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

Abstract

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.

From zero surgeries to candidates for exotic definite four-manifolds

“…It is shown in [37,Corollary 1.5] and that the set of H-slice knots can detect exotic smooth structures on some 4-manifold with indefinite intersection form. See [37,31] for more obstructions to H-sliceness in such manifolds.…”

Diffeomorphisms of 4-manifolds with boundary and exotic embeddings

“…Proof. The proof is essentially parallel to that of Theorem 6.2 An unparametrized version of Theorem 6.4 is proven in [32,Theorem 1.19 (i)]. We use the same perturbations used in the proof of [32,Theorem 1.19 (i)].…”

Involutions, knots, and Floer K-theory

“…locally flat) null-homologous disk in X \ Int D 4 . There are various known smooth obstructions to H-sliceness in both definite 4-manifolds (for examples, see [47,57,66]) and indefinite 4-manifolds [34,61]. For topological obstructions, see [20,42,56,67].…”

“…Note that a family of topologically H-slice but not smoothly H-slice knots in the punctured # 3 K3 are given in [34]. However, the Bauer-Furuta type invariant used in [34] vanishes for # n K3 when n ≥ 4. On the other hand, our invariant κ(K) may be used to give such examples.…”