On sections of configurations of points on orientable surfaces

TL;DR

This paper investigates the conditions under which one can continuously add points to existing configurations on orientable surfaces, linking the problem to surface braid groups and providing algebraic and geometric results.

Contribution

It establishes necessary conditions for the existence of sections in configuration spaces on surfaces, connecting the problem to surface braid groups and offering new algebraic and geometric insights.

Findings

For genus g ≥ 1 and m ≥ 2, a necessary condition is that n is a multiple of m+(2g-2).

For genus g ≥ 1 and m=1, a section exists for all n.

The work links the section problem to the splitting problem in surface braid groups.

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 $n$.