The Classification Theorem for Compact Surfaces

Abstract

The classification theorem for compact surfaces is a formidable result. This result was obtained in the early 1920's, and was the culmination of the work of many. The theorem gives a simple way of obtaining all compact 2-manifolds, moreover, as a result of the theorem, it's possible to decide whether or not any two compact surfaces are homeomorphic rather easily. Before the statement of the theorem, quite a bit of basic topological concepts are first introduced, including connectivity, compactness and quotient topology. In addition to that, a rigorous proof requires, among other things, a precise definition of a surface, orientability, a notion of generalized triangulation, and a precise way of determining whether two surfaces are homeomorphic, which requires some notion of algebraic topology. All of the above brings together the final proof of the theorem.