2025/09/11 by Gazull, Yann-Situ, Gonzalez-Lorenzo, Aldo, Bac, Alexandra
#Algebraic Topology (math.AT) #Computational Geometry (cs.CG) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.09350
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.