Homotopy of Braids on Surfaces: extending Goldsmith's Answer to Artin

Abstract

In 1947, in the paper "Theory of Braids", Artin raised the question of whether isotopy and homotopy of braids on the disk coincide. Twenty seven years later, Goldsmith answered his question: she proved that in fact the group structures are different, exhibiting a group presentation and showing that the homotopy braid group on the disk is a proper quotient of the Artin braid group on the disk $B_{n}$, denoted by $\widehat{B}_{n}$. In this paper, we extend Goldsmith's answer to Artin for closed, connected and orientable surfaces different from the sphere. More specifically, we define the notion of homotopy generalized string links on surfaces, which form a group which is a proper quotient of the braid group on a surface $B_{n}(M)$, denoting it by $\widehat{{B}}_{n}(M)$. We then give a presentation of the group $\widehat{B}_{n}(M)$ and find that the Goldsmith presentation is a particular case of our main result, when we consider the surface $M$ to be the disk. We close with a brief discussion surrounding the importance of having such a fixed construction available in the literature.