A closed Riemannian manifold whose curvature operator has positive average of its lowest n-p eigenvalues has zero p-th and (n-p)-th Betti numbers, with quantitative diameter-dependent estimates.
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New Curvature Conditions for the Bochner Technique
A closed Riemannian manifold whose curvature operator has positive average of its lowest n-p eigenvalues has zero p-th and (n-p)-th Betti numbers, with quantitative diameter-dependent estimates.