Kirby Problem 3.51

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 $Y$ be a closed oriented rational homology $3$-sphere, so $H_1(Y;\mathbb Z)$ is finite. The group $\operatorname{SU}(2)$ consists of complex $2\times2$ unitary matrices of determinant one. A representation is abelian if its image is an abelian subgroup. Write $I^\#(Y;\mathbb C)$ for framed instanton Floer homology over $\mathbb C$, the version defined via the standard admissible bundle on $Y\#T^3$; the framing includes reducible representations that ordinary instanton homology does not include. An instanton L-space has $\dim_{\mathbb C}I^\#(Y;\mathbb C)=|H_1(Y;\mathbb Z)|$.

Statement recorded in the supplied source.

For every rational homology $3$-sphere $Y$, does \[ \left(\forall\rho:\pi_1(Y)\to\operatorname{SU}(2),\ \rho(\pi_1(Y))\text{ abelian}\right) \quad\Longrightarrow\quad \dim_{\mathbb C}I^\#(Y;\mathbb C)=|H_1(Y;\mathbb Z)|? \] No cyclic-finiteness or Morse--Bott nondegeneracy assumption is added. [S2]

Short English statement

Must a rational homology sphere with no nonabelian SU(2) representation have the minimum possible framed instanton Floer dimension?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2849 (KP-3.51). 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 3.51, p. 167 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.