Fiber of Persistent Homology on Morse functions

TL;DR

This paper characterizes the fiber of the persistent homology map for Morse functions on manifolds, showing it corresponds to diffeomorphism orbits and analyzing its topological structure across various surfaces and dimensions.

Contribution

It establishes a geometric description of the fiber of persistent homology for Morse functions, linking it to diffeomorphism orbits and computing its homotopy type for many surfaces.

Findings

Fiber of PH is the orbit of Morse functions under diffeomorphisms isotopic to identity.

Homotopy type of fibers computed for many compact surfaces.

In 1D, fibers are contractible or circular components.

Abstract

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