Stratified Morse critical points and Brasselet number on non-degenerate locally tame singularities

Abstract

The generalization of the Morse theory presented by Goresky and MacPherson is a landmark that divided completely the topological and geo\-me\-tri\-cal study of singular spaces. Let \{$X_t\}_t$ be a suitable family of germs at $0$ of complete intersection varieties in $\mathbb{C}^n$ and $\{f_t\}_t, \{g_t\}_t$ families of non-constant polynomial functions on $X_t$. If the germs $X_t$, $X_t \cap f_t^{-1}(0)$ and $X_t\cap f_t^{-1}(0) \cap g_t^{-1}(0)$ are non-degenerate, locally tame, complete intersection varieties, for each $t,$ we prove that the difference of the Brasselet numbers, ${\rm B}_{f_t,X_t}(0)$ and ${\rm B}_{f_t,X_t\cap g_t^{-1}(0)}(0)$, is related with the number of Morse critical points {on the regular part of the Milnor fiber} of $f_t$ appearing in a morsefication of $g_t$, even in the case where $g_t$ has a critical locus with arbitrary dimension. This result connects topological and geometric properties and allows us to determine some interesting formulae, mainly in terms of the combinatorial information from Newton polyhedra.