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Local Reeb stability for compact leaves with finite holonomy

Statement

Assume ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). Let $F$ be a regular foliation of a smooth manifold $M$ and let $L$ be a compact leaf whose holonomy group is finite. Then $L$ is stable (Stable leaves): for every open neighbourhood $W$ of $L$ there is a saturated neighbourhood $U \subseteq W$ of $L$ and a foliated diffeomorphism of $U$ onto an open neighbourhood of the central leaf in the finite-holonomy normal model $(\hat{L} \times D) / H$ of The finite-holonomy normal model of a compact leaf, carrying $L$ to the central leaf. Moreover, after shrinking, the neighbourhood $U$ admits a retraction $π : U \to L$ such that for every leaf $L^{'} \subseteq U$ the restriction $π ∣_{L^{'}} : L^{'} \to L$ is a finite covering and $π^{- 1} (y)$ is a transverse disk for every $y \in L$ ; every leaf of $F ∣_{U}$ is compact with finite holonomy group and is finitely covered by the holonomy cover $\hat{L}$ . The hypothesis consumed is finiteness of the holonomy group, not finiteness of $π_{1} (L)$ ; no orientability of $F$ or $M$ is required.

Facts & Assumptions

Given: A regular foliation $F$ of a smooth manifold $M$ and a compact leaf $L$ with finite holonomy group $H$ , and an open neighbourhood $W$ of $L$ .

[F1]

A compact leaf with finite holonomy admits an $H$ -invariant transverse disk $D$ on which the finite holonomy group acts by diffeomorphisms, and the finite-holonomy normal model $(\hat{L} \times D) / H$ is defined with central leaf canonically diffeomorphic to $L$ (Finite holonomy acts on a small transverse disk, The finite-holonomy normal model of a compact leaf).

[F2]

The normal model map restricts to a foliated diffeomorphism of some model $(\hat{L} \times D^{'}) / H$ , $D^{'} \subseteq D$ , onto a saturated open neighbourhood $U$ of $L$ , and $U$ can be taken inside any prescribed neighbourhood of $L$ ; every leaf of $F ∣_{U}$ is compact with finite holonomy and is finitely covered by $\hat{L}$ (The normal model map restricts to a diffeomorphism onto a saturated neighbourhood).

[F3]

A leaf is stable when every neighbourhood of it contains a saturated neighbourhood; the neighbourhoods form a fundamental system under the model description (Stable leaves, Saturated neighbourhoods of a leaf).

[F4]

The model carries the product foliation by the slices modulo the finite group action; the leafwise covering projection gives a smooth model retraction $[(\hat{y}, t)] \mapsto p (\hat{y})$ onto $L$ , because $p$ is invariant under deck transformations (The finite-holonomy normal model of a compact leaf, Regular foliation atlases).

[F5]

The holonomy cover $\hat{L} \to L$ is a finite covering when $H$ is finite, of degree $∣ H ∣$ (The finite-holonomy normal model of a compact leaf, The normal model map restricts to a diffeomorphism onto a saturated neighbourhood).

[F6]

Compactness of $L$ supplies the uniform transverse size in the model construction (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

Proof

technique · direct

1.1F1F2F3F6

(The model neighbourhood.) Since $H$ is finite, [F1] provides the invariant transverse disk $D$ and the model $(\hat{L} \times D) / H$ ; applying [F2] gives an $H$ -invariant $D^{'} \subseteq D$ and a foliated diffeomorphism $\overline{Φ}$ of the model $(\hat{L} \times D^{'}) / H$ onto a saturated open neighbourhood $U$ of $L$ , with $U$ contained in the prescribed neighbourhood $W$ of $L$ because the model construction can be shrunk uniformly, using compactness of $L$ [F2, F6]. Thus $U \subseteq W$ is a saturated neighbourhood of $L$ , and $L$ is stable in the sense of [F3].

1.2F1F2F4F5

(The retraction and the finite-covering description.) On the model define $π_{0} ([(\hat{y}, t)]) : = p (\hat{y})$ . Deck invariance of $p$ makes this well defined, and covering trivializations show it is smooth. It is the identity on the central leaf under its identification with $L$ , so composing $π_{0}$ with the inverse model diffeomorphism gives a retraction $π : U \to L$ . For $y \in L$ , choose one lift $\hat{y}$ in the finite covering fibre; the map $t \mapsto [(\hat{y}, t)]$ identifies $D^{'}$ diffeomorphically with $π_{0}^{- 1} (y)$ , since the deck group acts freely and transitively on that fibre. Thus the retraction fibres are transverse disks in the actual codimension, not necessarily intervals. The leaf represented by $t$ is $\hat{L} / H_{t}$ , and its projection to $L = \hat{L} / H$ is the covering of degree $[H : H_{t}]$ . This gives the claimed finite covering on each leaf; compactness and finite holonomy follow from F2. Projection to the transverse factor itself does not define this retraction.

2.1F3step 1.1step 1.2∎

(Conclusion.) Every neighbourhood $W$ of $L$ contains the saturated neighbourhood $U$ constructed above, so $L$ is stable; the foliated diffeomorphism with the finite-holonomy normal model, the retraction with finite-covering leaf intersections, and the compactness and finite holonomy of the leaves of $F ∣_{U}$ are established in steps 1.1 and 1.2. The only hypothesis used beyond compactness of $L$ is finiteness of the holonomy group, not finiteness of $π_{1} (L)$ .

Depends on

Used by

Dependency tree · two levels

43 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ieke Moerdijk and Janez Mrčun, Introduction to Foliations and Lie Groupoids (Cambridge Studies in Advanced Mathematics 91, 2003) (standard reference, not scraped)
Matias del Hoyo and Rui Loja Fernandes, On deformations of compact foliations (Proc. AMS 147, 2019, 4555–4561) (standard reference, not scraped)