Equisingularity of families of map germs from the plane to 3-space

TL;DR

This paper characterizes Whitney equisingularity for families of map germs from the plane to three-space using the constancy of a specific Milnor number invariant, providing a solution to a question posed in 1994.

Contribution

It introduces a criterion based on the Milnor number of a particular curve to determine Whitney equisingularity in families of map germs from ^2 to ^3.

Findings

Whitney equisingularity is characterized by the constancy of the Milnor number of a specific curve.

Provides a solution to Ruas's 1994 question on equisingularity criteria.

Establishes a link between invariants of map germs and their equisingularity properties.

Abstract

In this work, we consider a finitely determined map germ $f$ from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We characterize the Whitney equisingularity of an unfolding $F=(f_t,t)$ of $f$ through the constancy of a single invariant in the source. Namely, the Milnor number of the curve $W_t(f)=D(f_t)\cup f_t^{-1}(\gamma)$, where $D(f)$ denotes the double point curve of $f_t$. This gives an answer to a question by Ruas in 1994.