Spherical braid group

In mathematics, the spherical braid group or Hurwitz braid group is a braid group on {{mvar|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.{{Citation |last=Ihara |first=Yasutaka |title=Automorphisms of Pure Sphere Braid Groups and Galois Representations |date=2007 |url=https://doi.org/10.1007/978-0-8176-4575-5_8 |work=The Grothendieck Festschrift: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck |pages=353–373 |editor-last=Cartier |editor-first=Pierre |access-date=2023-11-24 |series=Modern Birkhäuser Classics |place=Boston, MA |publisher=Birkhäuser |language=en |doi=10.1007/978-0-8176-4575-5_8 |isbn=978-0-8176-4575-5 |editor2-last=Katz |editor2-first=Nicholas M. |editor3-last=Manin |editor3-first=Yuri I. |editor4-last=Illusie |editor4-first=Luc}}

Definition

The spherical braid group on {{mvar|n}} strands, denoted $SB_n$ or $B_n(S^2)$ , is defined as the fundamental group of the configuration space of the sphere:{{Cite journal |last1=Chen |first1=Lei |last2=Salter |first2=Nick |date=2020 |title=Section problems for configurations of points on the Riemann sphere |url=https://www.mendeley.com/catalogue/75cd35b0-fb2a-3986-a5b6-e486dcc723ee/ |journal=Algebraic and Geometric Topology |language=en-GB |volume=20 |issue=6 |pages=3047–3082 |doi=10.2140/agt.2020.20.3047|s2cid=119669926 |arxiv=1807.10171 }}{{Cite journal |last1=Fadell |first1=Edward |last2=Buskirk |first2=James Van |date=1962 |title=The braid groups of E2 and S2 |url=https://www.mendeley.com/catalogue/5fe4ec1d-e4e9-3dc4-98af-cfbe28d6ce24/ |journal=Duke Mathematical Journal |language=en-GB |volume=29 |issue=2 |pages=243–257 |doi=10.1215/S0012-7094-62-02925-3}}

$B_n(S^2) = \pi_1(\mathrm{Conf}_n(S^2)).$

The spherical braid group has a presentation in terms of generators $\sigma_1, \sigma_2, \cdots, \sigma_{n - 1}$ with the following relations:{{Cite journal |last1=Klassen |first1=Eric P. |last2=Kopeliovich |first2=Yaacov |date=2004 |title=Hurwitz spaces and braid group representations |url=https://www.mendeley.com/catalogue/040a97b9-ac59-3411-abf4-e02c0191e13f/ |journal=Rocky Mountain Journal of Mathematics |language=en-GB |volume=34 |issue=3 |pages=1005–1030 |doi=10.1216/rmjm/1181069840|doi-access=free }}

$\sigma_i \sigma_j = \sigma_j \sigma_i$ for $|i-j| \geq 2$
$\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ for $1 \leq i \leq n - 2$ (the Yang–Baxter equation)
$\sigma_1 \sigma_2 \cdots \sigma_{n-1} \sigma_{n-1} \sigma_{n-2} \cdots \sigma_{1} = 1$

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

References

Category:Braid groups