Timeline for How many simple closed geodesics in a given primitive homology class?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Jul 3, 2018 at 21:33 | comment | added | Igor Rivin | @AknazarKazhymurat No, it does not matter if you have boundary (with the minor difference of whether or not you consider the boundary geodesic). As for the homology classes, it's because the minimal length of a homology class defines a norm on homology (and homology is a $2g$ dimensional vector space (if considered over $\mathbb{R}),, but the minimal length is sometimes represented by a multi-curve (for a punctured torus, it's always a connected curve). | |
| Jul 3, 2018 at 20:25 | comment | added | user74900 | also, could you please explain why 'the number of homology classes where there is a simpel closed geodesic...' grows no faster than $L^{2g}$ for a closed surface? | |
| Jul 3, 2018 at 20:22 | vote | accept | CommunityBot | ||
| Jul 3, 2018 at 20:22 | comment | added | user74900 | Dear Dr. Rivin, I also wanted to ask you: is the answer different if we consider surfaces with boundary as opposed to punctures? For instance, does every primitive homology class contain a unique simple geodesic for a torus with one boundary component? | |
| Jul 3, 2018 at 19:53 | history | answered | Igor Rivin | CC BY-SA 4.0 |