Digital Manifolds and the Theorem of Jordan-Brouwer

2011/11/13 by Martin Hünniger, Hünniger, Martin
Computer Science · Mathematics · #Computer Vision and Pattern Recognition (cs.CV) #Digital Image Processing Techniques #FOS: Computer and information sciences #FOS: Mathematics #Geometric Topology (math.GT) #Interconnection Networks and Systems #Topological and Geometric Data Analysis #cs.CV #math.GT

paper · pdf · doi:10.48550/arxiv.1111.3000

34 pages, 12 figures

arxiv created 2011/11/13 · openalex publication_date 2011/11/13 · arxiv updated 2011/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Zⁿ 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 Zⁿ 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.

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