The profinite completion of the fundamental group records exactly the finite quotients of a 3-manifold's fundamental group, and the profinite rigidity program asks which topological properties of a 3-manifold survive in this purely algebraic shadow. Cheetham-West and Lê (2026) pose Question 1.1: is being Haken — containing an essential embedded surface, the property underlying essentially all inductive arguments in 3-manifold topology — a profinite property of compact 3-manifolds? Prior work settles every case except closed hyperbolic rational homology 3-spheres, where the essential surfaces (typically quasiFuchsian) have no known finite-quotient signature; the source paper proves that five profinitely detectable conditions force a profinitely equivalent partner to be Haken, but these conditions do not settle the general question: each of them already implies Hakenness, so every non-Haken hyperbolic rational homology 3-sphere satisfies none of them, and it is not known whether every Haken one satisfies at least one. The problem asks whether Hakenness is always preserved under profinite equivalence; a negative answer would be witnessed by an explicit pair of profinitely equivalent compact 3-manifolds of opposite Haken type.

Background

The profinite completion Γ^\widehat{\Gamma} of a finitely generated group Γ\Gamma is the inverse limit of its finite quotients; equivalently, it packages exactly the finite quotients of Γ\Gamma. Two compact 3-manifolds MM and NN are profinitely equivalent if π1(M)^≅π1(N)^\widehat{\pi_1(M)}\cong\widehat{\pi_1(N)}, and a property of 3-manifolds is a profinite property if every profinitely equivalent pair either both has it or both lack it. This is the basic notion of the profinite rigidity program, which asks how much of a 3-manifold is visible from the finite quotients of its fundamental group alone.

A compact, oriented, irreducible, boundary-irreducible 3-manifold is Haken if it contains an essential embedded surface: a properly embedded surface whose inclusion is injective on fundamental groups and which admits no compression. Haken manifolds are the setting for essentially all inductive arguments in 3-manifold topology (normal surface theory, hierarchies), and by a classical correspondence an essential embedded surface is the same data as a non-trivial splitting of π1(M)\pi_1(M) as a graph of groups, i.e. a non-trivial action of π1(M)\pi_1(M) on a simplicial tree. Agol's proof of the Virtual Haken Conjecture (2013) shows every closed hyperbolic 3-manifold has a finite cover that is Haken; the question here concerns the manifold itself, not its covers.

Several structural properties are already known to be profinite: admitting a Thurston geometry and having a non-trivial JSJ decomposition (the canonical decomposition of an irreducible 3-manifold along essential tori into Seifert-fibered and atoroidal pieces; Wilton–Zalesskii), and fibering over the circle with compact fiber (Jaikin-Zapirain). Combining these with Wilkes's classification of profinite behavior for Seifert fibered spaces and Hempel's examples of distinct Seifert-fibered 3-manifold groups with identical finite quotients, Cheetham-West and Lê observe a clean reduction: outside the world of closed hyperbolic rational homology 3-spheres, Hakenness is already a profinite property. A rational homology 3-sphere (QHS³) is a compact 3-manifold MM with H∗(M;Q)≅H∗(S3;Q)H_*(M;\mathbb{Q})\cong H_*(S^3;\mathbb{Q}), i.e. first Betti number b1(M)=0b_1(M)=0; such manifolds cannot fiber over S1S^1, and their essential surfaces (necessarily homologically trivial) are far harder to detect algebraically.

The 2026 source paper proves (Theorem 1.3) that if a Haken hyperbolic QHS³ MM satisfies any of five profinitely detectable conditions — (1) containing an embedded virtual fiber, a surface that becomes the fiber of a fibration over the circle in some finite-sheeted cover; (2) admitting a regular finite-sheeted cover M′→MM'\to M with a non-trivial class in H2(M′;Z)H_2(M';\mathbb{Z}) fixed up to sign by the deck group; (3) having π1(M)\pi_1(M) surject onto the infinite dihedral group Z/2∗Z/2\mathbb{Z}/2*\mathbb{Z}/2; (4) having a positive-dimensional curve in its (P)SL2(P)SL_2-character variety (the algebro-geometric space of conjugacy classes of SL2(C)SL_2(\mathbb{C})- or PSL2(C)PSL_2(\mathbb{C})-representations of π1(M)\pi_1(M), a classical mechanism for detecting essential surfaces) over some algebraically closed field; or (5) having finitely many conjugacy classes of SL2(C)SL_2(\mathbb{C})-representations together with at least one algebraic representation of non-integral trace — then every 3-manifold NN with π1(M)^≅π1(N)^\widehat{\pi_1(M)}\cong\widehat{\pi_1(N)} is also Haken. They verify these conditions for many examples (surgeries on the 525_2 knot and census manifolds such as m019(3,4)m019(3,4), m140(4,1)m140(4,1), m188(2,3)m188(2,3)), but the criteria do not settle the general question. Each condition already implies Hakenness, so every non-Haken hyperbolic QHS³ satisfies none of them, and the paper's census experiments — 2295 Haken rational homology 3-spheres from the Hodgson–Weeks census, of which only 286 admit an infinite dihedral quotient, with the character-variety and trace conditions verified for a few dozen more — leave most known Haken examples unclassified by the criteria; whether every Haken hyperbolic QHS³ satisfies at least one condition is itself open. Two obstructions are identified. First, extending the homology-in-cover argument to a quasiFuchsian essential surface (a geometrically finite surface, quasiconformally conjugate to a lattice in PSL2(R)<PSL2(C)PSL_2(\mathbb{R})<PSL_2(\mathbb{C}), the standard type of essential surface in hyperbolic manifolds) fails: integral homology classes alone cannot ensure that the least-area representatives of a class and its deck translates in a finite cover are pairwise disjoint. Second, the naive group-theoretic generalization is false — admitting a non-trivial splitting is not a profinite property of finitely generated residually finite groups (Cheetham-West–Lubotzky–Reid–Spitler; Bridson) — so any positive proof must use genuinely 3-dimensional structure such as geometrization, LERF (the property that finitely generated subgroups of π1(M)\pi_1(M) are separable, which holds for hyperbolic 3-manifold groups), and hyperbolic geometry.

Cheetham-West and Lê also record two open questions raised by this problem (Questions 6.1 and 6.4 of the source), which are not part of the research target: whether φ∘Φ−1(π1(N))\varphi\circ\Phi^{-1}(\pi_1(N)) must be free for an epimorphism φ:π1(M)↠Fr\varphi:\pi_1(M)\twoheadrightarrow F_r to a non-abelian free group and an isomorphism Φ:π1(M)^→π1(N)^\Phi:\widehat{\pi_1(M)}\to\widehat{\pi_1(N)} of profinitely equivalent finite-volume hyperbolic 3-manifold groups, and whether a knot profinitely equivalent to a small knot — one whose exterior contains no closed embedded essential surface — must itself be small.

Problem Statement

Is Hakenness a profinite property of 3-manifolds (Cheetham-West–Lê, Question 1.1)? That is: if MM and NN are compact 3-manifolds with π1(M)^≅π1(N)^\widehat{\pi_1(M)}\cong\widehat{\pi_1(N)}, must MM be Haken if and only if NN is Haken?

By the reduction recalled in the Background, every case except closed hyperbolic rational homology 3-spheres is already settled affirmatively, so the open core is: let MM be a Haken closed hyperbolic rational homology 3-sphere and let NN be a compact 3-manifold with π1(M)^≅π1(N)^\widehat{\pi_1(M)}\cong\widehat{\pi_1(N)}; must NN be Haken? By Theorem 1.3 one may assume that none of its five conditions applies to MM: then MM contains no embedded virtual fiber, and by the dichotomy for essential surfaces in closed hyperbolic 3-manifolds (a geometrically infinite essential surface is a virtual fiber, a geometrically finite one is quasiFuchsian), the Hakenness of MM is witnessed by a quasiFuchsian essential surface. Whether Haken hyperbolic rational homology 3-spheres to which none of the five conditions applies actually exist is itself part of what is open: if every Haken hyperbolic rational homology 3-sphere satisfied one of the five, Question 1.1 would follow affirmatively from Theorem 1.3 together with the reduction recalled above.

References

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