Kirby Problem 4.125

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 hyperbolic four-manifold is a connected smooth manifold with a complete Riemannian metric of constant sectional curvature $-1$. Finite volume means $\int_M d\operatorname{vol}<\infty$. For an oriented four-manifold, a spin structure is a lift of the oriented orthonormal frame bundle through the double cover $\operatorname{Spin}(4)\to\operatorname{SO}(4)$; it exists exactly when the second Stiefel--Whitney class $w_2(TM)\in H^2(M;\mathbb Z/2)$ vanishes.

Statement recorded in the supplied source.

Does there exist a noncompact finite-volume orientable hyperbolic four-manifold $M$ with $w_2(TM)\ne0$, and hence with no spin structure? [S2]

Short English statement

Can a cusped finite-volume oriented hyperbolic four-manifold fail to admit a spin structure?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3001 (KP-4.125). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 4.125, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Stefano Riolo and Edoardo Rizzi, A cusped hyperbolic 4-manifold without spin structures, arXiv:2510.12657v2 (2026; first posted 2025), abstract and main theorem. The manuscript supplies the requested example. https://arxiv.org/abs/2510.12657.