Digital Manifolds and the Theorem of Jordan-Brouwer

TL;DR

This paper establishes a digital analog of the Jordan-Brouwer theorem by translating digital sets into simplicial complexes, thus defining digital manifolds and proving their separating properties in discrete spaces.

Contribution

It introduces a method to define digital manifolds in Z^n and proves the digital Jordan-Brouwer theorem using simplicial complexes.

Findings

Digital manifolds of dimension n-1 are well-defined in Z^n.

The digital Jordan-Brouwer theorem is proven for these manifolds.

A technique to translate digital sets into simplicial complexes is developed.

Abstract

We give an answer to the question given by T.Y.Kong in his article "Can 3-D Digital Topology be Based on Axiomatically Defined Digital Spaces?" In this article he asks the question, if so called "good pairs" of neighborhood relations can be found on the set Z^n such that the existence of digital manifolds of dimension n-1, that separate their complement in exactly two connected sets, is guaranteed. To achieve this, we use a technique developed by M. Khachan et.al. A set given in Z^n is translated into a simplicial complex that can be used to study the topological properties of the original discrete point-set. In this way, one is able to define the notion of a (n-1)-dimensional digital manifold and prove the digital analog of the Jordan-Brouwer-Theorem.