Modul: MAT772 Geometrie-Seminar

## Gromov's original approach to bounded cohomology

Dr. Roberto Frigerio talk

**Date:** 08.05.19 **Time:** 15.45 - 18.00 **Room:** ETH HG G 43

In his pioneering paper "Volume and bounded cohomology" (1982), Gromov introduced the notion of multicomplex in order to study the bounded cohomology of topological spaces. Multicomplexes are simplicial structures that generalize simplicial complexes without allowing all the degeneracies appearing in simplicial sets. After developing the general homotopy theory of multicomplexes, I will describe how multicomplexes can be exploited in the context of the bounded cohomology. In particular, I will discuss Gromov's Mapping Theorem (which implies that the bounded cohomology of a space only depends on its fundamental group) and of Gromov's Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity. If time is left, I will briefly describe further applications to the study of the simplicial volume of open manifolds.