We define an invariant of contact structures in dimension three from Heegaard Floer homology. Thi... more We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set Zě0 Y t8u. It is zero for overtwisted contact structures, 8 for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. As an application, we obstruct Stein fillability on contact 3-manifolds with non-vanishing Ozsváth-Szabó contact class.
We give a necessary and sufficient condition for the addition of a collection of disjoint bypasse... more We give a necessary and sufficient condition for the addition of a collection of disjoint bypasses to a convex surface to be universally tight-namely the nonexistence of a polygonal region which we call a virtual pinwheel.
We take a first step towards understanding the relationship between foliations and universally ti... more We take a first step towards understanding the relationship between foliations and universally tight contact structures on hyperbolic 3-manifolds. If a surface bundle over a circle has pseudo-Anosov holonomy, we obtain a classification of "extremal" tight contact structures. Specifically, there is exactly one contact structure whose Euler class, when evaluated on the fiber, equals the Euler number of the fiber. This rigidity theorem is a consequence of properties of the action of pseudo-Anosov maps on the complex of curves of the fiber and a remarkable flexibility property of convex surfaces in such a space. Indeed this flexibility may be seen in surface bundles over an interval where the analogous classification theorem is also established.
We present an alternate description of the Ozsváth-Szabó contact class in Heegaard Floer homology... more We present an alternate description of the Ozsváth-Szabó contact class in Heegaard Floer homology. Using our contact class, we prove that if a contact structure (M, ξ) has an adapted open book decomposition whose page S is a once-punctured torus, then the monodromy is rightveering if and only if the contact structure is tight.
We describe the natural gluing map on sutured Floer homology which is induced by the inclusion of... more We describe the natural gluing map on sutured Floer homology which is induced by the inclusion of one sutured manifold (M',\Gamma') into a larger sutured manifold (M,\Gamma), together with a contact structure on M-M'. As an application of this gluing map, we produce a (1+1)-dimensional TQFT by dimensional reduction and study its properties.
We take a first step towards understanding the relationship between foliations and universally ti... more We take a first step towards understanding the relationship between foliations and universally tight contact structures on hyperbolic 3-manifolds. If a surface bundle over a circle has pseudo-Anosov holonomy, we obtain a classification of "extremal" tight contact structures. Specifically, there is exactly one contact structure whose Euler class, when evaluated on the fiber, equals the Euler number of the fiber.
We outline Hutchings's prescription that produces an ECH analog of Latschev and Wendl's algebraic... more We outline Hutchings's prescription that produces an ECH analog of Latschev and Wendl's algebraic $k$-torsion in the context of $ech$, a variant of ECH used in a proof of the isomorphism between Heegaard Floer and Seiberg-Witten Floer homologies; and we explain how it translates into Heegaard Floer homology.
... the complex Projective Plane. Trans. Amer. Math. Soc. 303(2) (1987), 707-731 Received Novembe... more ... the complex Projective Plane. Trans. Amer. Math. Soc. 303(2) (1987), 707-731 Received November l, 1991 Peter J. Braam, Mathematical Institute, Oxford, EnglandGordana Matic, Harvard University, Cambridge, USA Page 14.
1. Introduction The main purpose of this paper is the study of rational homology cobordisms of ra... more 1. Introduction The main purpose of this paper is the study of rational homology cobordisms of rational homology 3-spheres. In particular it is shown that the p -invariants of Atiyah-Patodi-Singer [2] which can be defined as spectral invariants are, under some extra ...
We use convex decomposition theory to (1) reprove the existence of a universally tight contact st... more We use convex decomposition theory to (1) reprove the existence of a universally tight contact structure on every irreducible 3-manifold with nonempty boundary, and (2) prove that every toroidal 3-manifold carries infinitely many nonisotopic, nonisomorphic tight contact structures.
We discuss several applications of Seiberg-Witten theory in conjunction with an embedding theorem... more We discuss several applications of Seiberg-Witten theory in conjunction with an embedding theorem (proved elsewhere) for complex 2-dimensional Stein manifolds with boundary. We show that a closed, real 2-dimensional surface smoothly embedded in the interior of such a manifold satisfies an adjunction inequality, regardless of the sign of its self-intersection. This inequality gives constraints on the minimum genus of a smooth surface representing a given 2-homology class. We also discuss consequences for the contact structures existing on the boundaries of these Stein manifolds. We prove a slice version of the Bennequin-Eliashberg inequality for holomorphically tillable contact structures, and we show that there exist families of homology 3-spheres with arbitrarily large numbers of homotopic, nonisomorphic tight contact structures. Another result we mention is that the canonical class of a complex 2-dimensional Stein manifold with boundary is invariant under self-diffeomorphisms fixing the boundary.
We describe an invariant of a contact 3-manifold with convex boundary as an element of Juhász's s... more We describe an invariant of a contact 3-manifold with convex boundary as an element of Juhász's sutured Floer homology. Our invariant generalizes the contact invariant in Heegaard Floer homology in the closed case, due to Ozsváth and Szabó.
We are grateful to Yasha Eliashberg, Emmanuel Giroux, Bob Gompf, Dusa McDu and Tom Mrowka for int... more We are grateful to Yasha Eliashberg, Emmanuel Giroux, Bob Gompf, Dusa McDu and Tom Mrowka for interesting discussions, and happy to acknowledge the help of Peter Kronheimer for suggesting an idea to avoid a stumbling block in our original argument. The authors wish to thank the Max±Planck Institut fuÈ r Mathematik for support during the time when this work was started. Thanks from the ®rst author also go to the geometry group of the University of Georgia for travel support
We continue our study of the monoid of right-veering diffeomorphisms on a compact oriented surfac... more We continue our study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary, introduced in [HKM2]. We conduct a detailed study of the case when the surface is a punctured torus; in particular, we exhibit the difference between the monoid of right-veering diffeomorphisms and the monoid of products of positive Dehn twists, with the help of the Rademacher function. We then generalize to the braid group B n on n strands by relating the signature and the Maslov index. Finally, we discuss the symplectic fillability in the pseudo-Anosov case by comparing with the work of Roberts [Ro1, Ro2].
We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surfac... more We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary. The monoid strictly contains the monoid of products of positive Dehn twists. We explain the relationship to tight contact structures and open book decompositions.
We show the equivalence of several notions in the theory of taut foliations and the theory of tig... more We show the equivalence of several notions in the theory of taut foliations and the theory of tight contact structures. We prove equivalence, in certain cases, of existence of tight contact structures and taut foliations.
... the complex Projective Plane. Trans. Amer. Math. Soc. 303(2) (1987), 707-731 Received Novembe... more ... the complex Projective Plane. Trans. Amer. Math. Soc. 303(2) (1987), 707-731 Received November l, 1991 Peter J. Braam, Mathematical Institute, Oxford, EnglandGordana Matic, Harvard University, Cambridge, USA Page 14.
We define an invariant of contact structures in dimension three from Heegaard Floer homology. Thi... more We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set Zě0 Y t8u. It is zero for overtwisted contact structures, 8 for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. As an application, we obstruct Stein fillability on contact 3-manifolds with non-vanishing Ozsváth-Szabó contact class.
We give a necessary and sufficient condition for the addition of a collection of disjoint bypasse... more We give a necessary and sufficient condition for the addition of a collection of disjoint bypasses to a convex surface to be universally tight-namely the nonexistence of a polygonal region which we call a virtual pinwheel.
We take a first step towards understanding the relationship between foliations and universally ti... more We take a first step towards understanding the relationship between foliations and universally tight contact structures on hyperbolic 3-manifolds. If a surface bundle over a circle has pseudo-Anosov holonomy, we obtain a classification of "extremal" tight contact structures. Specifically, there is exactly one contact structure whose Euler class, when evaluated on the fiber, equals the Euler number of the fiber. This rigidity theorem is a consequence of properties of the action of pseudo-Anosov maps on the complex of curves of the fiber and a remarkable flexibility property of convex surfaces in such a space. Indeed this flexibility may be seen in surface bundles over an interval where the analogous classification theorem is also established.
We present an alternate description of the Ozsváth-Szabó contact class in Heegaard Floer homology... more We present an alternate description of the Ozsváth-Szabó contact class in Heegaard Floer homology. Using our contact class, we prove that if a contact structure (M, ξ) has an adapted open book decomposition whose page S is a once-punctured torus, then the monodromy is rightveering if and only if the contact structure is tight.
We describe the natural gluing map on sutured Floer homology which is induced by the inclusion of... more We describe the natural gluing map on sutured Floer homology which is induced by the inclusion of one sutured manifold (M',\Gamma') into a larger sutured manifold (M,\Gamma), together with a contact structure on M-M'. As an application of this gluing map, we produce a (1+1)-dimensional TQFT by dimensional reduction and study its properties.
We take a first step towards understanding the relationship between foliations and universally ti... more We take a first step towards understanding the relationship between foliations and universally tight contact structures on hyperbolic 3-manifolds. If a surface bundle over a circle has pseudo-Anosov holonomy, we obtain a classification of "extremal" tight contact structures. Specifically, there is exactly one contact structure whose Euler class, when evaluated on the fiber, equals the Euler number of the fiber.
We outline Hutchings's prescription that produces an ECH analog of Latschev and Wendl's algebraic... more We outline Hutchings's prescription that produces an ECH analog of Latschev and Wendl's algebraic $k$-torsion in the context of $ech$, a variant of ECH used in a proof of the isomorphism between Heegaard Floer and Seiberg-Witten Floer homologies; and we explain how it translates into Heegaard Floer homology.
... the complex Projective Plane. Trans. Amer. Math. Soc. 303(2) (1987), 707-731 Received Novembe... more ... the complex Projective Plane. Trans. Amer. Math. Soc. 303(2) (1987), 707-731 Received November l, 1991 Peter J. Braam, Mathematical Institute, Oxford, EnglandGordana Matic, Harvard University, Cambridge, USA Page 14.
1. Introduction The main purpose of this paper is the study of rational homology cobordisms of ra... more 1. Introduction The main purpose of this paper is the study of rational homology cobordisms of rational homology 3-spheres. In particular it is shown that the p -invariants of Atiyah-Patodi-Singer [2] which can be defined as spectral invariants are, under some extra ...
We use convex decomposition theory to (1) reprove the existence of a universally tight contact st... more We use convex decomposition theory to (1) reprove the existence of a universally tight contact structure on every irreducible 3-manifold with nonempty boundary, and (2) prove that every toroidal 3-manifold carries infinitely many nonisotopic, nonisomorphic tight contact structures.
We discuss several applications of Seiberg-Witten theory in conjunction with an embedding theorem... more We discuss several applications of Seiberg-Witten theory in conjunction with an embedding theorem (proved elsewhere) for complex 2-dimensional Stein manifolds with boundary. We show that a closed, real 2-dimensional surface smoothly embedded in the interior of such a manifold satisfies an adjunction inequality, regardless of the sign of its self-intersection. This inequality gives constraints on the minimum genus of a smooth surface representing a given 2-homology class. We also discuss consequences for the contact structures existing on the boundaries of these Stein manifolds. We prove a slice version of the Bennequin-Eliashberg inequality for holomorphically tillable contact structures, and we show that there exist families of homology 3-spheres with arbitrarily large numbers of homotopic, nonisomorphic tight contact structures. Another result we mention is that the canonical class of a complex 2-dimensional Stein manifold with boundary is invariant under self-diffeomorphisms fixing the boundary.
We describe an invariant of a contact 3-manifold with convex boundary as an element of Juhász's s... more We describe an invariant of a contact 3-manifold with convex boundary as an element of Juhász's sutured Floer homology. Our invariant generalizes the contact invariant in Heegaard Floer homology in the closed case, due to Ozsváth and Szabó.
We are grateful to Yasha Eliashberg, Emmanuel Giroux, Bob Gompf, Dusa McDu and Tom Mrowka for int... more We are grateful to Yasha Eliashberg, Emmanuel Giroux, Bob Gompf, Dusa McDu and Tom Mrowka for interesting discussions, and happy to acknowledge the help of Peter Kronheimer for suggesting an idea to avoid a stumbling block in our original argument. The authors wish to thank the Max±Planck Institut fuÈ r Mathematik for support during the time when this work was started. Thanks from the ®rst author also go to the geometry group of the University of Georgia for travel support
We continue our study of the monoid of right-veering diffeomorphisms on a compact oriented surfac... more We continue our study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary, introduced in [HKM2]. We conduct a detailed study of the case when the surface is a punctured torus; in particular, we exhibit the difference between the monoid of right-veering diffeomorphisms and the monoid of products of positive Dehn twists, with the help of the Rademacher function. We then generalize to the braid group B n on n strands by relating the signature and the Maslov index. Finally, we discuss the symplectic fillability in the pseudo-Anosov case by comparing with the work of Roberts [Ro1, Ro2].
We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surfac... more We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary. The monoid strictly contains the monoid of products of positive Dehn twists. We explain the relationship to tight contact structures and open book decompositions.
We show the equivalence of several notions in the theory of taut foliations and the theory of tig... more We show the equivalence of several notions in the theory of taut foliations and the theory of tight contact structures. We prove equivalence, in certain cases, of existence of tight contact structures and taut foliations.
... the complex Projective Plane. Trans. Amer. Math. Soc. 303(2) (1987), 707-731 Received Novembe... more ... the complex Projective Plane. Trans. Amer. Math. Soc. 303(2) (1987), 707-731 Received November l, 1991 Peter J. Braam, Mathematical Institute, Oxford, EnglandGordana Matic, Harvard University, Cambridge, USA Page 14.
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