On sections of configurations of points on orientable surfaces

Abstract

We study the configuration space of distinct, unordered points on compact orientable surfaces of genus $g$, denoted $S_g$. Specifically, we address the section problem, which concerns the addition of $n$ distinct points to an existing configuration of $m$ distinct points on $S_g$ in a way that ensures the new points vary continuously with respect to the initial configuration. This problem is equivalent to the splitting problem in surface braid groups. With an algebraic approach, for $g\geq 1$ and $m\geq 2$, we establish a necessary condition for the existence of a section, showing that if a section exists, then $n$ must be a multiple of $m+(2g-2)$. For $g\geq 1$ and $m=1$, we take a geometric approach to demonstrate that a section exists for all values of $n$.