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Timeline for Classification of surfaces and the TOP, DIFF and PL categories for manifolds

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Feb 25, 2022 at 20:16 history edited Victor CC BY-SA 4.0
Added a reference to the result in item 3.
Jun 28, 2019 at 21:29 comment added annie marie cœur maybe you know the answer for this: math.stackexchange.com/questions/3277380/… ? please
Jun 28, 2019 at 19:46 comment added annie marie cœur Just to make sure @all, is smooth manifold = differentiable manifold? iff? How to show that --- or is this a definition? –
Apr 13, 2017 at 12:58 history edited CommunityBot
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May 19, 2012 at 16:45 vote accept Victor
May 15, 2012 at 15:54 history edited Victor CC BY-SA 3.0
Title change
May 15, 2012 at 15:53 answer added Victor timeline score: 53
May 11, 2012 at 18:01 history edited Victor CC BY-SA 3.0
Defining the term "surface"
May 11, 2012 at 17:19 comment added Victor John, I think you are right about that. In Thurston/Levy's book Theorems 3.10.2 (Triangulating smooth manifolds) and 3.10.8 (Smoothings of PL manifolds of dimension up to three), and also Problem 3.10.19 (canonical smoothing of a triangulated surface) address my questions, including the non-compact case.
May 11, 2012 at 16:15 comment added John Klein I am somewhat familiar with smoothing theory, especially in high dimensions. The way one proves smoothing theory statements involves a choice of combinatorial structure on the manifold: either a handle structure or a triangulation. So, without being familiar with what Thurston does, I'm betting a nickel that you can't avoid a combinatorial structure on the manifold.
May 11, 2012 at 15:41 comment added Victor @John Klein: Thank you, I am familiar with the Morse theoretic proof, but it is not quite what I am looking for. What I am looking for goes more along the lines of what HW mentions in his comment. I wonder if a Morse theoretic approach is possible in the non-compact case.
May 11, 2012 at 14:03 comment added Johannes Ebert Yes, smooth surfaces can be classified by Morse theory, see Hirsch 'Differential topology', the last chapter.
May 11, 2012 at 14:03 comment added HJRW IIRC, smoothings of topological manifolds in low dimensions (certainly 3, probably 2) are discussed in Thurston/Levy's book '3-dimensional geometry and topology'.
May 11, 2012 at 13:44 comment added John Klein It seems to me one can use Morse theory to do the classification. Morse theory gives handlebody structures, rather than triangulations. That seems to satisfy your requirements.
May 11, 2012 at 13:40 history asked Victor CC BY-SA 3.0