nLab spherical braid group

Definition

By the spherical braid group $Br_n(S^2)$, for $n \in \mathbb{N}$, one means the sphere braid group, hence the surface braid group, where the surface in question is the 2-sphere $S^2$. More concretely, the spherical braid group

$$Br_n(S^2) \;\simeq\; \pi_1 Conf_n(S^2) \,,$$

is the fundamental group $\pi_1(-)$ of the configuration space of $n$-points, $Conf_n(-)$, on the 2-sphere.

Illustrated on the right is the element $b_2 b_1 \in Br_3(S^2)$ (where $b_i$ denote the Artin braid generators, cf. Prop. .)

Properties

(from math.SE:q/136821)

Proposition

The spherical braid group is the quotient group of the ordinary braid group by one further relation:

$$Br_n(S^2) \;\simeq\; Br_n/ \big( (b_1 b_2 \cdots b_{n-1})(b_{n-1} \cdots b_2 b_1) \,,$$

where the $b_i$ denote the Artin braid generators.

Moreover, the canonical map from the plain braid group to the symmetric group $Sym_n$ factors through the corresponding quotient coprojection:

(Fadell & Van Buskirk 1961 p 245, 255, cf. Tan 2024 §3.1)

References

• Edward Fadell, James Van Buskirk: On the braid groups of $E^2$ and $S^2$, Bull. Amer. Math. Soc. 67 2 (1961) 211-213 [euclid:bams/1183524083]

• Cindy Tan: Smallest nonabelian quotients of surface braid groups, Algebr. Geom. Topol. 24 (2024) 3997-4006 [arXiv:2301.01872, doi:10.2140/agt.2024.24.3997]