The Pontrjagin - Thom construction for singular maps and its applications to elimination of singularities
by
Andras Szucs
Etvosh Loran University

Given a stable map of a closed surface in the plane the number of cusps has the same pairity as the Euler characteristics of the surface. If the surface has even Euler characteristics then it has a stable map in the plane without cusps. These are well - known old theorems by Whitney , Levine and Eliashberg. Our aim is to generalize these statements to higher dimensions and arbitrary stable singularities.

Arnold and his coauthers posed the following question (in [AVGL]):

Question 1: Suppose that a stable map has vanishing Thom polynomials for a given singularity type \eta. Is then the map homotopic to a map having no singular points of type \eta ? (Vanishing of the Thom polynomials is clearly necessary for the elimination of \eta points.)

We shall consider the following similar question:

Question 2. Let \eta be a maximal singularity of a stable f: Nⁿ --> P^n+k. Under what conditions is the map cobordant to an \eta-free map g ? By cobordism we mean such a stable map in the cylinder P^n+k ×[0, 1] of a compact n+1 manifold with boundary, which has no other singularities than the map f has.

We give necessary and sufficient conditions for the positive answer. They differ only by a finite factor.

1. Necessary conditions: At this time we do not need vanishing of the Thom polynomials of \eta for the map f, but the Gysin map must annihilate it. Let us suppose that the manifold P is stably paralellizable. Then the higher Thom polynomials of \eta (expressing the images of the characteritic classes of the submanifold formed by the \eta-singular points in the cohomologies of the source of f) also must be mapped into zero by the Gysin map f_!.

2. Sufficient conditions: We shall show that up to a finite factor these necessary conditions are also sufficient. That is the following theorem holds.

Theorem. Let f: Nⁿ --> P^n+k be a stable map, where P^n+k is stably paralellizable, let \eta be a maximal singularity of f and suppose that the Gysin map f_! anihilites the Thom polynomial of the \eta singularity, and also all the higher Thom polynomials. Then there is a non-zero integer L such that L ·f is cobordant to an \eta-free map by a cobordism having no other singularities than f has.

The proof relies on the construction of the universal \tau-map f_\tau:Y_\tau --> X_\tau. Here \tau is a set of multisingularities and the universal \tau map is universal among the maps having only such multisingularities, that belong to \tau. The map f_\tau is universal in the same sense as the inclusion of the Grassmann manifold BO(k) in the Thom space MO(k) is universal among the (codimension k) embeddings.

Considering the cohomologies of the source space Y_\tau of f_\tau one can find the Thom polynomials mentioned above, see [R].

Investigating the homotopy type of the target space - combined with a so called stabilization trick - one can prove the sufficient condition formulated in the theorem. (The special case, when \eta = \Sigma^{1, 0} and so \eta-free means non-singular, has been considered in [Sz1] and [Sz2].)

Finally we give examples showing that the integer L can not be omitted, that is the map f itself may not be cobordant to an \eta-free map. For example if \eta = A_2r+1, Nⁿ is oriented, and k = 0, then the obstruction to the elimination of \eta-points lies in the stable homotopy groups of the target manifold P.

References

[AVGL] V.I. Arnold, V.A. Vasil'ev, V.V. Goryunov, O.V. Lyashko: Singularities, Local and Global Theory, Dynamical Systems VI., Encyclopaedia of Mathemetical Sciences, vol. 6.

[R] R. Rimányi: Thom polynomials via symmetries of singularities, Submtted to Inventiones Math.

[Sz1] A. Szücs: Analogue of the Thom space for mapping with singularity of type \Sigma¹, Math. Sbornik 108 (150) (1979) No 3. 438 - 456 (in Russian) English translation: Math. USSR-Sb. 36 (1979) No 3. 405 - 426 (1980)

[Sz2] A. Szücs: Immersions in bordism classes. Math. Proc. Camb. Phil. Soc. (1988) 103, 89 - 95.

Date received: June 1, 1999