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.