Semeon Bogatyi, Daciberg Gonçalves, Elena Kudryavtseva, Heiner Zieschang
June 1, 2003
Let V be a closed surface, H⊑π1(V) a subgroup of finite index l and D=[A 1,...,A m] a collection of partitions of a given number d≥2 with positive defect v(D). When does there exist a connected branched covering f:W→V of order d with branch data D and f∶W→V It has been shown by geometric arguments  that, for l=1 and a surface V different from the sphere and the projective plane, the corresponding branched covering exists (the data D is realizable) if and only if the data D fulfills the Hurwitz congruence v(D)э0 mod 2. In the case l>1, the corresponding branched covering exists if and only if v(D)э0 mod 2, the number d/l is an integer, and each partition A i ∈D splits into the union of l partitions of the number d/l. Here we give a purely algebraic proof of this result following the approach of Hurwitz . The realization problem for the projective plane and l=1 has been solved in [7,8]. The case of the sphere is treated in [1, 2, 12, 7].