Kirby Problem 4.42

Definitions and mathematical statement

For a closed smooth four-manifold $X$, choose a smoothly embedded ball and write $X^\circ=X\setminus\operatorname{int}(B^4)$, with its boundary identified with $S^3$. A smooth knot $K\subset S^3$ is slice in $X$ if it bounds a smoothly and properly embedded disk in $X^\circ$. No condition is imposed on that disk's relative homology class; requiring it to be null-homologous is the stronger notion of $H$-sliceness.

(a) Is every knot slice in every closed simply connected smooth four-manifold $X$ with $b_2(X)\ge2$, where $b_2(X)=\dim H_2(X;\mathbb Q)$?

(b) The source singles out the tests $X=\mathbb{CP}^2\#\overline{\mathbb{CP}}^{2}$ and $X=K3$. Here the bar reverses orientation and $K3$ denotes the standard smooth K3 surface, for example a nonsingular quartic hypersurface in $\mathbb{CP}^3$. Determine sliceness of arbitrary knots in these punctured manifolds, distinguishing any settled subclass from the universal assertion.

(c) Are there homeomorphic smooth four-manifolds $X,X'$ and a knot $K$ that is slice in $X$ but not in $X'$? Thus the sets of slice knots, without imposing null-homology of the disks, would distinguish their smooth structures.

The knot may depend on the exotic pair in (c). Agreement of intersection forms or cohomology rings alone is not the required homeomorphism hypothesis. All disks in these questions are smooth; a locally flat disk supplies only the separate topological version.

Short English statement

Which punctured simply connected four-manifolds slice every knot, and can ordinary sliceness detect exotic smooth structures?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2918 (KP-4.42), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.42. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.