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A Few Remarks About Symplectic Filling

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A Few Remarks About Symplectic Filling ISSN 1364-0380 (on line) 1465-3060 (printed) 277 eometry & opology VolumeG 8 (2004)T 277–293 G G T T T G T T Published: 14 February 2004 G T G G T G T T G T G T G T G T To Ada G G G G T T A few remarks about symplectic filling Yakov Eliashberg Department of Mathematics, Stanford University Stanford CA 94305-2125, USA Email: [email protected] Abstract We show that any compact symplectic manifold (W, ω) with boundary embeds as a domain into a closed symplectic manifold, provided that there exists a contact plane ξ on ∂W which is weakly compatible with ω, i.e. the restriction ω does not vanish and the contact orientation of ∂W and its orientation as |ξ the boundary of the symplectic manifold W coincide. This result provides a useful tool for new applications by Ozsv´ath–Szab´oof Seiberg–Witten Floer homology theories in three-dimensional topology and has helped complete the Kronheimer–Mrowka proof of Property P for knots. AMS Classification numbers Primary: 53C15 Secondary: 57M50 Keywords: Contact manifold, symplectic filling, symplectic Lefschetz fibra- tion, open book decomposition Proposed: Leonid Polterovich Received: 25 November 2003 Seconded: Peter Ozsv´ath, Dieter Kotschick Revised: 13 January 2004 c eometry & opology ublications G T P 278 Yakov Eliashberg 1 Introduction All manifolds which we consider in this article are assumed oriented. A contact manifold V of dimension three carries a canonical orientation. In this case we will denote by V the contact manifold with the opposite orientation. Con- − tact plane fields are assumed co-oriented, and therefore oriented. Symplectic manifolds are canonically oriented, and so are their boundaries. We prove in this article the following theorem: Theorem 1.1 Let (V,ξ) be a contact manifold and ω a closed 2–form on V such that ω > 0. Suppose that we are given an open book decomposition of |ξ V with a binding B. Let V ′ be obtained from V by a Morse surgery along B with a canonical 0–framing, so that V ′ is fibered over S1 . Let W be the corresponding cobordism, ∂W = ( V ) V ′ . Then W admits a symplectic form − ∪ Ω such that Ω = ω and Ω is positive on fibers of the fibration V ′ S1 . |V → Remark 1.2 Note that the binding B has a canonical decomposition of its tubular neighborhood given by the pages of the book. The 0–surgery along B is the Morse surgery associated with this decomposition. If the binding is disconnected then we assume that the surgery is performed simultaneously along all the components of B. We will deduce the following result from Theorem 1.1: Theorem 1.3 Let (V,ξ) and ω be as in Theorem 1.1. Then there exists a ′ ′ ′ ′ symplectic manifold (W , Ω ) such that ∂W = V and Ω V = ω. Moreover, ′ 1 −′ ′ | one can arrange that H1(W ) = 0, and that (W , Ω ) contains the symplectic cobordism (W, Ω) constructed in Theorem 1.1 as a subdomain adjacent to the boundary. In particular, any symplectic manifold which weakly fills (see Sec- tion 4 below) the contact manifold (V,ξ) can be symplectically embedded as a subdomain into a closed symplectic manifold. Corollary 1.4 Any weakly (resp. strongly) semi-fillable (see [10]) contact manifold is weakly (resp. strongly) fillable. Remark 1.5 Theorem 1.1 serves as a missing ingredient in proving that the Ozsv´ath–Szab´ocontact invariant c(ξ) does not vanish for weakly symplectically fillable (and hence for non-existing anymore semi-fillable) contact structures. 1This observation is due to Kronheimer and Mrowka, see [20]. eometry & opology, Volume 8 (2004) G T A few remarks about symplectic filling 279 This and other applications of the results of this article in the Heegaard Floer homology theory are discussed in the paper of Peter Ozsv´ath and Zolt´an Szab´o, see [27]. The observation made in this paper also helped to streamline the program of Peter Kronheimer and Tomasz Mrowka for proving the Property P for knots, see their paper [20]. Acknowledgements This article is my answer to the question I was asked by Olga Plamenevskaya and David Gay during the workshop on Floer homology for 3–manifolds in Banff International Research Station. I want to thank them, as well as Selman Akbulut, Michael Freedman, Robion Kirby, Peter Kronheimer, Robert Lipshitz, Tomasz Mrowka, Peter Ozsv´ath, Michael Sullivan and Zolt´an Szab´ofor stimulating discussions, and Augustin Banyaga, Steven Kerckhoff, Leonid Polterovich and Andr´as Stipsicz for providing me with the necessary information. I am grateful to John Etnyre, Tomasz Mrowka, Andr´as Stipsicz and Leonid Polterovich for their critical remarks concerning a preliminary ver- sion of this paper. I also want to thank Peter Kronheimer and Tom Mrowka with sharing with me their alternative “flux-fixing argument” (see Lemma 3.4 below). This research was partially supported by NSF grants DMS-0204603 and DMS- 0244663. 2 Proof of Theorem 1.1 We begin with the following lemma which is a slight reformulation of Proposi- tion 3.1 in [6]. A similar statement is contained also in [19]. Lemma 2.1 Let (V,ξ) and ω be as in Theorem 1.1. Then given any contact form α for ξ and any C > 0 one can find a symplectic form Ω on V [0, 1] × such that a) Ω = ω; |V ×0 b) Ω = ω + Cd(tα), where t [1 ε, 1] and 0 <ε< 1; V ×[1−ε,1] ∈ − c) Ω induces the negative orientation on V 0 and positive on V 1. × × Proof By assumption ω = fdα = d(fα) |ξ |ξ |ξ eometry & opology, Volume 8 (2004) G T 280 Yakov Eliashberg for a positive function V R. Setα ˜ = fα. Then ω = dα˜ +α ˜ β. Take a → ∧ smooth function h: V [0, 1] R such that × → Ct dh h = 0, h = , > 0, |V ×0 |V ×[1−ε,1] f dt where t is the coordinate corresponding to the projection V [0, 1] [0, 1]. × → Consider the form Ω = ω + d(hα˜). Here we keep the notation ω andα ˜ for the pull-backs of ω andα ˜ to V [0, 1]. Then we have × dh Ω= dα˜ +α ˜ β + d h α˜ + dt α,˜ ∧ V ∧ dt ∧ where dV h denotes the differential of h along V . Then dh Ω Ω = 2 dt α˜ dα>˜ 0 . ∧ dt ∧ ∧ Hence Ω is symplectic and it clearly satisfies the conditions a)–c). Let us recall that a contact form λ on V is called compatible with the given open book decomposition (see [14]) if a) there exists a neighborhood U of the binding B, and the coordinates (r, ϕ, u) [0, R] R/2πZ R/2πZ such that ∈ × × U = r R and λ = h(r)(du + r2dϕ) , { ≤ } |U where the positive C∞ –function h satisfies the conditions h(r) h(0) = r2 near r =0 and h′(r) < 0 for all r> 0 ; − − b) the parts of pages of the book in U are given by equations ϕ = const; c) dα does not vanish on the pages of the book (with the binding deleted). Remark 2.2 An admissible contact form α defines an orientation of pages and hence an orientation of the binding B as the boundary of a page. On the other hand, the form α defines a co-orientation of the contact plane field, and hence an orientation of B as a transversal curve. These two orientations of B coincide. Remark 2.3 By varying admissible forms for a given contact plane field one can arrange any function h with the properties described in a). Indeed, suppose we are given another function h which satisfies a). We can assume without loss of generality that α has the presentation a) on a bigger domain U ′ = r R′ e { ≤ } for R′ > R. Let us choose c > 0 such that h(R) > ch(R) and extend h to [0, R′] in such a way that h′(r) < 0 and h(r)= ch(r) near R′ . Then the form e e hα on U ′ extended to the rest of the manifold V as cα is admissible for the e e given open book decomposition. e eometry & opology, Volume 8 (2004) G T A few remarks about symplectic filling 281 Now we are ready to prove Theorem 1.1. Proof of Theorem 1.1 Take a constant a > 0 and consider a smooth on [0, 1) function g : [0, 1] R such that g = a, g(t) = √1 t2 for t near → |[0,1/2] − 1 and g′ < 0 on (1/2, 1). In the standard symplectic R4 which we identify with C 2 with coordinates iϕ1 iϕ2 (z1 = r1e = x1 + iy1, z2 = r2e = x2 + iy2) let us consider a domain P = r g(r ), r [0, 1] . { 1 ≤ 2 2 ∈ } The domain P is containede in the polydisc P = r a, r 1 and can be { 1 ≤ 2 ≤ } viewed as obtained by smoothing the corners of P . e Let us denote by Γ the part of the boundary of P given by Γ= r = g(r ), r [1/e2, 1] . {{ 1 2 1 ∈ } Note that Γ is C∞ –tangent to ∂P near its boundary. The primitive 1 γ = r2dϕ + r2dϕ 2 1 1 2 2 of the standard symplectic form ω = dx dy + dx dy = r dr dϕ + r dr dϕ 0 1 ∧ 1 2 ∧ 2 1 1 ∧ 1 2 2 ∧ 2 restricts to Γ as a contact form 2 2 r2 g (r2) γ Γ = 2 dϕ1 + dϕ2 . | 2 r2 2 2 Consider the product G = S D with the split symplectic structure ω0 = × 2 σ1 σ2 , where the total area of the form σ1 on S is equal to 2π and the total ⊕ 2 2 2 2 area of the form σ2 on the disc D is equal to πa .
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