On the Topology of Conormal Complexes and Posets of Matroids

Abstract. We introduce the poset of biflats of a matroid [Formula: see text], a Lagrangian analogue of the lattice of flats of [Formula: see text], and study the topology of its order complex, which we call the biflats complex. This work continues the study of the Lagrangian combinatorics of matroids, which was recently initiated by the work of Ardila, Denham, and Huh. We show that the biflats complex contains two distinguished subcomplexes: the conormal complex of [Formula: see text] and the simplicial join of the Bergman complexes of [Formula: see text] and [Formula: see text], the matroidal dual of [Formula: see text]. Our main theorems give sequences of elementary collapses of the biflats complex onto the conormal complex and the join of the Bergman complexes of [Formula: see text] and [Formula: see text]. These collapses give a combinatorial proof that the biflats complex, the conormal complex, and the join of the Bergman complexes of [Formula: see text] and [Formula: see text] are all simple homotopy equivalent. Although simple homotopy equivalent, these complexes have many different combinatorial properties. We collect and prove a list of such properties.