Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 20F / Solution

At a noncritical point a holomorphic map has a local holomorphic inverse. Fibres of a nonconstant map between compact connected Riemann surfaces are finite: zeros are isolated and compactness precludes infinitely many. Around a nonbranch value $w$, choose disjoint inverse neighbourhoods of every point in its fibre. Compactness of the remaining source ensures a sufficiently small disc about $w$ has no additional preimages. Its preimage is therefore a disjoint union of open sets each mapped biholomorphically onto that disc. This proves a finite-sheeted unbranched covering map after removing the branch fibres.

Here the word “regular” must mean locally unbranched or evenly covered, rather than the stronger modern normal-cover condition. A general holomorphic map does not give a normal covering, and the example below itself shows the distinction.

Lift a loop based at $w$ uniquely from each point of $f^{-1}(w)$. Its endpoint defines a permutation; reversing the loop gives the inverse permutation, and concatenation gives composition. Homotopic loops yield the same endpoints. The monodromy group of a covering consists of these permutations. Removing finitely many points from a connected surface leaves it path-connected. Any two points in the same fibre can therefore be joined upstairs; the projection of that path is a loop whose lift joins them. Thus the monodromy action is transitive.

For the rational map of degree three, differentiation givesAt $z=0$, the local degree is three, so a small loop around its value $0$ acts as a 3-cycle. At $z=\pm \sqrt{3}$, the local degree is two; their distinct branch values are $\mp 3\sqrt{3}/2$, and a loop around either acts as a transposition. The poles $\pm 1$ and the point at infinity are simple and unramified. A 3-cycle and a transposition generate the full symmetric group, soThe corresponding degree-three connected cover is not normal: a normal cover would have a regular monodromy action, whereas $S_3$ on three letters has nontrivial point stabilizers. Thus the stronger interpretation of the printed terminology cannot hold.