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2003 | 1 | 2 | 184-197
Tytuł artykułu

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

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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].
Wydawca
Czasopismo
Rocznik
Tom
1
Numer
2
Strony
184-197
Opis fizyczny
Daty
wydano
2003-06-01
online
2003-06-01
Twórcy
Bibliografia
  • [1] I. Berstein and A.L. Edmonds: “On the construction of branched coverings of low-dimensional manifolds”, Trans. Amer. Math. Soc., Vol. 247, (1979), pp. 87–124. http://dx.doi.org/10.2307/1998776
  • [2] I. Berstein and A.L. Edmonds: “On the classification of generic branched coverings of surfaces”, Illinois. J. Math., Vol. 28, (1984), pp. 64–82.
  • [3] S. Bogatyi, D.L. Gonçalves, E. Kudryavtseva, H. Zieschang: “Minimal number of roots of surface mappings”, Matem. Zametki, Preprint (2001).
  • [4] S. Bogatyi, D.L. Gonçalves, E. Kudryavtseva, H. Zieschang: “Realization of primitive branched coverings over closed surfaces”, Kluwer Academic Publishers, Preprint (2002).
  • [5] S. Bogatyi, D.L. Gonçalves, H. Zieschang: “The minimal number of roots of surface mappings and quadratic equations in free products”, Math. Z., Vol. 236, (2001), pp. 419–452.
  • [6] A.L. Edmonds: “Deformation of maps to branched coverings in dimension two”, Ann. Math., Vol. 110, (1979), pp. 113–125. http://dx.doi.org/10.2307/1971246
  • [7] A.L. Edmonds, R.S. Kulkarni, R.E. Stong: “Realizability of branched coverings of surfaces”, Trans. Amer. Math. Soc., Vol. 282, (1984), pp. 773–790. http://dx.doi.org/10.2307/1999265
  • [8] C.L. Ezell: “Branch point structure of covering maps onto nonorientable surfaces”, Trans Amer. Math. Soc., Vol. 243, (1978), pp. 123–133. http://dx.doi.org/10.2307/1997758
  • [9] D. Gabai and W.H. Kazez: “The classification of maps of surfaces”, Invent. math., Vol. 90, (1987), pp. 219–242. http://dx.doi.org/10.1007/BF01388704
  • [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
  • [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] 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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_BF02476007
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