Alexander Duality for Parametrized Homology

TL;DR

This paper generalizes Alexander duality to parametrized homology, linking the homological features of a compact set and its complement in a parametrized setting, with implications for understanding level set changes.

Contribution

It introduces a duality theorem for parametrized homology, extending classical Alexander duality to a new, dynamic context involving level set variations.

Findings

Established a relationship between parametrized homology of a set and its complement.

Proved the existence of well-defined reduced parametrized homology for the complement.

Extended Alexander duality to parametrized homology in a new setting.

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).