Persistence and Topological Complexity

Facundo M\'emoli; Ling Zhou

REVIEW 2 major objections 5 minor 14 references

Persistence and Topological Complexity

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Persistent topological complexity and zero-divisor-cup-length are stable, and they separate real projective space from a wedge of spheres more sharply than persistent homology.

desk verdict Useful new persistent invariants, but the TC stability proof leans on a weak-equivalence invariance that is not verified and is likely false in general. read the letter →

arxiv 2506.17888 v2 pith:PNABGFP4 submitted 2025-06-22 math.AT

Facundo Mémoli , Ling Zhou This is my paper · ORCID

classification math.AT MSC 55M3055N31

keywords persistenttopologicalcomplexityzero-divisor-cup-lengthVietoris–RipsfiltrationGromov–Hausdorffdistanceerosionhomotopyinterleavingmotionplanninginvariants

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 lifts topological complexity and its cohomological lower bound, zero-divisor-cup-length, from spaces to persistent families of spaces. It proves that these persistent invariants are stable: moving one persistent space to another by a homotopy interleaving of size $\epsilon$ changes the invariant by at most $\epsilon$ in erosion distance, and for Vietoris–Rips filtrations of compact metric spaces this gives a $2\,d_{\mathrm{GH}}$ bound. It then shows that for the pair consisting of real projective space $\mathbb{RP}^n$ and the wedge of round spheres $\vee^n S$, the erosion distance between the persistent invariants is at least $\pi/3$, which forces the Gromov–Hausdorff distance between the two spaces to be at least $\pi/6$. Persistent homology only yields $\pi/8$ for this pair, so the new invariants are more discriminating in this case. The reason to care is that the invariants encode the complexity of continuous motion planning, not just homology.

What carries the argument

The central machinery is the categorical invariant formalism for persistent objects: for a persistent space $X_\bullet$, the persistent invariant $I(X_\bullet)$ assigns to each interval $[a,b]$ the value $I(X_a\to X_b)$ of the invariant on the structure map. Topological complexity and zero-divisor-cup-length qualify because neither increases under composition, making them functors from the interval poset to $(\mathbb{N}\cup\{\infty\},\ge)$. Stability is measured with erosion distance and homotopy interleaving distance. For the numerical lower bound, Lemma 4.5 pins down the scale intervals on which the two invariants are constant using known homotopy types of Vietoris–Rips complexes of projective space, spheres, and metric gluings, and Proposition 4.6 converts those intervals into an erosion-distance lower bound of $\pi/3$ by comparing two step functions.

What would settle it

Check whether each Vietoris–Rips inclusion $VR_s\to VR_t$ in the ranges $0<s<t<2\pi/3$ (for $\mathbb{RP}^n$) and $0<s<t<\zeta_n$ (for $\vee^n S$) is actually a homotopy equivalence; if any one is not, Lemma 4.5's constant values fail and the $\pi/3$ erosion lower bound collapses. A second check is to test Proposition 1.1 on a pair of persistent CW complexes with small homotopy interleaving distance but large erosion distance for TC.

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Extended reading notes

Core claim

The paper establishes that topological complexity and zero-divisor-cup-length, when lifted to persistent spaces by evaluating them on the structure maps $X_a \to X_b$, are stable under the homotopy interleaving distance. Concretely, Proposition 1.1 gives $d_E(I(X_\bullet),I(Y_\bullet)) \le d_{HI}(X_\bullet,Y_\bullet)$ for persistent CW complexes, and for Vietoris–Rips filtrations of compact metric spaces this becomes a $2\,d_{\mathrm{GH}}$ bound. The paper then computes the two invariants along the Vietoris–Rips filtrations of $\mathbb{RP}^n$ and $\vee^n S$: both take constant values $I(\mathbb{RP}^n)>2$ and $I(\vee^n S)=2$ on a scale interval and vanish after the complexes become contractible. Feeding these values into a two-step erosion-distance estimate yields $d_E(I(VR_\bullet(\mathbb{RP}^n)),I(VR_\bullet(\vee^n S))) \ge \pi/3$, and combining this with stability yields $d_{\mathrm{GH}}(\mathbb{RP}^n,\vee^n S)\ge \pi/6$, improving the $\pi/8$ lower bound that persistent homology alone provides.

Load-bearing premise

The stability theorem inherits a hypothesis that the invariants are unchanged when a space is replaced by a weakly homotopy equivalent one; the proof verifies invariance only under ordinary homotopy equivalences, so the full weak-equivalence condition is assumed rather than checked.

Editorial extensions

If this is right

Persistent TC and persistent zcl are quantitative shape signatures for compact metric spaces, with the same $2\,d_{\mathrm{GH}}$ stability guarantee as persistent homology.
For the pair $(\mathbb{RP}^n,\vee^n S)$, the persistent invariants give $d_{\mathrm{GH}}\ge\pi/6$, strictly improving the persistent-homology bound of $\pi/8$.
The same lower bound can be recovered from persistent LS-category and persistent cup-length, so the improvement is not specific to motion-planning invariants.
Neither persistent TC nor persistent zcl dominates the other: static examples (spheres and wedges) show each can separate spaces the other cannot.

Reading between the lines

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

The paper's template suggests a general recipe: any integer-valued homotopy invariant that is non-increasing under composition and takes distinct constant values on two spaces with known Vietoris–Rips homotopy types yields a stable lower bound on $d_{\mathrm{GH}}$; the hard part is usually determining the scale interval where the structure maps are homotopy equivalences.
Because persistent Steenrod modules and these new invariants produce the same $\pi/6$ bound for the same pair, one could test whether all such stable invariants are detecting a single geometric obstruction; if so, combining invariants would not increase the bound beyond $\pi/6$ without a new idea.
An immediate testable extension would be to apply the same proof to lens spaces or other quotient metric spaces with known Vietoris–Rips homotopy types, and to see whether the erosion-distance gap grows with the size of the fundamental group.

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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

Two things to know. The paper introduces persistent topological complexity and persistent zero-divisor-cup-length, proves a stability inequality against the homotopy interleaving distance, and shows these invariants separate RP^n from a wedge of n spheres with a GH lower bound of pi/6, matching what persistent Steenrod modules give. That is a genuinely useful addition to the TDA toolbox. The stability proof, however, has a load-bearing gap: Proposition 1.1 verifies invariance only under homotopy equivalences between CW complexes, while the MSZ theorem it invokes requires invariance under pre/post-composition with weak homotopy equivalences of arbitrary spaces. TC is not invariant under weak equivalences in general - a weakly contractible non-contractible space like the Warsaw circle is weakly equivalent to a point but has TC > 0. Since the homotopy interleaving distance is defined through spans of weakly equivalent persistent spaces that may be non-CW, the quoted theorem does not apply to TC as written. zcl is fine, because cohomology rings are weak-homotopy invariant. This is not a nitpick; the gap is exactly what lets the stability argument go through.

What is good: the definitions are natural and the paper is the first to write them down; the categorical framework makes the lift to persistence almost automatic once the invariant is checked. The distinguishing example is worked carefully, and the authors are appropriately modest about the bound - they acknowledge the diameter bound pi/2 is stronger. Proposition 4.6, abstracting the erosion-distance estimate, is clean and reusable. Lemma 4.5 cites known results for the homotopy types of the Rips complexes, and the computation from those is sensible. One smaller soft spot: the lemma asserts the structure maps themselves are homotopy equivalences in the relevant scale ranges, but the cited sources support the levelwise homotopy type, not the map. That is likely true in these examples but needs a sentence or two.

The paper deserves a serious referee. It is a solid contribution to the growing literature on persistent invariants beyond persistent homology, and the main example is a meaningful comparison. The stability proof needs repair - either by formulating a version for CW complexes that respects the spans in dHI, or by proving TC is weak-equivalence invariant in the needed generality, which I suspect is false. The more plausible fix is to restrict the stability statement to a setting where TC is well-behaved and to adjust the proof accordingly. My recommendation: send to peer review, with the expectation of revision.

Referee Report

2 major / 5 minor

Summary. The paper introduces persistent analogues of two classical invariants in topological robotics: topological complexity (TC) and its cohomological lower bound, the zero-divisor-cup-length (zcl). For a persistent space X_•, these are defined by evaluating the map-level invariants on the structure maps X_a→X_b, yielding functors from the interval poset to the extended natural numbers. The main theoretical result is Proposition 1.1, which claims that, for persistent CW complexes, the erosion distance between these persistent invariants is bounded above by the homotopy interleaving distance, and hence, for Vietoris–Rips filtrations of compact metric spaces, by twice the Gromov–Hausdorff distance. The main application is Theorem 1.2, which gives a lower bound of π/3 on the erosion distance between the persistent TC (and zcl) of the Vietoris–Rips filtrations of RP^n and of the wedge sum ∨_{i=1}^n S^i, yielding d_GH(RP^n,∨_nS) ≥ π/6, matching the bound previously obtained via persistent Steenrod modules. The paper also compares the discriminative power of TC, zcl, cat, and cl on simple examples.

Significance. If the main stability statement is correct, the paper makes a useful contribution: it extends the categorical persistence framework of [MSZ24] to topological complexity, establishes a natural stability bound, and gives a concrete pair of metric spaces where persistent TC and zcl are strictly more discriminative than persistent homology. The example in Theorem 1.2 is worked out in detail and connects nicely to the existing literature on Gromov–Hausdorff lower bounds. The paper is clearly written and the definitions are reasonable. However, the central stability proof for TC has a gap that is load-bearing for the advertised claim, and a second gap appears in the verification that the structure maps in Lemma 4.5 are homotopy equivalences. These issues are potentially repairable, but they require substantive additional argument rather than cosmetic changes.

major comments (2)

[§4.2, proof of Proposition 1.1 (Eq. (1))] The proof invokes [MSZ24, Theorem 2.21], whose hypothesis is that the categorical invariant I is invariant under post- and pre-composition with weak homotopy equivalences. The proof verifies only invariance under homotopy equivalences, using Whitehead's theorem to identify weak equivalences between CW complexes with homotopy equivalences. This does not cover the arbitrary compactly generated weakly Hausdorff spaces that appear as W_• in the definition of the homotopy interleaving distance (Definition 2.4): the weak equivalences W_•→X_• and W_•→Y_• need not have CW domain, so Whitehead's theorem does not apply. TC is not invariant under weak equivalences with arbitrary domain: a weakly contractible non-contractible space such as the Warsaw circle is weakly equivalent to a point but has TC differing from TC(point), since it does not admit a global motion planner. Consequently, Equation (1) for I=TC is unsupported as written. The issue does not affect zcl, since singular cohomology is invariant under weak equivalences, but Proposition 1.1 and Theorem 1.2 state the result for both invariants, and the stability of persistent TC is a central advertised contribution.
[§4.2, Lemma 4.5] The lemma concludes that the structure maps VR_s→VR_t are homotopy equivalences for s,t in the indicated ranges, based on the fact that each individual complex VR_t is homotopy equivalent to RP^n (respectively ∨_nS). The cited results — [AHP22, Theorem 4.5 and the following remark], [LMO24, Theorem 7.1], and [Ada+20, Proposition 1] — support the homotopy type of the individual levels and, in the wedge case, the homotopy equivalence of the natural inclusion VR_t(X)∨VR_t(Y)→VR_t(X∨Y) for fixed t. They do not explicitly establish that the inclusions VR_s→VR_t for varying scales are homotopy equivalences. This distinction matters because persistent invariants are defined on the structure maps f_a^b, so the assertion I(VR_•)(J)=I(RP^n) requires the map, not only the endpoint spaces, to induce the stated value. An inclusion between two spaces each homotopy equivalent to RP^n can fail to be a homotopy equivalence. The proof should either cite a result that explicitly gives the maps as homotopy equivalences or provide a direct argument.

minor comments (5)

[§2.2.2, Definition 2.13] There are two typos in this definition: 'homotpy' should be 'homotopy', and 'or + inf' should read 'or +∞'.
[§2.3, Definition 2.19] The word 'ϵ-erosed' should be 'ϵ-eroded'. Also, in the displayed definition the codomain of the functors is written as (R≥0,≤), while the persistent invariants take values in N∪{∞} with the opposite order; the intended meaning is clear but the notation should be made consistent.
[§3.3, Definition 3.9] There is an inconsistency in the displayed definition of TC(f): the first sentence correctly says that an f-motion planner is a map U→Path(Y), but the displayed bullet writes s_i^f : U_i→Path(X) and then p_X∘s_i^f = id_{U_i}. The target should be Path(Y) and the fibration should be p_Y, matching the definition of the pullback topological complexity.
[§4.1, Definition 4.3] The arrow in the displayed formula appears to have the wrong variance. For a map f_a^b : X_a→X_b, the induced map on cohomology is (f_a^b×f_a^b)^* : H^*(X_b×X_b)→H^*(X_a×X_a), so it sends ker Δ^*_{X_b} to ker Δ^*_{X_a}. As written, 'ker Δ^*_{X_a} → ker Δ^*_{X_b}' suggests the opposite direction.
[§1 and §4.2, notation] The notation ∨_nS is used in the abstract and introduction before being defined as ∨_{i=1}^n S^i in Section 4.2. A brief parenthetical definition on first use would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stability theorem is cited from prior published work, and the example computations use known homotopy types and known invariant values rather than fitting the target conclusions.

full rationale

The paper's central stability claim, Proposition 1.1, invokes [MSZ24, Theorem 2.21] as a black-box theorem for categorical invariants. This is a self-citation by two of the present authors and it is load-bearing, but the cited theorem is published with a proof and does not assume the conclusions of this paper. The proof of Proposition 1.1 verifies invariance under homotopy equivalences of CW complexes via Whitehead's theorem, while Theorem 2.21 asks for invariance under weak homotopy equivalences for the wider class of spaces appearing in the homotopy interleaving distance. That discrepancy is a possible hypothesis-verification gap for TC, but it is not a circular reduction: the paper does not define TC or zcl in terms of the target erosion distance, nor does it fit any parameter to force the inequality. In the example, Lemma 4.5 combines known homotopy types of Vietoris-Rips complexes (from AHP22, LMO24, and Ada+20) with known values of TC and zcl, and Proposition 4.6 is an abstract erosion-distance estimate with constants chosen from those known scales. The lower bound pi/3 is not encoded in the definitions or in the cited machinery; it follows from the stated structure of the invariants on intervals. No fitted-input-called-prediction, self-definitional, or renaming pattern is present, so the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; thresholds such as 2π/3 and ζ_n come from cited results. The main additional assumptions are the map-level homotopy equivalence assertions and the weak-homotopy invariance required by the quoted stability theorem. No new entities are introduced.

assumptions (5)

domain assumption VR_t(RP^n) is homotopy equivalent to RP^n for t in (0, 2π/3) and contractible for t > π, and the inclusions between Rips complexes in this range are homotopy equivalences.
Used in Lemma 4.5(1); the space-level statement is cited to [AHP22, remark after Theorem 4.5], but the map-level homotopy equivalence is asserted without a specific citation.
domain assumption VR_t(S^d) is homotopy equivalent to S^d for t < ζ_d, and VR_t(X∨Y) is homotopy equivalent to VR_t(X)∨VR_t(Y) under the gluing metric.
Used in Lemma 4.5(2); cited to [LMO24, Theorem 7.1] and [Ada+20, Proposition 1]. The wedge factor combination and the map-level statement are assembled by the authors.
ad hoc to paper TC and zcl are invariant under post- and pre-composition with the weak homotopy equivalences that appear in the zig-zag defining the homotopy interleaving distance.
This is the requirement of [MSZ24, Theorem 2.21] that Proposition 1.1 must satisfy; the proof verifies the weaker property of invariance under homotopy equivalences of CW complexes, leaving a gap for non-CW spaces.
standard math Whitehead's theorem and the Künneth formula over a field hold in the stated generality.
Used in the proof of Proposition 1.1 and in the definition of zcl; standard background.
domain assumption The reduced topological complexity convention is used throughout, shifting Farber's original value by -1.
Stated in Section 3.1; affects all numerical values of TC.

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Pith. "Pith review of Persistence and Topological Complexity." pith.science (2026). https://pith.science/paper/PNABGFP4

@misc{pith2026250617888,
  author       = {Pith},
  title        = {Pith review of: Persistence and Topological Complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PNABGFP4}},
  note         = {Machine review of arXiv:2506.17888}
}

read the original abstract

Topological complexity is a homotopy invariant that measures the minimal number of continuous rules required for motion planning in a space. In this work, we introduce persistent analogs of topological complexity and its cohomological lower bound, the zero-divisor-cup-length, for persistent topological spaces, and establish their stability. For Vietoris-Rips filtrations of compact metric spaces, we show that the erosion distances between these persistent invariants are bounded above by twice the Gromov-Hausdorff distance. We also present examples illustrating that persistent topological complexity and persistent zero-divisor-cup-length can distinguish between certain spaces more effectively than persistent homology.

Figures

Figures reproduced from arXiv: 2506.17888 by the authors.

**Figure 1.** The persistent invariants I(VR•(RP n )) (left) and I(VR•(∨ nS)) (right), where I = TC or zcl, n ≥ 2, and ζn := arccos − 1 n+1 . Gray regions indicate undetermined values. See Lemma 4.5 for details. implying that persistent homology cannot yield a lower bound greater than π 8 for the Gromov–Hausdorff distance dGH(RP n , ∨ nS). To improve upon this, [MZ25] employed the persistent Steenrod modules introduced in [LM… view at source ↗

**Figure 2.** The functions f (left) and g (right). In each figure, white indicates function value 0, and gray represents unspecified values. Proof of Theorem 1.2. The first inequality in Equation (4) follows from Lemma 4.5 and Proposition 4.6 Item (1). The second inequality follows from Proposition 1.1. Remark 4.7 (Using LS-category and cup-length). It is known that cat(RP n ) = n for all n (see, for instance, [Opr14, Theorem 3… view at source ↗

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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On homotopy types of Vietoris–Rips complexes of metric gluings

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Some novel constructions of optimal Gromov-Hausdorff-optimal correspondences between spheres

Summer. 1999, 503–532 (2001). [Rod24a] Sa´ ul Rodr´ ıguez Mart´ ın. “Gromov-Hausdorff distances from simply connected geodesic spaces to the circle”. In: Proceedings of the American Mathematical Society. Series B 11 (2024), pp. 624–637. doi: 10.1090/bproc/243. [Rod24b] Sa´ ul Rodr´ ıguez Mart´ ın. “Some novel constructions of optimal Gromov-Hausdorff-opti...

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doi: 10.1016/j.topol.2022.108094. [Sho18] Robert Short. “Relative topological complexity of a pair”. In: Topology Appl. 248 (2018), pp. 7–23. doi: 10.1016/j.topol.2018.07.015. [Sma87] Steve Smale. “On the topology of algorithms, I”. In: Journal of Complexity 3.2 (1987), pp. 81–89. doi: https://doi.org/10.1016/0885-064X(87)90021-5. [SS89] Edwin H Spanier a...

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Limite inf´ erieure pour le nombre des valeurs critiques d’une fonction, donn´ ee sur une vari´ et´ e

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International Society for Optics and Photonics. 1992, pp. 122–133. doi: 10.1117/12.57059. 23 [FTY03] Michael Farber, Serge Tabachnikov, and Sergey Yuzvinsky. “Topological robotics: motion planning in projective spaces”. In: Int. Math. Res. Not. 34 (2003), pp. 1853–1870. doi: 10.1155/S1073792803210035. [Giu+21] Barbara Giunti, John S Nolan, Nina Otter, and...

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