Title: | Symmetric groups, Hurwitz spaces and moduli spaces of surfaces |

Speaker: | Andrea Bianchi |

Time: | Thursday, 13.01.2023, 14:00 |

Place: | SR 3.061 |

**Abstract:**

Let d>=2, and consider the symmetric group S_d. For k>=0, the classical Hurwitz space hur_k(S_d) parametrises d-fold branched covers of the complex plane C with precisely k branch points. We introduce an amalgamation of all Hurwitz spaces, for varying k, into a single space Hur(S_d). The construction relies on the notion of "partially multiplicative quandle", an algebraic structure slightly weaker than the structure of a group, and we will see how to consider S_d as a PMQ in a convenient way.

The main motivation to consider the amalgamated Hurwitz space Hur(S_d) is the following. For all g>=0 and n>=1, let M_{g,n} denote the moduli space of Riemann surfaces of genus g with n ordered and parametrised boundary components. Our main result ensures that if d is large enough (with respect to g and n), then there exists a connected component of Hur(S_d) which is homotopy equivalent to M_{g,n}.

Moreover, the space Hur(S_d) carries a natural structure of topological monoid, and we will briefly sketch the computation of the stable, rational cohomology of its connected components. The result is very explicit in degrees up to roughly d. Letting d go to infinity, one can in particular recover the Mumford conjecture on the stable, rational cohomology of moduli spaces of Riemann surfaces, originally proved by Madsen and Weiss.