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The compact leaf produced by a vanishing cycle bounds a Reeb component

Statement

Assume Countable Choice ${A C}_{ω}$ . In the situation of A nonzero limitwise-nullhomotopy class yields a compact boundary leaf, the compact leaf $L_{1}$ obtained is diffeomorphic to the torus $T^{2}$ , and there is a Reeb component $R \subseteq M$ with $\partial R = L_{1}$ : $R$ is a compact saturated submanifold diffeomorphic to $D^{2} \times S^{1}$ with boundary leaf $L_{1}$ , every interior leaf of $R$ is a plane, and $(R, F ∣_{R})$ is foliated-homeomorphic to the standard Reeb component (Reeb components of a codimension-one foliation). Moreover, on the side $j$ of the nonzero $Π_{1}^{j}$ class (the side approached by the vanishing-cycle family when one is given), $L_{1}$ is the limit set of every sufficiently nearby displaced leaf: for a corresponding one-sided normal fence based at the supporting leaf, there is $ε > 0$ such that the leaf $A_{t}$ through its displaced base point has limit set exactly $L_{1}$ for $0 < t < ε$ .

Facts & Assumptions

Given: The situation of A nonzero limitwise-nullhomotopy class yields a compact boundary leaf: a compact leaf $L_{1}$ produced by a vanishing cycle, on the side $j$ of a nonzero $Π_{1}^{j}$ class.

[F1]

The in-pair item A no-transversal leaf is a torus via the finite accessibility boundary sum identifies the no-transversal compact leaf as homeomorphic to a torus, and Finite C2 surface carriers have smooth normal forms and relative cap approximations upgrades its actual compact oriented $C^{2}$ carrier to a $C^{2}$ torus normal form, and the in-pair item A nonzero pi class on a torus has a primitive embedded pi root extracts a primitive embedded $Π$ meridian whose fence is an embedded annulus with actual embedded disk caps.

[F2]

The in-pair item A primitive pi torus collar has contracting longitude and exhausting plane caps supplies the contracting complementary longitude holonomy $H$ , the embedded leafwise longitude annuli $A_{t}$ with $D_{H (t)} = D_{t} \cup A_{t}$ , and the exhaustion of nearby leaves as planes with limit set exactly the original torus.

[F3]

The in-pair item The primitive pi cap block embeds and gives the global Reeb model assembles the paired quotient with compatible $C^{2}$ signed-flow seam collars and actual disk parametrizations into an embedded solid torus diffeomorphic to $D^{2} \times S^{1}$ whose foliation is foliated-homeomorphic to the standard Reeb component with continuous inverse at the boundary, and a Reeb component is a compact saturated solid torus with boundary mapped to a leaf (Reeb components of a codimension-one foliation, Saturated neighbourhoods of a leaf, The two-dimensional torus $T^{2} = (R / Z)^{2}$ , Euclidean spheres and closed balls as subspaces of $R^{n}$ ).

[F4]

The standing assumption is Countable Choice ${A C}_{ω}$ as recorded for this pair (The countable-choice principle used in the foliation pair).

Proof

technique · direct

1.1F1given

The original compact leaf $L_{1}$ obtained from the vanishing-cycle situation has no closed transversal, since a closed transversal through it would contradict the displaced nullness of the vanishing-cycle family on the approached side; its strict positive accessible region has finite compact inward boundary, and finite plane-bundle Euler boundary evaluation together with the finite oriented-surface normal forms identify the topological genus of $L_{1}$ as one; the finite $C^{2}$ carrier and smooth disk-band normal form in [F1] then give a $C^{2}$ diffeomorphism $L_{1} ≅ T^{2}$ .

2.1F1step 1.1

Extract a primitive embedded $Π$ meridian on $L_{1}$ by [F1]: increasing finite-order holonomy is the identity and torsion-free nearby surface groups turn the displaced root null, so the full primitive fixed-flow fence is embedded and its positive circles bound actual embedded Jordan disks. A complementary longitude has no small fixed point, since otherwise the compact graph lemma would contradict meridian nullness; choosing its inverse gives a contraction $H$ .

3.1F2step 2.1

Finite collar suspension over a cut fundamental polygon gives the embedded leafwise longitude annuli and the relations $D_{H (t)} = D_{t} \cup A_{t}$ of [F2], and the iterates exhaust each nearby leaf as a plane with limit set exactly $L_{1}$ .

4.1F2F3step 3.1

The fundamental cap sweep paired quotient has embedded boundary torus $C$ by [F2]; proper local inverse preimage counts, zero on the original-leaf side and jumping by one across $C$ , prove that the entire quotient is globally embedded as a solid torus, whose compact collar is attached to the original leaf $L_{1}$ by [F3]. The supplied signed-flow seam atlas makes this an actual $C^{2}$ submanifold, the disk and one-handle comparison gives $R ≅ D^{2} \times S^{1}$ diffeomorphically, and normal collar absorption preserves this type.

5.1F2F3F4step 4.1∎

Saturation follows because the boundary $L_{1}$ is a leaf and all block and collar points lie in the exhausting plane leaves; interval contraction conjugacy, compatible disk and annulus extension and uniform forward and inverse collar control produce the global foliated homeomorphism to the standard Reeb model by [F3]. Hence there is a Reeb component $R \subseteq M$ with $\partial R = L_{1}$ , every interior leaf of $R$ is a plane, and on the side $j$ of the nonzero $Π_{1}^{j}$ class the limit set of every sufficiently nearby displaced leaf is exactly $L_{1}$ by [F2]. The finite surface, index, spherical-stability and explicit disk-extension lemmas provide the local prerequisites, no source sentence substitutes for these constructions, and only the standing countable choice from [F4] is used.

Depends on

Used by

Novikov's Reeb component theorem Theorem

Dependency tree · two levels

51 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

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)