Structure of the mapping class groups of surfaces: a survey and a prospect

Abstract

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.

$\mathcal{K}_{g}$ is not finitely generated

“…[BC,Jo1]) and Morita [Mo4] have found large abelian quotients of K g . We would also like to remark that Morita has discovered (see, e.g., [Mo4,Mo2,Mo3]) a strong connection between the algebraic structure of K g and the Casson invariant for homology 3-spheres. For example, Morita proved in [Mo4] that every integral homology 3-sphere can be obtained by gluing two handlebodies along their boundaries via a map in K g ; further, he has been able to express the Casson invariant as a homomorphism K g −→ Z (see, e.g., [Mo1]).…”

The second Johnson homomorphism and the second rational cohomology of the Johnson kernel

“…Johnson [12] showed that K g coincides with the kernel of what is now called the first Johnson homomorphism τ g (1) of I g (see [10]), namely we have an exact sequence The group K g plays an important role in topology. For example, it has some relationships to the Casson invariant of homology 3-spheres and secondary characteristic classes of surface bundles as we see in Morita's papers [18,21]. However, we still do not have enough information on K g .…”

“…Moreover, in [21], Morita announced that Remark 2.1. Hain showed in [9] that as Lie algebras, Im τ Q g,1 , Im τ Q g, * , Im τ Q g are generated by their degree 1 parts, and that…”

Cohomological structure of the mapping class group and beyond

“…First we recall the following problem, because of its importance, which was already mentioned in [80] (Conjecture 3.4). Problem 2.…”

“…In [80], we defined a series of secondary characteristic classes for the mapping class group. However there was ambiguity coming from possible odd dimensional stable cohomology classes of the mapping class group.…”