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Universal cover of Salvetti's complex and topology of simplicial arrangements of hyperplanes

Author: Luis Paris
Journal: Trans. Amer. Math. Soc. 340 (1993), 149-178
MSC: Primary 52B30; Secondary 32S25
MathSciNet review: 1148044
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Abstract: Let $ V$ be a real vector space. An arrangement of hyperplanes in $ V$ is a finite set $ \mathcal{A}$ of hyperplanes through the origin. A chamber of $ \mathcal{A}$ is a connected component of $ V - ({ \cup _{H \in \mathcal{A}}}H)$. The arrangement $ \mathcal{A}$ is called simplicial if $ { \cap _{H \in \mathcal{A}}}H = \{ 0\} $ and every chamber of $ \mathcal{A}$ is a simplicial cone. For an arrangement $ \mathcal{A}$ of hyperplanes in $ V$, we set

$\displaystyle M(\mathcal{A}) = {V_\mathbb{C}} - \left({\bigcup\limits_{H \in \mathcal{A}} {{H_\mathbb{C}}} } \right),$

where $ {V_\mathbb{C}} = \mathbb{C} \otimes V$ is the complexification of $ V$, and, for $ H \in \mathcal{A}$ , $ {H_\mathbb{C}}$ is the complex hyperplane of $ {V_\mathbb{C}}$ spanned by $ H$.

Let $ \mathcal{A}$ be an arrangement of hyperplanes of $ V$. Salvetti constructed a simplicial complex $ \operatorname{Sal}(\mathcal{A})$ and proved that $ \operatorname{Sal}(\mathcal{A})$ has the same homotopy type as $ M(\mathcal{A})$. In this paper we give a new short proof of this fact. Afterwards, we define a new simplicial complex $ \hat{\operatorname{Sal}}(\mathcal{A})$ and prove that there is a natural map $ p:\hat {\operatorname{Sal}}(\mathcal{A}) \to \operatorname{Sal}(\mathcal{A})$ which is the universal cover of $ \operatorname{Sal}(\mathcal{A})$. At the end, we use $ \hat{\operatorname{Sal}}(\mathcal{A})$ to give a new proof of Deligne's result: "if $ \mathcal{A}$ is a simplicial arrangement of hyperplanes, then $ M(\mathcal{A})$ is a $ K(\pi ,1)$ space." Namely, we prove that $ \hat{\operatorname{Sal}}(\mathcal{A})$ is contractible if $ \mathcal{A}$ is a simplicial arrangement.

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