Identifying cobordisms using kernel persistence

Anne Marie Svane; Lisbeth Fajstrup; S{\o}ren Strandskov S{\o}rensen; Teresa Heiss; Yossi Bokor Bleile

REVIEW 3 major objections 7 minor 17 references

Identifying cobordisms using kernel persistence

T0 review · 3 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper defines open cobordisms as classes in a cokernel, proves their births and deaths follow eight kernel-persistence cases, and gives a column-reduction algorithm that pairs them.

desk verdict The cobordism invariant is a good idea, but Algorithm 2's proof skips the one case that matters: classes born in Case F have no column in DPhi, so the death-pairing cannot recover their true birth time. read the letter →

arxiv 2505.17858 v1 pith:WMECIJKR submitted 2025-05-23 math.AT

Yossi Bokor Bleile , Lisbeth Fajstrup , Teresa Heiss , Anne Marie Svane , S{o}ren Strandskov S{o}rensen This is my paper · ORCID

classification math.AT MSC 55N31

keywords persistenthomologykernelpersistenceopencobordismstunnelsbottleneckthicknessalphacomplexcolumnreductionpointcloud

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 gives a homological definition of tunnels—open cobordisms—connecting two disjoint parts $A$ and $B$ of a finite cell complex $X$. A tunnel is counted as a class in $\mathrm{Cok}(\Phi)$, where $\Phi$ compares cycles that become trivial inside $A$ or $B$ with cycles that become trivial in their union. When $X$ is filtered, this cokernel is a persistence module, and the paper proves that births and deaths in it occur in exactly eight configurations visible in the kernel persistence of $A$, $B$, and $A\cup B$. The main result is an algorithm that pairs each birth with its death; in the point-cloud setting, the death time is the bottleneck thickness of the tunnel, the radius of the largest ball that can roll through.

What carries the argument

The load-bearing object is the cokernel $\mathrm{Cok}(\Phi)$ of the map $\Phi_k\colon \mathrm{Ker}(\iota^A_k) \oplus \mathrm{Ker}(\iota^B_k) \to \mathrm{Ker}(\iota^{A\cup B}_k)$; equivalently, by the relative homology long exact sequence, a class is a $(k+1)$-chain with boundary in $A\cup B$ modulo chains whose boundary lies entirely in $A$ or entirely in $B$. The algorithm works by reordering boundary-matrix rows so that cells of $A$ (or $B$, or $A\cup B$) come first, then applying standard column reduction so the lowest 1 in a column marks a birth or death. Algorithm 2 assembles the stored relative-cycle columns into a matrix $D_\Phi$, and double columns in its reduced form mark the deaths.

What would settle it

Take a filtered triangulated cylinder with A and B the two boundary circles, and arrange the filtration so that a class born directly in $\mathrm{Ker}(\iota^{A\cup B}_*)$ later dies. Run Algorithm 2 and compare the birth time it pairs to the death time with the birth time read off from the defining persistence module $\mathrm{Cok}(\Phi)$; a mismatch would refute Theorem 4.1.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that open cobordisms between two disjoint subcomplexes are exactly the nonzero classes in $\mathrm{Cok}(\Phi)$, with $\Phi$ the map $\mathrm{Ker}(\iota^A_*) \oplus \mathrm{Ker}(\iota^B_*) \to \mathrm{Ker}(\iota^{A\cup B}_*)$, and that these classes form a persistence module when the complex is filtered. Theorem 3.1 states that a class is born only when a class appears in $\mathrm{Ker}(\iota^{A\cup B}_*)$ without appearing in either kernel for $A$ or $B$, and dies when births occur in the kernels in the combinations called Cases C, G, and H. Algorithm 2 takes birth and death information from the three kernel persistence modules and returns birth–death pairs in $\mathrm{Cok}(\Phi)$; Theorem 4.1 asserts this pairing is correct. For a point cloud with $\alpha$ filtration, the resulting pairs give tunnel birth times and bottleneck thicknesses.

Load-bearing premise

The pairing algorithm is correct only if every element of $\mathrm{Cok}(\Phi)$ that dies at a cell $\tau$ can be represented by adding the stored columns lying to the left of $\tau$, so that column reduction can locate the earliest birth cell.

Editorial extensions

If this is right

For a point cloud in a box, Algorithm 2 turns the alpha filtration into a list of tunnels, each with a birth radius and a death radius equal to the bottleneck thickness.
The same construction applies to any pair of disjoint subcomplexes of any filtered regular cell complex, so it does not rely on Euclidean structure.
Corollary 4.2 provides representatives of each tunnel at birth and just before death, so the cells surrounding the narrowest passage can be inspected directly.
Because the input consists of the kernel persistence modules of three inclusion maps, the algorithm can run on top of existing kernel persistence implementations.

Reading between the lines

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

One natural extension, left implicit by the paper, is to three or more boundary regions: the pairwise cokernel could be iterated, or a single cokernel defined from all boundary components at once, using the same relative-homology description.
Periodic boundary conditions could be handled by applying the construction after quotienting the box to a torus; the paper notes the torus creates extra homology classes, so the inference is that one would need to quotient those out, perhaps by working relative to the homology of the torus.
The representatives found by Algorithm 2 could be used to define a distance between porous materials, for example by counting tunnels whose death representative touches a given atom or bond.
Empirically, the sensitivity to slab thickness could be studied by running the algorithm for increasing slab size and tracking when the set of tunnels stabilizes; the paper's suggestion to delete slab 2-cells is a concrete preprocessing step to test.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper's core object, Cok(Phi), is a sensible homological way to count open cobordisms between disjoint subcomplexes, and Theorem 3.1's case analysis is a clean reduction to kernel persistence. The examples and the discussion of the chemistry motivation are helpful, and the authors are honest about the slice-thickness problem without pretending to solve it. That part of the paper is solid.

The soft spot is Theorem 4.1. The proof includes phrases like "It is now straightforward to check" and then a final assertion that adding columns to the left of tau gives all equivalent representatives, so the reduction finds the earliest birth. That is not demonstrated, and in fact it seems false for Case F. DPhi is built only from columns of V_A_im and V_B_im (plus absolute cycles from zero-columns of R_A_im). A class born in Case F is a birth in Ker(iota_A∪B) with no simultaneous birth in Ker(iota_A) or Ker(iota_B). Therefore, at its birth time there is no column in DPhi representing it. When the class later dies, the algorithm adds new columns from the simultaneous births in Ker(A) and/or Ker(B), but those columns represent the class only at the death time. The reduction can push the lowest 1 down through older columns, but the original birth cell is not present in any column of DPhi, so the lowest 1 cannot point to it. The final paragraph of the proof assumes exactly what needs proof: that the stored columns form a compatible basis covering all classes, including those born in Case F. This is a load-bearing gap, not a minor omission. As stated, Algorithm 2 would either leave Case F classes unpaired or pair them with the wrong birth time.

There is no implementation or worked small example, which would have helped the authors (and referees) test this. The paper is still worth reading for the definition and the classification, but the algorithmic claim needs major revision. I would send it out for review with a clear request to fix or restrict Theorem 4.1, ideally with a concrete counterexample or a revised algorithm that handles Case F births.

Referee Report

3 major / 7 minor

Summary. This paper introduces a homological model for 'open cobordisms' between two disjoint subcomplexes A and B of a filtered cell complex X, defined as elements of the cokernel of a map Φ between kernels of inclusion maps. It proves (Theorem 3.1) a classification of birth/death events in Cok(Φ) in terms of kernel persistence of the inclusions of A, B, and A∪B into X, and presents Algorithm 2 for pairing these birth and death times (Theorem 4.1). The motivating application is detecting tunnels connecting the top and bottom of a box in a point cloud, with birth and death times interpreted as tunnel separation and bottleneck thickness. The paper is primarily theoretical; it does not include an implementation.

Significance. The algebraic setup and the birth/death classification are clean and likely correct. The paper explicitly ships a matrix-reduction algorithm with a claimed correctness theorem; if Theorem 4.1 can be fully proved, the algorithm would be a useful contribution for computing a novel persistence invariant. The connection to kernel persistence is well motivated and the relative-homology formulation is instructive. However, the correctness proof of Algorithm 2 is incomplete at a load-bearing point, so the computational claim should be treated as conditional. The authors are honest about the sensitivity of the construction to the slice thickness ε and about the lack of discussion of representatives, which is positive.

major comments (3)

[§4.2, Theorem 4.1] The proof does not justify recovery of Case F births. The columns stored in DΦ are V_A_im and V_B_im columns plus absolute cycles; at filtration value t their span lies in the image of H_{*+1}(X_t,A_t) ⊕ H_{*+1}(X_t,B_t) → H_{*+1}(X_t,A_t∪B_t). A Case F class is born outside this image and enters it only at a later death time. The sentence 'by adding the columns to the left of τ we can get all equivalent representations of the tunnel that dies at time τ' asserts, but does not prove, that the stored columns form a basis of this image at every intermediate filtration step and that left-to-right reduction of the new τ-columns yields a lowest 1 at the true birth cell rather than at a later cell. Since Theorem 3.1 explicitly does not address pairing, this missing argument leaves the correctness of Algorithm 2 for Case F births unproved.
[§4.2, Algorithm 2, Line 1] The global row ordering for DΦ is not filtration-compatible. The line orders only the X\(A∪B) block by filtration value; the A∪B block is not globally ordered by filtration (cells of A and B can interleave in the filtration). The proof of Theorem 4.1 repeatedly invokes 'pushing the lowest 1 as far up as possible' to obtain the earliest born representation, which presupposes that the row order is the filtration order. As written, the reduction on DΦ may therefore select a lowest 1 that does not correspond to the earliest birth time among equivalent representatives, so the pairing rule in Lines 19–20 is not guaranteed to return the true birth–death pair. The algorithm should specify a total filtration-compatible ordering of all rows, or explain why the block ordering suffices.
[§4.2, Proof of Theorem 4.1] Several assertions in the case analysis are made without proof. For example, item 7 claims that a τ-column with τ ∈ A can never be added only in Line 12 because then ∂τ would be trivial and τ would have been added in Line 2; this presumes that a column with lowest 1 in the B-block necessarily corresponds to a cycle in X, a fact that is not established. The phrase 'It is now straightforward to check which cases can occur' is not a substitute for deriving the case analysis from the reduction properties established in Algorithm 1. These gaps, together with the missing Case F argument, make Theorem 4.1 an outline rather than a complete proof.

minor comments (7)

[Abstract] There is a typo: 'homlogical' should be 'homological'.
[Example 1] There is a typo: 'graoups' should be 'groups'.
[§2.2 and elsewhere] The name 'Delauney' is misspelled; it should be 'Delaunay' in several places.
[§2.1, Equation (4)] The justification that Cok(Φ) is isomorphic to the cokernel of the map in (4) is abbreviated; a short diagram chase would make the quotient argument rigorous and rule out potential issues with the image of j^{A∪B}_*.
[Theorem 3.1] The statement 'there is no class born in Ker(iota_A) and Ker(iota_B)' should be clarified to mean 'no new class born in either', to avoid ambiguity about simultaneous births in both kernels.
[§4.2] The algorithm is described in prose and pseudocode, but the construction of DΦ from the reduced matrices is not given as a detailed listing; a step-by-step pseudocode block would aid reproducibility and verification.
[General] The paper contains no implementation or experiments. While the theoretical contribution can stand alone, including a small worked example (e.g., the cylinder from Example 1 or a synthetic point cloud) would substantially increase confidence in Algorithm 2 and help readers understand the pairing rule.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained, with external kernel-persistence results and self-citations used only in non-load-bearing contexts.

full rationale

The paper's central object Cok(Phi) is defined independently of the algorithm (Definition 1 and Equation (4)), and the birth/death characterization in Theorem 3.1 is derived from the long exact sequence and the external kernel persistence framework of Cohen-Steiner et al. [3], not from the paper's own conclusions. Algorithm 2 is a column-reduction procedure whose correctness is argued from the structure of DPhi; while the proof of Theorem 4.1 contains a nontrivial asserted step ('by adding the columns to the left of tau we can get all equivalent representations') that is not fully demonstrated for Case F births, this is a proof completeness issue, not circularity: the target pairing is not an input to the algorithm, and no fitted parameter or renamed input is presented as a prediction. The self-citation [2] appears only in Appendix A for the Voronoi-dual deformation-retraction statement, which is independent support and not load-bearing for the main pairing theorem. No step reduces, by construction or by self-citation, to its own inputs, so the appropriate score is 0.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central algorithm has no fitted parameters. The only hand-chosen quantity is the slice thickness epsilon in the chemistry application. The mathematical claims rest on standard homology and persistent homology machinery: Z/2Z coefficients, regular CW complexes, injective filtrations, and column reduction. No new physical or mathematical entities are postulated beyond the defined equivalence classes of open cobordisms, which are a mathematical construction rather than an invented entity with independent evidence.

free parameters (1)

slice thickness epsilon
Top and bottom slice thickness in the point-cloud tunnel model; no value is fixed and the authors state results are highly sensitive to it (Section 2.2, Section 5). It is a modeling choice, not fitted to data.

assumptions (6)

standard math Coefficients are in Z/2Z throughout.
Stated at the start of Section 2; standard in TDA but restricts the invariant to mod-2 homology.
domain assumption The filtration map f is injective with finite image {1,...,N}.
Section 3 assumes this for simplicity; it ensures at most one birth or death per filtration step. Real alpha-filtrations may have ties and would require perturbation or tie-breaking.
domain assumption X is a finite regular CW complex with finitely many cells.
Section 2 restricts to regular CW complexes so boundaries are unions of lower-dimensional cells; alpha complexes from point clouds satisfy this.
standard math Standard column reduction correctly computes persistent homology pairings.
Algorithms 1 and 2 rely on the reduction described in [4, Sec. VII.1]; this is background taken from the literature, not proved in the paper.
standard math The kernel persistence algorithm from [3] correctly computes births and deaths in kernel modules.
Algorithm 1 recalls this algorithm and its output is used as input to Algorithm 2; correctness is attributed to [3].
domain assumption Point clouds are in general position for the Voronoi/Delaunay duality.
Appendix A assumes no k+2 points lie on a common sphere; needed for the dual filtered complex statement, which supports the alternative Voronoi formulation in the discussion.

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Pith. "Pith review of Identifying cobordisms using kernel persistence." pith.science (2026). https://pith.science/paper/WMECIJKR

@misc{pith2026250517858,
  author       = {Pith},
  title        = {Pith review of: Identifying cobordisms using kernel persistence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WMECIJKR}},
  note         = {Machine review of arXiv:2505.17858}
}

read the original abstract

Motivated by applications in chemistry, we give a homlogical definition of tunnels, or more generally cobordisms, connecting disjoint parts of a cell complex. For a filtered complex, this defines a persistence module. We give a method for identifying birth and death times using kernel persistence and a matrix reduction algorithm for pairing birth and death times.

Figures

Figures reproduced from arXiv: 2505.17858 by the authors.

**Figure 2.** The triangulated cylinder from Example 1. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Examples of tunnels. The blue surface is [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Examples of cobordisms in Cok (Φ1). The blue surface is X, the part in the top slice is A and the part in the bottom slice is B. The typical case is a) where the tunnel exits through the top and bottom. In both b) and c) we count 1 tunnel, while d) is not counted as a …

**Figure 5.** Figure 5: The 8 cases in Table 1. The cell that is inserted when a birth or death happens [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: With periodic boundaries, this would not be counted as a tunnel. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Banerjee, A

R. Banerjee, A. Phan, B. Wang, C. Knobler, H. Furukawa, M. O’Keeffe, and O. M. Yaghi. High-throughput synthesis of zeolitic imidazolate frameworks and application to co2 capture. Science, 319:939–943, 2008

work page 2008

[2] [2]

Bleile, A

B. Bleile, A. Garin, T. Heiss, K. Maggs, and V. Robins. The persistent homology of dual digital image constructions. In Research in Computational Topology 2, pages 1–26. Springer, 2022

work page 2022

[3] [3]

Cohen-Steiner, H

D. Cohen-Steiner, H. Edelsbrunner, J. Harer, and D. Morozov. Persistent homology for kernels, images, and cokernels. In Proceedings of the twentieth annual ACM-SIAM symposium on Discrete algorithms , pages 1011–1020. SIAM, 2009

work page 2009

[4] [4]

Edelsbrunner and J

H. Edelsbrunner and J. Harer. Computational Topology. AMS, Providence, RI, 2010

work page 2010

[5] [5]

Merge Trees of Periodic Filtrations

H. Edelsbrunner and T. Heiss. Merge trees of periodic filtrations. arXiv:2408.16575, 2024

work page Pith review arXiv 2024

[6] [6]

J. I. Gersten and F. W. Smith. The Physics and Chemistry of Materials . Wiley, 2001. 14

work page 2001

[7] [7]

A. Hatcher. Algebraic Topology. Cambridge University Press, online edition, 2002

work page 2002

[8] [8]

Hiraoka, T

Y. Hiraoka, T. Nakamura, A. Hirata, E. G. Escolar, K. Matsue, and Y. Nishiura. Revealing hidden medium-range order in amorphous materials using topological data analysis. Proceedings of the National Academy of Sciences , 113:7035–7040, 2016

work page 2016