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

• Autorzy: Semeon Bogatyi , Daciberg Gonçalves , Elena Kudryavtseva , Heiner Zieschang
• Języki publikacji: EN

Abstrakty

EN

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].

Słowa kluczowe

EN

covering; branched covering of surfaces; branching order; Hurwitz problem; representations to the summetry groups ∑ d; 55M20; 57M12; 20F99

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2003
• Tom: 1
• Numer: 2
• Strony: 184-197

Daty

wydano

2003-06-01

online

2003-06-01

Twórcy

autor

Semeon Bogatyi

• Moscow State Lomonossov-University

autor

Daciberg Gonçalves

• Agência Cidade de São Paulo

autor

Elena Kudryavtseva

• Moscow State Lomonossov-University

autor

Heiner Zieschang

Bibliografia

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• [4] S. Bogatyi, D.L. Gonçalves, E. Kudryavtseva, H. Zieschang: “Realization of primitive branched coverings over closed surfaces”, Kluwer Academic Publishers, Preprint (2002).
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• [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
• [13] H. Seifert and W. Threlfall: Lehrbuch der Topologie, Teubner, Leipzig, 1934.
• [14] R. Skora: “The degree of a map between surfaces”, Math. Ann., Vol. 276, (1987), pp. 415–423. http://dx.doi.org/10.1007/BF01450838
• [15] R. Stöcker and H. Zieschang: Algebraische Topologie, B.G. Teubner, Stuttgart, 1994.

Identyfikatory

DOI

10.2478/BF02476007