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). Octonionic projective plane is also called Cayley projective plane.

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

Last revised on March 28, 2021 at 13:07:47. See the history of this page for a list of all contributions to it.