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].
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References
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.
I. Berstein and A.L. Edmonds: “On the classification of generic branched coverings of surfaces”, Illinois. J. Math., Vol. 28, (1984), pp. 64–82.
S. Bogatyi, D.L. Gonçalves, E. Kudryavtseva, H. Zieschang: “Minimal number of roots of surface mappings”, Matem. Zametki, Preprint (2001).
S. Bogatyi, D.L. Gonçalves, E. Kudryavtseva, H. Zieschang: “Realization of primitive branched coverings over closed surfaces”, Kluwer Academic Publishers, Preprint (2002).
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.
A.L. Edmonds: “Deformation of maps to branched coverings in dimension two”, Ann. Math., Vol. 110, (1979), pp. 113–125.
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.
C.L. Ezell: “Branch point structure of covering maps onto nonorientable surfaces”, Trans Amer. Math. Soc., Vol. 243, (1978), pp. 123–133.
D. Gabai and W.H. Kazez: “The classification of maps of surfaces”, Invent. math., Vol. 90, (1987), pp. 219–242.
D.L. Gonçalves and H. Zieschang: “Equations in free groups and coincidence of mappings on surfaces’, Math. Z., Vol. 237, (2001), pp. 1–29.
A. Hurwitz: “Über Riemannische Fläche mit gegebenen Verzweigungspunkten”, Math. Ann., Vol. 39, (1891), pp. 1–60.
D.H. Husemoller: “Ramified coverings of Riemann surfaces”, Duke Math. J., Vol. 29, (1962), pp. 167–174.
H. Seifert and W. Threlfall: Lehrbuch der Topologie, Teubner, Leipzig, 1934.
R. Skora: “The degree of a map between surfaces”, Math. Ann., Vol. 276, (1987), pp. 415–423.
R. Stöcker and H. Zieschang: Algebraische Topologie, B.G. Teubner, Stuttgart, 1994.
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Bogatyi, S., Gonçalves, D.L., Kudryavtseva, E. et al. Realization of primitive branched coverings over closed surfaces following the hurwitz approach. centr.eur.j.math. 1, 184–197 (2003). https://doi.org/10.2478/BF02476007
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DOI: https://doi.org/10.2478/BF02476007
Keywords
- covering
- branched covering of surfaces
- branching order
- Hurwitz problem
- representations to the summetry groups ∑ d