Partially multiplicative quandles and generalised Hurwitz spaces
Moduli and Friends seminar
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Tuesday 30 November 2021

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 Sdgeo, 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.

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