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

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 a compact quotient manifold of $\mathbb{S}^{n-1} \times \mathbb{R}$ by diffeomorphisms, 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.Comment: 32 pages, proof of Theorem 1.1 and Corollary 1.2 clarifie