  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 $5
\leq p \leq \frac{n}{2}$ we exhibit a $C(p,n)$-positive algebraic curvature
operator of the second kind with negative Ricci curvatures.
