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A counterexample to the extension space conjecture for realizable oriented matroids

Algebra & Discrete Mathematics

Speaker: Gaku Liu, MIT
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Location: 1147 MSB
Start time: Mon, Oct 31 2016, 4:10PM

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. The talk will focus on the connection of this problem with polytopes and tilings of polytopes; no knowledge of matroids or oriented matroids is required.