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2004 | 2 | 1 | 112-122
Tytuł artykułu

The minimizing of the Nielsen root classes

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M 2)−χ(M 10/(1−χ(M 2)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M 2)−χ(M 2)/(1−χ(M 2)). Also we construct, for each integer n≥3, an example of a map f: K n→N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.
Wydawca
Czasopismo
Rocznik
Tom
2
Numer
1
Strony
112-122
Opis fizyczny
Daty
wydano
2004-03-01
online
2004-03-01
Twórcy
Bibliografia
  • [1] [An] C. Aniz: Raízes de funções de um Complexo em uma variedade, Phd Thesis, São Carlos-SP-Brazil, 2002.
  • [2] [Ar] M.A. Armstrong: Basic Topology. Undergraduate Texts in Mathematics, Springer-Verlag, New York Inc., 1983.
  • [3] [BGKZ1] S. Bogatyi, D.L. Gonçalves, E.A. Kudryavtseva and H. Zieschang: “Minimal number of roots of surface mappings”, Matem. Zametki, (to appear).
  • [4] [BGKZ2] S. Bogatyi, D.L. Gonçalves, E.A. Kudryavtseva and H. Zieschang: “On the Wecken property for the root problem of mappings between surfaces”, Moscow Math. Journal, (to appear).
  • [5] [BGKZ3] S. Bogatyi, D.L. Gonçalves, E.A. Kudryavtseva and H. Zieschang: “Realization of primitive branched coverings over surfaces following the Hurwitz approach”, Central Europ. J. of Mathematics, Vol. 2, (2003), pp. 184–197. http://dx.doi.org/10.2478/BF02476007
  • [6] [BGKZ4] S. Bogatyi, D.L. Gonçalves, E.A. Kudryavtseva and H. Zieschang: “Realization of primitive branched coverings over closed surfaces”, Kluwer, Preprint, (to appear).
  • [7] [BGZ] S. Bogatyi, D.L. Gonçalves and H. Zieschang: “The minimal number of roots of surface mappings and quadratic equations in free products”, Math. Z., Vol. 236, (2001), pp. 419–452.
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  • [13] [GKZ] D.L. Gonçalves, E.A. Kudryavtseva and H. Zieschang: “Roots of mappings on nonorientable surfaces and equations in free groups”, Manuscr. Math., Vol. 107, (2002), pp. 311–341. http://dx.doi.org/10.1007/s002290100238
  • [14] [GZ] 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
  • [15] [Kn] H. Kneser: “Die kleinste Bedeckungszahl innerhalb einer Klasse von Flächen abbildungen”, Math. Ann., Vol. 103, (1930), pp. 347–358. http://dx.doi.org/10.1007/BF01455699
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Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_BF02475955
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