Past exam of the geometry course of the University of Oxford / 2010 / A / 1 / iii

Use a square with boundary word $ab{a}^{-1}{b}^{-1}$, identifying opposite sides with matching labels and arrows. All four vertices become one, the four sides form two edge classes, and there is one face. Therefore the torus has

A polygonal model of the real projective plane is a bigon with boundary word $aa$, with the equally labelled arrows identified. Its two vertices become one, its sides form one edge, and it has one face. ThusEquivalently, glue antipodal boundary points of a disc.

A square with boundary word $ab{a}^{-1}b$ gives the Klein bottle. The side identifications produce one vertex, two edges and one face. ConsequentlyUnlike the torus, this surface is nonorientable; its equal Euler characteristic alone does not distinguish them.

Glue two triangular discs along their boundaries to obtain a sphere. The resulting CW complex decomposition has three vertices, three edges and two faces, givingThe two triangles represent the two closed hemispheres, with an equator subdivided into three edges.