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

Open Mathematics (2003)

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

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## Abstract

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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].

## How to cite

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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 -

## References

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10. [10] D.L. Gonçalves and H. Zieschang: “Equations in free groups and coincidence of mappings on surfaces’, Math. Z., Vol. 237, (2001), pp. 1–29. http://dx.doi.org/10.1007/PL00004856 Zbl0983.55003
11. [11] A. Hurwitz: “Über Riemannische Fläche mit gegebenen Verzweigungspunkten”, Math. Ann., Vol. 39, (1891), pp. 1–60. http://dx.doi.org/10.1007/BF01199469
12. [12] D.H. Husemoller: “Ramified coverings of Riemann surfaces”, Duke Math. J., Vol. 29, (1962), pp. 167–174. http://dx.doi.org/10.1215/S0012-7094-62-02918-6 Zbl0196.34001
13. [13] H. Seifert and W. Threlfall: Lehrbuch der Topologie, Teubner, Leipzig, 1934.
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15. [15] R. Stöcker and H. Zieschang: Algebraische Topologie, B.G. Teubner, Stuttgart, 1994.

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