Kirby Problem 1.15

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a closed oriented connected $3$-manifold $Y$, let $s(Y)$ be the minimum number of components of an integrally framed link in $S^3$ whose Dehn surgery gives $Y$. An integral homology sphere has the integral homology of $S^3$; irreducible means every embedded $2$-sphere bounds a ball. For an orientation-preserving smooth cyclic action $\varphi$ of order $n$, let $s_{\mathrm{eq}}(Y,\varphi)$ be the minimum number of components in an integral surgery description equivariant for that action: an invariant framed surgery link and its ambient cyclic action induce the specified action on the surgered manifold.

Statement recorded in the supplied source.

(a) Are the values $s(Y)$ unbounded among integral homology spheres, and also among irreducible ones? For nontrivial integral homology spheres $Y_1,\ldots,Y_n$, is $s(Y_1\#\cdots\#Y_n)\geq n$? (b) Are the differences $s_{\mathrm{eq}}(Y,\varphi)-s(Y)$ unbounded over cyclic actions? (c) Is this difference positive for some action of order $2$? [S1][S2]

Short English statement

Determine how many surgery components homology spheres require and whether requiring a cyclic symmetry can increase that number, including involutions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2674 (KP-1.15). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, author preliminary edition (2026), Problem 1.15 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.