Editing Triangulation (topology) (section)

==== Formula of Riemann-Hurwitz ====
{{Main|Riemann-Hurwitz formula}}
The Riemann-Hurwitz formula allows to determine the genus of a compact, connected [[Riemann surface]] <math>X  </math> without using explicit triangulation. The proof needs the existence of triangulations for surfaces in an abstract sense: Let <math>F:X \rightarrow Y  </math> be a non-constant holomorphic function on a surface with known genus. The relation between the genus <math>g  </math> of the surfaces <math>X  </math>  and <math>Y  </math> is

<math>2g(X)-2=\deg(F)(2g(Y)-2)+\sum_{x\in X}(\operatorname{ord}(F)-1)</math>

where <math>\deg(F)</math> denotes the degree of the map. The sum is well defined as it counts only the ramifying points of the function.

The background of this formula is that holomorphic functions on Riemann surfaces are ramified coverings. The formula can be found by examining the image of the simplicial structure near to ramifiying points.<ref>{{citation|surname1=Otto Forster|periodical=Heidelberger Taschenbücher|title=Kompakte Riemannsche Flächen|publisher=Springer Berlin Heidelberg|publication-place=Berlin, Heidelberg|at=pp.&nbsp;88–154|isbn=978-3-540-08034-3|date=1977|language=German
}}</ref>