Monodromy theorem

If a function element can be analytically continued along every path in a domain, continuation along two endpoint-fixed homotopic paths gives the same terminal germ.

For a covering $p:X\to Y$ and base point $y$, lifting loops based at $y$ permutes the fibre $p^{-1}(y)$. The image ofis the monodromy group.

A connected three-sheeted cover has transitive monodromy. If it also has a simple branch point, its monodromy contains a transposition and must be the full symmetric group on three letters. A local triple branch supplies a three-cycle explicitly.

The degree-four rational map $F(z)=z^2+z^{-2}$ has branch values $-2,2,\infty$. Away from their inverse images, the transformationsare commuting deck involutions. They generate a Klein four-group acting transitively on each fibreConsequently the monodromy group is $V_4$, acting by the identity and the three double transpositions.