23 - Bi-Lipschitz equisingularity

Summary

Abstract

Much of the recent work of both Terry Gaffney and Maria Ruas has centred on problems of equisingularity. I discuss recent results on bilipschitz equisingularity including some important results obtained by my former student Guillaume Valette, in particular a bilipschitz version of the Hardt semiagebraic triviality theorem and the resolution of a conjecture of Siebenmann and Sullivan dating from 1977.

An oft-heard slogan

Stratifications are often used in singularity theory via the slogan that follows:

“Given an analytic variety (or semialgebraic set or subanalytic set) take some Whitney stratification. Then the stratified set is locally topologically trivial along each stratum by the first isotopy theorem of Thom-Mather.”

Statements like this have been made hundreds of times, after the publication in 1969 of René Thom's foundational paper “Ensembles et morphismes stratifiés” [35], and John Mather's 1970 Harvard notes on topological stability [19]. (A detailed published proof of the Thom-Mather isotopy theorem, somewhat different to those of Thom and Mather, can be found in the write-up of the 1974–75 Liverpool Seminar published by Springer Lecture Notes in 1976, in the second chapter written by Klaus Wirthmüller [7].)

It seems to be not yet so well-known that in the above statements one may replace “Whitney stratification” by “Mostowski stratification” and “locally topologically trivial” by “locally bi-Lipschitz trivial”, despite the fact that the corresponding theorems were published by Tadeusz Mostowski over twenty years ago in 1985 [22].