We now introduce the counting problem for maps of Riemann Surfaces: fixing a compact Riemann Surface Y and a finite number of points b1, …, bn ∊ Y, how many maps to Y have a specified ramification behavior over the chosen points, and are unramifed elsewhere?

Natural questions that arise are:

1. 1. Is the number of such maps finite?

2. 2. Does it depend on the Riemann Surface Y?

3. 3. Does it depend on the configuration of the points bi?

As luck would have it, the answers are about as good as possible: the number is always finite, it does depend only on the genus of Y ; it also depends on the choice of ramification over the bi but not on the position of the points. We call the answers to the question in the first paragraph Hurwitz numbers and we will spend the rest of this book becoming well acquainted with them.

A key reason for the favorable answers to the above questions is that maps of Riemann Surfaces are essentially “controlled by topology”. We saw in Chapter 4 that maps of compact Riemann Surfaces are covering spaces away from a finite number of points of ramification: in this section we call them ramified covers. The Riemann Existence Theorem essentially says that any ramified cover corresponds to a map of Riemann Surfaces, and allows us to immediately witness that Hurwitz numbers are independent of the complex structure on Y or on the configuration of the branch points.

In subsequent chapters we will count ramified covers by analyzing the behavior of lifts of loops on Y. We conclude this chapter by looking at the simplest example: when the cover has degree 2, a loop winding around a branch point must lift to a path connecting the two inverse images of the base point. This allows us to compute all hyperelliptic Hurwitz numbers, counting ramified covers of degree 2 of ℙ1(ℂ).

Hurwitz Numbers

We begin this section by defining the notion of isomorphism and automorphims of a map of Riemann Surfaces.

Definition 6.1.1. Two holomorphic maps of Riemann Surfaces f : X → Y and are called isomorphic if there is an isomorphism of Riemann Surfaces such that.