We show three basic properties of the image Milnor number µI(f) of a germ $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ 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 states that µI(f) = 0 if and only if f is stable. Finally, we show a conjecture by Houston that any family $f_t\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with $\mu_I(\,f_t)$ constant is excellent in Gaffney’s sense. For technical reasons, in the last two properties, we consider only the corank 1 case.

We characterise the Whitney equisingularity of families of corank one map germs ft : (C n , S) → (C n+1 , 0) with isolated instabilities in terms of the constancy of the µ * I -sequences of ft and the projections π :The µ * I -sequence of a map germ consist of the image Milnor number (in Mond's sense) of the map germ and all its successive transverse slices. Since D 2 (f ) is an icis in C n ×C n (when f has corank one and has isolated instability), we need to develop the notions of image Milnor number and the image computing spectral sequence for map germs defined on icis.

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We prove that a map germ $$f:(\mathbb {C}^n,S)\rightarrow (\mathbb {C}^{n+1},0)$$ f : ( C n , S ) → ( C n + 1 , 0 ) with isolated instability is stable if and only if $$\mu _I(f)=0$$ μ I ( f ) = 0 , where $$\mu _I(f)$$ μ 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 $$\ge 2$$ ≥ 2 , provided that $$(n,n+1)$$ ( n , n + 1 ) are nice dimensions in Mather’s sense (so $$\mu _I(f)$$ μ I ( f ) is well defined). Our result can be seen as a weak version of a conjecture by Mond, which says that the $$\mathscr {A}_e$$ A e -codimension of f is $$\le \mu _I(f)$$ ≤ μ 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.

We give the definition of the Thom condition and we show that given any germ of complex analytic function $$f:(X,x)\rightarrow ({\mathbb {C}},0)$$ f : ( X , x ) → ( C , 0 ) on a complex analytic space X, there exists a geometric local monodromy without fixed points, provided that $$f\in {\mathfrak {m}}_{X,x}^2$$ f ∈ m X , x 2 , where $${\mathfrak {m}}_{X,x}$$ m X , x is the maximal ideal of $${\mathcal {O}}_{X,x}$$ O X , x . This result generalizes a well-known theorem of the second named author when X is smooth and proves a statement by Tibar in his PhD thesis. It also implies the A’Campo theorem that the Lefschetz number of the monodromy is equal to zero. Moreover, we give an application to the case that X has maximal rectified homotopical depth at x and show that a family of such functions with isolated critical points and constant total Milnor number has no coalescing of singularities.