Timeline for Convergence of a sum with the ranks of homotopy groups
Current License: CC BY-SA 3.0
15 events
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| May 27, 2015 at 2:33 | vote | accept | CommunityBot | moved from User.Id=62675 by developer User.Id=36770 | |
| May 25, 2015 at 19:35 | history | edited | Eric Wofsey |
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| May 25, 2015 at 19:35 | answer | added | Qiaochu Yuan | timeline score: 6 | |
| May 25, 2015 at 19:32 | comment | added | user62675 | @Eric I was looking at fiber bundles (that explains why I'm using $F$ :-P) and wanted to measure how far the (nice) fibers were from looking like other (nicer) fibers. I realized one way to do this was to look at the ranks of the homotopy groups. It seemed like "normalizing" by taking $\mathrm{Rank}(\pi_q(F))/q$ instead of simply $\mathrm{Rank}(\pi_q(F))$ made more sense. I wanted to see if that was really the case; but if it really is hopeless then perhaps the unnormalized version is more appropriate. | |
| May 25, 2015 at 19:26 | comment | added | Eric Wofsey | If $F$ is simply connected and a finite CW-complex, then $I(F)<\infty$ iff $F$ has only finitely many nontrivial rational homotopy groups (see Theorem 2.33 here, for instance; I'm not sure whether the simply connected hypothesis is really necessary). If you don't demand $F$ to be a finite CW-complex, the question seems hopeless in full generality (for instance, $F$ could be an arbitrary product of spheres of different dimensions). Do you have any particular motivation for looking at $I(F)$? | |
| May 25, 2015 at 19:26 | comment | added | user62675 | I have found this that relates the ranks of the $q$th homotopy group of a 1-connected space to the rational cohomology of the $q$th term of its Whitehead tower. | |
| May 25, 2015 at 19:23 | history | edited | Eric Wofsey |
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| May 25, 2015 at 19:10 | answer | added | David C | timeline score: 3 | |
| May 25, 2015 at 19:03 | comment | added | user62675 | @Dylan Fixed (I forgot (again) to say that its homology groups are finitely generated, which coupled with the other assumption says that the homotopy groups are finite (it's some result of Serre, I think)). Edit: see Theorem 1.7 here. | |
| May 25, 2015 at 19:02 | history | edited | user62675 | CC BY-SA 3.0 |
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| May 25, 2015 at 19:00 | comment | added | Dylan Wilson | Finite generation is a condition not a consequence of your other assumption... Edit: Ok now it is. | |
| May 25, 2015 at 18:59 | comment | added | user62675 | @Espen Yeah, I did have that as an example. I hadn't phrased my question properly. It's fixed now! | |
| May 25, 2015 at 18:58 | history | edited | user62675 | CC BY-SA 3.0 |
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| May 25, 2015 at 18:51 | comment | added | Espen Nielsen | For the second question, you have $S^1$, for example. It is easy to construct more examples. | |
| May 25, 2015 at 18:45 | history | asked | user62675 | CC BY-SA 3.0 |