A symplectic manifold (M, ω) is a smooth manifold M endowed with a nondegenerate and closed 2-form ω. By Darboux's Theorem such a manifold looks locally like an open set in some R 2n ∼ = C n with the standard symplectic formand so symplectic manifolds have no local invariants. This is in sharp contrast to Riemannian manifolds, for which the Riemannian metric admits various curvature invariants. Symplectic manifolds do however admit many global numerical invariants, and prominent among them are the so-called symplectic capacities.Symplectic capacities were introduced in 1990 by I. Ekeland and H. Hofer [19,20] (although the first capacity was in fact constructed by M. Gromov [40]). Since then, lots of new capacities have been defined [16,30,32,44,49,59,60,90,99] and they were further studied in [1,2,8,9,17,26,21,28,31,35,37,38,41,42,43,46,48,50,52,56,57,58,61,62,63,64,65,66,68,74,75,76,88,89,91,92,94,97,98]. Surveys on symplectic capacities are [45,50,55,69,97]. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. This is illustrated in § 2, where we discuss some examples of symplectic capacities and describe a few consequences of their existence. In § 3 we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities. In § 4, we describe several new relations between certain symplectic capacities on ellipsoids and polydiscs. Throughout the discussion we mention many open problems.As illustrated below, many of the quantitative aspects of symplectic geometry can be formulated in terms of symplectic capacities. Of course there are other numerical invariants of symplectic manifolds which could be included in *

We extract a nonnegative integer-valued invariant, 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 zero 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 one (though the converse is not true). We also construct examples for each nonnegative k of contact 3-manifolds that have algebraic torsion of order k but not k - 1, and derive consequences for contact surgeries on such manifolds. The appendix by Michael Hutchings gives an alternative proof of our cobordism obstructions in dimension three using a refinement of the contact invariant in Embedded Contact Homology.Comment: 53 pages, 4 figures, with an appendix by Michael Hutchings; v.3 is a final update to agree with the published paper, and also corrects a minor error that appeared in the published version of the appendi

We show that for every closed Riemannian manifold X there exists a positive number ε 0 > 0 such that for all 0 < ε Ϲ ε 0 there exists some δ > 0 such that for every metric space Y with Gromov-Hausdorff distance to X less than δ the geometric ε-complex |Y ε | is homotopy equivalent to X. In particular, this gives a positive answer to a question of Hausmann [4]. Introduction.This note grew out of an attempt to understand which properties of a given Riemannian manifold X are reflected in metric spaces Y Gromov-Hausdorff close to it. Even though the topology of Y may be arbitrarily complicated, we will prove that by a very simple procedure one can recover the homotopy type of X from Y , provided they are close enough in a sense made explicit below.Given a pseudometric space (X, d) and a positive number ε > 0, one can define an abstract simplicial complex X ε whose vertices are the points of X and whose k-simplices consist of all k + 1-tuples of points in X with pairwise distances less than ε. Apparently this complex was first introduced in the twenties by Vietoris [7]; it was used in an attempt to define a homology theory for general metric spaces (cf.[5], pp. 240 and 271). A version different from ours was studied by Dowker [1]. More recently the complex has found applications in the theory of hyperbolic groups, where it is known as the Rips complex.Hausmann [4] proved that for a closed Riemannian manifold X and ε sufficiently small the geometric realization |X ε | of this complex is homotopy equivalent to X. In fact, for ε > 0 sufficiently small he constructed an explicit homotopy equivalence T : |X ε | −→ X as follows. One chooses a total order on the points of X and defines the map T by induction on the dimension of the simplices, starting with the identity on the 0-skeleton. Assuming T is already defined on all k − 1-simplices, we define it on a k-simplex {x 0 , ..., x k } with x 0 < x 1 < ... < x k by mapping the straight line between x k and a point p ∈ {x 0 , ..., x k−1 } in the front face to the geodesic between the (already known) image points T(x k ) and T( p). This works for example when ε is smaller than the convexity radius of X. Because of the total order, these maps are compatible with each other and so T is continuous.Hausmann then raised the question whether there would always be a finite subset F ⊂ X and an ε > 0 such that the geometric realization |F ε | of the finite simplicial complex F ε is

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In this article we describe an algebraic framework which can be used in three related but different contexts: string topology, symplectic field theory, and Lagrangian Floer theory of higher genus. Contents 1. Introduction 2. Involutive Lie bialgebras up to infinite homotopy 3. Obstructions 4. Homotopy of morphisms 5. Homotopy inverse 6. Canonical model 7. Relation to differential Weyl algebras 8. Filtered IBL ∞ -structures 9. Maurer-Cartan elements 10. The dual cyclic bar complex of a cyclic cochain complex 11. The dIBL structure associated to a subcomplex.12. The dual cyclic bar complex of a cyclic DGA 13. The dual cyclic bar complex of the de Rham complex Appendix A. Orientations on the homology of surfaces