Fiber of persistent homology on morse functions

Abstract Let f be a Morse function on smooth compact manifold $$M$$ M with boundary. The path component $$\mathrm {PH}^{-1}_f(D)$$ PH f - 1 ( D ) containing of the space functions giving rise to same Persistent Homology $$D=\mathrm {PH}(f)$$ = is shown as orbit under pre-composition $$\phi \mapsto f\circ \phi $$ ϕ ↦ ∘ by diffeomorphisms which are isotopic identity. Consequently we derive topological properties fiber : In particular compute its homotopy type for many surfaces . 1-dimensional settings where unit interval or circle extend analysis continuous and show that fibers made contractible circular components respectively.