A counterexample to the extension space conjecture for realizable oriented matroids

摘要

The extension space conjecture of oriented matroid theory states that the space of all one-element, non-loop, non-coloop extensions of a realizable oriented matroid of rank d has the homotopy type of a sphere of dimension d-1. We disprove this conjecture by showing the existence of a realizable uniform oriented matroid of high rank and corank 3 with disconnected extension space. This also resolves the generalized Baues conjecture for cubes and the combinatorial Grassmannian conjecture of MacPherson, Mnev and Ziegler. Our proof uses probabilistic methods.