most recent 30 from http://mathoverflow.net 2013-05-23T23:40:24Z http://mathoverflow.net/feeds/question/17578 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17578/triangulating-surfaces Andy Putman 2010-03-09T05:36:15Z 2013-03-01T00:48:52Z

<p>I've had a few undergraduate students ask me for references for the classical fact (due to Rado) that closed topological surfaces can be triangulated. I know two sources for this, namely Ahlfors's book on Riemann surfaces and Moise's book "Geometric topology in dimensions 2 and 3". Both of these strike me as being a bit much for a bright undergraduate. Question : in the 30+ years since Moise's book, has anyone written a more accessible account?</p>

http://mathoverflow.net/questions/17578/triangulating-surfaces/17580#17580 Ryan Budney 2010-03-09T05:51:32Z 2010-03-09T05:51:32Z

<p>If you're okay going the extra step and assuming a smooth structure, the standard argument of Whitehead goes like this: take a smooth embedding of your manifold (of any dimension) into euclidean space. Triangulate Euclidean space, perturb the embedding to make it transverse to the skeleta of the triangulation. Refine the triangulation (barycentric subdivision) to the point where the embedding "looks linear" in each top-dimensional simplex. The triangulations of the simplices pulls-back to a polyhedral decomposition of the manifold, which you can subdivide to be a triangulation. </p> <p>If you insist on going the extra step to topological manifolds you could smooth the topological structure. I believe much of that argument appears in Thurston's 3-dimensional geometry and topology book but I don't have it at home at the moment, and I don't remember. </p>

http://mathoverflow.net/questions/17578/triangulating-surfaces/17582#17582 Tony Pantev 2010-03-09T06:13:54Z 2010-03-09T06:32:23Z

<p>Try this <a href="http://www.cis.upenn.edu/~jean/gbooks/surftop.html" rel="nofollow">book</a> by Jean Gallier and Diana Xu. It is aimed at undergraduates and has a nice account of Thomassen's elementary proof of the triangulation theorem in the last appendix. Or you can refer the students to Thomassen's original paper which is also quite readable.</p>

http://mathoverflow.net/questions/17578/triangulating-surfaces/17583#17583 Rafe Mazzeo 2010-03-09T06:24:23Z 2010-03-09T06:24:23Z

<p>There is a forthcoming undergraduate differential geometry book by Christian Baer (at the University of Potsdam, Germany) which is very nice; it gives a careful account of this theorem and a very accessible proof. If the book is not publically available, it should be soon.</p>

http://mathoverflow.net/questions/17578/triangulating-surfaces/123279#123279 Marvin Jay Greenberg 2013-03-01T00:48:52Z 2013-03-01T00:48:52Z

<p>The notes (in French) for Gramain's 1969-70 course, available at</p> <p><a href="http://www.math.u-psud.fr/~biblio/numerisation/docs/G_GRAMAIN-55/pdf/G_GRAMAIN-55.pdf" rel="nofollow">http://www.math.u-psud.fr/~biblio/numerisation/docs/G_GRAMAIN-55/pdf/G_GRAMAIN-55.pdf</a></p> <p>seem to include a proof using Morse theory, based on a quick glance I made.</p>