1.1. Topological classification of ramified covering of the sphere. For a compact connected genus g complex curve C let f : c → CP~1 be a meromorphic function. We treat this function as a ramified covering of the sphere. Two ramified coverings (C_1; f_1), (C_2; f_2) are called topologically equivalent if there exists a homeomorphism h : C_1 → C_2 making the following diagram commutative: C_1 → C_2 → CP~1 The critical values of topologically equivalent functions, i.e., the ramification points of the coverings, coincide, as do the genera of the covering curves. In his famous paper [H] Hurwitz initiated the topological classification of such coverings in the case when exactly one of the ramification points is degenerate, and the remaining points are nondegenerate. Below we refer to the degenerate ramification point as "infinity", and its preimages are called "poles".
展开▼
