of a topological space $X, \; \pi_n(X)$, is the set of continous maps $[0,1]^n \rightarrow X$ that take the boundary of the $n$ dimensional cube $[0,1]^n$ to a fixed point in $X$, the base point, where two such maps are identified if one can be continuously deformed into another, by a deformation that fixes the base point. To describe the group operation by analogy with the case of the first homotopy group (the fundamental group), note that “the concatenation” of two $n$-cubes in $X$ means that they are glued along a face. For $n\ge 2$, $\pi_n(X)$ is abelian, yet very hard to compute.

\[high\] A topological space $X$ is called $K(\pi,1)$ (or [*aspherical*]{}) if all homotopy groups $\pi_{\geq 2}(X)$ are trivial.

1.  \[wedgeS12\] The bouquet of $n\geq 1$ circles, $\bigvee_{n}S^1$, is a $K(\pi,1)$ space.

2.  \[Kpi2\] The complement of a fiber-type arrangement is also aspherical.

Deligne showed that a large family of arrangements, the [*complexified simplicial arrangements*]{} ([@De]) have $K(\pi,1)$ complements. Among them are the Coxeter arrangements, which we describe next. A [*Coxeter group*]{} is a finite group generated by real, orthogonal reflections. Irreducible Coxeter groups are classified. Their list is: $A_{l},\;B_{l},\;D_{l},\;E_{6-8},\;F_{4},\;G_{2},\;H_{3-4},\;I_{2}(p)$. A [*Coxeter arrangement*]{} is the set of reflecting 