Novikov's theorem in higher dimensions?

TL;DR

This paper investigates whether higher-dimensional analogues of Novikov's theorem hold for strong symplectic foliations, providing a counterexample on a five-dimensional manifold that challenges the expected rigidity properties.

Contribution

The paper constructs a five-dimensional example of a strong symplectic foliation that does not satisfy the analogue of Novikov's theorem, highlighting limitations of such rigidity results in higher dimensions.

Findings

Counterexample on a 5D manifold with a strong symplectic foliation

Shows failure of Novikov's theorem analogue in higher dimensions

Introduces a foliated Lefschetz fibration as the example

Abstract

Novikov's theorem is a rigidity result on the class of taut foliations on three-manifolds. For higher dimensional manifolds, foliations with a strong symplectic form have been suggested as the class of foliations having similar rigidity properties to taut foliations on three-manifolds. This leads to the natural question of whether strong symplectic foliations satisfy an analogue of Novikov's theorem. In this paper, we construct a five-dimensional manifold with a smooth foliation and a strong symplectic form that does not satisfy the expected analogue of Novikov's theorem. Our example is a foliated Lefschetz fibration.

Figures

Equations

$$\pi :X\to (B,{\mathcal{F}}_B)$$

$${\omega }_{pre}:=\zeta +\sum _{b\in \mathcal{U}}d({\eta }_b{\alpha }_b).$$

$${\pi }_{std}:{\mathbb{C}}^2\to \mathbb{C},(z_1,z_2)\mapsto z_1{z}_2.$$

$$X_0:=(X_{triv}\backslash R_{triv}){\cup }_{\varphi :A_{triv}\to A_{std}}(A_{std}\cup R_{std}).$$

$$\omega :=\{\begin{array}{l}{\omega }_{triv},\text{on }{X}_{triv}\backslash (A_{triv}\cup R_{triv})\\ {\omega }_{std},\text{ on }{R}_{std},\\ {\omega }_{triv}+d(\eta \alpha ),\text{ on }{A}_{triv},\end{array}$$

$$\Phi :=(\varphi ,\mathrm{Id}):(-ϵ,ϵ)\times W\to (-ϵ,ϵ)\times W$$

$$\overline{X}:=\tilde{X}/(\partial X_{+}{\sim }_{\varphi }\partial X_{-}).$$

$$\varphi :{\mathbb{T}}_b^2\times {\mathbb{T}}_f^2\to {\mathbb{T}}_b^2\times {\mathbb{T}}_f^2,(b,f)\to (f,b),$$

$$i_{\pm }:[0,ϵ)\times {\partial }_{\pm }\tilde{X}\to \mathcal{O}p(\tilde{X}),i_{\pm }(0,x)=x,$$

Full text

1. Introduction

1.1. Taut foliations in three-manifolds

A codimension one foliation on a closed three-manifold is taut if the manifold has a closed $2$-form inducing an area form on each leaf of the foliation. Equivalently, by a theorem of Sullivan [16], the foliation is taut if, through every point, there is a loop everywhere transverse to the leaves. This characterization shows that a taut foliation does not contain any Reeb components.

Reeb components are a source of flexibility in foliation theory in the following sense. If one is allowed to use Reeb components, existence questions about foliations can be reduced to topological questions of existence of distributions and Haefliger structures. See results of Thurston [17], Eynard-Bontemps [5]. The following theorem of Novikov is a rigidity result for taut foliations.

Proposition 1.1**.**

(Novikov’s theorem, [14])* Suppose $X$ is a compact $3$-manifold and $\mathcal{F}$ is a taut foliation. Then,*

1. (a)

For any leaf $\mathcal{L}$ of $\mathcal{F}$, the map $\pi_{1}(\mathcal{L})\to\pi_{1}(X)$ is injective. 2. (b)

For any loop $\gamma:\mathbb{S}^{1}\to X$ transverse to $\mathcal{F}$, the homotopy class $[\gamma]$ is non-trivial in $\pi_{1}(X)$.

In particular, if $X$ admits a taut foliation, it must have infinite fundamental group. For example, $\mathbb{S}^{3}$ has a foliation with a Reeb component, but Novikov’s result implies that it does not have any taut foliation. A non-trivial homology/homotopy class in a leaf is called a vanishing cycle if it becomes homotopically trivial when it is displaced to nearby leaves. By Novikov’s theorem, taut foliations in dimension $3$ possess no vanishing cycles.

1.2. Rigidity in higher dimensions

What is the right analogue of tautness in higher dimensional manifolds which imparts a similar kind of rigidity? Existence of a transversal to the foliation is not a good answer in higher dimensions. Indeed, Meigniez [12] showed that for a manifold $M$ of dimension greater than $3$, a pair comprised of a distribution and a Haefliger structure (both of codimension one) is homotopic to a pair arising from a taut foliation.

Martinez-Torres [9, 10] proposed that the analogue of a taut foliation in higher dimensions is a strong symplectic foliation. A $(2n+1)$-dimensional manifold with a codimension one foliation $\mathcal{F}$ is strong symplectic if there is a closed two-form $\omega\in\Omega^{2}(X)$ that restricts to a symplectic form on the leaves. This suggestion has been further explored in [11] via a study of the transverse geometry of strong symplectic foliations. This leads us to the question of whether strong symplectic foliations satisfy an analogue of Novikov’s theorem. Unfortunately, the answer to this question is negative, and in this paper, we present a counter-example.

Theorem 1**.**

There is a manifold $X^{5}$ with a strong symplectic foliation $(\mathcal{F}^{4},\omega)$ for which

1. (a)

the map $\pi_{1}(\mathcal{F})\to\pi_{1}(X)$ is not injective. 2. (b)

There is a loop which is transverse to the foliation $\mathcal{F}$ and which is contractible in $X$.

In spite of this result, it is still likely that strong symplectic foliations are fairly rigid objects similar to taut foliations. The example we construct is not simply connected, and the question of whether a Novikov-type theorem holds for simply connected manifolds remains open.

In contrast, weak symplectic foliations are more flexible objects. A codimension one foliation $(X,\mathcal{F})$ is weak symplectic if there is a two-form that restricts to a symplectic form on leaves, but is not required to be closed in $X$. Ibort and Martínez–Torres showed in [8] that any finitely presented group can be made to be the fundamental group of a weak symplectic foliation. Mitsumatsu [13] showed that, in particular, $\mathbb{S}^{5}$ admits a symplectic foliation. It is even possible that a foliation is simultaneously taut and weak symplectic, but not strong symplectic. This is the case of a symplectic mapping torus whose return map is a symplectomorphism that is smoothly, but not symplectically, isotopic to the identity. This has been explored in the work of Seidel [15].

Foliations with other geometric structures in the leaves have been studied in the literature. For example, holomorphic foliations on complex surfaces have been classified by Brunella [2]. Riemannian foliations are foliations with a transverse structure of a Riemannian manifold. This condition is very over-determined, and a manifold with a generic metric doesn’t have such a foliation. Such foliations exist in specific cases, such as manifolds with constant curvature and have been studied in [7]. These examples are more rigid than strong symplectic foliations.

We present our example in Section 3, and a slight variation in 4. In Section 2, we review results about symplectic Lefschetz fibrations and adapt them to the foliated setting. Some of the results about foliations and Lefschetz fibrations that are given in this paper are standard in the literature. We include these to make the paper self-contained.

Acknowledgements. I am indebted to Fran Presas for introducing me to the question of whether Novikov’s theorem extends to higher dimensions. I thank both him and Álvaro del Pino for discussions on this topic. I also acknowledge support of the grant 612534 MODULI within the 7th European Union Framework Programme.

2. Symplectic Lefschetz fibrations

The building block of our example is a foliated symplectic Lefschetz fibration. In this section, we construct a strong symplectic form on a Lefschetz fibration with a foliated base. This is a mild extension of Gompf’s construction [6]. All foliations are assumed to have codimension one.

A topological Lefschetz fibration consists of an oriented four-manifold $X$, and a proper map $\pi:X\to B$ to an oriented surface $B$ satisfying the following property: For any $x_{0}\in X$, either $d\pi_{x_{0}}$ is surjective, or there are coordinates $(z_{1},z_{2})$ in a neighbourhood of $x_{0}$ that respect orientation and satisfy $(z_{1},z_{2})(x_{0})=(0,0)$ and $f=f(x_{0})+z_{1}^{2}+z_{2}^{2}$. The projection $\pi$ is injective on the set of its critical points.

A topological Lefschetz fibration on a foliated base manifold consists of a five-manifold $X$ and a proper map to a foliated three manifold $(B,\mathcal{F}_{B})$

[TABLE]

such that the pullback $\mathcal{F}_{X}:=\pi^{*}\mathcal{F}_{B}$ is an oriented foliation on $X$, and each leaf $L$ of $\mathcal{F}_{X}$ is a Lefschetz fibration over the leaf $\pi(L)$ in $B$. Further, singular points in $X$ are circles transverse to the foliation.

Proposition 2.1**.**

Suppose $B$ is a manifold with a codimension one foliation $\mathcal{F}_{B}$ and a strong symplectic form $\omega_{B}$. The manifold $B$ can have boundary that is tangent to the foliation. Suppose $X\to B$ is a topological Lefschetz fibration on $(B,\mathcal{F}_{B})$, such that for any $b\in B$, the homology class of the fiber is non-trivial, i.e. $[F_{b}]\in H_{2}(X)$ is non-zero. Then there is a strong symplectic form $\omega\in\Omega^{2}(X)$ for which the fibers are symplectic, and have volume $\lambda>0$.

Remark 2.2*.*

If the base manifold $B$ is three-dimensional, then the existence of a strong symplectic form is equivalent to the foliation $\mathcal{F}_{B}$ being taut.

Proof of Proposition 2.1.

We start the construction by choosing two $2$-forms. One is a closed $2$-form $\zeta\in\Omega^{2}(X)$ that integrates to $\lambda$ on each fiber. Indeed, $\zeta$ exists because the homology class of the fiber is on-trivial and $H^{2}(X,\mathbb{R})=\operatorname{Hom}(H_{2}(X,\mathbb{R}),\mathbb{R})$. The second is a fiber symplectic form $\omega_{F}\in\Omega^{2}(F)$ that integrates to $\lambda$, where $F$ is a regular fiber of $X\to B$.

For any point $b\in B$, we will now define a two-form $\omega_{b}$ on $\pi^{-1}(U_{b})$ for a neighbourhood $U_{b}\subset B$ of $b$, that is symplectic on the fibers and has area $\lambda$ on each fiber. If $b$ is a regular value of $\pi$, we can choose a trivialization $\pi^{-1}(U_{b})\simeq U_{b}\times F$ and define $\omega_{b}\in\Omega^{2}(\pi^{-1}(U_{b}))$ to be the pullback of a symplectic form $\sigma\in\Omega^{2}(F)$. If $b\in B$ is a singular value of the fibration, we take a foliated chart $U_{b}\simeq V_{b}\times(-\epsilon,\epsilon)$ on $B$ such that $b\in V_{b}$ and $\{b\}\times(-\epsilon,\epsilon)$ is the set of singular values in $U_{b}$. A symplectic form is constructed on $\pi^{-1}(V_{b})$ by Lemma 2.3, and $\omega_{b}$ is defined to be its pullback to $\pi^{-1}(U_{b})$.

The global form $\omega$ is obtained by patching. On any open set the difference $\omega_{b}-\zeta$ is exact, because the fibration $\pi^{-1}(U_{b})$ retracts to the fiber $F_{b}$ and $\omega_{b}-\zeta$ integrates to zero on $F_{b}$. So, $\omega_{b}-\zeta=d\alpha_{b}$ for some $\alpha_{b}\in\Omega^{1}(\pi^{-1}(U_{b}))$. The base $B$ can be covered by a finite subset of $\{U_{b}\}_{b}$, denoted by $\mathcal{U}$. Let $\{\eta_{b}:B\to[0,1]\}_{b\in\mathcal{U}}$ be a partition of unity for the finite cover. Define a form

[TABLE]

This form is symplectic on each fiber $F_{b_{0}}$ and has the right volume, because on $F_{b_{0}}$, the form $\omega_{pre}=\sum_{b\in\mathcal{U}}\eta_{b}(b_{0})\omega_{b}$ is a convex combination of the forms $\omega_{b}$. Finally, for a large $C>0$, $\omega:=\omega_{pre}+C\pi^{*}\omega_{B}$ is a symplectic form on the leaves of $X$, finishing the proof of the proposition. ∎

Lemma 2.3**.**

(Symplectic form near a singular fiber)* Suppose $X\to B$ is a topological Lefschetz fibration over a two-ball $B\subset\mathbb{R}^{2}$ whose regular fiber is a compact Riemann surface $F$. The fibration has one singular point $p$ which projects to $0\in B$. Then, for any $\lambda>0$, there is a closed two-form $\omega\in\Omega^{2}(X)$ for which the fibers are symplectic, and have volume $\lambda>0$.*

Proof.

We construct the symplectic form by starting with a product fibration $B\times F$ with a product symplectic form. We delete a neighbourhood of $B\times V$ where $V$ is the vanishing cycle corresponding to the singularity $p$. In its place we glue in a neighborhood of the origin in $\mathbb{C}^{2}$, which models a singular point in the fibration. The details are as follows.

Let $X_{triv}:=B\times F$ be the product fibration equipped with the two-form $\omega_{triv}:=\pi_{2}^{*}\omega_{F}$, where $\omega_{F}$ is a symplectic form on $F$ that integrates to $\lambda$. For a small constant $\delta>0$, let $A\subset F$ consist of a pair annuli, both homotopic to the vanishing cycle $V$, each having area $\delta$, and enclosing an annulus of area $\delta$ between them. Denote the region in between by $R\subset F$. See Figure 1. Let $A_{triv}:=B\times A$, $R_{triv}:=B\times R$ be product regions in $X_{triv}$.

The neighbourhood of a singular point has oriented coordinates $(z_{1},z_{2})\in\mathbb{C}^{2}$ that map $p$ to the origin and such that the projection map is

[TABLE]

Let $\omega_{std}$ be the standard symplectic form on $\mathbb{C}^{2}$. Embed $B$ as a ball in the base space $\mathbb{C}$ around the origin. Let $A_{std}\subset\pi_{std}^{-1}(B)$ be a region whose intersection with any fiber of $\pi_{std}$ consists of a pair of homotopic non-intersecting annuli, each of area $\delta$, the area between them is an annulus of area $\delta$ for regular fibers, and a pair of disks for the singular fiber. The region between the two components of $A_{std}$ is denoted by $R_{std}$ and is a neighbourhood of the origin in $\mathbb{C}^{2}$.

There is a map $\phi:A_{triv}\to A_{std}$ that commutes with projection to $B$, and which is area-preserving on the fiber, i.e. $\phi^{*}\omega_{std}-\omega_{triv}$ vanishes in the fiber direction on $A_{triv}\to B$. Therefore, this form is exact on $A_{triv}$. There is a one-form $\alpha\in\Omega^{1}(A_{triv})$ such that $\phi^{*}\omega_{std}-\omega_{triv}=d\alpha$.

Finally we glue parts of the two fibrations using the identification $\phi$ constructed above. Define

[TABLE]

The space $X_{0}$ fibers over $B$. Define a form $\omega\in\Omega^{2}(X_{0})$ as

[TABLE]

where $\eta:A_{triv}\to[0,1]$ is $1$ in a neighbourhood of the inner boundary, and [math] in a neighbourhood of the outer boundary. The form $\omega$ is closed in $X_{0}$ and integrates to $\lambda$ on the fiber. Finally, we see that $X_{0}\to B$ is diffeomorphic to $X$ as a fibration. ∎

Lemma 2.4**.**

Assume the setting of Proposition 2.1. Additionally, assume that the bundle $X\to B$ has a section $s$, and that $L_{B}\subset B$ is a compact leaf in the base. Then, for any $\lambda_{L}$, $\lambda_{F}>0$, the strong symplectic form $\omega$ on $X$ can be constructed so that $\int_{L_{B}}s^{*}\omega=\lambda_{L}$ and $\int_{F_{b}}\omega=\lambda_{F}$ for any $b\in B$.

Proof.

The section $s$ can be thickened to obtain an embedding $\overline{s}:B\times\mathcal{O}\to X$, where $\mathcal{O}\subset\mathbb{R}^{2}$ is a ball, $\overline{s}(\cdot,0)=s$, and $\overline{s}(b,\cdot)$ is tangent to the fiber for any $b\in B$. For any compactly supported non-negative form $\eta\in\Omega^{2}(\mathcal{O})$, and a strong symplectic form $\omega$ on $X$, the sum $\omega^{\prime}+\eta$ is also a strong symplectic form whose integral on $L_{B}$ is same as that $\omega^{\prime}$. Therefore, we can increase the integral on $F_{b}$ without affecting that on $L_{B}$. Together with the operations of scaling $\omega$ by a positive scalar, and adding a positive multiple of the base form $\omega_{B}$, we can ensure that both conditions are satisfied. ∎

3. The example

In this section, we construct the example that proves Theorem 1. The following proposition constructs a building block, which satisfies part (a) of Theorem 1, but not (b). Given a Riemann surface $F$ and an essential loop $V\subset F$, the proposition constructs a Lefschetz fibration whose regular fiber is $F$ and $V$ is a vanishing cycle. The loop $V$ is essential, by which we mean there is a set of $2g$ generators of $\pi_{1}(F)$ that contains $V$.

Proposition 3.1**.**

Let $F$ be a Riemann surface with genus $g\geq 1$ and let $V\subset F$ be an essential loop. There is a foliation on the three-dimensional torus $B:=\mathbb{T}^{3}$, and a symplectic Lefschetz fibration $\pi:X\to B$ whose regular fiber is $F$, and there is a loop $\gamma$ in a fiber of $\pi$ that is

1. (a)

homotopic to $V$, 2. (b)

is non-contractible in $\mathcal{F}$, and 3. (c)

is contractible in $X$.

Proof.

We use an almost horizontal foliation on the base manifold, which we first describe. Denote the base manifold as a product $B=\mathbb{S}^{1}_{x}\times\mathbb{S}^{1}_{y}\times\mathbb{S}^{1}_{z}$, and let the maps $x$, $y$, $z$ be projection to the respective components. The foliation $\mathcal{F}_{B}$ is the pullback of a foliation $\mathcal{F}_{\mathbb{T}^{2}}$ on $\mathbb{S}^{1}_{x}\times\mathbb{S}^{1}_{y}$, which in turn is the quotient of a foliation $\mathcal{F}_{strip}$ on the strip $[-1,1]\times\mathbb{R}$. The foliation on the strip $\mathcal{F}_{strip}$ has the lines $\{x=\pm 1,0\}$ as closed leaves, and the non-closed leaves are $\{(y+c)\tan(\pi x)=\operatorname{sign}(x)\}_{c\in\mathbb{R}}$, see Figure 2. This foliation is invariant under translation in the $\mathbb{R}$-direction, and so, descends to a foliation $\mathcal{F}_{\mathbb{T}^{2}}$ on the quotient $([-1,1]\times\mathbb{R}/\mathbb{Z})/(-1,y)\sim(1,y)$. Finally, $\mathcal{F}_{B}$ is defined by pullback.

The form $\omega_{B}:=dy\wedge dz$ is a strong symplectic form on the base $B$.

We next describe the topology of the Lefschetz fibration. As prescribed, the regular fiber $F$ is a genus $g$ Riemann surface. The singular point set of the fibration consist of two loops $P_{\pm}$ transverse to the foliation such that $P_{\pm}$ is contained in the region $\{0<\pm x<1\}$ and intersects each leaf in that region once. As a result the compact leaves do not have any singular points. The fibration is trivial on the compact leaf $\{x=\pm 1\}$. On the leaf $\{x=0\}$, the fibration has trivial monodromy along the loop $\mathbb{S}^{1}_{y}$, and along the loop $\mathbb{S}^{1}_{z}$ the monodromy map is a $-1$ Dehn twist along the essential loop $V\subset F$. Now, consider non-compact leaves in the region $\{0<x<1\}$. These leaves project to cylinders in the base $B$ whose positive resp. negative end is asymptotic to the compact leaf $\{x=1\}$ resp. $\{x=0\}$, and are homotopic to the loop $\mathbb{S}^{1}_{z}$. As a result, the fibration is trivial near the positive end, and on the negative end, there is a $-1$ Dehn twist about $V$. This Dehn twist is induced by the singularity in each leaf at its intersection with the transversal $P_{+}$. See Figure 3. The fibration is analogous in the region $\{-1<x<0\}$.

By Proposition 2.1, the space $X$ can be given a strong symplectic form. To apply the Proposition, we need to check that the fiber class is non-trivial in $H_{2}(X)$. Indeed if we remove a neighbourhood of the vanishing cycle $V$ in all the fibers, the resulting bundle is trivial – it is diffeomorphic to $B\times(F\backslash V)$. Therefore, there is a section $s:B\to X$, which has a non-zero intersection with the fiber class.

The map $\pi_{1}(\mathcal{F})\to\pi_{1}(X)$ is not injective. Indeed consider a loop $\gamma$ in the compact leaf $\{x=1\}$ that lies in a fiber of $\pi$ and is homotopic to $V$. This loop is non-contractible in the compact leaf, but becomes contractible when it is displaced to a nearby non-compact leaf since it is a vanishing cycle of the Lefschetz fibration in the non compact leaf. ∎

To finish the proof of Theorem 1, the following definition is required.

Definition 3.2**.**

(Holonomy) Suppose $\gamma:\mathbb{S}^{1}\to L$ is a loop in the leaf $L$ of a foliated manifold $(X,\mathcal{F})$, where the foliation is of codimension one. Suppose $\tau:(-\epsilon,\epsilon)\to\mathbb{R}$ is a transversal through $\gamma(0)$. The transversal can be transported along $\gamma$ via foliated charts of $W$, and the return map $h:\tau\to\tau$ is a homotopy invariant, called the holonomy $\pi_{1}(\mathcal{F})\to\operatorname{Homeo}(\tau)$ of the loop $\gamma$. See p.48 in Candel-Conlon [3].

Proof of Theorem 1.

By Proposition 3.1 there is a Lefschetz fibration $\pi:X\to B$ that satisfies part (a) of Theorem 1. That is, there is a loop $\gamma$ in the leaf $L_{1}:=\{x=1\}$ of $X$ that is non-contractible in $L_{1}$ but contractible in $X$. The loop $\gamma$ has trivial holonomy because it is contained in a fiber of $\pi$, and the Lefschetz fibration is trivial in a neighbourhood of the leaf $L_{1}$.

In order to obtain a contractible transversal, we first perturb the foliation so that the holonomy of the loop $\gamma$ becomes non-trivial. We replace the single compact leaf $L_{1}$ by a family of compact leaves $L_{1}\times[-\epsilon,\epsilon]$ via a $C^{\infty}$-small perturbation of the foliation. Next, we perform a cut and shear operation (Lemma 3.4) at the leaf $L_{1}\times\{0\}$ to make the holonomy of $\gamma$ non-trivial. This requires a hypersurface $W$ in the leaf $L_{1}$ that is transverse to $\gamma$ and on which the holonomy is trivial. The region $L_{1}\times[-\epsilon,\epsilon]$ has the product foliation and trivial Lefschetz fibration. Therefore, we can take $W$ to be $\pi(L_{1})\times\gamma^{\prime}$, where $\gamma^{\prime}\subset F$ is a loop that intersects $\gamma$ transversely once. The cut and shear operation (Lemma 3.4) produces a $C^{\infty}$-small perturbation of the foliation such that $L_{1}\times\{0\}$ is still a leaf, and the loop $\gamma\subset L_{1}\subset\{0\}$ has non-trivial holonomy. We observe that $\omega$ continues to be a strong symplectic form after a $C^{\infty}$-small perturbation of the foliation.

We now have a loop $\gamma$ in the leaf $L_{1}\times\{0\}$ with non-trivial holonomy and which is contractible in $X$. By moving $\gamma$ off the leaf, it becomes transverse to the foliation, see Lemma 3.3. This proves part (b) of the theorem. ∎

Lemma 3.3**.**

Suppose $(X,\mathcal{F})$ is a foliated manifold, and $\gamma$ is a loop in a leaf $L$ with non-trivial holonomy. Then, there is a loop $\gamma^{\prime}$ that is $C^{0}$-close to $\gamma$ and transverse to the foliation.

Proof.

The proof is visual. We move the loop $\gamma$ in a transverse direction to obtain an embedded cylinder $[0,\epsilon)\times S^{1}$ in $X$. That is, the segments $[0,\epsilon)\times\{pt\}$ are transverse to the foliation. The non-trivial holonomy implies that the foliation restricted to the cylinder is as in Figure 4. The loop $\gamma$ can be moved to $\gamma^{\prime}$ as in the figure.

∎

Lemma 3.4**.**

(A cut and shear surgery)* Suppose $(X,\mathcal{F})$ is a foliated manifold containing*

1. (a)

a compact leaf $L$ and an embedded loop $\gamma:\mathbb{S}^{1}\to L$ with trivial holonomy, 2. (b)

and a compact embedded hypersurface $W$ in the leaf $L$ that intersects $\gamma$ transversely at a single point. Further, the restriction of the holonomy of $L$ to $W$ is trivial.

Then, there exists a $C^{\infty}$-perturbation of the foliation, called $\mathcal{F}^{\prime}$, such that $L$ is still in a leaf of $\mathcal{F}^{\prime}$ and $\gamma$ has non-trivial holonomy in the new foliation.

Proof.

Since the holonomy of the leaf $L$ is trivial over $W$, there is an embedding of $(-\epsilon,\epsilon)\times W$ into $X$ such that for any $w$, $(0,w)$ maps to $w$, $(-\epsilon,\epsilon)\times\{w\}$ is a segment transverse to the foliation, and for each $t$, $\{t\}\times W$ is mapped to a leaf.

We cut and shear along $W\times(-\epsilon,\epsilon)$. That is, we cut the manifold $X$ along $W\times(-\epsilon,\epsilon)$ and glue by a diffeomorphism

[TABLE]

where $\phi:(-\epsilon,\epsilon)$ is a diffeomorphism that is identity near $\pm\epsilon$, $\phi(0)=0$ and $\phi^{\prime}(0)>1$. Call the resulting foliated manifold $(X^{\prime},\mathcal{F}^{\prime})$. There is a homeomorphism $i:X\to X^{\prime}$ that is identity away from a small neighbourhood of $W\times(-\epsilon,\epsilon)$, and the pullback foliation $i^{*}\mathcal{F}^{\prime}$ is smooth and is $C^{\infty}$-close to $\mathcal{F}$. ∎

4. Another example

In this section, we provide a different modification of the Lefschetz fibration of Proposition 3.1 that satisfies parts (a) and (b) of Theorem 1. The new space will be denoted by $\overline{X}$.

By Proposition 3.1, we first produce a Lefschetz fibration $X\to B$ whose regular fiber $F$ is a Riemann surface of genus $1$. Recall that on the compact leaf $L_{1}:=\{x=1\}\subset X$, the fibration is trivial, and so the leaf is a product of tori :$L_{1}\simeq\mathbb{T}^{2}_{f}\times\mathbb{T}^{2}_{b}$. Here $\mathbb{T}^{2}_{f}$ is the fiber and $\mathbb{T}^{2}_{b}$ is the base of the fibration. Further, recall that there is a loop $\gamma\subset\mathbb{T}^{2}_{f}$ that is contractible in $X$, and there is a loop $\mathbb{S}^{1}_{y}\subset\mathbb{T}^{2}_{b}$ which has non-trivial holonomy.

The construction of $\overline{X}$ is as follows. Cut the space $X$ along $L_{1}$ to produce a foliated manifold $\tilde{X}$ with boundary. The boundary components, called $\partial_{+}\tilde{X}$ and $\partial_{-}X$, are both $L_{1}$. Define the new foliated manifold $\overline{X}$ as

[TABLE]

Here the diffeomorphism $\phi:\partial_{+}X\to\partial_{-}X$ is given by

[TABLE]

using an identification of the base and fiber $\mathbb{T}^{2}_{b}\sim\mathbb{T}^{2}_{f}$ where the loop $\gamma$ in the fiber is mapped to the loop $\mathbb{S}^{1}_{y}$ in the base. The loop $\gamma$ is contractible in $\overline{X}$. Via the identification to $\mathbb{S}^{1}_{y}$, it has non-trivial holonomy. Therefore, it can be made transverse to the foliation by ‘pushing’ it off the leaf, see Lemma 3.3.

It remains to describe the strong symplectic form on $\overline{X}$, which is done via the following Claim.

Claim*.*

There is a symplectic form on $\tilde{X}$ for which the boundary leaves are products of symplectic manifolds, and both factors have equal volume.

Proof.

We point out the modifications in the proof of Proposition 2.1 to achieve the Claim. In a neighbourhood of each of the boundary components, the Lefschetz fibration is trivial, and therefore, there is a projection $\pi_{F}:\mathcal{O}p(\partial_{\pm}\tilde{X})\to F$ to the fiber. Let $\omega_{F}$ be a symplectic form on the fiber $F$, and let $\omega_{\partial}:=\pi_{F}^{*}\omega_{F}$ be its pullback to $\mathcal{O}p(\partial_{\pm}\tilde{X})$. On open sets $U_{b}$ intersecting the boundary, we take $\omega_{b}$ to be equal to $\omega_{\partial}$. This ensures that the form $\omega_{pre}$, and consequently $\omega$, respect the product structure on the boundary leaves. Finally, using the Lemma 2.4, we can ensure that the areas on both factors are equal. ∎

As a consequence of the claim, we can choose the map $\phi$ in (1) to be a symplectomorphism – this is done by Moser’s theorem applied to the components of the map $\mathbb{T}^{2}_{b}\to\mathbb{T}^{2}_{f}$ and $\mathbb{T}^{2}_{f}\to\mathbb{T}^{2}_{b}$. Further, there is an embedding in the neighbourhood of the boundary components

[TABLE]

such that $i_{\pm}^{*}\omega=\pi_{2}^{*}(\omega|_{\partial_{\pm}X})$. The collar neighbourhoods can be glued along $\partial_{\pm}\tilde{X}$ to produce a strong symplectic form on $\overline{X}$.