nLab octonionic projective space

Idea

The notion of projective space $\mathbb{O}P^n$ over the octonions $\mathbb{O}$ makes sense for $n \,\in\, \{ 0,1,2 \}$ (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

$$\mathbb{O}P^1 \,\simeq\, S^8$$

between the octonionic projective line and the 8-sphere.

Cell structure on octonionic projective plane

Proposition

There is a homeomorphism

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

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

• Malte Lackmann, The octonionic projective plane (arXiv:1909.07047)
• Konrad Voelkel, Motivic cell structures for projective spaces over split quaternions, 2016 (freidok:11448, pdf)
• Rowena Held, Iva Stavrov, Brian VanKoten, (Semi-)Riemannian geometry of (para-)octonionic projective planes, Diff. Geom. & its Appl. 27:4 (2009) 464-481 doi:/10.1016/j.difgeo.2009.01.007