nLab
octonionic projective space

Contents

Contents

Idea

The notion of projective space 𝕆P n\mathbb{O}P^n over the octonions 𝕆\mathbb{O} makes sense for n{0,1,2}n \,\in\, \{ 0,1,2 \} (but not beyond, see e.g. Voelkel 16, Sec. 1.3).

Properties

Octonionic projective line and the 8-Sphere

Proposition

We have a homeomorphism

𝕆P 1S 8 \mathbb{O}P^1 \,\simeq\, S^8

between the octonionic projective line and the 8-sphere.

Cell structure on octonionic projective plane

Proposition

There is a homeomorphism

𝕆P 2S 15h 𝕆𝕆P 1 \mathbb{O}P^2 \,\simeq\, S^{15} \underset{h_{\mathbb{O}}}{\cup} \mathbb{O}P^1

between the octonionic projective plane and the attaching space obtained from the octonionic projective line along the octonionic Hopf fibration.

(Lackman 19, Lemma 3.4)

See also at cell structure of projective spaces.

References

See also

Last revised on December 17, 2020 at 04:29:17. See the history of this page for a list of all contributions to it.