# embedding of quaternionic projective spaces

Let $\mathbb{H}P^m$ be the $m$-th quaternionic projective space. What is the smallest integer $N$ such that there exists an embedding $$\mathbb{H}P^2\longrightarrow \mathbb{R}^N?$$ Are there any references?

• If I'm not mistaken, $12 \leq N \leq 16$, the lower bound coming from a standard calculation involving Stiefel-Whitney classes, and the upper bound coming from the Whitney embedding theorem. – Michael Albanese Jan 7 '16 at 5:26
• S. Feder and D. M. Segal Proceedings of the American Mathematical Society Vol. 35, No. 2 (Oct., 1972), pp. 590-592 shows that $N>12$. – user83633 Jan 7 '16 at 9:19

I. M. James, Lectures on algebraic and differential topology, pp. 134–174, Lecture Notes in Math., Vol. 279, Springer, Berlin, 1972, Theorems 1.2 and 1.3 show that $$N=13.$$
• The embedding can be geometrically described as follows. Let $V$ be the real vector space of $3\times3$ Hermitian quaternionic matrices with fixed (real) trace, say $1$. Note that $\dim V=14$. $\mathbb HP^2$ embeds into $V$ by mapping each quaternionic line in $\mathbb H^3$ to the matrix representing the corresponding orthogonal projection onto it. The image of the embedding in $V$ consists of the idempotent matrices, and it is also an orbit of the action of the group $Sp(3)$ on $V$ by conjugation. The image sits in the unit sphere of $V$, so it can be stereographically projected to $R^{13}$. – Claudio Gorodski Jan 13 '16 at 14:07