Classification of surfaces

Remarks

Idea of the proof: by Radó's theorem every surface has a triangulation, so $S$ is obtained from a polygon by gluing its edges in pairs. Cutting and regluing the polygon brings the gluing word to a normal form: $aa^{-1}$ for the sphere, $a_1 b_1 a_1^{-1} b_1^{-1} \cdots a_g b_g a_g^{-1} b_g^{-1}$ for $g$ tori, or $c_1 c_1 \cdots c_k c_k$ for $k$ projective planes. The surfaces in the list are different because orientability and Euler characteristic are topological invariants that take different values on them.

The theorem at work: the Klein bottle is a closed non-orientable surface with Euler characteristic $0$, so it is homeomorphic to $\R P^2 \# \R P^2$. Also $T^2 \# \R P^2$ is non-orientable with Euler characteristic $-1$, so it is homeomorphic to $\R P^2 \# \R P^2 \# \R P^2$. Compactness is needed: the open disc and the open annulus are connected surfaces without boundary that are not in the list.

Going deeper

For a student

The classification is the model of what one hopes for in manifold topology: a complete list with computable invariants. The orientable number $g$ is the genus of the surface. The proof through polygons and gluing words also computes the fundamental group of each surface. Every closed surface other than $S^2$ and $\R P^2$ is aspherical, and every closed surface carries a metric of constant curvature, with sign given by the Euler characteristic. In dimension $3$ no list this simple exists, and the analogue is the geometrization conjecture.