Determining the components of the complement of a Digital (n−1)-manifold in ℤn

  • R. Ayala
  • E. Domínguez
  • A. R. Francés
  • A. Quintero
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1176)


The goal of this paper is to determine the components of the complement of digital manifolds in the standard cubical decomposition of Euclidean spaces for arbitrary dimensions. Our main result generalizes the Morgenthaler-Rosenfeld's one for (26, 6)-surfaces in ℤ3 [9]. The proof of this generalization is based on a new approach to digital topology sketched in [5] and developed in [2].


Digital space digital n-manifold digital and topological index Jordan-Brouwer Separation Theorem 


  1. 1.
    R. Ayala, E. Domínguez, A.R. Francés, A. Quintero, J. Rubio. On surfaces in digital topology. Proceedings of the 5th Colloquium on Discrete Geometry for Computer Imagery DGCI'5. 271–276 (1995).Google Scholar
  2. 2.
    R. Ayala, E. Domínguez, A.R. Francés, A. Quintero, J. Rubio. A polyhedral approach to n-dimensional digital topology. Preprint.Google Scholar
  3. 3.
    R.H. Bing. The geometric topology of 3-manifolds. Coll. Pub., 40. Amer. Math. Soc. 1983.Google Scholar
  4. 4.
    A. Dold. Algebraic Topology. Die Grundlehren der Math., 200. Springer, 1972.Google Scholar
  5. 5.
    E. Domínguez, A.R. Francés, A. Márquez. A Framework for Digital Topology. Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, vol. 2, 65–70, 1993.Google Scholar
  6. 6.
    E. Khalimsky, R. Kopperman, P.R. Meyer. Computer Graphics and connected topologies on finite ordered sets. Topology and its Applications, 36 (1990), 1–17.Google Scholar
  7. 7.
    T.Y. Kong, A.W. Roscoe. Continuous Analogs of Axiomatized Digital Surfaces. Computer Vision, Graphics, and Image Processing 29, 60–86 (1985).Google Scholar
  8. 8.
    W.S. Massey. Homology and Cohomology Theory. Marcel Dekker, 1978.Google Scholar
  9. 9.
    D.G. Morgenthaler, A. Rosenfeld. Surfaces in Three-Dimensional Digital Images. Information and Control 51, 227–247 (1981).Google Scholar
  10. 10.
    F. Raymond. Separation and union theorems for generalized manifolds with boundary. Michigan Math. J. 7 (1970) 1–21.Google Scholar
  11. 11.
    A. Rosenfeld. Digital Topology. Amer. Math. Monthly, 86 (1979), 621–630.Google Scholar
  12. 12.
    C.P. Rourke, B.J. Sanderson. Introduction to Piecewise linear topology. Ergebnisse der Math., 69. Springer, 1972.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • R. Ayala
    • 1
  • E. Domínguez
    • 2
  • A. R. Francés
    • 2
  • A. Quintero
    • 1
  1. 1.Dpt. de Algebra, Computación, Geometría y Topología. Facultad de MatemáticasUniversidad de SevillaSevillaSpain
  2. 2.Dpt. de Informática e Ingeniería de Sistemas. Facultad de CienciasUniversidad de ZaragozaZaragozaSpain

Personalised recommendations