Does anyone know of a proof of the fact that any 2-manifold can be triangulated that does not use the Jordan-Curve Theorem or the Jordan-Schoenflies Theorem? Thanks for your help
Take the proof that any compact smooth manifold admits triangulations, and set the dimension to two.
The idea goes like this:
Paraphrasing Allen Hatcher:
If you're interested in topological surfaces, the paper
A.J.S. Hamilton, The triangulation of 3-manifolds, Oxford Quart. J. Math. 27 (1976), 63-70
takes the Kirby-Siebenmann machinery and scales it down to 3 dimensions where it becomes somewhat simpler, so one can prove existence and uniqueness of triangulations of 3-manifolds using only standard PL techniques, such as results of Waldhausen. Presumably the same approach would work for surfaces. Since the method works in 3 dimensions it can't be using the topological Shoenflies theorem since this fails in 3 dimensions. On the other hand, it would use some PL (or smooth) surface theory so it wouldn't be entirely "from scratch".
edit: Allen wrote this argument up in a recent paper. See this thread for details http://mathoverflow.net/a/151760/353