Kirby Problem 3.77

Definitions and mathematical statement

A closed rational homology $3$-sphere is a connected, compact, boundaryless $3$-manifold $Y$ with $H_i(Y;\mathbb Q)\cong H_i(S^3;\mathbb Q)$ for every $i$. Thus its rational Betti numbers are $1,0,0,1$. A hyperbolic metric has constant sectional curvature $-1$. For a compact Riemannian manifold $W$ with smooth boundary, the boundary is totally geodesic if its second fundamental form vanishes; a geodesic initially tangent to the boundary remains in it. The induced metric on a totally geodesic hypersurface of a hyperbolic manifold is hyperbolic.

Do there exist an orientable compact connected hyperbolic $4$-manifold $W$ and a hyperbolic rational homology $3$-sphere $Y$ such that $\partial W$ is isometric to $Y$ with its induced metric? The boundary must be the whole of $Y$, and the whole boundary of $W$ must have this single component. The metric on $W$ is required to extend smoothly to its boundary with the stated totally geodesic condition.

This is a geometric filling question, stronger than asking for a smooth or topological manifold with boundary $Y$. Orientability is required of the filling, not only its boundary. A nonorientable hyperbolic filling does not meet that hypothesis. A cusp cross-section of a noncompact finite-volume hyperbolic manifold is also a different object: it is not a totally geodesic boundary of a compact hyperbolic manifold. The problem imposes no arithmeticity condition.

Short English statement

Can a hyperbolic rational homology three-sphere be the entire totally geodesic boundary of a compact orientable hyperbolic four-manifold?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2875 (KP-3.77), 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 3.77. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.