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The quaternionic Hopf fibration is the Hopf fibration induced by the quaternions, hence it is the fibration
of the 7-sphere over the 4-sphere with fiber the 3-sphere, which is induced via the Hopf construction from the product operation
on the quaternions, or else from
to match standard conventions.
This means that if $S^7$ is regarded as the unit sphere $\{(x,y) | {\vert x\vert}^2 + {\vert y\vert}^2 = 1\}$ in $\mathbb{H}\times \mathbb{H}$ and $S^4$ is regarded as the quaternionic projective space, then $p$ is given (on points $(x,y)$ with $y \neq 0$) simply by
Since the automorphism group of the quaternions, as an $\mathbb{R}$-algebra, is the special orthogonal group $SO(3)$
acting by rotation of the imaginary quaternions, via the Hopf construction it follows that the 7-sphere and 4-sphere inherit $SO(3)$-actions under which the quaternionic Hopf map is equivariant.
Notice that this means that $SO(3)$ acts on $S^7$ here diagonally on the two copies of the imaginary octonions in $S^7 \hookrightarrow \mathbb{H} \oplus \mathbb{H}$ (as opposed to, say, via any one of the embeddings $SO(3) \hookrightarrow SO(8)$ and the following canonical action of $SO(8)$ on $S^7 \hookrightarrow \mathbb{R}^8$).
(see also Cook-Crabb 93)
But in fact more is true:
(Spin(5)-equivariance of quaternionic Hopf fibration)
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 $Spin(5) \simeq Sp(2) = U(2,\mathbb{H})$ (this Prop. ) by the canonical left action of $U(2,\mathbb{H})$ on $\mathbb{H}^2$ via $S^7 \simeq S(\mathbb{H}^2)$.
Then the quaternionic Hopf fibration $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$ is equivariant with respect to these actions.
This appears as (Gluck, Warner & Ziller (1986), Prop. 4.1).
The statement is also almost explicit in Porteous 95, p. 263
A way to make the $Spin(5)$-equivariance of the quaternionic Hopf fibration fully explicit is to observe that the quaternionic Hopf fibration is equivalently the following map of coset spaces:
Of the resulting action of Sp(2)$\times$Sp(1) on the 7-sphere (from Prop. ), only the quotient group Sp(2).Sp(1) acts effectively.
The quaternionic Hopf fibration gives an element in the 7th homotopy group of the 4-sphere
and in fact it is a generator of the non-torsion factor in this group.
Stably, i.e. as a generator for the stable homotopy groups of spheres in degree $7-4 = 3$, the quaternionic Hopf map becomes a torsion generator:
The third stable homotopy group of spheres (the third stable stem) is the cyclic group of order 24:
where the generator $[1] \in \mathbb{Z}/24$ is represented by the quaternionic Hopf fibration $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$.
Under the Pontrjagin-Thom isomorphism, identifying the stable homotopy groups of spheres with the bordism ring $\Omega^{fr}_\bullet$ of stably framed manifolds (see at MFr), this generator is represented by the 3-sphere (with its left-invariant framing induced from the identification with the Lie group SU(2) $\simeq$ Sp(1) )
Moreover, the relation $2 4 [S^3] \,\simeq\, 0$ is represented by the bordism which is the complement of 24 open balls inside the K3-manifold (Wang-Xu 10, Sec. 2.6).
Fix a finite subgroup $G \hookrightarrow SO(3)$ which does not come from $SO(2) \hookrightarrow SO(3)$ – i.e. not a cyclic group, but one of the dihedral groups or else the tetrahedral group or octahedral group or icosahedral group (by the ADE classification).
Regard both $S^7$ and $S^4$ as pointed topological G-spaces via the $SO(3)$-action induced via automorphisms of the quaternions, as above. Write
for the corresponding equivariant suspension spectra.
Notice that if we took trivial $G$, then in the stable homotopy category
by the above. In contrast:^{1}
In $G$-equivariant homotopy theory this becomes a non-torsion group, i.e.
with the quaternionic Hopf fibration, regarded as a $G$-equivariant map, representing a non-torsion element.
First use the Greenlees-May decomposition which says that for any two $G$-equivariant spectra $X,Y$ and writing $\pi_\bullet(X), \pi_\bullet(Y)$ for their equivariant homotopy groups, organized as Mackey functors $H \mapsto \pi_n^H(X)$ for all subgroups $H \subset G$, then the canonical map
is rationally an isomorphism.
With this we are reduced to showing that there exists $n \in \mathbb{Z}$ and a morphism of Mackey functors of equivariant homotopy groups $\pi_n(\Sigma^\infty_G S^7) \to \pi_n(\Sigma^\infty_G S^4)$ which is not a torsion element in the abelian hom-group of Mackey functors.
To analyse this, we use the tom Dieck splitting which says that the equivariant homotopy groups of equivariant suspension spectra $\Sigma^\infty_G X$ contain a direct summand which is simply the ordinary stable homotopy groups of the naive fixed point space $X^H$:
Now observe that the fixed points of the $SO(3)$-action on the quaternionic Hopf fibration that we are considering is just the real Hopf fibration:
since $SO(3)$ acts transitively on the quaternionic quaternions and fixes the real quaternions. By our assumption that $G \subset SO(3)$ does not come through $SO(2) \hookrightarrow SO(3)$ it follows that this statment is still true for $G$:
But the real Hopf fibration defines a non-torsion element in $\pi_0^S \simeq \mathbb{Z}$.
In conclusion then, at $n = 1$ and $H = G$ we find that the $G$-equivariant quaternionic Hopf fibration contributes a non-torsion element in
which appears as a non-torsion element in
and hence in $[\Sigma^\infty_G S^7, \Sigma^\infty_G S^4]_G$.
See also at equivariant stable cohomotopy
The original article:
Further discussion:
Herman Gluck, Frank Warner, Wolfgang Ziller, The geometry of the Hopf fibrations, L’Enseignement Mathématique, 32 (1986), 173-198 [ResearchGate, pdf]
Reiko Miyaoka, The linear isotropy group of $G_2/SO(4)$, the Hopf fibering and isoparametric hypersurfaces, Osaka J. Math. 30 2 (1993) 179-202 [Euclid]
Ian Porteous, Clifford Algebras and the Classical Groups, Cambridge Studies in Advanced Mathematics, Cambridge University Press (1995) (doi:10.1017/CBO9780511470912)
Machiko Hatsuda, Shinya Tomizawa, Coset for Hopf fibration and Squashing, Class. Quant. Grav. 26 225007 (2009) [arXiv:0906.1025, doi:10.1088/0264-9381/26/22/225007]
Rustam Sadykov, §8.3 in: Elements of Surgery Theory, 2013 (pdf, pdf)
Tyrone Cutler, p. 23 of: Fibrations IV (2020) [pdf, pdf]
Discussion in parameterized homotopy theory includes
A. L. Cook, M. C. Crabb, Fiberwise Hopf structures on sphere bundles, J. London Math. Soc. (2) 48 (1993) 365-384 (pdf)
Kouyemon Iriye, Equivariant Hopf structures on a sphere, J. Math. Kyoto Univ. Volume 35, Number 3 (1995), 403-412 (Euclid)
Discussion in homotopy type theory is in
Noteworthy fiber products with the quaternionic Hopf fibration, notably exotic 7-spheres, are discussed in
The proof of prop. profited from Charles Rezk, who suggested here that the reduction to fixed points will make the real Hopf fibration give a non-torsion contribution, and from David Barnes who amplified the use of the Greenless-May splitting theorem. ↩
Last revised on June 23, 2023 at 12:18:30. See the history of this page for a list of all contributions to it.