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.



Hopf invariant and the Freudenthal theorem
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. 


