I am currently reading Sullivan's Geometric Topology: Localization, Periodicity, and Galois Symmetry, on page 34 Sullivan claims that the degree 2 map $2:S^4 \to S^4$ induces the map $\left(\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix}\right)$ on $\pi\_8(S^4) \cong \mathbb{Z}/2 \oplus \mathbb{Z}/2$. I don't see how one would perform this computation. Sullivan attributes this contribution to David Frank, I skimmed a few of his papers and saw no mention of this computation.


I have a few ideas about how I might try to attack this, but none of them seem very palatable, I thought that I could look at the induced map on postnikov towers and see what happens in cohomology, but that doesn't seem to be very easy to work with. 


I would appreciate it if anyone had a direct method to attack this computation, or if someone had a reference about how similar computations are performed. I'm waiting on my library to retrieve a copy of Toda's "Composition Methods..." to see if this has any input on this computation.



Action of the degree 2 map on $\pi_8(S^4)$
Use Theorem 8.9 from Whitehead's "Elements of Homotopy Theory" on page 537. Take $k=2$, $\beta=\iota\_4$ and $\alpha\in\pi\_8S^4$. Use $[\iota\_4,\iota\_4]=\Sigma\nu'-2\nu\_4$, $h\_0(\nu\_4\circ\eta\_7)=h\_0(\nu\_4)\circ\eta\_8=\iota\_7\circ\eta\_8=\eta\_8$ and $h\_0(\Sigma\nu'\circ\eta\_7)=h\_0(\Sigma(\nu\circ\eta\_6))=0$. The triple product disappears and you are left with David Frank's answer.


Just a reminder that $\pi\_8S^4=\mathbb{Z}\_2\{\Sigma\nu'\circ\nu\_7\}\oplus\mathbb{Z}\_2\{\nu\_4\circ\eta\_7\}$. The Hopf invariants and whitehead products (and some formulas to compute them) are available in Oguchi's paper "Generators of 2-Primary Components of Homotopy Groups of Spheres, Unitary Groups and Symplectic Groups".


