Discrete Morse theory for the collapsibility of supremum sections

03/26/2018 ∙ by Balthazar Bauer, et al. ∙ 0

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 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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1 Introduction

The order dimension (also known as the Dushnik-Miller dimension) of a poset has been introduced by Dushnik and Miller [4]. It is defined as the minimum number of linear extensions of such that is the intersection of these extensions i.e. . See [16] for a comprehensive study of this topic. This notion is important because, for example, of a theorem of Schnyder [14] which states that a graph is planar if and only if the Dushnik-Miller dimension of the inclusion poset of the associated simplicial complex is at most . Representations were introduced by Scarf [13]. A -representation on a set is a set of linear orders on . Given a representation on a set , we can define a simplicial complex associated to this representation that we call its supremum section. Scarf proved that every supremum section of a representation satisfying some additional properties, so called “standard”, is the inclusion poset of a -polytope with one face removed. Ossona de Mendez [12] proved that every abstract simplicial complex of Dushnik-Miller dimension at most is contained in a complex which is shellable and has a straight line embedding in . Supremum sections also appeared in commutative algebra: Bayer et al. [2] studied monomial ideals which are linked to supremum sections by what they call Scarf complexes. They are used by Felsner et al. [7] in order to study orthogonal surfaces. They also appear in the study of Gonçalves et al. [9] of a variant of Delaunay graphs and in the study of empty rectangles graphs by Felsner [6]. Furthermore, they also appear in spanning-tree-decompositions and in the box representations problem as shown by Evans et al. [5].

The goal of our article is to generalize the result of Ossona de Mendez about the shellability of standard supremum sections to every supremum sections. As there exists supremum sections which are not shellable, for instance the simplicial complex characterized by its facets and , we will replace shellability by collapsibility which is a similar notion. A collapse is a topological operation on simplicial complexes, and more generally on CW-complexes, introduced by Whitehead [18] in order to define a simple homotopy equivalence which is a refinement of the homotopy equivalence. A complex is said to be collapsible if it collapses to a point. See [10] for a comprehensive study of this topic. The discrete Morse theory introduced by Forman [8]

is based on this notion and has numerous applications in applied mathematics and computer science. Homotopy equivalence is a topological notion of topological spaces introduced to classify topological spaces. Roughly speaking, two spaces are said to be homotopy equivalent if there exists a continuous deformation from one to the other. A topological space is said to be contractible if it is homotopy equivalent to a point. Collapsible spaces form an important subclass of contractible spaces. While contractibility is algorithmically undecidable by a result of Novikov 

[17], the subclass of collapsible spaces is algorithmically recognizable. More precisely Tancer [15] showed that it is NP-complete to decide whether a simplicial complex is collapsible. Furthermore, every -dimensional contractible complex is collapsible but the house with two rooms [1] and the dunce hat [19] show that there are complexes which are contractible but not collapsible. Finally, the conjecture of Zeeman [19], which implies the Poincarré conjecture, states that for every finite contractible -dimensional CW-complex , the space is collapsible.

2 Notations

In the following, is a finite set. An (abstract) simplicial complex is a subset of closed by inclusion (i.e. ). We call faces the elements of and facets the maximal faces of according to the inclusion order.

Definition 1 (Ossona de Mendez [12]).

Given a linear order on a set , an element , and a set , we say that dominates in , and we denote it , if for every . A -representation on a set is a set of linear orders on . Given a -representation , an element , and a set , we say that dominates in if dominates in some order . We define as the set of subsets of such that every dominates in . The set is called the supremum section of .

It is easy to show that if is a -representation on a set , then is a simplicial complex. An example is the following -representation on : , , and . The corresponding complex , depicted on the left of Figure 1, is characterized by its facets , and . For example is not in as does not dominate in any order.

Definition 2.

Let be a simplicial complex. We say that a face of is a free face of if it is non-empty, non-maximal and contained in only one facet of .

Let and be two simplicial complexes. We say that collapses to if there exists simplicial complexes and a free face of for every such that , for every and . We say that is collapsible if it collapses to a point.

The Hasse diagram of a poset is the transitive reduction of the digraph of the poset. Let be a representation on a set , we denote the Hasse diagram of the inclusion poset of .

Definition 3.

Let be a poset and let be a matching of the Hasse diagram of . For an arc of the Hasse diagram of , we denote and the elements of such that and . A matching of the Hasse diagram of is said to be acyclic if, when reversing the orientation of the arcs of , the Hasse diagram remains acyclic.

It is known that if is the poset of inclusion of a simplicial complex and is a matching of the Hasse diagram of then is acyclic if and only if there is no sequence of arcs of such that is in the Hasse diagram for all as well as .

Theorem 4 (Chari [3]).

Let be a simplicial complex. If the Hasse diagram of the inclusion poset of admits a complete (i.e. perfect) acyclic matching, then is collapsible.

3 Our contribution

Theorem 5.

Let be a representation on a set . Then is collapsible.

Because of Theorem 4, it is enough to show that if is a representation on a set , then admits a complete acyclic matching.

3.1 Proofs

The proof relies on an induction on the dimension of the representation . Let be a -representation on a set . We denote the -representation on obtained from by deleting the order .

Lemma 6.

The simplicial complex is a subcomplex of .

Proof.

Let be a face of . As every element of dominates in at least one of the orders , the element also dominates in at least one of the orders . We conclude that . ∎

See Figure 1 to see what is for the example representation of Definition 1.

Figure 1: denotes the representation from the example. From left to right: where a grey section corresponds to a face with elements, where is the representation obtained from by deleting the order , the Hasse diagram of where the crossed-out faces are the faces from and where the fat edges correspond to a complete acyclic matching of .
Lemma 7.

We define the function by

Then the function is well-defined.

Proof.

Let be a face of . We denote for every . As , there exists an element such that . So the minimum is taken in a non-empty set. ∎

We define the sets and . The goal is to find a complete acyclic matching between and .

Lemma 8.

For every , we have , and .

For every , we have and .

Proof.

Let be in , we denote . For every , we denote (resp. ) the maximum of (resp. ) in the order . By definition of , for every . Furthermore, , otherwise would not dominate . Thus, for every and . Suppose that . Then there would exist such that does not dominate in any order. Thus for every and which contradicts the minimality of . We deduce that .

As for every , does not dominate in , and . If then we would have for every as . We deduce that . Finally, we conclude that .

The second property can be proved in the same manner. ∎

Lemma 9.

The Hasse diagram of the inclusion poset of admits a complete acyclic matching.

See Figure 1 to see an example of a complete acyclic matching.

Proof.

We define the function defined by for every . Let us show that is a bijection. To do so, we define the function by where . Lemma 8 implies that is well defined, that , and that . Thus is a bijection and defines a complete matching between and .

Suppose that is not acyclic: there exists a sequence of arcs of where for a for every such that is in the Hasse diagram for every as well as . As for every , and , we deduce that . Therefore for every and thus . As is in the Hasse diagram, we show in the same way that which contradicts the fact that . We conclude that is a perfect acyclic matching of . ∎

We can now prove Theorem 5.

Proof.

We prove the result by induction on the number of orders. Let be a -representation on . We denote the minimum on in . Let be a face of which contains an element different from . Then does not dominate in as . The set is a face of as every element of dominates in the order . Thus and admits a complete acyclic matching . The base case is therefore true.

Let , we now suppose that the result is true for any -representation on . Let be a -representation on . We denote the -representation on obtained from by deleting the order . We define as . Because of Lemma 9, the Hasse diagram of the inclusion poset of admits a complete acyclic matching .

By induction hypothesis, admits a complete acyclic matching . Thus is a complete matching of . Furthermore, is the union of and the Hasse diagram of the inclusion poset of with some arcs between and . If an arc between and is oriented from to , then there would be a face of that would contain a face of . As is closed by inclusion, then is also in which contradicts the definition of . Therefore the arcs between and are oriented from to and we deduce that is a perfect acyclic matching of . We conclude by induction. ∎

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