Deloopings of Hurwitz spaces

TL;DR

This paper studies the topological structure of Hurwitz spaces associated with partially multiplicative quandles, computing their group completions and rational cohomology under certain finiteness and rationality conditions.

Contribution

It introduces a new framework for analyzing Hurwitz spaces via topological monoids and computes their group completions and rational cohomology in specific cases.

Findings

Group completion of Hurwitz space monoid is a product involving the enveloping group and a double loop space component.

Rational cohomology ring of certain Hurwitz spaces is explicitly computed under finiteness and rationality assumptions.

Provides new insights into the topological and algebraic structure of Hurwitz spaces related to PMQs.

Abstract

For a partially multiplicative quandle (PMQ) $\mathcal{Q}$ we consider the topological monoid $\mathring{\mathrm{HM}}(\mathcal{Q})$ of Hurwitz spaces of configurations in the plane with local monodromies in $\mathcal{Q}$. We compute the group completion of $\mathring{\mathrm{HM}}(\mathcal{Q})$: it is the product of the (discrete) enveloping group $\mathcal{G}(\mathcal{Q})$ with a component of the double loop space of the relative Hurwitz space $\mathrm{Hur}_+([0,1]^2,\partial[0,1]^2;\mathcal{Q},G)_{1\!\!\,1}$; here $G$ is any group giving rise, together with $\mathcal{Q}$, to a PMQ-group pair. Assuming further that $\mathcal{Q}$ is finite and rationally Poincare and that $G$ is finite, we compute the rational cohomology ring of $\mathrm{Hur}_+([0,1]^2,\partial[0,1]^2;\mathcal{Q},G)_{1\!\!\,1}$.