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 spherical space form. The assumption sharp as both \({\mathbb{CP}\mathbb{}}^2\) and \({\mathbb {S}}^3 \times {\mathbb {S}}^1\) have \(4\frac{1}{2}\)-nonnegative kind. In higher dimensions \(n\ge 5\), we Riemannian manifolds are homeomorphic forms. These results proved by showing implies positive isotropic Ricci curvature. Rigidity for also obtained.