On a Generalization of Zaslavsky's Theorem for Hyperplane Arrangements

TL;DR

This paper generalizes Zaslavsky's theorem to arrangements of codimension-1 submanifolds in smooth manifolds, providing formulas to count the resulting regions based on intersection combinatorics.

Contribution

It introduces a new framework for arrangements of submanifolds and extends Zaslavsky's theorem to this broader setting.

Findings

Derived formulas for counting regions in submanifold arrangements

Extended Zaslavsky's theorem to smooth manifolds

Showed the count depends on intersection combinatorics

Abstract

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.