for any continuous functions via the approximation theorem. Brouwer's fixpoint theorem treats the case where is an endomorphism of the unit-ball. For all its homology groups vanishes, and is always the identity, so , so has a fixpoint. Formula of Riemann-Hurwitz The Riemann- Hurwitz formula allows to determine the gender of a compact, connected Riemann surface without using explicit triangulation. The proof needs the existence of triangulations for surfaces in an abstract sense: Let be a non-constant holomorphic function on a surface with known gender. The relation between the gender of the surfaces and is where 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. Citations Literature Allen Hatcher: Algebraic Topology, Cambridge University Press, Cambridge/New York/Melbourne 2006, ISBN 0-521-79160-X James R. Munkres: . Band 1984. Addison Wesley, Menlo Park, California 1984, ISBN 0-201-04586-9 Marshall M. Cohen: A course in Simple-Homotopy Theory . In: Graduate Texts in Mathematics. 1973, ISSN 0072-5285, doi:10.1007/978-1-4684-9372-6. Topology Algebraic topology Geometric topology Structures on manifolds Triangulation (geometry)