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    Measuring and Localing Homology Classes

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    We develop a method for measuring and localizing homology classes. This involves two problems. First, we define relevant notions of size for both a homology class and a homology group basis, using ideas from relative homology. Second, we propose an algorithm to compute the optimal homology basis, using techniques from persistent homology and finite field algebra. Classes of the computed optimal basis are localized with cycles conveying their sizes. The algorithm runs in O(β4n3log2n)O(\beta^4 n^3 \log^2 n) time, where nn is the size of the simplicial complex and β\beta is the Betti number of the homology group.Comment: References 8 and 22 are fixe
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