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The Las Vergnas strong map conjecture asserts that any strong map of oriented matroids $$f{:}\,{\mathcal {M}}_1\rightarrow {\mathcal {M}}_2$$ f : M 1 → M 2 can be factored into extensions and contractions. This conjecture is known to be false due to a construction by Richter-Gebert, who finds a strong map that is not factorizable; however, in his example, $${\mathcal {M}}_1$$ M 1 is not realizable. The question of whether there exists a non-factorizable strong map between realizable oriented matroids remains open. In this paper, we provide a counterexample to the strong map conjecture on realizable oriented matroids, which is a strong map $$f:{\mathcal {M}}_1\rightarrow {\mathcal {M}}_2$$ f : M 1 → M 2 , where $${\mathcal {M}}_1$$ M 1 is an alternating oriented matroid of rank 4 and f has corank 2. We prove that the map is not factorizable by showing that there is no uniform oriented matroid $${\mathcal {M}}^{\prime }$$ M ′ of rank 3 such that $${\mathcal {M}}_1\rightarrow {\mathcal {M}}^{\prime }\rightarrow {\mathcal {M}}_2$$ M 1 → M ′ → M 2 .

A Counterexample to Las Vergnas’ Strong Map Conjecture on Realizable Oriented Matroids