If $X$ is a Riemann surface and $p: Y \rightarrow X$ is a covering map of topological spaces, show that there is a conformal structure on $Y$ such that $p: Y \rightarrow X$ is analytic.

Let $f(z)$ be the complex polynomial $z^{5}-1$. Consider the subspace $R$ of $\mathbb{C}^{2}=\mathbb{C} \times \mathbb{C}$ given by the equation $w^{2}=f(z)$, where $(z, w)$ denotes coordinates in $\mathbb{C}^{2}$, and let $\pi: R \rightarrow \mathbb{C}$ be the restriction of the projection map onto the first factor. Show that $R$ has the structure of a Riemann surface which makes $\pi$ an analytic map. If $\tau$ denotes projection onto the second factor, show that $\tau$ is also analytic. [You may assume that $R$ is connected.]

Find the ramification points and the branch points of both $\pi$ and $\tau$. Compute also the ramification indices at the ramification points.

Assuming that it is possible to add a point $P$ to $R$ so that $X=R \cup\{P\}$ is a compact Riemann surface and $\tau$ extends to a holomorphic map $\tau: X \rightarrow \mathbb{C}_{\infty}$ such that $\tau^{-1}(\infty)=\{P\}$, compute the genus of $X .$