I would like to review the classification of compact surfaces (orientable or not) and without boundary (first). I know that they are classified topologically by Euler characteristic (and therefore by genus), but I don't know how to proceed rigorously. Is this approach correct:

  1. Show that any surface admits a triangulation (difficult theorem based on Jordan's theorem), then use (singular) homology to define the genus. Or use Morse theory to define genus (the fact that genus is a topological invariant is not "directly visible").
  2. Study the model surfaces: sphere, torus, projective space and the connected sum of these surfaces. Study the presentation of the fundamental group of each of these surfaces (exhibit generators and relations). The surface genus is the half of the rank of the abélianization of the fundamental group: $$g=\frac{1}{2}\mathrm{rank}\ H_1(S,\mathbb{Z})=\frac{1}{2}\mathrm{rank}\ \pi_1(S)/[\pi_1(S),\pi_1(S)].$$
  3. Show that any closed surface (compact, without boundary) is homemomorphic to one of the above surfaces. I don't know how to proceed (maybe: triangulation + some operations to bring it back to the above normal forms ...)

Thanks for commenting on this steps and directing me to nice references (with detailed proofs if possible). Is the fundamental group of a compact surface with boundary, a free group? (as in the case of the Mobiüs strip).

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    $\begingroup$ Do you know Massey's book "A Basic Course In Algebraic Topology"? It has a very hands-on proof of the classification theorem for both oriented and non-oriented surfaces without boundary in the first chapter, using cut-and-paste arguments (this is where I first learned it). However, it still uses the triangulability theorem as a black box. $\endgroup$ – William Mar 31 at 13:30
  • $\begingroup$ @William, Thanks for your useful comment, I will take a look at Massey's book. $\endgroup$ – user56980 Mar 31 at 16:33

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