In this talk, we present a relative adjunction inequality for 4-manifolds with boundary. We begin by constructing generalized Heegaard Floer tau-invariants associated to a knot in a 3-manifold and a nontrivial Floer class. Given a 4-manifold with boundary, the invariant associated to a Floer class provides a lower bound for the genus of a properly embedded surface provided that the Floer class is in the image of the cobordism map induced by the 4-manifold. We sketch a proof of the theorem and discuss several applications to links and contact manifolds.
This is joint work with Matt Hedden.