Past exam of the topology course of the University of Oxford / 2021 / 3 / c / i

A complete pairing of the six polygon edges leaves three edges and one face as CW cells, together with the vertex equivalence classes. If $V$ denotes their number, then $V\ge 1$ and the polygonal-schema Euler count givesA closed orientable surface of genus two has Euler characteristic $2-2\cdot 2=-2$. Since the Euler characteristic is a topological invariant, no complete pairing of a hexagon can give that surface. Equivalently, the polygonal-schema Euler count bounds the genus of an orientable result by $2g\le 3$, so $g\le 1$.