Kirby Problem 2.15

Definitions and mathematical statement

Fix the standard genus-$g$ Heegaard splitting $S^3=H_g\cup_{S_g}H'_g$, where both $H_g$ and $H'_g$ are handlebodies (regular neighborhoods of bouquets of $g$ circles). The Goeritz group $\mathcal G_g$ consists of isotopy classes of orientation-preserving homeomorphisms of $S^3$ that preserve $H_g$ setwise. Isotopies must also preserve that handlebody at every time. Equivalently, it is the subgroup of $\operatorname{Mod}(S_g)$ whose classes extend over both handlebodies.

For every integer $g\geq4$, is $\mathcal G_g$ finitely generated? If so, give an explicit finite collection of homeomorphism classes $\phi_1,\ldots,\phi_k$ and prove \[ \mathcal G_g=\langle\phi_1,\ldots,\phi_k\rangle. \] The number and description of the generators may depend on $g$. The chosen side of the splitting must be preserved, so elements exchanging the two handlebodies are outside this convention. Likewise, isotopy as arbitrary maps of $S^3$ would erase the information being studied; the isotopy is an isotopy of the Heegaard pair. An infinite generating family alone does not settle finite generation.

Short English statement

Is the symmetry group of a standard Heegaard splitting of the three-sphere finitely generated in every genus at least four?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2763 (KP-2.15), 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 2.15. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.