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 punctured elliptic surfaces E(2n). In addition, we give obstructions to codimension-0 embedding of weak symplectic fillings with b 3 = 0 into closed symplectic 4-manifolds with b 1 = 0 and b + 2 ≡ 3 mod 4. We also show that any weakly symplectically fillable 3-manifold bounds a 4-manifold with at least two smooth structures.All of these results follow from our main result that gives an adjunction inequality for embedded surfaces in certain 4-manifolds with contact boundary under a non-vanishing assumption on Bauer-Furuta type invariants.

We give several criteria on a closed, oriented 3 3 -manifold that will imply that it is the boundary of a (simply connected) 4 4 -manifold that admits infinitely many distinct smooth structures. We also show that any weakly fillable contact 3 3 -manifold, or contact 3 3 -manifold with non-vanishing Heegaard Floer invariant, is the boundary of a simply connected 4 4 -manifold that admits infinitely many distinct smooth structures each of which supports a symplectic structure with concave boundary, that is there are infinitely many exotic caps for any such contact manifold.

In this note we show that any closed, oriented 3-manifold is the boundary of a simply connected 4-manifold that admits infinitely many distinct smooth structures. We also show that any fillable contact 3-manifold is the boundary of a simply connected 4-manifolds that admits infinitely many distinct smooth structures each of which supports a symplectic structure with concave boundary, that is there are infinitely many exotic caps for any contact manifold.

We establish the existence of a pair of exotic surfaces in a punctured K3 which remains exotic after one external stabilization and have diffeomorphic complements. A key ingredient in the proof is a vanishing theorem of the family Bauer-Furuta invariant for diffeomorphisms on a large family of spin 4-manifolds, which is proved using the Tom Dieck splitting theorem in equivariant stable homotopy theory. In particular, we prove that the S 1 -equivariant family Bauer-Furuta invariant of any orientation-preserving diffeomorphism on S 4 is trivial and that the Pin(2)equivariant family Bauer-Furuta invariant for a diffeomorphism on S 2 × S 2 is trivial if the diffeomorphism acts trivially on the homology. Therefore, these invariants do not detect exotic selfdiffeomorphisms on S 4 or S 2 ×S 2 . En route, we observe a curious element in the Pin(2)-equivariant stable homotopy group of spheres which could potentially be used to detect an exotic diffeomorphism on S 4 .