Exercise 2, § 3

We denote by $\mathbf{H}$ the quaternion algebra of Hamilton (TG, VIII, p. 4) and by $(1, i, j, k)$ its canonical basis. We denote by N the quadratic form norm of $\mathbf{H}$; the set of quaternions of norm 1 is the sphere $\mathbf{S}_3$. Let $\mathbf{H}_0$ be the vector subspace over $\mathbf{R}$ of quaternions of trace zero and $N_0$ the restriction to $\mathbf{H}_0$ of N.

a) Prove that the homomorphism of prop. 4 of TG, VIII, p. 4, makes the sphere $\mathbf{S}_3$ a covering of degree 2 of $\mathbf{S}\mathbf{O}(N_0)$.

b) Deduce that the Poincaré group of SO(3$,\mathbf{R}$) is isomorphic to $\mathbf{Z}/2\mathbf{Z}$. Give explicitly a loop at $I_3$ in SO(3$,\mathbf{R}$) whose strict homotopy class is the unique nontrivial element of this Poincaré group.

c) Let $\varphi$ be the mapping from $\mathbf{S}_3\times \mathbf{S}_3$ into SO(4$,\mathbf{R}$) given by $\varphi (\mathbf{q}_1,\mathbf{q}_2)(\mathbf{x}) =$ $\mathbf{q}_1\mathbf{x}\mathbf{q}^{-1}_2$. Prove that $\varphi$ makes $\mathbf{S}_3\times \mathbf{S}_3$ a covering of degree 2.

d) Deduce that the Poincaré group of SO(4$,\mathbf{R}$) is isomorphic to $\mathbf{Z}/2\mathbf{Z}$.