Some notes on the local topology of a deformation of a function-germ with a one-dimensional critical set

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 Le-Iomdin formula for the Brasselet number.