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Generalized Braid group on the Sphere
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}}
or
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})).}
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}}
with the following relations:
[4]
σ
i
σ
j
=
σ
j
σ
i
{\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}}
for
|
i
−
j
|
≥
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}}
for
1
≤
i
≤
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}
The last relation distinguishes the group from the usual braid group.
References
^ Ihara, Yasutaka (2007), Cartier, Pierre; Katz, Nicholas M.; Manin, Yuri I.; Illusie, Luc (eds.),
"Automorphisms of Pure Sphere Braid Groups and Galois Representations" , 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
^ 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 .
^ 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 .
^ 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 .