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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
##### Keywords:
arrangement; simplicial complexes; homotopy type
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##### References:
 [1] A. Björner, 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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