TY - JOUR

T1 - Two constructions of oriented matroids with disconnected extension space

AU - Mnëv, Nicolai E.

AU - Richter-Gebert, Jürgen

PY - 1993/12/1

Y1 - 1993/12/1

N2 - 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 element f is not contained in any mutation of a rank 4 oriented matroid ℳ, then ℰ(ℳ\f) 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. © 1993 Springer-Verlag New York Inc.

AB - 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 element f is not contained in any mutation of a rank 4 oriented matroid ℳ, then ℰ(ℳ\f) 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. © 1993 Springer-Verlag New York Inc.

UR - http://www.scopus.com/inward/record.url?scp=51649132245&partnerID=8YFLogxK

U2 - 10.1007/BF02573981

DO - 10.1007/BF02573981

M3 - Article

AN - SCOPUS:51649132245

VL - 10

SP - 271

EP - 285

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 1

ER -