Given a complete analytic function $\mathcal{F}$ on a domain $G \subset \mathbb{C}$, define the germ of a function element $(f, D)$ of $\mathcal{F}$ at $z \in D$. Let $\mathcal{G}$ be the set of all germs of function elements in $G$. Describe without proofs the topology and complex structure on $\mathcal{G}$ and the natural covering map $\pi: \mathcal{G} \rightarrow G$. Prove that the evaluation map $\mathcal{E}: \mathcal{G} \rightarrow \mathbb{C}$ defined by

$$\mathcal{E}\left([f]_{z}\right)=f(z)$$

is analytic on each component of $\mathcal{G}$.

Suppose $f: R \rightarrow S$ is an analytic map of compact Riemann surfaces with $B \subset S$ the set of branch points. Show that $f: R \backslash f^{-1}(B) \rightarrow S \backslash B$ is a regular covering map.

Given $P \in S \backslash B$, explain how any closed curve in $S \backslash B$ with initial and final points $P$ yields a permutation of the set $f^{-1}(P)$. Show that the group $H$ obtained from all such closed curves is a transitive subgroup of the group of permutations of $f^{-1}(P)$.

Find the group $H$ for the analytic map $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ where $f(z)=z^{2}+z^{-2}$.