Fiber of Persistent Homology on Morse functions

Let f be a Morse function on a smooth compact manifold M with boundary. The path component PH 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 7→ f ◦ φ. Consequently we derive topological properties of the fiber PH 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.

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