# Kappa classes

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