Given a genus surface , consider the
universal -bundle over
and let be
its vertical tangent bundle with Euler class
. We
define cohomology classes
on
by
where
denotes the Gysin push-forward map (i.e. integration over
the fiber). Mumford's conjecture states that these classes freely
generate the stable cohomology ring of the mapping class group. A
natural extension of these classes to the Deligne-Mumford compactified
moduli space was given by Arbarello and Cornalba.

Jeffrey Herschel Giansiracusa
2005-06-27