Closed convex hypersurfaces in (n+1)-manifolds with ⌈n/2⌉-positive curvature operator are rational homology spheres with finite fundamental group; similar results hold for q-convex hypersurfaces under mean curvature pinching.
Sha,p-convex Riemannian manifolds, Invent
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On $q$-convex hypersurfaces in Riemannian manifolds
Closed convex hypersurfaces in (n+1)-manifolds with ⌈n/2⌉-positive curvature operator are rational homology spheres with finite fundamental group; similar results hold for q-convex hypersurfaces under mean curvature pinching.