Problem
Problem 5.17 (Hopf Bundles) Set
$$
\begin{aligned}
& S^{1}=\{x \in \mathbb{C}:|x|=1\}, \\
& S^{2}=\left\{(x, t) \in \mathbb{C} \times \mathbb{R}:|x|^{2}+t^{2}=1\right\}, \\
& S^{3}=\left\{(x, y) \in \mathbb{C}^{2}:|x|^{2}+|y|^{2}=1\right\}, \\
& S^{7}=\left\{(x, y) \in \mathbb{H}^{2}:|x|^{2}+|y|^{2}=1\right\} .
\end{aligned}
$$
The spheres $S^{1}$ and $S^{3}$ are Lie groups with respect to the multiplication induced from $\mathbb{C}$ and $\mathbb{H}$, respectively (see Problem 4.103). Let $S^{1}$ act on $S^{3}$ (resp., $S^{3}$ on $S^{7}$ ) by the formula
$$
(x, y) \cdot z=(x z, y z), \quad(x, y) \in S^{3}, z \in S^{1}\left(\operatorname{resp} .(x, y) \in S^{7}, z \in S^{3}\right)
$$
Let
$$
\pi_{\mathbb{C}}: S^{3} \rightarrow \mathbb{C} \times \mathbb{R}, \quad \pi_{\mathbb{H}}: S^{7} \rightarrow \mathbb{H} \times \mathbb{R}
$$
be the maps given by
$$
\begin{array}{ll}
\pi_{\mathbb{C}}(x, y)=\left(2 y \bar{x},|x|^{2}-|y|^{2}\right), & (x, y) \in S^{3}, \\
\pi_{\mathbb{H}}(x, y)=\left(2 y \bar{x},|x|^{2}-|y|^{2}\right), & (x, y) \in S^{7}
\end{array}
$$
Prove:
(i) $\pi_{\mathbb{C}}\left(S^{3}\right)=S^{2}$.
(ii) $\pi_{\mathbb{H}}\left(S^{7}\right)=S^{4}$
(iii) The induced map $\pi_{\mathbb{C}}: S^{3} \rightarrow S^{2}$ is a principal $S^{1}$-bundle with respect to the action of $S^{1}$ on $S^{3}$ defined above.
(iv) The induced map $\pi_{\mathbb{H}}: S^{7} \rightarrow S^{4}$ is a principal $S^{3}$-bundle with respect to the action of $S^{3}$ on $S^{7}$ defined above.
(v) $\mathbb{C P}^{1} \cong S^{2}$.
(vi) $\mathbb{H} \mathrm{P}^{1} \cong S^{4}$.