Families of regular matroids

TL;DR

This paper explores the topological properties of the category of regular oriented matroids, comparing them with binary matroids, and discusses implications for algebraic topology and graph theory.

Contribution

It provides a comparison of homotopy types between regular and binary matroids and investigates their topological differences, especially in higher homotopy groups.

Findings

Regular and binary matroids share the same fundamental group in the unoriented case.

Higher homotopy groups differ for rank three regular and binary matroids.

Speculations on the impact of Galatius's theorem on graph cohomology and torsion applications.

Abstract

This is an introductory paper about the category of regular oriented matroids (ROMs). We compare the homotopy types of the categories of regular and binary matroids. For example, in the unoriented case, they have the same fundamental group but we show that the higher homotopy groups are different for rank three regular and binary matroids. We also speculate on the possible impact of a recent theorem of Galatius [Gal] computing the stable cohomology of the category of graphs and on possible applications to higher Reidemeister torsion.