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

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 $M$.

arXiv commentKeywords: Algebraic topology, homology, Alexander duality, Mayer-Vietoris sequences, persistent homology, point calculus

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