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

摘要

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 : S~(n+1) → [0,1] and a decomposition S~(n+1) =UUV into two (n + l)-manifolds with common boundary M, we prove elementary relationships between the persistence diagrams of f restricted to U, to V, and to M.