Timeline for Triangulating surfaces
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Sep 1, 2021 at 22:49 | comment | added | Ryan Budney | @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. | |
| Sep 1, 2021 at 22:24 | comment | added | horned-sphere | 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? | |
| Mar 16, 2010 at 3:49 | comment | added | Andy Putman | 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. | |
| Mar 10, 2010 at 23:44 | comment | added | Ryan Budney | Oh, sorry. Thurston only talks about the PL <-> Smooth relations. | |
| Mar 9, 2010 at 6:14 | comment | added | Ryan Budney | 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. | |
| Mar 9, 2010 at 5:58 | comment | added | Andy Putman | 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... | |
| Mar 9, 2010 at 5:57 | comment | added | Andy Putman | 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? | |
| Mar 9, 2010 at 5:51 | history | answered | Ryan Budney | CC BY-SA 2.5 |