Kirby Problem 3.54

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.

For a closed oriented $3$-manifold $M$, $\widehat{\operatorname{HF}}(M;\mathbb Z)$ denotes hat Heegaard Floer homology using the canonical isomorphism class of coherent orientation systems, as specified in the source. For a knot $K\subset S^3$, $\widehat{\operatorname{HFK}}(S^3,K;\mathbb Z)$ is the integral hat knot Floer homology, the homology of the associated Alexander-graded knot Floer complex. All gradings are included. An abelian group has torsion if it has $x\ne0$ with $nx=0$ for some integer $n\ge2$. A rational homology $3$-sphere satisfies $H_*(M;\mathbb Q)\cong H_*(S^3;\mathbb Q)$.

Statement recorded in the supplied source.

Determine separately whether there exists:

1. a closed $3$-manifold $M$ with torsion in $\widehat{\operatorname{HF}}(M;\mathbb Z)$;
2. a rational homology $3$-sphere $M$ with torsion in $\widehat{\operatorname{HF}}(M;\mathbb Z)$;
3. a knot $K\subset S^3$ with torsion in $\widehat{\operatorname{HFK}}(S^3,K;\mathbb Z)$.

The orientation-system convention is essential: torsion from a noncanonical choice or from a plus/infinity Floer variant does not answer these questions. [S2]

Short English statement

Can torsion occur in canonical integral hat Floer homology of a closed three-manifold, of a rational homology sphere, or of a knot in the three-sphere?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2852 (KP-3.54). 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 3.54, p. 170 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.