Contents

# Contents

## Idea

The Cayley plane is the projective plane over the octonions, i.e. the octonionic projective space $\mathbb{O} P^2$.

## Definition

The Cayley plane can’t be constructed using homogeneous coordinates the way it works for projective planes over division rings, since multiplication of octonions is not associative. However, it can be constructed in several other ways that generalize the approach for division rings:

• One can start with $\mathbb{O}^2$ and give it the structure of an affine plane? in the obvious way, and then add “points at infinity” in the usual way to obtain a projective plane. This is the most straightforward approach, but as always it has the defect that it makes the line at infinity appear special.

• One can consider the space of $3\times 3$ matrices over $\mathbb{O}$ that are “Hermitian” and idempotent, hence can be imagined as “projections onto dimension-1 subspaces of $\mathbb{O}^3$”.

• Writing out the components of such a matrix explicitly, one obtains a Veronese vector $(x_1,x_2,x_3;\xi_1,\xi_2,\xi_3)$ where $x_i\in \mathbb{O}$ and $\xi_i\in \mathbb{R}$, such that $\xi_i \overline{x_i} = x_j x_k$ and $\Vert x_i\Vert^2 = \xi_j \xi_k$ for all cyclic permutations $(i,j,k)$ of $(1,2,3)$. The Cayley plane can be identified with the space of nonzero such vectors modulo the scalar action of $\mathbb{R}$.

## Properties

### General

The octonionic projective plane is a non-Desarguesian plane, that is, Desargues' theorem does not hold. See projective plane.

### Isometries

• F4 is the isometry group of $\mathbb{O} P^2$, with the stabilizer of a point being Spin(9). Hence $\mathbb{O} P^2 \cong F_4/Spin(9)$.

### Cell structure

###### 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.

### AKM-Theorem

###### Proposition

(octonionic AKM-theorem)

The 13-sphere is the quotient space of the (right-)octonionic projective plane by the left multiplication action by Sp(1):

$\mathbb{O}P^2 / \mathrm{Sp}(1) \simeq S^{13}$

### General

The AKM-theorem for $\mathbb{O}P^2$: