Nielsen fixed point theory on manifolds

Robert Brown

Banach Center Publications (1999)

  • Volume: 49, Issue: 1, page 19-27
  • ISSN: 0137-6934

Abstract

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The study of fixed points of continuous self-maps of compact manifolds involves geometric topology in a significant way in topological fixed point theory. This survey will discuss some of the questions that have arisen in this study and indicate our present state of knowledge, and ignorance, of the answers to them. We will limit ourselves to the statement of facts, without any indication of proof. Thus the reader will have to consult the references to find out how geometric topology has contributed to our knowledge in this area. But we hope this overview can supply a framework for a more detailed investigation of this important and, as we shall see, very active branch of fixed point theory.

How to cite

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Brown, Robert. "Nielsen fixed point theory on manifolds." Banach Center Publications 49.1 (1999): 19-27. <http://eudml.org/doc/208959>.

@article{Brown1999,
abstract = {The study of fixed points of continuous self-maps of compact manifolds involves geometric topology in a significant way in topological fixed point theory. This survey will discuss some of the questions that have arisen in this study and indicate our present state of knowledge, and ignorance, of the answers to them. We will limit ourselves to the statement of facts, without any indication of proof. Thus the reader will have to consult the references to find out how geometric topology has contributed to our knowledge in this area. But we hope this overview can supply a framework for a more detailed investigation of this important and, as we shall see, very active branch of fixed point theory.},
author = {Brown, Robert},
journal = {Banach Center Publications},
keywords = {Nielsen theory; Wecken map; Reidemeister number},
language = {eng},
number = {1},
pages = {19-27},
title = {Nielsen fixed point theory on manifolds},
url = {http://eudml.org/doc/208959},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Brown, Robert
TI - Nielsen fixed point theory on manifolds
JO - Banach Center Publications
PY - 1999
VL - 49
IS - 1
SP - 19
EP - 27
AB - The study of fixed points of continuous self-maps of compact manifolds involves geometric topology in a significant way in topological fixed point theory. This survey will discuss some of the questions that have arisen in this study and indicate our present state of knowledge, and ignorance, of the answers to them. We will limit ourselves to the statement of facts, without any indication of proof. Thus the reader will have to consult the references to find out how geometric topology has contributed to our knowledge in this area. But we hope this overview can supply a framework for a more detailed investigation of this important and, as we shall see, very active branch of fixed point theory.
LA - eng
KW - Nielsen theory; Wecken map; Reidemeister number
UR - http://eudml.org/doc/208959
ER -

References

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  1. [1] D. Anosov, The Nielsen number of maps of nilmanifolds, Russian Math. Surveys 40 (1985), 149-150. Zbl0594.55002
  2. [2] M. Bestvina and M. Handel, Train tracks for surface homeomorphisms, Topology 34 (1995), 109-140. Zbl0837.57010
  3. [3] R. Brown, The Lefschetz Fixed Point Theorem, Scott-Foresman, 1971. 
  4. [4] R. Brown, Wecken properties for manifolds, in: Proceedings of the Conference on Nielsen Theory and Dynamical Systems, Contemp. Math. 152, 1993, 9-21. Zbl0816.55001
  5. [5] R. Brown and B. Sanderson, Fixed points of boundary-preserving maps of surfaces, Pacific J. Math. 158 (1993), 243-264. Zbl0786.57005
  6. [6] O. Davey, E. Hart and K. Trapp, Computation of Nielsen numbers for maps of closed surfaces, Trans. Amer. Math. Soc. (to appear). Zbl0861.55003
  7. [7] D. Dimovski, One-parameter fixed point indices, Pacific J. Math. 164 (1994), 263-297. Zbl0796.55001
  8. [8] D. Dimovski and R. Geoghegan, One-parameter fixed point theory, Forum Math. 2 (1990), 125-154. Zbl0692.55002
  9. [9] E. Fadell and S. Husseini, The Nielsen number on surfaces, in: Proceedings of the Special Session on Fixed Point Theory, Contemp. Math. 21, 1983, 59-98. Zbl0563.55001
  10. [10] R. Geoghegan and A. Nicas, Parametrized Lefschetz-Nielsen fixed point theory and Hochschild homology traces, Amer. J. Math. 116 (1994), 397-446. Zbl0812.55001
  11. [11] J. Harrison and J. Stasheff, Families of H-spaces, Quart. J. Math. 22 (1971), 347-351. Zbl0219.55008
  12. [12] B. Jiang, Estimation of the Nielsen numbers, Chinese Math. 5 (1964), 330-339. 
  13. [13] B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, 1983. Zbl0512.55003
  14. [14] B. Jiang, Fixed points and braids, II, Math. Ann. 272 (1985), 249-256. Zbl0617.55001
  15. [15] B. Jiang, Commutativity and Wecken properties for fixed points of surfaces and 3-manifolds, Topology Appl. 53 (1993), 221-228. Zbl0791.55002
  16. [16] B. Jiang and J. Guo, Fixed points of surface diffeomorphisms, Pacific J. Math. 160 (1993), 67-89. Zbl0829.55001
  17. [17] M. Kelly, Minimizing the number of fixed points for self-maps of compact surfaces, Pacific J. Math. 126 (1987), 81-123. Zbl0571.55003
  18. [18] M. Kelly, Minimizing the cardinality of the fixed point set for selfmaps of surfaces with boundary, Mich. Math. J. 39 (1992), 201-217. Zbl0767.55001
  19. [19] M. Kelly, The relative Nielsen number and boundary-preserving surface maps, Pacific J. Math. 161 (1993), 139-153. Zbl0794.55002
  20. [20] M. Kelly, The Nielsen number as an isotopy invariant, Topology Appl. 62 (1995), 127-143. Zbl0839.55002
  21. [21] M. Kelly, Nielsen numbers and homeomorphisms of geometric 3-manifolds, Topology Proc. 19 (1994), 149-160. Zbl0851.55002
  22. [22] M. Kelly, Computing Nielsen numbers of surface homeomorphisms, Topology 35 (1996), 13-25. Zbl0855.55002
  23. [23] E. Keppelmann and C. McCord, The Anosov theorem for exponential solvmanifolds, Pacific J. Math. 170 (1995), 143-159. Zbl0856.55003
  24. [24] T. Kiang, The Theory of Fixed Point Classes, Springer, 1989. Zbl0676.55001
  25. [25] C. McCord, Computing Nielsen numbers, in: Proceedings of the Conference on Nielsen Theory and Dynamical Systems, Contemp. Math. 152, 1993, 249-267. 
  26. [26] J. Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, Acta Math. 50 (1927), 189-358. Zbl53.0545.12
  27. [27] J. Nolan, Fixed points of boundary-preserving maps of punctured discs, Topology Appl. (to appear). Zbl0899.55001
  28. [28] H. Schirmer, A relative Nielsen number, Pacific J. Math. 122 (1986), 459-473. Zbl0553.55001
  29. [29] E. Spanier, Algebraic Topology, McGraw-Hill, 1966. 
  30. [30] J. Wagner, Classes of Wecken maps of surfaces with boundary, Topology Appl. (to appear). Zbl1001.55005
  31. [31] J. Wagner, An algorithm for calculating the Nielsen number on surfaces with boundary, preprint. Zbl0910.55001
  32. [32] F. Wecken, Fixpunktklassen, III, Math. Ann. 118 (1942), 544-577. 
  33. [33] P. Wong, Equivariant Nielsen numbers, Pacific J. Math. 159 (1993), 153-175. Zbl0739.55001
  34. [34] P. Wong, Fixed point theory for homogeneous spaces, Amer. J. Math. 120 (1998), 23-42. Zbl0908.55002

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