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arXiv · 2509.09350

Characterization of the computed homology and cohomology bases -- technical report

Yann-Situ Gazull·Aldo Gonzalez-Lorenzo·Alexandra Bac
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Abstract

Computing homology and cohomology is at the heart of many recent works and a key issue for topological data analysis. Among homological objects, homology generators are useful to locate or understand holes (especially for geometric objects). The present paper provides a characterization of the class of homology bases that are computed by standard algorithmic methods. The proof of this characterization relies on the Homological Discrete Vector Field, a combinatorial structure for computing homology, which encompasses several standard methods (persistent homology, tri-partitions, Smith Normal Form, discrete Morse theory). These results refine the combinatorial homology theory and provide novel ideas to gain more control over the computation of homology generators.

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Yann-Situ Gazull, Aldo Gonzalez-Lorenzo, Alexandra Bac. 2025-09-11. Characterization of the computed homology and cohomology bases -- technical report. https://doi.org/10.1007/978-3-032-09544-2_12

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