Four-orbifolds with positive isotropic curvature

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.