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In this paper, we survey recent works on the structure of the mapping class groups of surfaces mainly from the point of view of topology. We then discuss several possible directions for future research. These include the relation between the structure of the mapping class group and invariants of 3-manifolds, the unstable cohomology of the moduli space of curves and Faber's conjecture, cokernel of the Johnson homomorphisms and the Galois as well as other new obstructions, cohomology of certain infinite dimensional Lie algebra and characteristic classes of outer automorphism groups of free groups and the secondary characteristic classes of surface bundles. We give some experimental results concerning each of them and, partly based on them, we formulate several conjectures and problems.

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Abstract.We consider the condition when bounded cohomology injects into ordinary cohomology and prove the vanishing of bounded cohomology of the group of all compactly supported homeomorphisms of R".Introduction. In this note we consider relations among bounded cohomology, ordinary real cohomology and Z1 homology of spaces or groups. In particular we present a necessary and sufficient condition under which bounded cohomology injects into ordinary cohomology and by using it prove the vanishing of bounded cohomology and lx homology of Horneo^R", the group of all homeomorphisms of R" with compact support. We also determine the second bounded cohomology of SL2R.

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