Brasselet Number and Function-Germs with a One-Dimensional Critical Set

Abstract

The Brasselet number of a function f hnonisolated 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) → (C, 0) and g : (X , 0) → (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.

Euler obstruction, Brasselet number and critical points

“…The reader may refer to [9,32] for further works on the Brasselet number. In the rest of the paper, we will explain how to obtain a result in the spirit of the result of Seade, Tibȃr and Verjovsky mentioned above, i.e.…”

The geometrical information encoded by the Euler obstruction of a map

“…In this section, we recall the definition of the local Euler obstruction, the Euler obstruction of a function and the Brasselet number, both generalizations of the local Euler obstruction. The main references for this section are [19,17,6,24].…”

“…In [24], the author generalized the last formula to the case where the critical locus of g, Σ V g, is one-dimensional and that Σ V g∩{f = 0} = {0}. We can decompose Σ V g into branches b j , Σ V g = q α=1 Σg| Vα ∪ {0} = b 1 ∪ .…”

“…In the last result of this section, we apply formulas of [24] to compare the Brasselet number and the Euler obstruction of a map. Proposition 3.4.…”

“…Consider the perturbation f 1 = f 1 + f N 2 of f 1 , where N >> 1 is an integer. In [23], the author proved that, if N >> 1 is sufficiently large and δ is a regular value of f 2 , the number n reg of Morse critical points of a stratified Morsefication of f 1 | X∩f −1 2 (δ) appearing on X reg ∩f −1 2 (δ)∩B ε and the number n reg of Morse critical points of a stratified Morsefication of…”