Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl

PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
• # Artykuł - szczegóły

## Open Mathematics

2004 | 2 | 1 | 112-122

## The minimizing of the Nielsen root classes

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.

EN

112-122

wydano
2004-03-01
online
2004-03-01

autor
• IME-USP
autor

### 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.
• [8] [Br] R.B.S. Brooks: “On the sharpness of the Δ2 and Δ1 Nielsen numbers”, J. Reine Angew. Math., Vol. 259, (1973), pp. 101–109.
• [9] [BSc] R. Brown and H. Schirmer: “Nielsen root theory and Hopf degree theory”, Pacific J. Math., Vol. 198 (1), (2001), pp. 49–80. http://dx.doi.org/10.2140/pjm.2001.198.49
• [10] [Ep] D.B.A. Epstein: “The degree of a map”, Proc. London Math. Soc., Vol. 3 (16), (1966), pp. 369–383.
• [11] [GZ1] 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
• [12] [GZ2] D. Gabai and W.H. Kazez: “The classification of maps of nonorientable surfaces”, Math. Ann., Vol. 281, (1988), pp. 687–702. http://dx.doi.org/10.1007/BF01456845
• [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
• [16] [Ki] T.H. Kiang: The theory of fixed point classes, Springer-Verlag, Berlin-Heidelberg-New York, 1989.
• [17] [Sc1] H. Schirmer: “Mindestzahlen von Koinzidenzpunkten”, J. Reine Angew. Math., Vol. 194(1–4), (1955), pp. 21–39.