arXiv 1201.13062012-01math.COmath.AT

On Arrangements of Pseudohyperplanes

Priyavrat Deshpande

备注：: 19 pages, 3 figures. Third version: some more typos fixed, minor changes, final version. To appear in Proceedings Mathematical Sciences. In the second version: typos fixed, exposition improved, contact information updated. arXiv admin note: text overlap with arXiv:1110.1520

英文摘要：

To every realizable oriented matroid there corresponds an arrangement of real hyperplanes. The homeomorphism type of the complexified complement of such an arrangement is completely determined by the oriented matroid. In this paper we study arrangements of pseudohyperplanes; they correspond to non-realizable oriented matroids. These arrangements arise as a consequence of the Folkman-Lawrence topological representation theorem. We propose a generalization of the complexification process in this context. In particular we construct a space naturally associated with these pseudo-arrangements which is homeomorphic to the complexified complement in the realizable case. Further we generalize the classical theorem of Salvetti and show that this space has the homotopy type of a cell complex defined in terms of the oriented matroid.

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