Topology of closed manifolds with positive isotropic curvature

Determine whether every closed connected manifold admitting a positive isotropic curvature metric has virtually free fundamental group and, more strongly, whether a finite cover of the manifold is diffeomorphic to a sphere or to a connected sum of finitely many copies of S^{n-1} × S^1.

Background

Positive isotropic curvature imposes strong topological restrictions. Micallef–Moore proved that a closed simply connected manifold with positive isotropic curvature is homeomorphic to a sphere, while connected-sum constructions show that connected sums of copies of S^{n-1} × S^1 admit positive isotropic curvature metrics. These results motivate a broader classification conjecture for all closed connected manifolds with positive isotropic curvature.

The conjecture predicts both virtual freeness of the fundamental group and a precise finite-cover classification by spheres and connected sums of S^{n-1} × S^1. The paper notes that related finite-cover diffeomorphism results are known in dimensions five and six, but the general conjecture remains unresolved. The separate width and stable-minimal-disk conjectures stated later in the introduction are disproved by the paper's constructions and therefore are not included here as unresolved problems.