Kirby Problem 3.63

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 hyperbolic $3$-manifold with its curvature-$-1$ metric, and let $s$ be a spin structure, a lift of its oriented orthonormal frame bundle to $\operatorname{Spin}(3)$. The associated complex spinor bundle carries the self-adjoint Dirac operator $D_s$, formed by composing the spin connection with Clifford multiplication. Its real discrete spectrum is counted with multiplicity. For $\operatorname{Re}z$ sufficiently large put \[ \eta_{D_s}(z)=\sum_{\lambda\in\operatorname{Spec}(D_s)\setminus\{0\}} \operatorname{sign}(\lambda)|\lambda|^{-z}. \] Its meromorphic continuation is regular at $z=0$; the APS Dirac eta invariant is $\eta_{\rm Dirac}(Y,s)=\eta_{D_s}(0)$. This is the unreduced eta invariant, not $(\eta_{D_s}(0)+\dim\ker D_s)/2$.

Statement recorded in the supplied source.

Give a method to compute $\eta_{\rm Dirac}(Y,s)$ from the hyperbolic manifold and its spin structure, for every closed spin hyperbolic $3$-manifold. The requested quantity is the full real invariant; calculating only its class modulo integers, the signature-operator eta invariant, or examples forced to have zero by an orientation-reversing symmetry does not settle the general computation problem. [S2]

Short English statement

Give a general way to calculate the spectral asymmetry of the Dirac operator for every spin hyperbolic three-manifold.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2861 (KP-3.63). 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 3.63, p. 176 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.