  We prove a vanishing and estimation theorem for the $p^{\text{th}}$-Betti
number of closed $n$-dimensional Riemannian manifolds with a lower bound on the
average of the lowest $n-p$ eigenvalues of the curvature operator. This
generalizes results due to D. Meyer, Gallot-Meyer, and Gallot. For example, in
dimensions $n=5,6$ we obtain vanishing of the Betti numbers provided that the
curvature operator is $3$-positive. As B\"ohm-Wilking observed, $3$-positivity
of the curvature operator is not preserved by the Ricci flow.
