Isomorphisms in `1-homology

Taking the `1-completion and the topological dual of the singular chain complex gives rise to `1-homology and bounded cohomology respectively. Unlike `1-homology, bounded cohomology is quite well understood by the work of Gromov and Ivanov. We derive a mechanism linking isomorphisms on the level of homology of Banach chain complexes to isomorphisms on the level of cohomology of the dual Banach cochain complexes and vice versa. Therefore, certain results on bounded cohomology can be transferred to `1-homology. For example, we obtain a new proof that `1-homology depends only on the fundamental group and that `1-homology admits a description in terms of projective resolutions. Mathematics Subject Classification (2000) – 55N35, 46M10, 57N99