  We consider a generalized Gambaudo--Ghys construction on bounded cohomology
and prove its injectivity. As a corollary, we prove that the third bounded
cohomology of the group of area-preserving diffeomorphisms on the 2-disk is
infinite-dimensional. We also prove similar results for the case of the
2-sphere, the 2-torus and the annulus.

Geometric Topology;Group Theory