Brunnian braids on surfaces

Abstract

We determine a set of generators for the Brunnian braids on a general surface M for M 6 D S 2 or RP 2 . For the case M D S 2 or RP 2 , a set of generators for the Brunnian braids on M is given by our generating set together with the homotopy groups of a 2-sphere.57M07, 57M99; 20F36, 55Q40

Delta-structures on mapping class groups and braid groups

“…Since M has boundary, Conf k (M ) is a K(π, 1) by Lemma 3.4 in [1], and it follows from Theorem 4.4 that Diff 0 (M, m k ) is contractible.…”

Brunnian subgroups of mapping class groups and braid groups

“…Here and in the rest of the paper we use T to denote the (2-dimensional) torus and K the Klein bottle. 1 The Brunnian braid group is usually defined to be the subgroup of the full braid group B k+1 (M ) consisting of all braids that become trivial when any one of the strands is removed. By Proposition 2.6 in [3], this coincides with Brun(P k+1 (M )) for all k ≥ 2.…”

“…The non-exceptional cases are covered by Lemma 3.4 in [1], as are the cases π 2 (Conf k (S 2 )) for k ≥ 3 and π 2 (Conf k (RP 2 )) for k ≥ 2. The remaining cases follow since Conf 1 (M ) ∼ = M for any M and S 2 ≃ Conf 2 (S 2 ) via the map z → (z, −z).…”

Book Review: Braid groups (With the graphical assistance of Olivier Dodane); Ordering braids

“…Since generators for image(ι * ) are now known [4], the remaining problem to the soughtfor uniform approach to understanding π n−1 (S 2 ), n ≥ 5, is to obtain generators for Br n (S 2 ).…”