The Engel-Lutz twist and overtwisted Engel structures
Álvaro del Pino Thomas Vogel
math.SG math.GT 58A30
 



[2010]Primary: 58A30.





Utrecht University, Department of Mathematics, Budapestlaan 6, 3584 Utrecht, The Netherlands
a.delpinogomez@uu.nl


Mathematisches Institut der LMU, Theresienstr. 39, 80333 München, Germany. 
tvogel@math.lmu.de



We introduce a modification procedure for Engel structures that is reminiscent of the Lutz twist in 3-dimensional Contact Topology. This notion allows us to define what an Engel overtwisted disc is, and to prove a complete h-principle for overtwisted Engel structures with fixed overtwisted disc.


The Engel-Lutz twist and overtwisted Engel structures
    Thomas Vogel
    December 30, 2023
=====================================================





§ ABOUT THIS PAPER


A maximally non-integrable 2-plane field  on a 4-manifold M is called an Engel structure. Maximal non-integrability means that = [,] is a distribution of rank 3 that satisfies [,] = TM. Engel structures hold a privileged position in the taxonomy of distributions: they are one of the four topologically stable families of distributions (i.e. distributions described by an open condition and having a local model). In this paper we study them from the perspective of the h-principle, viewing them as sections of the Grassmann bundle Gr(TM,2) which satisfy a certain differential relation of order 2. The goal is to study the space of such sections, comparing it with the space of formal solutions. In the case when M and  are orientable and oriented, formal solutions are trivializations of TM in which the first two components of the framing span .

Gromov's method of flexible continuous sheaves <cit.> shows that Engel structures, which are given by an open and Diff-invariant partial differential relation, satisfy the h-principle in open manifolds. This does not prove the analogous result for closed manifolds. In <cit.> it is stated: “On the other hand, it is unknown whether the h-principle holds for Engel structures on closed 4-manifolds. In particular, it is an outstanding open question whether any closed parallelizable 4-manifold admits an Engel structure”.

It was proven in <cit.> that every parallelizable 4-manifold does admit an Engel structure. The proof relies on the interplay between Engel and contact structures: First, the ambient manifold is decomposed into round handles, then one can proceed handle by handle constructing the desired Engel structure. During this process, the boundary of each handlebody inherits a contact structure. The heart of the argument is to be able to manipulate these contact structures to ensure that the handles can indeed be glued. 

More recently, the existence problem for Engel structures (in every formal class) was solved <cit.>. It was shown that the inclusion 

    i: (M) →(M)

of the Engel structures into the formal Engel structures is a surjection in homotopy groups whenever M is closed. The key contribution of <cit.> is a method for constructing and manipulating Engel structures locally. As such, the approach differs from the one in <cit.> and is more of an h-principle. However, the proof of h-principle in <cit.> does not work relative to subsets U of the manifold. This is because it relies on adding twisting (see Sections <ref> and <ref>) along the leaves of the characteristic foliation. This is not possible when the characteristic foliation 
is tangent to the boundary of ∂ U (as sketched in Figure <ref>).


Currently, the outstanding open question in Engel Topology is, whether i might in fact be a homotopy equivalence. While answering this question is beyond the scope of this article, we are able to define what an overtwisted Engel structure is and show that Engel structures with fixed overtwisted disc satisfy the complete h-principle. In particular, the result we obtain is relative in the parameter, and the domain. An important corollary, which does not follow from <cit.> nor <cit.>, is that every Engel germ in ∂^4 extends to the interior of ^4 if it extends formally.




 §.§ Outline of the paper


Throughout the paper we use Gromov's notation (A) to denote an arbitrarily small neighborhood of the set A. We write ∫(A) for the interior of the set A.



  §.§.§ The Engel-Lutz twist and overtwisted Engel structures


In 3-dimensional Contact Topology one defines the Lutz twist <cit.> as a surgery operation along a transverse knot. One replaces the contact structure on a standard neighborhood of the knot by a contact structure which “twists” more. While this operation does not change the homotopy class as a plane field, it can change the homotopy class as a contact structure. In particular, the resulting contact structure is always overtwisted <cit.>.

The main focus of this article are Engel-Lutz twists. This construction is the Engel analogue of the classical Lutz twist. One replaces the Engel structure in a neighborhood of a 2-torus transverse to  by another Engel structure which has more “twisting”. This operation preserves the formal type, but we do not know whether the resulting Engel structure is Engel homotopic to the original one. This is explained in Section <ref>. 

Before defining the Engel-Lutz twist, we show that there are many transverse 2-tori in a given Engel manifold (Section <ref>). More precisely, we prove that transverse 2-tori can be constructed along knots transverse to the even-contact structure [,]. Moreover, the transverse surfaces we construct are quite flexible and can be isotoped effectively.

When one attempts to construct an Engel structure from a formal Engel structure, a certain family of Engel germs along ∂^4 is obtained (Proposition <ref>). A key observation is that the Engel-Lutz twist can be used to extend these germs to the interior (Subsection <ref>). Motivated by this fact, in Section <ref> we endow the 4-ball with a specific Engel structure _, which is a portion of Engel-Lutz twist satisfying a certain numerical constraint. We say that the Engel manifold Δ_ = (^4,_) is an overtwisted disc. If an Engel embedding Δ: Δ_ (M,) exists, we say that (M,) is an overtwisted Engel manifold.



  §.§.§ Statement of the main results


Let Δ: ^4  M be a smooth embedding. We define _(M,Δ) ⊂(M) to be the subspace of those Engel structures on M such that their pullback by Δ is _. Its formal analogue is (M,Δ) ⊂(M), the subspace of those formal Engel structures which are Engel on ⊷(Δ) and that pullback to _ under Δ. 

In families, the particular embedding of the overtwisted disc may vary and might even be twisted: Let K be a compact manifold. We will say that a K-family of Engel structures : K (M) is overtwisted if there is a locally trivial fibration of overtwisted discs Δ_k ⊂ (M,(k)), k ∈ K. We will say that the family Δ = (Δ_k)_k ∈ K is the certificate of overtwistedness of . In Subsection <ref> we discuss a different (but equivalent up to homotopy) way in which overtwistedness can appear in the parametric setting.

The terminology “overtwisted” is justified by our main result, which states that overtwisted families of Engel structures are flexible.
 
Let M be a smooth 4-manifold (possibly non-compact or with boundary). Let U ⊂ M be a closed subset such that M ∖ U is connected. Let K' ⊂ K be compact CW-complexes.

Suppose _0: K (M) is a family of formal Engel structures satisfying: 


  * _0(k) is Engel at p ∈ M if p ∈(U) or k ∈(K'),

  * _0 has a certificate of overtwistedness (Δ_k)_k ∈ K with Δ_k ⊂ M ∖ U.


Then, there is homotopy : K × [0,1] (M) such that


  * (·,0) = _0,

  * (k,s)(p) = (k,0)(p) whenever k ∈(K') or p ∈(U) ∪Δ_k,

  * (·,1): K (M). 


Its proof spans Section <ref>. We invite the reader to check that our approach provides an alternate proof of Eliashberg's classification of overtwisted contact structures in dimension 3 <cit.>.

From the theorem, the complete h-principle for Engel structures with fixed overtwisted disc follows immediately (Subsection <ref>):
 
The inclusion 

    _(M,Δ) (M,Δ)

is a homotopy equivalence.


As is often the case, the particular trivialization of the overtwisted disc does not matter for π_0 statements. Write _(M) ⊂(M) for the subspace of overtwisted Engel structures. The following statement follows from Theorem <ref>:
 
The inclusion

    _(M) (M)

induces a bijection between path components.


More generally, the homotopy type of the certificate is not relevant for lower dimensional families:
 
Let K be a CW-complex of dimension at most 3. Two overtwisted families _0,_1: K (M) are homotopic if and only if they are formally homotopic.

Corollaries <ref> and <ref> are worked out in Subsection <ref>.

Theorem <ref> addresses one of the shortcomings in <cit.>, namely that the result proved there is not relative in the domain (see Figure <ref>).
 
Let K be a compact CW-complex and let _0: K (^4) be a family of formal Engel structures in the 4-ball with _0(k)|_∂^4 Engel. Then, there is a family _1: K (^4) which, relative to ∂^4, is formally homotopic to _0.

In the non-parametric case, this is shown in Subsection <ref> without relying on the full force of Theorem <ref>. The parametric version is deduced from Theorem <ref> in Subsection <ref>.

Finally, in Subsection <ref> we briefly explain how an h-principle for foliations endowed with a leafwise Engel structure can be deduced from our main result.




  §.§.§ Looseness and overtwistedness


In <cit.> (see also the upcoming paper <cit.>) an alternative characterization of Engel flexibility is introduced and, using it, a result similar to Corollary <ref> is obtained for a different subclass of Engel structures, which we called loose in <cit.>. Normally, any two different notions of overtwistedness are equivalent up to homotopy. Indeed, suppose we have two different definitions, the first given by the presence of some local model, which we call the disc of type I, and the second one given by another local model, which we call the disc of type II. Suppose  is an overtwisted structure of type I. We modify it away from its overtwisted disc to introduce a disc of type II, and then we use the parametric nature of Theorem <ref> to construct a homotopy between them through overtwisted structures of type I. This shows that an overtwisted structure of type I contains a disc of type II up to homotopy.

This argument cannot be applied here because Engel looseness is not given by a local model but by a global property of the structure. Then, an intriguing open question is how Engel overtwistedness and looseness might be related. A closely related question is whether there exist Engel families that are not overtwisted nor loose.

A relevant remark is that the extension problem from Engel structures (Corollary <ref>) cannot be deduced either from the results in <cit.>. This is due to the fact that looseness is a global property and therefore not well-suited for performing constructions relative in the domain.

Acknowledgments: The work in this article has profited from an AIM-Workshop on Engel structures which was held in the spring of 2017. We would like to thank all the participants of this workshop for their questions and all the discussions we had. In particular, we would like to thank Roger Casals, Yakov Eliashberg, Dieter Kotschick, Emmy Murphy, and Fran Presas. Á. del Pino is supported by the NWO Vici Grant no. 639.033.312 of Marius Crainic.





§ DEFINITIONS AND STANDARD RESULTS


We start by reviewing the relevant definitions and going over basic results. Unless stated otherwise, all distributions we consider are smooth. 



 §.§ Engel structures
 


Let M be a smooth 4-manifold.

	
  * An even-contact structure is a smooth hyperplane field  such that [,]=TM.
	
  * An Engel structure is a smooth plane field  such that [,] is an even-contact structure.	


When U⊂ M is closed, we say that a plane field  is Engel on U if it is an Engel structure on some open neighborhood of U.

For an even-contact structure  and any p ∈ M, we can consider the map

    (p)×(p)     T_p M/(p) 
    
    X,Y    ⟼ [X,Y](p).

The commutator is defined using local extensions of the vectors X,Y to local sections of , but the result is independent of choices. Thus, we have an antisymmetric bilinear form on (p) taking values in a real vector space of dimension one. Therefore, this form has a kernel, which we denote by (p). It defines a foliation of rank one ⊂ which we will call the kernel foliation[Other common names for  include characteristic foliation and isotropic foliation.] of . It is characterized by the property [,]⊂, i.e. all flows tangent to  preserve .

If  is an Engel structure satisfying = [,], it follows that ⊂, because otherwise [,]⊄. Consequently, from every Engel structure  we obtain a flag of distributions 

    ⊂⊂⊂ TM

where the rank increases by one at every step. Whenever =[,] is induced by an Engel structure,  has a canonical orientation defined by {X,Y,[X,Y]}, where {X,Y} is any local frame of . However, there are no canonical orientations for the other elements of the flag.


A formal  Engel structure is a complete (non-oriented) flag ⊂⊂⊂ TM, together with two  isomorphisms

    ()    ≅/, 
    (/)   ≅ TM/.


The space of formal Engel structures on M, endowed with the C^0-topology, is denoted by (M). Under the natural inclusion

    ι : (M) (M)

it induces the C^2-topology on the space of Engel structures (M). 

If  is an Engel structure, the first isomorphism in Equation (<ref>) is induced by 

    Λ^2    /
    
    X∧ X'    ⟼ [X,X'].

The induced map being an isomorphism is a reformulation of the fact that  inherits a canonical orientation from . The second isomorphism is induced by the map

    Λ^2     TM/
    
    Y∧ Y'    ⟼ [Y,Y']

whose kernel is ∧. We will often write (,,) when we talk about a formal Engel structure, omitting the morphisms (<ref>).

Lastly, consider the case when  and  are oriented and fix a framing {W ∈,X ∈} of  compatible with the orientations. Then, the following expression yields a framing of TM:

    { W,X,[W,X],[X,[W,X]] }

This framing is unique, up to homotopy, once the orientations of  and  have been fixed. We call it the Engel framing. Under these orientation assumptions the Engel condition can be rephrased in terms of a pair of forms α,β defining =(α)∩(β) and (α)= as follows

    α∧ dα   ≠ 0,  
    α∧β∧ dα   =0, 
    α∧β∧ dβ   ≠ 0.

In this oriented case, we can consider oriented flags as the formal counterparts of Engel structures.



 §.§ The development map
 

Following <cit.> we now recall an interpretation of the condition [,]= in terms of the holonomy of .

Fix a point p ∈ M and write _p for the leaf of  through p. Let _p be its universal cover. Given a point q ∈_p, we can find a vector field W which is tangent to  and is non-vanishing on the segment of _p connecting p and the image of q in _p (we abuse notation and still denote it by q). The flow φ_t of W preserves  and . Then, the differential of φ_-t induces a linear map

    h_p(q):=Dφ_-t  : (q)/(q) ⟶(p)/(p)

where q=φ_t(p). This map is independent of the choice of W and it represents the linearized holonomy of the foliation . Therefore, one can consider the image of (q) for each q in _p. This defines the development map

    h_p : _p    ⟶ℙ((p)/(p)) 
    
    q    ⟼ Dφ_-t((q))  with φ_t(p)=q  as above.

Because of the Engel condition [,]=, we deduce that h_q is an immersion. 

When  is oriented, one can replace the projective space ℙ((p)/(p)) by the space ℙ((p)/(p)) of oriented lines in (p)/(p). 




 §.§ Curves on spheres and the Engel condition


Fix coordinates (p,t) in ^3×. A plane field  in ^3× containing ∂_t can be described in terms of a smooth family of curves (H_p : ^2)_p∈^3. Indeed, once we fix a framing {X(p),Y(p),Z(p)} of T^3, we can identify (H_p)_p ∈^3 with the vector field

    (p,t) ⟼(H_p(t))_1· X+(H_p(t))_2· Y  +  (H_p(t))_3· Z,

where the subscript i corresponds to taking the i-th coordinate. Using this identification, we construct a plane field:

    (p,t) = ⟨∂_t, H_p(t) ⟩.

One can now compute

    [,](p,t)     = ⟨∂_t, H_p(t), Ḣ_p(t) ⟩
    where Ḣ_p(t) = [∂_t,H_p(t)]. Similarly,[,[,]](p,t)     = ⟨∂_t, H_p(t), Ḣ_p(t), Ḧ_p(t), [H_p(t),Ḣ_p(t)] ⟩

where Ḧ_p(t) = [∂_t,Ḣ_p(t)].

The following characterization of Engel structures tangent to ∂_t will be fundamental. Its proof follows from the discussion above. For additional details we refer the reader to <cit.>.
 
A ^3-family of curves (H_p)_p ∈^3 in ^2 determines an Engel structure near (p,t)∈^3× if and only H_p is an immersion at time t and at least one of the following conditions holds.


  (i) The vectors H_p(t),Ḣ_p(t),Ḧ_p(t) are linearly independent. 

  (ii) The vector fields H_q(t) and Ḣ_q(t) span a contact structure in a neighborhood of (p,t) in ^3×{t}.  


We remark that H_p being an immersion is equivalent to the fact that  is non-integrable (i.e. [,]⊄). Note that conditions (i) and (ii) are both open and not mutually exclusive. Moreover, we do not require for the contact structure in (ii) to be positive with respect to a fixed orientation of ^3.  

By definition, the kernel foliation is tangent to ∂_t if and only if condition (i) fails. If condition (i) is satisfied, then H_p is strictly convex or strictly concave at t. 




 §.§ Model structures
  

In Section <ref> we will perform certain local manipulations of formal Engel structures. We will now describe a family of local models that is suitable for this purpose. The reader should compare the construction presented here to condition (ii) in Proposition <ref>.

Let I be a 1-manifold. We write (y,x) for the product coordinates in Γ = I × [0,1]. A pair of smooth functions f_±: Γ satisfying f_- ≤ 0 < f_+ determines a domain 

    D(I,f_-,f_+) = { (y,x,z) ∈Γ×  |   f_-(y,x)≤ z ≤ f_+(y,x)}

which we endow with the contact structure 

    ξ = (cos(z)dy-sin(z)dx) = ⟨∂_z, cos(z)∂_x+sin(z)∂_y ⟩.


Let J=[a,b] be a 1-manifold with coordinate w. A function c: D(I,f_-,f_+)× J determines a complete flag in D(I,f_-,f_+)× J: 

    = ⟨∂_w⟩
    _c     =⊕⟨ X_c= cos(c)∂_z + sin(c)(cos(z)∂_x+sin(z)∂_y)⟩
        = ⊕ξ,

where the distribution  is even-contact, does not depend on c, and has  as its characteristic foliation.

Using the flag we can describe a formal Engel structure by fixing bundle isomorphisms as in Equation (<ref>). First note that, since  is even-contact, the isomorphism (/) ≅ TM/ is automatically given by Equation (<ref>). The other bundle isomorphism () ≅/ is unique, up to homotopy, once we impose for {∂_w,∂_z,cos(z)∂_x+sin(z)∂_y} to be a positive framing of . It is given by an identification

    Λ^2   /
    ∂_t ∧ X_c    ⟼ b·(-sin(c)∂_z + cos(c)(cos(z)∂_x+sin(z)∂_y)),

where b is a positive function. Often, _c will be Engel on some subset U of the model. Then, the isomorphism is prescribed U by Equation (<ref>)  on, and we will assume that b|_U = (∂_t c)|_U.

 
The formal Engel manifold 

    M(I,J,f_-,f_+,c) = ((D(I,f_-,f_+)× J),,_c,)

is said to be a model structure with


1.3=10pt

angular function     c 

height function     c(y,x,z,b) - c(y,x,z,a) 

height     min_(y,x,z) ∈∂ D(I,f_-,f_+) (c(y,x,z,b) - c(y,x,z,a)) 

bottom boundary     {w=a} 

top boundary     {w=b} 

vertical boundary     {(y,x,z,w) |  (y,x,z) ∈∂ D(I,f_-,f_+)}

projection     π: D(I,f_-,f_+) × J  D(I,f_-,f_+) 

 


The model is solid if ∂_w c>0 everywhere. 

It is immediate that parametric families of models can be considered by smoothly varying the functions f_-, f_+, c, and the parametrisation of I and J.

Condition (ii) of Proposition <ref> implies that a model structure is Engel at (y,x,z,w) if and only if ∂_w c(y,x,z,w) ≠ 0. Having chosen {∂_w,∂_z,cos(z)∂_x+sin(z)∂_y} as a positive framing of , we need to further require ∂_w c(y,x,z,w) > 0 to obtain the correct formal class.
 
Let M(I,J,f_-,f_+,c) be a model that is Engel along the boundary and has strictly positive height function everywhere. Then _c can be homotoped to a solid model through model structures and relative to the boundary.


Since c(y,x,z,b) > c(y,x,z,a), one can construct a function 

    c^: D(I,f_-,f_+)× J

which coincides with c on a neighborhood of the boundary and satisfies ∂_w c^>0. Then M(I,J,f_-f_+,tc^sol+(1-t)c), t∈[0,1], provides the desired homotopy through model structures.







 §.§ Loops transverse to the even-contact structure
 

Let (M,) be an Engel manifold. In Sections <ref> and <ref>, we will construct surfaces and hypersurfaces transverse to . These submanifolds will lie in standard neighborhoods of curves transverse to =[,]. In this subsection we will explain some elementary facts about such curves.


 A transverse curve is an embedding of a 1-manifold into (M,) which is transverse to =[,].

When a transverse curve is closed, we will sometimes call it a transverse loop. 



  §.§.§ Normal form of  close to a transverse loop
 

Let I be a compact 1-manifold. We consider I×^3 with coordinates (y,x,z,w) and endowed with the Engel structure 

    _ = (α_=dy-zdx,β_=dx-wdz).

The following proposition states that any Engel structure  is isomorphic to this model in the vicinity of a transverse curve.
 
Let (M, ) be an Engel manifold,  the associated even-contact structure, and γ: I  M a transverse curve. Then, the map γ can be extended to an Engel diffeomorphism

    φ : (I×{0}) ⊂ (I ×^3 , _) (⊷(γ)) ⊂ (M,).

The space of such diffeomorphisms is weakly contractible. 



Since γ is transverse to , there is a hypersurface N≃ I×^2 containing γ which is transverse to . Then TN∩ is a contact structure ξ on N, γ is transverse to it, and the Engel structure induces a Legendrian line field = TN∩⊂ TN∩ on N.

We choose a vector field H spanning  and a surface Γ = I × (-ε,ε) ⊂ N which is transverse to H and contains γ. Using the characteristic foliation ξ(Γ) one obtains coordinates (x,y) on a neighborhood of γ within Γ such that γ={x=0} and ξ(Γ) is defined by dy. Then we use the flow of H and the contact condition to obtain coordinates (x,y,z) in N such that ξ=(dy-zdx). By construction,  is spanned by ∂_z.

Finally, we choose a vector field W spanning  near N. Due to the condition [,]= there are coordinates (x,y,z,w) on a neighborhood of γ in which the even-contact structure  is defined by (dy-zdx) and =∩(dx-wdz).  

If φ is a diffeomorphism with the desired property, then there are submanifolds Γ⊂ N and a vector field H which produce φ as above: we simply pick N=φ({w=0}), Γ=φ({z,w=0}), and H=φ_*(∂_z). Now assume K is a compact space and {φ_k}_k∈ K is a family of diffeomorphisms with the desired properties. Then the germs of Γ_k ⊂ N_k and H_k along γ as above form a weakly contractible space. Thus, the same is true for the space of germs of diffeomorphisms satisfying the hypotheses of the proposition.  




  §.§.§ Existence and classification of transverse curves


The following lemma proves the h-principle in π_0 for embedded transverse curves. It is an elementary consequence of the corresponding h-principle for immersed transverse curves in contact 3-manifolds.
  
Let γ: [0,1]  M be a smooth map that, in a neighborhood of the endpoints, is embedded and positively transverse to . Then it can be homotoped, relative to the boundary, to an embedded transverse arc. 

Let γ_0, γ_1 : [0,1]  M be embedded arcs, transverse to , that agree in a neighborhood of their endpoints, and which lie in the same homotopy class as maps. Then they are isotopic, relative to the boundary, as embedded transverse arcs.  
 

Let us prove the second statement, which is slightly more involved. The proof of the first is analogous.

Since γ_0 and γ_1 lie in the same homotopy class, we can choose a smooth homotopy (γ_s)_s∈[0,1] between them. Since γ_0 and γ_1 are transverse to  and agree at their endpoints, we can view γ_s as family of formal immersions transverse to  (that is, as a family of curves together with an injective bundle map T[0,1] TM/). These bundle maps are unique up to homotopy. Using the Smale-Hirsch theorem <cit.> we can assume that the (γ_s)_s∈[0,1] are immersions, but not necessarily transverse to . For dimensional reasons, standard transversality theorems imply that the (γ_s)_s∈[0,1] can be assumed to never be tangent to . Similarly, after a perturbation we may assume that γ_s is embedded for all s. 

We can pick a family of embeddings ψ_s : [0,L]×^2 ⟶ M which are transverse to  and satisfy ψ_s(y,0,0)=γ_s(y). Intersecting the tangent space of the image of ψ_s with  we obtain a contact structure ξ_s on N_s = ψ_s([0,1]×^2). The h-principle for transverse immersions in contact manifolds (see <cit.>) implies that there is a smooth family of immersions 

    γ_s : [0,L]  (N_s,ξ_s) ⊂ M
 
which are transverse to ξ_s (and therefore transverse to ) interpolating between γ_0 and γ_1. We may assume that γ_s coincides with γ_0 and γ_1 on a neighborhood [0,l]∪[1-l] of the points {0,1}. 

In order to turn γ_s into a family of transverse embeddings, we use the extra dimension complementary to N_s. Let g: [0,1]  [0,1] be a smooth function such that 


  * g≡ 0 on [0,l/2]∪[1-l/2,1],

  * g is strictly monotone outside of [l,1-l]. 

Locally in a neighborhood of γ_s we can choose a vector field W orienting ; denote its flow by h_r. Let λ : [0,1] [0,1] be a non-decreasing surjective function which is constant close to its endpoints. If ε>0 is small enough

    γ_s : [0,1]    ⟶ M 
    
    y    ⟼ h_ελ(s)g(y)(γ_s(y))

is a family of transverse embeddings interpolating between the curves γ_0 and γ_1. 




 §.§ Curves tangent to the even-contact structure
 

One of the technical steps in Section <ref> requires us to manipulate curves that are tangent to the even-contact structure but transverse to its kernel. The following statement provides standard neighborhoods for such curves:
 
Let (M,) be a 4-dimensional even-contact manifold and ν: I  M an embedded curve tangent to  but transverse to its kernel.

Then, there is an even-contact embedding:

    ((I ×{0}) ⊂ I ×^3, (dz-ydx))  ((⊷(ν)) ⊂ M, ),

where the coordinates in I ×^3 are (y,x,z,w).


This follows from the standard neighborhood theorem for Legendrians in a contact 3-manifold and the fact that any even-contact structure is semi-locally given by a contact slice stabilized by .


The following proposition states the existence h-principle for curves tangent to an even-contact structure.
 
Let K be a contractible set and (M,_k)_k ∈ K a K-family of 4-dimensional even-contact manifolds with corresponding kernels (_k)_k ∈ K. Let J ⊂ I be intervals.

Fix a family of embeddings (ν_k: I  M)_k ∈ K with ν_k|_(J) tangent to _k and transverse to _k. Then, there is a C^0-small perturbation ν̃_k of ν_k such that


  * ν̃_k is an embedded curve everywhere tangent to _k but transverse to _k, and

  * ν̃_k ≡ν_k over the interval J. 



For its proof, we need two standard h-principle results. The first one provides embeddings transverse to :
 
Let K be a contractible set, M a 4-dimensional manifold, and (_k)_k ∈ K a family of line fields in M. Let J ⊂ I be intervals.

Fix a family of embeddings (ν_k: I  M)_k ∈ K with ν_k|_(J) transverse to _k. Then, there is a C^0-small perturbation ν̃_k of ν_k which satisfies:


  * ν̃_k is an embedded curve everywhere transverse to _k, and

  * ν̃_k ≡ν_k over the interval J. 



For a fixed k_0 ∈ K, standard transversality implies that ν_k_0 can be C^∞-perturbed to yield an embedded curve transverse to _k_0. The homotopy between the two, as embeddings, endows ν_k_0 with the structure of an embedding that is formally transverse to _k_0. Then, since the parameter space K is contractible, it follows that the whole family (ν_k)_k ∈ K can be regarded as a family of embeddings that are formally transverse to the corresponding line fields (_k)_k ∈ K.

For all points p ∈ M, the subset T_pM ∖_k(p) is connected and its convex hull is T_pM, i.e. it is an ample subset. Then the result follows from the complete h-principle for ample, Diff-invariant, 1-dimensional differential relations (see <cit.>).


The second ingredient we need is a method for producing transverse knots in contact 3-manifolds. 
 
Let K be a contractible set and (N_k,ξ_k)_k ∈ K a K-family of 3-dimensional contact manifolds. Let J ⊂ I be intervals.

Fix a family of knots (ν_k: I  N_k)_k ∈ K with ν_k|_(J) tangent to ξ_k. Then, there is a C^0-small perturbation ν̃_k of ν_k which satisfies:


  * ν̃_k is a Legendrian knot for ξ_k, and

  * ν̃_k ≡ν_k over the interval J. 
 


If (N_k,ξ_k) ≃ (^3,ξ_), we can use the front projection and view ν_k as a (possibly singular) curve in the plane together with a slope at every point recording the missing coordinate. Then ν̃_k is drawn in the front projection as a cuspidal curve such that the slope approximates the given one. In the general case, for each t ∈ [0,1], we can find, parametrically in k, a Darboux ball ((ν_k(t)),ξ_k). We may then use the corresponding local front projections and the relative nature of the statement (in the domain, not the parameter) to conclude.



First we apply Lemma <ref> to yield a new family of embedded curves (ν_k')_k ∈ K which are transverse to the line fields (_k)_k ∈ K. Transversality with respect to _k implies that each ν_k' can be thickened to an embedded transverse 3-manifold endowed with a contact structure (N_k,ξ_k = TN_k ∩_k). Then we apply Lemma <ref>, parametrically in k, to homotope ν_k' to an embedding ν̃_k which is still contained in N_k but is additionally transverse to _k. This concludes the proof.







§ TRANSVERSE SURFACES IN ENGEL MANIFOLDS
 

Let (M,) be an Engel manifold, I a compact 1-manifold, and γ: I  M a transverse curve. The purpose of this section is to construct embedded surfaces transverse to  and contained in a tubular neighborhood of γ. Later, in Section <ref>, we will describe how to perform an Engel-Lutz twist along such a surface. 

Throughout this section we will assume that M, , and all surfaces appearing in the discussion are oriented. In particular,  is trivial as a vector bundle. 



 §.§ Transverse surfaces
  

The following notion is the central object in the discussions that follow.

Let (M,) be an Engel manifold. A transverse surface in M is an immersed surface S whose tangent space at each point is transverse to .

It follows that the even-contact structure = [,] intersects a transverse surface in a line field.

An immersed surface S⊂ (M,=(α)∩(β)), with = (α), is transverse if and only if:


  * (α)∩ TS = ∩ TS has rank 1 everywhere,

  * (β)∩ TS has rank 1 everywhere, and

  * these two line fields are everywhere transverse to each other.   




  §.§.§ Profiles and transverse cylinders


Let γ : I  M be a transverse curve.  We will construct transverse surfaces in (γ) using the normal form from Proposition <ref>. A reference concerning  transverse knots in contact 3-manifolds is <cit.>. 

Our first example is an explicit  transverse cylinder/torus:
 
Let r: I be a positive function. The map 

    S : I ×^1   ⟶      (I ×^3,_) 
    
    (y,θ)    ⟼     ([                         y; x(y,θ)=r(y)^2sin(θ)cos(θ);         z(y,θ)=r(y)sin(θ);        w(y,θ)=2r(y)cos(θ) ])

is an embedding of I ×^1. We now determine a condition which ensures that the 1-forms S^*(dy-zdx) and S^*(dx-wdz) are linearly independent everywhere, i.e. that S is transverse to _. We compute

    S^*(dy-zdx) 
                    = (1-2r(y)^2r'(y)sin(θ)^2cos(θ))dy 
        + r(y)^3sin(θ)(sin(θ)^2 - cos(θ)^2)dθ
    
    S^*(dx-wdz) 
                
    			
                    = -r(y)^2dθ.

Therefore, S is transverse to _ if |2r(y)^2r'(y)| < 1. In particular, the construction yields a transverse surface whenever the function r(y) is constant. 


Let us explain in more geometric terms the role of the non-integrability of _ in Equation (<ref>). We will work under the simplified assumption that r(y) is constant: The 2-torus S intersects the level {y=y_0} in a curve η that is transverse to (β_) (and in particular transverse to ∂_w). The front projection of η (i.e. its projection to the (x,z)-plane) is a planar curve that does not depend on y_0; it is a figure-eight, as depicted in Figure <ref>. The short line segments in Figure <ref> indicate how the line field T_∩{w=w(y,θ)} varies along the knot. Using β_ we can recover the missing coordinate w from the line field.

If _ was integrable and tangent to ⟨∂_w⟩, the line field T_∩{w=w_0} would be independent of w_0 and it would project to a non-singular line field in the (x,z)-plane. Since the projection of η is a closed curve, it would necessarily be tangent to this line field somewhere. The non-integrability of _ precisely implies that T_∩{w=w_0} varies with w_0, providing enough flexibility to construct embedded transverse curves in the projection and therefore embedded transverse tori in the model.



Example <ref> can be generalized as follows.
 
Let η be a transverse knot in (^3,ξ_=(β_)). The surface 

    S = I ×η⊂ (I ×^3,_)
  
is embedded and transverse to .


Since S is tangent to ∂_y, it is transverse to (α_). It is also transverse to (β_) because η itself is transverse to this hyperplane field. Using β_(∂_y)=0 we have: 

    α_∧β_(∂/∂ y,η̇)  
     = α_(∂_y)β_(η̇)-α_(η̇)β_(∂_y) 
     = β_(η̇) > 0.


In order to generalize this construction we fix the following terminology. 
 
Let γ be a transverse curve. Suppose (η_y)_y∈ I is an isotopy of transverse knots in (^3,ξ_) such that

    S: I ×^1   ⟶ (I ×^3,_) 
    
    (y,θ)       ⟼(y,η_y(θ))

is an embedded transverse surface lying in a standard neighborhood of γ. We say that γ is the core of S and the family (η_y)_y∈ I is its profile.  

As stated before, we think of the profile as a family of planar curves in the front projection (x,z) to which we add an oriented line every point to record the missing coordinate w. The explicit expression we gave in Example <ref> corresponds to the constant profile shown in Figure <ref>.



  §.§.§ Scaling of profiles


In order to build more general surfaces using gluing constructions, we will use diffeomorphisms of the form

    ψ_λ : I ×^3    ⟶ I ×^3 
    
    (y,x,z,w)    ⟼ (y, λ^2(y) x, λ(y) z, λ(y) w)

where λ is a positive function of y.


Let (η_y)_y ∈ I be an isotopy of transverse knots in (^3,ξ_) for a compact 1-manifold I. The embedded surface 

    S: I ×^1   ⟶ (I ×^3,_) 
    
    (y,θ)       ⟼ψ_λ(y,η_y(θ))

is transverse to _ if λ(y)=λ>0 is small enough.


For λ=1 there is no guarantee that S is transverse to _ because of the uncontrolled behavior of η_y as y varies. We use the notation

ψ_λ^*α_  =         dy-λ^3zdx     =:        α_λ

ψ_λ^*β_  =         λ^2(dx-wdz)   =:        β_λ.

Again,  we compute

    α_∧β_(∂ S/∂ y(y,θ),∂ S/∂θ(y,θ) )     = 
    α_λ∧β_λ(∂_y+∂η_y/∂ y(θ), η̇_y(θ)) 
        = (1+O(λ^3))λ^2C + O(λ^5).

Here C>0 is a lower bound for β(η̇_y) and the constants implicit in the Landau symbols depend only on the family (η_y)_y∈ I. Therefore, S is transverse if λ>0 is sufficiently small.


Finally, we show how to isotope a cylinder, relative to the boundary and through transverse surfaces, so that its profile η_y can become arbitrarily small for some values of y if a quantitative condition is satisfied.

Let η be a transverse knot in (^3,ξ_). Then there exists a constant τ_0  depending only on η with the following property.

For every function λ: I  (0,1] satisfying |λ'(y)| < τ_0, the surface

    S: I ×^1    ⟶ (I ×^3, _) 
    
    (y,θ)    ⟼ψ_λ(y,η(θ))

is transverse to .


The proof is almost identical to the one of Lemma  <ref>, and we use notation similar to the one introduced there. The pullbacks of α_ and β_ under ψ_λ are  

ψ_λ^*α_  =        (1-2λ^2λ'xz)dy - λ^3zdx      =:        α_λ

ψ_λ^*β_  =        λ^2(dx-wdz) + λλ'(2x-zw)dy   =:        β_λ.

Again, we seek a condition on λ which ensures that S is a transverse surface: 

    α_∧β_(∂ S/∂ y(y,θ),∂ S/∂θ(y,θ) )     =
    α_λ∧β_λ(∂_y, η̇(θ)) 
         = (1-2λ^2λ'xz)λ^2C - O(λ^4λ').

As above, C is a lower bound for β(η̇) and the constants implicit in the Landau symbols depend only  on η. Hence,  if λ^2λ' is sufficiently small, then S is transverse to _. This is immediate if |λ'|<τ_0 for a sufficiently small constant τ_0. 

Thus, if the core of a cylinder is sufficiently long, it is possible to shrink its profile as much as we want over some subinterval. This will be crucial when we define the Engel overtwisted disc.


 
Consider an isotopy (γ_t)_t∈[0,1] of curves transverse to . Let S_0 be a transverse surface with core γ_0 and profile (η_y)_y ∈ I. Suppose that the profile (and thus S_0) can be shrunk into an arbitrarily small tubular neighborhood of γ_0 using the dilation from Equation (<ref>) (as in Lemma <ref>). Then one obtains a family of surfaces (S_t)_t∈[0,1] transverse to  by shrinking (η_y)_y ∈ I until it fits into a tubular neighborhood of γ_0 that can be identified with the standard neighborhoods of all the γ_t.

Moreover, once the profile is sufficiently thin, one can apply Lemma <ref> to interpolate through transverse surfaces between S_0 and some other S_1, lying in a standard neighborhood of γ_1, which is given in terms of some other profile (η̃_y)_y ∈ I. 





 §.§ Universal twist systems
 

We finish the section describing a particular collection of transverse surfaces. This collection of cylinders will play an important role in the proof of our main theorem, and the properties we require are motivated by that proof.

Let I be the interval [a,b]. Fix coordinates (x,z,w) in ^3 and write ^3_- = {z≤ 0}. Suppose we are given constants λ_0,δ_0,ε_0 > 0 which are potentially very small. We will now construct a t-dependent family (_t)_t ∈ [0,1] of embedded transverse surfaces in (I ×^3, _), depending also smoothly on the parameters λ_0, δ_0, and ε_0. Each _t will be a collection of infinitely many cylinders.

The cylinders comprising _t are all copies of a single cylinder shifted in the x-direction by   consecutive applications of the translation

    T_λ : I ×^3     I ×^3_- 
    
    (y,x,z,w)    ⟼ (y,x+λ^2,z,w).
 
The core of each cylinder will be a curve parallel to the y-axis. The construction is done in several steps and it is only at the very end that we will achieve transversality with _. We abbreviate T := T_1.



  §.§.§ A nice transverse knot


Let η_1⊂ (^3_-,ξ_=(β_)) be a transverse unknot with self-linking number -3 satisfying the following properties:


  * η_1 is disjoint from {z=0} in the complement of η_1^+ = η_1 ∩{z,w=0}.

  * ∪_n∈ T^n(η_1^+) is a covering of the x-axis. 

  * η_1 is disjoint from T^n(η_1) for n ≠± 1.

  * η_1 and T(η_1) intersect in a non-degenerate interval P which is contained in the x-axis, and they are otherwise disjoint. P and T(P) are disjoint.

  *  There is an isotopy (η_t)_t∈[1/2,1] of η_1, supported in a neighborhood of P, such that η_t is disjoint from T(η_t) for t ∈ [1/2,1). This isotopy is purely vertical: only the w-coordinate varies. The following condition holds for t<1:


  (B) Given any two points (x,z,w)∈η_t and (x,z,w')∈ T(η_t), it holds that w<w'.   
In particular, the link ∪_n∈ T^n(η_t) is an unlink. 


  *  There is an isotopy (η_t)_t∈[0,1/2] of η_1/2 such that η_0 is contained in {x_0-1/2<x<x_0+1/2} for some x_0. This isotopy induces a T-equivariant isotopy of unlinks (∪_n∈ T^n(η_t))_t∈[0,1/2] such that all the components of ∪_n∈ T^n(η_0) have disjoint front projection. We further assume that the isotopy is constant for t ∈ [0,1/4].

The isotopy (η_t)_t∈[0,1] can be assumed to be smooth. A knot η_1 with the desired properties exists: its front projection is shown in Figure <ref>, along with the front projection of the translation T(η_1). The isotopy (η_t)_t∈[0,1/2] separating the front projections simply makes the upper half of the knot progressively smaller.






  §.§.§ An infinite family of cylinders
 

Write ^3_-(y_0) for the half-hyperplane {y=y_0}⊂ I ×^3_-. Recall the parameters λ_0,δ_0,ε_0 > 0 introduced at the beginning of the subsection. We now fix a smooth family of bump functions (χ_t)_t ∈ [0,1]: [a,b]  [0,1] with the following properties.


  * χ_t(y) ≡ t if y ∈ [a+δ_0,b-δ_0].

  * χ_t(y) ≡ 0 if y ∈ [a,a+2δ_0/3] ∪ [b-2δ_0/3,b].

  * χ_t is non-decreasing if y∈[a,a+δ_0].

  * χ_t is non-increasing if y∈[b-δ_0,b].

We may construct another smooth family of functions (ρ_t)_t ∈ [0,1]: [a,b]  [λ_0,1] satisfying:


  * ρ_t(y) ≡ 1 if y ∈ [a+2δ_0/3,b-2δ_0/3] and t ∈ [1/4,1].

  * ρ_t(y) ≡λ_0 if y ∈ [a,a+δ_0/3] ∪ [b-δ_0/3,b] or t ∈(0).

  * ρ_t is non-decreasing if y∈[a,a+2δ_0/3].

  * ρ_t is non-increasing if y∈[b-2δ_0/3,b].

  * t →ρ_t(y) is non-decreasing.

 
The functions ρ_t and χ_t depend on the parameters δ_0 and λ_0. We may assume that this dependence is smooth. Further, we assume that the derivative ρ_t' remains uniformly bounded as λ_0 goes to zero. 


These two families of functions allow us to define a family of (not yet transverse) cylinders (S_t)_t ∈ [0,1] by imposing 

    S_t ∩^3_-(y) = ψ_ρ_t(y)(η_χ_t(y))

where ψ_λ is the scaling function defined in Equation <ref>. What we achieve is the following: For t ∈ [1/4,1] and y ∈ [a+δ_0,b-δ_0], the profile of S_t is precisely η_t. As y approaches 0 or 1, the profile becomes η_0 and then it decreases in size; the precise scaling is given by the constant λ_0. Similarly, the surfaces (S_t)_t ∈ [0,1/4] have a constant profile which is a rescaling of η_0 which becomes smaller as t goes to 0.




  §.§.§ Scaling


Consider the collections of infinite cylinders (⋃_n∈T^n(S_t))_t ∈ [0,1]. The proof of Lemma <ref> can be applied parametrically in t. Using this fact we claim that there is a constant λ >0, depending smoothly on λ_0, δ_0, and ε_0, such that the collection of surfaces

    _t = ψ_λ(⋃_n∈T^n(S_t)) =  ⋃_n∈T^n_λ(ψ_λ(S_t))

satisfies:


  * _t is a collection of transverse cylinders,

  * if t≠ 1, the cylinders in _t are disjoint.

  * _t ⊂{|z|,|w| < ε_0}.

The first fact follows from Lemma <ref>,  the second is immediate from the construction. Finally, the third holds for any λ sufficiently small (depending on ε_0).








 
The collection of transverse cylinders (_t)_t∈[0,1] is a universal twist system. 


 
The family (_t)_t∈[0,1] depends on the three parameters λ_0 (an upper bound for the size of the profiles at y=0,1 and t=0), δ_0 (the size of the interval in the y-direction in which the profiles perform the unlinking and the shrinking), and ε_0 (an upper bound for the size of the cylinders in the (x,z,w)-coordinates). Let us clarify how they depend on each other.

We fix δ_0 and ε_0 first. The constant λ that determines the final scaling of the profiles depends on both; as they converge to zero so does λ. However, Lemma <ref> and Remark <ref> imply that λ does not depend on λ_0. We will fix λ_0 last in our constructions.






§ THE ENGEL-LUTZ TWIST
 

Having constructed transverse tori, we immediately obtain many examples of transverse hypersurfaces in (M,): Consider the boundary of a suitable tubular neighborhood of a given transverse torus. If the torus is contained in the vicinity of its core γ, so is the 3-manifold we obtain.

In this section we describe how to add Engel torsion along a transverse 3-manifold; this is analogous to adding Giroux torsion along a transverse torus in 3-dimensional Contact Topology. The special case when this operation is performed along a transverse 3-torus obtained from a transverse loop γ is what we call an Engel-Lutz twist with core γ. These procedures are well-defined up to homotopy through Engel structures.

Additionally, we show that if the core is contractible, then the Engel-Lutz twist preserves the homotopy type of the Engel framing. Similarly, the formal type of the Engel structure is unchanged when one introduces Engel torsion along the same transverse 3-manifold twice. Again, all of this is very reminiscent of the Lutz twist.




 §.§ Hypersurfaces transverse to the Engel structure
 


Let (M,) be an Engel manifold. An immersed 3-dimensional submanifold N⊂ M is a transverse hypersurface if its tangent space at each point is transverse to . 

Given a closed, orientable, embedded, transverse 3-manifold N ⊂ (M, ), we want to describe the germ |_N. First, we note that  imprints the following data on N:


  * a line field _N = TN ∩, and 

  * a 2-plane field ξ_N = TN ∩ which contains _N.

Of course, if N is not only transverse to  but in addition transverse to , then ξ_N is a contact structure and _N is a Legendrian line field. 

Once an orientation of N is fixed, we can define subsets 

    N^+     = {x∈ N | ξ_N  is a positive contact structure close to x}
    
    N^-     = {x∈ N | ξ_N  is a negative contact structure close to x}
    
    N^0     = N∖(N^+∪ N^-).

Equivalently, N^0 is the set where  is tangent to N. By definition, the regions N^± are open while N^0 is closed. If N is chosen C^∞-generically, N^+ and N^- are open 3-dimensional manifolds separated by the possibly disconnected surface N^0.

Let T be a vector field transverse to N and tangent to ; denote its flow by ϕ_t. Using ϕ_t for small times t ∈ [-ε,ε], we obtain a tubular neighborhood (N) ≅ N×(-ε,ε) of N ≅ N×{0}. We use coordinates (p,t) ∈ N×(-ε,ε) in the model, i.e. ∂_t=T. Choosing T suitably we may assume that the following conditions hold: 


  * T is tangent to  in the complement of an arbitrarily small neighborhood of N^0 × (-ε,ε),

  * If T(p,0) is tangent to  for p∈ N, then the T-orbit ∪_t∈(-ε,ε)ϕ_t(p) is part of the leaf of  through p. 

  * Conversely, if T(p,0) ∉ for p∈ N, then the segment ∪_t∈(-ε,ε)ϕ_t(p) is never tangent to .  


Write N_t=N×{t}. Fix a vector field H on N×(-ε,ε) such that ∩ TN_t is spanned by the restriction of H to N_t and let 

    H'     = [∂_t,H]    H”    = [∂_t,H'].

We can associate a sign to each connected component of N^0 indicating whether the framing ⟨ H,H',H”⟩ of TN is positive or negative. We write N^0,+ (respectively N^0,-) for the union of those connected components of N^0 in which it is positive (respectively negative).

 
Let (M,) be an Engel manifold and N ⊂ M a transverse hypersurface with framing {X ∈∩ TN,Y ∈∩ TN,Z}. Then, the Engel structure in (N) can be  written as 

    (p,t) = ⟨∂_t, H = X+tY+g(p,t)Z ⟩,

where g:(N) ≅ N × (-ε,ε) is a function with the following properties:


  * g(p,t) = 0 if ∂_t ∈(p,0),

  * g(p,t) is convex if p∈ N^0,+, 

  * g(p,t) is concave if p∈ N^0,-.



Consider the framing {T,X,Y,Z} of TM along N. Using the flow of T we can extend this framing to a translation invariant framing on  (N). The Engel structure  can now be described by a smooth family of curves (H_p: (-ε,ε) ^2)_p∈ N such that (p,t) is spanned by  

    T=∂_t    and    H(p,t) = (H_p(t))_1· X + (H_p(t))_2 · Y +(H_p(t))_3· Z.

Since  is an Engel structure, the curves H_p satisfy at least one of the conditions in Proposition <ref>. The conditions on T stated above ensure that each curve H_p is an immersion whose image is either contained in a great circle or is convex/concave everywhere. In either case, H_p(t) is graphical over the equator 

    t ⟼ (H_p(t))_1· X + (H_p(t))_2· Y.
 
Shrinking ε and scaling T appropriately one obtains the claim using the implicit function theorem.


The following elementary constructions provide many examples of transverse 3-mani­folds.
 
Let (M,) be an Engel manifold and let S be a transverse surface. A trivialized tubular neighborhood U≃ S×^2 of S is thin if the 3-manifolds S×∂^2_ε are transverse to  for all 0<ε≤ 1. 


 
Any transverse surface in an Engel manifold has a thin tubular neighborhood.


The transverse submanifolds obtained in this way are confined to small neighborhoods of S, but once a transverse hypersurface is available it can be isotoped through transverse hypersurfaces.
 
Let N be a hypersurface transverse to . Let X be a vector field tangent to  and transverse to N with flow φ_t. Then φ_t(N) is a transverse hypersurface for all t.





 §.§ Adding Engel torsion along a transverse hypersurface
 

Let us continue using the notation introduced in the previous section. We want to modify  on a neighborhood (N) of N such the resulting Engel structure () satisfies:


  * () coincides with  outside of (N),  

  * () is tangent to T, and

  * in the region where T(p,0) is tangent to (p,0), the even-contact structures associated to () and  are the same, but () performs one additional turn along the flow lines of T.

This is achieved by replacing the family of curves (H_p)_p∈ N by a family (λ_p)_p∈ N. First, we construct a C^2-family of C^2-curves (η_p)_p∈ N that has all the desired properties except that the curves are only piecewise smooth. In a second step we smooth (η_p)_p∈ N (both individually and as a family) to obtain (λ_p)_p∈ N. During the argument we do not keep track of the paremetrisation of the curves since any regular paremetrisation works. 

We now describe (η_p: (-ε,ε) ^2)_p∈ N. Each curve η_p consists of three pieces. The first and third pieces are, respectively, H_p((-ε,0]) and H_p([0,ε)). The middle piece consists of the (unique, possibly non-maximal) circle in ^2 which


  * passes through (1,0,0), 

  * is tangent to the maximal circle spanned by (1,0,0) and (0,1,0), and

  * has the same geodesic curvature as H_p at t=0.

These three pieces depend smoothly on p, each one of them is smooth and, at the gluing points, the assumption on the curvature guarantees C^2-regularity.

The smoothing step from η_p to λ_p is clear. Indeed, since the relevant properties of the curves depend only on their 2-jet and the segments η_p are already C^2-smooth, any C^2-small smoothing yields a smooth family of curves with the same convexity properties as the family (η_p)_p∈ N.


The plane field () is an Engel structure.


Consider the curves (λ_p)_p∈ N from the construction. Whenever H_p is convex (resp. concave), so is λ_p. Therefore, condition (i) from Proposition <ref> implies that () is Engel whenever this is the case. Let U be the complement, i.e. the set of those p such that H_p is everywhere tangent to the maximal circle C given by (1,0,0) and (0,1,0). The curves λ_p are tangent to C on U. Since U is closed, we cannot invoke condition (ii) of Proposition <ref>. However, one notes that the maximal circles tangent to the curves λ_p C^∞-converge to C as (p,t) approaches U. This implies that condition (ii) holds in small neighborhood of U. This concludes the proof.



We say that the plane field (), obtained from  and N by the procedure we just described, is the result of introducing Engel torsion along N.

If N is a 3-torus arising from a 2-torus S, as in Definition <ref>, we say that () is obtained from  by introducing an Engel-Lutz twist along S. The tubular neighborhood in which the Engel-Lutz twist is performed is an Engel-Lutz tube carried by S.

The construction is shown pictorially in Figure <ref>. In this figure, the kernel foliation is vertical, and each segment transverse to N represents a fibre of the tubular neighborhood (N). The horizontal segment on the right goes through the region N^0,+. The vertical ones on the left are away from N^0 and are therefore tangent to the kernel. For each segment we draw two spheres showing the corresponding curves H_p (left) and λ_p (right). For the horizontal one, H_p is a short convex curve and λ_p is a convex loop with the same endpoints. For the vertical ones, H_p is a short piece of equator and λ_p is a curve running around that same equator with the same endpoints but describing one more turn; we draw λ_p not overlapping with H_p to show  the additional turn and the boundary condition more clearly.


Since N is still transverse to (), Engel torsion can be introduced several times along the same hypersurface N.




 
The construction described above is well-defined up to deformations through Engel structures. To see this, it suffices to note that the vector fields T satisfying the required properties in a neighborhood of N form a contractible space. More generally, note that the construction can also be carried out parametrically when (N_k)_k ∈ K is a family of closed manifolds transverse to a family of Engel structures (_k)_k ∈ K. In the proof of the h-principle for overtwisted Engel structures, it will be essential to introduce Engel-Lutz twists parametrically along a family of transverse surfaces. 


The addition of Engel torsion along a hypersurface has already appeared in the literature in less generality, which we now recall. The first example deals with Engel structures obtained from contact structures which are trivial as bundles.
 
Let (N,ξ) be a contact 3-manifold admitting a global framing C_1,C_2. Let t be the coordinate ^1=/2π. For each positive integer k, we define an Engel manifold 

    (N×^1, _k = ⟨∂_t, X_k=cos(kt)C_1+sin(kt)C_2 ⟩).

The even-contact structure of _k is the preimage of ξ under the bundle projection and the characteristic foliation  is spanned by ∂_t. The hypersurfaces N×{t} are clearly transverse to _k (indeed, they are transverse to their kernel ). Introducing Engel torsion to _k along N×{t} produces _k+1. It is easy to see that the homotopy type of the Engel framing changes when one passes from _k to _k+1, but it only depends on the parity of k (see <cit.>).


The second example deals with the Engel structures on mapping tori. It is very similar, but not equivalent, to the previous example.

In <cit.> H. Geiges constructs an Engel structure _k on any mapping torus with trivial tangent bundle.  The Engel structures _k produced in this way are tangent to the suspension vector field. Trivializing the suspension vector field allows us to express _k as a 3-dimensional family of curves, as in Proposition <ref>. The key property of his construction is that these curves become convex if k is large enough. The fibers are 3-manifolds transverse to _k and passing from _k to _k+1 amounts to introducing Engel torsion along one of them.






 §.§ Modification of the formal data when adding Engel torsion
 

Example <ref> shows that an Engel-Lutz twist sometimes does change the homotopy type of the Engel framing.  We first show that the Engel-Lutz twist applied twice along the same hypersurface results in an Engel structure with an Engel framing that is homotopic to the original one. This is analogous to the case of contact structures. 

 
Let  be an Engel structure on M and let N be a transverse hypersurface. The Engel framing of  is homotopic to the Engel framing of .  


We will make use of the notation introduced in Subsections <ref> and <ref>. In the first part of this proof we compare the pairs (,) and (,[,]). The need for the double Lutz twist will come into play only in the second part. 

The even-contact structure =[,] has basis {T,H_p(t),Ḣ_p(t)} at (p,t)∈ N(U). We claim that the even-contact structure associated to  is homotopic to . Recall the curves (η_p)_p ∈ N, those appearing in the intermediate step of the construction of an Engel-Lutz twist, before smoothing. It suffices to see that one can homotope the middle piece of η_p, through non-maximal circles, to the great circle spanned by H_p(0) and Ḣ_p(0). The plane field _ associated to the homotoped curves is not integrable and the hyperplane field _ = [_,_] is homotopic to the even contact structure of  by construction. It is clear that _ is in turn homotopic to . Iterating this argument, we see that the even-contact structure associated to  is homotopic to  as well.

Notice that T is tangent to all the distributions throughout all homotopies that have been discussed so far. In the following, we may therefore pretend that the hyperplane fields underlying the even-contact structures associated to , , and  are the same (as long as we do not use assumptions on their characteristic foliation). 

In the model, the plane field  is coorientable within the oriented hyperplane field , so we may fix a coorientation. Then, one can homotope  inside of  to a plane field which coincides with  in a neighborhood of ∂(N) and which is transverse to T in the interior of the model. Choose the neighborhood of ∂(N) so small that , , and  coincide there. Then, we tilt  away from T (staying within ) in the direction prescribed by its coorientation until it coincides with ∩(N×{t}). The same can be done, exactly by the same argument, with  and . Therefore, the pairs 

    (,), (,[,]), and (,[,])
 
are homotopic as oriented pairs of hyperplane fields containing a plane field.

Now we pass to the second part of the proof, which focuses on . Let us abbreviate [,]= and write  for its kernel. It is difficult to compare the characteristic foliation of  with the characteristic foliation of , but we can homotope the pair (,) to (,) and carry the characteristic foliation of  along (as a family of oriented vector fields contained in the plane fields). 

By construction, we can view  as obtained from  by inserting two copies of the same layer N×(-δ,δ) carrying the same Engel structure (that is, two copies of the Engel torsion model). We can then choose the homotopy connecting (,) and (,) in such a way that the two layers always coincide at every step, including the homotopy of . Denote by _1 ⊂ the line field at the end of the homotopy.

Then, _1 is a line field in  which coincides with  near ∂(N) and, in the interior, describes twice the same loop within  as one moves along the fibers of (N) = N×(-ε,ε) N. We have shown that the Engel framing of  is homotopic to a framing whose last two components (i.e the components complementary to ) coincide with those of the Engel framing of , while the first two describe a certain loop twice. Since π_1((4)) = _2,  the Engel framings of  and  are homotopic.


Without topological assumptions on N it is not true, in general, that the Engel framings of  and  are homotopic. However, the following statements are corollaries of the first part of the proof of the proposition:
 
Assume that N is contained in a ball. Then the Engel framing of  is homotopic to the Engel framing of  relative to the boundary of the ball.


In view of the first half of the proof of Proposition <ref> we may assume that, after a homotopy, the last two components of the Engel framings of  and of  agree, and thus, the first two components span the same plane field. Since the ball is simply connected, the framings are homotopic. 


 
Assume that N is the boundary ∂(S ×^2) of a thin neighborhood of a transverse torus S. Then the Engel framing of  is homotopic, relative to the boundary of a slightly bigger neighborhood, to the Engel framing of .

In particular, introducing an Engel-Lutz twist does not change the formal type of the Engel structure.


We argue as in the previous corollary, noting this time that  π_1(∂(S ×^2)) π_1(S ×^2) is surjective. This implies that any two framings differing only in the first two components and agreeing along the boundary must be homotopic relative to the boundary.






§ OVERTWISTED ENGEL STRUCTURES
 

In 3-dimensional Contact Topology, a Lutz twist along a knot transverse to the contact structure gives rise to a ^1-family of overtwisted discs.  Motivated by this, we will define an Engel overtwisted disc to be a portion of an Engel-Lutz tube that satisfies an additional quantitative property.



 §.§ The overtwisted disc
 

Let I=[0,L] and fix coordinates (y,x,z,w) on I ×^3. On I×^3 we consider the Engel structure

    _ = (α = dy-zdx)∩(β = dx-wdz) = ⟨∂_w, ∂_z + w(∂_x+z∂_y)⟩.

Of course, this is the standard form of an Engel structure in the vicinity of a transverse curve γ≅ I×{0} obtained in Proposition <ref>.

Let η⊂ (^3,(β)) be a transverse unknot with self-linking number -3, as shown in Figure <ref>. According to Lemma <ref>, the cylinder Σ = I ×η with profile η is transverse to _. Let τ_0 be the scaling constant obtained from η by applying Lemma <ref>.

 
Consider the Engel structure (_) obtained from _ by an Engel-Lutz twist along Σ = I ×η. If the length of I satisfies L > 2/τ_0, then

    Δ_ = (I ×^3, _ = (_))

is an overtwisted disc. The curve I ×{0}⊂ I ×^3 is said to be its core.

An Engel structure is overtwisted if it admits an Engel embedding of an overtwisted disc.




 §.§ Self-replication of overtwisted discs


As we showed in Lemma <ref>, the requirement on L ensures that we can shrink the transverse cylinder Σ 


  * through transverse embedded surfaces and 

  * relative to the boundary
 
so that a piece of the resulting surface is an arbitrarily thin cylinder of small (but fixed) length. The existence of this homotopy guarantees the self-replicating property of the overtwisted disc, i.e. in the presence of an overtwisted disc we can produce new ones by an Engel homotopy. This is a property that appeared in a similar form in E. Murphy's definition of a loose chart for higher dimensional Legendrians <cit.>.
 
Let Δ_=([0,L]×^3,_) be an overtwisted disc. Then, there are:


  * a path of Engel structures ([0,L] ×^3,_r)_r∈ [0,r_0],

  * a path of Engel embeddings F_r: ([0,L] ×^3,_) ⟶ ([0,L] ×^3,_r), and

  * an Engel embedding G: ([0,L] ×^3,_) ⟶ ([0,L] ×^3,_r_0)

such that _r=_ on a neighborhood of the boundary, _0=_, F_0 is the identity, and the images of G and F_r_0 are disjoint.


The Engel structure _ is obtained from the standard Engel structure ([0,L]×^3,_ = (α=dy-zdx)∩(β=dx-wdz)) by applying an Engel-Lutz twist along the surface Σ = I ×η. By assumption, L and the constant τ_0 from Lemma <ref> satisfy 2<Lτ_0. 

Recall the y-dependent rescaling of the ^3-factor

    ψ_λ : [0,L] ×^3    ⟶ [0,L] ×^3 
    
    (y,x,z,w)    ⟼ (y, λ(y)^2x, λ(y)z, λ(y)w),

from Equation <ref>. Since L > 2/τ_0, we can construct a path of functions λ_r: [0,L]  (0,1], r ∈ [0,1), such that


  * λ_0(t) = 1 for all t,

  * λ_r(t) = 1 for all t in a neighborhood of ∂[0,L],

  * λ_r(t) = 1-r if t ∈ [Lr/2,L(1-r/2)], and

  * |λ_r'(t)| < τ_0.

By Lemma <ref>, the cylinders Σ_r = ψ_λ_r([0,L] ×η) are embedded and transverse. Adding an Engel-Lutz twist to _ along Σ_r yields a path of Engel structures _r in [0,L] ×^3 with _0 = _. All of them agree near the boundary of the model.

The embeddings F_r are 

    F_r : ([0,L] ×^3,_)    ⟶ ([Lr/2,Lr/2+L(1-r)^3] ×^3,_) 
    
    (y,x,z,w)    ⟼(y(1-r)^3 + Lr/2, x(1-r)^2, z(1-r), w(1-r)).

A simple computation shows that F_r is a well-defined embedding, since Lr/2+L(1-r)^3 ≤ L(1-r/2) ≤ 1 for all r. Additionally, the conformal nature of the Engel structure implies that F_r is an Engel embedding. Since F_r^-1(Σ_r)=Σ, we can invoke Remark <ref> and regard F_r as an Engel embedding of ([0,L] ×^3,_) into ([0,L] ×^3,_r).

When r is sufficiently close to 1, the map 

    G_r: ([0,L] ×^3,_)    ⟶ ([L/2,Lr/2+L(1-r)^3] ×^3,_) 
    
    (y,x,z,w)    ⟼(y(1-r)^3 + L/2, x(1-r)^2, z(1-r), w(1-r)).

is well-defined and its image is disjoint from F_r(I×^3). We fix such a value r_0 for r and set G = G_r_0.

We do not know whether this result still holds when the condition on L is dropped.


Mimicking the proof of the lemma one can show that the choice of constant L in the definition of overtwisted disc is not important as long as Lτ_0 > 2.





 §.§ Overtwisted Engel structures


Finally, we define what an overtwisted family of Engel structures is.
 
Let K be a compact manifold. A family of Engel structures : K (M) is overtwisted if there is a submanifold Δ⊂ M × K satisfying:


  * (Δ,)  K is a locally trivial fibration of Engel manifolds

  * whose fiber is Engel diffeomorphic to Δ_.

The manifold Δ is said to be the certificate of overtwistedness of .

That is, we require that there exists a family of overtwisted discs compatible with the family of Engel structures. We will often write Δ_k for an explicit paremetrisation of the overtwisted disc of (k). If Δ is not a globally trivial fibration, we cannot choose Δ_k parametrically in k globally.

Examples of overtwisted Engel structures arise from Engel-Lutz twists.
 
Let (M,) be an Engel manifold. Let Σ be a transverse 2-torus with core transverse to [,] and profile as depicted in Figure <ref>. Then, the Engel manifold (M,()) obtained by Engel-Lutz twisting along Σ is overtwisted up to Engel homotopy. 


We homotope Σ to make its profile arbitrarily small using Lemma <ref>. This homotopy of transverse surfaces provides, by parametric Engel-Lutz twisting, a homotopy of Engel structures (_s)_s ∈ [0,1] which starts on _0 = () and finishes on _1 overtwisted. This follows from the fact that the shrinking allows us to ensure that the length constant L is large enough.

It is unclear to the authors whether this result still holds when Σ is not obtained from a transverse knot.

An immediate consequence of the lemma is as follows:
 
Fix a smooth embedding Δ: [0,L] ×^3  M. The following inclusion induces a surjective map

    _(M,Δ) _(M,Δ).




Given any ^m-family _0 of formal Engel structures in _(M,Δ), we apply the existence h-principle from <cit.> to produce a family of Engel structures _1/3 which is formally homotopic to _0. Doing so destroys the overtwisted disc of _0 because <cit.> is not an h-principle relative in the domain (as pointed out in the introduction). Still, after a homotopy, we may assume that ⊷(Δ) is a Darboux ball for each _1/3(k), k ∈^m.

Choose a family of knots (γ_k ⊂⊷(Δ) ⊂ M)_k ∈^m with γ_k transverse to [_1/3(k),_1/3(k)]. From this we obtain a ^m-family of 2-tori. Parametrically introducing an Engel-Lutz twist yields an overtwisted family _2/3 which is formally homotopic to _1/3 by Corollary <ref>. We isotope the structures _2/3 to yield a family _1 such that Δ is an Engel embedding of the overtwisted disc.





 §.§ Replication of the certificate


The following is a corollary of Lemma <ref>:
 
Let K be a compact manifold and K' ⊂ K a smooth ball. Let _0: K (M) be a family of Engel structures with certificate Δ^0 ⊂ M × K. Then, there is a homotopy of Engel structures (_s)_s ∈ [0,1]: K (M) supported in a neighborhood of Δ^0 such that


  * _s is overtwisted with certificate Δ^0,

  * there is a K'-family of Engel embeddings of the overtwisted disc 

    Δ_k^1: Δ_(M,_1(k)),      k ∈ K'
 
such that Δ_k^1(Δ_) is contained in (Δ^0) ∖Δ^0.



The desired Engel homotopy will be constant on the complement of (K') in the parameter. Let us assume that K = (K') is a smooth ball. Then Δ^0 is a globally trivial fibration by overtwisted discs Δ_k^0: Δ_ (M,_0(k)), k ∈ K.

We can apply Lemma <ref> to _0 to obtain a homotopy of Engel families (_s)_s ∈ [0,1] supported near Δ^0. It yields

_s:      K             (M),                     _0    = _0, 

F_k,s:                Δ_   (M,_s),         F_k,0              = Δ_k^0,

and a certificate G_k: Δ_ (M,_1) which is disjoint from F_k,1.

By the isotopy extension theorem applied to F_k,s there are isotopies ψ_k,s of M such that  F_k,s= ψ_k,s∘Δ_k^0. We set _s(k) = ψ_k,s^*_s(k). The certificate Δ_k^1 is  ψ_k,1^-1∘ G_k. We can cut-off the replication as k goes from ∂ K' to ∂(K') to ensure that the homotopy is relative to the complement of (K'). 

Lemma <ref> yields copies of the certificate in the vicinity of Δ^0. We will use such copies whenever a certain Engel homotopy requires a certificate of overtwistedness. This way, Engel homotopies can be performed relative to the original certificate Δ^0.




 §.§ Homotopies of overtwisted discs
 

We conclude this section showing how homotopies of the core can be used to construct homotopies of the overtwisted disc. A key observation was explained in Remark <ref>: adding Engel torsion is parametric as the Engel structure and the transverse 3-manifold vary in families. 

Let (M,) be an overtwisted Engel manifold with Δ: Δ_ (M,) an Engel embedding of the overtwisted disc. We denote its core by γ: [0,L]  M, the corresponding transverse cylinder by Σ, and its (constant) profile by η. Apart from L (the length of Δ_ in the y-coordinate) we fix a constant l satisfying L/2 > l >1/τ_0. Here τ_0 is the constant from Lemma <ref> measuring the allowed speed of scaling for the profile η (which already appeared in the definition of the overtwisted disc). We use the terminology

    A_0 = Δ({y=0})     is the lower boundary, 
    
    A_L = Δ({y=L})     is the upper boundary.

The endpoint γ(0) (respectively γ(L)) of the core of the overtwisted disc lies in A_0 (respectively A_L).

We cut M along A_0 and A_L to produce a manifold with boundary and corners M^Δ. The boundary ∂ M^Δ has two connected components, each of which consists of two copies of A_0 (respectively, A_L) glued to one another along their common boundary. M^Δ inherits an Engel structure, which we still denote by . Similarly, we write γ and Δ for their lift to M^Δ. Observe that (M^Δ, ) is, by definition, obtained from some other Engel structure (M^Δ, ^-1()) by Engel-Lutz twisting along the cylinder Σ with core γ and profile η. As such, the structures  and ^-1() differ from one another only on  ⊷(Δ). ^-1() is given instead by an Engel embedding of the standard model around γ:

    φ: ([0,L] ×^3,_)  (M^Δ,^-1()).

We will refer to the region φ({y ∈ [0,l] ∪ [L-l,L]} as the scaling region.

Let (γ_t)_t ∈ [0,1] be a homotopy of embedded transverse arcs in (M^Δ, ^-1()) with γ_0 being the core γ. By Proposition <ref> this homotopy extends to a homotopy of standard neighborhoods

    φ_t: (U_t,_)  (M^Δ,^-1())  with φ_0 = φ.

Here U_t is a neighborhood of [0,L_t]×{0} in [0,L_t]×^3, L_t is a constant that depends smoothly on t, and L_0 = L. We will henceforth assume that L_t>2l, and we require that the curves γ_t agree with γ on the scaling region [0,l]∪[L_t-l,L_t], that is:


  * γ_t(y) = γ(y) and φ_t(y,·) = φ(y,·)  for all y ∈ [0,l], and

  * γ_t(L_t-y) = γ(L-y) and φ_t(L_t-y,·) = φ(L-y,·) for all y ∈ [0,l].


We consider a homotopy of profiles η_t(y) with t ∈ [0,1] and y∈ [0,L_t] such that


  * η_0(y) = η,

  * η_t(y) = η and η_t(L_t-y) = η for all y ∈ [0,l].

Recall the y-dependent rescaling from Equation <ref>

    ψ_λ : [0,L_t] ×^3    ⟶ [0,L_t] ×^3 
    
    (y,x,z,w)    ⟼ (y, λ(y)^2x, λ(y)z, λ(y)w).


The following result is an immediate consequence of Lemma <ref>.
 
In the situation described above, there is a homotopy of overtwisted Engel structures (_t)_t ∈ [0,1] satisfying:


  * _0 =,

  * the overtwisted disc Δ_t of _t has γ_t as core and a rescaling of (η_t(y))_y ∈ [0,L_t] as profile.



As shown in Lemma <ref>, there is a number λ_0∈(0,1] such that all the surfaces 

    ⋃_y∈[0,L_t]{y}×ψ_λ_0(η_t(y)) ⊂ [0,L_t] ×^3

are transverse to _ and contained in U_t for all t∈[0,1]. Our choice of l readily implies that there exists a smooth family of functions ρ_t : [0,L_t] ⟶(0,1] satisfying


  * ρ_0 ≡ 1,

  * |ρ_t'| < τ_0, 

  * ρ_t ≡ 1 in  ({0,L_t}), and 

  * ρ_t ≡λ_0 on [l, L_t-l] for all t in the complement of ({0}).

It follows from the properties of ρ_t that all the surfaces in the homotopy

    Σ_t = φ_t(⋃_y∈ [0,L_t]{y}×ψ_ρ_t(y)(η_t(y))),      t ∈ [0,1],

are transverse and, by construction, Σ_0 = Σ. One of these surfaces is shown in Figure <ref>.



We then add an Engel-Lutz twist to (M^Δ,^-1()) along the surface Σ_t. Doing this parametrically in t yields a homotopy of overtwisted Engel structures (_t)_t∈[0,1] starting at . Since this is a homotopy relative to the lower and upper boundaries of the overtwisted disc, it can be regarded as a homotopy in M instead of M^Δ.







§ H-PRINCIPLE FOR OVERTWISTED ENGEL STRUCTURES
 

The rest of the article is concerned with the proof of Theorem <ref> and its corollaries. First, we go over the general setup, applying standard methods to reduce the proof to a simplified problem which will be treated in Subsections <ref> and <ref>.



 §.§ Setup
 

We fix a 4-manifold M, a smooth compact manifold K of arbitrary dimension, and a K-parametric family of formal Engel structures

    _k⊂_k⊂_k⊂ TM,      k ∈ K.

When we say that this triple is Engel we mean that it is a formal Engel structure arising from a genuine Engel structure. Often we will just write _k for the formal Engel structure, even if it is not genuine, to keep the notation less cluttered. 

The h-principle we want to show will be valid for overtwisted Engel structures. As such, we require that:


  * the family (_k)_k ∈ K has a certificate of overtwistedness Δ⊂ M × K.

Since the h-principle we want to prove is parametric and relative in the parameter and the domain, we also assume that:


  * there is a CW-complex K' ⊂ K such that _k is Engel for all k ∈ K',

  * there is a submanifold U ⊂ M such that (_k)|_U is Engel for all k,

  * the manifold M ∖ U is connected,

  * Δ is contained in (M ∖ U) × K.

We denote M_k = M×{k}. For ease of notation, we will write V = (M × K') ∪ (U × K) ⊂ M × K.

To deal with the parametric nature of the statement it is sometimes more convenient to view (_k)_k∈ K as a formal Engel structure on the foliation _M × K by fibers of M × K  K. We use the notation _M× K for the plane field on M× K which coincides with _k on each fibre M_k. The distributions _M× K and _M× K are defined analogously. We say that _M× K is an Engel structure on an open subset A of M× K if, for each k, _k is an Engel structure on M_k∩ A.

In this setting, the h-principle for overtwisted Engel structures is
 
_M× K is homotopic to a fiberwise Engel structure _M× K through a family of fibrewise formal Engel structures, relative to V and Δ.

Note that the parametric and relative nature of the statement yields the analogous result when K is a CW-complex (as stated in Theorem <ref>).

In order to make quantitative statements on angles and distances, we fix Riemannian metrics on M and K. 




 §.§ Reductions using standard h-principle methods
  

In this section we reduce the proof of Theorem <ref> to an extension problem given in a standardized form. To describe this extension problem, we use the language of model structures (Subsection <ref>), which we recall briefly.

A model structure M(I,J,f_-,f_+,c) is a formal Engel structure with domain 

    {y∈ I, x∈ [0,1], f_-≤ z≤ f_+, w ∈ J}
 
where I is a 1-manifold and J is an interval. The functions f_-, f_+, and c define a formal Engel structure with flag

    =⟨∂_w⟩
    _c     = ⊕⟨cos(c)∂_z+sin(c)(cos(z)∂_x+sin(z)∂_y) ⟩
        = ⊕ker(cos(z)dy-sin(z)dx).
 
A simple computation shows that _c is an Engel structure with the correct formal class if and only if ∂_w c>0 everywhere.
 
Let ε < 0 be a small constant and K_0 a smooth manifold, possibly with boundary. A family of model structures 

    B_k = M(I,[-ε,1],-ε,f_k,+,c_k)

with k∈ K_0 satisfying


  * ∂_w c_k > 0 if k ∈∂ K_0,

  * ∂_w c_k > 0 on a 2ε-neighborhood of the boundary of the model,

  * c_k(y,x,z,w) ≡arcsin(w) if w ∈ [-ε,ε], and

  * c_k(y,x,z,w) > -π

is a shell if I is a closed interval, and a circular shell if I = ^1. 


The following proposition is the main result of this subsection.
 
There is a K-family of formal Engel structures

    _M× K' ⊂_M× K' ⊂_M× K'

satisfying the following conditions:
 

  (i) The family is homotopic to the original formal data _M× K⊂_M× K⊂_M× K, relative to Δ and V.

  (ii) _M× K' is a K-family of even-contact structures whose family of characteristic line fields is _M× K'.

  (iii) _M× K' is an Engel structure (with associated even-contact structure _M× K') in the complement of a finite collection of shells {B_i ⊂ M × K ∖ (V ∪Δ)}_i.

  (iv) For every subset of indices {i_j}_j, the intersection 

    ⋂_j B_i_j/_M × K
 
is either contractible or empty. 


Condition (iv)  will be used in Subsection <ref>, Proposition <ref>, to connect each ball B_i with the overtwisted disc.

The rest of the subsection is dedicated to the proof of Proposition <ref>. The result will be achieved in several steps, none of which use the overtwistedness of _M× K.[The proof is relatively technical but nonetheless standard. In particular, we introduce notation whose sole purpose is establishing claim (iv). The proof of Theorem <ref> can be understood  treating Proposition <ref> as a black box.]



  §.§.§ Reduction to balls and genuine even-contact structures


The Engel condition is open and Diff-invariant as a relation in the space of 2-jets of plane fields on M. Therefore, Gromov's h-principle (see <cit.>) for open, Diff-invariant relations on open manifolds implies that the given formal Engel structure is homotopic to a genuine Engel structure in the complement of a family of balls (D_k)_k∈ K disjoint from V and Δ. These balls can be assumed to vary smoothly with k (for instance, by placing them in a neighborhood of Δ). This h-principle is also parametric and relative, i.e. we do not modify the distributions on V and Δ. 

We may consider the neighborhoods (Δ_k) that contain D_k as a (smoothly) trivial fibration over K with ^4 fibres. This allows us to assume, for the rest of the proof, that the manifold M is just ^4 (so it is in particular compact) and that the formal Engel structure is a genuine Engel structure near ∂^4. It follows that all distributions are orientable and coorientable.

According to <cit.> (see also <cit.>) the formal even-contact structure (_k,_k) is homotopic to a family of honest even-contact structures (_k,_k). By this we mean that _k is a smooth family of even-contact structures whose characteristic foliations are _k. The relative nature of this h-principle allows us to assume that this homotopy is constant on V and Δ. 

From now on, we assume that (_k,_k) is an even-contact structure containing _k for all k∈ K. This even-contact structure will change in the course of the argument. We have achieved condition (ii) from Proposition <ref>, condition (i) is satisfied by all homotopies introduced so far.



  §.§.§ Adapted triangulations
 

A basic ingredient in our constructions is a triangulation  of M× K which is adapted to several distributions on M× K. This means that every simplex σ is so small that each distribution is almost constant with respect to the affine coordinates σ≃{x_0+…+x_n=1  and x_i≥ 0}⊂^n+1. The following definitions are minor adaptations of definitions from Thurston's paper <cit.>.
  
 
Let N be a manifold of dimension n and ξ a smooth distribution of codimension q. A top-dimensional simplex σ⊂ N is in general position with respect to ξ if, in the coordinates provided by σ and for all points p∈σ, the linear projection σ⟶^n/ξ_p along ξ_p maps each q-dimensional subsimplex of ∂σ to a non-degenerate simplex of ^q≃^n/ξ_p.

A triangulation  of N is in general position with respect to ξ if every top-dimensional simplex is in general position.

In particular, being in general position guarantees that all simplices are transverse to ξ in the sense that the intersection of the tangent space of each simplex with ξ has minimal dimension everywhere.

 
A top-dimensional simplex σ⊂ M × K is adapted to _M× K⊂_M× K⊂_M× K if the following conditions are satisfied:


  i. σ is in general position with respect to the foliation _M × K by fibres of M× K  K,

  ii. σ is in general position with respect to the distributions _M× K, _M× K, and _M× K,

  iii. the plane field _M× K|_σ describes less than one projective turn with respect to the line field _M× K|_σ.
  

A triangulation  is adapted, if:


  * it is a triangulation of the pair (M× K, (V ∪Δ)), where (V ∪Δ) is an Engel neighborhood of V ∪Δ,

  * every top simplex σ∈ is adapted.
 

By a triangulation of (M× K, (V ∪Δ)) we mean that there is a triangulation _∂ of ∂((V ∪Δ)) which  extends to the rest of (M× K) ∖(V ∪Δ).


Condition (iii) for an adapted simplex should be understood in terms of the development map: We defined it for Engel structures in Equation (<ref>) p. e:development map, and this definition extends to the present formal setting. The only difference with the Engel case is that now the development maps of _M× K|_σ are not immersions in general (as that would imply that _M× K|_σ is Engel). Condition (iii) is then equivalent to the assumption that the (oriented) development maps do not take antipodal values. This does not follow from Condition (ii), since the leaves of the kernel foliation may still twist around each other, as indicated in Figure <ref>.

 


Let us introduce some notation:  We will write _σ⊂_σ⊂_σ for the restriction of the formal Engel structure to a top-dimensional simplex σ. If σ is adapted, the subsimplices of σ are transverse to _σ, _σ, and _σ. In particular, a codimension-1 simplex is never tangent to _σ, intersects _σ in a line field, and _σ in a plane field. 

Fix auxiliary orientations of _σ and _σ. Then, we write σ_- for the union of those faces where _σ points into σ. Similarly, the line field Tσ_- ∩_σ divides the boundary of σ_- into the regions ∂_-σ_- and ∂_+σ_-, depending on whether it points inwards or outwards, respectively. Both ∂_±σ_- are codimension-2 complexes and homeomorphic to closed balls.

Given a subset of M × K, in this case σ, we write K_σ for the subset of those k∈ K such that M_k meets σ. For k∈ K_σ let σ_k=σ∩ M_k and σ_k,-=σ_-∩ M_k. When σ is adapted, _σ induces a contact structure ξ_k on σ_k,-. This contact structure contains the line field _k = Tσ_k,-∩_σ.

Figure <ref> illustrates the various parts of σ in the non-parametric case: σ is shown as a 3-dimensional simplex and _σ corresponds to the vertical direction. The line field in the bottom face represents the imprint _k of _σ on σ_-.



Some immediate properties of adapted simplices are summarized in the next lemma:
 
Let σ be an adapted simplex. Then:


  * the set K_σ is homeomorphic to a closed ball of dimension (K),

  * for every k ∈∂(K_σ), σ_k is a point,

  * for every k ∈∫(K_σ), σ_k is a polyhedron homeomorphic to a closed 4-ball, 

  * for every k ∈∫(K_σ), the union of the codimension-1 faces of σ_k which are positively transverse to _M× K is homeomorphic to a closed 3-ball. The same is true for the negatively transverse faces.





  §.§.§ Sequences of triangulations and coverings


Conditions (i), (ii), and (iii) in Proposition <ref> can be achieved by modifying the development map of _M × K in the vicinity of the codimension-1 skeleton of an adapted triangulation. This is the content of Lemma <ref>. It yields a structure _M × K' that is Engel in the complement of finitely many balls, each of which is obtained by slightly shrinking one of the top-dimensional simplices of the triangulation. However, condition (iv) does not follow immediately from this argument. To achieve it, we will need to consider an infinite sequence of triangulations, each of which is a subdivision of the previous one. For a sufficiently fine subdivision condition (iv) is satisfied. This will be shown in the upcoming lemmas. 

The first auxiliary lemma constructs the triangulations.
 
There is


  * a finite cover {U_i} of (M × K) ∖(V ∪Δ) by balls that are simultaneously flowboxes of _M × K and foliation charts of _M × K,

  * an infinite sequence {_j}_j=0^∞ of adapted triangulations, and

  * a finite list of model top-dimensional simplices {τ_k ⊂^(M × K)},

with the following properties.


  * _j+1 is a subdivision of _j.

  * Every top-dimensional simplex σ of a triangulation _j is adapted and contained in some U_i. In the coordinates provided by U_i, σ agrees with one of the model simplices {τ_k} up to rescaling and translation.
 


There is a finite covering of M × K by foliation charts of _M × K = ∪_k ∈ K M_k that are also flowboxes of _M × K; this follows from the compactness of M × K. We fix a triangulation that is subordinate to this covering. We apply to it Thurston's argument <cit.> of perturbing and carefully subdividing using a crystalline subdivision scheme. This yields only finitely many shapes of simplices up to rescaling for all subdivisions; note that this is not the case for the barycentric subdivision. Such a scheme is described in detail in <cit.>. This method can be applied simultaneously to all the distributions involved. Let us provide some additional details. 

There are only finitely many shapes of simplices in the following sense: Once a first adapted triangulation _0 has been found, each top-dimensional simplex σ∈_0 is in general position with respect to _M × K and _M × K. This implies that


  * σ can be identified with the standard simplex σ' in ^(M × K),

  * under this identification _M × K is mapped to a linear foliation which is transverse to all the simplices in the standard triangulation of the unit cube,

  * _M × K is mapped to a linear line field which is transverse to all the simplices in the standard triangulation of the unit cube.

This identification can be extended to a diffeomorphism between a neighborhood U ⊃σ and a neighborhood of σ'. By construction, U can be assumed to be a foliation chart of _M × K and a flowbox of _M × K. Doing this for each top-dimensional simplex σ_i ∈_0 we find the covering of M × K by flowboxes/foliation charts {U_i}. Because M× K is compact, there are only finitely many of them.

Every subsequent triangulation _j is obtained from _0 by subdividing each top-simplex σ_i ∈_0 in a standard manner in the coordinates of σ_i (which are the same as the coordinates of U_i). This yields finitely many shapes independently of j after sufficiently many subdivisions have been performed.


For condition (iv) in Proposition <ref> to hold, we need to achieve the Engel condition in a neighborhood of uniform size of the codimension–1 skeleton (that is, once the top-simplices are identified with the model simplices, the size of the neighborhood should not depend on j). For this purpose, we construct coverings adapted to the triangulations we are working with:
 
Let  be an adapted triangulation, which we regard as a collection of simplices {σ}. A covering {(σ)}_σ∈ of (M × K) ∖(V ∪Δ) is associated to  if:


  i. Every simplex σ is contained in the union ∪_σ' ⊂σ(σ'), where the union ranges over all the subsimplices of σ (including σ itself). 

  ii. Each (σ) is both a flowbox of _M × K and foliation chart of _M × K. In particular, the leaves of _M × K|_(σ) are connected non-degenerate intervals. The boundary of (σ) decomposes into its horizontal part ∂^h(σ), which is transverse to _M × K, and its vertical part ∂^v(σ), which is tangent to _M × K.

  iii. (σ) and (σ') only intersect if one is a subsimplex of the other. Moreover, if  σ' ⊂σ, then (σ) intersects the boundary of (σ') in its vertical part. 


These properties will allow us to inductively achieve the Engel condition on (σ) for all simplices  σ of positive codimension (see Lemma <ref>).

In the following lemma we construct neighborhoods of the codimension-1 skeleton with uniform size.
 
There are:


  * coverings {(σ)}_σ∈_j of (M × K) ∖ (V ∪Δ) associated to each triangulation _j, and

  * simplices {τ_k' ⊂τ_k} obtained from the model simplices {τ_k} by scaling with respect to the barycenter using a constant smaller than 1,

such that, after identifying the top-dimensional simplex σ∈_j with τ_k, 

    ∂τ_k' ⊂⋃_σ' ⊊σ(σ').




A similar result was proven in <cit.>. We now give a sketch of the proof.

Every simplex σ∈_j arises from the subdivision of some top-dimensional simplex σ_i ∈_0 and is therefore contained in the _M × K-flowbox U_i. We proceed by induction on their dimension (σ) < (M × K).

Let us start with the 0-simplices: Given σ_i ∈_0, we fix neighborhoods (σ) ⊃σ for every j and every vertex σ∈_j obtained in the subdivision of σ_i. We require (σ) to be a foliation chart and a flowbox as in the statement. If we fix the length of (σ) along _M × K and we shrink the other directions, every 1-simplex incident to σ will enter through the vertical boundary ∂^v(σ). This follows from the fact that every face is transverse to _M × K. 

If (σ) > 0, we first shrink σ to a slightly smaller simplex σ' whose boundary is still contained in the union ∪_τ⊊σ(τ). Then, we thicken σ' to (σ) in such a way that the thickening along _M × K is much greater than in the complementary directions. It follows that the simplices containing σ will enter (σ) through the vertical boundary ∂^v(σ). 

All sets we construct can be assumed not to depend on the simplex itself but on the model simplex they are identified with. In particular, once we have dealt with all the simplices in the codimension-1 skeleton, we can choose a constant (which is smaller than but sufficiently close to 1, and does not depend on j) to scale τ_k and obtain the desired τ_k'. 





  §.§.§ Achieving the Engel condition in the codimension-1 skeleton.
 

Given any adapted triangulation  with associated cover {(σ)}_σ∈, we can deform _M× K into a family of formal Engel structures _M× K' which is Engel over the neighborhood 

    ⋃_{σ∈ | (σ) < (M × K)}(σ)

of the codimension-1 skeleton of .  
 
Let _M× K⊂_M× K⊂_M× K be a family of formal Engel structures such that condition (ii) of Proposition <ref> holds. Let  be an adapted triangulation and {(σ)}_σ∈ an associated covering.

Then, there is a plane field '_M× K such that


  * _M× K' is homotopic to _M × K through plane fields contained in _M× K and containing _M× K,

  * _M× K'=_M× K on V and Δ,

  * _M× K' is Engel in (σ) for every simplex σ∈ with (σ) < (M × K).




Recall that there is a subcomplex _∂⊂ which triangulates the boundary ∂((V ∪Δ)). The structure _M × K is already Engel along _∂. The proof is by induction on the dimension of the simplices of ∖_∂. The relative nature of our claim follows by not modifying _M × K along _∂. 

Let σ∈ be a vertex. We fix a trivialization W of _M× K and an oriented W-invariant framing {W,X,Y} of _M× K on a neighborhood ((σ)) such that _M× K is spanned by {W,cos(c)X+sin(c)Y}, where c: ((σ)) is some function. We then homotope c, relative to the boundary of ((σ)), to yield a function c̃: ((σ)) with the property that _Wc̃>0 in (σ). This implies that the corresponding plane field is an Engel structure on (σ). This can be iterated over all the 0-simplices. The resulting plane field is again denoted by _M× K. 

Because  is adapted, a 1-simplex is nowhere tangent to _M× K. Hence, we can choose a W-invariant framing {W,X,Y} of _M× K on a neighborhood ((σ)) of the simplex. Then _M× K is spanned by {W,cos(c)X+sin(c)Y}, where c is a smooth function. For every vertex σ' ⊂σ it holds that _W c>0 on (σ) ∩(σ'). Condition (iii) of Definition <ref> implies that each leaf of _M × K|_(σ) is either contained or   disjoint from ∪_σ' ⊊σ(σ'). Therefore, we can replace c by a function c̃ which coincides with c on ∂((σ)) and on (σ') and satisfies _Wc̃>0. We thus obtain a plane field which is Engel on (σ). 

This procedure can be iterated until we have dealt with all simplices of codimension at least 1, always relative to the neighborhoods of simplices of lower dimension. In the regions where _W c ≤ 0, we can choose _Wc̃ to be positive but arbitrarily close to zero. This ensures that  is still adapted to the resulting  formal Engel structure _M × K'.





  §.§.§ Achieving the contractibility hypothesis
 

We now explain how to achieve claim (iv) from Proposition <ref>.

Recall that each top-dimensional simplex σ in _j is contained in a flowbox/foliation chart U_i_0. Under the identification of U_i_0 with the standard model, σ is mapped to (a scaled and translated copy of) some model simplex τ_l_0 (from a finite list {τ_l}). 

Apply Lemma <ref> to each adapted triangulation _j (produced by Lemma <ref>) with associated covering {(σ)}_σ∈_j (constructed in Lemma <ref>). Under the identification of σ with τ_l_0, the neighborhood ∪_σ' ⊊σ(σ') of ∂σ≅∂τ_l_0  covers some rescaling τ_l_0' with fixed scaling factor. Fix some strictly convex neighborhood τ_l_0' ⊂τ_l_0⊂τ_l_0 and write σ for its image in σ. 

Consider the simplex σ in the coordinates provided by the foliation charts {U_i}. In terms of U_i_0, σ is a genuine linear simplex. However, this might not the case for the other charts. Still, as finer subdivisions are considered (i.e. j goes to infinity) σ converges (in the C^∞ topology and after rescaling to have fixed diameter) to an affine simplex in all U_i. This follows from the fact that the change of coordinates between U_i and U_i' converges to linear map when we rescale it in progressively smaller neighborhoods of a point.

This argument implies that the set σ, which was convex in the coordinates of U_i_0, is also convex in terms of all the other charts U_i if j is large enough. The same is true for their projections to the leaf space K. That is, the set σ/_M × K is strictly convex in the coordinates provided by each U_i/_M × K for j large enough. By compactness, this can be achieved for all σ∈_j simultaneously. This implies that any finite intersection of them is either empty or contractible, as desired.

We fix a sufficiently large number j, and henceforth we just write = _j.





  §.§.§ Construction of shells
 

The last step in the proof is to produce a shell B = (B_k)_k ∈ K_σ from each σ, where σ is a top-dimensional simplex of . We consider a rescaling σ' ⊃σ of σ with respect to the barycenter by a factor smaller than but very close to 1; B_k will be a smoothing of σ_k'. The coordinates (y,x,z,w) provided in the simplex by B_k are shown roughly in Figure <ref>, p. b:sigma notation.

Fix a vector field W spanning _σ. Since σ' is adapted, W is transverse to the faces of each σ_k'. We write σ'_- for the union of those faces where W points into σ', and we restrict our attention to the subset (σ_k,-'=M_k∩σ_-')_k ∈ K_σ. Each face of σ_k,-' is either positively or negatively transverse to the line field _σ∩σ_-' and the union of the positive (resp. negative) faces is homeomorphic to a 2-ball. It follows that σ_k,-' can be smoothed, yielding a hypersurface N_k with the following properties:


  * N_k is a smooth hypersurface in M_k, transverse to _k, and contained in a small neighborhood of σ'_k,-. In particular, N=∪_k∈ K_σ N_k is a smooth hypersurface in M× K transverse to _σ.  

  * The holonomy of _σ yields an embedding  : σ_k,-'  N_k such that the image is disjoint from ∂ N_k.

  * The boundary ∂ N_k is a smooth 2-sphere. It decomposes into two discs, depending on whether TN_k∩_k is inwards or outwards pointing. The positive disc Γ_k is a smoothing of ∂_-σ_k,-'.  


Since σ' is adapted, we can assume that ∂Γ_k consists of two intervals: the characteristic foliation Γ_k∩_k is transverse to ∂Γ_k on the interior of both intervals and its holonomy identifies the two. Hence, we can equip Γ_k with coordinates (x,y) such that 

    TΓ_k∩_k=⟨∂_x⟩.


Using the flow of a vector field spanning TN_k∩_k we obtain a family of contact embeddings

    ψ_k: (N_k,_k ∩ TN_k)  (^3,(cos(z)dy ±sin(z)dx))

mapping Γ_k to a smoothing of [0,1] × [0,1] ⊂{z=0} (because of (<ref>)) and taking the leaves of TN_k∩_k to lines parallel to the z-axis. 

To obtain the correct sign for the contact structure in Equation <ref>, we might need to apply the reflection along the {x=0} hyperplane. Therefore, we assume that we have a contact embedding 

    ψ_k: (N_k,∩ TN_k)  (^3,(cos(z)dy - sin(z)dx)).

Since N_k is a compact manifold with boundary,  can extend these embeddings slightly so that their image is 

    {y∈ [0,1], x∈[0,1], z∈ [f_-,k,f_+,k]}

for some smooth functions f_-,k<0<f_+,k. We may choose f_-,k≡-ε with ε>0 small enough. 

The embedding ψ_k extends to an embedding Ψ_k: B_k = N_k × [-ε,1] ^3 × such that 


  * N_k is mapped to {w=0},

  * each leaf of _σ intersecting N_k is mapped to the interval {ψ_k(p)}× [-ε,1] for p ∈ N_k.

Since the development map of _σ does not describe a projective turn (condition (iii) in Definition <ref>), the extended embedding Ψ_k determines an angular function c_k vanishing along {w=0} such that its values lie in (-π,π). 

It might happen that ∂_w c_k < 0 along the boundary of the model we have constructed. To fix this, one applies the contact transformation (x,y,z)↦ (-x,y,-z). After applying this transformation, ∂_w c_k > 0 along the boundary of the model. In particular, we can reparametrize _σ so that c_k≡arcsin(w) if w ∈ [-ε,ε] (by possibly reducing ε and the domain of Ψ_k). 

This construction goes through for every top-dimensional simplex σ of , yielding a shell (B_k)_k ∈σ. The proof of Proposition <ref> is complete. 








 §.§ Extension in the non-parametric case


The π_0-version of Corollary <ref> states that any Engel germ ((∂^4),) with formal extension to the interior can also be extended to the interior as an Engel structure. With the tools introduced so far we can give a streamlined proof of this fact without invoking Theorem <ref>. The proof is a simplified version of the proof of our main theorem and showcases an important part of the argument.

The first ingredient is the following corollary of Proposition <ref>.
 
Let (M,) be a formal Engel manifold with  Engel over some closed region U ⊂ M. Assume that M ∖ U is connected and that there exists a formal Engel embedding of the overtwisted disc Δ: Δ_ (M ∖ U,).

Then,  is homotopic to a formal Engel structure ' satisfying:


  * '= on U and Δ,

  * ' is Engel in the complement of a finite collection of circular shells.



It can be proven directly mimicking the last step (Subsection <ref>) in the proof of Proposition <ref>. 

Alternatively, we apply Proposition <ref> to reduce to the case where the germ (^4,) is described by a shell B. We find a slightly smaller shell B' ⊂ B. Then we connect its boundary components {y=0} and {y=1} using a curve γ⊂ B transverse to [,] which is otherwise disjoint from B'. We can extend the parametrisation of B' to a neighborhood of γ, yielding the claim. We leave the details to the reader.


Then, the π_0-version of Corollary <ref> follows from an application of:
 
Any circular shell is formally homotopic, relative to the boundary, to a genuine Engel structure.


Let B = M(^1,[-ε,1],-ε,f_+,c) be the circular shell. We first construct a transverse surface ⊂ B as in Section <ref>. The core is parametrised by y ↦ (y,x=1/2,z=0,w=0) and the profile η (a family of transverse loops in ^1 ≡ [0,1]/_∼ (^3,(dx-wdz)) parametrised by y) is independent of y. Moreover, we require 


  * η(θ) = (x=θ,z=0,w=0), for θ∈ [ε,1-ε],

  * z ∘η(θ) < 0, for θ∉ [ε,1-ε], and

  * |w ∘η(θ)| < ε.

The last requirement implies the following condition on the slope: |dx(η̇)/dz(η̇)| < ε. Therefore, it is easy to construct a curve η with the desired properties if we introduce sufficiently many stabilizations, c.f. Figure <ref>. By Lemma <ref>,  is a torus transverse to _c. 



Consider the rank 1-foliation = ∪_w_0 (_c ∩ T{w=w_0}). In the region {|w_0|≤ε} we may orient it using the vector field ∂_z + w_0(∂_x + z∂_y). We pick a constant ρ>0 so that U_ρ()={q∈ B | dist(p,)≤ρ} is a thin tubular neighborhood satisfying:


  * U_ρ() is contained in {|w|≤ε},

  * N=∂ U_ρ() is transverse to _c, and

  * the set of tangencies N_0 = ∩ TN is a disjoint union of two tori. Each torus is obtained from  by pushing along the line field  and is, in particular, transverse to _c. 


Let η^+ be the part of the profile η where η(θ) = (x=θ,z=0,w=0) and 

    ^+=⋃_y∈[ε,1-ε]η^+ ⊂.

We denote  the region within N_0 obtained from ^+ by flowing along  positively by N_0^+. According to Lemma <ref>, one can isotope a transverse hypersurface using the flow of a vector field that is both tangent to the Engel structure and transverse to the hypersurface. In the present situation, we use the leaves of  to isotope N (note that  is transverse to N at least in the vicinity of N_0).

We want the isotopy to push a small neighborhood of N_0^+ ⊂ N positively while leaving the rest of the 3-manifold fixed. We denote the flow of ∂_z + w_0(∂_x + z∂_y) by φ. We then choose a function χ : N such that χ|_N_0^+ = f_+ - ε, and χ≡ 0 in the complement of (N_0^+), allowing us to define:

    N' =  {φ_χ(p)(p)  .|  p∈ N }.

This is a transverse 3-torus which is isotopic to N through transverse hypersurfaces.

The restriction of _c to 

    M' = {2ε≤ x ≤ 1-2ε, 0 ≤ z ≤ f_+ - 2ε, w ≥ε}

is a circular shell. The hypersurface N' is disjoint from M' and, by construction, its projection π(N') to the (y,x,z)-hyperplane contains π(M'). Now we perform an Engel-Lutz twist along N'. This takes place away from M', so the resulting formal Engel structure (_c) is still a circular shell in M'. Let c': M' be the corresponding angular function. A priori, c' agrees with c, but the presence of the Engel-Lutz twist allows us lower its value by at most 2π, effectively homotoping (_c) in the band { w ∈  (0,ε]}. Doing this, we can set c'(y,x,z,ε) < c(y,x,z,1) = c'(y,x,z,1) everywhere. 

Lemma <ref> shows that (M',_c') is homotopic to a solid model relative to its boundary. This implies that (_c) is homotopic to a genuine Engel structure relative to the boundary of the original circular shell.

Observe that the proof we have just presented is clearly parametric (although we avoided this for simplicity). However, we do not know whether it is relative in the parameter: is there an Engel homotopy that adds Engel torsion? An affirmative answer to this question would prove that all Engel structures are overtwisted up to homotopy (proving the complete h-principle for Engel structures).





 §.§ Setup for the extension



The rest of the paper is dedicated to completing the proof of Theorem <ref> (equivalently, Theorem <ref>). After the reduction argument from Proposition <ref>, we may assume that _M × K is Engel in the complement of a finite collection of pairwise disjoint shells {B^i = (B^i_k)_k ∈ K_i ⊂ K}_i. We have to construct a homotopy, relative to V and Δ, between _M × K and a K-family of Engel structures.

The construction can be carried out independently for each shell, so let us focus on the particular shell B = B_k^0 = (B_k = B_k^0)_k ∈ K_0. The construction has two main steps.


  * First, we homotope _M× K inside the region where _M× K is already Engel. Doing so, we make overtwisted discs with specific properties appear within B. We will describe them using the language of universal twist systems from Subsection <ref>. Their properties are motivated by the next step.

  * The second step is similar to the strategy outlined in Proposition <ref>: The presence of the overtwisted discs allows us to modify the angular functions (c_k)_k ∈ K_0 of B close to its bottom boundary. Some further manipulations allow us apply Lemma <ref> and conclude the proof. Since this takes place within B, it is relative to V and Δ.  

This process is relative to the other shells, so the argument can be iterated. The current Subsection introduces certain definitions and constructions needed for the proof, which we then carry out in Subsection <ref>.



  §.§.§ Overtwisted discs associated to shells
  

We want all subsequent formal Engel homotopies to be relative to the certificate and yet, we do need a certificate to complete the proof. The replication Lemma <ref> can be used to obtain new overtwisted discs as follows.

We use Lemma <ref> to modify _M × K in the Engel region (Δ) ∩ (M ×(K_0)). This modification is an Engel homotopy that introduces an additional family of overtwisted discs 

    Δ_k': Δ_ (M,_k),       k ∈ K_0.

It is relative in the parameter to the complement of (K_0) ⊂ K, and relative in the domain to Δ and the complement of (Δ). 

Any further operations involving overtwisted discs and the shell B will make use of Δ' = (Δ_k')_k ∈ K_0, so that our constructions do not affect the original certificate Δ.



  §.§.§ Connections between the shell and the overtwisted disc


The following definition packages the notion of a smooth family of paths connecting B with the certificate Δ' associated to it:
 
A submanifold ν: [0,1] × K_0  M × K is a connection between Δ' and B if


  * ν(·,k) ⊂ M_k = M is an embedded path transverse to _k and tangent to _k,

  * ν(·,k) is disjoint from V, Δ, and all shells except B,

  * ν(·,k) intersects B_k only at the endpoint ν(1,k). In the coordinates of B_k this corresponds to (0,0,-ε,0).

  * ν(·,k) is disjoint from the scaling region of Δ_k'. It intersects Δ_k' only at the endpoint ν(0,k). In the coordinates of Δ_k' this corresponds to (L/2,0,0,0).


In particular, a connection is contained in the region in which _M × K is Engel. A key fact, which follows from the first two items in the definition and Proposition <ref>, is that the Engel structure is unique up to diffeomorphism on a neighborhood of ν(·,k).

We will construct a connection between B and Δ' in Proposition <ref>. For this we will use the following lemma (which rephrases the parametric ambient connected sum lemma in <cit.>).

 
Let P_0 be a contractible set. Let {P_i ⊂ P_0}_i=1,…,m be a finite collection of subsets such that

    ⋂_j∈ J P_j is contractible or empty for all J⊂{1,…,m}.

Let M be a manifold and fix a collection of families of embeddings of the disc

    ℬ^i: ^(M)× P_i  M × P_0 
    ℬ^i(p,k) = (ℬ_k^i(p),k).


Then, there is a collection {D_i}_i=0,…,m of disjoint embeddings of ^(M) into M, and a P_0-family of isotopies (ψ_k: M  M)_k ∈ P_0 such that

    ψ_k ∘ℬ_k^i(p) = D_i(p)    for all i, k, and p.




We argue by induction on m. The inductive hypothesis is that the lemma holds for all collections of size at most m-1 satisfying the contractibility hypothesis and for all manifolds (not just M).

In the base case m=1 we reason as follows: Since P_0 is contractible, the isotopy extension theorem ensures the existence of a P_0-family of isotopies ψ_k^0: M  M satisfying ψ_k^0 ∘ℬ_k^0(p) = D_0(p), for some embedding of the disc D_0. Then, we regard ℬ^1 as a family of embeddings into M ∖ D_0, and we use the same argument to find a P_1-family of isotopies ψ_k^1: (M,D_0)  (M,D_0) (i.e. relative to D_0) satisfying ψ_k^1 ∘ℬ_k^1(p) = D_1(p).

We now explain the inductive step. First, we may assume that ℬ_k^0 = D_0 for all k ∈ P_0 using the same reasoning as above. Then, the sets P_1, {P_1 ∩ P_i}_i>1, the families {(ℬ_k^i)_P_1 ∩ P_i}_i=1,…,m, and the manifold M ∖ D_0 satisfy the inductive hypothesis. 

Therefore, there is a family of isotopies (ψ_k': (M,D_0)  (M,D_0))_k ∈ P_1 with ψ_k' ∘ℬ_k^i = D_i for k∈ P_1 and all i. Now we choose a neighborhood  of D_0 such that ℬ_k^i ∩ = ∅ for all k and all i > 0. It is then possible to modify the isotopies ψ_k' so that

    ψ_k' ∘ℬ_k^1     = D_1 ⊂
    
     (ψ_k' ∘ℬ_k^i) ∩    = ∅,      for  i > 1.

Then we conclude by noting that the inductive hypothesis applies once again to P_0, {P_i}_i>1, the families {(ℬ_k^i)_k∈ P_i}_i=2,…,m, and the manifold M ∖. 


For i>0 the embeddings ℬ^i in the lemma correspond to the shell B, the restrictions (B_k^i)_k ∈ K_0 ∩ K_i of the shells B_i, and the copy of the overtwisted disc (Δ_k')_k ∈ K_0. Note that assumption (<ref>) follows from condition (iv) in Proposition <ref>. 
 
 
There is a connection ν: [0,1] × K_0  M × K_0 between the certificate Δ' and the shell B. 



Let B_k^i with k ∈ K_i, i=1,…,m, be the collection of shells other than B. Recall that V is of the form (U × K)∪ (M × K'). In particular, K_0 and K' are disjoint. Using Lemma <ref> we find a K_0-family ψ_k: M  M of isotopies of M with support in the complement of U such that 

    D     = ψ_k∘ B_k         D_Δ    = ψ_k ∘Δ_k 
     D_Δ'    = ψ_k∘Δ_k'     D_i      = ψ_k∘ B_k^i

are embeddings of balls in M ∖ U that do not depend on k ∈ K_0. 

In particular, after applying the isotopy ψ_k, we may assume that


  * the region {z ≤ 0, w ≤ε}⊂ B_k, as a parametrized subset of M,  and

  * the overtwisted disc Δ_k', as a parametrized subset of M, 

are independent of k∈ K_0. 
Since M ∖ (U ∪ D_Δ∪ D_1 ∪…∪ D_m) is connected manifold, there is an embedded path ν: [0,1]  M such that


  * ν is disjoint from the balls U ∪ D_Δ∪ D_1 ∪…∪ D_m,

  * ν(0) is mapped to Δ_k'(L/2,0,0,0) = D_Δ'(L/2,0,0,0),

  * ν(1) is mapped to B_k(0,-ε,0,0) = D(0,-ε,0,0),

  * ν is transverse to _k and tangent to _k at times t ∈({0,1}).

The last property follows from the explicit models we have around the endpoints, allowing us to prefix the path ν there. Embeddedness follows from C^∞-genericity, since the dimension of M is 4. 

We can regard ν as a submanifold ν: [0,1] × K_0  M × K that does not actually depend on the parameter k ∈ K_0. Do note, however, that each ν_k maps into a different even-contact manifold (M,_k). Still, since K_0 is a ball, we can apply Proposition <ref> to obtain a C^0-deformation of the family ν = (ν_k)_k ∈ K_0 relative to the endpoints. In this manner, we make ν tangent to  _M × K, while preserving embeddedness and transversality with _M × K. Therefore, the resulting manifold is a connection.

We henceforth fix a connection ν between B and Δ'. Schematically, this is shown in Figure <ref>.





  §.§.§ Twist systems
 

We want to place a portion of the universal twist system inside of the shell B. Let us recall some notation from Section <ref>, p. ssec:universalTwistSystem. 

The universal twist system is a 1-parametric family of  surfaces (_t)_t ∈ [0,1] contained in I ×^3_-, we consider the case when I is a closed interval. Each of the surfaces _t consists of infinitely many cylinders stacked side to side along the plane {z=0}:

    _t = ψ_λ(⋃_n∈T^n(S_t)) = ⋃_n∈T^n_λ(ψ_λ(S_t)),

where S_t is a single cylinder, T_λ is the translation in x of length λ^2, and T = T_1. The scaling constant λ in the definition depends on two parameters (see Remark <ref>):


  * ε_0, the size of S_t in the x, z, and w coordinates, and

  * δ_0, the length of the interval at the ends of the cylinder in which the unlinking and shrinking take place.

A third parameter, λ_0 ∈ (0,1], controls the shrinking of S_t at its ends. It does not affect λ. In particular, once we fix λ we are allowed to further shrink the ends of the universal twist system (while remaining transverse to the Engel structure and embedded). We single out the region  

    S_t^+ = S_t ∩{z=0}⊂ S_t.

By construction, the union

    ⋃_n∈ψ_λ(T^n(S_t^+))

contains the infinite band {y∈ [δ_0,1-δ_0],  z=w = 0 }. 

Observe that the bottom part of any shell, i.e. a sufficiently small collar of the bottom boundary {w=a},  is itself a solid shell of the form 

    M([0,1],[-ε,ε],-ε,f_+,arcsin(w)).

By construction, the Engel structure in this region is the one we used to define the universal twist system:

    _ = (α_=dy-zdx)∩(β_=dx-wdz).

 
Let _t be a universal twist system with scaling constant λ>0. Consider a shell M([0,1],[-ε,1],-ε,f_+,c) and a pair of integers m_-,m_+ ∈ satisfying


  (a) ψ_λ(⋃_m_-≤ n≤ m_+ T^n(S_t)) is contained in [0,1] × (0,1) × (-ε,0] × (-ε,ε),

  (b) ψ_λ(⋃_m_-≤ n≤ m_+ T^n(S_1^+)) contains the square [ε,1-ε]^2 ×{(0,0)}.

Then, the collection of surfaces described in item (a) can be regarded as a subset of the shell and is called a twist system.


 
Any shell M([0,1],[-ε,1],-ε,f_+,c) admits a twist system. This holds parametrically for compact families of shells.



We choose constants ε_0 and δ_0 in the construction of the universal twist system to be much smaller than ε. Then there are integers m_- and m_+ that satisfy the claim. The parametric statement follows by taking a sufficiently small constant ε suitable for all the models in the family.


We apply the Lemma to the shell B = (B_k)_k ∈ K_0. To each model B_k we assign a twist system _k,t. This surface does not actually depend on k when we use the coordinates provided by B_k. Similarly, we write _k,t^+ ⊂_k,t for the corresponding collection of surfaces as defined in item (b) of Definition <ref>. The cores of the cylinders in _k,t will be denoted by {α_k^n}_m_-≤ n≤ m_+.


Twist systems will play a role analogous to the one of the 2-torus  in the proof of Proposition <ref>. Let us briefly compare the two.

The profile of the torus  is a transverse unknot η with many stabilizations. The precise number of stabilizations needed depends on how large the Engel region of the circular shell is and, as such, we do not have an a priori bound. However, we do not need such a bound for proving statements that are not relative in the parameter space. 

However, for a complete h-principle, one has to define what the overtwisted disc is. In our approach this means that one has to fix the profile of a transverse surface. We choose η_-3, the unknot with a single stabilization. This implies that we cannot introduce in our shells a single, wide surface like , but rather a thin cylinder coming from the overtwisted disc. In order to be  able to use these cylinders as in the proof of Proposition <ref> we stack many of them next to each other. This is exactly what a twist system is.

It is worth noting that   we could have chosen as profile a knot with more stabilizations as well. However, we do not know whether an unstabilized unknot can be used.








  §.§.§ Homotopies of cores after undoing the Engel-Lutz twist
 

In this section we show that the twist systems we have introduced can be obtained by homotoping the core of an overtwisted disc and transporting the transverse surface and the Engel-Lutz twist along. First, we describe how to move the core itself. We make extensive use of the notation introduced in Subsection <ref>, p. ssec:homotopyOTdisc, which we briefly review. 
We continue using the notation from the previous subsection as well.

Given the shell B = (B_k)_k ∈ K_0, we produce a copy Δ' = (Δ_k')_k ∈ K_0 of the certificate using Lemma <ref>.  The core of Δ' is denoted by γ: [0,L] × K_0  M × K. Additionally, we fix a connection ν: [0,1] × K_0  M× K between B and Δ'. Let K_0' ⊂ K_0 be a subset such that B_k is a solid shell for k∈ K_0 ∖ K_0'.

As in Section <ref> we write (M× K_0)^Δ' = M^Δ'× K_0 = (M_k^Δ')_k ∈ K_0 for the manifold obtained from M × K_0 by cutting along the upper {y=L} and lower {y=0} boundaries of the certificate Δ'. 

The restriction of the formal Engel structure _M × K to the manifold (M× K_0)^Δ' is obtained from some other structure ^-1(_M × K) by an Engel-Lutz twist along a K_0-family of cylinders Σ_k with core γ. All elements we have been working with (shells and connections) are disjoint from the upper and lower boundary of Δ', so we can regard them as lying in (M× K_0)^Δ'. In particular, the shell B inherits the same formal Engel structure from ^-1(_M × K).

We write ^-1(_k) for the restriction of ^-1(_M × K) to M_k and ^-1(_k) for the induced even contact structure. In  ⊷(Δ') the structure ^-1(_k) is given by an Engel embedding

    φ: ([0,L] ×^3,_) (M^Δ'_k,^-1(_k))

of the standard neighborhood of the core γ(·,k). We also recall that the length of the scaling region l satisfies L/2 > l > 1/τ_0 (where τ_0 is the constant from Lemma <ref>) and that the cores of the surfaces forming a twist system (_k,t) are independent of t. 

 
In this setting, there is a path of families of embedded transverse curves

    γ_s: [0,L] × K_0 ((M × K_0)^Δ', ^-1(_M × K))

with s∈ [0,1] such that 


  * γ_0 is the core γ of Δ',

  * γ_s is disjoint from V, Δ, and all shells other than B,

  * for k∈ K_0', γ_1 contains the cores {α_k^n}, m_-≤ n≤ m_+, of the twist systems (_k,t), and 

  * γ_s=γ_0 outside of ({L/2}) × K_0. In particular, they agree in the scaling region.




Consider the family of open manifolds

    C_k = (γ∪ν(·,k) ∪ B_k({|w| < ε})

with k∈ K_0. By construction, (C_k,^-1(_k))_k ∈ K_0 is a family of genuine Engel manifolds diffeomorphic to a  ball. Moreover, the even-contact structure ^-1(_k) on C_k does not depend on k∈ K_0: First, note that we have explicit models in the vicinity of γ (according to Proposition <ref>), in B_k({|w| < ε}) (provided by the model structure), and in a neighborhood of ν(·,k) (Proposition <ref>). These three models glue, so we can identify all the manifolds (C_k,^-1(_k))_k ∈ K_0 with a fixed even-contact manifold (C ≅^4,_C).

Under this identification, the curves γ(·,k) are all identified with the same curve γ̃_0 in (C,_C). Similarly, for a given n, the cores α_k^n are all identified with a single curve α_n. By  the existence part of the h-principle for transverse knots  Lemma <ref> (on p.  ssec:transverseKnots) there is a transverse embedded curve γ̃_1: [0,1]  (C,_C) such that


  * γ_1(y) = γ_0(y) for y outside of ({L/2}), and

  * γ_1 ⊃α_n for all n.

According to the classification part of Lemma <ref> there is a homotopy (γ_s)_s ∈ [0,1], as embedded curves transverse to _C, between γ_0 and γ_1.

Now we reintroduce k in the discussion. Fix a bump function χ: K_0  [0,1] such that


  * χ(k) ≡ 0 in the complement of (K_0'), and

  * χ(k) ≡ 1 in K_0'.

Using the identification between γ(·,k) ⊂ C_k ⊂ M_k^Δ' and γ_0 ⊂ C, we define the desired homotopy γ_s of γ 

    γ_s: [0,1] × K_0     (M × K_0)^Δ'
     
    (y,k)    ⟼γ_s(y,k)=γ_χ(k)s(y).

This homotopy takes place, in the domain, in the region ({1/2}) ×(K_0'). Similarly, in the target space, it takes place within (C_k)_k ∈ K_0. 


Now that we can move the core effectively, we explain how to move the overtwisted disc along the homotopy (γ_s)_s ∈ [0,1]. 
 
Fix a constant t_0 arbitrarily close to but smaller than 1. There is a homotopy of formal Engel structures (_s)_s ∈ [0,1] in M × K satisfying 


  * _0 = _M × K,

  * _1 has an Engel-Lutz twist along the twist systems (_k,t_0) for k ∈ K_0', and

  * _s differs from _0 only in a neighborhood of the overtwisted disc Δ', the connection ν, and the region {|w| < ε}⊂ B.




We apply Proposition <ref> to _M × K and (γ_s)_s ∈ [0,1], parametrically in k. This provides a homotopy of formal Engel structures (_s)_s ∈ [0,1] such that


  * _0 = _M × K,

  * _s differs from _0 only in a neighborhood of Δ', ν, and {|w| < ε}⊂ B, and

  * _s is obtained from ^-1(_M × K) by an Engel-Lutz twist along a family of cylinders Σ_s with cores γ_s. In the regions where γ_1(·,k) agrees with the cores of _k,t_0, the profiles describing Σ_1 can be assumed to be a  rescaling of the profile of the twist system.

Recall now that the size of the ends of a twist system were controlled by a constant λ_0. In particular, we choose them to be arbitrarily small. This implies that we can enlarge the profiles of each (Σ_s)_s ∈({1}) along the cores of the twist system until Σ_1 contains 

    ⋃_k ∈ K_0'_k,t_0.

The resulting cylinders are still transverse. Adding an Engel-Lutz twist to ^-1(_M × K) along this path of families of cylinders yields the result.





 §.§ Extension
 

To conclude the proof of Theorem <ref>, we need one more ingredient. The following proposition contains the main geometric ideas in this paper, most of which are essentially a refinement of the method shown in Subsection <ref>. It will become apparent during the proof that the properties of a twist system (Subsection <ref>) are precisely what is needed for the argument to go through.
 
Fix a shell

    (B,) = M([0,1],[-ε,1],-ε,f_+,c)

and a twist system

    = ⋃_m_-≤ n≤ m_+ψ_λ(T^n(S_t_0)) ⊂{z ≤ 0; |w| < ε}

with t_0 sufficiently close to but smaller than 1. Write () for the formal Engel structure obtained from  by performing an Engel-Lutz twist along the surfaces .

Then () is homotopic through formal Engel structures to an honest Engel structure ' satisfying:


  * the homotopy is relative to the boundary ∂ B,

  * in {|w| > ε}, the even-contact structure remains fixed and the homotopy only modifies the angular function.



Before giving the proof, we want to emphasize that when we speak of introducing an Engel-Lutz twist in the shell B along the twist system, we do not mean that a new Engel-Lutz twist is introduced. This Engel-Lutz twist arises via a homotopy of Engel structures as explained in Subsection <ref>, so we are effectively comparing the formal Engel structure before () and after (()) the homotopy of the overtwisted disc. In particular, note that B is a shell with respect to the original formal Engel structure, but not with respect to the formal Engel structure obtained after the homotopy.


Let S_n = ψ_λ(T^n(S_t_0)), n=m_-,…,m_+,  be  the individual cylinders in the twist system, and 

    S_n^+     = S_n ∩{y ∈ [ε,1-ε], z=0}
    ^+     = ⋃_m_-≤ n≤ m_+ S_n^+.

Let δ>0 be a small constant which will be determined later. 

Because of property (5) of the profile of the universal twist system (Section <ref> p. isotopy prop 1), there is a function ω: {z ≥ 0; w=0} [0,ε) such that


  * ω(y,x,z) ≡ 0 whenever z > δ,

  * the points in S_m_+^+ are of the form (y,x,0,ω(y,x,0)), and

  * ω(y,x,0) > w for all j < m_+ and (y,x,0,w) ∈ S_j.

That is, the hyperplane L(y,x,z) = (y,x,z,ω(y,x,z)) lies above the (S_j)_j < m_+, contains S_m_+^+, and agrees with {z ≥ 0; w=0} in the complement of ({z,w=0}). Let us single out the following regions (c.f. Figure <ref>).


  * The portion of B lying above L

    B' = {(y,x,z,w)  |  z ≥ 0; w ≥ω(y,x,z) },


  * The surface contained in L and lying directly above S_j^+

    L_j = ⋃_(y,x,0,w) ∈ S_j^+{(y,x,0,ω(y,x,0))},


  * The strip connecting S_j^+ with L_j

    A_j = ⋃_(y,x,0,w) ∈ S_j^+{(y,x,0)}× [w,ω(y,x,0)].




We claim that there is a path of formal Engel structures (_s)_s ∈ [0,1] with the following properties.


  * _0 is obtained from  by an Engel-Lutz twist  along (S_j)_j < m_+.

  * _s = _0 in the complement of a small neighborhood of ⋃_j < m_+ A_j.

  * _s is transverse to S_m_+.

  * (B',_s) is a shell with an angular function c_s such that

    c_0   (y,x,z,w) - π/2 < c_s(y,x,z,w) ≤ c_0(y,x,z,w) = c(y,x,z,w)
    
    c_1    ≡ -π/2+δ   on ⋃_j < m_+ L_j.


In particular, note that the leftmost region of S_m_+^+ (the region with smaller x-coordinate) is contained in ⋃_j < m_+ L_j.

For the construction of _s we will first modify S_0 inductively over the regions (A_j) for j=m_-,…,m_+-1. Let us start with j=m_-: Consider the structure _m_-,0 obtained from  by adding an Engel-Lutz twist along S_m_-. By construction, _m_-,0 describes more than one turn along the ∂_w-flowlines contained in the strip A_m_- (these flowlines are leaves of the kernel). 

Using an isotopy tangent to  we can push this turning of the structure upwards. When we push approximately a quarter of a turn in terms of the framing {∂_z, ∂_x + z∂_y} upwards, we obtain a path of formal Engel structures _m_-,s,s ∈ [0,1], satisfying the following conditions.


  * _m_-,s = _m_-,0 in the complement of a small neighborhood of A_m_-.

  * On the complement of a small neighborhood of S_m_- the formal Engel structure _m_-,s is determined by an angular function c_m_-,s such that

    c(y,x   ,z,w) - π/2 < c_m_-,s(y,x,z,w) ≤ c(y,x,z,w), 
    
    c_m_-,1   ≡ -π/2+δ   on  L_m_-.


We now use property (B) from the definition of the (universal) twist system crucially (c.f. p. condition B): If the neighborhood of A_m_- containing the support of the isotopy is small enough, then _m_-,s is transverse[This can be seen in Figure <ref>: If η is the profile of the twist system and T(η) is its translate, the regions η^+ and T(η^+) overlap over some interval P. It is in a neighborhood of this interval (or rather, its counterpart for the twist system) in which the angular function is changing. This corresponds to turning the line field depicted in the figure clockwise until it becomes almost parallel to the x-axis. It follows that T(η) remains transverse.] to S_j for all j>m_-. Since S_m_-+1 lies below L_m_-, it follows that c_m_-,s∈ (-π/2,-π/2 +δ) on S_m_-+1.

This construction can be iterated. For each n=m_-+1,…,m_+-1 we consider the family (_n-1,s)_s ∈ [0,1] constructed in the previous inductive step. Using the fact that _n-1,s is transverse to S_n we add an Engel-Lutz along S_n, parametrically in s. We denote the resulting family by _n,s = (_n-1,2s) for s ∈ [0,1/2]. By hypothesis, _n,1/2 is described  by an angular function c_n,1/2 that is precisely -π/2+δ in the region ∪_j < n L_j (away from a neighborhood of the (S_j)_j ≤ n).  

As above we define a formal Engel homotopy (_n,s)_s ∈ [1/2,1], of _n,1/2 using an isotopy along ∂_w in a neighborhood of the strip A_n (i.e. along the kernel foliation). This allows us to modify the angular function so that it is precisely -π/2+δ in the region ∪_j ≤ n L_j. This completes the induction.

As above, it follows that S_m_+ is transverse to the path of formal Engel structures (_s)_s ∈ [0,1] that we have constructed. Now we thicken S_m_+ to a family (N_s)_s∈ [0,1] of transverse 3-dimensional manifolds. We may assume that


  * N_s lies in an arbitrarily small neighborhood of S_m_+,

  * N_s fails to be transverse to the kernel of _s along two disjoint surfaces N_s^0,± which are themselves transverse to _s,

  * N_s^0,+ and N_s^0,- are obtained from S_m_+ by an isotopy tangent to _s, and

  * in the vicinity of S_m_+^+, this flow is along the Legendrian marking TL ∩_s.

We can now define an isotopy of 3-manifolds (N_s)_s∈ [1,2] transverse to _1 by pushing N_1^0,+ further along the line field TL ∩_1. By construction, the Legendrian marking TL ∩_1 is given by the line field

    cos(-π/2+δ)∂/∂ z + sin(-π/2+δ)(∂/∂ x+ z∂/∂ y)

in a neighborhood of ∪_j < m_+ L_j. In general, TL ∩_1 is a line field lying in the tangent cone given by turning clockwise from -(∂_x + z∂_y) to ∂_z.



If we flow N_1 along TL ∩_1 for sufficiently long times, and δ was chosen to be sufficiently small, we will obtain a transverse 3-manifold N_2 which contains  

    L ∩{y,x ∈ [ε,1-ε]; z ∈ [ε,f_+-ε]}.

In particular, note that the region where the Engel condition fails lies above this hypersurface. The continuous lines in Figure <ref> represent TL ∩_1. The area contained in the pointed square lies below the region in which _1 is not Engel. The thick line on the left corresponds to S_m_+. The dashed line depicts N_2^0,+, which is obtained from S_m_+ by pushing along the Legendrian marking. The manifold N_2 covers the region between the solid and the dashed curves.

Now we can finish the proof as in the Subsection <ref>. First, we construct a path of formal Engel structures (_s())_s ∈ [0,1] by adding Engel torsion to each _s along the corresponding N_s. This corresponds to adding an Engel-Lutz twist along S_m_+. Therefore, _0() is precisely (), as desired. We define (_s())_s ∈ [1,2] to be the structures obtained by adding Engel torsion to _1 along (N_s)_s ∈ [1,2]. In particular, we obtain a family for formal Engel structure  _2() that has Engel torsion below the region where the formal Engel structure is not necessarily induced by an Engel structure. As above, this implies that we can push this additional turning upwards, thereby modifying the angular function c_2 of _2() over the region B'. 

By definition,  c_2(y,x,z,1) = c(y,x,z,1) > -π and an application of Lemma <ref> produces a homotopy (_s())_s ∈ [2,3] of _2() with _3() Engel. This homotopy preserves the even-contact structure, since it simply changes the angular function.


The parametric version of Proposition <ref> is stated in the next corollary.
 
Fix a shell

    (B_k,_k) = M([0,1],[-ε,1],-ε,f_k,+,c_k)      k ∈ K_0

and a twist system

    _k = ⋃_m_-≤ n≤ m_+ψ_λ(T^n(S_t_0)) ⊂{z ≤ 0; |w| < ε}

where t_0 is sufficiently close to but smaller than 1. Let (_k) be the formal Engel structure obtained from _k by performing an Engel-Lutz twist along the surfaces _k.

Then (_k) is homotopic through formal Engel structures to an honest Engel structure _k' such that


  * the homotopy is relative to a ε-neighborhood of ∂ (∪_K B_k), and

  * the even-contact structure remains fixed on {|w| > ε}. There, the homotopy only affects the angular function.



The argument is identical to the one in the proof of Proposition <ref>, as one must simply add parameters and suitable cut–off functions. We leave the details to the reader, but we will briefly sketch the main points.

We fix a family of hypersurfaces (L_k)_k ∈ K_0 lying slightly above {w=0} and extending the last component ψ_λ(T^m_+(S_t_0)) of the twist system (which, in the paremetrisation of the shell, does not depend on k). The proof has two parts. In the first one, the angular function is modified close to the hyperplane {z=0} so that it agrees with -π/2 + δ in the region of L_k lying above the other components of the twist system. This has to be done relative to ∂ K_0 in the parameter. Since the argument boils down to isotoping the formal Engel structure using an isotopy of the angular function, this process can be capped off close to the boundary ∂ K_0.

In second part of the argument we use flows along TL_k ∩_k to thicken the surface ψ_λ(T^m_+(S_t_0)) to a transverse 3-manifold covering most of the hypersurface L_k. Again, this flow can be capped off as k approaches ∂ K_0, so that the resulting 3-manifold is instead the boundary of a thin tubular neighborhood of the surface. In this manner, for k ∈(∂ K_0), the resulting Engel structure is simply (_k).

Lastly, we apply Lemma <ref>, parametrically on k, to modify the angular function in the region lying above L_k. For k ∈(∂ K_0), the angular function is already increasing and, hence, the relative nature of the lemma yields the claim.


We can finally prove the main theorem.

After applying the reduction from Proposition <ref>, we obtain a family of formal Engel structures _M × K that is Engel in the complement of a collection of shells. Given one such shell (B = B_k)_k ∈ K_0 we associate to it


  * a copy Δ' of the certificate (Subsection <ref>),

  * a connection ν between B and Δ' (Proposition <ref>), and

  * a K_0-family of twist systems (_k)_k ∈ K_0 (Proposition <ref>).

Write K_0'⊂ K_0 for a ball such that the formal Engel structure in B_k is Engel for k∈ K_0∖ K_0'.

Using Proposition <ref> we construct a homotopy (γ_s)_s ∈ [0,1] of the core γ_0 of the overtwisted disc Δ' so that γ_1 contains the cores of the twist systems (_k)_k ∈ K_0'. Proposition <ref> implies that there is a homotopy of formal Engel structures (_M × K, s)_s ∈ [0,1], starting from _M × K, where the overtwisted disc moves along (γ_s)_s ∈ [0,1]. The final structure _M × K, 1 is still only a formal Engel structure, but it has an Engel-Lutz twist along _k for all k ∈ K_0'.

On B_k, k ∈ K_0', the structure _M × K, 1 is obtained from _M × K, 0 by adding an Engel-Lutz twist along _k, k ∈ K_0'. Then Corollary <ref> yields a formal Engel homotopy (_M × K, s)_s ∈ [1,2] between _M × K, 1 and an Engel structure  _M × K, 2. This homotopy is relative to the boundary of the shell (B_k)_k ∈ K_0', so the argument can be iterated for the other shells. 





 §.§ Proofs of Corollaries of Theorem <ref>
 

In this Subsection we  prove some consequences of Theorem <ref>.



  §.§.§ Proof of Corollary <ref>
 

In Corollary <ref> we already showed that the inclusion _(M,Δ) (M,Δ) induces a surjection in homotopy groups. Alternatively, one can apply Theorem <ref> taking K' to be the empty set and K to be a sphere. 

For the general statement we need to show the vanishing of the relative homotopy groups π_k((M,Δ),_(M,Δ)). This is also an application of Theorem <ref> with K = ^k and K' = ∂^k. 




  §.§.§ Proof of Corollary <ref>
 

The non-parametric statement was already proven in Subsection <ref>. The general statement is proven as follows: Set M = ^4 and U = ∂^4. Take a point p in the interior of ^4 and find a family of intervals (γ_k)_k ∈ K with γ_k passing through p and transverse to _0(k) = [_0(k),_0(k)]; the space of such families is contractible. We can then deform _0 in ({p}) × K to yield a formal Engel family _1/2 such that (({p}),_1/2(k))_k ∈ K is a locally trivial fibration with fiber ([0,L] ×^3, _), where γ_k corresponds to [0,L] ×{0}. This is a certificate Δ⊂^4 × K. An application of Theorem <ref> yields a homotopy (_s)_s ∈ [1/2,1] relative to U = ∂^4 with _1 honestly Engel. 




  §.§.§ Proof of Corollaries <ref> and <ref>
 

The proof of Corollary <ref> goes as follows: Consider _0 and _1 overtwisted Engel structures with overtwisted discs Δ_0 and Δ_1, respectively. We first homotope them in (Δ_i) arguing as in Lemma <ref>: this allows us to assume that both overtwisted discs have the same length L. Then, using an isotopy of the manifold we set Δ_0 = Δ_1. A homotopy between _0 and _1 is then provided by Corollary <ref>.

The proof of Corollary <ref> is slightly more involved. Consider the K-family _0 with certificate Δ^0 = (Δ_k^0)_k ∈ K. We can isotope _0, parametrically in k, to assume that ⊷(Δ_k^0) is an arbitrarily small ball. If we assume that (K) < 4, the union 

    ⋃_k ∈ K⊷(Δ_0(k)) ⊂ M

does not cover the whole of M; choose a point p disjoint from it. We can argue similarly for _1 and assume that the certificate Δ^1 misses the same point p.

Since _0 and _1 are formally homotopic, there are formal Engel families _0 and _1 satisfying:


  * _0 and _1 have a certificate Δ in U × K (where U is a small neighborhood of p),

  * _i is formally homotopic to _i and agrees with it in the complement of (U).

An application of Theorem <ref> provides a formal homotopy between _0 and some _1/3 which is honestly Engel. This homotopy is relative to the complement of (U) and to Δ (and uses Δ as certificate). A second application of Theorem <ref> states that _0 and _1/3 are Engel homotopic (using Δ^0 as certificate). Similarly, we produce a formal homotopy between _1 and some _2/3 genuinely Engel, which is itself Engel homotopic to _1 (using Δ^1 as certificate) and to _1/3 (using Δ as certificate). This concludes the proof. 



  §.§.§ Foliated results
 

A well-known observation due to Gromov says that any complete h-principle (that is, relative in the parameter and the domain) automatically yields a foliated h-principle. When this h-principle requires some extra data to be fixed (the certificate), the foliated analogue requires slightly more work, see <cit.>. Still, the methods used in the proof of Theorem <ref> (or  a careful application of the Theorem itself), readily imply the following.

Let (W^4+m, ^4) be a manifold endowed with a smooth foliation of rank 4. Let ⊂⊂⊂ be a complete flag for the foliation and assume that we are additionally given bundle isomorphisms

    ()    ≅/, 
    (/)   ≅/.

This data is a formal foliated Engel structure. We want to homotope a formal foliated Engel structure so that  is a leafwise Engel structure and ⊂⊂ is the corresponding leafwise Engel flag. We may suppose that there is a closed subset V ⊂ W where the Engel condition already holds.

Let K be a compact, possibly disconnected, m-manifold. Suppose we are given a (embedded) foliation chart

    Δ: (^4 × K, ∐_k ∈ K^4 ×{k})  (W ∖ V,)

satisfying Δ^i(·,k)^*  = _ and, additionally, every leaf of (W ∖ V,) intersects the image of Δ. We say that Δ is a certificate of overtwistedness for the formal leafwise Engel structure. 


The case where (W,) is a trivial fibration is precisely the usual parametric setting for the h-principle. In this case, the two definitions of certificate are different but equivalent up to homotopy. Indeed, one may use Theorem <ref> to produce a certificate in the usual sense if we are given a certificate in the foliated sense, and vice versa. We leave this to the reader. 


The h-principle in the foliated setting reads:

Let (W,,_0,_0,_0) be a formal foliated Engel manifold. Suppose that the formal structure is already Engel over some closed subset V and that there is a certificate of overtwistedness Δ⊂ W ∖ V.

Then, there is a homotopy of formal foliated Engel structures (_s ⊂_s ⊂_s)_s ∈ [0,1] with _1 ⊂_1 ⊂_1 a foliated Engel flag. This homotopy is relative to Δ and V.

In particular, this statement recovers Theorem <ref>.





CPP15

[BEM]BEM M.S. Borman, Y. Eliashberg, E. Murphy, Existence and classification of overtwisted contact structures in all dimensions.  Acta Math. 215 (2015), no. 2, 281–361.

[CPPP]cppp R. Casals, J.L. Pérez, Á. del Pino, F. Presas, Existence h-principle for Engel structures. To appear in Invent. Math. arXiv:1507.05342.

[CPP15]cpp15 R. Casals, Á. del Pino, F. Presas, h-Principle for Contact Foliations. Int. Math. Res.
Not. 20 (2015), 10176–10207.

[CPP]cpp R. Casals, Á. del Pino, F. Presas, Loose Engel structures. Preprint. arXiv:1712.09283

[El]El89 Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds. Invent. Math. 98 (1989), no. 3, 623–637. 

[EM]em Y. Eliashberg, N. Mishachev, Introduction to the h-principle. Graduate Studies in Mathematics, 48. American Mathematical Society, Providence, RI, 2002.



[Et]etnyre J. Etnyre, Legendrian and transversal knots,  Handbook of knot theory, 105–185, Elsevier B. V., Amsterdam, 2005.

[Ge]ge-review H. Geiges, Review of R. Montgomery's paper Engel deformations and contact structures. MathSciNet: MR1736216 (2001h:53127).



[Gr]gr M. Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1986.

[KS]ks M. Klukas, B. Sahamie. On prolongations of contact manifolds, Proc. Amer. Math. Soc. 141 (2013), no. 9, 3257–3263.

[Lu]lu R. Lutz, Structures de contact sur les fibrés principaux en cercles de dimension trois, Ann. Inst. Fourier (Grenoble) 27 (1977), no. 3, ix, 1–15.

[McD]mcd D. McDuff, Applications of convex integration to symplectic and contact geometry, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 1, 107–133. 

[Mo]montgomery R. Montgomery, Engel deformations and contact structures,  Northern Calif. Sympl. Geometry Seminar, 103–117, Amer. Math. Soc. Transl. Ser. 2, 196, Adv. Math. Sci., 45, Amer. Math. Soc., Providence, RI, 1999.  

[Mur]Mur E. Murphy, Loose Legendrian Embeddings in High Dimensional Contact Manifolds, arXiv:1201.2245.

[dP]dp Á. del Pino, On the classification of prolongations up to Engel homotopy, to appear in Proc. Amer. Math. Soc.

[dP2]dp2 Á. del Pino, Engel structures and symplectic foliations, PhD thesis, Universidad Autónoma de Madrid.

[Th]th W. Thurston, The theory of foliations of codimension greater than one, Comment. Math. Helv. 49 (1974), 214–231.

[Vo]vo T. Vogel, Existence of Engel structures, Ann. of Math. (2) 169 (2009), no. 1, 79–137.

[Wh]whitney H. Whitney, Geometric Integration Theory, Princeton Univ. Press 1957. 


