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Conference Papers Year : 2018

Discrete Morse theory for the collapsibility of supremum sections

Abstract

The Dushnik-Miller dimension of a poset $\le$ is the minimal number $d$ of linear extensions $\le_1, \ldots , \le_d$ of $\le$ such that $\le$ is the intersection of $\le_1, \ldots , \le_d$. Supremum sections are simplicial complexes introduced by Scarf and are linked to the Dushnik-Miller as follows: the inclusion poset of a simplicial complex is of Dushnik-Miller dimension at most $d$ if and only if it is included in a supremum section coming from a representation of dimension $d$. Collapsibility is a topoligical property of simplicial complexes which has been introduced by Whitehead and which resembles to shellability. While Ossona de Mendez proved in that a particular type of supremum sections are shellable, we show in this article that supremum sections are in general collapsible thanks to the discrete Morse theory developped by Forman.
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Dates and versions

hal-01867246 , version 1 (06-06-2019)

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Balthazar Bauer, Lucas Isenmann. Discrete Morse theory for the collapsibility of supremum sections. ICGT: International Colloquium on Graph Theory and combinatorics, Jul 2018, Lyon, France. ⟨hal-01867246⟩
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