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Let $\mathcal{H}(f)$ be the Hopf invariant of a map $f:\mathbb{S}^{4n-1}\to \mathbb{S}^{2n}$.

When is the suspension map $$ \sigma:\{f\in \pi_{4n-1}(\mathbb{S}^{2n}):\, \mathcal{H}(f)\neq 0\}\to \pi_{4n}(\mathbb{S}^{2n+1}) $$ a non-zero map? When is it a surjection?

Clearly it is a surjection when $n=1$, but I am interested in the higher dimensional cases. In other words, I would like to know if there exist non-trivial elements in $ \pi_{4n}(\mathbb{S}^{2n+1})$, $n\geq 2$, that are suspensions of a map with a non-zero Hopf invariant.

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I'm on shaky ground here, this is what I think happens. There are the Hopf invariant one maps $\nu:S^7\to S^4$ and $\sigma:S^{15}\to S^8$, and these suspend to generators of the stable stems $\pi^S_3\cong \mathbb{Z}/24$ and $\pi^S_7\cong\mathbb{Z}/240$. So you also have surjectivity when $n=2, 4$, also.

The Hopf invariant is a homomorphism $H:\pi_{4n-1}(S^{2n})\to \mathbb{Z}$, so it vanishes on torsion elements. It takes the value $\pm 2$ on the Whitehead square $[\iota_{2n},\iota_{2n}]$, which generates the $\mathbb{Z}$ summand in $\pi_{4n-1}(S^{2n})$. The suspension of $[\iota_{2n},\iota_{2n}]$ is trivial. These facts together imply that the answer to your final question is no in the other dimensions.

The facts above are stated in Hatcher's book on pp 473-474, and no doubt are proved in Toda's book.

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