Local systems on the complexification of an oriented matroid.

Abstract

The dissertation is concerned with topological invariants of arrangements of hyperplanes in complex affine space, particularly those that are defined over the real numbers, in relation to the matroid structure of the arrangement. In the first part, we consider a sheaf on the complement M of a real arrangement whose total space is an infinite cyclic cover. We do so combinatorially, using Salvetti's complex and an open cover due to Paris, both of which can be defined abstractly for any oriented matroid. For hyperplane arrangements, the cohomology of the sheaf is that of the arrangement's (weighted) Milnor fibre <italic>F</italic>. This construction used in an attempt to understand whether, or to what extent, the (co)homology of the Milnor fibre is determined by the matroid of the arrangement. It is known that the cohomology of certain local systems on M can be calculated from the Orlik-Solomon algebra, which is a matroid invariant; however, these methods do not apply to the situation described above when the hyperplanes intersect nongenerically away from the origin. A partial solution is obtained by examining integer torsion: we find a spectral sequence that compares a filtration of the integral homology of <italic>F </italic> with the cohomology of the Orlik-Solomon algebra, over <bold>Z</bold>, using a suitable Koszul boundary map. In the second part, we consider ordered matroids <italic>M</italic> and a double complex obtained from an exterior algebra, graded both by corank and nullity. The first row is the simplicial chain complex of <italic>IN</italic>(<italic> M</italic>), a simplicial complex whose simplices are indexed by the independent sets of the matroid. Dually, the first column is the cochain complex of <italic> IN</italic>(<italic>M</italic>*). We give a combinatorial description of a basis of eigenvectors for the combinatorial Laplacian of a family of boundary maps on the double complex, extending work by Kook, Reiner, and Stanton (1998) on <italic>IN</italic>(<italic>M</italic>). The eigenvalues are enumerated by a weighted version of the Tutte polynomial. This is applied to prove a duality statement for the (co)homology of Orlik-Solomon algebras.