On the topology of conormal complexes and posets of matroids

We introduce the poset of biflats of a matroid $M$, a Lagrangian analog of the lattice of flats of $M$, 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 work of Ardila, Denham and Huh. We show the biflats complex contains two distinguished subcomplexes: the conormal complex of $M$ and the simplicial join of the Bergman complexes of $M$ and $M^{\perp }$, the matroidal dual of $M$. Our main theorems give sequences of elementary collapses of the biflats complex onto the conormal complex and the join of the Bergman complexes of $M$ and $M^{\perp }$. These collapses give a combinatorial proof that the biflats complex, conormal complex and the join of the Bergman complexes of $M$ and $M^{\perp }$ 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.