The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y, equipped with a Spin c structure t. Given a Heegaard splitting of Y -- U0 tie U1, these theories are variants of the Lagrangian Floer homology for the g-fold symmetric product of Y relative to certain totally real subspaces associated to U0 and U1.

̂HF(Y, s),and HFred(Y, s) associated to closed, oriented three-manifolds Y equipped with a Spin c structures s ∈ Spin c (Y). In the present paper, we give calculations and study the properties of these invariants. The calculations suggest a conjectured relationship with Seiberg-Witten theory. The properties include a relationship between the Euler characteristics of HF ± and Turaev’s torsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to three-manifold topology. 1.

Abstract not found

Since Gromov introduced his pseudo-holomorphic curve theory in 80’s, pseudo-holomorphic curve soon became a predominant technique in symplectic topology. Many important theorems in symplectic topology have been proved by the technique of pseudo-holomorphic curve, for example, the squeezing theorem [Gr], the rigidity [E], the classifications of rational and ruled symplectic 4-

This paper is a continuation of our previous paper [RT]. In [RT], among other things, we build up the mathematical foundation of quantum cohomology ring on semi-positive symplectic manifolds.

We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension-two symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of [IP4]. The main step is the construction of a compact space of ‘V-stable ’ maps. Simple special cases include the Hurwitz numbers for algebraic curves and the enumerative invariants of Caporaso and Harris. Gromov-Witten invariants are invariants of a closed symplectic manifold (X, ω). To define them, one introduces a compatible almost complex structure J and a perturbation term ν, and considers the maps f: C → X from a genus g complex curve C with n marked points which satisfy the pseudoholomorphic map equation ∂f = ν and represent a class A =[f] ∈ H2(X). The set of such maps, together with their limits, forms the compact space of stable maps Mg,n(X, A). For each stable map, the domain determines a point in the Deligne-Mumford space Mg,n of curves, and evaluation at each marked point determines a point in X. Thus there is a map (0.1) Mg,n(X, A) → Mg,n × X n. The Gromov-Witten invariant of (X, ω)isthe homology class of the image for generic (J, ν). It depends only on the isotopy class of the symplectic structure. By choosing bases of the cohomologies of Mg,n and X n, the GW invariant can be viewed as a collection of numbers that count the number of stable maps satisfying constraints. In important cases these numbers are equal to enumerative invariants defined by algebraic geometry. In this article we construct Gromov-Witten invariants for a symplectic manifold (X, ω) relative to a codimension two symplectic submanifold V. These invariants are designed for use in formulas describing how GW invariants The research of both authors was partially supported by the N.S.F. The first author was also

The aim of this paper is to establish a connection between the topology of the automorphism group of a symplectic manifold (M,ω) and the quantum product on its homology. More precisely, we assume that M is closed and connected, and consider the group Ham(M,ω) of Hamiltonian automorphisms

The purpose of this article is to explain how pseudo-holomorphic curves in a symplectic 4-manifold can be constructed from solutions to the Seiberg-Witten equations. As such, the main theorem proved here (Theorem 1.3) is an existence theorem for pseudo-holomorphic curves. This article thus provides a proof of roughly half of

cases M carries a family of symplectic forms ωλ, where λ> −1 determines the cohomology class [ωλ]. This paper calculates the rational (co)homology of the group Gλ of symplectomorphisms of (M, ωλ) as well as the rational homotopy type of its classifying space BGλ. It turns out that each group Gλ contains a finite collection Kk, k = 0,..., ℓ = ℓ(λ), of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups “asymptotically commute”, i.e. all the higher Whitehead products that they generate vanish as λ → ∞. However, for each fixed λ there is essentially one nonvanishing product that gives rise to a “jumping generator” wλ in H ∗ (Gλ) and to a single relation in the rational cohomology ring H ∗ (BGλ). An analog of this generator wλ was also seen by Kronheimer in his study of families of symplectic forms on 4-manifolds using Seiberg–Witten theory. Our methods involve a close study of the space of ωλ-compatible almost complex structures on M. 1

Abstract not found