Pith. sign in

REVIEW 4 major objections 5 minor 49 references

Unknottedness of symplectic submanifold fillings

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 Any symplectic filling of the standard contact sphere lying linearly inside a standard ball is smoothly unknotted.

desk verdict Plausible and important result, but the proof has a load-bearing gap: the existence of an almost complex structure making W holomorphic while preserving the moduli-space counts is asserted in a corrupted passage, and the pi_1-argument is sketched. read the letter →

arxiv 2506.06807 v1 pith:PIOLMX64 submitted 2025-06-07 math.SG

Zhengyi Zhou This is my paper
classification math.SG MSC 53D3553D4053D4257R17
keywords symplecticfillingscontactsubmanifoldsunknottednesspuncturedholomorphiccurvesintersectiontheoryL-simplealmostcomplexstructureslinearizedhomologyS1-equivariantcohomology
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

This paper proves a relative version of filling rigidity: if $W$ is a symplectic filling of the standard contact sphere $(S^{2n-1},\xi_{\mathrm{std}})$ sitting linearly inside $(S^{2n+1},\xi_{\mathrm{std}})$ in the standard ball $(D^{n+1},\omega_{\mathrm{std}})$, then for every $n\ge 2$ the pair $(D^{n+1},W)$ is smoothly unknotted, meaning diffeomorphic relative to the boundary to the standard linear disk. The result rules out many knotted codimension-two balls: those smooth models cannot be symplectic fillings of the simplest contact submanifold. The proof turns the filling into a holomorphic hypersurface for a carefully chosen almost complex structure and then probes the complement with punctured holomorphic curves, using intersection theory to show those curves stay away from the filling. A degree-one evaluation map from a moduli space of holomorphic planes forces the complement to have the fundamental group of a circle; in the sphere case the complement is homotopy equivalent to $S^1$, and a codimension-two unknotting theorem concludes smooth unknottedness. The same machinery gives general statements about the complements of symplectic submanifold fillings in any Liouville domain.

What carries the argument

The engine of the proof is a higher-dimensional intersection theory for punctured holomorphic curves against holomorphic hypersurfaces, set up in the L-simple framework: near the relevant Reeb orbits the almost complex structure is chosen so that the Cauchy–Riemann equation splits into a hypersurface direction and a normal direction, and the normal part has a linear asymptotic expansion in eigenfunctions of a self-adjoint operator. Winding numbers of those eigenfunctions define hidden intersections at punctures, and the intersection formula $$u\bullet H = \sum_{p:u(p)\in H}\delta(p,u,H) + \sum_{p}\delta_\infty(p,u,H)$$ expresses the total intersection number as a sum of positive local terms; in particular $u\bullet H=0$ forces the curve either to lie in $H$ or to avoid it. A holomorphic foliation of $\hat V\times \mathbb{C}$ confines every holomorphic plane asymptotic to the maximal Reeb orbit $\gamma_{p_{\max}}$ to a fixed homology class, and a filtered isomorphism between linearized contact homology and $S^1$-equivariant positive symplectic cohomology for $V\times D$ yields the one-point count $\#\mathcal{M}_J(\gamma_{p_{\max}},q)=1$ for generic $q$. That degree-one evaluation, applied to a loop in the complement, is what turns homology information into the fundamental-group surjection.

What would settle it

Find, for some $n\ge 2$, a symplectic filling $W$ of the standard $(S^{2n-1},\xi_{\mathrm{std}})$ inside $(D^{n+1},\omega_{\mathrm{std}})$ whose complement $D^{n+1}\setminus W$ has fundamental group not isomorphic to $\mathbb{Z}$; the paper proves the complement is homotopy equivalent to $S^1$, so such an example would refute the unknottedness theorem.

Watch

Extended reading notes

Core claim

The central claim is that relative topological rigidity holds for the simplest contact submanifold: for $n\ge 2$, every symplectic filling $W$ of $(S^{2n-1},\xi_{\mathrm{std}})$ inside $(S^{2n+1},\xi_{\mathrm{std}})$ in $(D^{n+1},\omega_{\mathrm{std}})$ is smoothly unknotted, so the pair $(D^{n+1},W)$ is diffeomorphic, relative to the boundary, to the standard linear disk. More generally, for a Liouville domain $V$ and a codimension-two symplectic submanifold $U\subset V$, any symplectic filling $W$ of $\partial(U\times D)$ in $V\times D$ has complement homology isomorphic to that of $V\setminus U$, and the inclusion induces a surjection on fundamental groups; when $\pi_1(V\setminus U)$ is abelian, $V\times D\setminus W$ is homotopy equivalent to $V\setminus U$. The route taken is to choose almost complex structures making $\hat W$ a holomorphic hypersurface in the completion, and to let intersection theory for punctured holomorphic curves show that the curves probing the complement never meet $\hat W$, so their evaluation maps survive and control the topology of the complement.

Load-bearing premise

The proof rests on the existence of one geometric structure that both turns the filling into a holomorphic hypersurface and keeps the holomorphic-plane counting spaces regular, with degree-one evaluation maps; the paper asserts that such a structure can be arranged and then suppresses the requirement, and this assertion carries the main theorem.

Editorial extensions

If this is right

  • For $n\ge 2$, every symplectic filling of the standard $(S^{2n-1},\xi_{\mathrm{std}})$ inside $(D^{n+1},\omega_{\mathrm{std}})$ is smoothly the standard linear disk, so the smooth knot type of such a filling is unique.
  • For any Liouville domain $V$ and codimension-two symplectic submanifold $U\subset V$, any symplectic filling of $\partial(U\times D)$ in $V\times D$ has complement with the same homology as $V\setminus U$, and $\pi_1(V\setminus U)$ maps onto $\pi_1(V\times D\setminus W)$; if $\pi_1(V\setminus U)$ is abelian, the complement is homotopy equivalent to $V\setminus U$.
  • A symplectic filling of the binding $\partial V\times\{0\}$ of the trivial open book in $\partial(V\times D)$ induces an isomorphism on homology and a surjection on fundamental groups; if $\pi_1(V)$ is abelian, the inclusion is a homotopy equivalence.
  • Since the classical exact-filling theorem already forces $W$ to be a disk, the unknottedness statement is about the pair, and the proof works with the standard filling of $(S^{2n+1},\xi_{\mathrm{std}})$ replaced by any exact filling.
  • Smoothly knotted codimension-two balls are abundant, so the theorem implies that none of those knotted models can be symplectic fillings of the standard contact submanifold.

Reading between the lines

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

  • The paper proves smooth unknottedness only; symplectic unknottedness remains open, and a natural conjecture is that the same intersection-theoretic control can be upgraded to a symplectomorphism statement.
  • The self-contained proof of the intersection formula in the L-simple setup is likely to be reusable in other constructions of holomorphic hypersurfaces, such as contact-homology computations with intersection information, which the paper mentions but does not develop.
  • The fundamental-group surjection is established only for the specific contact pairs $\partial(U\times D)$ inside $\partial(V\times D)$; the paper expects this rigidity to be special, and a test in the lowest case $n=2$, where the filling is a 4-manifold inside a 6-ball, would show how far the regularity assumption reaches.
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.

Referee Report

4 major / 5 minor

Summary. The paper studies symplectic fillings W of a codimension-two contact submanifold H = ∂(U × D) inside ∂(V × D), where V is a Liouville domain, with the main application being the standard contact submanifold (S^{2n−1}, ξ_std) in (S^{2n+1}, ξ_std) inside the standard ball. The central claims are that, for n ≥ 2, every such filling of the standard contact submanifold is smoothly unknotted in the ball, and more generally that the complement inclusion V × {(0,1)} \ U × {(0,1)} → V × D \ W induces a surjection on π1 and, when π1(V \ U) is abelian, a homotopy equivalence. The method is pseudoholomorphic: the paper constructs L-simple almost complex structures and a holomorphic foliation, develops a Siefring-type intersection formula for punctured curves and holomorphic hypersurfaces in the L-simple setting, and uses holomorphic planes asymptotic to a maximal Reeb orbit to push loops in the complement to the boundary. A separate argument using holomorphic spheres proves an analogous rigidity statement for fillings of the binding of a trivial open book (Theorem 1.5).

Significance. If the proof is completed, the main theorem is a strong relative filling-rigidity result: it shows that symplectic fillings of the standard codimension-two contact submanifold are smoothly standard, a first concrete step toward Casals' question about symplectic unknottedness. The paper also contains useful technical contributions: the explicit holomorphic foliation and uniform homology bound of Proposition 4.6, the filtered Bourgeois–Oancea comparison in Theorem 4.3, and the pseudocycle arguments in Section 6. These are well-motivated and appear to be on the right track. However, the manuscript as written has load-bearing gaps, most importantly an asserted but unproved simultaneous existence of an almost complex structure that makes the arbitrary filling holomorphic and keeps the relevant moduli spaces regular with the required counts; this is used in the proof of Proposition 5.3 and hence in Theorems 1.4 and 1.2.

major comments (4)
  1. [§4, passage after Proposition 4.9] The passage after Proposition 4.9 asserts that for a symplectic filling W of H in V × D there exists an almost complex structure J on the completion such that W is J-holomorphic and Propositions 4.8 and 4.9 hold, with the explanation that one can perturb J near p and ν, which are outside W. This is not justified. Propositions 4.8 and 4.9 require J, at least on the positive end, to be a generic perturbation of the class considered in Proposition 4.6 so that M_J(γ_pmax, q) is regular with signed count 1 and the evaluation map on M_J(γ_pmax, ν)/∼ has degree 1. The condition that the arbitrary filling W be J-holomorphic is a closed constraint on J, and perturbing J only near the marked points p and ν does not automatically preserve transversality of the moduli spaces or the stated counts and degree. The proof of Proposition 5.3, and hence of Theorems 1.4 and 1.2, depends on this simultaneous-existence statement. A complete proof of this compatibility assertion is required.
  2. [§5, Proposition 5.3] The loop-pushing argument in Proposition 5.3 is too compressed and is not sufficient as written. After defining the intervals I_i and the reparameterizations φ_i ∈ Aut(C,0), the text states that the C∞_loc limits from the two sides of a glued point may be different, and then defines paths p_{i,s} connecting s and φ_i^{-1}(s) with lim_{s→∞} |p_{i,s}| = ∞. It is not established that these paths can be chosen continuously in i and s, nor that the concatenated maps u_i(t)(s) converge on the glued boundary points to an actual holomorphic curve. The conclusion that ν_s for s ≫ 0 lies in R_+ × (Y \ H) requires uniform or at least controlled convergence of the parametrized curves, which is not proved. Since this is the mechanism that proves π1-surjectivity, the argument needs to be written out in full.
  3. [§4.1, Proposition 4.1] The proof of Proposition 4.1 cites 'Theorem 2.10', but no such theorem exists in the manuscript; the closest statement is Example 2.10, which records that for f = −ε(x² + y²) the relevant eigenvalue has winding number 0. The conclusion u • W = 0 relies on the hidden intersection δ∞ being zero for this special f. The intended lemma should be stated and proved explicitly rather than referencing a nonexistent theorem. As written, the proof of Proposition 4.1, which is essential for ensuring that the probing curves do not intersect the filling W, is not complete.
  4. [§6.2, list of almost complex structure conditions] In Section 6.2 the paper assumes an almost complex structure on V × CP^1 that is compatible with λ_V ⊕ ω_CP1, is product-type on the end, makes the arbitrary filling W holomorphic, and makes V × {(2,0)} and V × {∞} holomorphic hypersurfaces. The simultaneous existence of such a J is asserted without proof, and Proposition 6.5 further requires generic J for pseudocycle transversality. This is the same type of compatibility issue as the gap after Proposition 4.9 and underpins Theorem 1.5. The existence of such a J with all the required properties needs to be established.
minor comments (5)
  1. [§2, abstract/introduction] The abstract and introduction promise a self-contained proof of the Siefring intersection formula, but Theorem 2.6 is stated without proof and the text refers to Wendl's book for the chain of arguments; Remark 2.8 further defers a needed variant to the in-preparation work [ABDRZ]. The self-containedness claim should be either fulfilled by a proof of Theorem 2.6 or softened.
  2. [Throughout] There are many cross-reference errors: for example 'Theorem 2.10' in Proposition 4.1, 'Theorem 3.1' for Proposition 3.1, 'Theorem 3.4' and 'Theorem 3.5' for Propositions 3.4 and 3.5, and 'Theorem 4.6'/'Theorem 4.8' in the proof of Proposition 4.8 for Propositions 4.6 and 4.8.
  3. [§4, after Proposition 4.9] The sentence after Proposition 4.9 contains corrupted text 'such that ?? 4.8?? 4.9 hold' and must be repaired; as printed it does not state a precise assertion.
  4. [References] The proofs rely on the in-preparation works [ABDRZ] and [Sie] for essential ingredients (Remark 2.8 and the higher-dimensional intersection theory in Section 2). For a journal submission these dependencies should be disclosed explicitly and, ideally, the relevant statements should be proved or stated as assumptions.
  5. [§5, Remark 5.4] Remark 5.4 discusses extensions to subcritical surgeries and explicitly says 'To rigorously prove those claims is non-trivial.' This is fine as a remark, but it should be separated more clearly from the main proof so that the reader does not confuse a stated extension with a proved theorem.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central unknottedness proof is a derivation from independent pseudo-holomorphic-curve and algebraic-topology inputs, not a repackaging of its own conclusion.

full rationale

The paper's derivation chain is not circular. The Siefring intersection formula is proved self-containedly in the L-simple setup rather than imported wholesale from [Sie] or [MS19], so the geometric tool is an independent input. The curve counts used later (Proposition 4.8) are obtained from the Bourgeois–Oancea isomorphism and the author's preceding computation [Zho23, Proposition 2.9]; these are parameter-free published results whose assumptions do not include the unknottedness or complement-homotopy conclusions of this paper. Proposition 5.1 uses [Zho23, Theorem 1.1] to identify the homology of W with that of U × {(0,1)}, but that theorem concerns fillings of ∂(V × D) in the absolute setting and does not already assert the present relative complement statement; it is a building block, not the target. No fitted parameter is renamed as a prediction, and no ansatz is smuggled in through citations: the foliated almost complex structures and moduli spaces are constructed explicitly in Section 3. The main genuine caveat is a technical existence claim, not a circular one: the passage after Proposition 4.9 states that one can find compatible almost complex structures such that Propositions 4.8 and 4.9 hold for p and ν that do not intersect hat W, as we can perturb J near p and ν, which are outside of hat W. This is an omitted regularity argument on which the proof of Proposition 5.3 depends; the manuscript even says the requirement will be suppressed. That makes Theorem 1.4 conditional on a nontrivial technical assertion, but it is not an equivalence between an output and an input. The in-preparation references [ABDRZ] and [Sie] are not load-bearing in the main chain: the paper develops its own L-simple intersection theory and cites [Sie] only as an announcement of a more general result. Self-citations to [Zho21], [Zho22], [Zho23] and [Zho24b] support specific technical steps but do not assume the theorem being proved. Overall, the central claim has independent content and the derivation does not reduce to its inputs by construction, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no free parameters or new entities. It imports several published theorems and one unproved regularity assertion; the latter is flagged as the weakest point.

assumptions (7)
  • standard math Gromov compactness and SFT compactness for alpha-tame almost complex structures
    Used to define and compactify moduli spaces of holomorphic curves; invoked via [BH18, Section 3.4] in Remark 2.2.
  • standard math Asymptotic expansion and self-adjoint asymptotic operators for L-simple almost complex structures
    Basis for the intersection theory; drawn from [Wen16, Section 7.2] and [Wen20].
  • domain assumption Bourgeois-Oancea isomorphism between filtered linearized contact homology and S^1-equivariant positive symplectic cohomology
    Theorem 4.3 relies on [BO09a] and [BO17]; it is imported from the literature.
  • domain assumption [Zho23, Theorem 1.1]: any symplectic filling of ∂(V×D) is homotopy equivalent to V
    Used in the proof of Theorem 5.1 to get the homology isomorphism of the filling W with U.
  • domain assumption Existence of L-simple contact forms whose short Reeb orbits are fibers over critical points of a Morse function
    Constructed in Section 3.1 following [BH23] and [Zho21, Section 6]; enables the holomorphic curve setup.
  • standard math Levine's theorem: a codimension-2 sphere in a sphere with complement homotopy equivalent to S^1 is unknotted
    Used in the proof of Theorem 1.2; cited as [Lev65].
  • ad hoc to paper For any symplectic filling W of H in V×D, there exists an L-simple compatible almost complex structure J making W holomorphic and making the moduli spaces M_J(γ_pmax, q) regular with degree-one evaluation
    Asserted in the note after Proposition 4.9, but the passage is corrupted ('?? 4.8?? 4.9') and no proof is supplied; this is the load-bearing regularity assumption.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Unknottedness of symplectic submanifold fillings." pith.science (2026). https://pith.science/paper/PIOLMX64

@misc{pith2026250606807,
  author       = {Pith},
  title        = {Pith review of: Unknottedness of symplectic submanifold fillings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PIOLMX64}},
  note         = {Machine review of arXiv:2506.06807}
}
abstract

We show that any symplectic filling of the standard contact submanifold $(\mathbb{S}^{2n-1},\xi_{\mathrm{std}})$ of $(\mathbb{S}^{2n+1},\xi_{\mathrm{std}})$ in $(\mathbb{D}^{n+1},\omega_{\mathrm{std}})$ is smoothly unknotted if $n\ge 2$. We also give a self-contained proof of the Siefring intersection formula between punctured holomorphic curves and holomorphic hypersurfaces used in the proof using the $L$-simple setup of Bao-Honda.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

49 extracted references · 47 canonical work pages

  1. [1]

    Chain level contact homology

    Russell Avdek, Erkao Bao, Georgios Dimitroglou Rizell, and Zhengyi Zhou. Chain level contact homology. in preparation

    work page

  2. [2]

    An algebraic generalization of Giroux's criterion

    Russell Avdek. An algebraic generalization of giroux's criterion. arXiv preprint arXiv:2307.09068 , 2023

    work page Pith review arXiv 2023

  3. [3]

    Stabilization of divisors in high-dimensional contact manifolds

    Russell Avdek. Stabilization of divisors in high-dimensional contact manifolds. arXiv preprint arXiv:2311.00435 , 2023

    work page Pith review arXiv 2023

  4. [4]

    Bourgeois' contact manifolds are tight

    Russell Avdek and Zhengyi Zhou. Bourgeois' contact manifolds are tight. arXiv preprint arXiv:2404.16311 , 2024

    work page Pith review arXiv 2024

  5. [5]

    Effect of L egendrian surgery

    Fr\' e d\' e ric Bourgeois, Tobias Ekholm, and Yakov Eliashberg. Effect of L egendrian surgery. Geom. Topol. , 16(1):301--389, 2012. With an appendix by Sheel Ganatra and Maksim Maydanskiy

    work page 2012

  6. [6]

    The diffeomorphism type of symplectic fillings

    Kilian Barth, Hansj\"org Geiges, and Kai Zehmisch. The diffeomorphism type of symplectic fillings. J. Symplectic Geom. , 17(4):929--971, 2019

    work page 2019

  7. [7]

    Definition of cylindrical contact homology in dimension three

    Erkao Bao and Ko Honda. Definition of cylindrical contact homology in dimension three. J. Topol. , 11(4):1002--1053, 2018

    work page 2018

  8. [8]

    Semi-global Kuranishi charts and the definition of contact homology

    Erkao Bao and Ko Honda. Semi-global Kuranishi charts and the definition of contact homology. Adv. Math. , 414:148, 2023. Id/No 108864

    work page 2023

Show all 49 references
  1. [9]

    An exact sequence for contact- and symplectic homology

    Fr \'e d \'e ric Bourgeois and Alexandru Oancea. An exact sequence for contact- and symplectic homology. Invent. Math. , 175(3):611--680, 2009

    2009

  2. [10]

    Symplectic homology, autonomous Hamiltonians , and Morse - Bott moduli spaces

    Fr \'e d \'e ric Bourgeois and Alexandru Oancea. Symplectic homology, autonomous Hamiltonians , and Morse - Bott moduli spaces. Duke Math. J. , 146(1):71--174, 2009

    2009

  3. [11]

    The Gysin exact sequence for \(S^1\) -equivariant symplectic homology

    Fr \'e d \'e ric Bourgeois and Alexandru Oancea. The Gysin exact sequence for \(S^1\) -equivariant symplectic homology. J. Topol. Anal. , 5(4):361--407, 2013

    2013

  4. [12]

    \(S^1\) -equivariant symplectic homology and linearized contact homology

    Fr \'e d \'e ric Bourgeois and Alexandru Oancea. \(S^1\) -equivariant symplectic homology and linearized contact homology. Int. Math. Res. Not. , 2017(13):3849--3937, 2017

    2017

  5. [13]

    Roger Casals and John B. Etnyre. Non-simplicity of isocontact embeddings in all higher dimensions. Geom. Funct. Anal. , 30(1):1--33, 2020

    2020

  6. [14]

    Cieliebak, Y

    K. Cieliebak, Y. Eliashberg, and N. Mishachev. Introduction to the \(h\) -principle , volume 239 of Grad. Stud. Math. Providence, RI: American Mathematical Society (AMS), 2nd edition edition, 2024

    2024

  7. [15]

    Homological invariants of codimension 2 contact submanifolds

    Laurent C \^o t \'e and Fran c ois-Simon Fauteux-Chapleau. Homological invariants of codimension 2 contact submanifolds. Geom. Topol. , 28(1):1--125, 2024

    2024

  8. [16]

    Contact homology and linearization without dga homotopies

    Julian Chaidez. Contact homology and linearization without dga homotopies. arXiv preprint arXiv:2412.20676 , 2024

    2024

  9. [17]

    Symplectic homology and the Eilenberg - Steenrod axioms

    Kai Cieliebak and Alexandru Oancea. Symplectic homology and the Eilenberg - Steenrod axioms. Algebr. Geom. Topol. , 18(4):1953--2130, 2018

    1953

  10. [18]

    Eliashberg, A

    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. , pages 560--673. Basel: Birkh \"a user, 2000

    2000

  11. [19]

    Unique holomorphically fillable contact structure on the 3-torus

    Yasha Eliashberg. Unique holomorphically fillable contact structure on the 3-torus. Int. Math. Res. Not. , 1996(2):77--82, 1996

    1996

  12. [20]

    Eliashberg and L

    Y. Eliashberg and L. Polterovich. Local Lagrangian 2-knots are trivial. Ann. Math. (2) , 144(1):61--76, 1996

    1996

  13. [21]

    M. Gromov. Pseudo holomorphic curves in symplectic manifolds. Invent. Math. , 82:307--347, 1985

    1985

  14. [22]

    Haefliger

    A. Haefliger. Differentiable embeddings of \(S^ n\) in \(S^ n+q \) for \(q > 2\) . Ann. Math. (2) , 83:402--436, 1966

    1966

  15. [23]

    Hofer, K

    H. Hofer, K. Wysocki, and E. Zehnder. Properties of pseudoholomorphic curves in symplectisation. IV . A symptotics with degeneracies. In Contact and symplectic geometry ( C ambridge, 1994) , volume 8 of Publ. Newton Inst. , pages 78--117. Cambridge Univ. Press, Cambridge, 1996

    1994

  16. [24]

    Hofer, K

    H. Hofer, K. Wysocki, and E. Zehnder. Properties of pseudoholomorphic curves in symplectisations. I . A symptotics. Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire , 13(3):337--379, 1996

    1996

  17. [25]

    Contact manifolds with flexible fillings

    Oleg Lazarev. Contact manifolds with flexible fillings. Geom. Funct. Anal. , 30(1):188--254, 2020

    2020

  18. [26]

    J. Levine. Unknotting spheres in codimension two. Topology , 4:9--16, 1965

    1965

  19. [27]

    Lagrangian tori in \( R ^ 4\)

    Karl Murad Luttinger. Lagrangian tori in \( R ^ 4\) . J. Differ. Geom. , 42(2):220--228, 1995

    1995

  20. [28]

    Symplectic manifolds with contact type boundaries

    Dusa McDuff. Symplectic manifolds with contact type boundaries. Invent. Math. , 103(3):651--671, 1991

    1991

  21. [29]

    J -holomorphic curves and symplectic topology , volume 52 of American Mathematical Society Colloquium Publications

    Dusa McDuff and Dietmar Salamon. J -holomorphic curves and symplectic topology , volume 52 of American Mathematical Society Colloquium Publications . American Mathematical Society, Providence, RI, second edition, 2012

    2012

  22. [30]

    Holomorphic curves in the presence of holomorphic hypersurface foliations

    Agustin Moreno and Richard Siefring. Holomorphic curves in the presence of holomorphic hypersurface foliations. arXiv preprint arXiv:1902.02700 , 2019

    1902 arXiv

  23. [31]

    Lagrangian Klein bottles in \( R ^ 2n \)

    Stefan Nemirovski. Lagrangian Klein bottles in \( R ^ 2n \) . Geom. Funct. Anal. , 19(3):902--909, 2009

    2009

  24. [32]

    Stipsicz

    Burak Ozbagci and Andr \'a s I. Stipsicz. Contact 3-manifolds with infinitely many Stein fillings. Proc. Am. Math. Soc. , 132(5):1549--1558, 2004

    2004

  25. [33]

    On the topology of fillings of contact manifolds and applications

    Alexandru Oancea and Claude Viterbo. On the topology of fillings of contact manifolds and applications. Comment. Math. Helv. , 87(1):41--69, 2012

    2012

  26. [34]

    Contact homology and virtual fundamental cycles

    John Pardon. Contact homology and virtual fundamental cycles. J. Am. Math. Soc. , 32(3):825--919, 2019

    2019

  27. [35]

    A biased view of symplectic cohomology

    Paul Seidel. A biased view of symplectic cohomology. In Current developments in mathematics, 2006 , pages 211--253. Somerville, MA: International Press, 2008

    2006

  28. [36]

    Symplectic field theory and stable hamiltonian submanifolds: Intersection theory

    Richard Siefring. Symplectic field theory and stable hamiltonian submanifolds: Intersection theory. in preparation

  29. [37]

    Relative asymptotic behavior of pseudoholomorphic half-cylinders

    Richard Siefring. Relative asymptotic behavior of pseudoholomorphic half-cylinders. Commun. Pure Appl. Math. , 61(12):1631--1684, 2008

    2008

  30. [38]

    Intersection theory of punctured pseudoholomorphic curves

    Richard Siefring. Intersection theory of punctured pseudoholomorphic curves. Geom. Topol. , 15(4):2351--2457, 2011

    2011

  31. [39]

    Functors and computations in Floer homology with applications

    Claude Viterbo. Functors and computations in Floer homology with applications. I . Geom. Funct. Anal. , 9(5):985--1033, 1999

    1999

  32. [40]

    Lectures on symplectic field theory

    Chris Wendl. Lectures on symplectic field theory. arXiv preprint arXiv:1612.01009 , 2016

    2016 arXiv

  33. [41]

    Lectures on contact 3-manifolds, holomorphic curves and intersection theory , volume 220 of Camb

    Chris Wendl. Lectures on contact 3-manifolds, holomorphic curves and intersection theory , volume 220 of Camb. Tracts Math. Cambridge: Cambridge University Press, 2020

    2020

  34. [42]

    Cylindrical contact homology of subcritical S tein-fillable contact manifolds

    Mei-Lin Yau. Cylindrical contact homology of subcritical S tein-fillable contact manifolds. Geom. Topol. , 8:1243--1280, 2004

    2004

  35. [43]

    Periodic symplectic cohomologies

    Jingyu Zhao. Periodic symplectic cohomologies. J. Symplectic Geom. , 17(5):1513--1578, 2019

    2019

  36. [44]

    Symplectic fillings of asymptotically dynamically convex manifolds I

    Zhengyi Zhou. Symplectic fillings of asymptotically dynamically convex manifolds I . J. Topol. , 14(1):112--182, 2021

    2021

  37. [45]

    Symplectic fillings of asymptotically dynamically convex manifolds II - \(k\) -dilations

    Zhengyi Zhou. Symplectic fillings of asymptotically dynamically convex manifolds II - \(k\) -dilations. Adv. Math. , 406:62, 2022. Id/No 108522

    2022

  38. [46]

    On fillings of (V D )

    Zhengyi Zhou. On fillings of (V D ) . Math. Ann. , 385(3-4):1493--1520, 2023

    2023

  39. [47]

    A note on contact manifolds with infinite fillings

    Zhengyi Zhou. A note on contact manifolds with infinite fillings. Arnold Math. J. , 10(3):423--430, 2024

    2024

  40. [48]

    On the intersection form of fillings

    Zhengyi Zhou. On the intersection form of fillings. J. Lond. Math. Soc., II. Ser. , 110(3):30, 2024. Id/No e12981

    2024

  41. [49]

    Pseudocycles and integral homology

    Aleksey Zinger. Pseudocycles and integral homology. Trans. Am. Math. Soc. , 360(5):2741--2765, 2008

    2008

Pith tools

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