Kirby Problem 3.74

Definitions and mathematical statement

Let $Y$ be a closed oriented rational homology three-sphere and $f:Y\to Y$ an orientation-preserving diffeomorphism. An extension is an orientation-preserving diffeomorphism $F:W\to W$ of a compact oriented smooth four-manifold with identified boundary $\partial W=Y$, satisfying $F|_Y=f$. The filling $W$ may depend on the boundary data.

Its intersection form $Q_W$ is the integral pairing on $H_2(W;\mathbb Z)/\mathrm{torsion}$ given by oriented intersections of representative cycles. It is nondegenerate over $\mathbb R$ because the boundary is a rational homology sphere. Positive or negative definiteness refers to this real pairing; rank zero is allowed and is definite vacuously.

(a) What restrictions does existence of $F$ place on $Q_W$? (b) Is there a pair $(Y,f)$ such that $Y$ bounds a smooth integral homology ball, but $f$ extends over no positive-definite or negative-definite smooth filling of $Y$? (c) Develop restrictions also for indefinite, non-spin fillings. Indefinite means both signs occur, and non-spin means $w_2(W)\ne0$.

No condition is imposed on the action of $F$ on homology unless a refinement explicitly adds one. In (b), merely obstructing an extension over one chosen homology ball, or only over fillings of one sign, is insufficient. The issue is a simultaneous restriction on every filling admitting an extension, although the same boundary manifold has some homology-ball filling without that requirement.

Short English statement

How does extension of a boundary diffeomorphism constrain a filling's intersection form, including both definite signs?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2872 (KP-3.74), 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 3.74. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.