Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 303640, 7 pages
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.


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.