16:00
A cohomology class on the diffeomorphism group Diff(M) of a manifold M
can be thought of as a characteristic class for smooth M-bundles.
I will survey a technique for producing examples of such classes,
and then explain how the signature (of 4-manifolds) provides an
obstruction to this technique in dimension 3.
I will define Miller-Morita-Mumford classes and explain how we can
think of them as coming from classes on the cobordism category.
Madsen and Weiss showed that for a surface S of genus g all cohomology
classes
of the mapping class group MCG(S) (of degree < 2(g-2)/3) are MMM-classes.
This technique has been successfully ported to higher even dimensions d= 2n,
but it cannot possibly work in odd dimensions:
a theorem of Ebert says that for d=3 all MMM-classes are trivial.
In the second part of my talk I will sketch a new proof of (a part of)
Ebert's theorem.
I first recall the definition of the signature sign(W) of a 4 manifold W,
and some of its properties, such as additivity with respect to gluing.
Using the signature and an idea from the world of 1-2-3-TQFTs,
I then go on to define a 'central extension' of the three dimensional
cobordism category.
This central extension corresponds to a 2-cocycle on the 3d cobordism
category,
and we will see that the construction implies that the associated MMM-class
has to vanish on all 3-dimensional manifold bundles.