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Here is what I know about Whitehead products in homotopy groups of spheres:

  • $[\mathrm{id}_{S^{2n}},\mathrm{id}_{S^{2n}}]$ has Hopf invariant (EDIT: $\pm$) two.
  • No element that survives into the stable range can be a Whitehead product, since the suspension of a Whitehead product is trivial.
  • If $\alpha \in \pi_m(S^n)$ with $m$ odd, then $[\alpha,\alpha]$ has order at most 2. This is because the Whitehead product is a graded Lie bracket.

I have two questions, one open-ended and one more specific.

The open-ended question: What are some known examples of nontrivial Whitehead products in finite homotopy groups of spheres? Are there other situations, besides the ones mentioned, where Whitehead products must be trivial or have order at most 2?

The specific question: Is there an example of $\alpha \in \pi_m(S^n)$, for some $m>n$, such that $2\cdot[\Sigma\alpha,\Sigma\alpha] \neq 0$?

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    $\begingroup$ For your specific question take $\alpha=id_{S^1}$. Then $[id_{S^2},id_{S^2}]=-2\eta$. $\endgroup$ – Tyrone Nov 13 '18 at 22:10
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    $\begingroup$ Also, it is not true in general that $[id_{S^n},id_{S^n}]$ has Hopf invariant $2$. $\endgroup$ – Tyrone Nov 13 '18 at 22:12
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    $\begingroup$ This is why I specified $m>n$. $\endgroup$ – Fedya Nov 13 '18 at 22:12
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    $\begingroup$ 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. $\endgroup$ – skd Nov 13 '18 at 22:21
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    $\begingroup$ @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)? $\endgroup$ – Fedya Nov 13 '18 at 22:46
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James proves a great number of things about the Whitehead product in his paper On the suspension sequence (though a number of results in that paper are stated in terms of cases rather than the stronger results that hold 2-locally). For example, he shows that there is a 2-local pairing $$ \{-,-\}: \pi_p(S^k) \times \pi_q(S^k) \to \pi_{p+q+1}(S^{2k+1}) $$ such that $\{\alpha, \beta\} = (-1)^{pq + k} \{\beta, \alpha\}$ and such that the composite with the "Whitehead product" map $$ P: \pi_{p+q+1}(S^{2k+1}) \to \pi_{p+q-1}(S^k) $$ that appears in the EHP sequence is the ordinary Whitehead product. In particular, since the Whitehead product is graded-commutative we get an identity $2 [\alpha, \alpha] = 0$ whenever $k$ is odd, complementing the fact that $2[\alpha, \alpha] = 0$ by graded-commutativity whenever the source degree $p$ of $\alpha$ is odd. (In particular, this shows that when $n$ is even, $2 [\Sigma \alpha, \Sigma \alpha] = 0$ for any element $\alpha$ in the homotopy groups of $S^n$.)

James also shows naturality for the EHP sequence in a sense, and this has the following consequence. Writing the element $[\Sigma \alpha, \Sigma \alpha]$ as a composite of $[id_{m+1}, id_{m+1}]: S^{2m+1} \to S^{m+1}$ with the map $\Sigma \alpha: S^{m+1} \to S^{n+1}$, James' naturality shows that $$ \{\Sigma \alpha, \Sigma \alpha\} = \Sigma^{m+1} \alpha \circ \Sigma^{n+1} \alpha \circ \{id_{m+1}, id_{m+1}\} = \pm 2 (\Sigma^{m+1} \alpha \circ \Sigma^{n+1} \alpha) = \pm 2 \Sigma^{n+1}(\Sigma^{m-n} \alpha \circ \alpha). $$ and thus that $$ [\Sigma \alpha, \Sigma \alpha] = \pm 2 P(\Sigma^{n+1} (\Sigma^{m-n} \alpha \circ \alpha)). $$

In particular, for this element to not be 2-torsion, the element $4\Sigma^{n+1} (\Sigma^{m-n} \alpha \circ \alpha)$ must not be in the kernel of $P$. However, the EHP spectral sequence tells you that this is the same as asking that the element $4\Sigma^{n+1} (\Sigma^{m-n} \alpha \circ \alpha)$ must not be in the image of the Hopf invariant map $H$. (Stably, this element is $4 \alpha^2 = (2 \alpha)^2$.)

(This is the point where I say that I've exhausted most of my knowledge of unstable theory. I don't know if there are any examples of non-2-torsion self-Whitehead squares.)

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