  We find many examples of compact Riemannian manifolds $(M,g)$ whose closed
minimal hypersurfaces satisfy a lower bound on their index that is linear in
their first Betti number. Moreover, we show that these bounds remain valid when
the metric $g$ is replaced with $g'$ in a neighbourhood of $g$. Our examples
$(M,g)$ consist of certain minimal isoparametric hypersurfaces of spheres;
their focal manifolds; the Lie groups $SU(n)$ for $n\leq 17$, and $Sp(n)$ for
all $n$; and all quaternionic Grassmannians.

Differential Geometry