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.
This paper is dedicated to Robion C
Kirby on the occasion of his 60th birthday.
Keywords
mapping class group, Torelli group,
Johnson homomorphism, moduli space of curves
