Singularities of families of mappings

In this section we discuss some of the theory of singularities of maps used in the rest of the paper.

For smooth manifolds *N* and *P*, (*N*, *P*) is
the space of infinitely differentiable mappings from *N* to *P*,
with the Whitney topology [GolG, p. 42]. Two maps
*f*, *g* (*N*, *P*)
are *isomorphic* if there exist diffeomorphisms of *N* and of *P* such that *g* = ° f ° ^{-1}. A map *f*
(*N*, *P*) is *stable* if *f* has a
neighborhood *U* in (*N*, *P*) such that every map
*g* *U* is isomorphic to *f*. In other
words, the orbit of *f* under the action of the group = Diff *N* x Diff *P* is open in
the Whitney topology [GolG, p. 72].

A *q*-parameter *unfolding* of the mapping
*f*_{0} (*N*, *P*) is a mapping
*F*: **R**^{q} x *N* -> **R**^{q} x *P*
of the form
*F*(*u*, *x*) = (*u*, *f*(*u*, *x*)),
with
*f*(0, *x*) = *f*_{0}(*x*).
If *h* (**R**^{r}, **R**^{q}) with
*h*(0) = 0, the *pull-back*
*h*^{*}*F* is the *r*-parameter unfolding of
*f*_{0} given by (h^{*}F)(v, x) = (v, f(h(v),
x)). Two q-parameter unfoldings *F*_{1} and
*F*_{2} of the same function *f*_{0} are
*isomorphic* if *F*_{2} = ° *F*_{1} ° ^{-1}, where (resp. ) is a
*q*-parameter unfolding of the identity of *N*
(resp. *P*). More generally, *F*_{1} and
*F*_{2} are *equivalent* if there is a
diffeomorphism *h* of **R**^{q} such that
*F*_{2} is isomorphic to
*h*^{*}*F*_{1}. An unfolding *F* of
*f*_{0} is *versal* if every other unfolding of
*f*_{0} is isomorphic to a pull-back of *F*. (These
definitions are taken from [Mar, p. 6].) A
theorem of Thom and Mather asserts that a *q*-parameter unfolding
of *f*_{0} is versal if and only if the associated map
**R**^q -> (*N*, *P*) is
transverse to the -orbit of f_{0} at
f_{0} (cf. [Mar, th. 3.3]).
A *family* of mappings from *N* to *P*
parametrized by a smooth manifold *q* is a map
*F*: *Q* x *N* -> *Q* x *P* of the form
*F*(*u*, *x*) = (*u*, *f*(*u*, *x*)). If
*f*_{u}(*x*) = *f*(*u*, *x*), the introduction of a coordinate
chart about *u* in *Q* determines an unfolding of *f*_{u}
defined by *F*. Different charts about *u* determine equivalent
unfoldings of *f*_{u} In other words, a family of mappings defines an unfolding of each of its members.

The *pull-back* of a family *F* by a map *h*: *Q*' -> *Q*
is defined by
(*h*^{*}*F*)(*v*, *x*) = (*v*, *f*(*h*(*v*), *x*)). Two
families *F*_{1} and *F*_{2} parametrized by *Q*
are *equivalent* if there is a diffeomorphism *h* of *Q*, a
family of diffeomorphisms of *N*
parametrized by *Q*, and a family of diffeomorphisms of *P* parametrized by *Q*, such that
*F*_{2} =
° (*h*^{*}*F*_{1}) ° ^{-1}. A family is *versal* if it defines a
versal unfolding of each of its members.

For *x* *N* and *y* *P*, consider the space of all maps *f*:*N* -> *P* such that *f*(*x*) = *y*. The
space of *germs* (*N*, *P*)_{(x, y)}
is the quotient of this space by the equivalence relation:
*f* ~ *g* if there is a neighborhood *U* of *x* in
*N* such that *f*|_{U} = *g*|_{U}. The
equivalence class of *f* is called the *germ* of *f* at *x*.
Introducing coordinate charts about *x* and *y*, (*N*, *P*)_{(x, y)} is identified with (**R**^{n}, **R**^{p})_{(0,
0)} = (*n*, *p*)_{0}. Stable germs
and versal unfoldings of germs are defined in a similar manner as for
functions (see [Mar, p. 6]).

A central problem of singularity theory is the classification of
germs up to isomorphism. Important examples of stable germs are the
*cuspoids* (or *Morin singularities*) in (*n*, *k*+1), *n* > *k*> 0, with normal form

where

A germ *f* (*n*, *p*)_{0} is *finitely determined* if
there is an integer *k* such that any germ with the same *k*-jet
(i.e. the same partial derivatives through order *k*) is isomorphic
to *f*. A germ is finitely determined if and only if it has a versal
unfolding. The minimum number *c* of parameters in a versal
unfolding of *f* is the *codimension* of *f*, and a
*c*-parameter versal unfolding of *f* is called a *universal
unfolding* of *f* (cf. [Mar]). The codimension of *f* equals the
codimension of the -orbit of *f* in (*n*, *p*)_{0}. Techniques for classifying
finitely determined germs have been recently work out (e.g. by
Gaffney, DuPleissis, and Martinet).

The classification of finitely determined germs leads to a
classification of the singularities of versal families of mappings.
Thom and Mather's transversality criterion for versal unfoldings
implies that only finitely determined germs of codimension of most
*q* can appear as members of a versal *q*-parameter unfolding. The
classification of finitely determined germs of codimension *q* allows one to write down normal forms
for the members of any *q*-parameter unfolding , and in fact one can
get a normal form for the unfolding itself.

Thom-Zeeman *catastrophe theory* is the study of singularities
of families of real-valued functions
*F*: *Q* x *N* -> *Q* x R. (an introduction to
the geometry of catastrophes is given in Callahan's survey article
``Sketching catastrophes'' [C2].) The *critical set* (or
*catastrophe manifold*) of the family *F* is the set
*C* = {(*u*, *x*)
*Q* x *N*| *F*/*x* (*u*, *x*) = 0}. If *F*
is versal, then *C* is a smooth manifold of the same dimension *q*
as *Q*. The *catastrophe map* of the family *F* is the
projections : *C* -> *Q*. The *bifurcation
set* of *F* is the image in *Q* of the singular set of (the set of points of *C* at which has rank less than *q*). As the parameter *u*
crosses the bifurcation set *F*, the number of critical points of the
function *f*_{u} changes
(*F*(*u*, *x*) = (*u*, *f*_{u}(*x*))).

A family of real-valued functions
*F*: *Q* x *N* -> *Q* x **R** is versal if and
only if *F* is a stable mapping (cf. [Mar]). However, the catastrophe
map of a versal family is *not* necessarily stable. Let (*q*, *q*)_{0} be the germ at zero of a
catastrophe map of a versal *q*-parameter unfolding of a germ *f*
(*n*, 1). If
*q* = 1, is either regular (i.e. rank *q*)
or a Morse singularity. If *q* = 2, is
either regular, a fold, or a cusp. If *q* = 3, is regular, a fold, a cusp, a swallowtail, or an
umbilic. Umbilic germs in (3, 3)_{0} are
not stable. Descriptions of these catastrophes can be found in
several sources: [T4, Chapt. 5], [C1], [C2], [PosS, Chapt. 9], [Wo].
The name of the germ is also used to refer to
the germ *f* being unfolded (e.g. *f*(*x*) = *x*^{4} is
called a cusp, because the catastrophe map of the universal unfolding
has a cusp singularity.) In chapter 5 we shall see why Thom chose the
name "umbilic" for the simplest unstable catastrophe map germs.

Since finitely determined germs can be classified, there is no
reason to consider only families of real-valued functions from the
viewpoint of catastrophe theory. Given a family
*F*: *Q* x *N* -> *Q* x *P*
and a closed subset *S* of (*n*, *P*)_{0} such that *S* = *S*, one can consider the
set *C* = {(*u*, *x*) *Q* x *N*| the germ of
*f*_{u} at *x* is in *S*}, and the associated
projection map : *C* -> *Q*. In other words,
there is a ``catastrophe map'' for (*n*, *p*)_{0}. Finally, since
the theory is local, it applies equally well to a ``twisted'' family
parametrized by *Q*, i.e. a mapping
*F*:*A* -> *B* where
*a*:*A*-> *Q* and
*B*:*B* -> *Q* are smooth fiber bundles over
*Q* , and
*B* ° *F* = *a*. These
generalizations of catastrophe theory have been developed by Gaffney
and Ruas [GafR], Kergosien [Ke3], and Arnold et al. [AGV]. These generalizations will be used in
chapter 7.