Timeline for Homotopy groups of smooth manifolds?
Current License: CC BY-SA 2.5
16 events
| when toggle format | what | by | license | comment | |
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| Jan 10, 2010 at 0:02 | vote | accept | Ilya Nikokoshev | ||
| Jan 9, 2010 at 21:08 | history | edited | Ilya Nikokoshev | CC BY-SA 2.5 |
upadte
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| Nov 30, 2009 at 17:00 | comment | added | Ryan Budney | The answer to your question will depend on how complicated a "formula" you allow. | |
| Nov 21, 2009 at 10:07 | answer | added | Thomas Riepe | timeline score: 1 | |
| Nov 21, 2009 at 9:32 | answer | added | Somnath Basu | timeline score: 4 | |
| Nov 19, 2009 at 23:29 | comment | added | Joel Fine | @Ryan Budney. Sorry, I think you misunderstood what I meant by "relations". Of course higher homotopy groups are abelian. I don't need Wikipedia to find that out. What I was thinking of was more along the lines of "is there some general formula expressing pi_n in terms of all pi_m for m<n?" If it is hard to write down a relation of this sort for the 2-sphere, then it's probably a lost cause in general too. But this point seemed so simple that I assumed I must have misunderstood the question. Hence my comment "I think I'm misunderstanding the question". | |
| Nov 18, 2009 at 8:31 | comment | added | Ryan Budney | Go to wikipedia and look up "homotopy groups of spheres", similarly look at what's in Hatcher's Algebraic Topology notes, esp the spectral sequences notes. There's quite a bit known, in particularly they're algorithmically computable, they're finite, they're abelian.... for more see the references. | |
| Nov 17, 2009 at 10:25 | comment | added | Joel Fine | I think I'm misunderstanding the question. As you say in dim 2 only the 2-sphere has non-trivial higher homotopy groups, but as Mariano says understanding the n-th homotopy groups of the 2-sphere is an extremely hard (unsolved?) question. Does anyone know of any relations they must satisfy? | |
| Nov 17, 2009 at 6:24 | comment | added | Ilya Nikokoshev | @Mariano, that's one of my points. | |
| Nov 16, 2009 at 23:41 | comment | added | Mariano Suárez-Álvarez | While for $d=2$ there is only the sphere to consider... is that simple? | |
| Nov 16, 2009 at 23:19 | history | edited | Ilya Nikokoshev | CC BY-SA 2.5 |
updated
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| Nov 16, 2009 at 23:16 | comment | added | Ilya Nikokoshev |
@Richard, a relationship between homotopy groups. E.g. homology groups must vanish for dimension >d -- now I'm looking for something analogous.
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| Nov 16, 2009 at 23:07 | answer | added | Ryan Budney | timeline score: 10 | |
| Nov 16, 2009 at 22:52 | comment | added | Steven Sivek | For d=2, every closed orientable manifold is a sphere, disk, annulus, torus, or something hyperbolic -- the sphere is the only one of these which even has nontrivial homotopy groups beyond \pi_1. (For d >= 3 this should be much harder and I don't know what to tell you.) | |
| Nov 16, 2009 at 22:41 | comment | added | Autumn Kent | I don't understand, a relationship to what? | |
| Nov 16, 2009 at 22:37 | history | asked | Ilya Nikokoshev | CC BY-SA 2.5 |