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In a transversely oriented codimension-one foliation a compact leaf with finite fundamental group has trivial holonomy

Statement

Assume ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). Let $F$ be a transversely oriented codimension-one foliation of a smooth manifold $M$ (Transversely oriented codimension-one foliations) and let $L$ be a compact leaf with finite fundamental group (Based loops and the fundamental group). Then the holonomy group of $L$ is trivial, and consequently $L$ has a fundamental system of product foliated neighbourhoods $L \times D$ and every leaf in such a neighbourhood is compact and diffeomorphic to $L$ .

Facts & Assumptions

Given: A transversely oriented codimension-one foliation $F$ of a smooth manifold $M$ and a compact leaf $L$ with finite $π_{1} (L, x)$ .

[F1]

A local transversal $T$ to a codimension-one foliation at $x \in L$ is one-dimensional; transverse orientability orients it, and the holonomy representation $ρ_{x}$ takes values in the germs of orientation-preserving local diffeomorphisms of $(R, 0)$ , that is, in ${Diff}_{0}^{+} (R, 0)$ (Transversely oriented codimension-one foliations, Local transversals to a regular foliation, The holonomy representation and the holonomy group of a leaf).

[F2]

Every finite subgroup of ${Diff}_{0}^{+} (R, 0)$ is trivial; equivalently the group of orientation-preserving one-dimensional germs is torsion-free (Germs of orientation-preserving diffeomorphisms of the line at zero are torsion-free).

[F3]

The image of a finite group under a homomorphism is finite (Images of finitely generated and of finite groups are finitely generated and finite).

[F4]

Trivial holonomy on a compact leaf gives a fundamental system of product foliated neighbourhoods $L \times D$ , whose leaves are compact and diffeomorphic to $L$ (Trivial holonomy gives a product foliated neighbourhood).

Proof

technique · direct

1.1F1F3

(The holonomy group is finite.) The holonomy group is $H = ρ_{x} (π_{1} (L, x))$ [F1]. Since $π_{1} (L, x)$ is finite, its image $H$ is finite by [F3].

2.1F1F2step 1.1

(It is trivial.) By [F1] the finite group $H$ is a subgroup of ${Diff}_{0}^{+} (R, 0)$ ; by torsion-freeness [F2] every finite subgroup of that group is trivial, so $H$ is the trivial group. Hence the holonomy of $L$ is trivial.

3.1F4step 2.1∎

(Product neighbourhoods.) Since $L$ is compact and its holonomy is trivial, [F4] provides a fundamental system of saturated neighbourhoods foliated-diffeomorphically as products $L \times D$ with the product foliation; every leaf of such a neighbourhood is a slice, hence compact and diffeomorphic to $L$ .

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Sources

Ieke Moerdijk and Janez Mrčun, Introduction to Foliations and Lie Groupoids (Cambridge Studies in Advanced Mathematics 91, 2003) (standard reference, not scraped)
Danny Calegari, Foliations and the Geometry of 3-Manifolds (Oxford Mathematical Monographs; complete author-hosted PDF) (standard reference, not scraped)