ALEXANDER DUALITY FOR PARAMETRIZED HOMOLOGY

作者：Kalisnik Sara^*

来源：Homology, Homotopy and Applications, 2013, 15(2): 227-243.

DOI：10.4310/HHA.2013.v15.n2.a14

• Alexander duality
• persistent homology
• zigzag persistence
• levelset zigzag persistence

This paper extends Alexander duality to the setting of parametrized homology. Let X subset of R-n x R with n >= 2 be a compact set satisfying certain conditions, let Y = (R-n x R)\X, 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 vertical bar X) has a well-defined parametrized homology, then the pair (Y; p vertical bar Y ) has a well-defined reduced parametrized homology. We also establish a relationship between the parametrized homology of (X; p vertical bar X) and the reduced parametrized homology of (Y; p vertical bar Y ).

• 出版日期2013