Past exam of the geometry of surfaces course of the University of Oxford / 2023 / 1 / e

Take a hexagonal polygonal schema with boundary wordIdentify equally labelled polygon edges in the indicated directions. The two consecutive occurrences of $a$ identify the first three vertices; the two $b$ edges extend this class to the next two vertices, and the $c$ edges include the remaining vertex. Thus all six polygon vertices become one, while the six edges become three and the interior is one face. Its Euler characteristic isThis is the usual polygonal schema for the connected sum of surfaces of three real projective planes. The same-sign edge pairs force a nonorientable surface, consistently with $\chi (N_3)=2-3=-1$. A closed orientable surface cannot have odd Euler characteristic.