Index estimate by first Betti number of minimal hypersurfaces in compact symmetric spaces

We show that the Morse index of unstable closed minimal hypersurface $\mathrm{\Sigma }$ in a compact semi-simple Riemannian symmetric space $M=G\mathrm{/}K$ is bounded from below by constant times the first Betti number of $\mathrm{\Sigma }$. Our proof is based on a natural extension of the previous method and this also provides a novel approach for the index estimate.