nLab octonionic Hopf fibration






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

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


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topological homotopy theory

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

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

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see also algebraic topology



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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 \coloneqq \{(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 12, 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- and Spin(9)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( 9)S^8 = S(\mathbb{R}^9) induced from the canonical action of Spin(9)Spin(9) on 9\mathbb{R}^9, and on S 15=S( 16)S^{15} = S(\mathbb{R}^{16}) induced from the canonical inclusion Spin(9)Spin(16)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):

S 7 fib(h 𝕆) S 15 h 𝕆 S 8 = = = Spin(8)Spin(7) Spin(9)Spin(7) Spin(9)Spin(8) \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)} }



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 December 11, 2020 at 01:52:24. See the history of this page for a list of all contributions to it.