Let $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ be a rational function. What does it mean for $p \in \mathbb{C}_{\infty}$ to be a ramification point? What does it mean for $p \in \mathbb{C}_{\infty}$ to be a branch point?

Let $B$ be the set of branch points of $f$, and let $R$ be the set of ramification points. Show that

$$f: \mathbb{C}_{\infty} \backslash R \rightarrow \mathbb{C}_{\infty} \backslash B$$

is a regular covering map.

State the monodromy theorem. For $w \in \mathbb{C}_{\infty} \backslash B$, explain how a closed curve based at $w$ defines a permutation of $f^{-1}(w)$.

For the rational function

$$f(z)=\frac{z(2-z)}{(1-z)^{4}}$$

identify the group of all such permutations.