Timeline for What is the easiest way to classify all possible smooth orientable closed 2-manifolds?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 15, 2010 at 0:06 | comment | added | Andy Putman | @Scott : Ah, you want maximal collections of disjoint curves whose union doesn't separate the surface! This is different from maximal collections of disjoint non-pairwise homotopic nonseparating curves, which is what I parsed your answer to mean. As you can see, I spend far too much of my time thinking about configurations of curves on surfaces... | |
| Apr 14, 2010 at 20:16 | comment | added | Ryan Budney | Deane, if you're looking for an analytic proof why not go back to the original proofs of classification of surfaces? I'm referring to the arguments of Poincare and Koebe. Depending on which way you want to go, you're dealing with Greene's functions or harmonic functions and various other standard analytical things most of the way to your destination. | |
| Apr 14, 2010 at 18:23 | comment | added | Scott Carter | @ Andy: Yes, of course, I meant non-homotopic, but if the curves are homotopic, then the second one will separate won't it? @Sam: Thanks for elaborating. My point was, of course, that the homeom. type is determined by the 1st homology. There is the Conway zipper proof. | |
| Apr 14, 2010 at 17:35 | comment | added | Andy Putman | If you go back to my list of proofs above, the main point of the proof in Armstrong's "Basic Topology" is to show that if a compact surface has no nonseparating embedded simple closed curves on it, then it is a sphere. However, this takes a fair amount of work. | |
| Apr 14, 2010 at 17:32 | comment | added | Deane Yang | Thanks, Andy, for saying out loud what I didn't have the nerve to ask about. So does this approach lead to a proof or not? | |
| Apr 14, 2010 at 17:09 | comment | added | Andy Putman | I think the OP wanted a PROOF of the classification, not merely an invariant distinguishing the different surfaces. And I also think you want your embedded circles to be non-homotopic. | |
| Apr 14, 2010 at 15:38 | comment | added | Sam Nead | A smooth closed orientable (connected) 2-manifold S is determined, up to diffeomorphism, by its genus: the number of disjoint embedded curves needed to cut S into a planar surface. The two-sphere is planar, by convention. <p> So the sphere has genus zero. the torus has genus one, and so on... One way to think of this: the genus is the number of "handles" the surface has. A suitcase with g handles has genus g. | |
| Apr 14, 2010 at 15:11 | comment | added | Deane Yang | Oh, man. That sounds good to me. But I'm an aging geometric analyst short on both time and ability to think geometrically (I prefer formulas and PDE's). Any chance you could add a few more sentences? Or provide a reference? | |
| Apr 14, 2010 at 14:57 | history | answered | Scott Carter | CC BY-SA 2.5 |