Embedded topological triviality of separable families of singularities

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.