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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 $$g'$$g′ in a neighbourhood of g. Our examples (M, g) consist of certain minimal isoparametric hypersurfaces of spheres, their focal manifolds, the Lie groups $${\text {SU}}(n)$$SU(n) for $$n\le 17$$n≤17 and $${\text {Sp}}(n)$$Sp(n) for all n, and all quaternionic Grassmannians.

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