\(K(\pi,1)\) arrangements
The paper is mainly devoted to the topological theory of arrangements. An \(l\)-arrangement \(A\) is a finite set of hyperplanes in \(C^ l\), and its complement \(C = C(A)\) is the open \(2l\)-manifold \(C^ l \smallsetminus \bigcup A\). If the defining linear forms have real coefficients, then \(A\) is said to be complexified, and the arrangement \(A_ R\) defined by the same equations in \(R^ l\) is said to be the real part of \(A\). If \(C\) is an aspherical space, then \(A\) is called a \(K(\pi, 1)\) arrangement. First, the author presents a survey of the known results and unsolved problems about \(K(\pi, 1)\) arrangements. The main part of the paper then refers to necessary and/or sufficient conditions for \(C\) to be aspherical. A new approach to this problem is given, and this is e.g. applied to derive a weight test for a complexified 3-arrangement \(A\) to be \(K(\pi, 1)\). Further on, a combinatorial test for \(A\) to be \(K(\pi, 1)\) is developed. Various further results are contained, and interesting relations between parts of that theory are discussed.
scientific article; zbMATH DE number 722022