Topological equisingularity of function germs with 1-dimensional critical set

TL;DR

This paper investigates topological equisingularity of holomorphic function germs with 1-dimensional critical sets, introducing new concepts and conditions that ensure topological stability and characterizing the deformation behavior of critical sets.

Contribution

It introduces the notion of equisingularity at the critical set, proves its equivalence to topological equisingularity under certain conditions, and refines the concept of singularity stems in the literature.

Findings

Families with constant generic Le numbers are topologically equisingular.

Introduces and characterizes topological stems related to Arnold's singularities.

Provides examples of non-flat deformations with topologically equisingular critical sets.

Abstract

We focus on topological equisingularity of families of holomorphic function germs with 1-dimensional critical set. We introduce the notion of equisingularity at the critical set and prove that any family which is equisingular at the critical set is topologically equisingular. We show that if a family of germs with 1-dimensional critical set has constant generic Le numbers then it is equisingular at the critical set, and hence topologically equisingular (answering a question of D. Massey). We use this to modify the definition of singularity stem present in the literature, introducing and characterising topological stems (being this concept closely related with Arnold's series of singularities). We provide another sufficient condition for topological equisingularity for families whose reduced critical set is deformed flatly. Finally we study how the critical set can be deformed in a topologically equisingular family and provide examples of topologically equisingular families whose critical set is a non-flat deformation with singular special fibre and smooth generic fibre.