Skip to main content
SpringerLink
Log in

Relative preimage problem

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We pose the problem of minimization for the relative preimage problem and prove a minimization theorem. The relative common preimage problem for a finite system of maps, whose particular cases are the relative coincidence and common root problems for finitely many maps, is reduced to the relative preimage problem. As corollaries, results concerning the relative coincidence problems for two maps, problems of fixed points and roots are obtained; these results, with certain distinctions, can be found in known papers by other authors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Bibliography

  1. R. F. Brown, The Lefschetz Fixed Point Theorem, Scott, Foresman and Co., Glenview, Illinois, 1971.

    MATH  Google Scholar 

  2. B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemporary Mathematics, vol. 14, Amer. Math. Soc., Providence, RI, 1983.

    MATH  Google Scholar 

  3. T. H. Kiang, The Theory of Fixed Point Classes, Springer-Verlag, Berlin-Heidelberg-New York, 1989.

    MATH  Google Scholar 

  4. J. Nielsen, “Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, I,” Acta Math., 50 (1927), 189–358.

    Article  MATH  Google Scholar 

  5. S. A. Bogatyi, D. L. Gonsalves, and H. Zieschang, “Coincidence problem: the problem of minimization, ” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 225 (1999), 52–86.

    MATH  MathSciNet  Google Scholar 

  6. C. K. McCord, “A Nielsen theory for intersection numbers,” Fund. Math., 152 (1997), no. 2, 117–150.

    MATH  MathSciNet  Google Scholar 

  7. R. Dobreńko and Z. Kucharski, “On the generalization of the Nielsen number,” Fund. Math., 134 (1990), no. 1, 1–14.

    MathSciNet  Google Scholar 

  8. H. Hopf, “Zur Topologie der Abbildungen von Mannigfaltigkeiten. II. Klasseninvarianten von Abbildungen,” Math. Ann., 102 (1929), no. 4, 562–623.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. Brooks, Coincidences, Roots and Fixed Points, Doctoral Dissertation, Univ. of California, Los Angeles, 1967.

    Google Scholar 

  10. H. Schirmer, “A relative Nielsen number,” Pacific J. Math., 122 (1986), no. 2, 459–473.

    MATH  MathSciNet  Google Scholar 

  11. H. Schirmer, “On the location of fixed points on pairs of spaces,” Topology Appl., 30 (1988), no. 3, 253–266.

    Article  MATH  MathSciNet  Google Scholar 

  12. H. Schirmer, “A survey of relative Nielsen fixed point theory,” in: Nielsen Theory and Dynamical Systems, Contemporary Mathematics, vol. 152, ed. C. McCord, Amer. Math. Soc., Providence, RI, 1993, pp. 291–309.

    Google Scholar 

  13. C. G. Jang and S. Lee, “A relative Nielsen number in coincidence theory,” J. Korean Math. Soc., 32 (1995), no. 2, 171–181.

    MATH  MathSciNet  Google Scholar 

  14. J. Jezierski, “The relative coincidence Nielsen number,” Fund. Math., 149 (1996), no. 1, 1–18.

    MATH  MathSciNet  Google Scholar 

  15. J. Guo and P. R. Heath, “Coincidence theory on the complement,” Topology Appl., 95 (1999), no. 3, 229–250.

    Article  MATH  MathSciNet  Google Scholar 

  16. K.-Y. Yang, “A relative root Nielsen number,” Comm. Korean Math. Soc., 11 (1996), no. 1, 245–252.

    MATH  Google Scholar 

  17. K.-Y. Yang, “The minimum theorem for the relative root Nielsen number,” Comm. Korean Math. Soc., 12 (1997), no. 3, 701–707.

    MATH  Google Scholar 

  18. R. F. Brown and H. Schirmer, “Nielsen theory of roots of maps of pairs,” Topology Appl., 92 (1999), no. 3, 247–274.

    Article  MATH  MathSciNet  Google Scholar 

  19. H. Schirmer, “Nielsen numbers for maps of triads,” Topology Appl., 52 (1993), no. 2, 181–202.

    Article  MATH  MathSciNet  Google Scholar 

  20. J. Jezierski, “The Nielsen coincidence theory on topological manifolds,” Fund. Math., 143 (1993), no. 2, 167–178.

    MATH  MathSciNet  Google Scholar 

  21. S. Sternberg, Lectures on Differential Geometry, Prentice Hall, Englewood Cliffs, NJ, 1964.

    MATH  Google Scholar 

  22. C. K. McCord, “The three faces of Nielsen: Coincidences, intersections and preimages,” Topology Appl., 103 (2000), no. 2, 155–177.

    Article  MATH  MathSciNet  Google Scholar 

  23. B. Jiang, “Fixed points and braids,” Invent. Math., 75 (1984), no. 1, 69–74.

    Article  MATH  MathSciNet  Google Scholar 

  24. B. Jiang, “Fixed points and braids, II,” Math. Ann., 272 (1985), no. 2, 249–256.

    Article  MATH  MathSciNet  Google Scholar 

  25. S. A. Bogatyi, D. L. Gonsalves, E. A. Kudryavtseva, and H. Zieschang, “Minimum number of preimages under maps between surfaces,” Mat. Zametki [Math. Notes], 75 (2004), no. 1, 13–19.

    MathSciNet  Google Scholar 

  26. U. Koschorke, Nielsen Coincidence Theory in Arbitrary Codimensions, Preprint 306/2003, Universität Siegen; http://www.math.uni-siegen.de/topology/publications.html.

  27. U. Koschorke, Geometric and Homotopy Theoretic Methods in Nielsen Coincidence Theory, Preprint 308/2004, Universität Siegen; http://www.math.uni-siegen.de/topology/publications.html.

  28. H. Schirmer, “Bemerkungen zur Homotopietheorie der Koinzidenzen mehrerer Abbildungen,” Arch. Math., 11 (1960), 56–64.

    Article  MATH  MathSciNet  Google Scholar 

  29. O. D. Frolkina, “On the coincidence problem for finitely many maps,” in: Proc. 26th Conf. of Young Scientists of Mechanics and Mathematics Dept. of M. V. Lomonosov Moscow State Univ., vol. II, Mech.-Math. Dept. of MSU Publ., Moscow, 2004, pp. 234–238.

    Google Scholar 

  30. J. Guo, Relative and Equivariant Coincidence Theory, Ph. D. Thesis, Memorial University of Newfoundland, 1996.

  31. J. Vick, Homology Theory, Graduate Text Math., vol. 145, Springer-Verlag, Berlin-Heidelberg-New York, 1994.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. M. V. Lomonosov Moscow State University, Moscow

    O. D. Frolkina

Authors
  1. O. D. Frolkina
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Matematicheskie Zametki, vol. 80, no. 2, 2006, pp. 282–295.

Original Russian Text Copyright © 2006 by O. D. Frolkina.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Frolkina, O.D. Relative preimage problem. Math Notes 80, 272–283 (2006). https://doi.org/10.1007/s11006-006-0136-1

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11006-006-0136-1

Key words

Search

Navigation