Characterization of the Homology and Cohomology Bases Computed by Standard Methods

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 and cohomology 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.

Yann-Situ Gazull, Aldo Gonzalez-Lorenzo, Alexandra Bac, "Characterization of the Homology and Cohomology Bases Computed by Standard Methods." Journal of Mathematical Imaging and Vision, 2026.
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