Elimination of Singularities: Thom Polynomials and Beyond

Summary

Dedicated to Professor C. T. C. Wall on his 60th birthday.

Introduction

This is a survey article concerning the authors' papers [47], [33], [50], [51], [52], [53], where the following problem is considered: Given a smooth map f: M → N between smooth manifolds, does there exist a smooth map homotopic to f which has no singularities of a prescribed type Σ? It is known that this problem is almost equivalent to the existence problem of a corresponding jet section M → J^r(M, N) covering f, which is homotopy theoretic in nature (for example, see [30], [28], [20], [21], [15], [16], [17] etc.). However, this homotopy theoretical problem is usually very difficult to solve, since even the homotopy type of the corresponding fiber in the jet bundle is difficult to determine.

The most easily computed part of an obstruction to the existence of a corresponding jet section is the Thom polynomial, which is the homology class represented by the closure of the set Σ(f) of the singular points of f of type Σ [58], [31]. In fact, in some cases, it has been shown that the vanishing of the Thom polynomial implies the existence of a map homotopic to f without the prescribed singularities: for example, H. Levine's cusp elimination theorem for maps into the plane [35] and its generalizations to some 0-dimensional singularities by Ando [5], [6].