  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!
