Realizable but not Strongly Euclidean Oriented Matroids †

The extension space conjecture of oriented matroid theory claims that the space of all (non-zero, non-trivial, single-element) extensions of a realizable oriented matroid of rank r is homotopy equivalent to an (r − 1)-sphere. In 1993, Sturmfels and Ziegler proved the conjecture for the class of strongly Euclidean oriented matroids, which includes those of rank at most 3 or corank at most 2. They did not provide any example of a realizable but not strongly Euclidean oriented matroid. Here we produce two such examples for the first time, one with rank 4 and one with corank 3. Both have 12 elements.