Kirby Problem 3.70

Definitions and mathematical statement

An integral homology three-sphere is a closed connected oriented smooth manifold $Y$ with $H_*(Y;\mathbb Z)\cong H_*(S^3;\mathbb Z)$. Two such manifolds are integrally homology cobordant if they bound a compact oriented smooth cobordism for which the inclusions of both boundary components induce isomorphisms on integral homology. Their classes form the abelian group $\Theta^3_{\mathbb Z}$ under connected sum, with identity $[S^3]$ and inverse $[-Y]$.

For a knot $K\subset S^3$, choose the meridian $\mu$ and preferred zero-linking longitude $\lambda$. The notation $S^3_{1/n}(K)$ means Dehn filling its exterior with solid-torus meridian attached along $\mu+n\lambda$. For $n\ne0$ this is surgery coefficient $1/n$; $n=0$ means meridional filling and gives $S^3$. All these fillings are integral homology spheres.

Do their classes generate the whole group? \[ \Theta^3_{\mathbb Z} =\left\langle [S^3_{1/n}(K)]:n\in\mathbb Z,\ K\subset S^3\right\rangle. \] Equivalently, must every class be a finite integer linear combination of these surgery classes? Both the knots and denominators may vary among the finitely many terms.

The question does not require one surgery to represent each class. It allows connected sums and inverses, followed by homology cobordism. No bound on the number of terms or the genera of their knots is stipulated; imposing either would give a stronger assertion.

Short English statement

Do finite sums of reciprocal-integer knot-surgery classes generate the integral homology cobordism group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2868 (KP-3.70), 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 3.70. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.