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 canβ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 βpoints at infinityβ 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 βHermitianβ and idempotent, hence can be imagined as βprojections onto dimension-1 subspaces of $\mathbb{O}^3$β.

• 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

$\mathbb{O}P^2 \,\simeq\, D^{16} \underset{h_{\mathbb{O}}}{\cup} \mathbb{O}P^1$

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.

### Homotopy groups

###### Proposition

The homotopy groups of octonionic projective plane are

(1)$\pi_k \big( \mathbb{O}P^2 \big) \;=\; \left\{ \array{ 1 &\vert& k \leq 7 \\ \pi_k \big( S^8 \big) &\vert& 8 \leq k \leq 14 } \right.$

Further homotopy groups are

$\pi_{15}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_{120}$
$\pi_{16}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_2^3$
$\pi_{17}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_2^4$
$\pi_{18}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_{24}\times\mathbb{Z}_2$
$\pi_{19}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_{504}\times\mathbb{Z}_2$
$\pi_{20}\big(\mathbb{O}P^2\big) \cong 1$
$\pi_{21}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_6$
$\pi_{22}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_4$
$\pi_{23}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}\times\mathbb{Z}_{120}\times\mathbb{Z}_2^2\,.$

(While $\pi_{15}\big(S^8\big) \cong\mathbb{Z}\times\mathbb{Z}_{120}$, which includes the homotopy class of the octonionic Hopf fibration.)

### Cohomology

###### Proposition

For $A \in$ Ab any abelian group, then the ordinary cohomology groups of octionionic projective plane $\mathbb{O}P^2$ with coefficients in $A$ are

$H^k(\mathbb{O}P^2,A) \simeq \left\{ \array{ A & for \; k=0,8,16 \\ 0 & otherwise } \right. \,.$

### 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):

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

## References

### General

• Wikipedia, Cayley plane

• Salzmann et. al., Compact Projective Planes, with an introduction to Octonion Geometry

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

### In string theory

Discussion of the Witten genus of Cayley plane-fiber bundles is in

Indications that M-theory in 10+1 dimensions may be understood as the KK-compactification on Cayley-plane fibers of some kind of bosonic M-theory in 26+1 dimensions:

Last revised on February 3, 2024 at 02:34:31. See the history of this page for a list of all contributions to it.