Past exam of the geometry course of the University of Oxford / 2008 / A / 1 / i

The classification of compact surfaces says that every compact connected surface without boundary is homeomorphic either to the sphere with $g\ge 0$ handles, with Euler characteristic $2-2g$, or to a connected sum of $k\ge 1$ real projective planes, with Euler characteristic $2-k$. The first family is orientable and the second is nonorientable; the displayed invariants uniquely determine the type. With boundary one additionally removes a uniquely determined number of open disks.

Number the vertices of the fourteen-gon $v_0,\dots ,v_{13}$ in boundary order. The printed edge identifications implyThus there is one vertex, seven edges and one face in the quotient. ConsequentlyThe two occurrences of $a_1$ have the same sign, as do those of $a_5$, so the orientation on the polygon cannot extend consistently across those paired edges. Thus the surface is nonorientable, and classification identifies it as a connected sum of seven projective planes. The inverse on the first occurrence of $a_4$ and the positive second occurrence have been retained in the vertex calculation.