Existence of conformal symplectic foliations on closed manifolds

Abstract

We study the existence of symplectic and conformal symplectic (codimension-1) foliations on closed manifolds of dimension $\ge 5$. Our main theorem, based on a recent result by Bertelson–Meigniez, states that in dimension at least 7 any almost contact structure is homotopic to a conformal symplectic foliation. In dimension 5 we construct explicit conformal symplectic foliations on every closed, simply-connected, almost contact manifold, as well as honest symplectic foliations on a large subset of them. Lastly, via round-connected sums, we obtain, on closed manifolds, examples of conformal symplectic foliations which admit a linear deformation to contact structures.

1 Introduction

Manifolds are often studied through geometric structures living on them. This approach requires structure with a suitable balance of "flexibility" and "rigidity". More precisely, we want a structure which exists in large classes of examples, while at the same time carrying enough geometric data to reflect the properties of the underlying manifold.

This phenomenon is nicely illustrated by the tight-overtwisted dichotomy for contact structures. Overtwisted contact structures satisfy an h-principle [5, 16] showing their flexibility. The classification of tight contact structures is much more intricate, reflecting their rigidity.

In this line of thought we study here (codimension-1) foliations whose leaves are endowed with a symplectic-type geometric structure. (Without explicit mention of the contrary, the word “foliation” will always mean “codimension 1 foliation” in this paper.) We consider the following variations:

• A symplectic foliation is a foliation ${\mathcal {F}}$ endowed with a leafwise symplectic form, i.e. $\omega \in \Omega ^2({\mathcal {F}})$ which is closed and non-degenerate.

• A strong symplectic foliation is a symplectic foliations $({\mathcal {F}},\omega )$ for which $\omega$ admits an extension to a closed 2-form on M.

• A conformal symplectic foliation $({\mathcal {F}},\eta ,\omega )$ on $M^{2n+1}$ consists of a smooth foliation ${\mathcal {F}}$, endowed with differential forms $\eta \in \Omega ^1({\mathcal {F}})$ and $\omega \in \Omega ^2({\mathcal {F}})$ satisfying:

  $$\begin{aligned} {{\text {d}}}\eta = 0, \quad \omega ^n > 0, \quad {{\text {d}}}_\eta \omega := {{\text {d}}}\omega - \eta \wedge \omega = 0.\end{aligned}$$

  Analogously to the non-foliated case, one considers such pairs up the equivalence relation, called conformal equivalence, described by $(\eta ,\omega ) \sim (\eta + d f, e^f \omega )$ for any $f \in C^\infty (M)$. Also, the non-degeneracy of $\omega$ implies that $\eta$ is uniquely defined by the equation ${{\text {d}}}\omega = \eta \wedge \omega$ if $n\ne 2$. (Hence, strictly speaking it is not an additional datum in the dimensions we are interested in but we include it for clarity.)

The formal counterpart of all three structures above is an almost contact structure, i.e. a pair $(\zeta ,\mu )$ consisting of a hyperplane field $\zeta$ endowed with a non-degenerate form $\mu \in \Omega ^2(\zeta )$. Thus, it is natural to ask which almost contact manifolds admit foliations of each of the types above.

Symplectic foliations naturally generalize symplectic structures to the foliated setting. Although there are existence h-principles for symplectic foliations on ambient open manifolds [6, 7, 20], their existence on closed manifolds is still an important open problem. For instance, it is not known which spheres $\mathbb {S}^{2n+1}$, with $n \ge 3$, admit such foliations, and even the fact that $\mathbb {S}^5$ admits one is a non-trivial result [34].

Strong symplectic foliations share the same problem and are in fact even more rigid. For example, their leaf spaces are isomorphic to those of taut foliations in dimension 3 [37]. (Recall that a foliation is taut if it admits a closed transversal intersecting every leaf.)

Notice that Meigniez showed that in higher dimensions taut foliations satisfy an existence h-principle. Hence, the apparent rigidity of strong symplectic foliations could suggest they provide the “right” generalization of taut foliations to higher dimensions.

The main result of this paper states that the existence of an almost contact structures is not only necessary, but also sufficient for the existence of a conformal symplectic foliation in dimension at least 7:

Theorem 1.1

Let M be a closed manifold of dimension $2n+1 \ge 7$, endowed with an almost contact structure $(\zeta ,\mu )$. Then, there exists an exact conformal symplectic foliation $({\mathcal {F}},\eta ,{{\text {d}}}_\eta \lambda )$ on M homotopic to $(\zeta ,\mu )$ through almost contact structures.

The main result of [31] implies that any almost contact structure in dimension $\ge 5$ is homotopic to a taut and almost symplectic foliation. As such our main theorem is an immediate consequence of the following:

Proposition 1.2

Let $n \ge 3$, and $M^{2n+1}$ be a closed manifold with a taut almost symplectic foliation $({\mathcal {G}},\mu )$. Then, M admits an exact conformal symplectic foliation $({\mathcal {F}},\eta ,{{\text {d}}}_\eta \lambda )$ such that

• (i) $({\mathcal {F}},\eta ,{{\text {d}}}_\eta \lambda )$ is homotopic to $({\mathcal {G}},\mu )$ among almost contact structures.

• (ii) ${\mathcal {F}}$ is obtained from ${\mathcal {G}}$ by turbulizing along (nested) hypersurfaces diffeomorphic to $\mathbb {S}^1 \times \mathbb {S}^{2n-1}$.

The idea of the proof of Proposition 1.2 is as follows. First, we interpret the complement of (tubular neighborhoods of) two transverse curves as a foliated almost symplectic cobordism between almost contact foliations. Using the h-principle result from [9], this cobordism can be homotoped to a conformal symplectic foliation with foliated overtwisted contact boundaries in the same almost contact class as the tight contact structure. Then, we extend the conformal symplectic foliation over the neighborhoods of the transverse curves using the following result:

Corollary 1.3

Let $\xi _{ot}$ be any overtwisted contact structure on $\mathbb {S}^{2n-1}$, with $n >2$, in the same almost contact class as the standard tight contact structure. Then, there exists a foliated conformal symplectic cobordism

$$\begin{aligned} \emptyset \rightarrow \mathbb {S}^1 \times (\mathbb {S}^{2n-1},\xi _{ot}), \end{aligned}$$

smoothly diffeomorphic to $\mathbb {S}^1\times D^{2n}$; here, the boundary is intended as foliated by the contact spheres $(\mathbb {S}^{2n-1},\xi _{ot})$ given by the second factor.

Remark 1.4

Alternatively, Theorem 1.1 also follows from [9, Theorem C] and using Corollary 1.3 only once. As far as existence of conformal symplectic foliations is concerned, the advantage of our approach is the fact that Proposition 1.2 does not require the foliation to be holonomous (as defined in [9, Theorem C]). On the other hand, the drawback is twofold: we are forced to change (even if in a controlled way) the underlying smooth foliation, and we lose the fact that the Lee form can be arranged to be a holonomy form for the foliation.

It is well known that a ball does not admit any symplectic structure filling an overtwisted contact structure on the boundary sphere, and hence no conformal symplectic structure either (since the ball is contractible). On the other hand, the corollary says that in the foliated setting this is possible. Its proof is based on the fact that the foliated conformal symplectic cobordism relation between (unimodular) contact foliations is symmetric (c.f. Proposition 6.3). This is in stark contrast with the symplectic case, where the cobordism relation is far from symmetric.

The dimensional assumption in the above corollary is essential. Indeed, its proof uses [13, 18] to construct a symplectic cobordism from $(\mathbb {S}^{2n-1},\xi _{ot})$ to $(\mathbb {S}^{2n-1},\xi _{st})$. Such a cobordism does not exist for $n=2$, see [27, 35]. This is also the reason for the dimensional assumption in Theorem 1.1.

In dimension 5 we take a different approach and focus on simply connected manifolds. Here, using turbulization of conformal symplectic foliations together with Barden’s classification [4] we prove:

Theorem 1.5

Any simply connected 5-manifold admits a conformal symplectic foliation.

In many cases, the foliations in Theorem 1.5 contain closed leaves diffeomorphic to $\mathbb {S}^1 \times \mathbb {S}^3$, and cannot be symplectic. This being said, using a surgery construction on contact open books from [17], we prove the following:

Theorem 1.6

Every simply-connected almost contact 5-manifold M satisfying at least one of the following conditions admits a symplectic foliation:

• (i) ${{\text {rk}}}(H^{2}(M;\mathbb {Z}))\ge 2$,

• (ii) M is spin (i.e. with trivial second Stiefel–Whitney class $w_2$) and ${{\text {rk}}}(H^{2}(M;\mathbb {Z}))= 1$,

• (iii) M is spin and $H^2(M;\mathbb {Z}) = \mathbb {Z}_p \oplus \mathbb {Z}_p$ for p relatively prime to 3,

• (iv) M is spin and $H^2(M;\mathbb {Z}) = \mathbb {Z}_q \oplus \mathbb {Z}_q$ for q relatively prime to 2.

In fact, manifolds satisfying (i) and the (only) manifold with ${{\text {rk}}}(H^2(M;\mathbb {Z})) = 1$ satisfying (ii), admit infinitely many, pairwise distinct symplectic foliations. Here, by the rank ${{\text {rk}}}(G)$ of a finitely generated $\mathbb {Z}-$module G we mean the rank of its free part.

Going back to the general setting of ambient dimension $2n+1\ge 5$, we prove a conformal symplectic foliated analogue of an observation due to Milnor [29, Corollary 6] stating that if a manifold admits a smooth foliation then so does its connect sum with any exotic sphere. This gives an “explicit” (at least at the level of underlying smooth foliations) method of constructing conformal symplectic foliations on connected sums with exotic spheres:

Theorem 1.7

If M admits a conformal symplectic foliation, then so does the connect sum $M\#\Sigma$ for any exotic sphere $\Sigma$. More precisely, the underlying smooth foliation on $M\#\Sigma$ is obtained from the one on M by (iterated) turbulization along a transverse curve, and cutting and regluing an open disc in a leaf via an exotic diffeomorphism.

The proof uses the conformal symplectic turbulization procedure in order to introduce open leaves which are symplectomorphic to the symplectization of an overtwisted sphere, together with results from [12, Appendix by Courte] that allow to extend any smoothly exotic diffeomorphism of a disk to a compactly supported symplectomorphism of these open leaves.

Lastly, we make a few remarks concerning deformations of conformal symplectic foliations to contact structures, in the spirit of [19]. More precisely, we consider type I linear contact deformations (see Definition 6.11) which are a higher dimensional analogue of the linear deformations from [19].

We show that some of the compact leaves contained in the foliation from Corollary 1.3obstruct the existence of a type I deformation (c.f. Lemma 6.13). As a result, the foliations from Theorem 1.1 cannot be deformed. However, if we avoid the use of Corollary 1.3, these problematic leaves do not appear, and we obtain examples of (taut) conformal symplectic foliations which admit type I linear contact deformations on closed manifolds of dimensions $4m+3\ge 7$.

In order to give the precise statement, recall that given closed curves $\gamma _\pm \subset M_\pm$, the round-connected sum $M_- \#_{\mathbb {S}^1} M_+$ is obtained by removing tubular neighborhoods of $\gamma _\pm$ inside $M_\pm$ respectively, and gluing along the resulting boundary components.

Theorem 1.8

Let $M^{4m+3}$, $m \ge 1$, be an almost contact manifold, and $\gamma \subset M$ a closed curve. Then, there are infinitely many $N>0$ for which the iterated round-connected sum

$$\begin{aligned} M \#_{\mathbb {S}^1}^N \left( \bigsqcup _{i=1}^N T^2 \times \mathbb {S}^{4m+1}\right) ,\end{aligned}$$

admits a conformal symplectic foliation having a Type I deformation to contact structures. Here, the round-connected sum pairs a parallel copy $\gamma _i$ of $\gamma$ with a loop in the $T^2$-factor of the i-th copy of $T^2 \times \mathbb {S}^{4m+1}$.

Remark 1.9

In the above statement, if one already has a taut almost symplectic foliation on M, by choosing a transverse curve $\gamma$ intersecting every leaf, the resulting smooth foliation on the round connect sum manifold can be arranged to coincide with the one on M away from the round connect sum region.

Remark 1.10

The foliations obtained in Theorem 1.8 are taut. However, tautness is not a necessary condition for the existence of Type I deformations. For instance, gluing two mapping tori of Liouville domains (after turbulizing at their boundary), yields a conformal symplectic foliation on a closed manifold admitting a Type I deformation [45, Theorem 2.6.31].

In fact, we show (c.f. Proposition 6.14) that on the complement of any three curves, two of which are parallel copies of each other, an almost contact structure is homotopic to a non-taut exact conformal symplectic foliation admitting a linear deformation to contact structures.

1.1 Outline of the paper

In Sect. 2 we give some preliminary statements and considerations. More precisely, in Sect. 2.1 we recall the results by Smale [42] and Barden [4] which classify closed simply-connected $5-$manifolds. In Sect. 2.2 we give some normal forms for conformal symplectic manifolds near their boundary, which naturally generalize to the foliated setting. Then, in Sect. 2.3 we describe the symplectic turbulization and resulting gluing statement for turbulized foliations. In Sect. 2.4 we describe how the results from [18] give homotopies relative to the boundary in the conformal symplectic setting, and recall one of the main results of [9].

In Sect. 3 we first describe a version of Eliashberg’s capping construction for 3-dimensional open books, that allows us to pass from a 5-dimensional contact open book to an open book with cosymplectic type boundary. We then use this to give a proof of Theorem 1.6, i.e. we give explicit symplectic foliations on some simply-connected 5-manifolds.

In Sect. 4 we give some general constructions of conformal symplectic foliations. More precisely, in Sect. 4.1 we describe the main procedures we use in order to construct conformal symplectic foliations, namely conformal symplectic turbulization and the resulting gluing statement. In Sect. 4.2 we describe a conformal symplectic foliation on $\mathbb {S}^1\times [0,1]\times M$ which is tangent to the two boundary components and restricts on it to the conformal symplectization of any two contact structures on M which are homotopic as almost contact structures; this will be used in the later sections. In Sect. 4.3 we describe how to construct conformal symplectic foliations on round connect sums of certain manifolds, which will be useful in the proof of Theorem 1.6. Lastly, in Sect. 4.4 we prove Theorem 1.7, stating that, on the connected sum of a conformal-symplectically foliated manifold with an exotic sphere, one can construct (rather explicitly in terms of the original foliation) a conformal symplectic foliation.

In Sect. 5 we prove Theorem 1.5 on the existence of conformal symplectic foliations on closed simply-connected almost contact 5-manifolds explicitly given by gluing turbulized symplectic mapping tori.

In Sect. 6 we deal with the general existence statement for conformal symplectic foliations in dimensions $\ge 7$. More precisely, in Sect. 6.1 we describe some useful conformal symplectic foliations on manifolds with non-empty boundary, which will be used in the sections that follow. In Sect. 6.2 we describe the conformal symplectic Reeb components used in the proof of the main result of Sect. 6. In Sect. 6.3 we prove Theorem 1.1, on the existence of a conformal symplectic foliation in each almost contact class in dimension at least 7, using Proposition 1.2. The latter, stating the existence, in dimensions $\ge 7$, of a conformal symplectic leafwise structure on any taut almost symplectic foliation after adding some (homotopically trivial) Reeb components, is then proved in Sect. 6.4. Lastly, in Sect. 6.5 we collect some observations regarding contact deformations of the conformal symplectic foliations described in the previous sections. More precisely, we start the subsection by recalling the definition of type I linear deformation, as well as a general existence theorem, and prove Lemma 6.13 stating that a model we introduce in the proof of Theorem 1.1 obstructs the existence of type I deformations. In Sect. 6.5.1 we describe how the foliations from Theorem 1.1 on the complement of some codimension 0 submanifolds give examples of non-taut conformal symplectic foliation on manifolds with boundary which are deformable to contact structures. We also describe how to extend this deformation to the closed manifold as a deformation to hyperplane fields which are contact structures away from three hypersurfaces. Lastly, in Sect. 6.5.2 we prove Theorem 1.8, which gives examples of linearly contact-deformable conformal symplectic foliations on closed manifolds of dimension $4k+3\ge 7$, as well as give details concerning Remark 1.10, dealing with linearly deformable non-taut conformal symplectic foliations on manifolds with non-empty boundary.

2 Preliminaries

2.1 Simply connected 5-manifolds

The classification of simply connected $5-$manifolds was started by Smale in [42], who dealt with the case of vanishing second Stiefel–Whitney class, and was completed by Barden in [4], who covered the general case. Recall that, by the universal coefficient theorem, for a simply connected manifold M we can regard the second Stiefel–Whitney class as a map $w_2(M):H_2(M)\rightarrow \mathbb {Z}_2$.

Theorem 2.1

([4]) Two simply connected $5-$manifolds $M_1$ and $M_2$ are diffeomorphic if and only if there exists an isomorphism of groups $\phi :H_2(M_1) \rightarrow H_2(M_2)$ preserving the linking product and such that $w_2(M_1) = w_2(M_2)\circ \phi$.

As a consequence, one can prove that any closed (oriented) simply connected 5-manifold M can be decomposed uniquely into prime manifolds

$$\begin{aligned} M = X_j \# M_{k_1} \# \cdots \# M_{k_s}\end{aligned}$$

with $-1 \le j \le \infty$, $s \ge 0$, $k_1$ and $k_i$ dividing $k_{i+1}$ or $k_{i+1} = \infty$. Moreover, the decomposition contains at most one summand of type $X_j$ and possibly no summand of type $M_k$. The prime manifolds satisfy:

• (i) $X_{-1}= SU_3/SO_3$ is the Wu-manifold, the homogeneous space obtained from the standard inclusion of $SO_3$ into $SU_3$, and it has $H_2(X_{-1})=\mathbb {Z}_2$, $w_2(X_{-1})\ne 0$ and $W_3(X_{-1})\ne 0$, where $W_3$ is the third integral Stiefel-Whitney class.

• (ii) $X_0=\mathbb {S}^5$, having $H_2(\mathbb {S}^5)=0$, $w_2(\mathbb {S}^5)=0$ and $W_3(\mathbb {S}^5)= 0$. Furthermore, $X_0$ is only needed in the decomposition if $M\simeq \mathbb {S}^5$.

• (iii) $X_\infty =\mathbb {S}^2\overset{\sim }{\times }_\gamma \mathbb {S}^3$ the total space of the non-trivial $\mathbb {S}^3-$bundle over $\mathbb {S}^2$, which has $H_2(X_\infty )=\mathbb {Z}$, $w_2(X_\infty )\ne 0$ and $W_3(X_\infty )= 0$.

• (iv) $M_\infty = \mathbb {S}^2 \times \mathbb {S}^3$ which has $H_2(M_\infty )=\mathbb {Z}$, $w_2(M_\infty )=0$ and $W_3(M_\infty )= 0$.

• (v) for $1<k<\infty$, $M_k$ has $H_2(M_k)=\mathbb {Z}_k\oplus \mathbb {Z}_k$, $w_2(M_k)=0$ and $W_3(M_k)= 0$.

• (vi) for $1\le j <\infty$, $X_j$ has $H_2(X_j)=\mathbb {Z}_{2^j}\oplus \mathbb {Z}_{2^j}$, $w_2(X_j)=0$ and $W_3(X_j)\ne 0$.

As (conformal) symplectic foliations are in particular almost contact structures, the prime manifolds which are relevant to us have vanishing $W_3$ according to the following.

Lemma 2.2

([21, Lemma 7]) A simply connected $5-$manifold admits an almost contact structure if and only if its third integral Stiefel–Whitney class $W_3$ vanishes.

As $w_2$ is additive under connected sums and $W_3$ vanishes if and only if $w_2$ is in the image of the reduction mod 2 morphism $H_2(\cdot ,\mathbb {Z})\rightarrow H_2(\cdot ,\mathbb {Z}/2\mathbb {Z})$, according to [4], any simply connected $5-$manifold which admits a (conformal) symplectic foliation is hence of the form

$$\begin{aligned} M=\mathbb {S}^5, \text { or } M = M_{k_1}\# \cdots \# M_{k_s}, \text { or } M = X_\infty \# M_{k_1} \# \cdots \# M_{k_s}, \end{aligned}$$

(1)

with $s \ge 0$, $k_i$ dividing $k_{i+1}$ or $k_{i+1} = \infty$. Moreover, a complete set of invariants for manifolds in this class are the homology group $H_2$ and the fact whether $w_2$ is trivial or not.

2.2 Normal form around the boundary

In the next two sections we discuss the behaviour of symplectic foliations on manifolds with boundary near the boundary, see also [11, 39, 45, 49]. Let $(M^{2n},\eta ,\omega )$ be a conformal symplectic manifold with boundary $\partial M$, and denote by $(\eta _\partial ,\omega _\partial )$ the restriction of the forms to the boundary. Notice that $\omega _\partial$ is ${{\text {d}}}_{\eta _\partial }$-closed, and that it has one-dimensional kernel. Moreover, one can always find an admissible form for $\omega _\partial$, by which we mean a $\beta \in \Omega ^1(\partial M)$ such that

$$\begin{aligned} \omega _\partial ^{n-1} \wedge \beta > 0. \end{aligned}$$

(2)

For instance, for any choice of vector field X on M transverse to the boundary, one can take $\beta := \iota _X\omega |_{\partial M}$. Note that the converse is also true: a choice of admissible form $\beta$ determines naturally a vector field, namely as the $\omega$-dual of $\beta$, that is transverse to the boundary.

Recall that conformal equivalence of conformal symplectic structures is given by $(\eta ,\omega ) \sim (\eta + {{\text {d}}}f, e^f\omega )$ for any function f.

Theorem 2.3

Let $(M,\eta ,\omega )$ be a conformal symplectic manifold, with $\partial M \ne \emptyset$. Let also $\beta \in \Omega ^1(\partial M)$ be any admissible form for $(\eta _\partial ,\omega _\partial )$. Then, a neighborhood of the boundary is conformally equivalent to:

$$\begin{aligned} \Big ( (-\varepsilon ,0] \times \partial M,\, \eta = \eta _\partial ,\, \omega = \omega _\partial + {{\text {d}}}_{\eta _{\partial }}(t \beta )\Big ). \end{aligned}$$

The following elementary observation will be used in the proof.

Lemma 2.4

Let $\eta$ be a closed nowhere vanishing 1-form on M, such that $\ker \eta \pitchfork \partial M$. If $\alpha \in \Omega ^k(M)$ satisfies

$$\begin{aligned} {{\text {d}}}_\eta \alpha = 0, \quad \alpha |_{\partial M} = 0,\end{aligned}$$

then, locally around the boundary, $\alpha$ is ${{\text {d}}}_\eta$-exact, with primitive vanishing at points of $\partial M$.

Proof of Lemma 2.4

Fix a collar neighborhood $(-\varepsilon ,0]\times \partial M$ of the boundary. In these coordinates we can write

$$\begin{aligned} \alpha = \alpha _t + {{\text {d}}}t \wedge \beta _t,\end{aligned}$$

for $\alpha _t \in \Omega ^k(\partial M)$, $\beta _t \in \Omega ^{k-1}(\partial M)$, and $t \in (-\varepsilon ,0]$.

We may assume $\partial _t \in \ker \eta$ which implies (also using that $\eta$ is closed) that $\eta = \eta _\partial$ does not have a ${{\text {d}}}t$ component. Then, it follows from ${{\text {d}}}_\eta \alpha =0$ and the above equation that $\alpha _0 = 0$, $\overline{{{\text {d}}}}_\eta \alpha _t = 0$, and $\dot{\alpha }_t = \overline{{{\text {d}}}}_\eta \beta _t$, where $\overline{{{\text {d}}}}$ denotes the differential on the $\partial M$ factor of $(-\epsilon ,0]\times \partial M$, and $\overline{{{\text {d}}}}_\eta$ its twisted version with respect to $\eta =\eta _\partial$. Therefore we have

$$\begin{aligned} \alpha _t = \alpha _t -\alpha _0 = \int _0^t \dot{\alpha }_s {{\text {d}}}s = \overline{{{\text {d}}}}_\eta \left( \int _0^t \beta _s {{\text {d}}}s\right) , \end{aligned}$$

which in turn implies

$$\begin{aligned} \alpha = \overline{{{\text {d}}}}_\eta \left( \int _0^t \beta _s {{\text {d}}}s\right) + {{\text {d}}}t\wedge \beta _t = {{\text {d}}}_\eta \left( \int _0^t \beta _s {{\text {d}}}s\right) . \end{aligned}$$

Note in particular that the primitive of $\alpha$ in the above formula vanishes at points of $\partial M$. $\square$

Proof of Theorem 2.3

Fix a collar neighborhood $(-\varepsilon ,0] \times \partial M$ of the boundary. With respect to this splitting, we define

$$\begin{aligned} \widetilde{\eta }:= \eta _\partial ,\quad \widetilde{\omega }:= \omega _\partial + {{\text {d}}}_{\widetilde{\eta }}(t\beta ),\end{aligned}$$

which is easily checked to be a conformal symplectic structure for $\varepsilon >0$ small enough. Since $\eta - \widetilde{\eta }$ is a ${{\text {d}}}$-closed form whose restriction to $\partial M$ vanishes, it follows from a leafwise version of Poincaré lemma (or Lemma 2.4 above directly) that

$$\begin{aligned} \eta = \widetilde{\eta } + {{\text {d}}}f, \end{aligned}$$

for some function f satisfying $f|_{\partial M} = 0$. Note that $(\eta ,\omega )$ is conformally equivalent to $(\widetilde{\eta },e^{-f}\omega )$. Similarly to before, as $e^{-f}\omega - \widetilde{\omega }$ is ${{\text {d}}}_{\widetilde{\eta }}$-closed and its restriction to $\partial M$ vanishes, Lemma 2.4 implies that

$$\begin{aligned} \widetilde{\omega } = e^{-f}\omega + {{\text {d}}}_{\eta _{\partial }} \gamma ,\end{aligned}$$

for some $\gamma \in \Omega ^1(M)$ vanishing at points of $\partial M$. As such, the 1-parameter family of conformal symplectic structures

$$\begin{aligned} \eta _s:= \eta _\partial ,\quad \omega _s:= s e^{-f} \omega + (1-s) \widetilde{\omega } = e^{-f}\omega + (1-s) {{\text {d}}}_{\eta _\partial } \gamma ,\end{aligned}$$

satisfies the conditions of [3, Theorem 4] (or [8, Corollary 3.2]), thus obtaining an isotopy $\phi _s$ and a family of functions $g_s$ such that $\phi _s^*\omega _s = e^{g_s} \omega _0$, and hence $\phi _s^*\eta _s = \eta _0 + d g_s$. We point out that, inspecting the proof of [8, Corollary 3.2], one can readily see that the isotopy $\phi _s$ can be taken to be the flow of the (unique by non-degeneracy of $\omega _s$) vector field $X_s$ solving the equation $\iota _{X_s}\omega _s = \gamma$ (notice $\gamma$ is a ${{\text {d}}}_{\eta _\partial }$-primitive of $-\dot{\omega }_s$ by construction of $\omega _s$). In particular, as $\gamma$ vanishes at points of $\partial M$, so does $X_s$; hence the flow $\phi _s$ of $X_s$ is the identity for all s on $\partial M$ and is then well defined up to time 1 on a sufficiently small neighborhood of $\partial M$ as needed. (Notice that $g_s$ is then automatically also well defined near $\partial M$ for all $s\in [0,1]$.) Then, $\phi _1$ is the desired conformal symplectomorphism. $\square$

Just as in the symplectic setting we distinguish special types of boundaries.

Definition 2.5

A conformal symplectic manifold $(M,\eta ,\omega )$ is said to have convex (resp. concave) boundary if there exists a vector field X defined on ${\mathcal {O}p}(\partial M)$, transverse to $\partial M$, pointing outwards (resp. pointing inwards) and which is $\eta -$Liouville for $\omega$, i.e. satisfies

$$\begin{aligned} \omega = {{\text {d}}}_\eta \iota _X \omega . \end{aligned}$$

Of course, when $\eta = 0$ this agrees with the usual definition for symplectic manifolds. An immediate corollary of the normal form of Theorem 2.3 is that being of contact type can be detected on the boundary (as in the honest symplectic case):

Lemma 2.6

Let $(M,\eta ,\omega )$ be a conformal symplectic manifold with boundary. Then the following are equivalent:

• (i) $(\eta ,\omega )$ is convex/concave at the boundary;

• (ii) There exists a vector field X on M transverse to $\partial M$, pointing outwards/inwards such that for all $p \in \partial M$

  $$\begin{aligned} \omega _p = ({{\text {d}}}_\eta \iota _X\omega )_p,\end{aligned}$$

• (iii) There exists an admissible form $\alpha \in \Omega ^1(\partial M)$ for $\omega _\partial := \omega |_{\partial M}$ satisfying

  $$\begin{aligned} \omega _\partial = \pm {{\text {d}}}_{\eta _\partial }\alpha .\end{aligned}$$

For boundaries of contact type it is often useful to slightly rewrite the normal form from Theorem 2.3. We state it here as a corollary for later reference.

Corollary 2.7

Let $(M,\eta ,\omega )$ be a conformal symplectic manifold with contact type boundary. Then, locally around the boundary, it is conformally equivalent to

$$\begin{aligned} \Big ( (-\varepsilon ,0] \times \partial M,\eta = \eta _\partial , \omega = {{\text {d}}}_{\eta _\partial }( e^{\pm t} \alpha )\Big ), \end{aligned}$$

where $\alpha \in \Omega ^1(M)$ is an admissible form such that $\omega _\partial = {{\text {d}}}_{\eta _\partial } \alpha$. The sign is a plus (resp. minus) in the convex (resp. concave) boundary case.

The entire discussion above carries over to the foliated case. Consider a conformal symplectic foliation $(M,{\mathcal {F}},\eta ,\omega )$ transverse to the boundary, and denote by $({\mathcal {F}}_\partial ,\eta _\partial ,\omega _\partial )$ its restriction to $\partial M$. As in Eq. (2), we define a (${\mathcal {F}}_\partial -$leafwise) admissible form for $\omega _\partial$ to be $\alpha \in \Omega ^1({\mathcal {F}}_\partial )$ such that

$$\begin{aligned} \alpha \wedge \omega _\partial ^{n-1} > 0. \end{aligned}$$

(3)

The key point in the proof of Theorem 2.3 is the Moser trick which produces the desired diffeomorphism as the flow of a vector field. The equation defining this vector field can be solved leafwise, that is, in the foliated case the vector field can be chosen tangent to the leaves. Thus, the Moser trick can be carried out leafwise, giving the following normal form.

Theorem 2.8

Let $({\mathcal {F}},\eta ,\omega )$ be a conformal symplectic foliation on M transverse to the boundary. Denote by $({\mathcal {F}}_\partial ,\eta _\partial ,\omega _\partial )$ the induced conformal symplectic foliation on the boundary, and by $\alpha$ a ${\mathcal {F}}_\partial -$leafwise admissible form for $\omega _\partial$. Then there exists a collar neighborhood $(-\varepsilon ,0]\times \partial M$ on which

$$\begin{aligned} {\mathcal {F}}= (-\varepsilon ,0] \times {\mathcal {F}}_{\partial },\quad \eta = \eta _\partial ,\quad \omega = \omega _\partial + {{\text {d}}}_{\eta _\partial }(t \alpha ).\end{aligned}$$

Here, the notation $(-\varepsilon ,0] \times {\mathcal {F}}_\partial$ refers to taking the product foliation:

$$\begin{aligned} \bigcup _{\mathcal {L}_\partial \in {\mathcal {F}}_\partial } (-\varepsilon ,0] \times \mathcal {L}_\partial ,\end{aligned}$$

where the union is over the leaves $\mathcal {L}_\partial$ of ${\mathcal {F}}_\partial$. We also point out that, when $\eta _\partial = 0$ in the above statement, we obtain the usual normal form for symplectic foliations.

Remark 2.9

According to our usual conventions, this means that if ${\mathcal {F}}_\partial = \ker \gamma _\partial$ and $\mu _\partial \in \Omega ^{n-1}({\mathcal {F}}_\partial )$ is a leafwise positive volume form, then $\gamma _\partial \wedge {{\text {d}}}t \wedge \mu _\partial$ is a positive volume form on the collar neighborhood from the theorem. In particular this implies that $\gamma _\partial \wedge \mu _\partial$ is a negative volume form on $\partial M$.

2.3 Symplectic gluing and turbulization

Constructing foliations often involves gluing foliated manifolds. As such it is useful to change foliations transverse to the boundary into ones tangent to the boundary.

Let ${\mathcal {F}}$ be a foliation on M transverse to the boundary. Then there exists a collar neighborhood $(-\varepsilon ,0] \times \partial M$ on which ${\mathcal {F}}$ is a product foliation, i.e. ${\mathcal {F}}= \ker \gamma _\partial$ with $\gamma _\partial \in \Omega ^1(\partial M)$. To turn ${\mathcal {F}}$ into a foliation tangent to the boundary we need to assume that $\gamma _\partial$ is closed. Choose a smooth function $f:(-\varepsilon ,0] \rightarrow [0,1]$ which is zero near $-\varepsilon$, $f(0) = 1$ and such that it can be smoothly extended as the constant function on $[0,\varepsilon )$. Then, in the collar we can also define

$$\begin{aligned} \widetilde{\gamma } = (1-f(t)) \gamma _\partial + f(t) {{\text {d}}}t.\end{aligned}$$

The foliation $\widetilde{{\mathcal {F}}}:= \ker \widetilde{\gamma }$ is said to be obtained from ${\mathcal {F}}$ by turbulization. Note that $\widetilde{{\mathcal {F}}}$ is tangent to the boundary and agrees with ${\mathcal {F}}$ away from the boundary. The condition that f can be smoothly extended ensures that the gluing of two turbulized foliations is again smooth.

Observe that $\widetilde{\gamma } = {{\text {d}}}t$ at points in the boundary $\partial M$. This means that the boundary leaf of $\widetilde{{\mathcal {F}}}$ is cooriented in the same way as the boundary of M. The turbulization construction can also be performed with $-f$ instead of f in the definition of $\widetilde{\gamma }$, in which case we still say that the resulting foliation is obtained from ${\mathcal {F}}$ by turbulization. In this case, the boundary leaf of the resulting foliation has the opposite coorientation.

We point out that the construction of smooth foliations just described is well known in smooth foliation theory. Turbulization in the neighborhood of a transverse curve dates back at least to the work of Reeb [41], and is now explained in detail in many standard textbooks in foliation theory, e.g. [10, Example 3.3.11] and [43, Section 1.2.5]. What we described above is a slight generalisation, which is found for instance in [29, Lemma 1]. We will now describe the symplectic analogue of this smooth construction.

For the turbulization construction in the realm of symplectic foliations, it is in fact useful to require some additional control over the leafwise symplectic structure near the boundary in order to have some clear conditions under which gluing at the boundary of turbulized symplectic foliations can be performed. To make this precise, consider a symplectic foliation $({\mathcal {F}},\omega )$ on M tangent to the boundary. Choose a collar neighborhood of the boundary $k: (-\varepsilon ,0] \times \partial M \rightarrow M$, and use it to define

$$\begin{aligned} M_\infty := M \cup _{\partial M} [0,\infty ) \times \partial M.\end{aligned}$$

On $[0,\infty ) \times \partial M$ we define an (a priori only continuous) extension of $({\mathcal {F}},\omega )$ by:

$$\begin{aligned} {\mathcal {F}}' :=\bigcup _{t \in [0,\infty )} \{t \} \times \partial M,\quad \omega ' :=\omega _\partial .\end{aligned}$$

If this extension is smooth we say that the collar neighborhood is adapted.

Definition 2.10

A symplectic foliation is said to be tame at the boundary if it admits an adapted collar neighborhood as above.

It follows immediately that such manifolds can be glued by "matching admissible collars". To be precise we have:

Proposition 2.11

Let $(M_i,{\mathcal {F}}_i,\omega _i)$, $i=1,2$ be symplectic foliations tame at the boundary. If there exists an orientation reversing diffeomorphism $\phi :U_1\rightarrow U_2$ from a union $U_1$ of connected components of $\partial M_1$ to a union $U_2$ of connected components of $\partial M_2$ satisfying $\phi ^*\omega _{2,\partial } = \omega _{1,\partial }$, then $M_1\cup _\phi M_2$ admits a symplectic foliation.

Note that this means that the orientation on the boundary is not necessarily the one induced by the symplectic form.

To state the symplectic version of the turbulization construction we need one more definition. Recall that a cosymplectic structure on a manifold $M^{2n+1}$ is a pair $(\gamma ,\mu )$ where $\gamma \in \Omega ^1(M)$ and $\mu \in \Omega ^2(M)$ are closed forms satisfying

$$\begin{aligned} \gamma \wedge \mu ^n > 0. \end{aligned}$$

Similarly, $({\mathcal {F}},\omega )$ is said to have boundary of cosymplectic type if the restriction at the boundary $(\mathcal {F}\vert _{\partial },\omega _\partial )$ admits a leafwise admissible form (see Eq. (3)) which is additionally leafwise closed; we will also call the latter an admissible form of cosymplectic type. If additionally ${\mathcal {F}}_\partial$ is unimodular (i.e. defined by a closed 1-form) then we say $({\mathcal {F}},\omega )$ has boundary of unimodular cosymplectic type.

Theorem 2.12

([45, Section 1.7], [11]) Let $(M,{\mathcal {F}},\omega )$ be a symplectic foliation with boundary of unimodular cosymplectic type. Then M admits a symplectic foliation $(\widetilde{{\mathcal {F}}},\widetilde{\omega })$ tame at the boundary. Moreover:

• 1. $(\widetilde{{\mathcal {F}}},\widetilde{\omega })$ and $({\mathcal {F}},\omega )$ agree away from the boundary;

• 2. Let $\gamma _\partial$ be any closed form defining ${\mathcal {F}}_\partial$, and $\alpha$ any admissible form of cosymplectic type for $\omega _\partial$. Then we can arrange that

  $$\begin{aligned} \widetilde{\omega }_\partial = \omega _\partial + \gamma _\partial \wedge \alpha ,\end{aligned}$$

  on the boundary leaf $\partial M$.

The symplectic foliation $(\widetilde{{\mathcal {F}}},\widetilde{\omega })$ is said to be obtained from $({\mathcal {F}},\omega )$ by symplectic turbulization.

One can apply this theorem to suitable decompositions of a manifold (as explicitly stated in Proposition 2.14 below) to obtain symplectic foliations. In order to describe this, first recall that an abstract open book is a pair $(\Sigma ,\phi )$ consisting of a manifold with boundary $\Sigma$, and a diffeomorphism $\phi : \Sigma \xrightarrow {\sim }\Sigma$ which is the identity near the boundary. Out of this data one can explicitly construct the manifold

$$\begin{aligned} M_{(\Sigma ,\phi )}:= (B \times \mathbb {D}^2) \cup _{B\times \mathbb {S}^1} \Sigma _\phi , \end{aligned}$$

where $B:= \partial \Sigma$ and $\Sigma _\phi$ denotes the mapping torus associated to $\phi$, i.e. $\Sigma \times {\mathbb {R}}/\sim$ where $(x,t) \sim (\phi (x), t-1)$. We will refer to $B\times \mathbb {D}^2$, and $\Sigma _\phi$ respectively as the inside and outside components of the open book.

Definition 2.13

An abstract open book $(\Sigma ,\phi )$ is said to be of:

• 1. contact type if $(\Sigma ,{{\text {d}}}\lambda )$ is a Liouville domain, and $\phi$ is an exact symplectomorphism;

• 2. cosymplectic type if $(\Sigma ,\omega )$ is a symplectic manifold with boundary of cosymplectic type, and $\phi$ is a symplectomorphism.

It is well-known, due to work of Thurston–Winkelnkemper [46] in dimension 3 and of Giroux–Mohsen [22] in higher dimensions, that if $(\Sigma ,{{\text {d}}}\lambda ,\phi )$ is an open book of contact type, then $M_{(\Sigma ,\phi )}$ admits a contact structure supported by this open book. The analogue for symplectic foliations using open books of cosymplectic type follows directly from Theorem 2.12:

Proposition 2.14

Let $(\Sigma ,\omega )$ be a $2n-$dimensional symplectic manifold with cosymplectic boundary B and $\phi :\Sigma \rightarrow \Sigma$ a symplectomorphism which is the identity near the boundary. Then, the open book $M_{(\Sigma ,\phi )}$ admits a symplectic foliation.

Proof

The proposition follows from applying Theorem 2.12 to both components of the open book. On the outside component $\Sigma _\phi$ we take the symplectic foliation $({\mathcal {F}}:= \ker {{\text {d}}}\theta ,\omega )$ where ${{\text {d}}}\theta$ denotes the pullback of the angular form on $\mathbb {S}^1$ under the projection $\pi :\Sigma _\phi \rightarrow \mathbb {S}^1$. After turbulization the boundary leaf equals

$$\begin{aligned} (B \times \mathbb {S}^1, \omega _\partial + {{\text {d}}}\theta \wedge \alpha ),\end{aligned}$$

where $\alpha \in \Omega ^1(B)$ is any admissible form of cosymplectic type for $\omega _\partial$.

Similarly, on the inside component $B\times \mathbb {D}^2$ we take the symplectic foliation $({\mathcal {F}}:= \ker \alpha , \omega := \omega _\partial + 2r{{\text {d}}}r \wedge {{\text {d}}}\theta )$, where $(r,\theta ) \in \mathbb {D}^2$ denote polar coordinates. Again one can apply Theorem 2.12, and the resulting boundary leaf equals

$$\begin{aligned} (B \times \mathbb {S}^1, \omega _\partial + \alpha \wedge {{\text {d}}}\theta ).\end{aligned}$$

Then $\phi :B \times \mathbb {S}^1 \rightarrow B \times \mathbb {S}^1$ defined by $(x,\theta ) \mapsto (x,-\theta )$ is an orientation reversing diffeomorphism between the boundaries preserving the symplectic forms. Thus both pieces can be glued using Proposition 2.11. $\square$

The hypotheses of the previous proposition are quite special. In most cases the (symplectic) page of an open book has boundary of contact type. Nevertheless, sometimes it is possible to perturb the symplectic form on the page to one which has cosymplectic boundary (c.f. [39, Lemma 6.4.6]):

Lemma 2.15

Let $(\Sigma ,\omega )$ be a symplectic manifold, and $(\gamma ,\mu )$ a cosymplectic structure on $\partial \Sigma$ such that:

• 1. $\gamma \wedge \omega _\partial = 0$;

• 2. $[\mu ]$ is in the image of the restriction $H^2(\Sigma ) \rightarrow H^2(\partial \Sigma )$.

Then there exists a symplectic form $\widetilde{\omega }$ on $\Sigma$ with boundary of cosymplectic type.

Proof

Using Hypothesis 2, one can find a closed extension $\widetilde{\mu } \in \Omega ^2(\Sigma )$ of $\mu$. Then, for $\varepsilon > 0$ sufficiently small,

$$\begin{aligned} \widetilde{\omega }:= \omega + \varepsilon \widetilde{\mu },\end{aligned}$$

defines a symplectic form on $\Sigma$. Lastly, by Hypothesis 1, $\gamma$ is a closed admissible form for $\widetilde{\omega }_\partial = \omega _\partial + \varepsilon \mu$, i.e. $\widetilde{\omega }$ has boundary of cosymplectic type. $\square$

2.4 Conformal symplectic cobordisms

To construct (conformal) symplectic cobordisms we will often use the h-principle results from [9, 18]. The formal data underlying such a cobordism consists of the following.

Definition 2.16

An almost symplectic cobordism $(W,\omega ):(M_-,\alpha _-,\omega _-) \rightarrow (M_+,\alpha _+,\omega _+)$ between (positive) almost contact manifolds $(M_\pm ,\alpha _\pm ,\omega _\pm )$ consists of:

• (i) a smooth oriented cobordism W from $M_-$ to $M_+$;

• (ii) an almost symplectic structure $\omega$ such that $\omega |_{\partial _\pm W} = \omega _\pm$.

By a homotopy of almost symplectic cobordisms, we mean a homotopy $\omega _{s}$, with $s\in [0,1]$ so that $(W,\omega _s)$ is an almost symplectic cobordism. This induces a homotopy of almost contact forms on the boundary.

In the case where the boundary is honest contact, i.e. $\omega _\pm ={{\text {d}}}\alpha _\pm$ in the definition above, we will also consider homotopies relative to the boundary.

Definition 2.17

A homotopy of almost symplectic cobordisms $(W,\omega _s)$ between positive almost contact manifolds $(M_\pm ,\alpha _\pm ,\omega _\pm )$ is said to be:

• (i) relative to the boundary if $\omega _s|_{\partial _\pm W} = {{\text {d}}}\alpha _\pm$.

• (ii) weakly relative to the boundary if $\omega _s|_{\partial _\pm W} = e^{f_{\pm ,s}} {{\text {d}}}\alpha _\pm$, for functions $f_{\pm ,s}\in C^\infty (M_\pm )$.

Thus, a homotopy relative to the boundary preserves the contact form while a weakly relative homotopy only preserves the contact structure and hence the conformal class of the symplectic structure at the boundary. (Note that the notion of “homotopy of almost symplectic cobordisms relative to the boundary” in [18] is in fact what we call here “weakly relative to the boundary”.)

Theorem 2.18

([18, Theorem 1.1]) Let $(W^{2n},\Omega )$, $n \ge 2$, be an almost symplectic cobordism between (non-empty) contact manifolds $(M_\pm ,\xi _\pm )$. Assume that $\xi _-$ is overtwisted. If $n=2$, additionally assume that $\xi _+$ is overtwisted. Then, $\omega$ is homotopic, weakly relative to $\partial W$, to an (exact) symplectic structure.

Remark 2.19

For homotopies of symplectic structures, being relative or weakly relative to the boundary are rather different conditions. For example, the above theorem cannot be strengthened to produce homotopies relative to the boundary. Indeed, by Stokes theorem, this would allow us to obtain symplectic cobordisms with negative volume.

For conformal symplectic structures no such obstruction exist, in fact any homotopy weakly relative to the boundary is, up to conformal equivalence, relative to the boundary. To see this let $(W,\omega _s)$ be weakly relative to the boundary as in Definition 2.17, so that $\omega _s|_{\partial _\pm W} = e^{f_\pm ,s}{{\text {d}}}\alpha _\pm$. Choose $g_s \in C^\infty (W)$ satisfying $g_s|_{\partial \pm } = f_{\pm ,s}$, and observe that $e^{-g_s}\omega _s$ a conformally equivalent to $\omega _s$ and relative to the boundary.

Thanks to the possibility of rescaling conformally near the boundary as just explained in Remark 2.19, Theorem 2.18 then has the following consequence:

Theorem 2.20

([18]) Let $(W^{2n},\omega )$ be an almost symplectic cobordism

$$\begin{aligned} (M_-,\alpha _-) \sqcup (\mathbb {S}^{2n-1},\alpha _{ot}) \rightarrow (M_+,\alpha _+),\end{aligned}$$

where $\xi _{ot}=\ker \alpha _{ot}$ is any overtwisted contact structure in the almost contact class of $(\mathbb {S}^{2n-1},\xi _{st})$, and $M_+$ is non-empty. If $n=2$ additionally assume $\xi _+=\ker \alpha _+$ is overtwisted. Then, $\omega$ is homotopic, relative to the boundary, to a conformal symplectic structure on W.

Using foliated Morse theory, Bertelson and Meigniez extended Theorem 2.18 to the foliated setting. In order to state their result, we recall that a foliation is taut if three is a transverse loop intersecting every leaf.

Theorem 2.21

([9, Theorem B]) Let $(M,{\mathcal {F}},\omega )$ be a taut, almost symplectic foliation, and $\eta \in \Omega ^1(\mathcal {F})$ any leafwise closed form. Suppose that ${\mathcal {F}}$ is transverse to the boundary, and $\partial M$ splits as the disjoint union of two non-empty compact subsets $\partial _\pm M$, each intersecting every leaf of ${\mathcal {F}}$. Then there exists $\lambda \in \Omega ^1({\mathcal {F}})$ such that

• (i) ${{\text {d}}}_\eta \lambda$ is leafwise non-degenerate, and homotopic (among non-degenerate 2-forms) to $\omega$;

• (ii) $\lambda$ restricts to a positive (resp. negative) overtwisted contact structure on every leaf of ${\mathcal {F}}|_{\partial _+ M}$ (resp ${\mathcal {F}}|_{\partial _- M}$).

Recall that the foliated analogue of an overtwisted disk is an overtwisted basis. This is defined in [5] as a collection of foliated (almost) contact embeddings

$$\begin{aligned} h_i:[0,1] \times B^{2n}_{ot} \rightarrow (M,{\mathcal {F}}),\quad i=1,\dots ,N\end{aligned}$$

where $B^{2n}_{ot}$ denotes a 2n-dimensional overtwisted ball, such that each leaf of ${\mathcal {F}}$ intersects at least one of the embeddings. An (almost) contact foliation is then called overtwisted if it admits an overtwisted basis.

In the previous theorem, the homotopy ${{\text {d}}}_\eta \lambda$ to $\omega$ induces a homotopy of almost contact foliations on $\partial W$. An inspection of the proof in [9] shows that, under a natural assumption, these almost contact foliations may be assumed to admit an overtwisted basis. More precisely:

Corollary 2.22

Let $(M,{\mathcal {F}},\omega )$ satisfy the conditions of Theorem 2.21. Suppose $({\mathcal {F}},\omega )$ defines an overtwisted almost contact foliation $({\mathcal {F}}_\partial ,\alpha _\partial ,\omega _\partial )$ on $\partial W$. Then, the induced homotopy from $({\mathcal {F}}_\partial ,\alpha _\partial ,\omega _\partial )$ to $({\mathcal {F}}_\partial ,\lambda |_{\partial W})$ is through overtwisted almost contact foliations. More precisely, these overtwisted almost contact foliations, seen as codimension 2 almost contact foliations on $\partial M \times [0,1]_s$ (with $[0,1]_s$ being the parameter space of the homotopy) admit an overtwisted basis.

3 Symplectic foliations on some simply-connected 5-manifolds

The usefulness of Proposition 2.14 relies on producing symplectic abstract open books with boundary of cosymplectic type. In some situations (for instance as in [34], reinterpreted as in [39], or in Sect. 3.1 below) one can obtain them by modifying open books supporting contact structures via Lemma 2.15. We now describe another explicit procedure to obtain an open book of cosymplectic type from one of contact type.

Suppose $(M,\xi )$ is a 5-dimensional contact manifold (for the higher dimensional case see Remark remark 3.2 below). Recall that, according to [22], $(M,\xi )$ admits a supporting open book whose page is a Weinstein domain; in particular, one can represent any contact manifold via an abstract open book $(\Sigma ,\phi )$ of contact type (as in Definition 2.13).

Since the 3-dimensional boundary $\partial \Sigma$ of the 4-dimensional page $\Sigma$ is of contact type, it also admits a supporting open book decomposition $(\Sigma ',\phi ')$. Moreover, up to stabilizing the open book, one can assume that the binding is connected. Its outside component, i.e. the mapping torus $\Sigma _{\phi '}$, is a symplectic fibration over the circle with total space having non-empty boundary.

Hence we can change $\partial \Sigma$, by removing a tubular neighborhood $\partial \Sigma '\times \mathbb {D}^2$ of the binding $\partial \Sigma '$, and capping off the outside component as described by Eliashberg [17, Theorem 1.1]. More precisely this amounts to attaching a symplectic 2-handle $\mathbb {D}^2 \times \mathbb {D}^2$ to $\partial \Sigma$, producing a symplectic cobordism from the contact manifold $\partial \Sigma$ to a cosymplectic manifold. Topologically speaking, the discs $\{pt\} \times \mathbb {D}^2 \subset \mathbb {D}^2 \times \mathbb {D}^2$, have the effect of “capping the pages $\Sigma '$” of the open book decomposition $(\Sigma ',\phi ')$ of $\partial \Sigma$. This means that the resulting cosymplectic manifold admits a symplectic fibration $\mathbb {S}^1$, whose fibers are precisely $\Sigma '\cup _{\partial \Sigma '} \mathbb {D}^2$.

On the level of 5-manifolds this implies the following:

Proposition 3.1

([17, Theorem 1.1]) Let $(\Sigma ^4,{{\text {d}}}\lambda ,\phi )$ be an abstract open book of contact type. Then there exists an abstract open book of cosymplectic type $(\widetilde{\Sigma },\widetilde{\omega },\widetilde{\phi })$, where

$$\begin{aligned} \widetilde{\Sigma }:= \Sigma \cup _{\mathbb {S}^1 \times \mathbb {D}^2} \mathbb {D}^2 \times \mathbb {D}^2,\end{aligned}$$

is obtained from $\Sigma$ by attaching a symplectic 2-handle, and $\phi$ is extended as the identity.

In other words, one can associate to each contact 5-manifold $(M,\xi )$ (or more precisely, to each open book supporting $\xi$) a different 5-manifold $\widetilde{M}$ which admits a symplectic foliation.

Remark 3.2

Although we will not use it here, a similar construction is possible for ambient contact manifolds $(M,\xi )$ of dimension greater than 5. More precisely, the first thing to point out is that in these dimensions the binding of the open book of $\partial \Sigma$ is always connected if the page of the open book is Weinstein, which is satisfied with the construction in [22] for instance. Secondly, a cap C for the binding of $\partial \Sigma$ always exists by [30, Theorem Corollary 1.14]. Then, using such cap C, the analogous of [17, Theorem 1.1] in higher dimensions is given in [14, Section 4]. Lastly, an analogous of Proposition 3.1 holds with a symplectic handle of the form $\mathbb {D}^2\times C$.

In general, there is no clear way to identify $\widetilde{M}$ other than being the manifold obtained from M by surgery along a curve $\mathbb {S}^1 \subset M$. However, if M is simply connected one can identify $\widetilde{M}$ with $M\#\mathbb {S}^2\times \mathbb {S}^3$ using the classification recalled in Sect. 2.1, thus leading to the following:

Proposition 3.3

Let $(M,\xi )$ be a simply connected contact 5-manifold. Then, $M \# \, \mathbb {S}^2\times \mathbb {S}^3$ admits infinitely many, pairwise smoothly distinct, symplectic foliations.

Proof of Proposition 3.3

The proof follows from combining Proposition 2.14, Proposition 3.1, applied to some contact open book and infinitely many of its positive stabilizations, and Lemma 3.4 below.

In order to use Lemma 3.4, we argue that every simply connected contact $5-$manifold admits an open book decomposition whose page is simply connected. This follows from the following facts. First, recall that every simply connected $5-$dimensional manifold is generated, under connected sum, by those in the list in Eq. (1). Each of these have open books with simply connected pages, according to [48]. Lastly, taking connected sums of open books (see e.g. [47, Section 5.1.1]) preserves the simply connectedness of the page.

We now show how to produce infinitely many distinct symplectic foliations. Eliashberg’s argument [17] shows that $\partial \widetilde{\Sigma }$ is a symplectic fibration over $\mathbb {S}^1$ whose fiber is the closed surface obtained by capping the page of an open book decomposition of $\partial \Sigma$. That is, the fiber equals $\Sigma _{g_0}$, the genus-$g_0$ surface, for some $g_0 \in {\mathbb {N}}$.

Positive stabilizations yield open books supporting the same contact structure. Hence, by stabilizing the open book of $\partial \Sigma$, we see that $\partial \widetilde{\Sigma }$ admits a symplectic fibration over $\mathbb {S}^1$ whose fiber equals $\Sigma _g$ for any $g \ge g_0$. Different choices of $\widetilde{g}$ yield non-isomorphic boundaries of $\widetilde{\Sigma }$. Therefore, since the symplectic foliation constructed by Proposition 2.14 has a single compact leaf diffeomorphic to $\partial \widetilde{\Sigma } \times \mathbb {S}^1$, the resulting foliations will be non-isomorphic for different choices of $\widetilde{g}$. $\square$

Lemma 3.4

Suppose that $\Sigma$ is simply connected and 4-dimensional. Let $\widetilde{\Sigma }$ be obtained from $\Sigma$ by attaching a 2-handle along the binding of an open book decomposition of $\partial \Sigma$ w.r.t. the page framing. Furthermore, let $\phi$ be a diffeomorphism of $\Sigma$, and $\widetilde{\phi }$ its extension to $\widetilde{\Sigma }$ by the identity. Then

$$\begin{aligned} \pi _1(M_{(\widetilde{\Sigma },\widetilde{\phi })}) = 0, \quad H_2(M_{(\widetilde{\Sigma },\widetilde{\phi })}) = H_2(M_{(\Sigma ,\phi )}) \oplus \mathbb {Z}, \text { and } w_2(M_{(\widetilde{\Sigma },\widetilde{\phi })}) = w_2(M_{(\Sigma ,\phi )}). \end{aligned}$$

That is, by Smale-Barden’s classification,

$$\begin{aligned} M_{(\widetilde{\Sigma },\widetilde{\phi })} = M_{(\Sigma ,\phi )} \# \mathbb {S}^2 \times \mathbb {S}^3. \end{aligned}$$

Proof of Lemma 3.4

Observe that the boundary of $\widetilde{\Sigma }$ is obtained from that of $\Sigma$ by doing a surgery:

$$\begin{aligned} \partial \widetilde{\Sigma } = \left( \partial \Sigma \setminus \mathbb {S}^1 \times \mathbb {D}^2\right) \cup \mathbb {D}^2 \times \mathbb {S}^1.\end{aligned}$$

Using also that $\widetilde{\phi }$ is the identity on the 2-handle we obtain topologically:

$$\begin{aligned} M_{(\widetilde{\Sigma },\widetilde{\phi })}&= \partial \widetilde{\Sigma }\times \mathbb {D}^2 \cup \widetilde{\Sigma }_{\widetilde{\phi }} \\ {}&= \left( (\partial \Sigma \setminus \mathbb {S}^1\times \mathbb {D}^2)\cup \mathbb {D}^2\times \mathbb {S}^1\right) \times \mathbb {D}^2 \cup (\Sigma \cup \mathbb {D}^2\times \mathbb {D}^2)_{\widetilde{\phi }} \\ {}&= (\partial \Sigma \times \mathbb {D}^2 \cup \Sigma _\phi \setminus \mathbb {S}^1\times \mathbb {D}^2\times \mathbb {D}^2)\cup (\mathbb {D}^2\times \mathbb {S}^1\times \mathbb {D}^2 \cup \mathbb {D}^2\times \mathbb {D}^2 \times \mathbb {S}^1) \\ {}&= M_{\Sigma ,\phi }\setminus \mathbb {S}^1\times \mathbb {D}^2 \times \mathbb {D}^2 \cup \mathbb {D}^2\times (\mathbb {S}^1\times \mathbb {D}^2\cup \mathbb {D}^2\times \mathbb {S}^1) \\ {}&= M_{\Sigma ,\phi }\setminus \mathbb {S}^1\times \mathbb {D}^4 \cup \mathbb {D}^2\times \mathbb {S}^3, \end{aligned}$$

where each of the gluing maps is the identity. That is, $M_{(\widetilde{\Sigma },\widetilde{\phi })}$ is obtained from $M_{(\Sigma ,\phi )}$ by surgery along an $\mathbb {S}^1$. This proves the claims about the fundamental and second homology groups.

It’s then only left to prove the desired formula for the second Stiefel-Whitney class $w_2$. By the formula for $H_2$, we have that $w_2(M_{(\widetilde{\Sigma },\widetilde{\phi })})$ can be seen as a map $H_2(M_{(\widetilde{\Sigma },\widetilde{\phi })})=H_2(M_{(\Sigma ,\phi )})\oplus \mathbb {Z}\rightarrow \mathbb {Z}_2$. Moreover, by naturality of the Stiefel-Whitney classes, $w_2(M_{(\widetilde{\Sigma },\widetilde{\phi })})$ restricts on the factor $H_2(M_{(\Sigma ,\phi )})$ simply to $w_2(M_{(\Sigma ,\phi )})$. In particular, if the latter is non-trivial, so must be the former. It is hence enough to prove that $w_2(M_{(\widetilde{\Sigma },\widetilde{\phi })})$ is trivial on the additional $\mathbb {Z}$ factor.

According to Mayer-Vietoris long exact sequence, a generator of this additional factor is simply given by the union of a page of the open book at the $3-$dimensional boundary of $\Sigma$ with the core of the $2-$handle $\mathbb {D}^2\times \mathbb {D}^2$ which is attached as in the statement. What’s more, this union, which is an embedded submanifold with corners, can be smoothened to an embedded oriented surface F with trivializable normal bundle $\nu F$. (In fact, such a smoothing F is just a parallel copy of a fiber of the $\mathbb {S}^1-$fibration at the new $3-$dimensional boundary after handle attachment as described in [17, Theorem 1.1].) Notice also that, because it is an oriented surface, F has stably trivial tangent bundle TF. In other words, all the Stiefel-Whitney classes of both TF and $\nu F$ are trivial. Then, $w_2(M_{(\widetilde{\Sigma },\widetilde{\phi })})$ evaluated on [F] is simply given by

$$\begin{aligned} w_2(T M_{(\widetilde{\Sigma },\widetilde{\phi })}\vert _{F} ) = w_2(TF\oplus \nu F) = w_2(TF) + w_1(TF) w_1(\nu F) + w_2(\nu F) =0. \end{aligned}$$

$\square$

3.1 Proof of Theorem 1.6

Case ${{\text {rk}}}(H_2(M;\mathbb {Z}))\ge 2$:

By Barden’s classification (Theorem 2.1) there is a simply connected manifold $\widetilde{M}$ such that $M = \widetilde{M} \# \mathbb {S}^3 \times \mathbb {S}^2$. More precisely, $\widetilde{M}$ is the unique (up to diffeomorphism) simply-connected $5-$manifold satisfying $H^2(\widetilde{M},\mathbb {Z}) \oplus \mathbb {Z}= H^2(M,\mathbb {Z})$, and $w_2(\widetilde{M}) = w_2(M)$. Since $\widetilde{M}$ admits a contact structure by [21, Theorem 8], it follows from Proposition 3.3 that M admits infinitely many symplectic foliations. $\square$

Cases $\mathbb {S}^5$ and $\mathbb {S}^3 \times \mathbb {S}^2$: A leafwise symplectic structure on the Lawson foliation [29] of $\mathbb {S}^5$ has already been found by Mitsumatsu [34]. The existence of infinitely many symplectic foliations on $\mathbb {S}^3 \times \mathbb {S}^2$ follows from applying Proposition 3.3 to $\mathbb {S}^5$. $\square$

For the next two cases we use the observation from [28] that the Brieskorn manifold

$$\begin{aligned} \Sigma (a) = \Sigma (a_0,\dots ,a_3):= \{z \in \mathbb {C}^4 \cap \mathbb {S}^7 \mid \sum _{j=0}^3 z_j^{a_j} = 0 \},\end{aligned}$$

comes equipped with a contact form

$$\begin{aligned} \alpha := \frac{i}{2} \sum _{j=0}^3(z_j {{\text {d}}}\overline{z}_j - \overline{z}_j {{\text {d}}}z_j). \end{aligned}$$

(4)

Moreover, the projection $\pi : \mathbb {C}^4 \rightarrow \mathbb {C}$ given by $(z_0,\dots ,z_3) \mapsto z_0$ induces an open book decomposition supporting $\alpha$, whose page is symplectomorphic to the Brieskorn variety

$$\begin{aligned} V_\varepsilon (a_1,a_2,a_3) = \{ z \in \mathbb {C}^3 \mid \sum _{j=1}^{3} z_j^{a_j} = \varepsilon \}.\end{aligned}$$

As usual we denote by $\alpha _B$ the restriction of $\alpha$ to the binding $B:= \Sigma (a) \cap \{z_0 = 0\} = \Sigma (a_1,a_2,a_3)$.

Case $H_2(M;\mathbb {Z})=\mathbb {Z}_p\oplus \mathbb {Z}_p$ for p relatively prime to 3.

Consider $\Sigma (p,3,3,3)$ and observe it is simply connected, with $H_2(\Sigma (p,3,3,3))=\mathbb {Z}_p\oplus \mathbb {Z}_p$ and vanishing second Stiefel Whitney class (c.f. for instance [28]). In other words, $\Sigma (p,3,3,3)$ is nothing else than the manifold $M_p$ from Sect. 2.1.

The binding of the open book described above is $\Sigma (3,3,3) \subset \mathbb {S}^5$. The restriction of the Hopf fibration $h:S^5 \rightarrow \mathbb {C}\mathbb {P}^2$ induces a fibration of $\Sigma (3,3,3)$ over the complex curve $S:= \{\sum _{j=0}^3 z_j^3\} \subset \mathbb {C}\mathbb {P}^2$. According to the genus formula (see e.g. [2, Page 53]), S is diffeomorphic to a torus and we denote by $\theta _1,\theta _2 \in \Omega ^1(\Sigma (3,3,3))$ the pullback of the two angular forms on $\mathbb {T}^2$. Since the Reeb vector field $R_B$ of $\alpha _B$ is tangent to the fibers of the Hopf fibration, it follows that ${{\text {d}}}\alpha _B$ is a multiple of $\theta _1 \wedge \theta _2$. Hence,

$$\begin{aligned} \gamma :=\theta _2,\quad \eta := \alpha _B \wedge \theta _1,\end{aligned}$$

defines a cosymplectic structure on B satisfying $\gamma \wedge {{\text {d}}}\alpha _B =0$. Furthermore, by Lemma 3.5 below, $[\eta ]$ is in the image of the restriction map $\iota ^*:H^2(V(3,3,3)) \rightarrow H^2(\Sigma (3,3,3))$. Then it follows from Proposition 2.14 and Lemma 2.15 that $\Sigma (p,3,3,3)$ admits a symplectic foliation. $\square$

Lemma 3.5

Let $\textbf{a}=(a_1,a_2,a_3)$, and A the least common multiple of the $a_i$’s. Consider the $\mathbb {S}^1-$action on $\mathbb {C}^3$ defined by

$$\begin{aligned} \lambda \cdot (z_1,z_2,z_3) = (\lambda ^{A/a_1} z_1, \lambda ^{A/a_2} z_2,\lambda ^{A/a_3} z_3), \end{aligned}$$

and assume that the basic cohomology group $H^2_b(\Sigma (\textbf{a});\mathbb {R})$ is generated by the differential of the contact form on $\Sigma (\textbf{a})$ as in Eq. (4). Then, the restriction map

$$\begin{aligned} \iota ^*:H^2(V(\textbf{a})) \rightarrow H^2(\Sigma (\textbf{a})),\end{aligned}$$

is surjective.

Notice that, as the Reeb vector field R of the contact form $\alpha$ in Eq. (4) is the infinitesimal generator of the $\mathbb {S}^1$-action, ${{\text {d}}}\alpha$ is a basic form, and hence defines a class in $H^2_b(\Sigma (\textbf{a}))$. However, as $\alpha$ itself is not a basic form, $[{{\text {d}}}\alpha ]$ is a priori not zero in $H^2_b(\Sigma (\textbf{a}))$; here we make the additional assumption that it generates the whole basic cohomology group.

Proof of Lemma 3.5

Consider the $5-$dimensional Brieskorn manifold $\Sigma (A,\textbf{a})= \Sigma (A,a_1,a_2,a_3) \subset \mathbb {S}^7$. The restriction of the projection

$$\begin{aligned} \pi :\mathbb {C}^{4} \rightarrow \mathbb {C}, \quad (z_0,\dots ,z_3) \mapsto z_0, \end{aligned}$$

(5)

defines an open book decomposition of $\Sigma (A,\textbf{a})$, with page $V_\varepsilon (\textbf{a})$ and binding $\Sigma (\textbf{a})$. Furthermore, there is a weighted $\mathbb {S}^1$-action on $\Sigma (A,\textbf{a})\subset \mathbb {C}^4$ given by

$$\begin{aligned} \lambda \cdot (z_0,z_1,z_2,z_3) = (\lambda z_0, \lambda ^{A/a_1}z_1,\lambda ^{A/a_2}z_2, \lambda ^{A/a_3}z_3). \end{aligned}$$

(6)

Notice that this action restricts to the one described in the statement on the binding $\Sigma (3,3,3)=\{z_0=0\}$.

Let $\tau (\Sigma (\textbf{a}))$ be a tubular neighborhood of the binding invariant under the $\mathbb {S}^1$-action, and consider (part of) the Mayer-Vietoris sequence in $\mathbb {S}^1$-basic cohomology:

$$\begin{aligned} H_b^2(\Sigma (A,\textbf{a}) \setminus \Sigma (\textbf{a})) \oplus H_b^2(\tau (\Sigma (\textbf{a}))) \xrightarrow {i^* - j^*} H_b^2(\partial \tau (\Sigma (\textbf{a}))) \rightarrow H^3_b(\Sigma (A,\textbf{a})). \end{aligned}$$

Recall that, for any proper G-manifold M, we have $H_b^\bullet (M) = H^\bullet (M/G)$, where the latter denotes singular cohomology, (see e.g. [32, Theorem 30.36]), and that G-equivariant maps which are G-equivariantly homotopic induce the same map in basic cohomology (see e.g. [32, Lemma 30.34]). This implies that

$$\begin{aligned}H^3_b(\Sigma (A,\textbf{a})) = H^3(\Sigma (A,\textbf{a})/\mathbb {S}^1) = 0,\end{aligned}$$

where the last equality follows from the general computation in [47, Equation 4.2].

Moreover, since the map in Eq. (5) is equivariant with respect to the action defined in Eq. (6) and the action has weight 1 on the first coordinate $z_0$, it follows that each $\mathbb {S}^1$-orbit intersects the page in a single point. Hence, we also have that

$$\begin{aligned} H^2_b(\Sigma (A,\textbf{a}) \setminus \Sigma (\textbf{a})) = H^2(V_\varepsilon (\textbf{a})), \end{aligned}$$

and that, because $\partial \tau (\Sigma (\textbf{a}))$ is foliated by $\mathbb {S}^1-$orbits and by copies of the binding given by its intersection with the pages of the open book,

$$\begin{aligned} H_b^2(\partial \tau (\Sigma (\textbf{a}))) = H^2(\Sigma (\textbf{a})). \end{aligned}$$

Lastly, note that $\tau (\Sigma (\textbf{a}))$ retracts, in a $\mathbb {S}^1$-equivariant way, onto the binding $\Sigma (\textbf{a})$. Summarizing, the relevant part of the Mayer-Vietoris sequence can be rewritten as:

$$\begin{aligned} H^2(V_\varepsilon (\textbf{a})) \oplus H^2_b(\Sigma (\textbf{a})) \xrightarrow {i^*-j^*} H^2(\Sigma (\textbf{a})) \rightarrow 0. \end{aligned}$$

Hence, to prove that $i^*$ is surjective it suffices to show that $j^*=0$. For this, consider on $\Sigma (A,\textbf{a})$ the contact form as in Eq. (4). As already observed before, because Reeb vector field R is the infinitesimal generator of the $\mathbb {S}^1$-action, ${{\text {d}}}\alpha$ defines a class in $H^2_b(\Sigma (1,\textbf{a}))$. Moreover, since $\alpha$ is supported by the open book, the restriction ${{\text {d}}}\alpha _B$ of ${{\text {d}}}\alpha$ to the binding $B=\Sigma (\textbf{a})$ is again of the form in Eq. (4), hence $[{{\text {d}}}\alpha _B] \in H^2_b(\Sigma (\textbf{a}))$ is, by assumption, a generator.

Hence, in order to prove that

$$\begin{aligned} j^*:H^{2}_b(\Sigma (\textbf{a})) =H^2_b(\tau \Sigma (\textbf{a})) \rightarrow H^{2}_b(\partial \tau \Sigma (\textbf{a})) = H^2(\Sigma (\textbf{a})) \end{aligned}$$

is the trivial map, one has to prove that the generator $[{{\text {d}}}\alpha _B]$ is sent to 0. But this simply follows from the fact that $j^*[{{\text {d}}}\alpha _B]$, seen in $H^2(\Sigma (\textbf{a}))$, is represented by the differential of $\alpha _B\in \Omega ^1(\Sigma (\textbf{a}))$. $\square$

Case $H_2(M;\mathbb {Z})=\mathbb {Z}_q\oplus \mathbb {Z}_q$ for q relatively prime to 2. This case is analogous to the one above. Here, we look instead at the Brieskorn manifold

$$\begin{aligned} \Sigma (q,4,4,2)=\{(z_0,z_1,z_2,z_3)\in \mathbb {C}^4\cap \mathbb {S}^{7} \, \vert \, z_0^q + z_1^4 + z_2^4 + z_3^2= 0 \}, \end{aligned}$$

which is, as observed in [28], a simply-connected spin manifold with $H_2$ equal to $\mathbb {Z}_q\oplus \mathbb {Z}_q$. In other words, $\Sigma (q,4,4,2)$ is diffeomorphic to $M_q$ of Sect. 2.1.

The open book we consider is again induced by the ambient projection $\mathbb {C}^4\rightarrow \mathbb {C}$, $(z_0,z_1,z_2,z_3)\rightarrow z_0$. This time, however, the pages are naturally symplectomorphic to the Brieskorn variety

$$\begin{aligned} V(4,4,2)=\{(z_1,z_2,z_3)\in \mathbb {C}^3 \, \vert \, z_1^4 + z_2^4 + z_3^2 = \varepsilon \}, \end{aligned}$$

with boundary $\Sigma (4,4,2)$, which naturally lives in $\mathbb {S}^5$. Now, there is a natural orbi-bundle projection $\mathbb {S}^5\rightarrow \mathbb {C}P^2(1,1,2)$, $(z_1,z_2,z_3)\rightarrow [z_1:z_2:z_3]$, where $\mathbb {C}P^2(1,1,2)$ is the weighted projective space given by the quotient of $\mathbb {C}^3$ by the action $\lambda \cdot (z_1,z_2,z_3)=(\lambda z_1,\lambda z_2,\lambda ^2 z_3)$ for every $\lambda \in \mathbb {C}\setminus \{0\}$. We are interested at the restriction of this orbi-bundle to $\Sigma (4,4,2)\subset \mathbb {S}^5$. The complex orbi-surface $S=\{z_1^4+z_2^4+z_3^2=0\}\subset \mathbb {C}P(1,1,2)$ (which is well defined because the polynomial is invariant under the action of $\mathbb {C}\setminus \{0\}$ on $\mathbb {S}^5$) is actually smooth, so that the restriction $\Sigma (4,4,2)\rightarrow S$ is a smooth fibration. Moreover, according to the genus formula for surfaces in weighted projective spaces [26, Theorem 5.3.7], S has genus 1, and is hence diffeomorphic to a $2-$torus $\mathbb {T}^2$. The proof now can be concluded word by word as in the case above. $\square$

4 Constructions of conformal symplectic foliations

We describe here the turbulization procedure, as well as the cobordisms and surgeries needed for the constructions of (conformal) symplectic foliations in the subsequent sections.

4.1 Gluing and turbulization

Let $({\mathcal {F}},\eta ,\omega )$ be a symplectic foliation on M tangent to the boundary. To glue two such manifolds, the boundaries must be isomorphic as conformal symplectic manifolds. Moreover, to ensure the resulting foliation is smooth, we also need to control the variation of $({\mathcal {F}},\eta ,\omega )$ in the direction normal to $\partial M$. (Recall that we did in Sect. 2.3 the simplifying assumption that the foliation is tame near the boundary. We will need however to work in this more general setting for the conformal symplectic case in order to study the problem of deformations to contact structures later on.)

Recall that the linear holonomy of ${\mathcal {F}}$ is encoded in a leafwise cohomology class. If ${\mathcal {F}}= \ker \gamma$ it can be computed as follows. The integrability condition of ${\mathcal {F}}$ implies

$$\begin{aligned} {{\text {d}}}\gamma = \mu \wedge \gamma ,\end{aligned}$$

for some $\mu \in \Omega ^1(M)$. The restriction $\mu _{\mathcal {F}}:= \mu |_{\mathcal {F}}$ is closed and the cohomology class $[\mu _{\mathcal {F}}] \in H^1({\mathcal {F}})$ depends only on ${\mathcal {F}}$. We refer to $\mu _{\mathcal {F}}$ as a holonomy form of ${\mathcal {F}}$.

To encode the behaviour of $({\mathcal {F}},\eta ,\omega )$ near the boundary, fix two closed 1-forms $\mu ,\nu$ on $\partial M$, and a collar neighborhood $k:(-\varepsilon ,0] \times \partial M \rightarrow M$. Using the collar neighborhood we define

$$\begin{aligned} M_\infty := M \cup _{\partial M} [0,\infty ) \times \partial M,\end{aligned}$$

and extend $({\mathcal {F}},\eta ,\omega )$ over $[0,\infty ) \times \partial M$ by:

$$\begin{aligned} {\mathcal {F}}:= \ker \, (-t \mu + {{\text {d}}}t),\quad \eta := \eta _\partial + \nu ,\quad \omega := \omega _\partial .\end{aligned}$$

If this extension is smooth we say that the collar is $(\mu ,\nu )-$adapted.

Definition 4.1

A conformal symplectic foliation is $(\mu ,\nu )$-tame at the boundary, if it admits a $(\mu ,\nu )-$adapted collar neighborhood as above. If $\mu = \pm \nu$, we will say positive/negative $\mu$-tame, and if $\mu = \nu = 0$, simply tame at the boundary.

Although the above definition makes sense for any choice of $\mu$ and $\nu$, in terms of deformations to contact structures (c.f. Theorem 6.12, i.e. [9, Lemma 6.1] and [45, Theorem 2.2.13]) is it most significant when the Lee form $\eta$ of the leafwise conformal symplectic structure coincides (at least up to sign) with the holonomy form $\mu$. In fact, this condition makes sense globally, and not only around the boundary.

Definition 4.2

A conformal symplectic foliation $({\mathcal {F}},\eta ,\omega )$ is said to be holonomy-like if $\eta$ equals a holonomy form $\mu _{\mathcal {F}}$ of ${\mathcal {F}}$. If the equality only holds up to sign we call the foliation $\vert$holonomy$\vert$-like .

Note that in the definition of $\vert$holonomy$\vert$-like we do not ask the sign to be the same at every point. That is, in the region where $\mu _{\mathcal {F}}= 0$, the sign can change.

As stated above, the tameness condition ensures that when gluing manifolds along their boundaries the resulting conformal symplectic foliation is smooth. The precise statement is as follows:

Proposition 4.3

Let $(M_i,{\mathcal {F}}_i,\eta _i,\omega _i)$, $i=1,2$ be (conformal) symplectic foliations $\mu _i-$tame at the boundary. If there exists an orientation reversing diffeomorphism $\phi :\partial M_1\rightarrow \partial M_2$ respecting the induced (conformal) symplectic structures on the boundary and the holonomy forms $\mu _i$, then $M_1\cup _\phi M_2$ admits a (conformal) symplectic foliation.

As in the symplectic setting (Sect. 4.1) we use turbulization to produce conformal symplectic foliations that are tame at the boundary. Unlike symplectic structures, exact conformal symplectic structures also exist on closed manifolds; consider for example the conformal symplectization of a contact manifold. Thus it is no surprise that in addition to boundaries of cosymplectic type, conformal turbulization also applies to boundaries of contact type, where these notions are defined as follows:

Definition 4.4

A conformal symplectic foliation $(M,{\mathcal {F}},\eta ,\omega )$ transverse to the boundary $\partial M$ is:

• cosymplectic if there exist an admissible form $\alpha \in \Omega ^1({\mathcal {F}}_\partial )$ for $\omega _\partial$ satisfying ${{\text {d}}}_{\eta _\partial } \alpha = 0$;

• convex/concave if there exists an admissible form $\alpha \in \Omega ^1({\mathcal {F}}_\partial )$ for $\omega _\partial$ satisfying ${{\text {d}}}_{\eta _\partial } \alpha = \pm \omega _\partial$ (c.f. Lemma 2.6).

In both cases we can use turbulization to change the foliation to become tangent to the boundary. Since the proofs are slightly different in the two cases, the constructions are stated in separate theorems.

Theorem 4.5

Let $(M,{\mathcal {F}},\eta ,\omega )$ a conformal symplectic foliation with contact type boundary, and such that ${\mathcal {F}}_\partial =\ker \gamma _\partial$ is unimodular. Then, there exists a conformal symplectic foliation $(\widetilde{{\mathcal {F}}},\widetilde{\eta },\widetilde{\omega })$ on M such that the following holds.

• (i) If $({\mathcal {F}},\eta ,\omega )$ has convex contact boundary, the boundary leaf of $(\widetilde{F},\widetilde{\eta },\widetilde{\omega })$ can be arranged to be

  $$\begin{aligned} (\partial M,\widetilde{\eta }_\partial = \eta _\partial \pm \gamma _\partial , \widetilde{\omega }_\partial = {{\text {d}}}_{\eta _\partial \pm \gamma _\partial } \alpha ),\end{aligned}$$

  for any $\alpha \in \Omega ^1(\partial M)$ satisfying $\omega _\partial = {{\text {d}}}_{\eta }\alpha$. Moreover, $(\widetilde{{\mathcal {F}}},\widetilde{\eta },\widetilde{\omega })$ is either of the following:

  • 1. tame at the boundary;

  • 2. positive/negative $\gamma _\partial$-tame at the boundary, where the sign corresponds to the sign in the equation above (which itself is arbitrary). If $\eta _\partial = 0$ then the resulting foliation is also holonomy-like.

• (ii) If $({\mathcal {F}},\eta ,\omega )$ has concave contact boundary, the boundary leaf of $(\widetilde{F},\widetilde{\eta },\widetilde{\omega })$ can be arranged to be

  $$\begin{aligned} (\partial M, \widetilde{\eta }_\partial = \eta _\partial {\mp } \gamma _\partial , \widetilde{\omega }_\partial = {{\text {d}}}_{\eta _\partial {\mp } \gamma _\partial } \alpha ),\end{aligned}$$

  for any $\alpha \in \Omega ^1(\partial M)$ satisfying $\omega _\partial = {{\text {d}}}_{\eta }\alpha$. Moreover, $(\widetilde{{\mathcal {F}}},\widetilde{\eta },\widetilde{\omega })$ is either of the following:

  • (a) tame at the boundary;

  • (b) positive/negative $(-\gamma _\partial )$-tame at the boundary, where the sign corresponds to the one in the equation above (which is arbitrary). If $\eta _\partial = 0$ then the resulting foliation is |holonomy|-like.

Moreover, in both cases above, on an arbitrary small neighborhood of the boundary we have $\widetilde{\omega } = {{\text {d}}}_{\widetilde{\eta }} \widetilde{\lambda }$ with $\widetilde{\lambda }_\partial = \alpha$, and $(\widetilde{{\mathcal {F}}},\widetilde{\eta },\widetilde{\omega })$ agrees with $({\mathcal {F}},\eta ,\omega )$ away from this neighborhood. Furthermore, if $\omega = d_\eta \lambda$ globally on $(M,{\mathcal {F}})$, then one can arrange $\widetilde{\omega }={{\text {d}}}_{\widetilde{\eta }}\widetilde{\lambda }$, with $\widetilde{\lambda }$ agreeing with $\lambda$ on the region where $(\widetilde{{\mathcal {F}}},\widetilde{\eta },\widetilde{\omega })$ and $({\mathcal {F}},\eta ,\omega )$ agree.

Theorem 4.6

Let $(M,{\mathcal {F}},\eta ,\omega )$ be a conformal symplectic foliation with boundary of cosymplectic type, and such that ${\mathcal {F}}_\partial = \ker \gamma _\partial$ is unimodular. Then, there exists a conformal symplectic foliation $(\widetilde{{\mathcal {F}}},\widetilde{\eta },\widetilde{\omega })$ on M tame at the boundary and such that:

• (i) $({\mathcal {F}},\eta ,\omega )$ and $(\widetilde{{\mathcal {F}}},\widetilde{\eta },\widetilde{\omega })$ agree away from an arbitrarily small neighborhood of $\partial M$.

• (ii) The conformal symplectic structure on the boundary leaf is given by

  $$\begin{aligned} \widetilde{\omega }_\partial = \omega _\partial \pm \alpha \wedge \gamma _\partial ,\end{aligned}$$

  where $\alpha \in \Omega ^1({\mathcal {F}}_\partial )$ is any admissible form satisfying ${{\text {d}}}_{\eta _\partial } \alpha = 0$, and the sign can be chosen arbitrarily.

We will say that the conformal symplectic foliations $(\widetilde{{\mathcal {F}}},\widetilde{\eta },\widetilde{\omega })$ of both Theorems 4.5 and 4.6 are obtained from $({\mathcal {F}},\eta ,\omega )$ via conformal symplectic turbulization.

Even though we will not directly need it in the rest of the paper, it is interesting to point out that by adding a further interval-worth of compact leaves near the boundary one can always obtain a conformal symplectization as conformal symplectic structure at the boundary leaf:

Corollary 4.7

Let $(M,{\mathcal {F}},\eta ,\omega )$ be a conformal symplectic foliation with ${\mathcal {F}}_\partial$ unimodular and of (convex or concave) contact type, with $\eta _\partial$ admitting a closed extension to $\partial M$. Then, there exists a conformal symplectic foliation $(\widetilde{{\mathcal {F}}},\widetilde{\eta },\widetilde{\omega })$ on M, tame at the boundary, such that the following holds.

• 1. $({\mathcal {F}},\eta ,\omega )$ and $(\widetilde{{\mathcal {F}}},\widetilde{\eta },\widetilde{\omega })$ agree away from an arbitrarily small neighborhood of $\partial M$.

• 2. The conformal symplectic structure on the boundary leaf equals:

  $$\begin{aligned} \widetilde{\omega }_\partial = {{\text {d}}}_{\pm \gamma _\partial } \alpha ,\end{aligned}$$

  where the sign can be chosen, $\gamma _\partial \in \Omega ^1(\partial M)$ is the closed form defining ${\mathcal {F}}_\partial$ and $\alpha \in \Omega ^1(\partial M)$ is any form satisfying $\omega _\partial = {{\text {d}}}_{\eta }\alpha$.

• 3. $\widetilde{{\mathcal {F}}}$ restricts to the foliation by closed leaves $\{pt\}\times \partial M$ on a sufficiently small neighborhood of the boundary $(-\delta ,0]\times \partial M$.

We now give the proof of Theorems 4.5 and 4.6 and Corollary 4.7 above.

Proof of Theorem 4.5

We start by proving the statement in the case that $({\mathcal {F}},\eta ,\omega )$ has convex boundary. Using Theorem 2.8 and the contact type boundary condition we fix a collar neighborhood $(-\varepsilon ,0] \times \partial M$ on which

$$\begin{aligned} {\mathcal {F}}= \ker \gamma _\partial , \quad \eta = \eta _\partial ,\quad \omega = {{\text {d}}}_{\eta _{\partial }}(e^t\alpha ),\end{aligned}$$

where $\gamma _\partial$ is a closed 1-form defining ${\mathcal {F}}_\partial$ (which exists by assumption of unimodularity), and, with an abuse of notation, $\eta _\partial$ is seen as a closed 1-form defined on $\partial M$ (which exists by assumption). Furthermore, we choose smooth functions $f,g:(-\varepsilon ,0] \rightarrow [0,1]$ satisfying $(f,g)\ne (0,0)$ everywhere and

$$\begin{aligned} f(t) = {\left\{ \begin{array}{ll} 1 &{} \text {for} \,\,t \text { near} -\varepsilon \\ -t &{} \text {for } \,\,t \,\, \text {near}\,\, 0 \end{array}\right. },\quad g(t) = {\left\{ \begin{array}{ll} 0 &{} \text {for}\,\, t \,\,\text { near}\,\, -\varepsilon \\ 1 &{} \text {for} \,\, t\,\, \text { near}\,\, 0 \end{array}\right. }, \end{aligned}$$

with $\dot{f}\le 0$ and $\dot{g}\ge 0$.

Then we define

$$\begin{aligned} \widetilde{\gamma } = f(t)\gamma _\partial \pm g(t){{\text {d}}}t,\quad \widetilde{\eta }:= \eta _\partial + \frac{\dot{f}}{f} {{\text {d}}}t, \quad \widetilde{\omega } = {{\text {d}}}_{\eta _\partial +\frac{\dot{f}}{f}{{\text {d}}}t}( e^{h(t)}\alpha ), \end{aligned}$$

(7)

where $h:(-\epsilon ,0]\rightarrow \mathbb {R}$ equals $e^t$ near $t=-\epsilon$, has $\dot{h}(t)>0$ for $t<0$, and has all derivatives zero at $t=0$.

Notice that $\widetilde{\eta },\widetilde{\omega }$ are not well defined at $t=0$ as forms on M, but are well defined as leafwise forms on $\widetilde{{\mathcal {F}}}=\ker \widetilde{\gamma }$. Indeed,

$$\begin{aligned} \widetilde{\gamma }\wedge \widetilde{\eta } = \dot{f}\gamma _\partial \wedge {{\text {d}}}t \pm g {{\text {d}}}t \wedge \eta _\partial \end{aligned}$$

extends in an obvious way to $t=0$. Alternatively, one can see this as follows: $\widetilde{{\mathcal {F}}}$ is spanned by $\ker (\gamma _\partial )$ and by ${\mp } gX+f\partial _t$, for any fixed vector field X on $\partial M$ such that $\gamma _\partial (X)=1$, and

$$\begin{aligned} \widetilde{\eta }({\mp } gX+f\partial _t) = {\mp } g \, \eta _\partial (X) + \dot{f} \quad \text {and} \quad \widetilde{\eta }(V)=\eta _\partial (V), \end{aligned}$$

for any $V\in \ker (\gamma _\partial )$, which are both well defined quantities even for $t=0$.

To see that $(\widetilde{{\mathcal {F}}},\widetilde{\eta },\widetilde{\omega })$ is a conformal symplectic foliation observe that $\widetilde{\eta }$ and $\widetilde{\omega }$ are, respectively, ${{\text {d}}}$- and ${{\text {d}}}_{\widetilde{\eta }}$-closed on $\widetilde{{\mathcal {F}}}$. The leafwise non-degeneracy is checked as follows (noticing again that the following differential form of maximal degree extends smoothly to $t=0$):

$$\begin{aligned} \widetilde{\gamma } \wedge \widetilde{\omega }^n&= (f \gamma _\partial \pm g {{\text {d}}}t) \wedge ({{\text {d}}}_{\eta _\partial }(e^{h}\alpha ) - e^{h}\frac{\dot{f}}{f} {{\text {d}}}t \wedge \alpha )^n \\&= f \gamma _\partial \wedge {{\text {d}}}_{\eta _\partial }(e^h\alpha )^n \pm g dt \wedge d_{\eta _\partial }(e^h\alpha )^n - n\dot{f} e^h \gamma _\partial \wedge {{\text {d}}}t \wedge \alpha \wedge {{\text {d}}}_{\eta _\partial }(e^h \alpha )^{n-1} \\&= n f \dot{h} e^{nh} \gamma _\partial \wedge {{\text {d}}}t \wedge \alpha \wedge {{\text {d}}}\alpha ^{n-1} \pm g e^{nh} {{\text {d}}}t \wedge ({{\text {d}}}_{\eta _\partial }\alpha )^n +ne^{n h}\dot{f} {{\text {d}}}t \wedge \gamma _\partial \wedge \\&\quad \, \alpha \wedge {{\text {d}}}\alpha ^{n-1} \\&\overset{(*)}{=}\ n e^{nh} ( f\dot{h} -\dot{f} ) \, \gamma _\partial \wedge {{\text {d}}}t \wedge \alpha \wedge {{\text {d}}}\alpha ^{n-1} > 0, \end{aligned}$$

as desired. Here, for $(*)$ we used that ${{\text {d}}}_{\eta _\partial }\alpha ^n = \omega _\partial ^n = 0$ on $\partial M$. The foliation $\widetilde{{\mathcal {F}}}:= \ker \widetilde{\gamma }$ is positive/negative $\gamma _\partial$-tame at the boundary and agrees with ${\mathcal {F}}$ away from the boundary. Moreover, the restriction of $(\widetilde{\eta },\widetilde{\omega })$ to ${\mathcal {F}}$ agrees with $(\eta ,\omega )$ near $t = -\varepsilon$.

Lastly, the statement about exactness of $\widetilde{\omega }$ under the assumption of exactness of $\omega$, as well as the fact that $\widetilde{\eta }$ is a holonomy form for $\widetilde{\gamma }$ in the case where $\eta _\partial =0$, are clear for the explicit formulas above. Moreover, ${\mathcal {F}}$ can be arranged to be tame at the boundary by choosing f constant and equal to 0 on a neighborhood of $t=0$. In this case we define the leafwise conformal symplectic structure by:

$$\begin{aligned} \widetilde{\eta } :=\eta _\partial \pm \gamma _\partial , \quad \widetilde{\omega } :={{\text {d}}}_{\eta _\partial \pm \gamma _\partial }(e^t\alpha ). \end{aligned}$$

Observe that $\eta$ is not a holonomy form for $\widetilde{\gamma }$ in this case.

If instead $({\mathcal {F}},\eta ,\omega )$ has concave boundary, then Theorem 2.8 provides a collar neighborhood $(-\varepsilon ,0] \times \partial M$ on which

$$\begin{aligned} {\mathcal {F}}= \ker \gamma _\partial ,\quad \eta = \eta _\partial , \quad \omega = {{\text {d}}}_{\eta _\partial } (e^{-t}\alpha ).\end{aligned}$$

Here $\alpha \in \Omega ^1(\partial M)$ satisfies ${{\text {d}}}_{\eta _\partial } \alpha = \omega$ and $\gamma _\partial \wedge \alpha \wedge \omega _\partial ^{n-1} < 0$. The same computations as above then show that defining

$$\begin{aligned} \widetilde{\gamma }:= f(t) \gamma _\partial \pm g(t) {{\text {d}}}t,\quad \widetilde{\eta }:= \eta _\partial - \frac{\dot{f}}{f} {{\text {d}}}t, \quad \widetilde{\omega }:= {{\text {d}}}_{\widetilde{\eta }}(e^{h(t)}\alpha ),\end{aligned}$$

gives the desired conformal symplectic foliation. Moreover, the explicit formula shows that the conformal symplectic leafwise structure is exact if $\omega$ is, with primitive $\lambda$ as in the statement, and that $\eta$ is minus a holonomy form. The same argument as in the convex case shows that ${\mathcal {F}}$ can be made tame at the boundary. $\square$

Proof of Theorem 4.6

For the case that $({\mathcal {F}},\eta ,\omega )$ has cosymplectic boundary, we use again Theorem 2.8 to find a collar neighborhood $(-\varepsilon ,0] \times \partial M$ on which

$$\begin{aligned} {\mathcal {F}}= \ker \gamma _\partial ,\quad \eta = \eta _\partial , \quad \omega = \omega _\partial + {{\text {d}}}_{\eta _\partial }(t \alpha ),\end{aligned}$$

for an admissible form $\alpha$ satisfying ${{\text {d}}}_{\eta _\partial } \alpha = 0$. Then a straightforward computation shows that

$$\begin{aligned} \widetilde{\gamma } = f \gamma _\partial \pm g {{\text {d}}}t, \quad \widetilde{\eta }:= \eta _\partial , \quad \widetilde{\omega } = \omega _\partial + {{\text {d}}}t \wedge \alpha \pm \alpha \wedge \gamma _\partial , \end{aligned}$$

defines the desired conformal symplectic foliation. $\square$

Proof of Corollary 4.7

Consider the trivial cobordism $[0,1] \times \partial M$ endowed with the conformal symplectic foliation

$$\begin{aligned} \left( {\mathcal {F}}:= \bigcup _{t \in [0,1]} \{t\} \times \partial M,\quad \eta := f(t) \eta _\partial + \gamma _\partial ,\quad \omega := {{\text {d}}}_{\eta } \alpha \right) ,\end{aligned}$$

where $f:[0,1]\rightarrow {\mathbb {R}}$ is a smooth function satisfying

$$\begin{aligned} f(t) = {\left\{ \begin{array}{ll} 1 &{} \text {for }\,\, t \,\,\text {near } 0 \\ 0 &{} \text {for}\,\, t \,\,\text {near } 1. \end{array}\right. } \end{aligned}$$

First applying Theorem 4.5 and then gluing the above cobordism to the boundary completes the proof. $\square$

4.2 Conformal symplectization of homotopic contact structures.

Lemma 4.8

Let $\alpha _0$ and $\alpha _1$ be homotopic (through almost contact forms) contact forms on a manifold N of dimension at least 5. Then $\mathbb {S}^1 \times [0,1] \times N$ admits a conformal symplectic foliation tame at the boundary such that:

• (i) The boundary leaves are isomorphic to $(\mathbb {S}^1 \times N, {{\text {d}}}_{\pm {{\text {d}}}\theta } \alpha _i)$, $i=0,1$, where the signs can be chosen freely.

• (ii) The foliation contains at most one closed leaf in the interior isomorphic to $(\mathbb {S}^1 \times N, {{\text {d}}}_{\pm R {{\text {d}}}\theta } \alpha _{ot})$, where the sign and $R> 0$ can be chosen arbitrarily, and $\alpha _{ot}$ is any overtwisted contact form in the almost contact class of $\alpha _i$. If one of the $\alpha _i$ is overtwisted we may arrange the foliation to have no closed interior leaves.

Proof

Suppose first that $\alpha _0$ and $\alpha _1$ are tight. According to [18, Theorem 1.1] there is a Liouville structure on $[0,1] \times N$ which restricts to $\alpha _{ot}$ and $\alpha _{0}$ on the concave and convex boundary respectively. Thus, the product $\mathbb {S}^1 \times [0,1] \times N$ admits a symplectic foliation which can be turbulized using Theorem 4.5 (with the choice of $\gamma _\partial$ equal to $\pm R{{\text {d}}}\theta$ and $\pm {{\text {d}}}\theta$ at the two ends). The resulting boundary leaves equal

$$\begin{aligned} (\mathbb {S}^1 \times N, {{\text {d}}}_{\pm R{{\text {d}}}\theta } \alpha _{ot}),\quad \text {and} \quad (\mathbb {S}^1 \times N,{{\text {d}}}_{\pm {{\text {d}}}\theta } \alpha _0).\end{aligned}$$

Repeating the argument for $\alpha _1$ and gluing the resulting manifolds using Proposition 2.11 proves the claim.

In the case where $\alpha _0$ is overtwisted, one can just apply the first part of the argument above with $\alpha _0$ instead of $\alpha _{ot}$. If $\alpha _1$ is overtwisted, one can apply the above proof to $\alpha _0'=\alpha _1$ and $\alpha _1'=\alpha _0$, then apply the orientation preserving diffeomorphism

$$\begin{aligned} \mathbb {S}^1\times [0,1]\times M \rightarrow \mathbb {S}^1\times [0,1]\times M, \quad (\theta ,t,p)\mapsto (-\theta ,-t,p) \end{aligned}$$

in order to conclude. $\square$

4.3 Round connected sum

The classification of simply connected $5-$manifolds (Sect. 2.1) is in terms of connected sums. In light of this, we would like a connected sum construction for (conformal) symplectic foliations. However, such a construction is not available even in the setting of smoothly foliated manifolds, namely it is not obvious how to naturally construct, on the connected sum of two foliated manifolds, a foliation which coincides with the original ones away from the connect sum region. That being said, there is a notion of round connected sums in the smoothly foliated setting, which results in a smooth foliation on the round connected sum of two foliated manifolds along two positively transverse curves, in such a way that the resulting leaves are connected sums of the leaves of the original two foliations.

As the connected sum of two symplectic manifolds is not symplectic in general (cf. [36, Lemma 7.2.2]), one cannot expect to be able to do exactly the same with symplectically foliated manifolds. To avoid this problem we slightly modify the construction using turbulization. Let us give more details.,

We work with round connected sums of manifolds, defined as follows. Let $\gamma _i \subset M_i^m$, $i=1,2$ be (embedded) closed curves and identify their tubular neighborhoods with $\mathbb {S}^1 \times \mathbb {D}^{m-1}$. Then we define the round connected sum as:

$$\begin{aligned} M_1 \#_{\mathbb {S}^1} M_2:= \left( M_1 \setminus {{\,\textrm{Int}\,}}(\mathbb {S}^1 \times \mathbb {D}^{m-1}) \right) \cup _\partial \left( M_2 \setminus {{\,\textrm{Int}\,}}(\mathbb {S}^1 \times \mathbb {D}^{m-1})\right) , \end{aligned}$$

where the boundaries are glued by identifying

$$\begin{aligned} (\theta ,x) \sim (-\theta ,x),\quad \theta \in \mathbb {S}^1, x \in \partial \mathbb {D}^{m-1}. \end{aligned}$$

Proposition 4.9

Let $\gamma _i \subset M_i$ be a closed curve (positively) transverse to a conformal symplectic foliation $({\mathcal {F}}_i,\eta _i,\omega _i)$, for $i = 1,2$. Then $M_1 \#_{\mathbb {S}^1} M_2$ admits a conformal symplectic foliation.

Proof

Around each of the transverse curves we fix a tubular neighborhood diffeomorphic to $\mathbb {S}^1 \times \mathbb {D}^{2n}$, where $\dim M_i = 2n+1$, in which the foliation equals

$$\begin{aligned} {\mathcal {F}}= \bigcup _{\theta \in \mathbb {S}^1} \{\theta \} \times \mathbb {D}^{2n},\end{aligned}$$

and $\gamma _i$ is identified with $\mathbb {S}^1 \times \{0\}$. The leafwise symplectic normal bundle of $\gamma _i$ a trivial symplectic vector bundle over $\mathbb {S}^1$. To see this recall that the the linear symplectic group $SP({\mathbb {R}},2n)$ retracts onto U(n), and that U(n)-bundles over $\mathbb {S}^1$ are classified by $\pi _0(U(n)) = 0$.

Thus, by a leafwise linear change of coordinates we may assume that $\omega = \omega _{st}$ at points $(\theta ,0) \in \mathbb {S}^1 \times \mathbb {D}^{2n}$. Using a standard Moser trick [3, Theorem 4] (as in the proof of Darboux theorem) we can then assume that, after shrinking the tubular neighborhood if necessary, the conformal symplectic foliation is equivalent to

$$\begin{aligned} {\mathcal {F}}= \bigcup _{ \theta \in \mathbb {S}^1} \{ \theta \} \times \mathbb {D}^{2n},\quad \eta = 0, \quad \omega = \omega _{st}.\end{aligned}$$

Thus after removing $\mathbb {S}^1 \times \mathbb {D}^{2n}$ the foliations are transverse to the boundary and of contact type (Definition 4.4). As such they can be turbulized using Theorem 4.5. The conformal symplectic structure on the resulting boundary leaves $\mathbb {S}^1 \times \mathbb {S}^{2n-1}$ are

$$\begin{aligned} \eta _\partial = \pm {{\text {d}}}\theta ,\quad \omega _\partial = {{\text {d}}}_{\pm {{\text {d}}}\theta } \alpha _{st}.\end{aligned}$$

Hence, if we choose opposite signs on each of the pieces we can glue using Proposition 4.3, giving a conformal symplectic foliation on $M_1 \#_{\mathbb {S}^1} M_2$. $\square$

It turns out that connected sums of simply connected 5-folds can also be obtained by round connected sums. This will be key in Sect. 5.

Lemma 4.10

Consider two 5-dimensional manifolds $M_1$ and $M_2$ such that $M_1$ is simply connected, while $\pi _1(M_2) = \mathbb {Z}$ and is generated by a closed curve $\gamma _2$. Let also $\gamma _1$ be a closed curve in $M_1$. Then the round connected sum $M_1\#_{\mathbb {S}^1} M_2$ of $M_1$ and $M_2$ along $\gamma _1$ and $\gamma _2$ satisfies:

• 1. $H_2(M_1 \#_{\mathbb {S}^1} M_2) = H_2(M_1) \oplus H_2(M_2)$;

• 2. $\pi _1(M_1 \#_{\mathbb {S}^1} M_2) = 1$;

• 3. $w_2(M_1 \#_{\mathbb {S}^1} M_2) = 0$ if and only if $w_2(M_1) = 0$ and $w_2(M_2) = 0$.

Proof

For $i=1,2$, denote $M_i^\circ := M_i\setminus \gamma _i$. Then, applying Mayer–Vietoris we find

$$\begin{aligned} \rightarrow H_2(\mathbb {S}^1\times \mathbb {S}^{3}) \rightarrow H_2(M_1^\circ )\oplus H_2(M_2^\circ ) \rightarrow H_2(M_1 \#_{\mathbb {S}^1} M_2) \rightarrow H_1(\mathbb {S}^1\times \mathbb {S}^{3})\\ \rightarrow H_1(M_1^\circ )\oplus H_1(M_2^\circ )\rightarrow , \end{aligned}$$

which reduces to

$$\begin{aligned} 0 \rightarrow H_2(M_1^\circ )\oplus H_2(M_2^\circ )\rightarrow H_2(M_1 \#_{\mathbb {S}^1} M_2) \rightarrow \mathbb {Z}\xrightarrow {h} H_1(M_1^\circ )\oplus H_1(M_2^\circ ). \end{aligned}$$

(8)

Furthermore, we claim that $H_1(M_i^\circ )=H_1(M_i)$ and $H_2(M_i^\circ )=H_2(M_i)$, $i=1,2$. Indeed, again applying Mayer–Vietoris gives:

$$\begin{aligned}{} & {} H_2(\mathbb {S}^1\times \mathbb {S}^{3})\rightarrow H_2(M_i^\circ )\oplus H_2(\mathbb {S}^1\times \mathbb {D}^{4})\rightarrow H_2(M_i) \rightarrow H_1(\mathbb {S}^1\times \mathbb {S}^{3})\xrightarrow {f} \\{} & {} \quad H_1(M_i^\circ )\oplus H_1(\mathbb {S}^1\times \mathbb {D}^4) \rightarrow H_1(M_i) \rightarrow H_0(\mathbb {S}^1\times \mathbb {S}^{3})\xrightarrow {g} H_0(M_i^\circ )\oplus H_0(\mathbb {S}^1\times \mathbb {D}^{4}). \end{aligned}$$

As the maps f and g are injective, this implies:

$$\begin{aligned} 0\rightarrow H_2(M_i^\circ )\rightarrow H_2(M_i) \rightarrow 0, \quad 0\rightarrow H_1(\mathbb {S}^1\times \mathbb {S}^{3}) \\ \rightarrow H_1(M_i^\circ )\oplus H_1(\mathbb {S}^1\times \mathbb {D}^{4}) \rightarrow H_1(M_i) \rightarrow 0. \end{aligned}$$

In particular, $H_1(M_i^\circ )=H_1(M_i)$ and $H_2(M_i^\circ )=H_2(M_i)$ as desired. Going back to Eq. (8), since $\gamma _2$ is a generator of $H_1(M_2)$, h is injective. Thus we find $H_2(M_1 \#_{\mathbb {S}^1} M_2) = H_2(M_1) \oplus H_2(M_2)$.

That $M_1 \#_{\mathbb {S}^1} M_2$ is simply connected follows immediately from Van Kampen’s theorem using the same covering by open subsets as above, so we omit the details. Hence it remains to show the relation for the second Stiefel-Whitney class.

Consider the following portion of the Mayer-Vietoris sequence:

$$\begin{aligned}{} & {} H^1(M_1 \#_{\mathbb {S}^1} M_2)\rightarrow H^1(M_1^\circ )\oplus H^1(M_2^\circ ) \rightarrow H^1(\mathbb {S}^1\times \mathbb {S}^3)\rightarrow \nonumber \\{} & {} \quad H^2(M_1 \#_{\mathbb {S}^1} M_2)\rightarrow H^2(M_1^\circ )\oplus H^2(M_2^\circ )\rightarrow H^2(\mathbb {S}^1\times \mathbb {S}^3) \end{aligned}$$

(9)

Here, $H^2(\mathbb {S}^1\times \mathbb {S}^3)=0$, as well as $H^1(M_1 \#_{\mathbb {S}^1} M_2)=H^1(M_1^\circ )=0$, because both $M_1 \#_{\mathbb {S}^1} M_2$ and $M_1^\circ$ are simply-connected. Moreover, the restriction map $H^1(M_2^\circ ) \rightarrow H^1(\mathbb {S}^1 \times \mathbb {S}^3)$ is surjective. Thus, Eq. (9) reduces to

$$\begin{aligned} 0\rightarrow H^2(M_1 \#_{\mathbb {S}^1} M_2) \xrightarrow {\sim } H^2(M_1^\circ ) \oplus H^2(M_1^\circ ) \rightarrow 0. \end{aligned}$$

In particular, by naturality of the Stiefel-Whitney classes, $w_2(M)=w_2(M_1^\circ ) \oplus w_2(M_2^\circ )$, so that $w_2(M)$ is trivial if and only if both $w_2(M_1^\circ )$ and $w_2(M_2^\circ )$ are.

Hence, it is left to prove that $w_2(M_i)$ is trivial if and only if $w_2(M_i^\circ )$ is. This can be argued using the interpretation of $w_2$ as the obstruction to find a trivialization of the tangent bundle over the $2-$skeleton of a cellular decomposition of the manifold. Start by choosing a cellular decomposition of $\mathbb {S}^1\times \mathbb {S}^3$ without $2-$cells, and extend it to a cellular decomposition of $\mathbb {S}^1\times \mathbb {D}^4$ also without $2-$cells; this can be further extended to a cellular decomposition of $M_i$. Hence, by construction the induced cellular decomposition on $M_i^\circ =M_i\setminus {{\,\textrm{Int}\,}}(\mathbb {S}^1\times \mathbb {D}^4)$ has the same $2-$skeleton as that of $M_i$. In particular, as $TM_i^\circ = TM_i\vert _{M_1^\circ }$, the obstruction to find a trivialization of the tangent bundle of $M_i$ and $M_i^\circ$ over their respective (coinciding) $2-$skeleton is the same; in other words, $w_2(M_i)$ is trivial if and only if $w_2(M_i^\circ )$ is, as desired. $\square$

4.4 Connected sums with exotic spheres

Let $f:\mathbb {S}^{2n-1} \rightarrow \mathbb {S}^{2n-1}$ be an (orientation preserving) diffeomorphism. The twisted sphere with twist f is defined by

$$\begin{aligned} S_f:= \mathbb {D}^{2n} \cup _f \mathbb {D}^{2n}.\end{aligned}$$

The diffeomorphism type of $S_f$ depends only on the isotopy class of f. Thus we may assume that f fixes a neighborhood of a point of the sphere, i.e. that it is a compactly supported diffeomorphism of ${\mathbb {R}}^{2n+1}$, extended as the identity to $\mathbb {S}^{2n+1}$.

Milnor [33] showed that suitable choices of f yield exotic spheres. As a consequence, Lawson [29, Corollary 6] showed that exotic spheres admit (smooth) foliations. To be precise, consider an embedding of ${\mathbb {R}}^{2n}$ into a leaf of a foliation on $\mathbb {S}^{2n+1}$. Cutting $\mathbb {S}^{2n+1}$ along this region and gluing back using a diffeomorphism f we obtain $S_f$. Since the cut is made inside a leaf, $S_f$ naturally inherits a foliation.

In turn, we use the results in [12, Appendix by S. Courte] to adapt this statement to conformal symplectic foliations. That is, we prove that if M admits a conformal symplectic foliation then so does the connected sum with any exotic sphere:

Proof of Theorem 1.7

Suppose M admits a conformal symplectic foliation $({\mathcal {F}},\eta ,\omega )$, and consider a loop $\gamma \subset M$ transverse to ${\mathcal {F}}$. On a neighborhood $\mathbb {S}^1 \times \mathbb {D}^{2n}$ of $\gamma$ the conformal symplectic foliation is equivalent to

$$\begin{aligned} {\mathcal {F}}= \bigcup _{\theta \in \mathbb {S}^1 } \{\theta \} \times \mathbb {D}^{2n},\quad \eta = 0,\quad \omega = \omega _{st}.\end{aligned}$$

We then cut M along $\mathbb {S}^1 \times \mathbb {S}^{2n-1}$ contained in the above local model, and turbulize both boundaries using Theorem 4.5. The resulting conformal symplectic foliation has two boundary leaves each isomorphic to $\left( \mathbb {S}^1 \times \mathbb {S}^{2n-1},\, \eta _\partial = {{\text {d}}}\theta ,\, \omega _\partial = {{\text {d}}}_{{{\text {d}}}\theta } \alpha _{st}\right)$. We insert the trivial cobordism $[0,1]\times \mathbb {S}^1 \times \mathbb {S}^{2n-1}$ with the conformal symplectic foliation from Lemma 4.8. This smoothly recovers M but with a different conformal symplectic foliation, which contains a compact leaf

$$\begin{aligned} \left( \mathbb {S}^1 \times \mathbb {S}^{2n-1}, \eta = R {{\text {d}}}\theta ,\, \omega = {{\text {d}}}_{R {{\text {d}}}\theta } \alpha _{ot}\right) ,\end{aligned}$$

for $R> 0$ arbitrarily large. Now, if $R\gg R_0>0$, the piece of symplectization $\left( ( -R_0,R_0) \times \mathbb {S}^{2n-1}, {{\text {d}}}(e^t \alpha _{ot}) \right)$ can be embedded in this compact leaf.

Consider now the symplectization ${\mathbb {R}}^{2n}$ of the standard overtwisted contact structure on ${\mathbb {R}}^{2n-1}$. By [12, Theorem 9.2] any given compactly supported diffeomorphism $\psi$ of ${\mathbb {R}}^{2n}$ can be realized, up to isotopy, by a compactly supported symplectomorphism $\psi _s$ of the symplectization $\left( \mathbb {R}\times \mathbb {S}^{2n-1}, {{\text {d}}}(e^t\alpha _{ot})\right)$. Notice that, once $\psi _s$ is fixed, there exists $R_0>0$ very big, so that the whole isotopy (hence $\psi$ in particular) is compactly supported in $(-R_0,R_0)\times \mathbb {S}^{2n-1}$. To conclude the proof, one can then simply cut M along $(-R_0,R_0) \times \mathbb {S}^{2n-1}$, and glue it back via $\psi _s$.

As the starting exotic diffeomorphism $\psi$ varies, the resulting conformally foliated manifold can then be made diffeomorphic to the connected sum of M with any given exotic sphere $\Sigma$. $\square$

5 Conformal symplectic foliations on simply-connected 5-manifolds

We prove here Theorem 1.5, stating the existence of conformal symplectic foliations on every simply-connected $5-$manifold. The strategy we follow goes along the lines of that in [1], although with the needed adaptations in order to ensure that the constructed foliations are conformal symplectic.

First, as a direct consequence of Theorem 4.5 and Proposition 4.3 we obtain the following:

Corollary 5.1

Let $(W_i,\omega _i)$, $i=1,2$ be symplectic manifolds with the same (convex/concave) contact boundary B, and $\phi _i:W_i\rightarrow W_i$ a symplectomorphism which is the identity near the boundary. Then the gluing

$$\begin{aligned} (W_1)_{\phi _1} \cup _{B \times \mathbb {S}^1} (W_2)_{\phi _2},\end{aligned}$$

of the mapping tori $(W_i)_{\phi _i}$ admits a conformal symplectic foliation.

The foliation is obtained by turbulizing the natural symplectic foliation on each of the mapping tori, and then gluing along their common boundary.

We use this result to construct symplectically foliated mapping tori $N_k$ satisfying $H_2(N_k)=\mathbb {Z}_k\oplus \mathbb {Z}_k$. Then, we describe explicit conformal symplectic foliations on $\mathbb {S}^5$, $\mathbb {S}^2\times \mathbb {S}^3$, $X_\infty$ and $X_\infty \# \,\mathbb {S}^2\times \mathbb {S}^3$. Lastly, in order to get a conformal symplectic foliation on any simply connected $5-$manifold, we take round-connect sum (Sect. 4.3) of properly chosen $N_k$’s, together with a symplectically foliated $\mathbb {S}^5$, $X_\infty$ or $X_\infty \#\,\mathbb {S}^2\times \mathbb {S}^3$ (c.f. Sect. 2.1 for the definition of $X_\infty$), with possibly one symplectically foliated $\mathbb {S}^2\times \mathbb {S}^3$, and with as many copies as needed to get the right free part of $H_2$ of an explicit mapping torus P with $H_2(P)=\mathbb {Z}\oplus \mathbb {Z}$.

Constructing the mapping tori with $H_2 = \mathbb {Z}_k\oplus \mathbb {Z}_k$. Let $k\ge 1$, and consider the Brieskorn manifold $S=\Sigma (2,2,k,1)=\{z_0^2+z_1^2+z_2^k+z_3=0\} \cap \mathbb {S}^7\subset \mathbb {C}^4$. Notice that S is naturally diffeomorphic to $\mathbb {S}^5$. Consider moreover the contact open book decomposition

$$\begin{aligned} S\rightarrow \mathbb {C}, (z_0,z_1,z_2,z_3) \mapsto z_3. \end{aligned}$$

This open book decomposition has pages which are naturally diffeomorphic to the Brieskorn variety $V:=V(2,2,k)=\{z_0^2+z_1^2+z_3^k=\epsilon \}\subset \mathbb {C}^3$. Denote then the symplectic monodromy of this open book by $\phi :V \rightarrow V$, and its mapping torus by $V_\phi$. Consider also the double $N_k$ of $V_\phi$, i.e. two copies of $V_\phi$ (one with the orientation reversed) glued along their boundary via the identity map. As done for instance in [15, Section 4], one can explicitly compute that $H_2(N_k)=\mathbb {Z}_k\oplus \mathbb {Z}_k$. Moreover, by the exact sequence of homology groups for mapping tori (see e.g. [25, Example 2.48]), $\pi _1(N_k)=\mathbb {Z}$ as V is simply connected. Lastly, notice that, by Corollary 5.1, $N_k$ admits a conformal symplectic foliation which just coincides with the fibers of the mapping tori away from the gluing region in the construction of the double.

Constructing a mapping torus with $H_2 = \mathbb {Z}\oplus \mathbb {Z}$. Consider this time the link of singularity $S=\{(z_0+z_1^2)(z_0^2+z_1^5)+z_2^2+z_3=0\}\cap \mathbb {S}^7\subset \mathbb {C}^4$. Again, S is naturally diffeomorphic to $\mathbb {S}^5$. Moreover, the projection

$$\begin{aligned} S\rightarrow \mathbb {C}, \quad (z_0,z_1,z_2,z_3) \mapsto z_3 \end{aligned}$$

describes an open book decomposition of $\mathbb {S}^5$ with pages which are naturally diffeomorphic to $T:=\{(z_0+z_1^2)(z_0^2+z_1^5)+z_2^2=\epsilon \}\subset \mathbb {C}^3$. Let $\psi :T \rightarrow T$ be the symplectic monodromy, and $T_\psi$ its mapping torus. Let also P be the double of $T_\psi$, constructed analogously to the $N_k$’s above. As explained again in [15, Section 4], one can explicitly compute that $H_2(P)=\mathbb {Z}\oplus \mathbb {Z}$. What’s more, as in the previous case, one gets $\pi _1(P)=\mathbb {Z}$. Lastly, by Corollary 5.1, P also admits a conformal symplectic foliation, coinciding with the fibers of the mapping tori away from the gluing region of the double.

Conformal symplectic foliations on $\mathbb {S}^5$, $\mathbb {S}^2\times \mathbb {S}^3$ and $X_\infty \# \mathbb {S}^2\times \mathbb {S}^3$. These manifolds actually admit symplectic foliations. The case of $\mathbb {S}^5$ is dealt with in [34] (and in [39, Chapter 6], where Mitsumatsu’s proof has been rephrased in a way which is more similar to the strategy adopted in Sect. 3.1). On $\mathbb {S}^2\times \mathbb {S}^3$ one can just use the symplectic foliation defined by the product of the Reeb foliation on $\mathbb {S}^3$ by the $\mathbb {S}^2-$factor. Lastly, on $X_\infty \# \mathbb {S}^2\times \mathbb {S}^3$ we just use the symplectic foliations constructed in Theorem 1.6.

Conformal symplectic foliation on $X_\infty$. Recall from the classification in Sect. 2.1 that $X_\infty = \mathbb {S}^2 \widetilde{\times } \mathbb {S}^3$ is the nontrivial $\mathbb {S}^3$-bundle over $\mathbb {S}^2$. Such bundles can be constructed by choosing a loop of diffeomorphisms $\phi : \mathbb {S}^1 \rightarrow {{\,\textrm{Diff}\,}}(\mathbb {S}^3)$ and using it to glue two trivial $\mathbb {S}^3-$bundles over $\mathbb {D}^2$ along their common boundary $\mathbb {S}^1\times \mathbb {S}^3$:

$$\begin{aligned} \mathbb {D}^2 \times \mathbb {S}^3 \cup _\phi \mathbb {D}^2 \times \mathbb {S}^3.\end{aligned}$$

Up to isomorphism the resulting bundle only depends on $[\phi ] \in \pi _1({{\,\textrm{Diff}\,}}(\mathbb {S}^3))$. Moreover, according to Hatcher’s result that $\textrm{Diff}(\mathbb {S}^3)\simeq O(4)$ [24], the latter group is isomorphic to $\mathbb {Z}/2$ (and generated by the loop of diffeomorphisms given by flowing along the Hopf fibers). If $[\phi ]$ is the non-trivial element of $\pi _1({{\,\textrm{Diff}\,}}(\mathbb {S}^3))$, the above gluing gives $X_\infty$.

Consider now a loop in $\mathbb {S}^5$, transverse to the symplectic foliation constructed by Mitsumatsu [34]. On a neighborhood $\mathbb {S}^1\times \mathbb {D}^4$ of this loop the symplectic foliation looks like

$$\begin{aligned} \left( \mathbb {S}^1 \times \mathbb {D}^4, {\mathcal {F}}= \bigcup _{z \in \mathbb {S}^1} \{z\} \times \mathbb {D}^4,\omega = \omega _{st}\right) ,\end{aligned}$$

where $\omega _{st}$ is the standard symplectic structure on $\mathbb {D}^4$.

Removing this neighborhood yields a symplectic foliation on $\mathbb {S}^5\setminus \mathbb {S}^1\times \mathbb {D}^4$ which on a neighborhood $(-\varepsilon ,0] \times \overline{\mathbb {S}^3} \times \mathbb {S}^1$ looks like

$$\begin{aligned} {\mathcal {F}}= \ker {{\text {d}}}\theta ,\quad \omega = {{\text {d}}}(e^{-t}\alpha _3), \end{aligned}$$

(10)

where $t\in (-\varepsilon ,0]$, $\theta \in \mathbb {S}^1$, and $\alpha _3 \in \Omega ^1(\mathbb {S}^3)$ denotes the standard contact structure. Notice that, as any two loops in $\mathbb {S}^5$ are isotopic, then one simply has $\mathbb {S}^5\setminus \mathbb {S}^1\times \mathbb {D}^4 = \mathbb {D}^2\times \mathbb {S}^3$ (because this is true for the standard unknot). In other words, we get a symplectic foliation on $\mathbb {D}^2\times \mathbb {S}^3$ which on a neighborhood¹$(-\varepsilon ,0] \times \overline{\mathbb {S}^3} \times \mathbb {S}^1$ of the boundary looks like in Eq. (10).

Now, applying the turbulization construction from Theorem 4.5 we obtain a conformal symplectic foliation on $\mathbb {S}^5\setminus \mathbb {S}^1\times \mathbb {D}^4$ which is tame at the boundary, and has as boundary leaf

$$\begin{aligned} \left( \{0\}\times \overline{\mathbb {S}^3} \times \mathbb {S}^1, \, \eta = {{\text {d}}}\theta ,\, \omega = {{\text {d}}}_{{{\text {d}}}\theta } \alpha _3\right) . \end{aligned}$$

(11)

If we consider $\mathbb {S}^3 \subset \mathbb {C}^2$ then a generator of $\pi _1({{\,\textrm{Diff}\,}}(\mathbb {S}^3))$ is given by the path

$$\begin{aligned} \phi _\theta (z_1,z_2) = (e^{i\theta }z_1, z_2),\end{aligned}$$

for $\theta \in \mathbb {S}^1$; see for example [40, p.263]. Moreover it is easily checked that $\phi _t$ preserves the standard contact form on $\mathbb {S}^3$. This implies that the gluing map

$$\begin{aligned} \phi :\mathbb {S}^1 \times \mathbb {S}^3 \rightarrow \mathbb {S}^1 \times \mathbb {S}^3,\quad (\theta ,x) \mapsto (\theta ,\phi _\theta (x)),\end{aligned}$$

used to construct $X_\infty$, is an isomorphism of the conformal symplectic structure from Equation (11). Hence, we obtain a conformal symplectic foliation on $X_\infty$.

Proof of Theorem 1.5. Let now M be the almost contact simply-connected $5-$manifold on which we need to construct the conformal symplectic foliation. According to Smale-Barden’s statement, it is enough to construct a conformal symplectic foliation on an almost contact, simply-connected manifold $M'$ such that it has the same $H_2$ as M and $w_2(M')$ is trivial if and only if $w_2(M)$ is.

Let’s start by dealing with the case of M spin, i.e. $w_2(M)=0$. Write then $H_2(M)=\mathbb {Z}^r \oplus \bigoplus _j \mathbb {Z}_{k_j}\oplus \mathbb {Z}_{k_j}$. We now further distinguish two cases: $r=2k$ or $r=2k+1$, for some $k\ge 0$.

• In the case of $r=2k$, we consider the round connected sum $M'$ of the following conformal-symplectically foliated spin manifolds. First, $\mathbb {S}^5$ with the Mitsumatsu foliation and a transverse curve $\gamma _0$ in it. Then, for each j, the mapping tori $N_{k_j}$ with curves $\gamma _{k_j}$ transverse to the foliations and such that they generate the (non-torsion) fundamental group $\pi _1(N_{k_j})$. Lastly, k copies $T_1,\ldots , T_k$ of the conformally symplectically foliated mapping torus T with k transverse curves $\delta _1,\ldots ,\delta _k$, each generating the fundamental group in the respective $T_i$.

• In the case of $r=2k+1$, we consider as $M'$ the round connected sum of the following conformal-symplectically foliated spin manifolds. First, $\mathbb {S}^2\times \mathbb {S}^3$ with the above described symplectic foliation, and a transverse curve $\gamma _0$ in it. Then, for each j, the mapping tori $N_{k_j}$ with curves $\gamma _{k_j}$ transverse to the foliations and such that they generate the (non-torsion) fundamental group $\pi _1(N_{k_j})$. Lastly, k disjoint copies $T_1,\ldots , T_k$ of the conformally symplectically foliated mapping torus T with r disjoint transverse curves $\delta _1,\ldots ,\delta _k$, each generating the fundamental group in the respective $T_i$.

Notice that all these transverse curves exist by construction of the foliations. Now, using Proposition 4.9 and Lemma 4.10 inductively (notice that in each set of generators for the connect sum, there is only one which is simply connected, while in the others the chosen curves are non-torsion and generate the $\pi _1$), $M'$ can be seen to be spin, satisfy $H_2(M')=H_2(M)$, and to admit a conformal symplectic foliation, as desired.

Let’s now deal with the non-spin case. Write here $H_2(M)=\mathbb {Z}^{r+1} \oplus \bigoplus _j \mathbb {Z}_{k_j}\oplus \mathbb {Z}_{k_j}$ (notice that in the non-spin almost contact case, ${{\text {rk}}}(H_2(M;\mathbb {Z}))\ge 1$ by the Smale-Barden’s classification). As before, we further distinguish two cases: $r=2k$ or $r=2k+1$ for some $k\ge 0$.

• In the case of $r=2k$, we consider the round connected sum $M'$ of the following conformal-symplectically foliated spin manifolds. First, $X_\infty$ with the conformal symplectic foliation described above and a transverse curve $\gamma _\infty$ in it. Then, for each j, the mapping tori $N_{k_j}$ with curves $\gamma _{k_j}$ transverse to the foliations and such that they generate the (non-torsion) fundamental group $\pi _1(N_{k_j})$. Lastly, k copies $T_1,\ldots , T_k$ of the conformally symplectically mapping torus T with k disjoint transverse curves $\delta _1,\ldots ,\delta _k$, each generating the fundamental group of the respective $T_i$.

• In the case of $r=2k+1$, we consider as $M'$ the round connected sum of the following conformal-symplectically foliated spin manifolds. First, $X_\infty \# \mathbb {S}^2\times \mathbb {S}^3$ with the above described conformal symplectic foliation, and a transverse curve $\gamma _\infty '$ in it. Then, for each j, the mapping tori $N_{k_j}$ with curves $\gamma _{k_j}$ transverse to the foliations and such that they generate the (non-torsion) fundamental group $\pi _1(N_{k_j})$. Lastly, k disjoint copies $T_1,\ldots , T_k$ of the conformally symplectically foliated mapping torus T with r disjoint transverse curves $\delta _1,\ldots ,\delta _k$, each generating the fundamental group of the respective $T_i$.

Using again Proposition 4.9 and Lemma 4.10 inductively, $M'$ can be seen to have non-trivial $w_2$, to satisfy $H_2(M')=H_2(M)$, and to admit a conformal symplectic foliation.

6 Existence of conformal symplectic foliations in high dimensions

6.1 Conformal symplectic foliated cobordisms

Consider an oriented manifold $M^{2n}$ endowed with a contact foliation $({\mathcal {F}},\xi :=\ker \alpha )$. That is, ${\mathcal {F}}$ is a (cooriented, codimension-1) foliation and $\alpha \in \Omega ^1({\mathcal {F}})$ satisfies

$$\begin{aligned} \alpha \wedge {{\text {d}}}\alpha ^{n-1} \ne 0.\end{aligned}$$

Since ${\mathcal {F}}$ is cooriented its leaves inherit an orientation from M. We say a contact foliation $({\mathcal {F}},\xi = \ker \alpha )$ is positive (resp. negative) if $\alpha$ is a positive (resp. negative) contact form on ${\mathcal {F}}$.

Recall that $\lambda \in \Omega ^1(M)$ is a Liouville form for the conformal symplectic structure $(\eta ,\omega )$ if

$$\begin{aligned} \omega = {{\text {d}}}_\eta \lambda .\end{aligned}$$

Such a form is equivalent to the choice of a Liouville vector field $Z \in {\mathcal {X}}(M)$, uniquely defined by $\lambda = \iota _Z \omega$. A cooriented hypersurface $\Sigma \subset (M,\eta ,\omega )$ is of positive (resp. negative) contact type, if there is a Liouville form $\lambda$ for $\omega$ whose restriction defines a positive (resp. negative) contact form on $\Sigma$. This is equivalent to the Liouville vector field being positively (resp. negatively) transverse to $\Sigma$, with respect to its coorientation.

Definition 6.1

We say that $(W,{\mathcal {F}},\eta ,\omega )$ is a foliated conformal symplectic cobordism between contact foliated manifolds $(M_\pm ,{\mathcal {F}}_\pm , \xi _\pm )$ if:

• 1. W is an oriented cobordism from $M_-$ to $M_+$, i.e. $\partial W = \overline{M}_- \sqcup M_+$.

• 2. $({\mathcal {F}},\eta ,\omega )$ is a conformal symplectic foliation transverse to the boundary which admits a (leafwise) Liouville form $\lambda$ near the boundary such that:

  • (a) ${\mathcal {F}}\cap M_\pm = {\mathcal {F}}_\pm$;

  • (b) $\lambda _\pm := \lambda |_{M_\pm }$ is a (positive) contact form defining $\xi _\pm$.

Remark 6.2

The second condition above is equivalent to the existence of a (leafwise) Liouville vector field Z for $(\eta ,\omega )$ which is transverse to $\partial W$. We denote by $\partial _+ W$ (resp. $\partial _- W$) the part of $\partial W$ where Z is pointing outwards (resp. inwards). Then, the above definition implies:

$$\begin{aligned} \partial _+ W = M_+, \quad \partial _- W = \overline{M}_-,\end{aligned}$$

and $\lambda _\pm = \iota _Z\omega |_{\partial _\pm W}$, defines (on $\partial _\pm W$) a positive/negative contact form for $\xi _\pm$. Also note that the equality in the first condition is not as cooriented foliations; c.f. Remark 2.9. Hence, comparing $\partial _-W = \overline{M}_-$ to $M_-$, it is the orientation of the leaves of ${\mathcal {F}}_-$ which changes. This is consistent with $\lambda _-$ being a (leafwise) negative contact form on $\partial _-W$ and a (leafwise) positive contact form on $M_-$.

In a categorical fashion, we will also denote a cobordism as in Definition 6.1 simply by an arrow:

$$\begin{aligned} (M_-,{\mathcal {F}}_-,\xi _-) \rightarrow (M_+,{\mathcal {F}}_+,\xi _+).\end{aligned}$$

Composition of morphisms is then just composition of cobordisms (using Theorem 2.8) and the identity morphism at $(M,{\mathcal {F}},\alpha )$ is given by the symplectization

$$\begin{aligned} \left( [0,1]\times M, \omega := {{\text {d}}}(e^t \alpha )\right) .\end{aligned}$$

Recall that a (contact) foliation is called unimodular if it can be defined by a closed 1-form. In this setting, we can use turbulization to “invert” cobordisms.

Proposition 6.3

The foliated conformal symplectic cobordism relation defines an equivalence relation, between manifolds with a unimodular contact foliation.

Proof

We can compose cobordisms using Theorem 2.8, and a cobordism from $(M,{\mathcal {F}}= \ker \gamma , \xi = \ker \alpha )$ to itself is given by the symplectization

$$\begin{aligned} \left( [0,1] \times M, {\mathcal {G}}= \ker \gamma , \omega = {{\text {d}}}(e^t\alpha )\right) .\end{aligned}$$

Hence the cobordism relation is transitive and reflexive.

To see it is symmetric consider a cobordism $(W,{\mathcal {F}}:= \ker \gamma , \eta , \omega )$ from $(M_-,{\mathcal {F}}_-:= \ker \gamma _-, \xi _-:= \ker \alpha _-)$ to $(M_+,{\mathcal {F}}_+:= \ker \gamma _+, \xi _+:= \ker \alpha _+)$. As a smooth manifold, the cobordism in the opposite direction can be described by

$$\begin{aligned} \widetilde{W}:= [-\varepsilon ,0] \times M_+ \cup _{M_+} \overline{W} \cup _{M_-} [0,\varepsilon ] \times M_-,\end{aligned}$$

where the gluing uses the (orientation reversing) identity map. Observe that

$$\begin{aligned} \partial \widetilde{W} = \overline{M}_+ \sqcup M_-,\end{aligned}$$

so that $\widetilde{W}$ is an oriented cobordism from $M_+$ to $M_-$.

On the middle piece $\overline{W}$ consider the conformal symplectic foliation $({\mathcal {F}}= \ker (- \gamma ), \eta , \omega )$. That is, we swap the coorientation of ${\mathcal {F}}$ to match changing the orientation on W. By turbulizing at both boundaries (Theorem 4.5) we obtain a foliation which is tame at the boundary and has compact leaves:

$$\begin{aligned} \left( M_+,\eta _+ + \gamma _+, {{\text {d}}}_{\eta _+ + \gamma _+} \alpha _+\right) ,\quad \text {and} \quad \left( M_-,\eta _-+\gamma _-,{{\text {d}}}_{\eta _- + \gamma _-} \alpha _-\right) ,\end{aligned}$$

where $\eta _\pm := \eta |_{M_\pm }$ and $\gamma _\pm :=\gamma \vert _{M_\pm }$. Note that the coorientation of the turbulized foliation is pointing inwards (into $\overline{W}$) along $M_+$ and outwards along $M_-$. On the left piece, $[-\varepsilon ,0] \times M_+$, consider the conformal symplectic foliation

$$\begin{aligned} \left( {\mathcal {F}}:= \ker \gamma _+,\eta _+,{{\text {d}}}_{\eta _+}(e^t \alpha _+)\right) .\end{aligned}$$

Again, using Theorem 4.5, we turbulize this foliation at $\{0\} \times M_+$ to make it tame with boundary leaf

$$\begin{aligned} \left( M_+, \eta _++\gamma _+, {{\text {d}}}_{\eta _+ + \gamma _+} \alpha _+\right) .\end{aligned}$$

The coorientation of the compact leaf is pointing outwards. This matches the negative boundary of the middle piece, so that the resulting (cooriented) foliation is smooth. The resulting left boundary is the contact foliated manifold

$$\begin{aligned} \left( \overline{M}_+, \widetilde{{\mathcal {F}}}_- = \ker \gamma _+,\widetilde{\xi }_- = \ker \alpha _+\right) \end{aligned}$$

Similarly we obtain a conformal symplectic foliation on the right piece $[0,\varepsilon ] \times M_-$, which glues smoothly to the middle piece and whose positive boundary is the contact foliated manifold

$$\begin{aligned} \left( M_- = \partial _- W,\widetilde{{\mathcal {F}}}_+ = \ker \gamma _-,\widetilde{\xi }_+ = \ker \alpha _-\right) . \end{aligned}$$

$\square$

The proof of Proposition 6.3 also shows that by attaching trivial cobordisms and turbulizing, we can transfer (connected components of) the positive boundary to the negative boundary and vice versa. More formally:

Lemma 6.4

Given two contact foliations $(M_i,{\mathcal {F}}_i,\xi _i)$, $i=1,2$, the following are equivalent:

• 1. $(M_0,{\mathcal {F}}_0,\xi _0) \rightarrow (M_1,{\mathcal {F}}_1,\xi _1)$;

• 2. $(M_0,{\mathcal {F}}_0, \xi _0) \sqcup (\overline{M}_1,\overline{{\mathcal {F}}}_1,\xi _1) \rightarrow \emptyset$;

• 3. $\emptyset \rightarrow (\overline{M}_0,\overline{{\mathcal {F}}}_0,\xi _0) \sqcup (M_1,{\mathcal {F}}_1,\xi _1)$.

Moreover, the corresponding cobordisms can be assumed to be diffeomorphic.

Proof

Suppose $(W,{\mathcal {F}},\eta ,\omega )$ is a cobordism as in the first statement. Then a neighborhood of the positive boundary is isomorphic to

$$\begin{aligned} \left( (-\varepsilon ,0] \times M_1, {\mathcal {F}}= \ker \gamma _1,\eta = \eta _1,\omega = {{\text {d}}}_{\eta _1} (e^t \alpha _1)\right) ,\end{aligned}$$

where ${\mathcal {F}}_1 = \ker \gamma _1$, and $\xi _1 = \ker \alpha _1$. By turbulizing we can make it tame at the boundary such that the boundary leaf is cooriented outwards and equal to $\left( M_1,{{\text {d}}}_{\eta _1 + \gamma _1}\alpha _1\right)$. Similarly, the trivial cobordism

$$\begin{aligned} \left( [0,1] \times M_1, {\mathcal {F}}:= - \ker \gamma _1, \eta := \eta _1,\omega := {{\text {d}}}_{\eta _1}(e^{-t} \alpha _1)\right) ,\end{aligned}$$

can be turbulized at $\{0\} \times M_1$. The resulting boundary leaf equals $(M_1,{{\text {d}}}_{\eta _1 + \gamma _1} \alpha _1)$ and is cooriented inwards (i.e. $\partial _t$ is positively transverse to it). As such the two cobordisms can be glued.

The Liouville vector field of the resulting cobordism points inwards at all boundary components, showing that

$$\begin{aligned} (M_0,{\mathcal {F}}_0,\xi _0) \sqcup (\overline{M}_1, \overline{{\mathcal {F}}}_1,\xi _1) \sim \emptyset .\end{aligned}$$

The other points can be proven analogously. $\square$

6.2 From almost symplectic to conformal symplectic

For simplicity, we introduce the following notation for product foliations. Given a cooriented positive contact structure $\xi$ on an oriented smooth manifold M, we denote by $\mathbb {S}^1\times (M,\xi )$ the contact foliation $({\mathcal {F}}=\ker ({{\text {d}}}\theta ),\xi )$, where $\theta \in \mathbb {S}^1$. The product foliation of $\mathbb {S}^1$ with of an (almost) symplectic manifold is defined analogously. In particular, Theorem 2.18 from [18] implies that products of $\mathbb {S}^1$ with almost symplectic foliated cobordisms from overtwisted to tight contact foliations can be homotoped to symplectic ones.

It is worth pointing out that in the (non-foliated) symplectic setting the assumption that the negative boundary is overtwisted is necessary, i.e. Theorem 2.18 is in general false without this assumption. This is simply a consequence of the fact that overtwisted contact manifolds are not symplectically fillable [5, 38]. In particular, tight and overtwisted structures play a very different role with respect to the symplectic cobordism relation. In the foliated setting this difference vanishes, at least in the conformal setting, as we shall see in this section.

Remark 6.5

Throughout this whole section, we will often use either Theorem 2.18 and Remark 2.19, or Theorem 2.20, on product almost symplectic foliations $(\mathbb {S}^1\times W, {\mathcal {F}}=\ker {{\text {d}}}\theta ,\Omega )$, with $\theta \in \mathbb {S}^1$. Notice that these results easily give a holonomy-like leafwise exact conformal symplectic structure on ${\mathcal {F}}$. Indeed, the resulting $(\eta ,\omega )$ satisfies $\omega ={{\text {d}}}_\eta \lambda$, for an exact $\eta$ which is 0 near the boundary. Then, $\eta$ can be realized as a holonomy form of a $1-$form $\gamma$ defining ${\mathcal {F}}$ which is just (a constant multiple of) ${{\text {d}}}\theta$ near the boundary.

Recall that Proposition 6.3 says that the conformal symplectic foliated cobordism relation is symmetric. In what follows, we will need a more precise version of Proposition 6.3 under the assumption of (topologically) trivial cobordisms, which allows to keep track of the underlying almost contact structure.

Lemma 6.6

Consider a homotopy of almost contact structures $(\alpha _t,\omega _t)$ between contact forms $\alpha _0$ and $\alpha _1$ on M. Suppose that $\alpha _1$ is overtwisted, and if $\dim M = 3$ additionally assume that $\alpha _0$ is overtwisted. Then, there exists a $\vert$holonomy$\vert$-like exact conformal symplectic foliated cobordism

$$\begin{aligned} \left( \mathbb {S}^1 \times [0,1]_t \times M, {\mathcal {F}},\eta , {{\text {d}}}_\eta \lambda \right) : \mathbb {S}^1 \times (M,\alpha _0) \rightarrow \mathbb {S}^1 \times (M,\alpha _1),\end{aligned}$$

Moreover:

• (i) $({\mathcal {F}},\eta ,{{\text {d}}}_\eta \lambda )$ is in the same almost contact class (relative to the boundary) as $\left( \ker {{\text {d}}}\theta , {{\text {d}}}t \wedge \alpha _t + \omega _t\right)$;

• (ii) Near the boundary component at $t=0$ (resp. $t=1$), $\lambda$ equals $e^t\alpha _0$ (resp. $e^t\alpha _1$) and $\eta =0$.

• (iii) $(\eta ,{{\text {d}}}_\eta \lambda )$ is holonomy-like away from a closed leaf.

Proof

Starting from the almost symplectic foliation

$$\begin{aligned} \left( \mathbb {S}^1 \times [0,1] \times M, {\mathcal {F}}_0:= \ker d\theta , \omega _0:= {{\text {d}}}t \wedge \alpha _t + \omega _t\right) ,\end{aligned}$$

the homotopy to $({\mathcal {F}},\eta ,\omega )$ is described in two steps. First we homotope the hyperplane distribution to the desired foliation, keeping track of the almost symplectic structure. This homotopy is depicted in Fig. 1b. The resulting foliation contains two closed leaves in the interior, diffeomorphic to $\mathbb {S}^1 \times M$. After this, we keep the foliation fixed and homotope the leafwise almost symplectic structure into a leafwise conformal symplectic structure. Let us now give the details.

First, to define the homotopy of almost symplectic foliations, consider the following 1-parameter families:

$$\begin{aligned} \gamma _s:= \cos (\varphi _s(t)) {{\text {d}}}\theta \!-\! \sin (\varphi _s(t)) {{\text {d}}}t, \quad \Omega _s:= \left( \cos (\varphi _s(t)) {{\text {d}}}t \!+\! \sin (\varphi _s(t)\right) {{\text {d}}}\theta ) \wedge \alpha _t \!+\! \omega _t, \end{aligned}$$

where $s \in [0,1]$, and $\varphi _s$ is a $[0,1]-$family of functions as in Fig. 1. We denote ${\mathcal {F}}_s=\ker \gamma _s$ for every $s\in [0,1]$. For later use, we also assume that $\varphi _1$ is such that $|\dot{\varphi }_1(t) \sin (\varphi _1(t))|$ is a smooth function, which can be arranged as the picture shows. It is easily seen that $(\gamma _s,\Omega _s)$ defines a homotopy of almost symplectic foliations since:

$$\begin{aligned} \gamma _s \wedge {{\text {d}}}\gamma _s = 0,\quad \gamma _s \wedge \Omega _s^n = n {{\text {d}}}\theta \wedge {{\text {d}}}t \wedge \alpha _t \wedge \omega _t^{n-1} > 0.\end{aligned}$$

Secondly, we want to homotope $\Omega _1$ to a leafwise exact conformal symplectic structure on ${\mathcal {F}}_1$ of the form

$$\begin{aligned} \Omega _2:= {{\text {d}}}_{\eta } \lambda ,\end{aligned}$$

where $\eta$ is defined by

$$\begin{aligned} \eta := - |\dot{\varphi }| {{\text {d}}}\theta , \end{aligned}$$

with $\varphi :=\varphi _1(t)$, and $\lambda$ satisfies

$$\begin{aligned} \lambda = {\left\{ \begin{array}{ll} e^{g(t)} \alpha _0 &{} \text {on the left component of }{{\,\textrm{supp}\,}}(\pi -\varphi ) \\ e^{g(t)} \alpha _1 &{} \text {on the right component of }{{\,\textrm{supp}\,}}(\pi -\varphi ), \end{array}\right. } \end{aligned}$$

(12)

for $g:[0,1] \rightarrow {\mathbb {R}}$ as in (the caption of) Fig. 2. Notice that

$$\begin{aligned} \gamma _1 \wedge ({{\text {d}}}_\eta e^g\alpha _i)^n = n e^{ng}(\vert \dot{\varphi } \sin \varphi \vert +\dot{g}\cos \varphi ) \, {{\text {d}}}\theta \wedge {{\text {d}}}t\wedge \alpha _i \wedge {{\text {d}}}\alpha _i^{n-1}, \end{aligned}$$

so that the pair $(\eta ,\lambda )$ indeed give, on their domain of definition, a leafwise conformal symplectic structure on the foliation ${\mathcal {F}}_1$. We also point out that, up to sign, $\eta$ equals the holonomy form of $\gamma _1$, so that $\Omega _2$ is $\vert$holonomy$\vert$-like . The conditions in Eq. (12) implies that, around the compact leaves, $\Omega _2$ equals the result of the turbulization construction in Theorem 4.5.

We then need to find a $\lambda$ such as above, and show that the resulting $\Omega _2$ is homotopic to $\Omega _{1}$ through ${\mathcal {F}}_1-$leafwise almost symplectic forms. To this end, observe that a homotopy $(\alpha _{t,s},\omega _{t,s})$ of 1-parameter families of almost contact structures on M induces a homotopy of ${\mathcal {F}}_1$-leafwise almost symplectic forms

$$\begin{aligned} \Omega _{1,s}:= \left( \cos (\varphi ) {{\text {d}}}t + \sin (\varphi ) {{\text {d}}}\theta \right) \wedge \alpha _{t,s} + \omega _{t,s}\end{aligned}$$

on $\mathbb {S}^1\times [0,1]\times M$. We apply this with any homotopy $(\alpha _{t,s},\omega _{t,s})_{s\in [0,1]}$ satisfying the following properties:

• (i) $(\alpha _{t,s}, \omega _{t,s}) = (\alpha _t,\omega _t)$ for $(s,t) \in \{0\} \times [0,1] \cup [0,1] \times \{0,1\}$. This ensures the homotopy starts at $(\alpha _t,\omega _t)$ and is relative to the boundary;

• (ii) The homotopy ends at $(\alpha _{t,1},\omega _{t,1})$ such that

  $$\begin{aligned} (\alpha _{t,1},\omega _{t,1}) = {\left\{ \begin{array}{ll} \left( e^{g(t)} \alpha _0, e^{g(t)} {{\text {d}}}\alpha _0\right) &{} \text {on the left component of }{{\,\textrm{supp}\,}}(\pi -\varphi ) \\ \left( e^{g(t)} \alpha _1, e^{g(t)} {{\text {d}}}\alpha _1\right) &{} \text {on the right component of }{{\,\textrm{supp}\,}}(\pi -\varphi ), \end{array}\right. }\end{aligned}$$

  with g as before.

We point out that such a homotopy clearly exists, and has simply the effect of arranging the desired form near the boundary and “pushing the unknown behavior” on the complement of ${{\,\textrm{supp}\,}}(\pi -\varphi )$ (which will in a moment be dealt thanks to the h-principle in [18]).

On ${{\,\textrm{supp}\,}}(\pi -\varphi )$ we then define $\lambda$ by Eq. (12). A straightforward computation shows that, in this region, the linear interpolation $\Omega _r:=(1-r)\Omega _{1,1}+r\Omega _2$ from $\Omega _{1,1}$ to $\Omega _2$, which is clearly relative to the boundary, is moreover through leafwise almost symplectic forms on ${\mathcal {F}}_1$, because

$$\begin{aligned} \gamma _1 \wedge \Omega _r^n = n e^{ng}[r(\vert \dot{\varphi } \sin \varphi \vert +\dot{g}\cos \varphi ) + 1-r] \, {{\text {d}}}\theta \wedge {{\text {d}}}t\wedge \alpha _i \wedge {{\text {d}}}\alpha _i^{n-1}, \end{aligned}$$

which is a positive volume form for $r\in [0,1]$. Lastly, as ${\mathcal {F}}_1$ is just a product foliation and $\eta =0$ on the middle piece $\mathbb {S}^1\times [0,1]\times M \setminus {{\,\textrm{supp}\,}}(\pi - \varphi )$, [18] and Remark 2.19 give a ${\mathcal {F}}_1$-leafwise homotopy relative to the boundary from $\Omega _{1,1}$ to a ${\mathcal {F}}_1$-leafwise exact symplectic structure ${{\text {d}}}_\lambda ={{\text {d}}}_\eta \lambda$, where $\lambda$ extends the one already defined on ${{\,\textrm{supp}\,}}(\pi -\varphi )$ (as the homotopy is relative to the boundary, and everything has been already previously arranged on ${{\,\textrm{supp}\,}}(\pi -\varphi )$). $\square$

Fig. 1

The initial homotopy to an almost symplectic foliation in the proof of Lemma 6.6

Fig. 2

The function g in Lemma 6.6 is obtained by smoothing the piecewise linear function in the picture, where $t_0$ and $t_1$ are the zeroes of $\cos \varphi$. More precisely, the smoothing is done so that $\dot{g}$ is strictly positive before $t_0$ and after $t_1$, strictly negative between them, and all the derivatives of g are zero at $t_0$ and $t_1$

An immediate consequence is the following more precise version of Corollary 1.3, which tells that the product foliation $\mathbb {S}^1 \times (\mathbb {S}^{2n+1},\xi _{ot})$ can be filled by a conformal symplectic foliation on the solid torus:

Corollary 6.7

Let $(\mathbb {D}^{2n},\Omega ):\emptyset \rightarrow (\mathbb {S}^{2n-1},\alpha _{ot})$, $n >2$, be an almost symplectic cobordism, with $\alpha _{ot}$ an overtwisted contact form in the almost contact class of $\alpha _{st}$. Then, there exists a foliated $\vert$holonomy$\vert$-like exact conformal symplectic cobordism

$$\begin{aligned}(W,{\mathcal {F}},\eta ,{{\text {d}}}_{\eta }\lambda ):\emptyset \rightarrow \mathbb {S}^1 \times (\mathbb {S}^{2n-1},\alpha _{ot})\end{aligned}$$

satisfying the following properties.

• (i) W is diffeomorphic to $\mathbb {S}^1 \times \mathbb {D}^{2n}$.

• (ii) $({\mathcal {F}},\eta ,{{\text {d}}}_\eta \lambda )$ is homotopic (through almost contact structures), relative to the boundary, to the almost symplectic foliation

  $$\begin{aligned} \big (\bigcup _{\theta \in \mathbb {S}^1} \{ \theta \} \times \mathbb {D}^{2n}, \,\Omega \big ).\end{aligned}$$

• (iii) On any collar neighborhood $(-\varepsilon ,0]\times \partial W$ we can arrange $\eta = 0$, and $\lambda = e^t\alpha _{ot}$.

• (iv) The foliation ${\mathcal {F}}$ contains two closed leaves, and $(\eta ,{{\text {d}}}_\eta \lambda )$ is holonomy-like away from one of the two.

Proof

Up to homotopy of $\Omega$, one can assume that, for $\mathbb {D}^{2n}_\epsilon$ a disk of very small radius $\epsilon$ centered at the origin, $\Omega$ restricts to $\mathbb {D}^{2n}_\epsilon$ simply as the standard symplectic structure $\omega _{std}$ on $\mathbb {R}^{2n}$. In particular, on the complement $\mathbb {D}^{2n}\setminus \mathbb {D}^{2n}_\epsilon$, $\Omega$ gives a homotopy of foliated contact forms from $\alpha _{std}$ to $\alpha _{ot}$. Consider now $(M:=\mathbb {S}^{1}\times \mathbb {D}^{2n},{\mathcal {F}}=\ker {{\text {d}}}\theta ,\Omega )$, and denote $T:=\mathbb {S}^1\times \mathbb {D}^{2n}_\epsilon$. One can then simply apply Lemma 6.6 on $M{\setminus } \mathring{T}\simeq \mathbb {S}^1\times [0,1]\times \mathbb {S}^{2n-1}$, which gives a homotopy, relative to the boundary, from $\Omega$ to a $\vert$holonomy$\vert$-like exact conformal symplectic structure on $M\setminus T$. This then glues well to $T= (\mathbb {S}^1\times \mathbb {D}^{2n}_\epsilon ,{\mathcal {F}}=\ker {{\text {d}}}\theta ,\omega _{std})$, which is holonomy-like exact conformal symplectic, along their common boundary, thus concluding the proof. $\square$

Although we do not need it for our applications, let us point out an explicit result confirming what we claimed above, namely that the fact that the concave boundary component of almost symplectic foliated cobordisms is overtwisted is not an essential assumption in order to obtain leafwise conformal symplectic structures on them. More precisely, we now prove that Lemma 6.6 remains true, in high dimensions, even if both the positive and negative boundary are tight:

Corollary 6.8

Let $\xi _0=\ker \alpha _0$ and $\xi _1=\ker \alpha _1$ be contact structures on $M^{2n-1}$, with $n\ge 3$, homotopic as almost contact structures via $(\alpha _t,\omega _t)$. Then, there exists an exact $\vert$holonomy$\vert$-like conformal symplectic foliated cobordism

$$\begin{aligned} \left( \mathbb {S}^1 \times [0,1] \times M, {\mathcal {F}},\eta ,{{\text {d}}}_\eta \lambda \right) ,\end{aligned}$$

from $\mathbb {S}^1\times (M,\xi _0)$ to $\mathbb {S}^1 \times (M,\xi _1)$. Furthermore:

• (i) $({\mathcal {F}},\eta ,{{\text {d}}}_\eta \lambda )$ is in the same foliated almost contact class (relative to the boundary) as $\left( \ker {{\text {d}}}\theta , {{\text {d}}}t \wedge \alpha _t + \omega _t\right)$, where t denotes the interval coordinate on $\mathbb {S}^1 \times [0,1] \times M$.

• (ii) Near the boundary, $\lambda$ equals $e^t\alpha _0$, resp. $e^t\alpha _1$, and $\eta =0$.

• (iii) $({\mathcal {F}},\eta ,{{\text {d}}}\lambda )$ is holonomy-like away from a closed leaf.

Proof

Consider $(V:=\mathbb {S}^1\times [0,1]_t\times M,{\mathcal {G}}=\ker {{\text {d}}}\theta ,\Omega = {{\text {d}}}t \wedge \alpha _t + \omega _t)$. Let also $N=\mathbb {S}^1_\theta \times \mathbb {D}^{2n}$ be a neighborhood of a (positively) transverse curve $\gamma$ where ${\mathcal {G}}=\ker {{\text {d}}}\theta$ and $\Omega = \omega _{std}$, with $\omega _{std}$ the standard symplectic structure on $\mathbb {R}^{2n}$. Up to $\mathbb {S}^1-$invariant homotopy (relative to the boundary of V) of $\Omega$, one can arrange that $\Omega$ restricts to ${{\text {d}}}r \wedge \alpha _{ot} + {{\text {d}}}\alpha _{ot}$ on a neighborhood $\mathbb {S}^1\times (-\epsilon ,\epsilon )_r\times \mathbb {S}^{2n-1}$ of $\partial N = \mathbb {S}^1\times \mathbb {S}^{2n-1}$ inside V (here, $\partial _r$ points outwards from N), with $\alpha _{ot}$ defining an overtwisted contact structure $\xi _{ot}$ on $\mathbb {S}^{2n-1}$ which is in the same almost contact class as the standard tight contact structure.

Then, Corollary 6.7 tells that $({\mathcal {G}},\Omega )$ on N can be homotoped, among almost contact structures and relative to the boundary (of N), to a $\vert$holonomy$\vert$-like exact conformal symplectic foliation on N, which is moreover holonomy-like away from a closed leaf. Moreover, on the complement $V\setminus N = \mathbb {S}^1\times ([0,1]\times M \setminus \mathbb {D}^{2n})$ there is a leafwise homotopy (relative to the boundary) of almost symplectic structures ending at an exact holonomy-like conformal symplectic leafwise structure according to Theorem 2.20 (c.f. Remark 6.5). Composing these two homotopies then give the desired homotopy on V. $\square$

We now state another corollary of the previous results, which is one of the main lemmas for the proof of Proposition 1.2 in Sect. 6.4.

Lemma 6.9

Let $(\alpha ,\omega )$ be an almost contact structure on $M^{2n+1}$, with $n> 2$. Let $\xi _\pm := \ker \alpha _\pm$ be overtwisted contact structures in the almost contact class $(\pm \alpha ,\omega )$, and $(\alpha _t,\omega _t)$ be an almost contact homotopy from $(\alpha _+,{{\text {d}}}\alpha _+)$ to $(-\alpha _-,{{\text {d}}}\alpha _-)$. Then, there is an exact $\vert$holonomy$\vert$-like conformal symplectic foliated cobordism

$$\begin{aligned} (W,{\mathcal {F}},\eta ,{{\text {d}}}_\eta \lambda ): \mathbb {S}^1 \times (M,\xi _+) \sqcup \mathbb {S}^1 \times (\overline{M},\xi _-) \rightarrow \emptyset \end{aligned}$$

satisfying the following properties.

• (i) The cobordism W is diffeomorphic to $\mathbb {S}^1 \times [0,1] \times M$;

• (ii) The conformal symplectic foliation $({\mathcal {F}},\eta ,\omega )$ is homotopic, relative to $\partial W$ and among almost contact structures, to the almost symplectic foliation

  $$\begin{aligned} \big ( \bigcup _{\theta \in \mathbb {S}^1} \{\theta \} \times [0,1]_t \times M, {{\text {d}}}t \wedge \alpha _t + \omega _t\big ). \end{aligned}$$

• (iii) Near the boundary components with $t=0$ and $t=1$, $\lambda$ coincides with ${{\text {d}}}(e^{\pm }\alpha _{\pm })$ respectively, and $\eta =0$.

• (iv) $({\mathcal {F}},\eta ,{{\text {d}}}_\eta \lambda )$ is holonomy-like away from two closed leaves.

Fig. 3

The picture depicts the several cobordisms which are stacked together in order to prove Lemma 6.9. (The foliated cobordism $\emptyset \sim (\mathbb {S}^{2n-1},\xi _{ot})\times \mathbb {S}^1$ used to fill the remaining boundary component is not depicted.)

Proof

The idea of the proof is completely analogous to that of Lemma 6.6, with an additional minor “complication”. First, one inverts the direction of the Liouville vector fields at both boundaries by conformal symplectic turbulization, as done in the proof of Lemma 6.6. However, this results here in the product of $\mathbb {S}^1$ with an almost symplectic filling of $(M,\xi _0)\sqcup (\overline{M},\xi _1)$, which has to be transformed to a foliated conformal symplectic filling, via a homotopy relative to the boundary. The idea is then to apply Theorem 2.20 (c.f. Remark 6.5) after removing a neighborhood of a transverse curve, and then use Corollary 6.7 to fill back such neighborhood. The building blocks are described in Fig. 3. For completeness, we also give the details of this proof via explicit formulas using what is done in the proof of Lemma 6.6.

As done with the first two homotopies in the proof of Lemma 6.6, one can first homotope rel. boundary (among almost contact structures) the almost symplectic foliation $(\ker {{\text {d}}}\theta ,\Omega )$ to an almost symplectic foliation $(\mathcal {F},\Omega ')$ such that:

• $\mathcal {F}$ is the desired (i.e. as in the conclusion of the statement) smooth foliation, obtained by turbulizing $\ker {{\text {d}}}\theta$ near both the boundary components;

• for some $\delta >0$, $\mathcal {F}=\ker (-{{\text {d}}}\theta )$ on $\mathbb {S}^1\times [\delta ,1-\delta ]\times M$;

• $\Omega '$ is $\theta -$invariant on $\mathbb {S}^1\times [\delta ,1-\delta ]\times M$;

• $\Omega '$ is a $\vert$holonomy$\vert$-like exact conformal symplectic leafwise structure $(\eta ,{{\text {d}}}_\eta \lambda )$ on $\mathbb {S}^1\times [0,\delta ]\times M \cup \mathbb {S}^1\times [1-\delta ,1]\times M$;

• $(\eta ,{{\text {d}}}_\eta \lambda )$ is holonomy-like away from two closed leaves, one in each connected component of $\mathbb {S}^1\times [0,\delta ]\times M \cup \mathbb {S}^1\times [1-\delta ,1]\times M$;

• near $\mathbb {S}^1\times \{\delta \}\times M$, $\eta =0$ and $\lambda = e^{-t}\alpha _0$;

• near $\mathbb {S}^1\times \{1-\delta \}\times M$, $\eta =0$ and $\lambda = e^{t}\alpha _1$.

Denote $X:=\mathbb {S}^1\times [\delta ,1-\delta ]\times M$ for simplicity. Consider now a neighborhood of a transverse curve $\mathbb {S}^1\times \{pt\}\times \{pt\}$ which is of the form $N:=\mathbb {S}^1_\theta \times \mathbb {D}^{2n}$, where ${\mathcal {F}}=\ker {{\text {d}}}\theta$. We also assume, up to a $\theta -$invariant homotopy, that $\Omega '$ looks like a piece of positive half-symplectization of $(\mathbb {S}^{2n+1},\xi _{ot})$ near the boundary of N. We then remove the interior of N from X, and denote $X'$ the resulting smooth manifold, as well as $(\mathcal {F}',\Omega '')$ the resulting almost symplectic foliation. Then, by Theorem 2.20 (c.f. Remark 6.5), $\Omega ''$ is homotopic, rel. the three boundary components of $X'$, to a holonomy-like exact conformal symplectic foliated structure on $(X',\mathcal {F}')$.

Lastly, the remaining piece is $\mathbb {S}^1 \times \mathbb {D}^{2n}$ with the almost symplectic foliation $(\ker {{\text {d}}}\theta ,\omega _{st})$. Using Corollary Corollary 6.7 we homotope it, relative to the boundary, to a $\vert$holonomy$\vert$-like exact conformal symplectic foliation, which is holonomy-like away from a closed leaf. This concludes the proof of Lemma 6.9. $\square$

6.3 Proof of Theorem 1.1

If $\dim M = 3$ the result follows from [44], so we assume $\dim M \ge 7$. Let $(\xi := \ker \alpha , \omega )$ be an almost contact structure. By [31] there exists a homotopy $\xi _t$, $t \in [0,1]$ from $\xi$ to a minimal foliation ${\mathcal {G}}$ (i.e. such that all leaves are dense). We claim that ${\mathcal {G}}$ admits a (leafwise) almost symplectic structure $\Omega$. Indeed, we can interpret $\xi _t$ as a vector bundle $\widehat{\xi }$ on $M \times [0,1]$, and using parallel transport it follows that

$$\begin{aligned}\widehat{\xi } \simeq \pi ^*\xi _0,\end{aligned}$$

where $\pi :M \times [0,1] \rightarrow M$ is the projection onto the first factor. Thus $\omega$ induces an almost symplectic structure on $\widehat{\xi }$, and in particular on ${\mathcal {G}}= \xi _1$. Now, since ${\mathcal {G}}$ is minimal, it is taut. Hence, applying Proposition 1.2 to $({\mathcal {G}},\Omega )$ concludes the proof.

6.4 Proof of Proposition 1.2

The strategy of the proof is to decompose the manifold into several pieces. We construct the desired homotopy on each of the pieces, and then show they can be glued together.

Since ${\mathcal {G}}$ is taut there exist closed transverse loops intersecting every leaf. Fix two (disjoint) such loops $\gamma _\pm$, positively transverse to ${\mathcal {G}}$. After a homotopy of $\mu$ we may assume that a small neighborhood of each curve is isomorphic to

$$\begin{aligned} \mathbb {S}^1 \times (\mathbb {D}^{2n},\omega _{st}).\end{aligned}$$

Let then $T_\pm$ be the complement of a slightly smaller open neighborhood of each curve compactly contained inside the previous one; in other words, $T_\pm$ are thickened tori “enclosing” $\gamma _\pm$ and entirely contained in the first neighborhood of $\gamma _\pm$ considered above. We choose (orientation preserving) splittings

$$\begin{aligned} T_- \simeq \mathbb {S}^1 \times [0,1] \times \overline{\mathbb {S}^{2n-1}},\quad \text {and,}\quad T_+ \simeq \mathbb {S}^1 \times [0,1] \times \mathbb {S}^{2n-1}. \end{aligned}$$

(13)

Here, we choose the $[0,1]_t$-factors in each of $T_\pm$ in such a way that $\partial _t$ points away from $\gamma _+$ and towards $\gamma _-$.

Let $\alpha _{ot,\pm } \in \Omega ^1(\mathbb {S}^{2n-1})$ be overtwisted contact forms in the almost contact class $(\pm \alpha _{st},{{\text {d}}}\alpha _{st})$. In particular, $\alpha _-$ is a positive contact form on $\overline{S^{2n-1}}$. Then, after another homotopy, we can arrange that

$$\begin{aligned} \mu |_{T_\pm } = {{\text {d}}}t \wedge \alpha _{ot,\pm } + {{\text {d}}}\alpha _{ot,\pm }.\end{aligned}$$

The complement of the two thickened tori $T_\pm$ consists of three connected components. These can be interpreted as foliated almost symplectic cobordisms between contact foliations:

$$\begin{aligned} \emptyset \rightarrow \mathbb {S}^1 \times (\mathbb {S}^{2n-1},\alpha _{ot,+}),\quad \mathbb {S}^1 \times (\mathbb {S}^{2n-1},\alpha _{ot,+}) \rightarrow \mathbb {S}^1 \times (\overline{\mathbb {S}^{2n-1}},\alpha _{ot,-}),\quad \nonumber \\ \text {and}\quad \mathbb {S}^1 \times (\overline{\mathbb {S}^{2n-1}}, \alpha _{ot,-}) \rightarrow \emptyset . \end{aligned}$$

(14)

On the first cobordism we use Corollary 6.7 to homotope $({\mathcal {G}},\mu )$, relative to the boundary, to an exact conformal symplectic foliation. Secondly, according to Corollary 2.22 the middle cobordism can be homotoped to an exact conformal symplectic foliated cobordism. Note that this homotopy is not relative to the boundary; instead, it induces a homotopy of overtwisted almost contact foliations on the boundary ending at a honest (overtwisted) contact foliation. Lastly, in the third cobordism we can homotope $({\mathcal {G}},\mu )$, relative to the boundary, to an exact conformal symplectic foliation using Lemma 6.9 and Corollary 6.7.

It remains to extend the homotopies over $T_\pm$. Since the arguments are the same on both cylinders we focus on $T_-$. In the coordinates of Eq. (13) any homotopy of $\Omega$ can be written as

$$\begin{aligned} \Omega _s = {{\text {d}}}t \wedge \alpha _{\theta ,t,s} + \omega _{\theta ,t,s},\end{aligned}$$

for a 3-parameter family of almost contact structures $(\alpha _{\theta ,t,s},\omega _{\theta ,t,s})$ on $\mathbb {S}^{2n-1}$, where $(\theta ,t,s) \in \mathbb {S}^1 \times [0,1]^2$.

Recall that an admissible form (Eq. (2)) is equivalent to the data of a transverse vector field. Hence the homotopy from Corollary 2.22 determines a homotopy of vector fields. We want that $\partial _t$ smoothly extends these vector fields. Although this need not be true a priori, one can always arrange it by reparametrizing $T_-$. Consequently, as the family of almost contact forms at the boundary coming from the application of Corollary 2.22 in the middle cobordism in Eq. (14) did, we may assume that the family of almost contact forms $(\alpha _{\theta ,t,s},\omega _{\theta ,t,s})$ for

$$\begin{aligned} (\theta ,t,s) \in \mathbb {S}^1 \times ([0,1] \times \{0\} \cup \{0,1\} \times [0,1]),\end{aligned}$$

admits an overtwisted basis (on each of the smooth pieces of the parameter space, coinciding at the corners). Using the h-principle from [5, Theorem 1.6], one can then extend it to a family of contact forms for $(\theta ,t,s) \in \mathbb {S}^1 \times [0,1]^2$. Finally, we apply Lemma 6.10 below to $\Omega _1$. This gives a homotopy, relative to the boundary of $T_-$ to a leafwise exact conformal symplectic structure. Repeating the same argument for $T_+$ then concludes the proof. $\square$

Lemma 6.10

Consider a family of contact forms $\alpha _{t,k}$, $(t,k)\in [0,1] \times K$ on M, where K is a compact space of parameters. Then, the K-family of almost symplectic structures

$$\begin{aligned}\Omega _k = d t \wedge \alpha _{t,k} + {{\text {d}}}\alpha _{t,k},\quad k \in K\end{aligned}$$

on $[0,1] \times M$ is homotopic relative to the boundary to a K-family of exact conformal symplectic structures. Moreover, if $\alpha _{t,k}$ is independent of t, for t near 0 and 1, the homotopy can be achieved relative to a neighborhood of the boundary $\{0,1\}\times M$.

Fig. 4

The picture describes the regions in the manifold and which result is used to obtain the homotopy. The middle unlabeled piece is the part where the $h-$principle from [9] is applied

Proof

We prove the case $K=\{pt\}$, thus suppressing the k from the notation for readability, as the proof in the general case is exactly the same. Now, unless this is not already the case, by a homotopy relative to t in $\{0,1\}$, one can arrange that the family $\alpha _t$ is constant for t near 0 and 1. Next, observe that

$$\begin{aligned} {{\text {d}}}t \wedge (C \alpha _t + s\dot{\alpha }_t) + {{\text {d}}}\alpha _t, \end{aligned}$$

is non-degenerate for any $s\in [0,1]$ provided that $C>0$ is sufficiently large, which we hence assume to be the case. Let now $f \in C^\infty ([0,1]_t)$, with $f\le -1$, be equal to $-C$ on ${{\,\textrm{supp}\,}}\dot{\alpha }_t$ and $-1$ near the boundary. Then ${{\text {d}}}_{f {{\text {d}}}t}\alpha _t$ is conformal symplectic, and the linear homotopy to ${{\text {d}}}t \wedge \alpha _t + {{\text {d}}}\alpha _t$ is among non-degenerate forms and relative to a neighborhood of the boundary. $\square$

6.5 Deformations to contact structures

A natural question is that of whether the foliations from Theorem 1.1 can be deformed into contact structures in the spirit of [19]. In higher dimensions there are several notions of convergence/deformation which are interesting to consider, but possibly the simplest is the following analogue of linear deformations in dimension three as defined in [19].

Definition 6.11

([45, Definition 2.2.5]) A foliation ${\mathcal {F}}= \ker \alpha$ on $M^{2n+1}$ is said to admit a Type I deformation if there exists a 1-parameter family $\alpha _t \in \Omega ^1(M)$ such that $\alpha _0 = \alpha$ and

$$\begin{aligned} \alpha _t \wedge {{\text {d}}}\alpha _t^n = t^n f_t {{\text {vol}}}_M,\end{aligned}$$

for $f_t \in C^\infty (M)$ satisfying $f_0 > 0$ and ${{\text {vol}}}_M$ a (positive) volume form.

The name refers to the fact that a deformation $\alpha _t$ is Type I if and only if its first order approximation is. More precisely, given $\alpha _t$ we can define its linearization:

$$\begin{aligned} \alpha _t^{lin}:= \alpha _0 + t \frac{{{\text {d}}}}{{{\text {d}}}t}\Big |_{t=0} \alpha _t,\end{aligned}$$

and a simple computation shows that $\alpha _t$ is Type I if and only if its linearization is. Note however, that linear contact deformations (i.e. $\alpha _t:= \alpha + t \beta$ for some $\alpha$ and $\beta$) are not necessarily of Type I.

Theorem 6.12

([9, Lemma 6.1], [45, Theorem 2.2.13]) Consider a foliation ${\mathcal {F}}$ with associated holonomy form $\mu _{\mathcal {F}}\in \Omega ^1({\mathcal {F}})$. Then ${\mathcal {F}}$ admits a Type I deformation if and only if there exists $\lambda \in \Omega ^1({\mathcal {F}})$ such that ${{\text {d}}}_{\mu _{\mathcal {F}}} \lambda$ is a leafwise conformal symplectic structure.

In particular, a conformal symplectic foliation $({\mathcal {F}},\eta ,\omega )$ admits a type I deformation if it is holonomy-like and exact.

The conformal symplectic foliations constructed in Theorem 1.1 do not admit a Type I deformation. To see this, first recall that given a contact manifold $(M,\xi )$, Theorem 4.5 and Proposition 4.3 allow us to construct a conformal symplectic foliation

$$\begin{aligned} \left( \mathbb {S}^1 \times [-1,1] \times M, {\mathcal {F}}, \eta , \omega \right) \end{aligned}$$

(15)

satisfying the following conditions:

• (i) The foliation ${\mathcal {F}}$:

  • (a) has a single closed leaf diffeomorphic to $\mathbb {S}^1 \times M$, and $\mu _{\mathcal {F}}$ restricts to $\pm {{\text {d}}}\theta$ on it;

  • (b) has every leaf accumulating onto this closed leaf;

  • (c) equals $\ker (-{{\text {d}}}\theta )$ near the boundary component with $t=-1$ (here, $\theta \in \mathbb {S}^1$), and $\ker {{\text {d}}}\theta$ near the one with $t=1$;

• (ii) The leafwise conformal symplectic structure has convex contact boundary (Definition 2.5) modeled on $(M,\xi )$. More precisely, this means $\omega ={{\text {d}}}_\eta \lambda$ such that, near the boundary, $\eta =0$ and $\lambda$ is $e^{-t}\alpha$ for t near $-1$ (because one needs to change coordinates in an orientation preserving way) and $e^t\alpha$ for t near 1, with $\xi = \ker \alpha$.

We will call any conformal symplectic foliation as above a concave-concave turbulization model for $(M,\xi )$.

Observe that near the boundary (where ${\mathcal {F}}= \ker {{\text {d}}}\theta$) we have $\mu _{\mathcal {F}}= 0$, and on the interior $(\eta ,\omega )$ is not prescribed by the model. Hence, at least a priori, by using a different model than that in the proof of Theorem 1.1, we might be able to avoid the problem that $\eta = -\mu _{\mathcal {F}}$. However, as shown in the following lemma, the existence of a model which admits a Type I deformation is often still obstructed. Recall, that a closed contact manifold is called semi-fillable if it is a connected component of a symplectic manifold with (possibly disconnected) convex contact boundary. (In fact, semi-fillable contact manifolds are also fillable; this follows from the capping construction in [17] in dimension 3, and from a combination of [14, Section 3] and [30, Corollary 1.14] in higher dimensions.)

Lemma 6.13

Let $(M,\xi )$ be a closed contact manifold which is not semi-fillable. Then, no concave-concave turbulization model for $(M,\alpha )$ admits a Type I deformation.

In particular, the lemma applies to the proof of Theorem 1.1, where we use a concave-concave model for $(\mathbb {S}^{2n-1},\alpha _{ot})$. Indeed, by [5, 38] these contact manifolds are not semi-fillable.

Proof

We give a proof for the case where $\mu _{\mathcal {F}}= {{\text {d}}}\theta$ along the closed leaf; the one of the other case is similar. Assume by contradiction that there exists a type I deformation, so that $\omega = d_{\mu _{\mathcal {F}}} \lambda$ for some $\lambda \in \Omega ^1({\mathcal {F}})$. In particular, on the closed leaf $\mathbb {S}^1 \times M$ we have $\omega = {{\text {d}}}\lambda - {{\text {d}}}\theta \wedge \lambda$ so that $\lambda$ defines a negative contact form on M.

Assume without loss of generality that $\{t=1\}$ is the boundary component where ${\mathcal {F}}=\ker {{\text {d}}}\theta$. Consider then a non-compact leaf $(-\infty ,0]_s \times M$ of ${\mathcal {F}}$, intersecting this boundary component on $\{s=0\}\times M$. This leaf accumulates onto the compact leaf. Hence, by continuity $\lambda |_{\{s_0\} \times M}$ is a negative contact form for some $s_0 \ll 0$. On the other hand, by definition of the model, $\lambda = e^t\alpha$ on $\mathbb {S}^1\times (1-\varepsilon ,1] \times M$, with $\xi =\ker \alpha$.

Thus, $(M,\xi )$ is a connected component of the conformal symplectic manifold $([s_0,0] \times M, {{\text {d}}}_{\mu _{\mathcal {F}}} \lambda )$. Since, $\mu _{\mathcal {F}}$ is exact away from the compact leaf, this implies that $(M,\xi )$ is (honest) symplectically semi-fillable, thus giving a contradiction. $\square$

6.5.1 Deformation to contact structures for non taut foliations

Consider a taut almost symplectic foliation on a manifold $M^{2n+1}$, $n \ge 2$. On the complement of two curves, [9, Theorem B] yields a taut conformal symplectic foliation that admits a Type I deformation. This being said, as the authors of [9, Remark 6.4] already expected, the tautness assumption is not necessary. More precisely, we prove so by removing an additional curve:

Proposition 6.14

Let $(M^{2n+1},\zeta ,\mu )$, $n \ge 2$, be an almost contact manifold. Fix any three curves, two of which are parallel copies of each other (w.r.t. any choice of framing). Then, on the complement of these curves, $(\zeta ,\mu )$ is homotopic to an exact conformal symplectic foliation, which is not taut and admits a Type I deformation.

Proof

The proof essentially follows from that of Proposition 1.2, noticing that the resulting exact conformal symplectic foliation constructed in its proof, i.e. in Sect. 6.4, is in fact not taut and can be arranged to be holonomy-like away from regions as claimed in the statement of Proposition 6.14. (We point out that, even though Proposition 1.2 assumes $\dim M = 2n+1 \ge 7$, this is only used when applying Corollary 6.7. As we now describe, these tubes will be removed from the ambient manifold in this proof, so that everything goes through in dimension 5.) Here are additional details.

The start of the proof of Proposition 1.2 should be modified as follows. We first choose two curves $\gamma _\pm$ and homotope the almost contact structure to a symplectic foliation near them which is positively transverse to them. Then, we apply [31] relatively to these foliated neighborhoods (c.f. [9, Remark 6.4]), in such a way as to deform the almost contact structure to a minimal almost symplectic foliation relative to the foliated neighborhoods of $\gamma _\pm$.

We now point out that all the results (namely Corollary 6.7, Corollary 2.22, and Lemma 6.9) used in the proof of Proposition 1.2 in Sect. 6.4 give $\vert$holonomy$\vert$-like exact conformal symplectic foliations. Moreover these are in fact holonomy-like away from closed leaves coming from the turbulization construction. To be precise, the only region of the manifold where this $\vert$holonomy$\vert$-like property has not been explicitly stated in the proof of Proposition 1.2 in Sect. 6.4 is in the construction of the homotopy in the regions $T_\pm$ defined as in Equation (13). However, there the holonomy-like property simply follows from the same consideration as in Remark 6.5.

So, summarizing, the argument in Sect. 6.4 produces a $\vert$holonomy$\vert$-like exact conformal symplectic foliation which is not holonomy-like just nearby three closed leaves, all resulting from the application of Corollary 6.7. Then, it is enough to consider, as candidate neighborhoods satisfying the conclusion of Proposition 6.14, the three tubular neighborhoods of transverse curves to which Corollary 6.7 is applied (two of which are parallel to $\gamma _+$) in the proof of Proposition 1.2 in Sect. 6.4; indeed, in the complement of these neighborhoods we have holonomy-likeness as just explained, and can thus conclude thanks to Theorem 6.12. $\square$

Instead of removing curves, we can try to deform the conformal symplectic foliations from Theorem 1.1 on the whole manifold. As we show now, this produces a Type I linear deformation to “singular contact structures”. That is, a hyperplane field which is a (positive) contact structure on the complement of a (not necessarily connected) embedded hypersurface.

Proposition 6.15

The conformal symplectic foliations constructed in Theorem 1.1 admit a Type I deformation to singular contact structures, with singularities along three closed leafs.

Proof

The Type I linear contact deformation has been already described in the proof of Proposition 6.14 away from solid tori in which Corollary 6.7 is applied. In fact, the same argument proves the existence of a Type I deformation everywhere except on neighborhoods of three closed leaves in these tori, which correspond to turbulizations where both sides are concave contact boundaries. Hence it suffices to describe how to extend the deformation near such closed leaves.

The concave-concave model in each of these neighborhoods is as follows. Consider the manifold $\mathbb {S}^1 \times [-1,1]\times \mathbb {S}^{2n-1}$ with coordinates $(\theta ,t,x)$, and $\alpha \in \Omega ^1(\mathbb {S}^{2n-1})$ a positive contact form on $\mathbb {S}^{2n-1}$. The turbulized smooth foliation obtained as in Theorem 1.1 can then be described in this model neighborhood of each of the closed curve by:

$$\begin{aligned} {\mathcal {F}}:=\ker \gamma , \quad \gamma :=f(t) {{\text {d}}}\theta - g(t) {{\text {d}}}t, \end{aligned}$$

where $f,g:{\mathbb {R}}\rightarrow {\mathbb {R}}$ satisfy the following properties:

• (i) f has a single zero at $t =0$ with $\dot{f}(0)=1$, and $f = {\left\{ \begin{array}{ll} -1 &{} \text {for t{ near}-1} \\ 1 &{} \text {for t{ near}1}\end{array}\right. }$.

• (ii) $g \ge 0$, supported inside $(-1/2,1/2)$, and $g(0) = 1$.

Moreover, the leafwise exact holonomy-like conformal symplectic structure $(\eta ,\lambda )$ satisfies $\eta =0$, and $\lambda$ is $e^{-t}\alpha$ for t near $-1$, and $e^{t}\alpha$ for t near 1.

We now describe an explicit deformation to a singular contact structure in this model neighborhood that glues well to the deformation on the rest of the manifold which has been described in the proof Proposition 6.14. For this, choose a function $k:[-1,1]\rightarrow \mathbb {R}$ as in Fig. 5, satisfying:

• $k=0$ at $t=0$ and $k>0$ otherwise,

• $k=e^{-t+1}$ near $t=-1$ and $k=e^{t-1}$ near $t=1$,

• $\dot{k} f - \dot{f}k\ge 0$, with $=0$ only for $t=0$.

In order to guarantee the last property, an explicit possible formula for k near $t=0$, where f can be assumed w.l.o.g. to be $f(t)=t$, is for given by $k(t)=1-\sqrt{1-t^2}$. It is then easy to extend such local choice away from this neighborhood of $t=0$ to a function satisfying the above conditions.

We then consider the linear deformation $\gamma _s:=\gamma + sk\alpha$. To see it defines the desired “degenerate Type I deformation” observe that:

$$\begin{aligned} \gamma _s \wedge {{\text {d}}}\gamma _s^n&= n \dot{f} s^{n} k^{n} {{\text {d}}}t\wedge {{\text {d}}}\theta \wedge \alpha \wedge {{\text {d}}}\alpha ^{n-1} + n f s^{n} \dot{k} k^{n-1} {{\text {d}}}\theta \wedge {{\text {d}}}t \wedge \alpha \wedge {{\text {d}}}\alpha ^{n-1} \\&= n s^{n}k^{n-1} (\dot{k} f - \dot{f}k) {{\text {d}}}\theta \wedge {{\text {d}}}t\wedge \alpha \wedge {{\text {d}}}\alpha ^{n-1}. \end{aligned}$$

By the previous choice of k, the local linear deformation $\gamma _s$ is then a contact deformation away from $t=0$, i.e. away from the closed leaf coming from this turbulization, and it glues well (up to rescaling) to the deformation away from this turbulization region described in the proof of Proposition 6.14. $\square$

Fig. 5

Parametric graph of the functions f and k

6.5.2 Proof of Theorem 1.8

Let $(\zeta ,\mu )$ denote the almost contact structure on $M^{2n+1}$, where $n=2m+1\ge 3$ by assumption. By a homotopy of $(\zeta ,\mu )$ we can assume that there is a neighborhood $N \simeq \mathbb {S}^1 \times \mathbb {D}^{2n}$ of the curve $\gamma$ on which

$$\begin{aligned} \zeta = \bigcup _{\theta \in \mathbb {S}^1} \{ \theta \} \times \mathbb {D}^{2n},\quad \mu = \omega _{st}.\end{aligned}$$

Applying [31] relative to this neighborhood (see [9, Remark 6.4]) we can homotope $\zeta$ to a taut foliation ${\mathcal {G}}$ (which restricts to the product foliation on the neighborhood above). Note that at this point we are in the hypothesis of Remark 1.9.

We fix two parallel copies $\gamma _\pm$ of the curve $\gamma$ inside N, and positively transverse to ${\mathcal {G}}$. Then, as done in the proof of Proposition 1.2, we apply [9] to the complement of $\gamma _\pm$, and compose it with the foliated cobordisms $T_\pm$ described in that proof, in order to obtain a foliated conformal symplectic cobordism

$$\begin{aligned} \mathbb {S}^1\times (\mathbb {S}^{2n-1},\alpha _{ot,+}) \rightarrow \mathbb {S}^1\times (\overline{\mathbb {S}^{2n-1}},\alpha _{ot,-}). \end{aligned}$$

Moreover, we can arrange the conformal symplectic foliation to be leafwise exact and holonomy-like.

Consider now an orientation reversing diffeomorphism $\psi :\mathbb {S}^{2n-1} \rightarrow \mathbb {S}^{2n-1}$. This induces an orientation preserving diffeomorphism

$$\begin{aligned}\Psi :\mathbb {S}^1 \times \mathbb {S}^{2n-1} \xrightarrow {\sim }\mathbb {S}^1 \times \overline{\mathbb {S}^{2n-1}},\quad (\theta ,x) \mapsto (\theta ,\psi (x)).\end{aligned}$$

As such, the above cobordism can also be interpreted as a foliated conformal symplectic cobordism

$$\begin{aligned} \mathbb {S}^1\times (\mathbb {S}^{2n-1},\alpha _{ot,+}) \rightarrow \mathbb {S}^1\times (\mathbb {S}^{2n-1},\alpha '_{ot,-}=\psi ^*\alpha _{ot,-}). \end{aligned}$$

(16)

Since $\Psi$ is orientation preserving, $\alpha _{ot,-}'$ is a positive, overtwisted contact form.

Recall now that the 0-th homotopy group of the space of almost contact structures on $\mathbb {S}^{2n-1}$, is a group under taking connected sum. The identity element is moreover given by the homotopy class of the standard tight contact structure (which is the same as that of $\xi _{ot,+}$). Because of the dimensional assumption $2n-1 = 4\,m+1 \ge 5$, it also follows from [23] that this group is finite. Therefore, there exist an integer $N >0$ such that $\#_{i=1}^N \xi _{ot,-}'$ is in the same almost contact class as $\xi _{ot,+}$. (Note that if $m=1$ we can choose $N=1$ as there is only one almost contact class up to homotopy.)

Consider then the standard symplectic ball $(\mathbb {D}^{2n},\omega _{st})$, and remove N smaller balls $\mathbb {D}^{2n}_\varepsilon$ from its interior. Then, using Theorem 2.18 we obtain a (non-foliated) conformal symplectic cobordism $(\mathbb {S}^{2n-1},\xi _{ot,-}') \rightarrow \sqcup _N(\mathbb {S}^{2n-1},\xi _{ot,-}')$. We compose this with N Weinstein 1-handle attachments to reach as convex boundary $(\#_N \mathbb {S}^{2n-1} = \mathbb {S}^{2n-1},\#_N \xi _{ot,-}')$. As pointed out before $\#_N \xi _{ot,-}'$ is homotopic to $\xi _{ot,+}$. Thus, by adding another (smoothly trivial) cobordism we reach $(\mathbb {S}^{2n-1},\xi _{ot,+})$ as the convex boundary. The resulting Liouville cobordism $(X,\lambda _X)$ is depicted in Fig. 6. Note that as a smooth manifold X is obtained from $[0,1]\times \mathbb {S}^{2n-1}$ by doing N self connected sums, which is topologically equivalent to performing, on $[0,1]\times \mathbb {S}^{2n-1}$, N disjoint connected sums with $\mathbb {S}^1\times \mathbb {S}^{2n-1}$.

Fig. 6

A schematic description of the two pieces composing the cobordism X

According to Remark 2.19, we can also fix any desired contact form at the boundary, at the expense of $(X,\lambda _X)$ becoming a conformal symplectic cobordism. Taking the product with $\mathbb {S}^1$ we then obtain a foliated conformal symplectic cobordism

$$\begin{aligned} \mathbb {S}^1 \times (\mathbb {S}^{2n-1},\alpha _{ot,-}') \rightarrow \mathbb {S}^1 \times (\mathbb {S}^{2n-1},\alpha _{ot,+}').\end{aligned}$$

Again, note that as a smooth manifold $\mathbb {S}^1 \times X$ is from $\mathbb {S}^1 \times [0,1] \times \mathbb {S}^{2n-1}$ by self round-connected sums, i.e. by performing, on $\mathbb {S}^1\times [0,1]\times \mathbb {S}^{2n-1}$, N disjoint round-connected sums with $\mathbb {T}^2\times \mathbb {S}^{2n-1}$.

Finally, we glue the above cobordism to the one of Eq. (16). This yields the desired exact, holonomy-like conformal symplectic foliation on the round connect sum of M with N disjoint copies of $T^2 \times \mathbb {S}^{4n+1}$. The existence of a Type I deformation lastly follows from Theorem 6.12. $\square$