Open Access

Estimating Nielsen Numbers on Wedge Product Spaces

Fixed Point Theory and Applications20082007:083420

https://doi.org/10.1155/2007/83420

Received: 8 May 2007

Accepted: 14 November 2007

Published: 24 January 2008

Abstract

Let be a self-map of a finite polyhedron that is an aspherical wedge product space . In this paper, we estimate the Nielsen number of . In particular, we study some algebraic properties of the free products and then estimate Nielsen numbers on torus wedge surface with boundary, Klein bottle wedge surface with boundary, and torus wedge torus.

[123456789101112131415123456789101112131415]

Authors’ Affiliations

(1)
Department of Mathematics, University of Connecticut
(2)
School of Mathematics, Korea Institute for Advanced Study

References

  1. Brown RF: The Lefschetz Fixed Point Theorem. Scott, Foresman, Glenview, Ill, USA; 1971:vi+186.MATHGoogle Scholar
  2. Jiang BJ: Lectures on Nielsen Fixed Point Theory, Contemporary Mathematics. Volume 14. American Mathematical Society, Providence, RI, USA; 1983:vii+110.View ArticleGoogle Scholar
  3. Kiang T-H: The Theory of Fixed Point Classes. Springer, Berlin, Germany; 1989:xii+174.MATHGoogle Scholar
  4. McCord CK: Computing Nielsen numbers. In Nielsen Theory and Dynamical Systems, Contemporary Mathematics. Volume 152. American Mathematical Society, Providence, RI, USA; 1993:249–267.View ArticleGoogle Scholar
  5. Magnus W, Karrass A, Solitar D: Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. 2nd edition. Dover, New York, NY, USA; 1976:xii+444.MATHGoogle Scholar
  6. Woo MH, Kim J-R: Note on a lower bound of Nielsen number. Journal of the Korean Mathematical Society 1992,29(1):117–125.MATHMathSciNetGoogle Scholar
  7. Ferrario D: Generalized Lefschetz numbers of pushout maps. Topology and Its Applications 1996,68(1):67–81. 10.1016/0166-8641(96)00040-5MATHMathSciNetView ArticleGoogle Scholar
  8. Fadell E, Husseini S: The Nielsen number on surfaces. In Topological Methods in Nonlinear Functional Analysis, Contemporary Mathematics. Volume 21. American Mathematical Society, Providence, RI, USA; 1983:59–98.View ArticleGoogle Scholar
  9. Hart EL: The Reidemeister trace and the calculation of the Nielsen number. In Nielsen Theory and Reidemeister Torsion, Banach Center Publications. Volume 49. Polish Academy of Sciences, Warsaw, Poland; 1999:151–157.Google Scholar
  10. Wagner J: An algorithm for calculating the Nielsen number on surfaces with boundary. Transactions of the American Mathematical Society 1999,351(1):41–62. 10.1090/S0002-9947-99-01827-9MATHMathSciNetView ArticleGoogle Scholar
  11. Kim SW: Computation of Nielsen numbers for maps of compact surfaces with boundary. Journal of Pure and Applied Algebra 2007,208(2):467–479. 10.1016/j.jpaa.2006.01.009MATHMathSciNetView ArticleGoogle Scholar
  12. Yi P: An algorithm for computing the Nielsen number of maps on the pants surface, Ph.D. thesis. University of California, Los Angeles, Calif, USA; 2003.Google Scholar
  13. Jiang B: Bounds for fixed points on surfaces. Mathematische Annalen 1998,311(3):467–479. 10.1007/s002080050195MATHMathSciNetView ArticleGoogle Scholar
  14. Kim S: Nielsen numbers of maps of polyhedra with fundamental group free on two generators. preprint, 2007Google Scholar
  15. Brooks RBS, Brown RF, Pak J, Taylor DH: Nielsen numbers of maps of tori. Proceedings of the American Mathematical Society 1975,52(1):398–400. 10.1090/S0002-9939-1975-0375287-XMATHMathSciNetView ArticleGoogle Scholar

Copyright

© N. Khamsemanan and S.W. Kim. 2007

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.