Instability of the fundamental group for non-collapsed Ricci-limits

Camillo Brena

REVIEW 4 major objections 5 minor 17 references

Instability of the fundamental group for non-collapsed Ricci-limits

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Two sequences of closed 4-manifolds with identical Gromov–Hausdorff limit have different fundamental groups.

desk verdict A genuinely new two-sequence counterexample to Pan's stability question, but the metric-existence proofs are too sketched to verify the main theorem as written. read the letter →

arxiv 2505.17263 v1 pith:WKI2GK5S submitted 2025-05-22 math.DG

Camillo Brena This is my paper · ORCID

classification math.DG MSC 53C2353C2053C21

keywords nonnegativeRiccicurvatureGromov–HausdorfflimitfundamentalgroupstabilityEguchi–Hansonspacesphericalsuspensionspacesdesingularizationofquotientsingularity4-manifolds

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 aims to settle a question posed in [9]: whether the fundamental group of a closed manifold is determined, for nearby metrics, by a non-collapsed Gromov–Hausdorff limit. It constructs two sequences of closed 4-dimensional manifolds with non-negative Ricci curvature, diameter at most 1, and volume bounded below by a positive constant, such that the two sequences converge to the same Gromov–Hausdorff limit, yet one manifold in each pair has fundamental group Z/2Z and the other is simply connected. If the construction is correct, the fundamental group is not invariant under Gromov–Hausdorff convergence even in the non-collapsed, uniformly bounded case.

What carries the argument

The construction rests on two models. The first is the Eguchi–Hanson space, a complete simply connected 4-manifold asymptotic to the cone over $RP^{3}$ = $S^{3}$/µ_2; a free involution on it, conjugated by a mapping of $R^{4}$, produces a manifold asymptotic to the cone over $S^{3}$/µ_4 with fundamental group Z/2Z. The second is a Berger-sphere type doubly warped product metric, used for the desingularization of the orbifold $C^{2}$/µ_4, which gives the simply connected model. The main mechanism is a gluing and rescaling procedure: the warped products are modified with concave 1-Lipschitz profiles (using a perturbation lemma and a gluing step from the literature) to make them extend smoothly over a compact piece with the spherical suspension as Gromov–Hausdorff limit, while preserving non-negative Ricci curvature through the curvature formulas in Lemma 3 and Lemma 7.

What would settle it

Compute the Ricci curvature of the metric $ds^{2}$_{M',c,d} of Theorem 2 and the metric $ds^{2}$_{N,c,d} of Theorem 5 on the gluing intervals (d/2, d) and (π−d, π−d/2) using Lemma 3 and Lemma 7; if any Ricci component is negative for all small c and d, the main theorem collapses. A concrete check would evaluate the three nonnegativity conditions of Lemma 7 at the seam points where the profiles switch between the original and the perturbed forms.

Watch

Extended reading notes

Core claim

The central claim is that there exist sequences (M_i) and (N_i) of closed 4-manifolds with d_GH(M_i, N_i) → 0 and uniform bounds on curvature, diameter, and volume, with π1(M_i) = Z/2Z and π1(N_i) = 1. Both sequences converge to the same singular space, the spherical suspension of a scaled quotient sphere c $S^{3}$/µ_4 (the sphere quotiented by the diagonal action of the fourth roots of unity). This gives a negative answer to both versions of the question posed in [9].

Load-bearing premise

The load-bearing premise is that the metrics described in Theorems 2 and 5 actually exist, are smooth, have non-negative Ricci curvature, and admit the stated quotient behavior with fundamental groups Z/2Z and trivial, especially after the gluing step.

Editorial extensions

If this is right

The fundamental group is not determined by the Gromov–Hausdorff limit of a non-collapsed sequence of closed manifolds with uniform Ricci lower bound, diameter bound, and volume lower bound.
Both forms of the question in [9] — an ε-stability of fundamental groups and the determination of π1 from the limit — have a negative answer.
The example lives in dimension 4 and produces a singular common limit that is a spherical suspension, showing that such instability can appear with a particularly simple limit space.
It highlights the gap between the known surjective homomorphism from π1(M_i) to the fundamental group of the limit and the absence of injectivity or uniqueness for such homomorphisms.

Reading between the lines

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

Editorial inference: the method likely extends to produce limits where the fundamental groups differ by other finite abelian groups, by replacing the quotients µ_2 and µ_4 with other group actions, although the paper does not claim this.
Editorial inference: a direct numerical or symbolic check of the three nonnegativity conditions in Lemma 7 at the gluing seams would convert the sketchy proof of Theorem 5 into a fully verified construction; this is the most natural testable follow-up.
Editorial inference: if the perturbative gluing steps in the paper fail, a plausible rescue would be to use explicit hyper-Kähler ALE metrics instead of the mollified warped products, but that alternative is not explored in the paper.

Share X Bluesky LinkedIn Reddit HN

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. First, this note answers Pan's question in the negative with a clean construction: two non-collapsed sequences of closed 4-manifolds with nonnegative Ricci curvature, uniformly bounded diameter and volume, same GH limit, but fundamental groups Z/2Z and {1}. That is a real advance over Otsu's earlier example, which only gives one sequence. Second, the construction is credible but the proofs of the two key metric-existence theorems are sketches, and one of them contains a concrete mismatch between its statement and proof.

What is new and good: the two-sequence counterexample is genuinely new; the strategy of combining the Eguchi-Hanson resolution with the quotient by a free involution and a desingularized C2/µ4 is elegant and the high-level picture is clear. The paper is honest about relying on known tools (Anderson's gluing, Otsu's concave 1-Lipschitz perturbations, Zhou's doubly warped products). The main theorem follows directly if Theorems 2 and 5 hold, and there is no circularity or data-fitting.

The soft spots are in those theorems. Theorem 5 is not proved; it is declared 'exactly as Theorem 2', but the two constructions are structurally different (warped products over Hopf fibrations vs a perturbed Eguchi-Hanson metric), so that analogy does not carry the Ricci-check. Theorem 2's proof has a specific mismatch: the statement claims ds^2 = dr^2 + c^2(sin r − sin(9d/20)+d)^2 ds_3^2 on (d, π−d), while the proof builds a warping function ψ with a c/2 factor, using 9d/10 and intervals (d/2, π−d/2). Either the statement has a typo or the construction differs from what is claimed; a referee will need this sorted out. The gluing step, the simple-connectivity after gluing, and the extension of the free involution are each asserted in a line or two, not demonstrated. Lemma 8's inequalities are plausible but the 'c small enough' is not quantified, and Theorem 5's Ricci verification is absent. The diameter bound in the main theorem needs a rescaling that is not stated, though that part is harmless.

Net: the idea is good and the answer is probably correct, but the proof is not referee-ready as is. The paper deserves a serious referee who can fill or fix these gaps; it should not be desk-rejected. I would recommend asking for complete proofs of Theorems 2 and 5 before publication.

Referee Report

4 major / 5 minor

Summary. The paper claims to construct two sequences of closed 4-dimensional manifolds, (M_i) and (N_i), with non-negative Ricci curvature, diameter bounded above by 1, volume bounded below by a positive constant v, and d_GH(M_i,N_i)→0, such that π_1(M_i)=Z/2Z and π_1(N_i)=1. If correct, this gives a negative answer to Pan's question of whether the fundamental group is determined by a non-collapsed Ricci limit. The construction combines an Eguchi–Hanson-type metric with a free involution quotient for M_i, and a desingularization of C^2/µ_4 for N_i, with the common limit being a spherical suspension of a rescaled S^3/µ_4. The main argument is presented as Theorems 2 and 5, which are asserted to produce the needed warped-product metrics, followed by a one-sentence proof of the main theorem.

Significance. The result, if fully established, would settle a natural open question in the stability theory of fundamental groups under non-collapsed Ricci limits. The paper is concise and the overall strategy is plausible, combining known tools (Eguchi–Hanson metrics, Anderson's gluing, Otsu's perturbation lemma, and a doubly warped product desingularization) in a explicit way. However, the current manuscript is best read as a well-motivated construction outline: the two key existence theorems (Theorems 2 and 5) are only sketched, and the proof of the main theorem omits the rescaling or conformal modification needed for the stated diameter bound. With the missing details supplied, the construction would be a valuable contribution; at present, the central claim is not yet verified to the standard expected for a journal publication.

major comments (4)

[Theorem 2] The statement of Theorem 2 and its proof are not compatible as written. The theorem asserts, for r in (d, π-d), the metric ds^2 = dr^2 + c^2(sin(r) - sin(9d/20) + d)^2 ds_3^2, but the proof constructs a function ψ on (d/2, π-d/2) with middle branch c/2 sin(r) - c/2 sin(9d/10) + cd and endpoint branches c(r + d/10) and c(π - r + d/10). The interval, the factor c/2, and the constant sin(9d/10) all differ from the theorem's displayed formula. Since this explicit warping is the basis for the claimed convergence to the spherical suspension, the metric that is actually constructed is not the one whose asymptotic form is used later.
[Theorem 5] The proof of Theorem 5 is delegated to the sentence 'The proof of Theorem 5 is exactly as the proof of Theorem 2,' but Theorem 5 concerns S^3/µ_4 and does not involve the free involution quotient that is central to Theorem 2. No verification is given that the analog of the ψ-gluing preserves non-negative Ricci curvature for the doubly warped product of Theorem 4, nor that the completion is simply connected in the absence of the quotient. Because Theorem 5 is load-bearing for the sequence (N_i), this is a genuine gap rather than a harmless repetition.
[Lemma 8] The verification of conditions (2) and (3) in Lemma 8 is asserted with 'provided that c is small enough' and the displayed lower bounds for r ∈ (3/4, 5/4) are not derived. In particular, the expressions such as (2ρ/φ)[(n/2)^3/(n+c/2)^3 - nc] and 4 - 2 - (n+c/2)/(n/2) nc - (n+c/2)c∥φ∥∞ - c^2 are introduced without justification, and the notation φ is used both for the mollification kernel and for the warping function, making the argument difficult to follow. Since the non-negative Ricci curvature of the metric in Theorem 4 rests on these inequalities, a complete proof of Lemma 8 is required.
[Section 1] The proof of the main theorem is a single sentence that does not address the stated diameter bound diam(M_i), diam(N_i) ≤ 1. The metrics in Theorems 2 and 5 have a radial variable r ∈ (0, π), so their diameters are comparable to π, not bounded by 1, unless an explicit rescaling or conformal modification is made. The overview mentions a conformal change and gluing 'cf. [1]', but Section 1 does not specify the rescaling, the resulting volume lower bound v, or why the convergence to the spherical suspension survives the modification. These are necessary steps for the theorem as stated.

minor comments (5)

[Notation] The symbol ds_3 is used for the round metric on S^3 and also for the induced metric on quotients S^3/µ_2 and S^3/µ_4; this should be clarified, since the quotient metrics are not literally the same as the round metric on S^3.
[Lemma 8] The notation φ is used for the mollification kernel and also for the warping function in the same paragraph; using different symbols (e.g., η for the kernel) would remove avoidable confusion.
[Lemma 8] The sentence 'Notice first that ˆρ = ρ on (0, 3/4) ∪ (5/4, ∞)' is missing the hat on one side; the intended meaning is that the mollified function agrees with the piecewise linear one away from the transition region.
[Proof of Theorem 2] The one-line claim that the glued space remains simply connected by the Seifert–van Kampen theorem and that the free involution extends naturally needs a more detailed justification, especially because the gluing regions are not explicitly described topologically.
[References] The paper repeatedly invokes [8, Lemma 1.5] for the existence of smooth concave 1-Lipschitz perturbations; since this lemma is central to both main constructions, it would be helpful to state its exact content or at least the precise conditions under which it applies.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction is an explicit existence example built from external, non-self-cited ingredients; Theorem 5's delegated proof is a completeness gap, not a circular reduction.

full rationale

The paper's central claim is an existence theorem: two sequences of closed 4-manifolds with nonnegative Ricci curvature and different fundamental groups but a common Gromov–Hausdorff limit. The derivation chain is constructive. The metrics are engineered from the Eguchi–Hanson space, Otsu's perturbation lemma, Anderson's gluing result, and Zhou's doubly warped product analysis. None of these ingredients is a self-citation of the present paper, none is fitted to the target conclusion, and the desired fundamental groups and common limit are genuinely verified consequences of the construction rather than restatements of the assumptions. The sentence in the proof of Theorems 4 and 5, 'The proof of Theorem 5 is exactly as the proof of Theorem 2,' is an omitted proof, not a circular reduction: it delegates an analogous construction, but does not define the desired conclusion into existence. Similarly, the compressed gluing argument in Theorem 2 is a verification gap, since the claimed nonnegative Ricci curvature and fundamental group behavior do not coincide by definition with the inputs. The paper also openly discloses the 'drawback' that the asymptotic cone is cS^3/µ4 with a small constant c, which is a design parameter, not a concealed circular input. I find no step in which an output equals an input by construction, no fitted quantity relabeled as a prediction, and no load-bearing self-citation chain. Hence the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical or geometric entities. The free parameters are smallness constants quantified existentially rather than fitted values. The central claim rests on several cited external results, most notably the Eguchi-Hanson construction, Anderson's conformal modification, Otsu's perturbation lemma, and Zhou's warped product Ricci criterion, all of which are treated as black boxes in the proof.

free parameters (2)

c
Positive smallness constant in Theorems 1, 2, 4 and 5, required to lie below an unspecified threshold c'. It is a construction parameter, not fitted to data.
d_i = 1/i
Sequence parameter used to make the caps shrink and force the two sequences to converge to the same spherical suspension as i tends to infinity.

assumptions (5)

domain assumption The Eguchi-Hanson metric exists as a smooth simply connected gravitational instanton with nonnegative Ricci curvature and the stated asymptotic cone structure.
Invoked in Section 2 as the starting point for the M' construction; the paper recalls the construction but relies on the classical properties from [5].
domain assumption Otsu's Lemma 1.5 provides smooth concave 1-Lipschitz perturbations of the piecewise-linear functions used in the warped products.
Used in the proofs of Theorems 1 and 2 to smooth the functions phi and psi while preserving nonnegative Ricci curvature; cited as [8, Lemma 1.5].
domain assumption Anderson's Proposition 3.1 gives a conformal modification h that keeps nonnegative Ricci curvature and the required asymptotics.
The metric modification in Section 2 follows [1, Proof of Proposition 3.1], and the nonnegative Ricci claim is imported from that reference.
domain assumption The doubly warped product metrics over S3/µ_n are smooth on the completion and have nonnegative Ricci curvature precisely when the three conditions in Lemma 7 hold.
Lemma 7 is taken from [17, Section 4.1] and [12, Exercise 1.6.23]; the N' construction depends on this criterion being correct.
standard math Standard covering space and Seifert-van Kampen facts: a free Z/2Z quotient of a simply connected manifold has fundamental group Z/2Z, and a glued space built from simply connected pieces remains simply connected.
Used to conclude that M_i has fundamental group Z/2Z and that N_i is simply connected after the gluing.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Instability of the fundamental group for non-collapsed Ricci-limits." pith.science (2026). https://pith.science/paper/WKI2GK5S

@misc{pith2026250517263,
  author       = {Pith},
  title        = {Pith review of: Instability of the fundamental group for non-collapsed Ricci-limits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WKI2GK5S}},
  note         = {Machine review of arXiv:2505.17263}
}

abstract

We construct two sequences of closed $4$-dimensional manifolds with non-negative Ricci curvature, diameter bounded from above by $1$, and volume bounded from below by $v>0$, with different fundamental groups but with the same Gromov-Hausdorff limit. This provides a negative answer to the question posed in [J. Pan. Ricci Curvature and Fundamental Groups of Effective Regular Sets. Journal of Mathematical Study, 58(1):3--21, 2025].

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 15 canonical work pages

[1]

M. T. Anderson. Short geodesics and gravitational instantons. J. Differential Geom., 31(1):265–275, 1990

work page 1990
[2]

Cheeger and T

J. Cheeger and T. H. Colding. On the structure of spaces with Ricci curvature bounded below. I. J. Differential Geom. , 46(3):406–480, 1997

work page 1997
[3]

Cheeger and T

J. Cheeger and T. H. Colding. On the structure of spaces with Ricci curvature bounded below. II. J. Differential Geom. , 54(1):13–35, 2000

work page 2000
[4]

Cheeger and T

J. Cheeger and T. H. Colding. On the structure of spaces with Ricci curvature bounded below. III. J. Differential Geom. , 54(1):37–74, 2000

work page 2000
[5]

Eguchi and A

T. Eguchi and A. J. Hanson. Self-dual solutions to Euclidean gravity. Ann. Physics, 120(1):82–106, 1979

work page 1979
[6]

Grove, P

K. Grove, P. Petersen, V, and J. Y. Wu. Geometric finiteness theorems via controlled topology. Invent. Math. , 99(1):205–213, 1990

work page 1990
[7]

P. B. Kronheimer. The construction of ALE spaces as hyper-K¨ ahler quotients. J. Differential Geom., 29(3):665–683, 1989

1989
[8]

Y. Otsu. On manifolds of positive Ricci curvature with large diameter. Math. Z. , 206(2):255–264, 1991

work page 1991

Show all 17 references

[9]

J. Pan. Ricci Curvature and Fundamental Groups of Effective Regular Sets. Journal of Mathematical Study , 58(1):3–21, 2025

2025
[10]

Pan and G

J. Pan and G. Wei. Semi-local simple connectedness of non-collapsing Ricci limit spaces. J. Eur. Math. Soc. (JEMS) , 24(12):4027–4062, 2022

2022
[11]

Perelman

G. Perelman. Alexandrov spaces with curvatures bounded from below II. Preprint, 1991

1991
[12]

Petersen

P. Petersen. Riemannian geometry , volume 171 of Graduate Texts in Mathematics . Springer, Cham, third edition, 2016

2016
[13]

Santos-Rodr ´ ıguez and S

J. Santos-Rodr ´ ıguez and S. Zamora-Barrera. On fundamental groups of RCD spaces. J. Reine Angew. Math. , 799:249–286, 2023

2023
[14]

Sormani and G

C. Sormani and G. Wei. Hausdorff convergence and universal covers. Trans. Amer. Math. Soc., 353(9):3585–3602, 2001

2001
[15]

J. Wang. Ricci limit spaces are semi-locally simply connected. J. Differential Geom., 128(3):1301–1314, 2024

2024
[16]

S. Zamora. First betti number and collapse. Preprint, arXiv: 2209.12628, 2022

2022 arXiv
[17]

S. Zhou. A family of 4-manifolds with nonnegative Ricci curvature and prescribed asymptotic cone. Preprint, arXiv: 2406.02279, 2024. School of Mathematics, Institute for Advanced Study, 1 Einstein Dr., Princeton NJ 08540, USA Email address : cbrena@ias.edu

2024 arXiv

Pith tools

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

[1] [1]

M. T. Anderson. Short geodesics and gravitational instantons. J. Differential Geom., 31(1):265–275, 1990

work page 1990

[2] [2]

Cheeger and T

J. Cheeger and T. H. Colding. On the structure of spaces with Ricci curvature bounded below. I. J. Differential Geom. , 46(3):406–480, 1997

work page 1997

[3] [3]

Cheeger and T

J. Cheeger and T. H. Colding. On the structure of spaces with Ricci curvature bounded below. II. J. Differential Geom. , 54(1):13–35, 2000

work page 2000

[4] [4]

Cheeger and T

J. Cheeger and T. H. Colding. On the structure of spaces with Ricci curvature bounded below. III. J. Differential Geom. , 54(1):37–74, 2000

work page 2000

[5] [5]

Eguchi and A

T. Eguchi and A. J. Hanson. Self-dual solutions to Euclidean gravity. Ann. Physics, 120(1):82–106, 1979

work page 1979

[6] [6]

Grove, P

K. Grove, P. Petersen, V, and J. Y. Wu. Geometric finiteness theorems via controlled topology. Invent. Math. , 99(1):205–213, 1990

work page 1990

[7] [7]

P. B. Kronheimer. The construction of ALE spaces as hyper-K¨ ahler quotients. J. Differential Geom., 29(3):665–683, 1989

1989

[8] [8]

Y. Otsu. On manifolds of positive Ricci curvature with large diameter. Math. Z. , 206(2):255–264, 1991

work page 1991