A Thorpe Trick for the Bochner Technique

Abstract: We prove that for $n \geq 6$ there exists $\varepsilon (n)>0$ such that every closed orientable Riemannian manifold with $$\left( \lceil \frac{n}{2} \rceil+ \varepsilon\right)$$-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.