Abstract
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. Nous étendons l'opérateur de Billera―Ehrenborg―Readdy entre le trellis d'intersection et la treillis de faces d'un arrangement hyperplans centraux aux arrangements affines et toriques. Pour les arrangements toriques, nous généralisons aussi les résultats fondamentaux de Zaslavsky sur le nombre de régions.
Highlights
Combinatorialists have studied topological objects that are spherical, such as polytopes, or which are homeomorphic to a wedge of spheres, such as those obtained from shellable complexes
We review important results about hyperplane arrangements, the cd-index and coalgebraic techniques that are essential for proving the main results of this paper
(i) What is the right analogue of a regular subdivision in order that it be polytopal? Can flag f -vectors be classified for polytopal subdivisions?
Summary
Combinatorialists have studied topological objects that are spherical, such as polytopes, or which are homeomorphic to a wedge of spheres, such as those obtained from shellable complexes. As in the case of a central hyperplane arrangement, our toric version of the Bayer–Sturmfels result determines the flag f -vector of the face poset of a toric arrangement in terms of its intersection poset. For non-central arrangements, we determine the cd-index of this complex in terms of the lattice of unbounded intersections of the arrangement. The techniques for studying toric arrangements and the unbounded complex of non-central arrangements are similar We present these results in the same paper. We end with many open questions about subdivisions of manifolds
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