Topological Methods in Nonlinear Analysis

Equivariant Nielsen fixed point theory

Joel Better

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Abstract

We provide an alternative approach to the equivariant Nielsen fixed point theory developed by P. Wong in [Equivariant Nielsen numbers, Pacific J. Math. 159 (1993), 153–175] by associating an abstract simplicial complex to any $G$-map and defining two $G$-homotopy invariants that are lower bounds for the number of fixed points and orbits in the $G$-homotopy class of a given $G$-map in terms of this complex. We develop a relative equivariant Nielsen fixed point theory along the lines above and prove a minimality result for the Nielsen-type numbers introduced in this setting.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 36, Number 1 (2010), 179-195.

Dates
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461251070

Mathematical Reviews number (MathSciNet)
MR2744838

Zentralblatt MATH identifier
1222.55001

Citation

Better, Joel. Equivariant Nielsen fixed point theory. Topol. Methods Nonlinear Anal. 36 (2010), no. 1, 179--195. https://projecteuclid.org/euclid.tmna/1461251070


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References

  • K. Borsuk, Theory of Retracts, Warszawa (1967) \ref\key 2
  • J. Better, Nielsen fixed point theory for $G$-compactly fixed maps of pairs , J. Geometry Topology, 4 (2004), 269–293 \ref\key 3
  • G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York (1972) \ref\key 4
  • R. F. Brown, The Lefschetz Fixed Point Theorem, Scott–Foresman (1971) \ref\key 5 ––––, Wecken properties for manifolds , Contemp. Math., 152 (1993), 9–21 \ref\key 6
  • A. Dold, Lectures on Algebraic Topology, Springer–Verlag, Heidelberg (1972) \ref\key 7
  • D. L. Ferrario, A fixed point index for equivariant maps , Topol. Methods Nonlinear Anal., 13 (1999), 313–340 \ref\key 8
  • E. Fadell and P. Wong, On deforming $G$-maps to be fixed point free , Pacific J. Math., 132 (1988), 272–281 \ref\key 9
  • J. Guo and P. Heath, Equivariant coincidence Nielsen numbers , to appear \ref\key 10
  • S. Illman, The equivariant triangulation theorem for actions of compact Lie groups , Math. Ann., 262 (1983), 487–501 \ref\key 11
  • J. W. Jawarowski, Extensions of $G$-maps and Euclidean $G$-retracts , Math. Z., 146 (1976), 143–148 \ref\key 12
  • B. Jiang, On the least number of fixed points , Amer. J. Math., 102 (1980), 749–763 \ref\key 13 ––––, Fixed point classes from a differential viewpoint , Lecture Notes, 886 , Springer–Verlag (1981), 163–170 \ref\key 14 ––––, Lectures on Nielsen Fixed Point Theory, Contemp. Math., 14 , Amer. Math. Soc., Providence (1983) \ref\key 15 ––––, Fixed points and braids , Invent. Math., 75 (1984), 69–74 \ref\key 16
  • H. Schirmer, A relative Nielsen number , Pacific J. Math., 122 (1986), 459–473 \ref\key 17 ––––, A survey of relative Nielsen fixed point theory , Contemp. Math., 152 (1993), 291–309 \ref\key 18
  • E. Spanier, Algebraic Topology, McGraw-Hill, New York (1966) \ref\key 19
  • N. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press (1951) \ref\key 20
  • T. tom Dieck, Transformation Groups, Walter de Gruyter, Berlin, New York (1987) \ref\key 21
  • F. Wecken, Fixpunktclassen \romI, Math. Ann., 117 (1941), 659–671 \moreref \paper \romII, 118 (1942), 216–234 \moreref \paper\romIII, 118 (1942), 544–577 \ref\key 22
  • D. Wilczyński, Fixed point free equivariant homotopy classes , Fund. Math., 123 (1984), 47–60 \ref\key 23
  • P. Wong, Equivariant Nielsen fixed point theory for $G$-maps , Pacific J. Math., 150 (1991), 179–200 \ref\key 24 ––––, Equivariant Nielsen numbers , Pacific J. Math., 159 (1993), 153–175 \ref\key 25 ––––, On the location of fixed points of $G$-deformations , Topology Appl., 39 (199), 159–165 \ref\key 26
  • X. Z. Zhao, A relative Nielsen number for the complement , Topological Fixed Point Theory and Applications (Proceedings, Tianjin 1988), Lecture Notes in Math., 1411 Springer–Verlag (1989), 189–199