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

The classification of compact surfaces says that every compact connected surface without boundary is homeomorphic to exactly one of the following: the sphere, an orientable surface obtained as the connected sum of $g\ge 1$ tori, or a nonorientable surface obtained as the connected sum of $k\ge 1$ real projective planes. The integers $g$ and $k$ are unique, and the orientable and nonorientable families do not overlap. Including $g=0$ makes the sphere the genus-zero member of the orientable family.

Their Euler characteristics are $2-2g$ and $2-k$, respectively. Thus orientability and the Euler characteristic determine the closed connected surface. If boundaries are allowed, one additionally specifies the number of boundary circles and removes that many open discs from a closed model.

For a finite polygonal CW complex decomposition of a compact surface, with $V$ vertices, $E$ edges and $F$ faces after all identifications,This is the Euler characteristic, independent of the chosen subdivision. In a polygonal presentation, count equivalence classes of vertices and paired edges, rather than their number before gluing.

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.

For $g\ge 1$, the standard commutator-word polygon has one face, $2g$ paired edge classes and a single vertex class. ThusAn orientable surface admits continuously compatible local orientations; for a smooth surface, this is equivalently a continuous choice of unit normal vector in an oriented ambient three-dimensional space when such an embedding is available. In a polygonal model, orientability means the faces can be oriented so that induced boundary directions on each glued edge oppose one another. Here each edge label occurs once positively and once negatively. The orientation of the polygon therefore descends consistently across every seam. Hence $X$ is orientable, and it is the closed orientable surface of genus $g$.