octonionic Hopf fibration




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


Under construction



The octonionic Hopf fibration is the fibration

S 7 S 15 p 𝕆 S 8 \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 7S^7 with its Moufang loop structure.

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

l m:={(x,mx)|x𝕆}l_m := \{(x, m x)|x \in \mathbb{O}\} and l :={(0,y)|y𝕆}l_{\infty} := \{(0, y)|y \in \mathbb{O}\}.

In this way the fibration 𝕆 2(0,0)S 8={m𝕆}{}\mathbb{O}^2 \setminus (0, 0) \to S^8 = \{m \in \mathbb{O}\} \union \{\infty\} is obtained, with fibers 𝕆0\mathbb{O} \setminus 0, and the intersection with the unit sphere S 15𝕆 2S^{15} \subset \mathbb{O}^2 provides the octonionic Hopf fibration (see OPPV, p. 7).

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


G 2G_2 action

The automorphism group G2 of the octonions, as a normed algebra, manifestly acts on the octonionic Hopf fibration.

(see also Cook-Crabb 93)

Spin(9) action

The subgroup Spin(9)SO(16)Spin(9) \subset SO(16) acts transitively on the octonionic Hopf fibration, which can be consider as a homogeneous fibration

Spin(9)Spin(7)Spin(8)Spin(7)Spin(9)Spin(8). \frac{Spin(9)}{Spin(7)} \stackrel{\frac{Spin(8)}{Spin(7)}}{\to} \frac{Spin(9)}{Spin(8)}.


  • Liviu Ornea, Maurizio Parton, Paolo Piccinni, Victor Vuletescu, Spin(9) geometry of the octonionic Hopf fibration, (arXiv:1208.0899)

  • Reiko Miyaoka, The linear isotropy group of G 2/SO(4)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)

Last revised on May 12, 2017 at 05:49:03. See the history of this page for a list of all contributions to it.