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Our main result gives an adjunction inequality for embedded surfaces incertain $4$-manifolds with contact boundary under a non-vanishing assumption onthe 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, wegive such knots in the boundaries of the punctured elliptic surfaces $E(2n)$.In addition, we give obstructions to codimension-0 orientation-reversingembedding of weak symplectic fillings with $b_3=0$ into closed symplectic4-manifolds with $b_1=0$ and $b_2^+\equiv 3$ mod $4$. From here we prove aBennequin type inequality for symplectic caps of $(S^3,\xi_{std})$. We alsoshow that any weakly symplectically fillable $3$-manifold bounds a $4$-manifoldwith at least two smooth structures.

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