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By Serre's theorem, we know the only nontorsion parts of the homotopy groups of spheres occur as $\pi_n(S^n)$ and $\pi_{4n-1}(S^{2n})$. The first of these are trivial to describe, but the second have very interesting, symmetric incarnations, they are the generalised hopf fibrations, at least for $n=1,2,4$, associated to the real normed division algebras.

Are there a similar explicit descriptions for representatives of these higher nontorsion elements too?

Even if we don't have explicit descriptions, do we know anything about the values of the hopf invariants associated to these maps?

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For $n \neq 1,2,4$, the minimal positive Hopf invariant of an element of $\pi_{4n-1}(S^{2n})$ is $2$.

An explicit element of Hopf invariant $2$ can be constructed as follows: consider the attaching map $S^{4n-1} \to S^{2n} \vee S^{2n}$ of the $4n$-cell of the CW-complex $S^{2n} \times S^{2n}$ and compose it with the codiagonal $S^{2n} \vee S^{2n} \to S^{2n}$.

That no elements of Hopf invariant $1$ exist outside the "Adams dimensions" $n = 1,2,4$ is known as the "Hopf invariant 1 problem". It was studied and solved by Adams. Later Adams and Atiyah found a shorter proof. For a very accessible write-up, see this REU paper.

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    $\begingroup$ The Hopf invariant 1 problem was first solved by Adams. A later simpler proof is due to Adams and Atiyah. The Adams Conjecture (which is not needed for the Hopf invariant 1 problem) was first proved by Quillen, with subsequent proofs by Sullivan and Becker-Gottlieb. $\endgroup$ Aug 27, 2021 at 14:00
  • $\begingroup$ Thanks a lot for the correction of my messed-up historical account, Prof. Hatcher! I edited my answer accordingly. $\endgroup$ Aug 27, 2021 at 14:51

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