Homological norms on nonpositively curved manifolds

TL;DR

This paper explores the relationship between the Gromov norm and harmonic norm on nonpositively curved manifolds, establishing bounds based on geometric properties and comparing various related norms.

Contribution

It introduces new bounds linking homological and harmonic norms on nonpositively curved manifolds, enriching the understanding of their geometric and topological interplay.

Findings

Established bounds relating Gromov and harmonic norms

Compared homological norms with other geometric quantities

Provided insights into the norms' dependence on volume and curvature

Abstract

We relate the Gromov norm on homology classes to the harmonic norm on the dual cohomology and obtain double sided bounds in terms of the volume and other geometric quantities of the underlying manifold. Along the way, we provide comparisons to other related norms and quantities as well.