On q-convex hypersurfaces in Riemannian manifolds

We prove that any closed, convex hypersurface in an (n+1)-dimensional Riemannian manifold with (n/2)-positive curvature operator is a rational homology sphere with finite fundamental group. The same conclusion holds for any (n/2)-convex hypersurface, provided that the mean curvature satisfies a sharp pinching condition. Both results follow from more general vanishing and estimation theorems for the Betti numbers of closed q-convex immersed hypersurfaces in (n+1)-dimensional Riemannian manifolds, under a lower bound on the average of the smallest (n-p) eigenvalues of the curvature operator.