On $q$-convex hypersurfaces in Riemannian manifolds

We prove that any closed, convex hypersurface in an $(n+1)$-dimensional Riemannian manifold with $\lceil \frac{n}{2} \rceil$-positive curvature operator is a rational homology sphere with finite fundamental group. The same conclusion holds for any $\lceil \frac{n}{2} \rceil$-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.