Singularities of mappings on ICIS and applications to Whitney equisingularity

Abstract

We study germs of analytic maps $f:(X,S)\rightarrow(\mathbb{C}^p,0)$, when $X$ is an ICIS of dimension $n<p$. We define an image Milnor number, generalizing Mond's definition, $μ_I(X,f)$ and give results known for the smooth case such as the conservation of this quantity by deformations. We also use this to characterise the Whitney equisingularity of families of corank one map germs $f_t\colon(\mathbb{C}^n,S)\to(\mathbb{C}^{n+1},0)$ with isolated instabilities in terms of the constancy of the $μ_I^*$-sequences of $f_t$ and the projections $π\colon D^2(f_t)\to\mathbb{C}^n$, where $D^2(f_t)$ is the ICIS given by double point space of $f_t$ in $\mathbb{C}^n\times\mathbb{C}^n$. The $μ_I^*$-sequence of a map germ consist of the image Milnor number of the map germ and all its successive transverse slices.

Figures (4)

• Figure 1: Illustration of how $\mu_I(X,f)$ works, i.e., of the homology of the image of a stable perturbation $(X_t,f_t)$.
• Figure 2: The first alternating Milnor number and its relationship with the deformations of $(X,f)$ and the $\mu_I(X,f)$. Here, one can also appreciate the conclusions of \ref{['conservation', 'uscontinuity']}.
• Figure 3: Depiction of the Lê-Greuel type formula for map germs.
• Figure 4: Representation of how the homology of the double point set of a stable perturbation works.

Theorems & Definitions (89)

• Definition \oldthetheorem: see Mond1994
• Definition \oldthetheorem
• Definition \oldthetheorem: cf. Mond1994
• Definition \oldthetheorem
• Definition \oldthetheorem: cf. Mond1994
• Definition \oldthetheorem
• Theorem \oldthetheorem
• Proposition \oldthetheorem
• proof
• Definition \oldthetheorem