Extension spaces of oriented matroids
Extension spaces \({\mathcal E}(M)\) of realizable oriented matroids \(M\) are of key importance in the theory of discrete models of Grassmannians (MacPhersonians). The main problem in this context is to prove sphericity of \({\mathcal E}(M)\), and this question is seen as a special case of the so- called ``Generalized Baues Problem'' of Billera, Kapranov and Sturmfels. The extension space is of course interesting in its own right, both as an invariant of \(M\) and because of its relations to other objects and constructions, say to higher Bruhat orders of Manin and Schechtman. The authors prove that the extension space is indeed spherical for the class of strongly Euclidean oriented matroids. This class includes the alternating matroids and all oriented matroids of rank at most 3 and corank at most 2. They also show that the subspace of realizable extensions is always connected but not necessarily spherical.
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