Editing Covering space (section)

=== Definitions ===
==== Holomorphic maps between Riemann surfaces ====
Let <math>X</math> and <math>Y</math> be [[Riemann surface|Riemann surfaces]], i.e. one dimensional [[Complex manifold|complex manifolds]], and let <math>f: X \rightarrow Y</math> be a continuous map. <math>f</math> is '''holomorphic in a point''' <math>x \in X</math>, if for any [[Chart (mathematics)|charts]] <math>\phi _x:U_1 \rightarrow V_1</math> of <math>x</math> and <math>\phi_{f(x)}:U_2 \rightarrow V_2</math> of <math>f(x)</math>, with <math>\phi_x(U_1) \subset U_2</math>, the map <math>\phi _{f(x)} \circ f \circ \phi^{-1} _x: \mathbb{C} \rightarrow \mathbb{C}</math> is [[Holomorphic function|holomorphic]].

If <math>f</math> is holomorphic at all <math>x \in X</math>, we say <math>f</math> is '''holomorphic.'''

The map <math>F =\phi _{f(x)} \circ f \circ \phi^{-1} _x</math> is called the '''local expression''' of <math>f</math> in <math>x \in X</math>.

If <math>f: X \rightarrow Y</math> is a non-constant, holomorphic map between [[Compact riemann surface|compact Riemann surfaces]], then <math>f</math> is [[Surjective function|surjective]] and an [[open map]],<ref name="Forster">{{Cite book|last=Forster|first=Otto|title=Lectures on Riemann surfaces|publisher=Springer Berlin|year=1991|isbn=978-3-540-90617-9|location=München}}</ref>{{rp|p=11}} i.e. for every open set <math>U \subset X</math> the [[Image (mathematics)|image]] <math>f(U) \subset Y</math> is also open.

==== Ramification point and branch point ====
Let <math>f: X \rightarrow Y</math> be a non-constant, holomorphic map between compact Riemann surfaces. For every <math>x \in X</math> there exist charts for <math>x</math> and <math>f(x)</math> and there exists a uniquely determined <math>k_x \in \mathbb{N_{>0}}</math>, such that the local expression <math>F</math> of <math>f</math> in <math>x</math> is of the form <math>z \mapsto z^{k_{x}}</math>.{{r|Forster|p=10}} The number <math>k_x</math> is called the '''ramification index''' of <math>f</math> in <math>x</math> and the point <math>x \in X</math> is called a '''ramification point''' if <math>k_x \geq 2</math>. If <math>k_x =1</math> for an <math>x \in X</math>, then <math>x</math> is '''unramified'''. The image point <math>y=f(x) \in Y</math> of a ramification point is called a '''branch point.'''

==== Degree of a holomorphic map ====
Let <math>f: X \rightarrow Y</math> be a non-constant, holomorphic map between compact Riemann surfaces. The '''degree <math>\operatorname{deg}(f)</math>''' of <math>f</math> is the cardinality of the fiber of an unramified point <math>y=f(x) \in Y</math>, i.e. <math>\operatorname{deg}(f):=|f^{-1}(y)|</math>.

This number is well-defined, since for every <math>y \in Y</math> the fiber <math>f^{-1}(y)</math> is discrete{{r|Forster|p=20}} and for any two unramified points <math>y_1,y_2 \in Y</math>, it is: <math>|f^{-1}(y_1)|=|f^{-1}(y_2)|.</math>

It can be calculated by: 
: <math>\sum_{x \in f^{-1}(y)} k_x = \operatorname{deg}(f)</math> {{r|Forster|p=29}}