The image Milnor number and excellent unfoldings

We show three basic properties on the image Milnor number ${\mu }_I(f)$ of a germ $f\text{\hspace{0.17em}⁣}:({\mathbb{C}}^n,S)\to ({\mathbb{C}}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one can deduce the upper semi-continuity and the topological invariance for families. Second, we prove the weak Mond’s conjecture, which says that ${\mu }_I(f)=0$ if and only if $f$ is stable. Finally, we show a conjecture by Houston that any family $f_t\text{\hspace{0.17em}⁣}:({\mathbb{C}}^n,S)\to ({\mathbb{C}}^{n+1},0)$ with ${\mu }_I(f_t)$ constant is excellent in Gaffney’s sense. By technical reasons, in the two last properties we consider only the corank 1 case.