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Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms

In: Algebra, Geometry and Software Systems

Author

Listed:
  • Jean-Guillaume Dumas

    (Laboratoire de Modélisation et Calcul)

  • Frank Heckenbach

    (Universität Erlangen-Nürnberg, Mathematisches Institut)

  • David Saunders

    (University of Delaware, Department of Computer and Information Sciences)

  • Volkmar Welker

    (Philipps-Universität Marburg, Fachbereich Mathematik und Informatik)

Abstract

We recall that the calculation of homology with integer coefficients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices and compare their running times for actual boundary matrices. Then we describe alternative approaches to the calculation of simplicial homology. The last section then describes motivating examples and actual experiments with the GAP package that was implemented by the authors. These examples also include as an example of other homology theories some calculations of Lie algebra homology.

Suggested Citation

  • Jean-Guillaume Dumas & Frank Heckenbach & David Saunders & Volkmar Welker, 2003. "Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms," Springer Books, in: Michael Joswig & Nobuki Takayama (ed.), Algebra, Geometry and Software Systems, pages 177-206, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-05148-1_10
    DOI: 10.1007/978-3-662-05148-1_10
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