Poisson structures of near-symplectic manifolds and their cohomology
aa r X i v : . [ m a t h . S G ] M a y POISSON STRUCTURES OF NEAR-SYMPLECTIC MANIFOLDS ANDTHEIR COHOMOLOGY
PANAGIOTIS BATAKIDIS AND RAM ´ON VERAA
BSTRACT . We connect Poisson and near-symplectic geometry by showing thatthere is an almost regular Poisson structure induced by a near-symplectic form ω when its singular locus is a symplectic mapping torus. This condition is auto-matically satisfied on any near-symplectic 4-manifold. The Poisson structure π is ofmaximal rank n and it drops its rank by 4 on a degeneracy set that coincides withthe singular locus of the near-symplectic form. We then compute its Poisson coho-mology in dimension 4. The cohomology spaces are finite dimensional and dependon the modular class. We conclude with a comment on the interaction between thePoisson structure π and an overtwisted contact structure. Introduction
It is well known that symplectic and Poisson structures are naturally related.A symplectic form on a smooth manifold determines a regular Poisson structure,whose symplectic leaf is the whole manifold. Relaxing the non-degeneracy con-dition of a symplectic form leads to a closed 2-form that is symplectic away fromits degeneracy locus, i.e it is singular with respect to the non-degeneracy. How-ever, this does not automatically imply that there is an induced Poisson structureas in the symplectic case. In this work we study this problem in relation to anear-sympectic form, a type of such singular symplectic structure. This is a closed2-form ω on a smooth n –manifold M that is positively non-degenerate outside acodimension-3 submanifold, where the rank of ω drops by 4. The idea of looking atnear-symplectic forms goes back to Taubes in relationship to J –holomorphic curves,Seiberg-Witten, and Gromov invariants [25, 27]. In dimension 4, these objects areequivalent to self-dual harmonic forms vanishing on circles for a generic metric[12, 27]. Near-symplectic forms have also been studied in the context of brokenLefschetz fibrations starting with the work of Auroux, Donaldson, and Katzarkov[2]. Deformations of these fibrations [15] and connections to overtwisted contactstructures [12] have also been considered in the near-symplectic context. Here, weprove the existence of Poisson structures on near-symplectic manifolds, and char-acterize them in terms of their Poisson cohomology. Our first main result is thefollowing. Theorem 1.1.
Let ( M, ω ) be a n –dimensional closed near-symplectic manifold. As-sume that the singular locus Z ω ⊂ M of the 2-form is a symplectic mapping torus.Denote by N Z ω its normal bundle, which splits into a line bundle L − and a rank-2bundle L , i.e N Z ω ∼ = L − ⊕ L . Then there is a a singular Poisson structure ofmaximal rank n on M , such that the degeneracy locus of π n − contains Z ω . Mathematics Subject Classification.
Primary: 53D17, 57R17, 17B63. Secondary: 16E45,17B56, 57M50.
Key words and phrases. near-symplectic forms, Poisson cohomology, harmonic self-dual 2-forms, Pois-son algebra, smooth –manifolds, almost regular Poisson structure. Propositions 3.1, 3.3 and 3.5 construct the Poisson structure. Afterwards, wecontinue by calculating its Poisson cohomology in dimension 4.Poisson cohomology was introduced by Lichnerowicz in 1977 [16]. It is an im-portant invariant of Poisson geometry, as it reveals features about deformations,normal forms, derivations, and other characteristics of a Poisson structure. In gen-eral it is hard to calculate, one of the reasons being that the complex used to defineit is it is elliptic only at the points where the Poisson bivector is non-degenerate. Inmany cases it is infinite-dimensional, and it is unknown for many types of Poissonstructures. For instance, it is known for Lie-Poisson structures of semi-simple andcompact type, yet it is unknown for semi-simple but non-compact. The linear Pois-son structure constructed in this work and whose Poisson cohomology in dimension4 is computed, is neither semi-simple, nor compact.Recently, Poisson cohomology has served as a valuable tool to understand certainsingular Poisson structures. For example, it was essential in the work of Radko[23] in order to classify topologically stable Poisson structures on smooth, compact,oriented, surfaces. These structures were later generalized under the name of logor b-symplectic structures. The Poisson cohomology of b -symplectic structures wasdetermined in the work of Guillemin, Miranda, and Pires [11], and Marcut andOsorno-Torres [19, 20].The results of Theorem 1.1 motivate us to define the notion of near-positivePoisson bivectors on a 4–manifold M (Section 4.1). By near-positive we mean asingular Poisson bivector π of maximal rank 4 such that π vanishes transversallyon a 2-dimensional subspace D π ⊂ M and π > on M \ D π . By Theorem 1.1these Poisson structures exist on any 4–manifold M admitting a near-symplecticform, that is, any smooth, oriented M with second positive Betti number satisfying b +2 ( M ) > [14, 12].In Section 4 we calculate the Poisson cohomology of the structure of Theorem1.1 in dimension 4. The Poisson cohomology presents an analogous behaviourwith log-symplectic structures: It is a mixture of the cohomology coming from theambient manifold and the one of its degeneracy locus. However, the analogy doesnot extend to the de Rham cohomology. For the Poisson structure treated here, thepart associated to the degeneracy locus is not its de Rham cohomology. Our secondmain result shows this characterization. Theorem 1.2.
Let ( M, ω ) be a near-symplectic 4–manifold with a near-positive Pois-son structure π . Denote by k the total number of circles in the singular locus Z ω . ThePoisson cohomology of ( M, π ) is given by H π ( M, R ) ∼ = R ∼ = span h i H π ( M, R ) ∼ = R k ⊕ H ( M ) ∼ = k M r =1 span D Y Ω r ( π ) , ∂ L r E ⊕ H ( M ) H π ( M, R ) ∼ = R k ⊕ H ( M ) ∼ = k M r =1 span D Y Ω r ( π ) ∧ ∂ L r E ⊕ H ( M ) H π ( M, R ) ∼ = H ( M, R ) H π ( M, R ) ∼ = H ( M, R ) OISSON STRUCTURES OF NEAR-SYMPLECTIC MANIFOLDS AND THEIR COHOMOLOGY 3 where the generators Y Ω1 ( π ) , . . . , Y Ω k ( π ) of H π ( M, R ) correspond to the modularvector field of π at each component of the singular locus Z ω , and ∂ L , . . . , ∂ L k arevector fields on the tubular neighbourhood of each component of Z ω . Section 4 contains the proof of Theorem 1.2. We start by computing Poissoncohomology with formal coefficients in Proposition 4.7. A key observation comesfrom the action of Hamiltonian vector fields on polynomial functions with respectto a certain notion of degree. We follow with Poisson cohomology with smoothcoefficients in Proposition 4.10. Section 4.4 finishes the proof by showing how topass to smooth global cohomology on M .We conclude with a note about Poisson and contact structures in a near-symplectic4-manifold in connection to Theorem 1.2. It is known that in dimension 4 there isan overtwisted contact structure in the boundary of the tubular neighbourhood ofthe singular locus of a near-symplectic form [12, 7]. In Section 5 we make someobservations regarding the orbits of the Reeb vector field in relation to the modularclass and the action of the anchor map on the contact form.The near-positive Poisson structure allows us to consider degeneracies in therank of a Poisson structure π which are different from regular and log-symplecticstructures. Let us explain this briefly: Let M be a smooth oriented 4-manifold and π ∈ Γ(Λ T M ) a Poisson bivector. In terms of distinct degeneracies in the rank of π , we have that at any point p ∈ M , π can have rank 4, 2, or 0 along symplecticleaves, so one has the following cases:( i ) Rank( π p ) = 4 , where π ( p ) = 0 ( ii ) Rank( π p ) = 2 , where π ( p ) = 0 , but π ( p ) = 0 ( iii ) Rank( π p ) = 0 .Regular Poisson structures are those with constant rank at all points of M . On oneend of the spectrum we find symplectic manifolds, which determine a regular Pois-son bivector satisfying condition ( i ) everywhere. On the other end, a trivial Poissonstructure corresponds to case ( iii ). If a Poisson structure is singular, there can be acombination of the three cases in the list, at different points of the manifold. Forinstance, log-symplectic structures are those equipped with a Poisson bivector π onan even dimensional manifold M such that π n is transverse to the zero section in Λ n T M . In dimension 4, they capture cases ( i ) and ( ii ); the rank of π is maxi-mal except at a codimension-1 submanifold, where π vanishes transversally. Thenear-positive structure that we introduce is an example for cases ( i ) and ( iii ).In a forthcoming paper, we will compute the Poisson cohomology of a brokenLefschetz fibration using the associated Poisson structure constructed in [6]. Thisstructure is a combination of the cases ( ii ) and ( iii ) in the previous list of possibledegeneracies on a 4-manifold. Together with the Poisson cohomology computed in[11] and this paper, one will then have available Poisson cohomology computationsfor large classes of singular Poisson structures on 4- manifolds. Acknowledgements
We warmly thank Pedro Frejlich and Ralph Klaasse for theircomments and feedback on the draft of this work. We are also very grateful toA¨ıssa Wade for fruitful discussions and interest in this work. Our thanks extendalso to Viktor Fromm, Luis Garc´ıa-Naranjo, Alexei Novikov, Tim Perutz and PabloSu´arez-Serrato.
PANAGIOTIS BATAKIDIS AND RAM´ON VERA Preliminaries
Poisson Geometry and Cohomology
We recall some basic facts about Poisson geometry, referring the reader to e.g.[13] for details. Let M be a smooth manifold and C ∞ ( M ) be the sheaf of smooth R -valued functions on M . A Poisson structure on M is a Lie bracket {· , ·} on C ∞ ( M ) obeying the Leibniz rule { f g, h } = f { g, h } + g { f, h } . Let X p ( M ) = Γ(Λ p T M ) bethe space of p - vector fields on M and [ · , · ] SN : X p ( M ) × X q ( M ) → X p + q − ( M ) the Schouten-Nijenhuis bracket. A Poisson structure on M can be equivalently de-scribed by a bivector field π ∈ X ( M ) , called Poisson bivector , satisfying [ π, π ] SN =0 . In local coordinates { x , . . . , x n } , a Poisson bivector is determined by an antisym-metric matrix π i,j , written explicitly as π = P ≤ i The pair ( X ( M ) , d π ) is called the Lichnerowicz-Poisson cochain com-plex , and H kπ ( M ) := ker (cid:0) d kπ : X k ( M ) → X k +1 ( M ) (cid:1) Im (cid:0) d k − π : X k − ( M ) → X k ( M ) (cid:1) , are called the Poisson cohomology spaces of ( M, π ) .For our purposes we recall the interpretation of the lower Poisson cohomologygroups: H π ( M ) = { Casimir functions } H π ( M ) = { Poisson vector fields }{ Hamiltonian vector fields } H π ( M ) = { infinitesimal deformations of π }{ trivial deformations of π } H π ( M ) = { obstructions to formal deformations of π } . OISSON STRUCTURES OF NEAR-SYMPLECTIC MANIFOLDS AND THEIR COHOMOLOGY 5 The map (1) is a chain map and defines a homomorphism of graded Lie algebras(3) ˆ π ♯ : H • dR ( M ) → H • π ( M ) . In general, ˆ π ♯ is neither injective nor surjective, however if ( M, ω ) is symplecticwith associated Poisson structure π ω , its Poisson cohomology is known, as ˆ π ♯ω is anisomorphism: H • dR ( M ) ≃ H • π ω ( M ) , and [ π ♯ ( ω )] = [ π ω ] . Another well studied case emerges from results of Lu [17],Ginzburg and Weinstein [9]. If g is a compact Lie algebra and W the Lie-Poissonstructure on g ∗ , one has H kW ( g ∗ ) = H k Lie ( g ∗ ) ⊗ Cas ( g ∗ , W ) , where H Lie ( g ∗ ) is the Lie algebra cohomology of g and Cas ( g ∗ , W ) denotes the spaceof Casimirs of ( g ∗ , W ) .The second cohomology group allows one to characterize certain Poisson struc-tures as exact. For any Posson structure, π is said to be exact if the fundamentalcohomology class vanishes, i.e. [ π ] = 0 . The first cohomology encompasses anotherdistinctive object of a Poisson structure, the modular class. To define it, consider anorientable Poisson manifold with positive oriented volume form Ω . The mapping Y Ω : C ∞ ( M ) → C ∞ ( M ) defined by L X f Ω = ( Y Ω f )Ω is a Poisson vector field. The vector field Y Ω is known as the modular vector field with respect to Ω . One can check directly that Y Ω = 0 if and only if Ω is invariantby the flows of all the Hamiltonian vector fields on ( M, π ) . On the other hand,for another choice Ω ′ = g · Ω , g ∈ C ∞ ( M ) , the vector fields Y Ω ′ and Y Ω differby a Hamiltonian vector field and thus there is a canonically defined Poisson coho-mology class (cid:2) Y Ω (cid:3) called the modular class of ( M, π ) . Modular vector fields andmodular classes are defined for non-orientable Poisson manifolds using densities.2.2. Near-symplectic forms Since we are interested in the connection between Poisson and near-symplecticgeometry, we briefly recall some facts about near-symplectic structures. We referthe reader to [2, 21, 25, 27, 30] and the references within for a detailed expositionon these structures.Let M be a smooth, oriented manifold of dimension n . Consider a 2-form ω ∈ Ω ( M ) with the property of being near-positive everywhere, that is ω n ≥ . For p ∈ M , let K p = { v ∈ T p M | ω p ( v, · ) = 0 } be the kernel of ω at a point. If ω issymplectic, K p is trivial. The collection of fibrewise kernels constitutes the kernel K := ker( ω ) ⊂ T M of the 2-form.Consider ω as a section ω : M → Λ T ∗ M . As any smooth map between mani-folds, we can consider the differential on tangent spaces ∇ ω p : T p M → Λ T ∗ p M .In the context of near-symplectic 4-manifolds this map is known in the litera-ture as the intrinsic gradient. Denote by ∇ ω : T M → Λ T ∗ M the derivative of ω : M → Λ T ∗ M , and by ¯ i : Λ T ∗ M → Λ K ∗ the map induced by the dual of the PANAGIOTIS BATAKIDIS AND RAM´ON VERA inclusion i : K ֒ → T M . Let ∇ ω | K : K → Λ T ∗ M be the restriction to K . Define themap D K : K → Λ K ∗ as the composition ¯ i ◦ ∇ ω | K .On any 4-dimensional vector space V the wedge product q : Λ V ∗ ⊗ Λ V ∗ → Λ V ∗ defines a quadratic form of signature (3 , on Λ V ∗ , giving a decomposition Λ V ∗ = Λ V ∗ ⊕ Λ − V ∗ , where Λ V ∗ = { α ∈ Λ V ∗ | α ∧ α ≥ } and Λ − V ∗ = { α ∈ Λ V ∗ | α ∧ α ≤ } . Going back to our original setting, if dim( K p ) = 4 , then Λ K ∗ = Λ K ∗ ⊕ Λ − K ∗ . If ω is near-positive, that is ω n ≥ , then dim(Im( D K )) ≤ since the image of D K : K → Λ K ∗ is a positive semi-definite subspace of Λ K ∗ .This can be summarized by the following lemma. Lemma 2.2. [30] If ω is near-positive and dim K p = 4 then Rank( D K ) ≤ . Definition 2.3. A near-symplectic form on an oriented manifold M n is a closed,near-positive 2-form ω such that at every point p ∈ M either(i) K p = 0 , i.e. ω p is symplectic, or(ii) dim( K p ) = 4 and Rank( D K ) = 3 .Its singular locus , Z ω = { p ∈ M | ω n − p = 0 } , is the submanifold of M consisting ofthe points where (ii) holds.Symplectic manifolds are those satisfying condition (i) everywhere. In order tostudy singular symplectic structures we look also at condition (ii) with Z ω havingnon-empty interior. Since we are considering near-positive forms, ω n ≥ , the con-dition Rank( D K ) = 3 gives an identification of the image of D K with the positivebundle of self-dual forms in K , i.e. Im( D K ) ∼ = Λ K ∗ ⊂ Λ K ∗ . Moreover, Z ω is asubmanifold of dim( Z ω ) = 2 n − as the following lemma shows. Lemma 2.4. [30] The singular locus Z ω of a near-symplectic form is a codimension-3smooth submanifold of M . Definition 2.3 is standard in the 4-dimensional case, where the intrinsic formu-lation is due to Donaldson. A near-symplectic 4-manifold ( M , ω ) is equipped witha closed 2-form ω that is symplectic on M \ Z ω and vanishes at Z ω . If M is closed,then Z ω = { p ∈ M | ω p = 0 } consists of a collection of embedded circles (see The-orem 2.6 below). It is possible to modify the number of zero components, but it hasbeen shown that Z ω is always non-empty unless the underlying manifold is sym-plectic [26, Section 5]. Furthermore, ω ≥ and there is no point p ∈ M where Rank( ω p ) = 2 . At a degenerate point p ∈ M , the kernel K p is T p M , and the map D K corresponds to the so called intrinsic gradient ∇ ω p : T p M → Λ T ∗ p M , whichpreserves the rank condition with Rank( ∇ ω p ) = 3 . Remark 2.5. Near-symplectic forms are related to self-dual harmonic 2-forms forsome Riemannian metric. This equivalent formulation appears in the work of dif-ferent authors [14, 12, 25, 26, 27] and can be seen by the following statement. Theorem 2.6. [25, Thm. 4] [2, Prop. 1] Let M be a smooth, oriented 4-manifold.For a near-symplectic form ω on M , there is a Riemannian metric g on M such that ω is self-dual harmonic with respect to g . Conversely, if M is compact and b ( M ) ≥ ,then for a generic Riemannian metric g there is a closed, self-dual harmonic form ω , that vanishes transversally as a section of Λ T ∗ M and defines a near-symplecticstructure. The zero set of ω is a finite, disjoint union of embedded circles. OISSON STRUCTURES OF NEAR-SYMPLECTIC MANIFOLDS AND THEIR COHOMOLOGY 7 Remark 2.7. Let ( M, ω ) be near-symplectic with dim( M ) = 2 n > and let ι : Z ω ֒ → M be the inclusion of its (2 n − -dimensional singular locus Z ω = { p ∈ M | ω n − p = 0 } . The 2-form ω Z := ι ∗ ω is closed and has positive constant rank n − ,hence ( ω Z ) n − p = 0 for all p ∈ Z ω . Recall that a presymplectic structure is a pair ( W, Ω) , W being a smooth manifold of dimension k + r and Ω a closed 2-form ofrank k on W . Thus, a near-symplectic form ω defines a presymplectic structure ( Z ω , ω Z ) .Given a presymplectic manifold ( W, Ω) , let V be its null distribution given by thecollection of null or vertical subspaces V p = { v ∈ T p W | Ω p ( υ, · ) = 0 } . There is aone-to-one correspondence between Ω -compatible Poisson structures and horizon-tal subbundles H ⊂ T W such that T W = V ⊕ H . Such a structure is Poisson if andonly if H is integrable [29].In our setting, the null distribution ε := ker( ω Z ) is a line subbundle of T Z ω .Since ω Z is of maximal constant rank on Z ω , ε is regular, thus integrable. Thehorizontal distribution H is equipped with a symplectic structure defined by ω Z ,and assuming that H is integrable then π Z = ω − Z defines a Poisson structure.The previous decomposition on T Z ω with respect to the presymplectic structure ( Z ω , ω Z ) is T Z ω = ε ⊕ H . Due to the regularity of ε there is a flat connection for which we can choose atrivialization and a vector field X = ∂∂θ ∈ Γ( ε ) .We now recall the local expression of a near-sympletic form. Thinking of Λ K ∗ as Λ R we have a splitting in two rank-3 subbundles Λ ± K ∗ . With coordinates ( θ, x , x , x ) , two bases of these bundles are given by the following elements Λ K ∗ : Λ − K ∗ : (4) β = dθ ∧ dx + dx ∧ dx β = dθ ∧ dx − dx ∧ dx β = dθ ∧ dx − dx ∧ dx β = dθ ∧ dx + dx ∧ dx β = dθ ∧ dx + dx ∧ dx β = dθ ∧ dx − dx ∧ dx . A Darboux-type theorem for near-symplectic forms tells us that around a point p ∈ Z ω ⊂ M , there is coordinate chart ( U, ( q , p , θ, x )) such that(5) ω = ω Z + x β − x β + x β , where ω Z = P n − i =1 dq i ∧ dp i . With respect to this model, Z ω is given by { x = x = x = 0 } . Remark 2.8. A near-symplectic form has the property of splitting the normal bun-dle N Z ω of its singular locus Z ω ⊂ M into two subbundles, a rank-1 bundle L − anda rank-2 bundle L [21, 30]. This can be seen through a self-adjoint, trace-free au-tomorphism F : N Z ω → N Z ω constructed through the geometric information from ω . Its representative matrix is symmetric, traceless, and has three eigenvalues, twopositive and one negative (for more details see [12, sections 3 & 4], [21, sec. 2.3],[27, sec. 2c], [30, sec. 4B]). The negative and positive eigensubspaces draw thecorresponding bundles L − and L . These properties are independent on the choiceof the metric g . There can be many conformal classes [ g ] for which ω is self-dual,yet they are all the same along Z ω because the ∇ ω identifies the normal bundle PANAGIOTIS BATAKIDIS AND RAM´ON VERA N Z ω with the bundle Λ K ∗ of self dual forms at each point of Z ω . Therefore, anear-symplectic form ω determines a canonical embedding of the intrinsic normalbundle N Z ω as a subbundle of T M | Z ω complementary to T Z ω [21, 27]. This issummarized for later use in the following Lemma. Lemma 2.9. [21, 27, 30] Let ( M, ω ) be a near-symplectic manifold with singularlocus Z ω . The normal bundle N Z ω of Z ω splits into a line bundle L − and a rank 2bundle L , i.e N Z ω ≃ L − ⊕ L . Remark 2.10. A comment regarding orientations. Let M be a n –dimensionalmanifold and Z ⊂ M a submanifold of codimension k . Consider a 2-form ω in M ,with the property that ω n | M \ Z > . It follows from a standard algebraic topologicalargument that M is oriented if codim ( Z ) = k ≥ . In particular, a near-symplecticform guarantees the orientability of M .Recall that from above the isomorphism N Z ω ≃ Λ K ∗ induced by ∇ ω | K . Since Λ K ∗ is oriented by the orientation of M , one obtaines an orientation on N Z ω with the declaration that ∇ ω | K is orientation reversing. This orientation of N Z ω induces one on Z ω by adopting the convention T M = T Z ω ⊕ N Z ω [27].A near-symplectic manifold M is naturally related to a broken Lefschetz fibration(bLf). A bLf is a submersion f : M → B to a codimension 2 base with indefinitefold singularities Γ and Lefschetz singularities C . The singular sets Γ , C ⊂ M aresubmanifolds of codimension 3 and 4 respectively. Example 2.11. Under a suitable cohomology condition on H ( M ) , given a bLfover a symplectic base, the total space M can be equipped with a near-symplecticstructure with Z ω = Γ . If dim( M ) = 4 , the converse is also true, i.e. given a near-symplectic form one can build a bLf on ( M, ω ) . Hence, examples of near-symplecticmanifolds arise from broken Lefschetz fibrations, as well as from symplectic fibra-tions as the next example shows. Example 2.12. Let g : M → S be a compact symplectic fibration with symplectictotal space M , and let ( V , ω V ) be a closed, near-symplectic, 4-manifold with abroken Lefschetz fibration f : V → S . Construct the pullback bundle W withinduced maps ˜ f : W → M , and ˜ g : W → V . The total space W is a 6-dimensionalmanifold, and carries a near-symplectic form ω W induced by ˜ g : W → V . Thesingular locus Z ω is a surface bundle over S with ω Z = σ F , where σ F is thesymplectic form of the fibre.Another prototypical example without using bLfs is the following product manifold. Example 2.13. Let N = Z ω = (( Q, ω Q ) × [0 , / φ ∼ denote a symplectic mappingtorus, where ( Q, ¯ ω ) is a symplectic manifold, φ : Q → Q a symplectomorphism,and the equivalence relation determined by ( x, ∼ ( φ ( x ) , . Since N fibres over S , there is a nowhere vanishing closed 1-form β ∈ Ω ( N ) . Consider a closed,connected, orientable, smooth 3-manifold Y , and let α ∈ Ω ( Y ) be a closed 1-form with indefinite (i.e. no maximum nor minimum) Morse singular points. By atheorem of Calabi [5], there is a metric such that α is harmonic. Set M = N × Y and define the 2-form ω ∈ Ω ( M ) by ω = β ∧ α + ¯ ω + ( ∗ Y α ) , OISSON STRUCTURES OF NEAR-SYMPLECTIC MANIFOLDS AND THEIR COHOMOLOGY 9 where ∗ Y denotes the Hodge ∗ –operator. This 2-form is near-symplectic on M andits singular locus is Z ω = N × Crit ( α ) .2.3. Euler-like vector fields and Tubular Neighbourhoods In this section we recall some notions on Euler-like vector fields and tubularneighbourhoods based on [4]. Let Z ⊂ M be a smooth submanifold and denote by N Z = ν ( M, Z ) = T M | Z /T Z the normal bundle of Z . Let also p : ν ( M, Z ) → Z , i : Z → M be the projection and inclusion maps.For a vector bundle F → Z , the normal bundle relative to the zero section is ν ( F, Z ) = F . The normal bundle of T M relative to T Z is canonically isomorphic tothe tangent bundle of the normal bundle. In particular, the normal and the tangentfunctors commute, and there is a canonical isomorphism ν ( T M, T Z ) ∼ = T ν ( M, Z ) .A smooth map of pairs ψ : ( M , N ) → ( M , N ) taking M to M , and N to N ,induces a map on normal bundles ν ( ψ ) : ν ( M , N ) → ν ( M , N ) . For instance,take a vector field X ∈ X ( M ) tangent to a submanifold Z . View X as a section M → T M . The condition of X being tangent to Z means that it takes Z to thesubmanifold T Z , i.e. ( M, Z ) → ( T M, T Z ) . Applying the normal functor, oneobtains ν ( X ) : ν ( M, Z ) → ν ( T M, T Z ) = T ν ( M, N ) . In this way, for a vector fieldtangent to Z , one can obtain a vector field on the normal bundle, called the linearapproximation . A linear approximation is then a coordinate-free way of defining atensor field, including Poisson bivectors and other multivector fields. Definition 2.14. [4, Def. 2.6] Let Z ⊂ M be a submanifold and E ∈ X ( ν ( M, Z )) an Euler vector field. A vector field R ∈ X ( M ) is called Euler-like if R is complete, R | Z = 0 , with linear approximation being the Euler vector field i.e ν ( R ) = E .Linear approximations serve in the following definition of tubular neighbourhoods. Definition 2.15. [4, Def. 2.3] A tubular neighbourhood embedding for Z ⊂ M is anembedding of the normal bundle ψ : ( ν ( M, Z ) , Z ) → ( M, Z ) such that: (i) it takesthe zero section of ν ( M, Z ) to Z , and (ii) its linear approximation is the identitymap, i.e. ν ( ψ ) = id.There is a direct connection between Euler-like vector fields and tubular neigh-bourhood embeddings. If E is the Euler vector field on the normal bundle, then anytubular neighbourhood embedding carries E to an Euler-like vector field defined ina neighborhood of Z in M . Proposition 2.16. [4, Prop. 2.7] Let Z ⊂ M be a submanifold and E ∈ X ( ν ( M, Z )) an Euler vector field. Any R ∈ X ( M ) Euler-like along Z , determines a unique tubularneighbourhood embedding ψ : ν ( M, Z ) → M with ψ ∗ E = R. In particular, Euler-like vector fields are always linearizable [4, Lemma 2.4]. Induced singular Poisson structures In this section we show that a near-symplectic manifold of any dimension n ≥ induces an almost regular Poisson structure. The next two propositions constructthe Poisson structure of Theorem 1.1 in dimension 4. Poisson structures in near-symplectic ManifoldsProposition 3.1. Let ( M, ω M ) be a closed near-symplectic 4-manifold. Denote by Z ω the singular locus of the 2-form and by U Z ⊂ M a tubular neighbourhood of Z ω . There is a Poisson structure π U on U Z ⊂ M such that the vanishing locus of π U contains Z ω .Proof. Assume that Z ω has only one connected component, that is only one circle S . Let X ∈ X ( S ) be the unit tangent vector field so that X = ∂∂θ . Recall thatgiven a near-symplectic form, the normal bundle splits into N Z ω = L ⊕ L , a rank1-bundle L and a rank 2-bundle L . We use this splitting property coming from ω to construct a Poisson structure on the tubular neighbourhood of Z ω .Let E − , E + be Euler vector fields on L − and L respectively. In bundle coordi-nates, with ( x ) ∈ L − , ( x , x ) ∈ L they are expressed as E − = x ∂∂x , E + = x ∂∂x + x ∂∂x . In particular, E − | L = 0 , E + | L − = 0 , and E ± | Z ω = 0 . Define on N Z ω the bivectorfield(6) P = X ∧ E + = x ∂∂θ ∧ ∂∂x + x ∂∂θ ∧ ∂∂x . Let ψ : N Z ω → M be a tubular neighbourhood embedding, which by proposition2.16 is uniquely determined by an Euler-like vector field R . Denote by U Z = ψ ( N Z ω ) the tubular neighbourhood of Z ω in M . Push forward P into M via ψ to define a bivector field η := ψ ∗ P = X ∧ R + on U Z ⊂ M , where R + ∈ X ( M ) isEuler-like on U Z with ψ ∗ ( E + ) = R + .Recall that given a near-symplectic form on a closed 4-manifold, there is a metric g such that ω is self-dual and vanishes on a collection of circles. In dimension 4, wehave K = T M . Let ∗ be the Hodge operator with respect to this g such that ∗ ω = ω .Using the orientation given by the volume form ω (Rem. 2.10), one can define aHodge duality isomorphism from the exterior algebra of the cotangent bundle tothe one of the tangent bundle, thus obtaining a transformation of bivector fields, ∗ : Λ T M → Λ T M . This Hodge operator is defined with respect to the volumeform and a metric that makes ω self-dual. The construction is independent on theparticular choice of ω and g , since given any near-symplectic form, we can finda Riemannian metric g such that ω is a self-dual harmonic 2-form vanishing on a1-submanifold of M (see Thm. 2.6).Define the following bivector field on U Z (7) π U = η + ∗ η = X ∧ R + + ∗ ( X ∧ R + ) . For a sufficiently small neighbourhood around Z , the linear model of π U is givenby(8) π U = x (cid:18) ∂∂θ ∧ ∂∂x + ∂∂x ∧ ∂∂x (cid:19) + x (cid:18) ∂∂θ ∧ ∂∂x + ∂∂x ∧ ∂∂x (cid:19) . This bivector vanishes on ψ ( L − ) , which includes the singular locus of ω . A calcula-tion shows that it satisfies the Poisson condition [ π U , π U ] SN = 0 , and π U ≥ . (cid:3) OISSON STRUCTURES OF NEAR-SYMPLECTIC MANIFOLDS AND THEIR COHOMOLOGY 11 To globalize the Poisson structure to M , we connect the 2-form dual to π U tothe near-symplectic form with a deformation path of near-symplectic forms. Forclarity we denote the near-symplectic form on M as ω M ∈ Ω ( M ) . The argumentthat we implement is due to Karl Luttinger and Carlos Simpson, who refer to thisphenomena as the birth/flight, as it perturbs a self-dual 2-form with degeneracyon a plane to a self-dual 2-form with degeneracy on a circle. Their work [18] hadbeen known and used in the literature, for example in [26, 21, 15]. Years later,Taubes and Perutz provided independent proofs of the theorems of Luttinger andSimpson [28, 21]. For completion we present the part relevant to this work here.The 2-form dual to the Poisson bivector is given by(9) ω U := π − U = 1( x + x ) (cid:0) x ( dθ ∧ dx + dx ∧ dx ) + x ( dθ ∧ dx + dx ∧ dx ) (cid:1) . This 2-form is symplectic outside the degeneracy locus of π U , where it is singular.Consider the 2-form ˆ ω = x ( dθ ∧ dx + dx ∧ dx ) − x ( dθ ∧ dx + dx ∧ dx ) , which is closed and non-degenerate outside the singularity set of ω U . The 2-forms ω U and ˆ ω lie in the same de Rham cohomology class as ω U − ˆ ω = dκ with κ = (cid:16) x − x − ln( x + x ) (cid:17) dθ + (cid:16) x x + arctan( x x ) (cid:17) dx . Consider the 1-parameterfamily ω r = r · ω U + (1 − r ) · ˆ ω with r ∈ [0 , . This family is generated by a linear combination of the basis el-ements β and β of Λ (see 4). Outside D π = { x = x = 0 } this familyof 2-forms is symplectic for each r . Moreover, the rank of the gradient remainsconstant, i.e. Rank( ∇ ω U ) = Rank( ∇ ˆ ω ) = 2 , and so thus the singular locus. As [ ω U ] = [ˆ ω ] ∈ H ( U Z \ D π ) there is an isotopy ρ s : ( U Z \ D π ) × R → U Z \ D π that such that ρ ∗ ω U = ˆ ω . Next we use ˆ ω and apply the birth/flight perturbation toobtain a near-symplectic form. Consider the 2-parameter family of 2-forms ω ( ǫ, t ) = x [ dθ ∧ dx + dx ∧ dx + ǫ ( θdθ ∧ dx + x dx ∧ dx )] − x [ dx ∧ dx + dθ ∧ dx + ǫ ( x dx ∧ dx − θdθ ∧ dx )]+ ǫ θ + x − t )( dθ ∧ dx − dx ∧ dx ) . (10)This path is a linear combination of the following three elements ω = dθ ∧ dx + dx ∧ dx + ǫ ( θdθ ∧ dx + x dx ∧ dx ) ω = dx ∧ dx + dθ ∧ dx + ǫ ( x dx ∧ dx − θdθ ∧ dx ) ω = dθ ∧ dx − dx ∧ dx . The forms ω , ω , ω are a small perturbation of the frame of self-dual forms Λ T ∗ M .For a sufficiently small ǫ and t ∈ [ − δ, δ ] they span a smooth wedge-positive rank-3subbundle Λ t of Λ T ∗ M that varies smoothly in t .The family ω ( ǫ, t ) has different degeneracy loci depending on the values of theparameters. The parameter ǫ is responsible for generating the extra basis element,and for a sufficiently small real value it keeps ω ( ǫ, t ) non-degenerate outside thesingular locus. Since we are interested in near-positive forms, it makes sense that ǫ only takes non-negative values. Note also that ω (0 , 0) = ˆ ω . Fix a sufficiently small ǫ > , and take t ∈ [ − δ, δ ] ⊂ R . For t < the degeneracylocus is empty, hence each form ω ( ǫ, t ) is non-degenerate. At t = 0 , the degeneracylocus become a point, with the special case ω (0 , , where it is a plane. For t > ,the degeneracy locus for each ω ( ǫ, t ) is circle. These observations follow directlyfrom the wedge square ω ( ǫ, t ) .For ǫ > , t > , the family ω ( ǫ, t ) has the following features: it has a 4-dimensional kernel K spanned by h ∂ θ , ∂ x , ∂ x , ∂ x i , it vanishes along a circle, and Rank( ∇ ω ( ǫ, t )) = 3 . Furthermore, for each path element one keeps the splittingof N Z ω = L ⊕ L − . Hence, away from the vanishing locus, for each ǫ, t > ,there is a diffeomorphism ϕ : U Z → U Z such that ϕ ∗ ω M = ω ( ǫ, δ ) , and which is ahomeomorphism on Z ω [30, Thm. 1.2]. One could even apply ϕ to bring ω ( ǫ, δ ) inDarboux-type form as in (5). We summarize the previous exposition in the follow-ing lemma. Lemma 3.2. For a sufficiently small ǫ, δ ∈ R and a sufficiently small neighbourhoodthe 2-parameter family ω ( ε, t ) has the following properties: • ω ( ǫ, t ) is nondegenerate for t < with Z ω = ∅ , • ω (0 , 0) = ˆ ω has degeneracy on a plane • ω ( ǫ, is degenerate with Z ω = { pt } , • ω ( ǫ, t ) for t > are near-symplectic with Z ω = S , and is near-symplectomorphicto ω M . Consequently, the 2-parameter family ω ( ǫ, t ) allows one to deform ˆ ω = ω (0 , to ω M . Consider the tubular neighbourhoods U Z ⊂ U ′ ⊂ U ′′ of Z ω in M . On U Z set π = π U , on the intersection ( U ′′ ∩ U ′ ) \ U Z apply the deformation given bythe 2-parameter family to perturb ρ ∗ ω U = ˆ ω = ω (0 , to ω ( ǫ, t ) for ǫ > , t > .This extends smoothly to U Z and to all M since ω ( ǫ, t ) is symplectomorphic to ω M outside Z ω . Thus we get the following proposition. Proposition 3.3. Let ( M, ω ) be a closed near-symplectic 4-manifold. The Poissonstructure π U of Proposition 3.1 extends to a Poisson structure on M . This concludes the construction of π on M in case L − is oriented. Recall that thedecomposition N Z ω ≃ L − ⊕ L has two possible splittings, as the line bundle L − can be oriented or not. In particular, on a tubular neighbourhood of a component of Z ω we have a splitting of S × D → S into a D -bundle and a D -bundle over S .These are classified by homotopy classes from S into RP . Since π ( RP ) = Z / Z ,there are two possible splittings.We have handled the oriented case, and it remains to be checked that the model π U is also valid on the non-trivial splitting on U Z = S × D for the non-orientedcase. This is shown in the next Lemma. Lemma 3.4. The bivector field π U of equation (8) is Poisson on the two homotopyclasses of splittings of S × D → S over each component of Z ω .Proof. The non-oriented model is given by the quotient of S × D by an involutionreversing the orientation on both summands of the splitting [12]. Explicitly, it iswritten as OISSON STRUCTURES OF NEAR-SYMPLECTIC MANIFOLDS AND THEIR COHOMOLOGY 13 ι : S × D → S × D (11) ( θ, x , x , x ) ( θ + π, − x , x , − x ) . We just need to check that if the normal bundle is non-orientable, the local model(8) still provides a Poisson structure. From the action of ι we obtain ι ∗ (cid:18) ∂∂θ (cid:19) = ∂∂θ , ι ∗ (cid:18) ∂∂x (cid:19) = − ∂∂x , ι ∗ (cid:18) ∂∂x (cid:19) = ∂∂x , ι ∗ (cid:18) ∂∂x (cid:19) = − ∂∂x . Thus, ι ∗ π = π and the involution ι is a Poisson map for π . (cid:3) Proposition 3.5. Let ( M, ω ) be a near-symplectic manifold of dim( M ) = 2 n withsingular locus being a symplectic mapping torus Z ω = (( Q, ω Q ) × [0 , / φ ∼ . There isa Poisson structure on M such that π n − vanishes on Z ω .Proof. We extend the construction of proposition 3.1 by first defining a Poissonbivector π U on the tubular neighbourhood U Z as in equation (7), and then addinga symplectic Poisson structure on Z ω . Since Z ω fibres over S and ε is an integrableline bundle, there is a non-vanishing section X ∈ Γ(Λ ε ) . The kernel K := ε ⊕ N Z ω ⊂ T M of ω splits as K = ε ⊕ L − ⊕ L due to the splitting of N Z ω . Thus, wehave an Euler vector field E + on L . By definition the kernel K ⊂ T M is a rank 4bundle, and since Z ω is a mapping torus, we can look at self-dual forms on Λ K ∗ vanishing on circles. Fix a metric g K on K such that ω is self-dual with respect to g K on Λ K ∗ . Using the orientation given by the volume form ω n , we can obtain atransformation of bivector fields, ∗ g K : Λ K → Λ K .Since the 2-form ω Q descends to the quotient and is well-defined and symplecticon Z ω , the horizontal distribution H ⊂ T Z ω is involutive. Thus, the bivector field π Z := ( ω ♭ ) − ( ω Q ) defines a symplectic Poisson structure on Z ω , where ( ω ♭ ) − is theinverse of ω ♭ : T M → T ∗ M . This Poisson bivector has the property that π n − Z = 0 , π n − Z = 0 . On the tubular neighbourhood U Z define the bivector field(12) π U = X ∧ R + + ∗ g K ( X ∧ R + ) + π Z . The assumption on the topology of Z ω allows us to apply the previous deformationargument of involving ω r and the 2-parameter family ω ( ǫ, t ) to extend the Poissonbivector field π U to M . (cid:3) Remark 3.6. We briefly comment on a log-symplectic structure on the line bundle L − . A log-symplectic manifold M , also known as b -symplectic , is a smooth even-dimensional manifold M equipped with a Poisson bivector π whose Pfaffian π n vanishes transversally on a codimension 1-submanifold D π . This is a Poisson hy-persurface foliated by symplectic leaves where π n − = 0 . Of particular interest toour setting are log-symplectic structures on line bundles [10].A Poisson vector bundle over a Poisson manifold is a vector bundle equipped witha flat Poisson connection [10, Def. 1.5]. On any real line bundle E one can find aflat connection ∇ , and a Poisson flat connection can be expressed as ∂ = π ◦ ∇ + V for a Poisson vector field V . Let ( D, π D ) be a (2 k − -dimensional Poisson man-ifold of rank (2 k − . The multivector field χ = V ∧ π k − D is called the residue of ( E, ∂ ) , and is independent of the choice of ∇ . A Poisson line bundle with non-vanishing residue χ ∈ Γ(Λ n − T D ) over such a Poisson manifold ( D, π D ) admitsa log-symplectic structure with degeneracy locus D [10, Prop. 1.9]. Back to ourcontext, assume the conditions of Theorem 1.1. We have then that the horizontaldistribution H ⊂ T Z ω of the presymplectic form ω Z is involutive and that Z ω fibresover S . Then π Z = ( ω Z ) − is Poisson, and there is a non-vanishing vector field V on the null distribution of ω Z complementary to H . The residue is χ = V ∧ π k − Z . Remark 3.7. The Poisson structure on a near-symplectic manifold that we con-structed in Propositionl 3.5, belongs to the class of almost regular Poisson structures[1] since it is generically symplectic. Almost regular Poisson structures include reg-ular Poisson and log-symplectic structures among others. The structure induced bya near-symplectic form is neither regular, nor log-symplectic. Poisson Cohomology on 4-manifolds. In this section we compute the Poisson cohomology of the Poisson structure de-scribed in Proposition 3.1 on a smooth 4–manifold. For computational reasons werelabel the variable θ as x , the local model of such Poisson bivector on the tubularneighbourhood U Z is(13) π = x ( ∂ ∧ ∂ + ∂ ∧ ∂ ) + x ( ∂ ∧ ∂ + ∂ ∧ ∂ ) . Since rank( π ) = 4 there are no nonconstant Casimirs. Our results show that thePoisson cohomology spaces vanish except for k − vector fields with constant coeffi-cients. Furthermore, we get that the modular field ∂∂ x has a nontrivial cohomologyclass, while [ π ] = 0 . For simplicity, we will make use of the reduced notation ∂ i := ∂∂x i . To begin, we introduce the notion of near-positive Poisson bivector. Thisnotion is independent of the cohomology results, yet we present it as a motivatingidea of a Poisson structure analogous to a near-symplectic form.4.1. Near-positive Poisson bivectors Let M be a smooth, oriented 4–manifold. We consider Poisson bivectors π on M that are near-positive , that is π ≥ , and such that π has maximal rank outside asubmanifold D π of M where it vanishes transversally. In contrast to log-symplecticmanifolds, where the transversality condition is in Λ T M , here the condition is in Λ T M .Recall that on a 4-dimensional vector space the wedge-product ∧ : Λ R ⊗ Λ R → Λ R ≃ R defines a quadratic form of signature (3, 3) on the exterior algebra Λ R . Thus, wehave a decomposition Λ R = Λ R ⊕ Λ − R into two rank-3 bundles. On M , thepositive subspace Λ T p M consists of bivectors χ , such that χ (vol) ≥ . The ele-ments in Λ − T p M are those such that χ (vol) ≤ . Using coordinates ( x , x , x , x ) OISSON STRUCTURES OF NEAR-SYMPLECTIC MANIFOLDS AND THEIR COHOMOLOGY 15 on R , a basis of the spaces described above is given by Λ R : Λ − R : χ = ∂ ∧ ∂ + ∂ ∧ ∂ χ = ∂ ∧ ∂ − ∂ ∧ ∂ χ = ∂ ∧ ∂ − ∂ ∧ ∂ χ = ∂ ∧ ∂ + ∂ ∧ ∂ χ = ∂ ∧ ∂ + ∂ ∧ ∂ χ = ∂ ∧ ∂ − ∂ ∧ ∂ . Regard π as a section of Λ T M . Let U ⊂ M be a neighbourhood of a point p ∈ M where π p = 0 . Consider the covariant derivative of π , namely ∇ π : X ( M ) → Γ(Λ T M ) , v 7→ ∇ v π . Since p is a zero of a smooth section of a bundle we have aderivative ( ∇ v π )( p ) ∈ Λ T p M in the direction of v ∈ T p M . The restriction of ∇ π at p is ∇ π p : T p M → Λ T p M , which map we call intrinsic gradient following theconvention in near-symplectic geometry [21, 27]. Expanding π with a Taylor serieson U about p = 0 , satisfying π (0) = 0 , we have that π ( t · v ) = π (0) + t · ∇ v ω (0) + O ( t ) = t · ∇ v ω (0) + O ( t ) . Thus if π p ≥ , then ( ∇ v π p ) ≥ , ∀ v ∈ T p U .Observe that the dimension of the image of ∇ π p can be at most 3 due to thenon-negative condition on π , thus it is a subset of the positive subbundle Λ R .Hence, at a point p ∈ M , where π p = 0 , the rank of ∇ π p , seen as a linear map R → R , can be at most 3, and one can set a transversality condition on Λ T M by fixing Rank ( ∇ π p ) . As Lemma 4.2 shows, this rank condition implies that thesingular locus D π is a submanifold of M . Next, we will study the behaviour of π along distinct singularities determined by the dimension of the image of ∇ π p . Definition 4.1. A bivector π ∈ Γ(Λ T M ) on an oriented 4-manifold M is said tobe near-positive if • π ≥ , • at each point p ∈ M we have that either π > or π p = 0 , and • at all points where π vanishes, the intrinsic gradient ∇ π p : T p M → Λ T p M is constant.If additionally, [ π, π ] SN = 0 , we say that π is a near-positive Poisson bivector . Lemma 4.2. Let π ∈ Γ(Λ T M ) be a near-positive bivector with singular locus D π = { p ∈ M | π p = 0 } . Assume that Rank( ∇ π p ) = r is constant for all points p ∈ D π ,where ∇ π p : T p M → Λ T p M , and r ∈ { , , } . Then D π is a (4 − r ) -submanifold of M .Proof. We proceed with a similar argument as in [21]. Let p ∈ D π and considera 4-ball B around p . Set k = dim(Λ T p M ) − dim(Im( ∇ π p )) . Consider a k -bundle E k complementary to the image of ∇ π p , where k = 3 , , or and regard π as alocal section of Λ T B . One has a natural p : Λ T B → Λ T B/E k and a section ¯ π : B → Λ T B/E k defined by ¯ π := p ◦ π .Recall that Λ T B is a rank 6 bundle, thus the quotient Λ T B/E k is of dimension3, 2, or 1 depending on the value of k . Regarding ∇ ¯ π : T B → Λ T B/E k as thedifferential of ¯ π , one can see that this is map is surjective, since it is of maximalrank on its codomain due to the assumption on the rank of ∇ π . Hence, ∇ ¯ π is asubmersion. The near-positive condition implies that Im( π ) is a subspace of Λ T B and so it includes the zero element, thus is a regular value of ¯ π . Let v ∈ T p B and consider a point π ( p ) ∈ Im( π ) ∩ E k such that ∇ π p ( v ) ∈ E k . Since E k hasbeen chosen to be the space complementary to Im( ∇ π p ) , the condition on the rankimplies that Im( π ) ∩ E k = 0 transversally in Λ T B . Thus ¯ π − (0) is a submanifoldof M of dimension − (6 − k ) as claimed. (cid:3) Example 4.3. Consider the phase space (cid:0) R , ( q , p , q , p ) (cid:1) . The bivector π = p ( ∂ q ∧ ∂ p + ∂ q ∧ ∂ p ) + p ( ∂ q ∧ ∂ p + ∂ p ∧ ∂ q ) is near-positive Poisson with singular locus D π = { p = p = 0 } ≃ R × . Wehave that π = ( p + p )(vol) ≥ . By looking at the matrix of partial derivatives J π coming from the linearization of ∇ π we can see that Rank( ∇ π ) = 2 at the singularpoints.When dim( M ) = 4 , the Poisson structure constructed in Proposition 3.1 is anexample of a near-positive Poisson structure with Rank( ∇ π p ) = 2 , at all points p ∈ D π . As a consequence of Theorem 1.1 and Proposition 3.1 we obtain thefollowing statement. Corollary 4.4. Let ( M, ω ) be a near-symplectic 4-manifold. There is a near-positivePoisson structure on M . The Poisson coboundary operator. We start by writing down the equations of the Poisson coboundary operator (2).The basic Hamiltonian vector fields of (13) are given by π ♯ (d x ) = − x ∂ − x ∂ , (14) π ♯ (d x ) = x ∂ − x ∂ , (15) π ♯ (d x ) = x ∂ − x ∂ , (16) π ♯ (d x ) = x ∂ + x ∂ (17)For simplicity in notation, we set X k := π ♯ (d x k ) , and then one may rewrite thePoisson bivector as π = 12 X k =0 X k ∧ ∂ k . Set d = [ π, · ] : X • → X • +1 to be the Poisson coboundary operator. For f ∈ C ∞ ( R ) , it is then(18) d ( f ) = − X k =0 X k ( f ) ∂ k . Let Y = P k =0 f k ∂ k ∈ X , and s be the index completing the triplet { , , } once i < j are chosen. Then d ( Y ) = X k =1 [ X k ( f ) − X ( f k ) − − ( − k f k ] ∂ k + X i For easiness in our upcoming computations, we write d ( Y ) in its expanded form d ( Y ) =[ X ( f ) − X ( f ) − f ] ∂ +[ X ( f ) − X ( f )] ∂ +[ X ( f ) − X ( f ) − f ] ∂ +[ X ( f ) − X ( f ) − f ] ∂ +[ X ( f ) − X ( f )] ∂ +[ X ( f ) − X ( f ) − f ] ∂ , (20)where we used the notation ∂ ij := ∂ i ∧ ∂ j for i < j , and write a bivector field as W = X i =0 Let V i = R i [ x , x , x , x ] be the space of homogeneous polynomials of degree i , r i := dim( V i ) and X mi be the space of m -vector fields on U Z whose coefficients areelements of V i . Let also V formal = R [[ x , x , x , x ]] and X m formal be the space of m − vector fields with coefficients from V formal . Restricting d to X • formal , since π is linear,the operator can be further decomposed as d m = P i d mi with d mi : X mi → X m +1 i .With the notation for polyvector fields and their coefficient functions as in equa-tions (18), (19), (21), and (22), the operators d • i are identified as the maps in thefollowing sequence representing the coefficients in the complex ( X • i , d • i ) :(23) −→ V i d i −→ V ⊗ i d i −→ V ⊗ i d i −→ V ⊗ i d i −→ V i −→ and more precisely(24) f d i −→ ( f , f , f , f ) d i −→ ( f , f , f , f , f , f ) d i −→ ( f , f , f , f ) d i −→ f . Each Ψ ∈ X • formal will then be a cocycle if and only if each of its homogeneouscomponents is itself a cocycle. Respectively, Ψ will be a coboundary if and only ifeach of its homogeneous components is itself a coboundary. Definition 4.5. Let deg x x denote the sum of degrees in the ( x , x ) - coordinatesof a monomial in V i , that is, deg x x ( x k x k x k x k ) = k + k and P s =0 k s = i . Remark 4.6. If deg x x ( f ) = c ∈ N , the action of Hamiltonian vector fields isrelated to this degree as follows:(25) X ( f ) = − cf, deg x x ( X ( f )) = c, deg x x ( X ( f )) = c + 1 , deg x x ( X ( f )) = c, deg x x ( X ( f )) = c + 1 . In the sequence, we denote by H • formal ( U Z , π ) the cohomology of the cochaincomplex ( X • formal , d • ) and H mi ( U Z , π ) will denote the m –th Poisson cohomologygroup with coefficients from V i . The next proposition computes the formal co-homology of (13) in the case where Z ω has only one component. The general casewith n components will be addressed in section 4.4. Proposition 4.7. Let ( M, ω ) be a near-symplectic 4–manifold. Consider the tubularneighbourhood ( U Z , π ) of the singular locus Z ω equipped with the Poisson bivector (13) . Assume Z ω has only one component. Then H formal ( U Z , π ) ∼ = R ∼ = h i H formal ( U Z , π ) ∼ = R ∼ = h ∂ , ∂ i H formal ( U Z , π ) ∼ = R ∼ = h ∂ ∧ ∂ i H formal ( U Z , π ) = 0 H formal ( U Z , π ) = 0 Proof. We will show that all cohomology groups vanish when the coefficient func-tions in (24) are homogeneous polynomials of fixed degree i > , in which case(23) becomes a short exact sequence. The cohomology with constant coefficientfunctions will then be computed at the end of the proof. This will yield the propo-sition; as the operators d • are linear, one can replace V i by V formal , the algebra offormal power series equipped with (13).We henceforth restrict (23) to some fixed i > . H ( U Z , π ) .Since Rank( π ) = 4 , there are no non-constant Casimirs, so ker(d i ) = 0 and Im(d i ) ∼ = V i , dim(Im(d i )) = r i . H ( U Z , π ) .The image of d i is spanned by vector fields of the following forms: d ( x k ) x k x k x k = k x k − x k x k x k X =: A d ( x k ) x k x k x k = k x k x k − x k x k X =: B d ( x k ) x k x k x k = k x k x k x k − x k X =: C d ( x k ) x k x k x k = k x k x k x k x k − X =: D OISSON STRUCTURES OF NEAR-SYMPLECTIC MANIFOLDS AND THEIR COHOMOLOGY 19 with P s =0 k s = i . We will show that any X ∈ Ker(d i ) is written as a linearcombination of vector fields of the form A, B, C, D .Let Y = f ∂ + f ∂ + f ∂ + f ∂ ∈ Ker(d i ) , i.e f k ∈ V i . Without loss ofgenerality, assume that f is a monomial with coefficient equal to 1, so let f = x k x k x k x k and assume deg x x ( f ) = c = 0 . The vanishing of the coefficient of ∂ in (20) together with (25) imply that f and f share the same deg x x and inparticular, X ( f ) = − cf . On the other hand, by the vanishing of the coefficient of ∂ in (20) together with(25), one gets X ( f ) = (1 − deg x x ( f )) f = − cf . Applying the same argument for the coefficient of ∂ we get X ( f ) = − cf . Given that k = c − k , k = i − c − k , a direct computation of the formulas f j = − c X j ( f ) gives f = − k c x k − x k +11 x i − c − k x c − k + i − c − k c x k x k x i − c − k − x c − k +13 , (26) f = − k c x k x k − x i − c − k x c − k +13 + c − k c x k x k +11 x i − c − k x c − k − , (27) f = − k c x k − x k x i − c − k x c − k +13 − i − c − k c x k x k +11 x i − c − k − x c − k . (28)Splitting the coefficient of f as c − k c + k c , Y = f ∂ + f ∂ + f ∂ + f ∂ isnow written as Y = (cid:18) − k c x k − x k x i − c − k x c − k (cid:19) X + (cid:18) i − c − k c x k x k x i − c − k − x c − k (cid:19) X + (cid:18) k c x k x k − x i − c − k x c − k (cid:19) X + (cid:18) c − k c x k x k x i − c − k x c − k − (cid:19) X (29)which is in Im(d i ) .If c = 0 , i.e if f does not depend on x , x , then X ( f ) = X ( f ) = X ( f ) = X ( f ) = 0 . Vanishing the coefficients of the bivectors in (20) and using a similarcomputation as before, one gets that Y is written as a linear combination of vectorfields of type A and C . We have thus proved that L i> H i ( U Z , π ) = 0 . H ( U Z , π ) .Since X ( f ) = deg x x ( f ) f , taking f = f = f = 0 at the coefficientof ∂ at (22), we get that dim(Im(d i )) = r i . Since dim(Ker(d i )) = r i , we havethat H i ( U Z , π ) = 0 and L i> H i ( U Z , π ) = 0 . H ( U Z , π ) .By the previous computations we have shown that dim(Ker(d i )) = r i and so dim(Im(d i )) = 3 r i . To prove that H i ( U Z , π ) = 0 , it is enough to prove that dim(Ker(d i )) = 3 r i . We do this by first examining the deg x x in the equationsdefining Ker(d i ) , that is, vanishing the coefficients of ∂ ijk in (21).Let W = P i Define a function f ∈ C ∞ ( U Z ) to be flat if all its derivatives andthe function itself vanish along the singular locus { x = x = 0 } of (13). Remark 4.9. Let X • flat ( U Z ) , X • formal ( U Z ) and X • smooth ( U Z ) be the multivector fieldswith flat, formal and smooth coefficients respectively. By a theorem of E. Borel, thesequence −→ X • flat ( U Z , π ) −→ X • smooth ( U Z , π ) −→ X • formal ( U Z , π ) −→ is exact. This shows that the cohomology of Proposition 4.7 is actually smooth in x , x .We now compute the smooth Poisson cohomology using an idea of Ginzburg [8]. Proposition 4.10. The smooth Poisson cohomology of the Poisson bivector (13) on U Z is given in Proposition 4.7.Proof. Because of Remark 4.9, it suffices to show that the flat cohomology H • flat ( U Z , π ) vanishes. Extend π ♯ to the chain map ∧ • π ♯ : (cid:0) Ω • ( U Z ) , d dR (cid:1) −→ (cid:0) X • ( U Z ) , d (cid:1) andthen consider the restriction to forms with flat coefficients ∧ • π ♯ flat : Ω • flat ( U Z ) −→ X • flat ( U Z ) .Away from the singular locus, π ♯ flat is an isomorphism. Indeed, π ♯ flat X i =0 f i d x i ! = 0 ⇔ x f + x f = 0 , − x f + x f = 0 , − x f + x f = 0 , − x f − x f = 0 . Using Taylor series in 4 dimensions, the first and third equations above, imply thatoutside the singular locus, f = f = 0 . Similarly the second and fourth equationsimply that f = f = 0 and so π ♯ flat is injective. On the other hand, if P i =0 g i ∂ i is inthe image of π ♯ flat then there is always a flat preimage P i =0 f i d x i with f = − x g − x g x + x , f = x g − x g x + x , f = x g − x g x + x , f = x g + x g x + x . Finally, the cohomology class of Y ∈ X • smooth ( U Z , π ) written as a convergent Taylorseries in a neighbourhood of the singular locus is 0, if and only if each i − homogeneousterm of the Taylor series is itself a coboundary. (cid:3) Remark 4.11. In terms of deformation quantization, the linearity of (13) impliesthat one has control on the polynomial degree of each term in the ⋆ − productcorresponding to π . As shown in [3] for the more general case of weight homo-geneous Poisson structures, if f, g are polynomials of weight k and n respectively,the i -th term B i ( f, g ) in the Taylor series defining the ⋆ − product will be of weight k + n − iω ( π ) where ω ( π ) is the weight of the given Poisson structur. Here it’s easyto see that for the weight vector ω = (1 , , , , it is ω ( π ) = − . However a globalexistence theorem for ⋆ − products over these singular spaces is more complicatedbecause of the singularities. With respect to near-symplectic manifolds, a reason-able approach would be through Fedosov’s deformation quantization and the useof Whitney functions [22] which are already used in the proof of Proposition 4.10.4.4. Global Cohomology This section contains the last step to prove Theorem 1.2. Our goal is to describehow to pass from the smooth semi-global cohomology on the tubular neighbour-hood U Z to the global cohomology on all M . We follow a similar argument asRadko [23] and Roytenberg [24], and start by assuming that Z ω has only one com-ponent, i.e one singular circle. Proof of Theorem 1.2 Consider an open cover D of ( M, ω ) . Let V = D \ Z ω . Letagain ˆ π ♯ : H • dR ( M ) → H • π ( M ) the homomorphism on cohomology induced by theanchor map, i.e (3). On V , the 2-form ω is symplectic, thus it induces a symplecticPoisson bivector. Denote by U the tubular neigbourhood of the single component OISSON STRUCTURES OF NEAR-SYMPLECTIC MANIFOLDS AND THEIR COHOMOLOGY 23 in Z ω . In this situation the Poisson cohomologies of V and U ∩ V are isomorphicto their corresponding de Rham cohomologies,(36) H • dR ( V ) ∼ = H • π ( V ) , H • dR ( U ∩ V ) ∼ = H • π ( U ∩ V ) . Observe that U ∩ V = ( S × D ) \ S is diffeomorphic to I × S × S and homotopyequivalent to S × S . For a fixed { t } ∈ S = [0 , π ] / ∼ , the product { t } × D \ { pt } retracts to S . Since all self-diffeomorphisms of D \{ pt } are isotopic to the identity,gluing { } × D \ { pt } back to { π } × D \ { pt } results to S × D \ { pt } whichretracts to S × S . Thus, H • π ( U ∩ V ) ˆ π ♯ ∼ = H • dR ( S × S ) . Associated to the cover given by U and V there is a short exact Mayer-Vietorissequence at the level of multivector fields and differential forms → X • ( M ) → X • ( U ) ⊕ X • ( V ) → X • ( U ∩ V ) → and → Ω • ( M ) → Ω • ( U ) ⊕ Ω • ( V ) → Ω • ( U ∩ V ) → . Each of them leads to a long exact sequence at the level of Poisson and de Rhamcohomology respectively → H π ( M ) α −→ H π ( U ) ⊕ H π ( V ) β −→ H π ( U ∩ V ) ρ −→ (37) → H π ( M ) α −→ H π ( U ) ⊕ H π ( V ) β −→ H π ( U ∩ V ) ρ −→→ H π ( M ) α −→ H π ( U ) ⊕ H π ( V ) β −→ H π ( U ∩ V ) ρ −→→ H π ( M ) α −→ H π ( U ) ⊕ H π ( V ) β −→ H π ( U ∩ V ) ρ −→→ H π ( M ) α −→ H π ( U ) ⊕ H π ( V ) β −→ H π ( U ∩ V ) → . and → H ( M ) a −→ H ( U ) ⊕ H ( V ) b −→ H ( U ∩ V ) r −→ (38) → H ( M ) a −→ H ( U ) ⊕ H ( V ) b −→ H ( U ∩ V ) r −→→ H ( M ) a −→ H ( U ) ⊕ H ( V ) b −→ H ( U ∩ V ) r −→→ H ( M ) a −→ H ( U ) ⊕ H ( V ) b −→ H ( U ∩ V ) r −→→ H ( M ) a −→ H ( U ) ⊕ H ( V ) b −→ H ( U ∩ V ) → . We start by describing the de Rham cohomology of V , since H • π ( V ) ∼ = H • dR ( V ) and this cohomology will be useful in subsequent calculations. Since M is con-nected, M \ S remains connected and H ( V ) ∼ = R . Removing an embeddedcircle from M amounts to attaching a 1-handle h to M . This implies that H ( V ) is given by the direct sum H ( M ) ⊕ H ( h ) . Then, H ( V ) ∼ = H ( M ) ⊕ h d θ i ,where d θ comes from the fundamental class of S of the handle attached. For thetop cohomology group, observe that since M is oriented (see remark 2.10), thefundamental cycle is no longer in V , thus H ( V ) ∼ = 0 . From a simple calculationon (38), it follows first that H ( V ) ∼ = H ( M ) ⊕ h ω S i , where ω S is the area formof S in U ∩ V , and secondly that H ( V ) ∼ = H ( M ) . H π ( M ) Observe that the first row of (37) is exact. On the symplectic region containing V ,a Casimir function is constant, thus by continuity it must be constant everywhere.Hence H π ( M ) ∼ = R ∼ = span h i . H π ( M ) Case: n = component Assume that Z ω consists of only one circle, and denote this component by ζ . Ex-actness of (37) leads to H π ( M ) = Im (cid:0) ρ (cid:1) ⊕ ker( β ) and similarly H ( M ) =Im (cid:0) r (cid:1) ⊕ ker( b ) . Since Z ω has only one component, V and U ∩ V are symplectic,hence H π ( V ) = ˆ π ♯ (cid:0) H ( V ) (cid:1) , H π ( U ∩ V ) = ˆ π ♯ (cid:0) H ( U ∩ V ) (cid:1) . Since the first row of (37) is short exact we have that Im( ρ ) = 0 . To determine thekernel of β : H π ( U ) ⊕ H π ( V ) → H π ( U ∩ V ) , ([ χ ] U , [ ν ] V ) [ χ − ν ] U ∩ V , recall that H ( V ) = H ( M ) ⊕ h d θ i , and H ( U ∩ V ) = H ( S × S ) = span h d θ i . Propositions 4.7 and 4.10 described the Poisson cohomology of the the tubularneighbourhood of Z ω . The first cohomology group is generated by the class of themodular vector field ∂ and a vector field ∂ . To simplify the notation, we relabelthe variables θ := x , and λ := x so that the Poisson vector fields are denoted by ∂ θ := ∂ and ∂ λ := ∂ , and H π ( U ) ∼ = span h ∂ θ , ∂ λ i ∼ = R . Then we can write β : span h ∂ θ , ∂ λ i ⊕ ˆ π ♯ (cid:0) H ( M ) ⊕ span h d θ i (cid:1) → ˆ π ♯ (span h d θ i ) . The image of the anchor map ˆ π ♯ on de Rham classes as determined by equations(14)–(17), implies that(39) ker( β ) = span h ∂ θ , ∂ λ i ⊕ ˆ π ♯ (cid:0) H ( M ) (cid:1) and thus(40) H π ( M ) ∼ = span h ∂ θ , ∂ λ i ⊕ H ( M ) . Case: n components Suppose that Z ω contains n components { ζ , . . . , ζ n } . We extend our previous ar-gument inductively as in [23]. Choose an open cover V = M , and set V i = V i − \ ζ i for i ∈ { , . . . , n } . The sequence (37) can now be read on each row as · · · −→ H kπ ( V i − ) α ki −→ H kπ ( U i ) ⊕ H kπ ( V i ) β ki −→ H kπ ( U i ∩ V i ) ρ ki −→ · · · and we can use a similar notation for the long exact sequence of the de Rhamcomplex. The regions V n and U i ∩ V i are symplectic for each i . Thus one has H • dR ( V n ) ∼ = H • π ( V n ) and H • dR ( U i ∩ V i ) ∼ = H • π ( U i ∩ V i ) . We also have that H π ( U i ) ∼ =span h ∂ θ i , ∂ λ i i , and H ( V i ) ∼ = H ( V i − ) ⊕ span h d θ i i . Recall that H π ( V i − ) = OISSON STRUCTURES OF NEAR-SYMPLECTIC MANIFOLDS AND THEIR COHOMOLOGY 25 Im (cid:0) ρ i (cid:1) ⊕ ker( β i ) with Im (cid:0) ρ i (cid:1) = 0 as in the case of one component. Then from i = n − to i = 0 , ker( β i − ) = span h ∂ θ i , ∂ λ i i ⊕ H π ( V i − )= span h ∂ θ i , ∂ λ i i ⊕ span h ∂ θ i +1 , ∂ λ i +1 i ⊕ ˆ π ♯ (cid:0) H ( V i − ) (cid:1) . Since H π ( V i − ) = Im( ρ i − ) ⊕ ker( β i − ) , we obtain H π ( V n ) ∼ = H ( V n ) H π ( V n − ) ∼ = span h ∂ θ n , ∂ λ n i ⊕ ˆ π ♯ (cid:0) H ( V n − ) (cid:1) H π ( V n − ) ∼ = span h ∂ θ n − , ∂ λ n − i ⊕ H π ( V n − ) ∼ = span h ∂ θ n − , ∂ λ n − i ⊕ span h ∂ θ n , ∂ λ n i ⊕ ˆ π ♯ (cid:0) H ( V n − ) (cid:1) ... ∼ = H π ( M ) ∼ = n M k =1 span h ∂ θ k , ∂ λ k i ⊕ H ( M ) . (41) H π ( M ) Case: n = component By exactness, it is H π ( M ) = Im( ρ ) ⊕ ker( β ) . Since H π ( U ∩ V ) ∼ = ˆ π ♯ (cid:0) H ( U ∩ V ) (cid:1) ,we get that Im( ρ ) ∼ = ˆ π ♯ (Im( r )) . To describe the kernel of β , we have from Propo-sitions 4.7 and 4.10 that H π ( U ) ∼ = 0 . On the de Rham complex we also have H ( U ) ∼ = 0 , since U ∼ = S × D . Together with H π ( V ) ∼ = ˆ π ♯ ( H ( V )) and H π ( U ∩ V ) ∼ = ˆ π ♯ ( H ( U ∩ V )) this leads to ker( β ) ∼ = ˆ π ♯ (ker( b )) . Consequently,(42) H π ( M ) ∼ = ˆ π ♯ (Im( r )) ⊕ ˆ π ♯ (ker( b )) ˆ π ♯ ∼ = H ( M ) . Case: n components This argument extends also to the general case when there are n circles { ζ , . . . , ζ n } in Z ω ; for every circle ζ i , the Poisson and de Rham cohomology groups of V i , U i ∩ V i , and U i are isomorphic, the latter following from Propositions 4.7, 4.10since H π ( U i ) ∼ = 0 ∼ = H ( U i ) . H π ( M ) We can apply the same line of reasoning as for the previous cohomology group,since around every component ζ i ∈ Z ω , we have H ( U i ) ∼ = 0 , and by Proposition4.7 we obtain H π ( U i ) ∼ = 0 . Similarly, this holds for H π ( V ) ∼ = ˆ π ♯ ( H ( V )) and H π ( U ∩ V ) ∼ = ˆ π ♯ ( H ( U ∩ V )) , thus(43) H π ( M ) ˆ π ♯ ∼ = H ( M ) . H π ( M ) Case: n = component The second cohomology space is H π ( M ) = Im (cid:0) ρ (cid:1) ⊕ ker( β ) . From (39) we know by exactness that ker( ρ ) ∼ = R and hence Im( ρ ) = 0 . On thesymplectic regions one has H π ( V ) ˆ π ♯ ∼ = H ( V ) ⊕ span h ω S i , and H π ( U ∩ V ) ˆ π ♯ ∼ = span h ω S i . Moreover, by Propositions 4.7, 4.10, it is H π ( U Z ) = span h ∂ θ ∧ ∂ λ i . Thenwe can write β : span h ∂ θ ∧ ∂ λ i ⊕ ˆ π ♯ (cid:0) H ( M ) ⊕ span h ω S i (cid:1) → ˆ π ♯ (span h ω S i ) . Since H π ( M ) ∼ = H ( M ) ∼ = H ( V ) , we have that α is injective and then byexactness ker( ρ ) ∼ = R . Thus(44) ker( β ) = span h ∂ θ ∧ ∂ λ i ⊕ ˆ π ♯ (cid:0) H ( M ) (cid:1) and(45) H π ( M ) ∼ = span h ∂ θ ∧ ∂ λ i ⊕ H ( M ) . Case: n components Extend this argument for n number of components { ζ , . . . , ζ n } in Z ω as in the caseof H π ( M ) . After choosing an open cover V = M and setting V i = V i − \ ζ i , weobtain H π ( V n ) ∼ = ˆ π ♯ (cid:0) H ( V n ) (cid:1) H π ( V n − ) ∼ = span h ∂ θ n ∧ ∂ λ n i ⊕ ˆ π ♯ (cid:0) H ( V n ) (cid:1) H π ( V n − ) ∼ = span h ∂ θ n − ∧ ∂ λ n − i ⊕ span h ∂ θ n ∧ ∂ λ n i ⊕ ˆ π ♯ (cid:0) H ( V n − ) (cid:1) ... H π ( M ) ∼ = n M k =1 span h ∂ θ k ∧ ∂ λ k i ⊕ H ( M ) . (cid:3) (46) Contact Structures In this section we comment on the interaction between the Poisson and con-tact structures in near-symplectic manifolds. In particular, we focus on the Poissonbivector π U of Proposition 3.1 and a contact structure on the tubular neighbour-hood of Z ω . In dimension 4, it has been shown that there is an overtwisted contactstructure on the boundary of U Z . Theorem 5.1. [7, 12] Let ( M, ω ) be a near-symplectic 4-manifold. There is an over-twisted contact structure ξ = ker( α ) on ∂U Z ∼ = S × S such that dα = i ∗ ω , where i : S × S ֒ → S × D . Recall that a contact structure on an (2 n − -dimensional manifold N is a max-imally non-integrable hyperplane distribution ξ ⊂ T N determined by the kernelof a globally defined 1-form α satisfying α ∧ dα n − = 0 . Contact structures on3-manifolds N are classified as tight or overtwisted . A contact structure is calledovertwisted if ( N , ξ ) contains an embedding of a disk D ֒ → N such that for itscharacteristic foliation ∆ = T p D ∩ ξ p : ( i ) the boundary ∂D is a closed leaf, and ( ii ) there is a unique elliptic singular point in the interior. If there is no such a disk,then the contact structure is said to be tight.With respect to the local model of ω on U Z = S × D as in (5) with ω Z = 0 [12,Sec. 5], the defining contact form for ξ = ker( α ) is(47) α = 12 (cid:0) x − x + x (cid:1) dθ + x ( x dx − x dx ) . OISSON STRUCTURES OF NEAR-SYMPLECTIC MANIFOLDS AND THEIR COHOMOLOGY 27 Now we look at the action of π ♯ on this contact form. Consider the Poisson bivector π U = η + ∗ η on U Z as in (7) and (8). Then π ♯ ( α ) = − (cid:0) x − x + x (cid:1) π ♯ ( dθ ) + x ( x + x ) ∂ . After a change of coordinates θ = θ, x = r cos( φ ) , x = z, x = r sin( φ ) , theprevious expression becomes π ♯ ( α ) = (cid:18) r − z (cid:19) V Ham + (cid:0) r · z (cid:1) ∂ z This vector field is clearly zero on Z ω = S × { } . As it moves to the boundary S × S the action of π ♯ on the contact form is a combination of the Hamiltonianvector field V Ham = π ♯ ( dθ ) and the Poisson vector field ∂ z .In [12] the author also provides the Reeb vector field of the contact structure,i.e. the unique vector field Y such that Y ∈ ker( dα ) and α ( Y ) = 1 . Up to a multiplethe Reeb vector field is given by Y = 1 f (cid:0) x − x + x (cid:1) ∂ θ + 3 x ( − x ∂ + x ∂ ) where f = − (cid:2) ( x + x )( x − x + x ) + 4 x (cid:3) . In terms of the Poisson structure π U , the Reeb vector field can be expressed using the modular vector field and aHamiltonian vector field(48) Y = 1 f (cid:0) x − x + x (cid:1) V mod + (3 x ) V Ham , where V mod = ∂ θ and V Ham = (cid:0) π ♯ ( dx ) (cid:1) = x ∂ − x ∂ .Denote by pt N := { (0 , , } , pt S := { (0 , − , } the north and south poles of S .The closed orbits of the Reeb vector field are(49) S × { pt N } , S × { pt S } , S × { ( x , , x ) } with x + x = 1 and x , x fixed. Hence, at the closed orbits, Y is a constant mul-tiple of the modular vector field and of a Hamiltonian vector field. Thus, by Propo-sition 4.7 one can summarize the previous observations in the following corollary. Corollary 5.2. Let ( M, ω ) be a near-symplectic 4–manifold. Along closed orbits, the Reeb vector fieldof the contact structure ( ∂U Z , ξ ) of Theorem 5.1 is in the Poisson cohomology class of [ ∂ θ ] ∈ H π . In higher dimensions, the situation is unknown. 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Taubes, A proof of a Theorem of Luttinger and Simpson about the Number of Vanishing Circles ofa Near-symplectic Form on a 4-dimensional Manifold , Math. Res. Lett. 13 (2006), no. 4, 557-570.[29] I. Vaisman, Geometric Quantization on Presymplectic Manifolds , Monatshefte fur Math., 96 (1983),293–310.[30] R. Vera, Near-symplectic n -manifolds , Alg. Geom. Topol. 16 no.3 (2016), 1403–1426.D EPARTMENT OF M ATHEMATICS & S TATISTICS , U NIVERSITY OF C YPRUS , N ICOSIA , 1678, C YPRUS E-mail address : [email protected] I NSTITUTE OF M ATHEMATICS - U NIVERSIDAD N ACIONAL A UT ´ ONOMA DE M´ EXICO , C IRCUITO E XTERIOR ,C IUDAD U NIVERSITARIA , C OYOAC ´ AN , 04510, M EXICO C ITY , M EXICO E-mail address ::