Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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COMPUTATION OF NIELSEN NUMBERS FOR CERTAIN MAPS OF HYPERBOLIC SURFACES

  • Received : 2014.12.18
  • Accepted : 2015.04.29
  • Published : 2015.05.15
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Abstract

Let X be a closed surface for which the Euler characteristic $_{\mathcal{X}}(X)$ is negative, and let $f:X{\rightarrow}X$ be a self-map that is not surjective. In this short paper, we prove that we can compute the Nielsen number of f, N(f), under some algebraic conditions.

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References

  1. R. Brown, The Lefschetz Fixed Point Theorem, Scott-Foresman, Grenview, IL, 1971.
  2. E. Hart, Algebraic techniques for calculating the Nielsen number on hyperbolic surfaces, in Handbook of Topological Fixed Point Theory, Springer, 2005, pp. 463-487.
  3. E. Hart, Reidemeister conjugacy for finitely generated free fundamental groups, Fund. Math. 199 (2008), 93-118. https://doi.org/10.4064/fm199-2-1
  4. E. Hart and S. Kim, The Nielsen number for free fundamental groups and maps without remnant, J. Fixed Point Theory Appl. 2 (2007), 261-275. https://doi.org/10.1007/s11784-007-0044-3
  5. B. Jiang, Lectures on Nielsen fixed point theory, Contemp. Math. vol.14, Amer. Math. Soc. Providence, RI, 1983.
  6. B. Jiang, Bounds for fixed points for surfaces, Math. Ann. 311 (1998), 467-479. https://doi.org/10.1007/s002080050195
  7. M. Kelly, Nielsen fixed point theory on surfaces, in Handbook of Topological Fixed Point Theory, Springer, 2005, pp. 647-658.
  8. T. Kiang, The Theory of Fixed Point Classes, Springer-Verlag, New York, 1989.
  9. S. Kim, The Nielsen number on aspherical figure-eight type polyhedra, Topology Appl. 181 (2015), 150-158. https://doi.org/10.1016/j.topol.2014.12.007
  10. S. Kim, Computation of Nielsen numbers for maps of compact surfaces with boundary, J. Pure Appl. Algebra 208 (2007), 467-479. https://doi.org/10.1016/j.jpaa.2006.01.009
  11. S. Kim, The WYK algorithm for maps of aspherical figure-eight type finite polyhedra, J. Pure Appl. Algebra 216 (2012), 1652-1666. https://doi.org/10.1016/j.jpaa.2011.10.032
  12. C. McCord, Computing Nielsen numbers, in Nielsen Theory and Dynamical Systems, Contemp. Math. vol.152, 1993, 249-267. https://doi.org/10.1090/conm/152/01327
  13. J. Wagner, An algorithm for calculating the Nielsen number on surfaces with boundary, Trans. Amer. Math. Soc. 351 (1999), 41-62. https://doi.org/10.1090/S0002-9947-99-01827-9
  14. P. Yi and S. Kim, Nielsen numbers of maps of aspherical figure-eight type polyhedra, Forum Math. 27 (2015), 1277-1308.