Timeline for Mapping a loop space to quaternionic projective space
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| May 12, 2020 at 4:27 | vote | accept | skd | ||
| May 12, 2020 at 4:27 | history | edited | skd | CC BY-SA 4.0 |
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| May 12, 2020 at 4:26 | answer | added | skd | timeline score: 2 | |
| May 9, 2019 at 0:36 | comment | added | user51223 | @user43326 I didn’t understand the point? If the map is a map of loop spaces then you may get a map as desired by the question. But, you don’t have such a map to start with!?! | |
| May 8, 2019 at 16:28 | comment | added | user43326 | What if you take the bar construction on the map $\Omega ^2S^5 \to S^3$? | |
| May 6, 2019 at 17:47 | comment | added | user51223 | At the prime $p=2$ there is no map $\Omega^2\Sigma^2 S^3\to S^3$ which is nonzero in homology. The existence of such a map will furnish 3-sphere with a commutative multiplication (up to homotopy) and that is known not to be the case at the prime $p=2$. I think this shows at this prime an extension as inin question cannot exist at $p=2$. I don’t know about off primes! | |
| May 6, 2019 at 14:06 | comment | added | Gustavo Granja | The inclusion of the bottom cell does not even extend to the 8-skeleton. The attaching map of the 8-cell in the James construction is the Whitehead product [i_4,i_4] which is twice the Hopf map plus the suspension of the Blakers-Massey element (the generator of \pi_6(S^3)). You would need the Whitehead product to be twice the Hopf map in order for the extension to the 8-skeleton to exist. | |
| May 5, 2019 at 23:49 | comment | added | Dylan Wilson | Maybe you mean 1.8? That says H-spaces X are a retract of \Loops\Sigma X, but surely the splitting is not unique. You can use that to build a map like the one you mention though. Anyway- I’m happy to forget about it, just wanted to make sure there wasn’t some contradiction lurking in my brain. | |
| May 5, 2019 at 23:43 | comment | added | Dylan Wilson | Theorem 1.11 in that paper says that the James construction is the free topological monoid with the basepoint of X acting as the identity, which is not the same... did you mean to cite a different theorem? I’m really having trouble believing the statement is true... | |
| May 5, 2019 at 20:38 | comment | added | skd | @DylanWilson This is in fact true (at least if X is path-connected), and was the original statement proved by James (Theorem 1.11 in his "Reduced product spaces"). | |
| May 5, 2019 at 20:21 | comment | added | Dylan Wilson | This isn’t relevant to your question, but you seem to claim that \Omega\Sigma X is the “free homotopy associative H-space” on X, which certainly isn’t true, right? Am I misunderstanding something? | |
| May 5, 2019 at 17:53 | history | edited | skd | CC BY-SA 4.0 |
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| May 5, 2019 at 17:24 | history | asked | skd | CC BY-SA 4.0 |