Fixed Simplex Property for Retractable Complexes
1 Institute of Mathematics, Jan Kochanowski University, 15 Świętokrzyska street, 25-406 Kielce, Poland
2 Institute of Computer Science, Polish Academy of Sciences, 21 Ordona street, 01-237 Warsaw, Poland
3 Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland
Fixed Point Theory and Applications 2010, 2010:303640 doi:10.1155/2010/303640Published: 14 September 2010
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.