Riemann–Hurwitz formula
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Riemann–Hurwitz formula, named after
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
and
Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory. Early life He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and died ...
, describes the relationship of the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
s of two
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s when one is a ''ramified covering'' of the other. It therefore connects ramification with
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, in this case. It is a prototype result for many others, and is often applied in the theory of
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s (which is its origin) and
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s.


Statement

For a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
,
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
,
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
surface S, the Euler characteristic \chi(S) is :\chi(S)=2-2g, where ''g'' is the
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
(the ''number of handles''), since the
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s are 1, 2g, 1, 0, 0, \dots. In the case of an (''unramified'')
covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete sp ...
of surfaces :\pi\colon S' \to S that is surjective and of degree N, we have the formula :\chi(S') = N\cdot\chi(S). That is because each simplex of S should be covered by exactly N in S', at least if we use a fine enough
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...
of S, as we are entitled to do since the Euler characteristic is a
topological invariant In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces ...
. 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' are
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s, and that the map \pi is
complex analytic Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
. The map \pi 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''''n'', 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 In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
at P'' and also denoted by ''e''''P''. In calculating the Euler characteristic of ''S''′ we notice the loss of ''eP'' − 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 :\chi(S') = N\cdot\chi(S) - \sum_ (e_P -1) or as it is also commonly written, using that \chi(X) = 2 - 2g(X) and multiplying through by ''-1'': :2g(S')-2 = N\cdot(2g(S)-2) +\sum_ (e_P -1) (all but finitely many ''P'' have ''eP'' = 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: :\chi(S')- r = N \cdot (\chi(S) - b) 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 In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, ...
). 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 \pi 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 \wp-function, considered as a
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), pole ...
with values in the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
, yields a map from an
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
(genus 1) to the
projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
(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 :0 = 2\cdot2 - \Sigma\ 1 with the summation taken over four values of ''P''. The formula may also be used to calculate the genus of
hyperelliptic curve In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dist ...
s. As another example, the Riemann sphere maps to itself by the function ''z''''n'', 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\cdot2 - (n - 1) - (e_\infty - 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 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 In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...
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 :\chi(S') = N\cdot\chi(S) \, denoting with \chi \, the orbifold Euler characteristic.


References

* , section IV.2. {{DEFAULTSORT:Riemann-Hurwitz formula Algebraic topology Algebraic curves Riemann surfaces