On the bounded cohomology of certain homeomorphism groups

We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional manifolds. In particular, for the group of orientation-preserving homeomorphisms of the circle and of the closed 2-disc, we show that the bounded cohomology is isomorphic to the polynomial ring generated by the bounded Euler class. We further prove that the group of orientation-preserving homeomorphisms of the line is boundedly acyclic. We also calculate the low-dimensional bounded cohomology of $\Homeo(S^2)$, of $\Homeo(S^3)$ and of homeomorphism groups of certain 3-manifolds.