nLab octonionic projective space

Contents

Contents

Idea

The notion of projective space 𝕆P n\mathbb{O}P^n over the octonions 𝕆\mathbb{O} makes sense for n{0,1,2}n \,\in\, \{ 0,1,2 \} (but not beyond, see e.g. Voelkel 16, Sec. 1.3). The octonionic projective plane 𝕆P 2\mathbb{O}P^2 is also called the Cayley projective plane.

Properties

Octonionic projective line and the 8-Sphere

Proposition

We have a homeomorphism

𝕆P 1S 8 \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

𝕆P 2D 16h 𝕆𝕆P 1 \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 19, Lemma 3.4.)

(Mimura 67, page 166)

See also at cell structure of projective spaces.

Homotopy groups

Proposition

The homotopy groups of octonionic projective plane are

(1)π k(𝕆P 2)={1 | k7 π k(S 8) | 8k14 \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

π 15(𝕆P 2) 120\pi_{15}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_{120}
π 16(𝕆P 2) 2 3\pi_{16}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_2^3
π 17(𝕆P 2) 2 4\pi_{17}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_2^4
π 18(𝕆P 2) 24× 2\pi_{18}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_{24}\times\mathbb{Z}_2
π 19(𝕆P 2) 504× 2\pi_{19}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_{504}\times\mathbb{Z}_2
π 20(𝕆P 2)1\pi_{20}\big(\mathbb{O}P^2\big) \cong 1
π 21(𝕆P 2) 6\pi_{21}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_6
π 22(𝕆P 2) 4\pi_{22}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_4
π 23(𝕆P 2)× 120× 2 2.\pi_{23}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}\times\mathbb{Z}_{120}\times\mathbb{Z}_2^2\,.

(While π 15(S 8)× 120\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 AA \in Ab any abelian group, then the ordinary cohomology groups of octionionic projective plane 𝕆P 2\mathbb{O}P^2 with coefficients in AA are

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

(Lackmann 19, Corollary 4.1.)

References

Last revised on February 3, 2024 at 02:36:20. See the history of this page for a list of all contributions to it.