Gromov's theory of multicomplexes and relative bounded cohomology

Simplicial volume, introduced by Gromov in the 80's, is a fundamental invariant of manifolds whose study is closely related to the dual theory of bounded cohomology. The first aim of this thesis is to investigate the relative bounded cohomology of pairs of topological spaces via Gromov’s original approach based on multicomplexes. Multicomplexes are simplicial structures generalizing simplicial complexes while avoiding the full range of degeneracies of simplicial sets, and they provide a flexible combinatorial framework for bounded cohomology. Within this setting, we establish both isometric and biLipschitz isomorphisms in relative bounded cohomology. In the second part of the thesis, we apply this framework to the simplicial volume of manifolds with boundary. We prove several additivity results under gluings, with a particular focus on boundary connected sums, and we extend to the relative case Gromov’s vanishing theorem, which guarantees the vanishing of simplicial volume in the presence of amenable open covers of small multiplicity.