Fixed Simplex Property for Retractable Complexes
© Adam Idzik and Anna Zapart. 2010
Received: 16 December 2009
Accepted: 9 September 2010
Published: 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.
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