  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.
