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 $S^3$ 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 $E\left(2n\right)$. In addition, we give obstructions to codimension-0 orientation-reversing embedding of weak symplectic fillings with $b_3=0$ into closed symplectic 4-manifolds with $b_1=0$ and $b_2^{+}\equiv 3\mathrm{mod}4$. From here we prove a Bennequin type inequality for strong symplectic caps of $(S^3,{\xi }_{std})$. We also show that any weakly symplectically fillable 3-manifold bounds a 4-manifold with at least two smooth structures.