I'm reading the chapter 17 of Fultons Book "Algebraic topology a first course" and I somehow have some problems in understanding the triangulation of a surface and what it has to do with computing the fundamental group of a surface. I don't know if I should post the pages of the book but maybe you have it also. It would be nice if we could discuss about it and maybe if you could do some examples because for example I don't see why we need the euler equation/characteristic to say something about the fundamental group of the surface.
I really would appreciate it if someone can discuss it with me, so you don't need to write a whole explanation maybe there are some useful and clear examples where we can apply the theory and then I maybe see where I still have problems. Because we didn't discussed it in the exercise sheets.
Thank you for your help!
Stackexchange is not a useful platform for having discussions or conversations. It works well when you have a specific, answerable question.
Triangulating a surface gives you concrete combinatorial data to work with. If the surface is compact, the triangulation can be chosen to be finite, so you have finitely many vertices, edges and triangles you can enumerate and study.
Every path on the surface is homotopic to a path the runs along the edges, and homotopy between paths can be described by sequences of sliding along triangles. By choosing a spanning tree, you can find a set of generators for the fundamental group, and the triangles yield relations. From there, the fundamental group can be computed.
The Euler characteristic will turn out to be one of the few governing factors of the topology of the surface. That’s not supposed to be obvious from the beginning: it will emerge once you compute sufficiently many fundamental groups.