Action of the degree 2 map on $\pi_8(S^4)$

Question

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.

Answer 1

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".