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Smallest weakly contractible non-contractible topological spaces

Part of: Ordered sets Homotopy theory

Published online by Cambridge University Press:  11 December 2019

Nicolás Cianci  and
Miguel Ottina
Nicolás Cianci
Affiliation:
Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Cuyo, Mendoza, Argentina (mottina@fcen.uncu.edu.ar)
Miguel Ottina
Affiliation:
Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Cuyo, Mendoza, Argentina (mottina@fcen.uncu.edu.ar)
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Abstract

We characterize the topological spaces of minimum cardinality which are weakly contractible but not contractible. This is equivalent to finding the non-dismantlable posets of minimum cardinality such that the geometric realization of their order complexes are contractible. Specifically, we prove that all weakly contractible topological spaces with fewer than nine points are contractible. We also prove that there exist (up to homeomorphism) exactly two topological spaces of nine points which are weakly contractible but not contractible.

Keywords

finite topological spaceweakly contractible topological spacepartially ordered setdismantlable poset

MSC classification

Primary: 55P15: Classification of homotopy type
Secondary: 06A07: Combinatorics of partially ordered sets
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 1 , February 2020 , pp. 263 - 274
Copyright
Copyright © Edinburgh Mathematical Society 2019

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