Timeline for How to rigorously prove that simple closed curves on a surface are primitive closed curves ?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 4, 2011 at 3:01 | vote | accept | Analysis Now | ||
| Nov 3, 2011 at 22:24 | answer | added | Sam Nead | timeline score: 10 | |
| Nov 3, 2011 at 15:33 | comment | added | Daniel Groves | Oh right, I was assuming separating. But as Richard says, there's a proof in the non-separating case in Farb and Margalit's book... | |
| Nov 3, 2011 at 15:32 | comment | added | Autumn Kent | There's a proof in Farb and Margalit's "Primer on mapping class groups". | |
| Nov 3, 2011 at 15:26 | comment | added | Analysis Now | Thanks , I have edited my question. But when you say " $c$ can be chosen to represent a basis element of the first homology group ", do you assume that c is a non-separating simple closed curve in X and use the result that given any two non-separating simple closed curves $c1,c2$ in an oriented surface $X$ , there exists a global homeomorphism of the surface carrying $c1$ onto $c2$, and then taking $c_1$ as $c$ and $c_2$ and generator of $H_1(X)$ ? Is there a similar result for separating closed simple curves ? | |
| Nov 3, 2011 at 15:13 | comment | added | Daniel Groves | First, you need to assume that c is essential, so that it doesn't represent the trivial element of \pi_1. Second, you need to be careful, since it's not true on non-orientable surfaces (since the boundary of a Mobius strip is twice the core curve). But once you're orientable, the first homology (abelianization) of the fundamental group is free abelian, and c can be chosen to represent a basis element of this free abelian group. Thus it is primitive in homology, and hence in the fundamental group. | |
| Nov 3, 2011 at 15:13 | history | edited | Analysis Now | CC BY-SA 3.0 |
edited tags; added 30 characters in body
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| Nov 3, 2011 at 14:42 | history | asked | Analysis Now | CC BY-SA 3.0 |