  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 $\mathbb{CP}^2$ and $\mathbb{S}^3 \times
\mathbb{S}^1$ have $4\frac{1}{2}$-nonnegative curvature operator of the second
kind. In higher dimensions $n\geq 5$, 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.
