Timeline for What is known about the strong Arnold conjecture?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jul 16, 2013 at 20:52 | vote | accept | Nathaniel Bottman | ||
| Jun 18, 2013 at 21:43 | comment | added | Thomas Kragh | @Jonny: Yes - absolutely relevant, and the stable Morse number is in fact the number of cells in a minimal cell structure of the suspension spectrum of the manifold. The reason these counter examples by Damian exists are because inherently Floer homology (at least in this example with cotangent bundles) is stable - i.e. it leaves room due to the possibility of having "negative dimensional cells" - that is negative Conley Zehnder index. In fact it is probably possible to construct similar counter examples in the case at hand ($M$ closed), so that the best possible bound is the stable Morseindex | |
| Jun 18, 2013 at 19:13 | comment | added | Jonny Evans | A related but not directly relevant point you may already be aware of: for the Arnold conjecture on cotangent bundles (that the number of intersections between the zero-section and a Hamiltonian pushoff is bounded below by the minimal Morse number of the zero section) one can show using generating functions (Chaperon, Laudenbach-Sikorav, Eliashberg-Gromov) that a lower bound is given by the stable Morse number. This is not necessarily the Morse number, there are examples due to Damian, but it sometimes gives a better bound than the sum of Betti numbers. | |
| Jun 18, 2013 at 14:49 | answer | added | Thomas Kragh | timeline score: 3 | |
| May 29, 2013 at 6:04 | history | edited | Nathaniel Bottman | CC BY-SA 3.0 |
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| May 29, 2013 at 5:17 | comment | added | Nathaniel Bottman | Hi Thomas! You are right, I was mixing up the versions with and without the "nondegenerate" hypothesis. I will edit my question. | |
| May 25, 2013 at 13:24 | comment | added | Thomas Kragh | Hi Nate. The reference you give seems (to me) to prove the cup length conjecture in those case. Ie that the number of fixed points is greater than or equal to the cup length (or something stronger). Does it mention anything about Morse functions? Or does that somehow follow from that? | |
| Apr 13, 2013 at 20:00 | history | edited | Nathaniel Bottman | CC BY-SA 3.0 |
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| Apr 12, 2013 at 1:22 | history | asked | Nathaniel Bottman | CC BY-SA 3.0 |