Skip to main content

Advertisement

You are viewing the new article page. Let us know what you think. Return to old version

Research Article | Open | Published:

Epsilon Nielsen fixed point theory

Abstract

Let be a map of a compact, connected Riemannian manifold, with or without boundary. For sufficiently small, we introduce an -Nielsen number that is a lower bound for the number of fixed points of all self-maps of that are -homotopic to . We prove that there is always a map that is -homotopic to such that has exactly fixed points. We describe procedures for calculating for maps of -manifolds.

[12345678]

References

  1. 1.

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

  2. 2.

    Brown RF: Nielsen fixed point theory on manifolds. In Nielsen Theory and Reidemeister Torsion (Warsaw, 1996), Banach Center Publ.. Volume 49. Polish Academy of Sciences, Warsaw; 1999:19–27.

  3. 3.

    Fadell E, Husseini S: Local fixed point index theory for non-simply-connected manifolds. Illinois Journal of Mathematics 1981,25(4):673–699.

  4. 4.

    Forster W: Computing "all" solutions of systems of polynomial equations by simplicial fixed point algorithms. In The Computation and Modelling of Economic Equilibria (Tilburg, 1985), Contrib. Econom. Anal.. Volume 167. Edited by: Talman D, van der Laan G. North-Holland, Amsterdam; 1987:39–57.

  5. 5.

    Hildebrand FB: Introduction to Numerical Analysis, International Series in Pure and Applied Mathematics. 2nd edition. McGraw-Hill, New York; 1974:xiii+669.

  6. 6.

    Jiang B: Fixed points and braids. Inventiones Mathematicae 1984,75(1):69–74. 10.1007/BF01403090

  7. 7.

    Milnor J: Morse Theory, Annals of Mathematics Studies, no. 51. Princeton University Press, New Jersey; 1963:vi+153.

  8. 8.

    Rourke CP, Sanderson BJ: Introduction to Piecewise-Linear Topology, Ergebnisse der Mathematik und ihrer Grenzgebiete. Volume 69. Springer, New York; 1972:viii+123.

Download references

Author information

Correspondence to Robert F Brown.

Rights and permissions

Reprints and Permissions

About this article

Keywords

  • Differential Geometry
  • Point Theory
  • Computational Biology
  • Fixed Point Theory
  • Nielsen Fixed Point Theory