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Fixed Simplex Property for Retractable Complexes

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.

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Correspondence to Adam Idzik.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Idzik, A., Zapart, A. Fixed Simplex Property for Retractable Complexes. Fixed Point Theory Appl 2010, 303640 (2010). https://doi.org/10.1155/2010/303640

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  • DOI: https://doi.org/10.1155/2010/303640

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