Question

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?

Answer 1

Toda's sequence

$$S^3 \to \Omega \widehat{S}^4 \to \Omega S^{11}$$

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.)