9 research outputs found
Affine and toric hyperplane arrangements
We extend the Billera-Ehrenborg-Readdy map between the intersection lattice
and face lattice of a central hyperplane arrangement to affine and toric
hyperplane arrangements. For arrangements on the torus, we also generalize
Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure
On a Generalization of Zaslavsky's Theorem for Hyperplane Arrangements
We define arrangements of codimension-1 submanifolds in a smooth manifold
which generalize arrangements of hyperplanes. When these submanifolds are
removed the manifold breaks up into regions, each of which is homeomorphic to
an open disc. The aim of this paper is to derive formulas that count the number
of regions formed by such an arrangement. We achieve this aim by generalizing
Zaslavsky's theorem to this setting. We show that this number is determined by
the combinatorics of the intersections of these submanifolds.Comment: version 3: The title had a typo in v2 which is now fixed. Will appear
in Annals of Combinatorics. Version. 2: 19 pages, major revision in terms of
style and language, some results improved, contact information updated, final
versio
Orientations, lattice polytopes, and group arrangements II: Modular and integral flow Polynomials of graphs
We study modular and integral flow polynomials of graphs by means of subgroup arrangements and lattice polytopes. We introduce an Eulerian equivalence relation on orientations, flow arrangements, and flow polytopes; and we apply the theory of Ehrhart polynomials to obtain properties of modular and integral flow polynomials. The emphasis is on the geometrical treatment through subgroup arrangements and Ehrhart polynomials. Such viewpoint leads to a reciprocity law on the modular flow polynomial, which gives rise to an interpretation on the values of the modular flow polynomial at negative integers and answers a question by Beck and Zaslavsky.Regal Entertainment Group (Competitive Earmarked Research Grants 600703)Regal Entertainment Group (Competitive Earmarked Research Grants 600506)Regal Entertainment Group (Competitive Earmarked Research Grants 600608
Affine and toric arrangements
We extend the Billera―Ehrenborg―Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For toric arrangements, we also generalize Zaslavsky's fundamental results on the number of regions
Arrangements of Submanifolds and the Tangent Bundle Complement
Drawing parallels with the theory of hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold we consider a finite collection \A of locally flat codimension submanifolds that intersect like hyperplanes. To such an arrangement we associate two posets: the \emph{poset of faces} (or strata) \FA and the \emph{poset of intersections} L(\A). We also associate two topological spaces to \A. First, the complement of the union of submanifolds in which we call the \emph{set of chambers} and denote by \Ch. Second, the complement of union of tangent bundles of these submanifolds inside which we call the \emph{tangent bundle complement} and denote by M(\A). Our aim is to investigate the relationship between combinatorics of the posets and topology of the complements.
We generalize the Salvetti complex construction in this setting and also charcterize its connected covers using incidence relations in the face poset. We also demonstrate some calculations of the fundamental group and the cohomology ring. We apply these general results to study arrangements of spheres, projective spaces, tori and pseudohyperplanes. Finally we generalize Zaslavsky\u27s classical result in order to count the number of chambers