Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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 didnâ€™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.
1 to 3 of 3