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, ${\mathrm{B}}_{f_t,X_t}(0)$ and ${\mathrm{B}}_{f_t,X_t\cap g_t^{-1}(0)}(0)$, 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.

21 pages