Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebraic interleavings of spaces over the classifying space of the circle

Katsuhiko Kuribayashi, Takahito Naito, Shun Wakatsuki, Toshihiro Yamaguchi

TL;DR

The paper develops a Cohomology Interleaving Distance (CohI) for spaces over the classifying space $BS^1$ by transporting spaces to persistence differential graded (dg) modules via singular cochains. It proves that CohI coincides with existing interleaving distances in the homotopy-theoretic framework ($d_{HC}=d_{IHC}=d_{HI}$), enabling computation of homotopy-consistent distances through bottleneck distances on cohomology barcodes. Extending to dg $\mathbb K[u]$-modules, the authors relate CohI to barcode bottlenecks at even/odd levels and show the overall CohI distance is the maximum of two parity-based distances. They also derive cup-length bounds for spaces over $BS^1$, analyze several classes of spaces (including BV-exact and formal spaces), and provide explicit toy computations for complex projective spaces and related bundles, illustrating the practical computability and interpretability of the CohI framework in persistence and rational homotopy contexts.

Abstract

We bring spaces over the classifying space $BS^1$ of the circle group $S^1$ to persistence theory via the singular cohomology with coefficients in a field. Then, the {\it cohomology} interleaving distance (CohID) between spaces over $BS^1$ is introduced and considered in the category of persistent differential graded modules. In particular, we show that the distance coincides with the {\it interleaving distance in the homotopy category} in the sense of Lanari and Scoccola and the {\it homotopy interleaving distance} in the sense of Blumberg and Lesnick. Moreover, upper and lower bounds of the CohID are investigated with the cup-lengths of spaces over $BS^1$. As a computational example, we explicitly determine the CohID for complex projective spaces by utilizing the bottleneck distance of barcodes associated with the cohomology of the spaces.

Algebraic interleavings of spaces over the classifying space of the circle

TL;DR

The paper develops a Cohomology Interleaving Distance (CohI) for spaces over the classifying space by transporting spaces to persistence differential graded (dg) modules via singular cochains. It proves that CohI coincides with existing interleaving distances in the homotopy-theoretic framework (), enabling computation of homotopy-consistent distances through bottleneck distances on cohomology barcodes. Extending to dg -modules, the authors relate CohI to barcode bottlenecks at even/odd levels and show the overall CohI distance is the maximum of two parity-based distances. They also derive cup-length bounds for spaces over , analyze several classes of spaces (including BV-exact and formal spaces), and provide explicit toy computations for complex projective spaces and related bundles, illustrating the practical computability and interpretability of the CohI framework in persistence and rational homotopy contexts.

Abstract

We bring spaces over the classifying space of the circle group to persistence theory via the singular cohomology with coefficients in a field. Then, the {\it cohomology} interleaving distance (CohID) between spaces over is introduced and considered in the category of persistent differential graded modules. In particular, we show that the distance coincides with the {\it interleaving distance in the homotopy category} in the sense of Lanari and Scoccola and the {\it homotopy interleaving distance} in the sense of Blumberg and Lesnick. Moreover, upper and lower bounds of the CohID are investigated with the cup-lengths of spaces over . As a computational example, we explicitly determine the CohID for complex projective spaces by utilizing the bottleneck distance of barcodes associated with the cohomology of the spaces.
Paper Structure (9 sections, 30 theorems, 84 equations, 1 figure)

This paper contains 9 sections, 30 theorems, 84 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Future work and perspective
  3. The interleaving distance and the bottleneck distance
  4. Interleavings up to homotopy
  5. Interleavings up to homotopy between persistence dg modules
  6. Interleavings of dg ${\mathbb K}[u]$-modules
  7. The cohomology interleaving of spaces over $BS^1$
  8. Toy examples
  9. Some rational homotopy invariants and the CohID

Key Result

Theorem 2.5

B-S The function $d_{\text{\em I}}$ defined above is an extended pseudometric on the class of objects in ${\mathcal{C}}^{({\mathbb R}, \leq)}$.

Theorems & Definitions (78)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Example 2.7
  • Lemma 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 68 more