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 ${\le }_1,\dots ,{\le }_d$ of $\le$ such that $\le$ is the intersection of ${\le }_1,\dots ,{\le }_d$. 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.