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.