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

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 \[ M(\mathcal {A}) = {V_\mathbb {C}} - \left ({\bigcup \limits _{H \in \mathcal {A}} {{H_\mathbb {C}}} } \right ),\] 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.