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Simplicial Perturbation Techniques and Effective Homology

  • Rocio Gonzalez-Díaz
  • Belén Medrano
  • Javier Sánchez-Peláez
  • Pedro Real
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4194)

Abstract

In this paper, we deal with the problem of the computation of the homology of a finite simplicial complex after an “elementary simplicial perturbation” process such as the inclusion or elimination of a maximal simplex or an edge contraction. To this aim we compute an algebraic topological model that is a special chain homotopy equivalence connecting the simplicial complex with its homology (working with a field as the ground ring).

Keywords

Simplicial Complex Chain Complex Betti Number Incremental Algorithm Ground Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Delfinado, C., Edelsbrunner, H.: An incremental algorithm for Betti numbers of simplicial complexes on the 3–sphere. Comput. Aided Geom. Design 12, 771–784 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    González–Díaz, R., Real, P.: Towards digital cohomology. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 92–101. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    González–Díaz, R., Real, P.: On the cohomology of 3D digital images. Discrete Applied Math. 147, 245–263 (2005)zbMATHCrossRefGoogle Scholar
  4. 4.
    Gonzalez-Diaz, R., Medrano, B., Real, P., Sánchez-Peláez, J.: Algebraic topological analysis of time-sequence of digital images. Lecture Notes in Computer Science, vol. 139, pp. 208–219 (2005)Google Scholar
  5. 5.
    Gugenheim, V.K.A.M., Lambe, L., Stasheff, J.: Perturbation theory in differential homological algebra, II. Illinois J. Math. 35(3), 357–373 (1991)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Huebschmann, J., Kadeishvili, T.: Small models for chain algebras. Math. Z. 207, 245–280 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    MacLane, S.: Homology. Classic in Math. Springer, Berlin (1995)Google Scholar
  8. 8.
    Munkres, J.R.: Elements of Algebraic Topology. Addison–Wesley Co, Reading (1984)zbMATHGoogle Scholar
  9. 9.
    Real, P.: Homological perturbation theory and associativity. Homology, Homotopy and its Applications 2(5), 51–88 (2000)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Sergeraert, F.: The computability problem in algebraic topology. Adv. Math. 104(1), 1–29 (1994)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Rocio Gonzalez-Díaz
    • 1
  • Belén Medrano
    • 1
  • Javier Sánchez-Peláez
    • 1
  • Pedro Real
    • 1
  1. 1.Departamento de Matemática Aplicada IUniversidad de SevillaSeville(Spain)

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