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 \Xt\t be a suitable family of germs at 0 of complete intersection varieties in Cn and \ft\t, \gt\t families of non-constant polynomial functions on Xt. If the germs Xt, Xt ft-1(0) and Xt ft-1(0) gt-1(0) are non-degenerate, locally tame, complete intersection varieties, for each t, we prove that the difference of the Brasselet numbers, Bft,Xt(0) and Bft,Xt gt-1(0)(0), is related with the number of Morse critical points on the regular part of the Milnor fiber of ft appearing in a morsefication of gt, even in the case where gt 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.