Topological properties of activity orders for matroid bases

Las Vergnas [9] introduced several lattice structures on the bases of an ordered matroid M by using their external and internal activities. He also noted [10] that when computing the Möbius function of these lattices, it was often zero, although he had no explanation for that fact. The purpose of this paper is to provide a topological reason for this phenomenon. In particular, we show that the order complex of the external lattice L(M) is homotopic to the independence complex of the restriction M|T where M is the dual of M and T is the top element of L(M). We then compute some examples showing that this latter complex is often contractible which forces all its homology groups, and thus its Möbius function, to vanish. A theorem of Björner [3] also helps us to calculate the homology of the matroid complex. 1 The external and internal orders In September of 2001, there was a conference on Tutte Polynomials and Related Topics at the Centre de Recerca Matemàtica in Barcelona, Spain. At the meeting, Michel Las Vergnas gave a talk about three lattice structures which he had imposed on the bases of an ordered matroid using external and internal activity [9]. During the question and answer period that followed, one of us (Sagan), asked if Las Vergnas knew anything about the Möbius function of these lattices. Las Vergnas replied that he had computed some examples and noted that the value was often zero, but did not have an explanation for that fact. In this paper, we will give a topological reason for Las Vergnas’ observation. The rest of this section will be devoted to developing the definition and some basic properties of the external lattice, L(M), of an ordered matroid M . In the next section, we derive some results about the structure of L(M) which will be useful in working with its order complex ∆(M). In particular, we give a simpler formula for the join operator than was given by Las Vergnas. The third section contains our main theorem, showing that ∆(M) is homotopic to the independence complex IN of the restriction M|T where M is the dual of M and T is the top element of L(M). In section 4, we compute some examples showing that IN is often contractible which forces all its homology groups, and thus its Möbius function, to be zero. A characterization of the homology of IN due to Björner [3] is recalled in the next section and used for the calculation of yet more examples. The final section contains a couple of open problems. Let M be a matroid on a finite set E. We denote the bases and independent sets of M by B = B(M) and I = I(M), respectively. We say that M is ordered if E is linearly ordered. From now on all matroids will be ordered. Given a set F ⊆ E we say that e ∈ E is active with respect to F if there is a circuit C(F ; e) ⊆ F ∪ {e} in which e is minimal with respect to the ordering on E. Let ActM(F ) = {e : e is active with respect to F}. Note that we include the possibility that e ∈ F . Note also that we will often write one-element sets without the set braces and drop M as a subscript if the matroid is clear from context. For F ⊆ E we define ExtM(F ) = ActM(F )− F. The elements of ExtM(F ) are called externally active with respect to F . This coincides with the usual notion of externally active elements with respect to an element of B. Las Vergnas defined the external lattice of M in a manner equivalent to the following. For A,B ∈ B, define A ≤ M B if and only if A ⊆ B ∪ ExtM(B).