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

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 fibres of isolated complete intersection singularity ( 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 µ 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 $p \gt n+1$ : stable perturbations with contractible image and homology of $\text{im} f_t$ in unexpected dimensions. We show that Houston’s conjecture, µ 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.