Moduli spaces of Riemann surfaces as Hurwitz spaces

Abstract

We consider the moduli space ${\mathfrak{M}}_{g,n}$ of Riemann surfaces of genus $g\ge 0$ with $n\ge 1$ 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.

52 pages, 5 figures, to appear in Advances in Mathematics

References in corpus (1)

• On Generators and Relations of the Rational Cohomology of Hilbert Schemes

Cited by in corpus (2)

• Deloopings of Hurwitz spaces
• The Mumford conjecture (after Bianchi)