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Spherical braid group

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In mathematics, the spherical braid group or Hurwitz braid group is a braid group on n strands. In comparison with the usual braid group, it has an additional group relation that comes from the strands being on the sphere. The group also has relations to the inverse Galois problem.[1]

Definition

The spherical braid group on n strands, denoted S B n {\displaystyle SB_{n}} {\displaystyle SB_{n}} or B n ( S 2 ) {\displaystyle B_{n}(S^{2})} {\displaystyle B_{n}(S^{2})}, is defined as the fundamental group of the configuration space of the sphere:[2][3]
B n ( S 2 ) = π 1 ( C o n f n ( S 2 ) ) . {\displaystyle B_{n}(S^{2})=\pi _{1}(\mathrm {Conf} _{n}(S^{2})).} {\displaystyle B_{n}(S^{2})=\pi _{1}(\mathrm {Conf} _{n}(S^{2})).}
The spherical braid group has a presentation in terms of generators σ 1 , σ 2 , ⋯ , σ n − 1 {\displaystyle \sigma _{1},\sigma _{2},\cdots ,\sigma _{n-1}} {\displaystyle \sigma _{1},\sigma _{2},\cdots ,\sigma _{n-1}} with the following relations:[4]

  • σ i σ j = σ j σ i {\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}} {\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}} for | i − j | ≥ 2 {\displaystyle |i-j|\geq 2} {\displaystyle |i-j|\geq 2}
  • σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 {\displaystyle \sigma _{i}\sigma _{i+1}\sigma _{i}=\sigma _{i+1}\sigma _{i}\sigma _{i+1}} {\displaystyle \sigma _{i}\sigma _{i+1}\sigma _{i}=\sigma _{i+1}\sigma _{i}\sigma _{i+1}} for 1 ≤ i ≤ n − 2 {\displaystyle 1\leq i\leq n-2} {\displaystyle 1\leq i\leq n-2} (the Yang–Baxter equation)
  • σ 1 σ 2 ⋯ σ n − 1 σ n − 1 σ n − 2 ⋯ σ 1 = 1 {\displaystyle \sigma _{1}\sigma _{2}\cdots \sigma _{n-1}\sigma _{n-1}\sigma _{n-2}\cdots \sigma _{1}=1} {\displaystyle \sigma _{1}\sigma _{2}\cdots \sigma _{n-1}\sigma _{n-1}\sigma _{n-2}\cdots \sigma _{1}=1}

The last relation distinguishes the group from the usual braid group.

References

  1. ↑ Ihara, Yasutaka (2007), “Automorphisms of Pure Sphere Braid Groups and Galois Representations”, in Cartier, Pierre; Katz, Nicholas M.; Manin, Yuri I.; Illusie, Luc (eds.), The Grothendieck Festschrift: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck, Modern Birkhäuser Classics, Boston, MA: Birkhäuser, pp. 353–373, doi:10.1007/978-0-8176-4575-5_8, ISBN 978-0-8176-4575-5, retrieved 2023-11-24
  2. ↑ Chen, Lei; Salter, Nick (2020). “Section problems for configurations of points on the Riemann sphere”. Algebraic and Geometric Topology. 20 (6): 3047–3082. arXiv:1807.10171. doi:10.2140/agt.2020.20.3047. S2CID 119669926.
  3. ↑ Fadell, Edward; Buskirk, James Van (1962). “The braid groups of E2 and S2”. Duke Mathematical Journal. 29 (2): 243–257. doi:10.1215/S0012-7094-62-02925-3.
  4. ↑ Klassen, Eric P.; Kopeliovich, Yaacov (2004). “Hurwitz spaces and braid group representations”. Rocky Mountain Journal of Mathematics. 34 (3): 1005–1030. doi:10.1216/rmjm/1181069840.

This article is adapted from “Spherical braid group” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

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