Measuring and Localing Homology Classes

Details Measuring and Localing Homology Classes Measuring and Localing Homology Classes

2007-05-21

arxiv.org/abs/0705.3061v2

Computer Science

Computational Geometry

References 8 and 22 are fixed

Scientific paper

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 $O(\beta^4 n^3 \log^2 n)$ time, where $n$ is the size of the simplicial complex and $\beta$ is the Betti number of the homology group.

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