On the equivariant cohomological rigidity of semi-free Hamiltonian circle actions

Liat Kessler; Nikolas Wardenski

REVIEW 3 major objections 4 minor 23 references

On the equivariant cohomological rigidity of semi-free Hamiltonian circle actions

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read Equivariant cohomology and fixed-point data determine the isomorphism type of semi-free Hamiltonian circle actions on six-manifolds.

desk verdict Real progress on a real problem, but the proof of the key extension theorem has a load-bearing gap at the modified-momentum-map step. read the letter →

arxiv 2509.04890 v1 pith:X6CZFCXV submitted 2025-09-05 math.SG

Liat Kessler , Nikolas Wardenski This is my paper · ORCID

classification math.SG MSC 53D3555N9153D2058D19

keywords semi-freeHamiltoniancircleactionsequivariantcohomologicalrigiditysix-dimensionalsymplecticmanifoldsfixedpointdatacanonicalclassesMorseflowreductionhomeomorphism

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 claims that equivariant cohomology, together with compatible data on the fixed point set, determines the isomorphism type of a compact, connected, semi-free Hamiltonian $S^1$-manifold of dimension six (semi-free meaning all stabilizers are connected). Two such manifolds with equivariantly isomorphic cohomology algebras, matching fixed-point components, matching equivariant normal bundles, and matching neighbourhoods of the minima must be equivariantly symplectomorphic, and in a topological version equivariantly homeomorphic. The result repairs a theorem of Gonzales that the authors showed was false: the earlier correction imposed an artificial restriction on where non-extremal fixed surfaces could appear, and this paper removes it by replacing it with a global equivariant-cohomology condition that any isomorphism would have to satisfy anyway. If the paper is right, classification of these six-manifold actions reduces to checking algebraic data rather than constructing maps by hand.

What carries the argument

The load-bearing object is the canonical class of a connected fixed-point component. For a fixed sphere or an isolated fixed point of index 2 at a critical level $\lambda$, Definition 1.9 asks for a class $c\in H^*_{S^1}(M)$ that vanishes on $M_{>\lambda}$, restricts on the component $C$ to the equivariant Euler class of its positive normal bundle, and vanishes on all other fixed components at $\lambda$; Theorem 1.10 proves such a $c$ exists and is unique, using the Tolman–Weitsman splitting of the long exact sequence in equivariant cohomology. The key property is that $c$ restricts in the reduced space $M_{\lambda-r}$ to the Poincaré dual of the Morse-flow preimage $[C']$ of $C$, so evaluating $c$ on spheres computes intersection numbers with $C'$. Under a $\mu$-isomorphism, $\eta$ must send canonical classes to canonical classes, which forces the map below $\lambda$ to preserve the subdivision of homology classes into point classes $D^{\mathrm{pt}}$ and sphere classes $D^{\mathrm{sph}}(k,l)$ and to preserve normal-bundle Euler classes. With that rigid homological control, the paper uses ambient-isotopy theorems for spheres in four-manifolds (Sunukjian's theorem in the topological setting, and the Lalonde–Pinsonnault result through the authors' earlier work in the symplectic setting) to isotope the map so that it matches the Morse-flow preimages, then pushes it across the critical level with the Morse flow. A pseudo momentum map—a perturbation of a momentum map that is constant near the fixed set—is the device that lets this argument isolate one fixed component at a time.

What would settle it

Take the pair of six-manifolds used in the paper's counterexample to Gonzales' theorem ([KW25, Example 2.1]) and compute whether their equivariant cohomology algebras admit a $\mu$-isomorphism $(\eta,\eta',(\eta_\lambda))$ that extends the given fixed-point isomorphism and the isomorphism of neighbourhoods of the minima. The paper's claim predicts that no such compatible $\eta$ exists, because the homology class of a fixed surface below the critical level obstructs the matching; if a compatible $\eta$ were found and the two manifolds were still not equivariantly symplectomorphic (or homeomorphic), Theorem 1.5 would be refuted.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.5: if $(M_1,\mu_1)$ and $(M_2,\mu_2)$ are compact, connected, semi-free Hamiltonian $S^1$-manifolds of dimension six satisfying Setting 1.1 (symplectic version) or Setting 1.3 (topological version), and if their equivariant cohomology algebras are $\mu$-isomorphic via compatible data $(\eta,\eta',(\eta_\lambda)_{\lambda\in C})$ with an isomorphism of neighbourhoods of the minima, then there is an equivariant homeomorphism (equivariant symplectomorphism) $f\colon M_1\to M_2$ with $f^*=\eta$ as an isomorphism $H^*_{S^1}(M_2)\to H^*_{S^1}(M_1)$. The workhorse is Theorem 1.6, an extension principle: an isomorphism defined on the sublevel sets right below a critical level $\lambda$ extends across $\lambda$ whenever the equivariant-cohomology diagram commutes and the fixed-point sets at $\lambda$ are matched by $\eta_\lambda$. Iterating this from the minimum to the maximum proves the main theorem, so the paper's truth rests entirely on the claim that the algebraic compatibility is strong enough to force the geometric extension at each critical level.

Load-bearing premise

The symplectic version of the theorem depends on a global rigidity condition that is hard to verify: for every interval of regular values, the reduced four-manifold must have a path-connected identity component of its symplectomorphism group, and any deformation of the symplectic form that stays in the same cohomology class must be homotopic, through such deformations, to an actual isotopy; if that condition fails anywhere, the proof's extension over a critical level no longer goes through.

Editorial extensions

If this is right

The distribution restriction from the authors' earlier correction is gone: fixed surfaces may occur at several non-extremal levels, and the rigidity conclusion still holds without assuming those levels are simple.
Cho's classification of six-dimensional monotone semi-free Hamiltonian $S^1$-manifolds can be obtained directly from the corrected statement, without the extra fixed-point-distribution assumption that was tailored to that application.
In the topological category, the criterion is checkable from standard data: compare equivariant cohomology, fixed-point components, and neighbourhoods of the minima, and verify that complements of fixed spheres have cyclic fundamental group.
Because a genuine equivariant homeomorphism inducing $\eta$ necessarily produces the $\mu$-isomorphism data, the theorem's hypotheses are not incidental: no classification of these actions can use weaker equivariant-cohomology information.

Reading between the lines

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

Inference beyond the paper: the canonical-class comparison is likely to generalize to Hamiltonian torus actions with higher-dimensional fixed components whenever the reduced spaces admit an ambient-isotopy theorem; the equivariant-cohomology mechanism does not itself use six-dimensionality except through the four-dimensional reduced spaces.
Inference beyond the paper: the topological version suggests a practical test for the manifolds in Cho's classification: compute the fundamental groups of complements of the fixed spheres in their reduced spaces; if the cyclic condition holds automatically there, the full equivariant classification of monotone semi-free actions follows from Theorem 1.5 alone.
Inference beyond the paper: the proof indicates that the rigidity assumption in the symplectic version may be stronger than needed; the places it is used are to ensure that homologous exceptional spheres are ambiently isotopic and that cohomologous deformations are homotopic to isotopies, so a version that checks these properties only at critical levels, rather than on every interval of regular va

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Let me give you the short version: this paper is a genuine advance on a question that matters in the field, and the main theorem is probably true, but the proof as written has a gap in the place where it matters most.

What's actually new: the authors replace the awkward distribution assumption from their earlier [KW25] (non-extremal fixed surfaces only at one level, etc.) with a global equivariant cohomology condition that is necessary for an isomorphism. That is the right kind of hypothesis. They also push the rigidity statement into the topological category, using a cyclic-fundamental-group condition instead of symplectic rigidity. The construction of canonical classes in Section 3 is careful and the evaluation lemmas (3.11, 3.12) look solid. The writing is clear and they are honest about Gonzales' error.

The soft spot is real, and it is exactly the one flagged in the stress test. In the non-extremal case of Theorem 1.6, the proof modifies the momentum/pseudo-momentum maps to mu'_i via Lemma 3.3, proves the extension statement for the modified maps, and then says 'rename mu'_i into mu_i.' The problem is that the extension g constructed for mu'_1, mu'_2 is an intertwining homeomorphism for the modified level sets, not for the original mu_1, mu_2. The theorem needs g to intertwine the original pseudo momentum maps near the new level lambda+delta, because Theorem 1.5 iterates Theorem 1.6 across critical levels and the next application requires the map to intertwine the original maps at the start level. The proof never returns from mu' to mu. The sentence in the theorem statement saying 'it may be assumed that g intertwines the pseudo momentum maps mu_1 and mu_2 near level lambda+delta' is exactly the missing step, and it is asserted without proof. This is not cosmetic: without it, the induction in Theorem 1.5 cannot continue.

I want to be fair: the gap looks fixable. Since mu' and mu agree outside a tubular neighborhood of the fixed set, one can probably straighten the level sets with a fiberwise reparametrization and adjust g in a collar of the new level. But it needs to be written, and it is load-bearing.

Two more concerns, in proportion. The proof relies heavily on the unpublished preprint [KW25] for several key statements (e.g., [KW25, Theorem 1.9] in the symplectic extension and Lemma 4.6 in the topological maximal case). For a definitive paper, those dependencies need to be resolved—either by a published version or by including the statements. Second, the rigidity assumption in Setting 1.1 is strong and hard to verify, so the symplectic theorem's range of applicability is narrower than it first appears; that is not a flaw, but it should be said plainly.

Who this is for: anyone working on Hamiltonian S^1-actions or equivariant cohomological rigidity. The canonical class machinery is a contribution on its own. The paper deserves a serious referee, but the referee should be asked to check the mu' to mu transition carefully and to demand a resolution of the [KW25] dependency before acceptance.

Referee Report

3 major / 4 minor

Summary. The paper investigates equivariant cohomological rigidity for semi-free Hamiltonian S^1-manifolds of dimension six. Its main result, Theorem 1.5, asserts that if two such manifolds (in a symplectic setting, Setting 1.1, or a topological setting, Setting 1.3) have μ-isomorphic equivariant cohomologies together with compatible fixed-point data and an isomorphism of neighbourhoods of the minima, then they are equivariantly homeomorphic (or equivariantly symplectomorphic) with the prescribed cohomology isomorphism. The proof is organized around an extension theorem, Theorem 1.6, which extends an isomorphism defined below a critical level across that level. The central technical tools are the construction of canonical classes associated to fixed-point components (Theorem 1.10), a modification of the momentum map to a pseudo momentum map with isolated critical levels (Lemma 3.3), and applications of isotopy theorems in four-manifolds in both the topological (Sunukjian) and symplectic (Lalonde–Pinsonnault) settings.

Significance. If the proof is completed, this would be a substantial and definitive result: it removes the restrictive distribution assumption on non-extremal fixed surfaces from the authors' previous work, extends the rigidity statement to the equivariant topological category, and provides the kind of fixed-point-data-to-isomorphism statement needed in applications such as Cho's classification of monotone semi-free Hamiltonian S^1-manifolds. The paper is also valuable for introducing canonical classes as a bridge between equivariant cohomology and geometric extension data, and for carefully formulating the rigidity hypotheses. The topological version, relying on complement-fundamental-group conditions and Sunukjian's isotopy theorem, is a meaningful new contribution. However, the proof as written has a load-bearing gap in the passage from modified pseudo momentum maps back to the original maps, and the verification of the main theorem is therefore incomplete in its current form.

major comments (3)

[§4, Case I (non-extremal λ), proof of Theorem 1.6] The proof never returns from the modified pseudo momentum maps μ'_i to the original maps μ_i. After stating 'It remains to show that the theorem holds with μ'_i replacing μ_i. To simplify notation, we rename μ'_i into μ_i,' the construction yields an extension g intertwining μ'_1 and μ'_2, but the theorem's conclusion requires g to intertwine the original μ_1 and μ_2 near level λ+δ. The maps μ_i and μ'_i differ by constants on neighbourhoods of the fixed-point set, so their level sets differ there, and the proof does not explain how to correct g so that it preserves the original level sets. The sentence in Theorem 1.6 asserting that 'it may be assumed that g intertwines the pseudo momentum maps μ_1 and μ_2 near level λ+δ' is not justified. This gap is load-bearing because Theorem 1.5 is proved by repeatedly applying Theorem 1.6 across critical levels; without the intertwinement near the new level, the induction cannot proceed past a non-extremal critical level. The issue affects both the symplectic and the topological versions, since both use the same modification argument.
[Definition 1.4 and Lemma 4.5] The isomorphism η is only assumed to be an isomorphism of algebras H^*_{S^1}(M_2)→H^*_{S^1}(M_1), but the proof of Lemma 4.5 uses that η restricts to the identity on H^*({pt.}×CP^∞) in order to conclude c_1 = ±η(c_2) with the correct sign. An arbitrary ring isomorphism of equivariant cohomology rings need not preserve the base ring H^*_{S^1}(pt); for example, the automorphism t↦-t can extend to an algebra automorphism. Since the desired conclusion f^* = η implies that η is base-preserving (f^* is induced by an equivariant map and hence is the identity on the base), Definition 1.4 should explicitly require η and η' to be isomorphisms of H^*_{S^1}(pt)-algebras, or the proof must justify that the μ-isomorphism data forces η to be base-preserving. Without this, the sign argument and the identification η(c_2)=c_1 are not fully supported.
[Throughout; dependence on [KW25]] The proofs of the main results rely at essential junctures on statements from the unpublished companion preprint [KW25]. Examples include [KW25, Theorem 1.9] and [KW25, Lemma 5.11] in the proof of Equation (1.6) for maximal λ, and [KW25, Lemma 5.20] in the symplectic non-extremal case. Because these results are load-bearing for the extension theorem, the manuscript's claim to give a definitive statement is conditional on the correctness and availability of [KW25]. The authors should state precisely which results from [KW25] are used, indicate whether they are proved there or in this paper, and ideally make the present paper self-contained for the statements on which Theorem 1.6 directly depends.

minor comments (4)

[Lemma 3.3, proof] The displayed definition of μ'_{ε,r} contains a typo: '(−|v_1|^2 + |v_1|^2)' should read '(−|v_1|^2 + |v_2|^2)'.
[Lemma 3.4] The statement begins 'Let M be a semi-free Hamiltonian S^1-manifold of dimension 6 and μ:M→S^1 a proper, pseudo momentum map'; the codomain of μ should be R, not S^1.
[Proof of Theorem 1.5] The proof jumps from the assumed isomorphism of neighbourhoods of the minima to applying Equation (1.6) on the whole sublevel set (M_1)_{≤λ_1-r}. It is not explained how the neighbourhood isomorphism is extended over the regular interval below the first critical value; in the symplectic version this appears to require a Moser-type argument using the rigidity assumption of Definition 2.4, and this extension should be stated explicitly.
[Proof of Theorem 1.10] The proof refers several times to 'fig. 1', but no figure appears in the manuscript text. Please ensure the figure is included and that the references are consistent with the displayed illustration.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the equivariant-cohomology data are genuine hypotheses, and the geometric isomorphism is constructed from them using independent prior results.

full rationale

The central derivation runs from the algebraic hypotheses in Theorem 1.5—mu-isomorphism of equivariant cohomologies with compatible fixed-point data and a neighbourhood isomorphism near the minima—to the existence of an equivariant homeomorphism or symplectomorphism f with f* = eta. No step of the proof takes the desired geometric isomorphism as an input. Theorem 1.6 is proved by constructing canonical classes using the independent Tolman–Weitsman result (1.12), identifying the Poincare-dual classes of Morse-flow preimages, and then using Sunukjian's isotopy theorem in the topological setting and the authors' prior [KW25] lemmas together with Lalonde–Pinsonnault in the symplectic setting. The closest concern is the passage in Section 4 where, after modifying momentum maps to pseudo momentum maps, the paper says 'To simplify notation, we rename mu'_i into mu_i'; if the proof never returns to the original momentum maps, that is a correctness gap, not a circular reduction. The self-citations to [KW25] are also not circular in the sense defined here: those prior statements carry their own hypotheses and do not presuppose Theorem 1.5. No fitted parameter is renamed as a prediction, and no defining equation is equivalent by construction to the conclusion. The derivation is therefore self-contained with respect to circularity.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claim rests on geometric hypotheses (Settings 1.1 and 1.3) and on the algebraic hypothesis of a mu-isomorphism of equivariant cohomologies. No numbers are fitted to data. The proof also invokes standard results in equivariant cohomology and four-manifold topology; these are unproved background, listed above. No new physical or mathematical entities are postulated; the pseudo momentum map is a definition, not an asserted entity.

assumptions (9)

domain assumption M is a compact, connected, semi-free Hamiltonian S^1-space of dimension 6 with a (pseudo) momentum map.
Setting 1.1 and 1.3; the whole paper is conditional on this class of manifolds.
domain assumption Every fixed sphere at a non-extremal critical value is exceptional, i.e., an embedded symplectic sphere of self-intersection -1.
Setting 1.1; used to ensure same homology class implies ambient symplectic isotopy, as stated in the introduction.
domain assumption Every connected component of M^{S^1} is simply connected.
Settings 1.1 and 1.3; implies fixed components are spheres and M is simply connected.
domain assumption Every reduced space of dimension 4 is a symplectic rational surface.
Setting 1.1; needed for [KW25, Lemma 5.20] in the symplectic extension argument.
domain assumption Rigidity assumption: for every interval of regular values, (M_{t0}, {omega_t}) is rigid as in Definition 2.4.
Setting 1.1, imported from Gonzales [Go11]; key to the symplectic isotopy step.
domain assumption For any non-extremal critical value lambda and any fixed sphere S, M_lambda \ S has cyclic fundamental group.
Setting 1.3; needed to apply [Su15, Theorem 6.1] in the topological version.
domain assumption H*_{S^1}(M1) and H*_{S^1}(M2) are (symplectically) mu-isomorphic via (eta, eta', (eta_lambda)), with the diagram and restriction conditions of Definition 1.4.
The algebraic input of Theorem 1.5; the conclusion is the geometric isomorphism.
standard math Tolman-Weitsman long exact sequence splitting and Kirwan injectivity over Z for semi-free actions.
Used to prove existence and uniqueness of canonical classes (Theorem 1.10) and to identify classes in Lemma 4.5.
standard math Sunukjian's theorem [Su15, Theorem 6.1]: homologous spheres in a four-manifold with same cyclic complement fundamental group are ambiently isotopic.
Used in the non-extremal topological extension proof.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On the equivariant cohomological rigidity of semi-free Hamiltonian circle actions." pith.science (2026). https://pith.science/paper/X6CZFCXV

@misc{pith2026250904890,
  author       = {Pith},
  title        = {Pith review of: On the equivariant cohomological rigidity of semi-free Hamiltonian circle actions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X6CZFCXV}},
  note         = {Machine review of arXiv:2509.04890}
}

abstract

We consider semi-free Hamiltonian $S^1$-manifolds of dimension six and establish when the equivariant cohomology and data on the fixed point set determine the isomorphism type. Gonzales listed conditions under which the isomorphism type of such spaces is determined by fixed point data. We pointed out in an earlier paper that this result as stated is erroneous, and proved a corrected version. However, that version relied on a certain distribution of fixed points that is not at all necessary. In this paper, we replace the latter assumption with a global assumption on equivariant cohomology that is necessary for an isomorphism. We also extend our result to the equivariant (non-symplectic) topological category. The variation in the earlier paper was tailored to suit the requirements of Cho's application of Gonzales' statement to classify semi-free monotone, Hamiltonian $S^1$-manifolds of dimension six. In the current paper, we aim to give the definitive statement relating fixed point data and equivariant cohomology to the isomorphism type of a semi-free Hamiltonian $S^1$-manifold.

Figures

Figures reproduced from arXiv: 2509.04890 by the authors.

**Figure 1.** A visualization of the homotopy from f to g, the image has to be understood as the momentum map of a Hamiltonian T 2 -action near C that fixes C. On the left is (a part of) the initial map f, on the right (a part of) the map g. By construction, g and gp agree on ∂D2 , and therefore piece together to a continuous map g : S 2 → (µ −1 ([λ − r, λ + r]) \ F)/S1 . This map is homotopic to f, because gp is homotopic rel ∂D… view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 9 canonical work pages

[1]

Atiyah, Convexity and commuting Hamiltonians, Bull

M. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1--15

1982
[2]

Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions I, Internat.\ J.\ Math.\ 30 (2019), no.\ 6, Paper No.\ 1950032, 71 pp

Y. Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions I, Internat.\ J.\ Math.\ 30 (2019), no.\ 6, Paper No.\ 1950032, 71 pp

2019
[3]

Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions II, Internat.\ J.\ Math.\ 32 (2021), no.\ 2, Paper No.\ 2050120, 47 pp

Y. Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions II, Internat.\ J.\ Math.\ 32 (2021), no.\ 2, Paper No.\ 2050120, 47 pp

2021
[4]

Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions III, Internat.\ J.\ Math.\ 32 (2021), no.\ 2, Paper No.\ 2050106, 58 pp

Y. Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions III, Internat.\ J.\ Math.\ 32 (2021), no.\ 2, Paper No.\ 2050106, 58 pp

2021
[5]

Delzant, Hamiltoniens p\'eriodiques et images convexes de l'application moment, Bull

T. Delzant, Hamiltoniens p\'eriodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315--339

1988
[6]

R. D. Edward and R. C. Kirby, Deformations of spaces of imbeddings. Ann. of Math. (2) 93 (1971), 63–88

work page 1971
[7]

Gonzales, Classifying semi-free Hamiltonian S^1 -manifolds, International Mathematics Research Notices 2011 (2011), no.\ 2, 387--418

E. Gonzales, Classifying semi-free Hamiltonian S^1 -manifolds, International Mathematics Research Notices 2011 (2011), no.\ 2, 387--418

2011
[8]

A.\ Higashitani, K.\ Kurimoto, and M.\ Masuda, Cohomological rigidity for toric Fano manifolds of small dimensions or large Picard numbers , Osaka J.\ Math.\ 59 (2022) no.\ 1, 177--215

work page 2022

Show all 23 references

[9]

M. W. Hirsch, Differential Topology, Grad. Texts in Math., No. 33 , Springer-Verlag, New York-Heidelberg, 1976, x+221 pp

1976
[10]

T. S. Holm , L. Kessler, and S. Tolman, Equivariant cohomological rigidity for four-dimensional Hamiltonian S^1 -manifolds, arXiv:2412.14310

arXiv
[11]

Karshon, Periodic Hamiltonian flows on four dimensional manifolds, Memoirs of the Amer.\ Math.\ Soc.\ 672 (1999)

Y. Karshon, Periodic Hamiltonian flows on four dimensional manifolds, Memoirs of the Amer.\ Math.\ Soc.\ 672 (1999)

1999
[12]

F. C. Kirwan, The cohomology of quotients in symplectic and algebraic geometry, Princeton University Press, 1984

1984
[13]

Kessler and N

L. Kessler and N. Wardenski, On isomorphisms of semi-free Hamiltonian S^1 -manifolds and fixed point data, arXiv:2505.14000v1

arXiv
[14]

Lalonde and M.Pinsonnault, The topology of the space of symplectic balls in rational 4-manifolds, Duke Math

F. Lalonde and M.Pinsonnault, The topology of the space of symplectic balls in rational 4-manifolds, Duke Math. J. 122 (2) (2004), 347--397

2004
[15]

J.\ Marsden and A.\ Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121--130

1974
[16]

Masuda, Equivariant cohomology distinguishes toric manifolds

M. Masuda, Equivariant cohomology distinguishes toric manifolds. Adv.\ Math.\ 218 (2008) no.\ 6, 2005--2012

2008
[17]

Topology 2 (3) (2009), 589--623

D.\ McDuff, Some 6 -dimensional Hamiltonian S^1 -manifolds, J. Topology 2 (3) (2009), 589--623

2009
[18]

American Mathematical Society, Providence, RI, 2005

A.\ Scorpan, The wild world of 4-manifolds. American Mathematical Society, Providence, RI, 2005. xx+609 pp. ISBN:0-8218-3749-4

2005
[19]

N.\ Sunukjian, Surfaces in 4-manifolds: concordance, isotopy, and surgery. Int. Math. Res. Not. IMRN 2015, no. 17, 7950–7978

2015
[20]

W. P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), no. 2, 467--468

1976
[21]

Tolman and J

S. Tolman and J. Weitsman, On the cohomology rings of Hamiltonian T-spaces. Northern California Symplectic Geometry Seminar, 251–258. Amer. Math. Soc. Transl. Ser. 2, 196 , American Mathematical Society, Providence, RI, 1999

1999
[22]

Tolman and J

S. Tolman and J. Weitsman, The cohomology rings of symplectic quotients. Comm. Anal. Geom. 11 (2003), no. 4, 751–773

2003
[23]

Weinstein, Symplectic manifolds and their lagrangian submanifolds, Advances in Mathematics

A. Weinstein, Symplectic manifolds and their lagrangian submanifolds, Advances in Mathematics. 6 (3): 329–346

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Atiyah, Convexity and commuting Hamiltonians, Bull

M. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1--15

1982

[2] [2]

Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions I, Internat.\ J.\ Math.\ 30 (2019), no.\ 6, Paper No.\ 1950032, 71 pp

Y. Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions I, Internat.\ J.\ Math.\ 30 (2019), no.\ 6, Paper No.\ 1950032, 71 pp

2019

[3] [3]

Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions II, Internat.\ J.\ Math.\ 32 (2021), no.\ 2, Paper No.\ 2050120, 47 pp

Y. Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions II, Internat.\ J.\ Math.\ 32 (2021), no.\ 2, Paper No.\ 2050120, 47 pp

2021

[4] [4]

Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions III, Internat.\ J.\ Math.\ 32 (2021), no.\ 2, Paper No.\ 2050106, 58 pp

Y. Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions III, Internat.\ J.\ Math.\ 32 (2021), no.\ 2, Paper No.\ 2050106, 58 pp

2021

[5] [5]

Delzant, Hamiltoniens p\'eriodiques et images convexes de l'application moment, Bull

T. Delzant, Hamiltoniens p\'eriodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315--339

1988

[6] [6]

R. D. Edward and R. C. Kirby, Deformations of spaces of imbeddings. Ann. of Math. (2) 93 (1971), 63–88

work page 1971

[7] [7]

Gonzales, Classifying semi-free Hamiltonian S^1 -manifolds, International Mathematics Research Notices 2011 (2011), no.\ 2, 387--418

E. Gonzales, Classifying semi-free Hamiltonian S^1 -manifolds, International Mathematics Research Notices 2011 (2011), no.\ 2, 387--418

2011

[8] [8]

A.\ Higashitani, K.\ Kurimoto, and M.\ Masuda, Cohomological rigidity for toric Fano manifolds of small dimensions or large Picard numbers , Osaka J.\ Math.\ 59 (2022) no.\ 1, 177--215

work page 2022