Handle attaching in symplectic homology and the Chord Conjecture

The Arnold Chord Conjecture

The symplectic Thom conjecture

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes’ basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the symplectic Thom conjecture: a symplectic surface in a symplectic fourmanifold is genus-...

The Symplectic Thom Conjecture

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes’ basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the Symplectic Thom Conjecture: a symplectic surface in a symplectic four-manifold is genus...

Symplectic Homology as Hochschild Homology

In the wake of Donaldson’s pioneering work [6], Picard-Lefschetz theory has been extended from its original context in algebraic geometry to (a very large class of) symplectic manifolds. Informally speaking, one can view the theory as analogous to Kirby calculus: one of its basic insights is that one can give a (non-unique) presentation of a symplectic manifold, in terms of a symplectic hypersu...

Rabinowitz Floer Homology and Symplectic Homology

The first two authors have recently defined RabinowitzFloer homology groups RFH∗(M,W ) associated to an exact embedding of a contact manifold (M, ξ) into a symplectic manifold (W,ω). These depend only on the bounded component V of W \ M . We construct a long exact sequence in which symplectic cohomology of V maps to symplectic homology of V , which in turn maps to Rabinowitz-Floer homology RFH∗...