A weak version of Mond's conjecture

Abstract

We prove that a map germ $f:(\mathbb{C}^n,S)\to(\mathbb{C}^{n+1},0)$ with isolated instability is stable if and only if $\mu_I(f)=0$, where $\mu_I(f)$ is the image Milnor number defined by Mond. In a previous paper we proved this result with the additional assumption that $f$ has corank one. The proof here is also valid for corank $\ge 2$, provided that $(n,n+1)$ are nice dimensions in Mather's sense (so $\mu_I(f)$ is well defined). Our result can be seen as a weak version of a conjecture by Mond, which says that the $\mathcal{A}_e$-codimension of $f$ is $\le \mu_I(f)$, with equality if $f$ is weighted homogeneous. As an application, we deduce that the bifurcation set of a versal unfolding of $f$ is a hypersurface.