Fiber of Persistent Homology on Morse functions

2021/08/17 by Leygonie, Jacob, Beers, David
#Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2108.07512

Abstract

Let f be a Morse function on a smooth compact manifold M with boundary. The path component PH_f^-1(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 ϕ↦ f∘ ϕ by diffeomorphisms of M which are isotopic to the identity. Consequently we derive topological properties of the fiber 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.

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