Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the topology of conormal complexes and posets of matroids

Anastasia Nathanson, Ethan Partida

Abstract

We introduce the poset of biflats of a matroid $M$, a Lagrangian analog of the lattice of flats of $M$, and study the topology of its order complex, which we call the biflats complex. This work continues the study of the Lagrangian combinatorics of matroids, which was recently initiated by work of Ardila, Denham and Huh. We show the biflats complex contains two distinguished subcomplexes: the conormal complex of $M$ and the simplicial join of the Bergman complexes of $M$ and $M^\perp$, the matroidal dual of $M$. Our main theorems give sequences of elementary collapses of the biflats complex onto the conormal complex and the join of the Bergman complexes of $M$ and $M^\perp$. These collapses give a combinatorial proof that the biflats complex, conormal complex and the join of the Bergman complexes of $M$ and $M^\perp$ are all simple homotopy equivalent. Although simple homotopy equivalent, these complexes have many different combinatorial properties. We collect and prove a list of such properties.

On the topology of conormal complexes and posets of matroids

Abstract

We introduce the poset of biflats of a matroid , a Lagrangian analog of the lattice of flats of , and study the topology of its order complex, which we call the biflats complex. This work continues the study of the Lagrangian combinatorics of matroids, which was recently initiated by work of Ardila, Denham and Huh. We show the biflats complex contains two distinguished subcomplexes: the conormal complex of and the simplicial join of the Bergman complexes of and , the matroidal dual of . Our main theorems give sequences of elementary collapses of the biflats complex onto the conormal complex and the join of the Bergman complexes of and . These collapses give a combinatorial proof that the biflats complex, conormal complex and the join of the Bergman complexes of and are all simple homotopy equivalent. Although simple homotopy equivalent, these complexes have many different combinatorial properties. We collect and prove a list of such properties.
Paper Structure (15 sections, 20 theorems, 60 equations, 1 table)

This paper contains 15 sections, 20 theorems, 60 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Posets and simplicial complexes
  4. Simple homotopy equivalence
  5. Matroids and their flats
  6. Lagrangian analogue to flats
  7. Unmixed biflats complex
  8. Conormal complex
  9. Examples
  10. Fan structures
  11. Unmixed biflats complex
  12. Conormal Complex
  13. Further Questions
  14. "Global" deformation retracts
  15. $\mathop{\mathrm{\mathrm{Aut}}}\nolimits(M)$-equivariant homotopy equivalences

Key Result

Theorem 1

Given a matroid $M$, there is an explicit sequence of elementary collapses from the biflats complex $\Delta(\mathbf{BF}_{M,M^\perp})$ onto the join of the Bergman complexes $\Delta_M * \Delta_{M^\perp}$.

Theorems & Definitions (59)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 49 more