Classification of compact manifolds with positive isotropic curvature

Abstract

We show the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\ge 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or a quotient manifold of ${\mathbb{S}}^{n-1}\times \mathbb{R}$ by a cocompact discrete subgroup of the isometry group of the round cylinder ${\mathbb{S}}^{n-1}\times \mathbb{R}$, or a connected sum of a finite number of such manifolds. This extends previous works of Brendle and Chen-Tang-Zhu, and improves a work of Huang. The proof uses Ricci flow with surgery on compact orbifolds, with the help of the ambient isotopy uniqueness of closed tubular neighborhoods of an isolated singular point in an orbifold.

arXiv comment28 pages, added more details on the proof of Proposition 2.2