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

Let R R be a connected 2-manifold without boundary obtained from a (possibly infinite) collection of polygons by identifying them along edges of equal length. Let V V be the set of vertices, and for every v ∈ V v ∈ V , let κ ( v ) κ (v) denote the (Gaussian) curvature of v v : 2 π 2 π minus the sum of incident polygon angles. Descartes showed that ∑ v ∈ V κ ( v ) = 4 π ∑ v ∈ V κ (v) = 4 π whenever R R may be realized as the surface of a convex polytope in R 3 R^3 . More generally, if R R is made of finitely many polygons, Euler’s formula is equivalent to the equation ∑ v ∈ V κ ( v ) = 2 π χ ( R ) ∑ v ∈ V κ (v) = 2 π χ (R) where χ ( R ) χ (R) is the Euler characteristic of R R . Our main theorem shows that whenever ∑ v ∈ V : κ ( v ) > 0 κ ( v ) ∑ v ∈ V : κ (v) > 0 κ (v) converges and there is a positive lower bound on the distance between any pair of vertices in R R , there exists a compact closed 2-manifold S S and an integer t t so that R R is homeomorphic to S S minus t t points, and further ∑ v ∈ V κ ( v ) ≤ 2 π χ ( S ) − 2 π t ∑ v ∈ V κ (v) ≤ 2 π χ (S) - 2 π t . In the special case when every polygon is regular of side length one and κ ( v ) > 0 κ (v) > 0 for every vertex v v , we apply our main theorem to deduce that R R is made of finitely many polygons and is homeomorphic to either the 2-sphere or to the projective plane. Further, we show that unless R R is a prism, antiprism, or the projective planar analogue of one of these that | V | ≤ 3444 |V| ≤ 3444 . This resolves a recent conjecture of Higuchi.