Open book foliations

Transcription

1 Open book foliations Tetsuya Ito (RIMS) joint with Keiko Kawamuro (Univ. Iowa) MSJ Autumn meeting Sep 26, 2013 Tetsuya Ito Open book foliations Sep 26, / 43

2 Open book Throughout this talk, every 3-manifold is assumed to be closed and oriented. S: Oriented compact surface with non-empty boundary Aut(S, S) = {ϕ : S Diff S ϕ S = id}/diffeotopy = MCG(S) = Mapping class group of S ϕ Aut(S, S): Surface automorphism Convention In this talk, we will often confuse ϕ Aut(S, S) and its representative diffeomorphism ϕ : S S. Tetsuya Ito Open book foliations Sep 26, / 43

3 Open book Throughout this talk, every 3-manifold is assumed to be closed and oriented. S: Oriented compact surface with non-empty boundary Aut(S, S) = {ϕ : S Diff S ϕ S = id}/diffeotopy = MCG(S) = Mapping class group of S ϕ Aut(S, S): Surface automorphism Convention In this talk, we will often confuse ϕ Aut(S, S) and its representative diffeomorphism ϕ : S S. Definition A pair (S, ϕ) is called an (abstract) open book. Tetsuya Ito Open book foliations Sep 26, / 43

4 Open book manifold For an open book (S, ϕ), the open book manifold M (S,ϕ) is an oriented, closed 3-manifold M (S,ϕ) = S [0, 1]/(x, 1) (ϕ(x), 0) }{{} Mapping torus (S 1 D 2 ) # S } {{ } Solid tori Ô Ë ½ Ô ¾ ˵ ÓÙÒ Ò Ò Ë ½ Ë ¼ Ë Ë Ø Ë Ø Ë ½ Tetsuya Ito Open book foliations Sep 26, / 43

5 Open book decomposition Definition For a closed, oriented 3-manifold M, an open book (S, ϕ) is called an open book decomposition of M if M = M (S,ϕ). An open book decomposition of M is nothing but a fibered link B (called a binding) in M with fibration M B S 1 specified. Ô Ë ½ Ô ¾ ˵ ÓÙÒ Ò Ò Ë ½ Ë ¼ Ë Ë Ø Ë Ø Ë ½ Tetsuya Ito Open book foliations Sep 26, / 43

6 Open book decomposition (Continued) Theorem (Alexander, 1923) Every closed, oriented 3-manifold M admits an open book decomposition. (Proof: Realize M as a branched covering of a closed braid in S 3 ) Thus, open book gives presentations of 3-manifolds (cf. Dehn surgery, Heegaard splitting, 3-fold branched covering) Tetsuya Ito Open book foliations Sep 26, / 43

7 Open book decomposition (Continued) Theorem (Alexander, 1923) Every closed, oriented 3-manifold M admits an open book decomposition. (Proof: Realize M as a branched covering of a closed braid in S 3 ) Thus, open book gives presentations of 3-manifolds (cf. Dehn surgery, Heegaard splitting, 3-fold branched covering) Example M (D 2,id) = S 3. M (A,id) = S 2 S 1 M (A,T m A ) = L(m, m 1) (m > 0). M (Σg,1,id) = (S 2 S 1 ) (S 2 S 1 ) (A: annulus, T A : Dehn twist along the core of A) Tetsuya Ito Open book foliations Sep 26, / 43

8 Contact structure α A 1 (M): 1-form on M ξ = Ker α TM: Plane field on M Definition ξ is called a (positive) contact structure on M if α dα > 0. A 3-manifold M equipped with a contact structure (M, ξ) is called a contact 3-manifold. Tetsuya Ito Open book foliations Sep 26, / 43

9 Contact structure α A 1 (M): 1-form on M ξ = Ker α TM: Plane field on M Definition ξ is called a (positive) contact structure on M if α dα > 0. A 3-manifold M equipped with a contact structure (M, ξ) is called a contact 3-manifold. c.f. Foliation ξ is a (tangent plane filed of) foliation (i.e, the plane field ξ is completely integrable) if and only if α dα = 0. Contact structure is regarded as an odd dimensional version of symplectic structure: The boundary of an symplectic 4-manifold is a contact 3-manifold. Tetsuya Ito Open book foliations Sep 26, / 43

10 Example I: Standard contact structure (r, θ, z): Cylindrical coordinate of R 3 S 3 = R 3 { } ξ std = Ker(dz + r 2 dθ) Tetsuya Ito Open book foliations Sep 26, / 43

11 Example I: Standard contact structure (r, θ, z): Cylindrical coordinate of R 3 S 3 = R 3 { } ξ std = Ker(dz + r 2 dθ) Fact (Darboux s theorem) Every contact structure is locally isomorphic to the standard contact structure (R 3, ξ std ) = (R 3, Ker(dz + xdy)). Tetsuya Ito Open book foliations Sep 26, / 43

12 Example II: Bennequin s exotic contact structure (r, θ, z): Cylindrical coordinate of R 3 ξ ot = Ker(cos rdz + sin rdθ) Tetsuya Ito Open book foliations Sep 26, / 43

13 Example II: Bennequin s exotic contact structure (r, θ, z): Cylindrical coordinate of R 3 ξ ot = Ker(cos rdz + sin rdθ) Fact (Bennequin, 1983) ξ std and ξ ot are homotopic as plane fields, but are not isotopic as contact structures (ξ std is tight, whereas ξ ot is overtwisted) Tetsuya Ito Open book foliations Sep 26, / 43

14 Giroux correspondence Giroux correspondence (Giroux, 2002) There is a one-to-one correspondence between two sets {Open books}/stabilization (+ conjugacy) and {Contact 3-manifolds}/contact isotopy Tetsuya Ito Open book foliations Sep 26, / 43

15 Giroux correspondence Giroux correspondence (Giroux, 2002) There is a one-to-one correspondence between two sets {Open books}/stabilization (+ conjugacy) and {Contact 3-manifolds}/contact isotopy c.f. (Markov Theorem) There is a one-to-one correspondence between two sets {(Closed) braids }/stabilization (+ conjugacy) and {Oriented links in S 3 }/isotopy Tetsuya Ito Open book foliations Sep 26, / 43

16 From open book to contact structure For an open book decomposition M (S,ϕ), take a singular (not continuous) plane field ξ OB so that: For p Int(S t ) (Page), ξ OB (p) = T p S t. For p B = S t (Binding), then ξ OB (p) transverse B. By perturbating ξ OB we get a contact structure. Tetsuya Ito Open book foliations Sep 26, / 43

17 Open book foliation: motivation Open book decomposition encodes both topology and contact structure, and provides an algebraic description of contact 3-manifolds. Tetsuya Ito Open book foliations Sep 26, / 43

18 Open book foliation: motivation Open book decomposition encodes both topology and contact structure, and provides an algebraic description of contact 3-manifolds. Motivating question Tetsuya Ito Open book foliations Sep 26, / 43

19 Open book foliation: motivation Open book decomposition encodes both topology and contact structure, and provides an algebraic description of contact 3-manifolds. Motivating question Understand the topology and (contact) geometry of 3-manifolds at one time (uniformally) by using open book decomposition. Tetsuya Ito Open book foliations Sep 26, / 43

20 Open book foliation: motivation Open book decomposition encodes both topology and contact structure, and provides an algebraic description of contact 3-manifolds. Motivating question Understand the topology and (contact) geometry of 3-manifolds at one time (uniformally) by using open book decomposition. Establish a relationship between a theory of mapping class group and (topology and) contact geometry of 3-manifolds. (c.f. Representation of braid groups Quantum invariants of knot and links) Tetsuya Ito Open book foliations Sep 26, / 43

21 Open book foliation: motivation Open book decomposition encodes both topology and contact structure, and provides an algebraic description of contact 3-manifolds. Motivating question Understand the topology and (contact) geometry of 3-manifolds at one time (uniformally) by using open book decomposition. Establish a relationship between a theory of mapping class group and (topology and) contact geometry of 3-manifolds. (c.f. Representation of braid groups Quantum invariants of knot and links) To this end, we introduce a new (topological and combinatorial, but having a quite elementary nature) technique, called an open book foliation. Tetsuya Ito Open book foliations Sep 26, / 43

22 Open book foliation: Idea (S, ϕ): Open book decomposition of (contact) 3-manifold (M, ξ) F M (S,ϕ) : (Possibly inessential) surfaces in M Take a singular foliation of F induced by intersection with pages, F = {F S t } t S 1 =[0,1]/0 1. Tetsuya Ito Open book foliations Sep 26, / 43

23 Open book foliation: Idea (S, ϕ): Open book decomposition of (contact) 3-manifold (M, ξ) F M (S,ϕ) : (Possibly inessential) surfaces in M Take a singular foliation of F induced by intersection with pages, F = {F S t } t S 1 =[0,1]/0 1. Foliation F encodes how F is embedded in M, and the contact structure near F. By putting F in a nice position, F can be simplified. Conversely, by making F simple in a reasonable manner, F is moved to better position (F reflects topology and geometry better) (c.f. Morse Theory proof of high-dimensional Poincaré conjecture) Open book foliation is a technique to manipulate the foliation F. Tetsuya Ito Open book foliations Sep 26, / 43

24 Open book foliation: Comments Open book foliation is a generalization of Birman-Menasco s braid foliation theory, which corresponds to the case (S 3, ξ std ) = M (D 2,id) and mainly treats (transverse) knots and links in S 3. Tetsuya Ito Open book foliations Sep 26, / 43

25 Open book foliation: Comments Open book foliation is a generalization of Birman-Menasco s braid foliation theory, which corresponds to the case (S 3, ξ std ) = M (D 2,id) and mainly treats (transverse) knots and links in S 3. Historically, an idea of open book foliation begun at Bennequin s work (1980 s): He used foliation F to distinguish the standard and exotic contact structure on S 3. Tetsuya Ito Open book foliations Sep 26, / 43

26 Open book foliation: Comments Open book foliation is a generalization of Birman-Menasco s braid foliation theory, which corresponds to the case (S 3, ξ std ) = M (D 2,id) and mainly treats (transverse) knots and links in S 3. Historically, an idea of open book foliation begun at Bennequin s work (1980 s): He used foliation F to distinguish the standard and exotic contact structure on S 3. Open book foliation is closely related to the characteristic foliation and convex surface theory in contact geometry. Open book foliation is, in some sense, topologically rigid version of characteristic foliation method. Tetsuya Ito Open book foliations Sep 26, / 43

27 Open book foliation: Comments Open book foliation is a generalization of Birman-Menasco s braid foliation theory, which corresponds to the case (S 3, ξ std ) = M (D 2,id) and mainly treats (transverse) knots and links in S 3. Historically, an idea of open book foliation begun at Bennequin s work (1980 s): He used foliation F to distinguish the standard and exotic contact structure on S 3. Open book foliation is closely related to the characteristic foliation and convex surface theory in contact geometry. Open book foliation is, in some sense, topologically rigid version of characteristic foliation method. In almost all cases, an open book folaition argument is indepedent of known techniques or theories, and provides a topological method to study contact structure. Tetsuya Ito Open book foliations Sep 26, / 43

28 Simplest case: Braid foliation Tetsuya Ito Open book foliations Sep 26, / 43

29 Open book folaition: setting and definition Definition Foliation F is called an open book foliation if (Fi) F transverse B, and v F B is an elliptic singular point of F. (Fii) The leaves of F transverse to F. (Fiii) S t transverse F for all but finitely many t. (Fiv) For exceptional t, S t tangent to F as a saddle tangency. Tetsuya Ito Open book foliations Sep 26, / 43

30 Open book folaition: setting and definition By taking signs of intersection (saddle), each singular point of F ob has sign. F B t B t t F t Tetsuya Ito Open book foliations Sep 26, / 43

31 Open book folaition: example Embedded sphere in S 3 = (D 2, id): Its open book foliation contains four elliptic points and two hyperbolic points. Tetsuya Ito Open book foliations Sep 26, / 43

32 Open book folaition: example II Overtwisted disc D in S 3 = (A, T 1 A ). ½ ¼ Tetsuya Ito Open book foliations Sep 26, / 43

33 Open book foliation: example II (Continued) The open book foliation of D consists of two positive hyperbolic and elliptic points, and one negative elliptic point. Tetsuya Ito Open book foliations Sep 26, / 43

34 Leaf There are three types of regular leaves of open book foliations. ¹ Ö ÒØ Ð ÙØ ÒÓØ ØÖÓÒ ÐÝ ÒØ Ð ¹ ÖÐ Ä ØÖÓÒ ÐÝ ÒØ Ð ¹ Ö Ë Ø Ò ÒØ Ð Ë Ø By homotopical property, b-arc is classified by three types: strongly essential, essential, and inessential. Tetsuya Ito Open book foliations Sep 26, / 43

35 Open book foliation: Basic facts Theorem (I-Kawamuro) 1 One can put every surface F so that it admits an open book foliation. Tetsuya Ito Open book foliations Sep 26, / 43

36 Open book foliation: Basic facts Theorem (I-Kawamuro) 1 One can put every surface F so that it admits an open book foliation. Moreover, one may further put F so that F ob contains no c-circles. Tetsuya Ito Open book foliations Sep 26, / 43

37 Open book foliation: Basic facts Theorem (I-Kawamuro) 1 One can put every surface F so that it admits an open book foliation. Moreover, one may further put F so that F ob contains no c-circles. 2 If F is incompressible, F can be put so that all b-arcs are essential. Tetsuya Ito Open book foliations Sep 26, / 43

38 Open book foliation: Basic facts Theorem (I-Kawamuro) 1 One can put every surface F so that it admits an open book foliation. Moreover, one may further put F so that F ob contains no c-circles. 2 If F is incompressible, F can be put so that all b-arcs are essential. 3 (Region decomposition) F is decomposed as a union of particular neighborhoods of hyperbolic points called regions. Tetsuya Ito Open book foliations Sep 26, / 43

39 Structural stability Structual stability Theorem (I-Kawamuro) If an open book foliation F ob has no c-circles, (this is always possible), then F ob can be identified with the characteristic foliation: foliation obtained by integrating vector filed TF ξ. Thus, open book foliation contain as much information as characteristic foliation. Tetsuya Ito Open book foliations Sep 26, / 43

40 Structural stability Structual stability Theorem (I-Kawamuro) If an open book foliation F ob has no c-circles, (this is always possible), then F ob can be identified with the characteristic foliation: foliation obtained by integrating vector filed TF ξ. Thus, open book foliation contain as much information as characteristic foliation. Remark Characteristic foliation (convex surface) theory is a machinery which provides cut-and-paste method in category of contact 3-manifolds, and extensively studied by many researchers. { Characteristic foliation } { Open book foliation }. Tetsuya Ito Open book foliations Sep 26, / 43

41 Good points for open book foliation 1 Compared to characteristic foliation, open book foliation is easy to visualize and is suited for explicit construction. Tetsuya Ito Open book foliations Sep 26, / 43

42 Good points for open book foliation 1 Compared to characteristic foliation, open book foliation is easy to visualize and is suited for explicit construction. 2 It is easy to see how open book foliation changes when we move surface F. Tetsuya Ito Open book foliations Sep 26, / 43

43 Good points for open book foliation 1 Compared to characteristic foliation, open book foliation is easy to visualize and is suited for explicit construction. 2 It is easy to see how open book foliation changes when we move surface F. 3 By using region decomposition and homotopical properties of leaves, one can encode open book foliation by combinatorial data so it is suited for algorithmic and algebraic treatment. Tetsuya Ito Open book foliations Sep 26, / 43

44 Good points for open book foliation 1 Compared to characteristic foliation, open book foliation is easy to visualize and is suited for explicit construction. 2 It is easy to see how open book foliation changes when we move surface F. 3 By using region decomposition and homotopical properties of leaves, one can encode open book foliation by combinatorial data so it is suited for algorithmic and algebraic treatment. 4 Open book foliation is topological, and applied for topological 3-manifolds we can treat both topology and geometry at one time. Tetsuya Ito Open book foliations Sep 26, / 43

45 Transverse knot and self-linking number Definition A knot K in (M, ξ) is a transverse knot if K positively transverse ξ. Transverse knot is represented by a closed braid. Tetsuya Ito Open book foliations Sep 26, / 43

46 Transverse knot and self-linking number Definition A knot K in (M, ξ) is a transverse knot if K positively transverse ξ. Transverse knot is represented by a closed braid. For null-homotopic transverse knot K with Seifert surface Σ, take a non-zero section X = X Σ of ξ Σ, and let K X = knot K slightly moved to the direction of X Σ=K. Definition The self-linking number of trasnverse knot K is sl(k, Σ) = lk(k, K X ) This depends on only [Σ] H 2 (M, K). Tetsuya Ito Open book foliations Sep 26, / 43

47 Self-linking number formula Bennequin s formula (1983) b: closed n-braid in S 3 = M (D 2,id) (= transverse knots in (S 3, ξ std )) sl( b) = n + exp(b) (exp : exponent sum) Tetsuya Ito Open book foliations Sep 26, / 43

48 Self-linking number formula Bennequin s formula (1983) b: closed n-braid in S 3 = M (D 2,id) (= transverse knots in (S 3, ξ std )) sl( b) = n + exp(b) (exp : exponent sum) Self-linking number and foliation (Eliashberg, 1992) K: Null-homologous transverse knot (M, ξ) Σ: Sefert surface of K e ±, h ± : The number of positive/negative elliptic and hyperbolic points of the characteristic foliation (open book foliation) of Σ sl(k, Σ) = [(e + h + ) (e h )] Tetsuya Ito Open book foliations Sep 26, / 43

49 Self-linking number formula Bennequin s formula (1983) b: closed n-braid in S 3 = M (D 2,id) (= transverse knots in (S 3, ξ std )) sl( b) = n + exp(b) (exp : exponent sum) Self-linking number and foliation (Eliashberg, 1992) K: Null-homologous transverse knot (M, ξ) Σ: Sefert surface of K e ±, h ± : The number of positive/negative elliptic and hyperbolic points of the characteristic foliation (open book foliation) of Σ how to count e ±, h ±? sl(k, Σ) = [(e + h + ) (e h )] Tetsuya Ito Open book foliations Sep 26, / 43

50 Self-linking number formula Theorem (I-Kawamuro) Let b be a closed n-braid in M (S,ϕ) which is null-homologous Then for a Seifert surface Σ of b, sl( b, Σ) = n + êxp(b) + ϕ (a Σ ) [b] + c(ϕ, a Σ ) where êxp(b) : Exponent sum (algebraic crossing number) a Σ = [Σ S 0 ] H 1 (S, S) = H 1 (S) : H 1 (S, S) H 1 (S) Z : Algebraic intersection c : MCG H 1 (S) Z : crossed homomorphism Moreover, [c] H 1 (MCG; H) = Z is non-trivial. Tetsuya Ito Open book foliations Sep 26, / 43

51 Idea of proof Use open book foliation technique, we explicitly construct the Seifert surface Σ of K and visualize open book foliation of Σ. Then n : number of positive elliptic points from a-arcs êxp(b) : signed count of the number of twisted bands ϕ (a Σ ) [b] : number of intersection with bands and Σ c(ϕ, a Σ ) : correction term for gluing to work Then by examing the properties of c(ϕ, ) we see [c] H 1 (MCG; H) = Z is non-trivial cohomology class. Tetsuya Ito Open book foliations Sep 26, / 43

52 Consequence of Self-linking number formula Self-linking number formula says that: Tetsuya Ito Open book foliations Sep 26, / 43

53 Consequence of Self-linking number formula Self-linking number formula says that: Self-linking number provides a non-trivial cohomology class [c] H 1 (MCG; H). (This is a contraction of the Johnson-Morita homomorphism, and appears in several ways) Thus, contact geometry is also related to (cohomology theory of) mapping class group in a reasonable, but surprising way!!! Tetsuya Ito Open book foliations Sep 26, / 43

54 Consequence of Self-linking number formula Self-linking number formula says that: Self-linking number provides a non-trivial cohomology class [c] H 1 (MCG; H). (This is a contraction of the Johnson-Morita homomorphism, and appears in several ways) Thus, contact geometry is also related to (cohomology theory of) mapping class group in a reasonable, but surprising way!!! Question future research Does (higher) cohomology class/johnson-morita homomorphisms of MCG provide invariants of transverse knot? (Explore a higher self-linking numbers) Tetsuya Ito Open book foliations Sep 26, / 43

55 Fractional Dehn twist coefficient For C S (boundary component of S) and ϕ Aut(S, S), the Fractional Dehn twist coefficient (FDTC) around C is a rational number c(ϕ, C) = how many times ϕ twists F near C Q µ ¾ FDTC has its origin in Nielsen-Thurston theory and foliation/lamination theory (though it is defined in a current general form in 2008) Tetsuya Ito Open book foliations Sep 26, / 43

56 FDTC and open book foliation A crucial observation is that open book foliation evaluates FDTC. FDTC Lemma (I-Kawamuro) If v C S is an elliptic point of F ob (F ) such that all leaves from v are essential b-arcs. If around v, there are p positive and n negative hyperbolic points then { n c(ϕ, C) p v is positive p c(ϕ, C) n v is negative Tetsuya Ito Open book foliations Sep 26, / 43

57 FDTC and open book foliation A crucial observation is that open book foliation evaluates FDTC. FDTC Lemma (I-Kawamuro) If v C S is an elliptic point of F ob (F ) such that all leaves from v are essential b-arcs. If around v, there are p positive and n negative hyperbolic points then { n c(ϕ, C) p v is positive p c(ϕ, C) n v is negative Remark This is a generalization of the relationship between the Dehornoy ordering of braid groups and braid foliation) Tetsuya Ito Open book foliations Sep 26, / 43

58 FDTC and incompressible surface in open book Idea of proof: One saddle = hyperbolic points twists b-arc at most once. Ø ¼ Ë Ø ¼ Ð Ú Ø ¼ Ø Ú Ë Ø Ø Tetsuya Ito Open book foliations Sep 26, / 43

59 FDTC and incompressible surface in open book With help of region decomposions, from FDTC lemma we conclude that incompressible surfaces in M (S,ϕ) give an estimation of c(ϕ, C). Tetsuya Ito Open book foliations Sep 26, / 43

60 FDTC and incompressible surface in open book With help of region decomposions, from FDTC lemma we conclude that incompressible surfaces in M (S,ϕ) give an estimation of c(ϕ, C). Theorem (I-Kawamuro) If c(ϕ, C) > 3 for all boundary component of S, then M (S,ϕ) is irreducible. If c(ϕ, C) > 4 for all boundary component of S and ϕ is irreducible, then M (S,ϕ) is atoroidal. Tetsuya Ito Open book foliations Sep 26, / 43

61 FDTC and incompressible surface in open book With help of region decomposions, from FDTC lemma we conclude that incompressible surfaces in M (S,ϕ) give an estimation of c(ϕ, C). Theorem (I-Kawamuro) If c(ϕ, C) > 3 for all boundary component of S, then M (S,ϕ) is irreducible. If c(ϕ, C) > 4 for all boundary component of S and ϕ is irreducible, then M (S,ϕ) is atoroidal. Sketch of Proof: If there exists an incompressible sphere or torus F, put F so that it admits an essential open book foliation. Euler characteristics and region decomposition show that the number of hyperbolic points around ellitpic point is not so large, so FDTC is small. Tetsuya Ito Open book foliations Sep 26, / 43

62 Geometric structure: open book manifold Open book foliation argument shows there is a one-to-one correspondence between Nielsen-Thurston classification (Dynamics of MCG) and Geometric structure. Theorem (I-Kawamuro) Assume that one of the following conditions are satisfied. Then, c(ϕ, C) > 4 for all boundary components C of S. S is connected, and c(ϕ, S) > 1. 1 ϕ is periodic M (S,ϕ) is Seifert-fibered 2 ϕ is reducible M (S,ϕ) is toroidal 3 ϕ is pseudo-anosov M (S,ϕ) is hyperbolic Tetsuya Ito Open book foliations Sep 26, / 43

63 Geometric structure: link complement L M (S,ϕ) : Closed n-braid S = S {n Discs} Then we have ϕ L : S = S 1 (L S 1 ) S 0 (L S 0 ) = S Theorem (I-Kawamuro) Assume that one of the following conditions are satisfied. Then, c(ϕ L, C) > 4 for all boundary components C of S S. S is connected, and c(ϕ L, S) > 1. 1 ϕ L is periodic M (S,ϕ) L is Seifert-fibered. 2 ϕ L is reducible M (S,ϕ) L is toroidal. 3 ϕ L is pseudo-anosov M (S,ϕ) L is hyperbolic. caution c(ϕ, C) and c(ϕ L, C) may different. Tetsuya Ito Open book foliations Sep 26, / 43

64 Tight vs. Overtwisted Definition A contact structure ξ of M is overtwisted if disc D M such that: 1 T p D = ξ p at p D. 2 For q Int(D), T q D transverse ξ q for all but exactly one exception Otherwise, ξ is called tight. Tetsuya Ito Open book foliations Sep 26, / 43

65 Tight vs. Overtwisted Definition A contact structure ξ of M is overtwisted if disc D M such that: 1 T p D = ξ p at p D. 2 For q Int(D), T q D transverse ξ q for all but exactly one exception Otherwise, ξ is called tight. Example: ξ ot = Ker(cos rdz + sin rdθ) is overtwisted. Tetsuya Ito Open book foliations Sep 26, / 43

66 Overtwisted contact structure is flexible Theorem (Lutz-Martinet 1971) Every plane filed of M is homotopic to an overtwisted contact structure. Tetsuya Ito Open book foliations Sep 26, / 43

67 Overtwisted contact structure is flexible Theorem (Lutz-Martinet 1971) Every plane filed of M is homotopic to an overtwisted contact structure. Theorem (Eliashberg, 1989) If two overtwisted contact strutures ξ and ξ are homotopic as plane fields, then ξ and ξ are isopotic as contact structures. Tetsuya Ito Open book foliations Sep 26, / 43

68 Overtwisted contact structure is flexible Theorem (Lutz-Martinet 1971) Every plane filed of M is homotopic to an overtwisted contact structure. Theorem (Eliashberg, 1989) If two overtwisted contact strutures ξ and ξ are homotopic as plane fields, then ξ and ξ are isopotic as contact structures. Classification of overtwisted contact structure is the same as the classification of plane field. Tetsuya Ito Open book foliations Sep 26, / 43

69 Overtwisted contact structure is flexible Theorem (Lutz-Martinet 1971) Every plane filed of M is homotopic to an overtwisted contact structure. Theorem (Eliashberg, 1989) If two overtwisted contact strutures ξ and ξ are homotopic as plane fields, then ξ and ξ are isopotic as contact structures. Classification of overtwisted contact structure is the same as the classification of plane field. It is a problem of classical algebraic/differential topology and easy to solve. Tetsuya Ito Open book foliations Sep 26, / 43

70 Overtwisted contact structure is flexible Theorem (Lutz-Martinet 1971) Every plane filed of M is homotopic to an overtwisted contact structure. Theorem (Eliashberg, 1989) If two overtwisted contact strutures ξ and ξ are homotopic as plane fields, then ξ and ξ are isopotic as contact structures. Classification of overtwisted contact structure is the same as the classification of plane field. It is a problem of classical algebraic/differential topology and easy to solve. Classification of overtwisted contact structure is done! Tetsuya Ito Open book foliations Sep 26, / 43

71 Tight contact structure On the other hand, tight contact structure is still mysterious. Theorem (Etnyre-Honda 2002) -(Poincaré homology 3-sphere) does not have tight contact structures. Tetsuya Ito Open book foliations Sep 26, / 43

72 Tight contact structure On the other hand, tight contact structure is still mysterious. Theorem (Etnyre-Honda 2002) -(Poincaré homology 3-sphere) does not have tight contact structures. Big open problem Which 3-manifold has a tight contact structure? It is often hard to know given contact structure is tight or not. Knowns tehchnique to show tightness are: (most of them require/are based on high-tech machinery like gauge theory) Tetsuya Ito Open book foliations Sep 26, / 43

73 Tight contact structure On the other hand, tight contact structure is still mysterious. Theorem (Etnyre-Honda 2002) -(Poincaré homology 3-sphere) does not have tight contact structures. Big open problem Which 3-manifold has a tight contact structure? It is often hard to know given contact structure is tight or not. Knowns tehchnique to show tightness are: (most of them require/are based on high-tech machinery like gauge theory) 1 Filling (realize (M, ξ) as a boundary of sympletic 4-manifold-) 2 Ozsvath-Szabo s Heegaard Floer homology invariant 3 Contact homology (Reeb flow and orbits) 4 Deformation of foliation (confoliation) 5 State traversal (cut-and-paste technique using convex surface theory) Tetsuya Ito Open book foliations Sep 26, / 43

74 Overtwistedness theorem Question How can we know that an open book (S, ϕ) corresponds to a tight/overtwisted contact structure? Tetsuya Ito Open book foliations Sep 26, / 43

75 Overtwistedness theorem Question How can we know that an open book (S, ϕ) corresponds to a tight/overtwisted contact structure? Theorem (Honda-Kazéz-Matić 2008, I-Kawamuro) Let (S, ϕ) be an open book decomposition of (M, ξ). Assume that ϕ is not right-veering (which is almost equivalent to saying that c(ϕ, C) < 0 for some boundary component C). Then ξ is overtwisted. Tetsuya Ito Open book foliations Sep 26, / 43

76 Overtwistedness theorem Question How can we know that an open book (S, ϕ) corresponds to a tight/overtwisted contact structure? Theorem (Honda-Kazéz-Matić 2008, I-Kawamuro) Let (S, ϕ) be an open book decomposition of (M, ξ). Assume that ϕ is not right-veering (which is almost equivalent to saying that c(ϕ, C) < 0 for some boundary component C). Then ξ is overtwisted. Unfortunately, converse is not true right-veering does not necessarily imply tight. Remark: partial converse If ξ is overtwisted, then there is a non-right-veering open book decomposition of (M, ξ). Tetsuya Ito Open book foliations Sep 26, / 43

77 Highly-nontrivial example of overtwisted open books Theorem (I-Kawamuro) Let S be a sphere minus four discs, and Φ = Φ h,i,k = T h a T i b T ct d T k 1 e h, i, k 1. Then the open book (S, Φ) supports an overtwisted contact structure ξ. This example is stimulating, since (S, Φ) is: Tetsuya Ito Open book foliations Sep 26, / 43

78 Highly-nontrivial example of overtwisted open books Theorem (I-Kawamuro) Let S be a sphere minus four discs, and Φ = Φ h,i,k = T h a T i b T ct d T k 1 e h, i, k 1. Then the open book (S, Φ) supports an overtwisted contact structure ξ. This example is stimulating, since (S, Φ) is: { Non-destabilizable (cannot be simplified in an obvious way) FDTC is very large (right-veering) : FDTC = (h, i, 1, 1) Tetsuya Ito Open book foliations Sep 26, / 43

79 Tightness problem As for tightness, several results are known for the case S is connected: Theorem (Honda-Kazez-Matić 2008) If S is connected and c(ϕ, S) 1, then (S, ϕ) is an open book decomposition of a tight contact 3-manifold. Tetsuya Ito Open book foliations Sep 26, / 43

80 Tightness problem As for tightness, several results are known for the case S is connected: Theorem (Honda-Kazez-Matić 2008) If S is connected and c(ϕ, S) 1, then (S, ϕ) is an open book decomposition of a tight contact 3-manifold. Theorem (Honda-Kazez-Matić 2009) If S is once-punctured torus then (S, ϕ) is an open book decomposition of a tight contact 3-manifold if and only if ϕ is right-veering (this is almost equivalent to c(s, ϕ) > 0) Tetsuya Ito Open book foliations Sep 26, / 43

81 Tightness problem As for tightness, several results are known for the case S is connected: Theorem (Honda-Kazez-Matić 2008) If S is connected and c(ϕ, S) 1, then (S, ϕ) is an open book decomposition of a tight contact 3-manifold. Theorem (Honda-Kazez-Matić 2009) If S is once-punctured torus then (S, ϕ) is an open book decomposition of a tight contact 3-manifold if and only if ϕ is right-veering (this is almost equivalent to c(s, ϕ) > 0) Theorem (Colin-Honda 2013) If S is connected, ϕ is pseudo-anosov and c(ϕ, S) > 2 k, then (S, ϕ) is an open book decomposition of a tight contact 3-manifold. Tetsuya Ito Open book foliations Sep 26, / 43

82 Tightness theorem Open book foliation gives new tightness criterion. Theorem (I-Kawamuro) Let (S, ϕ) be a planer open book decomposition of (M, ξ). If c(ϕ, C) > 1 for all boundary component C of S, then ξ is tight. Tetsuya Ito Open book foliations Sep 26, / 43

83 Tightness theorem Open book foliation gives new tightness criterion. Theorem (I-Kawamuro) Let (S, ϕ) be a planer open book decomposition of (M, ξ). If c(ϕ, C) > 1 for all boundary component C of S, then ξ is tight. Remark The condition c(ϕ, C) > 1 is best-possible. (We have seen that there is a four-punctured sphere open book (S, Φ) with FDTC = (h, i, 1, 1) (h, i Z), which supports an overtwisted contact structure.) A remarkable point is that, our proof is elementray (low-technology) and topological, and says something stronger: If (S, ϕ) is overtwisted open book, then the action of ϕ on the space of curves must satisfy certain special properties. Tetsuya Ito Open book foliations Sep 26, / 43

84 Reference 1 (Braid foliation and ordering theory) T. Ito, Braid ordering and the geometry of closed braids, Geom. Topol 15 (2011) T. Ito, Braid ordering and knot genus, J. Knot Theory Ramifications 20 (2011), (Self-linking number formula (based on characteristic foliation theory)) K. Kawamuro and E. Pavalescu, The self-linking number in annulus and pants open book decompositions, Algebr. Geom. Topol. 11 (2011) K. Kawamuro, The self-linking number in planer open book decompositions Math. Res. Lett 18 (2012) Tetsuya Ito Open book foliations Sep 26, / 43

85 Reference 2 (Open book foliation) T. Ito and Keiko Kawamuro, Open book foliation, arxiv: v1 T. Ito and Keiko Kawamuro, Visualizing overtwisted discs in open books, submitted T. Ito and Keiko Kawamuro, Essential open book foliation and fractional Dehn twist coefficient, arxiv: T. Ito and Keiko Kawamuro, Operations on open book foliation, arxiv: T. Ito and Keiko Kawamuro, Overtwisted discs in open books, in preparation. Tetsuya Ito Open book foliations Sep 26, / 43