Real \(K(\pi ,1)\) arrangements from finite root systems
The author studies the complement of certain classes of subspace arrangements in real \(n\)-space that are invariant under the action of \(S_n\) by permuting the coordinates and whose subspaces are defined by equality of certain coordinates. Such arrangements are called ``orbit arrangements'' and were introduced by \textit{A. Björner} [in `First European congress of mathematics', Prog. Math. 119, 321-370 (1994; Zbl 0844.52008)]. The author gives a positive answer to the question raised by Björner in the article mentioned, whether the complement of the arrangement consisting of all subspaces in real \(n\)-space defined by having three coordinates equal (``3-equal arrangement'') is a \(K(\pi,1)\)-space (resp., the classifying space of the discrete group \(\pi\) being the fundamental group of the complement). By a result of \textit{A. Björner} and the reviewer [Adv. Math. 110, No. 2, 277-313 (1995; Zbl 0845.57020)] only arrangements of subspaces of real codimension two can be \(K(\pi,1)\)-spaces. So there are only two other \(S_n\)-invariant arrangements defined by equality of coordinates, that are candidates for the complement being a \(K(\pi,1)\)-space. The first is the arrangement of subspaces defined by having two disjoint pairs of coordinates equal. The second is the union of this arrangement with the 3-equal arrangement. For the first, the author demonstrates that the complement is homotopy equivalent to a wedge of circles and hence \(K(\pi,1)\). The second space is also shown to be \(K(\pi,1)\) by the general result that is invoked to treat 3-equal arrangement. In all cases the fundamental groups are described. Then the author studies analogs of the previous construction for other root systems (i.e., arrangements of intersections of pairs of reflecting hyperplanes). The \(K(\pi,1)\) property is established in some cases (answering another question by Björner) and a general conjecture is raised.
scientific article; zbMATH DE number 913764