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 [4] 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 [11]. 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].
@article{bwmeta1.element.doi-10_2478_BF02476007,
author = {Semeon Bogatyi and Daciberg Gon\c calves and Elena Kudryavtseva and Heiner Zieschang},
title = {Realization of primitive branched coverings over closed surfaces following the hurwitz approach},
journal = {Open Mathematics},
volume = {1},
year = {2003},
pages = {184-197},
zbl = {1039.57001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02476007}
}
Semeon Bogatyi; Daciberg Gonçalves; Elena Kudryavtseva; Heiner Zieschang. Realization of primitive branched coverings over closed surfaces following the hurwitz approach. Open Mathematics, Tome 1 (2003) pp. 184-197. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02476007/
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