Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms

  • Jean-Guillaume Dumas
  • Frank Heckenbach
  • David Saunders
  • Volkmar Welker
Conference paper

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.

Keywords

Simplicial Complex Homology Group Minimal Polynomial Free Resolution Minimal Free Resolution 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Jean-Guillaume Dumas
    • 1
  • Frank Heckenbach
    • 2
  • David Saunders
    • 3
  • Volkmar Welker
    • 4
  1. 1.Laboratoire de Modélisation et CalculGrenobleFrance
  2. 2.Mathematisches InstitutUniversität Erlangen-NürnbergErlangenGermany
  3. 3.Department of Computer and Information SciencesUniversity of DelawareNewarkUSA
  4. 4.Fachbereich Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany

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