Abstract
We study the configuration space of distinct, unordered points on compact orientable surfaces of genus g, denoted Sg. Specifically, we address the section problem, which concerns the addition of n distinct points to an existing configuration of m distinct points on Sg 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.
For g≥1 and m=1, we take a geometric approach to demonstrate that a section exists for all values of n. With an algebraic approach, for g≥1 and m≥2, we establish necessary conditions for the existence of a section, showing that if a section exists, then n must be a multiple of m+(2g−2).
Using the theory of Jenkins--Strebel differentials and the corresponding ribbon graph on Sg, we construct, for g≥0, sections for a wide range of values of n, while we equip Sg with a Riemann structure, which we allow to vary. In particular, we construct sections for n=3km+3k(2g−2), n=2m+2(2g−2), and n=(3k+2)m+(3k+2)(2g−2), where k∈N. Furthermore, we focus specifically on the case of the 2-sphere, presenting for the first time a section for m=5. We make interesting observations comparing our results with those of Gonçalves--Guaschi and Chen--Salter on the section problem for the 2-sphere, where in their work, either the 2-sphere is considered without a Riemannian structure or the Riemannian structure is held fixed.
