Thom-Boardman singularities

Consider a smooth mapping , where and are smooth manifolds of dimension and , respectively (by smooth one understands: class ; cf. also Manifold). In order to understand the local structure of , it is natural to distinguish among points according to the rank of the derivative , where denotes the tangent space. For , set:

Using local coordinates on and , this set is defined locally by the vanishing of the -minors of the -matrix of first-order partial derivatives of . If one assumes that is a smooth submanifold of , for one can define

This can be visualized as follows: at a point there are two vector subspaces of , namely and . Then if and only if the intersection of these two subspaces has dimension .

Again, if one assumes that is a smooth submanifold, then one can define a subset , etc. At the end, one has partitioned the manifold into a collection of locally closed submanifolds, such that the restriction of to each submanifold is of maximal rank; in fact, if the local equations defining the various submanifolds of the collection are of maximal rank, it turns out that for the restriction of to each submanifold is an immersion, while for the same holds except at the points of rank , where it is a submersion.

This program has been initiated by R. Thom in his seminal paper [a9], inspired by earlier work of H. Whitney [a10]. Thom handles completely the first-order case, by showing that for a generic mapping (i.e. for mappings in a dense subset of all smooth mappings from to ), is a locally closed submanifold of codimension of , and that for the closure one has:

This is done by writing the mapping locally as , open, then associating to each the graph of the derivative . Clearly, if and only if , a condition defining a Schubert variety in the Grassmann manifold of -planes in . Thus, is seen locally as the pull-back of by the mapping .

This approach is exemplary, because it presents the singular locus as the pull-back of a universal situation, namely ; it is then straightforward to show that for most mappings , the induced local mappings are transversal to , and hence that is a locally closed smooth submanifold of codimension of . Moreover, this approach can serve as basis for the computation of the cohomology class that is Poincaré dual to , which can be interpreted as the first obstruction to having a homotopy from to a mapping for which (see [a9], p. 80, or [a5], Prop. 1.3; the dual classes for second-order singularities have been computed in [a7] and [a6]).

The complete proof that the process of decomposition of the source of a generic smooth mapping can be carried out successfully has been given by J.M. Boardman [a1]. See Singularities of differentiable mappings for the notions of jet space and -jet extension of a mapping , used below.

For smooth manifolds and , and integers , with , one defines the subsets of the space of -jets ; it can be proved that these are locally closed smooth submanifolds, and that if is a mapping whose jet extensions are transversal to , , then, setting

one obtains:

The codimension of equals

where is the number of sequences satisfying

Moreover, local equations for can be given explicitly, in terms of the ideal generated by the components of , in some local coordinates, and its Jacobian extensions, an operation which adds to an ideal of functions certain minors of the matrix of their first-order derivatives.

The are called Thom–Boardman singularities.

An alternative, more concise approach to Thom–Boardman singularities has been given later by J.N. Mather [a2], and an algebraic approach can be found in [a4].

Thom's transversality theorem [a8] implies that the set of mappings that are transversal to all possible Thom–Boardman singularities, that one may call generic mappings, is dense in the space of all mappings from to . So now one may ask how useful are Thom–Boardman singularities in the understanding of generic mappings.

In some cases, they allow a full classification. This is so, for example, if and , or and , , by a result of B. Morin [a3]; for and one finds the catastrophes of the fold, the cusp, the swallowtail, and the butterfly, respectively (see Thom catastrophes).

In general, Thom–Boardman singularities allow a very useful first approach to the understanding of the structure of a mapping; however, they are not fine enough to provide an, even coarse, classification. Indeed, as pointed out by I.R. Porteous [a6], a generic mapping can present the singularities and , both of dimension , and some isolated points of , called parabolic -points by Porteous, can be in the closure of ; the structure of such a mapping is definitely different at -parabolic and -non-parabolic points. Similar phenomena occur in other dimensions.

In fact, Thom–Boardman singularities provide a partition of the source of a generic mapping into locally closed submanifolds, but the closure of a submanifold is not necessarily a union of similar submanifolds.

When studying the equations of Thom–Boardman singularities, an interesting device shows up: the intrinsic derivative, first studied by Porteous (see [a5]). In general, derivatives of order higher than are not intrinsic, in the sense that are affected by higher derivatives of coordinate changes, not only the linear part of them. However, it turns out that if , then a suitable combination of the first derivatives, restricted to appropriate subspaces, is intrinsic. The simplest case is that of the second intrinsic derivative; if , then the bilinear mapping induced by the second derivative,

where , is intrinsic, as one can check easily. In the special case of a function , if is a critical point, then is the well-known Hessian bilinear form of at (cf. also Hessian matrix), whose signature determines completely the local structure of near .

The intrinsic derivative can be used to refine Thom–Boardman singularities; for example, for a generic mapping :

An inductive definition of the intrinsic derivatives is provided in [a1]; so far, it has not been tried to refine systematically Thom–Boardman singularities using them.

References

[a1]	J.M. Boardman, "Singularities of differentiable maps" Publ. Math. IHES , 33 (1967) pp. 383–419
[a2]	J.N. Mather, "On Thom–Boardman singularities" M.M. Peixoto (ed.) , Dynamical Systems, Proc. Symp. Univ. Bahia, 1971 , Acad. Press (1973) pp. 233–248
[a3]	B. Morin, "Formes canoniques des singularités d'une application différentiable" C.R. Acad. Sci. Paris , 260 (1965) pp. 5662–5665; 6503–6506
[a4]	B. Morin, "Calcul jacobien" Thèse Univ. Paris–Sud centre d'Orsay (1972)
[a5]	I.R. Porteous, "Simple singularities of maps" , Proc. Liverpool Singularities Symp. , Lecture Notes Math. , 192 , Springer (1971) pp. 286–312
[a6]	I.R. Porteous, "The second order decomposition of " Topology , 11 (1972) pp. 325–334
[a7]	F. Ronga, "Le calcul des classes duales aux singularités de Boardman d'ordre deux" Comment. Math. Helvetici , 47 (1972) pp. 15–35
[a8]	R. Thom, "Un lemme sur les applications différentiables" Bol. Soc. Mat. Mexicana (1956) pp. 59–71
[a9]	R. Thom, "Les singularités des applications différentiables" Ann. Inst. Fourier (Grenoble) , 6 (1955/6) pp. 43–87
[a10]	H. Whitney, "On singularities of mappings of euclidean spaces: I. Mappings of the plane into the plane" Ann. of Math. , 62 (1955) pp. 374–410