Fiber of Persistent Homology on Morse functions

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 $\varphi \mapsto f\circ \varphi$ 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.

arXiv commentVersion accepted in the Journal of Applied and Computational Topology