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?
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2$\begingroup$ If I'm not mistaken, $12 \leq N \leq 16$, the lower bound coming from a standard calculation involving StiefelWhitney classes, and the upper bound coming from the Whitney embedding theorem. $\endgroup$ – Michael Albanese Jan 7 '16 at 5:26

1$\begingroup$ S. Feder and D. M. Segal Proceedings of the American Mathematical Society Vol. 35, No. 2 (Oct., 1972), pp. 590592 shows that $N>12$. $\endgroup$ – user83633 Jan 7 '16 at 9:19
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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 $$N=13.$$

6$\begingroup$ 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 $R^{13}$. $\endgroup$ – Claudio Gorodski Jan 13 '16 at 14:07