Oriented matroids with few mutations

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 of n projective planes in ℝPd-1 contains at least n simplicial regions and every plane is adjacent to at least d simplicial regions [17], [18]. We construct a class of uniform pseudoarrangements of 4 n pseudoplanes in ℝP3 with only 3 n+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. © 1993 Springer-Verlag New York Inc.