Relative measure homology and continuous bounded cohomology of topological pairs (Q442045)

In order to exploit measure theory as a tool for computing the simplicial volume, one has to show that several homology theories are not only isomorphic, but also isometric (with respect to some appropriate seminorms). This paper is devoted to extending a result by \textit{C. Löh} [Math. Z. 253, No. 1, 197--218 (2006; Zbl 1093.55004)] to the context of relative homology of topological pairs. First, the authors develop some aspects of the theory of continuous bounded cohomology of topological pairs. They compare this theory with the usual bounded cohomology of pairs of groups and spaces. They show that the seminorm considered in the present paper does not coincide with the seminorm used by \textit{H. Park} [Topology Appl. 131, No. 3, 203--234 (2003; Zbl 1042.55003)] in order to provide the algebraic foundations to the theory of relative bounded cohomology. Finally, they show that Park and Gromov seminorms do not coincide in general.