Discrete Morse theory for the collapsibility of supremum sections

TL;DR

This paper demonstrates that supremum sections, a class of simplicial complexes linked to poset dimension, are generally collapsible using discrete Morse theory, extending previous shellability results.

Contribution

The paper proves that supremum sections are collapsible in general, utilizing discrete Morse theory, which broadens understanding beyond prior shellability results.

Findings

Supremum sections are collapsible in general.

Discrete Morse theory can be applied to prove collapsibility.

Extends previous shellability results to broader classes.

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.