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, cf. Voelkel 2016, Sec. 1.3). The octonionic projective plane $\mathbb{O}P^2$ is also called the 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

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

between the octonionic projective plane and the attaching space obtained from the octonionic projective line along the octonionic Hopf fibration.

(Lackmann 2019, Lemma 3.4., Mimura 1967, page 166)

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.)

(Lackmann 19, page 7, Mimura 67, Theorem 7.2.)

Cohomology

Proposition

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

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

(Lackmann 2019, Corollary 4.1.)

Related concepts

• octonionic line bundle

References

• Mamoru Mimura: The Homotopy groups of Lie groups of low rank, J. Math. Kyoto Univ. 6 2 (1967) 131-176 [doi:10.1215/kjm/1250524375]

• 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]

• Tevian Dray, Corinne Manogue, §12.2 and §12.5 of: The Geometry of Octonions, World Scientific (2015) [doi:10.1142/8456, web]

• Konrad Voelkel: Motivic cell structures for projective spaces over split quaternions (2016) [freidok:11448, pdf]

More on the octonionic projective plane:

• Malte Lackmann: The octonionic projective plane, in: 2019-20 MATRIX Annals, MATRIX Book Series 4, Springer (2021) [doi:10.1007/978-3-030-62497-2_6, arXiv:1909.07047]

• Daniele Corradetti, Alessio Marrani, Francesco Zucconi: A minimal and non-alternative realisation of the Cayley plane [arXiv:2309.00967]