Surface topology and incidence theorems over division rings

Abstract

Incidence theorems concern configurations of points, lines, and, more generally, higher-dimensional subspaces in projective space. Broadly speaking, such theorems fall into two classes: those that hold over an arbitrary division ring, such as Desargues' theorem, and those that hold only over fields, such as Pappus' theorem. In this paper, we explain the topological origin of this distinction. To this end, we extend to the noncommutative setting the surface-graph approach to incidence theorems developed by Richter-Gebert, Fomin, and Pylyavskyy. We then show that theorems associated with graphs embedded on the sphere, such as Desargues' theorem, hold over any division ring, whereas theorems corresponding to graphs embedded on surfaces of positive genus, such as Pappus' theorem, typically hold if and only if the ground ring is a field. We also extend these results to the setting of arbitrary rings, not necessarily admitting division.