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Journal of the London Mathematical Society 1987 s2-35(2):314-326; doi:10.1112/jlms/s2-35.2.314
© 1987 by London Mathematical Society
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© Oxford University Press

Boundary Complexes of Convex Polytopes cannot Be Characterized Locally

Bernd Sturmfels

Department of Mathematics, University of Washington Seattle, Washington 98195, USA

It is well known that there is no local criterion to decide the linear realizability of matroids or oriented matroids. We use the set-up of chirotopes or oriented matroids to derive a similar result in the context of convex polytopes. There is no local criterion to decide whether a combinatorial sphere is polytopal.

The proof is based on a construction technique for rigid chirotopes. These correspond, in the realizable case, to convex polytopes whose internal combinatorial structure is completely determined by its face lattice. So, a rigid chirotope is realizable over a field F if and only if its face-lattice is F-polytopal. Furthermore we prove that for every proper subfield F of the field A of real algebraic numbers there exists a 6-polytope which is not realizable over F.


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