Some tight contact foliations can be approximated by overtwisted ones

A contact foliation is a foliation endowed with a leafwise contact structure. In this remark we explain a turbulisation procedure that allows us to prove that tightness is not a homotopy invariant property for contact foliations.


Statement of the results.
Let M 2n+1+q be a closed smooth manifold. Let F 2n+1 be a smooth codimension-q foliation on M . We say that (M, F) can be endowed with the structure of a contact foliation if there is a hyperplane field ξ 2n ⊂ F such that, for every leaf L of F, (L, ξ| L ) is a contact manifold.
In [2,Theorem 1.1] it was shown that (M 4 , F 3 ) admits a leafwise contact structure if there exists a 2-plane field tangent to F. This was later extended in [1] to foliations of any dimension, of any codimension, and admitting a leafwise formal contact structure. In both cases, the foliations produced have all leaves overtwisted; therefore, the meaningful question is whether one can construct and classify contact foliations with tight leaves.

Instability of tightness under homotopies.
If the foliation F is fixed, the parametric Moser trick [2, Lemma 2.8] implies that any two homotopic contact foliations (F, ξ 0 ) and (F, ξ 1 ) are actually isotopic by a flow tangent to the leaves. In particular, if F is fixed, tightness is preserved under homotopies. Our main result states that this is not the case anymore if F is allowed to move: • the leaves of (N × S 1 , F 0 , ξ 0 ) are tight, • the leaves of (N × S 1 , F s , ξ s ) are overtwisted for all s > 0.

Foliations transverse to even-contact structures.
Given a codimension-1 contact foliation (M 2n+2 , F 2n+1 , ξ 2n ) and a line field X transverse to F, it is immediate that the codimension-1 distribution E = ξ ⊕ X is maximally non-integrable. Such distributions are called even-contact structures.
The kernel or characteristic foliation of E is a line field W ⊂ E uniquely defined by the expression [W, E] ⊂ E. Given an even-contact structure E, any codimension-1 foliation transverse to its kernel is imprinted with a leafwise contact structure. It is natural to study the moduli of contact foliations arising in this manner from E. Our second result states: Theorem 2. Let N be a closed orientable 3-manifold. There are foliations F 0 and F 1 and an even-contact structure E such that: Proof. During the proof of Theorem 1 we shall see that the contact foliations (F s , ξ s ) s∈ [0,1] are imprinted by the same even-contact structure E.
This result is in line with the theorem of McDuff [6] stating that evencontact structures satisfy the complete h-principle: one should expect this flexibility to manifest in other ways.

Turbulisation of contact foliations.
In this section we explain how to turbulise a contact foliation along a loop of legendrian knots.

Local model around a loop of legendrian knots.
Let (N, ξ) be a contact 3-manifold. Any legendrian knot K ⊂ (N, ξ) has a tubular neighbourhood with the following normal form: where (x, y, z) are the coordinates in D 2 × S 1 . A convenient way of thinking about the model is that it is the space of oriented contact elements of the disc. In particular, any diffeomorphism φ of D 2 relative to the boundary induces a contactomorphism C(φ) of the model, also relative to the boundary, as follows: Here we think of z as an oriented line in T (x,y) D 2 and we make dφ act by pushforward.
We can now define a contact foliation in the product with S 1 : Given a contact foliation (M, F, ξ) and an embedded torus K : Our aim now is to fix an even-contact structure E leg in (M leg , F leg ) imprinting ξ leg . The reason why we do not simply choose ξ leg ⊕ ∂ t is that E leg should allow us to turbulise. Take polar coordinates (r, θ) on D 2 . Fix a vector field h(r)∂ r with h(r) < 0 in the region r ∈ (1/2, 2/3) and h(r) = 0 everywhere else. Its flow (φ t ) t∈R is a 1-parameter subgroup of Diff(D 2 ); as explained above, it can be lifted to a 1-parameter subgroup (Φ t ) t∈R of contactomorphisms of D 2 × S 1 . We denote by X the (unique) contact vector field in D 2 × S 1 that generates (Φ t ) t∈R . By construction X is a lift of h(r)∂ r and, in particular, it has a negative radial component in the region r ∈ (1/2, 2/3).
Since X is a contact vector field, the 3-distribution E leg (x, y, z, t) = ξ leg ⊕ ∂ t + X(x, y, z) is an even-contact structure whose kernel is W leg = ∂ t + X(x, y, z) and whose imprint on (M leg , F leg ) is precisely ξ leg .

Turbulisation.
Consider the surface S = [0, 1]×S 1 with coordinates (r, t); the manifold M leg projects onto S in the obvious way. Under this projection the kernel W leg is mapped to the line field L = ∂ t +h(r)∂ r . Similarly, the foliation F leg is the pullback of F 1 = ∂ r . The line fields F 1 and L are transverse to one another. We can find a homotopy (F s ) s∈ [0,1] in S satisfying: • F 0 is as in the last frame of Figure 1: it has a closed orbit bounding a (half) Reeb component. This path of line fields lifts to a path of codimension-1 foliations F leg,s in M leg . The foliation F leg,1 is precisely F leg and F leg,s is isotopic to it for every positive s. The foliation F leg,0 has a single compact leaf, which is diffeomorphic to T 3 ; this leaf bounds a Reeb component whose interior leaves are diffeomorphic to R 2 × S 1 . Transversality of L with respect to F s implies that E leg imprints a contact foliation ξ leg,s on each F leg,s .
Let us package this construction: say that the homotopy (M, F s , ξ s ) s∈ [0,1] given by the procedure just described is the turbulisation of (M, F, ξ) along U . As explained in [3], any torus having a linear characteristic foliation T S ∩ ξ is quasi pre-lagrangian. Incompressibility means that π 1 (S) injects into π 1 (N ). Finally, we say that (N, ξ) is universally tight if the lift of ξ to the universal coverÑ is tight.

Lemma 4. Let (M, F, ξ) be a contact foliation with tight leaves. Denote its turbulisation along
Proof of Lemma 4. Let us write E for the even-contact structure imprinting ξ s and W for its characteristic foliation. E is chosen arbitrarily away from U , but in the model it is given by E leg .
Consider first the case s > 0. The foliations (F s ) s∈(0,1] are all isotopic to F 1 = F. This isotopy, by construction, can be realised by a flow tangent to W (which in particular preserves E). This immediately induces a contactomorphism between any leaf (L s , ξ s ) of F s and the corresponding leaf (L, ξ) of F to which it is isotopic.
Assume now s = 0. We can argue similarly to show that the open leaves of F 0 can be identified, as contact manifolds, with open subsets of leaves of F, proving tightness. We claim that the remaining leaf (T 3 , ξ 0 ), bounding the Reeb component, is also tight. Consider the 2-torus S = T 3 ∩ {t = t 0 } ⊂ T 3 . Due to the rotational symmetry of the model, S is quasi pre-lagrangian and also incompressible. Using a flow along W again, we can identify the contact manifold (T 3 \S, ξ) with an open subset in L∩U , where L is a leaf of F and U is the region where the turbulisation takes place. In particular, we are identifying it with a subset of the local model (D 2 × S 1 , ξ leg ), which is universally tight. An application of Proposition 5 concludes the claim.
In particular, the model (M leg , F leg , ξ leg ) and its turbulisation are tight.

Remark 6.
There is an alternate way to describe the turbulisation process. The contact foliation (M leg , F leg , ξ leg ) is the space of oriented contact elements of the foliation (D 2 ×S 1 , t∈S 1 D 2 ×{t}). Then, the turbulisation process upstairs amounts to turbulising (D 2 × S 1 , t∈S 1 D 2 × {t}) and applying the contact elements construction. This construction also works for higher dimensional contact foliations and highlights the fact that the resulting leaves (in the model) are tight. Vol. 110 (2018) Some tight contact foliations 417 3. Applications.

Proof of Theorems 1 and 2.
In [4] K. Dymara proved that there are legendrian links in overtwisted contact manifolds that intersect every overtwisted disc; that is, their complement is tight. Such a link is said to be non-loose. Let (N, ξ) be an overtwisted contact manifold with K a non-loose legendrian link. Consider the contact foliation where we abuse notation and write ξ for the leafwise contact structure lifting (N, ξ). Take U to be the tubular neighbourhood of K × S 1 ⊂ M and apply the turbulisation process to (M, F 1 , ξ 1 ) (on each component) to yield a path of contact foliations (M, F s , ξ s ) s∈ [0,1] . It is immediate that (M, F s , ξ s ) is diffeomorphic to (M, F 1 , ξ 1 ) if s is positive, because the foliations themselves are isotopic and Gray's stability applies. In particular, the leaves of all of them are overtwisted. We claim that (M, F 0 , ξ 0 ) has all leaves tight. This is clear for the leaves in the Reeb components we have introduced, as shown in Lemma 4. Similarly, the leaves outside of the Reeb components are tight because a neighbourhood of the non-loose legendrian link has been removed. We conclude by recalling that every closed overtwisted 3-manifold admits a non-loose legendrian link: the legendrian push-off of the binding of a supporting open book [5].
Remark 7. The foliation (M, F 1 ) is taut, since it admits a transverse S 1 . As pointed out by V. Shende during a talk of the author: we are trading tautness of the foliation to achieve tightness of the leaves.

A more general statement.
A slightly more involved argument shows: This allows us to choose a legendrian knot K ⊂ (D 3 , ξ std ) and lift it to K ×S 1 ⊂ (M std , F std , ξ std ) ⊂ (M, F, ξ). Turbulisation in a neighbourhood of K × S 1 yields a contact foliation (M, F , ξ ) whose leaves are still tight due to Lemma 4. The interior of the Reeb component we just inserted is diffeomorphic, as a contact foliation, to the model (M leg , F leg , ξ leg ). Given a homotopically essential transverse knot η ⊂ (D 2 × S 1 , ξ leg ), we may perform a Lutz twist along η