Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 303640, 7 pages
doi:10.1155/2010/303640
Research Article
Fixed Simplex Property for Retractable Complexes
1Institute of Mathematics, Jan Kochanowski University, 15 Świętokrzyska street, 25-406 Kielce, Poland
2Institute of Computer Science, Polish Academy of Sciences, 21 Ordona street, 01-237 Warsaw, Poland
3Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland
Received 16 December 2009; Revised 10 August 2010; Accepted 9 September 2010
Academic Editor: L. Górniewicz
Copyright © 2010 Adam Idzik and Anna Zapart. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Retractable complexes are defined in this paper. It is proved that they have the fixed simplex
property for simplicial maps. This implies the theorem of Wallace and the theorem of Rival and
Nowakowski for finite trees: every simplicial map transforming vertices of a tree into itself has a
fixed vertex or a fixed edge. This also implies the Hell and Nešetřil theorem: any endomorphism
of a dismantlable graph fixes some clique. Properties of recursively contractible complexes are
examined.