21 - Regularity of the transverse intersection of two regular stratifications

Summary

Abstract

We give a general theorem stating that transversely intersecting regular stratified sets have regularly stratified intersection (and union) for a large class of regularity conditions. Such a result was previously known only for Whitney regular stratified sets and for weakly Whitney stratified sets.

Introduction

It is often useful to know that the transverse intersection of two regularly stratified sets is again regular. That the transverse intersection of two Whitney regular stratified sets is again Whitney regular was apparently first published in 1976 by Chris Gibson [10] in the Liverpool notes on the topological stability of smooth mappings. In their book “Stratified Morse Theory” [11], perhaps following Teissier's account for complex analytic varieties in his La Rabida notes [19] where Teissier attributes the detailed proof given there to Denis Chéniot, Mark Goresky and Robert MacPherson cite the 1972 Comptes Rendus note by Chéniot [6] for a proof of this result, which is a mistake, as Chéniot does not prove the Whitney regularity of intersections or even discuss it; he is concerned with the frontier condition in the case of a complex variety, and moreover only for intersections with a smooth complex submanifold. This mistake in attribution has unfortunately been copied by many authors.

That the transverse intersection of two weakly Whitney regular stratified sets is again weakly Whitney regular was proved by Karim Bekka in his thesis [1] and published in [3].