- Research Article
- Open access
- Published:
Fixed point sets of maps homotopic to a given map
Fixed Point Theory and Applications volume 2006, Article number: 46052 (2006)
Abstract
Let be a self-map of a compact, connected polyhedron and a closed subset. We examine necessary and sufficient conditions for realizing as the fixed point set of a map homotopic to . For the case where is a subpolyhedron, two necessary conditions were presented by Schirmer in 1990 and were proven sufficient under appropriate additional hypotheses. We will show that the same conditions remain sufficient when is only assumed to be a locally contractible subset of . The relative form of the realization problem has also been solved for a subpolyhedron of . We also extend these results to the case where is a locally contractible subset.
References
Brown RF: The Lefschetz Fixed Point Theorem. Scott, Foresman, Illinois; 1971:vi+186.
Dold A: Lectures on Algebraic Topology, Die Grundlehren der mathematischen Wissenschaften. Volume 200. Springer, New York; 1972:xi+377.
Dydak Jprivate communication, 2003
Hu S-T: Theory of Retracts. Wayne State University Press, Michigan; 1965:234.
Jezierski J: A modification of the relative Nielsen number of H. Schirmer. Topology and Its Applications 1995,62(1):45–63. 10.1016/0166-8641(94)00039-6
Jiang BJ: On the least number of fixed points. American Journal of Mathematics 1980,102(4):749–763. 10.2307/2374094
Jiang BJ: Lectures on Nielsen Fixed Point Theory, Contemporary Mathematics. Volume 14. American Mathematical Society, Rhode Island; 1983:vii+110.
Ng CW: Fixed point sets of maps of pairs, M.S. thesis. University of California at Los Angeles, California; 1995.
Schirmer H: A relative Nielsen number. Pacific Journal of Mathematics 1986,122(2):459–473.
Schirmer H: Fixed point sets in a prescribed homotopy class. Topology and Its Applications 1990,37(2):153–162. 10.1016/0166-8641(90)90060-F
Schirmer H: A survey of relative Nielsen fixed point theory. In Nielsen Theory and Dynamical Systems (South Hadley, MA, 1992), Contemp. Math.. Volume 152. American Mathematical Society, Rhode Island; 1993:291–309.
Strantzalos P: Eine charakterisierung der fixpunktmengen bei selbstabbildungen kompakter mannigfaltigkeiten aus einer homotopieklasse. Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 1977,25(8):787–793.
Zhao XZ: Estimation of the number of fixed points on the complement. Topology and Its Applications 1990,37(3):257–265. 10.1016/0166-8641(90)90024-V
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Soderlund, C.L. Fixed point sets of maps homotopic to a given map. Fixed Point Theory Appl 2006, 46052 (2006). https://doi.org/10.1155/FPTA/2006/46052
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/FPTA/2006/46052