Spherical braid group explained

In mathematics, the spherical braid group or Hurwitz braid group is a braid group on 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.

Definition

The spherical braid group on strands, denoted

SB _{n} or B ^{2 )} _{n(S} , is defined as the fundamental group of the configuration space of the sphere:^{[1] [2]} $$B\_n(S^2)\; =\; \backslash pi\_1(\backslash mathrm\_n(S^2)).$$The spherical braid group has a presentation in terms of generators \sigma _{1,} \sigma _{2,} … , \sigma _{n} with the following relations:^[3] \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} … \sigma _{n-1} \sigma _{n-1} \sigma _{n-2} … \sigma _{1} = 1 The last relation distinguishes the group from the usual braid group.

Notes and References

1. Chen . Lei . Salter . Nick . 2020 . Section problems for configurations of points on the Riemann sphere . Algebraic and Geometric Topology . en-GB . 20 . 6 . 3047–3082 . 10.2140/agt.2020.20.3047. 119669926 . 1807.10171 .
2. Fadell . Edward . Buskirk . James Van . 1962 . The braid groups of E2 and S2 . Duke Mathematical Journal . en-GB . 29 . 2 . 243–257 . 10.1215/S0012-7094-62-02925-3.
3. Klassen . Eric P. . Kopeliovich . Yaacov . 2004 . Hurwitz spaces and braid group representations . Rocky Mountain Journal of Mathematics . en-GB . 34 . 3 . 1005–1030 . 10.1216/rmjm/1181069840. free .