Cohomological connectivity of perturbations of map-germs

Abstract

Let $f:({\mathbb{C}}^n,S)\to ({\mathbb{C}}^p,0)$ 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 $n\ge p$ 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 $f_{\delta }$.

32 pages, 6 figures