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, it is isomorphic to the polynomial ring generated by the bounded Euler class. These seem to be the first examples of groups for which the entire bounded cohomology can be described without being trivial.
We further prove that, contrary to ordinary cohomology, the diffeomorphisms groups of the circle and of the closed 2-disc have the same bounded cohomology as their homeomorphism groups and that both differ from the ordinary cohomology.
Finally, we determine the low-dimensional bounded cohomology of homeo- and diffeomorphism of the spheres Sⁿ and of certain 3-manifolds. In particular, we answer a question of Ghys by showing that the Euler class in H²(Homeo_o(S³)) is unbounded.