$v-e+f=2-2g$ as a topological invariant Euler's theorem was expanded to encompass polyhedrons homeomorphic to not only spheres but also $g$-holed toruses. I've tried to understand proofs about how $2-2g$ is a topological invariant but have always had trouble with the use of planar graphs.
Can anyone explain how $2-2g$ is a topological invariant without using the concept of planar graphs?

$2g-2$ is a topological invariant of any closed orientable surface $S$ of genus $g$, namely its Euler characteristic. There are several ways to see that the Euler characteristic is a topological invariant, see also this question, or this one.
