Postdoc day, Seminar 1: Symmetric groups, Hurwitz spaces and moduli spaces of surfaces

Date: 2022-02-11

Time: 13:30 - 14:00


Andrea Bianchi


Let d=2g>=2 be even. There is a graded commutative Q-algebra, denoted A(d), arising as the conjugation invariants of the associated graded of a multiplicative filtration on the group algebra Q[S_d] of the symmetric group S_d. In joint work with Alexander Christgau and Jonathan Pedersen, I computed a minimal system of generators for A(d) and a formula for the lowest degree relation among minimal generators.
There is a simply connected topological space B(d) enjoying the following two properties:
-the rational cohomology ring of B(d) is isomorphic to A(d); -a component of the double loop space of B(d) is homotopy equivalent to a component of the group completion of a certain topological monoid Hur(S_d^geo) of Hurwitz spaces. I will briefly describe the topological monoid Hur(S_d^geo) and its relation to the moduli space M_{g,1} of Riemann surfaces of genus g with one boundary curve. Time permitting, I will describe how the lowest degree relation gives rise to a cohomology class in H^{2g-1}(M_{g,1};Q), which I conjecture to be non-trivial.