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

TL;DR

This work develops a representation-theoretic framework for analyzing the homology of images of deformations of map germs $f:(\bC^n,S)\to(\bC^p,0)$ with $n<p$, focusing on the regime $p>n$ and ICIS Milnor fibers. It constructs the Image-Computing Spectral Sequence (ICSS) fed by multiple point spaces $D^k(f)$ and expresses the image homology of stable perturbations in terms of Milnor numbers of corank-1 ICIS, thereby generalizing Mond's image Milnor number $\mu_I$ to a broader setting, including the companion invariant $\nu_I$. The paper introduces and analyzes strongly contractible instabilities and unexpected homology, establishes conservation principles in families, and shows that Houston's conjecture on excellent unfoldings is false in general but can be repaired under precise hypotheses; a detailed dimension-based dichotomy in terms of $d_2$ governs the results. The approach leverages group actions, Alt-representations, and fixed-point data to compute isotypical components, yielding practical formulas for $\mu_I$, $\nu_I$, and their Alt variants via ICIS data. The work concludes with open problems about higher corank cases, coalescence phenomena, and potential icis-based inequalities linked to Mond-type conjectures, outlining a program for extending these techniques to broader singularity contexts.

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 fibers of 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 $p>n+1$: 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.

Figures (10)

• Figure 1: Diagrams of \ref{['ex: esfera con accion']}.
• Figure 2: Representation of a strongly contractible instability given by $f$ (left) with its locally stable perturbation $f_s$ (right).
• Figure 3: Plot of the pair of dimensions without strongly contractible germs of corank one, for $n\leq20$ (left) and for $n\leq250$ (right).
• Figure 4: Diagram of the tangent double point $f$ given in \ref{['ex: singularidad libre']} (cf. \ref{['fig:Tangency2']}).
• Figure 5: Possibly non-zero entries, with its rank, of the $E^\infty$-page of the spectral sequence $E^1_{r,q}= H^{\textnormal{Alt}_{r+1}}_q\left(D^{r+1}\left(F\right),D^{r+1}\left(f_s\right)\right)$ for a map germ $f:(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{p},0)$. Notice the shift $H_{m+1}(\operatorname{im} F ,\operatorname{im} f_s)\cong H_m(\operatorname{im} f_s)$ when $m>0$.

Theorems & Definitions (79)

• Lemma 2.1
• proof
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Example 2.6
• Definition 2.7