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

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 $f:(X,0)\to (\mathbb{C},0)$ and $g:(X,0)\to (\mathbb{C},0)$ 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.