Riemann–Hurwitz formula
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. Statement. For a compact, connected, orientable surface formula_1, the Euler characteristic formula_2 is where "g" is the genus (the "number of handles"), since the Betti numbers are formula_4. In the  case of an ("unramified") covering map of surfaces that is surjective and of degree formula_6, we have the formula That is because each simplex of formula_1 should be covered by exactly formula_6 in formula_10, at least if we use a fine enough triangulation of formula_1, 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"). Now assume that formula_1 and formula_10 are Riemann surfaces, and that the map formula_14 is complex analytic. The map formula_14 is said  to be "ramified" at a point "P" in "S"′ if there exist analytic coordinates near "P" and π("P") such that π takes the form π("z") = "z", and "n" > 1. An equivalent way of thinking about this is that there exists a small neighborhood "U" of "P" such that π("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 also denoted by "e". In calculating the Euler characteristic of "S"′ we notice the loss of "e  − 1 copies of P" above π("P") (that is, in the inverse image of π("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 formula_17 and multiplying through by "-1": (all but finitely many "P" have "e" = 1, so this is quite safe). This formula is known  as the "Riemann–Hurwitz formula" and also as Hurwitz's theorem. Another useful form of the formula is: where "r" is the number points in "S"' at which the cover has nontrivial ramification (ramification points) and "b" is the number of points in "S" that are images of such points (branch points). Indeed, to obtain this formula, remove disjoint disc neighborhoods of the branch points from "S" and disjoint disc neighborhoods of the ramification points in "S' " so that the restriction of formula_20 is a covering. Then apply the general degree formula to the restriction, use the fact that the  Euler characteristic of the disc equals 1, and use the additivity of the Euler characteristic under connected sums. Examples. The Weierstrass formula_21-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 Riemann–Hurwitz formula then reads with the summation taken over four values of "P". 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", 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. Consequences. 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 – 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. Generalizations. 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  Riemann–Hurwitz formula implies the usual formula for coverings denoting with formula_25 the orbifold Euler characteristic.