hyperplanes of the reflections in a Coxeter group.

Defining polynomials for Coxeter arrangements of type $A_l,  B_l$ and $D_l$ are, in this order: $\Pi_{1\le i <j\le l}(x_i-x_j)$, $\Pi_{1\le i< j\le l}(x_i\pm x_j)\Pi_{i \in \overline{1,l}}x_i$, respectively $\Pi_{1\le i <j\le l}(x_i \pm x_j)$.

Examples of arrangements with non-$K(\pi,1)$ complements are also known.

\[counterexp\] Let ${{\mathcal A}}\subset {\mathbb{C}}^r, \;r>2$, be a central arrangement with $n>r$ hyperplanes. ${{\mathcal A}}$ is called [*generic*]{} if any subarrangement $\mathcal{B} \subset {{\mathcal A}}$ of $r$ hyperplanes has rank $r$.

A fundamental result, due to Hattori [@Ha], on the homotopy type of the complement of a generic arrangement, implies $\pi_{r-1}(M_{{{\mathcal A}}}) \neq 0$, in other words generic arrangements are never $K(\pi,1)$, for $r \geq 3$.

[*So, one may ask: is the $K(\pi,1)$ property combinatorial? This is another interesting open question.*]{} The answer is positive for the class of hypersolvable arrangements (defined in Section \[sect:special\]).

\[test\] Let ${{\mathcal A}}$ be a hypersolvable arrangement. Then $M_{{{\mathcal A}}}$ is $K(\pi,1)$ if and only if $\mathcal L({{\mathcal A}})$ is supersolvable.

The hypersolvable class includes both generic and supersolvable arrangements, and more. For instance, it also contains the split solvable arrangements (also defined in the next section), which are in general neither generic nor 