For a degree-one map, the sum of local degrees in every fiber is one. Therefore each fiber contains exactly one point and its local degree is one. There is no ramification, and the <Riemann-Hurwitz formula> gives $2g-2=-2$, so \b[$g=0$]. The map is a continuous bijection from compact $C$ onto the Hausdorff <sphere> and hence \b[a <homeomorphism>]. In fact its local degree-one coordinate inverses are holomorphic, so it is biholomorphic as well.

Solution

Past exam of the geometry course of the University of Oxford / 2007 / B / 7 / v / Solution

 Solution (source code)