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.

We have a homeomorphism

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

between the octonionic projective line and the 8-sphere.

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.

See also at *cell structure of projective spaces*.

- 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

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