Fundamental groups on 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 pointwisequarter-pinched sectional curvatures and manifolds with positive curvature operator. By 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 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.