Timeline for Is there a homomorphism between $\pi_8(S^5)$ and $\pi_8(SO(6))$?
Current License: CC BY-SA 3.0
12 events
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| Jan 17, 2017 at 14:45 | history | edited | Michael Albanese | CC BY-SA 3.0 |
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| Aug 3, 2015 at 3:45 | comment | added | Yingfei Gu | @LennartMeier and Sean Tilson, Thanks for explanation. | |
| Jul 7, 2015 at 8:24 | comment | added | Sean Tilson | There is also a proof via rational homotopy theory which I prefer, but as, Lennart says, the argument is hardly short as it requires the set up of rational homotopy theory. | |
| Jul 4, 2015 at 8:49 | comment | added | Lennart Meier | The groups $\pi_k S^n$ are finite if $k\neq n$ and $k \neq 2n-1$ if $n$ is even. There is no short argument for this -- it is known as the Serre finiteness theorem. | |
| Jul 3, 2015 at 18:59 | vote | accept | Yingfei Gu | ||
| Jul 3, 2015 at 18:59 | comment | added | Yingfei Gu | @QiaochuYuan Do you mind providing a short argument/intuition on why it is fine?(And what's the general statement?) | |
| Jul 3, 2015 at 18:39 | comment | added | Michael Albanese | @YingfeiGu: You can use the fact that Qiaochu Yuan stated. Alternatively, it follows from the exact sequence (in particular, you don't need to know that $\pi_8(S^5)$ is finite). I have added some explanation. | |
| Jul 3, 2015 at 18:33 | history | edited | Michael Albanese | CC BY-SA 3.0 |
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| Jul 3, 2015 at 18:13 | comment | added | Qiaochu Yuan | @Yingfei: $\pi_8(S^5)$ is finite (e.g. by work of Serre) so the map to $\mathbb{Z}$ is necessarily zero. | |
| Jul 3, 2015 at 18:04 | comment | added | Yingfei Gu | Sorry, why does that sequence imply $p_*$ is an isomorphism? It is obviously injective, however, not obviously surjective to me. | |
| Jul 3, 2015 at 18:01 | history | edited | Michael Albanese | CC BY-SA 3.0 |
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| Jul 3, 2015 at 17:38 | history | answered | Michael Albanese | CC BY-SA 3.0 |