# Existence of Engel structures

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Existence of Engel structures

Annals of Mathematics, 169 (2009), 79–137 Existence of Engel structures By Thomas Vogel Abstract We develop a construction of Engel structures on 4-manifolds based on decompositions of manifolds into round handles. This allows us to show that all parallelizable 4-manifolds admit an Engel structure. We also show that, given two Engel manifolds M1 , M2 satisfying a certain condition on the characteristic foliations, there is an Engel structure on M1 #M2 #(S 2 × S 2 ) which is closely related to the original Engel structures. Contents 1. Introduction 2. Engel structures and related distributions 3. Contact topology 4. Round handle decompositions and model Engel structures 5. Attaching maps for round handles 6. Existence theorems References 1. Introduction Distributions are subbundles of the tangent bundle of a manifold. It is natural not to consider general distributions but to make geometric assumptions, for example, integrability. In this case the distribution defines a foliation on the manifold. Another possibility is to assume that a distribution is nowhere integrable. Important examples of this type are contact structures on odd-dimensional manifolds. On 3-dimensional manifolds, properties of contact structures reflect topological features of the underlying manifold in a surprising way. Engel structures form another class of nonintegrable distributions which is closely related to contact structures. By definition, an Engel structure is a smooth distribution D of rank 2 on a manifold M of dimension 4 which satisfies the nonintegrability conditions rank[D, D] = 3 rank[D, [D, D]] = 4, 80 THOMAS VOGEL where [D, D] consists of those tangent vectors which can be obtained by taking commutators of local sections of D. Examples of Engel structures can be obtained from contact structures on 3-manifolds. In this article we use results from contact topology to investigate Engel structures. This shows that Engel structures and contact structures are closely related. Similarly to contact structures and symplectic structures, all Engel structures are locally isomorphic. Every point of an Engel manifold has a neighborhood with local coordinates x, y, z, w such that the Engel structure is the intersection of the kernels of the 1-forms (1) α = dz − x dy, β = dx − w dy. This normal form was obtained first by F. Engel in [Eng]. Together with the fact that a C 2 -small perturbation of an Engel structure is again an Engel structure, this implies that Engel structures are stable in the sense of singularity theory. In [Mo1] R. Montgomery showed that the only distribution types with this stability property are • • • • foliations of rank one on manifolds of arbitrary dimension, contact structures on manifolds of odd dimension, even contact structures on manifolds of even dimension, Engel structures on manifolds of dimension 4. Thus Engel structures are special among general distributions and, even among the stable distribution types, they seem to be exceptional since they are a peculiarity of dimension 4. On the other hand Engel structures appear quite naturally. For example, a generic plane field on a 4-manifold satisfies the Engel conditions almost everywhere and one can construct an Engel structure from a contact structure on a 3-manifold. Certain nonholonomic constraints studied in classical mechanics also lead to Engel structures. An Engel structure D induces a distribution of hyperplanes E = [D, D], which is an even contact structure; i.e. [E, E] is the whole tangent bundle. Moreover, to the even contact structure E one can associate its characteristic foliation W and it turns out that W is tangent to D. This is explained in Section 2.2. Thus an Engel structure D on M induces a flag of distributions (2) W ⊂ D ⊂ E = [D, D] ⊂ T M such that each distribution has corank one in the next one. Hypersurfaces transverse to W carry a contact structure together with a Legendrian line field. We will refer to this line field as the intersection line field. The presence of the distributions in (2) leads to strong restrictions for the topology of Engel manifolds. The following result can be found in [KMS] where it is attributed to V. Gershkovich. EXISTENCE OF ENGEL STRUCTURES 81 Proposition 2.12. If an orientable 4-manifold admits an orientable Engel structure, then it has trivial tangent bundle. We develop a construction of Engel manifolds which allows us to prove the converse of Proposition 2.12. Theorem 6.1. Every parallelizable 4-manifold admits an orientable Engel structure. This solves a problem from [ElM] and [KMS]. For open manifolds, Theorem 6.1 can be proved using the h-principle for open, Diff-invariant relations; cf. [ElM]. In the literature one can find only two other constructions of Engel structures on closed manifolds. The first one is called prolongation and was introduced by E. Cartan. With this method one gets Engel structures on certain S 1 -bundles over 3-dimensional contact manifolds. The second construction is due to H. J. Geiges; cf. [Gei]. It yields Engel structures on parallelizable mapping tori. These two constructions cover only a small portion of all parallelizable 4-manifolds. Let us give a short outline of the proof of Theorem 6.1. The construction is based on decompositions of manifolds into round handles. A round handle of dimension n and index l ∈ {0, . . . , n − 1} is Rl = Dl × Dn−l−1 × S 1 . It is attached to a manifold with boundary using an embedding of ∂− Rl = S l−1 × Dn−l−1 × S 1 into the boundary of M . By a theorem of D. Asimov in [As] every 4-manifold with vanishing Euler characteristic can be obtained from R0 by attaching round handles of higher index successively. We fix a set of model Engel structures on round handles such that the characteristic foliation is transverse to ∂− Rl and ∂+ Rl = ∂Rl \ ∂− Rl . These model Engel structures are constructed using a perturbed version of the prolongation construction. The contact structure on each ∂− Rl depends essentially only on l while the intersection line field and the orientation of the contact structure are different for different model Engel structures on Rl . In the proof of Theorem 6.1 we fix a round handle decomposition and a framing of M . The proof of Proposition 2.12 shows that to each oriented Engel structure on M one can associate a framing which we will refer to as Engel framing. Now we attach the round handles successively and we extend to Engel structure at every step using one of our model Engel structures. The model Engel structures on R0 are chosen such that the contact structure on ∂+ R0 is overtwisted. Defining the model Engel structures appropriately we can achieve in addition that • round handles of index 1 are attached along Legendrian curves, 82 THOMAS VOGEL • round handles of index 2 are attached along convex tori in a certain standard form, • round handles of index 3 are attached along S 2 × S 1 with the unique (up to isotopy) overtwisted contact structure which extends to R3 = D3 × S 1 as a plane field. When we attach a round handle Rl with a model Engel structure to an Engel manifold M 0 with transverse boundary we have to ensure that the attaching map preserves the oriented contact structures and the intersection line field if we want to extend the Engel structure from M 0 to M 0 ∪ Rl by the model Engel structure. The given attaching map will not satisfy these assumptions in general. Thus we have to modify it and make a good choice of the model Engel structure on Rl . We assume that the contact structure on ∂+ M 0 is overtwisted and that the Engel framing extends from M 0 to a framing of M . Then the attaching map of Rl can be isotoped to a contactomorphism using • stabilizations of Legendrian curves in the case l = 1, • bypass attachments in the case l = 2 (this step makes essential use of the flexibility of overtwisted contact structures), • classification of overtwisted contact structures of Y. Eliashberg. Using the assumption that the Engel framing extends from M 0 to a framing of M we show that there is a model Engel structure such that the attaching map of Rl preserves the homotopy class of the intersection line field. After a modification of the Engel structure on M 0 (in order to adjust the intersection line field on ∂+ M 0 ) we can extend the Engel structure from M 0 to M 0 ∪ Rl such that the boundary is again transverse to the characteristic foliation. In order to iterate the construction it is necessary to ensure that the contact structure on the boundary of M 0 ∪ Rl is again overtwisted and that the Engel framing extends from M 0 ∪ Rl to M . Another result which can be obtained using round handles is the following theorem. Theorem 6.3. Let M1 , M2 be manifolds with Engel structures D1 , D2 such that the characteristic foliations admit closed transversals N1 , N2 . Then M1 #M2 #(S 2 × S 2 ) admits an Engel structure which coincides with D1 and D2 away from a neighborhood of N1 , N2 where all connected sums are performed. The characteristic foliation of the new Engel structure again admits a closed transversal. This theorem can be used to construct Engel manifolds — as for example (n + 1)(S 3 × S 1 )#n(S 2 × S 2 ) for n ≥ 1 — which are not covered by the construction of Geiges or by prolongation. EXISTENCE OF ENGEL STRUCTURES 83 This article is organized as follows. Section 2 contains several definitions and properties of Engel structures and related distributions. In Section 2.4 we introduce vertical modifications of the boundary of Engel manifolds. This construction will be used frequently in later sections. Our construction relies on several facts from 3-dimensional contact topology. We summarize them in Section 3 for the convenience of the reader. In Section 4 we discuss round handle decompositions of manifolds and model Engel structures on round handles. The various isotopies of attaching maps for round handles and the choice of the right model Engel structure are explained in Section 5. We give the proofs of Theorem 6.1 and Theorem 6.3 in Section 6. Acknowledgements. This article contains the main results of the author’s thesis. It is a pleasure for me to thank my advisor Dieter Kotschick for his continuous support and help. I would like to thank Kai Cieliebak, Yakov Eliashberg, Paolo Ghiggini and the referee for helpful discussions and suggestions. I am also grateful to the Studienstiftung des Deutschen Volkes for their financial support. 2. Engel structures and related distributions We first define all the distributions we will encounter in our constructions and the relations between them. This will lead to a proof of Proposition 2.12. To every Engel structure one can associate a foliation of rank 1. Hypersurfaces transverse to this foliation carry a contact structure together with a Legendrian line field. In Section 2.4 we explain vertical modifications of the boundary, a construction which will be used frequently. 2.1. Contact structures and even contact structures. Contact structures and even contact structures arise naturally on Engel manifolds. They will play an important role in all our constructions. Here we summarize the definitions and elementary properties. Definition 2.1. A contact structure C on a (2n − 1)-dimensional manifold N is a smooth subbundle of T N with corank 1 such that for every local defining 1-form α the rank of dαC is maximal, or equivalently α ∧ dαn−1 6= 0. If n is even the sign of α ∧ dαn−1 is independent of the choice of α. Thus a contact manifold of dimension 4n − 1 has a preferred orientation. In dimension 3 the orientability of M is the only obstruction to the existence of a contact structure by a result of J. Martinet in [Mar]. The following two theorems will be used later. We sketch the proofs because we will apply them in concrete situations. First we explain Gray’s theorem which states a remarkable stability property of contact structures. 84 THOMAS VOGEL Theorem 2.2 (Gray, [Gr]). Let Cs , s ∈ [0, 1] be a family of contact structures on N which is constant outside a compact subset of N . Then there is an isotopy ψs such that ψs∗ C0 = Cs . Proof. We assume that Cs is globally defined by a smooth family of 1-forms α(s). The proof without this assumption is slightly more complicated; cf. [Mar]. The desired isotopy is the flow of a time-dependent vector field Z(s). This is the unique vector field which is tangent to Cs = ker(α(s)) and satisfies (3) iZ(s) dα(s) = −α̇(s) on Cs . Because dα(s) is a nondegenerate 2-form on Cs , such a vector field exists and is uniquely determined. One can show by a direct calculation that the flow of Z(s) has the desired properties. An immediate consequence of Gray’s theorem is a normal form for the contact structure on neighborhoods of curves tangent to the contact structure, i.e. Legendrian curves. Corollary 2.3. If γ is a closed Legendrian curve in a 3-manifold with contact structure then there exist coordinates x, z, t on a tubular neighborhood of γ such that γ = {x = z = 0} and the contact structure is defined by dz −x dt. In particular, every point of a contact manifold admits a neighborhood with coordinates x, y, z such that the contact structure is defined by α = dz − x dy. Next we consider vector fields which preserve a given contact structure, so-called contact vector fields. First recall that to each contact form α one can associate a unique contact vector field R, the Reeb vector field, such that α(R) ≡ 1 and iR dα ≡ 1. Proposition 2.4. The map which assigns to each contact vector field X the function α(X) is a one-to-one correspondence isomorphism between contact vector fields and smooth functions on M . Proof. Injectivity follows immediately from Definition 2.1. Now let f be a smooth function on M and R the Reeb vector field. Since dαC is nondegenerate there is a unique vector field Y tangent to C such that (iY dα) C = −df C . Then f R + Y has the desired properties. This proves surjectivity. We will discuss several theorems from 3-dimensional contact topology in Section 3. Now we state the definition and some elementary properties of even contact structures. EXISTENCE OF ENGEL STRUCTURES 85 Definition 2.5. Let M be a 2n-dimensional manifold and E a distribution on M of corank one. If for every local defining 1-form α of E the 2-form dα has maximal rank on E, then E is an even contact structure. Equivalently, E is an even contact structure if for every local defining form α, the (2n−1)-form α∧dαn−1 has no zeroes. In dimension 4 this condition may be rephrased as [E, E] = T M . The definitions of contact structures and even contact structures are very similar. Still there are significant differences. One of them is the presence of a distinguished line field on a manifold with an even contact structure. Since E has rank 2n − 1, the rank of dαE is 2n − 2. Hence dαE has a kernel W ⊂ E of dimension 1 and it is easy to show that W does not depend on the choice of the defining form α. All flows which are tangent to W preserve E. Definition 2.6. The line field W is the characteristic line field of E. The foliation induced by this line field is called the characteristic foliation. A hypersurface is transverse if it is transverse to the characteristic foliation. The following lemma is a simple consequence of the definition of an even contact structure and its characteristic foliation. Lemma 2.7. Let E be an even contact structure on M and W be the characteristic foliation of E. If N is a transverse hypersurface, then T N ∩ E is a contact structure on N . If N 0 is another transversal such that two interior points p ∈ N and q ∈ N 0 lie on the same leaf Wp of the characteristic foliation, then the map obtained by following nearby leaves preserves the induced contact structures on neighborhoods of p and p0 As explained above, a contact structure on a manifold of dimension 2n − 1 induces an orientation of this manifold if n is even. In this situation an orientation of the characteristic foliation together with the contact orientation on a transversal define an orientation of M . Proposition 2.8. Let E be an even contact structure on a 4n-manifold M . Then an orientation of M induces an orientation of the characteristic line field W and vice versa. 2.2. Engel structures – Definition and first examples. If D is a distribution we denote the sheaf of all vectors which can be obtained as commutators of pairs of local sections of D by [D, D]. Definition 2.9. An Engel structure is a distribution D of rank two on a manifold M of dimension four with the following properties. 86 THOMAS VOGEL (i) E = [D, D] ⊂ T M is a subbundle of rank 3. (ii) T M = [E, E]. The second condition in Definition 2.9 implies that E = [D, D] is an even contact structure. To E we associate the characteristic foliation W of E. By the definition of W we have W ⊂ E. Lemma 2.10. If E = [D, D] is an even contact structure induced by Engel structure D, then W ⊂ D. Proof. Assume that Wp is not contained in Dp at p ∈ M . Let α be a local defining form for E on a neighborhood of p. If X, Y are two linearly independent local sections of D around p, then dα(Xp , Yp ) 6= 0 by the assumption on W. On the other hand dα(X, Y ) = −α([X, Y ]) = 0 since [X, Y ] is a local section of E = [D, D]. Thus the assumption Wp 6⊂ Dp leads to a contradiction. Together with Lemma 2.7 this allows us to give the following interpretation of the condition E = [D, D]. As one moves along the leaves of W, the plane field D rotates in E around the characteristic foliation without stopping. We have shown that an Engel structure D on M induces a flag of distributions (4) W ⊂ D ⊂ E ⊂ T M. Each of these distributions has corank one in the distribution containing it. By Proposition 2.8, an orientation of the characteristic foliation of an Engel structure induces an orientation of the underlying manifold and vice versa. In addition, E = [D, D] is oriented by the following proposition. Proposition 2.11. If D is an Engel structure, then the even contact structure E = [D, D] has a distinguished orientation. Proof. Let X, Y be local sections of D around p ∈ M such that X(p) and Y (p) are linearly independent. Then X(p), Y (p), [X, Y ](p) is an orientation of E(p) which is independent of the choice of X, Y . This leads to the following orientation conventions. (i) If W is oriented we orient M by Proposition 2.8. (ii) E = [D, D] is oriented according to Proposition 2.11. (iii) Transverse hypersurfaces are oriented by the induced contact structure. In particular transverse boundaries are oriented by this convention. (iv) If W is oriented, then we orient the contact structure on a transverse hypersurface such that the orientation of the contact structure followed by the orientation of W is the orientation of the even contact structure. EXISTENCE OF ENGEL STRUCTURES 87 Let M be an Engel manifold with oriented characteristic foliation and transverse boundary. We write ∂+ M for those connected components of ∂M where the characteristic foliation points out of M . The remaining boundary components are ∂− M = ∂M \ ∂+ M . It is clear from (4) that the existence of an Engel structure has strong implications for the topology of the underlying manifold. A proof of the following result can be found in [KMS]. It was known earlier by V. Gershkovic. Proposition 2.12. Let D be an oriented Engel structure on an oriented 4-manifold M . Then the tangent bundle of M is trivial. Proof. Consider the flag W ⊂ D ⊂ E ⊂ T M of subbundles of T M . According to our orientation conventions each of the quotient bundles D/W, E/D and T M/E is oriented. Thus (5) TM = W ⊕ D E TM ⊕ ⊕ W D E is isomorphic to the sum of four trivial line bundles. Under the assumptions of Proposition 2.12 we can choose oriented orthogonal complements in (4) after having fixed an auxiliary Riemannian metric. We obtain a trivialization of T M . Since we only chose the Riemannian metric, this framing is well defined up to homotopy. We will refer to such framings as Engel framings. A very simple example of an Engel structure is the standard Engel structure in the following Darboux theorem for Engel structures. Theorem 2.13 (Engel, [Eng]). Every point of an Engel manifold has a neighborhood with coordinates w, x, y, z such that the Engel structure is given by (6) ker(dz − x dy) ∩ ker(dx − w dy). Here, the characteristic foliation is spanned by ∂w and the even contact structure is defined by dz − x dy. We now discuss more interesting examples. Apart from the construction we present later, two other constructions of closed Engel manifolds can be found in the literature. Example 2.14. The starting point of the prolongation construction is a contact structure C on a 3-manifold N . The projectization PC of C is a circle bundle over N . The projection pr : PC → N of this fibration maps each Legendrian line to its base point in N . One can show that DC = v ∈ T[l] PC pr∗ (v) ∈ [l] 88 THOMAS VOGEL is an Engel structure whose characteristic foliation is tangent to the fibers of pr. The associated even contact structure is pr−1 ∗ C. Similarly, one obtains orientable Engel structures on the space of oriented Legendrian lines. The second previously known construction of closed Engel manifolds is due to H.-J. Geiges. It yields Engel structures on parallelizable mapping tori of compact 3-manifolds; cf. [Gei]. 2.3. Transverse hypersurfaces in Engel manifolds. Recall that transverse hypersurfaces carry a natural contact structure by Lemma 2.7. In this section we discuss a Legendrian line field induced by the Engel structure on a transverse hypersurface. This line field can be used to obtain a normal form for the Engel structure near a transverse hypersurface N . Definition 2.15. The intersection line field of D on N is the Legendrian line field T N ∩ D. If the Engel structure D is oriented, then we orient the intersection line field such that the orientation of W followed by the orientation of the intersection line field is the orientation of D. The following theorem, as well as Proposition 2.17, can be found in [Mo2] but according to that article they were known before. Both deal with Engel structures obtained by prolongation. If one applies the prolongation construction to the contact structure C on a transverse hypersurface N in an Engel manifold, then one obtains the manifold PC with its canonical Engel structure. Let σ be the section of PC which assigns to each point p of N the Legendrian line T N ∩ D(p). Theorem 2.16. Any sufficiently small tubular neighborhood of N in M is canonically diffeomorphic as an Engel manifold to a tubular neighborhood of σ in PC. The oriented contact structure and the intersection line field determine the Engel structure on a small neighbourhood of N up to isomorphy. Another property of prolonged Engel structures is that they admit many diffeomorphisms which preserve the Engel structure; cf. [Mo2]. Proposition 2.17. Let N1 , N2 be 3-manifolds with contact structures C1 , C2 and let ϕ : N1 → N2 be a contact diffeomorphism. The diffeomorphism between PC1 and PC2 which is induced by the action of ϕ on C1 preserves the Engel structures. Every diffeomorphism PC1 → PC2 with this property is of this form. We will mainly be concerned with the case that N is the transverse boundary of a compact Engel manifold. In the following section we will explain a modification of the Engel structure on a collar of the boundary which allows EXISTENCE OF ENGEL STRUCTURES 89 us to vary the intersection line field within its homotopy class. Therefore it suffices to control the homotopy class of the intersection line field and not the induced foliation. In the remaining part of this section we define simple invariants which determine the intersection line field on transverse boundaries of Engel manifolds. For this we assume that the characteristic foliation and the Engel structure are oriented. This induces orientations of the contact structure and the intersection line field on N . If we fix an oriented framing C1 , C2 of the contact structure, we can assign an element of H 1 (N ; Z) to a nowhere-vanishing Legendrian line field X as follows. There are unique functions f1 , f2 such that X = f1 C1 + f2 C2 . The element of H 1 (N ; Z) corresponding to X assigns the winding number of (f1 ◦ γ, f2 ◦ γ) : S 1 −→ R2 \ {0} around 0 to the homology class represented by γ. This depends only on the homotopy class of X as a nowhere vanishing Legendrian line field. In general we will not use a distinguished global framing but we will compare the homotopy classes of nowhere vanishing Legendrian vector fields using rotation numbers along Legendrian curves. Let X be a nowhere vanishing Legendrian vector field on N and let X, Y be an oriented framing of the contact structure on N . If γ is an immersed, closed Legendrian curve, then γ̇ is a Legendrian vector field along γ and there are unique functions f1 , f2 such that γ̇ = f1 X + f2 Y . Definition 2.18. The winding number of the map (f1 ◦ γ, f2 ◦ γ) : S 1 −→ R2 \ {0} around zero is the rotation number of X along γ. The sign of the rotation number changes if the orientation of γ, or if the orientation of the contact structure, is reversed. In particular, its sign changes if we change the orientation of the characteristic foliation of D, but it is independent of the orientation of D. The homotopy class of the intersection line field as a transverse hypersurface N is completely determined by the rotation numbera along a (finite) set of Legendrian curves representing a basis of H1 (N, R). 2.4. Vertical modifications of transverse boundaries. In this section we consider a manifold M with oriented Engel structure D with oriented characteristic foliation such that ∂M is transverse and compact. Vertical modifications of the boundary allow us to change the intersection line field on ∂M . In this construction we modify the Engel structure near the boundary such that the even contact structure remains unchanged. Assume that L is a Legendrian line field which is homotopic to the intersection line field LD of D. Then we 90 THOMAS VOGEL can modify D such that the intersection line field of the new Engel structure is L. This can be generalized to the case of a transverse hypersurface N , but then the possible changes of the intersection line fields depend on the behavior of the Engel structure along the leaves of the characteristic foliation through N . This problem will arise only in Theorem 6.5 at the end of this article. Let X0 be a Legendrian vector field which spans and orients the intersection line field on ∂M . Assume that the Legendrian vector field X1 on ∂M is homotopic to X0 through a family Xs , s ∈ [0, 1] of nowhere vanishing Legendrian vector fields. Our aim is to construct an Engel structure on M such that the intersection line field on ∂M is spanned and oriented by X1 without changing the even contact structure. We treat the boundary components ∂+ M where W points out of M . The components ∂− M = ∂M \ ∂+ M can be treated similarly. Choose a collar U ' ∂+ M × (−1, 0] of ∂+ M such that the second factor with its standard orientation corresponds to the oriented characteristic foliation of D. We write w for the coordinate on (−1, 0]. Furthermore let Y0 be a vector field such that X0 , Y0 is an oriented trivialization of the contact structure on ∂+ M . The even contact structure on U is spanned and oriented by X0 , Y0 , ∂w . There is a unique smooth function f : U ' ∂+ M × (−1, 0] −→ R which vanishes along ∂+ M such that ∂w and (7) X(p, w) = cos(f (p, w))X0 (p) + sin(f (p, w))Y0 (p) span and orient D(p, w). Because D is an Engel structure X, [∂w , X] and ∂w are linearly independent everywhere. This implies that f is strictly monotone along the leaves of W. By our orientation convention for the contact structure on transverse boundaries, f is strictly increasing. There is a unique family of real functions gs such that cos(gs (p))X0 + sin(gs (p))Y0 is a positive multiple of Xs and g0 = 0. Since ∂+ M is compact we can choose k ∈ Z such that g1 + 2πk is at least 2π. Now extend f from ∂+ M × (−1, 0] to a smooth function fb on ∂+ M × (−1, 1] such that fb is strictly increasing and coincides with g1 + 2πk along ∂+ M × {1}. We extend D from M to a distribution on M ∪∂ ∂+ M × [0, 1] by the span of the vector field ∂w and X(p, w) = cos(fb(p, w))X0 (p) + sin(fb(p, w))Y0 (p). The fact that ∂w fb > 0 implies that X and [∂w , X] are linearly independent. Moreover, ∂w together with X, [∂w , X], respectively, the horizontal lifts of X0 , Y0 , span the same even contact structure. Thus the extended distribution is an Engel structure whose intersection line field on ∂+ M × {1} is spanned by X1 . Using a flow tangent to the characteristic foliation one can identify M and M ∪∂ ∂+ M × [0, 1] so that the even contact structures are preserved. EXISTENCE OF ENGEL STRUCTURES 91 Definition 2.19. The modifications of an Engel manifold with boundary described above will be called a vertical modifications of the boundary. Let us summarize the discussion in the following theorem. Theorem 2.20. Let (M, D) be an oriented Engel manifold such that ∂M is transverse and compact. Assume that the Legendrian line field L on ∂+ M is homotopic to the intersection line field LD of D. Then L is the intersection line field of a suitable vertical modification of ∂+ M . There is an analogous construction in the context of contact structures. Let N be a compact 3-manifold carrying a contact structure C and a contact vector field V transverse to the boundary. For this, let X be a nowhere vanishing Legendrian vector field such that [V, X] and X are linearly independent. Assume that X1 is a Legendrian vector field along ∂N which is homotopic to the restriction of X to ∂N through nowhere vanishing Legendrian vector fields. Then we can modify X near the boundary of N such that the resulting Legendrian vector field X 0 coincides with X1 along ∂N and X 0 , [V, X 0 ] are linearly independent. Remark 2.21. In our applications the Legendrian vector field X1 is given only on a compact embedded submanifold U ⊂ ∂+ M . We then pick a positive function g1 + 2πk as above and extend it to a positive function ge which equals 2π outside of a small neighborhood of U . If U is not connected we can choose different values for k on each connected component of U . 3. Contact topology In this section we summarize facts from the theory of contact structures. Section 3.1 deals with Legendrian curves in contact manifolds. We discuss in particular stabilization of Legendrian curves. This will play an important role in the construction of attaching maps for round handles of index 1 carrying model Engel structures. In Section 3.2 we collect those facts about convex surfaces in contact manifolds which we will use in Section 3.3 for the construction of bypasses in overtwisted contact manifolds as well as in Section 4.5 and Section 4.6 for the construction of model Engel structures on round handles of index 2 and 3. Most of the material presented in this section (with the exception of the construction of bypasses in overtwisted contact manifolds in Section 3.3) can be found in [ABK], [EH1], [EH2], [Gi1], [Ho]. 92 THOMAS VOGEL 3.1. Properties of Legendrian curves. Throughout this section we consider an oriented contact structure C on a 3-manifold N . A curve is Legendrian if it is tangent to the contact structure. Definition 3.1. A framing (S, T ) of an embedded closed Legendrian curve γ is an oriented contact framing if (i) S is tangent to the contact structure and γ̇, S represents the orientation of the contact structure and (ii) T is transverse to the contact structure and γ̇, S, T represents the orientation induced by the contact structure. Every Legendrian curve has a contact framing which is well defined up to homotopy. (S, T ) and (S 0 , T 0 ) of γ are homotopic we write (S, T ) ∼ (S 0 , T 0 ). The homotopy class of framings of γ represented by contact framings is denoted by fr(γ). On the set of framings of γ there is a Z-action defined by m · (S, T ) (γ(t)) = cos(mt)S(γ(t)) + sin(mt)T (γ(t)), (8) − sin(mt)S(γ(t)) + cos(mt)T (γ(t)) . This action is transitive on the set of homotopy classes of framings such that γ̇ followed by the framing induces the contact orientation. When we reverse the orientation of γ the co-orientation of γ changes. Therefore the Z-action on the framings does not depend on the orientation of γ. Contact framings are related to the Thurston-Bennequin invariant of null-homologous Legendrian knots. Definition 3.2. Let γ be a null-homologous Legendrian curve in N . Fix a relative homology class [Σ] ∈ H2 (N, γ; Z) which is represented by an oriented surface Σ such that ∂Σ = γ and γ is oriented as the boundary of Σ. A new curve γ 0 is obtained by pushing γ slightly along a vector field which is transverse to C. The Thurston-Bennequin invariant tb(γ, [Σ]) is the homological intersection number of γ 0 with Σ. In Definition 3.2 the surface Σ singles out those framings of γ = ∂Σ whose first component is tangent to Σ. We write frΣ (γ) for the corresponding homotopy class of framings. The Thurston-Bennequin invariant measures the difference between frΣ (γ) and fr(γ) of γ (9) tb(γ, [Σ]) · frΣ (γ) = fr(γ). A simple application of Gray’s theorem (Theorem 2.2) shows that the contact framing of a Legendrian curve determines the isotopy type of the contact structure on a neighborhood of the Legendrian curve. Lemma 3.3. Let C0 , C1 be two contact structures on N which induce the same orientation of N and γ a curve tangent to both C0 and C1 . If the contact framings of γ with respect to C0 and C1 are homotopic, then there is an isotopy EXISTENCE OF ENGEL STRUCTURES 93 ψs , s = [0, 1], of N relative γ such that ψ1∗ C0 = C1 on a tubular neighborhood of γ. If C0 and C1 are oriented, then one can choose ψs such that ψ1 preserves also the orientations of the contact structures. Now we discuss the other classical invariant of null-homologous Legendrian curves. Definition 3.4. Let Σ be a connected orientable surface with ∂Σ = γ. Fix an oriented trivialization X, Y of C along Σ. There are unique functions fx , fy such that γ̇(t) = fX (t)X + fY (t)Y . The winding number of S 1 −→ R2 \ {0} t 7−→ (fX (t), fY (t)) around 0 is the rotation number rot(γ, [Σ]). Remark 3.5. In Definition 2.18 we defined the rotation number for Legendrian curves on transverse hypersurfaces in oriented manifolds with oriented Engel structures. In Definition 3.4 we fix an oriented trivialization of the oriented contact structure on a Seifert surface Σ of the Legendrian knot ∂Σ = γ and compare γ̇ with this trivialization. Now assume that C is the oriented contact structure on a transverse hypersurface of an oriented manifold with oriented Engel structure. In this case we can use the intersection line field as the first component of the trivialization of C over Σ. Thus in this situation the two rotation numbers in Definition 2.18 and Definition 3.4 are equivalent. Using contact framings and rotation numbers one can distinguish isotopy classes of Legendrian curves. This isotopy type can be changed using the following construction. Let γ be a closed Legendrian curve in a contact manifold. According to Corollary 2.3 there are coordinates x, z, t on a neighborhood of γ such that the contact structure is defined by dz − x dt and γ = {x = 0, z = 0}. The t-axis in both parts of Figure 1 represents the projection of a neighborhood of a segment of γ to the x, t-plane and ∂z is pointing outwards. Let σ be another smooth curve in the t, x-plane that coincides with the t-axis near its endpoints. Since the contact structure near γ is defined by dz − x dt the segment σ lifts to a Legendrian curve whose endpoints lie on γ if and only if the integral of the 1-form x dt along σ is zero. Then the lifts of curves σ + γ, respectively σ − γ, as in Figure 1, yield new closed Legendrian curves which are close to γ. They 94 THOMAS VOGEL represent the positive and negative stabilization of γ. x x σ+ γ γ t γ t σ− γ Figure 1 Up to isotopy through Legendrian curves the stabilized curves do not depend on the choices made above. In particular, positive and negative stabilizations commute up to Legendrian isotopy. The effect of stabilization on the rotation number with respect to a nowhere vanishing Legendrian vector field X is given by (10) rot(σ + γ, X) = rot(γ, X) + 1, rot(σ − γ, X) = rot(γ, X) − 1. Hence σ + γ, σ − γ and γ are pairwise nonisotopic as Legendrian curves. From the construction of σ ± γ it follows that (11) fr(σ + γ) = 1 · (ψ1∗ frC (γ)) = fr(σ − γ). This is coherent with the well known formula tb(σ ± γ) = tb(γ) − 1 for nullhomologous Legendrian curves by (9). 3.2. Facts from the theory of convex surfaces. Let N be a 3-manifold with an oriented contact structure C. Consider a properly embedded oriented surface Σ. The contact structure induces the singular foliation F = C ∩ T Σ on Σ. Usually this is called the characteristic foliation of Σ. We avoid this terminology since there is already a characteristic (nonsingular) foliation in the context of Engel structures. The singularities of F are those points p ∈ Σ where Cp = Tp Σ. From the orientations of Σ and C we obtain an orientation of F away from the singularities using the following convention: For each nonsingular point p of F we choose v ∈ Fp , vΣ ∈ Tp Σ \ Fp and vC ∈ Cp \ Fp such that (v, vC , vΣ ) is the contact orientation, (v, vΣ ) orients Σ and (v, vC ) orients C. Then v represents the orientation of Fp . Generically, singular points are nondegenerate. We say that a singular point is elliptic if its index is +1 and hyperbolic if the index is −1. When the orientation of C and the orientation of the surface coincide at a singular point of F, we say that this singularity is positive; otherwise it is negative. If we orient F according to our conventions, positive elliptic points are sources and negative elliptic points are sinks. EXISTENCE OF ENGEL STRUCTURES 95 Definition 3.6. Σ is called convex if there is a contact vector field transverse to Σ. Giroux studied closed convex surfaces in [Gi1]. If Σ has boundary we will usually assume that ∂Σ is Legendrian. In particular he showed that a closed embedded surface is generically convex (with respect to the C ∞ -topology). The analogous statement is in general not true when Σ has boundary (even when ∂Σ is Legendrian). For each boundary component γ ⊂ ∂Σ let t(γ, frΣ ) be the unique integer such that t(γ, frΣ ) · frΣ (γ) = fr(γ). If γ is a Legendrian knot and Σ is a Seifert surface for γ, then t(γ, frΣ ) is the Thurston-Bennequin invariant by (9). Proposition 3.7 (Kanda, [Ka1]). Let Σ be a compact, oriented, properly embedded surface with Legendrian boundary, and assume t(γ, frΣ ) ≤ 0 for all boundary components γ of Σ. There exists a C 0 -small perturbation near the boundary (fixing ∂Σ) which puts an annular neighborhood A of ∂Σ into a standard form, and a subsequent C ∞ -small perturbation keeping A fixed which makes Σ convex. Moreover, if V is a contact vector field defined on a neighborhood of A and transverse to A ⊂ Σ, then V can be extended to a contact vector field transverse to all of Σ. The singular foliation is enough to determine the contact structure on a small neighborhood of a surface up to conjugacy; i.e., if two contact structures induce the same singular foliation, then there is a diffeomorphism of a neighbourhood of Σ mapping one contact structure to the other ([Gi1]). If both contact structure are invariant under a vector field V which is transverse to Σ, then one has the following statement. Theorem 3.8 (Giroux, [Gi1]). Let Σ be a compact orientable convex surface with Legendrian boundary. Two R-invariant contact structures on Σ × R which induce the same orientation and the same singular foliation on Σ × {0} are isotopic. They are conjugate by a diffeomorphism ϕ × id, and ϕ is isotopic to the identity through diffeomorphisms of Σ that preserve the singular foliation. For a convex surface Σ with Legendrian boundary we fix a contact vector field V transverse to Σ. Definition 3.9. The dividing set ΓΣ of Σ is the set of points where V is tangent to the contact structure. 96 THOMAS VOGEL The dividing set contains much information about the contact structure near Σ. Giroux showed in [Gi1] that ΓΣ is a submanifold of Σ which is transverse to the singular foliation. Its isotopy class depends only on Σ and C but not on V . Definition 3.10. Let F be a singular foliation on Σ such that ∂Σ is tangent to F. A collection Γ ⊂ Σ of closed curves and arcs with end points on ∂Σ is said to divide F if there is a smooth area form ω on Σ such that on each connected component of the closure of Σ \ Γ there is a vector field X directing F such that (i) the divergence of X with respect to ω is positive everywhere, (ii) X points outward along those parts of the boundary which correspond to Γ. Theorem 3.11 (Giroux, [Gi1]). If Σ is a convex surface with Legendrian boundary, then ΓΣ divides the singular foliation on Σ. Conversely, let F be a singular foliation on a compact oriented surface Σ such that ∂Σ is tangent to F and Γ divides F. Then there is a positive R-invariant contact structure on Σ×R such that Σ×{0} is convex, the induced singular foliation on Σ × {0} is precisely F and such that Γ consists of those points where the contact structure is tangent to the second factor in Σ × R. This theorem implies in particular that the dividing set of a convex surface with Legendrian boundary is not empty. Assume that C is cooriented by α and that the orientation Σ followed by V is the contact orientation. Then the dividing set Γ separates the region Σ+ where α(V ) is positive from the region Σ− where α(V ) is negative. If Σ is closed, then the Euler number χ(C) of C viewed as an oriented bundle satisfies (12) χ(C) = χ(Σ+ ) − χ(Σ− ). If Σ is a Seifert surface for a Legendrian knot γ, then one can determine the Thurston-Bennequin invariant and the rotation number of γ using the following formulas from [Ka2], [Ho] 1 tb(γ) = − |Γ ∩ γ| (13) 2 rot(γ) = χ(Σ+ ) − χ(Σ− ). Next we consider deformations of the singular foliation. Let Σ be a convex surface with Legendrian boundary and fix a transverse contact vector field V . Definition 3.12. An isotopy Φs of a surface Σ is called admissible if Φs (Σ) is transverse to V for all s. The following theorem is a generalization of the Giroux flexibility theorem in [Gi1] where Σ is assumed to be closed. EXISTENCE OF ENGEL STRUCTURES 97 Theorem 3.13 (Giroux, Kanda, [Gi1], [Ka1]). Let F0 be the singular foliation on Σ induced by the contact structure and F1 a singular foliation which is divided by ΓΣ . Then there is an admissible isotopy Φs , s ∈ [0, 1], of Σ such that F1 is the singular foliation on Φ1 (Σ). Example 3.14. In this example we want to fix some terminology. Consider the R-invariant contact form α = cos(ϕ) dt+sin(ϕ) dx on T 2 ×R where x is the coordinate on the R-factor while ϕ and t correspond to T 2 = S 1 × S 1 . We say that the singular foliation F on T 2 × {0} is in standard form. The singularities of the singular foliation form two circles {ϕ = π/2} ∪ {ϕ = 3π/2}. These are called the Legendrian divides. The dividing set of T 2 × {0} has two connected components Γ = {ϕ = 0} ∪ {ϕ = π}. The curves tangent to ∂ϕ form the Legendrian ruling. By Theorem 3.13, the Legendrian ruling can be changed into any foliation by straight lines as long as the straight lines remain transverse to the dividing set. Using Theorem 3.13 we can isotope Σ such that a submanifold with certain properties becomes Legendrian. Definition 3.15. A union C of disjoint properly embedded arcs and closed curves on Σ is called nonisolating if (i) C is transverse to Γ and every arc begins and ends on Γ or on the boundary of Σ. (ii) the boundary of each component of Σ \ (Γ ∪ C) contains a segment of Γ. Theorem 3.16 (Kanda, Honda, [Ka1], [Ho]). Consider a collection C of properly embedded closed curves and arcs on a convex surface Σ with Legendrian boundary. If C is nonisolating there exists a C 0 -small admissible isotopy Φs , s ∈ [0, 1] such that (i) Φ0 = id and Φ1 (ΓΣ ) = ΓΦ1 (Σ) , (ii) Φ1 (C) is Legendrian. We finish this section with an important dichotomy of contact structures on 3-manifolds. Definition 3.17. An embedded disc with Legendrian boundary is an overtwisted disc if all singularities on the boundary have the same sign. A contact structure is called overtwisted if it contains an overtwisted disc. A contact structure is tight if it is not overtwisted. The following theorem shows that tight contact structures are more interesting than overtwisted ones in many aspects. 98 THOMAS VOGEL Theorem 3.18 (Eliashberg, [El1]). If two overtwisted contact structures on a closed manifold are homotopic as plane fields and induce the same orientation, then they are isotopic. For our purposes however, the flexibility of overtwisted contact structures will be very useful. Regions where a given contact structure is tight can be found using the following theorem which is probably due to E. Giroux; a proof can be found in [Col]. Theorem 3.19. If Σ 6= S 2 is a convex surface (closed or compact with Legendrian boundary) in a contact manifold (M, C), then Σ has a tight neighborhood if and only if the dividing set of Σ has no homotopically trivial closed curves. If Σ = S 2 , then Σ has a tight neighborhood if and only if ΓΣ is connected. 3.3. Bypasses in overtwisted contact structures. Consider a contact manifold (N, C) and a convex surface Σ ⊂ N which has empty or Legendrian boundary. Let ΓΣ be a dividing set of Σ. Definition 3.20. A bypass for Σ is an embedded half-disc D in N whose singular foliation has the following properties: (i) ∂D is the union of two Legendrian arcs γ1 , γ2 which intersect at their endpoints. (ii) D intersects Σ transversally along γ1 . There are no other intersection points. (iii) D admits an orientation such that – there are exactly two positive singularities along γ1 . These are the endpoints of γ1 . They are elliptic. – there is exactly one negative singularity on γ1 . It is elliptic. – there are only positive singularities along γ2 . They alternate between elliptic and hyperbolic. (iv) γ1 intersects ΓΣ in exactly three points. The intersections are transverse and correspond to the singularities of the singular foliation on D along γ1 . (v) The dividing set of D has exactly one connected component. A bypass for Σ allows us to isotope Σ such that the isotopy type of the dividing set changes as in Lemma 3.21. They where introduced by K. Honda in [Ho]. Requirement (v) in Definition 3.20 does not appear in [Ho] since all contact structures considered in that article are tight. In this situation, the dividing set ΓD of D is determined (up to isotopy) by (i)–(iv) but this is not true for overtwisted contact structures. Requirement (v) is necessary for the following bypass attachment lemma. EXISTENCE OF ENGEL STRUCTURES 99 Lemma 3.21 (Honda, [Ho]). Let D be a bypass for a convex surface Σ. There exists a neighborhood of Σ ∪ D which is diffeomorphic to Σ × [0, 1] such that Σ × {i} is convex for i = 0, 1. The dividing set of Σ × {1} can be obtained from the dividing set of Σ × {0} as in Figure 2. (The bypass is attached to the front. Figure 2 represents only a neighborhood of the attaching region of D.) γ1 Figure 2 Gluing together two bypasses along the boundary component containing the singularities with alternating signs, one obtains a disc with a singular foliation like the singular foliation on an overtwisted disc. Thus one can think of a bypass as one half of an overtwisted disc and it is not surprising that it is much easier to find bypasses if the contact structure is overtwisted rather than tight. In the following Proposition 3.22 we assume that γ1 is a Legendrian arc contained in the convex surface Σ. Since every arc γ10 which is transverse to Γ and intersects Γ as γ1 in Proposition 3.22 is nonisolating, we can apply Theorem 3.16 to find a C 0 small admissible isotopy of Σ such that γ10 becomes Legendrian, i.e. the assumption that γ1 is Legendrian is actually not restrictive. This result was certainly known to the experts and it might be due to K. Honda. It is implicitly mentioned in papers of J. Etnyre and K. Honda about Legendrian knots (note that a bypass attachment can be viewed as an inverse of stabilization in the context of Legendrian knots) and also in lectures by E. Giroux and K. Honda. However, it seems that a proof has not appeared in the literature yet. Proposition 3.22. Let Σ be a convex surface in a contact manifold and Dot an overtwisted disc disjoint from Σ. Assume that γ1 ⊂ Σ is an embedded Legendrian arc such that the endpoints lie on Γ and γ1 ∩ Γ contains three points. Then there is a bypass for Σ which intersects Σ along γ1 . (If Σ is separating, then the bypass lies in a component of the complement of Σ which is overtwisted .) Proof. Note that γ1 is automatically transverse to Γ. Let V be a contact vector field transverse to Σ. Consider the image R of γ1 under the flow ϕt of V for 0 ≤ t ≤ ε such that γ1 = R ∩ Σ. The curves ϕt (γ1 ), 0 ≤ t ≤ ε, are 100 THOMAS VOGEL Legendrian since ϕt is the flow of a contact vector field. If p ∈ γ1 ∩ Γ, then the segment ϕt (p), 0 ≤ t ≤ ε, is Legendrian. If q1 , q2 are points in different components of γ1 \ (γ1 ∩ Γ), then the union ΓR of the curves ϕt (qi ) with 0 ≤ t ≤ ε, i = 1, 2, divide the singular foliation on R. Since R has Legendrian boundary, Theorem 3.11 implies that R is convex. We orient R such that the singularities at the endpoints of γ1 are positive. By (13) tb(∂R) = −2, rot(∂R) = 1. The idea is to perform a Legendrian connected sum of the knots ∂R and ∂Dot together with a boundary connected sum of R and Dot . Doing this carefully enough, one obtains a bypass. (If Σ separates the manifold, then we find a bypass only on the overtwisted side of Σ.) Let us first explain the Legendrian connected sum of null-homologous Legendrian knots K1 , K2 in a contact manifold; cf. [EH2]. Let p1 ∈ K1 and p2 ∈ K2 . Choose a Legendrian curve γ between p1 and p2 such that γ does not meet the knots anywhere else and γ is not tangent to Ki in pi for i = 1, 2. By Corollary 2.3 there are coordinates x, z, t on a tubular neighborhood of γ such that γ corresponds to the t-axis and the contact structure is defined by dz − x dt. After a C 0 -small Legendrian isotopy of K1 and K2 we may assume that p1 , p2 are cusp points of the front projection, i.e the projection to the z, t-plane, and that the knots are oriented as in Figure 3. The Legendrian connected sum K1 #K2 is then formed using the dashed curves in Figure 3. z p2 p1 t K2 K1 Figure 3 Now consider Seifert surfaces Σ1 of K1 and Σ2 of K2 . We assume that Σ1 , respectively Σ2 , coincides with translates of K1 in the negative t-direction, respectively of K2 in the positive t-direction, on a neighborhood of p1 , respectively p2 . Such neighborhoods are shown in Figure 3. If we orient Σ1 and Σ2 such that K1 and K2 are oriented as boundaries, then p1 is a negative singularity and p2 is a positive singularity. EXISTENCE OF ENGEL STRUCTURES 101 We use the ribbon which is bounded by the dashed curves in Figure 3 to construct a Seifert surface Σ1 #Σ2 for the knot K1 #K2 . According to [EH2] (14) (15) tb(K1 #K2 , [Σ1 #Σ2 ]) = tb(K1 , [Σ1 ]) + tb(K2 , [Σ2 ]) + 1, rot(K1 #K2 , [Σ1 #Σ2 ]) = rot(K1 , [Σ1 ]) + rot(K2 , [Σ2 ]). Now we apply this to the Seifert surfaces R = Σ1 of ∂R = K1 and Dot = Σ2 of ∂Dot = K2 where Dot is oriented such that tb(∂Dot ) = 0 and rot(∂Dot ) = −1. Then we perturb R#Dot to a convex surface with boundary ∂R#∂Dot . The difficulty in showing that the perturbed surface is a bypass is to establish (v) of Definition 3.20. In order to do this we reduce the region where the Legendrian connected sum is performed such that the contact structure on this region is tight. By Theorems 3.16 and 3.13 we can assume that the singular foliation on Dot is of the form indicated in Figure 4 where the thickened circle is the dividing set. Then we can decompose Dot into two half-discs bounded by Legendrian unknots with Thurston-Bennequin invariant −1 and rotation number 0. The half-discs are separated by straight Legendrian arcs. The singular foliation near the unknots is in the standard form used in Proposition 3.7. p2 D1 − − + + Dot Figure 4 By the last part of Proposition 3.7 we can pretend that we form a boundary connected sum of the surfaces R with the left part Dl of Dot and that the perturbation of R#Dot only changes the half-disc Dl . The presence of the Legendrian curves in the middle of Dot prevents an interaction between the left and the right part of Dot . The union of tubular neighborhoods of R, of the Legendrian arc connecting R with Dot and of the left part Dl of Dot can be recovered in tight contact manifolds: Dl can be obtained applying Theorem 3.13 to a bypass in a tight contact manifold, hence a neighborhood of R#Dl is tight. The Thurston-Bennequin invariant of ∂R#∂Dl is −2 by (14). From (13) and Theorem 3.19 it follows that the dividing set on R#Dl (after this surface is perturbed to a convex surface) consists of exactly two arcs with endpoints on ∂R#∂Dl and no closed components. The notation R#Dl and R#Dot is 102 THOMAS VOGEL + − − + − Figure 5 slightly misleading because Dl , respectively Dot , is not a subset of R#Dl , respectively R#Dot , after these surfaces are smoothed and made convex. We are left with the two possibilities for the isotopy type of the dividing set of R#Dot shown in Figure 5. According to (13) the boundary of the left part of Figure 5 has rotation number 0 while in the right part its rotation number is −2. By (15) the rotation number of ∂R#∂Dot is 0. This shows that the dividing set on the Seifert surface of ∂R#∂Dot contains no closed components. The remaining conditions (i), (ii) and (iv) in Definition 3.20 are satisfied by construction. Condition (iii) can be achieved using Theorem 3.13. Thus R#Dot yields a bypass. 4. Round handle decompositions and model Engel structures This section provides the building blocks for our construction of Engel manifolds. We first recall several facts from [As] about round handle decompositions. In Section 4.2 we describe a perturbed version of the prolongation construction and first examples of Engel structures which are compatible with a round handle decomposition of the underlying manifold. Perturbed prolongation will be used for the construction of model Engel structures on round handles in Sections 4.3 to 4.6. 4.1. Round handle decompositions. Round handles were used by D. Asimov ([As]) for the study of flow manifolds. A flow manifold is a manifold M with a nonsingular vector field W transverse to the boundary which points outwards along ∂+ M and inwards along ∂− M . Definition 4.1. Rl = Dl × Dn−l−1 × S 1 is a round handle of dimension n and index l ∈ {0, . . . , n − 1}. The boundary ∂Rl is the union of the two subsets ∂− Rl = ∂Dl × Dn−l−1 × S 1 , ∂+ Rl = Dl × ∂Dn−l−1 × S 1 . A round handle is attached to a flow manifold by an embedding ψ : ∂− Rl −→ ∂+ M . One can extend W to a nonsingular vector field on M ∪ Rl which is again transverse to the boundary. This is impossible if one attaches an ordinary EXISTENCE OF ENGEL STRUCTURES 103 handle hl = Dl × Dn−l of index l ∈ {0, . . . , n} to M since this changes the Euler characteristic by (−1)l . Now assume that W spans the characteristic foliation of an Engel structure on M . If one extends the Engel structure from M to M ∪ Rl it is clear from the properties of transverse hypersurfaces that it is useful to ensure that the boundary of the new Engel manifold is again transverse to the characteristic foliation. Therefore round handles are suitable building blocks for the construction of Engel structures. Definition 4.2. If M is obtained from the disjoint union of finitely many round handles of index 0 by attaching round handles of higher index successively, i.e. [ M = ... R0 ∪ψ1 Rβ1 . . . ∪ψr Rβr with βi ∈ {1, . . . , n − 1} for i ∈ {1, . . . , r}, then we have a round handle decomposition of M . We can rearrange a given round handle decomposition of a manifold such that the round handles are ordered according to their index. Contrary to the case of ordinary handles, two round handles of the same index cannot be interchanged in general. If a closed manifold M admits a round handle decomposition, then its Euler characteristic χ(M ) has to vanish because we can use the round handle decomposition to find a nonsingular vector field on M . The converse direction is covered by the following theorem. Theorem 4.3 (Asimov, [As]). A closed, connected manifold of dimension n 6= 3 admits a decomposition into round handles if and only if χ(M ) = 0. This decomposition can be chosen such that there is only one round 0-handle and one round (n − 1)-handle. The analogous statement is wrong for manifolds of dimension 3; cf. [Mor]. In order to prove the last part of Theorem 4.3 one follows the proof in [As] starting with a decomposition of M into ordinary handles with only one handle of index 0, respectively n. One can decompose Rl into one ordinary handle of index l and another one of index l + 1 as follows (16) Rl = Dl × Dn−l−1 × S 1 = Dl × Dn−l−1 × (D1 ∪ D1 ) = Dl × (Dn−l−1 × D1 ) ∪ (Dl × D1 ) × Dn−l−1 = hl ∪ hl+1 . This allows us to obtain decompositions into ordinary handles from round handle decompositions. 104 THOMAS VOGEL 4.2. Perturbed prolongation. From now on we consider only 4-dimensional round handles. For the construction of Engel structures we will fix a set of particular Engel structures on Rl for each l = 0, 1, 2, 3. Definition 4.4. A model Engel structure on a round handle of index l is an Engel structure on Rl such that the characteristic foliation is oriented and transverse to ∂+ Rl and ∂− Rl . It points outwards along ∂+ Rl and inwards along ∂− Rl . The usual prolongation construction is described in Example 2.14 and deformations of certain prolonged Engel structures are discussed in [Mo2]. We now describe perturbations of Engel structures to be used later for the explicit construction of model Engel structures. Let α be a contact form on a compact manifold N such that there is a contact vector field V for C = ker(α) which is transverse to ∂N . Moreover we assume that C is trivial as a bundle and we fix a trivialization C1 , C2 of C. We use the same notation for the horizontal lifts to N × S 1 . Proposition 4.5. For k ∈ Z the distribution Dk on N × S 1 spanned by W = ∂t + εV and Xk = cos(kt)C1 + sin(kt)C2 is an Engel structure on N × S 1 for ε > 0 small enough and k 6= 0. For k = 0 we obtain an Engel structure if and only if C1 and [V, C1 ] are linearly independent everywhere. Then the characteristic foliation of Dk is spanned by W . In particular, the boundary is transverse and both C and the contact structure on ∂N × S 1 induce the same singular foliation on ∂N × {t0 } for every t0 ∈ S 1 . (17) Proof. In order to show that Dk is an Engel structure we first calculate [W, Xk ] = −k sin(kt)C1 + k cos(kt)C2 +ε(cos(kt)[V, C1 ] + sin(kt)[V, C2 ]). If k 6= 0 and ε > 0 is small enough, then W, Xk and [W, Xk ] are linearly independent. The case k = 0 is obvious. Since V is a contact vector field [Dk , Dk ] is generated by W, C1 and C2 . Hence E = [Dk , Dk ] is defined by (18) β = α − εα(V ) dt. Hence β and α induce the same singular foliation on surfaces with constant S 1 coordinate. From the fact that α is a contact form it follows that β defines an even contact structure for all ε > 0. This shows that Dk is an Engel structure. Notice that β(εV + ∂t ) = 0. Let h be the function with the property LV α = hα. Then L(εV +∂t ) (α − εα(V )dt) = εLV α − ε2 (LV α(V ))dt = εh(α − εα(V )) implies that W spans the characteristic foliation of Dk . EXISTENCE OF ENGEL STRUCTURES 105 If N = hl , then we obtain a model Engel structure on Rl = hl × S 1 if V points outwards along ∂+ hl = Dl × ∂D3−l and inwards along ∂− hl = ∂Dl × D3−l . Let us describe the orientation of E = [Dk , Dk ] : If k = 0 we obtain an Engel structure only if C1 and [V, C1 ] are linearly independent and then E is oriented by W, C1 , [V, C1 ]. For k > 0 the even contact structure E is oriented by W, C1 , C2 . If k < 0, then E has the opposite orientation. If p ∈ ∂N lies on the characteristic surface Σ = {α(V ) = 0} of V , then {p} × S 1 is tangent to the contact structure on ∂N × S 1 . For later use we determine the rotation number along these Legendrian curves. If k = 0 the rotation number is zero since the intersection line field along ∂N × S 1 is S 1 invariant. For k 6= 0 we may assume that C1 = V at p. Clearly (19) V = C1 = cos(kt)Xk − 1/k sin(kt)[W, Xk ] + Rε where the correction term Rε satisfies limε→0 Rε = 0. If we project all vector fields in (19) along W to ∂N × S 1 we obtain an analogous expression for ∂t . Notice that the projection of Xk to ∂N × S 1 spans the intersection line field. Hence the rotation number along {p} × S 1 is −|k| (cf. Definition 2.18). In order to obtain model Engel structures with positive rotation number along {p} × S 1 one can replace W by −∂t + εV , but doing so one changes the contact structure on ∂N × S 1 . In the cases N = hl for l = 0, . . . , 3 considered later we will construct a diffeomorphism f of Rl = hl × S 1 which preserves the contact structure on large parts of ∂− Rl and reverses the orientation of {p} × S 1 . Pushing forward the model Engel structures with f one obtains model Engel structures with nonnegative rotation numbers along {p} × S 1 . Before we discuss explicit model Engel structures let us first describe examples of Engel manifolds with a decomposition into round handles which carry model Engel structures. If N is decomposed into ordinary handles we obtain a round handle decomposition of N × S 1 such that the round handles are products of S 1 with ordinary handles from the decomposition of N . If V is transverse to the boundaries of all handles and points into the right directions, then N × S 1 carries an Engel structure such that all round handles carry model Engel structures. This situation arises in the context of convex contact structures defined in [ElG]. Definition 4.6. A contact structure C on a manifold N is convex if there is a proper Morse function g : N → [0, ∞) and a complete contact vector field V which is a pseudo-gradient for g, i.e. there is a Riemannian metric and a positive function s such that LV g ≥ skdgk2 . According to [Gi2] every contact structure on a 3-manifold is convex. From g one obtains a handle decomposition of N and applying Proposition 4.5 106 THOMAS VOGEL to convex contact structures one obtains examples of Engel structures with adapted round handle decomposition. 4.3. Model Engel structures on round handles of index 0. For our proof of Theorem 6.1 it is useful to have overtwisted contact structures on transverse boundaries of Engel manifolds. Since we will start our construction of Engel manifolds with a round handle of index 0 we define the model Engel structures on R0 such that the contact structure on ∂+ R0 is overtwisted. We use cylindrical coordinates on R3 . Consider the overtwisted contact structure ker(cos(r2 ) dz + sin(r2 ) dϕ) and the ball of radius r0 = 3π/2. The singular foliation on the boundary S(r0 ) admits a dividing set with three connected components. From Theorem 3.11 it follows that S(r0 ) is convex, i.e. there is a contact vector field V transverse to S(r0 ) which points out of the ball. Since the ball is contractible the contact structure admits a trivialization C1 , C2 . By Proposition 4.5 we obtain model Engel structures Dk , k ∈ Z \ {0}, on R0 . At this point we do not need to deal with the case k = 0. The singular foliation on S(r0 ) ⊂ ∂+ R0 is the same as the original singular foliation on S(r0 ). By Theorem 3.19 the contact structure on ∂+ R0 is overtwisted. Since π1 (SO(4)) = Z2 there are exactly two homotopy classes of oriented framings on R0 . The homotopy class of the Engel framing of Dk depends only on the parity of k. So we have shown Proposition 4.7. For each of the two homotopy classes of oriented framings on R0 there is a model Engel structure on R0 such that the Engel framing is homotopic to the given framing of the tangent bundle of R0 . Moreover, the induced contact structure on the boundary is overtwisted. Remark 4.8. J. Adachi asks in [Ad] whether it is possible to find an Engel structure on N × [0, 1] such that the boundary is transverse and the induced contact structures C0 on N × {0} and C1 on N × {1} are equivalent to given contact structures on N . One can show that C0 and C1 have to be homotopic as plane fields on N as in Lemma 5.7. The following example shows that C0 and C1 do not have to be equivalent. Consider the contact structure from above on B(r0 ). We remove a ball contained in a Darboux chart with convex boundary S 0 from the interior of B(r0 ) and choose the contact vector field V such that it is transverse to both S 0 and S(r0 ). By Proposition 4.5 we obtain an Engel structure on a manifold diffeomorphic to S 2 × [0, 1] × S 1 such that • the characteristic foliation is transverse to the boundary, • the contact structure on one boundary component is tight while it is overtwisted on the other boundary component. EXISTENCE OF ENGEL STRUCTURES 107 This answers the question completely for N = S 2 ×S 1 by the classification of contact structures on S 2 × S 1 . 4.4. Model Engel structures on round handles of index 1. On R1 = D1 × D2 × S 1 we denote the coordinate on D1 by x, the coordinates on D2 by y1 , y2 and the coordinate on S 1 by t. On h1 = D1 × D2 consider (20) C = ker(α = −dy1 − y2 dx − 1/2 x dy2 ), ∂ ∂ ∂ V = 1/2 y1 + y2 − 1/2 x . ∂y1 ∂y2 ∂x These choices are motivated by [Wei], [El2]: The resulting even contact structure on R1 is defined by the contraction of the symplectic form dy1 ∧dt+dx∧dy2 with the Liouville vector field ∂t + V . Notice that V is transverse to ∂± h1 and points outwards along ∂+ h1 = D1 × ∂D2 and inwards along ∂− h1 = ∂D1 × D2 . We apply Proposition 4.5 to (21) ∂ ∂ ∂ − + (y2 − 1/2 x) , ∂y2 ∂x ∂y1 ∂ ∂ ∂ C2 = [V, C1 ] = − − 1/2 + 1/2(y2 + x) , ∂y2 ∂x ∂y1 C1 = with ε = 1. The properties of the resulting distributions Dk are summarized in the following proposition. Proposition 4.9. The distributions Dk are model Engel structures on R1 for all k ∈ Z. They have the following properties. (i) The even contact structure E = [Dk , Dk ] is defined by β = α + 1/2 y1 dt. The orientation of the contact structure on ∂− R1 and ∂+ R1 with respect to the restriction of dβ is positive if k ≥ 0 and negative if k < 0. (ii) The curves γ± = {±1} × {0} × S 1 are Legendrian. The rotation number along them is −|k|. (iii) The rotation number of the intersection line field with respect to the Legendrian vector field ∂ ∂ Z = y2 + 1/2 y1 ∂t ∂x along {0} × {y1 = 0, y2 = 1} × S 1 is −|k| and it equals 0 along {0} × S 1 × {0}. This follows from the discussion of Proposition 4.5 and calculations using (20) and (21). Remark 4.10. The contact structure on ∂− R1 is defined by β− = −dy1 + 1/2 y1 dt − 1/2 x dy2 with x = ±1. It is invariant under the map fs induced by (y1 , y2 ) 7−→ (sy1 , sy2 ) for s ∈ (0, 1). In particular, if ψ : ∂− R1 −→ ∂+ M is an 108 THOMAS VOGEL embedding preserving contact structures, then the same is true for ψ ◦ fs . We can choose s so small that the image of ψ ◦ fs is contained in a given tubular neighborhood of ψ(γ± ). Up to now we have a model Engel structure on R1 for each orientation of the contact structure on ∂− R1 and each homotopy class of intersection line fields with negative rotation number along γ± . Moreover, we have a model Engel structure with rotation number 0 along γ± for one of the two possible orientations of the contact structure on ∂− R1 . In order to obtain the missing possibilities we extend the contact diffeomorphism f : (x, y1 , y2 , t) 7−→ (x, y1 , −y2 − 4xy1 , −t) of a neighborhood of γ± in ∂− R1 to a diffeomorphism of R1 . This map preserves γ± and the contact structure near ∂− R1 but it reverses the orientations of γ± and of the contact structure. We push forward the Engel structures Dk we have obtained so far using the extension of f . Then the rotation number along γ± changes its sign. Up to now all model Engel structures induce the same contact framing along γ± . In order to realize other contact framings one can push forward the model Engel structures obtained so far with the self diffeomorphisms Θm of R1 : (22) Θm : D2 × I × S 1 −→ D2 × I × S 1 (y1 , y2 , x, t) 7−→ (cos(mt)y1 − sin(mt)y2 , sin(mt)y1 + cos(mt)y2 , x, t) with m ∈ Z. This map preserves γ± as well as ∂− R1 . Now we have shown the following proposition. Proposition 4.11. For each oriented framing of γ± ⊂ ∂− R1 and k ∈ Z there is a model Engel structure on R1 such that the contact framing is homotopic to the given framing and the rotation number along γ± is k. Given one such model Engel structure, there is another model Engel structure which induces the same contact structure on ∂− R1 with the same rotation number along γ± but which induces the opposite orientation of the contact structure on ∂− R1 . For k, m ∈ Z let Dk,m for k, m ∈ Z be the push forward with Θm of a model Engel structure with rotation number k along γ± . The orientation of the contact structure on ∂± R1 does not appear in this notation. 4.5. Model Engel structures on round handles of index 2. The model Engel structures on R2 = D2 × D1 × S 1 depend on two parameters n, k which correspond to the homotopy class of the intersection line field near the torus EXISTENCE OF ENGEL STRUCTURES 109 T02 = ∂D2 × {0} × S 1 ⊂ ∂− R2 . The contact structure on ∂− R2 is essentially independent of the model Engel structure. Proposition 4.12. Given integers n ∈ Z and k ∈ Z\{0} there is a model Engel structure D = Dk,n on R2 with the following properties. (i) The characteristic foliation of D can be oriented such that it points outwards along ∂+ R2 and inwards along ∂− R2 . (ii) The singular foliation on T02 = ∂D2 × {0} × S 1 ⊂ ∂− R2 is in standard form. In particular, T02 is convex. The Legendrian ruling corresponds to the first factor of T02 = ∂D2 × {0} × S 1 . The dividing curves are tangent to the last factor. (iii) The rotation number of the intersection line field along γ = ∂D2 × {0} × {1} (with its orientation as boundary ∂D2 ) is 2n. (iv) The rotation number of the intersection line field along the Legendrian divides (with the canonical orientation of the last factor of ∂D2 ×{0}×S 1 ) is k 6= 0. (v) The orientation of the contact structure on ∂+ R2 can be chosen freely. All model Engel structures induce the same contact structure on a neighborhood of T02 ⊂ ∂− R2 . Proof. We choose the even contact structure on R2 = D2 × D1 × S 1 first. The starting point is a singular foliation F on a disc D2 where we use polar coordinates (r, ϕ). The coordinate on D1 is x. On A = {r > 1/2} we define F by cos(ϕ)dr. In addition we require that F admit a dividing set Γ with the following properties. (i) The straight arc γ0 from (r = 1, ϕ = 0) to (r = 1, ϕ = π) lies in Γ. In addition, Γ contains |n| closed circles. (ii) Except for γ0 , every component of Γ is closed and bounds a disc containing no other components of Γ. If n > 0 all closed components lie below γ0 in D2 \ γ0 and if n < 0 they lie in the upper half-disc. Figure 6 (below) shows a possible F for n = 2. The thickened curves divide F. Similar singular foliations can be found for other n. By Theorem 3.11 there is an R-invariant positive contact structure C on D2 × R, such that the induced singular foliation on D2 × {0} is F. We choose an R-invariant contact form α for C, such that α = cos(ϕ) dr + sin(ϕ) dx on A × R. The coordinate corresponding to the R-factor is x. This choice fixes an orientation of the contact structure. In order to find a contact vector field V and a 2-handle h2 ⊂ D2 × R such that V is transverse to ∂h2 we need to take some care since we know nothing about the region r < 1/2 except that ∂x is a contact vector field everywhere. We focus first on A × R. 110 THOMAS VOGEL + + − + Figure 6 Let g1 , g2 be smooth functions on A × R depending only on x. By the proof of Proposition 2.4 V = g1 (x) ∂ ∂ ∂ − g10 (x) cos2 (ϕ) + g20 (x) sin(ϕ) cos(ϕ) + g2 (x) ∂r ∂ϕ ∂x is the contact vector field associated to the function h = g1 (x) cos(ϕ)+g2 (x) sin(ϕ). We choose the functions g1 , g2 such that for x ≥ 43 a 0 for |x| ≥ 1 g1 (x) = g2 (x) = −a for x ≤ − 43 −1 for |x| ≤ 43 0 for − 12 ≤ x ≤ 12 . for a > 0. For these choices the contact vector field V associated to h can be extended by a · sgn(x)∂x on |x| ≥ 1 to a smooth contact vector field which we still denote by V . Finally, we extend V to a contact vector field on the whole of D2 × R. (For this it is enough to extend α(V ) and to apply Proposition 2.4.) The resulting contact vector field is transverse to ∂D2 × [−3/4, 3/4] and points inwards there. Now consider the pair of hypersurfaces defined by the equation |x| = 5/4 − r2 /2. Since r ≤ 1, both are contained in the region |x| ≥ 3/4 where g2 = ±a depending on the sign of x. If we fix a big enough, V is transverse to the hypersurfaces {|x| = 5/4 − r2 /2} and it points outwards. Then h2 = (r, ϕ, x) |x| ≤ 5/4 − r2 /2 is diffeomorphic to an ordinary handle of index 2 and V is transverse to both boundary components, Figure 7. Moreover V has the desired orientations along ∂± h2 . We chose F and C such that ∂D2 ⊂ h2 is Legendrian. Using (13) we can determine the rotation number of ∂D2 . We obtain (23) rotC (∂D2 ) = 2n. EXISTENCE OF ENGEL STRUCTURES 111 r x h2 Figure 7 By Remark 3.5 this is (up to sign) the rotation number of any model Engel structure on R2 whose even contact structure induces the singular foliation F on D2 ⊂ h2 × S 1 . Now fix an oriented trivialization C1 , C2 . We denote the horizontal lifts of C1 , C2 , V on R2 = h2 × S 1 by the same symbols. As usual t is the coordinate on S 1 . By Proposition 4.5 the distribution Dk spanned by W = ∂ + εV ∂t and Xk = cos(kt)C1 + sin(kt)C2 is a model Engel structure if ε > 0 is small enough and k ∈ Z \ {0}. The characteristic foliation of Dk is spanned by W . This vector field is transverse to ∂± R2 and it points in the desired directions. The even contact structure E = [Dk , Dk ] is defined by β = α − εα(V ) dt. Using the expressions for V, α, h and our choices of g1 , g2 we obtain β = cos(ϕ) dr + sin(ϕ) dx + ε cos(ϕ) dt e = A × {−1/2 ≤ x ≤ 1/2} × S 1 . The contact structure on ∂− R2 is defined on A by (24) β ∂− R2 = sin(ϕ) dx + ε cos(ϕ) dt − εg2 (x) sin(ϕ) dt. Restricting β to T02 = ∂D2 × {0} × S 1 we see that the singular foliation on T02 is in standard form. The curves ϕ = π/2 and ϕ = 3π/2 are the Legendrian divides and the Legendrian ruling is tangent to the foliation given by the first factor in T02 = ∂D2 × {0} × S 1 . For k > 0, the orientation of the even contact structure is W, C1 , C2 . This is the orientation used in (23); hence the rotation number of the intersection line field along ∂D2 is 2n. If k < 0 we obtain the opposite orientation: W, C1 , −C2 and therefore the rotation number of the intersection line field on ∂− R2 along ∂D2 has now the opposite sign, i.e. rot(∂D2 ) = −2n. The rotation number along the Legendrian divide is −|k| by Proposition 4.5. Let us summarize the properties of the model Engel structures we have obtained up to now. Recall that Dk depends not only on k but also on the choice of F at the beginning of the proof and that |n| is the number of closed components of the dividing set of F. In the following table |n| is the number of closed components of components. 112 THOMAS VOGEL k>0 k<0 Orientation of E/W C1 , C2 C1 , −C2 Rotation number ∂D2 × {0} × {1} 2n −2n Rotation number Legendrian divides −|k| −|k| Notice that since we fixed V and α on A × R, the contact structure on ∂− R2 is independent from k, n. The model Engel structures with positive rotation numbers along the Legendrian divides can be obtained by applying the involution f : R2 −→ R2 (r, ϕ, x, t) 7−→ (r, ϕ, −x, −t) to the model Engel structures we have obtained so far. The contact structure on {−1/2 ≤ x ≤ 1/2} ⊂ ∂− R2 is preserved by f but f reverses the orientation of the Legendrian divides; cf. (24). In particular, we can compare the orientations of the contact structure and the homotopy class of the intersection line fields with the corresponding properties of Dk . The model Engel structures f∗ Dk cover the cases missing in the table above (with the exception of k = 0). Using a singular foliation whose dividing set has more nonclosed components than F at the beginning of the proof of Proposition 4.12 one can construct model Engel structures on R2 such that the dividing set of T02 has 4, 6, . . . components. Remark 4.13. This remark is the analogue of Remark 4.10 for round handles of index 2. Equip R2 with a model Engel structure from Proposition 4.12. If ψ : ∂− R2 −→ ∂+ M is an embedding such that the image of T02 is convex and such that ψ preserves the isotopy class of the dividing set, then one can isotope ψ such that it preserves the contact structure. This follows almost immediately from Theorem 3.13 and Theorem 3.8, but notice that the contact structure on ∂− R2 is not invariant under ∂x . Since it is homotopic through contact structures to a ∂x -invariant contact structure, we can use Gray’s theorem (Theorem 2.2) to circumvent this problem. 4.6. Model Engel structures on round handles of index 3. We now come to the construction of model Engel structures on round handles of index 3. In the proof of Theorem 6.1 we will attach R3 to an Engel manifold M with transverse boundary such that the contact structure on ∂+ M is overtwisted. Hence the contact structure on ∂− R3 should also be overtwisted. Contrary to round handles of index 0 we will need all possible homotopy classes of intersection line fields on ∂− R3 . In particular we also need the case k = 0 in Proposition 4.5. EXISTENCE OF ENGEL STRUCTURES 113 If it were enough to have the unique positive tight contact structure on ∂− R3 ' S 2 × S 1 it would be easy to describe the model Engel structures explicitly. In the overtwisted case we were not able to find explicit formulas but we still can construct the model Engel structures. We use ordinary handles of dimension 3 to construct a contact structure together with a contact vector field V and a Legendrian vector field X such that X, [V, X] are linearly independent. The following construction is — up to a small modification — an example of Giroux’s construction of convex contact structures in [Gi1]. 4.6.1. Contact structures on ordinary handles of dimension 3. On h1 = let x be the coordinate on the first factor and y1 , y2 the coordinates on the second factor. Consider the contact structure C1 defined by α = dy1 +y2 dx and the contact vector field ∂ ∂ ∂ (25) V1 = 2y2 −x + y1 . ∂y2 ∂x ∂y1 D1 ×D2 A simple calculation shows that the Legendrian vector field X1 = ∂y2 + y2 ∂y1 − ∂x has the property that X1 , [V1 , X1 ] are linearly independent everywhere. The characteristic surface (i.e. the points where V is tangent to the contact structure) of V is the strip S1 = {(x, y1 , y2 ) ∈ h1 | y1 = xy2 }. The contact vector field V is transverse to ∂± h1 and it points outwards (respectively inwards) along ∂+ h1 = D1 × ∂D2 (respectively ∂− h1 = ∂D1 × D2 ). The intersection of S with ∂− h1 , respectively ∂+ h1 , consists of two arcs. On h2 = D2 × D1 ' h1 we use the same contact structure as on h1 and let X2 = X1 . The contact vector field is reversed, i.e. V2 = −V1 , so it points outwards along ∂+ h2 = ∂− h1 and inwards along ∂− h2 = ∂+ h1 . The characteristic surface S2 of V2 satisfies S2 = S1 . The singular foliation on ∂− h1 consists of parallel straight segments and Figure 8 shows the singular foliation on ∂+ h1 = ∂− h2 . Figure 8 On h3 = D3 consider the standard contact structure defined by α3 = dz + x dy and the contact vector field V3 = −x∂x − 2y∂y − 3z∂z . Let X3 = ∂x + x∂z − ∂y . Again a simple calculation shows that [V3 , X3 ] and X3 are linearly independent everywhere. The characteristic surface of V3 is S3 = (x, y, z) ∈ D3 z = −3/2 xy . In particular, the dividing set Γ on S 2 is connected. 114 THOMAS VOGEL Now we combine the handles with the contact vector fields and the Legendrian vector fields described above in order to construct a contact structure on D3 together with a contact vector field V and a Legendrian vector field C1 as in Proposition 4.5. We orient ∂± hi such that the orientation of ∂± hi followed by Vi gives the orientation of the contact structure. Moreover, we can associate a sign to each component of ∂± hi \ (Si ∩ ∂± hi ) as follows: If the orientation of Xi , [Vi , Xi ], Vi at p is the contact orientation, then the component containing p is positive. Otherwise this region is negative. Notice that along Si , the vector fields Xi , [Vi , Xi ], Vi are not linearly independent. Proposition 4.14. There is a contact structure C on D3 together with a contact vector field V and a Legendrian vector field X without zeroes such that [V, X] and X are linearly independent everywhere and V is transverse to ∂D3 . Moreover, the dividing set on ∂D3 has three connected components. Proof. Consider the overtwisted contact structure on R3 which is defined by α = cos(r2 ) dz + sin(r2 ) dϕp(in polar coordinates). By Theorem 3.11 the sphere S(r0 ) with radius r0 = 3π/2 around the origin is convex, i.e. there is a contact vector field V transverse to S(r0 ). We assume that V points inwards. Consider a collar U ' S(r0 ) × [0, ε], ε > 0, of S(r0 ) such that V corresponds to the standard vector field ∂τ induced by the second factor. Clearly, one can choose a nowhere vanishing section of ker(α) along S(r0 ) and extend it to a Legendrian vector field XU on U with the desired property. Since S(r0 ) is simply connected, all nowhere vanishing sections of C along S(r0 ) are homotopic. We orient the boundary of U such that the orientation of the boundary followed by V gives the orientation of the contact structure. In order to prove the proposition we want to extend the contact structure, XU and ∂τ to the interior of the sphere. To do so we attach the handles h1 , h2 , h3 . The dividing set ΓU on ∂+ U = S(r0 )×{ε} consists of three connected components γ0 , γ+ , γ− , as shown in the left part of Figure 9. Regions of ∂+ U where X, [V, X], V is the contact orientation are called positive and if X, [V, X], V represents the opposite orientation they are called negative. Now we attach h1 to U using an orientation preserving embedding ψ1 : ∂− h1 −→ ∂+ U such that ψ1 (p± ) ∈ γ± and such that ψ1 preserves the singular foliation. Moreover, we assume that ψ1 maps the positive, respectively negative, regions of ∂− h1 to the positive, respectively negative, regions of ∂+ U . Such a map exists because the dividing set on ∂− U is transverse to the singular foliation. It maps points where V1 is tangent to the contact structure to ΓU . Following [Gi1] one obtains a smooth contact structure C 0 and a smooth contact vector field on U 0 = U ∪ψ1 h1 . We can modify the boundary of U before we attach h1 such that there is a smooth section X 0 of C 0 with the 115 EXISTENCE OF ENGEL STRUCTURES + γ+ − + + p+ − γ0 γ0 − γ− − + p− Figure 9 property that X 0 , [V 0 , X 0 ] are linearly independent. Here ψ1 preserves positive and negative regions. The dividing set Γ0 on ∂+ U 0 contains γ0 . The other component consists of S ∩ ∂+ h1 together with the parts of γ+ , γ− which do not lie in the attaching region of h1 . The right part of Figure 9 shows Γ0 . In particular Γ0 has two connected components. The homotopy class of X 0 as a nowhere vanishing section of C 0 along the torus ∂+ U 0 is not fully determined by this construction. The restriction of X 0 to γ0 depends only on XU and it is therefore uniquely determined up to homotopy. On the other hand, the homotopy class of X 0 along a curve which intersects the belt sphere of h1 in exactly one point depends on the vertical modification of the boundary; cf. Remark 2.21. We will use this flexibility when we attach h2 . Using Theorem 3.11 we first deform the torus ∂+ U 0 by an admissible isotopy such that it is in standard form. The deformed space is still denoted by U 0 . We may vary the slope of the Legendrian ruling such that there is an orientation preserving embedding ψ2 : ∂− h2 −→ ∂+ U 0 which preserves singular foliations and which maps positive, respectively negative, regions of ∂− h2 to positive, respectively negative, regions of the boundary. Now we can extend the contact structure C 0 and the contact vector field 0 V from U 0 to U 00 = U 0 ∪ψ2 h2 using C2 and V2 . We denote the extensions by C 00 and V 00 . When we want to extend X 0 to U 00 using X2 we have to ensure that ψ2∗ (X2 ) is homotopic to X 0 and that the orientations of the contact structures given by X 0 , [V 0 , X 0 ] and X2 , [V2 , X2 ] are coherent. The latter requirement is satisfied since ψ2 maps positive, respectively negative, regions of ∂− h2 to positive, respectively negative, regions of ∂+ U 0 . In order to ensure that ψ2 (X2 ) is homotopic to X 0 we use the fact that the homotopy class of X 0 along ∂+ U 0 is not uniquely determined. Choosing different values for k in Remark 2.21 on the two connected components of the attaching region of h1 we can vary the homotopy class of X 0 along closed curves which intersect the belt sphere of h1 exactly once. In this way we can ensure 116 THOMAS VOGEL the ψ2∗ (X2 ) is homotopic to X 0 along the attaching curve of h2 . Now we can use vertical modifications of the boundary and we obtain a Legendrian vector field X 00 on U 00 such that X 00 and [V 00 , X 00 ] are linearly independent. The attaching curve of h2 intersects each component of Γ0 exactly once. The dividing set Γ00 of ∂+ U 00 contains both components of Γ0 with the segments contained in the image of ψ2 removed. The endpoints of the remaining curves are connected by the two segments S ∩ ∂+ h2 . Hence Γ00 is connected and ∂+ U 00 is diffeomorphic to a sphere. All nowhere vanishing Legendrian vector fields along ∂+ U 00 are homotopic. Using Theorem 3.11 again, we change the singular foliation on ∂+ U 00 such that it becomes equivalent to the singular foliation on ∂− h3 . We attach h3 using an embedding of ∂− h3 into ∂+ U 00 which satisfies the same conditions on the orientations as ψ1 and ψ2 . After a vertical modification of the boundary we can attach h3 to U 00 such that C 00 , V 00 , X 00 extend to a contact structure C on a ball, a contact vector field V and a nowhere vanishing section X of C with the desired properties. 4.6.2. Model Engel structures on R3 . We apply Proposition 4.5 to C, V and C1 = X constructed in the proof of Proposition 4.14. This yields model Engel structures Dk , k ∈ Z, on R3 such that the contact structure on ∂− R3 = ∂− U × S 1 is overtwisted by Theorem 3.19. Because S(r0 ) and the contact form α used in the proof of Proposition 4.14 are invariant under rotations around the z-axis we may assume that the same is true for V and α(V ). For s ∈ [0, 1] let βs = sr0 cos(r02 sin2 (θ)) dθ + sin(r02 sin2 (θ)) dϕ − α(V ) dt on ∂− R3 = S 2 × S 1 in spherical coordinates (θ, ϕ) on S 2 . According to (18), the contact structure on ∂− R3 is defined by β1 and βs is a contact form for all s. By Theorem 2.2 the contact structures defined by β0 and β1 are isotopic. The self diffeomorphism f of S 2 × S 1 with f (θ, ϕ, t) = (θ, −ϕ, −t) preserves β0 . Moreover f preserves the Legendrian curves {p} × S 1 when p ∈ {ϕ = 0} and α(V ) vanishes at p. The orientations are reversed by f . Hence f extends to a diffeomorphism fe of R3 which preserves the contact structure on ∂− R3 but reverses its orientation and which preserves a Legendrian curve on ∂− R3 but reverses its orientation. The rotation number of the intersection line field on ∂− R3 along {p} × S 1 is −|k| for the model Engel structures we have obtained so far. If we push these forward using fe, we obtain model Engel structures which represent the missing homotopy classes of intersection line fields on ∂− R3 . Thus we have proved the following theorem. Theorem 4.15. There are model Engel structures on R3 such that the induced contact structure on ∂− R3 is C = ker(β1 ) and all possible orientations of C and homotopy classes of intersection line fields are realized. EXISTENCE OF ENGEL STRUCTURES 117 5. Attaching maps for round handles In the following sections we explain how to extend Engel structures from M to M ∪ψ Rl using a suitable model Engel structure on Rl if ψ is an attaching map for a round handle of index l = 1, 2, 3. For this ψ has to have certain properties explained in Section 5.1. We will isotope ψ and apply vertical modification of the boundary to M to bring ψ into the desired form. This is discussed in Section 5.2 to Section 5.4. 5.1. Extending Engel structures. Let M be an Engel manifold with oriented characteristic foliation and transverse boundaries and Rl a round handle of index l ∈ {1, 2, 3} with a model Engel structure. Proposition 5.1. Assume that an embedding ψ : ∂− Rl → ∂+ M maps the oriented contact structure CR on ∂− Rl to the oriented contact structure CM on ∂+ M and preserves intersection line fields. Then the Engel structure extends from M to M 0 = M ∪ψ Rl by the model Engel structure on Rl . If the Engel structures are oriented, then one obtains an oriented Engel structure on M 0 if ψ preserves the orientation of the intersection line fields. Proof. Using the normal form theorem (Theorem 2.16) for neighborhoods of transverse hypersurfaces in Engel manifolds we can embed collars UR of ∂− Rl , respectively UM of ∂+ M , into PCR , respectively PCM . We identify UR and UM with their images. By the second part of Proposition 2.17 we obtain an embedding ψe : UR −→ PCM from ψ. Since ϕ preserves intersection line fields, ψe maps ∂− Rl ⊂ UR to ∂+ M ⊂ UM and because ψ preserves the orientation of the contact structure, e R ) and UM lie on opposite sides of the hypersurface ∂+ M ⊂ PCM . This ψ(U proves the claim. Strictly speaking M 0 is a manifold with corners. We smooth these corners by cutting out a piece of Rl such that the boundary of the resulting manifold is still transverse. Proposition 5.1 can of course be formulated with an Engel manifold instead of the round handle Rl . The following three sections describe how to choose model Engel structures on R1 , R2 , R3 and how to isotope the attaching map in order to satisfy the conditions in Proposition 5.1. It is enough to ensure that the attaching map preserves the homotopy class of the intersection line field since we can then apply vertical modifications of the boundary. Before we continue let us remark that we will always assume that the attaching map ψ : ∂− Rl −→ ∂+ M preserves the contact orientations. If this is not the case we can replace ψ by ψ ◦ f where f is the diffeomorphism of ∂− Rl 118 THOMAS VOGEL induced by complex conjugation of S 1 . We obtain diffeomorphic manifolds when we attach Rl to M using ψ or ψ ◦ f . 5.2. Attaching maps for round handles of index 1. The model Engel structures on round handles of index 1 induce contact structures on ∂− R1 such that the attaching curves γ± = {±1} × {0} × S 1 are Legendrian. Let ψ : ∂− R1 −→ ∂+ M be an embedding preserving the contact orientation. We want to isotope ψ such that the resulting map preserves oriented contact structures and the homotopy class of intersection line fields for a suitable choice of the model Engel structure Dk,m . After a vertical modification of ∂+ M we then obtain an Engel structure on M with R1 attached. It is a standard fact in contact topology that every embedded curve in a contact manifold can be isotoped to a Legendrian curve such that the isotopy never leaves an arbitrarily small tubular neighborhood of the original curve, [ABK]. Hence we may assume that ψ(γ± ) are Legendrian curves. Next we want to isotope ψ such that the resulting map preserves oriented contact structures and the homotopy class of intersection line fields along γ± for a suitable choice of model Engel structures on R1 . For this we stabilize the curves ψ(γ± ). We may assume that all stabilizations of ψ(γ± ) are carried out in disjoint tubular neighborhoods of the original curves. Then one can extend the isotopy of ψ from γ± to ∂− R1 and one obtains a stabilized attaching map. Let ψ+ and ψ− be the restrictions of ψ to {x = 1} × D1 × S 1 and {x = −1} × D2 × S 1 . By fr(γ± , m) we denote a contact framing of γ± if R1 carries the model Engel structure Dk,m . Since ψ is orientation-preserving there exist integers n+ , n− such that ψ maps a contact framing along γ± to a framing representing n± · fr(ψ± (γ± )) when R1 carries the model Engel structure Dk,0 . Proposition 5.2. One can choose a model Engel structure on R1 and stabilize ψ± such that the stabilized maps (i) send contact framings of γ± to framings of ψ± (γ± ) which are homotopic to a contact framing, and (ii) the rotation numbers of D along the stabilized Legendrian curves obtained from ψ+ (γ+ ) and ψ− (γ− ) are both equal, if and only if (26) n+ + rot(ψ+ (γ+ )) ≡ n− + rot(ψ− (γ− )) mod 2. Proof. We equip R1 with the model Engel structure Dk,0 . Let (S, T ) be a framing of γ. Because ψ is orientation-preserving, the framings m · ψ±∗ (S, T ) and ψ±∗ m · (S, T ) EXISTENCE OF ENGEL STRUCTURES 119 are homotopic. If we use the model Engel structure Dk,m instead of Dk,0 on R1 this implies ψ±∗ fr(γ± , m) = (m + n± ) · fr(ψ± (γ± )). In (11) we determined the effect of positive and negative stabilization on contact framings. Since we want the stabilized embeddings ψe± to map contact framings of γ± to framings of ψe± (γ± ) which are homotopic to contact framings, we have to apply positive or negative stabilization (n± +m)-times. Hence − + − n± + m has to be nonnegative. If n+ + , n+ , n− , n− ∈ N0 satisfy (27) − n+ + m = n+ + + n+ ≥ 0, − n− + m = n+ − + n− ≥ 0, then it follows from (10) that the rotation numbers of the stabilized Legendrian curves are given by − + − rot (σ+ )n+ (σ− )n+ ψ+ (γ+ ) = rot(ψ+ (γ+ )) + n+ + − n+ , + − − rot (σ+ )n− (σ− )n− ψ− (γ− ) = rot(ψ− (γ− )) + n+ − − n− . After sufficiently many stabilizations we also want the rotation numbers along the image of γ+ and γ− to be equal. This can be achieved if and only if we can solve (27) and (28) − + − n+ − − n− − n+ + n+ = rot(ψ+ (γ+ )) − rot(ψ− (γ− )) − + − with nonnegative integers n+ + , n+ , n− , n− and m ∈ Z. Then we can take − + − k = rot(ψ+ (γ+ )) + n+ + − n+ = rot(ψ− (γ− )) + n− − n− . From (27) and (28) we obtain the condition (26). Therefore this condition is necessary. Conversely, if (26) is satisfied, then the under-determined system of equations (10) and (28) admits solutions in Z. If we choose m large enough we can − + − achieve n+ + , n+ , n− , n− ∈ N0 . Note that in this proof we used only stabilization to modify Legendrian curves. Attaching a bypass to a convex surface along a segment of a Legendrian knot contained in the surface can be viewed as an inverse procedure of stabilization. Under the assumption that the contact structure on ∂+ M is overtwisted it is possible to prove Proposition 5.2 for model Engel structures with a fixed contact framing, i.e. without choosing m big enough as we did above. Let us explain the meaning of (26) in more topological terms. Consider an orientation-preserving attaching map ψ : ∂− R1 → ∂+ M . On T M we have the Engel framing and we may assume that its components except for the component spanning W form a framing of the tangent bundle of ∂+ M . Using 120 THOMAS VOGEL ψ we pull back this trivialization to ∂− R1 and add a vector field which points inwards and is transverse to ∂− R1 . We obtain a framing of T R1 along ∂− R1 . It is possible to isotope ψ such that we can extend the Engel structure from M to M with R1 attached only if we can extend the Engel framing from M to M ∪ψ R1 . This condition depends only on the Engel framing and the isotopy class of ψ. If ψ(γ± ) are Legendrian one can show that such an extension exists if and only if (26) is satisfied. On the other hand extending a framing from M to M ∪ψ R1 is a purely topological problem. Thus if we start with an attaching map ψ± and end up with a map violating (26) then it is not possible to construct an Engel structure on M ∪ψ R1 such that the Engel framing on M is homotopic to the Engel framing induced by D. Now if ψ : ∂− R1 −→ ∂+ M is an embedding such that ψ(γ± ) are Legendrian curves and ϕ preserves contact framings and rotation numbers along γ± then by Lemma 3.3 we can isotope ψ relative to γ± such that the resulting map preserves the contact structure on a tubular neighborhood of γ± . Using Remark 4.10 we can isotope the new attaching map such that its image lies in this tubular neighborhood. Theorem 5.3. Let M be an oriented Engel manifold with transverse boundary and assume that ψ : ∂− R1 −→ ∂+ M is an embedding and the trivialization of T M induced by the Engel structure can be extended to M ∪ϕ R1 . Then there is a model Engel structure on R1 such that ψ is isotopic to an embedding ψe which preserves oriented contact structures and the homotopy class of the intersection line field. 5.3. Attaching maps for round handles of index 2. All our model Engel structures on round 2-handles induce the same singular foliation on T02 = ∂D2 × {0} × S 1 ⊂ ∂− R2 . Now suppose that M is an Engel manifold with transverse boundary and ψ : ∂− R2 −→ ∂+ M is an attaching map which preserves the orientations induced by the contact structures. If we want to attach R2 to M and extend the Engel structure from M to M ∪R2 , we have to ensure that the attaching map preserves contact structures. By Theorem 3.11 and Theorem 3.8 together with Remark 4.13, it suffices to modify ψ such that after the deformation, the image of T02 is convex and the attaching map preserves singular foliation. Recall that the dividing set of T02 consists of two homotopically nontrivial circles. Let N = ∂+ M and T 2 = ψ(T02 ). We assume that the contact structure C on N is overtwisted. Using Lemma 3.21 and Proposition 3.22 we show how to bring T 2 into the desired form. It is clear how to obtain the desired isotopy of ψ from this. Theorem 5.4. Let T 2 be an embedded torus in an overtwisted contact manifold (N, C). Assume that C is orientable and that the Euler class of the 121 EXISTENCE OF ENGEL STRUCTURES restriction of C to T 2 is zero. Then we can isotope T 2 such that after the isotopy the singular foliation on the torus is in standard form. Moreover we can prescribe the slope of the dividing curves. After the isotopy, the complement of a tubular neighborhood of T 2 contains an overtwisted disc. Proof. It suffices to find a convex torus which is isotopic to the original one such that the dividing set consists of two homotopically nontrivial components which have the desired slope. Using the Giroux flexibility theorem (Theorem 3.13) one can arrange the singular foliation on T 2 such that it has standard form. We will frequently use Proposition 3.22. The following figures represent the dividing set on a torus before the bypass attachment. The thickened arc represents the attaching curve γ1 of the bypass. 1st Step. Let Dot be a convex overtwisted disc. We perturb the embedding of T 2 such that it becomes transverse to Dot . Using an extension of a radial vector field on Dot we can isotope T 2 such that after the isotopy T 2 ∩ Dot = ∅ and the resulting torus is convex. Since Dot is convex there is a neighborhood which is foliated by overtwisted discs. In the following we will always ensure that after each modification of the embedding of T 2 , there is an overtwisted disc disjoint from the deformed torus: If D is a bypass for T 2 , we choose the neighborhood of T 2 ∪ D in Lemma 3.21 so small that its complement still contains overtwisted discs. 2nd Step. In this step we remove all homotopically nontrivial components of the dividing set. If there are no such components we continue with step 3. If the dividing set contains more than two homotopically nontrivial components, then we reduce the number of its components of the dividing set using the bypass attachments in the left part of Figure 10 often enough. We end up with a dividing set which contains two homotopically nontrivial curves. We remove these components with the bypass attachment in the right part of Figure 10. γ1 γ1 Figure 10 3rd Step. Using the bypass attachment in Figure 11, we obtain two new components of the dividing set. Their slope depends on the attaching curve of the bypass. For a given identification T 2 ' S 1 × S 1 , we can achieve that the 122 THOMAS VOGEL γ1 Figure 11 new components of the dividing set are isotopic to {p} × S 1 for p ∈ S 1 . The dashed curve represents this circle. 4th Step. We are left with a convex torus whose dividing set contains exactly two homotopically nontrivial dividing curves σ1 , σ2 with the desired slope. If this is the entire dividing set we are done. Otherwise we consider the two annuli T 2 \ (σ1 ∪ σ2 ). We claim that if only one of these annuli contains other components of the dividing set Γ, then there is at least one component of Γ which bounds e containing another component of Γ. Assume that this is not true. a disc D Then T 2 \ Γ contains r > 0 discs, one annulus and one annulus with r holes. The Euler number χ(C, T 2 ) of the restriction of C to T 2 is (29) χ(C, T 2 ) = χ(T+2 ) − χ(T−2 ) = ±2r 6= 0. by (12). The sign depends on the orientations of T 2 and of the contact structure. But (29) contradicts our assumption on the Euler class of C. In order to reduce the number of connected components of Γ we perform a bypass attachment as indicated in the left part of Figure 12. Notice that this does not affect the homotopically nontrivial dividing curves. γ1 γ1 Figure 12 If both annuli T 2 \ (σ1 ∪ σ2 ) contain connected components of Γ we reduce the number of components using the bypass attachment in the right part of Figure 12. Done often enough, this yields the desired configuration of dividing curves on T 2 . A bypass attachment also affects framings. For our purpose, it is enough to show this for a particular bypass attachment. K. Honda described this effect in more detail; cf. Proposition 4.7 in [Ho]. The following lemma corresponds to the last part where one can find a proof of the following lemma. EXISTENCE OF ENGEL STRUCTURES 123 γ1 Figure 13 Lemma 5.5. Let T 2 = S 1 × S 1 be a torus in standard form contained in a contact manifold such that the Legendrian divides are isotopic to {1} × S 1 . Let X be a nowhere vanishing Legendrian vector field on the contact manifold. Assume that the rotation number of X along the Legendrian divides is zero and that it is even along S 1 × {0}. We attach a bypass as in Figure 13 to T 2 and bring the characteristic foliation in standard form such that the Legendrian ruling is still tangent to the foliation from the first factor in S 1 × S 1 . Then the rotation number along the Legendrian divides in the isotoped torus is odd (and therefore nonzero) while the rotation number along the Legendrian curve S 1 × {0} remains even. Proposition 5.6. Assume that M carries an oriented Engel structure with oriented characteristic foliation such that the boundary of M is transverse. Moreover, assume that the contact structure on ∂+ M is overtwisted. Let ψ : ∂− R2 −→ ∂+ M be an attaching map. We can extend the Engel structure from M to M ∪R2 using a model Engel structure if and only if the Engel framing on M extends to a trivialization of M ∪ R2 over the 2-cell e2 = D2 × {0} × {1} ⊂ R2 . The Engel structure on M ∪ R2 can be chosen such that the contact structure on ∂+ (M ∪ R2 ) is again overtwisted. Proof. The 2-cell e2 is of course attached when one uses the restriction of ψ to ∂e2 . Clearly, if the Engel framing cannot be extended over e2 , then it is impossible to extend the Engel structure to M ∪ψ2 R2 . The contact structure C on ∂+ M is oriented and the intersection line field yields a nowhere vanishing Legendrian vector field. Therefore the Euler characteristic of C viewed as a bundle is zero. We equip R2 with one of the model Engel structures from Section 4.5. By Theorem 5.4 and Remark 4.13 we can assume that ψ preserves contact structures. In particular, ψ(∂e2 ) is Legendrian. We first show that under the assumption of the proposition, the rotation number along ψ(∂e2 ) is even. For this we homotope the Engel framing on M such that the only component of the framing which is not tangent to ∂+ M spans the characteristic foliation. In this way we obtain a framing of ∂+ M . Using the attaching map to pull back this framing, we obtain a framing on ∂− R2 . Adding a vector field along ∂− R2 which spans and orients the characteristic foliation we obtain a framing of T R2 along ∂− R2 . 124 THOMAS VOGEL Since ψ preserves contact structures, the pull back of the component of the Engel framing which is orthogonal to the contact structure on ∂+ M is transverse to the contact structure on ∂− R2 . Without loss of generality, we assume that these components of the Engel framings are preserved by the attaching map. Then the pullback framing and the Engel framing on R2 have two components in common. When we want to compare the pull back framing with the Engel framing along e2 it is therefore enough to consider the rotation numbers along ∂e2 . By (iii) of Proposition 4.12, the rotation number along ∂e2 induced by the model Engel structure on R2 is even. If the rotation number of the pull back framing along ∂e2 is odd, then the pull back framing and the Engel framing are not homotopic along ∂e2 and the pull back framing cannot be extended over the disc D2 × {0} ⊂ h2 . This contradicts the assumption. Hence the rotation number along ψ(∂e2 ) is even. If the rotation number along the Legendrian divides of ψ(T02 ) is not zero, then by Proposition 4.12 we can choose a suitable model Engel structure such that the isotoped attaching map preserves the homotopy class of the intersection line fields. If the rotation number along the Legendrian divides of ψ(T02 ) is zero, then we attach a bypass as in Figure 13 to isotope ψ once again. By Lemma 5.5, the rotation number along the Legendrian rulings of the image of T00 remains even and the rotation number along the Legendrian divides is nonzero. The new attaching map preserves the singular foliation on T02 after one pushes forward the model Engel structures from Proposition 4.12 with the diffeomorphism Θ1 from (22). Restricting this diffeomorphism to ∂D2 ×{0}×S 1 we obtain a right-handed Dehn twist. Now we can choose the model Engel structure such that the new attaching map preserves oriented contact structures and the homotopy class of the intersection line field. After a vertical modification of the boundary of M , we can extend the Engel structure over R2 using a model Engel structure from Section 4.5. 5.4. Attaching maps for round handles of index 3. Let ψ : ∂− R3 −→ ∂+ M be an orientation-preserving attaching map for a round 3-handle carrying a model Engel structure (from Section 4.6). The model Engel structures on R3 induce overtwisted contact structures on ∂− R3 . By Theorem 3.18 we can isotope ψ to a contact diffeomorphism if and only if ψ is orientation-preserving and the image of the contact structure on ∂− R3 is homotopic to the contact structure on ∂+ M as a plane field. The contact structure on ∂− R3 is orientable and it extends to R3 as a plane field. The following lemma shows that this determines the homotopy class of the plane field completely. EXISTENCE OF ENGEL STRUCTURES 125 Lemma 5.7. There is a unique homotopy class of orientable plane fields on S 2 × S 1 = ∂D3 × S 1 which extends to D3 × S 1 . Proof. Recall from [HH] that the Grassmannian of oriented planes in R3 , respectively R4 , is Gr2 (3) ' S 2 , respectively Gr2 (4) ' S 2 × S 2 and that the inclusion R3 −→ R4 induces the diagonal map ∆ : Gr2 (3) ' S 2 −→ S 2 × S 2 ' Gr2 (4). Let C0 and C1 be two plane fields on S 2 × S 1 that extend to the interior of D3 × S 1 . We view C0 , C1 as maps from S 2 × S 1 to Gr2 (3) and their extensions as maps from D3 × S 1 to Gr2 (4). Because {0} × S 1 is a strong deformation retract of D3 × S 1 and Gr2 (4) is simply connected, there is a homotopy between the extensions of C0 and C1 through plane fields in T (D3 × S 1 ). Using the projection of Gr2 (4) ' S 2 × S 2 onto the first factor, we obtain a homotopy between C0 and C1 in T (S 2 × S 1 ). After we have isotoped ψ to a contact diffeomorphism we can choose a model Engel structure on R3 such that the orientation of the contact structure and the homotopy class of the intersection line field are preserved by the isotoped attaching map, cf. Theorem 4.15. Proposition 5.8. The Engel structure extends from M to M ∪ψ R3 if and only if the contact structure on ∂+ M extends to M ∪ψ R3 as a plane field. 6. Existence theorems Now we combine the tools from the previous sections to prove existence theorems for Engel structures. In Section 6.1 we show that every parallelizable manifold admits an orientable Engel structure. In Section 6.2 we explain the construction of Engel structures on the connected sum of two Engel manifolds with S 2 × S 2 if the characteristic foliations of the original Engel structures satisfy certain conditions. 6.1. Manifolds with trivial tangent bundle. In the proof of our main result Theorem 6.1 we use the model Engel structures from Section 4 and the results from Section 5 showing that we can extend an Engel structure from M 0 ⊂ M to M 0 ∪ Rl if M 0 has transverse boundary and the Engel framing extends to a framing of M 0 ∪ Rl . The remaining problem is to ensure that after we have constructed an Engel structure on M 0 ∪Rl , the new Engel framing extends from M 0 ∪ Rl to the whole of M . This allows us to perform the handle attachments successively. 126 THOMAS VOGEL Theorem 6.1. Every parallelizable 4-manifold admits an orientable Engel structure. Proof. Let M be a closed parallelizable manifold of dimension 4 and fix a trivialization T M ' M ×R4 of T M . We consider a round handle decomposition M = . . . . . . R0 ∪ψ11 R11 . . . ∪ψ1r1 R1r1 ∪ψ21 R21 . . . ∪ψ2r2 R2r2 ∪ψ3 R3 of M such that there is exactly one round 3-handle and one round 0-handle. Such a decomposition exists by Theorem 4.3. We will frequently isotope the attaching maps but this will not be reflected in the notation. We start with the round handle of index 0. By Proposition 4.7 we can choose a model Engel structure on R0 such that the Engel framing on R0 is homotopic to the original framing. In particular the Engel framing extends from R0 to M . The contact structure on ∂+ R0 is overtwisted by construction. Let M1i−1 be the round handle body obtained from R0 , R11 , . . . , R1i−1 . Assume that we have constructed an Engel structure on M1i−1 such that the contact structure on ∂+ M1i−1 is overtwisted. Assume moreover that throughout this process we have homotoped the trivialization of M chosen at the beginning such that it coincides with the Engel framing on M1i−1 . Then the Engel framing on M1i−1 can be extended to M1i−1 ∪ψ1i R1i . By Theorem 5.3, we can isotope ψ1i to an attaching map such that the Engel structure on M1i−1 extends to an Engel structure on M1i = M1i ∪ψ1i R1i using a model Engel structure on R1 from Section 4.4. In order to ensure that the contact structure on ∂+ M1i is again overtwisted, we show how to isotope ψ1i before the application of Theorem 5.3 such that its image is disjoint from an overtwisted disc in ∂+ M1i−1 . Let γ± = {±1} × {0} × S 1 be the attaching curves of R1i . Assume that ψ1i (γ± ) is transverse to an overtwisted disc Dot and let p be a point on Dot which does not lie on ψ1i (γ± ). Then use the flow of a radial vector field centered at p to isotope ψ1i such that the image of γ± becomes disjoint from Dot . The remaining steps in the modification of ψ, like making the attaching curves Legendrian and performing stabilizations, can be carried out in a small tubular neighborhood which is also disjoint from Dot . Next we compare the Engel framing and the original trivialization of M on M1i relative to M1i−1 . The cylinder D1 × {0} × S 1 ⊂ R1i = D1 × D2 × S 1 can be decomposed into a 1-cell e1 = D1 × {0} × {1} and a 2-cell e2 . The 1-cell is attached to M i−1 using the restriction of ψ1i to ∂e1 and e2 is attached to M1i−1 ∪ e1 . Since π1 (SO(4)) = Z2 there are two homotopy classes of oriented framings of T M along e1 . If necessary, we modify the model Engel structure on R1i such that the new Engel framing is homotopic to the given trivialization along e1 relative to EXISTENCE OF ENGEL STRUCTURES 127 the endpoints of e1 . For this let ρ : D1 −→ [0, 2π] be a smooth function which is constant near the boundary, ρ(−1) = 0 and ρ(1) = 2π. If we push forward the model Engel structure on R1 using the diffeomorphism F1 : R1i = D1 × D2 × S 1 −→ D1 × D2 × S 1 = R1i (x, y1 , y2 , t) 7−→ (x, cos(ρ(x))y1 + sin(ρ(x))y2 , − sin(ρ(x))y1 + cos(ρ(x))y2 , t), then we still obtain a smooth Engel structure on M1i by the choice of ρ. Because F1 interchanges the two homotopy classes of framings along e1 relative ∂e1 , the trivialization induced by the new Engel structure on R1 is now homotopic to the given trivialization of T M along e1 relative ∂e1 . We homotope the original framing so that it coincides with the Engel framing on M1i−1 ∪ e1 . Since π2 (SO(4)) is trivial the same is true for M1i−1 ∪ e1 ∪ e2 . Thus the Engel framing on M1i is now homotopic to the original framing relative to M1i−1 . In particular it extends from M1i to M . Hence we can iterate the procedure to attach round handles of index 1. During this process we ensure that the contact structure on ∂+ M1i is overtwisted for all i by choosing the attaching regions disjoint from Dot . In the next step we attach round 2-handles. Assume that we have already attached the first i − 1 round 2-handles such that on the resulting handle body M2i−1 there is an Engel structure extending the Engel structure on M1 . In addition we assume that the contact structure on ∂+ M2i−1 is overtwisted. Consider the attaching map ψ2i : ∂− R2i −→ ∂+ M2i−1 . of R2i . The contact structure on ∂+ M2i−1 is oriented and it has an orientable section, namely the intersection line field. Thus the Euler class of the contact structure, viewed as a bundle, vanishes. By assumption, the contact structure is overtwisted. According to Theorem 5.4 and Theorem 3.13 we can isotope ψ2i such that the resulting map preserves singular foliations. Recall that in Theorem 5.4 we also ensured that the attaching region of R2i is contained in a neighborhood of ψ2i (T02 ) which is disjoint from some overtwisted disc. In order to find a model Engel structure on R2i which extends the Engel structure on M2i−1 to an Engel structure on M2i = M2i−1 ∪ψ2i R2i we have to prove the following claim. We will deal with the extension of Engel framings in Claim 2. Claim 1. The Engel framing on M2i−1 extends to a trivialization of T M over e2 ⊂ R2i . The extension is unique up to homotopy relative ∂e2 . Proof of Claim 1. First we decompose each round 2-handle R2j , j ≤ i into one ordinary handle hj2 of index 2 and one ordinary handle hj3 of index 3 as in 128 THOMAS VOGEL (16). Then we rearrange the handles hj2 , hj3 , j ≤ i, such that the ordinary handles of index 2 are attached to M1 independently and the remaining ordinary 3-handles are attached to the resulting handle body. The Engel framing on M2i−1 extends over hi2 before we rearrange the handle decomposition if and only if the Engel framing extends from M1 over hi2 after the rearrangement. By construction the Engel framing on M1 is homotopic to the framing of M we chose at the beginning of the proof. In particular the Engel framing extends from M1 to hi2 after the rearrangement of the handles hj2 , hj2 , j ≤ i. This proves that the Engel framing can be extended from M2i−1 to M2i−1 ∪ ei2 . This extension is unique up to homotopy relative to ∂ei2 since π2 (SO(4)) = {0}. By Proposition 5.6 we can extend the Engel structure from M2i−1 to M2i using a model Engel structure from Section 4.5 on R2i . Since the contact structure on ∂+ M2i is again overtwisted we can iterate the procedure. When the last round 2-handle is attached we have constructed an Engel structure on M2 . In order to finish the construction we have to extend the Engel structure over the round 3-handle R3 . Claim 2. The Engel framing extends from M2 to M . Proof of Claim 2. We decompose all round 2-handles into ordinary handles hj2 , hj3 for 1 ≤ j ≤ r2 of index 2 and 3 and we rearrange the handles such that the 2-handles are attached to M1 . We also decompose R3 into an ordinary 3-handle b h3 and one ordinary 4-handle b h4 . In Claim 1 we have shown that the 1 Engel framing on M1 extends to M1 ∪ h2 ∪ . . . ∪ hr22 and that all such extensions are homotopic. Therefore, the Engel framing on M1 ∪ h12 ∪ . . . hr22 also extends to M . Next we reduce the problem to trivializations of bundles of rank 3. The first component W of the Engel framing is transverse to ∂M2 by construction. Thus W extends to a vector field without zeroes on M and we show next that we can find a trivialization of the entire manifold M whose first component is the extended vector field W . For this we equip M with an almost quaternionic structure such that the Engel framing and W, IW, JW, KW coincide on M1 ∪ h12 . . . ∪ hr22 . (This is possible because the Engel framing extends from M1 to M .) Then we can choose a trivialization of the orthogonal complement W ⊥ of W on the entire manifold M . (This trick can be found in [Gei].) For the proof of Claim 2 it suffices to show that we can extend the trivialization of W ⊥ induced by the Engel structure on M2 . On the 2-skeleton of M the SO(3)-bundle W ⊥ is trivial. Therefore we can lift it to an SU(2)-bundle. (Recall that Spin(3) = SU(2) = S 3 .) Since π2 (SU(2)) is trivial, the trivialization of W ⊥ induced by the Engel structure extends from M2 to b h3 . We fix such an extension. EXISTENCE OF ENGEL STRUCTURES 129 The obstruction for the extension of the trivialization from M2 ∪b h3 to M of induced by the Engel structure is a cocycle x in the cellular cochain group C 4 (M, M2 ∪ b h3 ; π3 (SU(2)) = Z) and x may depend on the choice of extensions of the trivialization over the 3-handles. The cocycle x represents a class [x] ∈ H 4 (M, M2 ∪ b h3 ; Z) which does not depend on the choice of trivializations on the 3-handles because it is a primary obstruction; cf. Theorem 34.2 in [Ste]. We have already shown that the Engel framing extends from the 2-skeleton to the whole of M . Thus [x] = 0. Since there is exactly one 4-handle we have C 4 (M, M2 ∪ b h3 ; Z) = H 4 (M, M2 ∪ b h3 ; Z) and hence x = 0. This implies that the Engel framing on M2 extends to M . W⊥ Since the Engel framing extends from M2 to M the same is true for the contact structure on ∂+ M2 viewed as a plane field. The contact structure on ∂+ M2 is overtwisted by construction. By Proposition 5.8 we can extend the Engel structure from M2 to M . This finishes the proof of the theorem. Let us compare the proof of Theorem 6.1 with the following characterization of parallelizable 4-manifolds. Theorem 6.2 (Hirzebruch, Hopf, [HH]). An orientable 4-manifold has trivial tangent bundle if and only if (i) the Euler characteristic vanishes, (ii) the second Stiefel-Whitney class w2 (M ) is zero and (iii) the signature of M is zero. Our construction relies on round handle decompositions and therefore condition (i) is used to apply Theorem 4.3 at all stages of the proof. The second Stiefel-Whitney class w2 (M ) of an orientable 4-manifold M is zero if and only if T M is trivial on the 2-skeleton of M . Thus condition (ii) of Theorem 6.2 is used when we attach round handles of index 1 and 2 in order to establish certain properties of rotation numbers. It is also used when we lift W ⊥ to an SU(2)-bundle in Claim 2. Finally the obstruction for the extension of a section of the SU(2)-bundle appearing at the end of the proof of Theorem 6.1 can be viewed as the primary obstruction to the construction of a section of the SU(2)-bundle over M since all sections of a SU(2)-bundle over M1 ∪ h12 . . . ∪ hr22 are homotopic. According to [GoS, p. 31], an SU(2)-bundle is trivial if and only if its second Chern class vanishes. On the other hand the second Chern class of the SU(2)-bundle is −p1 (W ⊥ )/4. Since p1 (T M ) = p1 (W ⊥ ), the vanishing of x corresponds to (iii) by the signature theorem of Hirzebruch. 6.2. Connected sums of Engel manifolds. Let M1 , M2 be manifolds with Engel structures D1 , D2 . The connected sum M1 #M2 does not admit an Engel 130 THOMAS VOGEL structure because its Euler characteristic is −2. On the other hand the Euler characteristic of M1 #M2 #(S 2 × S 2 ) is zero. Moreover, if M1 , M2 have trivial tangent bundles, then the same is true for M1 #M2 #(S 2 ×S 2 ) by Theorem 6.2. It is natural to try to construct an Engel structure on M1 #M2 #(S 2 × S 2 ) from D1 and D2 such that the resulting Engel structure coincides with the original Engel structures on large parts of M1 and M2 . In this section we show that this is possible under certain assumptions on D1 and D2 . For this we use model Engel structures on round handles of index 1 and 2. As model Engel structure D(1) on R1 we use D1 from Proposition 4.9. On R2 we will use a model Engel structure D(2) which has not appeared yet. We define D(2) as the span of ∂ ∂ ∂ ∂ − 1/2 y1 − y2 + 1/2 x , ∂t ∂y1 ∂y ∂x 2 ∂ ∂ ∂ ∂ X = cos(t) y2 + + + sin(t) 1/2 x . ∂y1 ∂x ∂y1 ∂y2 W= This model Engel structure can be obtained using Proposition 4.5. Its characteristic foliation is spanned and oriented by W . The model Engel structures D(1) and D(2) are very similar when one identifies R1 and R2 in the obvious way. The curves γ± = {0} × {±1} × S 1 in ∂+ R2 ' ∂− R1 are Legendrian and the rotation number of the intersection line field along these curves is −1 for both D(1) on R1 and D(2) on R2 . On a neighborhood of the boundary of R1 = R2 the even contact structures induced by D(1) and D(2) are homotopic through the family of even contact structures defined by (30) βs = −(1 − 2s)dy1 + 1/2 y1 dt − y2 dx − 1/2 x dy2 with s ∈ [0, 1]. By (20) the even contact structure on R1 is defined by β0 while the even contact structure on R2 is defined by β1 . The characteristic foliations of the even contact structures defined by βs are transverse to ∂− R1 ' ∂+ R2 and ∂+ R1 ' ∂− R2 for all s. Hence βs induces a family of contact forms on ∂− R1 and ∂+ R1 . We can also obtain contact embeddings of ∂+ R2 from attaching maps of R1 . Consider an Engel manifold M with transverse boundary and let ψ1 : ∂− R1 −→ ∂+ M be an embedding which preserves oriented contact structures. By Remark 4.10 we can isotope ψ1 such that the image of the resulting embedding lies in a small neighborhood of ψ1 (γ± ). The new embedding is again called ψ1 . For a suitable cut off function ρ (31) βes = −(1 − 2ρs)dy1 + 1/2 y1 dt − y2 dx − 1/2 x dy2 defines a family of contact structures on M which is constant away from a neighborhood of ψ1 (γ± ) and which is defined by βs near ψ1 (γ± ). This homotopy through contact structures has compact support. Using Gray’s theorem EXISTENCE OF ENGEL STRUCTURES 131 (Theorem 2.2) we obtain an isotopy fs of M such that f1∗ βe1 is a positive multiple of βe0 . Moreover, fs maps the Legendrian curves ψ1 (γ± ) to themselves for all s. The orientation of the contact structure induced by D(1) on ∂− R1 , respectively D(2) on ∂+ R2 , is given by the restriction of dβ0 , respectively dβ1 , to the contact structure. This implies that ψ2 = f1 ◦ ψ1 : ∂+ R2 −→ ∂+ M preserves oriented contact structures (but the orientations of the characteristic foliations do not match). Since fs preserves the Legendrian curves ψ1 (γ± ), the composition ψ2 = f1 ◦ ψ1 preserves the homotopy class of the intersection line field of D(2) . We will use the embedding ψ2 in the proof of the following theorem. Theorem 6.3. Let M1 , M2 be manifolds with Engel structures D1 , D2 such that both characteristic foliations admit closed transversals. Then there is an Engel structure on M1 #M2 #(S 2 × S 2 ) which coincides with D1 , respectively D2 , away from a neighborhood of the transversals where all connected sums are performed. The characteristic foliation of the new Engel structure again admits a closed transversal. Proof. Let us assume for the moment that the manifolds are oriented. Then the characteristic foliations are canonically oriented. For i = 1, 2 we cut Mi along a closed transversal Ni of the characteristic foliation. The resulting manifolds are still denoted by M1 and M2 . The boundary of Mi , i = 1, 2, has two connected components ∂+ Mi ' Ni ' ∂− Mi and there is a natural identification ϕ : ∂+ M1 ∪ ∂+ M2 −→ ∂− M1 ∪ ∂− M2 preserving oriented contact structures and the intersection line fields together with their orientations if D1 , D2 are both oriented. Let U1 ⊂ ∂+ M1 and U2 ⊂ ∂+ M2 be two balls. We will only need to modify intersection line fields on U1 , U2 and ϕ(U1 ), ϕ(U2 ). Therefore it is sufficient to orient D1 , D2 only along U1 , U2 . From this we obtain coherent orientations of the intersection line fields on U1 , ϕ(U1 ) and U2 , ϕ(U2 ). In U1 , U2 we choose Legendrian unknots K1 , K2 with Thurston-Bennequin invariant −2 and rotation number −1. One can obtain K1 , K2 by negative stabilization of the standard Legendrian unknot with Thurston-Bennequin invariant −1; cf. (10). Let ψ1 be an embedding of ∂− R1 into ∂+ M1 ∪ ∂+ M2 which preserves oriented contact structures such that ψ1 maps γ+ to K1 and γ− to K2 . Because the rotation numbers along γ± and K1 , K2 equal −1, we can apply vertical modification to ∂+ M1 ∪ ∂+ M2 such that ψ1 preserves oriented intersection line fields. 132 THOMAS VOGEL Using the discussion above we obtain an attaching map ψ2 : ∂+ R2 −→ ∂− M1 ∪ ∂− M2 for R2 with the model Engel structure D(2) which preserves oriented contact structures and the homotopy type of the intersection line field. Here we do not stick to the convention that R2 is attached using an embedding of ∂− R2 but notice that the orientation of the characteristic foliations match. After a suitable vertical modification of the boundary, we obtain an Engel structure on M = ((M1 ∪ M2 ) ∪ψ1 R1 ) ∪ψ2 R2 . Next we want to identify ∂+ M with ∂− M such that oriented contact structures and the homotopy class of the intersection line fields are preserved. Away from U1 ∪ U2 we can use ϕ. This map has an obvious extension to a map ∂− M −→ ∂+ M which coincides with the natural identification of ∂+ R1 ⊂ ∂+ M with ∂− R2 ⊂ ∂− M . Unfortunately the contact structures on ∂+ R1 and ∂− R2 are not preserved by this identification. However, the family of distributions defined by (31) away from ∂+ R1 together with the restriction of (30) to ∂+ R1 ' ∂− R2 , is a family of contact structures. By Theorem 2.2 there is an identification ϕ e : ∂− M −→ ∂+ M which preserves oriented contact structures and which coincides with ϕ away from a neighborhood of the attaching region of R1 . Next we compare the intersection line fields on ∂+ M and ∂− M . Near T02 = {x = 0} ⊂ ∂+ R1 the Gray isotopy constructed in the proof of Theorem 2.2 is induced by the Legendrian vector field ∂ 1 ∂ 8y2 y + y . (32) Z= 2 1 ∂t 2 ∂x 1 + y22 From (32) one sees that ϕ e maps the Legendrian curves {x = y1 = 0} ⊂ ∂+ R1 to themselves. The rotation number along these curves is −1 for both model Engel structures D(1) and D(2) . The rotation number of the intersection line field with respect to the Legendrian line field L spanned by y2 ∂t + 1/2 y1 ∂x is zero for both D(1) and D(2) (cf. Proposition 4.9). Since Z is tangent to L, the flow of this vector field preserves L. This implies that the identification ϕ e preserves the homotopy class of the intersection line field along T02 . Now let γ be a curve in ∂+ M which intersects T02 ⊂ ∂+ R1 exactly once such that the endpoints of γ lie outside of U1 and U2 . We have to ensure that the intersection line fields are homotopic along γ relative to the endpoints. For this we use the flexibility mentioned in Remark 2.21: Choosing suitable values of k for the vertical modification of the boundary when we attach R2 , 133 EXISTENCE OF ENGEL STRUCTURES we can ensure that ϕ e preserves the homotopy class of the intersection line field along γ relative to the endpoints. Then ϕ e preserves the homotopy class of the intersection line field. After a vertical modification of the boundary we can identify ∂+ M with ∂− M using ϕ e such that we obtain an Engel structure on the resulting manifold f M . The new Engel structure coincides with D1 and D2 away from sufficiently big neighborhoods of N1 and N2 . These transverse hypersurfaces are also f. contained in M f is diffeomorphic to M1 #M2 #(S 2 × S 2 ). In It remains to show that M order to prove this, we decompose R1 = h1 ∪ h2 and R2 = h02 ∪ R3 , where h1 , h2 , h02 , h3 are ordinary handles of index 1, 2, 2, 3, as in (16). The left part of Figure 14 shows the attaching curve and the framing of h2 after h1 is attached. One end of h1 lies in U1 and the other end lies in U2 . U1 0 1-handle U1 U2 U2 Figure 14 −4 Recall that R1 and R2 are attached to M1 ∪ M2 in a symmetric way. If we discard h2 and h02 and identify the boundary components of M1 ∪ M2 with h1 , h3 attached, then we obtain M1 #M2 . Using an isotopy of the attaching curves of h2 we can separate h2 , respectively h02 , from h1 , respectively h3 , such that h2 and h02 are attached along symmetric unknots in ∂+ M1 ∪ ∂+ M2 , respectively ∂− M1 ∪ ∂− M2 , which do not meet the attaching region of h1 , respectively h3 . Since the Thurston-Bennequin invariant of K1 and K2 is −2, the framing of the attaching curve of h2 is −4 with respect to the obvious Seifert surface of the attaching curve. The 2-handles h2 , h02 are attached by a doubling construction. The Kirby diagram therefore consists of spheres representing the attaching regions of h1 , the attaching curve of h2 and a zero-framed meridian of the attaching curve of h2 ; cf. [GoS, p. 133]. The ordinary handles of index 2 therefore account for the summand S 2 × S 2 . The right part of Figure 14 shows the Kirby diagram of M1 ∪ M2 with h1 , h2 , h02 attached. This proves the claim under the assumption that M1 and M2 are oriented. If this is not the case we can nevertheless orient Wi along the components of ∂Mi after we have cut along the hypersurfaces. For the proof it suffices to orient Wi along Ui such that it points outwards and along ϕ(U ) such that it points inwards. All constructions carry over to this situation. 134 THOMAS VOGEL In order to apply Theorem 6.3, one has to find Engel structures whose characteristic foliation admits a closed transversal. This is true for the Engel structures from the proof of Theorem 6.1. We can also apply Theorem 6.3 to Engel structures obtained by prolongation after we perturb them slightly in the neighborhood of a leaf of the characteristic foliation (as in Proposition 4.5). Corollary 6.4. If (N1 , C1 ) and (N2 , C2 ) are contact manifolds, then there is an Engel structure on PC1 #PC2 #(S 2 × S 2 ). Using this corollary one obtains Engel structures on Mn = n(S 3 × S 1 )#(n − 1)(S 2 × S 2 ). For n > 1 these manifolds can be viewed as circle bundles as follows. Take n−1 copies of S 2 × S 2 and let S 1 act by rotations on each factor. This action has four fixed points on each copy of S 2 × S 2 . In order to obtain Mn remove balls around the fixed points. The 4(n − 1) boundary components are all equivalent to the Hopf fibration on S 3 and we identify pairs of boundary components using orientation reversing diffeomorphisms. The resulting manifold is Mn and it is an S 1 -fibration over n(S 2 × S 1 ). One can show that it is impossible to construct an Engel structure on Mn , n > 1, using the method of Geiges or prolongation. Notice that since Mn is orientable, the characteristic foliation of all Engel structures on Mn are orientable. Using the Gysin-sequence one can show that the Euler class of all circle bundles with total space Mn , n > 1, is odd. Since the tangent bundle of an orientable 3-manifold is trivial, Mn cannot arise as a circle bundle of a subbundle of rank 2 of the 3-manifold. Let us return to the proof of Theorem 6.3 and discuss the meaning of the assumption that both Engel structures have characteristic foliations which admit a closed transversal. We do not make explicit use of the fact that N1 and N2 are closed transversals in the proof. But implicitly, this assumption is used when we apply vertical modification of the boundary. If N1 or N2 have boundary, then we have to ensure that everything is well defined at boundary points of Ni when we identify the boundary components of M . One can replace the existence of closed transversals in Theorem 6.3 by an assumption on the behavior of the Engel structure along leaves of the characteristic foliation. Let W(p) be a leaf of the characteristic foliation of an Engel structure D. We assume that W(p) is not closed. Using nearby leaves of the characteristic foliation we obtain identifications of Ep /Wp with Eq /Wq for all q on W(p) by Lemma 2.7. This allows us to define δ : W(p) −→ P(Ep /Wp ) ' S 1 q 7−→ [Dq ] EXISTENCE OF ENGEL STRUCTURES 135 where we identify Eq /Wq with Ep /Wp . This map is called the development map of W(p) and is an immersion since [D, D] = E; cf. [Ad], [Mo2]. For a given orientation of W(p) let W ± (p) be the segments of W which lie on the two sides of p. Now we define the twisting number tw+ (W(p)) = δ −1 ([Dp ]) ∩ W + (p) ; i.e., tw+ (p) is the number of half twists of D around W in E when one moves from p in the sense of the orientation along the leaf. We define tw− (W(p)) similarly. Finally, if W(p) is closed we define tw(W(p)) as the degree of δ : W(p) = S 1 −→ S 1 . Now we can state a modified version of Theorem 6.3 where the existence of closed transversals is replaced by a condition on twisting numbers. Theorem 6.5. Let M1 , M2 carry Engel structures D1 , D2 such that the characteristic foliation has nonclosed leaves W1 (p1 ) through p1 ∈ M1 and W2 (p2 ) through p2 ∈ M2 with the property (33) tw± (W1 (p1 )) ≥ C and tw± (W2 (p2 )) ≥ C for some positive constant C which is independent of the Engel structures. Then there is an Engel structure D on M1 #M2 #(S 2 ×S 2 ) which coincides with D1 , D2 outside of neighborhoods of p1 ∈ W1 and p2 ∈ W2 . Moreover one can choose D such that there is a leaf of the characteristic foliation which satisfies (33). If Wi (pi ) is closed one can replace (33) by tw(Wi (pi )) ≥ 2(C +1) for i = 1, 2. We just give a sketch of the proof and omit the details. Let p1 , p2 satisfy the assumption of the theorem. For i = 1, 2 we cut Mi along an embedded closed 3-ball Ni transverse to Wi , which is so small that the twisting numbers of all leaves of the characteristic foliation satisfy (33) after we have cut out Ni . Away from the boundary of Ni , the resulting space is a manifold with boundary. We can perform all constructions from the proof of Theorem 6.3 as long as we do not change anything near ∂Ni . Vertical modifications of the boundary are the only operations in the proof of Theorem 6.3 which affect ∂Ni . The problematic step is the vertical modification of the boundary used for the identification of the boundary components of M . The vertical modification of the boundary needed at this point is not allowed to change anything near ∂Ni . This means that the function fb used in Section 2.4 for the description of a vertical modification of the boundary has to be zero near boundary points of Ni . The condition (33) allows us to choose fb ≡ 0 on a collar of the boundary of Ni . We did not determine bounds for C. Since there are Engel structures (e.g. the standard Engel structure on R4 ) where the rotation number along all leaves of the characteristic foliation is zero, condition (33) is not fulfilled in general. 136 THOMAS VOGEL Ludwig-Maximilians University, Munich, Germany E-mail address: [email protected] References [Ad] J. Adachi, Engel structures with trivial characteristic foliations, Algebraic and Geom. Top. 2 (2002), 239–255. [ABK] B. Aebischer, M. Borer, M. Kälin, Ch. Leuenberger, and H. Reimann, Symplectic Geometry, Progr. Math. 124, Birkhäuser Verlag, Basel (1994). [As] D. Asimov, Round handles and nonsingular Morse-Smale flows, Ann. of Math. 102 (1975), 41–54. [Col] V. Colin, Recollement de variétés de contact tendues, Bull. Soc. Math. France 127 (1999), 43–69. [El1] Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989), 623–637. [El2] ——— , Topological characterization of Stein manifolds of dimension > 2, Internat. J. Math. 1 (1990), 29–46. [ElG] Y. Eliashberg and M. Gromov, Convex symplectic manifolds, Proc. Sympos. Pure Math. 52 (1991), 135–162. [ElM] Y. Eliashberg and N. Mishachev, Introduction to the h-Principle, Grad. Studies Math. 48, A. M. S., Providence RI (2002). [Eng] F. Engel, Zur Invariantentheorie der Systeme Pfaff’scher Gleichungen, Leipz. Ber. Band 41 (1889), 157–176. [EH1] J. Etnyre and K. Honda, Knots and contact geometry I: Torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001), 63–120. [EH2] ——— , On connected sums and Legendrian knots, Adv. Math. 179 (2003), 59–74. [Gei] H. J. Geiges, Review of [Mo2]; http://www.ams.org/mathscinet, MR 2001h:53127. [Gi1] E. Giroux, Convexité en topologie de contact, Comm. Math. Helv . 66 (1991), 637– 677. [Gi2] ——— , Géométrie de contact: de la dimension trois vers les dimensions supérieures, Proc. ICM 2002, Vol. III, 405–414. [GoS] R. E. Gompf and A. I. Stipsicz, 4-Manifolds and Kirby Calculus, Grad. Studies Math. 20, A. M. S., Providence, RI, 1999. [Gr] J. Gray, Some global properties of contact structures, Ann. of Math. 69 (1959), 421–450. [HH] F. Hirzebruch and H. Hopf, Felder von Flächenelementen in 4-dimensionalen Man- nigfaltigkeiten, Math. Ann. 136 (1958), 156–172. [Ho] K. Honda, On the classification of tight contact structures I, Geom. & Topology 4 (2000), 309–368. [Ka1] Y. Kanda, The classification of tight contact structures on the 3-torus, Comm. Anal. Geom. 5 (1997), 413–438. [Ka2] ——— , On the Thurston-Bennequin invariant of Legendrian knots and nonexactness of the Bennequin’s inequality, Invent. Math. 133 (1998), 227–242. [KMS] M. Kazarian, R. Montgomery, and B. Shapiro, Characteristic classes for the degenerations of two-plane fields in four dimensions, Pacific J. Math. 179 (1997), 355–370. EXISTENCE OF ENGEL STRUCTURES 137 [Mar] J. Martinet, Formes de contact sur les variétés de dimension 3, Proc. Liverpool Singularities Symposium II, Lecture Notes in Math. 209, 142–164, Springer-Verlag, New York (1971). [Mo1] R. Montgomery, Generic distributions and Lie algebras of vector fields, J. Differential Equations 103 (1993), 387–393. [Mo2] ——— , Engel deformations and contact structures, North. Calif. Sympl. Geom. Sem. A. M. S. Transl . 196 (1999), 103–117. [Mor] J. W. Morgan, Nonsingular Morse-Smale flows on 3-dimensional manifolds, Topology 18 (1979), 41–53. [Ste] N. Steenrod, The Topology of Fibre Bundles, Princeton Math. Series 14, Princeton Univ. Press, Princeton, NJ (1999). [Wei] A. Weinstein, Contact surgery and symplectic handle bodies, Hokkaido Math. J . 20 (1991), 241–251. (Received October 14, 2004)