Bounded cohomology of transformation groups.
Let MM be a closed Riemannian manifold and let μμ be the measure induced by the volume form. Denote by Homeo0(M,μ)Homeo0(M,μ) the group of all μμ -preserving homeomorphisms of MM isotopic to the identity. It is well-known that the second bounded cohomology of Diff0(M,μ)Diff0(M,μ) is infinite-dimensional due to existence of quasimorphisms on Diff0(M,μ)Diff0(M,μ) (Gambaudo-Ghys, Polterovich). In this talk, I will explain how to construct bounded classes in higher dimensions. As an application, we will show that under certain conditions on the fundamental group of MM , the third bounded cohomology of Diff0(M,μ)Diff0(M,μ) is infinite-dimensional. If time permits, I will discuss how this construction can be used to construct invariants of foliated fibre bundles. It is a joint work with Michael Brandenbursky and Martin Nitsche.