nLab 15-sphere

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 $\mathbb{O}$.

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

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

Other properties

• $S^{15}$ 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 $SU(8)$, $Sp(4)$ and $Spin(9)$ on $\mathbb{C}P^8$, $\mathbb{H}P^4$ and $\mathbb{O}P^2$, respectively (OPPV, p. 1)

References

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