Compact manifolds of dimension $n\geq 12$ with positive isotropic curvature

Abstract

We prove the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or the total space of an orbifiber bundle over $\mathbb{S}^1$ or $\mathcal{I}$ with generic fiber diffeomorphic to $\mathbb{S}^{n-1}/\Gamma$ such that the total space admits a metric with positive isotropic curvature, where $\Gamma$ is a finite subgroup of $O(n)$ acting freely on $\mathbb{S}^{n-1}$, and $\mathcal{I}$ is the one dimensional closed orbifold with two singular points both with local group $\mathbb{Z}_2$ and with $|\mathcal{I}|$ a closed interval, or a connected sum of a finite number of such manifolds. This extends a recent work of Brendle, and implies a conjecture of Schoen and a conjecture of Gromov in dimensions $n\geq 12$. The proof uses Ricci flow with surgery on compact orbifolds with isolated singularities.