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I would like to use that the fundamental group $π_1(M)$ of a compact surface $M$ possibly with boundary has a finite set of generators. According to some questions here and sources elsewhere², it sounds like this is well-known but I couldn't find a proof or reference anywhere. Do you know one?

²Consider only closed (i.e. boundary-less) surfaces

flukx
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  • Section 17c (p. 242) of Algebraic Topology A first course from 1995 by William Fulton states what I want but I am not sure if it considers manifolds with boundaries as well since this "normal form" that is used looks like only for boundaryless (closed) surfaces. – flukx Aug 21 '22 at 19:29
  • Generally speaking, the fundamental group of a compact manifold (possibly with boundary) is finitely presented, because a compact manifold is homotopy equivalent to a finite CW complex (see e.g. https://mathoverflow.net/questions/118429/fundamental-group-of-a-compact-manifold). In this case it suffices to show that a compact surface admits a (finite) triangulation, and for that see e.g. https://math.stackexchange.com/questions/33321/why-do-all-compact-connected-surfaces-have-a-triangulation. – Qiaochu Yuan Aug 21 '22 at 19:41
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    Does this answer your question? Is the fundamental group of a compact manifold finitely presented? For surfaces, it is a standard algebraic topology exercise to compute finite presentations of fundamental groups. You can find a proof in pretty much every AT textbook (Hatcher, Massey, etc). https://math.stackexchange.com/questions/995381/fundamental-group-of-surface-of-genus-g – Moishe Kohan Aug 21 '22 at 19:58
  • The fundamental group of any finite simplicial complex is finitely generated because it can be described as the "edge-path group": https://en.wikipedia.org/wiki/Fundamental_group#Edge-path_group_of_a_simplicial_complex. Any compact surface can be given the structure of a finite simplicial complex (i.e., triangulated), as the comment by @QiaochuYuan says. – John Palmieri Aug 21 '22 at 21:10
  • Thank you for the references and the link to the suitable question which I didn't find beforehand. Now I cannot "accept" your answer but thanks! – flukx Aug 21 '22 at 22:51

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