A topological space associated to corank 1 tropical phased matroids

TL;DR

This paper extends the Folkman-Lawrence topological representation theorem to tropical phased matroids, showing that their associated topological order complex is homeomorphic to a sphere of dimension 2n-3.

Contribution

It introduces a topological space associated with tropical phased matroids and proves a sphere homeomorphism analogous to classical oriented matroids.

Findings

Topological order complex for tropical phased matroids is homeomorphic to a (2n-3)-sphere.

Establishes a new topological representation theorem for tropical phased matroids.

Connects tropical geometry with topological combinatorics.

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.