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Persistent cohomology operations and Gromov-Hausdorff estimates

Anibal M. Medina-Mardones, Ling Zhou

TL;DR

This work builds a rigorous foundation for persistent cohomology operations, introducing Im_θ and Ker_θ as persistent invariants and proving their stability under Gromov--Hausdorff perturbations. It proves decomposition formulas for wedge sums and Cartesian products, and demonstrates that persistent cohomology operations can yield strictly sharper Gromov--Hausdorff lower bounds than persistent homology by leveraging critical radii and quotient geometry. The authors compute explicit radii for spheres, real projective spaces, and Lens spaces, and show, for example, that d_I(H_m^VR(RP^n)) < (π - ζ_n)/2 while d_I(Im_Sq^k^VR(RP^n)) ≥ π/3, yielding stronger GH estimates between RP^n and S_RP^n. Collectively, the paper advances the intersection of persistence, cohomology operations, and Riemannian geometry with concrete, geometry-driven invariants and stability results that sharpen space comparison in practice.

Abstract

We establish the foundations of the theory of persistent cohomology operations, derive decomposition formulas for wedge sums and products, and prove their Gromov-Hausdorff stability. We use these results to construct pairs of Riemannian pseudomanifolds for which the Gromov-Hausdorff estimates derived from persistent cohomology operations are strictly sharper than those obtained using persistent homology.

Persistent cohomology operations and Gromov-Hausdorff estimates

TL;DR

This work builds a rigorous foundation for persistent cohomology operations, introducing Im_θ and Ker_θ as persistent invariants and proving their stability under Gromov--Hausdorff perturbations. It proves decomposition formulas for wedge sums and Cartesian products, and demonstrates that persistent cohomology operations can yield strictly sharper Gromov--Hausdorff lower bounds than persistent homology by leveraging critical radii and quotient geometry. The authors compute explicit radii for spheres, real projective spaces, and Lens spaces, and show, for example, that d_I(H_m^VR(RP^n)) < (π - ζ_n)/2 while d_I(Im_Sq^k^VR(RP^n)) ≥ π/3, yielding stronger GH estimates between RP^n and S_RP^n. Collectively, the paper advances the intersection of persistence, cohomology operations, and Riemannian geometry with concrete, geometry-driven invariants and stability results that sharpen space comparison in practice.

Abstract

We establish the foundations of the theory of persistent cohomology operations, derive decomposition formulas for wedge sums and products, and prove their Gromov-Hausdorff stability. We use these results to construct pairs of Riemannian pseudomanifolds for which the Gromov-Hausdorff estimates derived from persistent cohomology operations are strictly sharper than those obtained using persistent homology.

Paper Structure

This paper contains 24 sections, 5 theorems, 90 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Persistence and Vietoris--Rips filtrations
  3. $\mathbb{R}$-diagrams and interleaving distance
  4. $\mathbb{R}$-spaces and homotopy interleaving distance
  5. Persistence modules and bottleneck distance
  6. Vietoris--Rips filtrations and stability
  7. Persistent Cohomology Operations
  8. Eilenberg--MacLane spaces
  9. Cohomology operations
  10. Steenrod operations
  11. Persistent $\theta$-modules
  12. Sums and products
  13. Stability
  14. Critical Radii and Spherical Quotients
  15. Kuratowski embedding and critical radii
  16. ...and 9 more sections

Key Result

Theorem A

For any two pointed metric spaces $\mathcal{X}$ and $\mathcal{Y}$, a linear cohomology operation $\theta$ satisfies: where the wedge sum $\mathcal{X} \vee \mathcal{Y}$ is equipped with the gluing metric.

Figures (3)

  • Figure 1: Let $\mathcal{M}$ be a closed Riemannian manifold. Top row: persistent reduced homology barcodes of $\mathcal{M}$. Bottom row:$\mathop{\mathrm{Im}}\nolimits_\theta$-barcodes of $\mathcal{M}$. In each figure, the gray region represents where additional bars could potentially exist within the corresponding barcode. See \ref{['ss:barcode_general']} for details.
  • Figure 2: Let $\mathbb{VS} = \mathbb{VS}^{u_1,\dots,u_n}$ for some tuple of non-negative integers. Top row: persistent reduced homology barcodes of $\mathbb{VS}$, where the dot $(0,\zeta_m)$ has multiplicity $u_m$ which can be zero. Bottom row:$\mathop{\mathrm{Im}}\nolimits_\theta$-barcodes of $\mathbb{VS}$ where $\theta \in \mathcal{O}(\ell,m)$ with $\ell \neq m$. In each figure, the gray region represents where additional bars could exist within the corresponding barcode. See \ref{['ss:barcode_Sn']} for the case when the wedge sum is a single sphere and see \ref{['ss:barcodes_VS']} for the general case.
  • Figure 3: Let $m \in \mathbb{N}$ and $\mathrm{Sq}^k \in \mathcal{O}(m-k, k)$. Top row: the persistent reduced homology barcode of $\mathbb{RP}^n$. Bottom row: the $\mathop{\mathrm{Im}}\nolimits_{\mathrm{Sq}^k}$-barcode of $\mathbb{RP}^n$. In each figure, the gray region represents where additional bars could exist within the corresponding barcode.

Theorems & Definitions (22)

  • Theorem A
  • Theorem B
  • Theorem : chazal2009gromovchazal2014geometric
  • Theorem C
  • Theorem D
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 12 more