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Persistent homology for functionals

Ulrich Bauer, Anibal M. Medina-Mardones, Maximilian Schmahl

Abstract

We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the Unstable Minimal Surface Theorem given by Morse and Tompkins in a modern and rigorous framework.

Persistent homology for functionals

Abstract

We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the Unstable Minimal Surface Theorem given by Morse and Tompkins in a modern and rigorous framework.

Paper Structure

This paper contains 10 sections, 20 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Filtered spaces and persistence theory
  3. Generalized Morse inequalities
  4. Existence of persistence diagrams
  5. Local connectivity and q-tameness
  6. Continuity
  7. Persistent homology and functional topology
  8. The unstable minimal surface problem
  9. Morse's local connectivity conditions
  10. Vietoris and ech homology for compact metric spaces

Key Result

Theorem 1

If the sublevel set filtration of a function $f \colon X \to \mathbb{R}$ is compact and $\mathrm{LHS}$, then its persistent homology is also q-tame.

Figures (1)

  • Figure 1: A closeup of the Hawaiian earring $\mathbb{H}^1$.

Theorems & Definitions (50)

  • Theorem
  • Theorem 2.1: Chazal.2016aChazal.2016b
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Definition 4.1
  • Definition 4.2
  • ...and 40 more