Timeline for Is there an effective way to calculate K-theory using Morse functions?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 12, 2014 at 16:04 | answer | added | Andreas Thom | timeline score: 2 | |
| Aug 12, 2014 at 13:42 | answer | added | Liviu Nicolaescu | timeline score: 1 | |
| Apr 5, 2013 at 21:16 | comment | added | John Klein | There are a few papers which deal with the providing the technical details which are lacking in the C-J-S paper (what is now known as the "associativity of gluing." There are some papers of Burghelea in this vein, and a different approach due to my student Lizhen Qin is also noteworthy (this is not merely to give Lizhen a plug---there are definitely difficult details which were not provided in the C-J-S program. | |
| Apr 3, 2013 at 23:48 | comment | added | Paul Siegel | You're both right - "critical point data" needs to include information about the gradient flow of the Morse function as well as the indeces of the critical points. I would allow whatever input from the Morse function which is required to recover the homotopy type of $M$. | |
| Apr 3, 2013 at 19:43 | comment | added | Jonny Evans | "Flow data" would be a better description. You need to know stuff like the homotopy type of the attaching map of the cell. Even for Morse homology you need flow data for the differential. I seem to recall that Cohen-Jones-Segal explain what you would need to reconstruct the stable homotopy theory in terms of manifolds of flow-lines. They pick this case because it's all you can hope to recover in Floer theory - having infinite-dimensional stable/unstable manifolds means that the "homotopy type of the attaching map" wouldn't be very interesting even if it made sense. | |
| Apr 3, 2013 at 17:47 | comment | added | Allen Knutson | What is your "critical point data"? Just the index isn't enough, right? I'm thinking about the blowup of CP^2 vs. S^2 x S^2, two nonhomotopic manifolds on which I have Morse functions with the same indices 0,2,2,4. Or were you ruling out the repetition of 2? | |
| Apr 3, 2013 at 17:15 | comment | added | Jonny Evans | I always wanted to know the answer to this question, ever since reading Cohen-Jones-Segal: "Floer's infinite dimensional Morse theory and homotopy theory" where they construct the flow category Dylan Wilson mentioned. You could ask the same question about your favourite generalised cohomology theory. | |
| Apr 3, 2013 at 15:02 | comment | added | Dylan Wilson | (Oh I guess I should say it's only a homeomorphism if $f$ is generic enough... in general it's a homotopy equivalence.) | |
| Apr 3, 2013 at 14:59 | comment | added | Dylan Wilson | Associated to any such $M$ and $f$ there is a topological category, $C_f$, whose realization is homeomorphic to $M$ (the objects are critical points and the morphisms are flows between them, basically). I imagine that continuous functors to the category with one object and $U$ as its morphism space might give $K$-theory? Unwinding we could probably give a more geometric description... Anyway, this is all sort of a guess. | |
| Apr 3, 2013 at 14:37 | history | asked | Paul Siegel | CC BY-SA 3.0 |