Stein fillings of contact 3-manifolds obtained as Legendrian sur

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1 Stein fillings of contact 3-manifolds obtained as Legendrian surgeries Youlin Li (With Amey Kaloti) Shanghai Jiao Tong University A Satellite Conference of Seoul ICM 2014 Knots and Low Dimensional Manifolds August 23, 2014

2 Contact 3-manifolds Suppose M is an oriented connected 3-dim smooth manifold, and α Ω 1 (M). If α dα > 0, then α is called a contact form, and ξ = ker α is called a contact structure.

3 Contact 3-manifolds Suppose M is an oriented connected 3-dim smooth manifold, and α Ω 1 (M). If α dα > 0, then α is called a contact form, and ξ = ker α is called a contact structure. Equivalently, a contact structure ξ on M is a nowhere integrable plane field in TM.

4 Examples of contact structures Example Suppose the 3-space R 3 has coordinate (x, y, z), then α = dz ydx is a contact form, since α dα = dx dy dz 0. ker α =< y, x + y z > is a contact structure on R 3. We denote it by (R 3, ξ st ).

5 Examples of contact structures Example Suppose the 3-space R 3 has coordinate (x, y, z), then α = dz ydx is a contact form, since α dα = dx dy dz 0. ker α =< y, x + y z > is a contact structure on R 3. We denote it by (R 3, ξ st ).

6 Examples of contact structures Example Suppose S 3 = {(x 1, x 2, x 3, x 4 ) R 4 : 4 xi 2 i=1 = 1}, then it has a contact structure ker(x 1 dx 2 x 2 dx 1 + x 3 dx 4 x 4 dx 3 ). We denote it by (S 3, ξ st ). Suppose S 3 = {(z 1, z 2 ) C 2 : z z 2 2 = 1}, then ξ st = TS 3 J(TS 3 ), where J is an almost complex structure on C 2.

7 Examples of contact structures Example Suppose S 3 = {(x 1, x 2, x 3, x 4 ) R 4 : 4 xi 2 i=1 = 1}, then it has a contact structure ker(x 1 dx 2 x 2 dx 1 + x 3 dx 4 x 4 dx 3 ). We denote it by (S 3, ξ st ). Suppose S 3 = {(z 1, z 2 ) C 2 : z z 2 2 = 1}, then ξ st = TS 3 J(TS 3 ), where J is an almost complex structure on C 2. Remark: point pt S 3, (S 3 \ {pt}, ξ st S 3 \{pt}) is contactomorphic to (R 3, ξ st ).

8 Legendrian knots Given a contact 3-manifold (M, ξ), a knot L M is called Legendrian if L is everywhere tangent to ξ, i.e., L (p) ξ p at every point p L.

9 Legendrian knots Given a contact 3-manifold (M, ξ), a knot L M is called Legendrian if L is everywhere tangent to ξ, i.e., L (p) ξ p at every point p L. Two Legendrian knots L 0 and L 1 are Legendrian isotopic if there is a continuous family L t, t [0, 1], of Legendrian knots starting at L 0 and ending at L 1.

10 Legendrian knots Given a contact 3-manifold (M, ξ), a knot L M is called Legendrian if L is everywhere tangent to ξ, i.e., L (p) ξ p at every point p L. Two Legendrian knots L 0 and L 1 are Legendrian isotopic if there is a continuous family L t, t [0, 1], of Legendrian knots starting at L 0 and ending at L 1. In this talk, we focus on the Legendrian knots in (S 3, ξ st ), which can be viewed clearly in (R 3, ξ st ) (S 3, ξ st ).

11 Front projection The projection of a Legendrian knot in (R 3, ξ st = ker(dz ydx)) on the xz-plane is the front projection.

12 Thurston-Bennequin invariant If a knot K S 3 is contained in a surface F, then the surface framing of K is a vector field which spans T p F with K (p) at every point p K. In particular, If F is the Seifert surface of K, then we call the surface framing as Seifert framing.

13 Thurston-Bennequin invariant If a knot K S 3 is contained in a surface F, then the surface framing of K is a vector field which spans T p F with K (p) at every point p K. In particular, If F is the Seifert surface of K, then we call the surface framing as Seifert framing. The contact framing of a Legendrian knot L (M, ξ) is a vector field which spans ξ p with L (p) at every point p L.

14 Thurston-Bennequin invariant If a knot K S 3 is contained in a surface F, then the surface framing of K is a vector field which spans T p F with K (p) at every point p K. In particular, If F is the Seifert surface of K, then we call the surface framing as Seifert framing. The contact framing of a Legendrian knot L (M, ξ) is a vector field which spans ξ p with L (p) at every point p L. The Thurston-Bennequin invariant of a Legendrian knot L (S 3, ξ st ) measures the contact framing with respect to the Seifert framing.

15 Rotation number Suppose L is a Legendrian knot in (S 3, ξ st ), and F is a Seifert surface of L, then ξ st F form a trivial two dimensional bundle. A trivialization of ξ st F induces a trivialization ξ st L = L R 2. Let v be the tangent vector field of L. The winding number of v along L w.r.t. the trivialization ξ st L = L R 2 is called the rotation number of L, denoted by rot(l).

16 Stabilization We define the local operation positive (negative) stabilization, by adding a downward (upward) zigzag in the front projection.

17 Stabilization We define the local operation positive (negative) stabilization, by adding a downward (upward) zigzag in the front projection. 1, S + S (L) = S S + (L).

18 Stabilization We define the local operation positive (negative) stabilization, by adding a downward (upward) zigzag in the front projection. 1, S + S (L) = S S + (L). 2, rot(s ± (L)) = rot(l) ± 1.

19 Stabilization We define the local operation positive (negative) stabilization, by adding a downward (upward) zigzag in the front projection. 1, S + S (L) = S S + (L). 2, rot(s ± (L)) = rot(l) ± 1. 3, tb(s ± (L)) = tb(l) 1.

20 Classification of Legendrian twist knots in (S 3, ξ st ) Twist knot, denoted by K m, is the knot of the following type. The box contains m right handed half twists if m 0, and m left handed half twists if m < 0.

21 Classification of Legendrian twist knots in (S 3, ξ st ) Twist knot, denoted by K m, is the knot of the following type. The box contains m right handed half twists if m 0, and m left handed half twists if m < 0.

22 Classification of Legendrian twist knots Theorem (Etnyre-Ng-Vértesi) For any integer m, the Legendrian twist knot K m can be classified up to Legendrian isotopy. In particular, for p 1, there is a unique Legendrian twist knot K 2p, denoted by L 0, with tb = 1 and rot = 0. If a Legendrian twist knot K 2p has the same tb and rot with some stabilization of L 0, then it is Legendrian isotopic to that stabilization of L 0.

23 Classification of Legendrian twist knots This is the Legendrian twist knot K 2p, L 0, with tb = 1 and rot = 0. The box contains 2p 2 S -shape Legendrian tangle.

24 Classification of Legendrian twist knots This is a stabilization of L, S n k + S k 1 (L 0), where n k 1.

25 Stein filling A 2-dimensional complex manifold S is a Stein surface if it admits a proper biholomorphic embedding S C 4.

26 Stein filling A 2-dimensional complex manifold S is a Stein surface if it admits a proper biholomorphic embedding S C 4. [Grauert] A 2-dimensional complex manifold S is Stein iff it admits a proper exhausting plurisubharmonic function ρ : S [0, ).

27 Stein filling A 2-dimensional complex manifold S is a Stein surface if it admits a proper biholomorphic embedding S C 4. [Grauert] A 2-dimensional complex manifold S is Stein iff it admits a proper exhausting plurisubharmonic function ρ : S [0, ). For any regular value c of ρ, the manifold S c := {p S : ρ(p) c} is a Stein domain. Example: D 4 C 2.

28 Stein filling A 2-dimensional complex manifold S is a Stein surface if it admits a proper biholomorphic embedding S C 4. [Grauert] A 2-dimensional complex manifold S is Stein iff it admits a proper exhausting plurisubharmonic function ρ : S [0, ). For any regular value c of ρ, the manifold S c := {p S : ρ(p) c} is a Stein domain. Example: D 4 C 2. Let M = S c, then ξ = TM J(TM) is a contact structure on M. We call the Stein domain S c a Stein filling of (M, ξ). Example: D 4 C 2 is a Stein filling of (S 3, ξ st ).

29 Legendrian surgery Given a Legendrian knot L in any contact 3-manifold (M, ξ), a Legendrian surgery on L yields the contact 3-manifold (M, ξ ), where (1) M is obtained from M by a Dehn surgery on L with framing 1 rel. to the contact framing, and (2) ξ is obtained by extending ξ M\N(L) to the reglued copy of N(L), where N(L) is the standard convex nbhd of L in (M, ξ).

30 Weinstein handle The Weinstein handle H is a 4-dimensional 2-handle D 2 D 2 = {(x 1, x 2, y 1, y 2 ) R 4 : x x 2 2 1, y y 2 2 1} with the symplectic form dλ, where λ = 1.5(x 1 dy 1 + x 2 dy 2 ) + 0.5(y 1 dx 1 + y 2 dx 2 ). Both S 1 D 2 and D 2 S 1 are contact solid tori. The attaching sphere S is Legendrian in S 1 D 2.

31 Legendrian surgery and Stein filling Theorem (Eliashberg) Suppose that W is a (complex) 2-dimensional Stein domain and L W is a Legendrian knot. By attaching a Weinstein handle H to W along L with framing 1 rel. to its contact framing to W along L, the Stein structure can be extended uniquely to W = W H.

32 Legendrian surgery and Stein filling Theorem (Eliashberg) Suppose that W is a (complex) 2-dimensional Stein domain and L W is a Legendrian knot. By attaching a Weinstein handle H to W along L with framing 1 rel. to its contact framing to W along L, the Stein structure can be extended uniquely to W = W H. Corollary A Legendrian link L (S 3, ξ st ) determines a Stein domain X L. Toplogically X L is given by 2-handle attachments along the link L with framings 1 rel. to the contact framings of the individual components. In particular, the Legendrian surgery on (S 3, ξ st ) along L yields a contact 3-manifold which has a Stein filling X L.

33 Classification of Stein fillings Stein fillings of a given contact 3-manifold are classified up to homeomorphism, up to diffeomorphism, up to Symplectic deformation, or up to symplectomorphism.

34 Classification of Stein fillings Stein fillings of a given contact 3-manifold are classified up to homeomorphism, up to diffeomorphism, up to Symplectic deformation, or up to symplectomorphism. Symplectomorphism Symplectic deformation diffeomorphism homeomorphism

35 Classification of Stein fillings Stein fillings of a given contact 3-manifold are classified up to homeomorphism, up to diffeomorphism, up to Symplectic deformation, or up to symplectomorphism. Symplectomorphism Symplectic deformation diffeomorphism homeomorphism [Eliashberg] There is a unique Stein filling, up to symplectic deformation, of (S 3, ξ st ).

36 Classification of Stein fillings [Stipsicz] There is a unique Stein filling, up to homeomorphism, of the Poincaré homology sphere Σ(2, 3, 5) and the 3-torus T 3.

37 Classification of Stein fillings [Stipsicz] There is a unique Stein filling, up to homeomorphism, of the Poincaré homology sphere Σ(2, 3, 5) and the 3-torus T 3. [Wendl] There is a unique Stein filling, up to symplectomorphism, of the 3-torus T 3.

38 Classification of Stein fillings [Stipsicz] There is a unique Stein filling, up to homeomorphism, of the Poincaré homology sphere Σ(2, 3, 5) and the 3-torus T 3. [Wendl] There is a unique Stein filling, up to symplectomorphism, of the 3-torus T 3. [McDuff] For a universally tight contact structure, there is a unique Stein filling of a lens space L(p, 1) (p 4), and there are two Stein fillings of the lens space L(4, 1), up to diffeomorphism. Remark: One of the two above Stein fillings of L(4, 1) can be obtained by attaching Weinstein handle.

39 Classification of Stein fillings [Plamenevskaya-Van Horn-Morris] There is a unique Stein filling, up to symplectic deformation, of L(p, 1) with any virtually overtwisted (i.e., not universally tight) tight contact structure.

40 Classification of Stein fillings [Plamenevskaya-Van Horn-Morris] There is a unique Stein filling, up to symplectic deformation, of L(p, 1) with any virtually overtwisted (i.e., not universally tight) tight contact structure. Remark: They used the fact that these contact 3-manifolds are obtained by Legendrian surgeries on Legendrian unknots in (S 3, ξ st ) with mixed stabilizations.

41 Classification of Stein fillings [Lisca] classified the Stein fillings of universally tight lens spaces L(p, q) up to diffeomorphism.

42 Classification of Stein fillings [Lisca] classified the Stein fillings of universally tight lens spaces L(p, q) up to diffeomorphism. [Kaloti] classified the Stein fillings of virtually overtwisted tight lens spaces L(pm + 1, m) up to symplectic deformation.

43 Classification of Stein fillings [Lisca] classified the Stein fillings of universally tight lens spaces L(p, q) up to diffeomorphism. [Kaloti] classified the Stein fillings of virtually overtwisted tight lens spaces L(pm + 1, m) up to symplectic deformation. [Ohta-Ono] classified the Stein fillings of some links of simple singularities up to diffeomorphism.

44 Classification of Stein fillings [Lisca] classified the Stein fillings of universally tight lens spaces L(p, q) up to diffeomorphism. [Kaloti] classified the Stein fillings of virtually overtwisted tight lens spaces L(pm + 1, m) up to symplectic deformation. [Ohta-Ono] classified the Stein fillings of some links of simple singularities up to diffeomorphism. [Starkston] gave finiteness results and some classifications, up to diffeomorphism, of minimal Stein fillings of certain contact Seifert fibered spaces over S

45 Main result I Theorem (Kaloti-L.) If L 0 is a Legendrian twist knot K 2p with tb = 1 and rot = 0, then the Legendrian surgery on (S 3, ξ st ) along any stabilization of L 0 yields a contact 3-manifold with unique Stein filling up to symplectic deformation.

46 Main result I Theorem (Kaloti-L.) If L 0 is a Legendrian twist knot K 2p with tb = 1 and rot = 0, then the Legendrian surgery on (S 3, ξ st ) along any stabilization of L 0 yields a contact 3-manifold with unique Stein filling up to symplectic deformation. Remark: If the stabilization on L 0 is S+ n k S k 1 (L 0), then the resulted contact 3-manifold is diffeomorphic to S n 1 3 (K 2p).

47 Open book decompositions EMBEDDED VERSION: An open book decomposition of M is a pair (B, π) where (1) B is an oriented link in M called the binding of the open book, and (2) π : M \ B S 1 is a fibration of the complement of B such that (i) π 1 (θ) is the interior of a compact surface Σ θ M and (ii) Σ θ = B for all θ S 1. The surface Σ = Σ θ, for any θ, is called the page of the open book.

48 Open book decompositions ABSTRACT VERSION: Suppose Σ is a compact bordered surface, and f is an orientation preserving automorphism of Σ fixing Σ. Let M f be Σ [0, 1]/, where (x, 1) (f (x), 0) for x Σ, and (y, t) (y, t ) for y Σ, t, t [0, 1]. If M f is diffeomorphic to M, then we call the pair (Σ, f ) is an abstract open book decomposition of M, and call f the monodromy.

49 Examples of open book decompositions of S 3

50 Contact structures and open book decompositions Definition (Thurston-Winkelnkemper) A contact structure ξ on M is supported by an embedded open book decomposition (B, π) of M if there is a contact 1-form α for ξ such that (1) dα is a positive area form on each page Σ θ of the open book and (2) α > 0 on B (Recall: B and the pages are oriented).

51 Contact structures and open book decompositions Definition (Thurston-Winkelnkemper) A contact structure ξ on M is supported by an embedded open book decomposition (B, π) of M if there is a contact 1-form α for ξ such that (1) dα is a positive area form on each page Σ θ of the open book and (2) α > 0 on B (Recall: B and the pages are oriented). Every embedded open book decomposition can be transformed to an abstract one. If (B, π) can be transformed to an abstract open book decomposition (Σ, f ), then we say that ξ is supported by (Σ, f ).

52 Main result II Σ : a cpct planar surface with n + p + q + 1 bdry cpnts. c 0, c 1,..., c n+p+q : components of Σ. Φ : = τ m 1 1 τ m τ m n+q 1 n+q 1 τ m n+q+1 n+q+1... τ mn+p+q n+p+q τ B 1 τ B2. τ i : positive Dehn twist along the bdry cpnt c i.

53 Main result II Theorem (Kaloti-L.) Let (M, ξ) be the contact 3-manifold supported by the open book decomposition (Σ, Φ). Then the contact 3-manifold (M, ξ) admits a unique Stein filling up to diffeomorphism.

54 Main result II Corollary (Kaloti-L.) Let (M, ξ) be the contact 3-manifold obtained by Legendrian surgery on the following Legendrian link. Then the contact 3-manifold (M, ξ) admits a unique Stein filling up to diffeomorphism.

55 Tools used in the proof i: Open book decom., Lefschetz fibration and Stein filling Theorem (Loi-Piergallini, Akbulut-Ozbagci, Giroux)

56 Tools used in the proof ii: Stein fillings of planar open book decompositions Theorem (Wendl) Suppose (M, ξ) is supported by a planar open book decomposition. Then every Stein filling of (M, ξ) is symplectic deformation equivalent to a positive allowable Lefschetz fibration compatible with the given open book decomposition of (M, ξ).

57 Tools used in the proof iii: Mapping class group techniques 1, Abelianization of mapping class groups for planar compact surfaces,

58 Tools used in the proof iii: Mapping class group techniques 1, Abelianization of mapping class groups for planar compact surfaces, 2, Stretch factors of pseudo-anosov diffeomorphisms.

59 Thank you!