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

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.