Deloopings of Hurwitz spaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866909394992627712 |
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| author | Bianchi, Andrea |
| author_facet | Bianchi, Andrea |
| contents | 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 $\mathrm{Hur}_+([0,1]^2,\partial[0,1]^2;\mathcal{Q},G)_{1\!\!\,1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_13081 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Deloopings of Hurwitz spaces Bianchi, Andrea Algebraic Topology 55N45, 55P35, 55P62, 55R80, 57T25 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 $\mathrm{Hur}_+([0,1]^2,\partial[0,1]^2;\mathcal{Q},G)_{1\!\!\,1}$. |
| title | Deloopings of Hurwitz spaces |
| topic | Algebraic Topology 55N45, 55P35, 55P62, 55R80, 57T25 |
| url | https://arxiv.org/abs/2107.13081 |