Fundamental groups of manifolds with positive isotropic curvature

Abstract

A central theme in Riemannian geometry is understanding the relationships between the curvature and the topology of a Riemannian manifold. Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes manifolds with pointwise quarter-pinched sectional curvatures and manifolds with positive curvature operator. By the results of Micallef and Moore there is only one topological type of compact simply connected manifold with PIC; namely any such manifold must be homeomorphic to the sphere. On the other hand, there is a large class of nonsimply connected manifolds with PIC. An important open problem has been to understand the fundamental groups of manifolds with PIC. In this paper we prove a new result in this direction. We show that the fundamental group of a compact manifold $M^n$ with PIC, $n \geq 5$, does not contain a subgroup isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$. The techniques used involve minimal surfaces.