Homotopy sphere representations for matroids

For any rank r oriented matroid M , a construction is given of a ”topological representation” of M by an arrangement of homotopy spheres in a simplicial complex which is homotopy equivalent to Sr−1. The construction is completely explicit and depends only on a choice of maximal flag in M . If M is orientable, then all Folkman-Lawrence representations of all orientations of M embed in this representation in a homotopically nice way. A fundamental result in oriented matroid theory is the Topological Representation Theorem ([FL78]), which says that every rank r oriented matroid can be represented by an arrangement of oriented pseudospheres in Sr−1. In [Swa03] Swartz made the startling discovery that any rank r matroid can be represented by an arrangement of homotopy spheres in a (r − 1)dimensional CW complex homotopic to Sr−1. The representation is far from canonical: it depends on, among other things, a choice of tree for each rank 2 contraction and choices of cells glued in to kill off homotopy groups. The present paper, inspired by Swartz’s work, gives a topological representation of any rank r matroid by an arrangement of homotopy spheres in a simplicial complex which is homotopy equivalent to Sr−1. The construction is completely explicit and depends only on a choice of maximal flag. For oriented matroids, there is a nice homotopy relationship between this representation and the representation given by the Topological Representation Theorem of Folkman and Lawrence. 1 Matroids background There are many equivalent characterizations of matroids. We will define matroids in terms of flats. 1 ar X iv :0 90 3. 27 73 v1 [ m at h. C O ] 1 6 M ar 2 00 9 Definition 1.1. A geometric lattice is a finite ranked poset L such that: 1. L is a lattice, i.e., every pair X, Y of elements has a meet X ∧ Y and a join X ∨ Y . (In particular, L has a least element, denoted 0̂, and a greatest element, denoted 1̂.) 2. Every element of L is a join of atoms of L. 3. The rank function is semimodular, i.e., for any X and Y in L, rank(X)+ rank(Y ) ≥ rank(X ∧ Y ) + rank(X ∨ Y ). Definition 1.2. Let E be a finite set. A rank r matroid on E is a collection M of subsets of E, viewed as a poset ordered by inclusion, such that • M is a rank r geometric lattice, • the meet of any two elements is their intersection, and • E is the join of all the atoms of L. The elements of M are called flats. The canonical example arises from a finite arrangement {He : e ∈ E} of hyperplanes (in an arbitrary vector space). Define a flat of the arrangement to be A ⊆ E such that, for every e ∈ E−A, (∩a∈AHa)∩He 6= ∩a∈AHa. Then the set of flats of the arrangement is a matroid on E. We note for future reference some standard facts about matroids: Notation 1.3. If L is a lattice and X, Y ∈ L then 1. coat(X) denotes the set of coatoms of L which are greater than or equal to X. 2. L≥X denotes {Y ∈ L : Y ≥ X}, and L≤X denotes {Y ∈ L : Y ≤ X}. 3. coatL≥Y (X) denotes the set of coatoms of L≥Y which are greater than or equal to X. Lemma 1.4. (cf. Proposition 3.4.2 in [Fai86]) 1. Every interval in a geometric lattice is a geometric lattice. 2. If L is a geometric lattice and 1̂ 6= X ∈ L then X = ∧ coat(X). Definition 1.5. A complete flag in a matroid L is a maximal chain in L. Lemma 1.6. (cf. Theorem 3.3.2 in [Fai86]) Let L be a geometric lattice. If F is a maximal chain in L and X ∈ L then {X ∨ F |F ∈ F} is a maximal chain in L≥X and {X ∧ F |F ∈ F} is a maximal chain in L≤X .