Relative genus bounds in indefinite four-manifolds

Abstract

Given a closed four-manifold $X$ with an indefinite intersection form, we consider smoothly embedded surfaces in $X\setminus$int$(B^4)$, with boundary a knot $K\subset S^3$. We give several methods to bound the genus of such surfaces in a fixed homology class. Our tools include adjunction inequalities and the $10/8+4$ theorem. In particular, we present obstructions to a knot being H-slice (that is, bounding a null-homologous disk) in a four-manifold and show that the set of H-slice knots can detect exotic smooth structures on closed $4$-manifolds.

22 pages, 5 figures; final version, to appear in Math. Annalen

References in corpus (2)

• Homotopy theoretical considerations of the Bauer-Furuta stable homotopy Seiberg-Witten invariants
• A relative version of Rochlin's theorem

Cited by in corpus (2)

• Homology concordance and knot Floer homology
• From zero surgeries to candidates for exotic definite four-manifolds