  By extending and generalising previous work by Ros and Savo, we describe a
method to show that the Morse index of every closed minimal hypersurface on
certain positively curved ambient manifolds is bounded from below by a linear
function of its first Betti number. The technique is flexible enough to prove
that such a relation between the index and the topology of minimal
hypersurfaces holds, for example, on all compact rank one symmetric spaces, on
products of the circle with spheres of arbitrary dimension and on suitably
pinched submanifolds of the Euclidean spaces. These results confirm a general
conjecture due to Schoen and Marques-Neves for a wide class of ambient spaces.
