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 Sarios 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!
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