The braid groups of E2 and S2

Braids and riemann surfaces

“…It can be defined (cf. [8]) by the generators a, , the relations (9), and the additional relation f2 = 1 (where Q is defined in (15)). It can be shown that R = 1 implies the relation 8 2 = 1, and all the relations 8, = 8.…”

Braid groups of compact 2-manifolds with elements of finite order

“…The groups ByiP2), B2(P2), B2(S2), B3(S2) are thus finite groups; while <jya2---a"-y has order 2n in Bn(P2) for n -3 and in Bn(S2) for n = 4 as shown in §111 and in [9] respectively. Combining these results with the theorem of Fadell and Neuwirth on the existence of elements of finite order, one obtains the following Theorem.…”

“…Theorem [9]. B2iS2) is cyclic of order 2, B3iS2) is a ZS-metacyclic group of order 12, while B"(S2) is infinite for n = 4.…”

“…In solving the Word Problem for B"(S2), Fadell [9] considered the isomorphic fundamental exact sequences 1->A"(S2)->Kn(S2) -> K".y(S2)-H of Bn(S2) and Gn(S2), and used the fact that the locally trivial fiber space (F0>n(S2), n, F0,"-y(S2)) admits cross sections for n -4. Recall that the fiber map X: sfn+y(S2) -* ¿¿"(S2) with fiber F*"A(S2) consisting of n distinct pairs of antipodal points of S2, and consider the following commutative diagram 1 -ny(F*2nA(S2))^ny(sin+y(S2)) -Jny(rfn(S2)) -1 <p* <t>n+l.…”

Topological aspects of spin and statistics in nonlinear sigma models

“…Later Fadell and Van Buskirk computed B n (S 2 ) [19], Van Buskirk computed B n (RP 2 ) [20], Birman computed B n (T 2 ) [18], and finally Scott computed B n (X) for X any compact 2-manifold [21]. For example, B n (S 2 ) is the quotient of B n by the additional relation…”