I am looking for a direct proof that the quaternionic hopf map generates (after suspension) the 3rd stable homotopy group of spheres. There are some related MO questions, for example:
- third-stable-homotopy-group-of-spheres-via-geometry
- explicitly-construct-generators-of-homotopy-groups-of-spheres
But the focus in these seems more to be on how to calculate that the image of the hopf map is 24-torsion, and I want to see why the hopf map generates this group.
background
The quaternionic Hopf map is a fibration $$S^3 \to S^7 \stackrel{\nu}{\to} S^4$$ which induces an isomorphism $$ \pi_k(S^4) \cong \pi_k(S^7) \oplus \pi_{k-1}(S^3).$$ The Freudenthal suspension theorem implies that the suspension map $$ \pi_7S^4 \to \pi_8 S^5 \cong \pi^{st}_3$$ is surjective. Thus there is a surjective map: $$ \mathbb{Z}\langle \nu \rangle \oplus \pi_6 S^3 \to \pi_3^{st}$$
In the analogous setting of the complex Hopf fibration (the classical case) this is enough to conclude that the image of the hopf map generates $\pi_1^{st}$ (the extra factor $\pi_2S^1 = 0$ vanishes). However in this case the extra factor is non-trivial. We can look up that $\pi_6 S^3 \cong \mathbb{Z}/12$.
The Question
Now if we know, by some argument, both facts that $\pi_6 S^3 \cong \mathbb{Z}/12$ and that $\pi_3^{st} \cong \mathbb{Z}/24$, then the above subjectivity implies that the image of $\nu$ must be a generator. However we can see by the previous MO questions that these calculations can be quite involved and non-obvious.
I will be teaching a topics course this summer and I would like to include a proof that the image Hopf map generates $\pi_3^{st}$. I want to use this in some later calculations. I would like to do this without having to explicitly calculate $\pi_6 S^3$ and $\pi_3^{st}$ since both those calculations seem like they will be quite involved and then the proof that the Hopf map generates is somewhat unsatisfying. I am hoping that there is a more direct argument, which hopefully doesn't use too much elaborate machinery. (Though what one considers elaborate machinery depends on ones perspective).
Some of the possible tools that will be available are: a direct geometric argument, Thom spectra, the Pontryagin-Thom correspondence between bordism groups and homotopy groups of these spectra, and the AHSS. I am considering later in the coure presenting the Galatius-Madsen-Tillmann-Weiss theorem, so it could be possible to use an h-principle argument in combination with those techniques if that helps, though I don't see that it would. Another possible tool is Whitehead products as they will show up in a limited form at the very end of the course (I seem to remember that there is a connection between Whitehead products and the kernel of the suspension homomorphism, so maybe these help).
