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A Reeb component has a compact boundary leaf with infinite holonomy

Statement refuted

In a codimension-one foliation every compact leaf has finite holonomy and is therefore stable, so that the finiteness hypothesis of local Reeb stability (Local Reeb stability for compact leaves with finite holonomy) is automatic for compact leaves.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). The Reeb foliation of the solid torus $X = {\overline{D}}^{2} \times S^{1}$ and its boundary torus $\partial X$ .

[F1]

The Reeb foliation of the solid torus is tangent to the boundary; $\partial X ≅ T^{2}$ is a single compact leaf with infinite holonomy, represented by the non-identity contraction germ $r \mapsto r^{'}$ with $u (r^{'}) = u (r) + 1$ , $u (r) = \exp (1 / (1 - r^{2}))$ , and every other leaf is a plane accumulating on $\partial X$ (The Reeb foliation of the solid torus has the boundary as a leaf, The two-dimensional torus $T^{2} = (R / Z)^{2}$ ).

[F2]

For a leaf in a smooth boundaryless foliation with a two-sided one-dimensional local transversal $T$ , its holonomy group is the image of $ρ_{x} : π_{1} (L, x) \to {Diff}_{x} (T)$ (The holonomy representation and the holonomy group of a leaf, Local transversals to a regular foliation). The original boundary holonomy uses the restrictions of these transport germs to the inward half-interval: equality means equality on some relatively open half-neighbourhood. Since transports preserve the boundary and its inward side, composition and inverses restrict, so these one-sided germs also form a group.

[F3]

A leaf is stable when every neighbourhood of it contains a saturated neighbourhood; a saturated neighbourhood is a union of leaves (Stable leaves, Saturated neighbourhoods of a leaf).

[F4]

Local Reeb stability requires a compact leaf with finite holonomy group (Local Reeb stability for compact leaves with finite holonomy).

[F5]

A nowhere-zero smooth one-form with $α \land d α = 0$ has integrable kernel and admits local foliation charts (The codimension-one Frobenius criterion, Frobenius local coordinate theorem).

Counterexample

technique · direct verification

1.1F1

(A compact leaf.) By [F1] the boundary $\partial X$ is a single leaf of the Reeb foliation and it is compact, diffeomorphic to the two-torus $T^{2}$ .

1.2F1F2F5

(Boundaryless extension and infinite holonomy.) Near $r = 1$ , the interior level foliation of F1 has kernel $d r - v (r) d t$ , where $v (r) = (1 - r^{2})^{2} \exp (- 1 / (1 - r^{2})) / (2 r)$ . Extend $v$ by zero for $r \geq 1$ . It is smooth and flat at one: every derivative is a smooth factor times a polynomial in $(1 - r^{2})^{- 1}$ times the rapidly decaying exponential. On the added collar in $Y = {r < 1 + ε} \times S^{1}$ , the form $α = d r - v (r) d t$ is nowhere zero and satisfies $α \land d α = 0$ . Its kernel agrees with the original foliation inside the collar, so F5 supplies a boundaryless extension with $r = 1$ still a torus leaf. A radial interval at fixed angles is now a two-sided transversal. The holonomy of the loop in the $S^{1}$ -factor of the Reeb component is computed in [F1] as the germ $r \mapsto r^{'}$ determined by $u (r^{'}) = u (r) + 1$ ; it is a non-identity one-sided contraction, so the holonomy group contains a non-identity element, and its iterates $u (r_{n}) = u (r) + n$ give infinitely many distinct germs near the boundary. These distinct inward restrictions force the two-sided germs in the extension to be distinct, so its compact torus leaf also has infinite holonomy in the exact sense of F2.

1.3F1F3

(The leaf is not stable.) Suppose the boundary leaf were stable. Then a small collar $W = {r > 1 / 2}$ of it would contain a saturated neighbourhood $U$ of the boundary leaf [F3]; but $U$ is open and contains the boundary leaf, hence contains a point $p$ with $r (p)$ close to $1$ , and being saturated it contains the entire leaf through $p$ ; by [F1] that leaf is a plane, and it meets the circle $r = 1 / 2$ and so is not contained in $W$ , a contradiction. Therefore the compact boundary leaf is not stable.

2.1F1F4step 1.3∎

(What this refutes.) A compact leaf can have infinite holonomy and can fail to be stable, so the finiteness hypothesis of [F4] cannot be dropped for compact leaves, and the failure is a failure of finiteness of holonomy, not of compactness of the ambient foliated manifold.

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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)