Trivialidade topológica relativa de família de funções

Abstract

In this work, we investigate the C^0-\mathcal{R}_X-triviality of families of germs h : (\mathbb{C}^n \times \mathbb{C}, 0) \to (\mathbb{C}, 0), extending the classical \mathcal{R}-equivalence to the context of the group \mathcal{R}_X, which preserves an analytic subvariety (X,0). We analyze sufficient conditions for topological triviality based on the integral closure of the tangent space to the group \mathcal{R}_X, with a particular focus on deformations in quasi-homogeneous varieties. We also study the constancy of the Bruce-Roberts number, \mu_{BR}, in families of functions with isolated singularities.