Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms

Jean-Guillaume Dumas, Frank Heckenbach, David Saunders and Volkmar Welker
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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

A chapter in Algebra, Geometry and Software Systems, 2003, pp 177-206 from Springer

Abstract: 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 (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-05148-1_10

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DOI: 10.1007/978-3-662-05148-1_10

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