Kirby Problem 3.9

Definitions and mathematical statement

For compact connected orientable irreducible three-manifolds, the Haken condition means the existence of a properly embedded two-sided incompressible surface which is not a sphere or boundary-parallel. Irreducibility means that every embedded sphere bounds a ball; incompressibility here requires $\pi_1$-injectivity and, when the surface has boundary, the customary boundary-incompressibility condition. In the closed case the witnessing surface is closed of positive genus.

For $G=\pi_1(M)$ let $\widehat G=\varprojlim G/N$, with $N$ ranging over finite-index normal subgroups. Ask whether the Haken condition is a profinite invariant: if two manifolds in the standard irreducible three-manifold setting have isomorphic profinite completions and one is Haken, must the other be Haken?

The core unresolved case identified by the source is particularly precise. Let $M,N$ be closed orientable hyperbolic three-manifolds with $H_1(M;\mathbb Q)=H_1(N;\mathbb Q)=0$. If $\widehat{\pi_1(M)}\cong\widehat{\pi_1(N)}$ and $M$ contains a closed embedded two-sided $\pi_1$-injective surface of positive genus, must $N$ contain one? The witnessing surfaces need not correspond under a given completion isomorphism; their existence is the proposed invariant.

Short English statement

Can finite quotients detect whether a three-manifold contains an incompressible surface, especially for closed hyperbolic rational homology spheres?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2807 (KP-3.9), 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 3.9. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.