An adjunction inequality for the Bauer–Furuta type invariants, with applications to sliceness and 4-manifold topology

Our main result gives an adjunction inequality for embedded surfaces in certain 4-manifolds with contact boundary under a non-vanishing assumption on the Bauer–Furuta type invariants. Using this, we give infinitely many knots in S³ that are not smoothly H-slice (that is, bounding a null-homologous disk) in many 4-manifolds but they are topologically H-slice. In particular, we give such knots in the boundaries of the punctured elliptic surfaces E(2n). In addition, we give obstructions to codimension-0 orientation-reversing embedding of weak symplectic fillings with b₃=0 into closed symplectic 4-manifolds with b₁=0 and b₂⁺≡3mod4. From here we prove a Bennequin type inequality for strong symplectic caps of (S³,ξ_std). We also show that any weakly symplectically fillable 3-manifold bounds a 4-manifold with at least two smooth structures.