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$C^0$-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces

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Pith's one-line read On positive symplectic rational surfaces of type D, the Hamiltonian diffeomorphism group is a closed connected component of the symplectomorphism group in the $C^0$-topology.

desk verdict First C0-closedness of Ham inside Symp with nontrivial symplectic mapping class group; the proof is mostly convincing but leans on an unpublished, overlapping preprint for its algebraic backbone. read the letter →

arxiv 2508.20285 v1 pith:BGRGJ76E submitted 2025-08-27 math.SG math.GT

Marcelo Atallah , Cheuk Yu Mak , Weiwei Wu This is my paper · ORCID
classification math.SGmath.GT MSC 53D3553D0557R1757S05
keywords C0-symplectictopologyHamiltoniandiffeomorphismgroupsymplecticrationalsurfacesmappingclasspurebraidgroupsJ-holomorphicfoliationinflationfillingdivisor
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 establishes that for positive symplectic rational surfaces of type D, the Hamiltonian diffeomorphism group is closed in the $C^0$-topology inside the full symplectomorphism group, and since the manifold is simply connected it is exactly the identity component of that group. This answers the long-standing Question 1.1 affirmatively for the first family of manifolds with nontrivial symplectic mapping class group. The strategy is to prove a quantitative statement: symplectomorphisms sufficiently $C^0$-close to the identity can be adjusted by a Hamiltonian diffeomorphism so that they fix a filling divisor pointwise, and pointwise fixing that divisor forces the map to be Hamiltonian. A non-Hamiltonian symplectomorphism therefore cannot be approximated by Hamiltonian diffeomorphisms, which is the content of the rigidity.

What carries the argument

The central object is a filling divisor $\Sigma\subset X$: a union of embedded symplectic spheres whose homology classes are configured as in Figures 2.1 and 2.2, chosen so that the complement is tractable. The proof studies the iterated fibration (2.3) linking compactly supported symplectomorphisms of $X\setminus\Sigma$, the stabilizers $\mathrm{Stab}(\Sigma)$ and $\mathrm{Stab}_0(\Sigma)$, and the symplectic Torelli group $\mathrm{Symp}_h(X,\omega)$, whose $\pi_0$ is identified with the pure braid group quotient $PB_k(S^2)/(\mathbb{Z}/2)$. Two mechanisms carry the argument: the bihopfian property of that quotient, meaning every epimorphism or monomorphism $G\to G$ is an isomorphism, which turns the diagram chase of Proposition 3.2 into an equality of image subgroups; and a family inflation procedure along $J$-holomorphic foliations in the fiber class $H-E_1$, which deforms the symplectic form until the previously known convex-complement case (2.4) applies. Theorem 4.1 uses a $J$-holomorphic ruling fibration $\pi_J:X\to D_1$ to control the motion of marked points on the divisor and prove that a $C^0$-small symplectomorphism cannot braid them.

What would settle it

Exhibit a type D positive symplectic rational surface $(X,\omega)$ and a sequence of Hamiltonian diffeomorphisms $f_n$ with $f_n\to f$ in the $C^0$-topology where $f\in\mathrm{Symp}(X,\omega)\setminus\mathrm{Ham}(X,\omega)$; Theorem 1.2 says no such sequence exists. A more local test of the quantitative engine: find $f$ with $d_{C^0}(f,\mathrm{id})<\epsilon$ such that for every Hamiltonian $\phi$, $\phi\circ f$ moves at least one point of a filling divisor $\Sigma$, contradicting Theorem 4.1.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1.2: for a positive symplectic rational surface $(X,\omega)$ of type D, $\mathrm{Ham}(X,\omega)$ is a connected component of $\mathrm{Symp}(X,\omega)$ in the $C^0$-topology, hence closed. The proof rests on Theorem 4.1, which produces an $\epsilon>0$ with the property that every symplectomorphism $f$ with $d_{C^0}(f,\mathrm{id})<\epsilon$ can be composed with a Hamiltonian diffeomorphism $\phi$ so that $\phi\circ f$ fixes the filling divisor $\Sigma$ pointwise. Theorem 3.1 then shows that any symplectomorphism fixing $\Sigma$ pointwise is Hamiltonian isotopic to the identity, so the $C^0$-ball around the identity in $\mathrm{Symp}(X,\omega)$ consists entirely of Hamiltonian diffeomorphisms. Since $X$ is simply connected, $\mathrm{Symp}_0(X,\omega)=\mathrm{Ham}(X,\omega)$, giving the affirmative answer to Question 1.1 for type D.

Load-bearing premise

The proof depends on an imported and not-yet-published classification of the symplectic mapping class group of type D rational surfaces — Theorem 2.7 from [LLW22b] — together with the associated claims that certain compactly supported symplectomorphism groups are weakly contractible and certain connecting maps are surjective; if those inputs are wrong, the chain from pointwise stabilizer to Hamiltonian breaks.

Editorial extensions

If this is right

  • On any positive symplectic rational surface of type D, a $C^0$-limit of Hamiltonian diffeomorphisms that happens to be smooth is automatically Hamiltonian, so the Hamiltonian group is closed in $\mathrm{Symp}(X,\omega)$.
  • Question 1.1 has a positive answer for type D: the identity component $\mathrm{Symp}_0(X,\omega)$ equals $\mathrm{Ham}(X,\omega)$ and is closed in the $C^0$-topology.
  • This is the first family of examples where $C^0$-closedness of the Hamiltonian group is known even though the symplectic mapping class group $\pi_0(\mathrm{Symp}_h(X,\omega))\cong PB_k(S^2)/(\mathbb{Z}/2)$ is nontrivial, so the obstruction to rigidity is genuinely nontrivial.
  • The proof gives an effective statement at the level of $C^0$-balls: there is a radius $\epsilon>0$ around the identity in $\mathrm{Symp}(X,\omega)$ inside which every element is Hamiltonian after composition with a Hamiltonian diffeomorphism.

Reading between the lines

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

  • Our inference: if the symplectic mapping class groups of type E rational surfaces can be described with the same fibration data, the inflation-plus-bihopfian strategy of this paper should carry over essentially unchanged, extending Theorem 1.2 to all positive rational surfaces.
  • Our inference: the paper's Question 5.2, if answered positively, would upgrade the rigidity result to local path-connectedness of $\mathrm{Symp}(X,\omega)$ in the $C^0$-topology, since the divisorial decomposition would then give short Hamiltonian paths moving divisors; this is the direction the authors indicate for a sequel.
  • Our inference: the role of the bihopfian property suggests the mechanism is not specific to rational surfaces; any closed symplectic 4-manifold whose Torelli mapping class group is a finite quotient of a braid group and that admits a filling divisor with a weakly contractible complement should satisfy the same $C^0$-rigidity.
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Editorial analysis

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

Referee Report

4 major / 5 minor

Summary. The paper proves that for positive symplectic rational surfaces of type D, the Hamiltonian diffeomorphism group Ham(X, ω) is a connected component of Symp(X, ω) in the C0-topology, and in particular is C0-closed. The proof has two main stages. First, an algebraic stage (Theorem 3.1) shows that the pointwise stabilizer Stab0(Σ) of a filling divisor Σ is contained in the Hamiltonian group; this is achieved via a diagram chase in symplectic mapping class groups, using the pure braid group PB_{n−1}(S2)/(Z/2) and its bihopfian property, together with an inflation/Moser transfer argument that reduces the statement to a parameter range where the relevant compactly supported symplectomorphism group is weakly contractible. Second, an analytic stage (Theorem 4.1) shows that any symplectomorphism sufficiently C0-close to the identity can be composed with a Hamiltonian diffeomorphism so that the resulting map fixes a filling divisor pointwise; this is proved using C0-controlled J-holomorphic foliation data, a two-step pseudo-holomorphic isotopy, and a projection estimate on the divisor. Theorem 1.2 follows by taking C0-limits. The paper also discusses a Floer-theoretic alternative and proposes open questions.

Significance. If the proof is completed and the imported mapping-class-group inputs are valid, the result is a significant advance: it provides the first examples with non-trivial symplectic mapping class group for which Question 1.1 has a positive answer, showing that Ham(X, ω) is C0-closed in Symp(X, ω) for positive type D rational surfaces. The strategy of combining symplectic mapping class group computations with C0-controlled foliation and inflation techniques is original and is likely to be influential. The paper is also careful in indicating where it relies on external results, and I found no circularity in the argument: the cited works do not contain the C0-conclusion. The main risk is not internal inconsistency but dependence on the unpublished preprint [LLW22b] and on a few compressed technical assertions that need to be substantiated before the paper can be accepted.

major comments (4)
  1. [§2.2, Theorem 2.7 and Proposition 2.8] The main theorem rests on an algebraic foundation that is almost entirely imported from the unpublished preprint [LLW22b], which has overlapping authorship with the present paper. Specifically, the diagram chase in Proposition 3.2 uses the isomorphism π0(Symph(X,ωλ)) ≅ PB_{n−1}(S2)/(Z/2) from Theorem 2.7 and the exactness of the rows in (3.1), while Proposition 2.8 uses the weak contractibility of Sympc(X\Σ) from [LLW22b, Lemmas 5.5 and 5.6] and the surjectivity of the connecting map from [LLW15, Lemma 2.9]. These inputs are load-bearing: if any of them fails, Theorem 3.1, Theorem 4.1, and Theorem 1.2 collapse. The present manuscript gives only a sketch of Proposition 2.8 and does not reproduce the arguments behind Theorem 2.7. The authors should either include proofs of these statements, state the main theorem as conditional on the verification of [LLW22b], or provide a detailed and self-contained account of the exact sequences and isomorphisms used in Proposition 3.2.
  2. [§4.1.2, Construction 4.2] The existence of the path (J_f^t)_{t∈[0,1]} in J^reg satisfying the three listed conditions is justified only by the sentence 'Such a path always exists since J\J^reg is a union of submanifolds of codimension 2 or higher.' This codimension statement is not proved and no reference is supplied. The path is essential to Step A, since it produces the Hamiltonian isotopy that moves the curves f(D_i) back to D_i. A proof of the codimension statement for the specific configuration classes appearing in the type D filling divisor, or a precise reference establishing it, must be provided.
  3. [§4.2.2, non-pure cases] The reduction from non-pure type D forms to pure type D forms is compressed into a single paragraph. It asserts that a C0-small symplectomorphism can first be adjusted to move the extra exceptional curves back, then descends to the blown-down pure type D surface, and that the conclusion follows from [LLW22b, Lemma 4.3]. Since Theorem 4.1 is stated for all type D forms, this step is load-bearing. The details of the isotopy of the extra exceptional curves, the C0-smallness control after blowing down, and the precise application of [LLW22b, Lemma 4.3] should be written out.
  4. [§4.1, Step A] The proof of Theorem 4.1 invokes the statement 'since f is in the Torelli part' in order to identify f(D_i) with the unique f_*J_0-holomorphic representative of [D_i]. The theorem statement, however, only assumes d_C0(f,id) < ε. This is fixable: for ε smaller than the injectivity radius of the fixed Riemannian metric, C0-closeness to the identity implies that f is homotopic to the identity and hence acts trivially on homology. The authors should state this explicitly in the proof; as written, the proof appears to use an unstated hypothesis.
minor comments (5)
  1. [§1, Abstract and Introduction] There is a typo in 'symplecitc surfaces' in the introduction; it should read 'symplectic surfaces.'
  2. [§2.2, diagram (2.3)] The arrows in the iterated fibration diagram (2.3) are not all labeled, making it hard to tell which maps are fibrations and which are inclusions. A short legend or a more explicit chain of fibrations would greatly improve readability.
  3. [§4.1.4, Lemma 4.7] Lemma 4.7 is stated as an 'elementary fact' with the proof omitted. A one-sentence proof, or a reference, would help the reader verify the criterion for triviality of the mapping class of φ_H^1 ∘ f on D_1.
  4. [§4.2.1, n=5 case] The notation J(X∗)^reg is introduced without an explicit definition; it should be defined analogously to J^reg_ω, and the regularity conditions for the classes B∗−∑ E_i∗ and E_1∗ should be stated.
  5. [§5.2, Question 5.2] The phrase 'C0-small Hamiltonian isotopy' is ambiguous: it could mean that the diffeomorphisms ϕ_t are C0-close to the identity for all t, or that the path itself is C0-close to the constant path. Please clarify.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the C0-closedness of Ham is not an input to any cited result; the paper's derivation chain is self-contained at the level of its own logic.

full rationale

The derivation chain is not circular. Theorem 1.2 is obtained by combining Theorem 3.1 (Stab0(Sigma) is contained in Ham) with Theorem 4.1 (C0-small symplectomorphisms can be Hamiltonian-corrected into Stab0(Sigma)). Theorem 4.1 is genuinely new and geometric: it uses C0-smallness to control J-holomorphic foliations, then Banyaga extensions and a Moser argument; no fitted constant or previously assumed rigidity enters. Theorem 3.1, in turn, is obtained from the exact-sequence diagram in Section 3.1; the only external inputs are the symplectic mapping class group computations pi0(Symph(X,omega)) isomorphic to PB_{n-1}(S^2)/(Z/2) and the bihopfian property quoted from [LLW22b]/[LLW22a], plus weak contractibility of Symp_c(X\Sigma) for the base case. These cited statements are parameter-free algebraic results about smooth symplectomorphism groups; none of them asserts or contains the C0-closedness of Ham in Symp, so using them is not assuming the target conclusion. The main verifiability concern, and the only sense in which self-citation enters, is that [LLW22b] is an unpublished arXiv preprint with overlapping authorship and Proposition 2.8's proof is sketched while Lemma 4.7's proof is omitted. These are gaps in independent verification, not circular reductions: no equation in the paper is equal by construction to its input, no parameter is fitted and then renamed a prediction, and the claimed rigidity is not embedded in the definitions of type D or of the filling divisor. Accordingly there is no circular step to report.

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

No free parameters: this is a pure theorem paper with no numerical fits. Seven axioms are listed; the four main ones are cited from prior work, chiefly the unpublished preprint [LLW22b], one genericity claim is asserted in the text without proof or citation (marked ad_hoc_to_paper), and two are standard theorems. No invented entities; the filling divisors and the Lagrangian RP2 are imported from the cited literature.

assumptions (7)
  • domain assumption Theorem 2.7 from [LLW22b]: for type D_k forms, pi0(Symph(X, omega)) is isomorphic to PB_k(S2)/Z2, generated by Dehn twists of Lagrangian spheres.
    Load-bearing input for the diagram chase in Proposition 3.2 and for the conclusion of Theorem 3.1; cited from arXiv preprint 2212.01873 with overlapping authorship (W.W.), not verified independently here.
  • domain assumption Fibration data (2.3) to (2.6) and Proposition 2.8: under condition (2.4) or n=5, Symp_c(X minus Sigma) is weakly contractible (or isomorphic to Z) and the relevant exact sequences hold.
    Inherited from [LLW22a], [LLW22b], and [Eva11]; the starting point of the inflation transfer in Section 3.2 is the triviality of v composed with u' under (2.4).
  • domain assumption Bihopfian property of G = PB_n(S2)/(Z/2) (Lemma 2.9, [LLW22a], [LLW22b]).
    Used in Proposition 3.2 to force im(u)=im(u') from the surjective quotient map; a cited lemma with proof in the cited works.
  • domain assumption Lemmas 2.5 and 2.6 ([LLW22b], [LZ15], [Zha17], [Pin08]): the classes H-E1 and the smallest exceptional class admit embedded J-holomorphic representatives for every J, yielding the foliation and fibration in class F = H-E1.
    The entire C0-control argument in Section 4.1 rests on these foliations and on the fibration projections chi_J.
  • ad hoc to paper The complement of the regular almost complex structures has codimension at least 2 in the space of compatible almost complex structures (stated without proof in Construction 4.2).
    Used to find the path J^f_t from f*J0 to J0 through regular almost complex structures; if false, Step A's construction collapses.
  • standard math Banyaga's extension theorem: symplectic isotopies of symplectic divisors extend to ambient Hamiltonian isotopies.
    Used in Corollaries 4.4, Lemma 4.5, and Section 3.2 to turn divisor isotopies into Hamiltonian diffeomorphisms; standard and cited through [Gro85] and [Eva11].
  • standard math Edwards-Kirby local path connectedness of Homeo(M): the C0-closure of Symp0 in Symp lies in the symplectic Torelli group Symph.
    Justifies Torelli membership of C0-limits in the proof of Theorem 1.2, which is needed for uniqueness of f*J0-holomorphic representatives in Step A.

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Pith. "Pith review of $C^0$-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces." pith.science (2026). https://pith.science/paper/BGRGJ76E

@misc{pith2026250820285,
  author       = {Pith},
  title        = {Pith review of: $C^0$-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BGRGJ76E}},
  note         = {Machine review of arXiv:2508.20285}
}
abstract

We investigate the $C^0$-topology of the group of symplectic diffeomorphisms of positive symplectic rational surfaces. For all but a few exceptions, we prove that the group of Hamiltonian diffeomorphisms forms a connected component in the $C^0$-topology. This provides the first nontrivial case in which the group of Hamiltonian diffeomorphisms is known to be $C^0$-closed inside the group of symplectic diffeomorphisms. The key to our approach is to build a bridge between techniques from symplectic mapping class groups and problems in $C^0$-symplectic topology. Via a careful adaptation of tools from $J$-holomorphic foliation and inflation, we establish the necessary $C^0$-distance estimates. We hope that this serves as an example of how these two subfields can interact fruitfully, and also propose several questions arising from this interplay.

Figures

Figures reproduced from arXiv: 2508.20285 by the authors.

Figure 2.1
Figure 2.1. A filling divisor Σ in X, n ≥ 6 2H − E1 − E2 − E3 − E4 − E5 E1 E2 E3 E4 E5 [PITH_FULL_IMAGE:figures/full_fig_p006_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. A filling divisor for n = 5 monotone case Such a symplectic configuration is called a filling divisor. Under additional assump￾tions, we will see that its complement is a Weinstein manifold, whose compactly supported symplectomorphism group is weakly contractible. The set of filling divisors is denoted by Cω. Take a subset C 0 ω ⊂ Cω, which consists of configurations whose components have pairwise symplectically ort… view at source ↗
Figure 3.1
Figure 3.1. Construction of ˆγ(t) from Moser techniques Proof. We can extend Jγ to an R-family Jγ : R → Jω (not J reg ω ) because Jω is contractible. Since H − E1 − E6 and E6 are exceptional classes of the minimal area, for every r ∈ R, there is a unique embedded Jγ(r)-holomorphic representative. By possibly perturbing Jγ(r), we can assume that the Jγ(r)-holomorphic representative of H − E1 − E6 and E6 intersect orthogonally. W… view at source ↗
Figures from the paper (2 more)
Figure 4.1
Figure 4.1. Figure 4.1: Isotopy of Di 4.1.3. Step B: moving the entire configuration back to D. Now define J A := (ϕ 1 HA )∗J f 0 . Let (J B t )t∈[0,1] be a path in J reg such that (1) when t = 0, we have J B 0 = J A, and (2) for all t ∈ [0, 1] and all i > 1, Di,JB 0 = Di,JA is J B t -holom…
Figure 4.2
Figure 4.2. Figure 4.2: Recognizing the mapping class of γ-projection Note that D′ JB 1 = DJ0 = D, so we have moved the whole configuration DJ f 0 back to D. 4.1.4. Trivial mapping class. Let ϕ t H be the concatenation of ϕ t HA and ϕ t HB (ϕ t HA goes first). The goal of this section is to…

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Works this paper leans on

47 extracted references · 45 canonical work pages

  1. [1]

    , Inventiones Mathematicae 131 (1998), 1--23

    Miguel Abreu, Topology of symplectomorphism groups of S^2 S^2 . , Inventiones Mathematicae 131 (1998), 1--23

    work page 1998

  2. [2]

    Atallah, J-P

    M.S. Atallah, J-P. Chassé, R. Leclercq, and E. Shelukhin, Weinstein exactness of nearby L agrangians: towards the L agrangian C ^ 0 flux conjecture , 2025, arXiv:2410.04158

    arXiv 2025

  3. [3]

    Miguel Abreu and Dusa McDuff, Topology of symplectomorphism groups of rational ruled surfaces, J. Amer. Math. Soc. 13 (2000), no. 4, 971--1009 (electronic)

    work page 2000

  4. [4]

    Topol (2002), 195--218

    S \' lvia Anjos, Homotopy type of symplectomorphism groups of S^2 S^2 , , Geom. Topol (2002), 195--218

    work page 2002

  5. [5]

    1-2, 245--292

    S \'e rgio Anjos and Martin Pinsonnault, The homotopy lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane, Mathematische Zeitschrift 275 (2013), no. 1-2, 245--292

    work page 2013

  6. [6]

    Atallah and E

    M.S. Atallah and E. Shelukhin, E xact L agrangian, H amiltonian fixed points, and the flux conjectures , Work in progress

    work page

  7. [7]

    Augustin Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Commentarii Mathematici Helvetici 53 (1978), 174--227

    work page 1978

  8. [8]

    2, 759--809

    Lev Buhovsky, Vincent Humilière, and Sobhan Seyfaddini, A C^0 counterexample to the A rnold conjecture , Inventiones Mathematicae 213 (2018), no. 2, 759--809

    work page 2018

Show all 47 references
  1. [9]

    1, 293--316

    , The action spectrum and C^0 symplectic topology , Mathematische Annalen 380 (2021), no. 1, 293--316

    2021

  2. [10]

    Olguta Buse and Jun Li, Symplectic isotopy on non-minimal ruled surfaces, Mathematische Zeitschrift 304 (2023), 44

    2023

  3. [11]

    1, 1--56

    Lev Buhovsky and Emmanuel Opshtein, Some quantitative results in C^0 symplectic geometry , Inventiones Mathematicae 205 (2016), no. 1, 1--56

    2016

  4. [12]

    6, 3493--3508

    Lev Buhovsky, Towards the C^0 flux conjecture , Algebraic & Geometric Topology 14 (2015), no. 6, 3493--3508

    2015

  5. [13]

    Olguta Buse, Negative inflation and stability in symplectomorphism groups of ruled surfaces, Journal of Symplectic Geometry 9 (2011)

    2011

  6. [14]

    1, 181--257

    Daniel Cristofaro-Gardiner, Vincent Humilière, and Sobhan Seyfaddini, Proof of the simplicity conjecture, Annals of Mathematics 199 (2024), no. 1, 181--257

    2024

  7. [15]

    1, 63--88

    Robert D Edwards and Robion C Kirby, Deformations of spaces of imbeddings, Annals of Mathematics 93 (1971), no. 1, 63--88

    1971

  8. [16]

    1, 45--82

    Jonathan David Evans, Symplectic mapping class groups of some S tein and rational surfaces , Journal of Symplectic Geometry 9 (2011), no. 1, 45--82

    2011

  9. [17]

    1, 45--93

    Albert Fathi, Structure of the group of homeomorphisms preserving a good measure on a compact manifold, Annales Scientifiques de l'École Normale Supérieure 13 (1980), no. 1, 45--93

    1980

  10. [18]

    2, 307--347

    Mikhail Gromov, Pseudo holomorphic curves in symplectic manifolds, Inventiones Mathematicae 82 (1985), no. 2, 307--347

    1985

  11. [19]

    1, 1--36

    Richard Hind, L agrangian unknottedness in S tein surfaces , Asian Journal of Mathematics 16 (2012), no. 1, 1--36

    2012

  12. [20]

    4, 767--799

    Vincent Humilière, Rémi Leclercq, and Sobhan Seyfaddini, Coisotropic rigidity and C^0 -symplectic geometry , Duke Mathematical Journal 164 (2015), no. 4, 767--799

    2015

  13. [21]

    , Reduction of symplectic homeomorphisms, Annales Scientifiques de l'École Normale Supérieure 49 (2016), 633--668

    2016

  14. [22]

    Alexandre Jannaud, Dehn-Seidel twist, C^0 symplectic topology and barcodes , 2021, arXiv:2101.07878

    2021 arXiv

  15. [23]

    672, viii+71

    Yael Karshon, Periodic H amiltonian flows on four-dimensional manifolds , Memoirs of the American Mathematical Society 141 (1999), no. 672, viii+71

    1999

  16. [24]

    1, 203--271

    Mikhail Khovanov and Paul Seidel, Quivers, F loer cohomology, and braid group actions , Journal of the American Mathematical Society 15 (2002), no. 1, 203--271

    2002

  17. [25]

    2, 319--333

    Jun Li, Tian-Jun Li, and Weiwei Wu, The symplectic mapping class group of CP ^2\#n CP ^2 with n 4 , Michigan Mathematical Journal 64 (2015), no. 2, 319--333

    2015

  18. [26]

    2, 1357--1410

    , Symplectic (-2) -spheres and the symplectomorphism group of small rational 4-manifolds II , Transactions of the American Mathematical Society 375 (2022), no. 2, 1357--1410

    2022

  19. [27]

    , Symplectic T orelli groups of rational surfaces , 2022, arXiv:2212.01873

    2022 arXiv

  20. [28]

    15, CRM Proceedings, 1995, pp

    François Lalonde, Dusa McDuff, and Leonid Polterovich, On the flux conjectures, Geometry, Topology, and Dynamics, vol. 15, CRM Proceedings, 1995, pp. 69--85

    1995

  21. [29]

    2, 347--397

    Francois Lalonde and Martin Pinsonnault, The topology of the space of symplectic balls in rational 4-manifolds., Duke Mathematical Journal 122 (2004), no. 2, 347--397

    2004

  22. [30]

    6, 2713--2825

    Frédéric Le Roux, Sobhan Seyfaddini, and Claude Viterbo, Barcodes and area-preserving homeomorphisms, Geometry & Topology 25 (2021), no. 6, 2713--2825

    2021

  23. [31]

    1, 71--91

    Tian-Jun Li and Michael Usher, Symplectic forms and surfaces of negative square, Journal of Symplectic Geometry 4 (2006), no. 1, 71--91

    2006

  24. [32]

    2, 1121--1169

    Tian-Jun Li and Weiwei Wu, Lagrangian spheres, symplectic surfaces and the symplectic mapping class group, Geometry and Topology 16 (2012), no. 2, 1121--1169

    2012

  25. [33]

    5, 1209--1256

    Tian-Jun Li and Weiyi Zhang, Almost K \"ahler forms on rational 4-manifolds , American Journal of Mathematics 137 (2015), no. 5, 1209--1256

    2015

  26. [34]

    4, 1119--1122

    Dusa McDuff, Symplectic embeddings of 4-dimensional ellipsoids: erratum, Journal of Topology 8 (2015), no. 4, 1119--1122

    2015

  27. [35]

    2, 167--219

    Stefan Müller and Yong-Geun Oh, The group of H amiltonian homeomorphisms and C^0 -symplectic topology , Journal of Symplectic Geometry 5 (2007), no. 2, 167--219

    2007

  28. [36]

    1, 231--286

    Dusa McDuff and Emmanuel Opshtein, Nongeneric J -holomorphic curves and singular inflation , Algebraic & Geometric Topology 15 (2015), no. 1, 231--286

    2015

  29. [37]

    Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, third ed., Oxford University Press, 2017

    2017

  30. [38]

    5, 981--1020

    Kaoru Ono, Floer- N ovikov cohomology and the flux conjecture , Geometric and Functional Analysis 16 (2006), no. 5, 981--1020

    2006

  31. [39]

    5, 857--864

    Emmanuel Opshtein, C^0 -rigidity of characteristics in symplectic geometry , Annales Scientifiques de l'École Normale Supérieure 42 (2009), no. 5, 857--864

    2009

  32. [40]

    3, 431--455

    Martin Pinsonnault, Maximal compact tori in the H amiltonian group of 4-dimensional symplectic manifolds , Journal of Modern Dynamics 2 (2008), no. 3, 431--455

    2008

  33. [41]

    in , Symplectic 4-Manifolds and Algebraic Surfaces, volume 1938 of Lecture N otes in M athematics , Springer, 2008, pp

    Paul Seidel, Lectures on four-dimensional D ehn twists. in , Symplectic 4-Manifolds and Algebraic Surfaces, volume 1938 of Lecture N otes in M athematics , Springer, 2008, pp. 231--268

    1938

  34. [42]

    21, 4920--4960

    Sobhan Seyfaddini, C^0 -limits of H amiltonian paths and the O h-- S chwarz spectral invariants , International Mathematics Research Notices 2013 (2013), no. 21, 4920--4960

    2013

  35. [43]

    , Géométrie symplectique C^0 , 2021, Diplôme d’habilitation à diriger des recherches en mathématiques de Sorbonne Université

    2021

  36. [44]

    6, 1514--1543

    Egor Shelukhin, Symplectic cohomology and a conjecture of V iterbo , Geometric and Functional Analysis 32 (2022), no. 6, 1514--1543

    2022

  37. [45]

    1, 321--373

    , V iterbo conjecture for Z oll symmetric spaces , Inventiones Mathematicae 230 (2022), no. 1, 321--373

    2022

  38. [46]

    1, 153--168

    Weiwei Wu, Exact L agrangians in A_n -surface singularities , Mathematische Annalen 359 (2014), no. 1, 153--168

    2014

  39. [47]

    6, 1227--1275

    Weiyi Zhang, The curve cone of almost complex 4-manifolds, Proceedings of the London Mathematical Society 115 (2017), no. 6, 1227--1275

    2017

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