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Chapter 4

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, (NP) is the space of infinitely differentiable mappings from N to P, with the Whitney topology [GolG, p. 42]. Two maps f, g (NP) are isomorphic if there exist diffeomorphisms of N and of P such that g =  ° f ° -1. A map f (NP) is stable if f has a neighborhood U in (NP) 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 f0 (NP) is a mapping FRq x N -> Rq x P of the form F(ux) = (uf(ux)), with f(0, x) = f0(x). If h (RrRq) with h(0) = 0, the pull-back h*F is the r-parameter unfolding of f0 given by (h*F)(v, x) = (v, f(h(v), x)). Two q-parameter unfoldings F1 and F2 of the same function f0 are isomorphic if F2 = ° F1 ° -1, where (resp. ) is a q-parameter unfolding of the identity of N (resp. P). More generally, F1 and F2 are equivalent if there is a diffeomorphism h of Rq such that F2 is isomorphic to h*F1. An unfolding F of f0 is versal if every other unfolding of f0 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 f0 is versal if and only if the associated map R^q -> (NP) is transverse to the -orbit of f0 at f0 (cf. [Mar, th. 3.3]). A family of mappings from N to P parametrized by a smooth manifold q is a map FQ x N -> Q x P of the form F(ux) = (uf(ux)). If fu(x) = f(ux), the introduction of a coordinate chart about u in Q determines an unfolding of fu defined by F. Different charts about u determine equivalent unfoldings of fu 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 hQ' -> Q is defined by (h*F)(vx) = (vf(h(v), x)). Two families F1 and F2 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 F2 = ° (h*F1) ° -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 (NP)(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, (NP)(xy) is identified with (RnRp)(0, 0) = (np)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 (nk+1), n > k> 0, with normal form

where r < k + 1 and j = ±1 (cf. [Mori]). If r = 1 the germ is a fold, r = 2 is a cusp, r = 3 is a swallowtail, and r = 4 is a butterfly. Much work has been done on classifying the stable map germs (e.g. by Mather and Damon).

A germ f (np)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 (np)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 FQ 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 = {(ux) Q x NF/x (ux) = 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 fu changes (F(ux) = (ufu(x))).

A family of real-valued functions FQ 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 (qq)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) = x4 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 FQ x N -> Q x P and a closed subset S of (nP)0 such that S = S, one can consider the set C = {(ux) Q x N| the germ of fu at x is in S}, and the associated projection map C -> Q. In other words, there is a ``catastrophe map'' for (np)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.

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