On a Generalization of Zaslavsky's Theorem for Hyperplane Arrangements

Details On a Generalization of Zaslavsky's Theorem for Hyperplane Arrangements On a Generalization of Zaslavsky's Theorem for Hyperplane Arrangements

2011-11-04

arxiv.org/abs/1111.1251v1

Mathematics

Combinatorics

18 pages, 2 figures

Scientific paper

We define arrangements of codimension 1 submanifolds in a smooth, real 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 an arrangement of submanifolds. 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.

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