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Gluing two Reeb components gives a foliation of the three-sphere

Statement

Assume Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). Let $X_{1}, X_{2}$ be two copies of the solid torus ${\overline{D}}^{2} \times S^{1}$ with their Reeb foliations (The Reeb foliation of the solid torus has the boundary as a leaf) and let $σ : \partial X_{1} \to \partial X_{2}$ be the diffeomorphism which interchanges the two circle factors, written in the boundary coordinates $\partial X_{j} ≅ S^{1} \times S^{1}$ as $σ (u, v) = (v, u)$ . Then the glued manifold $X_{1} \cup_{σ} X_{2}$ is diffeomorphic to the three-sphere $S^{3}$ (Euclidean spheres and closed balls as subspaces of $R^{n}$ ), the standard genus-one splitting, and the two Reeb foliations glue by Gluing manifolds with boundary along a boundary diffeomorphism to a codimension-one regular foliation $F_{R e e b}$ of $S^{3}$ . This foliation has exactly one compact leaf, the Heegaard torus $\partial X_{1} = \partial X_{2}$ (The two-dimensional torus $T^{2} = (R / Z)^{2}$ ), whose holonomy group is infinite; every other leaf is diffeomorphic to $R^{2}$ and accumulates on the torus leaf. Hence a compact manifold can carry a codimension-one foliation with non-compact leaves and a single unstable compact leaf.

Facts & Assumptions

Given: Two copies $X_{1}, X_{2}$ of the solid torus with their Reeb foliations, and the boundary diffeomorphism $σ (u, v) = (v, u)$ .

[F1]

The Reeb foliation of the solid torus is tangent to the boundary, has the boundary torus as a single compact leaf with infinite holonomy, and all other leaves are planes accumulating on the boundary leaf (The Reeb foliation of the solid torus has the boundary as a leaf).

[F2]

Assume ${A C}_{ω}$ . If $φ : \partial W_{1} \to \partial W_{2}$ is a diffeomorphism of boundaries and the foliations $F_{i}$ tangent to $\partial W_{i}$ induce boundary foliations matched by $φ$ , and their plane fields have matching jets in signed collar coordinates, then the foliations glue to a regular foliation of the quotient, restricting to $F_{i}$ (Gluing manifolds with boundary along a boundary diffeomorphism).

[F3]

The three-sphere is $S^{3} = {(z_{1}, z_{2}) \in C^{2} : ∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2} = 1}$ , and the closed unit disk and the solid torus ${\overline{D}}^{2} \times S^{1}$ are as in Euclidean spheres and closed balls as subspaces of $R^{n}$ ; the boundary of the solid torus is $T^{2} = S^{1} \times S^{1}$ (The two-dimensional torus $T^{2} = (R / Z)^{2}$ ).

[F4]

A diffeomorphism is a bijective smooth map with smooth inverse (Diffeomorphisms and local diffeomorphisms of manifolds). In the signed-collar smooth structure, smoothness across the seam is checked in those charts; agreement of the two boundary restrictions alone does not suffice.

Proof

technique · direct

1.1F3F4

(The genus-one Heegaard splitting of $S^{3}$ .) Put $V_{1} = {∣ z_{1} ∣^{2} \leq 1 / 2}$ and $V_{2} = {∣ z_{2} ∣^{2} \leq 1 / 2}$ in $S^{3}$ . At least one coordinate has square modulus at most $1 / 2$ , so $V_{1} \cup V_{2} = S^{3}$ ; their intersection is $∣ z_{1} ∣ = ∣ z_{2} ∣ = 1 / \sqrt{2}$ . The map $V_{1} \to {\overline{D}}^{2} \times S^{1}$ , $(z_{1}, z_{2}) \mapsto (\sqrt{2} z_{1}, z_{2} / ∣ z_{2} ∣)$ , is a diffeomorphism with inverse $(w, ζ) \mapsto (w / \sqrt{2}, \sqrt{1 - ∣ w ∣^{2} / 2} ζ)$ . The corresponding map for $V_{2}$ swaps the complex coordinates and has inverse $(w, ζ) \mapsto (\sqrt{1 - ∣ w ∣^{2} / 2} ζ, w / \sqrt{2})$ . The denominators are at least $1 / \sqrt{2}$ on their respective domains. On the shared boundary the disk-angle and longitude parameters interchange, giving $σ (u, v) = (v, u)$ . Write $s = ∣ z_{1} ∣^{2} - 1 / 2$ near the shared torus and let $(θ_{1}, θ_{2})$ denote its two angles. The map from the signed collar to $S^{3}$ is the single smooth formula $(θ_{1}, θ_{2}, s) \mapsto (\sqrt{1 / 2 + s} e^{i θ_{1}}, \sqrt{1 / 2 - s} e^{i θ_{2}})$ ; its inverse is given by those angles and $∣ z_{1} ∣^{2} - 1 / 2$ . Pulling these collars back to the two solid tori makes the piece identifications a smooth diffeomorphism across the seam, and F2 gives the same diffeomorphism type for any other smooth collars.

2.1F1F2step 1.1

(The glued foliation.) By F1 each boundary torus is itself a whole leaf; its induced foliation has codimension zero, not a circle decomposition. Use a signed radial collar $s$ with $r_{1} = 1 - s$ for $s \geq 0$ and $r_{2} = 1 + s$ for $s \leq 0$ , and let $(θ, t)$ be the meridional and longitudinal angles from the first boundary. Factor swapping makes the second longitude $θ$ . By the flat boundary form constructed in F1, the first plane field has annihilator $d s + v (1 - s) d t$ and the second has annihilator $d s - v (1 + s) d θ$ , after multiplication by a nonzero scalar. On $s = 0$ both forms equal $d s$ , and every derivative of either additional coefficient vanishes there. The plane fields, expressed as graphs over $span (\partial_{θ}, \partial_{t})$ , therefore have matching jets of every order. This checks the strengthened gluing hypothesis of F2 explicitly; its construction gives a smooth regular foliation restricting to both Reeb components on the glued manifold of step 1.1.

3.1F1step 2.1∎

(Leaves.) The common boundary torus is a single leaf of each Reeb foliation and survives the gluing as the Heegaard torus leaf, with infinite holonomy: a longitude loop from either component retains its nonidentity contracting one-sided germ on that side [F1]. Every other leaf lies entirely inside one of the two open Reeb components, hence is a plane accumulating on the boundary torus [F1]. Therefore $S^{3}$ carries a codimension-one foliation with exactly one compact leaf, that leaf being unstable because every collar meets plane leaves which leave the collar, as in F1, while all other leaves are non-compact planes.

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Sources

Leiden NCG seminar, Noncommutative Geometry of Foliations (2023 seminar notes; complete PDF) (standard reference, not scraped)
Danny Calegari, Foliations and the Geometry of 3-Manifolds (Oxford Mathematical Monographs; complete author-hosted PDF) (standard reference, not scraped)