We study the space of all extensions of a real hyperplane arrangement by a new pseudohyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the “Generalized Baues Problem” of Billera, Kapranov, and Sturmfels, via the Bohne-Dress theorem on zonotopal tilings.
We prove that the extension space is spherical for the class of strongly euclidean oriented matroids. This class includes the alternating matroids and all oriented matroids of rank at most 3 or of corank at most 2. In general it is not known whether the extension space is connected for all realizable oriented matroids (hyperplane arrangements). We show that the subspace of realizable extensions is always connected but not necessarily spherical. Nonrealizable oriented matroids of rank 4 with disconnected extension spaces were recently constructed by Mnëv and Richter-Gebert.
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This research was supported by the DFG Schwerpunkt at Augsburg and by the ARO through MSI Cornell (DAAG29-85-C-0018). The first author acknowledges partial support by the NSF (DMS-9002056).
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Sturmfels, B., Ziegler, G.M. Extension spaces of oriented matroids. Discrete Comput Geom 10, 23–45 (1993). https://doi.org/10.1007/BF02573961