in ${\mathbb{R}}^n$ also generates an arrangement over ${\mathbb{C}}$. Let ${\mathcal{Q}}({\mathcal{A}}_{{\mathbb{R}}})=\prod_{H \in {\mathcal{A}}_{{\mathbb{R}}}} \alpha_H$ be the defining (real) polynomial of ${\mathcal{A}}_{\mathbb{R}}$ in ${\mathbb{R}}^n$. The [*${\mathbb{C}}$-extended*]{} arrangement ${\mathcal{A}}_{\mathbb{C}}$ is the arrangement in ${\mathbb{C}}^n$ that consists of the hyperplanes that are the kernel of the polynomials $\alpha_H$ in ${\mathbb{C}}^n$ (instead of ${\mathbb{R}}^n$). The arrangement ${\mathcal{A}}_{\mathbb{C}}$ is also called the [*complexification*]{} of ${\mathcal{A}}_{\mathbb{R}}$.

#### Salvetti’s complex

As shown in [@Sal87], if the arrangement ${{\mathcal{A}}}_{{\mathbb{C}}}$ is the complexification of a real one ${{\mathcal{A}}}_{{\mathbb{R}}}$, there is a regular CW-complex ${\mathcal{S}}({{\mathcal{A}}}_{{\mathbb{R}}})$ having the homotopy type of the complement $M({{\mathcal{A}}}_{{\mathbb{C}}})$. We recall here briefly the construction of this complex, which is called [*Salvetti’s complex*]{}.

Let ${{\mathcal{A}}}_{{\mathbb{R}}}=\{H_{{\mathbb{R}}}\}$ be an essential finite affine hyperplane arrangement in ${\mathbb{R}}^n$. Let $M({{\mathcal{A}}}_{{\mathbb{C}}}) ={\mathbb{C}}^n \setminus \bigcup_{H_{{\mathbb{R}}} \in{\cal A}_{{\mathbb{R}}}} H_{{\mathbb{C}}}$ be the complement to the complexified arrangement. The CW-complex ${\mathcal{S}}({{\mathcal{A}}}_{{\mathbb{R}}})$ can be characterized as follows. Let $\mathbf{S}{\mathrel{\mathop:}=}\{F^k\}$ be the stratification of ${\mathbb{R}}^n$ into facets $F^k$ that is induced by the arrangement [@bourbaki68], where the exponent $k$ stands for codimension. Then $\mathbf S$ has a standard partial ordering defined by $$F^i \ <_{\mathbf S} F^j \quad \text{if}\quad \operatorname{clos}(F^i)\supset F^j$$ where $\operatorname{clos}(F^i)$ is the closure of $F^i$.

The $k$-cells of ${\mathcal{S}}({{\mathcal{A}}}_{{\mathbb{R}}})$ bijectively correspond to the pairs $[C<_{\mathbf 