Isotypical components of the homology of ICIS and images of deformations of map germs

TL;DR

This paper introduces a new approach to analyze the homology of simplicial complexes with group actions, applies it to Milnor fibers of ICIS, and investigates the behavior of images under deformations of complex map germs, revealing surprising traits and correcting existing conjectures.

Contribution

It provides a simple method to study isotypical components of homology in simplicial complexes with group actions and extends the understanding of image Milnor numbers for complex map germs.

Findings

New method for analyzing homology of simplicial complexes with group actions

Workable computation of image Milnor numbers in corank one cases

Counterexamples to Houston's conjecture and its corrected version

Abstract

We give a simple way to study the isotypical components of the homology of simplicial complexes with actions of finite groups, and use it for Milnor fibers of ICIS. We study the homology of images of mappings $f_t$ that arise as deformations of complex map germs $f:(\mathbb{C}^n,S)\to(\mathbb{C}^p,0)$, with $n<p$, and the behaviour of singularities (instabilities) in this context. We study two generalizations of the notion of image Milnor number $\mu_I$ given by Mond and give a workable way of compute them, in corank one, with Milnor numbers of ICIS. We also study two unexpected traits when $p>n+1$: stable perturbations with contractible image and homology of $\text{im} f_t$ in unexpected dimensions. We show that Houston's conjecture, $\mu_I$ constant in a family implies excellency in Gaffney's sense, is false, but we give a correct modification of the statement of the conjecture which we also prove.