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Deloopings of Hurwitz spaces
Part of:
Fiber spaces and bundles
Homotopy theory
Homology and homotopy of topological groups and related structures
Homology and cohomology theories
Published online by Cambridge University Press: 13 September 2024
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 
$\mathrm {Hur}_+([0,1]^2,\partial [0,1]^2;{\mathcal {Q}},G)_{\mathbb {1}}$; here 
$G$ is any group giving rise, together with 
${\mathcal {Q}}$, to a PMQ–group pair. Under the additional assumption that 
${\mathcal {Q}}$ is finite and rationally Poincaré and that 
$G$ is finite, we compute the rational cohomology ring of 
$\mathrm {Hur}_+([0,1]^2,\partial [0,1]^2;{\mathcal {Q}},G)_{\mathbb {1}}$.
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- Research Article
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- © 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
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