Deloopings of Hurwitz spaces

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 ${\mathrm{Hur}}_{+}([0,1{]}^2,\partial [0,1{]}^2;\mathcal{Q},G{)}_{11}$; 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 ${\mathrm{Hur}}_{+}([0,1{]}^2,\partial [0,1{]}^2;\mathcal{Q},G{)}_{11}$.

arXiv comment67 pages, 11 figures, to appear in Compositio Mathematica

References in corpus (3)

• A short exposition of the Madsen-Weiss theorem
• Moduli spaces of Riemann surfaces as Hurwitz spaces
• Hurwitz-Ran spaces

Cited by in corpus (1)

• The Mumford conjecture (after Bianchi)