Transversality theorems for the weak topology

TL;DR

This paper extends transversality theorems to the weak topology, establishing conditions under which openness of transverse maps implies regularity of stratifications, and explores analogous results in complex manifolds.

Contribution

It generalizes Trotman's transversality results to the weak topology and complex settings under weaker hypotheses.

Findings

Transversality theorems for the weak topology are established.

Conditions linking openness of transverse maps to stratification regularity are identified.

Analogous results are proved for complex manifolds and holomorphic maps.

Abstract

In his 1979 paper Trotman proves, using the techniques of the Thom transversality theorem, that under some conditions on the dimensions of the manifolds under consideration, openness of the set of maps transverse to a stratification in the strong (Whitney) topology implies that the stratification is $(a)$-regular. Here we first discuss the Thom transversality theorem for the weak topology and then give a similiar kind of result for the weak topology, under very weak hypotheses. Recently several transversality theorems have been proved for complex manifolds and holomorphic maps. In view of these transversality theorems we also prove a result analogous to Trotman's result in the complex case.