# Real, Complex, Quaternionic and Octonionic Projective spaces

Do Octonionic Projective spaces exist or defined similar to $$\Bbb RP^n$$, $$\Bbb CP^n$$, $$\Bbb HP^n$$? If so, are they symmetric spaces?

I am asking this question because I've never seen Octonionic Projective spaces as examples of the Compact Rank One Symmetric Spaces or the manifolds of positive sectional curvature except its 2-dimensional case i.e. Cayley plane $$\Bbb OP^2$$ (or $$\Bbb{Ca}P^2$$). What can be say about $$\Bbb OP^n$$ (or $$\Bbb{Ca}P^n$$)?

Update: Please give a proof or a reference for non-existence of Octonionic Projective spaces for $$n\geq 3$$.

• en.wikipedia.org/wiki/Cayley_plane I don't think there are octonionic projective spaces of higher dimension. – Angina Seng Dec 7 '19 at 7:51
• So, Why they aren't exist? – C.F.G Dec 7 '19 at 8:14
• This has nothing to do with symmetric groups. Also quaternion and octonion tags seem more relevant. – runway44 Dec 8 '19 at 21:54

Paul is right (and I upvoted him!) that the usual definition of projective spaces requires multiplication to be associative.

However, this seems to be counter to the fact that $$\mathbb{O}P^1$$ and $$\mathbb{O}P^2$$ exist. I have heard the explanation that "the octionions are associative enough" to support the existence of these two spaces, but honestly, I've never understood it. Instead, I understand projective spaces more geometrically and topologically.

Geometrically: Each projective space $$\mathbb{K}P^n$$ with $$\mathbb{K}\in\{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$$ has a canonical Riemannian metric, called the Fubini-Study metric.

With regards to this metric, the manifolds are homogeneous (better, they are symmetric spaces), with sectional curvatures lying between $$1$$ and $$4$$. In addition, for any $$p\in \mathbb{K}P^n$$, the cut locus is a copy of $$\mathbb{K}P^{n-1}$$ and the preimage of a point in $$\mathbb{K}P^{n-1}$$ in the unit sphere in $$T_p \mathbb{K}P^n$$ gives the expected Hopf fibration.

One can easily generalize this to what $$\mathbb{O}P^n$$ should be. It should be a symmetric space with curvature between $$1$$ and $$4$$, and with the cut locus working as expected. Of course, when $$n=1$$, $$\mathbb{O}P^1\cong S^8$$ fits kind of trivially. When $$n=2$$, there is such an example: $$F_4/Spin(9)$$ with normal homogeneous metric meets all the criteria.

However, we've classified symmetric spaces, and we've classified spaces with curvature between $$1$$ and $$4$$. From either of these classifications, there is no $$\mathbb{O}P^n$$ for $$n > 2$$.

Topologically: The cohomology ring of $$\mathbb{K}P^n$$ for $$\mathbb{K}\in \{\mathbb{C},\mathbb{H}\}$$ is a truncated polynomial algebra: $$H^\ast(\mathbb{K}P^n;\mathbb{Z})\cong \mathbb{Z}[\alpha]/\alpha^{n+1}$$ where $$|\alpha| = \dim_{\mathbb{R}} \mathbb{K}$$. One could reasonably define a "topological projective space" to be a manifold having a truncated polynomial ring as it's cohomology ring. This is not done, because there are essentially no other examples.

In Hatcher's Algebraic topology book, Theorem 4L.9 gives that $$|\alpha|$$ must be a power of $$2$$. Then, Corollary 4L.10 proves that there is no $$\mathbb{O}P^n$$ for $$n\geq 3$$.

Further, the ring $$\mathbb{Z}[\alpha]/\alpha^3$$ for $$|\alpha| = 2^k$$, $$k\geq 4$$ cannot arise by Adam's solution to the Hopf invariant one problem.

• Very nice, much better than my answer (+1). – Paul Frost Dec 9 '19 at 17:02
• @Paul: I appreciate the nice comment, but I would say it complements your answer, rather than saying it's better. – Jason DeVito Dec 9 '19 at 17:37
• @JasonDeVito: Very nice explanation as always. Honestly, I was waiting for your interesting answer!!! – C.F.G Dec 10 '19 at 4:44

The projective spaces $$\mathbb K P^n$$ are quotient spaces of $$\mathbb K^{n+1} \setminus \{0\}$$ under the equivalence relation $$v \sim \lambda v$$ for $$\lambda \in \mathbb K \setminus \{0\}$$. Now see Hatcher's Algebraic Topology p. 222:

Associativity of quaternion multiplication is needed for the identification $$v ∼ λv$$ to be an equivalence relation, so the definition does not extend to octonionic projective spaces, though there is an octonionic projective plane $$\mathbb OP^2$$ defined in Example 4.47.

Perhaps also the following is useful:

Dray, Tevian, and Corinne A. Manogue. The geometry of the octonions. World Scientific, 2015.

Also have a look at this, especially at Chapter 1.3 "Why Octonions are Bad".