Symplectic 2-handles and Transverse Links
نویسنده
چکیده
When constructing symplectic manifolds it is natural to wonder whether topological 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-byhandle” 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 LV ω = ω (in other words, V is a symplectic dilation) and pointing out of (respectively into) X. This induces a contact form α = ıV ω|∂X and a contact structure ξ = kerα 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, 2handles 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, we see boundaries which are partially convex and partially concave, so we develop a careful theory for such boundaries. For a weaker form of control, we say that ∂X is weakly convex if ∂X supports a positive contact structure ξ such that ω|ξ is nondegenerate; convexity implies weak convexity. In this paper we also show how to symplectically attach 2-handles along transverse knots in the convex boundary of a symplectic 4-manifold so that the new boundary becomes weakly convex.
منابع مشابه
R 65 ; Secondary 57 M 99 SYMPLECTIC 2 - HANDLES AND TRANSVERSE LINKS
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تاریخ انتشار 2000