Kirby Problem 3.53

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.

Write $\mathbb F_2=\mathbb Z/2\mathbb Z$. For a closed oriented $3$-manifold $Y$, $\widehat{\operatorname{HF}}(Y;k)$ denotes hat Heegaard Floer homology with coefficients $k$: it is the homology of $\widehat{\operatorname{CF}}=\operatorname{CF}^-/U\operatorname{CF}^-$, where $\operatorname{CF}^-$ is the Heegaard Floer chain complex over $k[U]$. All spin$^c$ summands are included unless one is specified. A rational homology sphere has $H_*(Y;\mathbb Q)\cong H_*(S^3;\mathbb Q)$. Its first integral homology is finite. An L-space satisfies $\dim_{\mathbb F_2}\widehat{\operatorname{HF}}(Y;\mathbb F_2)=|H_1(Y;\mathbb Z)|$. A pointed genus-$g$ Heegaard diagram is $H=(\Sigma,\boldsymbol\alpha,\boldsymbol\beta,z)$, with two systems of $g$ disjoint attaching circles on $\Sigma$. Its tori $T_\alpha,T_\beta\subset\operatorname{Sym}^g(\Sigma)$ are the images of the products of those circles. The hat chain group is freely generated by $T_\alpha\cap T_\beta$ after making the diagram transverse and admissible. A strong Heegaard diagram for $Y$ has \[ \#(T_\alpha\cap T_\beta)=|H_1(Y;\mathbb Z)|. \] A strong L-space is a rational homology sphere admitting such a diagram. An alternating link has a planar diagram in which each component alternates overcrossings and undercrossings. A link is non-split if no embedded $2$-sphere separates it into two nonempty sublinks. Write $\Sigma_2(L)$ for the double cover of $S^3$ branched along $L$.

Statement recorded in the supplied source.

For every strong L-space $Y$, does some alternating link $L\subset S^3$ satisfy $Y\cong\Sigma_2(L)$? The link can be taken non-split, as its branched double cover is a rational homology sphere. The assertion ranges over strong diagrams of every genus, not only genus two, and strongness means the generator-count equality above. [S2]

Short English statement

Is every rational homology sphere with a Heegaard diagram having the minimum possible number of Floer generators a double branched cover of an alternating link?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2851 (KP-3.53). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (editors), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), Problem 3.53, p. 169 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.