Index bounds for minimal hypersurfaces of the sphere

We consider a compact, orientable minimal hypersurfaces of the unit sphere and prove a comparison theorem between the spectrum of the stability operator and that of the Laplacian on 1-forms. As a corollary, we show that the index is bounded below by a linear function of the first Betti number; in particular, if the first Betti number is large, then the immersion is highly unstable