Bounded cohomology for groups and spaces was originally defined by Gromov in the '80s and it is intimately related to the geometric and dynamical properties of the groups. For example, Ghys used the bounded Euler class to classify certain group actions on the circle up to (semi)conjugacy. However, unlike the group cohomology, it is notoriously difficult to calculate bounded cohomology of groups. And in fact, there is no countably generated group known for which we can completely calculate the bounded cohomology unless it is trivial in all positive degrees like the case of amenable groups. In this talk, I will report on a joint work with Nicolas Monod on the bounded cohomology of certain homeomorphism groups.
In particular, we show that the bounded cohomology of $ \rm{Homeo}(\mathbb{S}^1)$ and $\rm{Homeo}(\mathbb{D}^2)$ are polynomial rings generated by the Euler class.


[IPM Youth Seminars on Topology and Dynamics]