, Volume 10, Issue 1, pp 251-269

Oriented matroids with few mutations

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Abstract

This paper defines a “connected sum” operation on oriented matroids of the same rank. This construction is used for three different applications in rank 4. First it provides nonrealizable pseudoplane arrangements with a low number of simplicial regions. This contrasts the case of realizable hyperplane arrangements: by a classical theorem of Shannon every arrangement ofn projective planes in ℝP d-1 contains at leastn simplicial regions and every plane is adjacent to at leastd simplicial regions [17], [18]. We construct a class of uniform pseudoarrangements of 4n pseudoplanes in ℝP3 with only 3n+1 simplicial regions. Furthermore, we construct an arrangement of 20 pseudoplanes where one plane is not adjacent to any simplicial region.

Finally we disprove the “strong-map conjecture” of Las Vergnas [1]. We describe an arrangement of 12 pseudoplanes containing two points that cannot be simultaneously contained in an extending hyperplane.