Digital Manifolds and the Theorem of Jordan-Brouwer

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.