On Cohen braids

Abstract

For a general connected surface M and an arbitrary braid α from the surface braid group B n−1 (M ), we study the system of equations d 1 β = . . . = d n β = α, where the operation d i is the removal of the ith strand. We prove that for M = S 2 and M = RP 2 , this system of equations has a solution β ∈ B n (M ) if and only if d 1 α = . . . = d n−1 α. We call the set of braids satisfying the last system of equations Cohen braids. We study Cohen braids and prove that they form a subgroup. We also construct a set of generators for the group of Cohen braids. In the cases of the sphere and the projective plane we give some examples for a small number of strands.

Simplicial structures and normal forms for mapping class groups and braid groups

“…From [2] we have degeneracy maps s j W B k .M / ! B kC1 .M / given by doubling the j th string, and we know that these maps satisfy the simplicial identities (1) Before checking that the degeneracy maps for mapping class groups behave as desired, we prove the following lemma.…”

“…Developing a theme that appears to originate in Cohen [5], Wu [12,Example 1.2.8] relates results in homotopy theory concerning Hopf invariants to the theory of braids. In particular, one can use normal forms on braids to determine the Cohen braids; see Bardakov, Mikhailov, Vershinin and Wu [1].…”

On groups Gnk, braids and Brunnian braids

“…where the parity is obtained from 5-th strand. For a pair (1,2), the value of MNinvariant for G 2 n of β is trivial, because Y is in a good condition and Y contains no a 12 . But w 5 12 (β) is not trivial.…”

“…But in principle, it is possible to go on enhancing the invariants coming from G k n (even from G n itself recognizes the non-triviality of such braids, and the corresponding invariants can be derived as maps from G 3 n to free products of Z 2 (Example 4.13). It would be interesting to compare it with lower central series [1], [2], [3].…”

“…Brunnian braids are those n-strand braids which become trivial after removal of each strand. Such braids are closely related to various problems in geometry and topology, cryptography, etc, in particular, to the homotopy groups of spheres [1], [3], [2]. The easy-to-calculate invariants from [10] G 3 n valued in the free product of Z 2 fail to recognize Brunnian braids(Lemma 4.5) when the number of strands is large.…”