Embedded topological triviality of separable families of singularities

TL;DR

This paper establishes conditions under which families of complex analytic singularities are topologically trivial in an embedded sense, advancing understanding of stability and the $mbda$-constant conjecture.

Contribution

It extends previous results to the embedded setting, providing new criteria for topological triviality and stability in singularity families.

Findings

Derived sufficient conditions for embedded topological triviality

Identified new classes of $mbda$-constant deformations

Provided insights into the $mbda$-constant conjecture

Abstract

Understanding how singularities behave under small perturbations is a central theme in singularity theory. In this paper we establish sufficient conditions for families of analytic function-germs on a germ of a complex analytic space to admit an embedded topological trivialization. Our results extend previous work of the third author and collaborators, moving from abstract triviality to the embedded setting. As an application, we obtain new instances of topological stability, including a broad class of $\mu$-constant deformations. These findings provide a new insight into the long-standing $\mu$-constant conjecture, one of the major open problems in the field.