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REVIEW 4 major objections 4 minor 31 references

Filtered Topology and Persistence in Stable Homotopy

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that the K-group of a persistence Spanier-Whitehead category of filtered CW complexes is isomorphic to the ring of Novikov polynomials, with the isomorphism induced by the weighted Euler characteristic.

desk verdict Plausible and attractive stable-homotopy packaging of persistence, but the main K-theory theorem is ill-posed on the category as defined and the invariance proof has a gap; worth referee time, not acceptance as is. read the letter →

arxiv 2505.02772 v1 pith:SEM5B5QU submitted 2025-05-05 math.AT math.SG

John Miller This is my paper · ORCID
classification math.ATmath.SG MSC 55N3155P4218G80
keywords filteredtopologicalspacespersistencecategoriesSpanier-WhiteheadcategoryweightedEulercharacteristictriangulatedNovikovpolynomialspersistenthomologyspectra
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 builds a category of filtered topological spaces, where each space carries a filtration by subspaces indexed by the real numbers, and develops filtered versions of CW approximation, smash products, and homotopy. It then stabilises this category into a persistence version of the Spanier-Whitehead category, whose objects are filtered CW complexes together with an integer suspension degree. The central claim is that the Grothendieck group (K-group) of this category is isomorphic to the ring of Novikov polynomials, with the isomorphism given by a weighted Euler polynomial that records the filtration level at which each cell appears. If true, this gives a numerical invariant of filtered spaces that refines the ordinary Euler characteristic and ties persistence theory to stable homotopy theory.

What carries the argument

The carrying object is the weighted Euler polynomial $\hat\chi_{CW}(X)=\sum_{a\in\mathrm{Cells}_*(X)}(-1)^{|a|}t^{w(a)}$, where $w(a)$ is the smallest filtration level at which cell $a$ appears, with eternal cells contributing zero. Its additivity over wedges and multiplicativity over the filtered smash product make it descend to a $\Lambda_P$-algebra map on the K-group of the persistence Spanier-Whitehead category $PSW_0$. The proof that this map is an isomorphism reduces every object to a linear combination of the zero-filtered sphere via cell attachments, giving $[(X,n)]=(-1)^n\hat\chi_{CW}(X)[(S^0_0,0)]$.

What would settle it

Filter the closed disk $D^2$ so that its unique 1-cell and 2-cell both appear at level 1, making $X(r)$ contractible for $r\ge 1$ and the basepoint below; compare with the constant point filtration $X'(r)=*$. The two filtered spaces are 0-filtered homotopy equivalent levelwise, but $\hat\chi_{CW}(X)=t^2-t$ while $\hat\chi_{CW}(X')=0$, directly contradicting Proposition 2.36 and thereby the invariance on which Theorem 3.22 depends.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 3.22: the weighted Euler polynomial induces an isomorphism of $\Lambda_P$-algebras $\mathcal{X}:K(PSW_0)\to\Lambda_P$, given by $[(X,n)]\mapsto(-1)^n\hat\chi_{CW}(X)$, where $\Lambda_P=\{\sum_{i=1,\dots,n}a_i t^{r_i}:a_i\in\mathbb{Z}, r_i\in\mathbb{R}\}$ is the ring of Novikov polynomials. The paper argues that $\hat\chi_{CW}$ is invariant under filtered homotopy equivalence, additive over filtered wedges, and multiplicative over the filtered smash product, so it descends to a ring map on the K-group. This is presented as a persistence analogue of the known isomorphism $K(SW)\cong\mathbb{Z}$ induced by the reduced Euler characteristic.

Load-bearing premise

The map $\mathcal{X}$ is well defined only if the weighted Euler polynomial is invariant under the filtered homotopy equivalences that identify objects in the K-group; the paper's proof of that invariance (Proposition 2.36) is load-bearing and is not valid, because a nonzero polynomial such as $t-t^2$ can evaluate to zero at $t=1$ through cancellation of terms.

Editorial extensions

If this is right

  • If Theorem 3.22 holds, the K-group of the persistence Spanier-Whitehead category is completely described by the ring of Novikov polynomials, so two filtered stable objects have the same K-class exactly when their weighted Euler polynomials agree.
  • The weighted Euler characteristic becomes a filtered homotopy invariant refining the ordinary Euler characteristic: evaluation at $t=1$ recovers the reduced Euler characteristic, and the derivative at $t=1$ gives a weighted count of cell birth levels.
  • The fragmentation metrics induced by the triangulated persistence structure have concrete geometric meaning: for a Morse function $f$ on a compact manifold, the size of the filtered space is bounded by $\sum_{x\in\mathrm{Crit}(f)} f(x)$.
  • The framework extends to filtered spectra, producing a triangulated persistence category of filtered spectra and a notion of generalised filtered homology theory represented by a filtered spectrum.
  • The matching number for two filtrations on the same total space is bounded by the $C^0$-distance between the defining functions, giving a stability-type statement for the weighted Euler characteristic.

Reading between the lines

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

  • A corrected proof of the invariance claim would need a finer comparison than levelwise Euler characteristics, because cell-weight polynomials can differ by terms such as $t^2-t$ that vanish at $t=1$ through cancellation; a chain-level or K-theoretic refinement is the natural repair.
  • If the isomorphism survives such a repair, $\hat\chi_{CW}$ becomes a complete Grothendieck-group invariant for filtered stable objects, analogous to the role of the reduced Euler characteristic in ordinary stable homotopy theory.
  • The appearance of Novikov rings suggests a connection to action filtrations in symplectic topology, where Novikov rings encode holomorphic disk areas; one testable extension is whether the isomorphism intertwines the $t$-adic filtration on $\Lambda_P$ with a filtration on $K(PSW_0)$ induced by spectral points.
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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

4 major / 4 minor

Summary. The paper introduces a category of filtered pointed topological spaces, develops filtered products, wedges, smash products, homotopy groups, and a filtered CW approximation theorem, and defines a weighted Euler polynomial \hat\chi_{CW}. It then constructs a persistence Spanier-Whitehead category PSW of filtered CW complexes, claims it is a triangulated persistence category, discusses fragmentation metrics, and states as its main theorem (Theorem 3.22) that the weighted Euler polynomial induces an isomorphism of \Lambda_P-algebras from K(PSW_0) to the ring of Novikov polynomials. A final section sketches a filtered stable homotopy category of filtered spectra and a notion of generalized filtered homology theory.

Significance. If the main theorems were established, the paper would provide a genuinely topological family of triangulated persistence categories and a clean K-theoretic computation identifying the K-group with Novikov polynomials, which would be a useful bridge between persistence theory and stable homotopy. The weighted Euler polynomial and the fragmentation metric discussion are natural and potentially interesting. However, the submitted version does not currently support these claims: the central invariance statement is proved by an invalid argument, the main theorem is not well-formed on the category as defined, and several auxiliary proofs are incomplete. I see no machine-checked proofs or reproducible code in the manuscript. The ideas are promising, but the technical execution needs substantial revision.

major comments (4)
  1. [§2.4, Proposition 2.36] The proof of filtered-homotopy invariance of \hat\chi_{CW} is invalid as written. From ev_{t=1}D^{\le r}=0 for a single r one cannot conclude D^{\le r}=0; for example D=t-t^2 evaluates to zero at t=1 while being nonzero. A valid argument would need a minimal-weight induction using that ev_{t=1}D^{\le s}=0 for every s together with a well-ordered set of spectral points. The proof also says 'the homotopy type of filtration levels can only change at finitely many points', but §2.3 only assumes countably many spectral points; with accumulating spectral points, D^{\le r} may itself be an infinite series, so the statement of the proposition is not even meaningful without a finiteness hypothesis.
  2. [§3.4, Theorem 3.22] Theorem 3.22 is not well-formed on the category as defined. Definition 2.20 allows arbitrary filtered CW complexes, while \Lambda_P is defined as finite sums. The object X=\bigvee_{n\ge 1} S^1_{1/n}, where S^1_a(r)=S^1 for r\ge a and a point otherwise, satisfies Definition 2.1 and the countable-spectral-point condition, and each level is a finite CW complex, yet \hat\chi_{CW}(X)=-\sum_{n\ge 1} t^{1/n} is not an element of \Lambda_P. Hence the map \mathcal{X}:K(PSW_0)\to \Lambda_P has no value on a legal object. The proof's finite-skeleton decomposition, with finite sets I_j of cells in each dimension, also does not apply. The theorem needs either a restriction to finite filtered CW complexes or a replacement of \Lambda_P by a completed ring of Novikov series with appropriate convergence conditions.
  3. [§2.4, Lemma 2.22] The proof of the filtered CW approximation lemma is incomplete for filtrations with countably many spectral points that accumulate. The proof enumerates Spec(F_X)=\{r_i: i\in \mathbb{N}\} and uses intervals [r_i,r_{i+1}), but if the r_i accumulate (for example r_i=1/i), these intervals do not cover points near the accumulation value, and the claimed decomposition of an arbitrary r<s into these intervals fails. The proof also identifies r_0=\lfloor X\rfloor with a lower stabilisation level, but \lfloor X\rfloor is defined through contractibility of levels, not through the level X_0 of Definition 2.1; these are different data unless X_0 is assumed contractible. The approximation statement may be salvageable, but the written proof does not establish it.
  4. [§3.1, Theorem 3.1] The proof that PSW is a triangulated persistence category is too sketchy in a load-bearing place. Part 2 asserts that the maps \eta^{\mathrm{Cone}(\eta^r_X)}_r, given on quotients by X(t+r)/X(t)\to X(t+2r)/X(t+r), are 'naturally nullhomotopic' after stabilization. This is not immediate and is not generally automatic for maps between successive filtration quotients, so a proof is needed. Part 1 appeals to [De] for the Spanier-Whitehead construction without verifying the required model-category hypotheses on (FTop_*)_0. Since Theorem 1.3 underlies the fragmentation distances and the K-group discussion, this gap affects the paper's central narrative even beyond Theorem 3.22.
minor comments (4)
  1. [§1.1] In the review of persistence modules, the sentence 'along with a collection of R-linear morphisms \iota_{r,s} for all s\le r' has the inequality backwards; it should read r\le s.
  2. [§2.4, Definition 2.20] The definition of a filtered CW complex would be clearer if it first required the total space X to be a CW complex and then required each X(r) to be a subcomplex with cellular inclusions; as written, 'the levels are all sub-CW-complexes' presupposes a CW structure on X that has not been stated.
  3. [§3.4, Proof of Theorem 3.22] The displayed triangle (S^{k-1}_0,0)\to *\to (S^k_0,0)\to (S^k_0,0) appears to have the wrong final object; the standard cofiber sequence would end with [1](S^{k-1}_0,0).
  4. [§3.3, Proposition 3.4] The proof of Proposition 3.4 states that 'each filtration step X(r) is equivalent to a finite CW-complex' and that Spec(X) is finite, but these are not standing assumptions in §2.3. They should be stated explicitly as hypotheses or proved, otherwise the finiteness of the fragmentation size is not established.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the K-theory isomorphism is derived from the cell-attachment structure and external TPC formalism, not by assuming the conclusion.

full rationale

I walked the paper's claimed derivation chain. The central objects are not defined in terms of the theorems they are used to prove: the filtered category FTop*, the weighted Euler polynomial \hat\chi_CW, and the persistence Spanier-Whitehead category PSW are introduced independently, and Theorem 3.22 is then proved by expressing K-classes of filtered CW complexes through sphere-attachment cofiber sequences. The additivity and multiplicativity identities for \hat\chi_CW are computed directly from cell counts, not imported from the K-group isomorphism. The TPC structure is taken from Biran-Cornea-Zhang's external work, and the Spanier-Whitehead triangulation from Dell'Ambrogio; these are not self-citations of the present author and are not invoked as a uniqueness device to forbid alternatives. No parameter is fitted and then renamed as a prediction. The main proof that K(PSW0) is generated by [(S0,0)] and that \hat\chi_CW records the generator's coefficients is a transparent cell-filtration argument, analogous to the classical computation K(SW) ≅ Z. There are genuine correctness concerns, but they are not circularity: Proposition 2.36's invariance proof silently upgrades the stated countable-spectral-point assumption to finitely many points and infers D^{≤r}=0 from ev_{t=1}D^{≤r}=0 without ruling out cancellation, and Theorem 3.22 is not well-formed for infinite filtered CW complexes because \hat\chi_CW can then be an infinite series outside \Lambda_P. These are unsupported steps in the proof, not cases where the conclusion is equivalent to the input by construction or where the argument reduces to a self-citation chain.

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

No free parameters are fitted to data; the paper introduces mathematical definitions (FTop*, PSW, filtered spectra) rather than hypothesized entities. The key unstated assumption is the finiteness of spectral points for filtered CW complexes.

assumptions (6)
  • domain assumption Filtrations stabilize below and above and have countably many spectral points
    Definition 2.1 and Section 2.3; without this, the category FTop* and the CW approximation are not defined.
  • domain assumption Filtered CW complexes are finite or have finitely many spectral points
    Needed for finite sums in \Lambda_P and for Proposition 2.36, but never stated as a standing assumption.
  • standard math CW approximation theorem for pairs (May's A Concise Course)
    Used in Lemma 2.22 to construct filtered CW approximations.
  • standard math Triangulated persistence category axioms from Biran-Cornea-Zhang
    The framework for TPCs and fragmentation metrics is taken from [BCZ1] and [BCZ2].
  • standard math Spanier-Whitehead category of a pointed model category is triangulated
    Used in Theorem 3.1 Part 1 via Dell'Ambrogio's thesis.
  • standard math Morse theory sublevel set cell-attachment
    Used in Lemma 3.6 and Section 3.3 to bound fragmentation sizes.

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Pith. "Pith review of Filtered Topology and Persistence in Stable Homotopy." pith.science (2026). https://pith.science/paper/SEM5B5QU

@misc{pith2026250502772,
  author       = {Pith},
  title        = {Pith review of: Filtered Topology and Persistence in Stable Homotopy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SEM5B5QU}},
  note         = {Machine review of arXiv:2505.02772}
}
read the original abstract

We define a category of filtered topological spaces and explore some of its homotopy theoretic properties, including a filtered analogue of CW approximation. With this, we define and study a filtered (weighted) variant of the Euler characteristic and show this is a `filtered homotopy invariant'. We then go on to use the recent work of Biran, Cornea and Zhang by considering a persistence Spanier-Whitehead category of filtered CW complexes and show this is a triangulated persistence category and discuss the fragmentation metrics induced by this structure. We go on to show that the K-group of this persistence category is isomorphic to the ring of Novikov polynomials and this isomorphism is induced by the weighted Euler characteristic. Finally, we discuss how these constructions extend to a filtered stable homotopy category and its relation to filtered/persistence homologies.

Figures

Figures reproduced from arXiv: 2505.02772 by the authors.

Figure 1
Figure 1. Diagrams representing the naive product X × Y (left) and the product XׯY (right). Definition 2.10. The naive filtered smash product is defined by (X Λ Y )(r) := X(r) Λ Y (r) (11) On morphisms, it is given simply by the levelwise smash product for morphisms. Note that this is simply the quotient of the naive product by the naive wedge; (X Λ Y )(r) = (X × Y )(r)/(X ∨ Y )(r). Definition 2.11. We define the filtered sm… view at source ↗
Figure 2
Figure 2. A depiction of (S 1 0ׯ X)(r) showing the eternal subcomplex S 1 ∨ X. 2.4 Filtered CW-complexes and weighted Euler characteristic In order to construct homotopical notions in the filtered setting we need to understand what a filtered CW complex should be and what filtered weak homotopy equivalences are. Definition 2.20. A filtered CW complex is a filtered space X such that the levelsets are all sub￾CW-complexes and … view at source ↗
Figure 3
Figure 3. A generic picture of T(r) in blue for various values of r with a deformation retract to a corresponding CW complex coloured in red/orange (left). Along with a depiction of a filtered CW complex with cell weights labeled which is filtered weak equivalent (right). The size polynomial and, in turn, its derivative depend on the ‘filtered cell decomposition’. We will see, however, that there is a weighted version of the … view at source ↗

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