Discrete Morse theory for the collapsibility of supremum sections

The Dushnik-Miller dimension of a poset ≤ is the minimal number d of linear extensions ≤1, . . . ,≤d of ≤ such that ≤ is the intersection of ≤1, . . . ,≤d. Supremum sections are simplicial complexes introduced by Scarf [13] 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 [17] and which resembles to shellability. While Ossona de Mendez [12] 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 [8].