I have Toda's book Composition methods in homotopy groups of sphere.
I have not found the generator of $\pi_9(S^3)=\mathbb{Z}_3$.
What is it?
I have Toda's book Composition methods in homotopy groups of sphere. I have not found the generator of $\pi_9(S^3)=\mathbb{Z}_3$. What is it? 


$$S^3 \to \Omega \widehat{S}^4 \to \Omega S^{11}$$ is a fibration $3$locally. I claim that $(\pi_9 \Omega\widehat{S}^4)_{(3)}=0$ in which case your element must be in the image of $\pi_{10}\Omega S^{11}=Z$ under the boundary map of the above sequence. To prove my claim, I appeal to (i) the $3$primary fibration sequence $\widehat{S}^4\to \Omega S^5\to \Omega S^{13}$, and (ii) wikipedia, who tells me that there is no $3$torsion in the $6$stem of $S^5$ (or, just use more Toda sequences to get up to the stable range.) 

