On the Homotopy Theory of Arrangements, Ii

In \On the homotopy theory of arrangements" published in 1986 the authors gave a comprehensive survey of the subject. This article updates and continues the earlier article, noting some key open problems. Let M be the complement of a complex arrangement. Our interest here is in the topology, and especially the homotopy theory of M, which turns out to have a rich structure. In the rst paper of this name 37], we assembled many of the known results; in this paper we wish to summarize progress in the intervening years, to reiterate a few key unsolved questions, and propose some new problems we nd of interest. In the rst section we establish some terminology and notation, and discuss general homotopy classiication problems. We introduce the matroid-theoretic terminology that has become more prevalent in the subject in recent years. In this section we also sketch Rybnikov's construction of arrangements with the same ma-troidal structure but non-isomorphic fundamental groups. In Section 2 we consider some algebraic properties of the fundamental group of the arrangement. Properties of interest include the lower central series, the Chen groups, the rational homotopy theory of the complement, and the cohomology of the group. At the time of our rst paper many questions in this area were in ux, so we make a special eeort here to clarify the situation. The group cohomology is naturally of interest in the third section as well, which focuses on when or if the complement is aspherical. It is this property which fostered much of the initial interest in arrangements (in the guise of the pure braid space); it is of interest that the determination of when the complement is aspherical is far from settled. Finally, in the fourth section we consider what one might call the topology of the fundamental group. We describe group presentations that have been discovered since the publication of 37], including the recent development of braided wiring diagrams. We also sketch the considerable progress in the study of the Milnor ber associated with an arrangement. In 1992 the long-awaited book Arrangements of Hyperplanes, by Peter Orlik and Hiroaki Terao appeared, to the delight of all of us working in arrangements. We refer the reader to this text as a general reference on arrangements, and adopt their notation and terminology except where speciied. We also mention that perhaps the most interesting development in arrangements in the last ten …