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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
Does the $Spin(5)$-equivariance of the quaternionic Hopf fibration lift to $Pin(5)$-equivariance?
(say for $Pin \coloneqq Pin^+$)
Theorem 4.1 in Gluck-Warner-Ziller says “No.” if the action on the ambient $\mathbb{R}^8 = \mathbb{H}^2$ is quaternionic linear. But may we drop this assumption?
What happens to the coset space as you replace Spin by Pin in $Spin(5)/Spin(3) \simeq S^7$? The 3- and 4-spheres still work (is that for any version of Pin?).
@David there’s issues with the number of connected components, if $Pin(4) \simeq Pin(3) \times Pin(3)$ (analogously to how $Spin(4) \simeq Spin(3)\times Spin(3)$. I thought this was the case, but I didn’t check the details, so I might be wrong. [Edit In fact this can’t be right, since by how $Pin(n)$ is defined it has two connected components.]
@Urs the definition of Pin(5) naturally involves some complex vector space underlying the Clifford algebra, IIRC, so my idea was to look at this using that representation.
No, hang on. The Wikipedia page on the $Pin$ groups says that $Pin_+(3) = SO(3)\times C_4$ and $Pin_-(3) = SU(2)\times C_2$. So there’s a convention mismatch, I think, if we want the version of $Pin$ that contains $Spin$.
(Edited earlier incorrect comments, was tired and not quite paying attention)
$Spin(3) = SU(2)$, and according to Wikipedia $Pin_{-}(3) = SU(2) \times C_2$, so that works.
Then it has $Pin_+(3)$ is isomorphic to $SO(3) \times C_4$.
[Didn’t update to see the editing above.]
From Table 5.1 of Matrix Groups: An Introduction to Lie Group Theory by Baker, for instance, $Cl_5 = M_4(\mathbb{C})$, so we should be looking for a (possibly squashed) 7-sphere that is preserved by $Pin(5) \subset M_4(\mathbb{C})$.
Thanks for all the reactions.
Meanwhile I was trying a different strategy, namely finding any orientation-reversing $\mathbb{Z}_2$-action under which the quaternionic Hopf fibration would be equivariant (a necessary condition for a full $Pin(5)$-equivariance).
A trap to beware of here is that the complex Hopf fibration is equivariant with respect to complex conjugation, with the latter being orientation reversing on the codomain 2-sphere. This gives the generator $\widehat \eta \in \pi^{st}_{1,0}$ from Araki-Iriye 82, p. 24.
This fact might make one feel that the quaternionic Hopf fibration should also be equivariant under quaternionic conjugation, which acts orientation-reversing on the codomain 4-sphere and which would evade the quaternion-linearity assumption in Gluck-Warner-Ziller, Theorem 4.1. But it is not the case: The relevant formula that works for $\mathbb{C}$ relies on commutativity. Fixing the formula for the quaternions requires performing an extra reflection, which makes everything be oriented again.
Indeed, the only way the quaternionic Hopf fibration appears with non-trivial $\mathbb{Z}_2$-action in Araki-Iriye 82 is with orientation-preserving action (their Prop. 10.1).
This doesn’t prove that there is no orientation-reversing equivariance, unstably, but it makes me worry.
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