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# Homology, Homotopy and Applications

## Volume 15 (2013)

### Number 2

### Alexander duality for parametrized homology

Pages: 227 – 243

DOI: http://dx.doi.org/10.4310/HHA.2013.v15.n2.a14

#### Author

#### Abstract

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. This paper extends Alexander Duality to this setting. Let $X \subset \mathbb{R}^n \times \mathbb{R}$ with $n\geq 2$ be a compact set satisfying certain conditions, let $Y = (\mathbb{R}^n \times \mathbb{R}) \setminus X$, and let $p$ be the projection onto the second factor. Both $X$ and $Y$ are parametrized spaces with respect to the projection. 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

Alexander duality, persistent homology, zigzag persistence, levelset zigzag persistence

#### 2010 Mathematics Subject Classification

55N05, 55U30

Published 4 December 2014