Homotopy and group cohomology of arrangements

It is well known that the complexification of the complement of the arrangement of reflecting hyperplanes for a finite Coxeter group is an Eilenberg-MacLane space. In general, the cohomology of the complement of a general complex arrangement is well behaved and well understood. In this paper we consider the homotopy theory of such spaces. In particular, we study the Hurewicz map connecting homotopy and homology. As a consequence we are able to derive understanding of the “obstructions” to such spaces being Eilenberg-MacLane spaces. In particular, in the case of arrangements in a three-dimensional vector space, we find that whether or not the complement is Eilenberg-MacLane depends solely on its fundamental group.