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

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

2011-09-23

arxiv.org/abs/1109.5052v1

Mathematics

Algebraic Topology

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

Scientific paper

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 $M$.

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