Kirby Problem 3.55

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.

Let $R=\mathbb F_2[U]$, where $\mathbb F_2=\mathbb Z/2\mathbb Z$. For a knot $K\subset S^3$, $\operatorname{HFK}^-(K)$ is minus knot Floer homology over $R$, retaining its $U$ action. For a rational homology $3$-sphere $Y$, define \[ \operatorname{HF}_{\rm red}(Y)=\operatorname{coker}\left(\operatorname{HF}^{\infty}(Y;\mathbb F_2)\longrightarrow\operatorname{HF}^{+}(Y;\mathbb F_2)\right), \] with its induced $R$ action. All spin$^c$ structures are included. To admit an $\mathbb F_2$ summand as an $R$-module means $M\cong R/(U)\oplus N$ for some $R$-module $N$; a one-dimensional submodule alone is insufficient.

Statement recorded in the supplied source.

(a) Does every nontrivial knot $K\subset S^3$ have an $R/(U)$ direct summand in $\operatorname{HFK}^-(K)$? (b) Does every rational homology sphere $Y$ with $\operatorname{HF}_{\rm red}(Y)\ne0$ have an $R/(U)$ direct summand in that reduced module? The source also asks about a geometric interpretation of the summand's generator and records the refinement: if $\operatorname{HF}_{\rm red}(Y)$ has a direct summand $R/(U^k)$, must it have a direct summand $R/(U^\ell)$ for each $1\le\ell<k$? [S2]

Short English statement

Must minus knot Floer homology of every nontrivial knot, and every nonzero reduced Floer group of a rational homology sphere, split off a one-dimensional summand annihilated by U? What geometric meaning and smaller-length summands accompany such splittings?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2853 (KP-3.55). 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 3.55, p. 170 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.