Stratification of Unfoldings of Corank 1 Singularities

TL;DR

This paper explores how image Milnor numbers can be used to stratify unfoldings of corank 1 singularities, ensuring excellent unfoldings and analyzing their topological properties.

Contribution

It introduces a new cohomological description of the disentanglement of corank 1 multi-germs and applies it to establish conditions for excellent unfoldings.

Findings

Image Milnor numbers can ensure conditions for excellent unfoldings.

The rational cohomology of the disentanglement is described for the first time.

Upper semi-continuity of the image Milnor number is established.

Abstract

In the study of equisingularity of families of mappings Gaffney introduced the crucial notion of excellent unfoldings. This definition essentially says that the family can be stratified so that there are no strata of dimension 1 other than the parameter axis for the family. Consider a family of corank 1 multi-germs with source dimension less than target. In this paper it is shown how image Milnor numbers can ensure some of the conditions involved in being excellent. The methods used can also be successfully applied to cases where the double point set is a curve. In order to prove the results the rational cohomology description of the disentanglement of a corank 1 multi-germ is given for the first time. Then, using a simple generalization of the Marar-Mond Theorem on the multiple point space of such maps, this description is applied to give conditions which imply the upper semi-continuity of the image Milnor number. From this the main results follow.