Abstract: We show that compact, ‐dimensional Riemannian manifolds with ‐nonnegative curvature operators of the second kind are either rational homology spheres or flat. More generally, we obtain vanishing of the th Betti number provided that the curvature operator of the second kind is ‐positive. Our curvature conditions become weaker as increases. For , we have , and for , we exhibit a ‐positive algebraic curvature operator of the second kind with negative Ricci curvatures.