  We study the Morse index of self-shrinkers for the mean curvature flow and,
more generally, of $f$-minimal hypersurfaces in a weighted Euclidean space
endowed with a convex weight. When the hypersurface is compact, we show that
the index is bounded from below by an affine function of its first Betti
number. When the first Betti number is large, this improves index estimates
known in literature. In the complete non-compact case, the lower bound is in
terms of the dimension of the space of weighted square summable $f$-harmonic
$1$-forms; in particular, in dimension $2$, the procedure gives an index
estimate in terms of the genus of the surface.
