A Thorpe Trick for the Bochner Technique

We prove that for $n\ge 6$ there exists $\epsilon (n)>0$ such that every closed orientable Riemannian manifold with $(\lceil \frac{n}{2}\rceil +\epsilon )$-positive curvature operator is a real homology sphere. For $n=6$ we show that the same result holds for manifolds with $4$-positive curvature operators.