1-Uryson width and covers

Arka Banerjee; Hannah Alpert; Panos Papasoglu

REVIEW 3 major objections 4 minor 13 references

1-Uryson width and covers

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A small universal cover bounds the 1-Uryson width of surfaces and virtually cyclic spaces.

desk verdict Two nice theorems with plausible proofs, but Proposition 4.5 has a gap that affects Theorem C. read the letter →

arxiv 2505.21126 v3 pith:EWHWRPM3 submitted 2025-05-27 math.MG math.DG

Hannah Alpert , Arka Banerjee , Panos Papasoglu This is my paper

classification math.MGmath.DG MSC 53C2353C20

keywords 1-UrysonwidthuniversalcoverRiemannianpolyhedronsurfacevirtuallycyclicfundamentalgroupD-separator2-dimensionalreduction4-manifold

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 asks whether a compact Riemannian polyhedron can have arbitrarily large 1-Uryson width even though its universal cover is narrow. Here 1-Uryson width is the smallest scale at which the space can be continuously crushed onto a 1-dimensional complex with all fibers of that diameter. The paper shows this cannot happen for two natural classes: if the fundamental group is virtually cyclic, the width of the space is at most six times the width of its universal cover (Theorem A), and if the space is a Riemannian surface with boundary, the width of the surface is at most that of its universal cover (Theorem B). It also proves that any counterexample in bounded dimension would already exist among Riemannian 2-complexes and among closed Riemannian 4-manifolds (Theorem C).

What carries the argument

The proof works at the level of D-separators: for a 2-dimensional Riemannian polyhedron, a D-separator is a 1-dimensional subpolyhedron Z such that every path component of Z and of its complement has diameter at most D, and UW1(X) is the infimum of such D. For surfaces, the separator on the universal cover is transported down to Σ by cutting along a sequence of length-minimizing geodesics, gluing in Euclidean strips, and modifying the separator inside each strip (Lemma 3.2), with Lemmas 3.3–3.5 showing the modification increases component diameters by at most a factor (1+ε); a (1+ε)^r-Lipschitz homeomorphism then returns the separator to Σ. For virtually cyclic groups, the load-bearing object is a map from the universal cover to a tree whose fibers have diameter at most UW1(\tilde X), together with Lemma 2.2: a continuous map from an n-gon to a tree has a fiber meeting three consecutive edges. Because virtually cyclic fundamental groups make suitable powers of loops homotopically trivial, those loops lift to the universal cover, and the small tree-fiber forces points on the corresponding sphere components of X to be close.

What would settle it

Take a flat rectangle [0,L]×[0,w] with the two vertical sides identified, forming an annulus with boundary, and compute the 1-Uryson width of this annulus and of its universal cover, the infinite flat strip of width w. Theorem B predicts the annulus width is no larger than the strip width; a computation giving the opposite inequality for any w,L would refute it.

Watch

Extended reading notes

Core claim

The central discovery is that the width gap cannot be realized by surfaces or by spaces with virtually cyclic fundamental group. More precisely, for a compact Riemannian polyhedron X with virtually cyclic fundamental group, the paper establishes UW1(X) ≤ 6·UW1(\tilde X); for a compact Riemannian surface Σ with boundary, it establishes UW1(Σ) ≤ UW1(\tilde Σ). The surface bound is sharp in the sense that no constant factor is lost. The paper also proves a reduction statement: if a bounded-dimensional sequence of compact Riemannian polyhedra has unbounded ratio UW1(X_n)/UW1(\tilde X_n), then a sequence with the same property exists among Riemannian 2-complexes and among closed Riemannian 4-manifolds. This is Theorem C, and it means a negative answer to the main question would already appear in very low dimensions.

Load-bearing premise

The surface argument depends on being able to choose the separator on the universal cover so that it meets each cutting geodesic transversely in only finitely many points, with the same separating constant; if that genericity assumption fails, the strip-gluing construction that carries the separator down to the surface does not go through.

Editorial extensions

If this is right

The virtually cyclic bound recovers and extends the earlier positive result for finite and cyclic fundamental groups, with the explicit factor 6.
For surfaces with boundary, the universal cover's 1-Uryson width is a true upper bound for the surface's, with no lost constant factor.
If a counterexample to the main question exists in bounded dimension, one exists among Riemannian 2-complexes and also among closed Riemannian 4-manifolds.
The periodic surface example shows the gap is real for non-universal covers: a regular cover can have width of order 1 while the base surface has width of order R.
For closed surfaces other than the sphere, the surface bound is vacuous because the universal cover has infinite 1-Uryson width; the substantive new case is surfaces with boundary, plus the projective plane.

Reading between the lines

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

An implication left implicit is that Question 1, in bounded dimension, becomes a question about the geometry of 2-dimensional skeleta; candidate counterexamples could be tested by cubulating 3- and 4-dimensional examples and measuring the 1-width of their intrinsic 2-skeleta.
The sharp constant in the surface theorem suggests that if a counterexample exists at all, it must use genuinely three- or four-dimensional geometry, not surface-like thickenings or cyclic covers.
The separator-and-strip construction is a natural template for the handlebody question the authors pose, where a system of compressing disks would play the role of the cutting geodesics.
The proof of the virtually cyclic case suggests the factor 6 may be improvable, since the finite-group case already yields factor 3 through the same polygon-to-tree lemma.

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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 does real work on a natural open problem. Theorems A and B are worth taking seriously, and the reduction result is a good idea, but the proof of Proposition 4.5 has a technical gap that looks fatal as written. I'd send it to a referee, but the referee should be asked to focus on Section 4.

What's new: Theorem A extends Katz's bound from finite/cyclic fundamental groups to virtually cyclic ones, with an explicit factor 6 and a different argument using tree fibers and sphere components. Theorem B gives the sharp constant for surfaces with boundary, using an elaborate strip-gluing construction. Theorem C's claim that counterexamples would already exist among 2-complexes and 4-manifolds is a genuinely useful reduction if it works.

What's good: Lemma 2.2 (a fiber hitting three consecutive edges) is clean. The separator perspective from [Pap20] is used well, and the surface argument in Section 3 is inventive, even if hard to read. The passage from a bad sequence to closed manifolds via regular neighborhoods in Proposition 4.1 is standard but well-executed.

Where it gets soft: the stress-test note on Proposition 4.5 lands. The proof subdivides X_n so every cube has diameter at most epsilon, then defines M_n as the number of cubes, then later says M_n is independent of epsilon. That is not consistent; M_n grows like epsilon^{-dim}. The term 2epsilon M_n in (†) and (††) cannot be absorbed by taking epsilon small, so the comparison between extrinsic and intrinsic metrics on the 2-skeleton is not established. The assertion that any geodesic passes through each cube at most once is also unjustified and is false for a flat torus. This undercuts the boundedness of UW1(Y~_n), so Theorem C's proof has a real gap.

In the surface section, Lemmas 3.2 and 3.6 assume Z meets the cutting geodesic in finitely many points and can be made transverse, with only a 'for convenience' and 'without loss of generality' handwave. That is a softer issue; standard perturbation arguments may close it, but as written it is an unproven genericity assumption. I'd ask the authors to either prove or cite a transversality statement that preserves the D-separator property.

Bottom line: the main results are plausible and the paper deserves a serious referee. But Section 4, as written, does not establish the 2-dimensional reduction, and Theorem C depends on it. This is fixable in principle, but it is a load-bearing gap, not a typo.

Referee Report

3 major / 4 minor

Summary. The paper studies the question whether the 1-Uryson width of a compact Riemannian polyhedron can be large while the 1-Uryson width of its universal cover is bounded. It proves three main results. Theorem A establishes UW1(X) ≤ 6·UW1(\tilde X) when π1(X) is virtually cyclic, via a sphere-component argument using maps to trees. Theorem B establishes UW1(Σ) ≤ UW1(\tilde Σ) for compact Riemannian surfaces with boundary, using a D-separator framework and a strip-gluing construction, with a separate argument for RP^2. Theorem C states that any sequence of bounded-dimensional polyhedra with unbounded ratio UW1(X_n)/UW1(\tilde X_n) must already contain such a sequence among Riemannian 2-complexes and among closed Riemannian 4-manifolds. The proof of Theorem C proceeds by a manifold reduction (Proposition 4.1), a 2-skeleton reduction (Proposition 4.5), and a final 4-manifold reduction (Theorem 4.6).

Significance. If the proofs are completed, the results are significant: they are among the first positive cases of the cover-to-base Uryson width question beyond finite or cyclic fundamental groups, and the low-dimensional reduction is a plausible route toward a full negative answer or a counterexample. The paper's use of the separator perspective from [Pap20] is natural and the explicit constants in Theorems A and B are useful. The main theorems are not equivalent to prior results; the separator machinery is used as a tool, not as the conclusion. The negative examples from [ABG21] are used appropriately for motivation and contrast. However, two load-bearing gaps, one in the surface proof and one in the low-dimensional reduction, currently prevent the theorems from being fully established as written.

major comments (3)

[§4, Proposition 4.5] The proof of Proposition 4.5 contains an internal inconsistency involving the dependence of M_n on ε. After subdividing X_n so that every cube has diameter at most ε, M_n is defined as the number of cubes in X_n. M_n therefore depends on ε (typically like ε^{-dim X_n}), yet the text states that M_n depends only on the initial cubulation and not on ε. In the displayed inequality (†) and the following estimate for d_{\tilde Y_n}, the terms 2εM_n and 2εM_n/(r_n−ε) appear; these do not tend to 0 as ε→0 when dim X_n ≥ 2 and the subdivision is fine. Thus the step 'choose ε small enough' is invalid. Moreover, the assertion that any geodesic in X_n passes through each cube at most once is not justified; it is false for cubulated flat tori, where a geodesic can wrap around and re-enter the same cube. Since the bookkeeping by 'at most M_n cubes' does not control the actual number of cube entries, the boundedness of UW1(\tilde Y_n) is not established. This gap affects both bullets of Theorem C, since Proposition 4.5 is used to produce the 2-complex sequence and then, via Proposition 4.1, the 4-manifold sequence.
[§3, Lemmas 3.2 and 3.6] The surface proof relies on two unproven genericity assumptions. Lemma 3.2 assumes that the D-separator Z contains only finitely many points of the cutting geodesic eγ, and Lemma 3.6 assumes, in addition, that each connected component of Z is a simple loop and that Z intersects eγ transversely. The paragraph beginning 'For convenience, we will assume that each connected component of Z is a simple loop' states these assumptions without proof. If these properties cannot be achieved by a perturbation that preserves the D-separator constant, the strip-gluing construction that produces the separator on the cut surface does not go through. In particular, the equivalence between UW1 and D-separators at the start of Section 3 gives no control over the intersection pattern of a separator with a given geodesic. The authors should either prove that such a perturbation exists or modify the argument to avoid the finiteness and transversality assumptions.
[§4, Lemma 4.4 and Proposition 4.5] The comparison between the extrinsic and intrinsic metrics on the 2-skeleton is the central technical point of Proposition 4.5, but the proof as written does not establish the required uniform closeness. The inequalities (†) and (††) give an upper bound that contains terms involving M_n, and the subsequent estimate for d_{\tilde Y_n}(a,b) increases the bound by factors involving kn and M_n. Even if M_n were independent of ε, the argument that a geodesic between two points in the 2-skeleton can be replaced by a path in the 2-skeleton whose length is controlled by the number of cubes is not proved for the case where the geodesic exits and re-enters cubes. The proof should provide a quantitative bound on the intrinsic distance in Y_n in terms of the extrinsic distance that is uniform over the sequence and over ε, without relying on the false claim that each cube is visited at most once.

minor comments (4)

[§2, Lemma 2.2] In the proof of the base case n=3, 'z ∈ e3' should read 'z ∈ e2'; also the sentence 'f(e0)∩[f(v0), f(v1)] and f(e1)∩[f(v0), f(v1)] intersects' is unclear and should be rephrased.
[§4, Proposition 4.5] In the first paragraph of the proof, the text says 'Since {UW1(Xn)} is unbounded and {UW1( fXn)} is unbounded' in two places; the second occurrence should be 'bounded'. Also, in the final estimate, 'p diam(Xn)' should be 'p dim(Xn)'.
[References] The reference [Bow20] is listed as 'B.H. Bowditch, Bilipschitz triangulations of Riemannian manifolds' without a venue or preprint identifier; the authors should provide a complete citation or explain why the result can be cited in this form.
[Throughout] The paper uses both 'Uryson' and 'Urysohn' spellings; the authors should choose one convention consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central theorems are derived from first principles, with external tools cited appropriately.

full rationale

The paper's central results are not circular. Theorem A rests on an elementary topological lemma (Lemma 2.2) proved in the text and on standard covering-space arguments; the citation to [GL83] supplies only a proof idea, not a premise. Theorem B introduces the D-separator perspective with an explicit proof of its equivalence to UW1, so the citation to [Pap20] is not load-bearing: the equivalence is established in the paper itself. The negative examples from [ABG21] are motivational and are not used as premises for Theorems A, B, or C. Theorem C depends on external cubulation results ([Bow20], [FO95]) and on internal propositions that do not assume the target conclusion. No fitted parameters are involved, and no cited 'uniqueness' result forces a choice. One correctness concern, distinct from circularity, appears in Proposition 4.5: the proof subdivides X_n so each cube has diameter at most ε and then defines M_n as the number of cubes, yet later asserts that 'rn and Mn depend only on the initial cubulation of Xn. In particular, they do not depend on ε.' Since M_n grows like ε^{-n}, the terms 2εM_n and 2εM_n/(r_n−ε) do not tend to zero as ε→0, so the displayed estimate for UW1(\tilde Y_n) is not justified. This is a proof gap in Theorem C, not a circular reduction: the conclusion does not equal an input by construction. The rest of the derivation chain is self-contained against external benchmarks and prior results are used only as tools or motivation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a pure mathematics proof with no experimentally fitted constants. The constants 6, 7, and (1+ε) in the estimates are derived from the arguments rather than chosen to match data. The listed external theorems are the main unproved background inputs; none of them is equivalent to the paper's central claims.

assumptions (4)

standard math Borsuk-Ulam theorem: any continuous map from S^2 to R^2 identifies some antipodal pair.
Invoked in the proof of Lemma 3.6 to show that diam(eγ) ≤ D, using a map from the universal cover of RP2, which is S^2, to R^2.
standard math Multi-jet transversality theorem from Golubitsky-Guillemin [GG73].
Used in the proof sketch of Theorem 3.1 to perturb the cut locus into a finite graph; the paper explicitly notes that full transversality details are omitted.
domain assumption Bowditch's cubulation theorem [Bow20, Theorem 1.1]: smooth closed Riemannian manifolds admit smooth cubulations whose path metric is bilipschitz equivalent to the original metric with a dimension-dependent constant.
Used in Proposition 4.5 to replace arbitrary Riemannian manifolds by cubulated complexes with standard Euclidean cubes, a key step in the reduction to 2-complexes.
domain assumption Ferry-Okun approximation theorem [FO95]: topological metrics on manifolds can be approximated by Riemannian metrics with prescribed closeness.
Used in Lemma 4.2 to endow the boundary of a regular neighborhood with a Riemannian metric that is within 1/i of the pullback metric from the polyhedron.

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Pith. "Pith review of 1-Uryson width and covers." pith.science (2026). https://pith.science/paper/EWHWRPM3

@misc{pith2026250521126,
  author       = {Pith},
  title        = {Pith review of: 1-Uryson width and covers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EWHWRPM3}},
  note         = {Machine review of arXiv:2505.21126}
}

abstract

We investigate the following question: Do there exist Riemannian polyhedra $X$ such that the 1-Uryson width of their universal covers $\mathrm{UW}_1(\widetilde{X})$ is bounded but $\mathrm{UW}_1(X)$ is arbitrarily large? We rule out two specific cases: when $\pi_1(X)$ is virtually cyclic and when $X$ is a Riemannian surface. More specifically, we show that if $X$ is a compact polyhedron with a virtually cyclic fundamental group, then its 1-Uryson width is bounded by the 1-Uryson width of its universal cover $\widetilde{X}$. Precisely: $$\mathrm{UW}_1(X) \leq 6 \cdot \mathrm{UW}_1(\widetilde{X}).$$ We show that if $X$ is a Riemannian surface with boundary then $$\mathrm{UW}_1(X) \leq \mathrm{UW}_1(\widetilde{X}).$$ Furthermore, we show that if there exist spaces $X$ for which $\mathrm{UW}_1(\widetilde{X})$ is bounded while $\mathrm{UW}_1(X)$ is arbitrarily large, then such examples must already appear in low dimensions. In particular, such $X$ can be found among Riemannian $2$-complexes.

Figures

Figures reproduced from arXiv: 2505.21126 by the authors.

**Figure 1.** If x1, x2, x3 are close to each other then a and b will also be close to each other. Proof. Note that d(x1, a) = d(x0, a) − d(x0, x1) ≤ d(x0, x3) + 2δ − d(x0, x1) ≤ d(x1, x3) + 2δ ≤ ε + 2δ [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Each fiber of the map in Theorem 3.1 is either far from the boundary and roughly parallel to it (gray region), or near the boundary and orthogonal to it, or a union of one piece of the first type with some pieces of the second type. In the general case, the rough picture, sketched in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** For each Zi , the corresponding Z ′ i connects to the same points along the left edge. If Zi connects to ∂Σ, then e Z ′ i connects to the top or bottom of the rectangle. (2) Let Z ′ 1 , . . . , Z′ k be the path components of Z ′ , except for {ε} × [0, L] if it is its own component, and let U ′ 1 , . . . , U′ ℓ be the path components of the complement of Z ′ in [0, ε] × [0, L]. Then every Z ′ i and every U ′ j contai… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: On the left are two typical instances of Z in the non– special case; on the right is the corresponding Z ′ . is { ε 2 i }×[min(Z ′ i ), max(Z ′ i )]. As in the special case, each Z ′ i is a rectilinear version of Zi , but we use the point p to determine which paths to …

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Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages

[1]

Hannah Alpert , Alexey Balitskiy , and Larry Guth , Macroscopic scalar curvature and codimension 2 width , arXiv e-prints (2021), arXiv:2112.04594, To appear in Journal of Topology and Analysis

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Pith tools

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

[1] [1]

Hannah Alpert , Alexey Balitskiy , and Larry Guth , Macroscopic scalar curvature and codimension 2 width , arXiv e-prints (2021), arXiv:2112.04594, To appear in Journal of Topology and Analysis

work page Pith review arXiv 2021

[2] [2]

780, 265--274

Alexey Balitskiy and Aleksandr Berdnikov, Local-to-global U rysohn width estimates , Journal f \"u r die reine und angewandte Mathematik (Crelles Journal) 2021 (2021), no. 780, 265--274

work page 2021

[3] [3]

Jean-Daniel Boissonnat, Ramsay Dyer, and Arijit Ghosh, Delaunay triangulation of manifolds, Found. Comput. Math. 18 (2018), no. 2, 399--431. 3777784

work page 2018

[4] [4]

Bowditch, Bilipschitz triangulations of R iemannian manifolds

B.H. Bowditch, Bilipschitz triangulations of R iemannian manifolds

work page

[5] [5]

Ferry and Boris L

Steven C. Ferry and Boris L. Okun, Approximating topological metrics by R iemannian metrics , Proceedings of the American Mathematical Society 123 (1995), no. 6, 1865--1872

work page 1995

[6] [6]

Golubitsky and V

M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, vol. Vol. 14, Springer-Verlag, New York-Heidelberg, 1973

work page 1973

[7] [7]

Blaine Lawson, Jr., Positive scalar curvature and the D irac operator on complete R iemannian manifolds , Inst

Mikhael Gromov and H. Blaine Lawson, Jr., Positive scalar curvature and the D irac operator on complete R iemannian manifolds , Inst. Hautes \' E tudes Sci. Publ. Math. (1983), no. 58, 83--196

work page 1983

[8] [8]

1, 1--147

Mikhael Gromov, Filling R iemannian manifolds , Journal of Differential Geometry 18 (1983), no. 1, 1--147

work page 1983