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Fixed point sets of maps homotopic to a given map

Abstract

Let be a self-map of a compact, connected polyhedron and a closed subset. We examine necessary and sufficient conditions for realizing as the fixed point set of a map homotopic to . For the case where is a subpolyhedron, two necessary conditions were presented by Schirmer in 1990 and were proven sufficient under appropriate additional hypotheses. We will show that the same conditions remain sufficient when is only assumed to be a locally contractible subset of . The relative form of the realization problem has also been solved for a subpolyhedron of . We also extend these results to the case where is a locally contractible subset.

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References

  1. 1.

    Brown RF: The Lefschetz Fixed Point Theorem. Scott, Foresman, Illinois; 1971:vi+186.

    Google Scholar 

  2. 2.

    Dold A: Lectures on Algebraic Topology, Die Grundlehren der mathematischen Wissenschaften. Volume 200. Springer, New York; 1972:xi+377.

    Google Scholar 

  3. 3.

    Dydak Jprivate communication, 2003

  4. 4.

    Hu S-T: Theory of Retracts. Wayne State University Press, Michigan; 1965:234.

    Google Scholar 

  5. 5.

    Jezierski J: A modification of the relative Nielsen number of H. Schirmer. Topology and Its Applications 1995,62(1):45–63. 10.1016/0166-8641(94)00039-6

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Jiang BJ: On the least number of fixed points. American Journal of Mathematics 1980,102(4):749–763. 10.2307/2374094

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Jiang BJ: Lectures on Nielsen Fixed Point Theory, Contemporary Mathematics. Volume 14. American Mathematical Society, Rhode Island; 1983:vii+110.

    Google Scholar 

  8. 8.

    Ng CW: Fixed point sets of maps of pairs, M.S. thesis. University of California at Los Angeles, California; 1995.

    Google Scholar 

  9. 9.

    Schirmer H: A relative Nielsen number. Pacific Journal of Mathematics 1986,122(2):459–473.

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Schirmer H: Fixed point sets in a prescribed homotopy class. Topology and Its Applications 1990,37(2):153–162. 10.1016/0166-8641(90)90060-F

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Schirmer H: A survey of relative Nielsen fixed point theory. In Nielsen Theory and Dynamical Systems (South Hadley, MA, 1992), Contemp. Math.. Volume 152. American Mathematical Society, Rhode Island; 1993:291–309.

    Google Scholar 

  12. 12.

    Strantzalos P: Eine charakterisierung der fixpunktmengen bei selbstabbildungen kompakter mannigfaltigkeiten aus einer homotopieklasse. Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 1977,25(8):787–793.

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Zhao XZ: Estimation of the number of fixed points on the complement. Topology and Its Applications 1990,37(3):257–265. 10.1016/0166-8641(90)90024-V

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Christina L Soderlund.

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Soderlund, C.L. Fixed point sets of maps homotopic to a given map. Fixed Point Theory Appl 2006, 46052 (2006). https://doi.org/10.1155/FPTA/2006/46052

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Keywords

  • Differential Geometry
  • Computational Biology