Pith. sign in

REVIEW 3 minor 72 references

The nearby Lagrangian conjecture for pinwheels

T0 review · 0 major / 3 minor · reviewed 2026-05-25 · grok-4.3

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

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

desk verdict The paper settles the nearby Lagrangian conjecture for (p,q)-pinwheels by classifying embeddings via neck-stretching and showing the compactly supported symplectomorphism group is generated by the pintwist. read the letter →

arxiv 2605.22473 v2 pith:24UNPMGY submitted 2026-05-21 math.SG math.AGmath.DGmath.GT

Nikolas Adaloglou , Gerard Bargalló i Gómez , Johannes Hauber This is my paper · ORCID
classification math.SGmath.AGmath.DGmath.GT
keywords LagrangianpinwheelsnearbyconjectureHamiltonianisotopyrationalhomologyballsymplectomorphismgroupneck-stretchingsymplecticblow-up
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 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.

What carries the argument

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

Watch

Extended reading notes

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.

Load-bearing premise

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

Editorial extensions

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).

Reading between the lines

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}.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

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 · score 0.0 of 10

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.

Assumptions & free parameters 0 free parameters · 1 assumptions · 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.

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

how reviews work

0 comments
Cite this review

Pith. "Pith review of The nearby Lagrangian conjecture for pinwheels." pith.science (2026). https://pith.science/paper/24UNPMGY

@misc{pith2026260522473,
  author       = {Pith},
  title        = {Pith review of: The nearby Lagrangian conjecture for pinwheels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/24UNPMGY}},
  note         = {Machine review of arXiv:2605.22473}
}
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 $(p,q)$.

Figures

Figures reproduced from arXiv: 2605.22473 by the authors.

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. 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 ∈ R>0, the… view at source ↗
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 p-pinwheel… view at source ↗
Figures from the paper (22 more)
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 Lagrangi…
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 de…
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 imag…
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…
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…
Figure 8
Figure 8. Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]
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]
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 ar…
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 …
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…
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 excep…
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 comp…
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 o…
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…
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 symplecto…
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]
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…
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 circl…
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 sta…
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 …
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(…
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 pas…

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

72 extracted references · 72 canonical work pages

  1. [1]

    Nearby Lagrangians with vanishing Maslov class are homotopy equivalent

    M. Abouzaid. “Nearby Lagrangians with vanishing Maslov class are homotopy equivalent”. In:Invent. Math.189.2 (2012), pp. 251–313.doi:10.1007/s00222-011-0365-0

    work page doi:10.1007/s00222-011-0365-0 2012

  2. [2]

    Normal invariant of nearby Lagrangians via twisted derivative.arXiv:2505.12515

    M. Abouzaid, D. ´Alvarez-Gavela, S. Courte, and T. Kragh.Normal invariant of nearby Lagrangians via twisted derivative. 2025. arXiv:2505.12515 [math.SG]

    work page arXiv 2025

  3. [3]

    Twisted generating functions and the nearby Lagrangian conjecture

    M. Abouzaid, S. Courte, S. Guillermou, and T. Kragh. “Twisted generating functions and the nearby Lagrangian conjecture”. In:Duke Math. J.174.5 (2025), pp. 949–1011.doi:10. 1215/00127094-2024-0052

    work page 2025

  4. [4]

    Simple homotopy equivalence of nearby Lagrangians

    M. Abouzaid and T. Kragh. “Simple homotopy equivalence of nearby Lagrangians”. In:Acta Math.220.2 (2018), pp. 207–237.doi:10.4310/ACTA.2018.v220.n2.a1

    work page doi:10.4310/acta.2018.v220.n2.a1 2018

  5. [5]

    Topology of symplectomorphism groups ofS 2 ×S 2

    M. Abreu. “Topology of symplectomorphism groups ofS 2 ×S 2”. In:Invent. Math.131.1 (1998), pp. 1–23.doi:10.1007/s002220050196

    work page doi:10.1007/s002220050196 1998

  6. [6]

    Modular invariance of characters of vertex operator algebras,

    M. Abreu and D. McDuff. “Topology of symplectomorphism groups of rational ruled sur- faces”. In:J. Am. Math. Soc.13.4 (2000), pp. 971–1009.doi:10.1090/S0894- 0347- 00- 00344-1

    work page doi:10.1090/s0894- 2000

  7. [7]

    Embeddings and disjunction of Lagrangian pinwheels via rational blow-ups

    N. Adaloglou. “Embeddings and disjunction of Lagrangian pinwheels via rational blow-ups”. In:J. Symplectic Geom.24.1 (2026), pp. 105–129.doi:10.4310/JSG.260423234950

    work page doi:10.4310/jsg.260423234950 2026

  8. [8]

    Uniqueness of Lagrangians inT ∗RP 2

    N. Adaloglou. “Uniqueness of Lagrangians inT ∗RP 2”. In:Ann. Math. Qu´ e.49.1 (2025), pp. 215–222.doi:10.1007/s40316-024-00238-3

    work page doi:10.1007/s40316-024-00238-3 2025

Show all 72 references
  1. [9]

    Adaloglou, J

    N. Adaloglou, J. Brendel, J. Evans, J. Hauber, and F. Schlenk.Markov staircases. 2025. arXiv:2509.03224 [math.SG]

    2025

  2. [10]

    Adaloglou and J

    N. Adaloglou and J. Hauber.Pinwheels in symplectic rational and ruled surfaces and non- squeezing of rational homology balls. 2025. arXiv:2503.16250 [math.SG]. REFERENCES 65

    2025

  3. [11]

    Symplectic staircases for domains in cotangent bundles

    N. Adaloglou and J. Hauber. “Symplectic staircases for domains in cotangent bundles”. Forthcoming

  4. [12]

    Atallah, C

    M. Atallah, C. Y. Mak, and W. Wu.C 0-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces. Preprint, arXiv:2508.20285 [math.SG]. 2025

    2025

  5. [13]

    Lagrangian skeletons, periodic geodesic flows and symplectic cuttings

    M. Audin. “Lagrangian skeletons, periodic geodesic flows and symplectic cuttings”. In: Manuscr. Math.124.4 (2007), pp. 533–550.doi:10.1007/s00229-007-0134-y

    2007 doi

  6. [14]

    Asymptotically holomorphic families of symplectic submanifolds

    D. Auroux. “Asymptotically holomorphic families of symplectic submanifolds”. In:Geom. Funct. Anal.7.6 (1997), pp. 971–995.doi:10.1007/s000390050033

    1997 doi

  7. [15]

    Sur la structure du groupe des diff´ eomorphismes qui preservent une forme symplectique

    A. Banyaga. “Sur la structure du groupe des diff´ eomorphismes qui preservent une forme symplectique”. In:Comment. Math. Helv.53 (1978), pp. 174–227.doi:10.1007/BF02566074

    1978 doi

  8. [16]

    Symplectic fillings of links of quotient surface singularities

    M. Bhupal and K. Ono. “Symplectic fillings of links of quotient surface singularities”. In: Nagoya Math. J.207 (2012), pp. 1–45

    2012

  9. [17]

    Spherical Lagrangians via ball packings and symplectic cutting

    M. S. Borman, T.-J. Li, and W. Wu. “Spherical Lagrangians via ball packings and symplectic cutting”. In:Sel. Math., New Ser.20.1 (2014), pp. 261–283.doi:10.1007/s00029- 013- 0120-z

    2014 doi

  10. [18]

    Compactness results in symplectic field theory

    F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder. “Compactness results in symplectic field theory”. In:Geom. Topol.7 (2003), pp. 799–888.doi:10.2140/gt.2003. 7.799

    2003 doi

  11. [19]

    Pinwheels as Lagrangian barriers

    J. Brendel and F. Schlenk. “Pinwheels as Lagrangian barriers”. In:Commun. Contemp. Math. 26.5 (2024). Id/No 2350020, p. 21.doi:10.1142/S0219199723500207

    2024 doi

  12. [20]

    M. R. Buck.Lagrangian Spheres and Cyclic Quotient T-singularities. 2025. arXiv:2509 . 18976 [math.SG]

    2025

  13. [21]

    Compactness for punctured holomorphic curves

    K. Cieliebak and K. Mohnke. “Compactness for punctured holomorphic curves”. In:J. Sym- plectic Geom.3.4 (2005), pp. 589–654.doi:10.4310/JSG.2005.v3.n4.a5

    2005 doi

  14. [22]

    Cristofaro-Gardiner, N

    D. Cristofaro-Gardiner, N. Magill, and D. McDuff.Curvy points, the perimeter, and the complexity of convex toric domains. Preprint, arXiv:2506.23498 [math.SG]. 2025

    2025

  15. [23]

    The classification of Lagrangians nearby the Whitney immersion

    G. Dimitroglou Rizell. “The classification of Lagrangians nearby the Whitney immersion”. In:Geometry & Topology23 (2019), pp. 3367–3458.doi:10.2140/gt.2019.23.3367

    2019 doi

  16. [24]

    Lagrangian isotopy of tori inS 2 ×S 2 andCP 2

    G. Dimitroglou Rizell, E. Goodman, and A. Ivrii. “Lagrangian isotopy of tori inS 2 ×S 2 andCP 2”. In:Geom. Funct. Anal.26.5 (2016), pp. 1297–1358.doi:10.1007/s00039-016- 0388-1

    2016 doi

  17. [25]

    Introduction to Symplectic Field Theory

    Y. Eliashberg, A. Givental, and H. Hofer. “Introduction to Symplectic Field Theory”. In: GAFA 2000. Visions in mathematics—Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25–September 3, 1999. Part II.Basel: Birkh¨ auser, 2000, pp. 560–673

    2000

  18. [26]

    Local Lagrangian 2-knots are trivial

    Y. Eliashberg and L. Polterovich. “Local Lagrangian 2-knots are trivial”. In:Ann. Math. (2) 144.1 (1996), pp. 61–76.doi:10.2307/2118583

    1996 doi

  19. [27]

    J. B. Etnyre, H. Min, L. Piccirillo, and A. Roy.Small symplectic caps and embeddings of homology balls in the complex projective plane. 2023. arXiv:2305.16207 [math.GT]

    2023

  20. [28]

    Evans.KIAS Lectures on Symplectic Aspects of Degenerations

    J. Evans.KIAS Lectures on Symplectic Aspects of Degenerations. Preprint, arXiv:2403.03519 [math.SG]. 2024. 66 REFERENCES

    2024

  21. [29]

    Evans.Lectures on Lagrangian Torus Fibrations

    J. Evans.Lectures on Lagrangian Torus Fibrations. London Mathematical Society Student Texts. Cambridge University Press, 2023.doi:10.1017/9781009372671

    2023 doi

  22. [30]

    Symplectic mapping class groups of some Stein and rational surfaces

    J. Evans. “Symplectic mapping class groups of some Stein and rational surfaces”. In:J. Symplectic Geom.9.1 (2011), pp. 45–82.doi:10.4310/JSG.2011.v9.n1.a4

    2011 doi

  23. [31]

    Bounds on Wahl singularities from symplectic topology

    J. Evans and I. Smith. “Bounds on Wahl singularities from symplectic topology”. In:Algebr. Geom.7.1 (2020), pp. 59–85.doi:10.14231/AG-2020-003

    2020 doi

  24. [32]

    Markov numbers and Lagrangian cell complexes in the complex projective plane

    J. Evans and I. Smith. “Markov numbers and Lagrangian cell complexes in the complex projective plane”. In:Geom. Topol.22.2 (2018), pp. 1143–1180.doi:10.2140/gt.2018.22. 1143

    2018 doi

  25. [33]

    Infinitely many exotic monotone Lagrangian tori inCP 2

    R. Ferreira de Velloso Vianna. “Infinitely many exotic monotone Lagrangian tori inCP 2”. In:J. Topol.9.2 (2016), pp. 535–551.doi:10.1112/jtopol/jtw002

    2016 doi

  26. [34]

    Exact Lagrangian submanifolds in simply-connected cotangent bundles

    K. Fukaya, P. Seidel, and I. Smith. “Exact Lagrangian submanifolds in simply-connected cotangent bundles”. In:Invent. Math.172.1 (2008), pp. 1–27.doi:10.1007/s00222-007- 0092-8

    2008 doi

  27. [35]

    Golla and B

    M. Golla and B. Owens.The Farey tree and embeddings of lens spaces and rational balls in CP2. Preprint, arXiv:2512.09183 [math.GT]. 2025

    2025

  28. [36]

    Locality of relative symplectic cohomology for com- plete embeddings

    Y. Groman and U. Varolgunes. “Locality of relative symplectic cohomology for com- plete embeddings”. In:Compos. Math.159.12 (2023), pp. 2551–2637.doi:10 . 1112 / S0010437X23007492

    2023

  29. [37]

    Pseudo holomorphic curves in symplectic manifolds

    M. Gromov. “Pseudo holomorphic curves in symplectic manifolds”. In:Inventiones mathe- maticae82 (1985), pp. 307–347

    1985

  30. [38]

    Flipping surfaces

    P. Hacking, J. Tevelev, and G. Urz´ ua. “Flipping surfaces”. In:J. Algebr. Geom.26.2 (2017), pp. 279–345.doi:10.1090/jag/682

    2017 doi

  31. [39]

    Lagrangian spheres inS 2 ×S 2

    R. Hind. “Lagrangian spheres inS 2 ×S 2”. In:Geom. Funct. Anal.14.2 (2004), pp. 303–318. doi:10.1007/s00039-004-0459-6

    2004 doi

  32. [40]

    Symplectomophism groups of non-compact manifolds, orbifold balls, and a space of Lagrangians

    R. Hind, M. Pinsonnault, and W. Wu. “Symplectomophism groups of non-compact manifolds, orbifold balls, and a space of Lagrangians”. In:J. Symplectic Geom.14(1) (2016), pp. 203– 226

    2016

  33. [41]

    On genericity for holomorphic curves in four- dimensional almost-complex manifolds

    H. Hofer, V. Lizan, and J.-C. Sikorav. “On genericity for holomorphic curves in four- dimensional almost-complex manifolds”. In:J. Geom. Anal.7.1 (1997), pp. 149–159.doi: 10.1007/BF02921708

    1997 doi

  34. [42]

    Karabas.Microlocal Sheaves on Pinwheels

    D. Karabas.Microlocal Sheaves on Pinwheels. 2018. arXiv:1810.09021 [math.SG]

    2018 arXiv

  35. [43]

    Keating, I

    A. Keating, I. Smith, and M. Wemyss.Splitting symplectic monodromy. 2026. arXiv:2601. 20438 [math.SG]

    2026

  36. [44]

    Khodorovskiy.Bounds on Embeddings of Rational Homology Balls in Symplectic 4- manifolds

    T. Khodorovskiy.Bounds on Embeddings of Rational Homology Balls in Symplectic 4- manifolds. 2013. arXiv:1307.4321 [math.SG]

    2013 arXiv

  37. [45]

    Khodorovskiy.Symplectic Rational Blow-up

    T. Khodorovskiy.Symplectic Rational Blow-up. 2013. arXiv:1303.2581 [math.SG]

    2013 arXiv

  38. [46]

    Threefolds and deformations of surface singularities

    J. Koll´ ar and N. I. Shepherd-Barron. “Threefolds and deformations of surface singularities”. In:Invent. Math.91.2 (1988), pp. 299–338.doi:10.1007/BF01389370. REFERENCES 67

    1988 doi

  39. [47]

    Parametrized ring-spectra and the nearby Lagrangian conjecture

    T. Kragh. “Parametrized ring-spectra and the nearby Lagrangian conjecture”. In:Geom. Topol.17.2 (2013), pp. 639–731.doi:10.2140/gt.2013.17.639

    2013 doi

  40. [48]

    The symplectic topology of some rational homology balls

    Y. Lekili and M. Maydanskiy. “The symplectic topology of some rational homology balls”. In:Comment. Math. Helv.89.3 (2014), pp. 571–596.doi:10.4171/CMH/327

    2014 doi

  41. [49]

    Symplectic cuts

    E. Lerman. “Symplectic cuts”. In:Math. Res. Lett.2.3 (1995), pp. 247–258.doi:10.4310/ MRL.1995.v2.n3.a2

    1995

  42. [50]

    Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

    T.-J. Li and W. Wu. “Lagrangian spheres, symplectic surfaces and the symplectic mapping class group”. In:Geom. Topol.16.2 (2012), pp. 1121–1169.doi:10.2140/gt.2012.16.1121

    2012 doi

  43. [51]

    On almost complex embeddings of rational homology balls

    P. Lisca and A. Parma. “On almost complex embeddings of rational homology balls”. In: Frontiers in geometry and topology. Summer school and research conference, The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, August 1–12, 2022. Providence, RI: Ame...

    2022 doi

  44. [52]

    On Stein rational balls smoothly but not symplectically embedded inCP 2

    P. Lisca and A. Parma. “On Stein rational balls smoothly but not symplectically embedded inCP 2”. In:Bull. Lond. Math. Soc.54.3 (2022), pp. 949–960.doi:10.1112/blms.12607

    2022 doi

  45. [53]

    McDuff and D

    D. McDuff and D. Salamon.J-holomorphic curves and symplectic topology. Vol. 52. American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004, pp. xii+669.doi:10.1090/coll/052

    2004 doi

  46. [54]

    McDuff and D

    D. McDuff and D. Salamon.Introduction to symplectic topology. 3rd edition. Vol. 27. Oxf. Grad. Texts Math. Oxford: Oxford University Press, 2016

    2016

  47. [55]

    Ellipsoidal superpotentials and singular curve counts

    D. McDuff and K. Siegel. “Ellipsoidal superpotentials and singular curve counts”. In:Int. Math. Res. Not.2025.21 (2025). Id/No rnaf285, p. 37.doi:10.1093/imrn/rnaf285

    2025 doi

  48. [56]

    Singular algebraic curves and infinite symplectic staircases

    D. McDuff and K. Siegel. “Singular algebraic curves and infinite symplectic staircases”. In: Invent. Math.242.2 (2025), pp. 387–459.doi:10.1007/s00222-025-01359-4

    2025 doi

  49. [57]

    On certain Lagrangian submanifolds ofS 2 ×S 2 andCP n

    J. Oakley and M. Usher. “On certain Lagrangian submanifolds ofS 2 ×S 2 andCP n”. In: Algebr. Geom. Topol.16.1 (2016), pp. 149–209.doi:10.2140/agt.2016.16.149

    2016 doi

  50. [58]

    Equivariant embeddings of rational homology balls

    B. Owens. “Equivariant embeddings of rational homology balls”. In:Q. J. Math.69.3 (2018), pp. 1101–1121.doi:10.1093/qmath/hay016

    2018 doi

  51. [59]

    Smooth, nonsymplectic embeddings of rational balls in the complex projective plane

    B. Owens. “Smooth, nonsymplectic embeddings of rational balls in the complex projective plane”. In:Q. J. Math.71.3 (2020), pp. 997–1007.doi:10.1093/qmathj/haaa013

    2020 doi

  52. [60]

    Milnor fibers and symplectic fillings of quotient surface singularities

    H. Park, J. Park, D. Shin, and G. Urz´ ua. “Milnor fibers and symplectic fillings of quotient surface singularities”. In:Adv. Math.329 (2018), pp. 1156–1230.doi:10.1016/j.aim.2018. 03.002

    2018 doi

  53. [61]

    Polterovich and F

    L. Polterovich and F. Schlenk.Lagrangian knots and unknots – an essay.With an appendix by G. Dimitroglou Rizell. 2024. arXiv:2406.15967 [math.SG]

    2024

  54. [62]

    Graded Lagrangian submanifolds

    P. Seidel. “Graded Lagrangian submanifolds”. In:Bull. Soc. Math. Fr.128.1 (2000), pp. 103– 149.doi:10.24033/bsmf.2365

    2000 doi

  55. [63]

    Lagrangian two-spheres can be symplectically knotted

    P. Seidel. “Lagrangian two-spheres can be symplectically knotted”. In:J. Differ. Geom.52.1 (1999), pp. 145–171.doi:10.4310/jdg/1214425219. 68 REFERENCES

    1999 doi

  56. [64]

    Symplectic triangle inequality

    V. Shevchishin and G. Smirnov. “Symplectic triangle inequality”. In:Proc. Am. Math. Soc. 148.4 (2020), pp. 1389–1397.doi:10.1090/proc/14842

    2020 doi

  57. [65]

    Four dimensions from two in symplectic topology

    M. Symington. “Four dimensions from two in symplectic topology”. In:Topology and geome- try of manifolds (Athens, GA, 2001). Vol. 71. Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 2003, pp. 153–208.doi:10.1090/pspum/071/2024634

    2001 doi

  58. [66]

    Generalized symplectic rational blowdowns

    M. Symington. “Generalized symplectic rational blowdowns”. In:Algebr. Geom. Topol.1 (2001), pp. 503–518

    2001

  59. [67]

    Symplectic rational blowdowns

    M. Symington. “Symplectic rational blowdowns”. In:J. Differ. Geom.50.3 (1998), pp. 505– 518.doi:10.4310/jdg/1214424968

    1998 doi

  60. [68]

    Urz´ ua.Negative continued Fractions in Birational Geometry: A guide to Degenerations of Surfaces with Wahl Singularities

    G. Urz´ ua.Negative continued Fractions in Birational Geometry: A guide to Degenerations of Surfaces with Wahl Singularities. 2025

    2025

  61. [69]

    Urz´ ua and J

    G. Urz´ ua and J. P. Z´ u˜ niga.The birational geometry of Markov numbers. 2025. arXiv:2310. 17957 [math.AG]

    2025

  62. [70]

    Wendl.Holomorphic curves in low dimensions

    C. Wendl.Holomorphic curves in low dimensions. From symplectic ruled surfaces to planar contact manifolds. Vol. 2216. Lect. Notes Math. Berlin: Springer, 2018.doi:10.1007/978- 3-319-91371-1

    2018 doi

  63. [71]

    Exact Lagrangians inA n-surface singularities

    W. Wu. “Exact Lagrangians inA n-surface singularities”. In:Math. Ann.359.1-2 (2014), pp. 153–168.doi:10.1007/s00208-013-0993-3

    2014 doi

  64. [72]

    On an exotic Lagrangian torus inCP 2

    W. Wu. “On an exotic Lagrangian torus inCP 2”. In:Compos. Math.151.7 (2015), pp. 1372– 1394.doi:10.1112/S0010437X14007945. Nikolas Adaloglou, imj-prg, Sorbonne Universit ´e et Universit´e Paris Cit´e, CNRS Email address:adaloglou@imj-prg.fr Gerard Bargall´o i G ´omez, Institut...

    2015 doi

Pith tools

Reviewed May 25, 2026 · model on record in the stance chip above.