Equisingularity and The Euler Characteristic of a Milnor Fibre

TL;DR

This paper investigates the Euler characteristic of Milnor fibres of hypersurface singularities, linking it to stratification invariants, and applies these insights to equisingularity criteria for families of complex mappings.

Contribution

It introduces a new approach to compute the Euler characteristic of Milnor fibres using stratification data and extends equisingularity criteria to certain families of complex mappings.

Findings

Euler characteristic expressed via stratification invariants

Extension of Massey and Siersma's results to one-dimensional critical loci

Equisingularity characterized by constancy of a sequence for corank 1 mappings

Abstract

We study the Euler characteristic of the Milnor fibre of a hypersurface singularity. This invariant is given in terms of the Euler characteristic of a fibre in between the original singularity and its Milnor fibre and in terms of the Euler characteristics associated to strata of the in-between fibre. From this we can deduce a result of Massey and Siersma regarding singularities with a one-dimensional critical locus. The result is also applied to the study of equisingularity. The famous Brian\c{c}on-Speder-Teissier result states that a family of isolated hypersurface singularities is equisingular if and only if its $\mu ^*$-sequence is constant. We show that if a similar sequence for a family of corank 1 complex analytic mappings from n-space to (n+1)-space is constant, then the image of the family of mappings is equisingular. For families of corank 1 maps from 3-space to 4-space we show that the converse is true also.