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The Reeb foliation of the three-sphere is not taut

Example

Assume Countable Choice ${A C}_{ω}$ . The Reeb foliation $F_{R e e b}$ of $S^{3}$ , obtained by gluing two Reeb solid tori along their boundary tori (Gluing two Reeb components gives a foliation of the three-sphere), contains two Reeb components meeting along the single compact leaf, the Heegaard torus $T^{2}$ . Consequently $F_{R e e b}$ is not taut: the torus leaf is the boundary leaf of both Reeb components, and no closed transversal meets it. Every other leaf is a plane accumulating on the torus.

Facts & Assumptions

Given: The Reeb foliation $F_{R e e b}$ of $S^{3}$ obtained by gluing two Reeb solid tori along their boundary tori.

[F1]

Gluing two Reeb components along their boundary tori gives a foliation of $S^{3}$ whose two solid tori are Reeb components with common boundary leaf the Heegaard torus (Gluing two Reeb components gives a foliation of the three-sphere); the standard Reeb foliation of the closed solid torus has the boundary as a single compact leaf diffeomorphic to $T^{2}$ and every interior leaf a plane accumulating on it (The Reeb foliation of the solid torus has the boundary as a leaf, The two-dimensional torus $T^{2} = (R / Z)^{2}$ ).

[F2]

A Reeb component is a compact saturated solid torus foliated homeomorphically by the standard Reeb model whose boundary is a single compact leaf (Reeb components of a codimension-one foliation), and a foliation containing a Reeb component is not taut (A Reeb component obstructs tautness).

[F3]

A foliation is taut when every leaf meets a closed transversal (Taut codimension-one foliations), and the standing assumption is Countable Choice ${A C}_{ω}$ as recorded for this pair (The countable-choice principle used in the foliation pair).

Verification

technique · direct

1.1F1F2

The gluing proposition [F1] supplies the foliation and identifies the two solid tori as Reeb components with common boundary leaf the Heegaard torus, so the definition of a Reeb component is satisfied with the foliated homeomorphism supplied by the model.

2.1F2F3step 1.1

By [F2] a foliation containing a Reeb component is not taut, and concretely the accessible manifold of the torus leaf is not all of $S^{3}$ : transverse curves crossing the torus into either solid torus are trapped by the accumulating plane leaves, so no closed transversal meets the torus leaf. Since tautness would require a closed transversal through that leaf, $F_{R e e b}$ is not taut.

3.1F1F3step 2.1∎

This realises Ranz's statement that a foliated manifold containing a Reeb component is not taut in the standard example, verifying the obstruction and supplying the negative model dual to the fibre-foliation example; every other leaf is a plane accumulating on the torus, and the argument uses only the two solid-torus models and the obstruction, hence only the standing countable choice from [F3].

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Sources

Danny Calegari, Foliations and the Geometry of 3-Manifolds (Oxford Mathematical Monographs; complete author-hosted PDF) (standard reference, not scraped)
S. P. Novikov, The Topology of Foliations (English translation by J. A. Zilber; complete PDF of the translation) (standard reference, not scraped)
Samuel Ranz, Approximately Holomorphic Techniques in Foliations: A Simple Proof of Novikov's Theorem (PhD thesis, Universidad Autonoma de Madrid, 2024; complete PDF) (standard reference, not scraped)