# nLab quaternionic Hopf fibration

Contents

### Context

#### Bundles

bundles

fiber bundles in physics

# Contents

## Idea

The quaternionic Hopf fibration is the Hopf fibration induced by the quaternions, hence it is the fibration

$\array{ S^3 &\hookrightarrow& S^7 \\ && \downarrow^{\mathrlap{p_{\mathbb{H}}}} \\ && S^4 }$

of the 7-sphere over the 4-sphere with fiber the 3-sphere, which is induced via the Hopf construction from the product operation

$\mathbb{H} \times \mathbb{H} \stackrel{(-)\cdot (-)}{\longrightarrow} \mathbb{H}$

on the quaternions, or else from

$\mathbb{H} \times \mathbb{H}^{\times} \stackrel{(-)\cdot (-)^{-1}}{\longrightarrow} \mathbb{H}$

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

$p_{\mathbb{H}} \colon (x,y) \mapsto [x;y] = [x/y; 1] \,,$

## Properties

### $SO(3)$- and $Spin(5)$-Equivariant structure

Since the automorphism group of the quaternions, as an $\mathbb{R}$-algebra, is the special orthogonal group $SO(3)$

$\mathrm{Aut}_{\mathbb{R}}(\mathbb{H}) \simeq SO(3)$

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 $S^7 \hookrightarrow \mathbb{R}^8$).

But in fact more is true:

###### Proposition

(Spin(5)-equivariance of quaternionic Hopf fibration)

Consider

1. the Spin(5)-action on the 4-sphere $S^4$ which is induced by the defining action on $\mathbb{R}^5$ under the identification $S^4 \simeq S(\mathbb{R}^5)$;

2. 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 86, 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:

$\array{ S^3 &\overset{fib(h_{\mathbb{H}})}{\longrightarrow}& S^{7} &\overset{h_{\mathbb{H}}}{\longrightarrow}& S^4 \\ = && = && = \\ \frac{Spin(4)}{Spin(3)} &\longrightarrow& \frac{Spin(5)}{Spin(3)} &\longrightarrow& \frac{Spin(5)}{Spin(4)} }$
###### Remark

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.

### Class in the homotopy groups of spheres

The quaternionic Hopf fibration gives an element in the 7th homotopy group of the 4-sphere

$[p_{\mathbb{H}}] \in \pi_7(S^4) \simeq \mathbb{Z} \times (\mathbb{Z}/12)$

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:

$[p_{\mathbb{H}}] \in \pi_3^S \simeq \mathbb{Z}/24 \,,.$

The third stable homotopy group of spheres (the third stable stem) is the cyclic group of order 24:

$\array{ \pi_3^s &\simeq& \mathbb{Z}/24 \\ [h_{\mathbb{H}}] &\leftrightarrow&  }$

where the generator $ \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) )

$\array{ \pi_3^s & \simeq & \Omega_3^{fr} \\ [h_{\mathbb{H}}] & \leftrightarrow & [S^3] \,. }$

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).

### Class in equivariant stable homotopy theory

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

$\Sigma^\infty_G S^7, \Sigma^\infty_G S^4 \in G Spectra$

for the corresponding equivariant suspension spectra.

Notice that if we took trivial $G$, then in the stable homotopy category

$[\Sigma^\infty S^7, \Sigma^\infty S^4] \simeq \mathbb{Z}/24$

by the above. In contrast:1

###### Proposition

In $G$-equivariant homotopy theory this becomes a non-torsion group, i.e.

$[\Sigma^\infty_G S^7, \Sigma^\infty_G S^4]_G \simeq \mathbb{Z} \oplus \cdots$

with the quaternionic Hopf fibration, regarded as a $G$-equivariant map, representing a non-torsion element.

###### Proof

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

$[X,Y]_G \longrightarrow \underset{n}{\oplus} Hom_{\mathcal{M}[G]}(\pi_n(X), \pi_n(Y))$

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 $X^H$:

$\pi_n^H(\Sigma^\infty_G X) \simeq \pi_n(\Sigma^\infty (X^H)) \oplus \cdots \,.$

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:

$(p_{\mathbb{H}})^{SO(3)} = p_{\mathbb{R}} \;\colon\; S^1 \longrightarrow S^1$

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 $G$:

$(p_{\mathbb{H}})^{G} = p_{\mathbb{R}} \;\colon\; S^1 \longrightarrow S^1 \,.$

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

$Hom_{Ab}(\pi_1^G(\Sigma^\infty_G S^7), \pi_1^G(\Sigma^\infty_G S^4))$

which appears as a non-torsion element in

$Hom_{\mathcal{M}[G]}( \pi_1(\Sigma^\infty_G S^7), \pi_1(\Sigma^\infty_G S^4) )$

and hence in $[\Sigma^\infty_G S^7, \Sigma^\infty_G S^4]_G$.

• Herman Gluck, Frank Warner, Wolfgang Ziller, The geometry of the Hopf fibrations, L’Enseignement Mathématique, t.32 (1986), p. 173-198 (ResearchGate)

• Reiko Miyaoka, The linear isotropy group of

$G_2/SO(4)$, the Hopf fibering and isoparametric hypersurfaces_, Osaka J. Math. Volume 30, Number 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)

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

• Llohann D. Sperança, Explicit Constructions over the Exotic 8-sphere (pdf, pdf)

1. 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.