Surface topology and incidence theorems over division rings

TL;DR

This paper explores the topological reasons behind which incidence theorems hold over division rings versus fields, extending surface-graph methods to noncommutative and ring settings.

Contribution

It extends the surface-graph approach to incidence theorems to noncommutative division rings and arbitrary rings, revealing topological conditions for theorem validity.

Findings

Desargues' theorem holds over any division ring.

Pappus' theorem holds only over fields, not general division rings.

Topological embedding surfaces determine theorem applicability.

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.