Realization of primitive branched coverings over closed surfaces following the hurwitz approach

Semeon Bogatyi, et al. "Realization of primitive branched coverings over closed surfaces following the hurwitz approach." Open Mathematics 1.2 (2003): 184-197. <http://eudml.org/doc/268918>.

@article{SemeonBogatyi2003,
abstract = {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].},
author = {Semeon Bogatyi, Daciberg Gonçalves, Elena Kudryavtseva, Heiner Zieschang},
journal = {Open Mathematics},
keywords = {55M20; 57M12; 20F99},
language = {eng},
number = {2},
pages = {184-197},
title = {Realization of primitive branched coverings over closed surfaces following the hurwitz approach},
url = {http://eudml.org/doc/268918},
volume = {1},
year = {2003},
}

TY - JOUR
AU - Semeon Bogatyi
AU - Daciberg Gonçalves
AU - Elena Kudryavtseva
AU - Heiner Zieschang
TI - Realization of primitive branched coverings over closed surfaces following the hurwitz approach
JO - Open Mathematics
PY - 2003
VL - 1
IS - 2
SP - 184
EP - 197
AB - 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].
LA - eng
KW - 55M20; 57M12; 20F99
UR - http://eudml.org/doc/268918
ER -