Tuesday 30 November 2021
15:00–16:30*

Andrea Bianchi (University of Copenhagen) — Partially multiplicative quandles and generalised Hurwitz spaces

Classically, a quandle Q is a set with a binary operation, called "conjugation", that behaves similarly as conjugation in a group. A "partially multiplicative quandle" (PMQ) is a slightly more elaborate algebraic structure: besides the conjugation, we have a partial multiplication (defined only for some pairs of elements of Q); again, there are some axioms governing the behaviour of these two operations, mimicking what happens in a group.

Given a finite subset P of the plane R², there is a notion of "fundamental PMQ of R²-P", containing small loops spinning clockwise around one point of P, or around no point of P (neutral element). For a subspace X of R², one can define a Hurwitz space Hur(X;Q) containing configurations P contained in X, together with a "monodromy", i.e. a morphism of PMQ from the fundamental PMQ of R²-P to Q.

In the special case of Q being a quandle with no partial multiplication, we recover classical Hurwitz spaces. My main motivating example comes however from symmetric groups, considered as a PMQ by restricting the multiplication to certain "geodesic" couples of permutations. The corresponding PMQ, denoted S_d^geo, has associated Hurwitz spaces which are strongly related to moduli spaces of Riemann surfaces with boundary.

When X is an open square and Q is any PMQ, Hur(X;Q) is up to homotopy a topological monoid. The group completion OmegaBHur(X:Q) has the following property: there is a space B²(Q) (which I will describe during the talk) such that a component Omega₀BHur(X;Q) of the group completion of the monoid is homotopy equivalent to a component Omega²₀B²(Q) of the double loop space of B²(Q). Moreover, under certain hypotheses on Q, the rational cohomology ring of B²(Q) can be computed explicitly solely in terms of Q.

* Eastern European Time, i.e. UTC+2 in winter and UTC+3 in summer.