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I have created an entry on the quaternionic Hopf fibration and then I have tried to spell out the argument, suggested to me by Charles Rezk on MO, that in $G$-equivariant stable homotopy theory it represents a non-torsion element in
$[\Sigma^\infty_G S^7 , \Sigma^\infty_G S^4]_G \simeq \mathbb{Z} \oplus \cdots$for $G$ a finite and non-cyclic subgroup of $SO(3)$, and $SO(3)$ acting on the quaternionic Hopf fibration via automorphisms of the quaternions.
I have tried to make a rigorous and self-contained argument here by appeal to Greenlees-May decomposition and to tom Dieck splitting. But check.
added the following fact, which I didn’t find so easy to see:
Consider
the Spin(5)-action on the 4-sphere $S^4$ which is induced by the defining action on $\mathbb{R}^5$ under the identification $S^4 \simeq S(\mathbb{R}^5)$;
the Spin(5)-action on the 7-sphere $S^7$ which is induced under the exceptional isomorphism $Spin(5) \simeq Sp(2) = U(2,\mathbb{H})$ by the canonical left action of $U(2,\mathbb{H})$ on $\mathbb{H}^2$ via $S^7 \simeq S(\mathbb{H}^2)$.
Then the complex Hopf fibration $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$ is equivariant with respect to these actions.
This is almost explicit in Porteous 95, p. 263
added the remark
Of the resulting action of Sp(2)$\times$Sp(1) on the 7-sphere (from this Prop.), only the quotient group Sp(n).Sp(1) acts effectively.
added pointer to Table 1 in
for the coset presentation
$\array{ S^3 &\overset{fib(h_{\mathbb{H}})}{\longrightarrow}& S^{7} &\overset{h_{\mathbb{H}}}{\longrightarrow}& S^4 \\ = && = && = \\ \frac{Spin(4)}{Spin(3)} &\longrightarrow& \frac{Spin(5)}{Spin(3)} &\longrightarrow& \frac{Spin(5)}{Spin(4)} }$added pointer to
which in its Prop. 4.1 explicitly states and proves the $Spin(5)$-equivariance of the quaternionic Hopf fibration
(but fails to mention the coset representation that makes this manifest)
added this in the list of references:
Noteworthy fiber products with the quaternionic Hopf fibration, notably exotic 7-spheres, are discussed in
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