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Persistent homology of function spaces

Jonathan Block, Fedor Manin, Shmuel Weinberger

TL;DR

The paper develops a persistent-homology framework for studying Lip$(X,Y)$ with the Lip filtration, revealing that nonpositive-curvature targets yield only infinite bars, while simply connected targets with positive weights or H-space structure exhibit uniformly bounded finite bars for fixed domain $X$. It leverages rational homotopy theory and the shadowing principle to translate algebraic obstructions into quantitative Lipschitz-control results for maps and homotopies, producing both general theorems and explicit unbounded-bar examples. The work thus connects geometric group theory, loop-space topology, and quantitative topology, highlighting both the power and limits of current methods. The results have potential implications for understanding the complexity of homotopies in function spaces and for computational topological analysis of Lipschitz-m constrained mappings.

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.

Persistent homology of function spaces

TL;DR

The paper develops a persistent-homology framework for studying Lip with the Lip filtration, revealing that nonpositive-curvature targets yield only infinite bars, while simply connected targets with positive weights or H-space structure exhibit uniformly bounded finite bars for fixed domain . It leverages rational homotopy theory and the shadowing principle to translate algebraic obstructions into quantitative Lipschitz-control results for maps and homotopies, producing both general theorems and explicit unbounded-bar examples. The work thus connects geometric group theory, loop-space topology, and quantitative topology, highlighting both the power and limits of current methods. The results have potential implications for understanding the complexity of homotopies in function spaces and for computational topological analysis of Lipschitz-m constrained mappings.

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.

Paper Structure

This paper contains 29 sections, 41 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Open questions
  5. Outline of the paper
  6. Definitions and general facts
  7. Persistent homology
  8. Distance between persistence modules
  9. Theory of infinite rank persistence modules
  10. Persistent homology of Lipschitz functions
  11. Maps to manifolds of nonpositive curvature
  12. Loop spaces of closed manifolds following Nabutovsky--Rotman
  13. Loop spaces following Gromov
  14. Tools from quantitative homotopy theory
  15. Rational homotopy theory
  16. ...and 14 more sections

Key Result

Theorem A

If $Y$ is a complete locally CAT(0) metric space (e.g. a complete nonpositively curved manifold) then all of the bars in $PH_*(\mathop{\mathrm{Lip}}\nolimits(X,Y), \mathop{\mathrm{Lip}}\nolimits)$ are of infinite length.

Figures (1)

  • Figure 1: Construct maps $u_{d,h}:S^2 \times S^3 \to Y$ by "budding off" a small sphere and then projecting the rest onto the $S^3$ factor.

Theorems & Definitions (65)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Remark 1.1
  • Example 2.1
  • Theorem 2.2: Isometry theorem CEH
  • Proposition 2.3
  • Theorem 2.4: CCS
  • ...and 55 more