Manifolds with $4\frac{1}{2}$-positive curvature operator of the second kind

We show that a closed four-manifold with $4\frac{1}{2}$-positive curvature operator of the second kind is diffeomorphic to a spherical space form. The curvature assumption is sharp as both ${\mathbb{C}\mathbb{P}}^2$ and ${\mathbb{S}}^3\times {\mathbb{S}}^1$ have $4\frac{1}{2}$-nonnegative curvature operator of the second kind. In higher dimensions $n\ge 5$, we show that closed Riemannian manifolds with $4\frac{1}{2}$-positive curvature operator of the second kind are homeomorphic to spherical space forms. These results are proved by showing that $4\frac{1}{2}$-positive curvature operator of the second kind implies both positive isotropic curvature and positive Ricci curvature. Rigidity results for $4\frac{1}{2}$-nonnegative curvature operator of the second kind are also obtained.