An incidence theorem in projective geometry states that certain configurations of points and lines satisfy certain intersection properties. Pappus's theorem, Desargues's theorem — these hold over the real projective plane. Some hold over all fields. Some hold over all division rings, including the noncommutative quaternions.
The question of when a theorem requires commutativity should be algebraic. Izosimov (arXiv:2603.00288) shows it is topological.
Each incidence configuration has a natural graph — vertices for points and lines, edges for incidence relations. That graph can be embedded on a surface. The key invariant is the genus: sphere (genus 0), torus (genus 1), higher. The result: configurations whose graphs embed on the sphere yield theorems valid over all division rings. Higher-genus embeddings require commutativity.
This connects three areas that have no obvious reason to be connected. The topology of surfaces, which is about holes and handles. The combinatorics of point-line configurations, which is about which things intersect. The algebra of division rings, which is about whether multiplication commutes. The genus — a property of shapes — determines the answer to an algebraic question.
The structural claim is that commutativity is not an intrinsic property of the theorem but a topological one of its configuration. The same combinatorial relations, drawn on different surfaces, do or don't require the underlying algebraic structure to commute. The algebra follows the geometry, not the other way around.
Izosimov, “Surface topology and incidence theorems over division rings,” arXiv:2603.00288 (March 2026).