Two constructions of oriented matroids with disconnected extension space (original) (raw)

The extension space g(~') of an oriented matroid .//is the poset of all one-element extensions of ~, considered as a simplicial complex. We present two different constructions leading to rank 4 oriented matroids with disconnected extension space. We prove especially that if an element f is not contained in any mutation of a rank 4 oriented matroid ~r then g(Jt'\f) contains an isolated point. A uniform nonrealizable arrangement ofpseudoplanes with this property is presented. The examples described contrast results of Sturmfels and Ziegler [12] who proved that for rank 3 oriented matroids the extension space has the homotopy type of the 2-sphere.