Cohomological connectivity of perturbations of map-germs

TL;DR

This paper investigates the cohomological properties of perturbations of map-germs, revealing how their topology is concentrated in specific degrees related to the instability locus, and explores associated monodromy phenomena.

Contribution

It establishes new results on the cohomology concentration of perturbed map-germs and their discriminants, extending classical topology to singularity theory.

Findings

Cohomology of $Y_\delta$ is concentrated in degrees related to the instability locus.

Analogous results hold for $n \geq p$ with $\mathcal{K}$-finiteness and discriminants.

Monodromy of perturbations is also analyzed.

Abstract

Let $f\colon (\mathbb C^n,S)\to (\mathbb C^p,0)$ be a finite map-germ with $n<p$ and $Y_\delta$ the image of a small perturbation $f_\delta$. We show that the reduced cohomology of $Y_\delta$ is concentrated in a range of degrees determined by the dimension of the instability locus of $f$. In the case $n\geq p$ we obtain an analogous result, replacing finiteness by $\mathcal K$-finiteness and $Y_\delta$ by the discriminant $\Delta(f_\delta)$. We also study the monodromy associated to the perturbation $f_\delta$.