Spin actions and Polygon spaces

In this article, we construct correspondences between polygon spaces in Euclidean spaces of dimension 2,3,5,9\and the quotient spaces of $2$-Steifel manifolds along the normed division algebra$\mathbb{F}$ real $\mathbb{R}$, complex $\mathbb{C}$, quaternions $\mathbb{H}$, octonions $\mathbb{O}$. For the purpose, we introduce Hopf map on \mathbb{F}^{2}\and consider the spin action of $SU(2,\mathbb{F})$ to spinor \mathbb{F}^{2}\and the induced SO\action to the Euclidean space $\mathbb{R}\oplus \mathbb{F}$. The correspondences are extension of the work of Hausmann and Knutson for polygon spaces of dimension 2,3\and $2$-Grassmannians over real and complex.