Topological Stability

Summary

This article is an introductory account of work by the authors which is to appear in a book entitled “The geometry of topological stability”.

A C^∞ map f : N → P between smooth manifolds is called C^r-stable if it has a neighbourhood W such that for any g ∈ W there exist C^r-diffeomorphisms p of N, λ of P with g = λ o f o p. If λ and p depend continuously on g, we call f strongly stable.

We will concentrate on the case when N is compact. The results can be extended to the general case, but matters become much more complicated, partly because we have a choice of topologies on the spaces C^∞(N,P) (of C^∞-mappings) etc - on the whole, the Whitney topologies are the most appropriate - leading to many variants of the definition of stability, which are demonstrably not all equivalent, so that it is harder to match up necessary with sufficient conditions for stability. To hint at the kind of considerations involved we draw attention to a generalisation of properness which plays an important rôle. We say f is quasi-proper if the discriminant Δ(f) has a neighbourhood U in P such that the restriction of f to f^-1U is proper. Then the condition of being quasi-proper is necessary for versions of C^°-stability requiring some kind of uniformity in the Whitney topology though not for C^°-stability per se.