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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?

Qiaochu Yuan
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Andy Putman
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7 Answers7

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[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 as a short expository paper "The Kirby torus trick for surfaces" 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.)

Allen Hatcher
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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
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    @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 '14 at 17:31
  • "The Osgood-Schoenflies theorem revisited" L C Siebenmann © 2005 Russian Academy of Sciences, (DoM) and London Mathematical Society, Turpion Ltd Russian Mathematical Surveys, Volume 60, Number 4

    Errata to the paper by L C Siebenmann "The Osgood-Schoenflies theorem revisited" Russian Mathematical Surveys, Volume 60 (2005), Number 4, Pages 645-672

     – Marvin Jay Greenberg Mar 14 '17 at 22:16
  • Abstract for Siebenmann: The very first unknotting theorem of a purely topological character established that every compact subset of the Euclidean plane homeomorphic to a circle can be moved onto a round circle by a globally defined self-homeomorphism of the plane. This difficult hundred-year-old theorem is here celebrated with a partly new elementary proof, and a first but tentative account of its history. Some quite fundamental corollaries of the proof are sketched, and some generalizations are mentioned. I currently can't access this and I wonder what Osgood had to do with the theorem. – Marvin Jay Greenberg Mar 14 '17 at 22:26
  • @Marvin Jay Greenberg: The Osgood reference Siebenmann gives is W. F. Osgood, “On the transformation of the boundary in the case of conformal mapping”, Bull. Amer. Math. Soc. vol. 9 no. 5 (1903), 233–235. This announces some technical results which Osgood says allow him to prove that, given a Jordan curve C, there is a conformal homeomorphism from the interior of C onto the interior of the unit circle that extends continuously to C. He doesn't seem to say the extension is a homeomorphism from C to the circle. This is three years before Schoenflies' paper in the 1906 Math. Annalen. – Allen Hatcher Mar 15 '17 at 03:25
  • The link in the answer seems to be down. Hopefully someone can fix this. In general, for such situations, it would be better to have the title of the paper included in the answer, so that one can put it into google. – M. Winter Aug 04 '20 at 12:51
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    @M.Winter: Fixed the link, added the title. – Allen Hatcher Aug 04 '20 at 19:12
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    @AllenHatcher: I think I suggested this to you a while ago, but I'll repeat it -- I think this note is a very valuable contribution to the literature, and you should consider submitting it to a journal that publishes semi-expository papers (e.g. L'Enseignment Math). – Andy Putman Aug 04 '20 at 19:44
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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.

Tony Pantev
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  • Thanks, I didn't know anyone had simplified the existence of triangulations proof that much. – Ryan Budney Mar 09 '10 at 06:23
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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 09 '10 at 06:28
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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 09 '10 at 07:04
  • @Greg: You'd probably be better off asking it as a separate question. – Harry Gindi Mar 09 '10 at 09:18
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    Thomassen's paper can be downloaded from Andrew Ranicki's website at http://www.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 08:03
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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
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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.

Ryan Budney
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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 09 '10 at 05:57
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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 09 '10 at 05:58
  • 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 09 '10 at 06:14
  • 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 03:49
  • What do you mean by "looks linear"? I suppose this property is what implies that the triangulations pull-back to a polyhedral decomposition of the manifold? – horned-sphere Sep 01 '21 at 22:24
  • @viniciuscantocosta: Correct on the latter question. For the former, the level of the refinement of the subdivision can be chosen so that forgetting some of the coordinates of the simplex gives a function that globally satisfies the implicit function theorem for the the manifold intersect the top-dimensional simplex path components. I imagine this is expressible as an upper bound on the diameter of the simplex, in terms of the extrinsic curvature of the surface in the euclidean space. – Ryan Budney Sep 01 '21 at 22:49
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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.

Rafe Mazzeo
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  • I looked inside this book by Christian Bar and did not see any account of triangulation in the table of contents. – Marvin Greenberg Jul 06 '15 at 04:32
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    He does give a very nice proof, in Chapter 6. He assumes that the surfaces are regularly embedded in R^3, so perhaps this is not what people are looking for here. – Tali Mar 07 '22 at 09:01
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What about the 1968 three-pages paper by Doyle and Moran?

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

Frédéric Le Roux
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This is not an answer that points to a more recent and more accessible account of the triangulability of surfaces, but rather a way to make the account in the first chapter of Ahlfors' & Sario's book more accessible, if sufficient time is available. It should be noted that the proof given by Ahlfors & Sario works for all (connected, 2nd countable) surfaces: compact or noncompact; with or without boundary. I will describe what I found to be the difficulties with Ahlfors' & Sario's presentation and how one can mitigate these difficulties, especially if the learning of this material is by self-study, as was the case for me, and not in the context of a university course. Disclaimer: I am a mathematician but not a topologist.

I found there were three main difficulties, all stemming from Ahlfors' & Sario's terse style of writing. The first difficulty is the absence of any references for background material. I found that the classic, self-contained book, Elements of the Topology of Plane Sets of Points (2nd ed.), by M.H.A. Newman, and the first two sections of the third chapter of the book, Algebraic Topology, by E. Spanier (for the basics of the theory of abstract simplicial complexes), provide sufficient background. The second difficulty is that almost every sentence resembles the statement of a lemma whose proof is left to the reader. The third difficulty is an intentionally omitted proof of a rather difficult result, "46C".

Regarding the second and third difficulties: after filling in the missing details, I decided to write them up in the form of a list of notes (as opposed to an article). I then created a website, on which I posted these notes. Included in them, is a proof of the result "46C" that relies heavily on the material in the cited book by Newman. Although I did not strive for either optimum mathematical efficiency or elegance, perhaps my notes will be useful to others, in making Ahlfors' & Sario's account more accessible.

While I was at it, I also posted some details for a proof of Schoenflies' Theorem via Complex Analysis; these are details for the presentation in the book, Boundary Behaviour of Conformal Maps, by C. Pommerenke. Note that this proof assumes the Riemann Mapping Theorem, proofs of which are more widely available. (These notes were written before I was aware of Newman's book, which happens to also contain a proof of Schoenflies' Theorem -- a proof that is purely topological in nature. As mentioned in Allen Hatcher's answer, an historical account of proofs of Schoenflies' Theorem, including a new one at the time, appeared in the paper and its errata. A preprint of the paper is freely available here.)

A careful accounting of all the prerequisite material for Ahlfors' & Sario's approach, reveals that it is a very substantial amount. Although such a proof of the triangulability of surfaces can be made accessible to undergraduates, it is not clear whether an undergraduate who embarks on such a project of self-study will still be an undergraduate upon completion of the project.

Ian Iscoe
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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.

Marvin Jay Greenberg
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  • This seems to assume the surfaces are smooth. – Ryan Budney Sep 30 '15 at 07:28
  • I did not see any discussion of triangulations, but perhaps triangulability is a consequence some of the results proved on the way to classification of surfaces (Ch. 5)? – j.c. May 25 '18 at 15:41