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 $SU\left( 2,\mathbb{F}\right)$ to spinor $\mathbb{F}^{2}\$and the induced $SO\$action to the Euclidean space $\mathbb{R\oplus F}$. 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.