Cohomological connectivity of perturbations of map‐germs.

Let f:(Cn,S)→(Cp,0)$f: (\mathbb {C}^n,S)\rightarrow (\mathbb {C}^p,0)$ be a finite map‐germ with n<p$n<p$ and Yδ$Y_\delta$ the image of a small perturbation fδ$f_\delta$. We show that the reduced cohomology of Yδ$Y_\delta$ is concentrated in a range of degrees determined by the dimension of the instability locus of f$f$. In the case n≥p$n\ge p$, we obtain an analogous result, replacing finiteness by K$\mathcal {K}$‐finiteness and Yδ$Y_\delta$ by the discriminant Δ(fδ)$\Delta (f_\delta)$. We also study the monodromy associated to the perturbation fδ$f_\delta$. [ABSTRACT FROM AUTHOR]