The fiber of persistent homology for trees

TL;DR

This paper explores the topology of the space of continuous functions on a tree with a fixed persistent homology barcode, revealing its connection to merge trees and configuration spaces, and computing homotopy types for simple barcodes.

Contribution

It establishes a correspondence between connected components of function spaces and merge trees, and characterizes their homotopy types, advancing understanding of persistent homology on trees.

Findings

Number of connected components equals the number of merge trees with barcode D.

Each component is homotopy equivalent to a constrained configuration space.

Homotopy types are computed explicitly for barcodes with one or two intervals.

Abstract

Consider the space of continuous functions on a geometric tree $X$ whose persistent homology gives rise to a finite generic barcode $D$. We show that there are exactly as many path connected components in this space as there are merge trees whose barcode is $D$. We find that each component is homotopy equivalent to a configuration space on $X$ with specialized constraints encoded by the merge tree. For barcodes $D$ with either one or two intervals, our method also allows us to compute the homotopy type of this space of functions.