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  • Research Article
  • Open Access

Geometric and homotopy theoretic methods in Nielsen coincidence theory

Fixed Point Theory and Applications20062006:84093

  • Received: 30 November 2004
  • Accepted: 21 July 2005
  • Published:


In classical fixed point and coincidence theory, the notion of Nielsen numbers has proved to be extremely fruitful. Here we extend it to pairs of maps between manifolds of arbitrary dimensions. This leads to estimates of the minimum numbers MCC (and MC , resp.) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are . Furthermore we deduce finiteness conditions for MC . As an application, we compute both minimum numbers explicitly in four concrete geometric sample situations. The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space into path components. Its higher-dimensional topology captures further crucial geometric coincidence data. An analoguous approach can be used to define also Nielsen numbers of certain link maps.


  • Differential Geometry
  • Theoretic Method
  • Computational Biology
  • Coincidence Theory
  • Nielsen Coincidence


Authors’ Affiliations

Universität Siegen, Emmy Noether Campus, Walter-Flex Street 3, Siegen, D-57068, Germany


  1. Bogatyĭ SA, Gonçalves DL, Zieschang H: Coincidence theory: the minimization problem. Proceedings of the Steklov Institute of Mathematics 1999,225(2):45–77.MathSciNetMATHGoogle Scholar
  2. Brooks RBS: On removing coincidences of two maps when only one, rather than both, of them may be deformed by a homotopy. Pacific Journal of Mathematics 1972,40(1):45–52.MathSciNetView ArticleMATHGoogle Scholar
  3. Brown RF: Wecken properties for manifolds. In Nielsen Theory and Dynamical Systems (Mass, 1992), Contemp. Math.. Volume 152. American Mathematical Society, Rhode Island; 1993:9–21.View ArticleGoogle Scholar
  4. Conner PE, Floyd EE: Differentiable Periodic Maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F.. Volume 33. Academic Press, New York; Springer, Berlin; 1964.Google Scholar
  5. Cornea O, Lupton G, Oprea J, Tanré D: Lusternik-Schnirelmann Category, Mathematical Surveys and Monographs. Volume 103. American Mathematical Society, Rhode Island; 2003.View ArticleMATHGoogle Scholar
  6. Dax J-P: Étude homotopique des espaces de plongements. Annales Scientifiques de l'École Normale Supérieure. Quatrième Série (4) 1972, 5: 303–377.MathSciNetMATHGoogle Scholar
  7. Dold A, Gonçalves DL: Self-coincidence of fibre maps. Osaka Journal of Mathematics 2005,42(2):291–307.MathSciNetMATHGoogle Scholar
  8. Gonçalves DL, Jezierski J, Wong P: Obstruction theory and coincidences in positive codimension. Bates College, preprint, 2002Google Scholar
  9. Koschorke U: Vector Fields and Other Vector Bundle Morphisms—a Singularity Approach, Lecture Notes in Mathematics. Volume 847. Springer, Berlin; 1981.MATHGoogle Scholar
  10. Koschorke U: Linking and coincidence invariants. Fundamenta Mathematicae 2004, 184: 187–203.MathSciNetView ArticleMATHGoogle Scholar
  11. Koschorke U: Selfcoincidences in higher codimensions. Journal für die Reine und Angewandte Mathematik 2004, 576: 1–10.MathSciNetView ArticleMATHGoogle Scholar
  12. Koschorke U: Nielsen coincidence theory in arbitrary codimensions. to appear in to appear in Journal für die Reine und Angewandte Mathematik, 2003,
  13. Koschorke U: Nonstabilized Nielsen coincidence invariants and Hopf-Ganea homomorphisms. preprint, 2005,
  14. Toda H: Composition Methods in Homotopy Groups of Spheres, Annals of Mathematics Studies, no. 49. Princeton University Press, New Jersey; 1962.Google Scholar


© Ulrich Koschorke. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.