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 Hörmander technique to produce (almost) holomorphic sections which we use to construct destabilizing sections.

arXiv commentSignificantly improved exposition and some arguments

References in corpus (3)

• Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, II
• Sphere Theorems in Geometry
• The fundamental group of manifolds of positive isotropic curvature and surface groups