Topological posets and tropical phased matroids

TL;DR

This paper extends McCord's theorem to topological posets, including those related to tropical phased matroids, establishing weak homotopy equivalences and aiming to address the tropical MacPhersonian Conjecture.

Contribution

It generalizes McCord's theorem to a broad class of topological posets, including those arising from tropical phased matroids, linking topology and tropical combinatorics.

Findings

Proved the analog of McCord's theorem for topological posets.

Established weak homotopy equivalences for order complexes of these posets.

Applied results to tropical phased matroids and the tropical MacPhersonian Conjecture.

Abstract

For a discrete poset $\mathcal X$ McCord proved that the natural map $|{\mathcal X}|\to {\mathcal X}$ from the order complex to the poset equipped with the Up topology is a weak homotopy equivalence. Much later, Zivaljevi\'{c} defined the notion of order complex for a topological poset. For a large class of such topological posets we prove the analog of McCord's theorem, namely that the natural map from the order complex to the topological poset with the Up topology is a weak homotopy equivalence. A familiar topological example is the Grassmann poset of proper non-zero linear subspaces of R^{n+1} partially ordered by inclusion. But our motivation in topological combinatorics is to apply the theorem to posets associated with tropical phased matroids over the tropical phase hyperfield, and in particular to elucidate the tropical version of the MacPhersonian Conjecture. This is explained in Section 2.