Matching polytopes, toric geometry, and the non-negative part of the Grassmannian

In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian (Gr_{kn})_{\geq 0}. This is a cell complex whose cells Delta_G can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell Delta_G we associate a certain polytope P(G). The polytopes P(G) are analogous to the well-known Birkhoff polytopes, and we describe their face lattices in terms of matchings and unions of matchings of G. We also demonstrate a close connection between the polytopes P(G) and matroid polytopes. We then use the data of P(G) to define an associated toric variety X_G. We use our technology to prove that the cell decomposition of (Gr_{kn})_{\geq 0} is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Gr_{kn})_{\geq 0} is 1.