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In [[mathematics]], the '''Riemann–Hurwitz formula''', named after [[Bernhard Riemann]] and [[Adolf Hurwitz]], describes the relationship of the [[Euler characteristic]]s of two [[surface]]s 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 surface]]s (which is its origin) and [[algebraic curve]]s.
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For an [[orientable]] surface ''S'' the Euler characteristic χ(''S'') is
 
:<math>2-2g \,</math>
 
where ''g'' is the [[genus (mathematics)|genus]] (the ''number of handles''), since the [[Betti number]]s are 1, 2''g'', 1, 0, 0, ... . In the case of an (''unramified'') [[covering map]] of surfaces
 
:<math>\pi:S' \to S \,</math>
 
that is surjective and of degree ''N'', we should have the formula
 
:<math>\chi(S') = N\cdot\chi(S). \,</math>
 
That is because each simplex of ''S'' should be covered by exactly ''N'' in ''S''&prime; &mdash; at least if we use a fine enough [[Triangulation (geometry)|triangulation]] of ''S'', 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 ''S'' and ''S&prime;'' are [[Riemann surface]]s, and that the map π is [[analytic function|complex analytic]]. The map π is said to be ''ramified'' at a point ''P'' in ''S''&prime; if there exist analytic coordinates near ''P'' and π(''P'') such that π takes the form π(''z'') = ''z''<sup>''n''</sup>, and ''n''&nbsp;>&nbsp;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''<sub>''P''</sub>. In calculating the Euler characteristic of ''S''&prime; we notice the loss of ''e<sub>P</sub>''&nbsp;&minus;&nbsp;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&prime;'' with vertices at the branch and ramification points, respectively, and use these to compute the Euler characteristics. Then ''S&prime;'' 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
 
:<math>\chi(S') = N\cdot\chi(S) - \sum_{P\in S'} (e_P -1) </math>
 
(all but finitely many ''P'' have ''e<sub>P</sub>'' = 1, so this is quite safe). This formula is known as the ''Riemann–Hurwitz formula'' and also as '''Hurwitz's theorem'''.
 
== Examples ==
The [[Weierstrass's elliptic functions|Weierstrass <math>\wp</math>-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 (topology)|double cover]] (''N'' = 2), with ramification at four points only, at which ''e'' = 2. The Riemann–Hurwitz formula then reads
 
:0 = 2&middot;2 &minus; &Sigma; 1
 
with the summation taken over four values of ''P''.  
 
The formula may also be used to calculate the genus of [[hyperelliptic curve]]s.
 
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
 
:2 = ''n''&middot;2 &minus; (''n'' &minus; 1) &minus; (''e''<sub>&infin;</sub> &minus; 1)
 
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 (mathematics)|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|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
:<math>\chi(S') = N\cdot\chi(S) \,</math>
denoting with <math>\chi \,</math> the orbifold Euler characteristic.
 
==References==
* {{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=[[Algebraic Geometry (book)|Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | oclc=13348052 | mr=0463157  | year=1977}}, section IV.2.
 
{{Algebraic curves navbox}}
 
{{DEFAULTSORT:Riemann-Hurwitz formula}}
[[Category:Algebraic topology]]
[[Category:Algebraic curves]]
[[Category:Riemann surfaces]]

Revision as of 22:39, 11 February 2014

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