Moduli spaces of Riemann surfaces as Hurwitz spaces

Abstract

We consider the moduli space $\mathfrak{M}_{g,n}$ of Riemann surfaces of genus $g\ge0$ with $n\ge1$ ordered and directed marked points. For $d\ge 2g+n-1$ we show that $\mathfrak{M}_{g,n}$ is homotopy equivalent to a component of the simplicial Hurwitz space $\mathrm{Hur}^{\Delta}(\mathfrak{S}_d^{\mathrm{geo}})$ associated with the partially multiplicative quandle $\mathfrak{S}_d^{\mathrm{geo}}$. As an application, we give a new proof of the Mumford conjecture on the stable rational cohomology of moduli spaces of Riemann surfaces. We also provide a combinatorial model for the infinite loop space $\Omega^{\infty-2}\mathrm{MTSO}(2)$ of Hurwitz flavour.