  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 $f_\delta$.
