Isomorphisms in `-homology

Taking the `1-completion and the topological dual of the singular chain complex gives rise to `1-homology and bounded cohomology respectively. In contrast to `1-homology, major structural properties of bounded cohomology are 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 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 of the fact that `1-homology depends only on the fundamental group and that `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. In the second part, we demonstrate how `1-homology can be used to get a better understanding of simplicial volume of non-compact manifolds.