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 realvalued 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 $M$.
 Publication:

arXiv eprints
 Pub Date:
 September 2011
 arXiv:
 arXiv:1109.5052
 Bibcode:
 2011arXiv1109.5052E
 Keywords:

 Mathematics  Algebraic Topology;
 Computer Science  Computational Geometry;
 Mathematics  Geometric Topology
 EPrint:
 Keywords: Algebraic topology, homology, Alexander duality, MayerVietoris sequences, persistent homology, point calculus