Four-orbifolds with positive isotropic curvature

TL;DR

This paper classifies 4-manifolds with positive isotropic curvature as connected sums of standard spaces and certain orbifold quotients, extending previous results using Ricci flow with surgery.

Contribution

It extends classification results of 4-orbifolds with positive isotropic curvature to include orbifolds and uses Ricci flow with surgery on orbifolds.

Findings

Classifies 4-manifolds with positive isotropic curvature as connected sums of standard spaces and orbifold quotients.

Extends classification to orbifolds with positive isotropic curvature.

Uses Ricci flow with surgery on orbifolds for the proof.

Abstract

We prove the following result: Let $(X,g_0)$ be a complete, connected 4-manifold with uniformly positive isotropic curvature and with bounded geometry. Then there is a finite collection $\mathcal{F}$ of manifolds of the form $\mathbb{S}^3 \times \mathbb{R} /G$, where $G$ is a discrete subgroup of the isometry group of the round cylinder $\mathbb{S}^3\times \mathbb{R}$ on which $G$ acts freely, such that $X$ is diffeomorphic to a possibly infinite connected sum of $\mathbb{S}^4,\mathbb{RP}^4$ and members of $\mathcal{F}$. This extends recent work of Chen-Tang-Zhu and Huang. We also extend the above result to the case of orbifolds. The proof uses Ricci flow with surgery on complete orbifolds.