Papers › A counterexample to Las Vergnas' strong map conjecture on realizable oriented matroids
A counterexample to Las Vergnas' strong map conjecture on realizable oriented matroids
Pei Wu
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The Las Vergnas' strong map conjecture, states that any strong map of oriented matroids f:ℳ₁→ℳ₂ 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:ℳ₁→ℳ₂, however in his example ℳ₁ 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:ℳ₁→ℳ₂, ℳ₁ 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 ℳ^' of rank $3$ such that ℳ₁→ℳ^'→ℳ₂.
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