Local topology of a deformation of a function-germ with a one-dimensional critical set

TL;DR

This paper investigates the local topology of deformations of functions with one-dimensional critical sets using the Brasselet number, providing new insights and a novel proof of the Lê-Iomdin formula.

Contribution

It introduces a method to analyze deformations of functions with non-isolated singularities using the Brasselet number, including a new proof of the Lê-Iomdin formula.

Findings

The Brasselet number effectively captures topological changes in deformations.

Deformations of functions with one-dimensional critical sets can be studied via the Brasselet number.

A new proof of the Lê-Iomdin formula is established.

Abstract

The Brasselet number of a function $f$ with nonisolated singularities describes numerically the topological information of its generalized Milnor fibre. In this work, we consider two function-germs $f,g:(X,0)\rightarrow(\mathbb{C},0)$ such that $f$ has isolated singularity at the origin and $g$ has a stratified one-dimensional critical set. We use the Brasselet number to study the local topology a deformation $\tilde{g}$ of $g$ defined by $\tilde{g}=g+f^N,$ where $N\gg1$ and $N\in\mathbb{N}$. As an application of this study, we present a new proof of the L\^e-Iomdin formula for the Brasselet number.