On rational K[π,1] spaces and Koszul algebras Original Research Article

The main result of the paper is that a formal topological space X is a rational K[π,1] space if and only if the graded algebra H*(X,Q) is Koszul. This implies the lower central series (LCS) formula for a formal rational K[π,1] space X:imageHere φn=rank(Γn/Γn+1), where {Γn}ngreater-or-equal, slanted1 is the lower central series of the fundamental group π1(X), and P(X,t) is the Poincaré polynomial of X. These results are applied to the complements of complex hyperplane arrangements that are known to be formal spaces. In particular, it is proved that the LCS formula implies the rational K[π,1] property for arrangements in C3.