Betti numbers and the curvature operator of the second kind

We show that compact, $n$-dimensional Riemannian manifolds with $\frac{n+2}{2}$-nonnegative curvature operators of the second kind are either rational homology spheres or flat. More generally, we obtain vanishing of the $p$-th Betti number provided that the curvature operator of the second kind is $C(p,n)$-positive. Our curvature conditions become weaker as $p$ increases. For $p=\frac{n}{2}$ we have $C(p,n)=\frac{3n}{2}\frac{n+2}{n+4}$, and for $5\le p\le \frac{n}{2}$ we exhibit a $C(p,n)$-positive algebraic curvature operator of the second kind with negative Ricci curvatures.