A counterexample to Las Vergnas' strong map conjecture on 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 $\mathcal{M}_1\rightarrow\mathcal{M}^{\prime}\rightarrow\mathcal{M}_2$.