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Algebraic interleavings of spaces over the classifying space of the circle

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Pith's one-line read The paper proves that, for persistence differential graded modules over a field, the homotopy interleaving distance, the homotopy-commutative interleaving distance, and the interleaving distance in the homotopy category all coincide with…

desk verdict A clean, well-written paper proving the equality of four interleaving distances on persistence dg modules and turning that into a computable barcode distance for spaces over BS^1, with only a minor repairable slip in the proof of Theorem 3.3. read the letter →

arxiv 2501.09257 v1 pith:PTKGMS7O submitted 2025-01-16 math.AT

Katsuhiko Kuribayashi , Takahito Naito , Shun Wakatsuki , Toshihiro Yamaguchi This is my paper · ORCID
classification math.AT MSC 55N3155U1555P6255U35
keywords interleavingdistancecohomologypersistencedifferentialgradedmodulebarcodebottleneckclassifyingspaceBS^1cup-lengthSullivanmodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that, for persistence differential graded (dg) modules over a field, three homotopy-level notions of interleaving distance are all equal, and that this common distance is the cohomology interleaving distance obtained by comparing homology barcodes degree by degree. The point is computational: homotopy-level comparisons, which are hard to access directly, become bottleneck distances between ordinary persistence barcodes. The paper applies this to spaces equipped with a map to the classifying space $BS^1$ of the circle, where singular cohomology has a natural module structure over the polynomial ring $K[u]$, so each space over $BS^1$ yields a persistence dg module. This gives a numerical way to compare spaces over $BS^1$ even when no morphism between them exists, with explicit computations for complex projective spaces and for two circle-orbit spaces that are hard to tell apart by rational homotopy invariants.

What carries the argument

The load-bearing object is the functor $C$ that converts a dg $K[u]$-module $M$ into a persistence dg module by placing $\Sigma^{2i}M$ at integer index $i$ and using multiplication by $u$ as the structure map $i\to i+1$; for spaces over $BS^1$, the singular cochain complex $C^*(X;K)$ becomes a $K[u]$-module via the classifying map. A second component is Lemma 3.5, the formality of single-parameter persistence dg modules, which lets the authors replace any module by its homology without changing homotopy interleavings. The computational engine is then the isometry between interleaving distance and bottleneck distance for barcodes of graded $K[t]$-modules, so $d^0_{CohI}$ and $d^1_{CohI}$ are read off directly from barcodes.

What would settle it

Independently compute the homotopy interleaving distance between the two dg $K[u]$-modules coming from the spaces $M_0$ and $M_1$ of Proposition 6.3, directly from the definition of $\varepsilon$-homotopy interleavings rather than from barcodes. The paper's Theorem 4.7 predicts the value $3$; any other value would falsify the claimed equality.

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Extended reading notes

Core claim

The central discovery is Theorem 3.3: on the class of persistence dg modules over a field, $d_{HC}=d_{IHC}=d_{HI}=d_{CohI}$. Here $d_{HI}$ is the homotopy interleaving distance, $d_{IHC}$ is the interleaving distance in the homotopy category, and $d_{CohI}$ is the supremum over homological degrees of the ordinary interleaving distances of the homology modules. For dg modules over $K[u]$, the persistence module is obtained by shifting by the action of $u$, and Theorem 4.7 refines the equality to $d_{CohI}=\max\{d^0_{CohI},d^1_{CohI}\}$, where the two superscripts track even and odd cohomology. The proof rests on Lemma 3.5, which says every such persistence dg module is formal: it is quasi-isomorphic to its own cohomology because $K[t]$ is a hereditary ring. The paper also establishes cup-length upper and lower bounds for the distance between spaces over $BS^1$, and works out explicit distances among complex projective spaces and the spaces $M_0$, $M_1$.

Load-bearing premise

The equality rests on the single fact that a one-parameter persistence dg module over a field can be replaced, up to quasi-isomorphism, by its own cohomology, because the polynomial ring $K[t]$ is hereditary; if that formality fails, the distances can diverge.

Editorial extensions

If this is right

  • The homotopy interleaving distance of persistence dg modules over a field is computable as the bottleneck distance of homology barcodes, degree by degree.
  • For spaces over $BS^1$, the cohomology interleaving distance is an extended pseudometric and simultaneously controls all three homotopy-level interleaving distances.
  • Two spaces over $BS^1$ whose cohomology barcodes have distance zero have associated persistence dg modules that are isomorphic in the homotopy category, even if the underlying spaces have different rational homotopy types.
  • Borel constructions of free loop spaces of formal spaces sit at distance $0$ or $1/2$, depending only on whether their cohomology $K[t]$-modules are isomorphic.
  • The distance between a space over $BS^1$ and the point is half the cup-length plus one half, and differences of cup-lengths give lower bounds on distances between spaces.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • This suggests a practical route to computing homotopy interleaving distances for finite-type spaces over $BS^1$: compute the cohomology $K[u]$-module structure, read off barcodes, and take a bottleneck distance, with no homotopy-level search required.
  • The equality is likely special to one parameter: over $K[t_1,\ldots,t_n]$ for $n\ge 2$ the formality lemma fails, so the cohomology interleaving distance should be viewed as a lower bound rather than a complete invariant for multiparameter persistence.
  • Zero-distance equivalence classes over $BS^1$ may serve as a coarse 'persistent shape' invariant for spaces that are otherwise hard to compare, analogous to the role of Gromov–Hausdorff distance in metric geometry.
  • The cup-length bounds invite refinement: sharper invariants such as rational toral rank could yield better lower bounds, since the examples $M_0,M_1$ separate by toral rank even when cup-length alone does not.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper introduces a cohomology interleaving distance dCohI for persistence dg modules over a field K and proves that, in the single-parameter setting, this distance coincides with the homotopy interleaving distances dHC, dIHC, and dHI introduced by Blumberg–Lesnick and Lanari–Scoccola (Theorem 3.3). The proof relies on a constructive formality lemma for persistence dg modules indexed by Z (Lemma 3.5), together with scaling arguments from Lanari–Scoccola. The paper then specializes to dg K[u]-modules, shows that the relevant distance is the maximum of an even and an odd cohomological bottleneck distance (Theorem 4.7), and applies this to spaces over BS^1 via singular cochains. Concrete computations include complex projective spaces, certain S^1-bundle orbit spaces, and free loop space Borel constructions, with cup-length bounds and a rational homotopy appendix.

Significance. If the main results stand, the paper gives a genuinely computable way to evaluate homotopy interleaving distances in a nontrivial class of persistence dg modules and spaces over BS^1, reducing them to bottleneck distances of cohomology barcodes. The constructive proof of formality for single-parameter persistence dg modules is a useful and clearly explained ingredient, and the authors are explicit that this formality is tied to K[t] being hereditary and fails in the multiparameter setting (Remark 3.6). The computational examples, especially the tetrahedron of distances and the CP^n calculations, are concrete and checkable. There are no fitted parameters or predictions, and the paper relies on standard external results.

major comments (2)
  1. [Section 4, Lemma 4.9] Lemma 4.9 is false as stated. Let H^* = K[t]/(t^2) with deg t = 1 and take the filtration F^0 = H^*, F^1 = F^2 = 0. Then H^* is non-negatively graded, dim H^i < infinity, tF^1 = 0 subset F^2, and all stated hypotheses hold. However, E^{0,0} = H^0, E^{0,1} = H^1, and E^{p,q} = 0 for p >= 1, so Tot E has trivial t-action, while t acts nontrivially from degree 0 to degree 1 in H^*. The proof uses the assertion 'F^i H^0 = 0 for i > 0', which does not follow from the stated hypotheses. The lemma becomes true if one adds the natural hypothesis that the filtration is by total degree, i.e. F^i subset H^{>=i}; this extra hypothesis is satisfied in the spectral-sequence applications of Propositions 5.10 and 6.3, but it must be stated and the proof adjusted.
  2. [Section 3, proof of Theorem 3.3] In the proof of Theorem 3.3, the equality H(Z) = direct_sum_{k>=0} eta_k H_*(Z) is incorrect for unbounded persistence dg modules; the direct sum should be over all k in Z, matching the definition of dCohI as a supremum over all integer cohomological degrees. As written, the proof of dHI <= dCohI would not cover objects with nonzero cohomology in negative degrees. This is a local fix, but it should be corrected because Theorem 3.3 is stated for all objects in Ch(R,<=)_K.
minor comments (4)
  1. [Section 3, Lemma 3.5] In the proof of Lemma 3.5, the generators b_lambda(i)_k should be explicitly chosen as cycles representing the homology classes [b_lambda(i)_k], since the map phi : F_0 -> Ker d is defined by sending each generator to its representative. This is presumably intended but should be stated.
  2. [Section 5, Example 5.7(1)] Example 5.7(1) states an assertion for arbitrary field K but the proof uses rational cohomology and the assumption that 1*t^l is nonzero in H^*(LM_hS1; Q). The statement should either be restricted to K = Q or the proof adapted to justify the claim for general K.
  3. [Section 5, Proposition 5.13] The proof of Proposition 5.13 is terse: the phrase '2(m/2)-trivial' is used without a definition, and the reduction to kernel/cokernel triviality via [4, Corollary 6.6] could be spelled out. This is a clarity issue rather than a mathematical gap.
  4. [Throughout] There are several typographical slips, e.g. 'fincor' in Section 4, 'as a consecuence' in Section 5, and 'more general BV-exact spaces' in Proposition 5.4 where 'more generally, BV-exact spaces' is meant. A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 2.0 of 10

The central derivation is not circular; the only self-citations are non-load-bearing.

full rationale

The main equality dHC = dIHC = dHI = dCohI (Theorem 3.3) is not a definitional collapse: dCohI is defined as a supremum of interleaving distances of cohomology persistence modules (Definition 3.1), and the nontrivial reverse inequality dHI ≤ dCohI is obtained through Proposition 3.7 and Lemma 3.5. Lemma 3.5, asserting formality of every persistence dg module, is proved constructively in the paper via a two-term free K[t]-module resolution and is independently supported by the hereditariness of K[t] (Remark 3.6). The paper explicitly notes that this formality fails for multiparameter persistence, which confirms that the equality is not tautological. No fitted parameter is renamed as a prediction: the computational results in Propositions 4.12, 4.13, 6.1, 6.3, and 6.9 are direct consequences of the isometry theorem and explicit barcode computations. The self-citations that appear ([29] in Remark 5.11, [30] in Proposition 5.4 and Example 5.7) concern supporting spectral-sequence remarks and the free-loop-space application; they are not load-bearing for the central Theorem 3.3 or Theorem 4.7. The proof of Theorem 3.3 contains a minor notational slip, writing H(Z) = ⊕_{k≥0} η_k H_*(Z) instead of a sum over k∈Z; since dCohI is defined as a supremum over all k∈Z and no boundedness is assumed, reading the sum over all k∈Z makes the argument valid. This is a typo-level correctness issue, not a circular step. No circular step can be quoted and exhibited, so the circularity burden is low; the score reflects only the presence of minor, non-load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard model category machinery, the formality of single-parameter persistence dg modules over a field, and the isometry theorem for barcodes. No parameters are fitted to data. No new physical or mathematical entities are postulated; the cohomology interleaving distance is a definition, not an entity requiring independent evidence.

assumptions (6)
  • standard math The category Ch_K^P with the projective model structure is a model category for any poset P.
    Used throughout Section 3 to define dHC, dIHC, and dHI; cited from Barthel-May-Riehl [3] and Hirschhorn [24].
  • standard math Every persistence dg module is formal: X is quasi-isomorphic to H(X) in Ch(Z,<=)_K.
    Lemma 3.5, proven using the fact that K[t] is a PID and hereditary, so the derived category of graded K[t]-modules is formal. This is load-bearing for Theorem 3.3.
  • standard math The isometry theorem identifies the interleaving distance with the bottleneck distance for locally finite persistence modules.
    Theorem 2.10, cited from Bubenik-Scott [8] and Chazal et al. [11], used in Proposition 4.5 and in all barcode computations.
  • standard math Lanari-Scoccola lemmas (Lemma 3.1 and 3.2) relate rescaled interleavings and homotopy interleavings.
    Used in the proof of Proposition 3.7 to pass from delta-interleaved cohomology to homotopy interleavings of dg modules.
  • domain assumption Spaces are connected and singular cohomology is locally finite.
    Section 5 states this as a standing assumption, needed for barcode decompositions and finiteness of distances for spaces in Class (III).
  • standard math The BV-exactness theorem and the statement that formal spaces are BV-exact.
    Theorem 5.2 and Corollary 5.3 from the authors' prior paper [30], used in Proposition 5.4 for free loop spaces. This is an independently published result and is not used in the main theorem.

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Pith. "Pith review of Algebraic interleavings of spaces over the classifying space of the circle." pith.science (2026). https://pith.science/paper/PTKGMS7O

@misc{pith2026250109257,
  author       = {Pith},
  title        = {Pith review of: Algebraic interleavings of spaces over the classifying space of the circle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PTKGMS7O}},
  note         = {Machine review of arXiv:2501.09257}
}
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.

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