Cohomological structure of the mapping class group and beyond

Abstract

Abstract. In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of free groups, the symplectomorphism group of surfaces.Based on these, we present several conjectures and problems concerning the cohomology of these groups. We are particularly interested in the possible interplays between these cohomology groups rather than merely the structures of individual groups. It turns out that, we have to include, in our considerations, two other groups which contain the mapping class group as their core subgroups and whose structures seem to be deeply related to that of the mapping class group. They are the arithmetic mapping class group and the group of homology cobordism classes of homology cylinders.

Lie algebras of symplectic derivations and cycles on the moduli spaces

“…We begin by determining the Sp(H Q )-irreducible decomposition of the degree 2 part a g (2) of the Lie algebra a g . Here we use the terminology of [26] to describe irreducible representations of the algebraic group Sp(2g, Q) as well as their highest weight vectors (see also Fulton and Harris [6] for the generality of the representation theory of Sp(2g, Q)). In particular, we use the symbol [b 1 b 2 · · · b k ] for expressing irreducible representations which are in one to one correspondence with Young diagrams.…”

The Johnson filtration of the McCool stabilizer subgroup of the automorphism group of a free group

“…There is a broad range of remarkable results for the Johnson homomorphisms of a mapping class group. (For example, see [7] and [15], [16], [17]. )…”

On the augmentation quotients of the IA-automorphism group of a free group

“…Using it, we can investigate the graded quotients gr k .A n / and L IA n .k/. Recently, good progress has been achieved by many authors, for example, Cohen and Pakianathan [6; 7], Farb [8], Kawazumi [12], Morita [14;15;16] and Pettet [18]. In particular, in our previous work [23], we determined the cokernel of the rational Johnson homomorphism…”