Macroscopic dimension and fundamental group of manifolds with positive isotropic curvature

TL;DR

This paper proves that manifolds with positive isotropic curvature are essentially 1-dimensional at large scales and have virtually free fundamental groups, advancing understanding of their geometric and topological structure.

Contribution

It confirms Gromov's conjecture and establishes that such manifolds have virtually free fundamental groups, using techniques inspired by Donaldson's holomorphic section construction.

Findings

Manifolds with isotropic curvature ≥ 1 are macroscopically 1-dimensional.

Compact manifolds with positive isotropic curvature have virtually free fundamental groups.

The proof employs Donaldson's method of constructing destabilizing sections.

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.