The degree of a map between surfaces (Q1077761)

Let f: (F,\(\partial F)\to (G,\partial G)\) be a proper map between surfaces. Then two classical definitions for the degree of f are the absolute degree and the geometric degree. It is proved that in several cases the two definitions coincide - in particular, in the case f is orientation-true. Then a map between surfaces is constructed for which the absolute degree is strictly less than the geometric degree. Finally, an inequality is derived which relates Euler characteristic and geometric degree.