An analogue of the Descartes-Euler formula for infinite graphs and Higuchi's conjecture

Let be a connected 2-manifold without boundary obtained from a (possibly infinite) collection of polygons by identifying them along edges of equal length. Let be the set of vertices, and for every , let denote the (Gaussian) curvature of : minus the sum of incident polygon angles. Descartes showed that whenever may be realized as the surface of a convex polytope in . More generally, if is made of finitely many polygons, Euler's formula is equivalent to the equation where is the Euler characteristic of . Our main theorem shows that whenever converges and there is a positive lower bound on the distance between any pair of vertices in , there exists a compact closed 2-manifold and an integer so that is homeomorphic to minus points, and further .

In the special case when every polygon is regular of side length one and for every vertex , we apply our main theorem to deduce that is made of finitely many polygons and is homeomorphic to either the 2-sphere or to the projective plane. Further, we show that unless is a prism, antiprism, or the projective planar analogue of one of these that . This resolves a recent conjecture of Higuchi.