Euler obstruction and generalizations

Author(s)

Hellen Monção de Carvalho Santana

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2019-12-10

Examining board members

Nivaldo de Góes Grulha Junior; Nicolas Andre Oliver Dutertre; Miriam da Silva Pereira; David John Angelo Trotman

• Advisor: Nivaldo de Góes Grulha Junior

Abstract

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. (AU)

• FAPESP's process: 15/25191-9 - The Euler obstruction and generalizations
• Grantee: Hellen Monção de Carvalho Santana
• Support Opportunities: Scholarships in Brazil - Doctorate