What is the generator of $\pi_9(S^2)$? This is more or less the same question as 
[ What is the generator of $\pi_9(S^3)$? ], except what I would like to know is if it is possible to describe this map in a way 
not only topologists can make sense of. 
[EDIT] (Following the advice of  Ryan Budney.)   A purely  geometric construction 
like the famous Hopf fibration for $\pi_3(S^2)$  would be perfect. 
(Something like a  map $S^9\to S^2$ or   $S^9\to S^3$  which may be written 
down in equations.)  But I understand that there is little hope for that. 
A less explicit but probably more reasonable approach is to try and  represent this homotopy class by a framed 7-manifold in ${\mathbb R}^9$  following Pontryagin. 
In fact, any information about such a manifold may be of help. Is it really complicated?
I  am not really familiar with  the work of Jie Wu, but  what I have read this far makes sense to me. So, the answer can also be along this lines, but if so I would like to see more then hints. (This  computation looks horrendous, and I probably  cannot handle it by myself.) 
 A: I posted an answer to the previous question describing one way to view a generator of $\pi_9 S^3$. From the Hopf fibration
$$S^1\to S^3 \overset{\eta}{\to} S^2$$ and fibration long exact sequence,
we know that $$\pi_9 S^1 = 0 \to \pi_9 S^3 \overset{\eta}{\to} \pi_9 S^2\to \pi_8 S^1=0 $$
is exact, so $\pi_9 S^3 \cong \pi_9 S^2$ via the Hopf map $\eta$ (comment made by @skd above). 
I'm not sure if these answers are only understandable by topologists, but all the information is available in Hatcher's book Algebraic Topology (however, some of the information is not proved there, e.g. Bott periodicity). 
A: Since my old answer is referenced here What is the generator of $\pi_9(S^3)$?, I spent a little time trying to figure out what it says about this.  "Toda's sequence" is a $p$-local fiber sequence ($p$ any prime) of the form 
$$ S^{2n-1} \to \Omega \widehat{S}^{2n} \to \Omega S^{2pn-1},
$$
where $\widehat{S}^{2n}$ is a certain space I don't need to care about yet; see https://en.wikipedia.org/wiki/EHP_spectral_sequence.   This backs up one step to a $p$-local fiber sequence 
$$
\Omega^2 S^{2pn-1} \xrightarrow{f} S^{2n-1} \to \Omega \widehat{S}^{2n}.
$$
The bottom non-trivial homotopy group of the fiber is $\mathbb{Z}$, and so we get a map
$$
\mathbb{Z}=\pi_{2pn-1}S^{2pn-1}=\pi_{2pn-3} \Omega^2 S^{2pn-1} \xrightarrow{f_*} \pi_{2pn-3}S^{2n-1}.
$$
When $p=3$ and $n=2$ this gives $\mathbb{Z}=\pi_{9}\Omega^2S^{11}\to \pi_9S^3$, which by the argument I gave in my other answer surjects with image $\mathbb{Z}/3$.  So we want to compute the image of $f_*$ in this dimension, i.e., the effect on $f$ on the bottom cell of $\Omega^2 S^{2pn-1}$.
Now, when $p=2$, "Toda's sequence" is the James's EHP sequence (with $\widehat{S}^{2n}=S^{2n}$).  In this case $f_*$ is $\mathbb{Z}\to \pi_{4n-3}S^{2n-1}$.  We know what the image of the generator is in this case: it is the "Whitehead square" $[\iota_{2n-1}, \iota_{2n-1}]$, which can be described geometrically as the attaching map of the $4n+2$-cell of $S^{2n-1}\times S^{2n-1}$.
I then tried to figure out what Toda's sequence actually is, in hopes of finding a geometric description of the map, and I failed.  Everyone seems to refer to Toda's Composition Methods book for this sequence, but the statement given on Wikipedia is not actually there.  However, the idea of the sequence (together with the other Toda sequence described on wikipedia) seems to be essentially 13.1 in Toda's book.  In turn, 13.1 is not proved in Toda's book, but rather in the paper

Toda, Hirosi: On the double suspension $E^2$, J. Inst. Polytech. Osaka City Univ. Ser. A. 7 (1956), 103–145.

The fiber sequence I am interested in is essentially Theorem 7.6 of Toda's paper.
