Alexander duality for functions: the persistent behavior of land and water and shore

Authors:
Herbert Edelsbrunner;Michael Kerber
Affiliations:
IST Austria, Klosterneuburg, Austria;IST Austria, Klosterneuburg, Austria
Venue:
Proceedings of the twenty-eighth annual symposium on Computational geometry
Year:
2012

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Abstract

This note contributes to the point calculus of persistent homology by extending Alexander duality from spaces to real-valued functions. Given a perfect Morse function f: Sspacen+1 - [0,1] and a decomposition Sspacen+1 = Uspace ∪ Vspace into two (n+1)-manifolds with common boundary Mspace, we prove elementary relationships between the persistence diagrams of f restricted to Uspace, to Vspace, and to Mspace.