Kirby Problem 3.10

Definitions and mathematical statement

A closed connected oriented three-manifold $M$ is a rational homology sphere if $H_*(M;\mathbb Q)\cong H_*(S^3;\mathbb Q)$, and an integral homology sphere if the analogous statement holds with $\mathbb Z$ coefficients. For the latter, in particular, $H_1(M;\mathbb Z)=0$.

An orientable hyperbolic three-manifold is arithmetic if it is $\mathbb H^3/\Gamma$, with $\Gamma$ a torsion-free lattice in $\operatorname{PSL}_2(\mathbb C)$ commensurable with the norm-one group of an order in a quaternion algebra over a number field with exactly one complex place, ramified at every real place. An order is a full integral lattice which is also a subring. Commensurability of lattices allows conjugation and means that their intersection has finite index in each. For manifolds this is equivalent to possessing a common finite-sheeted cover.

(a) Are there infinitely many commensurability classes containing an arithmetic hyperbolic rational homology three-sphere?

(b) Are there infinitely many arithmetic hyperbolic integral homology three-spheres, up to isometry?

The quantifiers differ: an infinite sequence of covers in one commensurability class would not settle (a), whereas (b) asks for infinitely many distinct manifolds and does not demand pairwise incommensurability.

Short English statement

Find infinitely many arithmetic hyperbolic rational homology sphere classes, and determine whether there are infinitely many arithmetic integral homology spheres.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2808 (KP-3.10), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.10. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.