Euler obstruction and generalizations

• Ano de defesa: 2019
• Autor(a) principal: Santana, Hellen Monção de Carvalho
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Brasselet number; Euler obstruction; Invariantes locais topológicos; Local topological invariants; Número de Brasselet; Obstrução de Euler; Pontos críticos de Morse estratificados; Stratified Morse critical points
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/55/55135/tde-19032020-161129/

Let f,g : (X, 0) → (C, 0) be germs of analytic functions defined over a complex analytic space X. The Brasselet number of a function f describes numerically the topology of its generalized Milnor fibre. In this thesis, we present formulas to compare the Brasselet numbers of f in X and of the restriction of f to X ∩ {g = 0}, in the case where g has a one-dimensional stratified critical set and f has an arbitrary critical set. If, additionally, f has isolated singularity at the origin, we compute the Brasselet number of g in X and compare it with the Brasselet number of f in X. As a consequence, we obtain formulas to compute the local Euler obstruction of X and of X ∩ {g = 0} at the origin, comparing these numbers with local invariants associated to f and g. We also study the local topology of a deformation of g, g = g + fN, for a positive integer number N ≫ 1. We provide a relation between the Brasselet number of g and g in X ∩ { f = 0}, in the case where f has isolated singularity at the origin. We also provide a new proof for the Lê-Iomdine formula for the Brasselet number.