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octonionic Hopf fibration

Contents

Context

Bundles

Topology

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

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

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Under construction

Contents

Idea

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 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.

Properties

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)} }

Subfibrations

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)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 November 14, 2019 at 09:45:22. See the history of this page for a list of all contributions to it.