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Fixed Simplex Property for Retractable Complexes

Fixed Point Theory and Applications volume 2010, Article number: 303640 (2010) Cite this article

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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Authors and Affiliations

  1. Institute of Mathematics, Jan Kochanowski University, 15 Świętokrzyska street, 25-406, Kielce, Poland

    Adam Idzik

  2. Institute of Computer Science, Polish Academy of Sciences, 21 Ordona street, 01-237, Warsaw, Poland

    Adam Idzik

  3. Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661, Warsaw, Poland

    Anna Zapart

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  1. Adam Idzik
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  2. Anna Zapart
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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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