Exercise 6, § 4

We take up again the notations of Exercise 5.

a) Prove that there exists for every integer $n\geqslant 1$ a covering $(E_n, p_n)$ of P, of degree $n$, such that $p^{-1}_n(C_n)$ is connected.

b) Let E be the disjoint union of the $E_n$ and let $f: E\rightarrow C\times \mathbf{N}$ be the canonical mapping. Prove that $(E, f)$ is a covering of $P\times \mathbf{N}$, that the mapping pr$_1$ makes $P\times \mathbf{N}$ a covering of P, but that $(E$, pr$_1\circ f)$ is not a covering of P.

c) Let $P^+$ and $P^-$ be the sets of the points $(x, y)$ of P satisfying $y\geqslant 0$ and $y\leqslant 0$ respectively. Now prove that $E|P^+$ is a trivial covering of $P^+$ and that $E|P^-$ is a trivial covering of $P^-$.