On rational \(K[\pi,1]\) spaces and Koszul algebras
The complement of an arrangement of hyperplanes in \(\mathbb C^n\) is known to be a formal space; that is, a space whose rational type is determined by its rational cohomology. Here, rational type must be understood as \(\mathbb Q\)-Bousfield-Kan completion because the fundamental group is non-nilpotent. A main focus in the study of these spaces has been to determine when a complement is a rational \(K(\pi,1)\); that is, when do all higher homotopy groups of the \(\mathbb Q\)-completion vanish. The reason for this interest is that many results on complements come from looking at the rational homotopy theoretic minimal (Sullivan) model of the space. For a space \(X\) with \(\mathbb Q_\infty(X)\) having higher homotopy, the minimal model is virtually impossible to compute. For a rational \(K(\pi,1)\), however, all essential information is contained in degree \(1\) and this simplifies things immensely. The main result of this paper is that a formal space \(X\) is a rational \(K(\pi,1)\) if and only \(H^*(X;\mathbb Q)\) is a Koszul algebra. An algebra \(A\) over \(\mathbb Q\) is Koszul if for every \(p \not = q\), \(\text{Ext}_A^{pq}(\mathbb Q,\mathbb Q) = 0\). Using this characterization, the authors show that complements in \(\mathbb C^3\) satisfying the so-called LCS formula are rational \(K(\pi,1)\)'s.
scientific article; zbMATH DE number 1398841