Finiteness of Mather's Canonical Stratification

Summary

Introduction

In [5, 6, 7], Mather showed how to construct a “canonical” Whitney-regular stratification of jet-space, with the property that a proper map whose jet-extension is multi-transverse to the stratification is topologically stable. (An alternative, rather simpler, presentation of this result is given in [2].)

Mather also claimed, in [7], that the stratification constructed has only finitely many connected strata. This is of considerable interest because the topological type of a map-germ transverse to a (connected component of a) canonical stratum is determined by the stratum, so the claim implies that there are only finitely many topological types of topologically stable map-germs in any given dimension-pair.

Unfortunately, Mather's proof [7, p.170] of this finiteness is not correct. The aim of this article is to give a correct proof.

We give this proof in §2. In §1 we give the necessary background; in particular we describe a construction for the canonical stratification rather simpler than both Mather's and that given in [2].

The arguments given here were originally worked out for the review section of the book [9], see in particular pp. 44–5, but in the event were not included there.

I thank Terry Wall for help with the preparation of this article.