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

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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.

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Correspondence to Paweł Pilarczyk.

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Communicated by: D. N. Arnold

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Pilarczyk, P., Real, P. Computation of cubical homology, cohomology, and (co)homological operations via chain contraction. Adv Comput Math 41, 253–275 (2015). https://doi.org/10.1007/s10444-014-9356-1

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