The topology of the external activity complex of a matroid

We prove that the external activity complex $\textrm{Act}_<(M)$ of a matroid is shellable. In fact, we show that every linear extension of LasVergnas's external/internal order $<_{ext/int}$ on $M$ provides a shelling of $\textrm{Act}_<(M)$. We also show that every linear extension of LasVergnas's internal order $<_{int}$ on $M$ provides a shelling of the independence complex $IN(M)$. As a corollary, $\textrm{Act}_<(M)$ and $M$ have the same $h$-vector. We prove that, after removing its cone points, the external activity complex is contractible if $M$ contains $U_{3,1}$ as a minor, and a sphere otherwise.


Introduction
Matroid theory is a combinatorial theory of independence which has its roots in linear algebra and graph theory, but which turns out to have deep connections with many fields. There are natural notions of independence in linear algebra, graph theory, matching theory, the theory of field extensions, and the theory of routings, among others. Matroids capture the combinatorial essence that those notions share.
A matroid can be described in many equivalent ways, arising from the many contexts where matroids are found: the bases, the circuits, the lattice of flats, and the matroid polytope, among others. One important approach, which is the most relevant one to this paper, has been to model a matroid in terms of a simplicial or polyhedral complex.

Motivation for this work.
The external activity complex Act < (M ) of a matroid is a simplicial complex associated to a matroid M and a linear order < on its ground set. This complex arose in work of the first author with Adam Boocher [1]. They started with a linear subspace L of affine space A n with a chosen system of coordinates. There is a natural embedding A n → (P 1 ) n into a product of projective lines, and they considered the closure L of L in (P 1 ) n . They proved that many geometric and algebraic invariants of the variety L are determined by the matroid of L.
As is common in combinatorial commutative algebra, a key ingredient of [1] was to consider the initial ideals in < L under various term orders. These initial ideals are the Stanley-Reisner ideals of the external activity complexes Act < (M ) under the different linear orders < of the ground set. This led them to consider and describe the complexes Act < (M ).
The ideals in < L are shown to be Cohen-Macaulay in [1], and the authors asked the stronger question: Are the external activity complexes Act < (M ) shellable? We prove they are, but furthermore, along the way we prove other results that we now describe.

Our results.
The facets of Act < (M ) are indexed by the bases B of M , and [1] suggested a possible connection between Act < (M ) and LasVergnas's internal order < int on B. Suprisingly, we find that it is the external/internal order < ext/int on B, also defined in [6], which plays a key role. Our main result is the following: ) be a matroid, and let < be a linear order on the ground set E. Any linear extension of LasVergnas's external/internal order < ext/int of B induces a shelling of the external activity complex Act < (M ).
As a corollary we obtain that these orders also shell the independence complex IN (M ), and in fact we show a stronger statement. These theorems are as strong as possible in the context of LasVergnas's active orders. We also obtain the following enumerative corollary. It is easy to see that Act < (M ) is a cone, and hence trivially contractible. It is more interesting to study the reduced external activity complex Act • < (M ), obtained by removing all the cone points of Act < (M ). Our main topological result is the following. Theorem 1.4 Let M be a matroid and < be a linear order on its ground set. The reduced external activity complex Act • < (M ) is contractible if M contains U 3,1 as a minor, and a sphere otherwise. In the present abstract we explain these statements. In the next section we introduce all necessary terminology and in the last section we illustrate all the above theorems in an extended example.

Background
In this section we collect the background information on matroids and shellability.

Matroids
Basic definitions. A simplicial complex ∆ = (E, I) is a pair where E is a finite set and I is a non empty family of subsets of E, such that if A ∈ I and B ⊂ A, then B ∈ A. Elements of I are called faces of the complex. The maximal elements of I are called facets. A complex is said to be pure if all facets have the same number of elements.
The following is one of many equivalent ways of defining a matroid: is a simplicial complex such that the restriction of M to any subset of E is pure.
Since there are several simplicial complexes associated to M , we will denote this one IN (M ) = (E, I). It is often called the independence complex of M .
The two most important motivating examples of matroids are the following.
• (Linear Algebra) Let E be a set of vectors in a vector space, and let I consist of the subsets of E which are linearly independent. Then (E, I) is a linear matroid.
• (Graph Theory) Let E be the set of edges of an undirected graph G, and let I consist of the sets of edges which contain no cycle. Then (E, I) is a graphic matroid.
For any matroid M = (E, I), it is customary to call the sets in I independent. The facets of a matroid are called bases. The set of all bases is denoted B.

Example 2.2
The simplest example of a matroid is the uniform matroid U k,n , whose ground set is [n] and whose independent sets are all the subsets of [n] of cardinality at most k. The uniform matroid U 1,3 is going to play an important role later.
The minimal non-faces of M , that is, the minimal dependent sets, are called circuits. The circuits of a matroid have a special structure [7]: Lemma 2.3 (Circuit Elimination Property) If γ 1 and γ 2 are circuits of a matroid and c ∈ γ 1 ∩ γ 2 , then there is a circuit γ 3 that is contained in γ 1 ∪ γ 2 − c.
Matroids have a notion of duality which generalizes orthogonal complements in linear algebra and dual graphs in graph theory.
Let M be a matroid with bases B. Then the set

Definition 2.5
The deletion M \e of a non-coloop e ∈ E is the matroid on E − e whose bases are the bases of M that do not contain e. We also call this the restriction of M to E − e. Dually, the contraction M/e of a non-loop e ∈ E is the matroid on E − e whose bases are the subsets B of E − e such that B ∪ e is a basis of M .
It is easy to see that any sequence of deletions and contractions of different elements commutes. We say that a matroid M is a minor of a matroid M if M is isomorphic to a matroid obtained from M by performing a sequence of deletions and contractions. Definition 2.6 Given a basis B and an element e ∈ E − B there is a unique circuit contained in B ∪ e, called the fundamental circuit of e with respect to B. It is given by Given a basis B and an element i ∈ B there is a unique cocircuit disjoint with B − i, called the fundamental cocircuit of i with respect to B. It is given by Note that the cocircuit Cocirc(B, i) in M equals the circuit Circ(E − B, i) in the dual M * .

Basis activities.
Let < be a linear order on the ground set E. For a basis B, define the sets: The following elegant result of Tutte [8] (for graphs) and Crapo [4] (for arbitrary matroids) underlies many of the results of [1] and this paper. It follows from the work of Crapo and Tutte [4,8] that this polynomial does not depend on the chosen order <. The Tutte polynomial is the most important matroid invariant, because it answers an innumerable amount of questions about the combinatorics, algebra, geometry, and topology of matroids and related objects. For more information, see [3]. The internal and external orders are consistent in the sense that A ≤ int B and B ≤ ext A imply A = B. Therefore the following definition makes sense.

Definition 2.10
The external/internal order < ext/int is the weakest order which simultaneously extends the external and the internal order. It is characterized by the following equivalent properties for two bases A and B: This poset is a lattice. It is not necessarily graded.
We have the following proposition.

Proposition 2.11
The lexicographic order < lex on B is a linear extension of the three posets < int , < ext , and < ext/int . In symbols, any of A < int B, A < ext B or A < ext/int B implies A < lex B.

Shellability and the h-vector.
Shellability. Shellability is a combinatorial condition on a simplicial complex that allows us to describe its topology easily. A simplicial complex is shellable if it can be built up by introducing one facet at a time, so that whenever we introduce a new facet, its intersection with the previous ones is pure of codimension 1. More precisely: Definition 2.12 Let ∆ be a pure simplicial complex. A shelling order is an order of the facets F 1 , . . . F k such for every i < j there exist k < j and f ∈ F j such that If a shelling order exists, then we call ∆ shellable.
Given a shelling order and a facet F j , there is a subset R(F j ) such that for every A ⊆ F j , we have A ⊆ F i for all i < j if and only if R(F j ) ⊆ A. Equivalently, when we add facet F j to the complex, the new faces that we introduce are precisely those in the interval [R(F j ), F j ]. The set R(F j ) is called the restriction set of F j in the shelling.
This polynomial is also known as the shelling polynomial h ∆ (x), due to the following description of the h-vector for shellable complexes. Note that it is not clear a priori that these numbers should be the same for any shelling order. Understanding the topology of a shellable simplicial complex is easy once we know the last entry of the h-vector, thanks to the following result.  IN (M ). Furthermore, the restriction set of a basis B in this shelling order is given by IP (B).
A straightforward consequence of the previous theorem is that the internal order poset is equal to the poset of bases of M where the order is given by inclusion of restriction sets of the lexicographic shelling order.

Example
Instead of proving the theorems, we want to illustrate them in an example. Consider the graphical matroid given by the graph of Figure 1. Its bases are all the 3-subsets of [5] except {1, 2, 3} and {1, 4, 5}. Under the standard order 1 < 2 < 3 < 4 < 5 on the ground set, Table 1 records the basis activity of the various bases. The resulting internal, external, and external/internal orders < ext , < int , < ext/int are shown in Figure  2 The active orders < ext , < int , and < ext/int , respectively. Table 1 lists the bases in lexicographic order < lex , and this is a shelling order for the independence complex  [3,134]; that is, faces 3, 13, 34, and 134.
Our goal is to shell the external activity complex Act < (M ) whose facets, listed in Table 2, are the sets F (B) = B ∪ EP (B) ∪ B ∪ EA(B). Since 1, 3, 4, and 5 are in all facets of Act < (M ), we remove them, and shell the resulting reduced external activity complex Act • < (M ). Our main result, Theorem 1.1, states that any linear extension of the external/internal order < ext/int gives a shelling order for this complex. For example, we may again consider the lexicographic order, which is indeed a linear extension of < ext/int . For each basis B, Table 2 lists the corresponding facet F (B) of Act < (M ), the corresponding facet F (B) • of Act • < (M ), and the restriction set of the facet F (B) in the shelling. This restriction set is R(F (B)) = IP (B). For example, when we add facet 1234 to the complex Act • < (M ) in the third step of the shelling, the new faces that appear are the eight sets in the interval [R(1234), 1234] = [3,1234].
Notice that we can embed IN (M ) −→ Act • < (M ) by sending 1 → 1, 2 → 2, 3 → 3, 4 → 4, 5 → 5. The latter complex has the same h-vector and is contractible. Therefore, it is no coincidence that the shellings of IN (M ) and Act < (M ) are related. In fact, we will prove that any shelling order for Act < (M ) is a shelling order for IN (M ). Theorem 1.1 then gives: any linear extension of < ext/int is a shelling order for IN (M ) and Act < (M ). (1) We conclude this section with two examples showing that the linear extensions of the internal and external orders < int and of < ext are not necessarily shelling orders for Act < (M ).