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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?

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6 Answers 6

up vote 18 down vote accepted

[Three years later …]

All the published proofs of triangulability of surfaces that I am aware of use the Schoenflies theorem, which is not exactly an easy thing to prove. There is however another line of proof which avoids the Schoenflies theorem and instead uses the Kirby torus trick that underlies Kirby-Siebenmann theory in higher dimensions. There is a 1974 paper by A.J.S.Hamilton that gives much simpler proofs of Moise's theorems on triangulability of 3-manifolds using the torus trick, and the same ideas can be applied even more simply for surfaces. Instead of the Schoenflies theorem one needs a few results about surfaces strictly in the PL (or smooth if one prefers) category. Namely, one needs to know that PL structures are unique up to PL homeomorphism in the following four cases: $S^1\times S^1$, $S^1\times{\mathbb R}$, $[0,1]\times{\mathbb R}$, and $D^2$. These can be regarded as special cases of the usual classification theorem for compact PL surfaces, extended to include a few noncompact cases.

I haven't seen this proof in the literature, so I've written it up and posted it on the arXiv here, working in the smooth category rather than the PL category.

It's not clear how suitable this proof would be for an undergraduate course. Besides the ingredients mentioned above, a little basic covering space theory is also needed. If one were in the fortunate position of already having covered these things, then this proof might be accessible to undergraduates. On the other hand, it could be of some interest to go through a proof of the often-quoted-but-seldom-proved Schoenflies theorem. (In this connection I might mention a paper by Larry Siebenmann on the Schoenflies theorem in the Russian Math Surveys in 2005, giving history as well as a proof.)

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Thank you Allen! –  Ryan Budney Dec 13 '13 at 16:46
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This proof is really great. I probably wouldn't try to teach it to undergraduates, but for people with a little sophistication it is clearly the "right" proof. Thanks for writing it up! –  Andy Putman Dec 13 '13 at 16:52
    
@Allen Hatcher: Hi Professor Hatcher. What I gather from your response is that it’s wrong to think that the Jordan-Schoenflies Theorem is solely responsible for the triangulability of surfaces — topological $ 3 $-manifolds are triangulable, yet the Jordan-Schoenflies Theorem doesn’t hold in dimension $ 3 $. It thus seems that Kirby’s torus trick is the correct way of approaching the problem of the triangulability of topological manifolds of dimension $ 2 $ or $ 3 $. However, I don’t see why the trick fails in higher dimensions, so I’m wondering if you can say something about it. Thanks! –  Leonard Jul 13 at 17:31

Try this book 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.

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This is exactly what I was looking for! I had never seen that paper of Thomassen, which from the looks of it is a great little gem to share with my students. The bibliography also mentions a very short paper of Doyle and Moran that triangulates surfaces. I had figured that classical material like this had easy proofs by now... –  Andy Putman Mar 9 '10 at 6:28
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Thomassen's paper is very nice, except that I had a lot of trouble with Theorem 3.1, the Jordan-Schoenflies theorem. Can someone give a crisp explanation of his argument here? –  Greg Kuperberg Mar 9 '10 at 7:04
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Thomassen's paper can be downloaded from Andrew Ranicki's website at maths.ed.ac.uk/~aar/jordan/index.htm . I think there's a fixable error in his main proof. He assumes that if a point is accessible from a region, then after one puts in a polygonal path based at the point then it is still accessible from the new regions. This is false: consider the "cuspidal cubic" curve. But one can retrieve the situation by redfining accessibility by insisting that a positive angle's worth of segments starting at the given point and going into the given region. –  Robin Chapman Mar 10 '10 at 8:03
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Rado's original triangulation paper is now also available from the website, along with sundry other items. –  Andrew Ranicki Nov 24 '11 at 21:33
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It should be pointed out that the Gallier-Xu book does not give a complete proof of triangulability of surfaces, contrary to what seems to be claimed in the preface to the book. The hardest step in the proof of Thomassen that they present in Appendix E is the Schoenflies theorem, and they only state this without including a proof. To make matters worse, immediately after the statement of the Schoenflies theorem they make the erroneous claim that it can be proved using tools of algebraic topology, but this is true only for the weaker Jordan curve theorem. –  Allen Hatcher Dec 13 '13 at 15:10

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.

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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.

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.

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The main reason they want to see a proof for a topological surface is that it is the first step in classifying surfaces. The way I like to arrange that proof immediately thickens the triangulation up to a handle decomposition; if I assumed that the manifold was smooth, then I could dispense with the triangulation and apply Morse theory. Do you know a proof that topological surfaces can be smoothed that doesn't pass through a triangulation? –  Andy Putman Mar 9 '10 at 5:57
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And thanks for pointing me towards Thurston's book! There are an amazing number of things in there, though I am loath to recommend it to undergraduates given Thurston's cavalier attitude toward rigor... –  Andy Putman Mar 9 '10 at 5:58
    
I'll check tomorrow or at latest Wednesday. I thought he was doing something more along the lines of Kirby-Siebenmann, smoothing the transition maps from the atlas. But it's been a long time and I haven't read that part of the book with any focus. –  Ryan Budney Mar 9 '10 at 6:14
    
Oh, sorry. Thurston only talks about the PL <-> Smooth relations. –  Ryan Budney Mar 10 '10 at 23:44
    
Yeah, that's what I thought I remembered, but there's so much stuff in Thurston's book that that I thought I might have missed it. –  Andy Putman Mar 16 '10 at 3:49

What about the 1968 three-pages paper by Doyle and Moran?

http://link.springer.com/article/10.1007%2FBF01425546

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It is freely available at digizeitschriften.de/dms/img/… –  BS. Sep 17 '13 at 9:22

The notes (in French) for Gramain's 1969-70 course, available at

http://www.math.u-psud.fr/~biblio/numerisation/docs/G_GRAMAIN-55/pdf/G_GRAMAIN-55.pdf

seem to include a proof using Morse theory, based on a quick glance I made.

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