Contractibility of a persistence map preimage

Abstract

This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of snapshots of solutions, what conclusions can be drawn about solutions of the original dynamical system? We address this challenge in the context of an N dimensional system of ordinary differential equation defined in $${\mathbb {R}}^N$$ R N . To each point in $${\mathbb {R}}^N$$ R N (e.g. an initial condition) we associate a persistence diagram. The main result of this paper is that under this association the preimage of every persistence diagram is contractible. As an application we provide conditions under which multiple time series of persistence diagrams can be used to conclude the existence of a fixed point of the differential equation that generates the time series.

Algorithmic Reconstruction of the Fiber of Persistent Homology on Cell Complexes

“…When K is a tree (Figure 6), we report these statistics both when the domain of PH consists of all filters and when it is restricted to lower star filters. For lower star filters on the interval the fiber is shown by [7] to consist of contractible components. Our computations indicate that this property holds as well for general filters on the interval, however it breaks for other trees where the fiber has loops as indicated by non-trivial Betti numbers β 1 .…”

“…In a section 5 dedicated to experiments, we apply the algorithm to multiple complexes and barcodes, and report statistics about the fiber PH ´1pDq, such as its number of polyhedra and its Betti numbers. Sometimes unexpectedly, most of the properties observed for the 1-dimensional complexes studied in [7,14,15] do not hold for more general complexes. For instance, already for a triangulated 2-sphere the fiber PH ´1pDq over some barcodes D has non-trivial homology in degree 3, unlike all the existing examples of graphs, for which PH ´1pDq has vanishing homology in dimension greater than 1.…”

“…However, understanding the fiber PH ´1pDq remains a challenge, in general. If the underlying simplicial complex K is a subdivision of the unit interval or of the circle, then the fiber PH ´1pDq is a disjoint union of contractible sets [7] and circles [15], respectively. Outside these and a few other 1-dimensional examples, however, little is known.…”

Fiber of Persistent Homology on Morse functions

“…In the appendix we prove the analogues of Propositions 4.2 and 4.3 for continuous functions. The analogues for lower-star filtrations on the subdivided interval and circle have been proved in [8] and [14] respectively.…”

“…In the discrete setting where X = K is a finite simplicial complex and f is compatible with face inclusions, the fiber PH −1 (D) is a complex of polyhedra [11]. In the restricted case where K is a line complex, each path connected component of PH −1 (D) is contractible [8], and it is homeomorphic to a circle in the case where K is a subdivision of the unit circle [14].…”