Abstract
We prove that all combinatorial differential manifolds involving only Euclidean oriented matroids are PL manifolds. In doing so we introduce a new notion of triangulations of oriented matroids, cand show that any triangulation of a Euclidean oriented matroid is a PL sphere. In Section 5 we adapt these results to get a new definition of triangulations of oriented matroid polytopes, and show that any triangulation of a Euclidean oriented matroid polytope is a PL ball.
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