Skip to main content


  • Research Article
  • Open Access

A base-point-free definition of the Lefschetz invariant

Fixed Point Theory and Applications20062006:34143

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


In classical Lefschetz-Nielsen theory, one defines the Lefschetz invariant of an endomorphism of a manifold . The definition depends on the fundamental group of , and hence on choosing a base point and a base path from to . At times, it is inconvenient or impossible to make these choices. In this paper, we use the fundamental groupoid to define a base-point-free version of the Lefschetz invariant.


  • Differential Geometry
  • Computational Biology


Authors’ Affiliations

Department of Mathematics, Fort Lewis College, Durango, CO 81301, USA


  1. Bass H: Euler characteristics and characters of discrete groups. Inventiones Mathematicae 1976,35(1):155–196. 10.1007/BF01390137MathSciNetView ArticleMATHGoogle Scholar
  2. Bass H: Traces and Euler characteristics. In Homological Group Theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser.. Volume 36. Cambridge University Press, Cambridge; 1979:1–26.Google Scholar
  3. Brown RF: The Lefschetz Fixed Point Theorem. Scott, Foresman, Illinois; 1971.MATHGoogle Scholar
  4. Geoghegan R: Nielsen fixed point theory. In Handbook of Geometric Topology. North-Holland, Amsterdam; 2002:499–521.Google Scholar
  5. Hatcher A: Algebraic Topology. Cambridge University Press, Cambridge; 2002.MATHGoogle Scholar
  6. Jiang BJ: Lectures on Nielsen Fixed Point Theory, Contemporary Mathematics. Volume 14. American Mathematical Society, Rhode Island; 1983.View ArticleGoogle Scholar
  7. Lück W: Transformation Groups and Algebraic K-Theory, Lecture Notes in Mathematics. Volume 1408. Mathematica Gottingensis, Springer, Berlin; 1989.Google Scholar
  8. Lück W: The universal functorial Lefschetz invariant. Fundamenta Mathematicae 1999,161(1–2):167–215.MathSciNetMATHGoogle Scholar
  9. Stallings J: Centerless groups—an algebraic formulation of Gottlieb's theorem. Topology. An International Journal of Mathematics 1965,4(2):129–134. 10.1016/0040-9383(65)90060-1MathSciNetMATHGoogle Scholar


© Vesta Coufal. 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.