Robust index bounds for minimal hypersurfaces of isoparametric submanifolds and symmetric spaces

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.