A counterexample to Las Vergnas' strong map conjecture on realizable oriented matroids

TL;DR

This paper presents a counterexample to Las Vergnas' strong map conjecture specifically for realizable oriented matroids, demonstrating that certain strong maps cannot be factored into extensions and contractions.

Contribution

It provides the first known counterexample of a non-factorizable strong map between realizable oriented matroids, addressing an open problem in the field.

Findings

Counterexample involves an alternating oriented matroid of rank 4

Proves the strong map cannot be factored through a rank 3 uniform oriented matroid

Addresses the open question about factorizability in realizable oriented matroids

Abstract

The Las Vergnas' strong map conjecture, states that any strong map of oriented matroids $f:\mathcal{M}_1\rightarrow\mathcal{M}_2$ can be factored into extensions and contractions. The conjecture is known to be false due to a construction by Richter-Gebert, he find a non-factorizable strong map $f:\mathcal{M}_1\rightarrow\mathcal{M}_2$, however in his example $\mathcal{M}_1$ is not realizable. The problem that whether there exists a non-factorizable strong map between realizable oriented matroids still remains open. In this paper we provide a counterexample to the strong map conjecture on realizable oriented matroids, which is a strong map $f:\mathcal{M}_1\rightarrow\mathcal{M}_2$, $\mathcal{M}_1$ is an alternating oriented matroid of rank $4$ and $f$ has corank $2$. We prove it is not factorizable by showing that there is no uniform oriented matroid $\mathcal{M}^{\prime}$ of rank $3$ such that $\mathcal{M}_1\rightarrow\mathcal{M}^{\prime}\rightarrow\mathcal{M}_2$.