On the topology of families of isolated singularities

TL;DR

This paper proves that for families of complex hypersurfaces with isolated singularities, being μ-constant is equivalent to having a uniform Milnor radius and constant embedded topological type, solving a longstanding problem.

Contribution

It establishes the equivalence between μ-constancy, uniform Milnor radius, and constant embedded topological type for such families, resolving the μ-constant problem.

Findings

μ-constant families admit a uniform Milnor radius

μ-constant families have constant embedded topological type

The μ-constant problem is solved

Abstract

We study the topology of analytic families of $n$-dimensional complex hypersurfaces having an isolated singularity at the origin. We prove that such a family is $\mu$-constant if and only if it admits an uniform Milnor radius, which happens if and only if it has constant embedded topological type. In particular, this solves the $\mu$-constant problem, formulated by D.T. L\^e and C.P. Ramanujam in 1976.