Kirby Problem 4.124

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 closed hyperbolic four-manifold is a compact boundaryless quotient $M=\mathbb H^4/\Gamma$, where $\mathbb H^4$ has constant sectional curvature $-1$ and $\Gamma$ is a torsion-free cocompact discrete isometry group. It is an integral homology four-sphere if $H_j(M;\mathbb Z)\cong H_j(S^4;\mathbb Z)$ for every $j$, namely $\mathbb Z$ for $j=0,4$ and zero otherwise. An arithmetic hyperbolic four-manifold is one whose lattice is arithmetic: up to commensurability and conjugacy, it arises from integral points of an orthogonal group of a quadratic form over a totally real number field, of signature $(4,1)$ at one real place and positive definite at the others. Commensurable groups have finite-index intersection. The Euler characteristic is $\chi(M)=\sum_j(-1)^j\operatorname{rank}H_j(M;\mathbb Z)$.

Statement recorded in the supplied source.

Does a closed hyperbolic integral homology four-sphere exist? Does such a manifold exist with arithmetic lattice? More generally, does any closed hyperbolic four-manifold with $\chi(M)=2$ exist, without requiring its integral homology to equal that of $S^4$? [S2]

Short English statement

Does a hyperbolic integral homology four-sphere exist, possibly arithmetic; and does any closed hyperbolic four-manifold have Euler characteristic two?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3000 (KP-4.124). 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 4.124, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.