Shigeyuki Morita

We discuss characteristic classes of surface bundles which we understand in a broad sense. In the case of usual differentiable fiber bundle with fiber a closed oriented surface of genus greater than one, characteristic classes are nothing other than the cohomology classes of the MCG (mapping class group) of the fiber. We first compare this case with that of flat surface bundles with holonomy groups in the diffeomorphism group or its subgroup consisting of area-preserving diffeomorphisms. This case of flat surface bundles is a widely open research area. We then proceed to enlarge the structure group from MCG in two different directions. One is the arithmetic MCG developed in number theory and the other is the group of homology cobordism classes of homology cylinders introduced by Garoufalidis and Levine based on the theories of Goussarov and Habiro. We consider this group in both smooth and topological categories. One feature of our earlier works may be phrased as follows. The first MMM class of surface bundles is rationally the generator of the Picard group of the moduli space of curves which has rank one by Harer. We constructed the "square root" of this class as a certain first cohomology class of MCG with twisted coefficient in the abelianization of the Torelli group determined by Johnson. It has the associated secondary invariant which is the unique MCG invariant homomorphism from the Johnson kernel to an infinite cyclic group. We interpreted this homomorphism as the manifestation of the Casson invariant in MCG. We mention our project of enhancing this picture in the above two enlargements of MCG. The starting point is the trace maps which are first cohomology classes of certain Lie algebra or group with twisted coefficients in odd symmetric powers of the homology group of surfaces. As the associated secondary invariants, we expect to obtain invariants for the absolute Galois group and homology 3-spheres. We also expect that the "Picard group" of the enlarged MCG has rank one in the smooth category while infinity in the topological category. As the first step towards the faraway goal, we mention recent joint work of Sakasai, Suzuki and myself.