Kirby Problem 3.49

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 manifold is prime if a connected-sum decomposition has an $S^3$ summand. Following the source, a left-orderable group must be nontrivial and have a strict total order $<$ with $a<b\Rightarrow ga<gb$ for all $g$. A coorientable taut foliation is a codimension-one foliation with a consistent transverse orientation such that each leaf meets some closed transverse curve. A weak symplectic filling of a positive cooriented contact structure $\xi$ on $Y$ is a compact connected symplectic $4$-manifold $(W,\omega)$ whose oriented boundary is $Y$ and with $\omega|_\xi$ positive. Let $b_2^+(W)$ be the maximal dimension of a positive-definite subspace of the intersection pairing on $H_2(W;\mathbb R)$; for rational-homology-sphere boundary the pairing is nonsingular.

Statement recorded in the supplied source.

For all prime rational homology $3$-spheres $Y$, determine whether each of the three conditions \[ \pi_1(Y)\text{ left-orderable},\quad Y\text{ not an L-space},\quad Y\text{ admits a coorientable taut foliation} \] is equivalent to the existence of a contact structure $\xi$ and a symplectic filling $(W,\omega)$ with $b_2^+(W)>0$. Weak and strong fillability are equivalent for these $Y$, as explained in the source. [S2]

Short English statement

For prime rational homology spheres, does existence of a contact structure with a symplectic filling having a positive intersection direction characterize each of the three L-space-conjecture conditions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2847 (KP-3.49). 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 3.49, p. 167 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.