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

Article

Abstract

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 http://www.pawelpilarczyk.com/chaincon/ 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.

Keywords

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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Copyright information

© 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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