Advances in Computational Mathematics

, Volume 41, Issue 1, pp 253–275 | Cite as

Computation of cubical homology, cohomology, and (co)homological operations via chain contraction

  • Paweł PilarczykEmail author
  • Pedro Real


We introduce algorithms for the computation of homology, cohomology, and related operations on cubical cell complexes, using the technique based on a chain contraction from the original chain complex to a reduced one that represents its homology. This work is based on previous results for simplicial complexes, and uses Serre’s diagonalization for cubical cells. An implementation in C++ of the introduced algorithms is available at together with some examples. The paper is self-contained as much as possible, and is written at a very elementary level, so that basic knowledge of algebraic topology should be sufficient to follow it.


Algorithm Software Homology Cohomology Computational homology Cup product Alexander-Whitney coproduct Chain homotopy Chain contraction Cubical complex 

Mathematics Subject Classification (2010)

55N35 52B99 55U15 55U30 55-04 


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Universidade do Minho, Centro de MatemáticaBragaPortugal
  2. 2.Departamento de Matemática Aplicada I, E. T. S. de Ingeniería InformáticaUniversidad de SevillaSevillaSpain

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