Alexander Duality for Parametrized Homology
Algebraic Topology
2018-10-09 v2
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).
Keywords
Cite
@article{arxiv.1303.1591,
title = {Alexander Duality for Parametrized Homology},
author = {Sara Kalisnik Verovsek},
journal= {arXiv preprint arXiv:1303.1591},
year = {2018}
}