Handle attaching in symplectic homology and the Chord Conjecture

Arnold conjectured that every Legendrian knot in the standard contact structure on the 3-sphere possesses a characteristic chord with respect to any contact form. I confirm this conjecture if the know has Thurston-Bennequin invariant −1. More generally, existence of chords is proved for a standard Legendrian unknot on the boundary of a subcritical Stein manifold of any dimension. There is also a multiplicity result which implies in some situations existence of infinitely many chords.?The proof relies on the behaviour of symplectic homology under handle attaching. The main observation is that symplectic homology only changes in the presence of chords. Received July 14, 2000 / final version received June 1, 2001?Published online August 1, 2001     

Tight contact structures with no symplectic fillings

We exhibit tight contact structures on 3-manifolds that do not admit any symplectic fillings. Oblatum 7-XII-2000 & 14-XI-2001?Published online: 9 April 2002     

Growth of maps, distortion in groups and symplectic geometry

In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism:?• The uniform norm of the differential of its n-th iteration;?• The word length of its n-th iteration, where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms.?We find lower bounds for the growth rates of these sequences in a number of situations. These bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism. They are obtained by using recent results of Schwarz on Floer homology. As an application, we prove non-existence of certain non-linear symplectic representations for finitely generated groups. Oblatum 6-XII-2001 & 19-VI-2002?Published online: 5 September 2002 RID="*" ID="*"Supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.     

K-area, Hofer metric and geometry of conjugacy classes in Lie groups

Given a closed symplectic manifold (M,ω) we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group Ham (M,ω) by means of the Hofer metric on Ham (M,ω). We use pseudo-holomorphic curves involved in the definition of the multiplicative structure on the Floer cohomology of a symplectic manifold (M,ω) to estimate this quantity in terms of actions of some periodic orbits of related Hamiltonian flows. As a corollary we get a new way to obtain Agnihotri-Belkale-Woodward inequalities for eigenvalues of products of unitary matrices. As another corollary we get a new proof of the geodesic property (with respect to the Hofer metric) of Hamiltonian flows generated by certain autonomous Hamiltonians. Our main technical tool is K-area defined for Hamiltonian fibrations over a surface with boundary in the spirit of L. Polterovich’s work on Hamiltonian fibrations over S ². Oblatum 23-II-2001 & 9-V-2001?Published online: 20 July 2001     

The parameterized algorithm for symplectic (butterfly) matrices

The algorithm is a structure-preserving algorithm for computing the spectrum of symplectic matrices. Any symplectic matrix can be reduced to symplectic butterfly form. A symplectic matrix in butterfly form is uniquely determined by parameters. Using these parameters, we show how one step of the symplectic algorithm for can be carried out in arithmetic operations compared to arithmetic operations when working on the actual symplectic matrix. Moreover, the symplectic structure, which will be destroyed in the numerical process due to roundoff errors when working with a symplectic (butterfly) matrix, will be forced by working just with the parameters.

     

Symplectic hypersurfaces in the complement of an isotropic submanifold

Using Donaldson's approximately holomorphic techniques, we construct symplectic hypersurfaces lying in the complement of any given compact isotropic submanifold of a compact symplectic manifold. We discuss the connection with rational convexity results in the K?hler case and various applications. Received: 9 January 2001 / Published online: 19 October 2001     

Algebraic Torsion in Contact Manifolds

We extract an invariant taking values in \mathbbNè{￥}{\mathbb{N}\cup\{\infty\}} , which we call the order of algebraic torsion, from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic torsion of order 0 if and only if it is algebraically overtwisted (i.e. has trivial contact homology), and any contact 3-manifold with positive Giroux torsion has algebraic torsion of order 1 (though the converse is not true). We also construct examples for each k ? \mathbbN{k \in \mathbb{N}} of contact 3-manifolds that have algebraic torsion of order k but not k − 1, and derive consequences for contact surgeries on such manifolds.     

On families of Lagrangian submanifolds

We study the stability of a compact Lagrangian submanifold of a symplectic manifold under perturbation of the symplectic structure. If X is a compact manifold and the ω _t are cohomologous symplectic forms on X, then by a well-known theorem of Moser there exists a family Φ _t of diffeomorphisms of X such that ω _t =Φ _t ^*(ω₀). If L⊂X is a Lagrangian submanifold for (X,ω₀), L _t =Φ _t ^-1(L) is thus a Lagrangian submanifold for (X,ω _t ). Here we show that if we simply assume that L is compact and ω _t | _L is exact for every t, a family L _t as above still exists, for sufficiently small t. Similar results are proved concerning the stability of special Lagrangian and Bohr–Sommerfeld special Lagrangian submanifolds, under perturbation of the ambient Calabi–Yau structure. Received: 29 May 2001/ Revised version: 17 October 2001     

Minimality of totally geodesic submanifolds in Finsler geometry

Using the symplectic definition of the Holmes-Thompson volume we prove that totally geodesic submanifolds of a Finsler manifold are minimal for this volume. Thanks to well suited technics the minimality of totally geodesic hypersurfaces (see álvarez Paiva and Berck in Adv Math 204(2):647–663, 2006) and 2-dimensional totally geodesic surfaces (see álvarez Paiva and Berck in Adv Math 204(2):647–663, 2006, Ivanov in Algebra i Analiz 13(1)26–38, 2001) had already been proved. However the corresponding statement for the Hausdorff measure is known to be wrong even in the simplest case of totally geodesic 2-dimensional surfaces in a 3-dimensional Finsler manifold (see álvarez Paiva and Berck in Adv Math 204(2):647–663, 2006).     

An index inequality for embedded pseudoholomorphic curves in symplectizations

Let Σ be a surface with a symplectic form, let φ be a symplectomorphism of Σ, and let Y be the mapping torus of φ. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in ℝ×Y, with cylindrical ends asymptotic to periodic orbits of φ or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces.?This paper establishes some of the foundations for a program with Michael Thaddeus, to understand the Seiberg-Witten Floer homology of Y in terms of such pseudoholomorphic curves. Analogues of our results should also hold in three dimensional contact topology. Received November 2, 2000 / final version received December 16, 2001?Published online November 19, 2002     

Hamiltonian loops from the ergodic point of view

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h:S¹→G is called strictly ergodic if for some irrational number α the associated skew product map T:S¹×Y→S¹×Y defined by T(t,y)=(t+α,h(t)y) is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected ones). Further, we find a restriction on the homotopy classes of smooth strictly ergodic loops in the framework of Hofer’s bi-invariant geometry on G. Namely, we prove that their asymptotic Hofer’s norm must vanish. This result provides a link between ergodic theory and symplectic topology. Received July 7, 1998 / final version received September 14, 1998     

Calabi-Yau cones from contact reduction

We consider a generalization of Einstein–Sasaki manifolds, which we characterize in terms both of spinors and differential forms, that in the real analytic case corresponds to contact manifolds whose symplectic cone is Calabi-Yau. We construct solvable examples in seven dimensions. Then, we consider circle actions that preserve the structure and determine conditions for the contact reduction to carry an induced structure of the same type. We apply this construction to obtain a new hypo-contact structure on S ² × T ³.     

The geography of spin symplectic 4-manifolds

In this paper we construct a family of simply connected spin non-complex symplectic 4-manifolds which cover all but finitely many allowed lattice points () lying in the region . Furthermore, as a corollary, we prove that there exist infinitely many exotic smooth structures on for all n large enough. Received: 29 August 2000 / in final form: 15 August 2001 / Published online: 28 February 2002     

Seidel–Smith cohomology for tangles

We generalize the “symplectic Khovanov cohomology” of Seidel and Smith (Duke Math J 134(3):453–514, 2006) to tangles using the notion of symplectic valued topological field theory introduced by Wehrheim and Woodward (arXiv:0905.1368).     

Actions symplectiques de groupes compacts (Symplectic Actions of Compact Groups)

For any symplectic action of a compact connected group on a compact connected symplectic manifold, we show that the intersection of the Weyl chamber with the image of the moment map is a closed convex polyhedron. This extends Atiyah–Guillemin–Sternberg–Kirwan's convexity theorems to non-Hamiltonian actions. As a consequence, we describe those symplectic actions of a torus which are coisotropic (or multiplicity free), i.e. which have at least one coisotropic orbit: they are the product of an Hamiltonian coisotropic action by an anhamiltonian one. The Hamiltonian coisotropic actions have already been described by Delzant thanks to the convex polyhedron. The anhamiltonian coisotropic actions are actions of a central torus on a symplectic nilmanifold. This text is written as an introduction to the theory of symplectic actions of compact groups since complete proofs of the preliminary classical results are given. An erratum to this article is available at .     

Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds

We assign to each nondegenerate Hamiltonian on a closed symplectic manifold a Floer-theoretic quantity called its “boundary depth,” and establish basic results about how the boundary depths of different Hamiltonians are related. As applications, we prove that certain Hamiltonian symplectomorphisms supported in displaceable subsets have infinitely many nontrivial geometrically distinct periodic points, and we also significantly expand the class of coisotropic submanifolds which are known to have positive displacement energy. For instance, any coisotropic submanifold of contact type (in the sense of Bolle) in any closed symplectic manifold has positive displacement energy, as does any stable coisotropic submanifold of a Stein manifold. We also show that any stable coisotropic submanifold admits a Riemannian metric that makes its characteristic foliation totally geodesic, and that this latter, weaker, condition is enough to imply positive displacement energy under certain topological hypotheses.     

Constructing the Kähler and the symplectic structures from certain spinors on 4-manifolds

We show that, on an oriented Riemannian 4-manifold, existence of a non-zero parallel spinor with respect to a spin structure implies that the underlying smooth manifold admits a Kähler structure. A similar but weaker condition is obtained for the 4-manifold to admit a symplectic structure. We also show that the structure in which the non-zero parallel spinor lives is equivalent to the canonical spin structure associated to the Kähler structure.

     

Lengths of Contact Isotopies and Extensions of the Hofer Metric

Using the Hofer metric, we construct, under a certain condition, a bi-invariant distance on the identity component in the group of strictly contact diffeomorphisms of a compact regular contact manifold. We also show that the Hofer metric on Ham(M) has a right-invariant (but not left invariant) extension to the identity component in the groups of symplectic diffeomorphisms of certain symplectic manifolds.Mathematics Subject classification (2000): 53C12, 53C15.     

Contact reduction and groupoid actions

We introduce a new method to perform reduction of contact manifolds that extends Willett's and Albert's results. To carry out our reduction procedure all we need is a complete Jacobi map from a contact manifold to a Jacobi manifold. This naturally generates the action of the contact groupoid of on , and we show that the quotients of fibers by suitable Lie subgroups are either contact or locally conformal symplectic manifolds with structures induced by the one on .

We show that Willett's reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter. Since a symplectic manifold is prequantizable iff the symplectic form is integral, this explains why Willett's reduction can be performed only at distinguished points. As an application we obtain Kostant's prequantizations of coadjoint orbits. Finally we present several examples where we obtain classical contact manifolds as reduced spaces.