On the Duality between l^1-Homology and Bounded Cohomology

TL;DR

This paper introduces a modified definition of l^1-homology that better captures its properties and establishes a duality with bounded cohomology, proving key theorems and conjectures in the process.

Contribution

It proposes a new axiomatic framework for l^1-homology, demonstrating its pre-duality with bounded cohomology and proving significant theorems within this context.

Findings

Reconstruction of the Hausdorffification of l^1-homology.

Establishment of a duality between l^1-homology and bounded cohomology.

Proof of Gromov's theorem and the Matsumoto-Morita conjecture.

Abstract

We modify the definition of l^1-homology and argue why our definition is more adequate than the classical one. While we cannot reconstruct the classical l^1-homology from the new definition for various reasons, we can reconstruct its Hausdorffification so that no information concerning semi-norms is lost. We obtain an axiomatic characterization of our l^1-homology as a universal delta-functor and prove that it is pre-dual to our definition of bounded cohomology. We thus answer a question raised by Loeh in her thesis. Moreover, we prove Gromov's theorem and the Matsumoto-Morita conjecture in our context.