Computing simplicial homology based on efficient Smith normal form algorithms Geometric properties of topological spaces are conveniently expressed by algebraic invariants of the space. This paper focuses on methods for the computer calculation of the homology of finite simplicial complexes and its applications. 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. First, the authors provide a review of several algorithms for the calculation of the Smith Normal Form of sparse matrices and compare their running times for actual boundary matrices, then they describe alternative approaches to the calculation of simplicial homology. In the last section they present motivating examples and actual experiments with the GAP package (implemented by the authors). There is an example with calculations of Lie algebra homology.

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  1. Lampret, Leon; Vavpetič, Aleš: (co)homology of Lie algebras via algebraic Morse theory (2016)
  2. Gameiro, Marcio; Hiraoka, Yasuaki; Izumi, Shunsuke; Kramar, Miroslav; Mischaikow, Konstantin; Nanda, Vidit: A topological measurement of protein compressibility (2015)
  3. Burton, Benjamin A.: A new approach to crushing 3-manifold triangulations (2014)
  4. Harker, Shaun; Mischaikow, Konstantin; Mrozek, Marian; Nanda, Vidit: Discrete Morse theoretic algorithms for computing homology of complexes and maps (2014)
  5. Mischaikow, Konstantin; Nanda, Vidit: Morse theory for filtrations and efficient computation of persistent homology (2013)
  6. Erickson, Jeff: Combinatorial optimization of cycles and bases (2012)
  7. Dłotko, Paweł; Kaczynski, Tomasz; Mrozek, Marian; Wanner, Thomas: Coreduction homology algorithm for regular CW-complexes (2011)
  8. Mrozek, Marian; Wanner, Thomas: Coreduction homology algorithm for inclusions and persistent homology (2010)
  9. Bayer, Dave; Taylor, Amelia: Reverse search for monomial ideals (2009)
  10. Hagen, Matthias: Lower bounds for three algorithms for transversal hypergraph generation (2009)
  11. Jonsson, Jakob: Five-torsion in the homology of the matching complex on 14 vertices (2009)
  12. Larrión, F.; Pizaña, M.A.; Villarroel-Flores, R.: The clique operator on matching and chessboard graphs (2009)
  13. Mrozek, Marian; Batko, Bogdan: Coreduction homology algorithm (2009)
  14. Mrozek, Marian; Pilarczyk, Paweł; Ẓelazna, Natalia: Homology algorithm based on acyclic subspace (2008)
  15. Peltier, Samuel; Ion, Adrian; Haxhimusa, Yll; Kropatsch, Walter G.; Damiand, Guillaume: Computing homology group generators of images using irregular graph pyramids (2007)
  16. Álvarez, V.; Armario, J.A.; Frau, M.D.; Real, P.: A Mathematica notebook for computing the homology of iterated products of groups. (2006)
  17. Chari, Manoj K.; Joswig, Michael: Complexes of discrete Morse functions (2005)
  18. Kavvadias, Dimitris J.; Stavropoulos, Elias C.: An efficient algorithm for the transversal hypergraph generation (2005)
  19. Matoušek, Jiří; Ziegler, Günter M.: Topological lower bounds for the chromatic number: a hierarchy (2004)
  20. Dumas, Jean-Guillaume; Heckenbach, Frank; Saunders, David; Welker, Volkmar: Computing simplicial homology based on efficient Smith normal form algorithms (2003)

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