Kirby Problem 3.75

Definitions and mathematical statement

Let $Y$ be a closed oriented rational homology three-sphere equipped with a smooth orientation-preserving action $\rho$ of a finite cyclic group $C_p=\mathbb Z/p\mathbb Z$, $p\ge2$. An extension over a compact oriented smooth four-manifold $W$ with $\partial W=Y$ is an action $\widetilde\rho:C_p\to\operatorname{Diff}^+(W)$ restricting to $\rho$ on the boundary.

Let $Q_W$ denote the intersection pairing on $H_2(W;\mathbb Z)$ modulo torsion, evaluated over $\mathbb R$ when discussing its signs. It is definite if it is positive or negative definite; include the rank-zero case. It is indefinite if it has both positive and negative directions. The filling is non-spin when $w_2(W)\ne0$.

(a) Determine constraints on $Q_W$ imposed by the extension of the given cyclic action. (b) Is there an integral homology sphere $Y$ bounding a smooth integral homology ball and a cyclic action on $Y$ that extends over no definite smooth filling of either sign? (c) Obtain methods imposing constraints also on indefinite non-spin fillings.

The extension must satisfy the group relation: if a generator acts by $f$ on $Y$, its extension $F$ must obey $F^p=\operatorname{id}_W$. Extending $f$ merely as a diffeomorphism without this relation does not answer the equivariant problem. The order, filling, and action on its homology are otherwise unrestricted, so a proposed obstruction must state any additional hypotheses it uses.

Short English statement

What intersection forms can bound a rational homology sphere while extending a specified cyclic group action?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2873 (KP-3.75), 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 3.75. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.