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octonionic Hopf fibration

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

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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, 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 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)}.

References

  • 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.