R 65 ; Secondary 57 M 99 SYMPLECTIC 2 - HANDLES AND TRANSVERSE LINKS

A standard convexity condition on the boundary of a symplectic manifold involves an induced positive contact form (and contact structure) on the boundary; the corresponding concavity condition involves an induced negative contact form. We present two methods of symplectically attaching 2-handles to convex boundaries of symplectic 4-manifolds along links transverse to the induced contact structures. One method results in concave boundaries and depends on a fibration of the link complement over S 1 ; in this case the handles can be attached with any framing larger than a lower bound determined by the fibration. The other method results in a weaker convexity condition on the new boundary (sufficient to imply tightness of the new contact structure), and in this case the handles can be attached with any framing less than a certain upper bound. These methods supplement methods developed by Weinstein and Eliashberg for attaching symplectic 2-handles along Legendrian knots. When constructing symplectic manifolds it is natural to wonder whether topo-logical techniques using handles can be made to work symplectically. Weinstein [6] and Eliashberg [2] have shown how to do this in certain cases; here we present two new symplectic " handle-by-handle " constructions in dimension four. In such constructions it is desirable to retain control of the symplectic form near the boundary; one form of control is the following: Given a symplectic manifold (X, ω) we say that ∂X is convex (respectively concave) if there exists a vector field V defined in a neighborhood of ∂X, satisfying the equation L V ω = ω (in other words, V is a symplectic dilation) and pointing out of (respectively into) X. This induces a contact form α = ı V ω| ∂X and a contact structure ξ = ker α on ∂X. Weinstein and Eliashberg show that, if (X, ω) is a symplectic 2n-manifold with ∂X convex, then one can attach k-handles to X, for 0 ≤ k ≤ n, and extend ω across the handles so that the new boundary is again convex. Conditions are imposed on the attaching spheres in relation to the contact structure ξ on ∂X and in particular, in dimension four, 2-handles must be attached along Legendrian knots (knots tangent to ξ). In this paper we show how to symplectically attach 2-handles along transverse knots (transverse to ξ) in the convex boundary of a symplectic 4-manifold so that the new boundary becomes concave. Along the way, …