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Giroux correspondence, confoliations, and symplectic structures on S^1 x M

Let M be a closed oriented 3-manifold such that S^1 x M admits a symplectic structure w. The goal of this paper is to show that M is a fiber bundle over S^1. The basic idea is to use the obvious S^1-action on S^1 x M by rotating the first factor, and one of the key steps is to show that the S^1-action on S^1 x M is actually symplectic with respect to a symplectic form cohomologous to w. We achieve it by crucially using the recent result or its relative version of Giroux about one-to-one correspondence between open book decompositions of M up to positive stabilization and co-oriented contact structures on M up to contact isotopy.


Introduction and statements of results
It was Thurston in [31] who first proved that any closed oriented smooth 4-manifold X which fibers over a Riemann surface admits a symplectic structure, unless the fiber class is torsion in H 2 (X, Z). Thus, if the genus of the fiber of such a closed oriented 4-manifold X is greater than or equal to 2, then the manifold X always admits a symplectic structure. Moreover, any fibration of a closed oriented 3-manifold M over a circle S 1 induces a symplectic structure on the 4-manifold S 1 ×M . (See [24] and [16].) Furthermore, it seems to have been widely believed that the converse also holds (see [30]). Conjecture 1.1. Let M be a closed oriented 3-manifold such that S 1 × M admits a symplectic structure. Then M fibers over S 1 .
The goal of this paper is that by taking a completely different but elementary approach from the previous work, we give a short and affirmative proof to the Conjecture 1.1. To do so, we shall use the S 1 -action on the symplectic 4-manifold S 1 × M obtained by rotating the first factor of S 1 × M . One crucial observation of the proof is that every symplectic class on S 1 × M can always be represented by a symplectic form invariant under the S 1 -action. We show this important fact by a recent result and its relative version of Giroux about one-to-one correspondence between contact structures up to contact isotopy and open book decompositions up to positive stabilizations (see [15], [18], [6], and [7] for more details).
To the author's knowledge, it is still unknown whether or not any action of a compact connected Lie group G on a symplectic 2n-manifold X always induces a G-invariant symplectic form on X, in general. On the other hand, in Riemannian geometry one can always obtain a G-invariant Riemannian metric by taking the average of a Riemannian metric over the Lie group G. As S 2 × S 2 with the product symplectic form shows, simply taking the average of a symplectic form over the Lie group G does not yield a G-invariant symplectic form [27].
Once we show that there exists a symplectic form invariant under an S 1 -action on S 1 × M , it is an easy and well-known procedure to complete the proof of the conjecture. In Section 4, for the sake of the reader we provide a proof of Theorem 4.2, using an argument of D. Tischler in [32] about fibering certain foliated manifolds over S 1 . We remark that Theorem 18 in [8] gives an alternative argument of the second half of the proof of our main Theorem 4.2.
As a generalization of the Theorem 4.2, Baldridge asked in [1] whether or not for every closed symplectic 4-manifold admitting a free S 1 -action whose orbit space is M the quotient manifold M fibers over S 1 . We think that our method of the present paper can be adapted to answer the following Conjecture 1.2. But we do not pursue it in this paper, for the sake of simplicity.
Conjecture 1.2. Let X be a closed symplectic 4-manifold admitting a free S 1 -action whose orbit space is M . Then the quotient manifold M fibers over S 1 .
Let us denote by M K a 3-manifold M K obtained by 0-surgery on a knot K in S 3 . As an interesting consequence, we can give an answer to the question in [20] of Kronheimer about symplectic structures on S 1 × M K . To be more precise, Fintushel and Stern proved in [9] that if S 1 × M K admits a symplectic structure then the symmetrized Alexander polynomial ∆ K (t) is monic. On the other hand, Kronheimer proved in [20] that if the knot has a genus g(K) of two or more a necessary condition for S 1 × M K to admit a symplectic structure is that its genus g(K) be equal to the degree of its symmetrized Alexander polynomial. It is a well-known fact ( [2] or [28]) that if a knot K is fibered then its symmetrized Alexander polynomial is monic and its genus g(K) is equal to the degree of is symmetrized Alexander polynomial. Since S 1 × M K is symplectic for fibered knots, Kronheimer raised a question whether or not S 1 × M K admits a symplectic structure for non-fibered knots such as the pretzel knot P (5, −3, 5). The symmetrized Alexander polynomial of the pretzel knot P (5, −3, 5) is t − 3 + t −1 and thus monic with its degree equal to the genus 1 of the knot. Our another main result is to give a negative answer to the question of Kronheimer as follows: Theorem 1.3. The product 4-manifold S 1 × M K admits a symplectic structure if and only if the knot K is always fibered.
According to the recent paper [10] of S. Friedl and S. Vidussi, the product of S 1 with the 0-surgery of S 3 along the pretzel knot P (5, −3, 5) does not admit a symplectic structure, which fits well with our result. We give the proof of Theorem 1.3 at the end of Section 4. As a corollary, as the pretzel knot P (5, −3, 5) shows, the statement that the symmetrized Alexander polynomial of a knot K is monic and its genus g(K) is equal to the degree of its symmetrized Alexander polynomial does not imply that S 1 × M K admits a symplectic structure. In summary, when we set the statements (A), (B), and (C) as follows, The symmetrized Alexander polynomial of a knot K with genus ≥ 2 is monic and its knot genus g(K) is equal to the degree of its symmetrized Alexander polynomial, we can establish the following table for the various implications:  Finally a few remarks are in order. During the preparation of our paper, two papers related to the Conjecture 1.1 have appeared. In their paper [21], Kutluhan and Taubes studied the Seiberg-Witten Floer homology of M , assuming that S 1 × M admits a symplectic form. As a consequence, by combining their results with Theorem 1 of Y. Ni in [26], they gave a different proof that M fibers over S 1 , in case that M has the first Betti number equal to 1 and the first Chern class of the canonical line bundle is not torsion. Friedl and Vidussi also posted a preprint [13] asserting the proof of Conjecture 1.1 modulo some technical step regarding the residually finite solvability of π 1 (M ) which allegedly depends on a work under preparation by M. Aschenbrenner and S. Friedl. Among other things, the twisted Alexander polynomials, algebraic group theory, and Stallings' characterization ( [29]) for the fibration of a 3-manifold over a circle play crucial roles in their proof.
We organize this paper as follows. In Section 2, we give some basic facts about open book decompositions for closed contact 3-manifolds, partial open book decompositions for compact contact 3-manifolds with convex boundary, and confoliations. Section 3 is one of the key sections for this paper.
In that section, we show that every symplectic class on S 1 × M can always be represented by a symplectic form invariant under the naturally defined S 1 -action. Finally Section 4 is devoted to the proofs of the main Theorems 4.2 and 4.3.

Giroux correspondence and confoliations
The aim of this section is to review some basic facts about open book decompositions for contact 3-manifolds, partial open book decompositions for compact contact 3-manifolds with convex boundary, and confoliations.
First we briefly review the definition of an open book decomposition of a closed 3-manifold M , and its extension to compact contact 3-manifolds with convex boundary can be easily obtained with an obvious modification (see the recent papers [18] and [6]). Let (F, h) be a pair consisting of an oriented surface F and a diffeomorphism h : F → F which is the identity on ∂F , and K be a link in M . An open book decomposition for M with binding K is the quotient space which is homeomorphic to M . Here the equivalence relation ∼ h is given by We will call F ×{t} for t ∈ [0, 1] a page of the open book decomposition. Two open book decomposition is equivalent if there is an ambient isotopy between them taking binding to binding and pages to pages. We can obtain a new open book decomposition (F, h ′ ) from (F, h) by a positive (resp. negative) stabilization. Namely, F ′ is obtained from F by attaching a 1-handle B along ∂F and h ′ is obtained by extending h by the identity map on the 1-handle B and taking the composition It is known that every closed 3-manifold has an open book decomposition, but it is not unique. A contact structure τ is said to be supported (or adapted) by the open book decomposition (F, h, K) if there is a contact 1-form λ satisfying the following properties: • λ induces a symplectic form dλ on each fiber F .
• K is transverse to τ , and the orientation on K given by λ is the same as the boundary orientation induced from F coming from the symplectic structure.
Thurston and Winkelnkemper showed in [33] that any open book decomposition (F, h, K) supports a contact structure. The contact planes constructed by them can be made arbitrary close to the tangent planes of the pages away from the binding. Recently E. Giroux showed in [15] that the converse also holds. To be more precise, the following theorem holds: In our situation, we do not use the full version of this theorem. Rather we will need the result of Giroux to choose a coordinate chart on S 1 ×M with which we can easily calculate the Lie derivative of the symplectic form for our purposes (e.g., see Lemma 3.4 for more details). Even if the above Theorem 2.1 is stated for closed contact 3-manifold, the construction of Giroux shows that the same result holds for contact 3-manifolds with contact boundary, as the papers [18] and [6] of Honda-Kazez-Matić and Etgü-Ozbagci show.
For the sake of reader's convenience, we briefly review the relative Giroux correspondence for compact contact 3-manifolds with convex boundary, although in the present paper we do not need the full strength of this correspondence. For more details, see [18] and [6], and most of what is presented here can be found in those two papers.
We first begin with the abstract version of a partial open book decomposition which is a triple (S, P, h) satisfying the following three properties: • S is a compact oriented connected surface with non-empty boundary ∂S, • P = P 1 ∪P 2 ∪· · ·∪P r where P 1 , P 2 , . . ., P r are 1-handles is a proper, but not necessarily connected, subsurface of S such that S is obtained from the closure of S\P by attaching 1-handles P 1 , P 2 , . . ., P r successively, • h : P → S is an embedding such that h| ∂P ∪∂S =identity.
Given a partial open book decomposition (S, P, h), we can construct a compact 3-manifold with boundary as follows.
for all x ∈ ∂P ∩ ∂S and t, t ′ ∈ [0, 1]. Again each component of N is a solid handlebody whose boundary can be described by the connected arcs of the closure of ∂P \∂S. In other words, let c 1 , c 2 , · · · , c n denote such connected arcs. Then each disk Recall that a closed oriented embedded surface Σ in a contact manifold (M, ξ) is called convex if there is vector field tansverse to Σ which preserves the contact structure ξ. A generic surface Σ inside a contact manifold can be made convex (cf. [14] and Section 2.2 of [17]). So the assumption that the boundary be convex can be imposed without loss of generality.
In [18], Honda-Kazez-Matić associated the isomorphism classes of compact contact 3-manifolds with convex boundary to the isomorphism classes of partial open book decompositions modulo positive stabilizations. Conversely, in [6] Etgü and Ozbagci constructed its inverse by describing a compact contact 3-manifold with convex boundary compatible with a given partial open book decomposition. As in the proof of Proposition 1.9 in [6], such a construction is essentially given by the explicit construction of Thurston and Winkelnkemper. Hence the property, as well as others, that for closed contact 3-manifolds the contact planes constructed by them can be made arbitrary close to the tangent planes of the pages away from the binding can also be used for compact 3-manifolds with convex boundary. Now we can state a relative version of Giroux correspondence as follows, which is a relative version of Giroux correspondence for closed contact 3manifolds (see Theorem 0.1 in [6]).

Theorem 2.2. There is a one-to-one correspondence between isomorphism classes of partial open book decompositions modulo positive stabilization and the isomorphism classes of compact contact 3-manifolds with convex boundary.
In what follows, we will also need to use the result of Eliashberg and Thurston (Theorem 2.4.1 in [4]). A plane field η = ker θ on an oriented 3-manifold is called positive (resp. negative) confoliation if θ ∧ dθ ≥ 0 (resp. θ ∧ dθ ≤ 0). Let us denote by ζ the product foliation of the manifold S 2 × S 1 by the spheres S 2 × {z} for z ∈ S 1 .

Theorem 2.3 (Eliashberg-Thurston).
Suppose that a C 2 -confoliation η on an oriented 3-manifold is different from the foliation ζ on S 2 × S 1 . Then η can be C 0 -approximated by contact structure. When η is a foliation it can be C 0 -approximated both by positive and negative contact structure.

Existence of S 1 -invariant symplectic structures
Recall that there exists an obvious circle action on the 4-manifold S 1 × M obtained by rotating the first factor of S 1 × M . The aim of this section is to show that every symplectic class on S 1 × M can be represented by a symplectic form which is invariant under the obvious action of S 1 .
In what follows, we assume that S 1 × M admits a symplectic structure ω. If M is S 2 × S 1 , then clearly M fibers over S 1 . So from now on we also assume that M is not S 1 × S 2 , unless stated otherwise. Let X be the fundamental vector field associated to the action of S 1 , and let α = ι X ω.
Since ω is a symplectic 2-form, α is clearly a nowhere vanishing 1-form on S 1 × M . Now choose an arbitrary point t in S 1 . Let j t denote the inclusion from M into S 1 × M given by x → (t, x). Then we obtain a nowhere vanishing 1-form β t by the pull-back of the 1-form α restricted to {t} × M via the inclusion j t . In other words, β t = j * t (α| {t}×M ). Then we have the following proposition. Proof. We divide the proof into the following three cases: (Case 1) First of all, assume that β t ∧ d M β t is non-zero for all t. Thus the 2-plane field ξ t = ker β t is a family of contact structures on M . It is obvious that d M β t is a nowhere vanishing 2-form on M . Moreover, the following holds. Proof. The proof follows from the Cartan's formula. Indeed, it suffices to note that This completes the proof of Lemma 3.2.
Next we can show the following Proof. To see it, first note that there exists a Reeb vector field Z t on M , depending on the parameter t, such that Since  \N (B)) satisfying the following two properties: • f is nowhere vanishing over S 1 × (M Bt \N (B t )) and • f · d M β t coincides with ω, when restricted to the contact plane ξ t .
Then the following lemma holds.  \N (B t )) which can be identified with S 1 × (M \N (B t )), the symplectic 2-form ω can be written locally as the form where δ is a 1-form on S 1 × (M \N (B t )) and x denotes a local coordinate on M .
Proof. To see it, notice first that ω(Z t , W t ) may be non-zero for the Reeb vector field Z t and W t ∈ ξ t , while ω(X, W t ) should be zero for W t ∈ ξ t . Thus in local coordinates the symplectic form ω should have only the terms involving π * (dt ∧ ds t ), d M β t and ds t ∧ δ. Due to the equation (3.2), the coefficient of π * (dt ∧ ds t ) should be 1, as stated. Thus we are done.
Finally, over S 1 × (M Bt \N (B t )) we compute the Lie derivative L X ω explicitly. To do so, note that we have However, since we have 1 = ω(X, (j t ) * (Z t )) = 1 − ι X δ by the equations (3.2) and (3.3), ι X δ should be zero. Thus it follows from (3.4) that L X ω = 0. This clearly contradicts to Lemma 3.2. That is, this case does not occur.
(Case 2) We next assume that, for some t = t 0 in S 1 , β t ∧ d M β t is non-zero for all x ∈ M . By the continuity of smooth differential forms, β t ∧ d M β t should be non-zero for all t in some sufficiently small open interval I of t 0 and all x ∈ M .
Then apply the arguments in (Case 1) to the manifold I × M instead of S 1 × M . Then we can also derive a contradiction in this case. In more detail, the manifold I × M admits a symplectic structure, denoted ω, by the restriction of the symplectic form ω on S 1 × M to I × M . There still exists the fundamental vector field X on I × M associated to the natural action of S 1 on S 1 × M , since the interval is regarded as an open submanifold of S 1 . However, clearly there is no S 1 -action on I. Using this fundamental vector field X on I × M and the fact that β t ∧ d M β t is non-zero on I × M , as in Lemma 3.2 we can show that the Lie derivative L X ω is nowhere vanishing on I × M . Furthermore, one can check that other arguments as well as Lemmas 3.3 and 3.4 go through without any modification. So, we can conclude that this case does not occur, either.
(Case 3) In this case we assume that, for some t = t 0 in S 1 and some x 0 ∈ M , β t ∧ d M β t is non-zero. Once again it follows from the continuity of smooth differential forms that β t ∧ d M β t should be non-zero for all t in some sufficiently small open interval I of t 0 and some x 0 ∈ M .
In order to apply the arguments of the previous cases, we need to take the contact part For simplicity, for each t ∈ I we shall denote by W t the closure of the connected component of the contact part V (β t ) which contains x 0 . Then for each t ∈ I, W t is a compact contact submanifold of M of codimension 0 with (possibly empty) boundary, since β t ∧ d M β t is a nowhere vanishing 3-form on V (β t ) and so is a volume form there. Fortunately, in the present paper we do not need to know the precise information of W t , unlike to the case in the paper [19].
Assume now that the boundary is non-empty. (Otherwise, we are reduced to (Case 2).) Then, as already mentioned in Section 2, we may assume without loss of generality that the boundary is convex. Thus for each t ∈ I we obtain a compact contact 3-manifold W t with convex boundary. It is also true that as in (Case 2) above there still exists the fundamental vector field X on ∪ t∈I {t} × W t ⊂ S 1 × M associated to the natural action of S 1 on S 1 × M , since the interval I is again regarded as an open submanifold of S 1 . But this case is slightly different from (Case 2) in that W t is a compact contact 3-manifold with convex boundary. So we need to use the relative Giroux correspondence (Theorem 2.2) instead of the Giroux correspondence (Theorem 2.1). In other words, for each t ∈ I apply Theorem 2.2 to W t in order to obtain its partial open book decomposition (S t , P t , h t ). Thus W t can now be described as the gluing of two handlebodies H t and N t by the map h t whose boundary is given as in (2.1). However, since all the arguments in (Case 1) and (Case 2) are essentially local, those arguments applied to the compact contact 3-manifold W t with convex boundary and symplectic 4-manifold ∪ t∈I {t} × W t equipped with the symplectic form induced from S 1 × M will again go through without any modification. This in turn gives rise to a contradiction for this case, which means that this case does not occur, either.
This completes the proof of Proposition 3.1.
With this understood, the following theorem will play a crucial role in the proof of Theorem 4.2. Proof. We prove this theorem by contradiction. That is, suppose that the cohomology class [ω] cannot be represented by any S 1 -invariant symplectic form ω under the obvious S 1 -action. Then we would have (3.5) L X ω = 0.
Note also that by Proposition 3.1 the differential 3-form β t ∧ d M β t vanishes identically for all t ∈ S 1 . Then there are two possibilities we have to consider: d M β t vanishes identically on {t} × M for all t ∈ S 1 or not. So, suppose first that d M β t vanishes identically on M as well. Then it can be shown that the Lie derivative L X ω vanishes identically. To see it, notice that it follows from the identity (3.1) that we have j * t (L X ω) = 0. Thus we have L X ω(Z t , W t ) = 0 for any vector fields Z t and W t on {t} × M for each t ∈ S 1 . Moreover, since ω is a symplectic form on S 1 × M , we can choose a Darboux chart in a neighborhood of a point (t, x) whose coordinate vectors are given by non-zero vector fields X 0 = X and X i (i = 1, 2, 3). With this coordinate chart, we have L X ω(X, X i ) = dι X ω(X, X i ) = X(ι X ω(X i )) = X(1) = 0, which implies that L X ω(X, Y t ) = 0 for all vector field Y t of {t} × M . Therefore, we can conclude that in this case L X ω vanishes identically. But this clearly contradicts to the assumption (3.5).
On the other hand, if d M β t does not vanish on {t} × M , we need to use the result of Eliashberg and Thurston about perturbing a confoliation into a contact structure. If the 3-manifold M is S 2 × S 1 , clearly M fibers over S 1 , as mentioned earlier. Thus we may assume that our foliation ξ t is different from the foliation ζ on S 2 × S 1 . Now if we apply Theorem 2.3 to ξ t then we have a contact structureξ t = kerβ t which is a C 0 -approximation to ξ t . Sinceξ t is a C 0 -approximation of ξ t , the symplectic 2-form ω can also be C 0 -approximated by a symplectic 2-formω on S 1 × M so thatβ t = j * t (ι Xω | {t}×M ). So we are essentially led to the (Case 1), (Case 2), or (Case 3) of Proposition 3.1, which has already shown not to occur. Therefore, for either case we have derived a contradiction under our assumption (3.5). This completes the proof of Theorem 3.5.
Remark 3.6. Note that Theorem 3.5 does not imply that any arbitrary symplectic form ω on S 1 × M is always S 1 -invariant under the S 1 -action on the first factor of S 1 × M . This can be easily seen by taking M to be the 3-dimensional torus T 3 . That is, if the theorem implies that any symplectic form ω on S 1 × T 3 is always S 1 -invariant under the obvious S 1 -action, the symplectic form on S 1 × T 3 = T 4 should also be invariant under the obvious S 1 -action of the last three S 1 -factors of T 4 . So we can conclude that every symplectic form on T 4 should be invariant under the componentwise T 4 -action on T 4 . But obviously this is not the case for T 4 .

Proofs of Theorems 4.2 and 4.3
In this section we present the proofs of main theorems of the present paper. Once we have established the existence of an S 1 -invariant symplectic structure on S 1 × M , it is a fairly standard procedure to complete the proof of Conjecture 1.1. For the sake of reader's convenience, we give its proof relatively in detail.
To do so, we begin with the following well-known lemma which says that when the cohomology class [ι X ω] is not integral and non-zero, by some suitable perturbation we can always make it integral. Proof. If N admits an S 1 -invariant symplectic form ω ′ such that [ω ′ ] is rational, then we can easily obtain an S 1 -invariant symplectic formω such that [ω] is integral by multiplying some suitable integer to ω ′ . Note also that the class [ι X ω ′ ] is rational if the class [ω ′ ] is.
So assume now that the class [ω] is not rational. It is clear that there exists an arbitrary small closed 2-form η such that ω + η represent a rational cohomology class. Letη be the average of η over the S 1 -action. Since S 1 is connected, for ν ∈ S 1 ν * η is a closed 2-form representing the same cohomology class as η. Thus ω+η and ω+η have the same rational cohomology class. Note also that ω ′ = ω +η is symplectic, provided that η is sufficiently small. By the openness of symplectic condition again, we can further choose ω ′ in such a way that the class [ι X ω ′ ] is non-zero. This completes the proof.
Finally we are ready to prove the main theorems. Proof. By Theorem 3.5, we may assume that the symplectic structure ω is S 1 -invariant. Further, we may also assume that the class [ι X ω] on S 1 × M is integral by Lemma 4.1.
In order to prove the theorem, we first consider the case where the class [ι X ω] is zero. In this case, there exists a function, called the moment map µ : S 1 × M → R such that ι X ω = dµ. Thus the S 1 -action is Hamiltonian. But it is clear that the S 1 -action on S 1 × M does not have any fixed points that are critical points of µ. This gives rise to a contradiction to the fact that any Hamiltonian function on a closed symplectic manifold should have at least two critical points (e.g., extremal points). Therefore we can conclude that the class [ι X ω] is actually non-zero. Under this condition, McDuff proved in [23] that by using an argument of D. Tischler in [32], there exists a generalized moment map µ : S 1 × M → S 1 satisfying ι X ω = µ * (dt). Thus by restricting the map µ to {a point} × M , we easily obtain a fibration of M over S 1 . This completes the proof of Theorem 4.2.
Now we close this section with a proof of Theorem 1.3 as follows.
Theorem 4.3. If S 1 × M K admits a symplectic structure, then K is always a fibered knot.