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 PH −1 f (D) containing f of the space of Morse functions giving rise to the same Persistent Homology D = PH(f ) is shown to be the same as the orbit of f under pre-composition by isotopies of M , i.e. φ → f • φ. Consequently we derive topological properties of the fiber PH −1 f (D): In particular we compute its homotopy type when M is a compact surface. In the 1dimensional 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.