In mathematics, the RiemannâÂÂHurwitz 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 a compact, connected, orientable surface , the Euler characteristic is
where g is the genus (the number of handles). This follows, as the Betti numbers are .
For the case of an (unramified) covering map of surfaces
that is surjective and of degree , we have the formula
That is because each simplex of should be covered by exactly in , at least if we use a fine enough triangulation of , as we are entitled to do since the Euler characteristic is a topological invariant. What the RiemannâÂÂHurwitz formula does is to add in a correction to allow for ramification (sheets coming together).
Now assume that and are Riemann surfaces, and that the map is complex analytic. The map is said to be ramified at a point P in S′ if there exist analytic coordinates near P and ÃÂ(P) such that ÃÂ takes the form ÃÂ(z) = z<sup>n</sup>, and n > 1. An equivalent way of thinking about this is that there exists a small neighborhood U of P such that ÃÂ(P) has exactly one preimage in U, but the image of any other point in U has exactly n preimages in U. The number n is called the ramification index at P and is denoted by e<sub>P</sub>. In calculating the Euler characteristic of S′ we notice the loss of e<sub>P</sub> − 1 copies of P above ÃÂ(P) (that is, in the inverse image of ÃÂ(P)). 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. Therefore, we find a "corrected" formula
or as it is also commonly written, using that and multiplying through by âÂÂ1:
(all but finitely many P have e<sub>P</sub> = 1, so this is quite safe). This formula is known as the RiemannâÂÂHurwitz formula and also as Hurwitz's theorem.
Another useful form of the formula is:
where b is the number of branch points in S (images of ramification points) and b' is the size of the union of the fibers of branch points (this contains all ramification points and perhaps some non-ramified points). 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 is a covering. 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.
We can also see that this formula is equivalent to the usual form, as we have
since for any we have
The Weierstrass -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). It is a double cover (N = 2), with ramification at four points only, at which e = 2. The RiemannâÂÂHurwitz formula then reads
with the summation taken over four ramification points.
The formula may also be used to calculate the genus of hyperelliptic curves.
As another example, the Riemann sphere maps to itself by the function z<sup>n</sup>, which has ramification index n at 0, for any integer n > 1. There can only be other ramification at the point at infinity. In order to balance the equation
we must have ramification index n at infinity, also.
Several results in algebraic topology and complex analysis follow.
Firstly, there are no ramified covering maps from a curve of lower genus to a curve of higher genus â and thus, since non-constant meromorphic maps of curves are ramified covering spaces, there are no non-constant meromorphic maps from a curve of lower genus to a curve of higher genus.
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.
An orbifold covering of degree N between orbifold surfaces S' and S is a branched covering, so the RiemannâÂÂHurwitz formula implies the usual formula for coverings
denoting with the orbifold Euler characteristic.