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

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 $M$.