  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.

Algebraic Topology