nLab octonionic Hopf fibration

Under construction

Idea

The octonionic Hopf fibration is the fibration

$$\array{ S^7 &\hookrightarrow& S^15 \\ && \downarrow^{\mathrlap{p_{\mathbb{O}}}} \\ && S^8 }$$

of the 15-sphere over the 8-sphere with fiber the 7-sphere. This may be derived by the Hopf construction on the 7-sphere $S^7$ with its Moufang loop structure.

Alternatively, we may construct a fibration by first decomposing $\mathbb{O}^2$ into the octonionic lines,

$l_m \coloneqq \{(x, m x)|x \in \mathbb{O}\}$ and $l_{\infty} := \{(0, y)|y \in \mathbb{O}\}$.

In this way the fibration $\mathbb{O}^2 \setminus (0, 0) \to S^8 = \{m \in \mathbb{O}\} \union \{\infty\}$ is obtained, with fibers $\mathbb{O} \setminus 0$, and the intersection with the unit sphere $S^{15} \subset \mathbb{O}^2$ provides the octonionic Hopf fibration (see OPPV 12, p. 7).

This second construction yields the standard parameterization of the octonionic Hopf fibration via $(x,y) \mapsto x y^{-1}$ (in one chart) and $(x,y ) \mapsto x^{-1} y$ (in the other), while the Hopf construction gives $(x,y) \mapsto x y$. The latter yields the generator $-1$ of $\pi_{15}(S^8) \cong \mathbb{Z}$, while the former yields $+1$.

Properties

$G_2$- and $Spin(9)$-equivariance

The automorphism group G₂ of the octonions, as a normed algebra, manifestly acts on the octonionic Hopf fibration, such that the latter is equivariant.

(see also Cook-Crabb 93)

But the octonionic Hopf fibration is equivariant even with respect to the Spin(9)-action, the one on $S^8 = S(\mathbb{R}^9)$ induced from the canonical action of $Spin(9)$ on $\mathbb{R}^9$, and on $S^{15} = S(\mathbb{R}^{16})$ induced from the canonical inclusion $Spin(9) \hookrightarrow Spin(16)$.

(Gluck-Warner-Ziller 86, Prop. 7.1)

This equivariance is made fully manifest by realizing the octonionic Hopf fibration as a map of coset spaces as follows (Ornea-Parton-Piccinni-Vuletescu 12, p. 7):

$$\array{ S^7 &\overset{fib(h_{\mathbb{O}})}{\longrightarrow}& S^{15} &\overset{h_{\mathbb{O}}}{\longrightarrow}& S^8 \\ = && = && = \\ \frac{Spin(8)}{Spin(7)} &\longrightarrow& \frac{Spin(9)}{Spin(7)} &\longrightarrow& \frac{Spin(9)}{Spin(8)} }$$

Subfibrations

• The complex and quaternionic Hopf fibrations are not subfibrations of the octonionic one (Parton & Piccinni 18, p. 4).

• The octonionic Hopf fibration does not admit any $S^1$ subfibration? (Parton & Piccinni 18, p. 13).

Related concepts

• real Hopf fibration, complex Hopf fibration, quaternionic Hopf fibration

• octonionic projective space

References

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

• Liviu Ornea, Maurizio Parton, Paolo Piccinni, Victor Vuletescu, Spin(9) geometry of the octonionic Hopf fibration, Transformation Groups (2013) 18: 845 (arXiv:1208.0899, doi:10.1007/s00031-013-9233-x)

• Maurizio Parton, Paolo Piccinni, The role of $Spin(9)$ in octonionic geometry, (arXiv:1810.06288).

• 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)

Discussion in parameterized homotopy theory is in

• 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)