Diagram Models for the Covers of the Salvetti Complex

To every affine real arrangement of hyperplanes AR we associate a family of diagrams of spaces over the face poset of the arrangement. We show that any cover of the complement of the complexification of AR is homotopy equivalent to the homotopy colimit of one of the diagrams. More precisely, we show that any cover of the Salvetti complex is isomorphic to the order complex of the poset limit of one of the diagrams. We thus obtain explicit simplicial models for covers of the Salvetti complex. Introduction Let V be a d-dimensional complex vector space. An arrangement of hyperplanes in V is a finite set A = {Hi}i=1,...,n of affine or linear codimension 1 subspaces of V . The arrangement induces a stratification of the ambient space by its hyperplanes and their intersections. The poset of strata, i.e. the intersections inA, is customarily perceived as the combinatorial data of the arrangement. On the topological side, it is interesting to study the link ⋃ Hi and the complement of the arrangement M(A) := V \ ⋃ Hi. One of the main questions in arrangement theory is to clarify to what extent the combinatorial data of the arrangement determine topological invariants of the complement or of the link of the arrangement. Our interest restricts now to the case where V is a complex vector space. A famous open question in this direction is the so-called K(π, 1)-problem. An arrangement is said to be K(π, 1) if the higher homotopy groups of the complement πi(M(A)) vanish for i > 1. There are two large classes of arrangements that were shown to be K(π, 1) in classical works by Deligne (complexified simplicial arrangements, see [D]) and Falk-Randell and Terao (supersolvable arrangements, see [FR], [T]): both these classes admit a purely combinatorial characterization. It is an open question whether being K(π, 1) is a combinatorial property in general. A very useful tool in studying the topology of arrangements are combinatorial models for M(A), i.e. cell complexes that are built from the combinatorial data of the arrangement and that model M(A) up to homotopy equivalence or even homeomorphism. Many combinatorial models for M(A) exist at present. Among them let us only mention the Salvetti complex, introduced by Mario Salvetti in [S]; it models the homotopy type of the complement of complexified arrangements (i.e., complex arrangements where the defining equations of the Hi have real coefficients), and The author acknowledges support for this project by ETH research grant TH-10/02-3. 1