Exercise 9, § 6

The complex projective line $\mathbf{P}^1(\mathbf{C})$ is identified with the set of lines of $\mathbf{C}^2$. Let E be the subspace of $\mathbf{C}^2\times \mathbf{P}^1(\mathbf{C})$ consisting of pairs $((u, v), x)$ such that $v\in x$ and $|u|^2+|v|^2= 1$. We equip E with the action of the group $\mathbf{S}_1$ given by $((u, v), x)\cdot z= ((zu, zv), x)$.

a) Exercise prove that the second projection pr$_2: E\rightarrow \mathbf{P}^1(\mathbf{C})$ endows E with a structure of principal fibre space with base $\mathbf{P}^1(\mathbf{C})$ and group $\mathbf{S}_1$.

b) Let $\varphi$ be a homeomorphism of $\mathbf{S}_2$ onto $\mathbf{P}^1(\mathbf{C})$; exercise prove that the group $\pi_1(\mathbf{S}_1, e)$ is generated by the class $c(\varphi^*E)$.