Is there a concrete description of the nontorsion elements in the homotopy groups of spheres?

Question

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?

Answer 1

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.