A counterexample to the extension space conjecture for realizable oriented matroids
Gaku Liu

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 has the homotopy type of a sphere of dimension . We disprove this conjecture by showing the existence of a realizable uniform oriented matroid of high rank and corank 3 with disconnected extension space.
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