A New Means for Investigating 3-Manifolds

  • Vladimir Kovalevsky
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1953)


The paper presents a new method of investigating topological properties of three-dimensional manifolds by means of computers. Manifolds are represented as finite cell complexes. The paper contains definitions and a theorem necessary to transfer some basic knowledge of the classical topology to finite topological spaces. The method is based on subdividing the given set into blocks of simple cells in such a way, that a k-dimensional block be homeomorphic to a k-dimensional ball. The block structure is described by the data structure known as “cell list” which is generalized here for the three-dimensional case. Some experimental results are presented.


Fundamental Group Hausdorff Space Dimensional Sphere Incidence Structure Classical Topology 
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  1. 1.
    Alexandroff, P.: Diskrete Räume, Mat. Sbornik, Vol. 2 (1937) 501–518zbMATHGoogle Scholar
  2. 2.
    Casler, B.G.: An Imbedding Theorem for Connected 3-Manifolds with Boundary, Proceedings of the American Mathematical Society, Vol. 16 (1965) 559–566Google Scholar
  3. 3.
    Dehn, M., Heegaard, P.: Analysis situs, Encyklopädie der mathematischen Wissenschaften, Vol. III, AB3, Leipzig (1907) 153–220Google Scholar
  4. 4.
    Fomenko, A.T., Matveev, S.V.: Algorithmic and Computer Methods for Three-Manifolds, Kluwer (1997)Google Scholar
  5. 5.
    Kong, T.Y., Kopperman, R., Meyer, P.R.: A topological approach to digital topology, Amer. Math. Monthly, Vol. 98 (1991) 901–917zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kovalevsky, V.A.: Finite Topology as Applied to Image Analysis, Computer Vision, Graphics and Image Processing, Vol. 45, No. 2 (1989) 141–161CrossRefGoogle Scholar
  7. 7.
    Kovalevsky, V.A.: Finite Topology and Image Analysis, in Advances in Electronics and Electron Physics, P. Hawkes ed., Academic Press, Vol. 84 (1992) 197–259Google Scholar
  8. 8.
    Kovalevsky, V.A.: A New Concept for Digital Geometry, in Shape in Picture, O, Ying-Lie, Toet, A., Foster, D., Heijmans, H.J.A.M., Meer, P. (Eds.), Springer-Verlag Berlin Heidelberg (1994) 37–51Google Scholar
  9. 9.
    Matveev, S.V.: Computer classification of 3-manifolds, in Russian, TR, University of Cheliabinsk, Russia, (1999)Google Scholar
  10. 10.
    Moise, E.E.: Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. Math. Vol. 56 (1952) 865–902MathSciNetCrossRefGoogle Scholar
  11. 11.
    Poincaré, H.: Analysis situs, J. de l’École Polyt. (2) Vol. 1 (1895) 1–123Google Scholar
  12. 12.
    Stillwell, J.: Classical Topology and Combinatorial Group Theory, Springer (1995)Google Scholar
  13. 13.
    Tietze, H.: Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatsh. Math. Phys. Vol. 19, (1908) 1–118CrossRefMathSciNetGoogle Scholar
  14. 14.
    Turaev, V.G., Viro, O.Y.: State sum invariants of 3-manifolds and quantum 6j-symbol, Topology, Vol. 31, N 4 (1992) 865–902zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Vladimir Kovalevsky
    • 1
  1. 1.Institute of Computer GraphicsUniversity of RostockRostockGermany

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