Measuring and Localizing Homology Classes

We develop a method for measuring and localizing homology classes. This involves two problems. First, we dene 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 nite eld algebra. Classes of the computed optimal basis are localized with cycles conveying their sizes. The algorithm runs in O( 4 n 3 log 2 n) time, where n is the size of the simplicial complex and is the Betti number of the homology group.

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