Luncheon Talk

Abstract

Probably more than 35 years have elapsed since the first time I heard about you. It was in Princeton in 1955 when Raoul Bott spoke to me about a very bright student, who had recently found a wonderful theorem about immersions. I was startled by the result, and by the straightforward geometric aspect of the proof (of which I gave an account at the Bourbaki Seminar a little later). Moe Hirsch gave us, in his opening lecture, a beautiful review of all the consequences of your theorem. If you will excuse me for some extra technicalities, I would like to add to these a further one, in another—not mentioned —direction; namely, the connection with the theory of singularities of smooth maps, which started to attract my interest after the founding theorems of H. Whitney on the cusp and the cuspidal point. The problem of classifying immersions can be generalized as follows: Given two smooth manifolds Xⁿ and Y^p, and some “singularity type” (s) of local smooth maps: g; ℝⁿ → ℝ^p, find conditions for a homotopy class h: X → Y such that no smooth map f of the class h exhibits the singularity (s). Following Ehresmann’s theory of jets, one may associate to any singularity of type (s) a set of orbits for the algebraic action of the group L^k(n) × L^k(p) of k-jets of local isomorphisms of source and target spaces in the space J^k(n, p) of local jets from ℝⁿ to ℝ^p. (See, for instance, H. Levine’s Notes on my Lectures as Gastprofessor in Bonn.) If it happens that this set of orbits is an algebraic cycle in homology theory, then it was proved by A. Haefliger that the dual cohomology class to this singular cycle s(f) for a given map f is some specific characteristic class (mod Z or Z₂) of the quotient bundle (T(X/f*(T(Y)) [this class is a polynomial characterizing the singularity (s)]. This may be seen as a faraway consequence of your immersion theorem.