Macroscopic dimension and fundamental group of manifolds with positive isotropic curvature

Abstract: We prove a conjecture of Gromov's to the effect that manifolds with isotropic curvature bounded below by 1 (after possibly rescaling) are macroscopically 1-dimensional on the scales greater than 1. As a consequence we prove that compact manifolds with positive isotropic curvature have virtually free fundamental groups. Our main technique is modeled on Donaldson's version of H\"ormander technique to produce (almost) holomorphic sections which we use to construct destabilizing sections.