Fiber of persistent homology on morse functions

Abstract

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

Deformations of circle-valued Morse functions on 2-torus

“…More details on these results can be found in [13]. We will also note that recently homotopy properties of orbits were applied to some questions on persistent homology of Morse functions by J. Leygonie and D. Beers [7].…”