Vector braids Original Research Article

In this paper we define a new family of groups which generalize the classical braid groups on image. We denote this family by {Bmn}n ≥ m + 1 where image. The family image is the set of classical braid groups on n strings. The group Bmn is related to the set of motions of n unordered points in imagem, so that at any time during the motion, each m + 1 of the points span the whole of imagem in the sense of affine geometry. There is a map from Bmn to the symmetric group on n letters. We let Pmn denote the kernel of this map. In this paper we are mainly interested in finding a presentation of and understanding the group P2n. We give a presentation of a group PLn which maps surjectively onto P2n. We also show the surjection PLn → P2n induces an isomorphism on first and second integral homology and conjecture that it is an isomorphism. We then find an infinitesimal presentation of the group P2n. Finally, we also consider the analagous groups where points lie in imagem instead of imagem. These groups generalize of the classical braid groups on the sphere.