Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 2 / 23G (Riemann Surfaces)

A point of the surface is a germ of a branch obtained by analytic continuation of the original function. A representative analytic branch on a small disc gives a neighbourhood of germs, one at each centre in that disc. Projection to the centre is a complex coordinate chart; overlaps have identity transition maps. Complete continuation along every path gives a Riemann surface with an unramified covering map $\pi$ to $U$. Lifting a closed path starting at each germ above $P$ analytically continues that branch back to $P$, permuting the fibre. Reverse paths give inverse permutations, and concatenation gives their composition. This is the monodromy action of a covering space.

Write $A=\sqrt{z^2-1}$, $B=\sqrt{z^2-4}$. There are four branches $(++),(+-),(-+),(--)$, which are distinct: neither square root vanishes on $U$, and $A^2-B^2=3$ prevents $A=\pm B$. Small loops about either $1$ or $-1$ change the sign of $A$ only; loops about either $2$ or $-2$ change the sign of $B$ only. Label the four branches in the displayed order. The corresponding permutations are $(13)(24)$ and $(12)(34)$, respectively. Every loop in this punctured plane is generated up to homotopy by these puncture loops, so no other generators arise. The resulting monodromy group of a covering isIt acts transitively and regularly on the four branches.