Persistent homology of function spaces

TL;DR

This paper explores the topological structure of the space of maps between manifolds by analyzing the Lipschitz constant as a height function, using persistent homology to understand the landscape of homotopies and higher-dimensional cycles.

Contribution

It introduces a novel application of persistent homology to study the Morse landscape of function spaces, extending previous work to higher-dimensional cycles.

Findings

Identifies topological features of the Lipschitz constant landscape.

Provides initial results on the persistence of homotopies.

Suggests new methods for analyzing function space topology.

Abstract

We can view the Lipschitz constant as a height function on the space of maps between two manifolds and ask (as Gromov did nearly 30 years ago) what its ``Morse landscape'' looks like: are there high peaks, deep valleys and mountain passes? A simple and relatively well-studied version of this question: given two points in the same component (homotopic maps), does a path between them (a homotopy) have to pass through maps of much higher Lipschitz constant? Now we also consider similar questions for higher-dimensional cycles in the space. We make this precise using the language of persistent homology and give some first results.