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 Y delta$Y_\delta$ the image of a small perturbation f delta$f_\delta$. We show that the reduced cohomology of Y delta$Y_\delta$ is concentrated in a range of degrees determined by the dimension of the instability locus of f. In the case n >= p$n\ge p$, we obtain an analogous result, replacing finiteness by K$\mathcal {K}$-finiteness and Y delta$Y_\delta$ by the discriminant Delta(f delta)$\Delta (f_\delta)$. We also study the monodromy associated to the perturbation f delta$f_\delta$.