Timeline for Whitehead products in homotopy groups of spheres
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 15, 2021 at 23:37 | comment | added | Piotr Hajlasz | I hope you do not mind I asked this question: mathoverflow.net/q/395418/121665. I hope I did correctly state the result. Correct me if I am wrong. | |
| Nov 19, 2018 at 19:57 | vote | accept | Fedya | ||
| Nov 14, 2018 at 11:36 | answer | added | Tyler Lawson | timeline score: 9 | |
| Nov 14, 2018 at 5:37 | history | edited | Fedya | CC BY-SA 4.0 |
deleted 6 characters in body
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| Nov 14, 2018 at 0:24 | comment | added | Fedya | Ah, gotcha. Yeah, I'm curious about Whitehead products not of the identity map, but thanks! | |
| Nov 14, 2018 at 0:13 | comment | added | skd | Yes, that was a typo. The symbol $\iota_n \in \pi_n(S^n)$ is the class of the identity map. In particular, $[\iota_{k+1}, \iota_{k+1}] \in \pi_{2k+1}(S^{k+1})$ where $k$ is as in my previous comment. The statement I claimed is Corollary 3.2 of Barratt-Jones-Mahowald, "The Kervaire invariant problem". | |
| Nov 13, 2018 at 22:46 | comment | added | Fedya | @skd I assume you mean the Kervaire element exists if and only if $[\iota_{k+1},\iota_{k+1}]$ is divisible by 2? What do you mean by $\iota_{k+1}$ (a link is fine)? | |
| Nov 13, 2018 at 22:21 | comment | added | skd | One family of examples containing both nontrivial and trivial Whitehead products comes from the Kervaire classes. If $k = 2^{j+1}-2$, then the Kervaire element $\theta_j\in \pi_{2^{j+1}-2}(\mathbb{S})$ if and only if $[\iota_{k+1},\iota_{k+1}]$ is divisible by 2. | |
| Nov 13, 2018 at 22:12 | comment | added | Fedya | This is why I specified $m>n$. | |
| Nov 13, 2018 at 22:12 | comment | added | Tyrone | Also, it is not true in general that $[id_{S^n},id_{S^n}]$ has Hopf invariant $2$. | |
| Nov 13, 2018 at 22:10 | comment | added | Tyrone | For your specific question take $\alpha=id_{S^1}$. Then $[id_{S^2},id_{S^2}]=-2\eta$. | |
| Nov 13, 2018 at 21:21 | history | asked | Fedya | CC BY-SA 4.0 |