Zariski.

The $K(\pi,1)$ property of $M_{{{\mathcal A}}}$ means by definition the vanishing of the higher homotopy groups $\pi_i(M_{{{\mathcal A}}})$, for $i>1$. It is easy to see that fiber-type implies $K(\pi,1)$, but the converse is not true, as shown by an example of Falk [@Fa1]. A basic result of Hattori [@Ha] implies that the $K(\pi,1)$ property does not hold, if the lattice $\mathcal L({{\mathcal A}})$ is [*generic*]{}. Jambu-Papadima introduced in [@JP1] the (combinatorially defined) [*hypersolvable*]{} class of arrangements, which (strictly) contains both the fiber-type and the generic arrangements. They also proved that the $K(\pi,1)$ property is combinatorial, within the hypersolvable class. In general, this question is open.

Section \[sect:special\] is devoted to the hypersolvable class. For the [*split solvable*]{} subclass, we highlight a remarkable perfect match, namely the fact that the lattice $\mathcal L({{\mathcal A}})$ and the homeomorphism type of the complement $M_{{{\mathcal A}}}$ determine each other [@CDP].

Rank $1$ complex locally constant sheaves (alias [*local systems*]{}) on $M_{{{\mathcal A}}}$ may also be viewed as characters $\rho$ belonging to the character torus, ${\mathbb{T}}_{\pi_1(M_{{{\mathcal A}}})}=({\mathbb{C}}^*)^n$. Combinatorial determinacy of the [*twisted cohomology*]{} $H^*(M_{{{\mathcal A}}}, {}_{\rho}{\mathbb{C}})$ is a major open problem in the field. We review in Section \[sect:VR\] several known facts in this 