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In the words of Milnor himself, the classification theorem for compact surfaces is a formidable result. According to Massey, this result was obtained in the early 1920’s and was the culmination of the work of many. Indeed, a rigorous proof requires, among other things, a precise definition of a surface and of orientability, a precise notion of triangulation, and a precise way of determining whether two surfaces are homeomorphic or not. This requires some notions of algebraic topology such as, fundamental groups, homology groups, and the Euler-Poincaré characteristic. Most steps of the proof are rather involved and it is easy to loose track.
The purpose of these notes is to present a fairly complete proof of the classification Theorem for compact surfaces. Other presentations are often quite informal (see the references in Chapter V) and we have tried to be more rigorous. Our main source of inspiration is the beautiful book on Riemann Surfaces by Ahlfors and Sario. However, Ahlfors and Sario’s presentation is very formal and quite compact. As a result, uninitiated readers will probably have a hard time reading this book.
Our goal is to help the reader reach the top of the mountain and help him not to get lost or discouraged too early. This is not an easy task!
We provide quite a bit of topological background material and the basic facts of algebraic topology needed for understanding how the proof goes, with more than an impressionistic feeling. We hope that these notes will be helpful to readers interested in geometry, and who still believe in the rewards of serious hiking!
Jean H. Gallier, "The Classification Theorem for Compact Surfaces and a Detour on Fractals", . October 2007.
Date Posted: 23 October 2007