nLab
15-sphere

Contents

Context

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

Manifolds and cobordisms

Contents

Idea

The sphere of dimension 15.

Properties

Octonionic Hopf fibration

The 15-sphere participates in the octonionic Hopf fibration, the analog of the complex Hopf fibration with the field of complex numbers replaced by the division ring of octonions 𝕆\mathbb{O}.

S 7 ↪ S 15 ↓ p S 8 \array{ S^7 &\hookrightarrow& S^15 \\ && \downarrow^\mathrlap{p} \\ && S^8 }

Here the idea is that S 15S^{15} can be construed as {(x,y)∈𝕆 2:|x| 2+|y| 2=1}\{(x, y) \in \mathbb{O}^2: {|x|}^2 + {|y|}^2 = 1\}, with pp mapping (x,y)(x, y) to x/yx/y as an element in the projective line ℙ 1(𝕆)≅S 8\mathbb{P}^1(\mathbb{O}) \cong S^8, with each fiber a torsor parametrized by octonionic scalars λ\lambda of unit norm (so λ∈S 7\lambda \in S^7).

Other properties

  • S 15S^{15} is the only sphere that admits three homogeneous Einstein metrics.
  • It is the only sphere that appears as a regular orbit in three cohomogeneity one actions on projective spaces, namely of SU(8)SU(8), Sp(4)Sp(4) and Spin(9)Spin(9) on ℂP 8\mathbb{C}P^8, ℍP 4\mathbb{H}P^4 and 𝕆P 2\mathbb{O}P^2, respectively (OPPV, p. 1)

References

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

Created on December 1, 2015 at 11:03:19. See the history of this page for a list of all contributions to it.