Lectures on the topology of symplectic fillings of contact 3-manifolds

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1 Lectures on the topology of symplectic fillings of contact 3-manifolds BURAK OZBAGCI (Last updated on July 11, 2016) These are some lectures notes for my course in the Summer School "Mapping class groups, 3 and 4-manifolds", Cluj-Napoca, Romania, July 27-August 1, Most of this manuscript is an adaptation of my survey article On the topology of fillings of contact 3-manifolds that appeared in the Geometry & Topology Monographs Vol 19, Contents 1 Generalities on symplectic/stein and contact manifolds What is a Stein manifold? Symplectic geometry of Stein manifolds What is a contact manifold? Existence of contact structures Fillings of contact manifolds What is a strong symplectic filling? What is a Stein filling? What is a weak symplectic filling? Appendix: Existence of almost complex structures Basic results about fillings of contact 3-manifolds Legendrian knots and contact Dehn surgery Weinstein handle attachment The fundamental dichotomy: Tight versus overtwisted Different types of fillings Appendix: Brieskorn spheres

2 2 Ozbagci 3 Complex dimension two: Stein surfaces and contact 3-manifolds Lefschetz fibrations Contact open books The standard contact S 1 S Stein domains and Lefschetz fibrations Appendix: The embedding theorem Contact 3-manifolds admitting only finitely many fillings The 3-torus T Lens spaces Monodromy substitutions and rational blowdowns The lantern relation A generalized lantern relation: The daisy Symplectic fillings of lens spaces and rational blowdowns Appendix: Links of isolated complex surface singularities Contact 3-manifolds admitting infinitely many fillings Infinitely many pairwise non-homeomorphic Stein fillings Infinitely many exotic Stein fillings Stein fillings with arbitrarily large Euler characteristics Appendix: Fintushel-Stern knot surgery Bibliography 69 1 Generalities on symplectic/stein and contact manifolds 1.1 What is a Stein manifold? An analogue of Whitney Embedding Theorem, which says that Every smooth n- manifold admits a proper smooth embedding into R 2n, is not true for all complex manifolds. Suppose that M is a compact, connected, complex submanifold of some C N. Then the restriction of each coordinate function on C N to M must be constant by the "maximum principle". Therefore, a compact complex manifold of dimension 1 can not be a complex analytic submanifold of any C N.

3 3 Definition 1.1 A Stein manifold is an affine complex manifold, i.e., a complex manifold that admits a proper holomorphic embedding into some C N. An excellent reference for Stein manifolds in the context of symplectic geometry is the recent book of Cieliebak and Eliashberg [19]. For our purposes, we give an equivalent definition of a Stein manifold below. Definition 1.2 An almost complex structure on an even-dimensional manifold X is a complex structure on its tangent bundle TX, or equivalently a bundle map J : TX TX with J J = id TX. The pair (X, J) is called an almost complex manifold. It is called a complex manifold if the almost complex structure is integrable, meaning that J is induced via multiplication by i in any holomorphic coordinate chart. Let φ: X R be a smooth function on an almost complex manifold (X, J). We set d C φ := dφ J (which is a 1-form) and hence ω φ := dd C φ is a 2-form which is skew-symmetric (by definition). In general, ω φ may fail to be J-invariant, i.e, the condition ω φ (Ju, Jv) = ω φ (u, v) may not hold for an arbitrary almost complex structure J. However, Lemma 1.3 If J is integrable, then ω φ is J-invariant. Proof [19, Section 2.2] The claim can be verified by a local computation. The Euclidean space R 2n with linear coordinates (x 1, y 1,..., x n, y n ) has a standard complex structure J defined as J( x j ) = y j and J( y j ) = x j. The space (R 2n, J) can be identified (C n, i) via z j = x j + iy j, where we use linear coordinates (z 1,..., z n ) for C n and i denotes the complex multiplication on C n. Let φ: R 2n = C n R be a smooth function. Since dφ = φ + φ, dz j i = idz j and dz j i = idz j we have d C φ = j ( φ z j dz j i + φ z j dz j i ) = j ( i φ z j dz j i φ z j dz j ) = i φ i φ.

4 4 Ozbagci Using d = + again, we get dd C φ = ( + )(i φ i φ) = 2i φ and hence ω φ = 2i φ where more explicitly we can write φ = j,k 2 φ z j z k dz j dz k. The fact that φ is i-invariant finishes the proof. Definition 1.4 Let (X, J) be an almost complex manifold. A smooth function φ: X R is called J-convex if ω φ (u, Ju) > 0 for all nonzero vectors u TX. The condition ω φ (u, Ju) > 0 is often described as ω φ being positive on the complex lines in TX, since for any u 0, the linear space spanned by u and Ju can be identified with C with its usual orientation. Let g φ be the 2-tensor defined by g φ (u, v) := ω φ (u, Jv). The J-convexity condition in Definition 1.4 is indeed equivalent to g φ being positive definite, i.e., g φ (u, u) > 0 for any nonzero vector u TX. Exercise 1.5 If ω φ is J-invariant, then show that g φ is symmetric and H φ := g φ iω φ is a Hermitian form. Combining Lemma 1.3 and Exercise 1.5, we conclude that A smooth function φ: X R on any complex manifold (X, J) is J-convex if and only if the Hermitian form H φ is positive definite. Lemma 1.6 A smooth function φ: C n R is i-convex if and only if the Hermitian matrix ( 2 φ ) is positive definite at every point. z j z k Proof Let u = (u 1,..., u n ) and v = (v 1,..., v n ) C n. We set h jk := 2 φ z j z k compute ω φ (u, v) = 2i h j,k dz j dz k (u, v) = 4 Im ( ) h jk u j v k. j,k j,k and

5 5 Similarly, since g φ (u, v) = ω φ (u, iv), we have g φ (u, v) = 4 Re ( ) h jk u j v k, and thus H φ (u, v) = g φ (u, v) iω φ (u, v) = 4 j,k j,k h jk u j v k. Therefore we conclude that the Hermitian form H φ is positive definite (i.e.,h φ (u, u) > 0 for all u 0) if and only if the Hermitian matrix ( 2 φ z j z k ) is positive definite. Definition 1.7 A real valued function on X is called exhausting if it is proper (i.e., preimage of a compact set is compact) and bounded below. Exercise 1.8 Show that every Stein manifold admits an exhausting J-convex function. (Hint: Using Lemma 1.6, show that the map φ: C N R defined as φ(z) = z 2 is an i-convex function on C N with respect to the standard complex structure i: C N C N. Then use the fact that a Stein manifold is properly embedded in some C N.) The converse of Exercise 1.8 is due to Grauert [52]: Theorem 1.9 (Grauert) A complex manifold (X, J) is Stein if and only if it admits an exhausting J-convex function φ: X R. Every exhausting J-convex function on a Stein manifold (X, J) becomes an exhausting J-convex Morse function by a C 2 -small perturbation. The following result puts strong restrictions on the topology of Stein manifolds. Proposition 1.10 (Milnor) If φ: X R is a J-convex Morse function on a Stein manifold (X, J) of real dimension 2n, then the index of each critical point of φ is at most equal to n. According to [19, Corollary 3.4], Proposition 1.10 holds true for any J-convex Morse function φ : X R on any almost complex manifold (X, J). Below we describe their proof.

6 6 Ozbagci Proof Let p be a critical point of φ of index > n. Then, by definition of a Morse index, there is a linear subspace S T p X of (real) dimension > n on which the Hessian of φ is negative definite. Now since dim S = dim JS > n, we observe that S JS can not be empty and therefore S contains a complex line L because if v S JS, then Jv S JS. Let C be a complex curve (i.e., a 1-dimensional complex submanifold) in X passing through p such that TC = L. Since the Hessian is negative definite on L S, φ C attains a local maximum at p. But this is impossible since, as we will show below, φ C is (strictly) subharmonic, and the maximum of a subharmonic function cannot be achieved in the interior of its domain unless the function is constant. Recall that the restriction of J to a complex curve C is always integrable. Let U C be an open domain in C, and let ψ = φ U. If we apply Lemma 1.6 to ψ : U R, we see that ψ is i-convex if and only if 4 2 ψ z z = 2 ψ x ψ = ψ > 0, y2 where z = x + iy, and denotes the Laplacian. We conclude that φ C is J-convex if and only if φ C is (strictly) subharmonic. This finishes the proof of the claim above by observing that φ : X R is J-convex if and only if its restriction to every complex curve is J-convex. Remark 1.11 In [19], the authors preferred to name strictly plurisubharmonic functions as J-convex functions. If X is a smooth manifold of real dimension 2n, then by Proposition 1.10, a necessary condition for X to carry a Stein structure is that its handle decomposition does not include any handles of indices greater than n. In particular, all homology groups H i (X, Z) vanish for i > n. Note that there is another obvious necessary condition the existence of an almost complex structure on X (see Section 1.4). Eliashberg proved that, for n > 2, these two necessary conditions are also sufficient for the existence of a Stein structure: Theorem 1.12 (Eliashberg [25]) Let X be a 2n-dimensional smooth manifold, where n > 2. Suppose that X admits an almost complex structure J, and there exists an exhausting Morse function φ: X R without critical points of index > n. Then J is homotopic through almost complex structures to a complex structure J such that φ = g φ is J -convex, for some increasing diffeomorphism g: R R. In particular, the complex manifold (X, J ) is Stein.

7 7 For the case n = 2, the corresponding result is described in Theorem Symplectic geometry of Stein manifolds In the following, we briefly explain how symplectic geometry is built into Stein manifolds. Definition 1.13 A symplectic form on a 2n-dimensional manifold X is a differential 2-form ω that is closed (dω = 0) and non-degenerate, meaning that for every nonzero vector u TX there is a vector v TX such that ω(u, v) 0. The pair (X, ω) is called a symplectic manifold. A submanifold S X is called symplectic if ω S is non-degenerate and isotropic if for all p S, and for all u, v T p S, ω p (u, v) = 0. The non-degeneracy condition in Definition 1.13 is equivalent to ω n 0, where ω n denotes the n-fold wedge product ω... ω. Therefore, a symplectic manifold (X 2n,ω) has a natural orientation defined by the non-vanishing top form ω n. We will always assume that a symplectic manifold (X 2n,ω) is oriented such that ω n > 0. Since the cohomology class of any symplectic form ω is nonzero in H 2 (X, R), an orientable closed manifold X 2n can carry a symplectic form only if H 2 (X, R) is non-trivial. For example, the sphere S n is not symplectic for n > 2. Definition 1.14 We say that a symplectic form ω on an even dimensional manifold X is compatible with an almost complex structure J if ω is J-invariant and ω tames J, i.e., ω(u, Ju) > 0 for all nonzero vectors u TX. It is well-known (see, for example, [74, Proposition 4.1],[17, Proposition 13.1]) that Theorem 1.15 For any symplectic manifold (X,ω), the space of almost complex structures compatible with ω is non-empty and contractible.

8 8 Ozbagci Remark 1.16 This statement in fact holds true for any symplectic vector bundle over a smooth manifold, since only the non-degeneracy of the form ω is used in the proof. In short, every symplectic vector bundle is complex. Exercise 1.17 Show that an (almost) complex manifold need not be symplectic. (Hint: It is clear that S 3 S 1 does not admit a symplectic structure. But it admits a complex structure: Identify S 3 S 1 as the quotient of C 2 {0} by the action (z 1, z 2 ) (2z 1, 2z 2 ). The complex structure is induced from C 2.) Suppose that (X, J) is a complex manifold. For any φ: X R, we have φ is J-convex g φ defines a Riemannian metric ω φ is non-degenerate The upshot is that ω φ is a symplectic form compatible with J for a Stein manifold (X, J,φ). Therefore, ω φ is a Kähler form (a symplectic form compatible with an integrable almost complex structure) and hence every Stein manifold is a Kähler manifold. Definition 1.18 A vector field V on a symplectic manifold (X,ω) is called a Liouville vector field if L V ω = ω, where L denotes the Lie derivative. Suppose that (X, J,φ) is a Stein manifold. Let φ denote the gradient vector field with respect to the metric g φ, which is uniquely determined by the equation dφ(u) = g φ ( φ, u). Let α φ be the 1-form ω φ -dual to the vector field φ, i.e., α φ := ι φ ω φ, that is, α φ (v) = ω φ ( φ, v). Exercise 1.19 Show that φ is a Liouville vector field for ω φ. (Hint: First show that ι φ ω φ = d C φ and then use Cartan s formula.) Definition 1.20 A Weinstein structure on a 2n-dimensional manifold X is a triple (ω, V,φ), where ω is a symplectic form, φ: X R is an exhausting Morse function and V is a complete Liouville vector field which is gradient-like for φ. The quadruple (X, ω, V, φ) is called a Weinstein manifold.

9 9 Definition 1.21 ([71]) A vector field V on any m-dimensional manifold X is called gradient-like for a Morse function φ: X R if V φ > 0 away from the critical points of φ and for each critical point p of φ of index k, there is a local coordinate system (x 1,..., x m ) such that f is of the form f (p) x 2 1 x 2 k + x 2 k+1 + x 2 m and V coincides with the gradient of φ in these local coordinates. Note that a generic J-convex function is a Morse function. Moreover, for an exhausting J-convex Morse function φ: X R on a Stein manifold (X, J), the gradient vector field φ may be assumed to be complete, after composing φ by a suitable function R R ([19, Proposition 2.11]). We conclude that Every Stein manifold (X, J,φ) is a Weinstein manifold (X,ω φ, φ,φ). Moreover, the symplectic structure defined above on a Stein manifold (X, J) is independent of the choice of the J-convex function in the following sense: Theorem 1.22 [19, Chapter 11] Let φ j be an exhausting J-convex Morse function on a Stein manifold (X, J) such that φ j is complete for j = 1, 2. Then (X,ω φ1 ) is symplectomorphic to (X,ω φ2 ). Definition 1.23 Two symplectic manifolds (X 1,ω 1 ) and (X 2,ω 2 ) are said to be symplectomorphic if there exists a diffeomorphism ϕ: X 1 X 2 such that ϕ ω 2 = ω 1. A major theme in [19] is the reconstruction of Stein structures from Weinstein structures. 1.2 What is a contact manifold? The reader is advised to turn to [45] for a thorough discussion about the topology of contact manifolds.

10 10 Ozbagci Definition 1.24 A contact structure on a (2n + 1)-dimensional manifold Y is a hyperplane field ξ = ker α TY, where α is a (local) 1-form such that α (dα) n 0. The 1-form α is called a contact form and the pair (Y,ξ) is called a contact manifold. Exercise 1.25 α defining ξ. Show that the condition α (dα) n 0 is independent of the choice of In these notes, we assume that every contact structure ξ is co-orientable, which, by definition, means that α is global 1-form. This is equivalent to the quotient line bundle TY/ξ being trivial. It follows that Y is necessarily orientable since α (dα) n is a non-vanishing top-dimensional form, i.e., a volume form on Y. Moreover, ξ is called co-oriented if an orientation for TY/ξ is fixed. When Y is equipped with a specific orientation, one can speak of a positive or a negative co-oriented contact structure ξ on Y, depending on whether the orientation induced by ξ agrees or not with the given orientation of Y. The contact condition in Definition 1.24 is equivalent to dα ξ being non-degenerate. In particular, (ξ, dα ξ ) is a symplectic vector bundle, where for any co-oriented contact structure ξ, the symplectic structure on ξ p is defined uniquely up to a positive conformal factor. Definition 1.26 Two contact manifolds (Y 1,ξ 1 ) and (Y 2,ξ 2 ) are said to be contactomorphic if there exists a diffeomorphism ϕ: Y 1 Y 2 such that ϕ (ξ 1 ) = ξ 2. Example 1.27 In the coordinates (x 1, y 1,..., x n, y n, z), the standard contact structure ξ st on R 2n+1 is given as the kernel of the 1-form n dz + x i dy i. The following theorem shows that contact topology is a global phenomenon. i=1 Theorem 1.28 (Darboux) Any point in a contact manifold has a neighborhood contactomorphic to a neighborhood of the origin in the standard contact R 2n+1.

11 11 An important class of submanifolds of contact manifolds is given by the following definition. Definition 1.29 A submanifold L of a contact manifold (Y 2n+1,ξ) is called an isotropic submanifold if T p L ξ p for all p L. An isotropic submanifold of maximal dimension n is called a Legendrian submanifold. Exercise 1.30 Show that an isotropic submanifold L of a contact manifold (Y 2n+1,ξ) is at most n-dimensional. This is why a contact structure is often described as a maximally non-integrable tangent hyperplane field. (Hint: If α is a contact 1-form and i : L Y denotes the inclusion, then i α 0 and thus i dα 0.) Existence of contact structures Theorem 1.31 (Lutz-Martinet) Every co-oriented 2-plane field on a closed orientable 3-manifold is homotopic to a contact structure. Exercise 1.32 Show that there are infinitely many homotopy classes of co-oriented 2-plane fields on any closed orientable 3-manifold. (Hint: Consider the co-oriented 2-plane fields with vanishing Euler class. These are homotopic over the 2-skeleton, but their extension to the top cell may be non-homotopic distinguished by the Hopf invariant Z.) Remark 1.33 The classification of homotopy classes of co-oriented 2-plane fields on closed oriented 3-manifolds is well-understood (cf. [50, 51, 84, 45]). Although every closed orientable 3-manifold admits infinitely many (non-homotopic) contact structures by Theorem 1.31, there is an obstruction to the existence of contact structures on odd-dimensional manifolds of dimension 5. Suppose that n > 0 and let ξ denote a co-oriented contact structure on a closed manifold Y 2n+1. Then the tangent bundle of Y has a splitting as TY = ξ R, where R is the trivial (and hence orientable) line bundle oriented by the co-orientation of ξ. The contact structure ξ = ker α can be viewed as a symplectic vector bundle on Y since dα is symplectic on ξ, whose conformal class is independent of choice of α. Therefore ξ admits a compatible complex vector bundle structure (see Remark 1.16).

12 12 Ozbagci Such a splitting of TY is called an almost contact structure which is equivalent to a reduction of the structure group of TY to U(n) 1 SO(2n + 1, R). Note that, since we assume Y is oriented, by a choice of Riemannian metric, the structure group of Y can be reduced from GL(2n + 1, R) to SO(2n + 1, R) to begin with. By the discussion above, one can speak about Chern classes of ξ viewed now as a complex bundle over Y. Moreover, the first Chern class c 1 (ξ) H 2 (Y, Z) reduces to the second Stiefel-Whitney class w 2 (Y) H 2 (Y, Z 2 ). We conclude that if Y admits an almost contact structure, then w 2 (Y) admits an integral lift, or equivalently, the third integral Stiefel-Whitney class W 3 (Y) H 3 (Y, Z) vanishes. Recall that the third integral Stiefel-Whitney class W 3 (Y) is the image of the second Stiefel-Whitney class w 2 (Y) H 2 (Y; Z 2 ) under the Bockstein homomorphism β H 2 (Y; Z) H 2 (Y; Z 2 ) associated to the short exact sequence β H 3 (Y; Z) 0 Z 2 Z Z 2 0. Therefore, W 3 (Y) = 0 is a necessary condition for Y to admit an almost contact (hence contact) structure. Exercise 1.34 The Lie group SO(3) is a closed subgroup of the Lie group SU(3) and hence the quotient homogeneous space SU(3)/SO(3) is a smooth manifold. Show that Y = SU(3)/SO(3) is a 5-dimensional smooth manifold (a.k.a. the Wu manifold). Also, use the homotopy exact sequence of the fibration SO(3) SU(3) Y to show that Y is simply-connected, π 2 (Y) = Z 2 and thus H 2 (Y, Z) = Z 2 = H 3 (Y, Z). By using Steenrod squares, one can show that W 3 (Y) is the generator of H 3 (Y, Z), and therefore Y does not admit an almost contact structure. If an oriented closed odd-dimensional manifold Y admits an almost contact structure, then a natural question arises as to whether Y admits a contact structure. This question was recently answered affirmatively: Theorem 1.35 (Borman, Eliashberg and Murphy [16]) There exists a contact structure in every homotopy class of almost contact structures.

13 13 In dimension three, this was known for a long time (see Theorem 1.31). Note that an almost contact structure on an oriented 3-manifold is nothing but a co-oriented (equivalently oriented) 2-plane field. Remark 1.36 For any n > 1, and any finitely presented group G, there exists a closed contact manifold of dimension 2n + 1 whose fundamental group is G. For n 3, this follows from the facts that the unit cotangent bundle of any Riemannian manifold admits a contact structure and that for any m 4, and any finitely presented group G, there exists an m-dimensional closed smooth manifold whose fundamental group is G. This argument above breaks down for n = 2, since Z and Z 3 are the only free abelian groups that are realized as the fundamental groups of some closed orientable 3-manifolds. Nevertheless, the statement holds true for n = 2 (see [8]), but it is definitely false for n = Fillings of contact manifolds What is a strong symplectic filling? Exercise 1.37 Suppose that Y is a hypersurface in a symplectic 2n-manifold (X, ω). If V is a Liouville vector field for ω defined in a neighborhood of Y, which is transverse to Y, then show that the 1-form α := ι V ω Y is a contact form on Y. Such hypersurfaces are said to be of contact type. (Hint: First observe, by Cartan s formula, that ω = dα. Now show that (ι V ω) ω n 1 = 1 n ι V(ω n ). It follows that α (dα) n 1 is a volume form on Y, since ω n is a volume form on X and V is transverse to Y.) Definition 1.38 Suppose that (W, ω) is a compact symplectic 2n-manifold whose boundary can be expressed as Y Y + (one of which is possibly empty). We say that Y + is the convex boundary if there exists a Liouville vector field V for ω defined in a neighborhood of Y +, which is transverse to Y + and pointing outwards. Similarly, we say that Y is the concave boundary if the Liouville vector field points inwards. We orient (W,ω) by ω n, and Y ± acquires the induced boundary orientation. By Exercise 1.37, there is a positive contact structure ξ + on Y +, and a negative contact structure ξ on Y. We say that (W,ω) is a symplectic cobordism from (Y,ξ ) to (Y +,ξ + ).

14 14 Ozbagci Definition 1.39 A strong symplectic filling of a contact manifold (Y, ξ) is a symplectic cobordism from the empty set to (Y, ξ). A strong symplectic filling is also called a convex filling. On the other hand, a symplectic cobordism from a contact manifold (Y, ξ) to the empty set is called a concave filling of (Y, ξ). Exercise 1.40 Show that a compact symplectic manifold (W, ω) is a strong symplectic filling of a closed contact manifold (Y,ξ) if and only if W = Y as oriented manifolds, ω is exact near the boundary and its primitive α can be chosen in such a way that ker(α Y ) = ξ. (Hint: Use Cartan s formula.) Example 1.41 Let ω st := dx 1 dy 1 +dx 2 dy 2 denote the standard symplectic 2-form on R 4 in the coordinates (x 1, y 1, x 2, y 2 ). Then λ := 1 2 (x 1dy 1 y 1 dx 1 + x 2 dy 2 y 2 dx 2 ) is a primitive of ω st. The standard contact structure on the unit sphere S 3 R 4 is defined as ξ st = ker(λ S 3). The vector field V = x 1 + y 1 + x 2 + y 2 x 1 y 1 x 2 y 2 is a Liouville vector field for ω st which is transverse to S 3 and pointing outwards. This shows that (D 4,ω st ) is a strong symplectic filling of its convex boundary (S 3,ξ st ). Exercise 1.42 ([45, Example 5.1.6]) Let ω st := dx 1 dy 1 = rdr dϕ denote the standard symplectic 2-form on R 2. Show that the vector field V + = 1 2 r r is a Liouville vector field for ω st (which is transverse to the unit circle S 1 pointing outwards). This shows that (D 2,ω st ) is a convex filling of S 1. Show that the vector field V = ( r 2 4) 2r r is also a Liouville vector field for ω st which is transverse to S 1 and pointing inwards. This shows that (D 2,ω st ) is also a concave filling of S 1. Remark 1.43 One can generalize Exercise 1.42 to prove that every connected boundary component of a symplectic 2-manifold is both convex and concave. However, no boundary component of symplectic manifold of dimension > 2 is both convex and concave [101, Proposition 7.9].

15 What is a Stein filling? Suppose that (X, J) is a Stein manifold with a J-convex function φ. Then, a regular level set φ 1 (t) is a compact hypersurface in X which is transverse to the Liouville vector field φ for the symplectic form ω φ := dd C φ. Therefore α φ = ι φ ω φ = d C φ restricts to a contact form on φ 1 (t) by Exercise 1.37 and the sublevel set φ 1 (, t] is a special kind of strong symplectic filling of the contact manifold (φ 1 (t), ker(α φ )), which leads to the following definition. Definition 1.44 A compact complex manifold (W, J) with boundary W = Y is a Stein domain if it admits an exhausting J-convex function φ: W R such that Y is a regular level set. Then we say that the contact manifold (Y,ξ = ker(α φ Y )) is Stein fillable and (W, J) is a called a Stein filling of it. It immediately follows that a Stein filling is a strong symplectic filling, where the symplectic form is exact. In the following, we give an equivalent formulation of the contact structure ξ with a different point of view. Let Y be an oriented smooth real hypersurface in a complex manifold (X, J). The complex tangencies ξ := TY J(TY) along Y form a unique maximal complex hyperplane distribution in TY. This just means that ξ is a hyperplane field in TY with a complex structure J. We would like to know under which condition ξ is a contact structure. Let V be a vector field on X transverse to Y such that JV TY. So JV is vector field transverse to ξ in TY. This gives a co-orientation to ξ by the choice α(jv) > 0 for any 1-form α such that ξ = ker α. (Recall that any co-oriented tangent hyperplane field can be expressed as the kernel of a global 1-form.) To sum up, V defines a co-orientation of Y in X and JV defines a co-orientation of ξ in Y. Then the 2-form ω Y (u, v) := dα ξ (u, v) is uniquely defined up to multiplication by a positive function. The hypersurface Y is called J-convex if ω Y (u, Ju) > 0 for every non-zero vector u ξ. This implies that ξ is a contact structure on Y since dα is non-degenerate on ξ.

16 16 Ozbagci If (X, J, φ) is a Stein manifold, then any regular level set Y is an oriented hypersurface in X with a co-orientation given by dφ. We can choose α = d C φ as the 1-form defining ξ so that ω Y (u, v) = dd C (u, v) = ω φ (u, v). The upshot is that Every regular level set of a J-convex function on a Stein manifold is a J-convex hypersurface equipped with the co-oriented contact structure given by the complex tangencies to the hypersurface. Example 1.45 Consider the standard complex structure J st on R 4 given by J st ( x j ) = y j and J st ( y j ) = x j for j = 1, 2. Note that J st is just the complex multiplication by i when R 4 is identified with C 2, by setting z j = x j + iy j. Let φ : R 4 R be defined by φ(x 1, y 1, x 2, y 2 ) = x y x y 2 2. Then φ is an exhausting J st -convex function on R 4, and the unit sphere S 3 is a regular level set. By the discussion above, ξ st = TS 3 J st (TS 3 ) = ker( (dφ J st ) S 3). This proves that (D 4, J st ) is a Stein filling of (S 3,ξ st ) What is a weak symplectic filling? There is also the notion of a weak symplectic filling which we will discuss only for 3-dimensional contact manifolds. We refer the reader to [70] for the detailed study of weak versus strong symplectic fillings of higher dimensional contact manifolds. Definition 1.46 A contact 3-manifold (Y, ξ) is said to be weakly symplectically fillable if there is a compact symplectic 4-manifold (W,ω) such that W = Y as oriented manifolds and ω ξ > 0. In this case we say that (W,ω) is a weak symplectic filling of (Y, ξ). 1.4 Appendix: Existence of almost complex structures We refer to [17] or [74] for basic facts about almost complex structures.

17 17 Remark 1.47 Every orientable 2-manifold admits an almost complex structure and as a matter of fact, any such almost complex structure is necessarily integrable. Exercise 1.48 Show that S 4 does not admit an almost complex structure. (Hint: Let X be any closed oriented 4-manifold. Then Hirzebruch signature theorem says that p 1 (X), [X] = 3σ(X), where σ(x) denotes the signature of X. We also know the following identities for the characteristic classes of any 2-dimensional complex bundle E X : p 1 (E) = c 2 1 (E) 2c 2(E) and c 2 (E) = e(e). Suppose that X admits an almost complex structure J, so that (TX, J) is a complex bundle over X. Now derive the formula c 2 1(X, J), [X] = 3σ(X) + 2χ(X) where χ(x) denotes the Euler characteristic of X.) How about CP 2 #CP 2 or (S 2 S 2 )#(S 2 S 2 )? (Hint: [51, Appendix 1.4].) Remark 1.49 The sphere S n admits an almost complex structure if and only if n {2, 6} (see, for example, [74]). Currently, it is not known whether or not S 6 admits a complex structure. The existence of an almost complex structure on a 2n-dimensional manifold is equivalent to a reduction of the structure group of the tangent bundle from GL(2n, R) to GL(n, C). If the manifold at hand is oriented and equipped with a Riemannian metric, an almost complex structure is a reduction of the structure group from SO(2n) to U(n). The existence question is then a purely algebraic topological one (cf. [69]). The obstruction to the existence of an almost complex structure on a compact orientable 6-dimensional manifold X lies in H i+1( X;π i (SO(6)/U(3)) ) for 0 i 5. Exercise 1.50 Use the fact that SO(6)/U(3) is diffeomorphic to CP 3 (see [45, Proposition 8.1.3]), to show that the only homotopy group of interest (in dimension 6) is π 2 (SO(6)/U(3)) = Z. (Hint: Consider the Hopf fibration S 1 S 7 CP 3.) The only obstruction to the existence of an almost complex structure on a compact orientable 6-dimensional manifold X is the third integral Stiefel-Whitney class W 3 (X) in H 3 (X; Z). Note that W 3 (X) = 0 if and only if w 2 (X) admits an integral lift. In this case, each integral lift of w 2 (X) in H 2 (X; Z) can be realized as c 1 (X, J) for some almost complex structure J on X. Remark 1.51 A closed orientable manifold M (of arbitrary dimension) admits a Spin c structure if and only if W 3 (M) = 0.

18 18 Ozbagci 2 Basic results about fillings of contact 3-manifolds 2.1 Legendrian knots and contact Dehn surgery Let K be knot in a 3-manifold Y. We get a new 3-manifold by removing a tubular neighborhood νk = S 1 D 2 of K from Y and regluing it back via a diffeomorphism f : νk (Y νk). This is called a Dehn surgery along K. Since we reglue S 1 D 2 along its boundary, the gluing map f (and hence surgery) is determined by the induced map f : H 1 (T 2 ; Z) H 1 (T 2 ; Z) = Z 2, which can be represented by an integral 2 2 matrix in any given basis of H 1 (T 2 ; Z). Actually, the surgery is determined by a rational number rather than four integers in the 2 2 matrix above: The solid torus S 1 D 2 can be viewed as the union of a 2-handle and a 3-handle, and one only has to describe how to glue the 2-handle, since the gluing of any 3-handle is unique. To glue a 2-handle, however, one needs to specify the gluing circle, which is an embedded simple closed curve in (Y int νk). Consequently, the Dehn surgery along K is determined by a simple closed curve in (Y int νk), which can be given (up to isotopy) by fixing its homology class a H 1 ( (Y int νk); Z ) = Z 2. A choice of a basis of H 1 ( (Y int νk); Z ) converts a into a pair of relatively prime integers. A canonical choice for one basis element is the meridian µ H 1 ( (Y int νk); Z ) of the knot K, which is a nontrivial primitive element that vanishes under the inclusion of νk into νk. By fixing an orientation on K, the element µ is uniquely determined by the requirement that its linking number with K is +1. The choice of a second basis element (called a longitude) in H 1 ( (Y int νk); Z ) is not canonical in general, but once such a choice is made, all the others can be parameterized by the set of integers. Any null-homologous knot K Y admits a canonical longitude λ obtained as follows: Since K is null-homologous, there is a compact, connected, orientable surface in Y whose boundary is K. This is called a Seifert surface for K. Now, λ (a parallel copy of K ) is obtained by pushing K along (any of) its Seifert surface. Here λ can also be viewed as the trivializing section of the normal bundle of K in Y, and thus it determines a framing the Seifert framing. Note that λ is uniquely determined by the fact that [λ] = 0 H 1 ( (Y int νk); Z ). Consequently, the aforementioned homology class a can be converted into a pair of relatively prime integers (p, q) by setting a = pµ + qλ. The ratio p/q does not depend on the chosen orientation of K (used in defining µ

19 19 and λ) and encodes all the gluing information for the surgery. Let Y p/q (K) denote the resulting 3-manifold. The following exercise shows that integral Dehn surgery is equivalent to ordinary surgery. Exercise 2.1 Show that Y p/q (K) can be given by a 4-dimensional 2-handle attachment to Y I along K D 2 Y {1}, provided that q = ±1. (Hint: To attach a 4- dimensional 2-handle D 2 D 2 to the boundary X of some 4-manifold X, one needs to specify an embedding of S 1 D 2 in X and glue D 2 D 2 to X by identifying D 2 D 2 D 2 D 2 with the image of that embedding in X. ) Theorem 2.2 (Lickorish and Wallace) Every closed, oriented 3-manifold can be given as a result of an integral Dehn surgery on some link in S 3. Now we are ready to describe the contact version of integral Dehn surgery. We first need to discuss contact framings of Legendrian knots. The standard contact structure ξ st on the 3-sphere S 3 is commonly defined as the extension of the standard contact structure ker(dz + x dy) on R 3. This makes sense since, for any p S 3, (S 3 \ {p},ξ st S 3 \{p} ) is contactomorphic to (R3, ker(dz + x dy)). Exercise 2.3 Show that the contact structures ker(dz+x dy) and ker(dz+x dy y dx) on R 3 are contactomorphic. (Hint: Use the diffeomorphism ψ : (x, y, z) = (x, y 2, z+ xy 2 ).) In the following, we take the standard contact structure ξ st in R 3 as the kernel of the contact 1-form dz + x dy. This is a convention commonly used in many sources such as [45, 50, 51], but be aware that different authors use different conventions. A knot in a contact 3-manifold is called Legendrian if it is everywhere tangent to the contact planes. Now consider a Legendrian knot L (R 3,ξ st ) and take its front projection, i.e., its projection to the yz-plane. Notice that the projection has no vertical tangencies since dz dy = x. It turns out that L can be approximated by another Legendrian knot for which the projection has only transverse double points and cusp singularities (see [45, Lemma 3.2.3], for a precise statement). Conversely, a knot projection with these properties gives rise to a unique Legendrian knot in (R 3,ξ st ) by defining x from the projection as. Since any knot projection can be isotoped to satisfy the above properties, every knot in S 3 can be isotoped (non uniquely) to a Legendrian knot. dz dy

20 20 Ozbagci z y Figure 1: (Front projection of) a right-handed Legendrian trefoil knot Note that z is a vector field transverse to the contact planes ξ st in R 3. Therefore, we can get a parallel copy L of any Legendrian knot L in (R 3,ξ st ), by slightly pushing it in the positive z-direction. The knot L defines the contact framing of L which, measured with respect to the canonical Seifert framing of L in R 3, translates into an integer the Thurston- Bennequin invariant tb(l) of L. Exercise 2.4 Define w(l) (the writhe of L) as the sum of the signs of the double points in a front projection of L. For this to make sense, we need to fix an orientation on L. Verify that w(l) is independent of the choice of the orientation but it depends on the projection. Once a projection is fixed, show that tb(l) = w(l) 1 2c(L), where c(l) is the number of cusps. Calculate the Thurston-Bennequin invariant of the right-handed Legendrian trefoil knot depicted in Figure 1. Remark 2.5 For any null-homologous Legendrian knot L in a contact 3-manifold (Y,ξ), the Thurston-Bennequin invariant tb(l) Z of L is defined as the twisting of its contact framing (obtained by pushing L in a direction transverse to ξ) relative to its Seifert framing (obtained by pushing L along a Seifert surface of L). Moreover, tb(l) is a Legendrian isotopy invariant and does not depend on an orientation of L. A contact (±1)-surgery on a Legendrian knot L in a contact 3-manifold (Y 3,ξ st ) is, first of all, an integral Dehn surgery along L whose framing is one more/less than the

21 21 contact framing. Even if L is not homologically trivial, it has a contact framing given by a vector field transverse to the contact planes and it still makes sense to talk about a framing which is one more/less than the contact framing. Moreover, the contact structure ξ on Y \ ν(l) can be uniquely extended to the solid torus S 1 D 2 that is glued in while performing the surgery. For the uniqueness, an essential requirement is that the contact structure on S 1 D 2 is tight (see Section 2.3 for its definition). It turns out that Theorem 2.2 holds true in the contact category: Theorem 2.6 (Ding and Geiges [22]) Every closed contact 3-manifold can be obtained by a contact (±1)-surgery on a Legendrian link in (S 3,ξ st ). Remark 2.7 Contact ( 1)-surgery is also called Legendrian surgery. 2.2 Weinstein handle attachment In [98], Weinstein described a handle attachment scheme in the symplectic category. We follow the exposition in [34] for the low-dimensional case in our discussion. A 4-dimensional 2-handle H can be described as the closure of the component of {{ R 4 x x } { 2 (y2 1 + y 2 2) = 1 x x 2 2 ε 6 (y2 1 + y 2 2) = ε } } 2 which contains the origin as depicted in Figure 2. Remark 2.8 The ε above is a positive real number which can be chosen to be arbitrarily small if needed. The standard symplectic structure ω st = dx 1 dy 1 + dx 2 dy 2 on R 4 restricts to a symplectic structure on the 2-handle H. Let f : R 4 R be the function given by and let f (x 1, y 1, x 2, y 2 ) = x x (y2 1 + y2 2 ) V := f = 2x 1 y 1 + 2x 2 y 2 x 1 y 1 x 2 y 2 with respect to the standard Euclidean metric on R 4.

22 22 Ozbagci V y 1, y 2 S 01 H x 1, x 2 f = 1 01 S Figure 2: A model for a 4-dimensional 2-handle Exercise 2.9 Verify that ι V ω st = 2x 1 dy 1 + y 1 dx 1 + 2x 2 dy 2 + y 2 dx 2. Observe that V = f is transverse to the level set {f = 1} (part of which is in H) pointing into H. It follows, by Exercise 1.37, that ξ = ker(ι V ω st ) is a contact structure on the hypersurface {f = 1}. Show that V is a Liouville vector field for ω st using Cartan s formula and conclude that the attaching region of the handle H is concave with respect to ω st. Finally, show that the attaching circle S = {x 1 = x 2 = 0, y y2 2 = 2} ( H,ξ) is Legendrian. Let (W, ω) be a compact symplectic 4-manifold with a convex boundary component (Y,ξ) and let L be a Legendrian knot in (Y,ξ). According to the Legendrian Neighborhood Theorem, L and S admit contactomorphic neighborhoods. The attaching region of H becomes a subset of this neighborhood of S by choosing ε sufficiently small. Then the contactomorphism between the neighborhoods of L X and S H allows us to attach H to W by Lemma Lemma 2.10 Suppose that (i) (Y 1,ξ 1 ) is a convex boundary component of a symplectic cobordism (W 1,ω 1 ),

23 23 (ii) (Y 2,ξ 2 ) is a concave boundary component of a symplectic cobordism (W 2,ω 2 ) and (iii) (Y 1,ξ 1 ) is contactomorphic to (Y 2,ξ 2 ). Then we can glue (W 1,ω 1 ) and (W 2,ω 2 ) symplectically by identifying (Y 1,ξ 1 ) with (Y 2,ξ 2 ). Proposition 2.11 (Weinstein [98]) Let (W, ω) be a compact symplectic 4- manifold with a convex boundary component (Y,ξ) and let L be a Legendrian knot in (Y,ξ). A 2-handle H can be attached to (W,ω) along L with framing 1 relative to its contact framing in such a way that the symplectic structure ω extends to a symplectic structure ω on W = W H. Moreover, (W,ω ) has a convex boundary component (Y,ξ ), where (Y,ξ ) is obtained from (Y,ξ) by contact ( 1)-surgery along L. Exercise 2.12 Explain why the attaching framing of H is 1 (relative to its contact framing) in Proposition (Hint: Consider the field of unit tangent vectors to S in the (y 1, y 2 )-plane. Now you need to compare two other vectors fields along S transverse to this vector field; one which lies in the contact planes ξ giving the contact framing and the other which lies in the (x 1, x 2 )-plane giving the attaching framing. See [84, Page 117] for the details.) Remark 2.13 A Weinstein 2-handle can also be attached along a Legendrian knot L in the boundary (Y, ξ) of a weak symplectic filling (W, ω) so that symplectic structure extends over the 2-handle and weakly fills the resulting contact 3-manifold. This is done by perturbing ξ in an arbitrary small neighborhood N of L so that N is strongly convex, which means that there is a vector field V defined on W near N so that V points transversely out of W, L V ω = ω and ξ N = ker(ι V ω) N. In particular, Legendrian surgery preserves weak fillability [38, Lemma 2.6]. 2.3 The fundamental dichotomy: Tight versus overtwisted An embedded disk D in a contact 3-manifold (Y, ξ) is called overtwisted if T p D = ξ p at each point p D. A contact 3-manifold which contains an overtwisted disk is called overtwisted, otherwise it is called tight which is the fundamental dichotomy in 3-dimensional contact topology.

24 24 Ozbagci Note that D of an overtwisted disk D is a Legendrian unknot with tb( D) = 0. We conclude that (Y, ξ) is overtwisted if and only if it contains a topologically unknotted Legendrian knot K with tb(k) = 0. For any null-homologous Legendrian knot K in an arbitrary contact 3-manifold, we can find a C 0 -small isotopy that decreases tb(k) by any integer, but it is not always possible to increase tb(k). Exercise 2.14 Let (Y, ξ) be an overtwisted contact 3-manifold. Show that, for any n Z, there exists a Legendrian unknot K (Y,ξ) such that tb(k) = n. (Hint: Take two parallel copies of any overtwisted disk and show that the Thurston-Bennequin invariant of the Legendrian connected sum of their boundaries is equal to 1.) The following is due to Eliashberg and Gromov [30]: Theorem 2.15 A weakly symplectically fillable contact 3-manifold (Y, ξ) is tight. Proof Here we give a sketch of a proof (cf. [84, Theorem ]) of Theorem 2.15 which is very different from the original proof. Suppose that (W, ω) is a symplectic filling of an overtwisted contact 3-manifold (Y, ξ). Then, by Exercise 2.14, there is an embedded disk D Y such that D is Legendrian with tb( D) = 2. By attaching a Weinstein 2-handle along D to (W,ω), we obtain a weak symplectic filling (W,ω ) of the surgered contact 3-manifold (Y,ξ ) (see Remark 2.13). Now we claim that W contains a sphere S with self-intersection +1. (This also guarantees that S is essential.) The sphere S is obtained by gluing D with the core disk of the 2-handle, and [S] 2 = 1 follows from the fact that the Weinstein 2-handle is attached with framing tb( D) 1 = 1. By Theorem 3.29, (W,ω ) can be embedded into a closed symplectic 4-manifold X with b + 2 (X) > 1. But if there is any essential sphere S of nonnegative self-intersection in a closed 4-manifold X with b + 2 (X) > 1 we know that SW X 0 (cf. [99], see also [41, Lemma 5.1]). This is a contradiction since, according to Taubes [92], for any closed symplectic 4-manifold (X,ω) with b + 2 (X) > 1, we have SW X(c 1 (X,ω)) 0. The first examples proving that the converse of Theorem 2.15 is false were discovered by Etnyre and Honda [38]. Soon after, a variety of such examples were constructed by Lisca and Stipsicz [66, 67] using Heegaard Floer theory.

25 25 We just include here a few selected important results about 3-dimensional contact manifolds: Darboux: All contact structures look the same near a point, i.e., any point in a contact 3-manifold has a neighborhood isomorphic to a neighborhood of the origin in the standard contact R 3. Martinet: Every closed oriented 3-manifold admits a contact structure. Lutz: In every homotopy class of oriented plane fields on a closed oriented 3-manifold there is an overtwisted contact structure. Eliashberg: Two overtwisted contact structures are isotopic if and only if they are homotopic as oriented plane fields. Combining the last three results we get: There is a unique overtwisted contact structure in every homotopy class of oriented plane fields. Etnyre & Honda: The Poincaré homology sphere with its non-standard orientation does not admit a tight contact structure. Colin & Giroux & Honda: Only finitely many homotopy classes of oriented plane fields carry tight contact structures on a closed oriented 3-manifold. Colin & Giroux & Honda + Honda & Kazez & Matić: A closed, oriented, irreducible 3-manifold carries infinitely many tight contact structures (up to isotopy or up to isomorphism) if and only if it is toroidal. Giroux: There are infinitely many non-isotopic tight contact structures on T 3, all of which are in the same homotopy class of oriented plane fields. 2.4 Different types of fillings A strong symplectic filling of a contact 3-manifold is a weak symplectic filling since dα is a symplectic form on the contact planes ξ = ker α for any contact 1-form α. The converse is shown to be true for rational homology spheres: Theorem 2.16 [27] [78] [29, Proposition 4.1] Suppose that (W, ω) is a weak symplectic filling of (Y,ξ), where Y is a rational homology sphere. Then ω can be modified to a new symplectic form ω, where this modification is supported in a neighborhood of W, so that (W, ω) becomes a strong symplectic filling of (Y, ξ). Proof We will give here the argument in [45, Lemma 6.5.5] which is essentially the same as in [29, Proposition 4.1]. We know that W = Y has a collar neighborhood

26 26 Ozbagci N = [0, 1] Y of Y in W with Y {1} Y. Since H 2 (N) = H 2 (Y) = 0, we can write ω = dη for some 1-form η defined in N. Then we can find a 1-form α on Y such that ξ = ker α and α ω Y > 0, by the assumption that (W,ω) is a weak symplectic filling of (Y, ξ). We would like to construct a symplectic form ω on [0, 1] Y which agrees with ω in a neighborhood of {0} Y and strongly fills {1} Y. Let ω = d(fη) + d(gα) = f dt η + fω + g dt α + gdα be a 2-form on [0, 1] Y, where we will impose some conditions on the smooth functions f : [0, 1] [0, 1] and g : [0, 1] R 0 to fit our purposes. First of all, we require that ω agrees with ω in a neighborhood of {0} Y. We simply fix a small ε > 0 and set f 1 on [0,ε] and g 0 on [0, ε 2 ]. More importantly, ω needs to be a symplectic form, and thus we compute ω 2 = f 2 ω 2 + 2fg dt α ω + 2gg dt α dα +2ff dt η ω + 2f g dt η dα + 2fg ω dα Since ω is symplectic, α ω Y > 0 and α is contact, we know that the first three terms are positive volume forms on [0, 1] Y provided that they have positive coefficients. Hence we impose the condition g > 0 on ( ε 2, 1]. Moreover, we can ensure that these positive terms dominate the other three terms, by choosing g to be very small on [0,ε] (where f 0) and g very large compared to f and g on [ε, 1]. Finally, to verify that ω strongly fills the contact boundary ({1} Y,ξ), we impose that f 0 near t = 1. By setting s := log g(t), we see that ω looks like d(e s α) near the boundary. It follows that s is a Liouville vector field for ω near the boundary: L d(e s α) = d ( ι (e s ds α + e s dα) ) = d(e s α) s s and clearly we have ξ = ker α = ker(e s α) = ι d(e s α) = ι ω on Y {1} Y. s s For an arbitrary closed contact 3-manifold we have {Stein fillings} {strong symplectic fillings} {weak symplectic fillings} where the inclusions are shown to be strict for some contact 3-manifolds: Eliashberg [28] showed that there are weakly but not strongly symplectically fillable contact structures on T 3, and further examples of this kind were given on torus bundles over the circle by Ding and Geiges [21].

27 27 Using the contact Ozsváth-Szabó invariants, Ghiggini [49] proved that for any even positive integer n, the Brieskorn 3-sphere Σ(2, 3, 6n + 5) admits a strongly symplectically fillable contact structure which is not Stein fillable. As a consequence of the above discussion, the following problem arises naturally in the study of the topology of fillings of contact 3-manifolds: Basic Problem: Given a closed contact 3-manifold, describe all of its Stein (or minimal strong/weak symplectic) fillings up to diffeomorphism (or symplectic deformation). Definition 2.17 We say that two symplectic 4-manifolds (W 1,ω 1 ) and (W 2,ω 2 ) with convex boundary are symplectically deformation equivalent if there is a diffeomorphism ϕ : W 1 W 2 such that ϕ ω 2 can be deformed to ω 1 through a smooth 1-parameter family of symplectic forms that are all convex at the boundary. The following result is due to Gromov [53] (see also [26, Theorem 5.1], [72, Theorem 1.7], [19, Theorem 16.6]). Theorem 2.18 Any weak symplectic filling of (S 3,ξ st ) is symplectically deformation equivalent to a blow-up of (D 4,ω st ). Remark 2.19 This result can be obtained as an easy corollary to Theorem 3.27 since (S 3,ξ st ) admits a planar open book whose page is an annulus and whose monodromy is a single positive Dehn twist along the core circle. This monodromy admits a unique positive factorization which proves Theorem However, some oriented closed 3-manifolds can not admit any symplectic fillings at all. For example, Theorem 2.20 (Lisca [61, 62] ) The Brieskorn sphere Σ(2, 3, n) does not admit any fillable contact structures for n {3, 4, 5}. Proof We first give the proof for Σ(2, 3, 3). Let E denote the 4-manifold with boundary obtained by plumbing oriented disk bundles of Euler number 2 over the