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

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

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.