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 embeds with trivial normal bundle inside a (hyper)tight 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.


Introduction
This paper is concerned with the systematic study of some explicit constructions of high dimensional co-oriented contact structures, i.e. of hyperplane fields ξ on oriented smooth manifolds M 2n−1 which are given by the kernel of α ∈ Ω 1 (M ) such that α ∧ dα n−1 is a positive volume form on M . More precisely, the focus is on the constructions due to Geiges [Gei97] and Bourgeois [Bou02]. In the first article, developing ideas from Gromov [Gro86], Geiges transposes some constructions from the symplectic world to the contact setting, introducing in particular the notion of contact branched coverings. Contact fiber sums and contact reductions are also constructed, but we will not deal with them in the following (see Gironella [Gir18,Section 5.3] for the case of contact fiber sums). In the paper [Bou02], taking inspiration from Lutz [Lut79], Bourgeois proves that, given a closed contact manifold (M 2n−1 , ξ) and an open book decomposition (B, ϕ) of M supporting ξ, there is a contact structure η on M × T 2 that is invariant under the natural T 2 -action, that restricts to ξ on each submanifold M × {pt} and that naturally deforms to the hyperplane field ξ ⊕ T T 2 on M × T 2 . Recall that, according to Giroux [Gir02], for any contact manifold (M 2n−1 , ξ), one can always find an open book decomposition (B, ϕ) on M supporting ξ, i.e. such that B is a positive contact submanifold and there is α ∈ Ω 1 (M ) defining ξ such that dα is a positive symplectic form on the fibers of ϕ : M \ B → S 1 .
The main motivation behind both [Gei97,Bou02] was the problem of the existence of contact structures, i.e. the question of which high dimensional manifolds admit a contact structure. This (big) problem in contact topology has now been solved by Borman-Eliashberg-Murphy [BEM15]: high-dimensional contact structures exist whenever the corresponding formal objects, i.e. almost contact structures, exists. As a consequence, the aim has now shifted from providing examples to providing "interesting" examples of contact structures.
The papers [Gei97,Bou02] fit well in this perspective because they actually give rather explicit contact manifolds, which can be studied in some detail and which (under the right conditions) manifest interesting properties of tightness, fillability, overtwistedness, etc. For instance, these two papers provided the first explicit methods of building PS-overtwisted (hence overtwisted, according to the posterior Casals-Murphy-Presas [CMP19] and Huang [Hua17]) contact manifolds in high dimensions. The interested reader can consult Presas [Pre07] for the case of the construction in [Bou02] and Niederkrüger-Presas [NP10,page 724] for the case of contact branched coverings; see also Niederkrüger [Nie13,Theorem I.5.1], attributed to Presas, which uses contact fiber sums. Compare also with Observation 5.9 in Section 5.2 below.
The aim of this article is hence to construct contact manifolds with particular properties starting from [Gei97,Bou02]. In order to do so, we need to pass from the construction procedures by Geiges and Bourgeois to definitions. We can then study the properties of these contact structures, without the need to rely on any auxiliary choice made in their actual constructions in [Gei97,Bou02].
As far as contact branched coverings are concerned, we point out that the uniqueness problem is not explicitly addressed in [Gei97], i.e. it is not shown that the objects obtained are independent of the auxiliary choices made to build them. We hence propose in this paper a definition of contact branched coverings that allows to naturally obtain a uniqueness (up to isotopy) statement. A definition and a uniqueness statement can also be given in the case of contact fiber sums; see Gironella [Gir18,Section 5.3].
We remark that in the literature there is already a definition of contact branched coverings that goes in this direction. Indeed, Öztürk-Niederkrüger [ON07] 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: 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 [Bou02].
As already recalled above, one can look at the examples in [Bou02] 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 [Gir18,Chapter 7], where the links between the construction in [Bou02] and the study of T 2 -invariant contact structures in Lutz [Lut79] are analyzed in detail. The second point of view shows that the examples in [Bou02] 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 [Ler04], which are contact fiber bundles on M × T 2 → T 2 . We point out that this contact bundle structure on the examples from [Bou02] has already been exploited successfully in Presas [Pre07], van Koert-Niederkrüger [KN07], Niederkrüger-Presas [NP10], Etnyre-Pancholi [EP11,EP16] 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 [Bou02].
In this paper we then use the theory of contact fiber bundles from Lerman [Ler04] 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 [Bou02] 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 [Bou02] 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 Section 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 [Bou02]. Indeed, the statement and the idea of the proof already appeared in Massot-Niederkrüger- These reinterpretations and generalizations of [Gei97,Bou02] 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 [Gom98]. The interested reader can also consult Giroux [Gir00] and Honda [Hon00], 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 [Bou02] and contact branched coverings preserve the weak fillability condition, and relies on the existence of supporting open books proven by Giroux [Gir02], on the Bourgeois construction [Bou02] and on the "large" neighborhood criterion for overtwistedness proven in [CMP19, 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 [Gei08, 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 r0 , 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-Presas [NP10], where it is shown that "big" neighborhoods of contact overtwisted submanifolds obstruct fillability of the ambient manifold. As reported in Niederkrüger [Nie13], 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-Eliashberg-Murphy [BEM15] of a definition of overtwisted structures in all dimensions, Casals-Murphy-Presas [CMP19] 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 [CMP19, 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-Presas-Sandon [CPS16], Etnyre-Furukawa [EF17] and Etnyre-Lekili [EL19]. More precisely, [CPS16] proves that each 3-dimensional overtwisted manifold can be contact-embedded with trivial normal bundle into an exact symplectically fillable closed contact 5manifold. In [EF17], the authors shows how to embed many contact 3-manifolds into the standard contact 5-sphere. Finally, it is proven in [EL19] 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: 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 [Hof93], Albers-Hofer [AH09] and Casals-Murphy-Presas [CMP19].
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-Presas-Sandon [CPS16, Proposition 11], and the standard tight 3-sphere (which is the unique tight contact structure on S 3 up to isotopy according to Eliashberg [Eli92]) 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 [Bou02]. 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 [Bou02]. 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-Gironella-Moreno [BGM19], where it is shown, among other things, that the Bourgeois construction [Bou02] 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 -Martín-Merchán -Presas [HMP18] (without any assumption on H 1 (M ; Q)), with completely different techniques. More precisely, there the authors deduce such a generalization from [HMP18, 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-Eliashberg [CE12], and is an analogue of Theorem G in all dimensions, with less control on the codimension.
Outline In Section 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 [Gir12], 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 Section 4 the construction by Bourgeois using the notion of contact fiber bundle introduced in Lerman [Ler04]. 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 Section 6 we analyze the Reeb dynamics of the contact forms in Bourgeois [Bou02] and we prove Theorem G and Corollary H.

Acknowledgements
This work is part of my PhD thesis [Gir18], written at Centre de mathématiques Laurent Schwartz of École polytechnique (Palaiseau, France). I would like to thank P. Massot, who encouraged me to look at known constructions from different perspectives and greatly improved this manuscript with many suggestions. I am also grateful to C. Margerin and K. Niederkrüger for useful discussions concerning, respectively, contact connections and the Bourgeois construction, which resulted in a big improvement in the exposition in Section 4, as well as to J. Bowden for pointing out a mistake in the assumptions of the previous version of Theorem G. Finally, I also wish to thank the anonymous referee for many useful comments that greatly improved the presentation of the results with respect to the original version of the paper.

Contact branched coverings
In Section 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 [Gei97]. 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 [Gir18, Section 5.3].
Then, Section 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-Thurston [ET98, 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, JX) ≥ 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: Fact 2.3. Let (η n ) 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 kj ) 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: Lemma 2.4. Given K a compact set and (η k ) k∈K a smooth K-family of confoliations on V adjusted to M , there is a smooth family of confoliations (η s k ) s∈[0,1], k∈K such that (η s k ) s∈[0,1] is contactization of η k , for each k ∈ K. Moreover, if η k is contact for all k in a closed subset H ⊂ K, then η s k can be chosen so that η s k = η k for all k ∈ H and s ∈ [0, 1].
Proof (Lemma 2.4). This proof follows almost step by step the construction and the computations made in Geiges [Gei97, 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 → 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 α s k := α k + s ρ(k)g(r)r 2 γ .
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) + 2g (r)] α k ∧ (dα k ) n−1 ∧ rdr ∧ γ = Q k vol. Let also R k ( ) := r 2 g (r) h( , k). Then, Now, Q k > 0 and R k ( ) = 0 along M , for all k ∈ K and ∈ [0, 1] (remark we allow here = 0). Hence, by compactness of M and [0, 1], there is an open 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 for all k ∈ K and all ∈ (0, 1]. 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).
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: Proposition 2.6. Let V → V be a smooth branched covering and η a contact structure on V . Then, contact branched coverings of η on V exist and are all isotopic (among contact branched coverings).
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.
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 Section 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 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 (Claim 2.9). We have ω = π * ω + (2f + rf ) 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: 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 (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 n is a positive volume form on O r0 , 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: Claim 2.12. There is 0 < 1 < 0 such that P 0 (τ ) + [P 1 (τ ) + P 2 (τ, )] > 0 on the complement of Or 0/2 , for all 0 ≤ < 1 and all τ ≥ 0.
We now give a proof of Claims 2.11 and 2.12 above. They are corollaries of the following fact, whose proof is easy and omitted: Fact 2.13. Consider a smooth manifold S and a continuous function p : S × R ≥0 → R such that, for each s ∈ S, p s : R ≥0 → R defined by p s (τ ) := p(s, τ ) is polynomial in τ . Suppose there is s 0 ∈ S and a neighborhood U of s 0 such that for all s ∈ U the followings are satisfied: Then, there is a neighborhood O of s 0 contained in U such that, for all s ∈ O, the minimum m s of p s exists and it depends continuously on s. In particular, if moreover m s0 > 0, then m s > 0 for s sufficiently near to s 0 .
Proof (Claim 2.11). We would like to use Fact 2.13, with S := V × [0, ) and which is positive because the restriction of ω to X weakly dominates ξ on M = ∂X. Thus, for (q, 0) ∈ K, P (q,0) has positive leading coefficient and One can then apply Fact 2.13, which, by compactness of K, tells that there is a neighborhood U of K in S such that m s exists and is positive for all s ∈ U. Now, U contains an open set of the form {r < r 0 , < 0 } ⊂ S = V ×[0, ), which concludes.
Proof (Claim 2.12). We use again Fact 2.13. Here, S : r 0/2 is the complement of Or 0/2 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 r 0/2 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 Section 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 [Gir12]. 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: be an open book decomposition of M 2n−1 supporting ξ. Denote by α a contact form defining ξ such that dα is symplectic on the fibers of ϕ. Then, there is a smooth function φ : M → R 2 defining (B, ϕ) such that the contact vector fields X and Y , associated via α respectively to the contact Hamiltonians φ 1 and −φ 2 , have Lie bracket [X, Y ] negatively transverse to ξ.
Recall from Giroux [Gir02] 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 Proof (Proposition 3.1). Let φ = (φ 1 , φ 2 ) : M → R 2 be a smooth function defining (B, ϕ). Consider then > 0 such that α ∧ dα n−2 ∧ dφ 1 ∧ dφ 2 is positive on { φ < }. Such an exists because α induces a contact form on B = φ −1 (0). Consider now a smooth function f : M → R >0 , depending only on φ in a non-decreasing 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.
Hence, ι X α ∧ d (α (Y )) ∧ dα n−1 = 0, which, using the graded Leibniz rule for the interior product, gives (Here, we adopted the notation Z · f = df (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, Equations (2) to (5) give Then, Equation (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 (M 2n−1 , ξ) be a closed contact manifold. Suppose X, Y are two contact vector fields with Lie bracket [X, Y ] everywhere negatively transverse to ξ. 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: (a) The set Σ θ := {X θ ∈ ξ} is a non-empty regular hypersurface, which is moreover ξ-convex.
(c) For each θ ∈ S 1 , consider the set is an open book decomposition of M , which is moreover adapted to ξ.
Recall from Giroux [Gir91] that a hypersurface Σ in M is called ξ-convex if there is a vector field Z which is contact for ξ and transverse to Σ.
The rest of Section 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 (Proposition 3.2.(a)). We start by proving that α (X θ ) is somewhere zero, i.e. that Σ θ = {α (X θ ) = 0} is non-empty. Suppose by contradiction this is not the case, i.e. α (X θ ) > 0 without loss of generality. If we define 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 Equation (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 Equation (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 Equation (9), we get dα We point out a direct consequence of Equation (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), Lemma 3.5. X θ is tangent to Σ θ . Moreover, it is transverse to ∂F θ = Σ θ ∩ Σ θ+π/2 and points outwards from F θ . Proof (Lemma 3.5). Evaluating the left identity in Equation (10) The second part of the statement follows from the fact that α (Y θ ) = 0 along ∂F θ = Σ θ ∩ Σ θ+π/2 (by definition of Σ θ+π/2 ), and that d (α (Y θ )) (X θ ) < 0 along ∂F θ by Corollary 3.4. Indeed, this means that X θ points in the region where α (Y θ ) < 0 along ∂F θ , being always tangent to Σ θ , i.e., by definition of F θ , that it points outwards from F θ along its boundary.
Proof (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: is the open unit disk in R 2 ) and a contact form α defining ξ such that: (i) ϕ restricted to N \ K is the angular coordinate of the projection on the second factor N = K × D 2 → D 2 ; (ii) ξ induces a positive contact structure on each submanifold K z := K × {z} of N (notice each K z is oriented because K is); (iii) dα induces a positive symplectic form on each fiber of ϕ |M \N .
Then, the open book decomposition (K, ϕ) supports the contact structure ξ.
Proof (Lemma 3.6). Let α be a contact form for ξ as in the statement. The aim is to find a function f : M → R + such that d(f α) is positively symplectic on the fibers of ϕ.
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 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 We then need to control its sign on K × D 2 1− /2 . Let's start by analyzing it on K × D 2 1− . Here, ∂f ∂r = −2kre −kr 2 , so that 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.
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 function D 2 δ \ {0} → S 1 : is a positive contact submanifold of (M, ξ).
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.
) (X)∂ y along K. According to Lemma 3.5 and Corollary 3.4, d (α (X)) (X) = 0 and 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).
Proof (Claim 3.8). Let's start with point (i). We can explicitly evaluate the differential dΨ at points of the form (p, 0, 0). On K × {0}, we simply have that dΨ(∂ x ) = Y , dΨ(∂ y ) = X and that dΨ(V ) = V for all vector fields V which are tangent to K × {0}. This shows that Ψ is a local diffeomorphism at each point (p, 0, 0). Hence, by compactness, Ψ is also a diffeomophism from K × D 2 δ onto its image, provided δ is small enough.
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 [Gir91]: indeed, X θ defines the characteristic foliation of Σ θ , and K is transverse to it.
Proof (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 M \ { φ < 2 }.
We then analyze dϕ ∧ d α on N c .
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 Equation (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 [Ler04]. More precisely, we start by recalling the Bourgeois construction in Section 4.1.
In Section 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 [Bou02]. In particular, in Section 4.4 we take a general fibration admitting a flat contact connection and we consider on it two non-trivial 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 [Bou02] in the setting of contact fiber bundles and is presented in Section 4.5. There Proposition D from the introduction is also proved using the results from Section 3. Lastly, in Section 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 [Gir02], Bourgeois constructs in [Bou02] explicit contact structures on M × T 2 . More precisely, the main statement of [Bou02] can be rephrased as follows: b. If φ is as in point a., then for any choice of coordinates (θ 1 , θ 2 ) on T 2 and for any contact form β defining ξ and adapted to the open book (B, ϕ), the 1-form α := β + φ 1 dθ 1 − φ 2 dθ 2 is a contact form on M × T 2 .
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 .

Generalities on contact fiber bundles
We recall in this section the notion of contact fiber bundle introduced by Lerman in [Ler04], 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: Definition 4.5. A fiber bundle with contact fibers is the data (π : V → Σ, ξ f ib ) of a fiber bundle π : V → Σ and a 3-codimensional distribution ξ f ib on V inducing, for all b ∈ Σ, a contact structure ξ b on the fiber M b .
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-Michor [KM97] 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 (pr Ui : 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 [KM97,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, X f ib (V, ξ f ib ) → Σ is just the adjoint bundle associated to E → 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 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, (Z b ) h0 and (Z b ) h give contactomorphisms between different fibers of (π, ξ f ib ).
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 ) h0 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 [KM97,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, W ] σ for all U, W vector fields on Σ and σ ∈ Γ(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-Michor [KM97, 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 formula d ∇ d ∇ ω = ω ∧ F for all ω ∈ Ω p (Σ, X f ib (V, ξ f ib )) (see for instance [KM97,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 X h , Z h their Hhorizontal component. By definition of curvature R ∈ Ω 2 (V ; ker(dπ)) of H, we have R(X, Z) = ω([X h , Z h ]). Introducing the potential A, one can further write Finally, remark that X h0 and Z h0 are just the H 0 -horizontal lifts U 0 and W 0 of U and W respectively. Hence, by Equation (13) W ] . Putting all the pieces together, we then get: for all X, Z vector fields on V , 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. Indeed, once fixed a certain fiber (M, ξ) of (π : V → Σ, ξ f ib ) over b ∈ Σ, one gets a representation ρ : π 1 (Σ) → Diff(M, ξ), where Diff(M, ξ) is the space of contactomorphisms of (M, ξ), as follows. Because H 0 is flat (hence a foliation according to Frobenius' theorem), the monodromy Ψ δ of the connection H 0 over a (smooth immersed) curve δ in Σ depends only on [δ] ∈ π 1 (Σ). Moreover, it is also a contactomorphism of the fibers, by Lemma 4.4. Hence, for each c ∈ π 1 (Σ), one can define ρ(c) := Ψ δ , where δ is any (smooth immersed) representative of c. 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, F induces an isomorphism f : M × ρ Σ → V of fiber bundles over Σ, where M × ρ Σ is the quotient of M × Σ by ρ. Notice also that on M × ρ Σ → Σ there are natural ξ ρ f ib and H ρ 0 induced, respectively, by ξ ⊕ {0} and {0} ⊕ T Σ on M × Σ. Because of the properties of F , we also have that the differential of f sends ξ ρ f ib and H ρ 0 respectively to ξ f ib and H 0 . 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 Section 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 Equation (14) (with R 0 = 0), we can rephrase Lerman [Ler04, Proposition 3.1] in the following computational-friendly way: Proposition 4.9. On a flat contact fiber bundle π : V 2n+1 → Σ 2 , ξ f ib , H 0 , 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 Section 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 α t := dθ + t cos θdx − sin θdy, with t ∈ [0, 1]. We point out that, by Giroux [Gir99, 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 [Vog16, 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 [Gir99]).

Bourgeois' construction revisited
The aim here is to use what we defined in the previous sections to generalize the construction by Bourgeois recalled in Section 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 ∂ θj = 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 Section 4.3 for the definition of ∇).
We can then give the following definition: Definition 4.11. Let (π : V → Σ, η 0 = ξ f ib ⊕ H 0 ) 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 (sA) 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 [Bou02]: 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 : M × T 2 → T 2 , ξ M ⊕ T T 2 ). 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. Then, the hyperplane field 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 : M × ρ T 2 → T 2 , η 0 ) equipped with the standard flat ∇ T 2 on T 2 .
Proof (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 [Bou02]. 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 : M × R 2 → M × ρ T 2 . 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 (Proposition 4.12). We recall that van Koert-Niederkrüger [KN05] 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 : M × ρ T 2 → T 2 , η 0 ). Here, η 0 is the flat contact bundle induced by ξ ⊕T R 2 on M ×R 2 → R 2 .
The only thing left to show is that (π : M × ρ T 2 → T 2 , η 0 ) is not isomorphic to the trivial flat contact bundle (p : M × T 2 → T 2 , ξ ⊕ T T 2 ). 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 : 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 [Bou02], 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 (π : M × ρ T 2 → T 2 , η 0 ), where η 0 is induced by ξ ⊕ T R 2 on the cover M × R 2 of M × ρ T 2 . 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)).
Recall from Section 4.4 that every flat contact bundle (π : V → T 2 , η 0 ) is isomorphic (as flat contact bundle) to (π : M × ρ T 2 → T 2 , η 0 ), where η 0 is induced by ξ ⊕ T R 2 on the cover M × R 2 of M × T 2 . Thus, Proposition 4.14 says that the examples given 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 (Proposition 4.14). Consider the following natural commutative diagram: with Q and q the natural quotients maps (see Section 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 dQ 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 d ∇ A = 0, so that R A = [ A, A]. Now, A is the potential associated to a contact structure on M × R 2 , because the same is true for A on M × ρ T 2 . Let then X := A(∂ x ) and Y := A(∂ y ), for any choice coordinates (x, y) ∈ R 2 . Proposition 4.9 then tells that [X, Y ] is negatively transverse to ξ everywhere on M . Moreover, as A is the pullback of A, both X and Y are Im(ρ)-invariant. 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 Section 1. Indeed, according to Equation (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), we have (∇A)(W, vector fields U, W on T 2 . Using the fact that A is ∇-parallel, we compute where T ∈ Ω 2 (T 2 ; T T 2 ) 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 := tA 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, oriented basis (u, v) of T b Σ and t > 0, This property is a generalization of the fact that strong Bourgeois structures (which includes the examples in [Bou02]) 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 Section 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 [Bou02] is covered by Massot-Niederkrüger-Wendl [MNW13, Example 1.1] and Lisi-Marinković-Niederkrüger [LMN18, Theorem A.a].) This stability of weak fillability is also true in a more general case, as stated in Proposition 5.4.
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) : ii. the map φy φy : 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 pr Z π f as the vector sub-bundle {W ∈ T (f * X)| d(pr X )(W ) ∈ η} of T (f * X), 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 (γ * pr * V, γ * pr * η) From the above result, we can deduce a more precise version of Proposition D stated in the introduction:  Proof (Corollary 4.17). Given b ∈ Σ, consider an ordered basis (u, v) of T b Σ and define ψ η (b) as the class of Ψ η (u, v) under the relation ∼. Here, Ψ η is the map given by Proposition D. We then need to show that this is well defined. Suppose (u , v ) is another ordered basis of T b Σ. We want to show that Ψ η (u, v) ∼ Ψ η (u , v ). Choose a curve γ : [0, 1] → F Σ with image contained in the fiber pr −1 (b) of pr : F Σ → Σ and such that γ(0) = (u, v) and γ(1) = (u , v ). Then, according to point ii. of Proposition 4.16, Ψ η • γ gives a smooth [0, 1]−family of open books on γ * pr * V adjusted to γ * pr * η. Now, Because a smooth path of open book decompositions 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 Section 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 . In other words, for all b ∈ T 2 , if (∂ θ1 , ∂ θ2 ) is the oriented base of T b T 2 coming from the choice of coordinates (θ 1 , θ 2 ) ∈ T 2 as in the statement of Theorem 4.1, then Ψ η (∂ θ1 , ∂ θ2 ) = (B, ϕ ). In particular, ψ η (b) is the isotopy class of (B, ϕ ), which coincides with that of (B, ϕ).
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 1form 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 2action 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 (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 2 -family 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 X = m,n∈Z e i(mx+ny) X m,n and Y = h,k∈Z 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 Equation (18) is equivalent to: mY m,n = nX 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 where Y m,n and Y m,n denote respectively the real and imaginary part of Y m,n . Moreover, (a) comes from the fact that X m,n is zero if m = 0, n = 0 and Y −m,−n is zero if n = 0, m = 0 by Equation (19), (b) comes from Equations (17) and (19)  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,

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: Proposition 4.20. Let (π : V 2n+1 → Σ, η 0 ) be a flat contact fiber bundle and p : Σ → Σ a branched covering map that lifts to a branched covering map p : V → V . Consider now the pull-back flat contact fiber bundle ( π : V → Σ, η 0 ) induced by p, i.e. η 0 := p * η 0 . If η is a contact deformation of η 0 , then there is a contact branched covering η of η to V that is a contact deformation of η 0 .
Proof. By definition of contact deformation, there is a smooth family of 1-forms (α t ) t∈[0,1] on V interpolating between η = ker α 1 and η 0 = ker α 0 , such that η t = ker α t is contact for t > 0 and the fibers of π : V → Σ have induced contact structures independent of t. For t ∈ [0, 1], define f t , h t : V → R by α t ∧ dα n−1 t | ker(dπ) = f t α 1 ∧ dα n−1 1 | ker(dπ) and α t ∧ dα n t = h t α 1 ∧ dα n 1 . Notice that f t > 0 everywhere for every t ∈ [0, 1], whereas h t > 0 everywhere for t > 0 and h 0 = 0. Moreover, as h t is a smooth function of t and V is compact, the function k(t) := min{h t (p) | p ∈ V } is smooth in t.
According to the proof of Lemma 2.4, we can choose η on V to be the kernel of α = p * α 1 + g(r)r 2 dθ, with the same notations as in that proof, using the particular choice of closed form γ = dθ as connection on the trivial unit normal bundle of M in V . Recall that > 0 can be chosen arbitrarily small here. Define ( α t ) t∈[0,1] by α t = p * α t + tk(t) g(r)r 2 dθ. In particular, ker( α 1 ) = η and ker( α 0 ) = η 0 . We then claim that α t is a contact deformation of η 0 to η. Now, α t gives on each fiber a contact structure independent of t, hence the only thing we need to show is that α t defines a contact structure for t > 0. We can explicitly compute Notice that p * (α t ∧ dα n t ) = h t p * (α 1 ∧ dα n 1 ). Moreover, 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 Section 5.1, we prove Proposition C from Section 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, Section 5.2 contains the proof of Theorem F, also stated in Section 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 : M × T 2 → T 2 , η 0 = ξ ⊕ T T 2 ), where π is the projection on the T 2 -factor.
Recall that the result is already known in the case of η obtained by the Bourgeois construction [Bou02]. The statement and the idea of the proof in that case already appeared in Massot-Niederkrüger-Wendl [MNW13, Example 1.1], and an explicit proof can be found in Lisi-Marinković-Niederkrüger [LMN18, Theorem A.a]. Notice also that, in a similar vein, [LMN18, Theorem A.b] studies the stability of (subcritical) Stein fillability under the Bourgeois construction.
Proof (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 := β + φ 1 dθ 1 − φ 2 dθ 2 , with φ 1 , φ 2 : M × T 2 → R and (θ 1 , θ 2 ) coordinates on T 2 . Recall from Section 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). Thus, up to isotopy, we suppose η = ker(α ), where α = β + φ 1 dθ 1 − φ 2 dθ 2 , for a certain > 0 that will be chosen very small in the following. 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 (Claim 5.3). We can compute Moreover, one has the following: Indeed, we have A = X ⊗ dθ 1 Y ⊗ dθ 2 , with X, Y the contact vector fields on (M, ξ) with contact hamiltonians −φ 1 , φ 2 w.r.t. β. By Equation (18), d ∇ A = 0 if and only if ∂ ∂θ1 Y − ∂ ∂θ2 X = 0. Now, because the latter is a contact vector field on each fiber (M, ξ), it is zero if and only if its contact hamiltonian w.r.t. β is zero, i.e. if and only if thus giving the equivalence in Equation (22). Because η is a Bourgeois contact structure, Equation (21) then becomes For dimensional reasons, we thus get A similar explicit computation (using again Equation (22)) shows that so that the second and third term in the right hand side of the last equality in Equation (23) give 2 τ n α 1 ∧ dα n−1 1 + 2 h Ω, 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: Proposition 5.4. Let (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 Section 4.4). Then, there is a symplectic form Ω on X × ρ Σ g that weakly fills η on M × ρ Σ g . 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).
Proposition 5.6. Let η be a Bourgeois contact structure on the flat contact bundle (M × T 2 , T 2 , M, η 0 ) and consider a contact branched covering η g of η with respect to (Id, π) : M × Σ g → M × T 2 . If (M, ξ) is weakly fillable and virtually overtwisted, then, for g ≥ 2 big enough, (M × Σ g , η g ) is weakly fillable and virtually overtwisted.
Starting for instance from the case of a holomorphically fillable virtually overtwisted contact structure on lens spaces, that exist by Gompf [Gom98, Proposition 5.1] (see also Giroux [Gir00, Theorem 1.1]), and using the construction in [Bou02], a proof by induction on the dimension 2n − 1 of M gives then the following: Corollary 5.7. Virtually overtwisted manifolds exist in all odd dimensions ≥ 3.
For the proof of Proposition 5.6 we will need the following result, which is essentially just a rephrasing of the discussion in Niederkrüger-Presas [NP10, 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 : M × D 2 → M × D 2 is just overtwisted, according to Casals-Murphy-Presas [CMP19] and Albers-Hofer [AH09], 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: Observation 5.9 (Massot-Niederkrüger). If (M, ξ) is overtwisted, the contact manifold (M × Σ g , η g ) is overtwisted already for g = 2.
The argument, which we now sketch, is due to Massot and Niederkrüger, and relies on the idea from Presas [Pre07] of using monodromy on contact fiber bundleswith overtwisted fibers in order to find embedded Plastikstufes.
Proof (sketch). Take an arc γ on T 2 going from one (downstairs) branching point of the cover Σ 2 → T 2 to the other and such that it is radial in a local model (trivializing the smooth branched covering) around the two branching points, in such a way that its double cover δ in Σ 2 is a smooth closed curve. The monodromy of the contact fiber bundle M × Σ 2 → Σ 2 over δ is trivial. Indeed, as the proof of Lemma 2.4 shows, the contact branched covering η 2 of a Bourgeois contact structure (M × T 2 → T 2 , η = ker β) can be chosen to be defined by a form β 2 on M × Σ 2 which is invariant under deck transformations of the branched covering π : M × Σ 2 → M × T 2 and C ∞ -close to π * β. Then, one can see that the monodromy of (M × Σ 2 → Σ 2 , η 2 ) over δ is obtained as the concatenation of the monodromy f γ of M × T 2 → T 2 , η = ker (β) over γ, plus a C ∞ −little perturbation h, and the monodromy (f γ ) −1 over −γ, plus the inverse h −1 of the same perturbation. Using the techniques from [Pre07], we can then find an embedded plastikstufe inside M × δ ⊂ M × T 2 . In practice, if p ∈ Σ 2 denotes one of the two upstairs branching points, this PS is obtained by parallel-transporting (w.r.t. the connection defined by η 2 ) an overtwisted disk in M × {p} M along δ. This procedure actually gives an embedded PS because the monodromy along the loop δ is the identity. Finally, Huang [Hua17] tells that each PS-overtwisted manifold is also overtwisted, which concludes.

Bourgeois construction and Reeb dynamics
The main aim of this section is to give a proof of Theorem G stated in Section 1. In order to do this, starting from a contact manifold (M 2n−1 , ξ) and an open book (B, ϕ) adapted to ξ, we consider in Section 6.1 a strong Bourgeois contact structure η on the flat contact bundle (M × T 2 → T 2 , ξ ⊕ T T 2 ) which admits a contact form α with very specific Reeb vector field. This η is actually one of the examples described in [Bou02]. We then show that the Reeb dynamics of α on M × T 2 is strictly related to the Reeb dynamics on the binding B of the open book (B, ϕ). This will give a criterion for the existence of closed contractible Reeb orbits of α on M × T 2 . Then, we show in Section 6.2 how to deduce Theorem G as a corollary of this study in the case of 3-dimensional M .

Bourgeois structures and contractible Reeb orbits
Proposition 6.1. Let (M, ξ) be a (2n−1)-dimensional contact manifold, (B, ϕ) an open book decomposition on M supporting ξ and β 0 a contact form for ξ adapted to (B, ϕ). Then, there is a strong Bourgeois contact structure η on the flat contact bundle (M ×T 2 → T 2 , ξ ⊕T T 2 ), which is obtained as in Theorem 4.1 and admits a contact form α with associated Reeb vector field of the form Here, Z, f and g are as follows: a. Z is a smooth vector field on M such that: For the proof of Proposition 6.1 we will need the following result, whose proof can be found for instance in Dörner-Geiges-Zehmisch [DGZ14, Section 3]: Lemma 6.2 (Giroux). Let D 2 ⊂ R 2 be the disk centered at the origin with radius 1 and β be a contact form on B × D 2 with the following properties: 1. β B := β| T B is a contact form on B = B × {0}.
3. With the orientations of B and Σ ϕ induced, respectively, by β B and dβ| T Σϕ , B is oriented as the boundary of Σ ϕ .
Proof (Proposition 6.1). We start by finding a convenient isotopic modification of the adapted form β 0 in the statement as well as a particular normal neighborhood N of the binding and a particular smooth map φ : M → R 2 defining (B, ϕ).
Take a normal neighborhood B × D 2 of the binding B in M such that ϕ : B× D 2 \ {0} → S 1 becomes the angular coordinate of D 2 \{0}. Such neighborhood exists by definition of open book decomposition. Then, Lemma 6.2 gives an isotopic modification β of β 0 , still adapted to the same open book, and of the form h 1 β B + h 2 dϕ in the neighborhood N := B × D 2 δ ⊂ B × D 2 . Moreover, β| T B = β 0 | T B , so that they induce the same Reeb vector field on B; as in the statement, we denote it R B . Consider now a function ρ : M → R which is smooth away from B, equal to the radial coordinate r of D 2 δ on the neighborhood {r ≤ δ /3} of B × {0} inside N = B×D 2 δ , equal to 1 on the complement in M of the open set {r < 2δ /3} ⊂ N , and depending only on r in a strictly increasing way on the set δ /3 < r < 2δ /3. Then, we define φ := ρ (cos ϕ, sin ϕ). Remark that such a φ is indeed well defined and smooth on all M , and defines the open 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-Goodman [GG06], essentially follows from Stallings' study in [Sta78]: Theorem 6.5. Let Σ be a compact surface with boundary in a manifold M and δ 0 a properly embedded arc in Σ. Let also Σ ⊂ M be obtained by plumbing a positive Hopf band to Σ, i.e. Σ = Σ ∪ A where A is an annulus in M such that 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 Figures 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. Now, δ is homotopic (in Σ ) to the concatenation (K 1 ) −1 * β g of the inverse of the boundary component K 1 of Σ = Σ ∪ A that passes through p and β g ; see Figure 3. The existence of the disk ∆ then tells that [K 1 ] = [β g ] in H 1 (M ; Z) and, because [β g ] is non-torsion, the same is true for [K 1 ]. Moreover, K = ∂Σ is cohomologous (in Σ ) to K 1 K 2 ; see again Figure 3. Working in H 1 (M ; Z) modulo torsion, it is then easy to check that [K] torsion and [K 1 ] non-torsion imply that [K 2 ] is also non-torsion, as desired. Corollary 6.3 and Proposition 6.4 have the following direct consequence: Proposition 6.6. Let (M 3 , ξ) be a contact manifold with H 1 (M ; Q) = {0}. Then, there is a hypertight strong Bourgeois contact structure on (M × T 2 → T 2 , ξ ⊕ T T 2 ). More precisely, given any open book (K, ϕ) supporting ξ, there is another supporting (K , ϕ ), obtained from (K, ϕ) by a sequence of positive stabilizations, such that the strong Bourgeois contact structure on M × T 2 obtained as in Theorem 4.1 from (M, ξ, (K , ϕ )) is hypertight.
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: