Skip to content


  • Research Article
  • Open Access

Epsilon Nielsen fixed point theory

Fixed Point Theory and Applications20062006:29470

  • Received: 11 October 2004
  • Accepted: 21 July 2005
  • Published:


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.


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


Authors’ Affiliations

Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA


  1. Brown RF: The Lefschetz Fixed Point Theorem. Scott, Foresman, Illionis; 1971:vi+186.Google Scholar
  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.Google Scholar
  3. Fadell E, Husseini S: Local fixed point index theory for non-simply-connected manifolds. Illinois Journal of Mathematics 1981,25(4):673–699.MathSciNetMATHGoogle Scholar
  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.Google Scholar
  5. Hildebrand FB: Introduction to Numerical Analysis, International Series in Pure and Applied Mathematics. 2nd edition. McGraw-Hill, New York; 1974:xiii+669.Google Scholar
  6. Jiang B: Fixed points and braids. Inventiones Mathematicae 1984,75(1):69–74. 10.1007/BF01403090MathSciNetView ArticleMATHGoogle Scholar
  7. Milnor J: Morse Theory, Annals of Mathematics Studies, no. 51. Princeton University Press, New Jersey; 1963:vi+153.Google Scholar
  8. Rourke CP, Sanderson BJ: Introduction to Piecewise-Linear Topology, Ergebnisse der Mathematik und ihrer Grenzgebiete. Volume 69. Springer, New York; 1972:viii+123.MATHGoogle Scholar