Two constructions of oriented matroids with disconnected extension space

The extension space ℰ(ℳ) of an oriented matroid ℳ is the poset of all one-element extensions of ℳ, 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 elementf is not contained in any mutation of a rank 4 oriented matroid ℳ, then ℰ(ℳf) contains an isolated point. A uniform nonrealizable arrangement of pseudoplanes 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.