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 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.
Loading