On some examples and constructions of contact manifolds

The first goal of this paper is to construct examples of higher dimensional contact manifolds with specific properties. Our main results in this direction are the existence of tight virtually overtwisted closed contact manifolds in all dimensions and the fact that every closed contact 3-manifold, which is not (smoothly) a rational homology sphere, contact-embeds with trivial normal bundle inside a hypertight closed contact 5-manifold. This uses known construction procedures by Bourgeois (on products with tori) and Geiges (on branched covering spaces). We pass from these procedures to definitions; this allows to prove a uniqueness statement in the case of contact branched coverings, and to study the global properties (such as tightness and fillability) of the results of both constructions without relying on any auxiliary choice in the procedures. A second goal allowed by these definitions is to study relations between these constructions and the notions of supporting open book, as introduced by Giroux, and of contact fiber bundle, as introduced by Lerman. For instance, we give a definition of Bourgeois contact structures on flat contact fiber bundles which is local, (strictly) includes the results of the Bourgeois construction, and allows to recover an isotopy class of supporting open books on the fibers. This last point relies on a reinterpretation, inspired by an idea by Giroux, of supporting open books in terms of pairs of contact vector fields.

We remark that in the literature there is already a definition of contact branched coverings that goes in this direction. Indeed, Öztürk and Niederkrüger [40] define this notion in terms of contact deformations verifying an additional condition at the branching locus. Removing this further constraint, we show here the following: Proposition A Let (V 2n−1 , η) be a contact manifold and π : V → V be a smooth branched covering map with downstairs branching locus M. Suppose that η ∩ T M is a contact structure on M. Then: 1. there is a [0, 1]-family of hyperplane fields η t on V such that η 0 = π * η and η t is a contact structure for all t ∈ (0, 1]; 2. if η t and η t are as in point 1, then η r is isotopic to η s for all r , s ∈ (0, 1].
Moreover, in point 1, η t can be chosen invariant under local deck transformations of π for all t ∈ (0, 1]. Similarly, the isotopy in point 2 can be chosen among contact structures invariant under local deck transformations, provided that η t and η t are invariant too.
We will hence call contact branched covering a contact structure η on V that is the endpoint of any path η t as above. Notice that Proposition A tells exactly that this object exists and is well defined up to isotopy.
At this point, we are able to give precise statements about the properties of contact branched coverings. For instance, we prove the following: Theorem B Consider a smooth branched covering π : V → V and a contact structure ξ on V and let η be a contact branched covering of η. Suppose that (V , η) is weakly filled by (W , Ω) in such a way that the downstairs branching locus M of π is filled by a symplectic submanifold X of (W , Ω). Suppose also that π extends to a smooth branched covering π : W → W branched over X . Then, there is a symplectic structure Ω on W weakly filling η on V = ∂ W .
We then devote a part of the paper to an analysis and a generalization of the Bourgeois construction in [3].
As already recalled above, one can look at the examples in [3] in two different and "orthogonal" ways, namely via the projections M × T 2 → M and M × T 2 → T 2 . The first one tells that these examples are T 2 -invariant contact structures on the total space of the T 2 -bundle M × T 2 → M. We will not deal with this point of view here and we invite the interested reader to consult Gironella [18,Chapter 7], where the links between the construction in [3] and the study of T 2 -invariant contact structures in Lutz [36] are analyzed in detail. The second point of view shows that the examples in [3] are contact structures on M × T 2 which moreover induce a contact structure on each fiber of M × T 2 → T 2 , i.e., using the language introduced by Lerman in [34], which are contact fiber bundles on M × T 2 → T 2 . We point out that this contact bundle structure on the examples from [3] has already been exploited successfully in Presas [41], van Koert and Niederkrüger [32], Niederkrüger and Presas [39], Etnyre and Pancholi [13,15] to obtain high dimensional contact manifolds with remarkable properties. This suggests that this second point of view might be the best one to analyze and generalize the construction in [3].
In this paper we then use the theory of contact fiber bundles from Lerman [34] in order to generalize the Bourgeois construction and define the notion of Bourgeois contact structures. More precisely, on a fiber bundle π : V 2n+1 → Σ 2 equipped with a reference contact fiber bundle η 0 , every contact fiber bundle η admits a potential form A with respect to η 0 , with a well defined curvature form R A . In the case where the reference contact bundle η 0 is flat, we call Bourgeois contact structure any contact fiber bundle structure on π : V → Σ that is also a contact structure on V and verifies 1 R A → 0 for → 0.
Beside the need to pass from the construction procedure in [3] to a definition, another motivation behind the introduction of this notion is the following: the condition on the curvature is, on one hand, weak enough to be satisfied by a class of contact structures strictly containing the results of the construction in [3] and, on the other hand, strong enough to ensure some nice properties, for instance from the points of view of weak fillings and adapted open book decompositions (other properties will also be analyzed in Sect. 4.5).
As far as the weak-fillability is concerned, we prove the following: Proposition C Let (M 2n−1 , ξ) be a contact manifold and η be a Bourgeois contact structure on the trivial fiber bundle M × T 2 → T 2 , that restricts to ξ on M × { pt} = M. If (M, ξ) is weakly filled by (X 2n , ω), then (M × T 2 , η) is weakly filled by (X × T 2 , ω + ω T 2 ), where ω T 2 is an area form on T 2 .
We point out that the result is already known in the case of the Bourgeois construction [3]. Indeed, the statement and the idea of the proof already appeared in Massot et al. [ A more detailed statement and a detailed proof of Theorem E are given in Sect. 3. We point out that this result does not only serve to prove Proposition D but also gives another point of view on adapted open book decompositions, which is of independent interest.
These reinterpretations and generalizations of [3,16] lead us to examples of high dimensional contact manifolds with interesting tightness, fillability or overtwistedness properties. As a byproduct, we obtain two new results, one concerning tight virtually overtwisted contact structures and one concerning codimension 2 embeddings with trivial normal bundle of contact 3-manifolds.
As far as the first result is concerned, we recall that a tight contact structure ξ on M is called virtually overtwisted if its pullback ξ on a finite cover M of M is overtwisted. In this paper, we prove the following: Theorem F Virtually overtwisted structures exist in all odd dimensions ≥ 3.
The proof of this result is by induction on the dimension. As far as the initialization step is concerned, the existence of tight virtually overtwisted contact structures is known in dimension 3 since Gompf [25]. The interested reader can also consult Giroux [21] and Honda [29], which present a classification result for this type of contact structures on particular 3-manifolds. The inductive step uses Propositions C and A above, i.e. the fact that both the construction in [3] and contact branched coverings preserve the weak fillability condition, and relies on the existence of supporting open books proven by Giroux [22], on the Bourgeois construction [3] and on the "large" neighborhood criterion for overtwistedness proven in [5,Theorem 3.1].
Another application concerns the following question: for a given contact manifold (M, ξ = ker α), is there > 0 such that M × D 2 , ker α + r 2 dϕ is tight? Here, D 2 is the disk of radius and centered at the origin in R 2 , and (r , ϕ) are its polar coordinates.
This is linked to the problem of finding codimension 2 contact-embeddings with trivial normal bundle in tight ambient manifolds. Indeed, having trivial normal bundle and trivial conformal symplectic normal bundle is equivalent in codimension 2. Hence, according to the contact neighborhood theorem (see for instance Geiges [17,Theorem 2.5.15]), if (M 2n−1 , ξ = ker α) embeds into (V 2n+1 , η) with trivial normal bundle then it admits a neighborhood M × D 2 r 0 , ker α + r 2 dϕ , for a certain r 0 > 0. In particular, if (V , η) is tight, so is this neighborhood.
Historically, the first motivation for addressing the above question on the "size" of the neighborhood of a codimension 2 submanifold is given by Niederkrüger and Presas [39], where it is shown that "big" neighborhoods of contact overtwisted submanifolds obstruct fillability of the ambient manifold. As reported in Niederkrüger [38], this led Niederkrüger and Presas to conjecture that the presence of a chart contactomorphic to a product of an overtwisted R 3 and a "large" neighborhood in R 2n with the standard Liouville form could be the correct generalization of overtwistedness to dimensions greater than 3. After the introduction in Borman et al. [2] of a definition of overtwisted structures in all dimensions, Casals et al. [5] confirmed this conjecture, proving that the presence of such a chart in a contact manifold is indeed equivalent to it being overtwisted. More precisely, this follows from [5,Theorem 3.1], which states that, if (M, ξ = ker α) is overtwisted, then M × D 2 R , ker α + r 2 dϕ is also overtwisted, provided that R > 0 is sufficiently large. In particular, this motivates the above question on the existence, for a given contact manifold (M, ξ = ker α), of an > 0 such that M × D 2 , ker α + r 2 dϕ is tight. The problem of finding codimension 2 embeddings in tight manifolds has already been explicitly addressed for instance by Casals et al. [6], Etnyre and Furukawa [11] and Etnyre and Lekili [14]. More precisely, [6] proves that each 3-dimensional overtwisted manifold can be contact-embedded with trivial normal bundle into an exact symplectically fillable closed contact 5-manifold. In [11], the authors shows how to embed many contact 3-manifolds into the standard contact 5-sphere. Finally, it is proven in [14] that each 3-dimensional contact manifold contact-embeds in the (unique) non-trivial S 3 -bundle over S 2 equipped with a Stein fillable contact structure.
In this paper, we prove the following result: embeds with trivial normal bundle in a hypertight closed (V 5 , η).
We recall that a contact structure is called hypertight if it admits a defining form with no contractible closed Reeb orbit. Recall also that each hypertight contact manifold is in particular tight, according to Hofer [28], Albers and Hofer [1] and Casals et al. [5].
Remark that, by the Poincaré's duality and the universal coefficients theorem, the condition H 1 (M; Q) = {0} is equivalent to M being a rational homology sphere. An analogue of Theorem G, with (V 5 , η) symplectically fillable, is actually already known both in the case of every contact structure on S 3 and in the case of overtwisted structures on any rational homology sphere. Indeed, the case of overtwisted rational homology spheres (which includes the overtwisted S 3 's) is covered in Casals et al. [6,Proposition 11], and the standard tight 3-sphere (which is the unique tight contact structure on S 3 up to isotopy according to Eliashberg [9]) naturally embeds in the strongly fillable standard contact 5-sphere with trivial normal bundle.
The main ingredients we use in the proof of Theorem G are the existence of adapted open book decompositions for contact 3-manifolds, due to Giroux, and a detailed study of the dynamics of the Reeb flow of the contact forms constructed in [3].
More precisely, under the assumption H 1 (M; Q) = {0}, we will show that, up to positive stabilizations, each open book decomposition (B, ϕ) of M can be supposed to have binding components of infinite order in H 1 (M; Z). We will then show that this allows us to get hypertight contact forms on M × T 2 using [3]. Finally, (M, ξ) naturally embeds in the contact manifold constructed by Bourgeois as a fiber of the fibration M × T 2 → T 2 given by the projection on the second factor.
We point out that an analogue of Theorem G for any M 3 and with (V 5 , η) tight (and not necessarily hypertight) follows from Bowden et al. [4], where it is shown, using part of the proof of Theorem G above, that the Bourgeois construction [3] on any 3-dimensional manifold results in a contact structure on its product with T 2 which is tight, no matter what the original contact structure and supporting open books are.
As far as Corollary H is concerned, notice that it has recently been generalized to all dimensions in Hernández-Corbato et al. [27] (without any assumption on H 1 (M; Q)), with completely different techniques. More precisely, there the authors deduce such a generalization from [27,Theorem 10], stating that every contact (2n − 1)-manifold embeds with trivial conformal symplectic normal bundle in a Stein-fillable contact (2n + 2m − 1)-manifold. This result relies on the h-principle from Cieliebak and Eliashberg [7], and is an analogue of Theorem G in all dimensions, with less control on the codimension.
Outline In Sect. 2, we give the announced new approach to contact branched coverings, thus proving in particular Proposition A. We also analyze the stability of the weak fillability condition under contact branched covering, thus proving Theorem B.
Section 3 describes the equivalent formulation, based on an idea by Giroux [23], of open book decompositions supporting contact structures in terms of pairs of contact vector fields and it contains the proof of Theorem E. Then, we rephrase and generalize in Sect. 4 the construction by Bourgeois using the notion of contact fiber bundle introduced in Lerman [34]. In particular, we give the definition of Bourgeois contact structures and prove Proposition D.
Section 5 contains the study of the weak fillability of Bourgeois contact structures, hence the proof of Proposition C, and the proof of Theorem F. Lastly, in Sect. 6 we analyze the Reeb dynamics of the contact forms in Bourgeois [3] and we prove Theorem G and Corollary H.

Contact branched coverings
In Sect. 2.1, we give a definition of contact branched coverings that allows to naturally obtain uniqueness statements; we will in particular prove Proposition A stated in the introduction. We point out that the proofs in this section are mainly a reformulation of those in Geiges [16].
An analogous analysis can be carried out in the case of contact fiber sums, but, as it is not necessary for our purposes, it will not be presented here and we redirect the interested reader to Gironella [18,Section 5.3].
Then, Sect. 2.2 contains a proof of Theorem B stated in the introduction, i.e. of the fact that, under some natural assumptions, contact branched coverings of a weakly fillable contact manifold are also weakly fillable.

Definition and uniqueness
Suppose π : V 2n+1 → V 2n+1 is a branched covering map of manifolds without boundary, branched along the codimension 2 submanifold M 2n−1 ⊂ V . Let M 2n−1 be the locus of points of V with branching index > 1 and M its projection π( M). In the following, we will also refer to M 2n−1 as upstairs branching set and to M as downstairs branching set. Consider now η a contact structure on V such that ξ := η ∩ T M is a contact structure on M.
The pullback π * η is a well defined hyperplane field on V , because if we fix a contact form α for η then π * α is nowhere vanishing. Though, π * η is not a contact structure, because at each point p of M we have π * (α ∧ dα n ) | p = 0. Nonetheless, the restriction of π * η to M is a honest contact structure on M. We then want to show that π * η gives a "natural" way to construct contact structures on V .
We start by considering a more general setting. Let Y 2n+1 be a smooth manifold, Z 2n−1 a codimension-2 submanifold and η a hyperplane field on Y . Definition 2. 1 We say that η is adjusted to Z if it is a contact structure away from Z and η ∩ T Z is a contact structure on Z . If that's the case, we also call contactization of η a contact structure ξ such that there is a smooth path {η s } s∈ [0,1] of hyperplane fields, all adjusted to Z , starting at η 0 = η and ending at η 1 = ξ , such that η s is a contact structure for all s ∈ (0, 1].

Proposition 2.2
Let η be a hyperplane field on Y adjusted to Z . Contactizations of η exist and are all isotopic.
Recall from Eliashberg and Thurston [10, Section 1.1.6] that a confoliation is a hyperplane field ζ = ker α that admits a complex structure J : ζ → ζ tamed by dα| ζ , i.e. such that dα(X , J X) ≥ 0 for all vector fields X tangent to ζ . We point out that, in our situation we can talk directly about confoliations adjusted to a certain codimension 2 submanifold. Indeed, if η is a hyperplane field on Y adjusted to a 2-codimensional submanifold Z , then η is in particular a confoliation. This follows from Proposition 2.2 and the following: n∈N be a sequence of contact structures on a compact manifold Y 2n+1 which C 1 -converges to a hyperplane field η on Y . Then, η = ker α admits a complex structure J tamed by dα| η .

Idea of proof (Fact 2.3)
A first attempt could be to take, for each k ∈ N, a complex structure J k on η k = ker α k tamed by dα k | η k (which exists because η k is a contact structure) and to define J as "the limit" of the sequence (J k ) k∈N . However, such a limit does not necessarily exist for a general choice of J k .
The solution is hence to ensure the orthogonality of each of the J k with respect to an auxiliary Riemannian metric g, using the polar decomposition of matrices. By the compactness of the space of vector bundle isomorphisms of T Y preserving the metric g, one can now find a subsequence (J k j ) j∈N converging to a certain J , which is hence a complex structure on η tamed by dα| η . Proposition 2.2 is a consequence of the following lemma, which deals with the more general situation of any number of parameters:

Proof of Proposition 2.2
The existence of contactizations follows directly from Lemma 2.4 with K a point. We then prove their uniqueness up to isotopy.
Given two contactizations ξ, ξ of η, we have by definition two associated paths of adjusted confoliations η t , η t , with t ∈ [0, 1], such that η 0 = η 0 = η, η 1 = ξ , η 1 = ξ and η t , η t contact for t ∈ (0, 1]. Then, the path is a continuous path of adjusted confoliations from η 0 = ξ to η 1 = ξ . Moreover, up to perturbing it smoothly at t = 1 /2, we can suppose that η t is smooth in t. of adjusted confoliations such that η s 0 = ξ , η s 1 = ξ for all s ∈ [0, 1] and such that η s t is contact for s > 0. The subfamily η 1 t is then a path of contact structures from ξ to ξ , and it can be turned into an isotopy by Gray's theorem.

Proof of Lemma 2.4
This proof follows almost step by step the construction and the computations made in Geiges [16,Section 2].
Because of the C 1 -openness of the contact condition, there is an open subset U of K which contains H and such that ξ k is contact for all k ∈ U . We then consider a smooth cut-off function ρ : K → [0, 1], equal to 0 on H and equal to 1 on the complement of U .
Take now an auxiliary Riemannian metric on V and consider the circle bundle S (N M) given by the vectors of norm 1 in the normal bundle N M of M inside V . Let γ be a connection form on S (N M), i.e. a nowhere vanishing 1−form defining a hyperplane field which is transversal to the fibers of the fibration S (N M) → M. Using the natural retraction R 2 \{0} → S 1 , γ can also be seen as a 1−form on N M\M. Moreover, the form r 2 γ , where r is the radial coordinate in N M\M, extends smoothly to N M.
We consider then a non-increasing cut-off smooth function g = g(r ) which is 1 near r = 0 and vanishes for r > 1 and we identify N M with a neighborhood of M inside V . If α k is a smooth K -family of 1-forms defining ξ k , set Here is a positive real constant which will be chosen very small later. Suppose, without loss of generality, that ≤ 1. Remark that ξ s k := α s k is a well defined hyperplane field. Moreover, it is adjusted to M, for all values of s, k.
We then need to show that, for an small enough, ξ s k is actually a contact structure on V for all s > 0, k ∈ K . We can compute where vol is the Riemannian volume form on V and h is a function of p ∈ V , k ∈ K , s ∈ [0, 1], ∈ R + and is polynomial in .
Consider the smooth functions P k , Q k : V → R such that α k ∧ (dα k ) n = P k vol and n rg (r ) . P k is independent of , s and is non-negative everywhere on V for all k. Moreover, P k is positive on the complement of O for all k ∈ K , and even on all Finally, for very small, P k dominates s ρ(k) [Q k + R k ( )] wherever it is positive, because the latter is bounded above in norm (recall we are working with ≤ 1). Hence, by compactness of V \O, P k + s ρ (k) [Q k + R k ( )] is also positive on the complement of O for all k ∈ K , for > 0 small enough.
Coming back to the specific case of branched coverings, the hyperplane field π * η on V is adjusted to M (and is then in particular a confoliation).

Definition 2.5
We say that a contact structure on V is a contact branched covering of η if it is a contactization of π * η and it is invariant under all the diffeomorphisms of V covering the identity of V .
We point out that, by definition of contactization, if η is a contact branched covering of η, the upstairs branching locus M is naturally a contact submanifold in ( V , η). Then, Proposition 2.2 easily implies the following: We point out that, in order to deduce this result from Proposition 2.2, the contactization in the statement Proposition 2.2 has to be invariant under deck transformations of π , as requested in Definition 2.5, and the isotopy has to be among invariant contactizations. From the explicit formula in the proof of Lemma 2.4 above, it's clear that both these conditions can be easily arranged.
Remark also that Proposition A stated in the introduction is a simple consequence of Gray's theorem and the fact that contact branched coverings exist and are unique up to isotopy. Indeed, the [0, 1]-families of hyperplane fields in points 1 and 2 in the statement of Proposition A are automatically adjusted to the upstairs branching locus for small parameters t ≥ 0.

Effects of branched coverings on weak fillings
We will use in this section the notion of weak fillability introduced in Massot et al. [37], in the following computation-friendly form: Definition 2.7 [37] We say that (W , ω) weakly fills (V , η), or that ω weakly dominates ξ , if, for one (hence every) 1-form α defining η, α ∧ (ω + τ dα) n is a positive volume form on V for all τ ≥ 0.
Consider now a branched covering π : W 2n+2 → W 2n+2 of even dimensional manifolds with non-empty boundaries V 2n+1 = ∂ W and V 2n+1 = ∂ W . Let also X 2n be the upstairs branching set, X the downstairs branch set, M, M the boundaries of X , X respectively and π the restriction π | V : V → V . Here's a more detailed version of Theorem B from Sect. 1: Theorem 2.8 Suppose we are in the following situation: (a) η is a contact structure on V and ξ := η ∩ T M is contact on M; (b) η on V is a contact branched covering of (V , η); (c) ω on W weakly dominates η on V ; (d) X is a symplectic submanifold of (W , ω) and it weakly fills (M, ξ).
Then, W admits a symplectic form ω that weakly dominates η on V .
Proof Consider the normal bundle of X inside W and view it as a neighborhood U of X . Similarly for a neighborhood O of M in V . In particular, we have a norm function on U and O, and we can denote by U r , O r the set of vectors of norm less than r . Fix now an arbitrary smooth function f : W → R ≥0 , compactly supported in U 1 , depending only on r , non-increasing in it, and equal to 1 on a neighborhood of X . Denote also by g its restriction to V = ∂ W . Notice that in particular f (r ) = 0, hence g (r ) = 0, for r = 0.
Let now δ be a connection 1-form on the circle bundle S U given by the vectors of norm 1 in U. Denote also by γ the restriction of δ to the sub-bundle S O given by the vectors of norm 1 in O. Notice that γ is in particular a connection form on S O. The explicit formula in the proof of Lemma 2.4 then shows that, up to isotopy, we can assume that the contact branched covering η is the kernel of α := π * α + g(r )r 2 γ , for every smaller than or equal to a certain constant 0 > 0.
As far as the symplectic structure on W is concerned, consider the closed 2-form ω := π * ω + d f (r )r 2 δ on W , where > 0.

Claim 2.9
There is 1 > 0 such that ω is symplectic on W for all 0 < < 1 .
Proof of Claim 2. 9 We have ω = π * ω + 2 f + r f rdr ∧ δ + f r 2 dδ, so that where vol is a volume form on W and h is a smooth function depending on p ∈ W and on > 0. Using that π * ω is symplectic on the complement of X and that the restriction of ω to X is symplectic on X , we can then conclude, as we did in the proof of Lemma 2.4, that ω n+1 > 0 for small enough.
We then want to show that ω weakly dominates η = ker( α ), provided that > 0 is small enough (and in particular such that < := min ( 0 , 1 )). In other words, we need to check that, if is small enough, the following is satisfied: where ω ,V denotes the pullback of ω via the inclusion V → W , i.e.
Using that d α = π * α + 2g + rg rdr ∧ γ + r 2 gdγ , we can compute where vol is a volume form on V and h is a smooth function of p ∈ V , and τ , which is moreover polynomial in and in τ , with deg τ h ≤ n. Denote now by P 0 (τ ) and P 1 (τ ) the polynomials in τ , with coefficients in the ring of functions V → R, defined respectively by the identities Similarly, denote by P 2 (τ, ) the polynomial in τ and given by P 2 (τ, ) = gr 2 h.

Claim 2.10
For all τ ≥ 0, P 0 (τ ) is non-negative everywhere on V and positive away from M.

Proof of Claim 2.10
This follows from the fact that (W , ω) is a weak filling of (V , η) and that π | V is a branched cover with (upstairs) branching locus M.
Notice that we allow = 0 in its statement. The proof of Claim 2.11 will follow after the end of the proof of Theorem 2.8.
According to Claims 2.10 and 2.11, we have that α ∧ ( ω V + τ d α ) n is a positive volume form on O r 0 , for all 0 < < 0 and all τ ≥ 0. (Notice that here = 0.) Then, the following result, whose proof is also postponed, concludes the proof of Theorem 2.8:
Proof of Claim 2. 12 We use again Fact 2.13. Here, S : is the complement of Or0 /2 in V and r 0 , 0 are given by Claim 2.11. Also, P : S × R ≥0 → R is here defined as for ( p, ) ∈ S. Notice that once again we allow = 0.
hence it is positive by Claim 2.10. In particular, P (q,0) has positive leading coefficient and positive minimum m (q,0) for all (q, 0) ∈ K . Moreover, deg τ P (q,0) = n ≥ deg τ P ( p, ) , for all q, p ∈ O c r0 /2 and ∈ [0, 0 ). Fact 2.13 then implies, by compactness of K , that P ( p, ) admits a minimum m ( p, ) , which is moreover positive in a neighborhood of K . More precisely, in Sect. 3.1 we describe how to obtain a pair of contact vector fields with Lie bracket everywhere transverse to ξ from the data of an open book decomposition supporting a contact structure. This is the part of Theorem E that has already been stated in [23]. Section 3.2 deals with the converse, i.e. contains the proof of the fact that it is also possible to recover a supporting open book from such a pair of contact vector fields.

From open books to contact vector fields
We have the following more precise version of the first part of Theorem E: Recall from Giroux [22] that an open book decomposition (B, ϕ) on M is said to support a contact structure ξ if B is a positive contact submanifold and there is a defining 1-form α for ξ such that dα is positively symplectic on the fibers of ϕ : M\B → S 1 . In the statement of Proposition 3.1 above, by "φ : Consider now a smooth function f : M → R >0 , depending only on φ in a nondecreasing way, equal to 1 for φ < /2 and equal to 1 / φ for φ > . Let then φ := f φ : M → R 2 ; in particular, φ defines (B, ϕ) too. Consider also ρ := φ and θ := φ /ρ : M\B → S 1 , and notice that θ = ϕ. We claim that is a volume form on M. Indeed, the first term is non-negative everywhere and positive away from B, because dα is symplectic on the fibers of θ = ϕ, and the second term is positive along B and non-negative everywhere, by choice of f . We then denote by X , Y the contact vector fields associated, respectively, to the contact hamiltonians φ 1 , −φ 2 via the contact form α given in the statement. Because which, using the graded Leibniz rule for the interior product, gives (Here, we adopted the notation Z · f = d f (Z ) for a smooth function f and a vector field Z .) Exchanging the roles of X and Y in Equation (3), we also get As X and Y are contact vector fields for ξ , one also has Then, Eqs. (2)- (5) give Again for dimensional reasons, dα n = 0 on M, so that ι X ι Y dα n = 0, i.e.
Then, Eq. (6) finally becomes As Ω is a volume form on M and α is a positive contact form, [X , Y ] must then be negatively transverse to ξ .

From contact vector fields to open books
We have the following converse to Proposition 3.1.

Proposition 3.2 Let
Then, if we denote X θ := cos θ X + sin θ Y and Y θ := X θ+π/2 for θ ∈ S 1 , we have the following: The rest of Sect. 3.2 is devoted to the proof of the above result, which is a more detailed version of the second part of Theorem E. To improve readability, each claim in this section will be proved right after the conclusion of the part of the proof in which it is used.
Let α be a contact form for ξ and denote by f , g : M → R the smooth functions given by L X α = f α and L Y α = gα respectively (these functions exist because X and Y are contact vector fields). For the proof of point (c) we will need to change this α conveniently.

Proof of Proposition 3.2 (a)
We start by proving that α (X θ ) is somewhere zero, i.e. that Suppose by contradiction this is not the case, i.e. α (X θ ) > 0 without loss of generality. If we define β : Here, for (i) we used the fact that β( by the formula for the exterior derivative of differential forms, and for (ii) we used that dβ(R β , .) = 0 and β(R β ) = 1. Now, β (Y θ ) is a function defined on the compact manifold M, hence it has at least one critical point. This contradicts Eq. (7) and the fact that β ([X θ , Y θ ]) < 0, thus proving that α (X θ ) is somewhere zero.
In order to conclude the proof, it is then enough to show that Indeed, this tells that α (X θ ) : M → R is transverse to {0} ⊂ R, i.e. Σ θ is a smooth hypersurface, and that, more precisely, the contact vector field Y θ is transverse to Σ θ , i.e. the latter is ξ −convex. We then proceed to prove Eq. (8).
Using the formula for the exterior derivative, we compute Also, by Fact 3.3 there are f θ , g θ : M → R such that Now, evaluating these last two equations respectively on Y θ and X θ gives Substituting inside Eq.
We point out a direct consequence of Eq. (8) and another lemma, which we will both need later.
In particular, along Σ θ ∩Σ θ+π/2 (which we will show below to be independent of θ and denote by K ),

Proof of Proposition 3.2.(b)
Σ θ ∩ Σ θ is non-empty because at the previous point we showed that Y θ is a contact vector field transverse to Σ θ , and we know from convex surface theory that {α(Y θ ) = 0} ∩ Σ θ ⊂ Σ θ is a dividing set for the characteristic foliation Σ θ (ξ ), and that dividing sets are always non-empty. This last statement is a consequence of the fact that there are no exact symplectic forms on closed manifolds due to Stokes' identity.
Finally, we prove that K is a codimension 2 submanifold of M. For that, it is enough to find a vector field tangent to Σ θ and transverse to Σ θ at every point of K . Because K = Σ θ ∩ Σ θ is independent of θ, θ , we can suppose that θ = 0 and θ = π/2. This being said, the contact vector field Y serves well to our purposes. In fact, in the proof of point (a), we showed that Y = X π/2 is transverse to Σ 0 ; moreover, it is also tangent to Σ π/2 , because X θ is tangent to Σ θ according to Lemma 3.5.
It now only remains to prove Proposition 3.2.(c). We use the following: Notice that Hypothesis (iii) implies that there is a very small > 0 such that dα is a symplectic form on each fiber of the restriction of ϕ to is the disk of radius 1 − in R 2 . We then search the function f of the following form: f is a smooth function that depends only on the radius coordinate r on D 2 inside N , non-increasing in r , which is equal to 1 on M\K × D 2 1− /2 and equal to 1 + e −kr 2 on K × D 2 1− , where k > 0 is a constant yet to determine. We can then compute By Hypothesis (ii), the form rdr ∧ dϕ ∧ α ∧ dα n−2 is positive on N , hence on K × D 2 1− /2 , and dϕ ∧ dα n−1 is bounded above in norm, even if we don't know its exact sign. This means that for k > 0 big enough, the second form will dominate the first, i.e. their sum will still be positive.
It then remains to study the sign on the open set K × D 2 1− /2 \D 2 1− . Here, the situation is easy because dϕ ∧ dα n−1 is positive and − ∂ f ∂r dr ∧ dϕ ∧ α ∧ dα n−2 is non-negative (remember f is a non-increasing function of r in this set), so their sum is also positive.
We are now ready to give a proof of the last part of Proposition 3.2. In order to improve the readability, the latter is split in three main claims, which are then proved separately right after the end of the proof of Proposition 3.2.

Proof of Proposition 3.2.(c) Consider the smooth map
In other words, Claim 3.7 tells that (K , ϕ) is an open book decomposition of M. We then need to prove that it moreover supports ξ . Notice that this is enough in order to prove point (c) of Proposition 3.2, because the ϕ in point (c) is just obtained from the ϕ of Claim 3.7 by post-composing with the rotation of S 1 of angle − π 2 , so they have the same pages.
Consider on K , F θ the orientations such that φ −1 (0) = K , ϕ −1 (θ ) = F −θ−π/2 as oriented manifolds. To show that (K , ϕ) is adapted to ξ , we then need to verify that ξ ∩ T K is a positive contact structure on K and that there is a contact form defining ξ whose differential is a positive symplectic form on each F θ . Thus, Lemma 3.6 together with the following two claims conclude the proof of Proposition 3.2.(c):

Claim 3.8 Let Ψ be the map defined by
where ψ 1 Z denotes the time-1 flow of the vector field Z on M and D 2 δ is the 2-disk of radius δ in R 2 . Then, for δ > 0 sufficiently small, we have the following: , then we have the following commutative diagram, where ν is the composition of the projection on D 2 δ \{0} and the natural angle

Claim 3.9
Let N be the neighborhood of K given by Claim 3.8. Then there is a contact form α defining ξ such that: (i) α induces a positive contact structure on each submanifold K z of N ; (ii) dα is a positive symplectic form on the fibers of ϕ| M\N .
We now prove the claims used in the above proof.

Proof of Claim 3.7 Clearly
In other words, φ is transverse to the origin of R 2 and the oriented couple (Y , X ) gives the positive coorientation of φ −1 (0).
Let's finish with point (iii). Because the contact condition is open, up to shrinking δ, it is enough to prove that K 0 = Ψ (K × {0}) is a positive contact submanifold. This follows from general results from Giroux [19]: indeed, X θ defines the characteristic foliation of Σ θ , and K is transverse to it.
Proof of Claim 3. 9 We search for a function f such that α := f α satisfies dϕ ∧ d α n−1 > 0 on M\Int(N ). We start by computing Let now > 0 be such that { φ < 2 } ⊂ N and choose a smooth non-increasing function f of φ , equal to 1/ on the set { φ < } and equal to 1/ φ on the set Moreover, recalling that φ = (α (X ) , −α (Y )), one has In particular, Notice then that the right hand side is exactly the same (up to a factor n) as the one of Eq. (2) in the proof of Proposition 3.1. Hence, the exact same computations made in that proof tell us that is negatively transverse to ξ by hypothesis, so d α is positively symplectic on the fibers of ϕ| M\N , as desired.

Bourgeois structures as contact fiber bundles
The aim of this section is to generalize the construction due to Bourgeois using the notion of contact fiber bundles introduced in Lerman [34].
More precisely, we start by recalling the Bourgeois construction in Sect. 4.1. In Sect. 4.2 we recall the definitions and the main properties of contact fiber bundles. Section 4.3 describes how to effectively compare two of them, which is then used to generalize the construction by Bourgeois [3]. In particular, in Sect. 4.4 we take a general fibration admitting a flat contact connection and we consider on it two nontrivial subclass of all its contact connections. The first class is characterized in terms of deformations to the flat contact connection, in a flavor similar to the notion of contactizations introduced in Definition 2.5. The second one, subclass of the first, is a direct generalization of the examples from [3] in the setting of contact fiber bundles and is presented in Sect. 4.5. There Proposition D from the introduction is also proved using the results from Sect. 3. Lastly, in Sect. 4.6 we study the stability of the first class under the operation of contact branched covering.

The Bourgeois construction
Using the notion of open book decompositions for contact manifolds (M 2n−1 , ξ) from Giroux [22], Bourgeois constructs in [3] explicit contact structures on M × T 2 . More precisely, the main statement of [3] can be rephrased as follows: We point out that the condition γ ∧ dγ n−2 ∧ dφ 1 ∧ dφ 2 ≥ 0 in point (a) of Theorem 4.1 is independent of the choice of form γ defining ξ . Indeed, it is equivalent to the fact that ξ induces by restriction a contact structure on φ −1 (z), for each z regular value of φ.
Notice, moreover, that the contact form α in point (b) clearly induces the original contact structure ξ on each fiber M × {pt} of π : M × T 2 → T 2 . φ = (φ 1 , φ 2 ) satisfies point (a) of Theorem 4.1, then, for all > 0, the same is true for φ = ( φ 1 , φ 2 ). In particular, the 1-forms α := β+ φ 1 dθ 1 − φ 2 dθ 2 always define positive contact structures by point (b) of Theorem 4.1, which are moreover all isotopic by Gray's theorem. Notice that α 0 = β defines the hyperplane field ξ ⊕ T T 2 , which is not a contact structure on M × T 2 , but still defines a contact structure on each fiber of the projection π : M × T 2 → T 2 .

Generalities on contact fiber bundles
We recall in this section the notion of contact fiber bundle introduced by Lerman in [34], focusing in particular on their description using contact connections. We specialize here to the case of fiber bundles over (closed) surfaces as this will be the case we are interested in for the following sections.
Vice versa, the data of ξ f ib := η ∩ ker(dπ) = (ξ b ) b∈Σ and H obviously allows to restore the hyperplane field η. For this reason, we introduce the following auxiliary object: Recall also that any Ehresmann connection H on a fiber bundle π : V → Σ is equivalent to a fiber-wise projection ω of T V onto ker(dπ), i.e. to a connection form ω ∈ Ω 1 (V ; ker(dπ)), defined on V and with values in ker(dπ) ⊂ T V , such that ω • ω = ω and ω| ker(dπ) = Id | ker(dπ) . More precisely, given H, each vector field Z on V can be uniquely decomposed as Z = Z h + Z v , where Z h is horizontal, i.e. everywhere tangent to H, and Z v is vertical, i.e everywhere tangent to the fibers of π . Then, for each Z vector field on V , one can define ω(Z ) := Z v . Vice versa, given an ω ∈ Ω 1 (V ; ker(dπ)) as above, H := ker(ω) is an Ehresmann connection.

Comparing contact fiber bundles
In this section, we are going to compare two contact fiber bundles having the same underlying structure of fiber bundles with contact fiber.
We start by showing that, given a fiber bundle with contact fibers (π : V → Σ, ξ f ib ), one can naturally associate to it a vector bundle X f ib (V , ξ f ib ) → Σ having as fiber, over a point b ∈ Σ, the Frechet vector space of contact vector fields for (M b , ξ b ). We invite the reader to consult Kriegl and Michor [33] for the foundations of analysis on manifolds locally modeled on Frechet vector spaces. We can explicitly construct X f ib (V , ξ f ib ) → Σ as follows.
The fiber bundle with contact fibers (π : V → Σ, ξ f ib ) is equivalent to the following data: an open cover (U i ) i∈I of Σ, trivial bundles with contact fibers ξ) is the space of contact vector fields on (M, ξ)), and by transition functions Remark 4. 6 We proposed here a very direct construction of the the vector bundle X f ib (V , ξ f ib ) → Σ, in order to keep this presentation simple and self contained. This being said, X f ib (V , ξ f ib ) → Σ can also be interpretated as an adjoint bundle as follows.
Analogously to [33,Paragraph 44.4] defining the principal ((nonlinear) frame bundle of a smooth fiber bundle, one can associate to a given fiber bundle with contact fibers (π : V → Σ, ξ f ib ) (and with model fiber (M, ξ)) its ((nonlinear) contact frame bundle E → Σ, which is a natural principal Diff(M, ξ)-bundle associated to (π, ξ f ib ). Then, up to isomorphism of vector bundles over Σ, We now want to show that the space of contact connections on a given fiber bundle with contact fibers (π : V → Σ, ξ f ib ) has naturally the structure of an affine space over the vector space Ω 1 (Σ; X f ib (V , ξ f ib )) of 1-forms defined on Σ and with values in the vector bundle X f ib (V , ξ f ib ) → Σ.
Let H 0 be a reference contact connection on (π : V → Σ, ξ f ib ). For simplicity, we call (π : V → Σ, ξ f ib , H 0 ) a referenced fiber bundle with contact fibers in the following. Denote also by ω 0 the connection form associated to H 0 .
Notice that, given a point p ∈ V and a vector v ∈ T p V , the vector ω 0 (v) − ω(v) is tangent to the fiber M p of π . Indeed, both ω and ω 0 are with values in ker(dπ). Moreover, it only depends on the vector u : ), which, by linearity of ω and ω , gives the desired equality. In other words, given a point b ∈ Σ, a vector u ∈ T b Σ and a vector field Z b on the fiber The last thing to show is then that A has actually values in the vector bundle of contact vector fields on the fibers of the fiber bundle with contact fiber (π : are the lifts of u which are horizontal for, respectively, H 0 and H. Then, Lemma 4.4 tells that both the flows ψ t 0 and ψ t of, respectively, With a little abuse of notation, for each b ∈ Σ and u ∈ T b Σ, we will just denote by u and u 0 the vector fields on M p given by the lifts of u which are horizontal for, respectively, H and H 0 , i.e. the (Z b ) h and (Z b ) h 0 above.

Definition 4.7
For any referenced fiber bundle with contact fibers (π : V → Σ, ξ f ib , H 0 ) and any contact connection H on (π : V → Σ, ξ f ib ), the 1-form A ∈ Ω 1 (Σ; X f ib (V , ξ f ib )) defined as above is called potential of H with respect to H 0 .

Remark 4.8
The fact that the space of contact connections on a given (π : V → Σ, ξ f ib ) is an affine space over Ω 1 (Σ; X f ib (V , ξ f ib )) has also the following more theoretical interpretation, in the spirit of Remark 4.6.
Analogously to what is explained in [33,Paragraph 44.5] in the case of smooth fiber bundles, connection forms on (π, ξ f ib ) correspond bijectively to principal connections on the (nonlinear) contact frame bundle E → Σ. Now, the space of principal connections on a principal bundle has naturally an affine structure over the vector space of 1-forms on the base with values in the adjoint bundle.
The potential A also allows to compare the curvature of H with that of H 0 . In order to explain how, we need to introduce two more objects.
Firstly, we show that the connection H 0 on (π : More precisely, given a vector field U on Σ and a section σ of X f ib (V , ξ f ib ) → Σ, consider the H 0 -horizontal lift U 0 of U and the vector field σ on V defined by σ ( p) = σ π( p) ( p), where σ π( p) denotes the image of π( p) via σ . Notice that their Lie bracket [ U 0 , σ ] is contained in ker(dπ) and is, moreover, a contact vector field on each fiber of π : V → Σ. An explicit computation also shows that [ U 0 , σ ] is C ∞ (Σ)-linear in U and satisfies the Leibniz rule in σ . In other words, ∇ U σ := [ U 0 , σ ] gives a well defined covariant derivative.
We point out that ∇ is flat. Indeed, the curvature F ∈ (X f ib (V , ξ f ib )) → Ω 2 (Σ; X f ib (V , ξ f ib )) of ∇ is, by definition, given by F(U , W )σ = ∇ U ∇ W σ − ∇ W ∇ U σ − ∇ [U ,W ] σ for all U , W vector fields on Σ and σ ∈ (X f ib (V , ξ f ib )). A direct computation using the Jacobi identity for the Lie bracket of vector fields on V then shows that F = 0, as desired.
The second object we need to introduce is the covariant exterior derivative d ∇ : Ω p (Σ; X f ib (V , ξ f ib )) → Ω p+1 (Σ; X f ib (V , ξ f ib )) naturally induced by ∇. As explained for instance in Kriegl and Michor [33,Paragraph 37.29], d ∇ is characterized by the formula for all ω ∈ Ω p (Σ; X f ib (V , ξ f ib )) and U 1 , . . . , U p+1 vector fields on Σ. Here, the notation Z denotes the fact that the vector field Z is omitted in the argument.
Notice that the flatness of ∇ implies that d 2 ∇ = 0. This is a consequence of the [33,Paragraph 37.29] for a proof of this identity). In other words, d ∇ is a differential on the chain complex Ω * (Σ, X f ib (V , ξ f ib )).
We are now ready to give an expression for the curvature R of H in terms of the curvature R 0 of H 0 .
Let X , Z be vector fields on V , and denote, as before, by  Z ). Finally, remark that X h 0 and Z h 0 are just the H 0 -horizontal lifts U 0 and W 0 of U and W respectively. Hence, by Eq. (13), we also have d Putting all the pieces together, we then get: for all X , Z vector fields on V , ( 1 4 ) As it will be useful for the following section, we also point out the following fact. Fix a covariant derivative ∇ Σ on the tangent bundle of Σ. Then, the ∇ introduced above naturally extends to a unique map satisfying the following property: for all W 1 , . . . , W p , U vector fields on Σ,

Flat contact bundles and contact deformations
Here we call flat contact bundle any referenced fiber bundle with contact fibers π : V → Σ, η 0 = ξ f ib ⊕ H 0 such that H 0 satisfies R 0 = 0. The first reason why flat contact bundles are interesting is because they admit a "nice" presentation in terms of their monodromy.
Let now π Σ : Σ → Σ be the universal cover of Σ, and consider the map F : M × Σ → V covering π Σ given by F(q, [γ ]) := ρ c (q). Here, we see Σ as the set of arcs γ on Σ starting at b, up to homotopy. The differential of F sends the connection {0} ⊕ T Σ of M × Σ → Σ to the connection H 0 of π : V → Σ, and the contact structure ξ ⊕ {0} on the fiber of Σ × M over p ∈ Σ to the contact structure ξ p of the fiber M p of V over p = π Σ ( p).
Moreover, if we denote by ρ the diagonal action of π 1 (Σ) on M × Σ induced by the natural action on the second factor and by the action ρ on the first factor, In other words, f gives the desired "nice" presentation of (π, ξ f ib , H 0 ) in terms of the monodromy ρ.
The second reason for restricting to the class of flat contact bundles is the following: using the notion of potential from Sect. 4.3, given a flat π : V → Σ, ξ f ib , H 0 , we can give an explicit criterion that tells whenever any other contact bundle on it (inducing the same ξ f ib , hence described by a contact connection H) defines a contact structure on the total space V . More precisely, using Eq. (14) (with R 0 = 0), we can rephrase Lerman [34, Proposition 3.1] in the following computational-friendly way:

a contact connection H with potential A gives a contact structure η on the total space if and only if, for all b in Σ and all oriented basis
Recall that a contact vector field is called negative if it is everywhere negatively transverse to the contact structure.
In Sect. 4.5, we will use Proposition 4.9 to study the following objects: Definition 4.10 Let π : V → Σ, η 0 = ξ f ib ⊕ H 0 be a flat contact bundle. We say that a contact fiber bundle η on π is a contact deformation of η 0 if it defines a contact structure on the total space V and if there is a smooth family of contact fiber bundles (η s ) s∈[0,1] starting at η 0 , ending at η 1 := η and satisfying: 1. for all s ∈ [0, 1], η s ∩ ker(dπ) = ξ f ib ; 2. for all s > 0, η s defines a contact structure on V .

By point 1, a contact deformation is equivalent to a path of contact connections H s interpolating between H 0 and H.
We point out that this definition is "non-empty", i.e. given a flat contact fiber bundle (π : V → Σ, η 0 = ξ f ib ⊕ H 0 ), not all the contact fiber bundles for the same underlying fibration with contact fibers (π, ξ f ib ) are contact deformations of η 0 .
For instance consider the contact fiber bundle structure on T 3 = S 1 × T 2 which is given by the kernel η of α = dθ + cos(θ )dx − sin(θ )dy, where θ ∈ S 1 and (x, y) are coordinates on T 2 . This contact fiber bundle structure is a contact deformation of the flat contact fiber bundle structure given by η 0 = ker (dθ ): the deformation is given by We point out that, by Giroux [20, Lemma 10], η admits prelagrangian tori only in the isotopy class of { pt} × T 2 . Take now a diffeomorphism ψ of T 3 sending (θ, x, y) to (θ + x, x, y). Then, ψ * η is still transverse to the S 1 factor, hence it is a contact fiber bundle on the chosen fibration, and obviously it still defines a contact structure on the total space. Though, it has prelagrangian tori in an isotopy class which is different from that of the prelagrangian tori of η. According to Vogel [43,Proposition 9.9], this implies that φ * η cannot be a contact deformation of We also remark that, even though the above definition is of a very similar flavor to Definition 2.5, the objects they define behave differently. For instance, there is no uniqueness up to isotopy for contact deformations.
Indeed, if we take again the fiber bundle π : T 3 = S 1 × T 2 → T 2 where we see the fibers as contact manifolds S 1 , ker (dθ ) , then the flat contact bundle defined by η 0 = ker(dθ) on π actually admits as contact deformations every contact structure on T 3 defined by α n := dθ + cos(nθ)dx − sin(nθ)dy. Though, these are not isotopic one to the other as contact fiber bundles defining contact structures on the total space. Indeed, they are not even isomorphic as contact structures on T 3 , because they have different Giroux torsion (see Giroux [20]).

Bourgeois' construction revisited
The aim here is to use what we defined in the previous sections to generalize the construction by Bourgeois recalled in Sect. 4.1. Let's start by reformulating it with this new terminology.
Let (M 2n−1 , ξ) be a contact manifold and (π : M × T 2 → T 2 , ξ ⊕ T T 2 ) be a flat contact bundle. Once an open book decomposition (B, ϕ) supporting ξ on M and a particular adapted contact form β are fixed, consider a function φ = (φ 1 , φ 2 ) : M → R 2 as in the statement of Theorem 4.1. Now take the contact vector fields X and Y on (M, ξ) associated, respectively, to the contact hamiltonians φ 1 and −φ 2 via the contact form β, and consider the potential A := −X ⊗ dθ 1 − Y ⊗ dθ 2 , where (θ 1 , θ 2 ) are coordinates on T 2 = R 2 /Z 2 . A direct computation shows that the contact fiber bundle associated to A is the kernel of the contact form α = β + φ 1 dθ 1 − φ 2 dθ 2 given by Theorem 4.1.
Notice also that T 2 has a natural (flat and) torsion-free ∇ T 2 , inherited by the standard ∇ R 2 on R 2 , and such that ∇ T 2 ∂ θ i ∂ θ j = 0. Then, because X and Y are independent from the point of T 2 in the product M × T 2 , it is easy to check that A is ∇-parallel, i.e. that ∇ A = 0 (see Sect. 4

.3 for the definition of ∇).
We can then give the following definition: be a flat contact bundle, and consider a torsion-free covariant derivative ∇ Σ on Σ. We call strong Bourgeois contact structure each contact structure on the total space V given by a contact fiber bundle structure η on V with ∇-parallel potential A.
Notice that, generalizing Remark 4.2, each strong Bourgeois contact structure η is a contact deformation of the underlying flat contact bundle η 0 . More precisely, if A is the potential associated to η with respect to η 0 , the deformation is just given by the family of potentials (s A) s∈ [0,1] .
We also point out that Definition 4.11 is a non-trivial generalization of the Bourgeois construction, i.e. the class of strong Bourgeois contact structures is not exhausted by the examples on M × T 2 from Bourgeois [3]:

Proposition 4.12
There is a flat contact fiber bundle (π : V → T 2 , η 0 ) that admits a strong Bourgeois contact structure (for the standard flat ∇ T 2 on T 2 ) and is non-trivial, i.e. not isomorphic, as flat contact fiber bundle, to (π :  Let's also denote by β a G-invariant contact form for ξ on M such that dβ is symplectic on the fibers of ϕ, and by η 0 the flat contact bundle induced on π : M × ρ T 2 → Σ by the flat contact bundle ξ ⊕ T R 2 on M × R 2 → Σ. Here, ρ is the action of π 1 (T 2 ) on M × R 2 given by ρ on the first factor and by the natural action on the universal cover R 2 → T 2 on the second factor.

Proof of lemma 4.13
The form β+φ 1 dθ 1 −φ 2 dθ 2 on M ×R 2 defines a contact structure η on M × R 2 . This follows from the same computations as those in [3]. Moreover, it is invariant under the action ρ. Hence, it induces a well defined contact structure η on the codomain M × ρ T 2 of the quotient map q : Finally, we need to prove that η is indeed a strong Bourgeois contact structure. Being ∇-parallel is a local condition, hence it is enough to prove that the potential A of η = q * η is parallel with respect to the connection q * ∇, pullback of ∇ to M ×R 2 via q. Now, an explicit computation gives A = X ⊗dx +Y ⊗dy, where (x, y) are coordinates on R 2 and X , Y are contact vector fields on (M, ξ) with contact hamiltonians (via β) respectively −φ 1 , φ 2 . Hence, X and Y , as functions from M × T 2 to the space X(M, ξ) of contact vector fields for (M, ξ), are independent of the coordinates on T 2 . This implies that A is (q * ∇)-parallel.
Proof of Proposition 4. 12 We recall that van Koert and Niederkrüger [31] exhibited an explicit open book decomposition for each Brieskorn manifold W 2n−1 k ⊂ C n+1 , with supporting form α k . We invite the reader to consult their article for the details. What's important for us is that the adapted open book decomposition is defined by a map φ : W 2n−1 k → R 2 which is invariant under the action of the subgroup SO(n) of the group of strict contactomorphisms for the strict contact manifold (W 2n−1 k , α k ). More precisely, if (z 0 , . . . , z n ) are the coordinates of C n+1 , the action of SO(n) on W 2n−1 k ⊂ C n+1 is given by the identity on the z 0 -coordinate and by matrix multiplication on  (z 1 , . . . , z n ). For simplicity, we denote here the couple (W 2n−1 k , α k ) by (M, β). Let now ρ : π 1 T 2 → SO(n) be defined by ρ(a, b) = a · f for each (a, b) ∈ Z 2 = π 1 T 2 , where f is any element of SO(n) of order 2. Then, Lemma 4.13 tells us that the η on M × ρ T 2 , induced by ker(β + φ 1 dθ 1 − φ 2 dθ 2 ) on M × R 2 , is a strong Bourgeois contact structure on the flat contact bundle (π : The only thing left to show is that (π : The connection H 0 associated to η 0 defines a foliation F 0 by tori T 2 on V , which is also transverse to the fibers of π : V → T 2 . Moreover, because of our particular choice of ρ : π 1 (T 2 ) → SO(n), each leaf L of F 0 intersects every fiber twice. Now, the connection T T 2 on the trivial bundle p : M × T 2 → T 2 gives a foliation F 1 with leaves { pt} × T 2 , which only intersects each fiber once. In particular, there is no isomorphism Ψ of fiber bundles (equipped with connections) over T 2 between (π : M × ρ T 2 → T 2 , H 0 ) and ( p : M × T 2 → T 2 , {0} ⊕ T T 2 ). Indeed such a Ψ would send F 0 to F 1 , but this contradicts the fact that their leaves intersect the fibers a different number of times.
Even though strong Bourgeois contact structures are a non-trivial extension of the examples from [3], we believe that this class of contact structures is still, in a certain sense, too "rigid". The first (somewhat philosophical) reason is that their definition depends on the choice of a torsion-free ∇ Σ on the base Σ, that is an auxiliary data with respect to the underlying flat contact bundle structure. The second (much more concrete) reason is given by the following converse to Lemma 4.13:

Proposition 4.14 Let (M, ξ) be a contact manifold and consider the flat contact bundle
Let also η be a strong Bourgeois contact structure on (π, η 0 ), equipped with the standard flat ∇ T 2 on T 2 . Then, there is a Im(ρ)-invariant function φ : M → R 2 that defines an open book decomposition of M supporting ξ , and such that the given η is the result of the application of Lemma 4.13 with these choices of (M, ξ), ρ and φ (and G = Im(ρ)).
by Lemma 4.13 are actually all the possible strong Bourgeois contact structures on (π, η 0 ), equipped with the standard flat ∇ T 2 .
Proof of Proposition 4.14 Consider the following natural commutative diagram: with Q and q the natural quotients maps (see Sect. 4.4). This induces the commutative diagram where X(M, ξ) denotes the space of contact vector fields of (M, ξ) and G is the restriction of d Q to the fibers of pr 2 . Consider the pullbacks ∇ and A of ∇ and A via (G, q). Because of the particular choice of ∇ T 2 , the fact that A is ∇-parallel translates to the fact that A is R 2 -invariant. A straightforward computation then shows that We then propose the following generalization of Definition 4.11.

Definition 4.15
Let (π : V → T 2 , η 0 = ξ f ib ⊕ H 0 ) be a flat contact fiber bundle. We call Bourgeois contact structure each contact structure on the total space V given by a contact fiber bundle structure η on V → Σ with potential A that is d ∇ -closed, i.e. such that d ∇ A = 0.
Notice that the condition d ∇ A = 0 in Definition 4.15 above is actually the same as the condition 1 R → 0, for → 0, used to introduce Bourgeois contact structures in Sect. 1.
Indeed, according to Eq. (14), the curvature R of a Bourgeois contact structure is just d ∇ A + [A, A], where A is its potential. In particular, this curvature has two terms which behave differently under rescaling A → A, for > 0. The term d ∇ A is rescaling linearly in , whereas [A, A] is rescaling quadratically in it. Then, if we denote by R the curvature associated to the connection H of potential A with respect to η 0 , the condition d ∇ A = 0 is equivalent to the fact that 1 R → 0 for → 0.
We point out that, as announced before Definition 4.15 (and as the terminology suggests), strong Bourgeois structures are also Bourgeois structures. Indeed, by Equation (15) for all vector fields U , W on T 2 . Using the fact that A is ∇-parallel, we compute is the torsion of ∇ T 2 , which is by assumption zero.
We point out, however, that a direct analogue of Proposition 4.14 is not true for Bourgeois contact structures. For instance, given a strong Bourgeois contact structure with potential A on any flat contact bundle (π : is any other d ∇ -closed potential, not necessarily inducing a contact structure on the total space V (these are not hard to find, for instance in the case of (π : M × T 2 → T 2 , ξ ⊕ T T 2 )), then, for > 0 small enough, the perturbation A + A 0 gives a Bourgeois contact structure η . Though, such η 's do not necessarily come from the construction in Lemma 4.13. In other words, the class of Bourgeois contact structures is bigger than the one of strong Bourgeois contact structures.
This being said, the motivation behind Definition 4.15 doesn't only consist in the fact that it's a strict generalization of Definition 4.11. Indeed, we now show that the condition d ∇ A = 0 above, while being general enough to be satisfied by a class of contact structures strictly larger than those given by the construction in Lemma 4.13, is also strong enough to ensure some nice properties from the points of view of contact deformations, weak fillability and adapted open book.
We start by showing that each Bourgeois contact structure η is in particular a contact deformation of the underlying flat contact bundle η 0 . Indeed, we have the natural path of contact bundle structures (η t ) t∈[0,1] that is given by the potential A t := t A with respect to H 0 , where A is the potential of η. This has the wanted starting and ending points and gives a contact structure η t for t > 0, according to Proposition 4.9, because d ∇ A t = td ∇ A is zero and, for any b ∈ Σ, This property is a generalization of the fact that strong Bourgeois structures (which includes the examples in [3]) are contact deformations of the trivial flat contact bundle on M × T 2 .
The study of weak fillability of Bourgeois contact structures is postponed to Sect. 5.1. There, Proposition 5.1 states that if (M, ξ) is weakly fillable then a Bourgeois contact structure η on the flat contact bundle (π : M × T 2 → T 2 , ξ ⊕ T T 2 ) is weakly fillable too. (Notice that the particular case of the contact structures obtained as in [3]  As far as adapted open book decompositions are concerned, we prove the following: given a Bourgeois contact structure η on the flat contact bundle (π : V → Σ, η 0 ), we can "naturally" associate to each point b of Σ an open book decomposition of the fiber M b supporting the contact structure ξ b . This crucially relies on Proposition 3.2 on pairs of contact vector fields and supporting open book decomposition. In order to give a precise statement, let's introduce some notations. Consider a smooth contact bundle η on X → Y , where X is not assumed to be closed. Denote by Λ the space of maps Φ : X → R 2 such that, for each y ∈ Y : i. the restriction φ y := Φ| π −1 (y) : which is an open book decomposition of π −1 (y) according to points i., ii., is moreover adapted to the contact structure η ∩ T π −1 (y) .
Notice that this space Λ comes endowed with a natural C ∞ -topology induced by that on the space of functions X → R 2 in which it is contained. Consider then the quotient Λ /∼ of Λ by the relation ∼ defined as follows: Notice that Λ /∼ inherits a natural topology as quotient of the topological space Λ. We then call smooth Y −family of open books on X (adjusted to η) each element of Λ /∼. Remark also that if we have a contact bundle η on a smooth fiber bundle π : X → Y and f : Z → Y is a smooth map, we can define the pullback contact bundle f * η on the pullback bundle where pr X , pr Z are the projections of Z × X on the first and second factors respectively. This f * η is indeed a contact bundle because its trace on each fiber (pr Lets now go back to the specific case of Bourgeois contact structure η on a flat contact bundle From the above result, we can deduce a more precise version of Proposition D stated in the introduction:  comes from an isotopy as described in the statement, this actually means that (K 0 , ϕ 0 ) is isotopic to (K 1 , ϕ 1 ). In other words, The last statement about the construction by Bourgeois follows directly from the definition of Ψ η and from point (c) of Proposition 3.2. Indeed, let η = ker(β +φ 1 dθ 2 − φ 2 dθ 2 ) a the Bourgeois contact structure on the flat contact bundle (π : M × T 2 → T 2 , ξ ⊕ T T 2 ) given by Theorem 4 .1 starting from open book (B, ϕ) of M adapted to ξ . As already observed in the beginning of Sect. 4.5, we can compute that A ∂ θ 1 and A ∂ θ 2 are respectively the contact vector fields on (M, ξ) of contact hamiltonians −φ 1 and φ 2 (via β), with φ = (φ 1 , φ 2 ) defining (B, ϕ). Then, we can see that point (c) of Proposition 3.2 with X := A ∂ θ 1 and Y := A ∂ θ 2 gives exactly the open book (B, ϕ ), where ϕ is obtained from ϕ by composition with the antipodal map S 1 → S 1 .
We now derive Proposition 4.16 as a consequence of Proposition 3.
Finally, we point out a somehow peculiar property: on the trivial flat contact bundle (π : M × T 2 → T 2 , ξ ⊕ T T 2 ) there is a natural way to associate a strong Bourgeois contact structure to each Bourgeois contact structure, in such a way that it gives a left inverse to the natural inclusion i : Let's give a precise statement.
The potential A of a contact bundle η, with respect to the natural flat connection {0} ⊕ T T 2 ⊂ T M × T 2 on π : M × T 2 → T 2 , can actually be seen as a 1-form defined on T 2 and with values in the vector space of contact vector fields of (M, ξ), thanks to the canonical identification of each fiber of π with M. Moreover, an explicit computation gives that A is ∇-parallel (with respect to the natural flat ∇ T 2 on T 2 ) if and only if η is invariant under the natural T 2 -action on the total space M × T 2 . Using this equivalence, we get: Proposition 4.18 Let η be a Bourgeois contact structure on the flat contact bundle (π : M × T 2 → T 2 , ξ ⊕ T T 2 ) and denote by A its potential. The average A of A via the natural T 2 -action is the potential of a strong Bourgeois contact structure with respect to the natural flat ∇ T 2 on T 2 .
In particular, taking the average of the potential gives a well defined map F : In other words, the space of Bourgeois contact structures on (π, ξ ⊕ T T 2 ) retracts to its subspace of strong Bourgeois contact structures. It is not clear to the author whether this is actually a deformation retract or not.
Proof of Proposition 4.18 As remarked above, because A is T 2 -invariant, it is also ∇parallel, hence satisfies d ∇ A = 0. By Proposition 4.9, what we need to show is then that [A, A] is with values in the negative contact vector fields for (M, ξ). Let's start by analyzing this condition more explicitly.
Write A = X ⊗dx+Y ⊗dy, with (x, y) coordinates on T 2 = ( R /2πZ) 2 and X , Y a T 2family of vector fields on M parametrized smoothly by (x, y). Here, for all (x, y) ∈ T 2 , [X , Y ] is everywhere negatively transverse to ξ . Because A = X ⊗ dx + Y ⊗ dy, it is then enough to show that their averages X , Y are such that [X , Y ] is also everywhere negatively transverse to ξ .
We point out that, if Z , W are T 2 -parametric vector fields on M, it is not true in general that the T 2 -average of [Z , W ] is equal to the Lie bracket of the averages of Z and W . This being said, what we are going to show here is that this is actually true for X , Y , because of the additional condition d ∇ A = 0.
Notice that X , Y can be seen as smooth functions from T 2 to the space of vector field on M, which has a natural structure of vector space over R. As such, they both admit a complex Fourier series expansion where, for all m, n, h, k ∈ Z, X m,n , Y h,k are complex vector fields on M, i.e. sections of the complexified tangent bundle T M ⊗ R C → M. Because X , Y are actually real, we have the following condition on the coefficients: where X m,n , Y h,k denote the complex conjugates of X m,n and Y h,k respectively. The condition d ∇ A = 0 also gives some information on the Fourier coefficients. More precisely, Indeed, we can explicitly compute where (i) comes from the fact that ∂ x and ∂ y commute (and from the definition of ∇), and (ii) follows from the expression in coordinates of the Lie bracket.
A straightforward computation shows that the right condition in Eq. (18) is equivalent to: mY m,n = n X m,n for all m, n ∈ Z .
Notice now that the averages of X and Y are, respectively, X 0,0 and Y 0,0 , which are in particular real vector fields on M. To avoid confusion with the conjugation, we will hence drop the notation X and Y for the averages and just denote them by X 0,0 and Y 0,0 instead. Let for r , s ∈ Z. In particular, its average is given by

Remark 4.19
In analogy with the case of Bourgeois contact structures, we could have also considered, on a flat contact fiber bundle (π : V → Σ, η 0 ), the class of contact structures η on V given by a potential A with [A, A] = 0.
For such an η, Proposition 4.9 tells us that d ∇ A is with values in the negative contact vector fields of the fibers. Such a condition, though, is not compatible with the fact that the surface Σ is closed. Indeed, by explicit computations (analogous to the ones in the proof of Claim 5.3 in the following) it can be proven that this condition on d ∇ A implies the existence of an exact volume form on Σ. Now, the latter can't exist if Σ is closed, according to Stoke's theorem.
Moreover, even if we allow Σ to have boundary, we do not recover all the informations on the fiber that we have with a Bourgeois contact structure. More precisely, we can't recover in general an (isotopy class of) open book decomposition supporting the contact structure on the fiber.
For instance, consider on the flat contact bundle (M × Σ → Σ, ξ M ⊕ T Σ) the contact fiber bundle structure η = ker (α + λ), with ξ M = ker α and dλ symplectic on Σ (that hence has non−empty boundary). Then, an explicit computation shows that A = −R α ⊗ λ, where R α is the Reeb vector field of α. In particular, [A, A] = 0 and d ∇ A = −R α ⊗ dλ, and we do not have any way to recover an (isotopy class of) open book decomposition on M from A.

Contact deformations and branched coverings
We show in this section that the class of contact fiber bundles that are contact deformations of a flat contact fiber bundle is stable under the operation of contact branched coverings: In particular, p * α t ∧ dα n−1 t ∧ rdr ∧ dθ is bounded below by a positive volume form independent of t. Using the fact that t k(t)/h t → 0 for t → 0, an argument analogous to the one in the proof of Lemma 2.4 allows to conclude that, if > 0 is small enough, α t ∧ d α n t > 0 for every t > 0.

Virtually overtwisted contact structures in high dimensions
In Sect. 5.1, we prove Proposition C from Sect. 1, stating that a Bourgeois contact structure on a fiber bundle with total space M × T 2 is weakly fillable provided that the same is true for the fiber (M, ξ). Then, Sect. 5.2 contains the proof of Theorem F, also stated in Sect. 1, about the existence of virtually overtwisted manifolds in all odd dimensions.

Bourgeois contact structure and weak fillability
Let (M 2n−1 , ξ) be a contact manifold and consider the flat contact bundle (π : where π is the projection on the T 2 -factor.

Proposition 5.1 Let η be a Bourgeois contact structure on
Recall that the result is already known in the case of η obtained by the Bourgeois construction [3]. The statement and the idea of the proof in that case already appeared in Massot et al. [37,Example 1.1], and an explicit proof can be found in Lisi et al. [35,Theorem A.a]. Notice also that, in a similar vein, [35,Theorem A.b] studies the stability of (subcritical) Stein fillability under the Bourgeois construction.
The proof of Proposition 5.1 uses the following fact about polynomials: of degree n, with P 1 (τ ) > 0 ∀ τ ≥ 0 and with P 2 having positive leading coefficient.
Proof of Proposition 5. 1 We first choose a convenient contact form for η. Let β is a form on M defining ξ . We can then write η = ker (α), where α : Recall from Sect. 4.5 that if A denotes the potential of η with respect to η 0 then, for each > 0, the family of potentials A := A defines a family η of Bourgeois contact structures that are all isotopic among Bourgeois contact structures (hence among contact structures).
The weak fillability condition for M implies that where ω M denotes the restriction of ω to M = ∂ X . We want to verify that, for > 0 small enough, we also have

Claim 5.3
Let Ω be an arbitrary volume form on M × T 2 . We then have where h is independent of and polynomial in τ , with deg τ (h) ≤ n − 1.
To improve readability, the proof of this claim is postponed. Denote now f and g the functions defined by f Ω = nβ ∧ (ω M + τ dβ) n−1 ∧ ω T 2 , g Ω = τ n α 1 ∧ dα n 1 . Then we need to show that f + 2 (g + h) > 0 on M × T 2 . Notice that for each p ∈ M ×T 2 , f ( p), g( p) and h( p) are polynomials in τ , by explicit computation in the case of f and g, and by Claim 5.3 in the case of h. Moreover, we have the following properties: for each p ∈ M × T 2 , (a) f ( p) > 0, because (X , ω) weakly fills (M, ξ); (b) g( p) > 0, because α 1 is a contact form for η; (c) h( p) has degree in τ strictly less than g( p), by Claim 5.3.
Proof of Claim 5. 3 We can compute Moreover, one has the following: thus giving the equivalence in Eq. (22). Because η is a Bourgeois contact structure, Eq. (21) then becomes For dimensional reasons, we thus get A similar explicit computation [using again Eq. (22)] shows that so that the second and third term in the right hand side of the last equality in Eq. (23) give 2 τ n α 1 ∧ dα n−1 where h is as in the statement. This conclude the proof of Claim 5.3.
Even if we will not use it in the following, we remark that the local nature of the condition d ∇ A = 0 and of all the computations in the above proof actually gives the following more general result: (M 2n−1 , ξ) be a contact manifold weakly filled by (X 2n , ω). Suppose that a representation ρ of π 1 (Σ g ) in the group of symplectomorphisms of (X , ω) gives, by restriction to the boundary, a representation ρ of π 1 (Σ g ) in the group of contactomorphisms of (M, ξ). Let also η be a Bourgeois contact structure on the flat contact bundle (π : M × ρ Σ g → Σ g , η 0 ) (as constructed in Sect. 4.4). Then, there is a symplectic form Ω on X × ρ Σ g that weakly fills η on M × ρ Σ g .

Proposition 5.4 Let
More precisely, if R 2 → Σ g denotes the universal covering map, Ω can be chosen to be the symplectic form on X × ρ Σ g induced by ω + ω g R 2 on X × R 2 and where ω g R 2 is a symplectic form on R 2 invariant by the action of π 1 (Σ g ) on R 2 by deck transformations.
Let's now come back to the results we need in order to exhibit examples of virtually overtwisted manifolds in all odd dimensions. Proposition 5.1 and (the proof of) Theorem 2.8 have the following immediate corollary: Proposition 5.5 Consider a branched covering Σ g → T 2 , where Σ g is the closed genus g ≥ 2 surface, and the naturally induced branched covering M ×Σ g → M ×T 2 . Let η g on M × Σ g be a contact branched covering of a Bourgeois contact structure η on the the contact bundle (π : M ×T 2 → T 2 , ξ ⊕ T T 2 ), where ξ is a contact structure on the fiber M. Then, if (M, ξ) admits a weak filling (X , ω), there is a symplectic form Ω on X × Σ g weakly dominating η g on M × Σ g = ∂ X × Σ g . More precisely, Ω can be chosen to be of the form ω + ω g , for a certain area form ω g on Σ g .

High dimensional virtually overtwisted manifolds
Let π : Σ g → T 2 be a branched covering map, branched along two points, and consider (Id, π): M × Σ g → M × T 2 . Notice that g is the branching index along each of the two connected components of the upstairs branching locus of (Id, π). For the proof of Proposition 5.6 we will need the following result, which is essentially just a rephrasing of the discussion in Niederkrüger and Presas [39,Page 724] for the local situation near the branching locus: Lemma 5.8 For k ∈ N >1 , let π k : V k → V 2n+1 be a branched covering map of branching index k. Suppose that all π k 's have same downstairs branching locus M and that the upstairs branching locus M k of π k is connected (in particular, π k | M k induces a diffeomorphism between M k and M). Suppose also that there is a tubular neighborhood N := M × D 2 (where D 2 is the 2−disk centered at 0 and of radius 1) of the downstairs branching locus M over which all the π k 's are trivialized at the same time, i.e. such that π k : Finally, let η be a contact structure on V inducing a contact structure ξ on M and η k on V k be a contact branched covering of η.
Then, there is > 0 such that, for all k ≥ 2, the upstairs branching locus ( M k , ξ k = ker π * k α) π k (M, ξ = ker α) has a neighborhood of the form we denote the open disk centered in 0 and of radius r inside R 2 ).
Proof of Proposition 5.6 Proposition 5.5 tells us that M × Σ g , η g is weakly fillable for all g ≥ 2. We then have to show that, for g sufficiently big, this contact manifold admits a finite cover which is overtwisted.
By hypothesis, there is a finite cover p : M → M such that (M, ξ := p * ξ) is overtwisted. Consider then the following commutative diagram of smooth maps: Consider also η := ( p, Id) * η on M ×T 2 and ζ g := ( p, Id) * η g on M ×Σ g . Notice that the restriction of ζ g to the upstairs branching locus of (Id, π): M × Σ g → M × T 2 is exactly ξ .
We now claim that (M × Σ g , ζ g ) is a branched contact covering of (M × T 2 , η). Indeed, we can see that ζ g is a contact deformation of the confoliation (Id, π) * η on M × Σ g as follows. If (η t g ) t∈[0,1] is a path of confoliations adapted to the upstairs branching locus of (Id, π): M × Σ g → M × T 2 starting at η 0 g = (Id, π) * η, ending at η 1 g = η g and such that η t g is contact for t ∈ (0, 1], then ( p, Id) * η t g is the path of confoliations on M × Σ g which shows that ζ g is a contact deformation of (Id, π) * η.
Notice that, letting g ≥ 2 vary, we get a sequence of branched coverings M × Σ g of M × T 2 , together with contact branched coverings ζ g of η. Lemma 5.8 then tells that each of the fibers (M, ξ = ker(α)) that belong to the (upstairs) branching set has a contact neighborhood of the form M × D 2 R g , ker α + r 2 dθ , with R g → +∞ for g → +∞. Because ξ on M is overtwisted, this implies, according to Casals et al. [5,Theorem 3.2], that if g is big enough then the upstairs branching set has an overtwisted neighborhood, so that (M ×Σ g , ζ g ) is also overtwisted. In other words, we just proved that, for g big enough, (M × Σ g , η g ) has a finite cover which is overtwisted.
Notice that taking g = 1 in the statement of Proposition 5.6, i.e. working directly on M × T 2 without taking a branched covering, is in general not enough to ensure the same conclusion.
For instance, this follows from Sect. 6, where we will show that for each contact manifold (M 3 , ξ), with π 1 (M) = 0, there is an open book decomposition of M supporting ξ such that the construction in [3] yields a hypertight contact form α on M × T 2 . In particular, even if (M, ξ) is virtually overtwisted, with (M, ξ) an overtwisted finite cover, the pullback α of α to M × T 2 will still define a tight contact structure η = ker α: indeed, if by contradiction η is overtwisted, according to Casals et al. [5] and Albers and Hofer [1], α admits a contractible Reeb orbit in M × T 2 , which then projects to a contractible Reeb orbit of α in M × T 2 , contradicting the hypertightness of α.
We also point out that we preferred to take a very big g in Proposition 5.6 in order not to enter too much in technical details and to keep the construction simple, but actually g = 2 is already enough. Indeed, it's enough to apply the following observation to the overtwisted cover (M, ξ) in the proof of Proposition 5.6 above. Here, Z , f and g are as follows: (a) Z is a smooth vector field on M such that: i. on M\B, it is tangent to the fibers of ϕ, ii. on the binding B, it is a (non-zero) multiple of the Reeb vector field R B of the restriction of β 0 to B; (b) f , g : M → R are smooth functions such that ( f , g) : M → R 2 is positively proportional to (cos ϕ, sin ϕ) on M\B and f = g = 0 on B.
For the proof of Proposition 6.1 we will need the following result, whose proof can be found for instance in Dörner et al.

With the orientations of B and Σ ϕ induced, respectively, by β B and dβ| T Σ ϕ , B is
oriented as the boundary of Σ ϕ .
Then, for a sufficiently small δ > 0, there is an embedding B × D 2 δ → B × D 2 (here, D 2 δ ⊂ R 2 denotes the disk centered at the origin of radius δ > 0) which preserves the angular coordinate ϕ on the second factor, is the identity on B × {0} and pulls back a convenient isotopic modification β of β (with an isotopy between contact forms that satisfy Hypothesis 1, 2 and 3 above) to a 1-form h 1 (r ) β B + h 2 (r ) dϕ, where:  book (B, ϕ).
We now define two functions λ, μ : M → R as follows: Notice that they are well defined smooth functions on all M × T 2 . Indeed, ρ smoothly extends as 1 at r = 0, h 1 = O(r ) near r = 0 (by point (i) of Lemma 6.2) and ρ h 1 − ρh 1 is positive for r > 0 and smoothly extends as h 1 (0) at r = 0. Consider then Z := λR B and ( f , g) := μ(cos ϕ, sin ϕ). Here, R B is seen as as a vector field on N = B × D 2 δ tangent to the first factor and λ has support contained inside N , hence λR B is well defined on all M. Similarly, f , g are well defined because μ is zero on B. It is also easy to check that such Z , f , g satisfy points (a) and (b) of Proposition 6.1.
Finally, we have to choose a contact form α defining a strong Bourgeois contact structure η on the flat contact bundle (M × T 2 → T 2 , ξ ⊕ T T 2 ), as desired in the statement of Proposition 6.1. Let α := β + φ 1 dx − φ 2 dy, i.e. the one obtained from Theorem 4.1.(b) with the choices of φ and N made above.
We already know that the contact structures given by Theorem 4.1 are in particular strong Bourgeois structures. An explicit computation also shows that Z + f ∂ x − g∂ y is indeed the Reeb vector field associated to α, as desired.
We have the following immediate consequence on the Reeb dynamics: Notice that, even if the closed orbits of R B are contained in B, we are interested here in their homotopy class as loops in M.

Embedding 3-manifolds in (hyper)tight 5-manifolds
We start with a proposition on (topological) open books of 3-manifolds: For the proof of Proposition 6.4 we need the following embedded version of the stabilization procedure for open book decompositions, which, as explained in Giroux and Goodman [24], essentially follows from Stallings' study in [42]:  Proof of Proposition 6. 4 We start by applying a sequence of stabilizations to reduce the number of connected components of the boundary of the pages to one. We can thus suppose that the open book decomposition (K , ϕ) has connected binding K . Notice that if [K ] ∈ H 1 (M; Z) is of infinite order, then we have nothing to prove. We can hence suppose that it is a torsion element. We now consider the following set of generators for H 1 (M; Z). Let p ∈ K , Σ be a page of (K , ϕ) inside M and α 1 , β 1 , . . . , α g , β g , K be curves on Σ that generate π 1 (Σ, p), as in Fig. 1. We can then use Van-Kampen theorem (see for instance Etnyre and Ozbagci [12]) with the following two open sets: U given by an open neighborhood K × D of the binding K and V := M\K . Because V is just the mapping torus of the monodromy ϕ : Σ → Σ, we then get that the inclusion Σ → M induces a surjection at the π 1 -level. Moreover, by Hurewicz theorem, the same is true at the H 1 -level, i.e. H 1 (M, Z) is generated by the homology classes of (the images in M of) α 1 , β 1 , . . . , α g , β g , K .
Now, by the hypothesis H 1 (M; Q) = {0} and by the universal coefficients theorem, at least one of the generators [α i ], [β i ], [K ] of H 1 (M; Z) is non-torsion; we can w.l.o.g. assume that this is the case for [β g ] (as we are assuming that [K ] is torsion).
Consider then a surface Σ obtained, as surface embedded in M, by plumbing a positive Hopf band A to Σ along a properly embedded arc δ 0 which is in the same class as β g in π 1 (Σ, ∂Σ), as shown in Figs. 2 and 3. According to Theorem 6.5, Σ is the page of an open book of M. In particular, we know that the core δ of A bounds a disk in M\Σ. Recall that a contact structure is called hypertight if it admits a contact form with non-contractible Reeb orbits.
Let's denote by D 2 R the disk of radius R > 0 centered at the origin in R 2 and by (r , ϕ) the polar coordinates on it. Theorem G and Corollary H then follow from Proposition 6.6: Theorem 6.7 Every closed 3-dimensional contact manifold (M, ξ) with non-trivial H 1 (M; Q) can be embedded, with trivial conformal symplectic normal bundle, in a hypertight closed 5-dimensional contact manifold (N , η). In particular, for each contact form α defining ξ on M, there is an > 0 such that M × D 2 , ker α + r 2 dϕ is tight.
As already remarked in the introduction, Hernández-Corbato et al. [27] deal with the higher dimensional case. More precisely, they give a generalization of the second part of this result, as well as an analogue (with less control on the codimension) of the first part of it.
Each (M × {pt}, η ∩ T (M × {pt}))) is then exactly (M, ξ) and it has topologically trivial normal bundle, hence trivial conformal symplectic normal bundle. Indeed, a symplectic vector bundle of rank 2 is symplectically trivial if and only if it is topologically trivial.
As far as the second part of the statement is concerned, according to the standard neighborhood theorem for contact submanifolds (see for instance Geiges [17,Theorem 2.5.15]), the contact submanifold (M, ξ = ker(α)) = (M × {pt}, η ∩ T (M × {pt}))) of (M × T 2 , η) has a contact neighborhood of the form M × D 2 , ker α + r 2 dϕ , for a certain real > 0. Moreover, each hypertight high dimensional contact manifold is in particular tight, according to Albers and Hofer [1] and Casals et al. [5]. In particular, (M × T 2 , η) is tight. Then, M × D 2 , ker α + r 2 dϕ is tight too, because it embeds (in codimension 0) in a tight contact manifold.