ec 2 00 6 Isomorphisms in l 1 - homology

Taking the ℓ 1-completion and the topological dual of the singular chain complex gives rise to ℓ 1-homology and bounded cohomology respectively. Unlike ℓ 1-homology, bounded cohomology is very well understood by the work of Gromov and Ivanov. Based on an observation by Matsumoto and Morita, we derive a mechanism linking isomorphisms on the level of homology of Banach chain complexes to isomorphisms on the level of co-homology of the dual Banach cochain complexes and vice versa. Therefore, certain results on bounded cohomology can be transferred to ℓ 1-homology. For example, we obtain a new proof of the fact that ℓ 1-homology only depends on the fundamental group and that ℓ 1-homology with twisted coefficients admits a description in terms of projective resolutions. The latter one in particular fills a gap in Park's approach.