Consider the multivalued complex function $f(z)= \sqrt{z-a}+\sqrt{z-b}$, where $a\neq b$, defined in the domain $U=\mathbb{C}-\{a,b\}$. The graph $W$ of $f$ defines a regular covering space $W \rightarrow U$ with discrete fiber $F$.
(a) How many sheets does this covering have?
(b) If $w_a$ is a small loop about $a$ and not containing $b$ in its interior and similarly $w_b$ is a small loop about $b$ and not containing $a$ in its interior, determine the action of $w_a$ and $w_b$ on the fiber $F$.
(c) There exists a group $G$ so that $W/G$ is homeomorphic.
Attempt:
(a) Since $U$ is connected and we are given that this is a covering we know that the cardinality of the fiber is constant over $U$. So wlog, assume $a\neq 0, b\neq 0$. Then the fiber at 0, $F_0$, is $\{f(0)\}=\{\sqrt{-a}+\sqrt{-b}, a\neq b\}=\{\pm z_a \mp z_b\}$. Since $a,b$ are distinct this has $4$ elements, so we have a $4-$ fold covering.
(b) I'm not sure about this one. I see that $U$ is homotopy equivalent to $S^1$v $ S^1$, but don't really know how to use this. I am guessing the loops $w_a$, $w_b$ would take the point $+z_a - z_b$ to $-z_a - z_b$ and $+z_a + z_b$, but this is just a hunch.
(c) Don't really know, but am guessing $\mathbb{Z}_2 \oplus \mathbb{Z}_2 $?