Minimal fixed point sets of relative maps

Xue Zhao

Fundamenta Mathematicae (1999)

  • Volume: 162, Issue: 2, page 163-180
  • ISSN: 0016-2736

Abstract

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Let f: (X,A) → (X,A) be a self map of a pair of compact polyhedra. It is known that f has at least N(f;X,A) fixed points on X. We give a sufficient and necessary condition for a finite set P (|P| = N(f;X,A)) to be the fixed point set of a map in the relative homotopy class of the given map f. As an application, a new lower bound for the number of fixed points of f on Cl(X-A) is given.

How to cite

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Zhao, Xue. "Minimal fixed point sets of relative maps." Fundamenta Mathematicae 162.2 (1999): 163-180. <http://eudml.org/doc/212417>.

@article{Zhao1999,
abstract = {Let f: (X,A) → (X,A) be a self map of a pair of compact polyhedra. It is known that f has at least N(f;X,A) fixed points on X. We give a sufficient and necessary condition for a finite set P (|P| = N(f;X,A)) to be the fixed point set of a map in the relative homotopy class of the given map f. As an application, a new lower bound for the number of fixed points of f on Cl(X-A) is given.},
author = {Zhao, Xue},
journal = {Fundamenta Mathematicae},
keywords = {fixed point class; minimal fixed point set; relative Nielsen number; bipartite graph; matching; minimal number; by-passing; Nielsen relation},
language = {eng},
number = {2},
pages = {163-180},
title = {Minimal fixed point sets of relative maps},
url = {http://eudml.org/doc/212417},
volume = {162},
year = {1999},
}

TY - JOUR
AU - Zhao, Xue
TI - Minimal fixed point sets of relative maps
JO - Fundamenta Mathematicae
PY - 1999
VL - 162
IS - 2
SP - 163
EP - 180
AB - Let f: (X,A) → (X,A) be a self map of a pair of compact polyhedra. It is known that f has at least N(f;X,A) fixed points on X. We give a sufficient and necessary condition for a finite set P (|P| = N(f;X,A)) to be the fixed point set of a map in the relative homotopy class of the given map f. As an application, a new lower bound for the number of fixed points of f on Cl(X-A) is given.
LA - eng
KW - fixed point class; minimal fixed point set; relative Nielsen number; bipartite graph; matching; minimal number; by-passing; Nielsen relation
UR - http://eudml.org/doc/212417
ER -

References

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  1. [1] R. F. Brown, The Lefschetz Fixed Point Theorem, Scott and Foresman, Glenview, IL, 1971. Zbl0216.19601
  2. [2] H P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935), 26-30. Zbl0010.34503
  3. [3] B. Jiang, On the least number of fixed points, Amer. J. Math. 102 (1980), 749-763. Zbl0455.55001
  4. [4] B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence, RI, 1983. 
  5. [5] H. Schirmer, A relative Nielsen number, Pacific J. Math. 122 (1986), 459-473. Zbl0553.55001
  6. [6] H. Schirmer, On the location of fixed points on pairs of spaces, Topology Appl. 30 (1988), 253-266. Zbl0664.55003
  7. [7] W P. Wolfenden, Fixed point sets of deformations of polyhedra with local cut points, Trans. Amer. Math. Soc. 350 (1998), 2457-2471. Zbl0897.54015
  8. [8] X. Z. Zhao, A relative Nielsen number for the complement, in: Topological Fixed Point Theory and Applications, B. Jiang (ed.), Lecture Notes in Math. 1411, Springer, Berlin, 1989, 189-199. 
  9. [9] X. Z. Zhao, Basic relative Nielsen numbers, in: Topology-Hawaii, World Sci., Singapore, 1992, 215-222. Zbl1039.55503

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