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 introduce a Floer homotopy version of the contact invariant introduced by Kronheimer-Mrowka-Ozváth-Szabó. Moreover, we prove a gluing formula relating our invariant with the first author's Bauer-Furuta type invariant, which refines Kronheimer-Mrowka's invariant for 4-manifolds with contact boundary. As applications, we give two constraints for a certain class of symplectic fillings using equivariant K and KO-cohomology. We also treat the extension property of positive scalar curvature metrics on 4-manifolds with boundary.NOBUO IIDA AND MASAKI TANIGUCHI 5.2. Equivariant K theory 65 5.3. Extension problem of positive scalar curvature 70 References 72

We give Bennequin-Plamenevskaya-Shumakovitch type lower bounds for the concordance invariant s # introduced by Kronheimer and Mrowka. The proof is a consequence of computations for torus knots and the cobordism inequality of s # due to Gong, combined with well-known arguments used for slice-torus invariants. Contents 1. Introduction 1 Acknowledgement 2 2. Background 3 3. Proof of the main theorem 4 References 6

We give a generalized Thurston-Bennequin-type inequality for links in S 3 using a Bauer-Furuta-type invariant for 4-manifolds with contact boundary. As a special case, we also give an adjunction inequality for smoothly embedded orientable surfaces with negative intersection in a closed oriented smooth 4-manifold whose non-equivariant Bauer-Furuta invariant is non-zero.

We give a generalized Thurston–Bennequin-type inequality for links in [Formula: see text] using a Bauer–Furuta-type invariant for four-manifolds with contact boundary introduced by the first author. As a special case, we also give an adjunction inequality for smoothly embedded orientable surfaces with negative intersection in a closed oriented smooth four-manifold whose non-equivariant Bauer–Furuta invariant is nonzero.