Moduli spaces of Riemann surfaces as Hurwitz spaces

We consider the moduli space M-g,M-n of Riemann surfaces of genus g > 0 with n >= 1 ordered and directed marked points. For d >= 2g +n- 1 we show that M-g,M-n is homotopy equivalent to a component of the simplicial Hurwitz space Hur(Delta)(S-geo d ) associated with the partially multiplicative quandle S-d(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 infinity-2MTSO(2) of Hurwitz flavour.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates the moduli space of Riemann surfaces with ordered and directed marked points. It shows a homotopy equivalence between the moduli space and a component of the simplicial Hurwitz space associated with a partially multiplicative quandle. Furthermore, it provides a new proof of the Mumford conjecture on the stable rational cohomology of moduli spaces of Riemann surfaces and presents a combinatorial model for the infinite loop space of Hurwitz flavor.