Perspective of characteristic classes of surface bundles
in various geometric contexts
We discuss characteristic classes of surface bundles which we understand
in a broad sense. In the case of usual differentiable fiber bundle with
a closed oriented surface of genus greater than one, characteristic
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
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
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
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
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
"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.