Kirby Problem 3.50

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 $\dim_{\mathbb F_2}\widehat{\operatorname{HF}}(Y;\mathbb F_2)=|H_1(Y;\mathbb Z)|$. An integral homology sphere has $H_*(Y;\mathbb Z)\cong H_*(S^3;\mathbb Z)$. Let $P=\Sigma(2,3,5)$ be the Poincaré homology sphere, the link in the unit $5$-sphere of $z_1^2+z_2^3+z_3^5=0$. Write $-P$ for its opposite orientation and $\#$ for oriented connected sum.

Statement recorded in the supplied source.

If an integral homology $3$-sphere $Y$ is an L-space, must there be an integer $m\ge0$ and signs $\epsilon_1,\ldots,\epsilon_m\in\{+1,-1\}$ such that \[ Y\cong\mathop{\#}_{j=1}^m\epsilon_jP? \] Here $\cong$ denotes oriented diffeomorphism and the empty connected sum is $S^3$. No primeness assumption is imposed on $Y$. [S2]

Short English statement

Is every integral homology sphere that is an L-space the three-sphere or a connected sum of arbitrarily oriented Poincaré spheres?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2848 (KP-3.50). 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 3.50, p. 167 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.