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arxiv: 2605.22473 · v2 · pith:24UNPMGYnew · submitted 2026-05-21 · 🧮 math.SG · math.AG· math.DG· math.GT

The nearby Lagrangian conjecture for pinwheels

Nikolas Adaloglou , Gerard Bargall\'o i G\'omez , Johannes Hauber This is my paper

Pith reviewed 2026-05-25 02:42 UTC · model grok-4.3

classification 🧮 math.SG math.AGmath.DGmath.GT
keywords Lagrangian pinwheelsnearby Lagrangian conjectureHamiltonian isotopyrational homology ballsymplectomorphism groupneck-stretchingsymplectic blow-up
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The pith

Any two embeddings of Lagrangian (p,q)-pinwheels in the rational homology ball B_{p,q} are related by compactly supported Hamiltonian isotopy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that all Lagrangian embeddings of a (p,q)-pinwheel into the rational homology ball B_{p,q} are connected by a compactly supported Hamiltonian isotopy. This establishes Arnold's nearby Lagrangian conjecture for these singular, immersed Lagrangians. The proof proceeds in two steps: neck-stretching combined with symplectic rational blow-up shows that embeddings are unique up to symplectomorphism, while a separate computation shows that the group of compactly supported symplectomorphisms is generated by a single twist about the pinwheel. The methods also yield three concrete applications, including non-squeezing for pin-balls and a new proof of local Lagrangian unknotting.

Core claim

Any two embeddings of Lagrangian (p,q)-pinwheels in B_{p,q} are related by a compactly supported Hamiltonian isotopy, establishing the nearby Lagrangian conjecture for this wide class of singular Lagrangians.

What carries the argument

The pintwist τ_{p,q}, the generator of the compactly supported symplectomorphism group Symp_c(B_{p,q}).

If this is right

  • Gromov non-squeezing holds for pin-balls.
  • The local Lagrangian unknotting theorem of Eliashberg--Polterovich receives a new proof.
  • The only Lagrangian (n,m)-pinwheel that embeds in B_{p,q} is the one of type (p,q).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The neck-stretching and blow-up techniques may apply to uniqueness questions for other singular Lagrangian skeletons in rational homology balls.
  • The generation result for Symp_c suggests that similar twist generators could control isotopy classes in related 4-dimensional symplectic manifolds.
  • The result constrains possible Lagrangian pinwheels in fillings of contact manifolds with the same boundary as B_{p,q}.

Load-bearing premise

The compactly supported symplectomorphism group of B_{p,q} is generated by the pintwist τ_{p,q}.

What would settle it

An explicit pair of (p,q)-pinwheel embeddings in B_{p,q} that cannot be connected by any compactly supported Hamiltonian isotopy.

Figures

Figures reproduced from arXiv: 2605.22473 by Gerard Bargall\'o i G\'omez, Johannes Hauber, Nikolas Adaloglou.

Figure 1
Figure 1. Figure 1: The Lefschetz fibrations defined on Ap−1 and Bp,q. Note that the central cylinder Kp,q is p-fold covered by a generic fibre of the Lefschetz fibration defined on Bp,q. 2.1.2. The almost toric fibration. The symplectic manifold Bp,q also admits an almost toric fibra￾tion, as is thoroughly explained in [28, 29]. Its almost toric base diagram Ap,q is as shown in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Representing Bp,q via its almost toric base diagram has many advantages. For example, it is easy to talk about geometrically interesting objects in Bp,q and to come up with potential ways to prove theorems about them. Domains in Bp,q of interest to us are so-called pin-ellipsoids and pin-balls, which were introduced in [10] and explored further in [9]. Definition 2.4 (Pin-ellipsoid and pin-ball). For α, β … view at source ↗
Figure 2
Figure 2. Figure 2: On the left the almost toric base diagram Ap,q of Bp,q. The branch cut points into the (p, q)-direction. Note that over the toric boundary lies the embedded symplectic cylinder Kp,q and the Lagrangian pinwheel’s core circle is an essential loop on the central cylinder Kp,q. On the right, the almost toric base diagram Ap,q(α, β) of the pin-ellipsoid Ep,q(α, β). 2.2. Pinwheels. Definition 2.7. A topological … view at source ↗
Figure 3
Figure 3. Figure 3: On the left the domain of parametrization of a Lagrangian pinwheel with a collar of its boundary (that becomes the core, in red) highlighted. On the right, the image of the collar neighbourhood for the (3, 1)- and (3, 2)-cases. Remark 2.10. Our definition of a Lagrangian pinwheel is stronger than the original one given by Khodorovskiy [45, Definition 3.1]. There, the way the flanges meet the core circle is… view at source ↗
Figure 4
Figure 4. Figure 4: The Lefschetz fibrations on Ap−1 and on Bp,q. Also shown are the vanishing thimbles and vanishing paths. 2.3. Pintwists. We now introduce the map τp,q ∈ Sympc (Bp,q) which will act as the analogue of the Dehn–Seidel twist about a Lagrangian sphere. It was originally defined by Buck [20, Propo￾sition 5.4], adapting arguments from [62], and we will follow his approach. Since τp,q has only recently been intro… view at source ↗
Figure 5
Figure 5. Figure 5: (a), for positive real numbers α and β. Since we will be interested mainly in uniqueness questions of Lagrangian pinwheels, we will fix for λ > 0 the “shape” ∆p,q(λ, λ) in the moment image ∆p,q and denote the corresponding compact orbifold by Bp,q(λ), whose moment image is shown in [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The relations between the spaces defined in Section 2.4. The upshot of this setup is the following lemma for pseudoholomorphic curves in the three manifolds (V , J),(X, b Jb) and (X, e Je). Lemma 2.31 ([9, Lemma 3.1.7/3.1.8]). There are bijections between the following collections of objects: (a) irreducible finite-energy punctured J-holomorphic curves C in V ; (b) irreducible orbifold Jb-holomorphic curve… view at source ↗
Figure 7
Figure 7. Figure 7: The steps taken to compactify the rational homology ball Bp,q to Xp,q. This introduces the compactifying divisor Dp,q = (D0, . . . , Dn). Remark 3.4. Note that in the computation of the type of the quotient singularity in Lemma 3.3 we went through a transformation with determinant equal to −1, i.e. the reflection along the x = y axis. One can avoid this and just apply a shear along the branch cut given by … view at source ↗
Figure 8
Figure 8. Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: The compactification X5,2. This almost toric base diagram is related to the one partially shown on the right in [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: This figure shows how the compactification X5,2 is birationally derived from the Hirzebruch surface F2 via a sequence of blow-ups. On the right we only display the blow-down sequence of the broken fibre. The homology classes of the divisors and the fibre on the left are easily calculated from the sequence of blow-ups, using the usual identification S 2 × S 2#CP2 ∼= X2. In the language of Lemma 3.10 this me… view at source ↗
Figure 10
Figure 10. Figure 10: The transformation of the regulation in the visible case. Lemma 3.20. The number of contractions that lead from Xe to the minimal model Y is equal to m + n − 1. In particular, the second Betti number of Xe is equal to m + n + 1. Proof. We know that a basis of H2(X, e Q) is given by Cp,q and Dp,q, and therefore we have dim(H2(X, e Q)) = n + (m + 1). Denoting the number of contractions that lead from Xe to … view at source ↗
Figure 11
Figure 11. Figure 11: (a) An example that illustrates the structure of the regulation in the class Ce proved in Theorem 3.21 in the case (p, q) = (4, 1). (b) The contraction process of the broken ruling for this example. In this example d0 = 5 and the figure shows how Xe4,1 is derived from F5. Remark 3.25. Note that the uniqueness of Lagrangian (p, q)-pinwheels in Bp,q up to symplecto￾morphism follows from this theorem, since … view at source ↗
Figure 12
Figure 12. Figure 12: The visible situation in the case (p, q) = (2, 1). The linear chain of spheres Tvis is formed by the Wahl chain Cvis, consisting of a single (−4)-sphere in this case, and the compactifying divisor Dvis, formed by a (+3)-sphere and a (−1)- sphere, connected by an exceptional divisor Evis. The boundary Σvis of a normal neighbourhood of the configuration Tvis = Cvis ∪Evis ∪ Dvis is a contact hypersurface. Σv… view at source ↗
Figure 13
Figure 13. Figure 13: The “visible” part of the constructions in Section 3.3. (a) shows how the Delzant polytope of Xevis \ Tvis embeds into the standard Delzant polytope of T ∗S 1 × R 2 , respecting the Liouville vector fields. How Σvis embeds is also shown. (b) shows how ψ(ν) can be completed to match the situation in (a). 4. Part 2: From symp to Ham Our goal in this section is to compute the homotopy groups of Sympc (Bp,q),… view at source ↗
Figure 14
Figure 14. Figure 14: (a) an almost toric base diagram of CP 2 and the same almost toric base diagram with the line ℓ∞ and the conic C removed. Moreover, the Liouville vector field emanating from the Whitney sphere is indicated. The Liouville vector field is not radial in a neighbourhood of the node, which is shown schematically in the figure. (b) The almost toric base diagram of CP 2\(ℓ∞ ∪C) extends to an almost toric base di… view at source ↗
Figure 15
Figure 15. Figure 15: The almost toric base diagram of Wp,q, as a subset of the toric base diagram of T ∗W h. Remark 4.5. For the special case (p, q) = (1, 1), the above proof shows that D∗W h, the unit cotangent disc bundle of the Whitney sphere, can be symplectically identified with B1,1(1, 1)\C1,1, which is exactly Rizell’s point of view in [23]. Combining Lemma 4.3 and Proposition 4.4 we immediately get: Corollary 4.6. The… view at source ↗
Figure 16
Figure 16. Figure 16: On the left, the almost toric base diagram Ap,q(∞, λ) defining Ep,q(∞, λ); on the right, the corresponding almost toric base diagram for Ep,q(λ, ∞). Remark 5.1. It is important to observe that if p ≥ 3, the two pin-cylinders Ep,q(λ, ∞) and Ep,q(∞, λ) are not symplectomorphic. That the usual standard cylinder E(λ, ∞) is symplecto￾morphic to E(∞, λ) is obvious by exchanging the coordinate planes and that E2… view at source ↗
Figure 17
Figure 17. Figure 17: (a) The almost toric base diagram of the compactification used in the proof of Theorem 5.2. The darker shaded region indicates how the embedded pin-ball ι [PITH_FULL_IMAGE:figures/full_fig_p049_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: (a) Schematic illustration of the initial situation of Theorem 5.7. The plane ∆ coincides with ∆τ outside B4 (1 − ε). (b) The situation after the symplectic cut as described in Section 5.2.1. The Lagrangian RP 2 s L and Lτ coincide in a small neighbourhood of the line at infinity CP 1 ∞ ⊆ CP 2 , which is shaded darker in the figure. On the complementary piece obtained by the symplectic cut we obtain two L… view at source ↗
Figure 19
Figure 19. Figure 19: On the left, the moment image of (G, H), with the origin missing. Note that it is clear from (31) that Lτ sits over the origin. On the right, the moment image after an affine SL2(Z) transformation. {z3 = 0} and, since it has real coefficients, intersects Lτ in a circle (an algebraic line of RP 2 ). Thus CP 1 ∞ splits into two discs in CP 2\Lτ , related by the involution τ . It is straightforward to check … view at source ↗
Figure 20
Figure 20. Figure 20: (a) A schematic illustration of the effect of the rational blow-up of L. Shaded in gray is the Weinstein neighbourhood discussed in Section 5.2.2. The “proper transform” of the line CP 1 ∞ is given by the two fibres F + and F −. (b) The reference model, namely the standard toric fibration on F4, with the configuration Tvis = (F + vis, Σvis, F − vis) and the neighbourhood νvis shaded in darker gray. 5.2.3.… view at source ↗
Figure 21
Figure 21. Figure 21: (a) A neighbourhood UΣ of the core of a Lagrangian (p, q)-pinwheel L admits a Lagrangian torus fibration as shown. Shaded in darker gray is a Weinstein neighbourhood of the annulus Γ. (b) The standard toric fibration of the ball and the visible Lagrangian disc Dst. A Weinstein neighbourhood of the annulus is shown in darker gray. Note that a neighbourhood of the centre of the disc contains a saturated nei… view at source ↗
Figure 22
Figure 22. Figure 22: (a) The base diagram of the Lagrangian fibration constructed on a neighbourhood of a Lagrangian pinwheel Lp,q ⊆ (X, ω). (b) The Lagrangian fibration on the same subset after a nodal slide and (c) a restriction of the neighbourhood given by taking the preimage of Ap,q(ϵ), which is contained in (b). To see that (b) contains a base of the form Ap,q(ϵ) change the branch cut in (b). bundle of the disc such tha… view at source ↗
Figure 23
Figure 23. Figure 23: (a) The reduced space. (b) The reduced space with the changed folia￾tion that induces the nodal slide. Remark 6.7. Note that this is a refinement of the Weinstein-type theorem that Khodorovskiy proves in [45, Section 3]. We emphasize that this theorem allows us to pass freely from embeddings of Lagrangian pinwheels to embeddings of small rational homology balls. References [1] M. Abouzaid. “Nearby Lagrang… view at source ↗
read the original abstract

The Lagrangian skeleton of the rational homology ball $B_{p,q}$, for $0<q<p$ coprime integers, is an immersed but not embedded Lagrangian, called a $(p,q)$-pinwheel. We show that any two embeddings of Lagrangian $(p,q)$-pinwheels in $B_{p,q}$ are related by a compactly supported Hamiltonian isotopy, establishing Arnold's nearby Lagrangian conjecture for this wide class of singular Lagrangians. Our proof has two largely independent parts: the first uses neck-stretching and the symplectic rational blow-up to understand embeddings of pinwheels up to symplectomorphism; the second computes that $\text{Symp}_c(B_{p,q})$ is generated by a twist about the pinwheel, which we call the pintwist $\tau_{p,q}$. We provide three applications of our methods: Gromov non-squeezing for pin-balls; a new proof of the local Lagrangian unknotting theorem of Eliashberg--Polterovich; and that the only Lagrangian $(n,m)$-pinwheel in $B_{p,q}$ is of type $(p,q)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves Arnold's nearby Lagrangian conjecture for (p,q)-pinwheels (0<q<p coprime) in the rational homology ball B_{p,q}: any two Lagrangian embeddings of such pinwheels are related by a compactly supported Hamiltonian isotopy. The argument splits into two largely independent parts: (i) neck-stretching combined with symplectic rational blow-up to classify embeddings up to symplectomorphism, and (ii) an explicit computation showing that Symp_c(B_{p,q}) is generated by the single element τ_{p,q} (the pintwist). Three applications are derived: Gromov non-squeezing for pin-balls, a new proof of the Eliashberg–Polterovich local Lagrangian unknotting theorem, and uniqueness of the (p,q)-type among (n,m)-pinwheels in B_{p,q}.

Significance. If the two parts hold, the result is a notable advance in symplectic geometry: it settles the nearby Lagrangian conjecture for a broad family of singular (immersed but not embedded) Lagrangians and supplies concrete applications that recycle the same techniques. The explicit group-generation computation and the separation of the classification step from the isotopy step are strengths; both reduce the scope for hidden assumptions. The work also supplies a new proof of an existing theorem, which is useful for the literature.

minor comments (3)
  1. [Introduction] The abstract states that the two parts are 'largely independent,' but the introduction should contain a short paragraph (perhaps after the statement of the main theorem) explaining why the classification up to symplectomorphism does not feed into the group-generation computation and vice versa.
  2. [§3 (neck-stretching section)] Notation for the rational blow-up and the almost-complex structures used in the neck-stretching limit should be introduced once, with a single consistent symbol, rather than redefined in each subsection.
  3. [Applications] The three applications are stated only in the abstract and introduction; a dedicated short section or subsection collecting the statements and indicating which parts of the main argument are reused would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's proof is structured as two explicitly independent parts: (1) classification of pinwheel embeddings up to symplectomorphism via neck-stretching and rational blow-up, and (2) direct computation that Symp_c(B_{p,q}) is generated by the pintwist τ_{p,q}. The central claim (nearby Lagrangian conjecture for these pinwheels) is the conjunction of these parts. No load-bearing step reduces by definition, fitted input, or self-citation chain to its own outputs; the second part is presented as an explicit group computation rather than an appeal to prior results by the same authors. The derivation is therefore self-contained against external benchmarks with no circular reduction visible from the given structure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background results in symplectic geometry; no free parameters, ad-hoc axioms, or new postulated entities are indicated in the abstract.

axioms (1)
  • standard math Standard properties of symplectic manifolds, Lagrangian submanifolds, and Hamiltonian isotopies
    The proof invokes established symplectic geometry as background.

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