Kappa classes

Given a genus $ g$ surface $ F$, consider the universal $ F$-bundle $ X$ over $ B\mathrm{Diff}(F)$ and let $ T^{v}X$ be its vertical tangent bundle with Euler class $ e \in H^{2}(X)$. We define cohomology classes $ \kappa_{i}$ on $ B\mathrm{Diff}(F)$ by

$\displaystyle \kappa_{i} = \int_{F} e^{i+1} \in H^{2i}(B \mathrm{Diff}(F))$

where $ \int_{F}$ 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