Topological complexity of visibility manifolds

TL;DR

This paper extends results relating volume and homology torsion to a broader class of manifolds with variable negative curvature that satisfy the visibility condition, providing new bounds and models.

Contribution

It generalizes existing theorems on homology control from pinched negative curvature to manifolds with sectional curvature near zero but satisfying the visibility axiom.

Findings

Constructed an efficient simplicial model for the thick part of these manifolds.

Extended homology volume control results to a wider class of curvature conditions.

Provided new proofs for classical theorems using the generalized models.

Abstract

A recent result of Bader, Gelander and Sauer shows that for manifolds of pinched negative curvature, the torsion part of the homology can be controlled by the volume. This is done by constructing an efficient simplicial model of the thick part, which also provides another proof of the analogous statement for the free part of the homology, a classical theorem due to Gromov. We will extend these results to more general curvature conditions, namely the case where the sectional curvature can get arbitrarily close to zero, but the visibility axiom still holds.