Contents

# Contents

## 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.