r(m~8..x~ ) . r m r,p N
Now the following diagram is commutative: K
mpSN•
PN
f* (mp)Sf
Tc~e
T ~1 N• (where so,
since
c)
K
Clearly
K
is defined by
~
T jr
~ § n o (l,f))
is clearly surjectlve,
T~N.INx P = ~
Thus
Z
@ T~N•
T~iNxP = ~r(f*(mp)Sf) .
; and , since
Z(z~f) z = T ( z ~ ) z + T(zer) z =
~r(tf(mNSN)
+
f*(mp)ef) .
Although the next is rather a side-lssue as far as this book is concerned, it seems worthwhile to include it for completeness: (6.8) If
L~mm, f
is~-determined by its r-jet, then r+l
ms
of = tf(mNe N) § f*(mp)ef .
Proof If let
f : (N,xO) + (P,yo)
isIC-determined by its r-jet, let
E = (pr+l'r)-Ipr+l'r(z) , so
E
z = jr+If , and
is the set of (r+l)-jets of all germs with
117
the same r-jet as TE z c T(z
f 9
Since
But clearly
)z '
f 9 is~-determined TE z = ~r+l(~+lsf)
by
jrf , E r z ~ +I ; so that
, so that by (6.7) c) we have
r+l. r+l^ . r+l. ~mN ~f) c ,,
.
By (i.i0), then, the CN-mOdule {tf(mNSN) + (f*(mp) + ~+l)sf}/{tf(mN%N) r+l m N 8f = if(raN%N) + f*(mp)Sf
is zero; so that
(6.9)
+ f*(mplSf}
, as required.
Corollary A map-germ
if it is of
f : (N,x O) -~ (P,yo)
isKrdetermined
by a finite jet if and only
FST .
Proof 'If v was proved in (5.7). VOnly if' :
if
f
is~-determined
r+l m N 8f c tf(mNSN) + f*(mp)Sf i.e.
f
(6.10)
is of
by its r-jet,
, so that certainly
of diffeomorphisms
all
< | ;
i s a subgroup of t h e pseudogroup ~ ' N , P = ~ N , N ' % x P , N x P =
{h.HJp N source (H) = s o u r c e (h)} , where ~ N , N
the form
dim~{Sf/tf(8 N) +f*(mp)Sf}
FST .
The group ~
diffeomorphism
then by (6.8)
N§ N
(N•
and CN•
i s t h e pseudogroup of p o i n t - g e r m s
i s t h e pseudogroup of germs
x y v) § ( N x P , x V X y '') , for any
H(x.y) = (x,}{x(y)) with
Hx
H of
x ! 9 N , yV y, 9 p , of
a diffeomorphism-germ
(p,yV) + (p,yV,)
x 9 (N,x v) . ~<~N,P
acts on point-germs
f : N -~ P
by
(IN,f. (h,H)) = H o (iN,f 0 h -1) ,
whenever this composition is defined. Also of interest is the subpseudogroup
~txO,P = which acts on germs
{h.H
9
I source (h) = target (h) = x O}
f ; (N,Xo) + P .
for
118
Taking r-jets, we obtain pseudogroup actions ~ rN,P jr(N,P)Xo .
Let
Z ~ , p c jr(N,p )
r under the action of % , p z
jr(N'P)x0
on
jr(N,P) p ~:0~ P
be the equivalence class of
, and let
Z~xo,P
on
z ~ jr(N,P)
be the equivalence class of
under the action of ~ r xo,P "
Choosing local co-ordinates, we see at once that, with respect to the trivialization Z~,pIUXV xO , y
jr(N,P) I U •
= UxVxzX~ r near
= U x V x J r , Z~:o,Pl{xo} x V = {x0} x V x z ~
(for the action of an element of
(~N,P)x,y
, for
x
near
YO ' may always be expressed as the composite of the action of an
element of ~F~ with translations (with respect to the local co-ordinates) x § x0 , YO § y )" respectively.
Thus
Z~:o,p , ~ , e
jr(N,P)xo , Jr (N,P)
are submanifolds of
It follows from (6.5) a) and (6,7) that T( Z~:o,p)z = ~ r (tf(mN% N) + wf(Sp) + (f*mp)Sf) .
At last we can state and prove our transversality theorem: (6.11)
Theorem
Let
f : (N,x O) + (P,yo)
Let
r ~ Xf + 1 .
Let
Y : ( N x U , x O•
where
be a map-germ of
+ (pxU,y OxuO)
g : (N x U,x 0 x Uo) + ( p , y o ) L e t t h e germ
FU(x) = F(x,u), and F
be d e f i n e d by
~u x
F
P(x,u)
i s a map-germ s u c h t h a t
J~F : (N • U,x 0 x u0) § ( j r ( N ~ P ) , j r f )
r .r-~u J,F(x,u) = j Fx , where
Then
FST .
is a representative of is the germ of
is stable if and only if
~u J,F
at
=
(x,g(x,u))
g(X,Uo) = f ( x )
,
.
b e d e f i n e d by
F , ~u
is defined by
x .
is transverse to
(jrf~,p
at
jrf .
Proof It is clear that (N•215 where So, if
a
is
is the composite jr
'forgetting
{Xl,...,xn}
J~F
jr(N•
derivatives
P) _~a j r ( N , p ) • in the U-directions'
is a system of local co-ordinates at
is a system of local co-ordinates at
b~ and
jr(N,p ) b
is projection.
(N,x0) ,
{Ul, ....uk}
(U,Uo) , then it follows easily from our
119 calculation (6,5) b) (applied to
g ) that, with repsect to the splittings
induced by the co-ordinates,
so that Im T(j,rF)xoXU O Thu s
jrF
=
I~< ~r(tf(
is transverse to
z~
~
r. a ,
i" )'~ ~8--u~.i g)> $ TNxo if and only if
,P
~<~r(tf( ~x~_~. .) ) , ~ r (~--~., a , g)> + T(z ~Cxro,p) = T (jr (N,P)xo) z 1 i But, by our formula for
T (Z~o,P1 z , and since
Kerr
= m Nr+l 8f , this is
equivalent to r+l . ' g> + (f*(mp) + m N )ef , 1
8f = tf(SN) + wf(Sp) + R < which in turn is equivalent to
ef = tf(ON) + wf(8p) + ~ < ~-~-. au. g> + f* (mp)ef
,
1 r+l
since
mN
8f c tf(eN) + f*(mp)ef
from our hypothesis and (I.12).
But the last equation i s e q u i v a l e n t to the s u r j e c t i v i t y of PF,f : T ( P x U ) since, if
yl,...,y P
pF,f(~-~i
§ ef/tf(eN) + f*(mp)ef
YoXUo
is a system of local co-ordinates at
ffi [wf(
. ]
By (2.61, this is equivalent to
and F
pF,f(
(P,yo) ,
.lUo) = [- ~-~j, g] ,
being stable, and so the theorem is
proved. (6.121
Remark Thus
F
is stable if and only if
J:F
acts as a 'slice' to t h e ~ , p -
equivalence classes; so we expect all 'nearby' classes to he represented by for some
(x,u) ~ N x U
'near'
~ux
(Xo,Uo) .
We take this as justification for our remarks about 'contact-versality ~ in w
120
w 7.
Genericit7
In this section we show that map-germs of
FST
precisely, we will show that the map-germs not of codimension,
are ~generlc'
FST
More
form a subset of infinite
in a certain sense, of the space of map-germs
(and this even with
respect to the VobviousV topology - comparing all partial derivatives at the source - which we dismissed in w
(7.1)
Let
z c jr(n,p)
as apparently too coarse,v).
, and let
f : (Rn,o) § (RP ,O)
be a map-germ such that
jrf = z . Define
X z = diml~{ef/tf(BN)
+ (f*mp + ~ ) S f }
To see that this is well-defined,
(where we write
suppose that
jrf = jrfv
.
N 9 An
).
Then, by
(5.4), r r tf'(eN) + (f'*mp + mN)e f -= tf(eN) + (f*mp + mN)e f , so that
di~{ef/tf(eN) + Define
(f*~ + ~)ef} = dim~{ef,/tf'(e N) + (f'* 89 + ~)ef, .
wr(n,p) = {z e jr(n,p) IX z > r} .
We observe the following facts about a)
If
z ~ wr(n,p)
jrf = z , then
f
, then
z
b)
f
f
is such that
ef ~ tf(e N) + (f*~)ef ,
is indeed /<,-determined by its r-jet).
is ~r-invar iant.
(For if
z = jrf , then, by (1.12), r+l + (f*mp + m N )8f) ~ r .
is the codimension of
(that is, if
, then, by (1.12),
wr (n,p)
dlm]R{ef/tf(ON)
:
by its r-jet.
Xz -< r - I r-i
whence, by (5.6),
is ~<.-determining
is ~-determined
This follows since, if
Wr(n,p)
Xz ~ r
if and only if
But, by (6.7) (and(6.4)) this number
zTCr , and so is preserved by I<~r ).
c)
wr(n,p)
is a homogeneous real-algebraic variety in
(For
wr(n,p)
is defined by the condition that a linear map with polynomial
coefficients has corank at least
r ).
jr(n,p).
121
Our
(7.2)
'genericlty'
theorem
is:
Theorem r lim § -
wr(n,p)
codim,
= |
Proof Let us write, Let codim.
for convenience,
r ~ s .
Wr
Then we have,
is non-decreasing
that for each
r
with
there exists
W r = wr(n,p) clearly,
that
It will
r .
. (pr,s)-lwS
therefore
such that
rv 9 r
codim.
D W r , so that
be sufficient W r' > codim.
to show
Wr .
To do this, C (p
of
it will be enough to show that for each irreducible component r ,r r there exists r c > r such that (p c )-Ic - W c + ~ ; for then
Wr
rc,r)_ic
Wr
n
by the above, codimension of
w~'
is a union of irreducible
is of smaller dimension
than
codim.
is of greater
that indeed
then,
C
> codim.
there exists
f : (JRn ,0) §
(~P ,0)
+I xI
In the case
~(r+l)
In the case r+l~.
xi
.
W r , so
Wr . rv > r
such that for any
- W r' .
p _> n
r+l r+l r+l+...+xr+l) ... ,x i +... +Xn_p+i, ... ,Xp n
f*(yi ) = x r+l
of the form
Thus
if
n Pi ~ x i=l I
~(r+l)of
for
i -< n , and so, since
, where
= (f*mp)Of
n [ Pi = n(r + I) i=l ; in particular
8 .
c tf(SN)
+ (f*mp)Sf
(i=l .... ,n ; j=l,,..,p)
it will he enough to show this in the case
if
p < n. -
~(r+l)
is
it follows
Xf < "
p ~ n : we claim that
w-rt%)
By symmetry,
of
be the germ defined by
p >- n ,
c f*mp
component
:
r+l +...+Xnrp§
by polynomials
r W c , each of which,
dim. (prc'r)'Ic , that is, of greater I~RK r' = c r each irreducible component c '
. r+l r+l ~x I ,...,x n ,0,...,0)
f(xl' .... Xn) =
that
if
of
than some irreducible
z' ~ (pr''r)-Iz
I(
generated
than
components
it will clearly be enough to find
We proceed as follows let
Thus,
codimension
w~'
codim.
Finally, z ~ Wr
c
.
j = 1 .
122
Let us write
Y~" = Yi - Yi-i
for
i=2,,.. ,p .
With respect to the
co-ordinates
(yl,yl, ..,,yp) , f has the form , r+l r+l r+l r+l r+l - x~+I ) f(x I ..... x n) = ~xI +.., +Xk+ I ..... Xk+i+ l - x i ,...,x n (where we write
First consider the case Let
q
be such that
k = n - p ).
i ~ i ~ k + i .
(q-l)(k+l)
+ i
, q(k+l)
+ i~p
.
We have
(o)
~r+l .I and, for
~) xitf(v~. = x ri+ l -(wf( ~ i ) i
- wf(~vW~ ) Yi+l
i s s ~ q - I ,
r+l 1 x .tf(9~) = Xs(k+l)+i{wf(~y , )-wf(~ ~, )} (s) r-~ ' s(k+l)+l ~Xs(k+l)+i (s-l) (k+l)+i+l Ys(k+l)+i+l an~ finally, 1 x . . . . if( ~-~-) = r+l 9 ~, ~ )} r+l ' q(k+i)+1 ~Xq(k+l)+i Xq(k+l)+itwr ~Y(--~-I) (k+l)+i+l
(q)
Adding these
q + I
equations, we obtain
r+l % = q Xs(k+l)+i~f( ~ qrI.- r+l r+l (^, ) x i wf(~-~l) 7 r+l ~ "~-~) - s~otX(s+l) (k+ll+i-Xs(k+l)+i)wf OYs(k+l)+i+l s=O s (k+l)+i Now the first sum on the
LHS
is an element of
r+l r+l x(s+l)(k+l)+i-Xs(k+l)+i = f*(Ys(k+l)+i§ ) the
LHS
is an element of
(f*mp)ef .
for
q 9 O
and
l_<j-
i9
q -I > s > 0 , the second sum on
So we have shown
r+l ~. 8 . x i w~t~-~l) c tf(eN) + (f*mp)Sf Now consider the case
tf(SN) ; while, since
for
; we may write
l-
.
i = q ( k + l ) + j , where
.
Then xir+lwf (~I)B = q;IsL=O 'x(s+' r+l I)(k+l)+j- xr(l+l)+j)w f (~i__) + x.r+13wf (~i)
But
r+l r+l X(s+l)(k+l)+j - Xs(k+l)+j = f* (Y 's (k+l ) +j +l ) ' and
r+l ~y~) x. wf( ~ tf(SN) + (f*mp)Sf , by the previous analysis, so that 3
.
123
r+l .. 8 . x i wzts-~ij ~ tf(8 N) + (f*mp)Sf It follows easily that _particular
~(r+l)sf
through
z ~
be any element of and
z0
is algebraic,
then, there exists W r'
rt = Xf + I , and let
z0 = jr~f ; so that
(by (5.6) and (7.1) a) ). Let
W r'
c tf(SN) + (f*mp)8f , so that in
Xf < =
In either case, then, let z0 ~
also, and thus the claim is proved.
But
constructed
N r , and let
is not contained in L n w r'
the required
W r~ , since
z 0 ~ Wrv
The line
such that
(since
z v = -i-% ~ - . Zo+ ~z
L
Hence, since
consist~ of at most finitely many points.
X ~ (0,i~
pr''r(z ,) = z
~z ~ (prV,r)-I z .
Certainly,
is not contained
p r',r(zo) = jrf = 0 ).
r' > r , and the proof is complete.
Thus we have
in
CHAPTER
IV
Proof of the Topological Stability Theorem by Eduar~ Looijeng~
125
Throughout this chapter are denoted by
n
and
p
N
and
P
are fixed manifolds and their dimensions
respectively.
w
Multi-transversality
In the next sections we will encounter the following situation: subset
(A,A)
section
of the jet space and a smooth mapping f : N-* P
jtf : N ~ J~(N, P)
(I, 1.4), (A, A)
is transversal to the strata of
then pulls back via
It would be nice if we could refine property that the equal.
jef
B
f - images of any two members of
This is possible if the
in a 'regular' way.
f-images
such that the jet A .
Following
to a stratified subset
to a stratification B'
a stratified
B'
of
(B, B) B
of
N.
with the
are either disjoint or
of the strata of
B
intersect in
P
In this section we will show that under fairly mild conditions
this last 'regular' intersection property holds if we impose a certain transversality condition on
f .
This transversality condition (usually called multi-
transversality) was introduced property. his.
by Mather, who also showed that it is a 'generic'
This corresponds to our theorem (I.i), ~hose proof is rather close to
It is perhaps good to emphasize here that multi-transversality is not
necessarily an open condition (even when the manifolds in question are compact), although in the last section we will meet a situation where this happens to be the case.
Another main result of this section, proposition (1.5), is also concerned
with this:
it asserts that a stable map-germ satisfies all 'natural' multi-
transversality conditions. We begin to give the term 'regular intersection' a precise meaning. that 8 family
(Vi)
of subspaces of a vector space
V
Recall
is said to he in 6eneral
ieI (or regular) position
if the canonical map V
is surjective, or equivalently, if
~
@ iEl
V/V i
codim (OiEl Vi) = s
this condition implies that there are at most V.
dim V
codim V.1.
subspaces
V. l
Clearly,
not equal to
This notion has its counterpart in the context of stratified sets.
126
Suppose we are given a smooth mapping (B, B)
of
X x.
N .
Then
Denote for any
(B, B)
and any subset position. I
I
p
situation any
of
f-l(y)
the subspaces
~ p + i .
y E P
f
is of rank
B
which contains
f ,
if for any
ITf(TxXx)~xElOf
< p .
x
by
y E f(B)
T P are in general Y
properties:
(i)
and
explicitly,
any two strata of
(ii) Bf
then contains at
Assume moreeVer that in this f - counterimage
This is for instance the case when Bf
~f
any refinement of
of
~
Bf
is
have either equal or disjoint images B
having property (i) refines
may be characterized by the property that a member of
Hence a member of
fib
with the following two
indecomposable element of the Boolean algebra generated by
X I, ..., X k E ~.
fiB
possesses a neighbourhood such that its
Then there exists a refinement
f
and a stratified subset
In particular, each fibre of
intersects only finitely many strata.
under
the stratum of
has re6ular intersections relative
points where
proper.
P
By the remark above it suffices to verify this condition for subsets
of cardinal
most
x E B
f : N*
is locally of the form
~f. Sf
More is an
IX ~ f-if(y) : X, Y E B~.
f-l(f(X I) ~ ... ~ f(Xk))
with
The regular intersection property implies "~hat the sets of this
form are smooth.
Clearly,
stratification of
B .
with respect to subsets
~f
is locally finite, so we conclude that
~f
is a
We note in passing the naturality of this construction V
in
P : (B[f-~)f[f_~
=
~f[f-iv .
Bf need not satisfy the frontier property, even if we replace strata by their connected components, as is illustrated below.
[
127
As mentioned earlier, the stratified subset
(B, B)
of
N
will usually be
obtained as the transversal counterimage of a stratified subset of the jet bundle. This suggests the following definition. Let f : N*
P
(A, A) be a stratified subset of
is called multi-transverse with respect to
transversal to the strata of subset
Je(N, P).
(jtf)-I (A, A)
of
A N
A smooth mapping (A, A)
if
~f
is
and moreover, the (hence defined) stratified has regular intersections relative
Our first aim is to show that if
(A, A)
satisfies a certain condition,
'most' mappings will be multi-transverse with respect to precise, we need a topology on the function space the Whitney topology is best.
f .
(A, A) .
Coo(N, P).
To make this
For our purposes
The subsets
M(O) = {f E C~(N, P) : Jtf(N) _COl , (t = 0, i, 2, ..., Q Following [MorletS
an open subset of C~176 P)
Jr(N, P))
form a basis for this topology.
with the Whitney topology is a Baire space, i.e.
countable intersection of open-dense subsets of
Coo(N, P)
a
is dense.
This leads to the following definition.
Let
U
be an open subset of
~eneric in
U
if it contains a countable intersection of open - dense subsets of
U.
(I.i)
C~(N, P).
A subset of
U
is called
Clearly, a countable interesection of such sets is again generic.
Theorem.
Let
(A, A)
property that for any stratum a submersion. respect to
be a stratified subset of X E A ,
Then the set of
(A, A)
r
r
of the r - f o l d
constructed as follows. points of
pr
Let
~
r
is
which are multi-transverse with
C~176 P).
We prove (I.I) via two lemmas. stratification
with the
th__enatural projection ~pIX : X ~ P
f E C~176 P)
is generic in
~(N, P)
In the first one we deal with an auxiliary
Cartesian product
A r c Jr(N, p)r
be the 'diagonal stratification' of
belong to the same stratum of
~
r
which is pr :
two
if and only if they have the
same orbit type with respect to the natural action of the symmetric group on
Nr.
128
It is not difficult to verify that this defines a stratification (see also the discussion at the end of I, 2)). submersively to to
pr .
Hence
P , ~
Ar
(~)-lDr
intersection, which we denote by
any
f e Coo(N, P)
~p : Jl(N, P) ~ P
: Jr(N, p ) r
and
We further put
Since
pr
maps the strata of
maps the strata of d r
submersively
are in regular position so that their
gr '
is a stratification of
A r.
N (r) = I(xI ..... x r) e Nr : i / j => x i ~ xjl,
we let
j~f : N(r) ~ jl(N ' p)r
and for
denote the restriction of
r
(j~f)r . (1.2)
Lemma.
Let
f E Coo(N, P)
end suppose that
jlf
is transverse to
r
(Ar, gr) to
for
r = i, .... p + i .
Then
f
is multi-transverse with respect
(A, A) .
Proof.
Since
rJlf
is transverse to
in particular transverse to
(Ar, Ar),
(Ar, gr)
(r = I, ..., p + 1),
it is
and the restriction of ( J~f)-l(Ar, A r) r
is transverse to
( ~ ) ~ i Dr .
the restriction of to
D r.
Since
~
o
w~
The latter is equivalent to %he condition that
rJ~f
o rJtf
to the strata of
is simply the restriction of
it is easily seen that this means that relative
f
at subsets of
(rJtf)-l(Ar, Ar)
N
(j~f)-l (A, A)
of cardinal
is transverse
fr : Nr ~ P r
to
N (r)
has regular intersections
r ~ p + 1.
In view of an earlier
remark this proves the lemurs. (1.3)
Lemma.
Let
(X1 ..... Xr~
e N (r)
N (r)
of
r
and
JtflU
V
be a submanifold of
in
M M
such that the set on
V
prove that
U1, ..., U r O(U, M)
U
D(U, V)
U
of
P are disjoint, and put
is dense in
C~176 P),
difficult to construct a k - parameter family
let
x ~ in a coordinate patch for P U = U I x ... x Ur c N (r) . To
pick an
f e C~(N, P).
(lii)
agree outside some compact subset of
ft
It is not
(ft s Coo(N, P))
such that
and
with
~ ( N , P).
k = dim Jl(N, p)r f
fo = f'
and
o_~f (xo I ..... x o r ) _in _
with te~
(i)
z E M
o_~f f e ~(N, P)
is o~en and dense in
Choose a compact neighbourhood
such that
J~(N, p)r
There exist neighbourhoods
transverse to
Proof. N
z
M
(ii)
k
(x, t) ~ ft(x) N
for all
is smooth, t,
129
and (iv)
(x 1 . . . . .
neighbourhood UI, ..., U
x r, t ) ~ r J t f t ( ~
W
U x {01 .
of
....
, x)
~ jr(N, p ) r
is a submersion on a
(This can be done by means of the coordinates on
and finitely many coordinate patches for
P
coverir~ the compact
r
set
f(Ul) U ... • f(Ur)
which so determine a finite number of coordinate
patches for the jet space
Je(N, P).
is referred to KMather
].)
Now, if then to
t
Then
W~
j-IM
is a submanifold of
is a regular value of the composite
W N M.
5
The reader who w~shes to see a precise proof
j-IM
is transverse to
W ~ j-IMc N (r) • l~k-~ ~k
N (r) x {tl and hence
rJtftlU
Since by Sard'e theorem such regular values are dense in
follows that is dense in
f
lies in the closure of
C~176 P).
Q~(U, M).
W .
is transverse I~k
This proves that
In particular, for any subset
V
of
it
,
O(U, M)
M, ~O(U, V)
is
co
dense in
C (N, P).
But if
Proof of (I.I). submanifold
M
of
V
is compact, then
~(U, V)
is also open.
By l.emma (1.2) it is sufficient to prove that for a
Jl(N, p)r
the set of
f E Coo(N, P)
with
jlf
transverse to
r
M
is generic in
Coo(N, P).
In view of lemma (1.3) and the fact that
has a countable basis we can cover IU i x V i ~ = l
with
~(Ui, Vi)
Q(N (r), M) = f)~=l ~(Ui' Vi)'
N (r) x M
by countably many set
open and dense in proving that
N (r) • M
Coo(N, P) .
~(N (r), M)
Clearly,
is generic in
As we have now shown that multi-transversality is a generic
~~
P).
property, we
may expect that any stable map-germ has a representative satisfying any finite number of natural multi-transversality conditions.
This is made precise by the
following (I.~)
Proposition.
Let
(A, A)
be a stratified subset of
Je(N, P)
invariant under the natural action of
,dill N • diff P
stable map-germ
has a representative which is multi-trans-
f : (N, x) , (P, y)
verse with resvect to
o_nn J~(N, P).
which is Then any
(A, A).
We remark here that by results of Mather a stronger statement is true. Namely that
f
admits a representative which is multi-transverse with respect to
the (possibly uncountably many)
diff N • diff P
orbits in
Jl(N, P).
130
Proof of (i.4).
Put
~' = max(i, p + I).
sufficient and according to (III, 6.11), diff N • diff P : U~
P
of
approximate
orbit of f .
~
respect to
j
t'
f
~ : U~P
diff N • diff P
orbit of
f.
fact that
and
equivalent to
g .
the fact that
g
of
(A,
(1.5)
is
intersects the
which is multi-transverse with
j
j g
such that
Put
J'f
jl' g(x')
g = gx'"
we may ~oreover lies in the
Since
j
f
lie in the same orbit implies that
is sufficient, the f
is smoothly
The required multi-transversality property now follows from
possesses this property and the
diff N •
P -invariance
~).
Corollary.
type.
f
Now choose a representative
By the transveraality property of x' E U
f
j
Since the hypotheses of (I.I) are satisfied we can
require that there exists an
j
jt'f : (N, x) ~ J~'(N, H)
transversally.
by a smooth mapping
(A, A).
Following (III, 5.~
Then
f
Let
f : (N, x) ~ (P, y)
has a representative
~
be a map-~erm of finite singularity
such that the restriction
of
~
to its
critical set has finite fibres. Proof.
.According to (III, 2.8) such a map-germ possesses a stable unfolding.
Of course, it suffices to prove (1.6) for this unfolding, so we may as well suppose that
f
is a stable map-germ.
: U ~ P J~
Following (1.4),
f
then admits a representative
which is multi-transverse with respect to the whole manifold
P) = N • P.
In other words
U
has regular intersections relative
~ .
As observed at the beginning of this section, this implies that each fibre of contains at most
p
points where
~
is of rank
< p
.
We close this section with a simple, but useful lemma concerning transversality. (1.6)
Lemma.
Le__t M
an__dd Q
be manifolds,
R c Q
a submanifcld and
(B, ~)
a stratified subset of
a stratificaticn~ and the restriction of t_~o R .
F
F : M~
Q M
a smooth mapping, such that
to an~ stratum of
Then the following two conditions are eguivalent.
B
BF
is
is transverse
ISI
(i)
Near
F-IR,
(B, B)
is transverse to (ii)
(N, B)IF-IR
has regular intersections relative
F-~
has regular intersections relative F-IR
FIF-IR.
is a smooth submanifold of
M
near
so that these conditions make sense.)
Proof. for any
and
S F.
(Note that our hypotheses imply that B ~ F-IR,
F
Let x E I,
y E R X
x
and let
I
be a subset of
denote the stratum of
~
B ~ F-IR.
which contains
As before, let, x .
Then (i)
correspondSto the condition that the canonical map TyQ
~
@ xsI
T Q/TF(TxX x) Y
is surjective and that its kernel is transverse to
T R Y
.
Likewise (ii) corresp-
onds to the condition that
xE is surjective.
I
Now we have canonical isomorphisms
~R/CTFCT x Xx) n ~y R) ~ (~R + TF(TxXx))/TF(TXx) = TyQITFC~xXx) (since
FIX x
is transversa~ to
w
R), so the two conditions are clearly equivalent.
A stratification of the ~et space
First a word of motivation. At(N,~P), of a big open subset of property:
if
t
Our aim is to construct a stratification Je(N, P), with the following rather strong
is sufficiently large, then, for any mapping
f : N -~ P
which
is multitrausverse with respect to
A~(N, P),
partition
is actually a Whitney stratification which
B = ((j~f)-l ~L(N ' p))f
extends to a Thorn stratification particular, the partition
(B, B' )
of
B' = If(X) : X E ~I
the locally finite manifold
f
in the sense of
(I, 3),
is a Whitney stratification.
in Once
132
we have established this in section 3, the topological ability theorem will be close at hand.
In this section, first the construction of
Umfortunately, that
is a manifold partition.
postponed to section 4.
The proof that
this ~s so, is also
Here we settle a partial result, which is however quite
crucial:
if
At(N, P)
has not only the above-mentioned f
is carried out.
the nature of this construction is such that it is not even obvious
At(N, P)
true;
A t (N, P)
f : (N, Xo) -+ (P, yo)
has a representative
which the stratification
is a stable map-germ and
~ : U ~ V
property near
jtf,
t ~ p + l,
then
but still more is
admitting a canonical stratification of
of the source coincides with
((j~)-I At(N, p))~ f
The last property clearly exhibits a certain universal character of
At(N, P).
The proof of these statements will occupy the largest part of this section. In the preceding chapter (III, 5.8) it has been shown that f6r any stable map-germ
f : (N, x o) ~ (P, yo )
(U; x I ..... : U-* V Let
Z
x n) of
f
at
we can find coordinate neighbourhoods
(N, xo) ,
such that
(V; Yl ..... yp)
yj o
denote the critical set of
B = Ix E U:
Zi xi(x)2 < ~I
we may, following
Z.j yj(y)2 < W~
(P, yo)
is a polynomial in ~ .
Let
9
is contained in
(1.5), also assume that
r ~ f-l(yo) ~ B c IXol . D = IY E V:
~
at
U .
is contained in
W > o V
x I ....... x n
(j = 1 ..... p).
be so small that the closure of By taking
~I Z ~ B
Next we choose
and a representative
~
even smaller,
has finite fibres and that such that the closure of
and
Z f~-l(D)naB
=~.
133
We claim that then
s
~ : ~ ~ ~-I(D) ~B-* D
is proper.
Indeed, if
K cD
is compac~
is a closed subset of the open relatively compact set ~-I(D)AB,
and hence compact.
We deduce that the restriction
T ~ : ~ -l(D) ~ B-~ D
of
~
is
a polynomial mapping between semialgebraic sets (with respect to the coordinates introduced above) which satisfies the Genericity Condition introduced in Chapter I. It has the additional property that According to (I, 3.5)
~
then admits a canonical Thorn stratification (A,A').
o
a sense the 'germ' of (A, A') if
~i : ~1-1 (D1) ~ B1 -~ D1
as
~
and if
o
(S,
B')
(I, 3.8) and (I, 3.9)
~ : l ( y o ) / { x o I contains no critical point.
at
(Xo, yo ) depends on
is a representative of
f
f
only.
In
More precisely,
obtained in the same way
denotes its canonical stratification, then it follows from
that
(A, A')
and
(~, ~')
coincide on the pair
(~-l(o r% Dl) D B r] BI, D r% D1) . Since we are mainly interested in the propertie s of neighbourhood of closure.
(Xo, yo ) we would like any stratum of
The strata of
realised as follows. closed set U~X' ; V'
A'
will then have
Let
V'
Yo ~ ~'
is an open subset of
and
D,
Yo
A
in a
to have
in their closure.
be the subset of X'
(A, A')
D
x~
in its
This can be
obtained by removing the
a connected component of a stratum of
so by (I, 3.8) the pair
A' I-
(AI~-I(v') ~ B, A'IV')
the canonical stratification of the corresponding restriction of
~.
is
Now the
stratification of the target is as required, so we turn our attention to the stratification of the source. which does not contain X c Z , as to
x
o
F ~ ~-l(D ) ~ B
Z~-l(D)
Z ~-l(yo)
~ B-~ D
~ X #,
a contradiction.
is a union of strata.
It follows, that if
AI~-l(v)~ B
Moreover
is open.
with
x~
f
like
a representative
F
: U' * V'
~'
~ :
if not, then
this implies
x ~ E X,
whence
is the union of connected components
By (I, 3.9) the pair
Representatives of
AIM-I(v' )~ B
it follows that
in their closure,
stratification of the restriction ~ '
a definition :
Yo E ~(X) ,
U'
of
Because the restriction of
~-I Since f (yo) ~ X c IXol ,
.
X
in its closure, necessarily avoids
is proper and
of strata of U'
We claim that any stratum
then
(AIU', A'IV') of
Z ~ U' = Z n ~ - I ~ ) ~ B . is then a canonical
f .
are sufficiently interesting to deserve
of a map-germ
f : (N, x o) -* (P, yo )
is
IS4
ca~led special if (ii)
~ - I (yo) /
(i) ixol
it satisfies the genericity condition, contains no critical point, and
canonical stratification such that
x
~
admits a
lies in the closure of any'connected
o
component of a stratum in the source. (2.1)
(iii)
For future reference we sum up:
Any stable m2p-~erm admits a s_~ecial representative.
The canonical
stratifications of two such representatives aoincide where both are defined. Let
f
be a stable map-germ,
~
(4, 4')
the canonical stratification of
stratum
X E ~
which contains
x
o
a special representative of ~.
we write
cod f.
f ;
is connected and that its codimension ~s
Clearly
cod f
cod f = o
if and only if f
let
F
and
F'
(say
F)
the same unfolding dimension. (III, 4.$ smoothly equivalent.
Both
F • id
and
f
and
is of maximal rank.
is only of f i l e
singularity t~pe. f .
To see
f .
By
with the identity mapping
(SRk, o), we can arrange that the unfoldings
cod (F x id) = cod F'
By (2.1) this
be stable unfoldin~s of
taking the cartesian product of one of them of some
f
be the codimensien of some stable unfolding of
that this is well-defined,
id
A .
we shall therefore call it the codimension of
This concept generalises to the ease when Then we let
and
It follows from (I, i.i) that the
strictly greater than the codimensien of any other stratum of number only depends on
f
F'
F • id
and
F'
have
are stable and hence by
Then it follows from (I, 3.7) that
and it is easy to see that
cod F = cod (F • id) - see
(i, 3.~D. Since contact-equivalent germs of finite singularity type have equivalent universal unfoldings by (III,$.4), class of
f .
cod f
only depends on the c e n t a ~ equivalence
In particular, any jet which is contact-sufficient,
defined codimension.
has a well
Our notion of codimension has of course little to do with,
for instance, the notion of contact-codimension. We recall from chapter IIl, that with
2(z) $ I .
closed subset of sufficient.
Wt(N, P)
stands for the set of
Among other things it was shown there that Je(N, P)
We let
and that any
A~(N, P)
z E J~(N, P)i/ # ( N ,
denote the partition of
We(N, P) P)
J~(N, P ) /
zEJt(N, P) is a
is contact#(N,
P)
135
whose members are
S. : Iz E Jl(N, P ) k We( N, P) : cod z : Jl , J : 0, l, 2 .... 0 In the remainder of this section we investigate the local properties of this
partition.
In particular we will show that
At(N, P)
is a Whitney stratification.
For Shis purpose it is convenient to have the following lemma at our disposal. (2.2)
Lemma.
Let
M
be a manifold and
is transverse to the contact classes. invariant subsets of
(i)
If
(ii)
r
~ : M~J~(N,
Let further
X
and
X
is a submanifold of
M ,
then
J~(N, P)
of the same codimension near
If
and
r
r
r
are submanifolds of
~-lx,
then
Y
X
Y
be contact -
M
We only prove the contact -case;
same way.
The natural projection
Since the contact group J ( n , p)
r
is Whitney
Jr(n, p)
near
r
~ X.
diff N x diff P -crbit.
x P
comes from a locally trivial
and structural group
contains
X
the other case can be treated in the
Jl(N, P) ~ N
e(n, p)
and
is Whitney regular over
Proof.
fibre bundle with typical fibre
is a submanifold of ~ X .
Similar statements hold if contact class is replaced by
by
a smooth mapping which
J$(N, P).
regular over
Je(N, P)
P)
Le(n) x L~(p),
Lg(n) • Lt(p). we may replace
t(n, p)-orbit, in the statements of
and contact class by
the lemma, and prove (2.2) for this case. Since mapping If
X
Jg(n, p)
g(n, p) • M ~
is now transverse to
Jg(n, p),
r X
is ~nooth, then so is is smooth near
@(M) ~ X.
g(n, p) - orbits,
$(g, m) = g.r
is a contact-invariant subset of
So if that
~ :
r : M~
r
J~(n, p), .
As
~
the
must be a submersion. we have
@-I(x) =
e(n~.p)x ~ - ~ .
is a submersion, this implies
This proves (2.2.i) . (2.2.ii)
can be dealt
with in a similar fashion. The key result of this chapter is (2.3)
Proposition.
~ x(f) . (a)
Let
f : (N, x o) ~ (P, yo ) be a stable map-~erm and let
Then Ag(N, P)
special representative
is a Whitney stratification near ~
such that
jgf,
and
f
admits a
136
(b)
~
(c)
if
is multi-transverse with respect to (A, A')
At(N, P).
is the canonical stratification of
f,
then
A
and
are equivalent, i.e. they have the same family of connected
((jt~)-l AZ(N, p))~ components of strata, (d)
the strata of
A
and
(jt~)-I A~(N, p)
We shall prove (2.3) in a number of steps. true if f
f
which contain
jtf ~ Wt(N, p),
~ : U -~ V .
we may suppose that
U
and
Following (2.1)
W~(N, P)
JeT(U) /~ Wt(N, P) # ~ .
is closed and
By the transversality
x E U , ~x
is a stable
In the course of the proof below it is sometimes necessary to shrink
V .
If we do so, we tacitly assume that these conditions are still
satisfied after shrinking. of
cod f > o.
Since
criterion (III, 6.11) we may also suppose that for any map-germ.
are equal.
The proposition is trivially
is of maximal rank, so we may assume that
admits a special representative
x~
We let
(A, A' )
denote the canonical stratification
~. Step I.
Let
X E A
and
X* E (Jt~)-IAt(N, P)
contain
x o.
Then the
following two conditions are equivalent
(i) x ~ X (ii) Proof. is
x E X*
and
~-l~(x) •
The critical set
Z~-l~(x).
Since
~
7[Z
covering map.
~(X),
~
is a union of strata of
is proper and finite-to-one,
also proper and finite - to - one. submersively to
of
Z = Ix~
Because the strata in
it follows that
Its fibre over
consists of
Hence so
~]r ~ - I ~ ( x )
Z ~ ~-I~(x)
~ : Z ~ ~-l~(x) -~ ~(X)
Yo e ~(X)
A .
IXo~.
is
are mapped
is actually a As
X
is connected,
it follows that all its fibres consist of one point, in other words ~ n~-l~(x)
= {xl
for all
Conversely, let
x E X .
x E Z
after removing strata of
A
representative of" ~x"
So if
This proves part of (i) => (ii).
have the property that
and
A' Y ~ A
Z n
~-l~(x) =
Ixl.
Then
if necessary, we obtain a special contains
In view of the preceding we may apply this to any
x ,
then
x E X .
cod ~x = co dim Y. Then
X = Y,
so that
137
cod ~
=codim
X .
Since
codim X = cod f ,
this
implies
x e X~ .
X
This completes the proof that (i) => (ii) . x
as in (ii).
Then we have
But we can also apply this to an
codim Y = cod ~
= cod f
(since
x 9 X*)
and hence
X
Y
and
X
have the same codimension.
x 9 X.
It follows that
Y = X,
in particular
This proves (ii) => (i). Step 2.
Let
T.
denote the set of
which have a stable map-germ
z 9 S.
i
I
Then
satisfying (2.3.d) for some special representative). diff N x diff P - invariant submanifold of Proof.
It is clear that
T.
is
T. l
is a
of codimension
J~(N, P)
diff N x diff P - invariant.
i . To prove that it
i
is smooth and of the correct codimension, that
z = j~f.
(j~)-i Tic
X ~.
Now, let
(i) => (ii) of step l, that X = X*
and
deduce that
z 9 Ti
and assume for convenience
This allows us to stick to the notation introduced above.
are given (after possibly shrinking have
let
~x
~
U
and
x 9 X.
V)
X = X* .
Since
is a special representative of
Using
T i c Si,
X = X~
~x
Since
Jt~(x) 9 T i.
again, we see that
We
X = (J~)-lT..
I
By (2.2.i)
T. z
I
is then a submanifold of
correct codimension, since
we
It follows from the implication
is a stable map-germ, this implies that
X c (Jt~)-IT..
So w e
Jr(N, P)
near
j~f.
It is of the
codim T i = codim X = cod f = i.
The following step will be only used to prove the next one.
Step 3. If
Put
UIX ~ : j ~> cod fl
~-1(y) n ~ , / ~ Proof.
for
Let
x E ~-l(y) ~ x.
Clearly
Zk , Zk
an open subset of y 9 W ~ Zk 9
X ~.J = (J~)-Is.j X ~ ~ , some
y e Vk~(X)
such that
k ~codf. y E ~(Xk)
with the property that for any
is contained in the stratum
is open in ~(~),
X**j = (Jt~)-ITj (j = O, I, 2, ...).
then there exists a
denote the set of Xk
and
~(Xk).
It is also dense in
then a point of
Zlcodim X x : x 9 ~-l(y) ~ ~ I
Xx E A
W,
(with ~(~)
x 9 Xx) :
if
near W
is
where
takes its minimum necessarily belongs to
138
We show that for and that
k > o,
Zk
~
Let
~-l(Zk) A X k c X
Using the fact that condition,
~I Z
is a submanifold of k
be
> o
is finite- t o - one
and
V
of dimension
n - k
x E ~-l(Zk) O X k
and that
A
satisfies the frontier
it is easily seen that we can find a neighbourhood
U
of
x
in
U
X
such that
~-l~(x') N Z N U
: Ix'l
for all
x' E X
X
special representative
g
N U . X
of
~
Now, choose a
X
defined on an open subset
U' c U
X
X k* N U'x C X x
~ U'.x
Since
X
~-l~(x') ~ ~ A Ux = Ix'l for all
it follows from Step 1 (applied to
~)
of the canonical stratification
~
also follows that g)
~(X~ N U~)
is a submanifold of
V
Now suppose that smallest number X~kXS
/ ~ .
subset of
Xk ~ U'x
which contains
x.
Hence
x E Xk
.
It
(being a stratum of the canonical stratification of
of the same dimension as
with this property.
Z.j
x' E X x O U'x : X~ N U~,
coincides with the stratum
Xk
UIX~ : j ~ cod fl k X / ~ .
~ cod f Since
of
that
such that X
Zj\
Then by step (I)
~(X) / ~ .
n - k .
We assume that
is an open-dense subset of
V, we then also have
, namely
~(X~)
Because
j
is the
(i) => (ii),
and Zk
~(X)
is an
a closed (n-k)-
manifold, it follows that
(Zjk~(X))\ku> J~(x~) ~ Since
Xk\
X = ~
for
(zj\~(x))\ u
~k /
k>j
cod f .< k < j, any
y
in the left hand side of the
inclusion above is as required. We now put
i = cod f
and assume Inductively that (2.3) holds for all
stable map-germs of smaller ~dimension.
J~(u)
So in particular
T. = 8. J J
near
for j < i. By (1.4) we may suppose that (after shrinking
U
and
is multi-transverse with respect to each of the manifolds, Step 4.
We have
X = X ~ , J~f e T.
and
A
V Tj,
if necessary)
?
j~0.
is equivalent to
l
((JT) -1 [so ..... sil) ~ 9 Proof.
Let
y E V k~(X )
the distinct points of U
of P
x
such that P
be such that
~-l(y) ~ ~
by
~IUp : Up -~ ~(Up)
~-l(y) n Z c U j % ~
x~ .....
&
x
r
Xj**.
We enumerate
and choose a neighbourhocd
is a special representative for
139
~Xp (p = ~IUp
1
.....
and let
r).
Let
Xp e Ap
(Ap, Ap)
contain
By hypothesis,
Xp E X** c ' P
X *~ Xp = Cp A U p
Since
Further put
Xp
V.
~.
~
easily deduced that Vy
of
y .
A~, ..., A' r
~,
..., A'r
F~AplVy
(Ay, Ay)
members of
X
have
~
= Ali~-~ y
y
the member of
Since
r X
are in general position in
are in general position in an open neighbourhood
the canonical stratification "'
it
are Whitney stratifications, it is now
then extends to the canonical
~ : Up (Up f3 ~ - ~ y ) -, Vy,
IX1 f ~ - I v , . .
ITj 1j ~o ,
has regular intersections
~(XI) , ..., ~(Xr)
Following (I, 3.10),
stratification of
IXl,... , Xrl
(= cod ~Xp).
Up )
is multi-transverse with respect to
Hence their images,
From the fact that
cP = codim XP
so that (after possibly shrinking
follows that the manifold collection relative
denote the canonical stratification of
N~-~ p
of I y~
(I, 3.9) it even extends to
~ : ~-iVy ~ Vy. are strata of
Observe that the
A . y
By (I, 3.8) we
= X** 0 U n ~ - ~ , it follows that c p y P which contains x coincides on
~ ?-i V
IXo , X 1 , X 2 , ...I
and by
y
P
U
P
f3 ~ - l v
y
with the corresponding stratum
implies that
X
x
of
A .
This in particular
P X
c = codim X ~* ~ codim X As ~ X (recall that y ~ ~(X)), p o x x P P P we have codim X < i , so it follows that c < i. We have now proved that x p P the strata of A which intersect ~-l(y) f3 and the corresponding strata of
i xo ' " "
((jt~)-l iSo,
. . . .
Si_ll
coincide (or are equivalent) on
~-lv
. This proves at least a part of the last clause of step 4. It remains Y then clearly X i = X (for we know already from to show that Uj~i Xj* = X : step 1
that
X~'I = X* D X),
which immediately implies
jlf E Ti,
while, in view
of the preceding, this also completes the proof of the last clause. that
Uj~ I X * j k X # ~ .
~-l(y) f3 T. C U k ~ "
and
By step 3 there then exists a f-l(y) f3 ~ "
~ ~
for some
Now suppose
y E Vk~(X)
k ~ i .
But we have just
seen that this cannot occur. Step 5.
For any
.i ,
S.
is Whitney regular over
S.
at
such that
jef.
140
Proof. in
U
According to (2.2) it suffices to prove this for their counter images under
J~
.
By step 4,
(j~)-I Si = X ,
union of connected components of strata of regular over
A.
while
(Jt~)-Isj
is
a
Each of these is Whitney
X , hence so is their unfon
(Jt~)-is.. J Step 5 implies (2.3-a), while it follows from step 4 that
(2.3-b, c, d).
~
satisfies
The proof of (2.3) is now complete.
w
Properties of the stratification
In this section we prove that were announced earlier.
Ae(N, P)
possesses the prqperties which
The main result here is (3.4), which states that under
suitable transversality conditions (with respect to
Ag(N, P))
a smooth family of
mappings N • T (from
N
to
P)
~
P x T ~
T
admits a stratification satisfying almost all the hypotheses of
Thom's Second Isotopy
Lemma (II, 5.8).
It is clear that this must bring us
quite close to a proof of the topological stability theorem.
But first we
partially generalise (2.3). (3.1)
Proposition.
(a)
A~(N, P)
Let
is a Whitne~ stratification.
f : (N, Xo) , (P, Yo)be a smooth map-germ such that
x(f) ~ ~ ,
together with a stable unfolding (N, Xo)
~
(P, yo)
F
(P' y;)
(N' x;) amd a special representative canonical stratification of (b)
i
~ : U' ~ V' ~ .
is transverse to
of
F .
Denote by
(A, A' ) the
Then A
if and pnly if
J~f
is transverse tq
At(N, P).
141
Moreover, if one of the conditions of (b) is satisfied, there exis~ representatives ? : U*V
of
f ,
~ : U~
(c)
(~-IA,
~-IA')
(d)
~
and
is ecuivalent to
(P', y~) ,
is of the form
o_~f j
such that
? .
At(U, V)
and
~-IA .
by choosing suitable coordinates for
we can write
(P', y'o ) = (P • T, Yo • to) F
S : V~V'
is a Thom stratification of
Following (III, 0.1 ) ,
(N', x~)
and
i and
is multi-transverse with respect to
((Jt~)-iAt(U, V))~ f Proof.
U' o f
(N', x~) = ( N x T , X'o •
such that
i
and
F(x, t) = (ft(x), t)
j
,
are the obvious embeddings
with
ft
= f "
Now it is clear
o that shrinking
U'
and
statements of (3.1). we may assume that (with that
U
and
V
V'
(such that
is still special) does not affect the
Therefore, as the reader will verify without much trouble, T, U'
and
V'
are such that
open neighbourhoods of
~(x, t) = (~t(x), t) .
jet extension
~
U' = U • T
x~ e N
and
Yo e P
V' = V x T
respectively)
so
Moreover by (III,6.11) we may suppose that the
J : U • T ~ J~(N, P), J(x, t) = Jilt(x),
contact classes in
and
is transverse to all the
Jr(N, P) .
Now it is almost immediate from our definition of codimension that j-IAI(N ' p) = (jt~)-i At(N x T, P • T). (after possibly shrinking follows from (2.2-ii) that
U • T
and
At(N, P)
By (I.4) V • T)
and (2.3-a)
the latter is
a Whitney stratification.
is a Whitney stratification near
It then jtf.
This proves (a). Let of
X E A
contain
(Jl~)-IAt(NxZ,
only if Since
J o i
PxZ)
Following (2.3-d),
(Xo, to).
= j-i At(N, p).
So
i
is transverse to the stratum of
jtf = j o i ,
Then
V x {tol
I : U-* U • T then
and
(~-lA, ~-IA' )
is also a stratum
is transverse to
At(N, P)
X
which contains
if and jtf .
this implies (b).
To prove the last two statements, we assume that A .
X
is clearly transverse to ~ : V-* V x T
A' .
(I, 3.10)).
~
is transverse to
So if we let
be the natural maps, and take
is a Thom stratification of
or consider this as a special case
U • Itcl
indeed
~ =
o (easy to check,
142
Finally, to prove that
~
is multi-transverse with respect to
we would like to apply (1.6) with the following substitutions R = V • [Zol (2.3 -b, c)
and
~ = (j~)-i At(N x Z, P x Z).
that the condition (L6-i)
is satisfied9
Hence according to
has regular intersections relative
((J~)-lAt(N, P))~
~-lA
(3.2)
Corollary.
M = U x T, Q = V • T,
Now it follows from
(1.6), (j~)-i At(u, v) an~
At(U, V)
~ .
Clearly,
are equivalent stratifications.
Let
f : (N, X o ) ~
(9, yo ) be a smooth map-~erm with
an unfolding
(N, ~o )
~
(P, yo )
$i
D
(N', ~)
~
(9', y'o)
Then the following two conditions are equivalent. (i) (ii)
Proof. finite.
jtf ~ Wt(N, p)
and
jtf
j~F ~ We(N ', P'), J~F
is transverse to
is transverse to
X E (JtF)-lAt(N ', P')
contains
By (III,2.5) we have
X(F) = x(f).
Then by (III,2.8) F
x'
AI(N, 9).
A~(N ', P') then
i
and if
is transverse to
Suppose that this number
admits a stable unfolding
G .
N'
is
If we assume that
condition (i) is satisfied, then (ii) follows by applying (3.1-b) to the pair (f, G)
first and then to the pair
(F, G).
Proposition (3.1) directly leads to a stratification for certain mappings. (3.3)
Proposition.
Let
f : N-* 9
multi-transverse with respect to
be a proper smooth mapping which is
A~(N, P)
and such that
J~f(N) ~ Wt(N, P) = ~.
Put
A -- ((jtf)-l At(N, p))f
and
A' =If(X) : X E A1 U IP\ f(N)l Then Proof.
(4, A')
Thorn stratifies
Let
y E P.
f.
Since the critical set
(j~f)-l A~(N, p)
each of dimension < p ,
that
contains at most
f-l(y) ~ ~
p
P. is a union of strata of
the multi-transversality property implies
points.
Let
Xl, ..., x r
be R e distinct
143
points of
f-l(y) ~ Z. VP
gp : Up
Following (3.1 - c, d),
Ap = ((J~ gp)-i At(N ' p))gp it is clear that that if
admits a representative P (Ap, A~) such that
with a Thom Stratification
Though it was not explicitly mentioned in (3.1),
A'p = Igp(X) : X e ~pl U I v p k gp(U p )I.
Xp E Ap
contains
(J~gp)-lAt(N, P).
Xp,
then
Xp
y .
stratification of
gl(X1) ..... g(Xr)
are
Because these are strata of the Whitney stratifications
position in an open nei~hbourhood
V
of
A~, "''' A'r must be in general
y
in
V 1 ~ ... A Vr.
!
( ~ ~ ... ~ ~r) IV
f : (U1 ~ ... ~ U r) ~ f - ~ *
stratification is just the restriction of (Alf-~, A'IV)
their images
it is easily shown that then
(I, 3.10), the intersection
We may moreover assume
is also a stratum of
By multi-transversality,
in regular position at A~, ..., A'r '
fx
is a stratification of
Following
then naturally extends to a Thom V .
It is easy to see that this
(A, ~').
f : f-~V
It follows that . This
The previous proposition generalises to families of such mappings. has the interesting corollary (3.5) below. (3.4)
Proposition.
Let
T
smooth mapping of the form
be a smooth manifold and let F(x, t) = (ft(x), t).
F : N • T , P x T
be a
Then the followin~ two
conditions are equivalent.
(•
For all
t E T,
with respect to
(ii)
J~ft(N) n W~(N, P) = ~
~T : N • T *
A~(N • T, P • T) T
ft
is multi-transverse
A~(N, P).
JtF(N x T) ~ W~(N x T, P • T) = ~ , respect to
and
F
is multi-transverse with
and the restriction of the projection
to any stratum of
((J~F) -1 AI(N x T, P x T))
is__~a F
submersion. In either case we have that for all ((J~F) -I / ( N Proof. that
t
• T, P • T))FIN • {tl = ((J~ft)-IAt(N,
Assume that condition (i) is satisfied. J~F
avoids
A~(N• T,
? • T).
WI(N x T, P x T) Now, define
and that
J : N • T~
J~F
P))ft • It~.
It then follows from (3.2) is transverse
J~(N, P)
by
to
J(x, t) = J~ft(x).
144
Then
J-IAt(N, p) = (JtF)-IAt(N • T, P • T).
At(N, P),
N x Itl
is transverse to
it follows that
~T
also given that
j-IAt(N, P)IN x Itl
all
t E T .
J-IA~(N, P)
Itl is transverse to
As this is so for all
t E T,
suhmersively to
We are
T .
has regular intersections relative
so that we may apply
j-IA~(N, P) = (JtF)-IAI(N x T, P x T) that the strata of
JINx
j-IAt(N, P).
maps the strata of
F : N • Itl ~ P x It~ ,
Since
(13, ii => i).
It follows that
has regular intersections relative
((JIF)-IAt(N • T, P • T)) F
are transverse to
N x Itl
This proves (ii) as well as the last statement.
(ii) implies (i) uses (1.7, i => li)
F
and for
The proof that
and is left to the reader.
A little more than the hypotheses of (3.4) is needed to ensure that the family
Ift~tE T
(3.5)
is topologically trivial.
Corollary (to (3.3), (3.4) and (I, 5.8))
Keep the notations of (3,4) a n d
assume that one of the conditions (3.4 - i, ii) ~hat
F
Suppose moreover
is proper and that there exists a proper smooth function
that the restriction of the composite ((Jtft)-IAe(N'-- P))ft
~ oft
is transverse to
and homeomorphisms
: N ~ ~
~ c I~
trivial family of mappings, i.e. for any t E T
is satisfied,
Then
@
t E T,
h : N x U-~ N x U,
$ : P ~ I~
such
to any stratum of F
defines a locall2
there exist a neighbourhood
h' : p x U-* P x U
U
of
such that the
diagram below commutes F
N•
NxU
)
P•
~
PxU
ft x idU
In particular I a~v
ft' (t' E U)
Observe that if ~(P) = 189 9 F
P
is topologically equivalent to
is compact such a
The existence of ,
at infinity in case
P
~
always exists;
ft" we simply let
is postulated in order to have some control on
is not compact.
145
Proof F
Let
(At, A~)
respectively,
N • It'l
AIN x It'l
innocent refinement of
any intersection if we let
@
(A, A' )
X ~ (N • It'J ;
and
19-~Z ,
position and hence so are B = A ~ (~ o N T o F)-l@ F
HT
then and
~T o F
(~ o ~
X E A, t' E T . X E A, of
A
I~
and
o F)-l~ Hence
is transverse
HT o F
maps
T .
Now,
whose strata are the connected ($ o ~
o F)-I@
.
The pair ~ (B,
maps the strata of B'.
are in regular
We put
B
B') Thom
submersively 9o
Moreover each stratum of
Since
~-l[n, n + l]
to the closure of any stratum of F
is transverse to
submersively to
(~ o NT)-l@ .
$-l[n, n + I] • T.
the restriction of
A
Our aim is to let an
B' = 4' ~ (* o N ~ - l @
B'
T, HT~ B'
is
is compact, the
is proper.
to the closure of any stratum of
B
As
F
is
is also
Hence the hypotheses of ~I, 5.8) are satisfied for the diagram (N x T, B)
The result follows from (11,5.8)
w
F
(p • T, B,) ~T and
f : N ~ P
summarised by (3.5).
.
(11,5.9).
~
sufficiently large) any proper smooth
which is multi-transverse
topologically stable. stability theorem.
T
Topological stability as a transversality property
We now aim to prove that (for mapping
At .
imply that
does the same with the strata of
restriction of
proper.
A' and
and since
contained in a subset
proper,
~
X F] (~ o Np o F) -1 2Z ,
~
Then by (3.4)
of ft' and
satisfy the hypotheses of the Second Isotopy
denote the stratification
components of
stratifies
denote the Thom stratifications
corresponds with
The assumptions regarding
to each intersection
$ o NT
(A, A')
obtained by applying (3.3).
and
Lemma.
and
with respect to
AI(N, P)
is
In view of (1, l) this will imply the topological The progress we have made so far in this direction is best With an application of this last result in mind the
following proposition must be crucial.
146
(4.1)
Proposition.
transverse with respect to an open subset
f : N-+ P
The proper smooth mappings
O~(N,
P)
AI(N, P) of
which are multi-
J~f(N) n W l(N, P) = r
and satisfy
form
C~(N, P).
The proof requires a bit of preparation in the form of lemmas (4.2) and (4.3) below. (4.2)
Le__~t f ~ Coo(N, P)
Lemma.
cO
and let
[fj e Coo(N, P)l j= 1
converge to
f.
Then a subse~uence of
Ifj ~ embeds in a one-parameter family, more precisel.z ~
there exists .a family
IFt : N-* P~tem
IF1/k]k--I Oo
is smooth and Proof.
of mappings such that
respect to this metric and do the same for metric
p~
on
Jr(N, P)
for which
as a submanifold of
first factor. Vp
P • P,
N
In particular~
such that
N
P .
Jl(N, P)
We choose a tube in the sense of P • P
Ifj l~__l.
is a subsequence of
Choose a Riemannian metric for
(x, t) b~ Ft(x )
-- f.
F~
is complete with
These determine for every
t
is complete.
(II, 1.4, 1.5) for the diagonal of
compatible with the projection onto the
This exists by (II, 1.6).
The tube furnishes us a neighbourhood
of the diagonal together with a smooth mapping
F : Vp • [0, 1] * P
such
that Y(YI' Y2' 0) --Yl and ~(Yl' Y2' I) = y2 Ifj (k )1oo k=l
We now select a subsequence
of
Ifjloo j=l
such that
supI~t(Jfj(k)(X), Jbf(x)) : x ~ N1 < k -k2 and (fj(~+1)(x), fj(k)(X)) E V~ for all
x e N.
(-~o, 0]
Let
o : I~ -* [0, l]
and equals I on [i, co).
be a smooth function which vanishes on
Then define
g(x, t) = w(fj(k+l)(X), fj(k)(X), a(t + 1 - k)) g(x, t) = f(x)
if
t < 0.
with the property that
Then
g
is a smooth
suplp~(Jgt(x),
g : N x I~ if
(x, 0) . (t / O)
extends smoothly over
It follows that the family and
Fo(X) = f(x)
P
Coo-mapping on
J~f(x)) : x e N I < k -k2
N x m
We use (4.2) to prove the following
and N x (I~\[01)
if
Itl > k .
(x, t) ~ g(x, t -1)
by giving it the value
IFt}tel~ defined by
is as required.
by
t ~ [k - l, k]
We leave it to the reader to deduce that the mapping N x (SR k 101)
a
f(x)
on in
Ft(x) = g(x, t-1)
147 (4.3)
Lemma.
the set
Let
O K of
A~(N, ?)
f E
.,
~t
(N, P)
f' E ~(N, P)
with
is a neighbourhood of
Proof.
Suppose not.
As
and let
f
K
f'If-lK
f
[fj E Coo(N, P)Ij= I
always refers to a neighbourhood of
a
subsequenoe of
an
multi-transverse with respect to
converging to
A~(H, P).
f-IK
in
there exists a smooth one-parameter family Ifj 1~176
-
Then
has a countable neighbourhood basis there
is not multi-transverse with respect to
{Fl/klk_ I
P.
i_.nn C~176 P).
oo
then exists a sequence
be a compact subse t of
f
such that
fjlf-IK
(As usual, such a condition
N.)
By the previous lemma (L~.2)
IF t : N -* ]~ItEl~
with
F~ = f
and
It follows from (3.2) that there exists
j=l"
E > o
such that
Following (3.2),
JIF
is transverse to
N x IO~
is transverse to
intersection coincides with
~ ,
f ,
at
K • [-~, ~].
Since
and their
(Jtf)-lA~(N, P)
it follows from (1.7) that for a possibly
(J~F)-IA~(N • 3,
intersections relative
P x 3)
(Jg)-IAt(N • I~, P x 2)
(Jlf)-lA~(N, P) x IOl 9
has regular intersections relative smaller positive
A~(N • 2,
P • 2) I F-I(K • [-e, ~])
has regular
F , and that the strata of
AK, e = (JeF)-IA~(N x I~, P • 2) I F-I(K x [-e, ~])F are transverse to
N • I0J .
we may then choose t E [-e, r to
AK, ~
so small that
for all
t e [-e, r
is a Whitney stratification by (3.3),
N • ItJ
Then (1.7) implies that
At(N, P)
I t = ~
~
Since
FtlF;IK
.
is transverse to
AK, r
for all
is multi-transverse with respect
In particular this is so for
-I k > ~ , which contradicts our assumption.
if
Proof of (4. i).
Let
f E O~(N, P).
Choose a pair
{Vi cWil
of iEl
locally finite coverings of ~i c W.
for all
i E I.
P
by relatively compact open subsets for which
The set
D
of
f'E Coo(N, P)
with
Jf'(N)~Wt(N,P)=~
i
and
f, (f-I ~i ) c W i
any
f' E 0
AlE I O~i
is proper.
(O~i
intersection
for all
i E I
is clearly open.
On the other hand it follows from (4.3) that
as there defined) is a neighbourhood of
0 ~ NiE I ~ i
By the last condition
is a neighbourhood of
f.
f .
Hence their
It is clear that this
148
intersection is contained in
Of(N, p).
(4.4)
Theorem.
Any
f E Or(N, P)
is topolg~ically stable.
(4.5)
Corollary.
(The Topological Stabili~ Theorem)
The topologically
C_oo_(N, P)
stable mappings intersect the set of proper smooth mappings
in a
9r
dense subset. Proof.
In view of (4.4) it suffices to show that
Coo (N, P) pr
for some
e
codim We(n~ p) > n.
By (III, 7o 2)
The density of
(1.1) once we have show that codimension
> n .
Of(N, P)
W e (N, P)
Je(n, p)
canonical stratification. structural group
there exists an in
Lt(n) x L~(p)
Proof of (4.4).
of
Y(YI' Y2' I) = Y2 " P)
(III, 7.1).
Recall that
of the Jet-bundle Wt(N, P)
Jl(N, P) -~ N x P
into strata of codimension
F : Vp x [0,I] -~ P Now let
a stratification
~ c ~.
N. = ($ o f)-iIj ~ and if f'j : ~.'-*jPj
Vp
close to
f ,
then
(4, 4')
For any
for
f .
j E Z~ we put
f' E C~176 P)
neighbourhood of
Nj
enables us to compare
onto
is close to Nj,
then
Nj , ~l~j'
$
Note that
so if
> n .
P x P
~(YI' Y2' 0) = Yl
Let
$ : P-* I~
be
to any stratum of
4'
P. = $-IIj~ , 0 = ($ o f')-IIj I and let
we set f'.
and so
Then (3.3) constructs out
It follows from (3.4) that ~.' ~
is a
of the diagonal in
such that
f E ~e(N, P).
denote the restriction of
compact submanifolds.
Wt(n, p)
Hence it admits by (I, 2.7) a
a proper smooth function such that the restriction of is transverse to
will follow from
By choosing a tube for the diagonal in
and a smooth mapping
(Jls
such that
admits a stratification into strata of
we obtain (as in the proof of (4.2)) a neighbourhood
and
is dense in
This stratification is then invariant under the
determines a stratification of
P x P
l
~pr(N, P)
This is easily done as follows.
semi-algebraic subset of
D ~(N, P)
N.0 and
P.j are
fj E D~(Nj, Pj). ~
If
f'
is
is a smooth retraction of a
will be a diffeomorphism.
f' with f . Then (4.1) applied to f J 0 yields a neighbourhood Of of f such that f' E Of implies
and
This f. (j ~ 2Z), J
149
(a)
For any t ~ [0, IS , the mappi~
f~ E C~(N, P),
defined by
f~(x) ~ ~(f(x), f'(x), t), belongs to ~(N, P). (b) and
The composite ~ oft' is transverse to ~ c 9 for any t E [0, IS f, f~,j E n2(Mj t, Pj) for any j E ~ . We prove that any f' E Qf is
topologically equivalent to (Jef~)-IA~(N, P)
f .
According to (3.3) property (a) implies that
determines a stratification
(At, A~)
and property (b) above it follows that the strata of A~ manifolds stratum of f'
Pj, At
J s 2~,
of
f~.
are transverse to the
in particular, that the restriction of
is transverse to
are topologically equivalent.
~.
From (3.4)
$
ft
I~ then follows from (3.5) that
to any f
and
Referenees
[Bo ]
Borel~ A.
Linear Algebraic Groups, W.A. Benjamin, Inc. New York, 1 969.
[B=]
~,
,a
.
Brocker I T. & Jamch~ K.
Einf~hrung in die Differential topologie.
Springer - Verlag, 1 973.
[Gi ]
Gibson. C.G,
Regularity of the Segre Stratification,
Math. ~roo.
Camb. Phil. Soc. (To appear)
[Go ]
@olubits~1 M. & Guillemin I V.
Stable Mappings and their Singularities,
Springer - Verlag, 1 973.
[Hi ]
Hironaka, H t
"Number Theory, algebraic geometry and commutative algebra".
Volume in honour of Y. Akizuki.
Published
by Kinokuniya, Tokyo, 1973.
[~o]
Lan~. S.
Differential Manifolds,
~o~asiewicz I S~
Ensembles Sumi-Analytiques,
Looi.~enga, E.J.N.
Structural Stability of Smooth Families of functions.
[Ma]
Addison-Wesley,
1 972.
IHES Lecture Notes, 1965. C~ -
Thesis, University of Amsterdam, 1974.
Morlet, C~
Seminaire H. Caftan, Expos~ 4, 1961-62.
Mather t J. N~
I. Notes on Topological Stability, Lecture Notes, Harvard University, 1 970~ II. Stratifications and Mappings, Proceedings of the Dynamical Systems Conference, Salvador, Brazil, July, 1 971, Academic Press. III. Finitely-determined map germs.
Publ. Math.
IHES 35 (i968) p p . i 2 7 - i 5 6 . IV. Classification of Stable Germs by
9
algebras.
Publ. Math. IHES 37 (1969) pp.223-248. V. Transversality.
Advances in Mathematics 4 (1970)
pp.301 - 336. VI. The Nice Dimensions.
Proc. Liverpool
Singularities Symposium I, Springer Lecture Notes in Maths. 192 (1971).
151
[Th]
Thorn,R.
Propri~tes Diff~rentielles Locales aes Ensembles Analytiques, Seminaire Bourbaki, 1964/5. exp. 28i.
[Wa]
Wall, C.T.C.
Regular Stratifications. Warwick 1974".
'~ynamical Systems -
Springer Lecture Notes in Math.
No. ~68, P.332-344.
[Wa]
Wass erman, C-.
Stability of Unfol~ings, Springer Lecture Notes in Mathematics, 393 (i974).
[~]
vait~,, ~.
I. Local Properties of Analytic Varieties, pp.205244 in Differential an~ Combinatorial Topolo~, Princeton, i 965. II. Tangents to an Analytic Variety, Annals. of Maths.
8~ (~ 965), ~-% - ~ 9 .
Index
associated Thom stratification
28
bad set
~9
canonical partial stratification
26
canonical Thom stratification
32
canonical Whitney stratification
16
cl-equivalent mappings cl-stable mapping
I I
codimension
134
commutation relations
42
compatible tube
38
contact equivalence
94
control conditions
50
controlled vector field
36
controlled vector field
50
cross-ratio example
41
determined map-germ
307
diagonal stratification
127
dimension
18,68
distance function
38
equivalent map-germs
94
filtration by dimension finite singularity type
9,15,46 (FST)
Frontier Condition general position Genericity Condition generic subset
85 17 12,13,125 25 127
germ
37
induced stratification
13
infinitesimally stable map-germ
8O
ISS
Isotopy Lemma, First
4,59
Isotopy Lemma, Second
4,62
integrable vector field r-jet equivalent germs
54 105 5O
lift Local Finiteness Condition
9
local ring
73
local trivialisation
58
locally trivial stratification
59
locally trivial stratified set
58
minimal partial stratification
26
minimal Whitney stratification
15
multi-transverse mapping
5,127
Nakayama's Lemma
77
partial stratification
26
pinch map
24
Preparation Theorem of Malgrange
78
product stratification
12
refinement
14
retraction
38
regular point
18
regular intersection
14
restriction semialgebraic
126
set
17
singular point
19
singular set
19
special representative
134
spray
39
stable map-germ
69
154
9,22
stratification stratified vector field
9
stratum Tarski-Seidenberg
50
Theorem
17
Thom mapping
23
Thom Regularity Condition
23
Thom regular strata
23
Thom stratification
4,23
Thom
(Whitney stratification)
jet space topological equivalence Topological Stability Theorem topologically trivial family
5,131 2 148 3
transverse mapping
13
trivial stratified set
58
trivial unfolding
68
tube
38
tube system
46
tube system, controlled
46
unfolding
66,67
versal unfolding
67,69
Whitney regular strata
10,11
Whitney regularity Whitney Regularity Condition
3 11
Whitney stratification
4,11
Whitney stratified set
11
Whitney's Theorem
19
Whitney topology
127
Whitney umbrella
9
Symbols A, A, A' w A' a, a' B(X,Y) (CR.), (CRp), (CRf) CN,x O' mN, Xor C N, m N f, f X jr(N,P) jr(n,p) jrf jrf L r (n) Mi,M ,i pr,S Qi, Qi(k,l). Qpi
46 45,46 19 42 73 112 111 112 112 105 112 45,46 111 47,48
Sg(f)
3O
ITil
47 75 50,51 2O
tf VF., VFp, VFf, VFpf W(A,B)
W r (n,p) wf xf A1 (N,P) 8f C Dr K Nf R
120 75 85 134 126 94,114 127 94,114 83 114
Xc
17
(U.V)
128
R I(N,P)
146 70
eN,x o, e N ef
74
