A topological space associated to corank 1 tropical phased matroids

Abstract

A consequence of the Folkman-Lawrence topological representation theorem is that the geometric realization of the order complex of the poset of non-zero covectors of a loopless rank $n-1$ oriented matroid on $[n]$ is homeomorphic to an $(n-2)$-sphere. In this paper, we begin the study of an analogous theorem for tropical phased matroids by proving that the topological order complex for a loopless rank $n-1$ tropical phased matroid on $[n]$ is homeomorphic to a $(2n-3)$-sphere.