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Topological posets and tropical phased matroids

Ulysses Alvarez, Ross Geoghegan

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, Zivaljević 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.

Topological posets and tropical phased matroids

Abstract

For a discrete poset McCord proved that the natural map from the order complex to the poset equipped with the Up topology is a weak homotopy equivalence. Much later, Zivaljević 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.

Paper Structure

This paper contains 36 sections, 39 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. A Short Outline
  3. Topological Posets
  4. The Order Complex and the Comparison Map.
  5. Polyhedral $M$-posets
  6. Convenient Choice of Bases
  7. Geometric Posets and Main Theorem
  8. Known theorem on weak homotopy equivalences
  9. Matroid Examples
  10. Introduction
  11. The tropical phase hyperfield
  12. Various relevant $M$-posets
  13. Products
  14. Covectors
  15. Strong matroids over hyperfields
  16. ...and 21 more sections

Key Result

Theorem 1.1

(Comparison Theorem) When the $M$-poset ${\mathcal{X}}$ is geometric and carries the Up topology, the map $f \colon \Delta ({\mathcal{X}})\to {\mathcal{X}}$ is a weak homotopy equivalence.

Theorems & Definitions (84)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Example
  • Definition 5
  • Example
  • Example
  • Definition 6
  • Definition 7
  • ...and 74 more