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Fixed Simplex Property for Retractable Complexes


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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Correspondence to Adam Idzik.

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Idzik, A., Zapart, A. Fixed Simplex Property for Retractable Complexes. Fixed Point Theory Appl 2010, 303640 (2010).

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