Lower bound on the number of faces of a polyhedron of genus g Is there a lower bound on the number of faces of a polyhedron of topological genus g? For example: it seems very reasonable that $g$ < $F$ i.e. the genus of a polehydron is less than the number of faces of the polyhedron, but i can't find a proof. To be clear what is meant by polyhedron let's use the definition from wikipedia: "A polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices." The genus can be calculated by $g = \frac{2-\chi}{2}$, where $\chi$ is the Euler characteristic of the polyhedron.

Example of polyhedron with 4096 faces and 4097 holes.

P. McMullen, C. Schulz, and J.M. Wills. "Polyhedral manifolds in E3 with unusually large genus". Israel Journal of Mathematics., 46 (1983), no. 1-2, pages 127–144

<