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155
Holomorphic disks, link invariants, and the multivariable Alexander polynomial
, 2007
"... We define a Floerhomology invariant for links in S 3, and study its properties. ..."
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Cited by 44 (9 self)
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We define a Floerhomology invariant for links in S 3, and study its properties.
Lefschetz fibrations and symplectic homology, preprint
, 2007
"... Abstract. We show that for each k> 3 there are infinitely many finite type Stein manifolds diffeomorphic to Euclidean space R 2k which are pairwise distinct as symplectic manifolds. ..."
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Cited by 42 (5 self)
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Abstract. We show that for each k> 3 there are infinitely many finite type Stein manifolds diffeomorphic to Euclidean space R 2k which are pairwise distinct as symplectic manifolds.
Gluing pseudoholomorphic curves along branched covered cylinders II
, 2007
"... This paper and its prequel (“Part I”) prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves U+ and U − in the symplectization of a contact 3manifold. We assume that for each embedded Reeb orbit γ, the total multiplicity of the negative ends of U+ at covers of γ ..."
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Cited by 41 (15 self)
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This paper and its prequel (“Part I”) prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves U+ and U − in the symplectization of a contact 3manifold. We assume that for each embedded Reeb orbit γ, the total multiplicity of the negative ends of U+ at covers of γ agrees with the total multiplicity of the positive ends of U − at covers of γ. However, unlike in the usual gluing story, here the individual multiplicities are allowed to differ. In this situation, one can often glue U+ and U − to an index 2 curve by inserting genus zero branched covers of Rinvariant cylinders between them. This paper shows that the signed count of such gluings equals a signed count of zeroes of a certain section of an obstruction bundle over the moduli space of branched covers of the cylinder. Part I obtained a combinatorial formula for the latter count and, assuming the result of the present paper, deduced that the differential ∂ in embedded contact homology satisfies ∂ 2 = 0. The present paper completes all of the analysis that was needed in Part I. The gluing technique explained here is in principle applicable to more gluing problems. We also prove some lemmas concerning the generic behavior of pseudoholomorphic curves in symplectizations, which may be of independent interest.
The Weinstein conjecture for planar contact structures in dimension three
 COMMENT. MATH. HELV
, 2005
"... In this paper we describe a general strategy for approaching the Weinstein conjecture in dimension three. We apply this approach to prove the Weinstein conjecture for a new class of contact manifolds (planar contact manifolds). We also discuss how the present approach reduces the general Weinstein ..."
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Cited by 33 (2 self)
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In this paper we describe a general strategy for approaching the Weinstein conjecture in dimension three. We apply this approach to prove the Weinstein conjecture for a new class of contact manifolds (planar contact manifolds). We also discuss how the present approach reduces the general Weinstein conjecture in dimension three to a compactness problem for the solution set of a first order elliptic PDE.
An exact sequence for contact and symplectic homology
, 2008
"... A symplectic manifold W with contact type boundary M = ∂W induces a linearization of the contact homology of M with corresponding linearized contact homology HC(M). We establish a Gysintype exact sequence in which the symplectic homology SH(W) of W maps to HC(M), which in turn maps to HC(M), by a ..."
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Cited by 31 (4 self)
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A symplectic manifold W with contact type boundary M = ∂W induces a linearization of the contact homology of M with corresponding linearized contact homology HC(M). We establish a Gysintype exact sequence in which the symplectic homology SH(W) of W maps to HC(M), which in turn maps to HC(M), by a map of degree −2, which then maps to SH(W). Furthermore, we give a description of the degree −2 map in terms of rational holomorphic curves with constrained asymptotic markers, in the symplectization of M.
AUTOMATIC TRANSVERSALITY AND ORBIFOLDS OF PUNCTURED HOLOMORPHIC CURVES IN DIMENSION FOUR
, 2008
"... We derive a numerical criterion for J–holomorphic curves in 4–dimensional symplectic cobordisms to achieve transversality without any genericity assumption. This generalizes results in [HLS97] and [IS99] to allow punctured curves with boundary that generally need not be somewhere injective or imme ..."
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Cited by 29 (2 self)
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We derive a numerical criterion for J–holomorphic curves in 4–dimensional symplectic cobordisms to achieve transversality without any genericity assumption. This generalizes results in [HLS97] and [IS99] to allow punctured curves with boundary that generally need not be somewhere injective or immersed. As an application, we combine this with the intersection theory of punctured holomorphic curves to prove that certain geometrically natural moduli spaces are smooth orbifolds, consisting generically of embedded curves, plus unbranched
BORDERED FLOER HOMOLOGY FOR SUTURED MANIFOLDS
, 2009
"... We define a sutured cobordism category of surfaces with boundary and 3–manifolds with corners. In this category a sutured 3– manifold is regarded as a morphism from the empty surface to itself. In the process we define a new class of geometric objects, called bordered sutured manifolds, that gener ..."
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Cited by 26 (2 self)
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We define a sutured cobordism category of surfaces with boundary and 3–manifolds with corners. In this category a sutured 3– manifold is regarded as a morphism from the empty surface to itself. In the process we define a new class of geometric objects, called bordered sutured manifolds, that generalize both sutured 3–manifolds and bordered 3–manifolds. We extend the definition of bordered Floer homology to these objects, giving a functor from a decorated version of the sutured category to A∞–algebras, and A∞–bimodules. As an application we give a way to recover the sutured homology SFH(Y,Γ) of a sutured manifold from either of the bordered invariants ĈFA(Y) and ĈFD(Y) of its underlying manifold Y. A further application is a new proof of the surface decomposition formula of Juhász.
Strongly fillable contact manifolds and Jholomorphic foliations
, 2008
"... We prove that every strong symplectic filling of a planar contact manifold admits a symplectic Lefschetz fibration over the disk, and every strong filling of T 3 similarly admits a Lefschetz fibration over the annulus. It follows that strongly fillable planar contact structures are also Stein fill ..."
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Cited by 25 (1 self)
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We prove that every strong symplectic filling of a planar contact manifold admits a symplectic Lefschetz fibration over the disk, and every strong filling of T 3 similarly admits a Lefschetz fibration over the annulus. It follows that strongly fillable planar contact structures are also Stein fillable, and (strengthening a result of Stipsicz [Sti02]), all Stein fillings of T 3 are symplectomorphic to star shaped domains in T ∗ T 2. These constructions result from a compactness theorem for punctured J–holomorphic curves that foliate a convex symplectic manifold. We use it also to show that the compactly supported symplectomorphism group on T ∗ T 2 is contractible, and to define an obstruction to strong fillability that yields a nongaugetheoretic proof of Gay’s recent nonfillability result [Gay06] for contact manifolds with positive
Relative Asymptotic Behavior of Pseudoholomorphic HalfCylinders
 Comm. Pure and Appl. Math
"... Abstract. In this article we study the asymptotic behavior of pseudoholomorphic halfcylinders which converge exponentially to a periodic orbit of a vector field defined by a framed stable Hamiltonian structure. Such maps are of central interest in symplectic field theory and its variants (symplecti ..."
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Cited by 23 (2 self)
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Abstract. In this article we study the asymptotic behavior of pseudoholomorphic halfcylinders which converge exponentially to a periodic orbit of a vector field defined by a framed stable Hamiltonian structure. Such maps are of central interest in symplectic field theory and its variants (symplectic Floer homology, contact homology, embedded contact homology). We prove a precise formula for the asymptotic behavior of the “difference ” of two such maps, generalizing results from [11, 3, 2, 8]. Using this result with a technique from [10], we then show that a finite collection of pseudoholomorphic halfcylinders asymptotic to coverings of a single periodic orbit is smoothly equivalent to