A counterexample to the extension space conjecture for realizable oriented matroids

TL;DR

This paper disproves the extension space conjecture for realizable oriented matroids by providing a counterexample where the extension space is disconnected, challenging previous assumptions about their topological structure.

Contribution

It introduces a specific realizable uniform oriented matroid of high rank with a disconnected extension space, refuting the conjecture.

Findings

Counterexample with disconnected extension space

Disproves the conjecture for high-rank realizable oriented matroids

Shows the extension space need not be homotopy equivalent to a sphere

Abstract

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.