Restrictions of aspherical arrangements (Q1800220)

An arrangement of hyperplanes \(\mathcal{A}\) in \(\mathbb{C}^\ell\) is a \(K(\pi, 1)\)-arrangement if the complement \(M(\mathcal{A}):=\mathbb{C}^\ell\setminus\bigcup_{H\in\mathcal{A}}H\) is a \(K(\pi, 1)\)-space, i.e. the universal covering space of \(M(\mathcal{A})\) is contractible and the fundamental group \(\pi_1(M(\mathcal{A}))\) of \(M(\mathcal{A})\) is isomorphic to the group \(\pi\). There are several classes of arrangements that are known to be \(K(\pi, 1)\)-arrangement. For example, simplicial real arrangements, fiber type arrangements and reflection arrangements. In this article, the authors present examples of \(K(\pi, 1)\)-arrangements which admit a restriction which fails to be \(K(\pi, 1)\). This shows that being \(K(\pi, 1)\)-arrangement is not hereditary among hyperplane arrangements. Specifically, the authors first prove that the arrangement \(\mathcal{A}\) with defining equation \(y(x-y)(x^2-z^2)(y^2-z^2)\) is not a \(K(\pi, 1)\)-arrangement, and then they obtain this arrangement as restriction of two ideal arrangements of the reflection arrangement \(D_n\), for any \(n\geq4\).