Complete Classification of Compact Four-Manifolds with Positive Isotropic Curvature

Abstract

In this paper, we completely classify all compact 4-manifolds with positive isotropic curvature. We show that they are diffeomorphic to $\mathbb{S}^4$ or $\mathbb{RP}^4$ or quotients of $\mathbb{S}^3 \times \mathbb{R}$ by a cocompact fixed point free subgroup of the isometry group of the standard metric of $\mathbb{S}^3 \times \mathbb{R}$, or a connected sum of them.