a non-constant, holomorphic map between compact Riemann surfaces. For every there exist charts for and and there exists a uniquely determined , such that the local expression of in is of the form . The number is called the ramification index of in and the point is called a ramification point if . If for an , then is unramified. The image point of a ramification point is called a branch point. Degree of a holomorphic map Let be a non-constant, holomorphic map between compact Riemann surfaces. The degree of is the cardinality of the fiber of an unramified point , i.e. . This number is well-defined, since for every the fiber is discrete and for any two unramified points , it is: It can be calculated by: Branched covering Definition A continuous map is called a branched covering, if there exists a closed set with dense complement , such that is a covering. Examples Let and , then with is branched covering of degree , whereby is a branch point. Every non-constant, holomorphic map between compact Riemann surfaces of degree is a branched covering of degree . Universal covering Definition Let be a simply connected covering. If is another simply connected covering, then there exists a uniquely determined homeomorphism , such that the diagram commutes. This means that is, up to equivalence, uniquely determined and because of that universal property denoted as the universal covering of the space . Existence A universal covering does not always exist, but the following properties guarantee its existence: Let be a connected, locally simply connected topological space; then, there exists a universal covering . is defined as and by . The topology on is constructed as follows: Let be a path with . Let be a simply connected neighborhood of the endpoint , then for every the paths inside from to are uniquely determined up to homotopy. Now consider , then with is a bijection and can be equipped with the final topology of . The fundamental group acts freely through on and with is a homeomorphism, i.e. . Examples with is the universal covering of the unit circle . with is the universal covering of the projective space for . with is the universal covering of the unitary group . Since , it follows that the quotient map is the universal covering of the . A topological space which has no universal