Torsion in the Braid Monodromy of Elliptic Fibrations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911160833409024 |
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| author | Jackson, Faye |
| author_facet | Jackson, Faye |
| contents | Given an elliptic fibration $π: M \to S^2$ with singular locus $Δ\subseteq S^2$, let $\operatorname{Br}(π) < \operatorname{Mod}(S^2,Δ)$ be the subgroup of the spherical braid group consisting of those braids that lift to a fiber-preserving diffeomorphism of $M$. We classify the order $n = |Δ|$ elements of $\operatorname{Br}(π)$ up to conjugacy in $\operatorname{Br}(π)$. To do so, we relate these conjugacy classes to special points on the $\operatorname{SL}_2$-character variety for $(S^2,Δ)$ that correspond naturally to the exceptional elliptic curves $\mathbb{C}/\mathbb{Z}[ω]$ and $\mathbb{C}/\mathbb{Z}[i]$ with their associated norms on $\mathbb{Z}[ω]$ and $\mathbb{Z}[i]$. We also show that there are no elements of order $n-1$ or $n-2$ in $\operatorname{Br}(π)$, as there are in $\operatorname{Mod}(S^2,Δ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14454 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Torsion in the Braid Monodromy of Elliptic Fibrations Jackson, Faye Geometric Topology Algebraic Geometry Group Theory Given an elliptic fibration $π: M \to S^2$ with singular locus $Δ\subseteq S^2$, let $\operatorname{Br}(π) < \operatorname{Mod}(S^2,Δ)$ be the subgroup of the spherical braid group consisting of those braids that lift to a fiber-preserving diffeomorphism of $M$. We classify the order $n = |Δ|$ elements of $\operatorname{Br}(π)$ up to conjugacy in $\operatorname{Br}(π)$. To do so, we relate these conjugacy classes to special points on the $\operatorname{SL}_2$-character variety for $(S^2,Δ)$ that correspond naturally to the exceptional elliptic curves $\mathbb{C}/\mathbb{Z}[ω]$ and $\mathbb{C}/\mathbb{Z}[i]$ with their associated norms on $\mathbb{Z}[ω]$ and $\mathbb{Z}[i]$. We also show that there are no elements of order $n-1$ or $n-2$ in $\operatorname{Br}(π)$, as there are in $\operatorname{Mod}(S^2,Δ)$. |
| title | Torsion in the Braid Monodromy of Elliptic Fibrations |
| topic | Geometric Topology Algebraic Geometry Group Theory |
| url | https://arxiv.org/abs/2509.14454 |