A contact structure on an oriented 3-manifold is called tight if it contains no overtwisted disk; while overtwisted contact structures exist on every closed 3-manifold and are classified by an h-principle, tight ones are rigid and their existence is delicate. The only closed irreducible 3-manifolds known to admit no tight contact structure are the Seifert fibered spaces obtained by (2n−1)(2n-1)-surgery on the torus knots T2,2n+1T_{2,2n+1} (Etnyre–Honda, Lisca–Stipsicz), and every closed irreducible 3-manifold with positive first Betti number admits a tight structure via taut foliations and confoliations. Li–Wan–Zhou's 2025 theorem that every negative rational surgery on every knot in S3S^{3} admits a tight contact structure re-foregrounds the question those authors explicitly call central: does every closed hyperbolic 3-manifold admit a tight contact structure, or does there exist one that admits none? As of August 2026 the question remains open: all known hyperbolic results are positive existence results on specific families or non-fillability obstructions, which are strictly weaker than non-tightness.

Background

A contact structure on a closed oriented 3-manifold YY is a cooriented tangent 2-plane field ξ=ker⁡(α)\xi=\ker(\alpha) for a 1-form α\alpha with α∧dα\alpha\wedge d\alpha nowhere zero. Eliashberg's 1992 classification showed that overtwisted contact structures — those containing an overtwisted disk, an embedded disk whose boundary is a Legendrian curve (a knot everywhere tangent to ξ\xi) with Thurston–Bennequin invariant (the contact framing of the boundary curve) zero — obey an h-principle: they exist on every closed 3-manifold and are classified by the homotopy class of the underlying plane field. Tight contact structures, those without overtwisted disks, are the rigid, geometrically interesting class: they are preserved under symplectic filling — a contact structure is (symplectically) fillable if it arises as the boundary of a compact symplectic 4-manifold, a Stein filling being the special case where the filling is a Stein domain — so every fillable contact structure is tight, but tightness is strictly weaker than fillability. Which closed 3-manifolds admit tight contact structures at all has been a guiding question of contact topology since Eliashberg posed it in 1992 (recorded as Problem 4.142 in Kirby's problem list; see Mj–Sen for this attribution).

The existence landscape is as follows. Etnyre and Honda produced the first closed 3-manifolds with no tight contact structure, including the reducible connected sum P#P‾P\#\overline{P} and the orientation-reversed Poincaré homology sphere −P=P‾-P=\overline{P} itself (PP being the Poincaré homology sphere and P‾\overline{P} its orientation reversal). Lisca and Stipsicz completed the Seifert fibered case: a closed oriented Seifert fibered 3-manifold (a 3-manifold foliated by circles) either is diffeomorphic to Mn=S2n−13(T2,2n+1)M_{n}=S^{3}_{2n-1}(T_{2,2n+1}) — the (2n−1)(2n-1)-surgery on the (2,2n+1)(2,2n+1)-torus knot, where Sr3(K)S^{3}_{r}(K) denotes Dehn surgery, i.e., removing a tubular neighborhood of the knot KK and regluing a solid torus so that the curve of slope rr on the boundary torus bounds a disk — or carries a positive tight contact structure, while each MnM_{n} carries none; these surgeries remain the only known irreducible examples without tight structures. For manifolds with positive first Betti number, Gabai's construction of taut foliations (codimension-one foliations in which some closed curve transverse to the leaves meets every leaf) together with the Eliashberg–Thurston theory of confoliations (perturbing a taut foliation yields a tight contact structure) gives a tight contact structure on every closed irreducible 3-manifold with b1>0b_{1}>0. By Agol's virtual fibering theorem, which resolved the virtual Haken conjecture, every closed hyperbolic 3-manifold — one admitting a complete Riemannian metric of constant sectional curvature −1-1 — has a finite-sheeted cover that fibers over the circle, hence (via the fibered monodromy's supporting open book) a cover with a tight structure; but tightness of a cover does not descend to the base (as summarized by Mj–Sen). The existence question for closed hyperbolic 3-manifolds is therefore the central open case, and any potential counterexample must be a hyperbolic rational homology sphere (a closed 3-manifold with H1(Y;Q)=0H_{1}(Y;\mathbb{Q})=0).

The problem was recently re-foregrounded by Li, Wan, and Zhou, who proved that for any knot K⊂S3K\subset S^{3} and any r∈Q>0r\in\mathbb{Q}_{>0}, the negative-surgery manifold S−r3(K)S^{3}_{-r}(K) admits a tight contact structure detected by a nonvanishing Heegaard Floer contact invariant. Heegaard Floer homology is a homology theory for 3-manifolds, defined by Ozsváth–Szabó from a Heegaard splitting via Lagrangian Floer homology of curves on the Heegaard surface; the contact invariant is an element c(ξ)∈HF^(−Y)c(\xi)\in\widehat{HF}(-Y), defined by Ozsváth–Szabó and Honda–Kazez–Matić from an open book decomposition supporting ξ\xi (a fibration of the complement of a link over S1S^{1} whose pages bound that link), whose nonvanishing certifies tightness. Since most Dehn surgeries on hyperbolic knots are hyperbolic, this exhibits tight structures on a large class of hyperbolic manifolds and removes all non-positive knot surgeries as potential counterexamples; positive-slope and non-surgery hyperbolic manifolds remain untouched. An LL-space in this context is a rational homology sphere whose Heegaard Floer homology HF^\widehat{HF} has the minimal possible rank; hyperbolic L-spaces are the natural testing ground, since fillability is already obstructed for many of them while tightness is not.

Problem Statement

Does every closed, orientable, hyperbolic 3-manifold — i.e., every closed 3-manifold admitting a complete Riemannian metric of constant sectional curvature −1-1 — admit at least one tight contact structure (a cooriented tangent 2-plane field ξ=ker⁡(α)\xi=\ker(\alpha) with α∧dα\alpha\wedge d\alpha nowhere zero, containing no overtwisted disk)? Equivalently: does there exist a closed hyperbolic 3-manifold that admits no tight contact structure whatsoever? This is the hyperbolic case of Eliashberg's 1992 tight-existence question for 3-manifolds; it is stated as a central open problem in Li–Wan–Zhou (2025), which settles the complementary non-positive-surgery classes, and as Question 1.1 in Nonino (2024). The related but distinct auxiliary question — whether every surgery on a knot in S3S^{3} outside the known Seifert exceptions S2n−13(T2,2n+1)S^{3}_{2n-1}(T_{2,2n+1}) admits a tight structure (i.e., the positive-slope case) — is a sub-problem of the positive resolution of this question, restricted to the hyperbolic surgery manifolds, and is treated as context, not as the target of this problem.

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