# The stable moduli space of Riemann surfaces: Mumford’s conjecture

### Abstract

D. Mumford conjectured in [33] that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes $\kappa_i$ of dimension $2i$. For the purpose of calculating rational cohomology, one may replace the stable moduli space of Riemann surfaces by $B\Gamma_{\infty}$, where $\Gamma_\infty$ is the group of isotopy classes of automorphisms of a smooth oriented connected surface of “large” genus. Tillmann’s theorem [44] that the plus construction makes $B\Gamma_{\infty}$ into an infinite loop space led to a stable homotopy version of Mumford’s conjecture, stronger than the original [24]. We prove the stronger version, relying on Harer’s stability theorem [17], Vassiliev’s theorem concerning spaces of functions with moderate singularities [46], [45] and methods from homotopy theory.