Timeline for Suspension of the third Hopf map
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Dec 17, 2015 at 18:12 | history | edited | Mark Grant | CC BY-SA 3.0 |
added argument
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| Nov 21, 2015 at 22:49 | comment | added | Gustavo Granja | The only thing you are missing in your next to last comment is that the component of $[\iota_8,\iota_8]$ along $h_3$ in the splitting is $\pm 2$. This can be seen by computing the classical $\mathbb Z$-valued Hopf invariant via the cohomology rings of the cones on these maps. The cone on $[\iota_8,\iota_8]$ is the 16-skeleton of $\Omega S^9$ by the James construction. | |
| Nov 21, 2015 at 22:37 | comment | added | Gustavo Granja | The suspension map is induced by the inclusion of the bottom cell in $\Omega S^8$, so it is the splitting map in the fiber sequence mentioned in my comment above. | |
| Nov 21, 2015 at 11:00 | comment | added | András Szűcs | But just the diagram does not seem to imply the answer, why the generator $\sigma$ of $Z$ goes onto the generator of $Z_{240}$ | |
| Nov 21, 2015 at 10:54 | comment | added | András Szűcs | The suspension map $\pi_{14}(S^7) \to \pi_{15N (S^8)$ provides the splitting. This follows from the fact that image of the double supsension $\pi_ {14}(S^7) \to \pi^{st}(7)$ has index $2$, and so injective. (The latter follows from the fact, that in dimension $7$ there is a Hopf invarinat one map. To see this use the butterfly lemmma to the "cross diagram" formed by the part of EHP sequence $\pi_{14}(S^7) \to \pi_{15}(S^8) \to Z$ and the Freudenthal exact sequence: $ 0 \to Z<[id_{S^8}, id_{S^8}]> \to \pi_{15}(S^8) \to \pi^{st}(7) \to 0$..) | |
| Nov 21, 2015 at 10:32 | comment | added | András Szűcs | The suspension map $\pi_{14}(S^7) \to \pi_{15N (S^8)$ provides the splitting. | |
| Nov 20, 2015 at 13:19 | comment | added | Gustavo Granja | A better way to see that the sequence splits is to note that the inclusion of the fiber is null and therefore its homotopy fiber is $\Omega S^8 \cong S^7 \times \Omega S^{15}$. | |
| Nov 20, 2015 at 9:10 | history | answered | Mark Grant | CC BY-SA 3.0 |