Moduli spaces of Riemann surfaces as Hurwitz spaces

TL;DR

This paper establishes a homotopy equivalence between moduli spaces of Riemann surfaces with marked points and certain Hurwitz spaces, offering new proofs and models for their topological and cohomological properties.

Contribution

It introduces a novel connection between moduli spaces and Hurwitz spaces via simplicial structures, providing new proofs of classical conjectures and combinatorial models.

Findings

Homotopy equivalence between moduli spaces and Hurwitz spaces for certain parameters.

New proof of the Mumford conjecture on stable rational cohomology.

A combinatorial model for the infinite loop space related to 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.