Homotopy sphere representations for matroids

TL;DR

This paper introduces a new topological method to represent oriented matroids using arrangements of homotopy spheres within a simplicial complex, providing explicit constructions that relate to existing Folkman-Lawrence representations.

Contribution

It presents a novel, explicit topological representation of oriented matroids via homotopy spheres, unifying and embedding existing Folkman-Lawrence models in a homotopically favorable manner.

Findings

Constructed explicit homotopy sphere arrangements for any oriented matroid.

Representation depends only on a choice of maximal flag in the matroid.

All Folkman-Lawrence representations embed homotopically nicely in this new model.

Abstract

For any rank $r$ oriented matroid $M$, a construction is given of a "topological representation" of $M$ by an arrangement of homotopy spheres in a simplicial complex which is homotopy equivalent to $S^{r-1}$. The construction is completely explicit and depends only on a choice of maximal flag in $M$. If $M$ is orientable, then all Folkman-Lawrence representations of all orientations of $M$ embed in this representation in a homotopically nice way.