Equivariant Chain Complexes, Twisted Homology and Relative Minimality of Arrangements

We show that the π-equivariant chain complex (π = π1(M(A))), C•(X̃), associated to a Morse-theoretic minimal CW -structure X on the complement M(A) of an arrangement A, is independent of X . The same holds for all scalar extensions, C•(X̃)⊗Zπ KZ, K a field, where X is an arbitrary minimal CW -structure on a space M . When A is a section of another arrangement Â, we show that the divisibility properties of the first Betti number of the Milnor fiber of A obstruct the homotopy realization of M(A) as a subcomplex of a minimal structure on M(Â). If  is aspherical and A is a sufficiently generic section of Â, then H∗(M(A);L) may be described in terms of π, L and χ(M(A)), for an arbitrary local system L; explicit computations may be done, when  is fiber-type. In this case, explicit KZpresentations of arbitrary abelian scalar extensions of the first non-trivial higher homotopy group of M(A), πp, may also be obtained. For nonresonant abelian scalar extensions, the CZ-rank of πp ⊗Zπ CZ is combinatorially determined.