Timeline for The existence of a fiber sequence involving $\mathrm{Spin}(9)$ and $\mathrm{SU}(2)$
Current License: CC BY-SA 4.0
5 events
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| May 3, 2020 at 16:26 | comment | added | skd | Yep, thanks! (I also just added a construction of a candidate for Y.) | |
| May 3, 2020 at 16:25 | history | edited | skd | CC BY-SA 4.0 |
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| May 3, 2020 at 11:04 | comment | added | Robert Bryant | You are quite right that $\mathrm{G}_2/\mathrm{SU}(2)$ is not a topological group. Since, as you point out, there is a fibration $S^5\to \mathrm{G}_2/\mathrm{SU}(2)\to S^6$, all the homotopy groups $\pi_i(\mathrm{G}_2/\mathrm{SU}(2))$ for $i =0,1,2,3$ are trivial, so it can't be a compact Lie group. Also, because $\pi_3\bigl(\mathrm{Spin}(9)/\mathrm{G}_2\bigr)$ is finite (probably trivial, but I haven't checked to be sure), it can't be a compact Lie group. For similar reasons, your $Y$, if it exists, cannot be a topological group either. | |
| May 2, 2020 at 20:41 | history | edited | skd | CC BY-SA 4.0 |
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| May 2, 2020 at 19:59 | history | asked | skd | CC BY-SA 4.0 |