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Complexified real arrangements of hyperplanes. (English) Zbl 0731.57011
Let V be a finite dimensional vector space over \({\mathbb{R}}\), or \({\mathbb{C}}\). A real (or complex) arrangement \({\mathcal A}\) is a finite collection of real (or complex) affine hyperplanes in V. Let \({\mathcal A}\) be a real arrangement in a real vector space V, and let M(\({\mathcal A})\) be the complement of the corresponding complex arrangement in the complexified vector space: \[ M({\mathcal A})=V\otimes {\mathbb{C}}-\cup_{H\in {\mathcal A}}H\otimes {\mathbb{C}} \] M. Salvetti and P. Orlik independently constructed finite simplicial complexes which carry the homotopy type of M(\({\mathcal A})\). Here, the author studies both of these simplicial complexes, and constructs explicit homotopy equivalence between them.

MSC:
57Q99 PL-topology
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References:
[1] A. Bj√∂rner, M. Las Vergnas, B. Sturmfels, N. White, G. M. Ziegler, ”Oriented Matroids”, Cambridge Univ. Press, to appear · Zbl 0944.52006
[2] M. Cohen, Simplicial Structures and Transverse Cellularity,Annals of Math. 85 (1967), 218–245 · Zbl 0147.42602 · doi:10.2307/1970440
[3] P. Orlik, Complements of Subspace Arrangements, to appear · Zbl 0795.52003
[4] M. Salvetti, Topology of the complement of real hyperplanes in \(\mathbb{C}\) N ,Invent. math. 88 (1987), 603–618 · Zbl 0594.57009 · doi:10.1007/BF01391833
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