Persistent homology with non-contractible preimages

TL;DR

This paper studies the topological structure of spaces of functions with fixed persistence diagrams, revealing they are often homotopy equivalent to circles or bouquets of circles, highlighting complex non-contractible features.

Contribution

It characterizes the homotopy types of spaces of functions with fixed persistence diagrams, including cases on the circle and star-shaped trees, showing they are often non-contractible.

Findings

Spaces of functions with fixed persistence diagrams are homotopy equivalent to S^1.

Spaces on Y-shaped and star-shaped trees are homotopy equivalent to S^1 and bouquets of circles.

The topology of these spaces is more complex than previously understood.

Abstract

For a fixed $N$, we analyze the space of all sequences $z=(z_1,\dots,z_N)$, approximating a continuous function on the circle, with a given persistence diagram $P$, and show that the typical components of this space are homotopy equivalent to $S^1$. We also consider the space of functions on $Y$-shaped (resp., star-shaped) trees with a 2-point persistence diagram, and show that this space is homotopy equivalent to $S^1$ (resp., to a bouquet of circles).