Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 22G / Solution

The monodromy theorem states that if a holomorphic germ can be analytically continued along every path in a domain, its continuations along paths homotopic with fixed endpoints coincide. In a simply connected domain this gives a single-valued holomorphic function on the whole domain.

Write ${\zeta }_j=e^{2\pi ij/n}$. Remove the radial rays $R_j=\{t{\zeta }_j:t\ge 1\}$ and set $U=\mathbb{C}\setminus {\bigcup }_j{R}_j$. This is an open star-shaped domain containing zero, hence simply connected, and $h=z^n-1$ never vanishes there. Locally its logarithm continues along all paths in $U$, so monodromy gives a holomorphic logarithm $L$ andEquivalently, integrate $h^{\prime }/h$ on the simply connected domain and choose the additive constant to obtain $e^L=h$.

Take $r$ copies of the slit domain, labeled by $j\in \mathbb{Z}/r\mathbb{Z}$, carrying values $e^{(L+2\pi ij)/r}$. Across each cut identify banks so that a positive loop around its simple zero increments the sheet index by one. This cyclic gluing produces the unbranched Riemann surface over $\mathbb{C}\setminus Z$, on which $w^r=z^n-1$ is single valued. Completing each finite puncture gives local coordinate $w$, withThere is one ramification point of index $r$ over each of the $n$ roots of unity.

At infinity a loop around all $n$ zeros acts on sheet labels by $j\mapsto j+n(\mathrm{mod}r)$. With $d=\gcd (n,r)$ this permutation has $d$ cycles, each of length $r/d$. In the compactification there are therefore $d$ points over infinity, each with ramification index $r/d$. Infinity is unbranched precisely when $d=r$, namely when $r\mid n$. Counting branch values in the base sphere,If “branch points” denotes ramification points upstairs instead, the count is $n$ or $n+d$ respectively, with exactly the same stated equivalence.