Kirby Problem 3.59

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.

A balanced sutured manifold $(M,\gamma)$ is a compact oriented $3$-manifold without closed components, with oriented sutures in annular neighborhoods $\gamma\subset\partial M$, whose complement is $R_+(\gamma)\sqcup R_-(\gamma)$, with compatible boundary orientations, $\chi(R_+)=\chi(R_-)$, and every boundary component meeting $\gamma$. The group $\operatorname{SFH}(M,\gamma;k)$ is sutured Heegaard Floer homology, the homology of its admissible sutured Heegaard chain complex. The group $\operatorname{SHI}(M,\gamma;\mathbb C)$ is sutured instanton homology: the distinguished surface-eigenspace of instanton Floer homology of an admissible closure of the sutured manifold. Their isomorphism classes are independent of diagram and closure choices. For a closed oriented $Y$, $I^\#(Y;\mathbb C)$ is framed instanton Floer homology and $\widehat{\operatorname{HF}}(Y;\mathbb C)$ is hat Heegaard Floer homology. The corresponding knot versions arise from the knot exterior with meridional sutures.

Statement recorded in the supplied source.

For every balanced sutured manifold $(M,\gamma)$, prove an isomorphism of complex vector spaces \[ \operatorname{SHI}(M,\gamma;\mathbb C)\cong\operatorname{SFH}(M,\gamma;\mathbb C). \] This includes the resulting closed-manifold comparison $I^\#(Y;\mathbb C)\cong\widehat{\operatorname{HF}}(Y;\mathbb C)$ and the corresponding non-equivariant knot-invariant comparison. This is the explicit sutured interpretation given in the source. [S2]

Short English statement

Prove that sutured instanton and sutured Heegaard Floer homology agree over the complex numbers for every balanced sutured manifold, including the closed-manifold and knot special cases.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2857 (KP-3.59). 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 3.59, p. 173 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.