Abstract
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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Acknowledgements
This paper is a part of the author’s doctoral thesis (Chapter 5 of [11]). The author would like to thank his supervisor Graham Denham for his support. The author would also like to thank Eric Babson, Emanuele Delucchi, Alex Papazoglou and Thomas Zaslavsky for fruitful discussions. The author would like to acknowledge the support of the Mathematics Department at Northeastern University for hosting a visit during 2011–12 during which the initial draft was written. He also sincerely thanks the anonymous referee for many helpful suggestions and for pointing out the connection with microbundles.
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Communicating Editor: B V Rajarama Bhat
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DESHPANDE, P. On arrangements of pseudohyperplanes. Proc Math Sci 126, 399–420 (2016). https://doi.org/10.1007/s12044-016-0286-3
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DOI: https://doi.org/10.1007/s12044-016-0286-3