Classification of symplectic non-Hamiltonian circle actions on 4-manifolds

Rei Henigman

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Classification of symplectic non-Hamiltonian circle actions on 4-manifolds

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Pith's one-line read Seven invariants completely classify symplectic non-Hamiltonian circle actions on 4-manifolds under a discreteness assumption.

desk verdict A serious classification theorem for non-Hamiltonian circle actions in dimension four, with genuinely new invariants; the main soft spot is a terse transfer from Karshon–Tolman that a referee should ask to be expanded. read the letter →

arxiv 2411.10157 v2 pith:R5L647L5 submitted 2024-11-15 math.SG

Rei Henigman This is my paper · ORCID

classification math.SG MSC 53D2057R17

keywords symplecticcircleactionsnon-Hamiltonianequivariantclassificationcircle-valuedHamiltoniansurfacebundlesoveramappingclassgroupDuistermaat–HeckmantheorySeifertfibrations

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 solves a classification problem for symplectic actions that are not Hamiltonian: it provides a complete list of invariants that decide when two symplectic non-Hamiltonian circle actions on compact connected symplectic 4-manifolds are equivariantly symplectomorphic. The list comprises the group of periods of $\iota_X\omega$, the constant Duistermaat–Heckman density, the genus of the reduced surfaces, the isotropy data of a level set, a monodromy class recording how non-free orbits braid around the circle of momentum values, and two cohomological invariants that control the $S^1$-bundle and the symplectic form. The classification requires the group of periods to be discrete, which the paper shows holds automatically for rational symplectic forms and for quotients with first Betti number one. If correct, it turns the equivalence problem into seven checkable data and exposes new topology, the braiding of non-free orbits, that cannot appear in the Hamiltonian case.

What carries the argument

The load-bearing object is the tight circle-valued Hamiltonian $\Phi: M \to \mathbb{R}/P$, whose level sets are connected precisely because the group of periods $P(\iota_X\omega)$ is discrete. Associated to it is a painted surface bundle over the circle $\mathbb{R}/P$: a surface bundle in which the non-free orbits are marked and labeled by their isotropy data. A sheaf-of-groupoids comparison shows that two spaces have $\Phi$-diffeomorphic quotients exactly when their associated painted surface bundles are isomorphic, and those bundles are then classified by the genus, the labeling, and the conjugacy class of the gluing map in the mapping class group $\mathrm{MCG}(\Sigma_{g,C_1,\ldots,C_k})$ of the labeled surface. Two further cohomological devices, the fibration invariant (an orbit of $H^2(M/S^1,\mathbb{Z})$ under $\Phi$-diffeomorphisms) and the de Rham invariant (an orbit in basic cohomology controlling the symplectic class), upgrade $\Phi$-diffeomorphisms first to equivariant diffeomorphisms and then to symplectomorphisms via a Moser-type argument.

What would settle it

A concrete counterexample would settle the completeness claim: find two compact connected symplectic 4-manifolds with non-Hamiltonian circle actions and discrete period groups that agree on all seven invariants yet are not equivariantly symplectomorphic. Alternatively, take the torus example with period group $\mathbb{Z}+\sqrt{2}\mathbb{Z}$ and perturb the symplectic form in two different ways to rational classes; if the five topologically defined invariants of the perturbed spaces differ, then the classification's conclusions for non-discrete periods are perturbation-dependent, which the paper itself flags as a possibility.

Watch

Extended reading notes

Core claim

The paper's central claim is that the classification of symplectic non-Hamiltonian circle actions on compact connected symplectic 4-manifolds reduces to seven computable invariants. Theorem 2.9 asserts that, when the group of periods of $\iota_X\omega$ is discrete, two such actions are equivariantly symplectomorphic if and only if the two spaces have the same group of periods, Duistermaat–Heckman constant, genus invariant, isotropy data invariant, monodromy invariant, fibration invariant, and de Rham invariant. Theorem 2.8 asserts that every combination of values satisfying the evident constraints is realized by some space, so the list is exhaustive. The geometric picture that makes the completeness proof work is that the quotient $M/S^1$ is a surface bundle over the circle $\mathbb{R}/P$ whose fibers are the reduced surfaces and whose non-free orbits appear as finitely many labeled points that can braid around the base direction; this braiding is captured by the monodromy invariant, a conjugacy class in the mapping class group of the labeled surface. This topological information is new in dimension four, because the analogous painting invariant for Hamiltonian complexity-one spaces is known to be trivial there.

Load-bearing premise

The classification assumes throughout that the group of periods of $\iota_X\omega$ is a discrete subgroup of $\mathbb{R}$; when the periods are dense, the space $\mathbb{R}/P$ is not a circle and the machinery collapses.

Editorial extensions

If this is right

Under the discreteness assumption, two spaces are equivariantly symplectomorphic if and only if all seven invariants agree, so the classification problem is solved and the data are checkable in practice (Theorem 2.9).
Every combination of values satisfying the relevant integrality and orbit constraints is realized by some constructed space, so the list is exhaustive (Theorem 2.8).
When the quotient space has first Betti number one, the fibration and de Rham invariants are trivial, so the first five invariants already determine the space (Remarks 10.5 and 11.6).
Every space with non-discrete periods is still equivariantly diffeomorphic to one of the constructed spaces after perturbing the symplectic form, although the invariants of the perturbed space are not shown to be independent of the perturbation (Corollary 2.10 and Remark 2.11).
The quotient of any such space is a surface bundle over a circle with finitely many labeled braids of non-free orbits, providing a concrete topological model for the action (Proposition 2.4).

Reading between the lines

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

The discreteness hypothesis may be genuinely necessary: the paper's Example 4.5 shows that when the periods are irrational the target $\mathbb{R}/P$ is not a manifold, and Remark 2.11 leaves open whether the invariants obtained from different perturbations agree; a concrete failure there would require new invariants for the non-discrete class.
The new monodromy invariant is a natural candidate for an obstruction: the paper raises, without resolving, whether it controls the existence of equivariant Kähler or complex structures and whether the circle action extends to a symplectic $T^2$ action.
The same strategy plausibly extends to higher-dimensional complexity-one actions with isotropic orbits, as the paper itself notes; the connectivity lemma (Lemma 2.2) is dimension-free, so the main obstacles are technical rather than conceptual.
Because the fibration and de Rham invariants are orbits of finite-dimensional cohomology actions, computing them for explicit examples (products of surfaces, mapping tori of the torus) is a finite linear algebra problem, even though describing the orbits in general is complicated.

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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

This is a substantial paper and probably the right answer to Problem 1.3 under the discreteness assumption. The seven invariants are genuinely new, and the monodromy invariant capturing braided non-free orbits is a real addition, not a repackaging of Karshon–Tolman's painting invariant. The paper is also honest: the discreteness assumption is stated up front, the non-discrete case is explicitly left open (Remark 2.11), and the appendix computes actual invariants for concrete examples rather than just asserting they exist.

The main theorems are independent of the prior classifications; they build on McDuff and Karshon–Tolman but do not reduce to them. The analytic lemmas that carry the paper — fiber connectivity and constant Duistermaat–Heckman measure — are proved in detail. The construction of spaces from invariants is concrete and uses a strengthened local uniqueness statement that is at least plausible from the KT01 proof.

The soft spot is Section 7. The isomorphism between the sheaf of circle-valued Hamiltonian pieces and the sheaf of painted surface bundles (Corollary 7.18) is asserted by transfer from [KT03], and the transfer is not automatic because the base is a circle rather than an interval. The local isotopy lemmas in [KT03] are local, so I suspect the adaptation goes through, but Lemma 7.11 needs a real check, especially for covers of the circle whose intersections have two components. This is load-bearing for the uniqueness theorem. It is not obviously fatal, but a referee should ask for the transfer to be proved or for precise references to the exact propositions with an explanation of why the circle base does not matter.

There are smaller terseness issues: several "proof is identical to [KT03]" claims are fine, but Proposition 7.20 is important enough to deserve a sketch. None of this looks like circularity or fitted parameters; the invariants are intrinsic, and the proof structure is coherent.

I agree with the conditional verdict. The paper deserves a serious referee. If the Section 7 gap is fixed — and I expect it can be — this should go in a strong journal. I would send it to review and ask for the transfer argument to be written out.

Referee Report

3 major / 5 minor

Summary. The paper defines seven invariants for compact connected symplectic 4-manifolds equipped with symplectic non-Hamiltonian circle actions whose group of periods is discrete: the group of periods, the Duistermaat-Heckman constant, the genus invariant, the isotropy data invariant, the monodromy invariant, the fibration invariant, and the de Rham invariant. It then proves an existence theorem (Theorem 2.8) realizing every valid combination of these invariants and a uniqueness theorem (Theorem 2.9) stating that two such spaces are equivariantly symplectomorphic if and only if the seven invariants agree. The main strategy is to choose a tight circle-valued Hamiltonian, reduce the classification to painted surface bundles over R/P via the sheaf-theoretic machinery of Karshon-Tolman, and then classify those bundles by the monodromy, fibration, and de Rham data.

Significance. If the main theorems are correct, the paper gives a complete solution to a natural extension of Karshon's classification of Hamiltonian circle actions and Pelayo's classification of symplectic T^2 actions, under a discreteness assumption that covers rational symplectic forms and quotients with first Betti number one. The paper is original and technically ambitious: it proves a fiber connectivity lemma for circle-valued Hamiltonians, establishes constancy of the Duistermaat-Heckman measure, develops a monodromy invariant capturing braiding of non-free orbits, and gives explicit computations in Appendix A. The self-contained proof in Appendix B that smooth and topological mapping class groups agree for surfaces with marked points is also a useful contribution. The main unresolved issue is whether the transfer of the Karshon-Tolman sheaf-theoretic classification from an interval base to the circle base R/P is justified; this is load-bearing for the uniqueness theorem.

major comments (3)

[§7, Lemma 7.11] The proof of Lemma 7.11 asserts a chain of isomorphisms ˇH^1(R/P, Q_{R/P}) ≅ ˇH^1(R/P, P_{R/P}) by saying that the objects can be regarded as proper tall grommeted complexity one spaces and therefore the propositions of [KT03] apply. This transfer is not automatic. In [KT03] the base of the sheaves is the momentum image, which is an interval, whereas here the base is the circle R/P. The local isotopy results [KT03, Props 13.19, 14.9, 14.12, 15.10] are used to verify the hypotheses of Lemma 7.12, and the verification of condition (3) in Lemma 7.12 involves finite unions W of open sets and may be more delicate when intersections of cover elements have two connected components. The paper gives no argument addressing this difference. Since Corollary 7.18, Proposition 7.20, Proposition 8.9, and ultimately Theorem 2.9 all depend on this isomorphism, this is a load-bearing gap that must be repaired.
[§7, Proposition 7.20] The proof of Proposition 7.20 is delegated to the statement that it is identical to the proof of Proposition 17.5 in [KT03]. That proposition is proved for proper tall complexity one spaces over an interval, but here the associated painted surface bundle lives over the circle R/P. The existence of a global object and the homeomorphism Ψ : M/S^1 → N require gluing over the two ends of the interval, and this is precisely where the circle-valued setting differs from the Hamiltonian setting. The author should either supply the full argument or state a precise reduction showing that the circle base causes no change to the proof.
[§9, Lemma 9.6] Lemma 9.6 is a strengthened version of the local uniqueness theorem [KT01, Theorem 1] and is used in Theorem 9.1 to realize an arbitrary prescribed monodromy class α. The proof is only a sketch. In particular, the sentence 'By applying Lemma B.11 to each labeled point, we get a rigid map f' : Σ → Σ′' is not immediate: Lemma B.11 produces an isotopy of a diffeomorphism to the identity near a fixed point, whereas a rigid map must rotate the slice by the prescribed rotation. This step needs a detailed justification, because the construction of the final gluing map Ψ and hence the claimed realization of the monodromy invariant depends on it.

minor comments (5)

[§9, first paragraph] The phrase 'Dusitermaat-Heckman invariant' should be 'Duistermaat-Heckman invariant'.
[§2, Remark 2.14] The phrase 'non-trivial principle S1 bundle' should be 'non-trivial principal S1 bundle'.
[§7, Lemma 7.12] The statement contains the condition 'X ∩ Y =,' which appears to have a missing symbol, likely ∅. As written it is difficult to parse.
[Appendix A.3] The computation of the de Rham invariant action would benefit from a displayed formula showing how the SL2(Z) matrix acts on (m,n), rather than only describing it in prose.
[§5, Example 5.11] The phrase 'there are inﬁnitely many pairs of non symplectomorphic spaces in this family' should be 'infinitely many isomorphism classes' or similar, since 'pairs' is not the intended meaning.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the seven invariants are intrinsic, and the classification rests on external published theorems (Karshon–Tolman, McDuff, Haefliger–Salem) rather than on the paper's own conclusions.

full rationale

The derivation chain is self-contained in the relevant sense. The invariants (group of periods, Duistermaat–Heckman constant, genus, isotropy data, monodromy, fibration, de Rham) are defined directly from a given space and are shown to be preserved by equivariant symplectomorphisms; no invariant is defined in terms of the classification it is later used to prove. The classification up to Φ-diffeomorphism is grounded in the standard mapping-torus classification of surface bundles over a circle (Proposition 8.7), and the fibration and de Rham invariants are genuine orbit-space obstructions defined relative to fixed model spaces (MJ and MK), not fitted parameters or renamed conclusions. The paper's heavy reliance on [KT01], [KT03], [KT14], [McD88], and [HS91] is citation of external, published work; the author is not an author of those papers, and the acknowledgements do not make the advisor's prior results a self-citation in the sense relevant to circularity. The only potentially delicate step is Lemma 7.11, where a chain of sheaf-cohomology isomorphisms from Karshon–Tolman's interval-based complexity-one theory is transferred to the circle-valued setting by saying the objects 'can be regarded as proper tall grommeted complexity one spaces.' That is an assertion about the applicability of an external theorem to a new base topology, and a reader may reasonably ask for a more detailed verification of the local-isotopy lemmas over R/P. But this is a rigor or completeness concern about a cited transfer, not a case in which the paper's classification reduces by construction to its own inputs, nor a fitted parameter being relabelled as a prediction. The paper also honestly flags the non-discrete-periods limitation in Remark 2.11. Overall, no circular step is exhibited, so the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central proof rests on published theorems in symplectic topology and low-dimensional topology, listed above. The only new hypothesis is the discreteness of the period group, which is a stated assumption of the main theorems rather than an ad hoc patch. No free parameters are fitted and no new entities are invented.

assumptions (5)

domain assumption McDuff's no-fixed-point theorem for symplectic non-Hamiltonian circle actions on compact 4-manifolds (Lemma 3.2).
Ensures the circle action has no fixed points, so the circle-valued Hamiltonian is a proper submersion and reduced spaces are surfaces. Invoked throughout, especially in Lemma 2.2 and Lemma 4.14.
domain assumption Karshon-Tolman classification and local uniqueness for tall complexity one spaces (KT01, KT03, KT14).
Provides the grommet, painted surface bundle, and local normal form tools that the paper adapts to the circle-valued Hamiltonian setting in Sections 7-9.
domain assumption Hae fliger-Salem correspondence between isomorphism classes of S1-orbibundles over the quotient and H^2(M/S^1, Z).
Used in Lemma 6.2 and Section 10 to detect when a quotient diffeomorphism lifts to a bundle diffeomorphism, defining the fibration invariant.
standard math Duistermaat-Heckman theorem for proper Hamiltonian S1-manifolds.
Gives the piecewise-linear Duistermaat-Heckman function and its derivative as the Seifert Euler number; used in Lemma 2.3 to prove constancy in the non-Hamiltonian case.
standard math Smooth and topological mapping class groups of surfaces with marked points are isomorphic (proved in Appendix B).
Needed to define the monodromy invariant using homeomorphisms; the author supplies a full proof because no reference was found.

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Pith. "Pith review of Classification of symplectic non-Hamiltonian circle actions on 4-manifolds." pith.science (2026). https://pith.science/paper/R5L647L5

@misc{pith2026241110157,
  author       = {Pith},
  title        = {Pith review of: Classification of symplectic non-Hamiltonian circle actions on 4-manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R5L647L5}},
  note         = {Machine review of arXiv:2411.10157}
}

abstract

We classify symplectic non-Hamiltonian circle actions on compact connected symplectic 4-manifolds, up to equivariant symplectomorphisms. Namely, we define a set of invariants, show that the set is complete, and determine which values are attainable by constructing a space for each valid choice. We work under the assumption that the group of periods of the one-form $\iota_X \omega$ is discrete, which allows us to define a circle-valued Hamiltonian for the action, and apply tools from Karshon-Tolman's work on the classification of complexity one spaces. This assumption is always satisfied if the symplectic form is rational, or if the quotient space has first Betti number one.

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Symplectic torus actions with non-contractible orbits
math.SG 2026-07 accept novelty 8.0 of 10

A symplectic T^{n−1} action on a closed 2n-manifold is Hamiltonian precisely when its orbits are contractible.

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Works this paper leans on

18 extracted references · 18 canonical work pages · cited by 1 Pith paper

[1]

4-dimensional symplectic S1-manifolds admitting mo- ment map

[AH91] K. Ahara and A. Hattori. “ 4-dimensional symplectic S1-manifolds admitting mo- ment map”. In: J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38.2 (1991), pp. 251–

work page 1991
[2]

Periodic Hamiltonian ﬂows on four-d imensional manifolds

Brandeis Lecture Notes. Brandeis University, Waltham, MA, 1983, pp. i+84+27. [Kar99] Yael Karshon. “Periodic Hamiltonian ﬂows on four-d imensional manifolds”. In: Mem. Amer. Math. Soc. 141.672 (1999), pp. viii+71. [KL15] Yael Karshon and Eugene Lerman. “Non-compact symple ctic toric manifolds”. In: SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015...

work page 1999
[6]

Math. Phys. Stud. Reidel, Dordrecht, 1984, pp. 161–1

work page 1984
[7]

Teichmüller theo ry for surfaces with bound- ary

Festival Yve s Colin de Verdière. 2007, pp. 2239–2327. [ES70] Cliﬀord J Earle and Alfred Schatz. “Teichmüller theo ry for surfaces with bound- ary”. In: Journal of Diﬀerential Geometry 4.2 (1970), pp. 169–185. [EE69] Cliﬀord J. Earle and James Eells. “A ﬁbre bundle descr iption of Teichmüller theory”. In: J. Diﬀerential Geometry 3 (1969), pp. 19–43. [FM12...

work page 1970
[25]

Actions symplectiques de groupes co mpacts

[Ben02] Yves Benoist. “Actions symplectiques de groupes co mpacts”. In: Geom. Dedicata 89 (2002), pp. 181–245. [BS85] F. Bonahon and L. Siebenmann. “The classiﬁcation of S eifert ﬁbred 3-orbifolds”. In: Low-dimensional topology (Chelwood Gate,

work page 2002
[37]

Centered complexity one Hamiltonian torus actions

[KT01] Yael Karshon and Susan Tolman. “Centered complexity one Hamiltonian torus actions”. In: Trans. Amer. Math. Soc. 353.12 (2001), pp. 4831–4861. [KT03] Yael Karshon and Susan Tolman. “Complete invariants for Hamiltonian torus ac- tions with two dimensional quotients”. In: J. Symplectic Geom. 2.1 (2003), pp. 25–

work page 2001
[49]

Classification of Hamiltonian $S^1$-actions on compact symplectic orbifolds with isolated cyclic singular points in dimension four

Princeton Mathematical Series. Princeton University Pres s, Princeton, NJ, 2012, pp. xiv+472. [FGG88] Marisa Fernández, Mark J. Gotay, and Alfred Gray. “C ompact parallelizable four- dimensional symplectic and complex manifolds”. In: Proc. Amer. Math. Soc. 103.4 (1988), pp. 1209–1212. [FGM91] Marisa Fernández, Alfred Gray, and John W. Morgan. “ Compact sy...

work page Pith review arXiv 1988
[59]

Hamiltonian Torus A ctions on Symplectic Orbifold and Toric Varieties

[LT97] Eugene Lerman and Susan Tolman. “Hamiltonian Torus A ctions on Symplectic Orbifold and Toric Varieties”. In: Transactions of the American Mathematical Society 349.10 (1997), pp. 4201–4230. [Liu] Yichen Liu. “Toric extension of complexity one spaces ”. in preparation. [LPT] Yichen Liu, Joseph Palmer, and Susan Tolman. “Extendi ng tall complexity one...

work page 1997

Show all 18 references

[75]

Actions of tori on o rbifolds

[HS91] André Haeﬂiger and Éliane Salem. “Actions of tori on o rbifolds”. In: Ann. Global Anal. Geom. 9.1 (1991), pp. 37–59. [Hat13] Allen Hatcher. “The Kirby torus trick for surfaces” . In: arXiv preprint arXiv:1312.3518 (2013). [Igl91] Patrick Iglésias. “Les SO(3)-variétés sy...

1991 arXiv
[82]

Classiﬁcation of Ham iltonian torus actions with two-dimensional quotients

[KT14] Yael Karshon and Susan Tolman. “Classiﬁcation of Ham iltonian torus actions with two-dimensional quotients”. In: Geom. Topol. 18.2 (2014), pp. 669–716. [Kim06] Min Kyu Kim. “Frankel’s theorem in the symplectic ca tegory”. In: Trans. Amer. Math. Soc. 358.10 (2006), pp. 4...

2014
[88]

Automorphisms of multiplicity fr ee Hamiltonian manifolds

Annals of Mathematics Studies. With notes by John Milnor and Michael Atiyah. Princ eton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 197 7, pp. vii+355. [Kno11] Friedrich Knop. “Automorphisms of multiplicity fr ee Hamiltonian manifolds”. In: J. Amer. Math...

2011
[93]

Birkhäuser Verlag, Basel, 2004, pp

Progress in Mathematics. Birkhäuser Verlag, Basel, 2004, pp. viii+3

2004
[95]

Topologie de certains espaces de plonge ments

London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, 1985, p p. 19–85. [Cer61] Jean Cerf. “Topologie de certains espaces de plonge ments”. In: Bull. Soc. Math. France 89 (1961), pp. 227–380. 66 REFERENCES [CKT] Isabelle Charton, Liat Kessler, and Susan Tolman....

1961
[98]

A normal for m for the moment map

Mathematical Surveys and Monographs. Appendix J by Maxim Braverman. American Mathematical Socie ty, Providence, RI, 2002, pp. viii+350. [GS84] Victor Guillemin and Shlomo Sternberg. “A normal for m for the moment map”. In: Diﬀerential geometric methods in mathematical physics ...

2002
[222]

Symplectic actions of 2-tori on 4-m anifolds

Progress in Mathematics. Birkhäuser Boston, Inc ., Boston, MA, 2004, pp. xxxiv+497. [Pel10] Álvaro Pelayo. “Symplectic actions of 2-tori on 4-m anifolds”. In: Mem. Amer. Math. Soc. 204.959 (2010), pp. viii+81. [Pel17] Álvaro Pelayo. “Hamiltonian and symplectic symmet ries: an ...

2010
[298]

Convexity and commuting Hamiltonian s

[Ati82] M. F. Atiyah. “Convexity and commuting Hamiltonian s”. In: Bull. London Math. Soc. 14.1 (1982), pp. 1–15. [Aud90] Michèle Audin. “Hamiltoniens périodiques sur les v ariétés symplectiques com- pactes de dimension 4”. In: Géométrie symplectique et mécanique (La Grande Motte,

1982
[312]

The classiﬁcation of transversal multiplicity-free group ac- tions

[Woo96] Chris Woodward. “The classiﬁcation of transversal multiplicity-free group ac- tions”. In: Ann. Global Anal. Geom. 14.1 (1996), pp. 3–42. School of Mathematical Sciences, Tel A viv University, Tel A vi v 69978, Israel Email address : rei.henigman@gmail.com

1996
[1416]

Springer, Berlin, 1990, p p

Lecture Notes in Math. Springer, Berlin, 1990, p p. 1–25. [Aud04] Michèle Audin. Torus actions on symplectic manifolds . revised. Vol

1990

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

4-dimensional symplectic S1-manifolds admitting mo- ment map

[AH91] K. Ahara and A. Hattori. “ 4-dimensional symplectic S1-manifolds admitting mo- ment map”. In: J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38.2 (1991), pp. 251–

work page 1991

[2] [2]

Periodic Hamiltonian ﬂows on four-d imensional manifolds

Brandeis Lecture Notes. Brandeis University, Waltham, MA, 1983, pp. i+84+27. [Kar99] Yael Karshon. “Periodic Hamiltonian ﬂows on four-d imensional manifolds”. In: Mem. Amer. Math. Soc. 141.672 (1999), pp. viii+71. [KL15] Yael Karshon and Eugene Lerman. “Non-compact symple ctic toric manifolds”. In: SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015...

work page 1999

[3] [6]

Math. Phys. Stud. Reidel, Dordrecht, 1984, pp. 161–1

work page 1984

[4] [7]

Teichmüller theo ry for surfaces with bound- ary

Festival Yve s Colin de Verdière. 2007, pp. 2239–2327. [ES70] Cliﬀord J Earle and Alfred Schatz. “Teichmüller theo ry for surfaces with bound- ary”. In: Journal of Diﬀerential Geometry 4.2 (1970), pp. 169–185. [EE69] Cliﬀord J. Earle and James Eells. “A ﬁbre bundle descr iption of Teichmüller theory”. In: J. Diﬀerential Geometry 3 (1969), pp. 19–43. [FM12...

work page 1970

[5] [25]

Actions symplectiques de groupes co mpacts

[Ben02] Yves Benoist. “Actions symplectiques de groupes co mpacts”. In: Geom. Dedicata 89 (2002), pp. 181–245. [BS85] F. Bonahon and L. Siebenmann. “The classiﬁcation of S eifert ﬁbred 3-orbifolds”. In: Low-dimensional topology (Chelwood Gate,

work page 2002

[6] [37]

Centered complexity one Hamiltonian torus actions

[KT01] Yael Karshon and Susan Tolman. “Centered complexity one Hamiltonian torus actions”. In: Trans. Amer. Math. Soc. 353.12 (2001), pp. 4831–4861. [KT03] Yael Karshon and Susan Tolman. “Complete invariants for Hamiltonian torus ac- tions with two dimensional quotients”. In: J. Symplectic Geom. 2.1 (2003), pp. 25–

work page 2001

[7] [49]

Classification of Hamiltonian $S^1$-actions on compact symplectic orbifolds with isolated cyclic singular points in dimension four

Princeton Mathematical Series. Princeton University Pres s, Princeton, NJ, 2012, pp. xiv+472. [FGG88] Marisa Fernández, Mark J. Gotay, and Alfred Gray. “C ompact parallelizable four- dimensional symplectic and complex manifolds”. In: Proc. Amer. Math. Soc. 103.4 (1988), pp. 1209–1212. [FGM91] Marisa Fernández, Alfred Gray, and John W. Morgan. “ Compact sy...

work page Pith review arXiv 1988

[8] [59]

Hamiltonian Torus A ctions on Symplectic Orbifold and Toric Varieties

[LT97] Eugene Lerman and Susan Tolman. “Hamiltonian Torus A ctions on Symplectic Orbifold and Toric Varieties”. In: Transactions of the American Mathematical Society 349.10 (1997), pp. 4201–4230. [Liu] Yichen Liu. “Toric extension of complexity one spaces ”. in preparation. [LPT] Yichen Liu, Joseph Palmer, and Susan Tolman. “Extendi ng tall complexity one...

work page 1997