Classification of compact manifolds with positive isotropic curvature
Hong Huang
TL;DR
The paper proves that compact connected $n$-manifolds with positive isotropic curvature, for $n\ge 12$, are diffeomorphic to spherical space forms, quotients of $\mathbb{S}^{n-1}\times\mathbb{R}$ by cocompact isometries, or connected sums thereof. It extends the work of Brendle and CTZ and sharpens Huang by employing Ricci flow with surgery on orbifolds, along with ambient isotopy uniqueness of tubular neighborhoods to control topology through surgeries. The authors develop a robust framework of $\varepsilon$-caps, orbifold canonical neighborhoods, and a refined pinch which yields a surgical solution (r,δ) that becomes extinct, enabling a topological decomposition into spherical and $\mathbb{S}^{n-1}\times\mathbb{R}$-type pieces. Consequences include vanishing of low-degree homotopy/cohomology groups, finite fundamental group implications, and a complete orbifold version of the classification in high dimensions, substantially advancing understanding of manifolds with PIC in dimensions $n\ge 12$. The methods also indicate validity for $9\le n\le 11$ under updated curvature pinching results.
Abstract
We show 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 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 compact embedded full suborbifolds.