Brasselet number and function-germs with a one-dimensional critical set

TL;DR

This paper explores the Brasselet number for functions with nonisolated singularities, providing formulas for cases where the critical set is one-dimensional and discussing applications to isolated singularities and linear forms.

Contribution

It introduces formulas for the Brasselet number in the context of functions with one-dimensional critical loci, expanding understanding of their topological invariants.

Findings

Formulas for the Brasselet number with one-dimensional critical sets

Applications to functions with isolated singularities

Results for generic linear forms

Abstract

The Brasselet number of a function $f$ with nonisolated singularities describes numerically the topological information of its generalized Milnor fibre. In this work, using the Brasselet number, we present several formulas for germs $f:(X, 0) \rightarrow (\mathbb{C}, 0)$ and $g : (X, 0)\rightarrow (\mathbb{C}, 0)$ in the case where g has a one-dimensional critical locus. We also give applications when f has isolated singularities and when it is a generic linear form.