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arXiv · 1303.1591

Alexander Duality for Parametrized Homology

Sara Kalisnik Verovsek
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Abstract

This paper extends Alexander duality to the setting of parametrized homology. Let X with be a compact subset of R^n x R (n \geq 2) satisfying certain conditions, let Y be its complement, and let p be the projection onto the second factor. Both X and Y are parametrized spaces with respect to the projection. The parametrized homology is a variant of zigzag persistent homology that measures how the homology of the level sets of the space changes as we vary the parameter. We show that if (X, p|_X) has a well-defined parametrized homology, then the pair (Y, p|_Y) has a well-defined reduced parametrized homology. We also establish a relationship between the parametrized homology of (X, p|_X) and the reduced parametrized homology of (Y, p|_Y).

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Sara Kalisnik Verovsek. 2013-03-07. Alexander Duality for Parametrized Homology. https://doi.org/10.4310/hha.2013.v15.n2.a14

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