Alexander Duality for Functions: the Persistent Behavior of Land and Water and Shore

TL;DR

This paper extends Alexander duality to real-valued functions within persistent homology, establishing relationships between persistence diagrams of functions on decomposed spheres and their intersections, enhancing understanding of land-water-shore topology.

Contribution

It introduces a novel extension of Alexander duality to the point calculus of persistent homology for real-valued functions on spheres.

Findings

Established relationships between persistence diagrams of functions on decomposed spheres.

Extended Alexander duality to the setting of persistent homology.

Provided elementary proofs connecting the persistence of functions on U, V, and M.

Abstract

This note contributes to the point calculus of persistent homology by extending Alexander duality to real-valued functions. Given a perfect Morse function $f: S^{n+1} \to [0,1]$ and a decomposition $S^{n+1} = U \cup V$ such that $M = \U \cap V$ is an $n$-manifold, we prove elementary relationships between the persistence diagrams of $f$ restricted to $U$, to $V$, and to $M$.