In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other.
It therefore connects ramification with algebraic topology, in this case.
It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves.
For the case of an (unramified) covering map of surfaces that is surjective and of degree
, as we are entitled to do since the Euler characteristic is a topological invariant.
What the Riemann–Hurwitz formula does is to add in a correction to allow for ramification (sheets coming together).
The number n is called the ramification index at P and is denoted by eP.
Now let us choose triangulations of S and S′ with vertices at the branch and ramification points, respectively, and use these to compute the Euler characteristics.
Then S′ will have the same number of d-dimensional faces for d different from zero, but fewer than expected vertices.
Indeed, to obtain this formula, remove disjoint disc neighborhoods of the branch points from S and their preimages in S' so that the restriction of
Removing a disc from a surface lowers its Euler characteristic by 1 by the formula for connected sum, so we finish by the formula for a non-ramified covering.
-function, considered as a meromorphic function with values in the Riemann sphere, yields a map from an elliptic curve (genus 1) to the projective line (genus 0).
As another example, the Riemann sphere maps to itself by the function zn, which has ramification index n at 0, for any integer n > 1.
In order to balance the equation we must have ramification index n at infinity, also.
Several results in algebraic topology and complex analysis follow.
As another example, it shows immediately that a curve of genus 0 has no cover with N > 1 that is unramified everywhere: because that would give rise to an Euler characteristic > 2.
For a correspondence of curves, there is a more general formula, Zeuthen's theorem, which gives the ramification correction to the first approximation that the Euler characteristics are in the inverse ratio to the degrees of the correspondence.