Combinatorial Invariants of Toric Arrangements

Abstract

An arrangement is a collection of subspaces of a topological space. For example, a set of codimension one affine subspaces in a finite dimensional vector space is an arrangement of hyperplanes. A general question in arrangement theory is to determine to what extent the combinatorial data of an arrangement determines the topology of the complement of the arrangement. Established combinatorial structures in this context are matroids and -for hyperplane arrangements in the real vector space- oriented matroids. Let X be the punctured plane C- 0 or the unit circle S 1, and a(1),...,a(n) integer vectors in Z d. By interpreting the a(i) as characters of the torus T=Hom(Z d,X) isomorphic to X d we obtain a toric arrangement in T by considering the set of kernels of the characters. A toric arrangement is covered naturally by a periodic affine hyperplane arrangement in the d-dimensional complex or real vector space V=C d or R d (according to whether X = C- 0 or S 1). Moreover, if V is the real vector space R d the stratification of V given by a finite hyperplane arrangement can be combinatorially characterized by an affine oriented matroid. Our main objective is to find an abstract combinatorial description for the stratification of T given by the toric arrangement in the case X=S 1 - and to develop a concept of toric oriented matroids as an abstract characterization of arrangements of topological subtori in the compact torus (S 1) d. Part of our motivation comes from the possible generalization of known topological results about the complement of "complexified" toric arrangements to such toric pseudoarrangements. Towards this goal, we study abstract combinatorial descriptions of locally finite hyperplane arrangements and group actions thereon. First, we generalize the theory of semimatroids and geometric semilattices to the case of an infinite ground set, and study their quotients under group actions from an enumerative and structural point of view. As a second step, we consider corresponding generalizations of affine oriented matroids in order to characterize the stratification of R d given by a locally finite non-central arrangement in R d in terms of sign vectors.