Manifolds with positive isotropic curvature of dimension at least nine

TL;DR

This paper extends Brendle's classification of compact manifolds with positive isotropic curvature to dimensions nine and above, showing they are diffeomorphic to connected sums of standard spaces.

Contribution

It improves the known dimension bound from twelve to nine for the classification of manifolds with positive isotropic curvature.

Findings

Manifolds with positive isotropic curvature in dimension n≥9 are diffeomorphic to connected sums of standard spaces.

The classification previously known for n≥12 now applies for n≥9.

The result applies to manifolds containing no nontrivial incompressible (n-1)-dimensional space form.

Abstract

In [Bre19], Simon Brendle showed that any compact manifold of dimension $n\geq12$ with positive isotropic curvature and contains no nontrivial incompressible $(n-1)-$dimensional space form is diffeomorphic to a connected sum of finitely many spaces, each of which is a quotient of $S^n$ or $S^{n-1}\times \mathbb{R}$ by standard isometries. We show that this result is actually true for $n\geq9$.