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

Semeon Bogatyi; Daciberg Gonçalves; Elena Kudryavtseva; Heiner Zieschang

Open Mathematics (2003)

- Volume: 1, Issue: 2, page 184-197
- ISSN: 2391-5455

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topSemeon 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 -

## References

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- [11] A. Hurwitz: “Über Riemannische Fläche mit gegebenen Verzweigungspunkten”, Math. Ann., Vol. 39, (1891), pp. 1–60. http://dx.doi.org/10.1007/BF01199469
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