Old and new results on density of stable mappings
Results on density of stable mappings
Maria Aparecida Soares Ruas
M. A. S. Ruas
Maria Aparecida Soares Ruas Instituto de Ciências
Matemáticas e de Computação - USP, Av. Trabalhador são-carlense,
400, 13566-590 - São Carlos - SP, Brazil<EMAIL_ADDRESS>
The analysis of the conditions for a map-germ to be finitely
determined and of the degree of determinacy involves the most important
of the local aspects of singularity theory.

Density of stable maps is the common thread of this
paper. We review Whitney's contribution to singularities of
differentiable mappings and Thom-Mather theories on $C^{\infty}$
and $C^{0}$-stability. Infinitesimal and algebraic methods are
presented in order to prove Theorem A and Theorem B on density of
proper stable and topologically stable mappings $f:N^{n}\to P^{p}.$
Theorem A states that the set of proper stable maps is dense in the
set of all proper maps from $N$ to $P$, if and only if the pair
$(n,p)$ is in nice dimensions, while Theorem B shows that
density of topologically stable maps holds for any pair $(n,p).$  A
short review of results by du Plessis and Wall on the range in
which proper smooth mappings are $C^{1}$- stable is given.
A Thom-Mather map is a topologically stable map $f:N \to P$ whose Map!Thom-Mather
associated $k$-jet map $j^{k}f:N \to P$ is transverse to
the Thom-Mather stratification in $J^{k}(N,P).$ We give a detailed
description of Thom-Mather maps for pairs $(n,p)$ in the boundary
of the nice dimensions.
The main open question on density of stable mappings is to determine
the pairs $(n,p)$ for which Lipschitz stable mappings are
dense.  We discuss recent results by Nguyen, Ruas and Trivedi  on
this subject, formulating conjectures for the density
of Lipschitz stable mappings in the boundary of the nice
dimensions. At the final section, Damon's results relating
$\mathcal{A}$-classification  of map-germs and $\mathcal{K}_{V}$
classification of sections of the discriminant $V=\Delta(F)$ of a
stable unfolding of $f$ are reviewed and open problems are discussed.

§ CONTENTS

§ INTRODUCTION

Although Riemann, Klein, Poincaré and other great mathematicians of
the nineteenth century  already used deep topological concepts in their work,
the birth of algebraic and differential topology as formal sub-areas
of Mathematics occurred in the first half  of the twentieth century.

After previous works of Whitehead, Veblen and others, the American
mathematician Hassler Whitney introduced fundamental concepts and
proved strong results in differential topology such as the well known
strong Whitney
embedding theorem and weak Whitney embedding
theorem. The first one states that any smooth real $m$-dimensional
manifold can be smoothly embedded in $\mathbb R^{2m}$, while the
latter says that any continuous mapping of an $n$-dimensional
manifold to an $m$-dimensional manifold may be approximated by a
smooth embedding provided that $m > 2n.$ Furthermore, replacing
embedding by immersion in this last  statement the result holds for
all $m\geq 2n.$  His survey paper Topological
properties of differentiable manifolds published in 1937
[111] contains many contributions he made in those early years
of differential topology.

In 1944, Whitney [113] studied the first pair of dimensions not
covered by his immersion theorem. For mappings $f$ from $\mathbb
R^{n}$ to $\mathbb R^{2n-1}$ Whitney proved that singularities cannot
be avoided in general. He introduced the semi regular mappings
as proper mappings $f:\mathbb R^{n}\to \mathbb R^{2n-1}$ whose only
singularities are the generalized cross-caps (Whitney umbrellas)
points.  Away from singular
points, $f$ is an immersion with transverse
double points, and when $n=2,$ a finite number of triple points may also
appear in the image of  $f.$ These are the only stable singularities in these
dimensions. However, only later, Whitney introduced the notion of
stable mappings.

Abstract spaces and their topological properties were known by then,
so that the notion of stability of systems and mappings appeared
naturally. It appeared first in dynamical systems, introduced by
A. Andronov and L. Pontryagin [1] for a class of autonomous
differential systems on the plane, under the name of “systèmes
grossiers”. The term “structural stability” appears in the english
language edition of the book by Andronov and Chaikin, edited under the
direction of Solomon Lefschetz in 1949 [2] (see also
[91]). It also appears in other pioneering papers on the
subject, among them the paper On structural stability by
Mauricio Peixoto [78], published in 1959.

The notion of stable mappings was formulated by Whitney in
[115] around the middle of last century. He characterized stable
mappings  from $\mathbb R^{n}$ to $\mathbb R^{p}$  with $p \geq
2n-1$ in [112] and stable mappings from the plane into the
plane in [114], showing in these cases  that stable mappings
form  a dense set in the space of smooth proper mappings.

The article Whitney [114] published in 1955 is a landmark, considered
by many to be the cornerstone of the theory of singularities. The
stable singularities of mappings of the plane into the plane are folds
and cusps and any proper smooth mapping $f:\mathbb R^{2}\to \mathbb
R^{2}$ can be approximated by a stable mapping. Whitney
conjectured that density of stable mappings would hold for any
pair $(n,p).$  However
René Thom showed, in his 1959 lecture at Bonn, that this is not the
case by given an example of a map $f:\mathbb R^{9}\to \mathbb R^{9}$
that appears generically in a $1$-parameter family of maps.

Thom conjectured that the topologically stable maps are always dense
and gave an outline of the proof. The complete proof was given  by John
Mather,  who from 1965 to 1975, solved almost completely the program drawn by René Thom for
the problem of stability.

Mather found several characterizations of stability and proved that
the set $\mathcal S^{\infty}(N,P)$ of stable mappings is dense in the
set $C_{pr}^{\infty}(N,P)$ of smooth proper mappings, from the $n$-dimensional manifold
$N$ to the $p$-dimensional manifold $P,$ if and only if $(n,p)$ is in the nice
dimensions, which he completely characterized in
[63]. Based on Thom's ideas, he  also proved in [65, 66] that the set of topologically  stable
mappings $\mathcal S^{0}(N,P)$ in $C_{pr}^{\infty}(N,P)$ is residual for all pairs $(n,p).$

The 70's was blooming period for singularity theory. Along with
Mather's work, René Thom's book on catastrophe theory [94]
and Arnold's  seminal classification of simple singularities of
functions [3] also had a great impact.
These works paved the intense development of the theory
of the following decades.
The deep understanding of stable
mappings, versal unfoldings and finite determinacy transformed
singularity theory into an organizing center for several areas of
mathematics and sciences.

The common thread of these notes is the question of density of stable
mappings in  $C^{\infty}_{pr}(N,P).$  We outline the solutions of the various formulations of this
problem: $C^{\infty}, C^{0}$ and $C^{l}, 1\leq l < \infty$
stability. The remaining  open problem in this setting is density
of Lipschitz stable mappings. Recent
progress in the solution of this problem  appear in [88, 75].

We give an account of tools for the proofs of the main theorems
including  the notion of infinitesimal stability, the generalized
Malgrange's theorem, Thom's transversality theorem, mappings of finite
singularity type and finite determinacy of Mather's groups. Whitney
and Thom's results on stratified sets and maps are fundamental
pieces of the theory.
For an account of these topics we refer to David Trotman's article in
Volume 1 of this Handbook .

In these notes we concentrate on the discussion of real
singularities. The infinitesimal methods discussed here also hold true
for holomorphic mappings. For an account on Mather's theory of
${\mathcal A}$-equivalence  and the description of the topology of
stable perturbations of ${\cal A}$-finitely determined holomorphic
germs the reader may consult the notes by David Mond and Juan José
Nuño-Ballesteros in this Handbook [70].

Related topics to those discussed in these notes, as well as new
developments of the theory, are given in the subsections Notes at the end of
each section. The final section includes a discussion of open problems
in the theory of singularities of smooth mappings.

§ SETTING THE PROBLEM

Let $C^{\infty}(N,P)=\{f:N\to P, f\in C^{\infty}\}$ be the set of
smooth mappings from $N$ to $P,$ where $N$ and $P$ are smooth
manifolds of dimension $n$ and $p$ respectively. The topology on
$C^{ \infty}(N,P)$  is the $C^{\infty}$-Whitney topology.

We  review here the contributions of
singularity theory to solve the following problem.

Find an open and dense set $\mathcal S$ in $C^{\infty}(N,P)$ and
describe all singularities of mappings $f\ \in \mathcal S.$

The relevant equivalence is $\mathcal{A}$-equivalence.

$\mathcal A$-equivalence
Two smooth maps $f,g:N \to P$ are $\mathcal
A$-equivalent if there
exist $C^{\infty}$ diffeomorphisms $h:N \to N$ and $k:P \to P$ such
that the following diagram commutes

\xymatrix{\ar @{} [dr] |{\circlearrowleft}
N \ar[d]_h \ar[r]^f &P  \ar[d]^k \\
N \ar[r]_g   &      P }

The map $f:N \to P$ is stable ($\mathcal A$-stable) if there exists a neighborhood $W$ of $f$ in
$C^{\infty}(N,P),$ such that  $g \widesim{\mathcal{A}}
f$ for every $g\in W.$

Replacing $C^{\infty}$-diffeomorphisms by homeomorphisms,  $C^{l}$-diffeomorphisms, $l>0$ or bi-Lipschitz
homeomorphisms in definitions <ref> and  <ref> we get respectively the definitions of $C^{0}$-$\mathcal
A$, $C^{l}$-$\mathcal A$ ($l> 0$), bi-Lipschitz-$\mathcal A$ equivalences
and of topological
stability, $C^{l}$-stability, or Lipschitz stability of maps in
$C^{0}$-$\mathcal A$-equivalence
$C^{l}$-$\mathcal A$-equivalence
Bi-Lipschitz! $\mathcal A$-equivalence
Map!topologically stable
Map!Lipschitz stable
Map!$C^{l}$- stable

Before starting the discussion of Problem <ref>, we review some notation
and definitions.

The Whitney $C^{\infty}-$ topology in $C^{\infty}(N,P)$ was defined by
John Mather in [57]. We review it here (more details can be
found in the book of Golubitsky and Guillemin [40]).

For $x\in N,$ $y\in P$ and for a non-negative integer $k,$ we denote by
$J^{k}(N,P)_{x,y}$ the set of $k$-jets of map-germs $(N,x)\to
(P,y).$ When $N=\mathbb R^{n},$ $P=\mathbb R^p,$ we denote
$J^{k}(n,p)$ the set of polynomial mappings $f:\mathbb R^n \to
\mathbb R^p$ of degree $\leq k,$ such that $f(0)=0.$

The set $J^{k}(N,P)=\bigcup_{x\in N,y\in P} J^{k}(N,P)_{x,y}$ is the $k$-jet
space of mappings from $N$ to $P.$ Jet space of mappings
The set
$J^{k}(N,P)$ is a smooth manifold (theorem 2.7 in
[40]). Moreover, it has the structure of a fibre bundle
with  basis  $N\times P.$

Let $U$ be an open set in $J^{k}(N,P)$ and

M(U)=\{ f\in C^{\infty}(N,P) |\; j^{k}f(N)\subset U \}.

The family of sets $\{M(U)\}$ where $U$ is an open set of $J^{k}(N,P)$
is a basis for a topology in $C^{\infty}(N,P)$ (note that $M(U)\cap
M(V)=M(U\cap V)$).
This topology is called the Whitney $C^{k}$-topology.Whitney! $C^{k}$-topology

Denote by $W_{k}$ the set of open subsets of $C^{\infty}(N,P)$ in the
Whitney $C^{k}$-topology. The Whitney $C^{\infty}$-topology is the
topology whose basis is $W=\cup_{k=0}^{\infty} W_{k}.$
Whitney! $C^{\infty}$-topology

Given a metric $d$ on $J^{k}(N,P),$ compatible with
its topology and a nonnegative continuous function $\delta: N\to
\mathbb R$ we can define a basic neighborhood of $f\in C^{\infty}(N,P)$ as

B_{\delta}(f)=\{g \in C^{\infty}(N,P) |\;
d(j^{k}f(x),j^{k}g(x))<\delta(x), \forall x \in N  \}.

When $N$ is compact, $f_{n}$ converges to $f$ in the Whitney
$C^{k}$-topology if and only if $j^{k}f_{n}$ converges uniformly to
$j^{k}f.$  On noncompact manifolds  $f_{n}$ converges to $f$ in the Whitney
$C^{k}$-topology if and only there exists a compact $K\subset N,$ such
$j^{k}f_{n}$ converges  to $j^{k}f$ uniformly in $K,$ and there
exists $n_{0}$ such that $f_{n}\equiv f$ in $N\setminus K$ for any $n\geq
n_{0}$ (for details see the book by Golubitsky and Guillemin [40]).

Thus we can see that there is a great difference in the Whitney
topology depending on whether or not the domain $N$ is a compact

When $N$ is not compact, the Whitney $C^{k}$-topology is a very fine
topology, with many open  sets. As a consequence, dense sets in
$C^{\infty}(N,P)$ are very large sets, and theorems characterizing
these sets in $C^{\infty}(N,P)$ are strong results.

§.§ The work of Hassler Whitney: from 1944 to 1958

The foundations of the theory were Whitney's work, in which he formulated
the problem of classifying singularities that can not be eliminated by
small perturbations, and completely succeeded  in solving it for maps
from $\mathbb R^{n}$ to $\mathbb R^{p}$ with $p\geq 2n-1$ in Whitney <cit.> and  from $\mathbb R^{2}$ to $\mathbb R^{2}$  in Whitney [114].

The article [114] published in 1955 is a magnificent work
dedicated to maps from the plane into the plane. In the introduction to the article, Whitney presents a
complete review of the existing results and future perspectives of the
theory. We reproduce it here:
“Let $f_{0}$ be a
mapping of an open set $R$ in $n$-space $E^{n}$ into $m$-space
$E^{m}.$ Let us consider, along with $f_{0},$ all the mappings $f$
which are sufficiently good approximations to $f_{0}.$ By the
Weierstrass Approximation Theorem, there are such mappings $f$ which
are analytic; in fact, (see <cit.>) we may make $f$
approximate to $f_{s}$ throughout $R$ arbitrarily well, and if
$f_{0}$ is $r$-smooth, (i.e., has continuous partial  derivatives of
order $\leq r$),  we may make corresponding derivatives of $f$
approximate those of $f_{0}.$

Supposing $f$ is smooth, (i.e., $1$-smooth), the Jacobian matrix $J(f)$
of $f$ is  defined (using fixed coordinate systems); we say the point
$p\in R$ is a regular point or singular point of $f,$
according as $J(f)$ is of maximal rank (i.e., of rank $\min(n,m)$) or
lesser rank. In general we cannot expect $f$ to be free of singular
points. A fundamental problem is to determine what sort of
singularities any good approximations $f$ to $f_{0}$ must have; what
sort of sets they occupy, what $f$ is like near such points, what
topological properties hold with references to them, etc.

Some special cases of this problem have been studied as follows:

a) For $m=1,$ we have a real valued function in $R.$ It was
shown by M. Morse in Theorem 1.6 of [73], that $f$ may be chosen
so that the singular points (called critical points here) are
isolated, the “Hessian” being non-zero at each.” “Moreover, each
critical point may be assigned a “type number”; topological relations
among these were given by Morse [72].

b) If $m\geq 2n,$ we may find an $f$ with no singular
points; see (a) and (b) of Theorem 2 in [110].

c) If $m=2n-1,$ we may obtain an $f$ with singular
points: see [112]. For each such point $p\in R,$ coordinate
systems $(x_{1}, x_{2}, \dots, x_{n})$ in $E^{n}$ and $(u_{1}, u_{2},
\dots, u_{m})$ in $E^{m}$ may be chosen, in which $f,$ near $p,$ has
the form

\ u_{1}= x_{1}^{2},\ \ \ \, u_{i}= x_{i},\,\,\,\,\,
u_{n+i-1}=x_{1}x_{i},\ \ \  (i=2,
\dots, n).

The singularities are studied from a topological point of view in

d) Some beginnings have been made for the other pairs of
values $(n.m)$ by N. Wolfsohn, [120], but no complete
classification of the singularities exist in these cases. Thus the
smallest pair of values for which the problem is open is the pair
$(2,2),$ i.e for mappings of the plane into the plane; it is this case
that we treat here. In this case, there can be “folds” lying along
curves and isolated “cusps” on the folds ”(Figure <ref>).

We review Whitney's results in this section.

Let $f:U\to \mathbb R^{2}$ be a smooth mapping defined on the open set
$U\subset\mathbb R^{2}.$ With coordinates systems $(x,y)$ in $U$ and
$(u,v)$ in the target, the Jacobian of $f$ is given by

A point $p\in U$ is regular or singular according as
$J(f)(p)\neq 0$ or $J(f)(p)= 0.$ A singular point $(x_{0},y_{0})$ is
good if the derivatives $\frac{\partial J(f)}{\partial
x}(x_{0},y_{0})$ and $\frac{\partial J(f)}{\partial
y}(x_{0},y_{0})$ do not vanish simultaneously. We say that $f$ is
good if every singular point of $f$ is good. This condition
implies that the set $S(f)$ of singular points of $f$ is a regular
If $f$ is good and $p$ is a singular point, let $\phi:(-\epsilon,
\epsilon) \to \mathbb R^{2}$ be a parametrization of the singular set
$\mathcal{S}(f)$ in a neighborhood of $p\in \mathcal{S}(f)$ such that
$\phi(0)=p.$ Then, we define
(i) If $(f\circ\phi)'(0) \neq 0,$ we say  $p$ is
fold point  of $f.$

(ii) $(f\circ\phi)'(0)= 0$ and $(f\circ\phi)''(0) \neq 0,$
we say  $p$ is a cusp point of $f.$
These definitions are independent of the parametrization chosen for
$S(f)$ in a neighborhood of $p.$

One can easily see that at a fold point, the restriction of $f$ to its
singular set is non singular, while a cusp point is a singular point  of this restriction.

It follows from the definition that cusp points are isolated.

Let $f$ be a good map.Map!excellent
We say that $p$ is an excellent point of $f$ if it is a
regular, fold or cusp point of $f.$ If  each point $p\in U$ is
excellent we say $f$ is excellent.

Any smooth map can be approximated in
the $C^{r}$-Whitney topology, $r\geq 3,$ by an excellent map.

Let $f_{0}$ be a mapping from $U\subset \mathbb R^{2}$ to $\mathbb
R^{2},$ where $U$ is an open set in $\mathbb R^{2}.$ Then
arbitrarily near $f_{0}$ there is an excellent mapping $f.$ If
$f_{0}$ is $r$-smooth and $\epsilon$ is a positive continuous
function in $U,$ we make $f$ an $(r,\epsilon)$-approximation of $f.$

Prior to Thom's transversality theorem ([92]), Whitney
introduced the method of characterizing in the jet space the set of
jets with degenerate singularities, the so called “bad set”.

In addition,  methods of producing generic $C^{r}$-perturbations of any
given map were also introduced by him. The goal was to find
sufficiently close perturbations that would avoid the bad set.

For polynomial maps from the plane into plane, the bad set are the
polynomial maps admitting singularities more degenerate than folds and

Folds and cusps have simple normal forms.

Folds and Cusps

* Let $p$ be a fold point of the $r$-smooth mapping $f$
of  $\mathbb R^{2}$ into $\mathbb R^{2},$ with $r\geq 3.$ Then
$(r-3)$-smooth coordinate systems $(x,y),\, (u,v)$ may be introduced
about $p$ and $f(p)$ respectively, in terms of which $f$ takes the

\begin{equation}
\label{eq:1.1}
u=x^2\, ,v=y
\end{equation}

* Let $p$ be a cusp point of the $r$-smooth mapping $f$
of  $\mathbb R^{2}$ into $\mathbb R^{2},$ with $r\geq 12.$ Then
$(\frac r2-5)$-smooth coordinate systems $(x,y),\, (u,v)$ may be introduced
about $p$ and $f(p)$ respectively, in terms of which $f$ takes the

\begin{equation}
\label{eq:1.2}
u=xy -x^3\, ,v=y
\end{equation}

While the proof of (1) is not hard, Whitney's proof of the normal
form in a neighborhood of a cusp point $p$ follows by an ingenious
sequence of changes of coordinates in the source and
target. The tool is essentially the implicit function theorem.

Today, there are simpler proofs of this result, based on current tools
of singularity theory: see for instance, Theorem 2.4, Chapter VI in
Golubitsky and Guillemin's book [40] or Example 3.6 in Mond
and Ballesteros [69].

The notion of stable mappings is due to Whitney. In order to
characterize them, in addition to the local behavior of stable
singularities, it is necessary to explain the behavior of multiple
points. For maps from the plane into the plane the following holds.

Let $f:N^{2}\to P^{2}$ be a smooth map, $N$ and $P$
$2$-dimensional manifolds, $N$ compact. Then $f$ is $C^{\infty}$-
stable if and only if the following conditions hold.

* $f$ is excellent and hence $\mathcal{S}(f)$ is a
regular curve, with at most a finite number of cusp points.

* If $p_{1}$ and $p_{2}$ are singular points of $f,$
$f(p_{1})=f(p_{2}),$ then $p_{1}$ and $p_{2}$ are not cusp
points. Moreover the fold lines intersect transversaly at

* The restriction of $f$ to $\mathcal{S}(f)$ has no
triple points.

Whitney  formulated in [115] a general approach to defining a
stratification in jet space and to define locally generic mappings as
those  whose $r$-jets were transversal to the strata of the
stratification, for every $r \in \mathbb N^{*}.$ The article contains
an explicit description of generic singularities for pairs $(n,p)$
such that $n,p\leq 5.$

He asked the question whether for any pair of dimensions
$(n,p),$ the stable maps could be characterized by transversality to a
finite collection of submanifolds in jet space, so that one could
apply Thom's transversality theorem to prove that a smooth map could
be always approximated by stable maps.

However, in a course taught at the University of Bonn in 1959, René
Thom showed with  an example that
it is not always possible to approximate a given map by $C^{\infty}$
stable mappings (See section <ref>, on Thom's
example). In fact, in the notes  Singularity of differentiable
mappings I, written by Harold Levine [96], Thom sketched
the proof that  $C^{2}$-stable mappings do not form an open set in
$C^{\infty}(N,P),$ when $n=p=9$ and  he formulated conjectures that
promoted a great development in the
theory in the following decades. In particular, Thom conjectured the density of topologically
stable mappings, proved by John Mather in 1971. We discuss René Thom and John
Mather's contributions in the next section.

§.§ René Thom and John Mather: from 1958 to 1970

We start by reviewing the subjects covered by R. Thom in his course at
the University of Bonn. H. Levine's notes are divided into three

Chapter I, named “Jets” introduces the notion of jet spaces, the
action of the group ${\mathcal A}$ in jet space and ${\mathcal
A}^{r}$-invariant manifolds, denominated, in the notes,
critical varieties in $J^{r}(n,p).$ The set ${\cal S}_{k}$
of $1-$jets of corank $k$ and its topological closure $\overline {\cal S}_{k}$
in $J^{1}(n,p)$ were defined.

In Chapter II, entitled “Singularities of mappings”, Thom's
transversality theorem was stated and proved. We remark however that
the topology in the space of mappings in Thom's proof was the weakest topology
making the mapping

\begin{align*}
j^{r}:C^{\infty}(N,P) &\to C^{\infty}(N,J^{r}(N,P))\\
f &\to j^{r}f
\end{align*}

continuous. The topology in the second space was the compact open
topology. The transversality theorem in
[96] was stated as follows:
For $s>r\geq 0,$ let $W$ be a codimension $q,$ $C^{s-r}$ submanifold
of $J^{r}(N,P),$ $s-r> \dim N -q.$ Then the set of mappings $f\in
C^{\infty}(N,P),$ such that $j^{r}f \pitchfork W$ is dense in
Thom!transversality theorem
The notion of second order
singularities  $S_{h,k}$ in $J^{2}(n,p)$ was introduced. These sets are connected to
the singular points $S_{h}\subset J^{1}(n,p)$ by the relation: if
$j^{1}f \pitchfork S_{k},$ then
$(j^{2}f)^{-1}(S_{k,h})=S_{h}(S_{k}(f)).$ The general definition of
the singular varieties $S_{k_{1},\dots,k_{r}}\subset J^{r}(N,P),$
introduced in [96] was
better formulated by J.M. Boardman, in 1967, in [11]. Mather's
account in [64] is the clearest.

In the following sections the sets $S_{k}$ and $S_{k,h}$ will be
denoted by $\Sigma^{k}$ and $\Sigma^{k,h},$ respectively.

In Chapter III, “Equivalence and stability”, Thom formulated the
problem of characterizing singularities determined by their
jet of some order.
The name finitely determined germs, was later given by John
Mather [58], who also gave
necessary and sufficient conditions for finite determinacy. The notion
of $C^{s}$-stable mappings and the example illustrating that $C^{2}$
stable mappings are not dense when $n=p=9$ were discussed in that

The notion of homotopic stability was also introduced. A mapping $f:N\to P$
is homotopically stable if for every homotopy $F:N\times I \to
P$ of $f,$ there exist $t_{0}$ and homotopies of diffeomorphisms
$\phi_{t}:N\to N,$ $0\leq t \leq t_{0},$  $\psi_{t}:P \to P,$ of
$1_{N}$ and $1_{P}$ such that $F_{t}=\psi_{t}\circ f \circ \phi_{t},\

The program for the theory of stable mappings originated from the
contributions of Whitney and Thom consisted of finding pairs of
dimensions $(n,p),$ for which there exists a set of mappings
$\mathcal{S}\subset C^{\infty}(N^{n},P^{p}),$ with the following

* $\mathcal{S}$ is a residual set in $C^{\infty}(N^{n},P^{p}),$

* The maps $f\in \mathcal{S}$ are $C^{\infty}$-stable,

* There exists a finite number of polynomial normal forms such that
every singular point of $f\in\mathcal{S}$ is equivalent to a
normal form in this list.

In a memorable series of six articles from 1968 to 1971,
John Mather found several characterizations of stability and provided
theorems answering almost completely the question of density of stable

The main results on density of stable mappings are stated below. The
proofs are based on ideas of René Thom  developed by Mather in the
sequence of papers, on Stability of $C^{\infty}$-mappings, I to VI,
[57, 60, 58, 61, 62, 63] and
[56, 65, 66]. In
these notes we review the main steps leading to the proofs of Theorems
A and B.

Let $C^{\infty}_{pr}(N,P)$ be the set of proper smooth mappings
$f:N\to P.$

Theorem A (Density of stable mappings in the
nice dimensions, Mather [63, 61])
The set $\mathcal{S}^{\infty}(N,P)$ of proper stable mappings $f:N\to
P$ is dense in $C_{pr}^{\infty}(N,P)$ if and only if $(n,p)$ is in the nice dimensions.

See section <ref> for the definition of the nice dimensions.

Theorem B (Density of topologically stable
mappings, Mather <cit.>)
The set $\mathcal{S}^{0}(N,P)$  of proper topologically stable mappings is
dense in $C_{pr}^{\infty}(N,P).$

The main tools in the proofs of theorems A and B are the notion of infinitesimal
stability, Thom's transversality theorem, the generalized
Malgrange theorem,
the notions of mappings of finite singularity type and contact
equivalence, finite determinacy and unfoldings of Mather's groups, properties of Whitney stratified sets and Thom's
isotopy theorems. Such notions and results form the framework of the
theory of singularities of differentiable mappings.

We organize the contents of the next sections as follows.

In section <ref> we introduce infinitesimally
stable and transverse stable mappings. The main goal of the section is
to discuss theorem <ref>  which establishes the equivalence
between these notions and stable mappings.

Section <ref>  gives a short presentation of
the infinitesimal machinery of singularity theory. We introduce the
contact group $\mathcal{K}$ defined by Mather as a tool to classify
stable singularities. For Mather's groups $\mathcal{G}=\mathcal{R},
\mathcal{L}, \mathcal{A}, \mathcal{C}$ and $\mathcal{K}$ we define
$\mathcal{G}$-finitely determined germs and prove the Infinitesimal
Criterion for $\mathcal{G}$-determinacy. We finish the section with a
discussion of maps of finite singularity type (FST), a global
version of $\mathcal{K}$-finitely determined germs, which plays a
central role in the proof of theorem B.

In section <ref> we define the nice dimensions
and give an outline of the proof of theorem A.

Section <ref> gives a detailed presentation of
Thom's example, illustrating  that the set of stable maps in
$C^{\infty}_{pr}(\mathbb R^{9},\mathbb R^{9})$ is not dense.

Section <ref> is dedicated to the proof of
density of topologically stable mappings $f:N \to P,$ when $N$ is
compact manifold. The general lines of the proof are discussed, although
the details are omitted.

Section <ref> gives a systematic
presentation of the topologically stable singularities in the boundary
of the nice dimensions. Much of the section is well known to experts,
however the organized presentation of the  Thom-Mather stratification in
jet space and the discussion of properties of topologically stable
mappings in these dimensions do not appear in the literature.

The question of the density of Lipschitz stable mappings is still
We report on section <ref> some recent results
of Ruas and Trivedi [88] and Nguyen, Ruas and Trivedi [75] on this subject.

In section <ref>, Damon's results relating
$\mathcal{A}$-classification  of map-germs and $\mathcal{K}_{V}$
classification of sections of the discriminant $V=\Delta(F)$ of a
stable unfolding of $f$ are reviewed and open problems are

§ EQUIVALENT NOTIONS OF STABILITY

Mather defined infinitesimally stable mappings
in [57], in order to introduce infinitesimal deformations
of a map as a tool to study stability. The main goal in this
section is to review Mather's result that, for proper mappings, stability and infinitesimal
stability are equivalent notions.

First, we introduce some notation.
Let $C^{\infty}(N)=\{\lambda:N\to \mathbb R\}$ be the ring of smooth
functions defined on the smooth manifold $N.$

We denote by $\Theta_{f}$ the $C^{\infty}(N)$-module of vector
fields along $f,$ defined as follows

\Theta_{f}=\{\sigma:N\to TP|\;\, \pi_{2}\circ \sigma=f\}

where $\pi_{2}:TP\to P$ is the projection of the tangent bundle $TP$
into $P.$

Let  $f^{*}(TP)$ denote the pull-back bundle over $N $ via $f.$ Then
the module $\Theta_{f}$ is the set of sections of this bundle.

\Theta_{N}=\{\xi:N\to TN|\;\, \pi_{1}\circ \xi=I_{N}\}

is the set of sections of the
tangent bundle of $N,$ and

\Theta_{P}=\{\eta:P\to TP|\;\, \pi_{2}\circ \eta=I_{P}\},

the set of sections of the tangent bundle of $P,$ where $I_{N}$ and
$I_{P}$ are the identities.,

The set $\Theta_{N}$ is a $C^{\infty}(N)$-module, while $\Theta_{P}$ is a
module over the ring $C^{\infty}(P).$

We have the following diagram and homomorphisms

\xymatrix{TN \ar[d]_{\pi_{1}} \ar[r]^{df} &TP  \ar[d]^{\pi_{2}} \\
N \ar[r]_f\ar[ru]^{\sigma}   &      P }.

\begin{align*}
tf:\Theta_{N}&\to \Theta_{f}\\
\xi &\mapsto tf(\xi)
\end{align*}

where $tf(\xi)(x)= df_{x}(\xi(x)),$

\begin{align*}
{\omega}f:\Theta_{P}&\to \Theta_{f}\\
\eta &\mapsto {\omega}f(\eta)=\eta\circ f
\end{align*}

The map $tf$ is a homomorphism of $C^{\infty}(N)$-modules. The map
$f:N\to P$ induces a ring homomorphism

\begin{align*}
f^{*}:C^{\infty}(P) &\to C^{\infty}(N) \\
\phi &\mapsto f^{*}(\phi) = \phi\circ f.
\end{align*}

We say that the map ${\omega}f$ is a homomorphism over
$f^{*}(C^{\infty}(P))$ (or alternatively a $C^{\infty}(P)$-module
homomorphism via $f$).

Notice that  ${\omega}f(\eta_{1}+ \eta_{2}) = (\eta_{1}+\eta_{2})\circ f={\omega}f(\eta_{1}) + {\omega}f(\eta_{2})$
and  $ {\omega}f(\alpha \eta) = (\alpha\circ f)(\eta\circ f)=(\alpha\circ f) {\omega}f(\eta),$
for any  $\alpha \in C^{\infty}(P)$ and any $\eta_{1}, \eta_{2} \in \Theta_{p}.$

Map!infinitesimally stable
The map $f:N \to P$ is infinitesimally stable if for any $\sigma \in
\Theta_{f},$ there are sections $\xi \in \Theta_{N}$ and $\eta \in
\Theta_{P}$ such that $\sigma=tf(\xi)+\eta\circ f.$

Equivalently, we can say that $\Theta_{f}=tf(\Theta_{N})+ {\omega}f(\Theta_{P}).$

If $N$ is compact, $1-1$ immersions and submersions $f:N\to P$ are
infinitesimally stable.

Infinitesimal stability has a local counterpart that we define now.
Recall that two maps $f,g:N^{n}\to P^{p}$ define the same germ at $x=a$ if they agree in some
neighborhood of $a.$ The point $x=a$ is the source of the germ
and $b=f(a)$ is its target. The analogues of the above notations
for a germ $f:(N,a) \to (P,b)$ can be obtained replacing  $N$ by
$(N,a)$ and $P$ by $(P,b)$ in the previous notation.  However to
simplify notation, we take local coordinates such that $a=0\in
\mathbb R^{n}$ and $f(a)=0\in \mathbb R^{p},$ denoting the
germ  $f:(\mathbb R^{n},0) \to (\mathbb R^{p},0).$ In this case, we
use the usual notation:

$\mathcal{E}_n=\{\lambda:(\mathbb R^{n},0)\to \mathbb R\}$ is the
local ring of $C^{\infty}$ function germs at the origin. Its unique
maximal ideal is $\mathcal{M}_{n}=\{\lambda \in
\mathcal{E}_{n}|\;\,\lambda(0)=0\}.$

$\mathcal{E}_{n}^{p}=\{f:(\mathbb R^{n},0) \to \mathbb R^{p}\}$ is a
free $\mathcal{E}_{n}$-module of rank $p,$ also denoted by $\mathcal{E}_{n,p}.$

The local version of  the previous diagram is

\xymatrix{{(T\mathbb R^{n},0)} \ar[d]_{\pi_{1}} \ar[r]^{df}
&{(T\mathbb R^{p},0)}  \ar[d]^{\pi_{2}} \\
{(\mathbb R^{n},0)} \ar[r]_f\ar[ru]^{\sigma}   &      {(\mathbb R^{p},0)} }.

The set

\Theta_{f}=\{\sigma:(\mathbb R^n,0)\to (T\mathbb R^{p},0)|\;\, \pi_{2}\circ \sigma=f\}

is the  $\mathcal{E}_{n}$-module of rank $p$ consisting of germs of vector
fields along $f.$ When $f$ is the identity in $\mathbb R^{n},$
respectively in $\mathbb R^{p},$ we obtain

\Theta_{n}=\{\xi:(\mathbb R^n,0)\to (T\mathbb R^n,0)|\;\, \pi_{1}\circ \xi=\textrm{id}_{\mathbb R^n}\}

\Theta_{p}=\{\eta:(\mathbb R^p,0)\to (T\mathbb R^p,0)|\;\, \pi_{2}\circ \eta=\mathrm{id}_{\mathbb R^p}\}

We now define the groups acting on $\mathcal{E}_{n}^P.$

Mather's groups

\begin{align*}
&\mathcal{R}=\{h:(\mathbb R^n,0)\to (\mathbb
R^n,0),\,\mathrm{ germs\;  of\; } C^\infty-\mathrm{diffeomorphisms\
in}\, (\mathbb R^n,0)\},\\
&\mathcal{L}=\{k:(\mathbb R^p,0)\to (\mathbb
R^p,0),\,\mathrm{ germs\;  of\; } C^\infty-\mathrm{diffeomorphisms\
in}\, (\mathbb R^p,0)\},
\end{align*}

and $\mathcal{A}=\mathcal{R}\times \mathcal{L}.$

The actions of the groups $\mathcal{R}, \mathcal{L}$ and $\mathcal{A}$
are as follows

\begin{align*}
\mathcal{R}\times\mathcal{E}_{n}^{p}&\to \mathcal{E}_{n}^{p}
&\mathcal{L}\times\mathcal{E}_{n}^{p}&\to \mathcal{E}_{n}^{p}   &\mathcal{A}\times\mathcal{E}_{n}^{p}&\to \mathcal{E}_{n}^{p}\\
(h,f) &\mapsto f\circ h^{-1},  &(k,f) &\mapsto k\circ f,  &((h,k),f) &\mapsto k\circ f\circ h^{-1}.
\end{align*}

These notions extend to multigerms. Let $ S=\{x_{1},x_{2},\dots
x_{s}\}$ be a finite subset of $\mathbb R^{n}.$

A multigerm at $\mathcal{S}=\{x_{1}, \dots,x_{s}\}$ is the germ of a smooth map

$$f=\{f_{1},f_{2},\dots f_{s}\}:(\mathbb R^{n},S)\to(\mathbb
R^{p},y),\; f_{i}(x_{i})=y,\ i=1,\dots, s.$$

By a local change of coordinates at each $x_{i}\in \mathcal{S},$ we
can take $f_{i}:(\mathbb R^{n},0)\to(\mathbb R^{p},0)$ and we let
$\mathcal{M}_{S}\mathcal{E}^{p}_{n,S}$ be the vector space of these map-germs, and call $f_{i}, i=1,\dots, s$ a branch of $f.$

The previous notations for monogerms extend naturally to multigerms. As before
$\Theta_{f}$ and $\Theta_{n,S}$ are $\mathcal{E}_{n,S}$-modules. The
map $ tf:\Theta_{n,S}\to \Theta_{f}$ is an $\mathcal{E}_{n,S}$-module
homomorphism defined by $tf(\xi)(x)=df_{x}(\xi(x)).$

The map-germ $f:(\mathbb R^{n},S)\to (\mathbb
R^{p},0)$ induces the ring homomorphism

\begin{align*}
f^{*}:\mathcal{E}_{p}&\to \mathcal{E}_{n,S} \\
\gamma &\mapsto f^{*}(\gamma)= \gamma\circ f,
\end{align*}

and  we say that the map

\begin{align*}
{\omega}f:\Theta_{p} &\to \Theta_{f}.\\
\eta &\mapsto {\omega}f(\eta)= \eta \circ f
\end{align*}

is a homomorphism over $f^{*}(\mathcal{E}_{p})$ (or
alternatively, an $\mathcal{E}_{p}$-module homomorphism via $f$).

Two germs $f,g:(\mathbb{R}^{n},S)\to (\mathbb{R}^{p},0)$ are
$\mathcal{A}$-equivalent ($f\widesim{\mathcal{A}}g$) if there exist
$h:(\mathbb{R}^{n},S)\to (\mathbb{R}^{n},S)$ and $k:(\mathbb{R}^{p},0)\to (\mathbb{R}^{p},0)$
such that $g=k \circ f\circ h^{-1}.$

Germ!infinitesimally stable
The germ   $f:(\mathbb{R}^{n},S)\to (\mathbb{R}^{p},0)$ is infinitesimally stable if

When we refer to an infinitesimally stable multigerm $f:(N,S)
\to (P,y),$ we use the notation

Tangent space!$\mathcal A$-equivalenceExtended tangent space!$\mathcal A$-equivalence
For the groups $\mathcal{G}=\mathcal{R},\mathcal{L},\mathcal{A},$
and any multigerm $f:(\mathbb{R}^{n},S)\to (\mathbb{R}^{p},0),$ we define the
tangent space $T\mathcal{G}_{f}$ and the extended
tangent space $T\mathcal G_{e}f$ as  follows:

\begin{align*}
T\mathcal{R}f&= tf(\mathcal{M}_{n}\Theta_{n,\mathcal{S}})
&T\mathcal{R}_{e}f&= tf(\Theta_{n,\mathcal{S}})\\
T\mathcal{L}f&= {\omega}f(\mathcal{M}_{p}\Theta_{p}) &T\mathcal{L}_{e}f&=
{\omega}f(\Theta_{p}) \\
T\mathcal{A}f&= tf(\mathcal{M}_{S}\Theta_{n,\mathcal{S}})+ {\omega}f(\mathcal{M}_{p}\Theta_{p}) &T\mathcal{A}_{e}f&= tf(\Theta_{n,\mathcal{S}})+{\omega}f(\Theta_{p})
\end{align*}

One can give a heuristic justification for the definition of the
tangent space for the groups $\mathcal G$ in the above
definition. They  can be seen as the set of
“tangent vectors” at the origin,
to “paths” $f_{t}$, such  that $f_{0}=f,$ and
$f_{t}$ is contained in the $\mathcal{G}-$orbit of $f.$ A careful
calculation in the case $\mathcal{G}=\mathcal{A},$ beginning with
$f_{t}=\psi_{t}\circ f_{t}\circ \phi_{t}$ and differentiating with
respect to $t,$ is done on pages
60-61 of the book of Mond and Nuño-Ballesteros [69].

For any group $\mathcal{G}$ acting on $\mathcal{E}_{n, S}$ the
$\mathcal{G}$-codimension and  the $\mathcal{G}_{e}$-codimension  to
the $\mathcal{G}$-orbit of $f,$ are given by

\mathcal{G}\textrm{-}{\cod f}=\dim_{\mathbb R}\frac{\mathcal{M}_{S}\Theta_{f}}{T\mathcal{G}f}\
\textrm{\ and\ }\
\mathcal{G}_e-{\cod f}=\dim_{\mathbb R}\frac{\Theta_{f}}{T{\cal G}_{e}f}.

Note that a map-germ $f\in \mathcal{E}_{n, S}$ is infinitesimally
stable if and only if $\mathcal{A}_e$-${\cod f}=0.$
Map!locally infinitesimally stable
A mapping $f:N\to P$ is locally infinitesimally stable at $\mathcal{S}=\{x_{1},
\dots, x_{s}\}\subset N$ if the germ of $f$ at $\mathcal{S}$ is
infinitesimally stable.

The next theorem shows that for proper mappings infinitesimal stability is locally a
condition of finite order. That is, if the equations can be solved
locally to order $p=\dim P,$ then they can be solved globally.

Let $f:N\to P$ be a smooth and  proper $C^{\infty}$ mapping. Then
$f$ is infinitesimally stable if and only if for every $y\in P$ and every
finite set $\mathcal{S} \subset f^{-1}(y),$  with no
more than $(p+1)$ points, we have

\Theta_{f}= tf(\Theta_{(N,\mathcal S)})+{\omega}f(\Theta_{(P,y)})+\mathcal
M^{p+1}_{\mathcal S}\Theta_{f}.

The proof of the necessity in theorem <ref> is obvious. To prove
the sufficiency, the main tool is the generalized Malgrange
Preparation Theorem proved by Mather in [57]. See
Proposition <ref> and Corollary <ref>.
A complete proof of this theorem is given in Chapter 5, section 1 of [40].

Our main goal in this section is to discuss the following theorem.

The following conditions are equivalent in $C^{\infty}_{pr}(N,P)$
for a proper mapping $f:N\to P.$

* $f$ is stable,

* $f$ is infinitesimally stable.

* $f$ is transverse  stable.

We present the main steps of the proof of Theorem <ref>. Initially we
discuss the notion of transverse stability.

§.§ Transverse stability and the proof of $2. \Leftrightarrow 3.$

The idea of transverse stability consists in defining a stratification in jet space, such
that the strata of this stratification are invariant by the action of the group $\mathcal{A}$ in jet
space. A map is transverse stable if its $k$-jet is transversal to this stratification. To make
this notion more precise, we introduce the $r$-fold $k$-jet bundle, following Mather [62].

Let $N$ and $P$ be manifolds. Let $N^{(r)}=\{(x_1,x_2,\dots, x_r)\in
N^{r} |\; x_i\neq x_j\
\textrm{if}\  i\neq j\}$. Let $\pi_{N}:J^{k}(N,P) \to N$ denote the
projection where $J^{k}(N,P)$ is the bundle of $k$-jets. We define
$_{r}J^{k}(N,P)=(\pi^{r}_{N})^{-1}(N^{(r)})$ where
$\pi^{r}_{N}:J^{k}(N,P)^{r}\to N^{r}$ is the projection.

It follows that

$$_{r}J^{k}(N,P)=\{(z_{1}, \dots, z_r)\in
J^{k}(N,P)^{r}, \mathrm{such\  that}\ \pi_{N}(z_{i})\neq \pi_{N}(z_{j}),
\ \mathrm{if}\  i\neq j\}.$$

The set $_{r}J^{k}(N,P)$ is a fibre bundle over $N^{(r)}\times P^{r},$
and we call it the r fold k-jet bundle of mappings of $N$ into

If $f:N \to P$ is a $C^{\infty}$ mapping, we define

\begin{align*}
_{r}j^{k}f:N^{(r)} &\to\  _{r}J^{k}(N,P)
\intertext{by}
_{r}j^{k}f(x_{1}, \dots, x_{r}) &=(j^{k}f(x_{1}), \dots, j^{k}f(x_{r}))
\end{align*}

The action of the group $\mathcal{A}$ in $_{r}J^{k}(N,P)$ is defined
as follows. If $(h,h') \in \mathcal{A},$
$z=(z_{1},\dots, z_{r}) \in\
_{r}J^{k}(N,P).$ $x_{i}=\pi_{N}z_{i},$ and
$j^{k}f_{i}(x_{i})=z_{i},$ then $(h,h')z=(z_{1}', \dots, z_{r}')$
where $z_{i}'=j^{k}(h'\circ f_{i}\circ h^{-1})h(x_{i}).$ We denote by
$\mathcal{A}^{k}$ the group of $k$-jets of elements in $\mathcal{A}.$

An $\mathcal{A}^{k}$ orbit $W$ in $_{r}J^{k}(N,P)$ is a submanifold.

$f:N\to P$ is transverse stable if $_{r}j^{k}f:N^{(r)} \to   {_{r}J}^{k}(N,P)$ is transverse to every $\mathcal{A}^{k}$ orbit $W$
in $_{r}J^{k}(N,P).$

An important remark is that in order to understand the local
structure of the orbits in $_{r}J^{k}(N,P)$ it is sufficient to
understand the structure of the orbits in $\pi^{r}_{P}(\Delta_{r}),$
where $\Delta_{r} \subset P^{r}$ is the diagonal (see Mather
[62] for details). In other words, it suffices to take jets
with sources $\mathcal S =\{x_{1}, \dots, x_{r}\}$ for which
$f(x_{1})= \dots =f(x_{r}).$

The next proposition gives a characterization of transversality of
$_{r}j^{k}f$ to $W;$ it is an important step in the proof of theorem <ref>.

$_{r}j^{k}f$ is transverse to $W$ at $x$ if and only if,

tf(\Theta_{(N,\mathcal S)})+{\omega}f(\Theta_{(P,y)})+\mathcal M^{k+1}_{S}\Theta_{f}=\Theta_{f},

where $y=f(x),$ $\mathcal{S}= f^{-1}(y)=\{x_{1}, \dots, x_{r}\}.$

From proposition <ref> and theorem <ref> we obtain the
proof of $2.\Longleftrightarrow 3.$ in theorem <ref>.

That $1.$ implies $3.$ in Theorem <ref> follows from a
general fact, and it is not hard to show.

In fact, let $f:N\to P$ be a stable mapping. It follows from the
transversality theorem that $f$ can be
well approximated by  a mapping $g:N\to P,$ such that $g$ is transverse
stable and $g\widesim{\mathcal A} f.$ That is, there is $(h,k)\in \mathcal{A}$
such that $g=k\circ f \circ h^{-1}.$ Now, transversality is preserved
by $\mathcal{A}$-equivalence,  hence $f$ is
transverse stable as well, as we wanted to show.

We have proved $1. \Rightarrow 2.\Leftrightarrow 3..$

Mather proved in [60], Theorem 1 that if $f$ is proper and
infinitesimally stable then it is stable, that is
$2.\Rightarrow 1..$

His proof follows from the following result.

If $f$ is proper and infinitesimally stable, then there exists a
neighborhood $U$ of $f$ in $C^{\infty}(N,P)$ and continuous mappings
$H_{1}:U\to \textrm{Diff}^{\infty}(N)$ and $H_{2}:U\to
\textrm{Diff}^{\infty}(P)$ such that $H_{1}(f)=1_{N},$
$H_{2}(f)=1_{P}$ and $g = H_{2}(g)\circ f \circ H_{1}(g),$ for $g\in U.$

Du Plessis and Wall [82] introduced the notion of
W-strongly stable mappings as stable mappings $F:N\to P$
admitting a neighborhood $U$ in $C^{\infty}(N,P)$ satisfying the
conditions stated in theorem <ref>.

The main difficult  to prove that stable mappings are $W$-strongly stable is that in the Whitney
$C^{\infty}$ topology, the composition of mappings is not
continuous. However continuity holds when one restricts to proper
mappings. The strong stability of non proper functions was recently
discussed by Kenta Hayano in [42].

It follows that the result $2. \Rightarrow 1.$ is an easy consequence of theorem <ref>.

The hypothesis that $f$ is proper cannot be omitted, as we see in the
following example.

[[60], pp.  267]
Let $N=(-1,1) \cup (1,2),$ $P=(-1,1),$ and

\begin{align*}
f|_{(-1,1)}:(-1,1) &\to (-1,1) &f|_{(1,2)}:(1,2) &\to (-1,1) \\
x &\mapsto x^{2}  &x\mapsto 2-x
\end{align*}

We can verify that $f$ is infinitesimally stable, as
the restrictions to $(-1,1)$ and $(1,2)$ are.

However, $f$ is not stable since it has the following non-stable
property: for any $a \in P,$ $f^{-1}(a)$ contains either $0,$ $1$ or
$3$ points.

The reader can find in [62] the discussion of which
implications in theorem <ref> depend on the hypothesis that
$f$ is proper.

In the next example we illustrate the role of the Whitney
$C^{\infty}$-topology in the characterization of stable mappings.

The cusp map

\begin{align*}
F:\mathbb R^{2} &\to \mathbb R^{2}\\
(x,y) &\mapsto F(x,y)=(x, y^{3}+xy)
\end{align*}

is a stable mapping when the topology  in $C^{\infty}_{pr}(\mathbb
R^{2},\mathbb R^{2})$ is the Whitney topology.
This follows from Whitney's theorem as we discussed in section
We can also apply Mather's result: the map $F$ is proper and
infinitesimally stable , hence it is stable

Let $F_{n}(x,y)=(x, y^{3}+xy+\frac{x^{2}}{n}y)$. The singular set of
$F_{n}$ is the set $\Sigma_{n}$ defined by $3y^{2}+x+\frac{x^{2}}{n}=0.$ For each $n$,
$F_{n}$ has two cusp points: $(0,0)$ and $(-n,0).$

We can easily see that $F_{n}\to F$ in $C^{\infty}_{pr}(\mathbb
R^{2},\mathbb R^{2})$ with the topology of uniform convergence on
compact sets. Hence $F$ is not stable when one considers this topology in
$C^{\infty}_{pr}(\mathbb R^{2},\mathbb R^{2}).$

§.§ Notes

The definitions and properties of infinitesimally stable mappings also
hold for real and complex analytic germs. However, care is necessary to
characterize stable maps $f:N \to P,$ when $f$ is a holomorphic map
between complex manifolds $N$ and $P.$ In fact,Thom's transversality
theorem does not hold in general in this case. See discussion by
F. Forstnerič, [34] and examples given by S. Kaliman and
M. Zaĭdenberg in [46]. In a recent paper, S. Trivedi [99]
proves that the set of maps between Stein manifolds and Oka manifolds,
transverse to a countable collection of submanifolds in the target is
dense in the space of holomorphic maps with the weak topology. The results
hold, in particular, for holomorphic maps $f:\mathbb C^{n} \to
\mathbb C^{p},$ as the complex spaces satisfy the hypothesis of the

A related problem is the characterization of topologically stable
polynomial mappings $f:\mathbb C^{n} \to \mathbb C^{p}.$ M. Farnick,
Z. Jeloneck and M.A. S. Ruas [32], characterize topologically stable
polynomial mappings $F:\mathbb C^{2}\to\mathbb C^{2}$ in the space
$\Omega_{\mathbb C^{2}}(d_{1}, d_{2})$ of polynomial mappings of degree
bounded by $(d_{1},d_{2}).$
Locally stable singularities are folds and cusps, but the behavior of
generic polynomial mappings at infinity imposes new restrictions. The
number of cusps of a topological stable $F\in \Omega_{\mathbb
C^{2}}(d_{1}, d_{2})$ is given by $c(F)= d_{1}^{2}+d_{2}^{2}+3
d_{1}d_{2}-6d_{1}-6d_{2}+7.$ In particular, when $d_{1}=1$ and
$d_{2}=3,$  $c(F)=2.$

§ FINITE DETERMINACY OF MATHER'S GROUPS

Mather's groups are the groups $\mathcal{G}=\mathcal{R},\mathcal{L},\mathcal{A},
\mathcal{K}$ and $\mathcal{C}.$

The contact group $\mathcal{K},$ defined by Mather in
[58] plays a fundamental role in the classification of
stable singularities. In subsections <ref> and <ref>
we define the group $\mathcal{K},$ discuss properties of $\mathcal{K}$-equivalence
and their role in the study of stable mappings.

The problem of classification of stable singularities motivated the
introduction of the notion of $\mathcal{G}$-finitely determined
germs [58].
For the groups $\mathcal{G}=\mathcal{R}$ or $\mathcal{K},$ finite
determinacy was studied by J. Tougeron in [97] and chapter II
of [98]. When $\mathcal{G}=\mathcal{A}$ or $\mathcal{L},$ the
first results are due to Mather's in [58].
Infinitesimal criteria of finite determinacy for
$\mathcal{G}=\mathcal{A}$ and $\mathcal{L}$ depend on the
Preparation Theorem. We discuss the infinitesimal criterion for
Mather's group in section <ref>. In section
<ref> we introduce the basic properties of maps
of finite singularity type.

§.§ The contact group

The contact group $\mathcal{K}$ is the set of pairs of germs of
diffeomorphisms $(h,H),$ where $h:(\mathbb R^{n},0)\to (\mathbb
R^{n},0),$ $H:(\mathbb R^{n}\times \mathbb R^{p},0)\to (\mathbb
R^{n}\times\mathbb R^{p},0)$ such that $\pi_{1}\circ H=h,$
$(\pi_{2}\circ H)(x,0)=0$ where $\pi_{1}$ and $\pi_{2}$ are the
projections into $\mathbb R^{n}$ and $\mathbb R^{p},$ respectively.

Notice that
$H(x,y)=(h(x),H_{2}(x,y)),\, H_{2}(x,0)=0.$

The set of pairs $(h,H)\in \mathcal{K},$ such that $h$ is the
identity $I_{\mathbb{R}^{n}}$ form a subgroup of $\mathcal{K},$ usually
denoted by $\mathcal{C}.$

Let $f,g\in \mathcal{E}_{n}^{p}.$ We say
that $f$ and $g$ are contact equivalent, $f\widesim{\mathcal{K}}g, $ if there is a pair $(h,H) \in
\mathcal{K}$ such that $H(x,f(x))=(h(x),g(h(x)).$

Notice that if $f \widesim{\mathcal{K}} g, $ then the diffeomorphism
$H:(\mathbb R^{n}\times \mathbb R^{p},0) \to (\mathbb
R^{n}\times\mathbb R^{p},0)$ sends $\mathrm{graph}(f)$ into
$\mathrm{graph}(g),$ leaving $\mathbb R^{n}\times\{0\}$ invariant (see
Figure <ref>). This geometric viewpoint of contact equivalence was
extended by Montaldi [71] as follows: two pairs of germs of submanifolds
of $\mathbb R^{m}$ have the same contact type if there is a
germ of diffeomorphism of $\mathbb R^{m}$ taking one  pair to the
other. Moreover,  he proved in [71], that the
contact type of a pair of germs of manifolds is completely characterized by
the $\mathcal{K}$-equivalence class of a convenient map. This result
is one the fundamental pieces of the applications of singularity theory
to differential geometry (see Bruce and Giblin [13] and Izumiya,
Romero-Fuster, Ruas and Tari,  [45]).

Tangent  space!$\mathcal{K}$-equivalenceExtended tangent space!$\mathcal{K}$-equivalence
The tangent space and the extended tangent space of
$\mathcal{K}$-equivalence are, respectively

\begin{align*}
T\mathcal{K}f &=tf(\mathcal{M}_{n}\Theta_{n})
T\mathcal{K}_{e}f &=tf(\Theta_{n})
\end{align*}

We also define $\mathcal{K}$-${\cod f}=\dim_{\mathbb
R}\frac{\mathcal{M}_{n}\Theta_{f}}{T\mathcal{K}f}$ and $\mathcal{K}_e$-${\cod
f}=\dim_{\mathbb R}\frac{\Theta_{f}}{T\mathcal{K}_{e}f}.$

Contact equivalence

The following result was first proved by Mather in [61].

The following statements are equivalent.

(1)  Two map-germs $f,g \in \mathcal{E}_{n}^{p}$ are $\mathcal{K}$-equivalent.

(2)  There exists a germ of diffeomorphism $h:(\mathbb R^{n},0) \to (\mathbb R^{n},0)$ such that

The local algebra we introduce now is an useful invariant of $\mathcal{K}$-equivalence.
For a given map-germ $f:(\mathbb R^{n},0) \to (\mathbb R^{p},0)$ we
define the local algebra of $f$ as Local algebra

\frac{\mathcal{E}_{n}}{f^{*}(\mathcal{M}_{p})\mathcal{E}_{n}}.

It follows from the previous proposition that the isomorphism class of
$Q(f)$ is a $\mathcal{K}$-
Furthermore, it is a complete invariant of
$\mathcal{K}$-equivalence for germs $f$ with finite
$\mathcal{K}$-codimension. More precisely, we have

If $f$ and $g$ are map-germs with finite $\mathcal{K}$-codimension it
follows that

$$  f\widesim{\mathcal{K}}g \mathrm{\ if\ and\ only\ if\ the\ local\
algebras\ } Q(f)\ \mathrm{\ and\ } Q(g) \mathrm{\ are\ isomorphic.\ } $$

For complex analytic germs the hypothesis of $\mathcal{K}-$determinacy
in Theorem <ref> is not needed.

Let $F:(\mathbb R^{n},0) \to (\mathbb R^{p},0)$ be a germ of rank
$r.$ Then, up to $\mathcal{A}$-equivalence, we can take $F$ in the
normal form $F(x,y)=(x,f(x,y)), x \in \mathbb R^{r}, y \in \mathbb
R^{n-r},$ with $f:(\mathbb R^{n},0) \to (\mathbb R^{p-r},0)$ and
$j^{1}f(0,0)\equiv 0.$ Let $f_{0}:(\mathbb R^{n-r},0) \to (\mathbb
R^{p-r},0)$ be the rank zero germ $f_{0}(y)=f(0,y).$ Then

If $\mathcal{K}$-$\cod f_{0} <\infty$ and $Q(F) \cong Q(f_{0})$ it follows
that $F$ is $\mathcal{K}$-equivalent to the suspension $F_{0}(x,y)=
(x,f_{0}(y))$ of $f_{0}.$

As we shall see in the next section, germs $f\in \mathcal{E}_{n}^{p}$ of
finite $\mathcal{K}$-codimension are finitely
$\mathcal{K}$-determined, and in this case
$\mathcal{K}(f)=\mathcal{K}(z),$ where $z=j^{k}f(0)$ for some $k.$

Now, for each positive integer $k,$ we set

$Q_{k}(f)$ is the local algebra of $z=j^{k}f(0).$ We can also write $Q_{k}(f)=Q(z).$

It is not hard to show that $z\widesim{\mathcal {K}^{k}}z'$  if and
only if $Q_{k}(z)$ and $Q_{k}(z')$ are isomorphic.

This definition can be extended to  $k$-jets of a multigerm $f:(\mathbb
R^{n},S)\to (\mathbb R^{p},0)$ $S=\{x_{1}, x_{2}, \dots, x_{s}\}.$
By a contact class in $J^{k}(N,P)$ we mean an equivalence class of
$ _{s}J^{k}(N,P)$ under the relation of $\mathcal{K}^{k}$-equivalence.

§.§ Finitely determined germs

Let $\mathcal{G}$ be a group acting in the space of germs $f:(\mathbb
R^{n},0) \to (\mathbb R^{p},0).$  We say that $f$ is
finitely $\mathcal{G}$-determined$\mathcal{G}$-determined if there exists a positive integer
$k$ such that for all $g:(\mathbb R^{n},0) \to (\mathbb R^{p},0)$ with
$j^{k}g(0)=j^{k}f(0),$  it follows that $f\widesim{\mathcal{G}} g.$ We
say that $f$ is $\mathcal{G}$-finitely determined if $f$ is
$k$-determined for some $k.$ The denomination ${\cal
G}$-finite germs is also widely used.

Finite determinacy has been an important subject in singularity theory
for many decades and the bibliography in this topic is extensive.

With regard to results on necessary and sufficient conditions of
finite determinacy and estimates of the order of determinacy we
refer  to Mather [58], Gaffney [36, 37], du
Plessis [79], Damon [24] and Du Plessis, Bruce and
Wall [14]. The survey article by Terry Wall
[108] is a complete account of the theory of finite
determinacy for Mather's groups $\mathcal{G}=\mathcal{A},
\mathcal{R}, \mathcal{L}, \mathcal{K}$ and $\mathcal{C}$ until
1981. See also the clear presentation (with examples) in Chapter 6
of the book of Mond and Nuño-Ballesteros [69].

An important advance appeared in [24] in which J. Damon  defined
the geometric subgroups of $\mathcal{K},$  a large class of
subgroups for which the theory of finite determinacy can be
formulated as for Mather's group.

The following theorem, known as infinitesimal
criterionFinite determinacy!infinitesimal criterion of gives
necessary and sufficient conditions for finite determinacy.
The original result is due to Mather [58]. We give here an
improved version due to Gaffney [37] and du Plessis
[79]. The statement and proof of Theorem <ref> are
slight modifications  of T. Wall <cit.>. The reader can also compare the statement for the group
$\mathcal{A}$ in section 1.2.3 (Theorem 1.2.12) of the article of Mond
and Nuño-Ballesteros in this Handbook [70].

For each $f\in \mathcal{E}^{p}_{n}$, $\mathcal{G}=\mathcal{R},
\mathcal{L}, \mathcal{A}, \mathcal{C}, \mathcal{K}$ the following conditions are

(1) $f$ is finitely $\mathcal{G}$-determined,

(2) for some $r,$ $T\mathcal{G}f \supset \mathcal{M}^r_n
\Theta_{f},$

(3) $\mathcal{G}$-$\cod f <\infty,$

(4) $\mathcal{G}_{e}$-$\cod f <\infty.$

More precisely, if we set $\epsilon =1$ for $\mathcal{G}=\mathcal{R},
\mathcal{C}$ or $ \mathcal{K}$ and $\epsilon=2$ for $\mathcal{G}= \mathcal{L}, \mathcal{A},$

(i) If $f$ is $k$-$\mathcal{G}$-determined then $T\mathcal{G}f \supset \mathcal{M}^{k+1}_n
\Theta_{f},$

(ii) If $T\mathcal{G}f \supset \mathcal{M}^{k+1}_n\Theta_{f},$ then
$f$ is  $(\epsilon k+1)$-$\mathcal{G}$-determined.

(iii)  If $T\mathcal{G}f + \mathcal{M}^{\epsilon
k+2}_n\Theta_{f} \supset \mathcal{M}^{k+1}_n\Theta_{f},$
then $T\mathcal{G}f \supset \mathcal{M}_n^{k+1}\Theta_{f}.$

This section is mainly devoted to describe this result. Although the
theory applies to multigerms,  for   simplicity we restrict our
discussion to monogerms $f:(\mathbb R^{n},0) \to (\mathbb R^{p},0).$

The successful approach to finite determinacy was inspired by the
action of a Lie group on finite dimensional manifolds.
The following lemma is due to Mather.

Let $G$ be a Lie group, $M$ a $C^{\infty}$ manifold and
$\alpha:G\times M \to M$ a $C^{\infty}$ action. Let $V$ be a
connected $C^{\infty}$-submanifold of $M.$ Then $V$ is contained in
an orbit of $\alpha$ if and only if

(a) For all $v\in V,$ $T_{v} G\cdot v \supseteq T_{v}V,$ and

(b) $\dim T_{v}(G\cdot v)$ is the same for all $v\in V.$

Our groups are not Lie groups, and our function spaces are not Banach
manifolds. But, the solution to the problem of finding sufficient
conditions for a germ $f\in \mathcal{E}^{p}_{n}$ to be finitely
determined, consists in reducing our infinitesimal approach to jet

Suppose $f$ is $k$-$G$-determined. Then, given $g\in
\mathcal{E}^{p}_{n},$ $j^{k}g(0)=j^{k}f(0),$ the one-parameter family

\begin{align*}
\bar f:(\mathbb R^{n}\times \mathbb R, 0 \times \mathbb R) &\to (\mathbb R^{p}\times \mathbb
(x,t) &\mapsto \bar f(x,t)=(1-t)f(x) + t g(x)
\end{align*}

has a constant $k$-jet $j^{k}\bar f_{t}(0)=j^{k}f(0) + t j^{k}(g-f)(0)=

We will identify $\bar f$ with a “line” $L_{t}$ in
$\mathcal{E}^{p}_{n}.$ Our problem is to show that $L_{t}$ is
contained in a unique orbit.

A sufficient condition is to find a $1$-parameter family $h_t$ of
elements in $\mathcal{G}$ such that $h_{0}=1 \in \mathcal{G},$
$h_{t}(0)=0,$ $h_{t}\cdot f_{t}=f,$ for any $t\in \mathbb R.$
Trivial family!$\mathcal{G}-$
These conditions say that the family $\bar f$ is
$\mathcal{G}$-trivial. As in the case of stable singularities,
the next step is to search for an infinitesimal condition, giving an
equivalent characterization of triviality in terms of vector fields.

This step, in principle, is not hard: the equation $h_{t}\cdot
f_{t}= f$ implies that $\frac{\partial}{\partial t}(h_{t}\cdot
f_{t})=0$ leading to the desired infinitesimal condition. The converse
follows from integration of vector fields.

For any group $\mathcal{G}$ acting on $\mathcal{E}_{n}^{p},$ we call
this result  “the Thom-Levine lemma.”  We now specialize to $\mathcal{G}=\mathcal{A},$ as this case
includes all difficulties  of the proof of the infinitesimal criterion.

Trivial family!$\mathcal{A}-$
A $1$-parameter family $\bar f:(\mathbb R^{n}\times \mathbb R, 0) \to
(\mathbb R^{p},0)$, $\bar f(x,0)=f(x)$ is $\mathcal{A}$-trivial if there is a pair $(h,k)$ of
$1$-parameter families of germs of diffeomorphisms

\begin{align*}
h:(\mathbb R^{n}\times \mathbb R, 0) &\to (\mathbb R^{n},0)
&k:(\mathbb R^{p}\times \mathbb R, 0) &\to (\mathbb R^{p},0)\\
(x,t) &\mapsto h(x,t)    &(y,t) &\mapsto k(y,t)
\end{align*}

such that $h(x,0)=x$, $k(y,0)=y,$ $ h_{t}(0)=0$, $k_{t}(0)=0$ and

k_{t}\circ f_{t}\circ h_{t} =f.

We also use the notation $F(x,t)=(\bar f(x,t),t),$
$H(x,t)=(h(x,t),t)$ and   $K(y,t)=(k(y,t),t)$  for the corresponding
$1$-parameter unfoldings. In this  notation $F$ is
$\mathcal{A}$-trivial if $K\circ F \circ H = f\times
\mathrm{Id}_{\mathbb R}.$
We denote by $\partial\cdot F$ the vector field in $(\mathbb
R^{n}\times\mathbb R, 0)$ with zero component in the
$\frac{\partial}{\partial t}$ direction, that is $dF(\frac{\partial}{\partial
t})=(\partial\cdot F, 1).$

The next result is known as the Thom-Levine lemma (see [58, 79, 69]).
Lemma!Thom-LevineThom-Levine|see Lemma

Let $f\in \mathcal{E}_{n}^{p}$ and $F:(\mathbb R^{n}\times \mathbb R, 0) \to
(\mathbb R^{p}\times \mathbb R,0),$ $F(x,t)=(\bar f(x,t),t),$ $\bar
f(0,t)=0, \,\bar f(x,0)=f(x),$ the germ at $0$ of a $1$-parameter unfolding of $F.$ Then $F$ is
$\mathcal{A}$-trivial if and only there exist vector fields
$V:(\mathbb R^{n}\times \mathbb R, 0) \to (\mathbb
R^{n}\times \mathbb R,0)$ with $V(x,t)=v(x,t)
+\frac{\partial}{\partial t},$ $v(x,t)=
\sum_{i=1}^{n}v_{i}(x,t)\frac{\partial}{\partial x_{i}},$
$v_{i}(0,t)=0$ for  $i=1,\dots,n$
and   $ W:(\mathbb R^{p}\times \mathbb R, 0) \to (\mathbb
R^{p}\times \mathbb R,0)$ with $W(y,t)=w(y,t)
+\frac{\partial}{\partial t},$ $w(y,t)=
\sum_{j=1}^{p}w_{j}(y,t)\frac{\partial}{\partial y_{j}},$
$w_{j}(0,t)=0$ for  $j=1,\dots,p.$
such that

\begin{equation}
\label{eq:2}
\partial\cdot F(x,t)=\sum_{i=1}^{n}\frac{\partial \bar f}{\partial
x_{i}}(x,t)\cdot v_{i}(x,t) + w \circ F(x,t).
\end{equation}

We give here an idea of the proof.  The reader may consult, for
instance, Mather <cit.>, du Plessis <cit.>, or
Mond and Nuño-Ballesteros <cit.> for a
complete proof.

If $F$ is a trivial unfolding of $f,$ $K\circ F \circ H = f\times 1_{\mathbb R}$
and then $\partial \cdot(K\circ F \circ H)=0$ and we apply the chain
rule to get (<ref>).

Conversely, if condition (<ref>) holds, we consider the systems of
differential equations in $(\mathbb R^{n}\times \mathbb R,0)$ and  $(\mathbb
R^{p}\times \mathbb R,0),$ respectively:

\begin{align}\label{eq:4}
&\begin{cases}\dot x = v(x,t)\\
\end{cases} \qquad \qquad \qquad \qquad
\begin{cases}
\dot y= w(y,t)  \\
\end{cases}\\
& \qquad \nonumber
\end{align}

We can integrate these vector fields to obtain $1-$parameter families
$h_{t}$ and $k_{t}$ of diffeomorphisms of $(\mathbb R^{n}\times \mathbb R,0)$ and  $(\mathbb
R^{p}\times \mathbb R,0),$ respectively, such that $h_{0}(x)=x,$
$h_{t}(0)=0;$ $k_{0}(y)=y,$ $k_{t}(0)=0$ and  $k_{t}\circ \bar
f_{t}\circ h_{t}=f.$

Condition (<ref>) in Proposition <ref> admits an useful
algebraic formulation. First, we introduce some notation.

Given the $1$-parameter unfolding $F:(\mathbb R^{n}\times \mathbb R,0)
\to (\mathbb R^{p}\times \mathbb R,0),$ $F(x,t)=(\bar f(x,t),t)$ with
$\bar f(x,0)=f(x),$ as before, $\Theta_{F}$ denotes the
$\mathcal{E}_{n+1}$ module of vector fields along $F.$ However, here
it will be more convenient to consider the submodule of $\Theta_{F}$
defined as:

\Psi_{F}=\{\sigma \in \Theta_{F}|\, \mathrm{the}\ \mathbb R\mathrm{-component\ of}
\ \sigma\ \mathrm{is\  zero}\}.

Similarly, $\Psi_{n+1}$ and $\Psi_{p+1}$ denote vector fields in
$(\mathbb R^{n}\times \mathbb R,0)$ and $(\mathbb R^{p}\times \mathbb
R,0)$ respectively, with zero $\mathbb R$-components.

The restrictions of the homomorphisms $tF$ and ${\omega}F$ give respectively
the $\mathcal{E}_{n+1}$-homomorphism $tF:\Psi_{n+1}\to \Psi_{F}$ and
the $\mathcal{E}_{p+1}$-homomorphism via $F^{*},$ ${\omega}F:\Psi_{p+1} \to \Psi_{F}.$

With this notation, we can see that (<ref>) holds if and
only if

\begin{equation}
\label{eq:6}
\partial \cdot F \in tF(\mathcal{M}_{n}\Psi_{n+1}) +
\end{equation}

We call $T\mathcal{A}_{un}(F)=tF(\mathcal{M}_{n}\Psi_{n+1}) +
{\omega}F(\mathcal{M}_{p}\Psi_{p+1}),$ the $\mathcal{A}$-tangent space of the
unfolding $F.$ Similarly $T\mathcal{K}_{un}(F)=tF(\mathcal{M}_{n}\Psi_{n+1}) +
F^{*}(\mathcal{M}_{p+1})\Psi_{p+1}$ is the
$\mathcal{K}-$tangent space of $F.$

We now turn to the algebraic tools we need in the proof of theorem

In the cases $\mathcal{G}=\mathcal{R},\,\mathcal{C}$ or $\mathcal{K}$
the proof of the infinitesimal criterion of $\mathcal{G}$-determinacy

will follow from the following elementary result.
Lemma!NakayamaNakayama|see Lemma

Let $R$ be a commutative ring, $M$ an ideal such that for $x\in M,$
$(1+x)$ is invertible. Let $C$ be a finitely generated $R$-module,
$A$ a submodule, then

(i) if $A+ M\cdot C=C,$ then $A=C,$

(ii)  if $R$ is a $k$-algebra, and
$\dim_{k}(\frac{C}{A+M^{d+1}C})\leq d $ then $M^{d}\cdot C
\subseteq A.$

An equivalent formulation of condition (i) in Lemma <ref> is the

(i')  If $M C = C,$ then $C=0.$

When $\mathcal{G}=\mathcal{L}$ or $\mathcal{A},$ we need
a fairly deep result, the generalized Malgrange preparation theorem
(see Golubitsky and Guillemin [40], Martinet [54, 55],
Wall [108]).

Malgrange preparation theorem

Let $f:(\mathbb R^{n},0) \to (\mathbb R^{p},0)$ be a $C^{\infty}$
map-germ, $E$ a finitely generated $\mathcal{E}_{n}$-module. If
$\dim_{\mathbb R}(\frac{E}{f^{*}(\mathcal{M}_{p})\cdot E})<\infty,$ then $E$ is
finitely generated as $\mathcal{E}_{p}$-module (via $f$).

The next proposition is a consequence of the Preparation theorem. It
is an useful tool to study $\mathcal{A}$-finite determinacy.

Let $C$ be a finitely generated $\mathcal{E}_{n}$-module,
${B} \subset {C}$ a finitely generated
${\mathcal E}_{n}$-submodule, ${A} \subset f^{*}(\mathcal{M}_{p}){C}$ a finitely generated
$\mathcal{E}_{p}$-submodule   (via $f$), and ${M}$ a proper,
finitely generated ideal in $\mathcal{E}_{n}.$ If

{M}{C} \subset {A} + {B} +
{M}(f^{*}(\mathcal{M}_{p}) + {M}){C}

then ${M}{C} \subset {A} + {B}.$

We are now ready to prove Theorem <ref>.

First we notice that (i) and (ii)  give respectively the implications
$(1)\Rightarrow (2)$ and  $(2)\Rightarrow (1).$
The implication  $(2)\Rightarrow (3)$ is trivial since
$\mathcal{M}^{k}_{n}\Theta_{f}$ has finite codimension.

It is easy to prove the equivalence between $(3)$ and $(4).$ The
implication $(3)\Rightarrow (2)$ will follow from (iii), as we now

For any

$$ c_{k}=\dim_{\mathbb K}\frac{\mathcal{M}_{n}\Theta_{f}}{T\mathcal{G}f
+\mathcal{M}^{k}_{n}\Theta_{f}}, \ k \geq 1.

Since $\mathcal{G}$-$\textrm{cod}\, f< \infty,$ the sequence

0=c_{1}\leq c_{2}\leq \dots \leq \mathcal{G}\textrm{-cod}\, f$$

is finite.

Then, there exists $s$ such that $c_{k}=c_{s}$ for all $k\geq s+1.$ It
follows that
$T\mathcal{G}f+\mathcal{M}^{s}_{n}\Theta_{f} =
T\mathcal{G}f+\mathcal{M}^{k}_{n}\Theta_{f}$ for all $k\geq s+1.$ In
particular $\mathcal{M}^{s}_{n}\Theta_{f}\subseteq
T\mathcal{G}f+\mathcal{M}^{k}_{n}\Theta_{f}$ for all $k\geq s+1.$
Taking $k=s+1,$ when
$\mathcal{G}=\mathcal{R},\mathcal{C},\mathcal{K}$ and $k=2s,$ when
$\mathcal{G}=\mathcal{A},\mathcal{L},$ we obtain the statement in
(iii) from which the result follows.

It suffices to prove (i), (ii) and (iii). For a clearer presentation,
we first prove (iii).

If $\mathcal{G}=\mathcal{R},\mathcal{C},\mathcal{K},$ the result
follows easily by Nakayama's Lemma. If $\mathcal{G}=\mathcal{A}$ (the
argument for $\mathcal{G}=\mathcal{L}$ is similar) we apply  Proposition <ref> taking $C=\Theta_{f},$
$M=\mathcal{M}^{k+1}_{n},$  $B=tf(\mathcal{M}_{n}\Theta_{n})$ and
$A=\omega f(\mathcal{M}_{p}\Theta_{p}).$

We leave the details as an exercise to the reader.
Finite determinacy!necessary condition for
(i) Necessary condition for finite determinacy.

This is not hard. A map-germ $f:(\mathbb
R^{n},0) \to (\mathbb R^{p},0)$ is $k$-$\mathcal{G}$-determined if
$\mathcal{G}  f$ contains all germs $g \in \mathcal{E}_{n}^{p},$ such
that $j^{k}g(0)=j^{k}f(0).$ Let us denote this set by $\mathtt{W}.$

\begin{align*}
\pi^{l}:\mathcal{E}_{n}^{p} &\to J^{l}(n,p) \\
g &\to j^{l}g(0).
\end{align*}

As $\mathcal{G} f\supset \mathtt{W},$  then
$\pi^{l}(\mathcal{G} f)\supset \pi^{l}(\mathtt{W}).$ Thus we also get that

\begin{equation}
\label{eq:3}
\textrm{the tangent space of } \pi^{l}(\mathcal{G}f)
\supset \textrm{the tangent space of } \pi^{l}(\mathtt{W}).
\end{equation}

Notice that for all $l>k,$ the set $\pi^{l}(\mathtt{W})$ is the
affine subspace of $J^{l}(n,p)$ consisting of all $l$-jets whose
$k$-jet is $j^{k}f(0).$ Hence we can rewrite  (<ref>) as

$$ T\mathcal{G}f + \mathcal{M}_{n}^{l+1}\Theta_{f}\supset
\mathcal{M}_{n}^{k+1}\Theta_{f},\, l>k.$$

The result now follows from (iii) taking $l=k+1$ for
$\mathcal{G}=\mathcal{R},\,\mathcal{C}$ or $\mathcal{K}$ and  $l=2k+1$
when $\mathcal{G}=\mathcal{A}$ or $\mathcal{L}.$

Finite determinacy!sufficient condition for
(ii) Sufficient condition for finite determinacy.

Let $f,g \in \mathcal{E}_{n}^{p},$ $j^{\epsilon k +1}f(0)=
j^{\epsilon k +1}g(0)$, $\epsilon= 1$ or $2,$ $F(x,t)=(\bar
f(x,t),t),$ where $\bar f(x,t)= (1-t)f(x) + tg (x), t\in [0,1].$

(I) $\mathcal{G}=\mathcal{R},\,\mathcal{C}$ or $\mathcal{K}.$

In these cases the hypothesis

\begin{equation}
\label{eq:7}
T\mathcal{G}f\supset \mathcal{M}_{n}^{k+1}\Theta_{f}
\end{equation}

\begin{equation}
\label{eq:8}
T\mathcal{G}_{un}(F)+\mathcal{M}^{k+2}_{n+1}\Psi_{F}\supseteq \mathcal{M}^{k+1}_{n}\Psi_{F}.
\end{equation}

The proof that (<ref>) implies (<ref>) is not hard, but we omit it (the reader may consult Wall
[108] or du Plessis [79]).

The tangent spaces $T\mathcal{G}_{un}(F),$ $\mathcal{G}=\mathcal{R},
\mathcal{C}$ or $\mathcal{K},$ are finitely generated
$\mathcal{E}_{n+1}$-modules, so we can apply Nakayama's lemma to
(<ref>) with $C= T\mathcal{G}_{un}(F)+\mathcal{M}^{k+1}_{n}\Psi_{n+1},$
$A=T\mathcal{G}_{un}(F)$ and  $M= \mathcal{M}_{n+1}$ to get
$T\mathcal{G}_{un}(F)\supseteq \mathcal{M}_{n}^{k+1}\Psi_{F}. $

Now, $\partial\cdot F = g-f \in \mathcal{M}_{n}^{k+2}\Psi_{F},$ and we
can apply the Thom-Levine lemma to prove that $F$ is
$\mathcal{G}$-trivial in some neighborhood of $t=0.$ For a proof of
the Thom-Levine lemma for $\mathcal{G}=\mathcal{K}$ see
du Plessis et all [39]. Notice that
$j^{k+1}\bar f_{t}(0)= j^{k+1}f(0), $ and the hypothesis (ii) holds
for $\bar f_{a},$ for any $a\in [0,1]$ , so
that the arguments of the proof also hold to prove that $F$ is
$\mathcal{G}$-trivial in a small neighborhood of $t=a$ for any $a \in
[0,1].$ Hence f is $(k+1)$-$\mathcal{G}$-determined, $\mathcal{G}=\mathcal{R},\,\mathcal{C}$ or $\mathcal{K}.$

(II) $\mathcal{G}=\mathcal{L}$ or $\mathcal{A}.$

In these cases, $T\mathcal{G}_{un}(F)$ is not an $\mathcal{E}_{n+1}$-module
in general. Let $\mathcal{G}=\mathcal{A}$ ( the case
$\mathcal{G}=\mathcal{L}$ follows as a particular case).

$$F(x,t)=(\bar f(x,t), t),\ \bar f(x,t)= (1-t)f(x)+tg(x),$$

First notice that if $F_{0}(x,t)=(f(x),t)$ is the suspension of $f,$
the hypothesis $\mathcal{M}^{k+1}_{n}\Theta_{f}\subseteq
tf(\mathcal{M}_{n}\Theta_{n}) +{\omega}f(\mathcal{M}_{p}\Theta_{p})$ implies
\mathcal{M}^{k+1}_{n}\Theta_{F_{0}}\subseteq
tF_{0}(\mathcal{M}_{n}\Psi_{n+1})+{\omega}F_{0}(\mathcal{M}_{p}\Psi_{p+1})+(t\mathcal{M}_{n}^{k+1} +
\mathcal{M}_{n}^{2k+2})\Psi_{F_{0}}

Notice that $\mathcal{M}_{n}^{k+1}\Psi_{F_{0}}\subset
\mathcal{M}_{n}^{k+1}\Theta_{f}+t\mathcal{M}_{n}^{k+1}\Psi_{F_{0}}.$

The next step is to verify that similar  inclusion holds replacing
$F_{0}$ by $F,$ $j^{2k+1}\bar f_{t}(0)=j^{2k+1}f(0), $ that is

\begin{equation}
\label{eq:9}
\mathcal{M}_{n}^{k+1}\Psi_{F} \subset  tF(\mathcal{M}_{n}\Psi_{n+1})
+ {\omega}F(\mathcal{M}_{p}\Psi_{p+1})+(t\mathcal{M}_{n}^{k+1}+ \mathcal{M}_{n}^{2k+2})\Psi_{F}
\end{equation}

(see sublemma 2.2 in du Plessis [79]).

If we can show that the term $(t\mathcal{M}_{n}^{k+1}+
\mathcal{M}_{n}^{2k+2})\Psi_{F}$ can be eliminated in (<ref>)
then the Thom-Levine lemma can be applied to prove that $F$ is

To achieve this goal Malgrange's  preparation theorem will be the
fundamental tool.

Multiplying (<ref>) by $\mathcal{M}_{n}^{k+1}$ and since $\mathcal{M}_{n}^{k+1}{\omega}F(\mathcal{M}_{p}\Psi_{p+1})\subset
F^{*}(\mathcal{M}_{p})\mathcal{M}_{n}^{k+1} \Psi_{F},$
we get

\begin{equation}
\label{eq:10}
\mathcal{M}_{n}^{2k+2}\Psi_{F}\subset
tF(\mathcal{M}_{n}^{k+2}\Psi_{n+1}) +
F^{*}(\mathcal{M}_{p})\mathcal{M}_{n}^{k+1}\Psi_{F} +(t +\mathcal{M}_{n}^{k+1})\mathcal{M}_{n}^{2k+2}\Psi_{F}.
\end{equation}

The $\mathcal{E}_{n+1}$-module

E=\frac{tF(\mathcal{M}_{n}^{k+2}\Psi_{n+1}) +
F^{*}(\mathcal{M}_{p})\mathcal{M}_{n}^{k+1}\Psi_{F} +
\mathcal{M}_{n}^{2k+2}\Psi_{F}.}
{tF(\mathcal{M}_{n}^{k+2}\Psi_{n+1}) +  F^{*}(\mathcal{M}_{p})\mathcal{M}_{n}^{k+1}\Psi_{F}}

is finitely generated, since it is a quotient of finitely generated modules.
Moreover, from (<ref>) we get that $E=(t +\mathcal{M}_{n}^{k+1})E,$ and by
Nakayama's lemma it follows that $E=0.$ Then, we get

\begin{equation}
\label{eq:11}
\mathcal{M}_{n}^{2k+2}\Psi_{F}\subset  tF(\mathcal{M}_{n}^{k+2}\Psi_{n+1}) +
\end{equation}

Using (<ref>) to replace part of the remainder term in
(<ref>), we get

\begin{equation}
\label{eq:12}
\mathcal{M}_{n}^{k+1}\Psi_{F} \subset tF(\mathcal{M}_{n}\Psi_{n+1})
+ {\omega}F(\mathcal{M}_{p}\Psi_{p+1})+ (t+F^{*}(\mathcal{M}_{p}))\mathcal{M}_{n}^{k+1}\Psi_{F}.
\end{equation}

Let $E'$ be the $F^{*}(\mathcal{E}_{p+1})$-module

E'=\frac{tF(\mathcal{M}_{n}\Psi_{n+1}) +{\omega}F(\mathcal{M}_{p}\Psi_{p+1})+
\mathcal{M}_{n}^{k+1}\Psi_{F}}{tF(\mathcal{M}_{n}\Psi_{n+1}) +{\omega}F(\mathcal{M}_{p}\Psi_{p+1})}.

Using (<ref>), it follows that
$E'=(t+F^{*}(\mathcal{M}_{p}))E'.$ Notice that the ideal $\langle
t\rangle + F^{*}(\mathcal{M}_{p})$ is contained in
$F^{*}(\mathcal{M}_{p+1}),$ so it follows that $E'=F^{*}(\mathcal{M}_{p+1})E'.$

To apply Nakayama's lemma, one has to show that $E'$ is a
$F^{*}(\mathcal{E}_{p+1})$-module finitely generated. For this, let
the finitely generated $\mathcal{E}_{n+1}$-module

E''=\frac{tF(\mathcal{M}_{n}\Psi_{n+1}) +\mathcal{M}_{n}^{k+1}\Psi_{F}}{tF(\mathcal{M}_{n}\Psi_{n+1})}.

Notice that the inclusion

+\mathcal{M}_{n}^{k+1}\Psi_{F}\subset tF(\mathcal{M}_{n}\Psi_{n+1}) +
{\omega}F(\mathcal{M}_{p}\Psi_{p+1}) +\mathcal{M}_{n}^{k+1}\Psi_{F}$$

induces an epimorphism of $F^{*}(\mathcal{E}_{n+1})$-modules $E'' \to E'$ so
that if $E''$ is a finitely generated $F^{*}
(\mathcal{E}_{p+1})$-module, then $E'$ also is.

From Malgrange preparation theorem, $E''$ is a finitely generated
$F^{*}(\mathcal{E}_{p+1})$-module if and only if

\begin{equation}
\label{eq:13}
\dim_{\mathbb R}\frac{E''}{F^{*}
(\mathcal{M}_{p+1})E''}< \infty.
\end{equation}

\frac{E''}{F^{*}(\mathcal{M}_{p+1})E''} \simeq
\frac{tF(\mathcal{M}_{n}\Psi_{n+1})+\mathcal{M}_{n}^{k+1}\Psi_{F}}
{tF(\mathcal{M}_{n}\Psi_{n+1}) +F^{*}(\mathcal{M}_{p+1})\mathcal{M}_{n}^{k+1}\Psi_{F}}

It follows from (<ref>) that

F^{*}(\mathcal{M}_{p+1})\mathcal{M}_{n}^{k+1}\Psi_{F} \supset \mathcal{M}_{n}^{2k+2}\Psi_{F}.

As $t\in F^{*}(\mathcal{M}_{p+1}),$ we also get that

F^{*}(\mathcal{M}_{p+1})\mathcal{M}_{n}^{k+1}\Psi_{F} \supset
\mathcal{M}_{n+1}^{k+1}\mathcal{M}_{n}^{k+1}\Psi_{F},

so that,

\dim_{\mathbb R} \frac{E''}{F^{*}(\mathcal{M}_{p+1})E''} \leq
\dim_{\mathbb R} \frac{\mathcal{M}_{n}^{k+1}\Psi_{F}}{\mathcal{M}_{n+1}^{k+1}\mathcal{M}_{n}^{k+1}\Psi_{F}} <\infty

Then we can
apply Nakayama's lemma to (<ref>) to get that $E'=0,$ so
that $\mathcal{M}_{n}^{k+1}\Psi_{F} \subset
tF(\mathcal{M}_{n}\Psi_{n+1}) + {\omega}F(\mathcal{M}_{p}\Psi_{p+1}).$

To conclude we proceed as in part (I).

The following result follows from Theorem <ref> and Mather's

Let $f\in \mathcal{E}_{n}^{p},$  $\epsilon =1$ when
$\mathcal{G}=\mathcal{R},\mathcal{C}$ or $\mathcal{K}$ and
$\epsilon=2$ when   $\mathcal{G}=\mathcal{L},\mathcal{A}.$
Then $f$ is $k$-$\mathcal{G}$-determined if and only if
$\mathcal{M}_{n}^{k+1}\Theta_{g}\subset T\mathcal{G}g+
\mathcal{M}_{n}^{\epsilon(k+1)}\Theta_{g}$ for all $g \in
\mathcal{E}_{n}^{p}$ such that $j^{k}g(0)=j^{k}f(0).$

We see in the next example that the converse of condition $(i)$ in theorem
<ref> does not hold, that is, the condition $T\mathcal{G}f\supseteq
\mathcal{M}_{n}^{k+1}\Theta_{f}$ does not imply in general that $f$ is $k$-$\mathcal{G}$-determined.

Let $f:(\mathbb R^{2},0) \to (\mathbb R, 0)$, $f(x,y)= x^{3}+y^{3},$ and
$\mathcal G= \mathcal R.$ Then
$$T\mathcal{R} f= \langle \frac {\partial f}{\partial x}, \frac
{\partial f}{\partial y}\rangle \mathcal{M}_{2}= \mathcal{M}_{2}^{3}$$

but $f$ is not $2$-$\mathcal{R}$-determined as $j^{2}f(0)\equiv 0.$

A successful approach to a necessary and sufficient condition for
finite determinacy appears in [14] where J. Bruce, A. du
Plessis and C.T.C. Wall prove this condition for unipotent subgroups of
\mathcal{K},\,\mathcal{L}\,$ or $\mathcal{A}.$

Let $\mathcal{G}_{s}=\{h\in \mathcal{G}|\, j^{s}h(0)=j^{s}1_{\mathcal{G}}\}$
where $1_{\mathcal{G}}$ is the identity of $\mathcal{G},$ and
$J^{s}\mathcal{G}$ the Lie group of $s$-jets
of elements of $\mathcal{G}.$ The sets $\mathcal{G}_{s},$ $s\geq 1$ are
unipotent subgroups of $\mathcal{G}.$ A special case of the main
result in [14] is the following:

A $C^{\infty}$ map-germ $f:(\mathbb R^{n},0) \to (\mathbb R^{p},0)$ is
$r$-$\mathcal{G}_{s}$-determined $(s\geq 1)$ if and only if
$\mathcal{M}_{n}^{r+1}\Theta_{f}\subset T\mathcal{G}_{s} (f).$

§.§ Classification of stable singularities

We consider here the problem of classification of stable germs with
respect to $\mathcal{A}$-equivalence. The main result is the following

If $f, g$ are  stable germs then
$f\widesim{\mathcal{A}}g$ if and only if the algebras $Q(f)$ and
$Q(g)$ are isomorphic.

The proof of this theorem follows from the following property holding
for infinitesimally stable germs: $\mathcal{A}^{p+1}z=
\mathcal{K}^{p+1}z \cap St^{p+1},$ where $z=j^{p+1}f(0),$ and
$St^{p+1}$ is the set of all stable jets in $J^{p+1}(n,p).$ We omit
the complete proof, however the main steps leading to the proof are given.

The hypothesis that $f$ and $g$ are stable is essential. For
instance, let $f(x,y)=(x,y^{3}+xy)$ and $g(x,y)= (x,y^{3}).$ Both
algebras $Q(f)$ and $Q(g)$ are isomorphic to
$\frac{\mathcal{E}_{1}}{(y^{3})},$ but $f$ and $g$ are not
$\mathcal{A}$-equivalent. In fact, $f$ is stable and
$g$ is not.

The condition that $f\in \mathcal{E}_{n}^{p}$ is infinitesimally
stable is determined by its $p+1$-jet. In fact
the following holds:

The map-germ  $f:(\mathbb R^{n},S)\to (\mathbb R^{p},0)$ is
stable if and only if

\begin{equation}
\label{eq:5}
tf(\Theta_{(n,S)})+{\omega}f(\Theta_{p}) + (f^{*}(\mathcal{M}_{p}) + \mathcal{M}_{S}^{p+1})\Theta_{f}=\Theta_{f}.
\end{equation}

We need to show that (<ref>) implies

tf(\Theta_{(n,S)})+{\omega}f(\Theta_{p})= \Theta_{f}.

The proof is similar to the proof of Proposition  <ref> but simpler.

Let $D=tf(\Theta_{(n,S)})+ f^{*}(\mathcal{M}_{p})\Theta_{f}.$ Note that

f^{*}(\mathcal{M}_{p})\Theta_{f}\subset D.$$

\dim_{\mathbb R}\frac{\Theta_{f}}{\mathcal{M}_{S}^{p+1}\Theta_{f}+D}\leq
\dim_{\mathbb R}\frac{{\omega}f(\Theta_{p})}{{\omega}f(\mathcal{M}_{p}\Theta_{p})}
\leq p.

The result then follows by Lemma <ref> (ii).

Mather gives  in [61], Proposition (I.6), a simple
geometric characterization of a stable multigerm $f:(\mathbb R^{n},S)\to (\mathbb R^{p},0),$ $S=\{x_{1},
x_{2}, \dots, x_{r}\}.$
Recall that if $V$ is a vector space and
$H_{1}, \dots, H_{r}$ are subspaces of $V,$ then  $H_{1},\dots,
H_{r}$ are in general position if for every sequence of
integers $i_{1}, \dots, i_{l}$ with $1\leq i_{1}\leq \dots \leq
i_{l}\leq r$, we have $\cod (H_{i_{1}}\cap \dots \cap H_{i_{l}})
=\cod (H_{i_{1}})+ \dots + \cod (H_{i_{l}}).$

Let $f_{i}:U_{i}\to \mathbb R^{p},$ $i=1, \dots,
r$ be a representative of the germ $f_{i}:(\mathbb R^{n}, x_{i})
\to (\mathbb R^{p},0).$ Denote by $X_{i}=\{x \in U_{i}|\,
(f_{i},x)\widesim{\mathcal{A}}(f_{i},x_{i})\}$ where $(f_{i},x) $
denotes the germ $f_{i}:(\mathbb R^{n}, x)  \to (\mathbb
R^{p},0),$ $i=1,\dots, r.$ Since $f$ is infinitesimally stable,
the sets $X_{i}$ are submanifolds.
Mather's result states that the multigerm $f$ is stable if and only if
each branch $f_{i}:(\mathbb R^{n},x_{i})\to (\mathbb R^{p},0),$
$i=1\dots r$ is infinitesimally stable and the images
$f_{i}(X_{i}), i=1, \dots, r$ are in general position.

An infinitesimally stable germ $f:(\mathbb R^{n},0)\to (\mathbb R^{p},0)$ is  $(p+1)$-$\mathcal{A}$-determined.

Notice that Proposition <ref> implies that if
$j^{p+1}g(0)=j^{p+1}f(0),$ then $g$ is also infinitesimally

It is also clear that every such $g$ is $\mathcal{A}$-finitely determined,
say $l$-$\mathcal{A}$-determined. Then, we can apply Proposition
<ref> to get the result.

As the local algebra is a complete invariant for the
classification of stable germs, we can ask:

– Can we provide a normal form of a stable germ whose
local algebra is a given algebra $Q?$

The answer  was given by  Mather [61] and we review
it here (see also section 1.2.5 of the Mond and Nuño-Ballesteros
in this Handbook [70]).

We start with a rank zero $\mathcal{K}$-finitely determined
$f:(\mathbb R^{n},0)\to (\mathbb R^{p},0)$, $f=(f_{1}, f_{2},\dots,
f_{p}).$ Let

{\langle f_{1},\dots,f_{p}\rangle\mathcal{E}_{n}}.

Since $f$ is $\mathcal{K}$-finitely determined, the quotient

\begin{equation}
\label{eq:33}
Nf =\frac{\Theta_{f}}{tf(\Theta_{n})+f^{*}(\mathcal{M}_{p})\Theta_{f}+{\omega}f(\Theta_{p})}
\end{equation}

is a finite dimensional $\mathbb R$-vector space of dimension $r$ and
we can choose $\sigma_{i}\in \mathcal{E}^{p}_{n}, i=1, \dots,r$ such

\begin{equation}
\label{eq:14}
Nf = \mathbb R\{\sigma_{1}, \dots, \sigma_{r}\},
\end{equation}

For practical purposes, note that the vector space $Nf$ admits the
following simpler characterization:

Nf\simeq \frac{\mathcal{M}_{n}\Theta_{f}}{tf(\Theta_{n}) +f^{*}\mathcal{M}_{p}\Theta_{f}}

Let $F:(\mathbb R^{n}\times \mathbb R^{r},0)\to (\mathbb R^{p}\times
\mathbb R^{r},0)$ be the linear $r$-parameter unfolding of $f$ defined

\begin{equation}
\label{eq:17}
F(x,u)=(f(x)+\sum_{i=1}^{n}u_{i}\sigma_{i}(x), u).
\end{equation}

Then $F$ is infinitesimally stable. In, fact from (<ref>) we get

\Theta_{f}=tf(\Theta_{n})+
R\{\sigma_{1}, \dots, \sigma_{r}\},

which implies that

\begin{equation}
\label{eq:15}
\Psi_{F}= tF(\Psi_{n+r})+
\dots, \sigma_{r}\},
\end{equation}

where $\mathcal{E}_{r}\{\sigma_{1},  \dots, \sigma_{r}\}$ denotes  the
$\mathcal{E}_{r}$-module generated by $ \{\sigma_{1}, \dots,
\sigma_{r}\}.$ Notice that
$F^{*}(\mathcal{M}_{p})\mathcal{E}_{n+r}\supset \langle u_{1},\dots,
u_{r}\rangle \mathcal{E}_{n+r}.$ Then, it follows from that

\Theta_{F}=tF(\Theta_{n+r})+

and it follows from Proposition <ref> that $F$ is
infinitesimally stable.

(a) $\mathcal{A}_{k}$ singularities

Let  $f:(\mathbb R,0)\to (\mathbb R,0),$ $f(x)=x^{k+1}.$ Then
$  Nf = \mathbb R\{1, x, \dots, x^{k-1}\}.$ From the above
construction, we obtain that

\begin{align*}
F:\mathbb R\times \mathbb R^{k-1} &\to \mathbb R\times
\mathbb R^{k-1}\\
(x,u) &\mapsto F(x,u)=(x^{k+1}+\sum_{i=1}^{k-1}u_{i} x^{i}, u),
\end{align*}

is infinitesimally stable.

(b) $\Sigma^{2,0}$ singularities $B^{\pm}_{2,2}=(x^{2}\pm
y^{2}, xy)$

(We use here du Plessis and Wall notation [82]. They are
denoted $\text{I}_{2,2}=(x^{2}+ y^{2}, xy)$ and $\text{II}_{2,2}= (x^{2}- y^{2}, xy)$
by Mather [61].)

Normal forms for infinitesimally stable singularities whose local algebra are
$B^{\pm}_{2,2}$ are

\begin{align*}
F:(\mathbb R^{2}\times \mathbb R^{2},0) &\to (\mathbb R^{2}\times
\mathbb R^{2},0)\\
(x,y,u,v) &\mapsto F(x,y,u,v)=(x^{2}\pm y^{2}+ ux+vy, xy, u, v).
\end{align*}

As a consequence of the  results of this section we can state the
following addendum to Theorem <ref>.

Let $r\leq p+1$ and $k\geq p.$ Let  $f:N\to P$ be a proper
$C^{\infty}$ mapping. Then the following conditions are equivalent

(a) $f$ is stable.

(b) $ _{r}j^{k}f$ is transversal to every contact class in $ _{r}j^{k}(N,P).$

(c) For every subset $S$ of $N$ having $r$ or fewer points,
such that $f(S)$ is a single point $y\in P,$ we have

$$ tf(\Theta_{(N,S)}) +{\omega}f(\Theta_{(P,y)} ) +\mathcal{M}_{S}^{k+1}\Theta_{f}=\Theta_{f}$$

§.§ Maps of finite singularity type

Finite singularity typeFST|see finite singularity type
Another fundamental notion introduced by Mather in [65] was the
notion of mappings of finite singularity type, denoted by
FST. Properties of such mappings are also discussed in [39].

A mapping $f:N\to P$ will be said of finite singularity type if
$E=\frac{\Theta_{f}}{tf(\Theta_{N})}$ is a finite module over
$C^{\infty}(P)$ via $f.$

We can also define similarly the notion of FST for multigerms
$f:(\mathbb R^{n}, S)\to (\mathbb R^{p}, 0).$

Local properties of mappings of finite singularity type follow from
our previous discussion. The critical set of $f$ is the set $\Sigma(f)$ of
non-submersive points  of $f.$

Let $F:(\mathbb R^{n}\times \mathbb R^{r},0) \to (\mathbb R^{p}\times
\mathbb R^{r},0)$ with $F(x,u)=(\bar f(x,u),u)$ and $\bar
f(x,0)=f(x).$ If $F$ is a stable germ, we say that $F$ is a
parametrized stable unfolding of $f.$

Let $f:(\mathbb R^{n}, S)\to (\mathbb R^{p}, 0).$ The following are equivalent.

(1)  $f$ is of FST.

(2) $f$ is $\mathcal{K}$-finitely determined.

(3)  $f$ admits a stable parametrized unfolding.

Moreover, these conditions imply

(4)  for every sufficiently small representative $f:U\to V,$
$f|_{\Sigma(f)}:\Sigma(f) \to V$ is proper and has finite fibers.

We say that $f:X\to Y$ has finite fibers (or, is finite-to-one) if
for every $y\in Y,$ $ f^{-1}(y)$ has a finite number of points.

The equivalence $(1)\Leftrightarrow (2)$ follows from the Preparation
Theorem. In fact $E=\frac{\Theta_{f}}{tf(\Theta_{(n,S)})}$ is a finitely
generated $f^{*}(\mathcal{E}_{p})$-module if and only if
$\mathcal{K}_{e}$-$\cod f=\dim_{\mathbb R}\frac {\Theta_{f}}{tf(\Theta_{(n,S)})+f^{*}(\mathcal{M}_{p})\Theta_{f}}
< \infty.$

We saw in section <ref> that a
$\mathcal{K}$-finitely determined germ has a stable unfolding; so
that $(2)\Rightarrow (3).$ We saw in Example <ref> that
$Q(f)=Q(f_{0}),$ so that $(3)\Rightarrow (2).$

It is sufficient  to prove  $(4)$ for infinitesimally stable
germs. In this case, the general position condition implies that  for any $y\in V,$
$f^{-1}(y) \cap \Sigma(f)$ has at most $p$ points (see Remark <ref>).

We shall need some extra conditions to formulate the theory of
FST mappings $f:N\to P.$ The condition that $f$ has a parametrized
stable unfolding  is fairly easily computable, but it does
not always have a global version (see Mather [65] for
counter examples).

Let $f:N\to P$ be  smooth. We say that $\{F,N', P', i, j\}$ is an unfolding
of $f$ if we have a commutative diagram

\xymatrix{
N'  \ar[r]^{F}    & P'  \\
N \ar[u]_{i} \ar[r]^{f}   & P\ar[u]_{j} }

where $N', P'$ are smooth manifolds, $F$ is a smooth mapping, $i, j$
are closed smooth embeddings, $i(N)= F^{-1}(j(P))$ and $F$ is
transverse to $j.$

Let $f:N \to P$ be smooth and $N$ compact. Then $f$ is of finite
singularity   type  if and only if there exists an unfolding
$\{F,N', P', i, j\}$ of $f$ such that $F$ is proper and
infinitesimally stable.

§.§ Notes

All the results in this section  remain
true if we replace smooth germs by real analytic or complex analytic germs.
Finite determinacy!of Mather's groups $\mathcal{G}$
In particular, the notion of $\mathcal{G}$-finite determinacy for $\mathcal{G}=
\mathcal{R}, \mathcal{L}, \mathcal{A}, \mathcal{C}$ and $\mathcal{K}$
is independent of whether we consider $f$ as a real analytic,
$C^{\infty}$ or complex analytic map-germ. The Infinitesimal Criterion
of $\mathcal{G}$-finite determinacy holds with essentially the same
proof replacing Malgrange Preparation Theorem by Weirstrass
Preparation Theorem.
Finite determinacy!infinitesimal criterion of
We use the same notation $\mathcal{O}_{n}$ for the local rings of real
analytic or complex analytic map-germs at the origin. The maximal
ideal in both cases is also denoted by $\mathcal{M}_{n}.$ The set
$\mathcal{O}_{n}^{p}$ denotes the $\mathcal{O}
_{n}$-module of real or complex analytic map-germs from $(\mathbb K^{n},0)
\to (\mathbb K^{p},0),$ $\mathbb K=\mathbb R$ or $\mathbb C.$ The
following result explains the relation among finite determined germs
in these different modules.

Let $f:(\mathbb R^{n},0) \to (\mathbb R^{p},0) $ be a real analytic
map-germ. The following are equivalent

(i)  $f$ is $k$-$\mathcal{G}$-determined in the space of real
analytic map-germs $\mathcal{O}_{n}^{p}.$

(ii)  $f$ is $k$-$\mathcal{G}$-determined in $\mathcal{E}_{n}^{p}.$

(iii)  The complexification of $f$,
$f_{\mathbb C}:(\mathbb C^{n},0) \to
(\mathbb C^{p},0),$ is $k$-$\mathcal{G}$-determined in the space
$\mathcal{O}_{n}^{p}$ of holomorphic map-germs.

In the complex case there are  useful geometric characterization of
$\mathcal{G}$-finite determinacy. The main result characterizes
$\mathcal{G}$-finite determined germs as map-germs with isolated
instability. The case $\mathcal{G}=\mathcal{A}$ was stated by Mather
and proved by Gaffney. For a complete account we refer to Wall
[108] or Mond and Nuño-Ballesteros [69]. See also
Mond and Nuño-Ballesteros article in this Handbook [70]
Finite determinacy!Geometric criterion of

A holomorphic map-germ $f:(\mathbb C^{n},0) \to
(\mathbb C^{p},0),$ is $\mathcal{A}$-finite  if and only if
there is a neighborhood $U$ of $0$ in $\mathbb C^{n}$ such that for
every finite subset $S\subset U\setminus \{0\},$ the multigerm of $f$
at $S$ is $\mathcal{A}$-stable.

The geometric condition of this theorem (isolated instability)
holds for any real $\mathcal{A}$-finite map-germ. However, the converse
statement does not hold. For a simple example, let $f(x,y)= (x^{2}
+y^{2})^{2}.$ As $\Sigma (f)=\{0\},$ the origin is an isolated
instability, but $f$ is not $\mathcal{A}$-finitely determined.

§ THE NICE DIMENSIONS
Nice dimensions|(

We discuss in this section the main steps in the proof of theorem A.
Mather proved in [61] that for a pair of positive integers $(n,p),$ there exists a
smallest Zariski closed $\mathcal{K}^{k}$-invariant set
$\Pi^{k}(n,p)$ in the set $J^{k}(n,p)$
such that
$J^{k}(n,p) \setminus \Pi^{k}(n,p)$  is the union of finitely many
The set $\Pi^{k}(n,p)$ is the “bad set.” It is in fact the set of
$k$-jets in  $J^{k}(n,p)$  of “modality” ( $\mathcal{K}$-modality)
greater than or equal to $1$ (see Section <ref>
for the definition of modality).

We review Mather's construction of $\Pi^{k}(n,p).$ For each $r, k \in
\mathbb N$ we define $W^{k}_{r}(n,p)$ as the set of $z\in J^{k}(n,p)$
such that ${\mathcal K}^{k}$-$\cod{z} \geq r.$ This set is a closed
algebraic subset of $J^{k}(n,p).$  Let $W^{k}_{r}(n,p)^{*}$ denote the
union of all irreducible components of $W^{k}_{r}(n,p)$ whose
codimension is less than $r.$ We let
$\Pi^{k}(n,p)=\cup_{r\geq 0}W^{k}_{r}(n,p)^{*}.$ The following
properties hold:

* $\Pi^{k}(n,p)$ is a closed algebraic subset of $J^{k}(n,p).$

* Let $\pi_{k}:J^{k+1}(n,p) \to
J^{k}(n,p)$ be the projection . It follows that
$\pi^{-1}_{k}(\Pi^{k}(n,p))\subset \Pi^{k+1}(n,p),$ hence $\cod
\Pi^{k+1}(n,p)\leq \cod \Pi^{k}(n,p).$

* There exists a $k$ big enough for which the codimension
of $\Pi^{k}(n,p)$ attains its minimum. For this $k$, $\cod
\Pi^{k}(n,p)$ is denoted $\sigma(n,p).$

Mather calculated $\sigma(n,p)$ in [63] and the result is as follows:

Case 1: $n\leq p$

\begin{align*}
\sigma(n,p) &=
\begin{cases}
6(p-n)+8 &\textrm{if } p-n \geq 4\, \textrm{ and }\, n\geq 4\\
6(p-n)+9 &\textrm{if } 3\geq p-n \geq 0 \, \textrm{ and }\, n\geq 4\,
\mathrm{ or\  if }\, n=3\\
7(p-n) +10 &\textrm{if } n=2\\
\infty &\textrm{if } n=1
\end{cases}
\intertext{Case 2: $n>p$}
\sigma(n,p) &=
\begin{cases}
9 &\textrm{if } n=p+1\\
8 &\textrm{if } n=p+2\\
n-p+7 &\textrm{if } n\geq p+3\\
\end{cases}
\end{align*}

Boundary of nice dimensions

Nice dimensions
A pair $(n,p)$ is in the nice dimensions if $n<\sigma(n,p).$

Suppose $k$ has the property that  $\cod \Pi^{k}(n,p)=\sigma(n,p).$
If $(n,p)$ is in the nice dimensions, then there exists an
analytically trivial
stratification $\mathcal{S}^{k}(n,p)$ of $J^{k}(n,p)\setminus \Pi^{k}(n,p)$ such that the
strata are a finite number of $\mathcal{K}$-orbits. To get a
stratification of the whole jet space $J^{k}(N,P),$ we add to
$S^{k}(n,p)$  a Whitney regular stratification of $\Pi^{k}(n,p)$ (it
exists since $\Pi^{k}(n,p)$ is an algebraic closed set of

This stratification of $J^{k}(n,p)$ induces a
partition of $J^{k}(N,P)$ by $\mathcal{K}$-orbit bundles whose
restriction to $J^{k}(N,P)\setminus \Pi^{k}(N,P)$ is
denoted by $S^{k}(N,P).$

As we saw in Theorem <ref>, stable mappings can be characterized
by transversality of the
$k$-jet extension $j^{k}f:N\to J^{k}(N,P)$ to the $\mathcal

When $ \sigma(n,p)>n,$ transversality  to the strata
of the stratification $J^{k}(N,P),$ implies that
$j^{k}f(N)\cap\Pi^{k}(N,P)=\emptyset.$  Hence Theorem A  follows
from Thom's transversality theorem.
Stable singularities!when $n=p\leq 8$
[Stable singularities when $n=p\leq 8$]
We refer to [69] for the list of stable singularities in
the nice dimensions.

When $n=p,$ $\sigma(n,p)=9$, then $(n,n)$ is a nice pair of
dimensions if and only if $n\leq 8.$
The set $\Pi^{k}(n,n)\subset J^{k}(n,n),$ $k\geq n+1,$ $n\leq 8$ is
the closure of all ${\cal K}^{k}$-orbits of ${\cal
K}^{k}$-codimension greater than or equal to $n+1.$ In particular,
$\Sigma^{3}(n,n) \subset \Pi^{k}(n,n),$   where $n\leq 8$ since
$\mathrm{cod}\, \Sigma^{3} =9.$ The strata of the stratification
${\cal S}^{k}(n,n),$ $k\geq n+1,$ $n\leq 8$ are presented  in
Table <ref>:

Type     Name     Normal form     Conditions
\leq n$

$\Sigma^{1}$    $A_{j}$            $(x^{j+1})$     $ 1\leq j\leq n$     $j$
$\Sigma^{2,0}$    $B^{\pm}_{p,q}$    $(xy, x^{p}\pm y^{q})$     $2\leq p,q \leq n-2$
$p+q $
$\Sigma^{2,0}$     $B^{*}_{p,p}$     $(x^{2}+y^{2}, x^{p})$      $3 \leq p\leq 4$
$\Sigma^{2,1}$    $C_{2k-1}$        $(x^{2}+y^{3},y^{3})$
$\Sigma^{2,1}$     $C_{2k}$         $(x^{2}+y^{3},  xy^{2})$

${\cal K}$-orbits of stable germs $n=p \leq 8$

Stable singularities! in the nice dimensions
Classification of stable singularities in the nice dimensions.
Mather classified the stable germs in the nice dimensions as an
application of results and arguments in [63].
He gave complete proofs of the classification of the local algebras
of singularities of type $\Sigma^{1}$ and $\Sigma^{2,0}$ and outlined
the classification of $\Sigma^{2,1}$ and $\Sigma^{n-p+1}$
singularities. Further classification of simple and unimodular
algebras were performed by Arnold [4], Wall <cit.>,
Dimca and Gibson [27, 28, 29] and Damon
[18, 20, 19].

A remarkable property  of stable map-germs in the nice dimensions is
that, with  respect  to suitable
coordinates, all singularities are weighted homogeneous. For many
years, this property was considered to be true but there was no
reference of a written proof.

This result was recently proved by Mond and Nuño-Ballesteros [69]
theorem 7.6. Their proof is based on Mather's classification of local
algebras of stable germs in the nice dimensions and on the direct
construction of the normal forms of their minimal stable
unfoldings. This property of the nice dimensions plays a crucial role
in the proof of Damon and  Mond [26] that the
$\mathcal{A}_{e}$-codimension is less than or equal to the rank of the
vanishing homology of the  discriminant ( the discriminant Milnor
number) for map germs $(\mathbb C^{n},0) \to (\mathbb C^{p},0)$ with
$n\geq p$ and $(n,p)$ nice dimensions.

Nice dimensions|)

§.§ Notes

Map!non-proper stable
Non proper stable mappings. $C^{\infty}$ non-proper stable mappings were discussed by du Plessis
and Vosegaard [81] and more recently by Kenta Hayano

For proper maps $f:N\to P,$ Mather proves that stability,
strong stability, infinitesimal stability an local infinitesimal
stability are equivalent notions.
In [81], du Plessis and Vosegaard prove that these notions
are equivalent  when $f$ is a quasi-proper map with closed discriminant.

The purpose of Hayano's paper, [42], is to give a sufficient
condition for strong stability of non-proper smooth functions $f:N\to
\mathbb R.$ He introduces the notion of end-triviality of smooth
mappings, which controls the behavior of $f$ around the ends of the
source manifold $N.$ He shows that a Morse function is stable if it is
end-trivial at any point in its discriminant.
Extra-nice dimensions
The extra-nice dimensions.  When the pair $(n,p)$ is in the
nice dimensions and
the source $N$ is compact, an important problem  in the applications
of singularity theory  to topology of manifolds is the
characterization of generic singularities of $1$-parameter  paths
between two stable maps; they are also known as pseudo-isotopies. A
$1$-parameter family $F:N\times[0,1] \to P$ connecting two
non equivalent stable maps always intersects the set of non stable maps
at a finite number of values of the parameter, the bifurcation
points. The classification of singularities of bifurcation points in
generic families of maps is an important  step in results on
elimination of singularities (see for instance [50, 7]) and
on results about the topology of the space of smooth maps such as
[16, 44, 104].

We say that a family $F:N\times[0,1] \to P$ is a locally stable family
if $F_{t}:N \to P$ is stable for all $t\in [0,1]$ except possibly a
finite number of values $\{t_{1}, \dots, t_{k}\}$ and the non stable
singularities of $F_{t}$ are a finite number of points $x_{j}$ at
which $\mathcal{A}_{e}$-cod$(F_{t_{i}})=1.$

In [6] Sinha, Ruas and
Atique obtain a result parallel to Mather's characterization of the
nice dimensions. They define the extra-nice dimensions and (see
Figure <ref>)
prove that the subset of stable $1$-parameter families in
$C^{\infty}(N\times [0,1],P)$ is dense if and only $(n,p)$ is in the
extra-nice dimensions.

Extra-nice dimensions

In section <ref> we relate the condition that
$(n,p)$ is in the extra-nice dimensions to the geometry of sections of
the discriminant of stable maps in dimensions $(n+1,p+1).$

§ THOM'S EXAMPLE

If a pair of dimensions $(n,p)$ is not in the nice range of dimensions,
then there exists an open non void subset $U$ of $C^{\infty}(N,P),$
such that $U$ is the union of an uncountable number of $\mathcal{A}_{e}$-orbits.
This property was first proved by René Thom when $n=p=9.$ We review
Thom's example ([96]) here.
Nice dimensions!boundary of
The pair $n=p=9$ is in the boundary of the nice dimensions,
which consists  of pairs $(n,p)$ such that $\sigma(n,p)=n.$

The construction of Thom's example was based on the following

* The set of mappings $F:N\to P,$ $\dim N= \dim P=n,$ such that $j^{k}F\pitchfork
\Sigma^{r}(N,P),$ where $\Sigma^{r}(N,P)=\{\sigma \in J^{k}(N,P)|\, \mathrm{corank}\,
\sigma =r\},$ $0\leq r\leq n$ is a residual set of $C^{\infty}(N,P).$

* $\cod_{J^{k}(N,P)}\Sigma^{r}(N,P)= r^{2}.$

* When $r=3,$ $n=9,$ there exists a $1$-parameter family of non
mappings $F_{\lambda}:\mathbb R^{9} \to \mathbb R^{9},$ such that
$j^{k}_{1}F:\mathbb R \times {\mathbb R}^{9} \to J^{k}(\mathbb R^{9},
\mathbb R^{9})$ is transversal to $\Sigma^{3}(\mathbb
R^{9},\mathbb R^{9}),$ where $j^{k}_{1}F$ denotes the $k-$jet with
respect do the variable $x.$

The sets $\Sigma^{r}$ are the first order Boardman symbols and it is
an easy exercise to prove that they are codimension $r^{2}$
submanifolds of $J^{k}(N,P)$ when $\mathrm{dim}(N)=\mathrm{dim}(P).$ Hence (1) follows from Thom's
transversality theorem.

It is sufficient to verify (3) for map-germs $F:(\mathbb R^{9},0)
\to (\mathbb R^{9},0),$ such that $\mathrm{corank}\;F(0)=3.$ By changing
coordinates in source and target, it follows that $F$ can be written
in the form $F(x,u)=(f(x,u),u),$ $x=(x_{1}, x_{2}, x_{3}),$
$u=(u_{1}, \dots, u_{6}), $ $ f_{0}(x)=f(x,0),$ where
$f_{0}:(\mathbb R^{3},0) \to (\mathbb R^{3},0)$ has zero rank.

The local algebras $Q(F)$ and $Q(f_{0})$ are isomorphic. As we saw
in Example <ref>, $F$ is $\mathcal{K}$-equivalent  to a suspension of $f_{0}.$
The $2$-jet $j^{2}f_{0}$ is a quadratic polynomial mapping
$q:\mathbb R^{3} \to \mathbb R^{3},$ which determines a net of real
Net of quadrics
Non degenerate nets of quadrics over the complex numbers were
classified by C. T. C. Wall in [107]. Over the reals, the
classification was given by Wall and Edwards in [30]. The
complete classification of real nets of quadrics can be found in [82]
chapter 8, table 8.21.

For our purpose here, it suffices to remark that the set
$\Sigma^{3,3}$ has a Zariski open set, denoted by
$W_{2},$   defined by the union of the
$J^{2}\mathcal{K}$-orbits of the unimodular family:

\begin{align}
\label{eq:1}
(f_{0})_{\lambda}:&(\mathbb R^{3},0) \to (\mathbb R^{3},0)\\
&(x_{1},x_{2},x_{3})\to (x_{1}^{2}+\lambda
x_{2}x_{3},x_{2}^{2}+\lambda x_{1}x_{3},x_{3}^{2}+\lambda x_{2}x_{3}) \nonumber
\end{align}

with $\lambda(\lambda^{3}+8)(\lambda^{3}-1)\neq 0.$

For each $\lambda,$ $(f_{0})_{\lambda}$ is a homogeneous polynomial
map of degree $2,$ hence the $J^{2}\mathcal{K}$-action in $W_{2}$
coincides with the action of the linear group
$\mathcal{G}=GL(3)\times GL(3)$ on $W_{2}.$ Notice that the dimension
of the linear group $\mathcal{G}$ is $18,$ as well as the dimension
of $W_{2}.$

However $\mathcal{G}$ contains a one dimensional subgroup which acts
trivially on $W_{2},$ namely $\{(cI_{\mathbb R^{3}}, \frac 1
{c^{2}}I_{\mathbb R^{3}})\},$ $c$ a non zero number. Hence the orbits
have codimension at least $1$ in $W_{2}.$

We can prove that the family (<ref>) is $2$-determined with respect
to $\mathcal{K}$-equivalence. It follows that $W_{2}$ determines the
$\mathcal{K}$-invariant sets $W_{2}^{k}=(\pi_{2}^{k})^{-1}(W_{2}),$
where $\pi^{k}_{2}:J^{k}(9,9)\to J^{2}(9,9).$ Moreover,
$\cod_{J^{k}(9,9)} W_{2}=9,$ and $\mathcal{K}$-$\cod

In other words, $\sigma(9,9)= \cod W_{2}=9,$ so that the
unimodular stratum $W_{2}$ cannot be avoided by a generic set of
proper mappings $F:\mathbb R^{9} \to \mathbb R^{9}.$ As a consequence,
stable mappings are not dense when $n=p=9.$

For each $\lambda \notin \{0, -2, 1\},$ $(f_{0})_{\lambda}$ admits the topologically stable

\begin{align}\label{nove}
F_{\lambda}:&(\mathbb R^{9},0) \to (\mathbb R^{9},0)\\
&(x,u) \mapsto (f_{\lambda}(x, u), u) \nonumber
\end{align}

where $f_{\lambda}(x,u)=( x_{1}^{2}+\lambda x_{2}x_{3} +u_{1}x_{2} +u_{2}x_{3},
x_{2}^{2}+\lambda x_{1} x_{3} + u_{3}x_{1} +u_{4}x_{3}, x_{3}^{2}+\lambda x_{1}x_{2} + u_{5}x_{1} +u_{6}x_{2}).$

We will discuss the topological stability of $F_{\lambda}$ in section <ref>.

§ DENSITY OF TOPOLOGICALLY STABLE MAPPINGS

From the previous example, it becomes clear that outside the nice
dimensions, one has to loosen the formulation of Problem <ref> to
obtain a solution. Mather considered in [64] two
possible ways.

One might hope that the space of mappings $f$ whose germ $f_{x}$ at
each point $x\in N$ is $\mathcal{A}$-finitely determined is an open
and dense subset in $C^{\infty}_{pr}(N,P).$ However, Mather gave in
[59] an example which shows that this set is not always
dense.  In [80] du Plessis defined the semi-nice dimensions as the
pairs $(n,p)$  for which finite determinacy holds in general (see
Definition <ref>).  The complement of the semi-nice dimensions is
essentially made of pairs $(n,p)$ where singularities of
$\mathcal{K}$-modality greater than or equal to $2$ occur generically (see
[80], <cit.>).

The second way to try to solve the problem is based on ideas due to
Thom, and led to Theorem B on density of $C^{0}$stable mappings
in $C_{pr}^{\infty}(N,P).$

In his article Local topological properties of differentiable
mappings [93], Thom describes the topological
structure of differentiable mappings, outlining the proof of the
topological stability theorem.

Let $z$ be any jet in $J^{r}(n,p).$ Then, there exists a positive
integer $s$ depending only on $r, n$ and $p,$ and a proper algebraic
variety $\Sigma$ in $\pi_{s}^{-1}(z) \subset J^{r+s}(n,p)$ such that
any jet in $\pi_{s}^{-1}(z)$ outside $\Sigma$ is
$C^{0}$-$\mathcal{A}$-finitely determined. Moreover, any two
mappings realizing such jet are locally weakly stratified and

A complete proof of this theorem  follows from the proof of the Main
Theorem in A. Varchenko's article with the same title, Local topological
properties of differentiable mappings [103] ( see also
[102, 101]). He also proves in [103] a
stratification theorem, although he states in the paper he
does not know
whether  Mather's density theorem follows from his
stratification theorem, or whether the stratification theorem can be
proved by Mather's methods.

Mather gave in 1970, an outline of a complete proof of Theorem
B. His proof was published in the Proceedings of the
Symposium of Dynamical Systems, held in Salvador, Bahia
[64]. As remarked by him, he expected to publish a book in
which the details of the proof would appear. In the Spring 1970, he
gave a series of lectures  and the notes appeared as a booklet
published  in the same year by the Harvard Printing Office.
The notes also discuss the Thom-Whitney theory of stratified sets
and stratified mappings. They were recently republished in the
Bulletin of the American Mathematical Society [56].

Complete proofs of Theorem B  were given in 1976, independently, by Gibson,
Wirthmüller, du Plessis and Looijenga in [39] and by
Mather in [66]. Both proofs are based on Thom's ideas and Mather's outline
[64]. In what follows we refer to Theorem B as the Thom-Mather

The book [39] comprises the notes of a seminar on
Topological Stability of Smooth Mappings held at the Department of
Pure Mathematics in the University of Liverpool, during the academic
year 1974-75. The main objective was to organize a complete proof of
the Topological Stability Theorem, for which no published complete
account existed.
The book has become a fundamental reference on the subject.

The proof in [39] and [66] are similar and they
rely on the following ingredients:

(1) Properties of Whitney regular stratifications

(2) Łojasiewicz theorem, giving the existence of Whitney
regular stratification of semialgebraic sets.

(3) Properties of stable mappings and mappings of finite
singularity type (FST). A fundamental property of mappings of FST is
the existence of a stable unfolding.

(4) Thom's second isotopy theorem, applied to show that
families of mappings transverse to the Thom-Mather stratification are
topologically trivial.

For a review of stratification theory and Thom's isotopy theorems
in the differentiable category, we also refer to the paper by David Trotman, in
Volume I of this Handbook. We only make a brief presentation of basic
concepts and results.

Let $V$ be a subset of a smooth manifold $N$ of class $C^{k}.$ A
$C^{k}$-stratification of $V$ is a filtration by closed subsets

V=V_{d}\supset V_{d-1}\supseteq \cdots \supseteq V_{1}\supseteq V_{0}

such that each difference $V_{i}\setminus V_{i-1}$ is $C^{k}$-manifold
of dimension $i$, or is empty. Each connected component of
$V_{i}\setminus V_{i-1}$ is a stratum of dimension $i.$
It follows that $V$ is disjoint union of
strata $\{X_{\alpha}\}_{\alpha\in A},$ and we say that $V$ is a
stratified set.

For the purposes of these notes we assume that the stratified sets
$V=\cup_{\alpha \in A} X_{\alpha}$ are locally finite and
satisfy the frontier condition (see Gibson et al. book
[39] or Trotman [100] for the definition).

Let $V$ be a subset of $\mathbb R^{n}$ and $\{X_{\alpha}\}_{\alpha\in
A}$ a stratification of $V.$ Whitney defined regularity conditions
(a) and (b), seeking for stratifications topologically trivial
along strata.
Whitney!conditions (a) and (b)

Let $X$ and $Y$ be strata of $\{X_{\alpha}\}_{\alpha\in  A},$ such
that $Y\subset \overline X \setminus X.$

(a) The pair $(X,Y)$ satisfies Whitney's condition (a) at
$y\in Y$ if: for all sequences $(x_{m}) \in X$ with $x_m \to y,$
such that $T_{x_{m}}X$ converges to a subspace $T \subset \mathbb
R^{n}$ ( in Grassmannian of $\dim X$- planes in $\mathbb R^n$),
then $T\supset T_{y}Y.$

(b) The pair $(X,Y)$ satisfies Whitney's condition (b) at
$y\in Y$ if: for all sequences $(x_{m}) \in X$ and $(y_{m}) \in Y,$ with $x_m \to y,$
$y_m \to y,$ such that $\{T_{x_{m}}X\}$ converges to  $T$ and the
lines $\overline{x_{m}y_{m}}$ converges to a line $\ell$ one has
$\ell \in T.$

It was pointed out by Mather in his notes on topological stability
that Whitney (b) implies Whitney (a). The reader may verify this as
an exercise. We say that the stratification is Whitney regular if every pair
of strata $(X_{\alpha},X_{\beta})$ satisfies  (b) ( hence also
satisfies (a)) at every point in $X_{\beta}.$

These regularity conditions are local and can be easily  extended to
stratified sets of a manifold $N.$

Whitney [117, 116] proved in 1965  that any analytic
variety in $\mathbb R^{n}$ or $\mathbb C^{n}$ admits a regular
stratification whose strata are analytic. This result was extended to
semi-analytic sets by Łojasiewicz [49], also in 1965. For the
purposes of this section, the relevant result is the existence theorem
for semialgebraic sets. We refer to Thom [95] and Wall
[105] for accessible proofs.

Let $f:N\to P$ be a smooth mapping and $A\subseteq N, B\subseteq P$
sets with $f(A)\subset B.$ A stratification of $f:A \to B$ is a
pair $(\mathcal{X},\mathcal{X}'),$ such that $\mathcal{X}$ is a
Whitney stratification of $A$, $\mathcal{X}'$ is a Whitney
stratification of $B,$ and the following conditions hold

* f maps strata to strata.

* If $X\in \mathcal{X},$ $X' \in \mathcal{X}'$, $f(X) \subset X'$
then $f:X \to X'$ is a submersion.

Let $f:N\to P$  and $\mathcal{X}$ and $\mathcal{X}'$ as in
definition <ref>. Given $X_{\alpha}, X_{\beta}$ strata of
$\mathcal{X},$ $x\in X_{\beta}$ we say that $X_{\alpha}$ is
Thom regular over $X_{\beta}$ at $x\in X_{\beta}$ relative
to $f$ when the following holds: for every sequence $(x_{i}) \in
X_{\alpha},$ $x_{i}\to x$ such that
$\textrm{ker\;}(d_{x_{i}}(f|_{X_{\alpha}}))$ converges to $T$ in the
appropriate Grasmannian, then $\textrm{ker\;}d_{x}(f|_{X_{\beta}})
\subseteq T.$ We say that $X_{\alpha}$ is Thom regular over
$X_{\beta}$ relative to $f$ when this condition hold for all
$x\in X_{\beta}.$The pair
$(\mathcal{X},\mathcal{X}')$ is a Thom stratification for $f$
when Thom's regularity condition holds for all pair of strata
$(X_{\alpha}, X_{\beta})$ with $X_{\beta}\subset \overline
X_{\alpha}.$ The triple $(f,\mathcal{X},\mathcal{X}')$ with $f$ a
smooth mapping and $(\mathcal{X},\mathcal{X}')$ a Thom
stratification for $f$ is called a Thom stratified mapping.Map!Thom stratified

§.§ How to stratify mappings and jet spaces

We first discuss the Thom-Mather stratification in jet space and how
to stratify stable mappings and mappings of finite singularity
type. Then, we discuss why mappings transverse to  the Thom-Mather
stratification are topologically stable.

The idea of the proof is to  construct a stratification
$\mathcal{A}^{l}(N,P),$ of  a big open subset of $J^{l}(N,P),$ with
the following property: if $l$ is sufficiently large, then for any
mapping $f:N\to P$ which is multitransverse to
$\mathcal{A}^{l}(N,P),$ then the locally finite manifold partition
$\mathcal{B}=((j^{l}f)^{-1} \mathcal{A}^{l}(N,P))$ is a Whitney
stratification which extends to a Thom stratification
$(\mathcal{B},\mathcal{B}')$ of $f.$

Let $z \in J^{l}(n,p)$ and let $f:(\mathbb R^{n},0) \to (\mathbb
R^{p},0)$ such that $j^{l}f(0)=z.$

Following Gibson et al. [39], we let

\chi_{z}=\dim_{\mathbb R}\frac{\Theta_{f}}{tf(\Theta_{n})+(f^{*}(\mathcal{M}_{p})+\mathcal{M}_{n}^{l})\Theta_{f}}

We define $W^{l}(n,p)=\{z\in J^{l}(n,p)\,|\, \chi_{z}\geq l\}.$
$W^{l}(n,p)$ is the bad set, and the following hold

(a) If $z \in J^{l}(n,p) \setminus W^{l}(n,p),$ then any $f\in
\mathcal{E}^{p}_{n}$ such that $j^{l}f(0)=z$ is

(b) $W^{l}(n,p)$ is  $\mathcal{K}$-invariant.

(c) $W^{l}(n,p)$ is a real algebraic variety in $J^{l}(n,p).$

To verify (a) notice that,  if $\chi_{z}\leq l-1,$ then

\begin{equation}
\label{eq:16}
\supset \mathcal{M}_{n}^{l-1}\Theta_{f}.
\end{equation}

Then we can multiply (<ref>) by $\mathcal{M}_{n}$ and the result
follows from Theorem <ref>.

It follows from (a) that map-germs $f\in \mathcal{E}_{n}^{p}$ such
that $z=j^{l}f(0)$ satisfy $\chi_{z}\leq l-1$ are of finite singularity
type. In the following proposition we prove that the property of
FST holds in general.

The following conditions hold:

(i) $\cod W^{l+1}(n,p) \geq \cod W^{l}(n,p).$

(ii)  $\lim_{l\to\infty}\cod W^{l}(n,p)= \infty.$

(iii)   There is a subbundle $W^{l}(N,P)\subset J^{l}(N,P)$
naturally associated to $W^{l}(n,p).$ Moreover, when $N$ is
compact, mappings $f:N\to P$ such that $j^{l}f(N)\cap W^{l}(N,P)=
\emptyset$ are of finite singularity type.

Properties holding in general

We say that a property $\mathcal{P}$ of map-germs holds in
general if the sets
$ W_{\mathcal{P}}^{l}(n,p)= \{z\in  J^{l}(n,p)|\; z \textrm{ does not
satisfy } \mathcal{P}\}, $ satisfy (i) and (ii) (see [108]).

While condition (i) in Proposition <ref> can be easily
verified, we can prove (ii) as  follows.

Given $z \in W^{l}(n,p), $ find $z' \in W^{l+q}(n,p),$ $\pi_{l}(z')
=z,$ where $\pi_{l}:W^{l+q}(n,p)\to W^{l}(n,p)$ is the projection,
such that $z' \notin W^{l+q}(n,p)$ (see Bruce, Ruas and Saia
[15], for a simpler proof of this result).

As $W^{l}(n,p)$ is a real algebraic variety, it follows from
Łojasiewicz's result [49] that it has a Whitney stratification
with semialgebraic strata. Condition (iii) is
immediate. Notice that conditions (i) and
(ii) imply that we can choose sufficiently high $l$ for which $\cod
W^{l}(n,p)>n.$ Then, the mappings $f:N \to P$ which are multitransverse
to $\mathcal A^{l}(N,P)$ satisfy the condition $j^{l}f(N)\cap

Our problem now is to construct a stratification $\mathcal{A}^{l}(n,p)$ of $J^{l}(n,p)
\setminus W^{l}(n,p)$ whose members are ${\cal K}$-invariant sets  $S_{j}=\{z \in J^{l}(n,p)
\setminus W^{l}(n,p)|\; {\mathbf{cod}\,}z =j\}$, for $j=0,1,2,\dots.$ The
definition of ${\mathbf{cod}\,}z$ will be given in the sequel.

We shall see that ${\cal K}^{l}$-equivalent jets $z$ and $z'$ have the
same codimension, i.e., ${\mathbf{cod}\,}z= {\mathbf{cod}\,}z'.$ This number does
not coincide with the ${\cal K}^{l}$-codimension.

Although  we know that contact classes are smooth submanifolds of the
jet spaces, it is not clear at this point that the collection $S_{j}$
defines a stratification of $J^{l}(n,p) \setminus W^{l}(n,p).$ To
define ${\mathbf{cod}\,}z$ and to understand the structure of the strata $S_{j}$ in  ${\cal
A}^{l}(n,p),$ we first discuss shortly  how to stratify
infinitesimally stable mappings and mappings of FST. Recall that
for any smooth map $f:N\to P,$ the critical set of $f$ is $\Sigma(f)=\{x \in N|\,
df_{x}:T_{x}N \to T_{f(x)}\mathrm{\ is\  not\  surjective} \}$ and the
discriminant of $f$ is $\Delta(f)=f(\Delta(f)).$
Critical set

We saw in section <ref> that if $f:N\to P$ is infinitesimally
stable, the restriction $f|_{\Sigma(f)}:\Sigma(f) \to P$ is proper and
uniformly finite-to-one. In fact for any $y \in P$, $\#(f^{-1}(y)\cap
\Sigma(f)) \leq p.$ Moreover, if $f^{-1}(y)\cap \Sigma(f)=\{x_{1},
x_{2}, \dots, x_{s}\}$ the  multigerm $f:(N,S) \to (P,y)$ has a
representative equivalent to a polynomial mapping $f:U \subset \mathbb R^{n} \to V
\subset \mathbb R^{p},$ where $U$ and $V$ are open sets in $\mathbb
R^{n}$ and  $\mathbb R^{p}$ respectively. In other  words $f$ is a
semialgebraic map defined on semialgebraic subsets.
Then we can apply the basic theorems of Whitney  and
Lojasiewicz  to construct Whitney stratifications
$\mathcal{S}$ of $N$ and $\mathcal{S}'$ of $P$ with the following properties

* For each stratum $X$ of $\mathcal{S},$ there is a stratum $Y$ of
$\mathcal{S}'$ such that $f(X)\subset Y.$

* For each stratum $Y$ of $\mathcal{S}',$ it follows that
$f^{-1}(Y) \setminus \Sigma(f)$ is a stratum of $\mathcal{S}.$

* For each stratum $X$ of $\mathcal{S},$ such that $X\subset
\Sigma(f),$   we have that $\dim X= \dim Y$ and $f:X \to Y$ is an
immersion, where $Y$ is the stratum of $\mathcal{S}'$ which contains $f(X).$

Notice that from 2. it follows that $N\setminus \Sigma(f)$ is a
union of strata. Hence, $\Sigma(f)$ is also a union of strata.

Now, if $f:(N,x_{0})\to (P,y_{0})$ is a stable germ, for any  small
representative that we also denote by $f,$ the stratum $X\in
\mathcal{S}$ which contains $x_{0}$ is connected  and its codimension
is strictly greater than the codimension of any other stratum of
$\mathcal{S}.$ This number depends only of $f.$ We call it the
codimension  of $f,$ and we write ${\mathbf{cod}\,}f.$ A germ $f$ has
codimension zero if and only if it is of maximal rank.

This notion generalizes to map-germs of finite singularity  type.

Let $f:(\mathbb R^{n}, 0) \to (\mathbb R^{p}, 0)$ be a map of
finite singularity type. We define ${\mathbf{cod}\,}f$ at $x=0$ as the
codimension of a stable unfolding of $f.$

Notice that this number is well defined. In fact, if $F:(\mathbb
R^{n}\times\mathbb R^{s}, 0) \to (\mathbb R^{p}\times\mathbb R^{s},
0)$ and $F':(\mathbb
R^{n}\times\mathbb R^{r}, 0) \to (\mathbb R^{p}\times\mathbb R^{r},
0)$ are stable unfoldings of $f$ and if, say, $r=s+k,$ then it follows
that $F\times Id$ is equivalent to $F',$ where $Id$ is the identity
map in $\mathbb R^{k}.$ Then ${\mathbf {cod}\,}(F\times Id) = {\mathbf{cod}\,}F',$ and it easy
to see that ${\mathbf{cod}\,}F ={\mathbf {cod}\,}(F\times Id).$ Now the following result
follows easily.

If $f \widesim{\mathcal{K}}f'$ then ${\mathbf{cod}\,}f= {\mathbf{cod}\,} f'.$

The properties of the stratification $\mathcal{A}^{l}(N,P)$ can be
summarized in the following results.

Let $f:(N,x_{0})\to (P,y_{0})$ be a smooth map-germ with an
unfolding $F:(N',x'_{0})\to (P',y'_{0})$, as in the diagram

\xymatrix{
(N',x'_{0})  \ar[r]^{F} &(P',y'_{0})   \\
(N,x_{0}) \ar[u]^{i}\ar[r]_f   &   (P,y_{0}). \ar[u]^{j} }

Then the following conditions are equivalent

(i) $j^{l}f \notin W^{l}(N,P)$ and $j^{l}f$ is transverse to

(ii)  $j^{l}F \notin W^{l}(N',P')$ and $j^{l}F$ is transverse to
$\mathcal{A}^{l}(N',P'),$ and in addition if $X\in
(j^{l}F)^{-1}\mathcal{A}^{l}(N',P')$ contains $x'_{0},$ then $i$ is
transverse to $N'.$

Let $f:N\to P$ be a proper smooth mapping multi-transverse to

and such that $j^{l}f(N)\cap W^{l}(N,P)=
\emptyset .$ Let $\mathcal{S}= (j^{l}f)^{-1}\mathcal{A}^{l}(N,P)$
and $\mathcal{S}'= \{f(X)\,|\,X\in \mathcal{S} \}\cup \{P\setminus
f(N) \}.$ Then $(\mathcal{S},\mathcal{S}')$ is a Thom stratification
of $f.$

The pair $(\mathcal S, \mathcal S')$ in Proposition <ref>
has a minimality property which uniquely characterizes it among all
possible pairs. We refer to Gibson et al., [39] or
Mather [66] for details.

§.§ Proof that topologically stable mappings are dense (
Mather, [66], 8)

Initially, we state the Thom-Mather topological stability theorem, whose
proof we outline in this section. Theorem B will follow from this
result and Thom's transversality theorem.
Thom!topological stability theorem

If $f:N\to P$ is proper and for some (and hence for all) $k\geq p+1,$ $j^kf$ is
multitransverse to the Thom-Mather stratification of $J^{k}(N,P),$ then $f$
is strongly $C^{\infty}$-stable.

Given $f:N\to P,$ we will show that we can approximate it by a
topologically stable mapping. First, we approximate $f$ by a mapping
$f_{1}:N \to P$ of finite singularity type (Proposition
<ref>). Then, we can choose an unfolding  $(F,N',P', i, j)$ of $f_{1}$
such that $F$ is proper and infinitesimally stable. Let
$\mathcal{S}'_{N'}$ and $\mathcal{S}'_{P'}$ be stratifications  of $N'$
and  $P',$ respectively satisfying conditions (1)-(3) in Section <ref>.
Map!Thom stratified

By Thom's transversality theorem, we can approximate $j$ by
$j_{2}:P\to P'$ such that $j_{2}$ is transverse to the strata of
$\mathcal{S}'_{P'}.$ Moreover we may suppose $j_{2}=j$ outside a compact
neighborhood of $f(N).$

Since $F$ is transverse to $j,$
it follows  that $F$ is transverse  to $j_{2}$ for $j_{2}$
sufficiently close to $j.$

The set $N_{2}=F^{-1}(j_{2}(P))$ is a smooth manifold. One can show
that there is a diffeomorphism $i_{2}:N\to N_{2}$ close
to $i:N\to N'.$

We let $f_{2}:j_{2}^{-1}\circ F\circ i_{2}:N\to P.$ It follows from construction that
$f_{2}$ is close to $f$ in the $C^{\infty}$ topology. We claim that
$f_{2}$ is topologically stable.

The proof is based in the
following facts from the construction we have made:

(i)  $(F,N',P',i_{2},j_{2})$ is an unfolding of $f_{2};$

(ii)  $F$ is proper and infinitesimally stable;

(iii)  $j_{2}$ is transverse to the stratification $\mathcal{S}'_{P'}$ of

Let $g$ be a small perturbation of $f_{2},$ so that we can suppose
$f_{2}$ and $g$ are connected by a small arc $g_{t}$ in
$C^{\infty}(N,P), t\in [0,1],$ $g_{0}=f_{2}, g_{1}=g.$ We can lift $g_{t}$ to an arc $G_{t}$ in
$C^{\infty}(N',P')$ such that $G_{0}=F$ and $(G_{t},N',P',i,j)$ is an
unfolding of $g_{t}.$ Moreover, we may suppose that $G_{t}=F$ outside
of a sufficiently small compact neighborhood of $i(N).$

From Theorem <ref>, it follows that there exist one parameter
families of diffeomorphisms $(H_{t},K_{t})\in \mathcal{A}, $
$H_{0}=Id_{N'},$ $K_{0}=Id_{P'},$ such that $F=K_{t}\circ G_{t}\circ
H_{t}^{-1},$ for all $t\in [0,1].$

Now consider the commutative diagram

\xymatrix{
N \ar[d]_{g_{t}} \ar[r]^{i} &N' \ar[d]_{G_{t}}\ar[r]^{H_{t}} &N'\ar[d]_{F}   \\
P \ar[r]^{j}   &   P'  \ar[r]^{K_{t}} &P'  }

Since $(G_{t},N',P',i,j)$ is an unfolding of $g_{t},$ it follows that
$(F,N',P',H_{t}\circ i,K_{t} \circ j)$ is also an
unfolding of $g_{t}.$ Let $G(x,t)=(g_{t}(x),t),$ $\tilde
H(x,t)=H_{t}(x)$ and $\tilde K(y,t)=K_{t}(y).$ Then we have the
following commutative diagram

\xymatrix{
{N\times I}\ar[d]_{G}  \ar[r]^{\tilde H} &N'\ar[d]^{F}   \\
{P\times I} \ar[r]^{\tilde K}   &   P'

So,we have that the triple $(F,  \mathcal{S}'_{N'},
\mathcal{S}'_{P'})$ is a Thom stratified map, and $i$ and $j$ are
transverse  respectively to $\mathcal{S}'_{N'}$ and
$\mathcal{S}'_{P'}.$ Then taking $g$ sufficiently close to $f_{2}$,
$H_{t}\circ i$ and $K_{t}\circ j$ are also transverse to  $\mathcal{S}'_{N'}$ and
$\mathcal{S}'_{P'},$  respectively.

It follows that these stratifications pull back to the Whitney's
stratifications $\tilde H^{*}(\mathcal{S}_{N'})$ and $\tilde
K^{*}(\mathcal{S}'_{P'})$ in $N\times I$ and $P\times I,$ respectively.

Moreover, each $N\times\{t\}, P\times\{t\}$ is transverse to $\tilde H^{*}(\mathcal{S}_{N'})$ and $\tilde
K^{*}(\mathcal{S}'_{P'}),$ and conditions (1)-(3) are satisfied.

Thom!second isotopy theorem
Then, we may apply the Thom's second isotopy  lemma (Gibson
et al., [39], theorem 5.8, Chapter II) and
conclude that $f_{2}=g_{0}$ is topologically equivalent to $g=g_{1}.$

§.§ The geometry of topological stability

Whether $C^{0}$-stability and $C^{\infty}$-stability are equivalent
notions in the nice dimensions is  a question not answered by
the Thom-Mather theory. The first steps towards such result appear in
Robert May's thesis [67, 68]. Mays's results were followed
by a series of papers by Damon [20, 21, 19], who
proved in [21] that $C^{\infty}$-stability is equivalent to a stronger notion
of $C^{0}$-stability.

Some of the ideas introduced in these papers form part of the basis
for Andrew du Plessis and Terry Wall's book on topological stability.
The book, The geometry of
topological stability, [82] published in 1995, is a deep
contribution to the subject of topological stability of smooth
mappings. They are  motivated by the problems left unanswered in the
Thom-Mather theory. One such problem is that it is very
difficult to determine explicitly the Thom-Mather stratification
$\mathcal{A}^k(n,p)$ in the complement of the nice dimensions and its
boundary. Another problem is that the transversality to the Thom-Mather
stratification is not a necessary condition for topological
stability. In fact, this follows from a combination of results of
Looijenga [51] and Bruce[12] as we see in examples
<ref> and <ref> below. du Plessis and Wall  give partial answers to the following two

Conjecture (i) (Conjecture 1.3 in [82]) The smooth map $f:N\to P$ is $W$-strongly
$C^{0}$-stable if and only if it is quasi-proper and locally $C^{0}$-stable.

Following [82], we say that a map $f$ is quasi-properMap!quasi-proper
if there is a neighborhood $V$ of the discriminant $\Delta (f)$ in $P$
such that the restriction of $f$ to $f^{-1}(V),$ $f:f^{-1}(V) \to V$, is
a proper map.

Conjecture (ii) If $N$ is compact, $f:N \to P$ is
$C^{0}$-stable if and only if it is locally $C^{0}$-stable.

Conjecture (iii) (Conjecture 1.4 in [82]) There
exist a $\mathcal{K}$-invariant semi-algebraic stratification
$\mathcal{B}^{k}(n,p)$ of $J^{k}(n,p)\setminus W^{k}(n,p)$ such that a
smooth map $f:N\to P$ is locally $C^{0}$-stable if and only if, for
$k$ such that $\mathrm{cod}\, W^{k}(n,p) > n,$ $j^k f$ avoids
$W^{k}(n,p)$ and is multitransverse to $\mathcal{B}^{k}(n,p).$

We summarize  now the main results of  [82].

(i) If $f:N\to P$ is $W$-strongly $C^{0}$-stable, then it is
quasi-proper and locally $C^{0}$-stable.

(ii) If $f:N\to P$ is quasi-proper, of a finite singularity
type over a neighborhood of its discriminant, and locally tamely
$P$-$C^{0}$-stable, then it is $W$-strongly $C^{0}$-stable.

The local $P$-$C^{0}$-stability is a very strong form of local
$C^{0}$-stability. We refer to <cit.>, for the
definition of tame $P$-$C^{0}$-stability.

There exist $\mathcal{K}$-invariant algebraic subsets $Y^{k}(n,k)$
in   $J^{k}(n,k)$ with  $W^{k}(n,k) \subseteq Y^{k}(n,k),$ and a
$\mathcal{K}$-invariant stratification $\mathcal{B}^{k}(n,p)$ of
$J^{k}(n,k)\setminus Y^{k}(n,k)$ with the following properties:

(a) If $f:N\to P$ is locally $C^{0}$-stable, or if $N$ is
compact and $f$ is $C^{0}$-stable, then $j^{k}f$ is
multitransverse to $\mathcal{B}^{k}(N,P);$ moreover, if
$\mathrm{codim\;} Y^{k}(n,p)\geq n,$ then $j^{k}f$ avoids $Y^{k}(N,P).$

(b) If $f:N\to P$ is such that $j^{k}f$ avoids $Y^{k}(N,P)$
and is multitransverse to $\mathcal{B}^{k}(N,P),$ then $f$ is
locally tamely $C^{0}$-stable.

As remarked by the authors,  in the range of dimensions $n <
\mathrm{codim\;} Y^{k}(n,p),$ the results imply that Conjectures 1.3 and
1.4, with $W^{k}$ replaced by $Y^{k},$ hold.

We finish this section with two examples illustrating two rather
delicate questions in the theory of $C^{0}$-stability.

Simple elliptic singularity
[The simple elliptic singularity $\tilde{E}_8$]
The simple elliptic singularities $\tilde{E}_8$ in $\mathbb{K}^{3}$,
$\mathbb{K}=\mathbb{R}$ or $\mathbb{C},$ is the
$\mathcal{K}$-unimodular family of hypersurfaces with isolated
singularities defined by

\tilde E_{8}:\ \ \ f_{\lambda}(x_{0},x_{1},x_{2})=x_{0}^{2}+x_{1}^{3}+x_{2}^{6}+\lambda

The family $f_{\lambda}$ is weighted homogeneous of type $(3,2,1;6),$
then the Milnor number $\mu(f_{\lambda})$ is constant and equal to
$10.$ When $\mathbb K=\mathbb C,$ it was shown by Looijenga
[51] that  the stable unfolding of
$f_{\lambda}$ is topologically trivial along the moduli parameter

From section <ref>, (<ref>), it follows
that the stable unfolding of $f_{\lambda}$ can be given as

\begin{align*}
F:(\mathbb C^{3} \times \mathbb C^{8}\times \mathbb C,0) &\to
(\mathbb C \times\mathbb C^{8}\times\mathbb C,0)\\
(x,u,\lambda) &\mapsto (\tilde f(x,u,\lambda), u,\lambda)
\end{align*}

with $x=(x_{0},x_{1},x_{2})$, $u=(u_{1}, \dots, u_{8}),$
$\tilde f_{\lambda}(x,u)=\tilde f(x,u,\lambda),$ $\tilde
f_{\lambda}(x,0)=f_{\lambda}(x),$ and

\begin{multline*}
\tilde f(x,u,\lambda)=x_{0}^{2}+x_{1}^{3}+x_{2}^{6}+\lambda
u_{3}x_{1}x_{2}\\ +u_{4}x_{2}^{2}
\end{multline*}

For all $\lambda$ sufficiently small, including
$\lambda=0,$ $F_{\lambda}:(\mathbb C^{11},0) \to (\mathbb C^{9},0)$ is
topologically stable. See Looijenga [51] and Bruce [12].

On the other hand, the construction of the  Thom-Mather
stratification $\mathcal{A}^{k}(n,p)$ in $J^{k}(n,p)\setminus
W^{k}(n,p)$ as discussed in sections <ref> and
<ref> reduces to the problem of finding a minimal
Whitney stratification of jets of finite singularity
type. However, Bruce proved that at $\lambda=0$ the Whitney condition
$(b)$ fails (see [12], Proposition 2 and Example 3(a)). The
failure of condition (b) can be geometrically detected as follows: the
number of cusps (${\cal A}_2$-singularities) of the intersection of
the discriminant $\Delta(F)$ with a family of $2$-planes transversal
to $\Delta(F)$ jumps from $12$ to $13$ at $\lambda=0.$ This number is
an invariant of the Thom-Mather stratification ([12],Proposition

If follows that the germ $F_{0}:(\mathbb C^{11},0) \to (\mathbb
C^{9},0)$ is topologically stable, but $j^k F_{0}$ is not transverse to
the Thom-Mather stratification.

[May [67] and  du Plessis and
Wall [82], Section 4.1 ]

Lef $f:\mathbb R \to \mathbb R$ be the proper map whose graph is
illustrated in Figure <ref>. Its singular set $\Sigma(f)$ is
$\mathbb Z \subset \mathbb R,$ and the critical values are $F(0)=0,$
$f(n)=n+1,$ for $n>0$ and odd, and $f(n)=n-1$ for $n>0$ and even;
while $f(-x)=-f(x).$ For example, we may define, as in du Plessis and
Wall [82],

\begin{cases}
x^{3}  & x \in [\frac{-1}{4}, \frac{1}{4}], \\
n+1 -(x-n)^{2}  &x\in[n-\frac{1}{4},n+ \frac{1}{4}], n\in \mathbb
N, n\ \mathrm{odd} \\
n-1 +(x-n)^{2}  &x\in[n-\frac{1}{4},n+ \frac{1}{4}], n\in \mathbb
N, n\ \mathrm{even}
\end{cases}

with $f$ defined on the remaining intervals so that it is monotone
(with $f'\neq 0$) on each interval and $C^{\infty}$ everywhere.

$C^{0}$-stable non transversal map

One can see that $f$ is $C^{0}$-stable. However it is not transverse to
the Boardman manifold $\Sigma^{1}$ at the origin. In fact, $f$ cannot
be transverse to any invariant stratification of jet space. Thus
$C^{0}$-stability  of proper maps $f:\mathbb R \to \mathbb R$ cannot be
characterized by multitransversality to any stratification.

Notice that $f$ is not locally $C^{0}$-stable, then it follows from
Theorem <ref>(i) that $f$ is not strongly stable.

§.§ Notes

Whitney!fibering conjecture
In the recent paper On the smooth Whitney fibering
conjecture [74] Murolo, du Plessis and Trotman give a