Kirby Problem 3.34

Definitions and mathematical statement

A Heegaard splitting of a closed connected orientable three-manifold $M$ is a decomposition into two handlebodies meeting along their common boundary surface. Its genus is the genus of that surface. A stabilization is the connected sum of the splitting with the standard genus-one splitting of $S^3$, performed in a ball. It raises genus by one without changing $M$.

Take any two genus-$g$ Heegaard splittings of $M$. Are they equivalent after at most $g$ stabilizations of each? Equivalently, does there exist a common stabilization of genus at most $2g$? Equivalence means ambient isotopy of the splitting surface, allowing the two handlebodies to be interchanged; a version remembering their order should be stated separately.

The quantifier includes splittings that are not of minimum genus for $M$. One may choose the stabilization arcs and isotopies freely, but must obtain a common stabilized splitting of the original manifold. The general stabilization theorem guarantees a finite common stabilization, without giving the particular bound $2g$ asked for here. Conversely, examples requiring many stabilizations must be compared using the same equivalence convention, since remembering which handlebody is first can affect the required number.

Short English statement

Do two Heegaard splittings of the same genus always become isotopic after adding at most that many stabilizing handles to each?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2832 (KP-3.34), 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 3.34. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.