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Novikov's Reeb component theorem

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a $C^{2}$ transversely oriented codimension-one foliation of a closed oriented $3$ -manifold $M$ . If either (a) some leaf $L$ of $F$ has non-injective inclusion-induced homomorphism $π_{1} (L) \to π_{1} (M)$ , or (b) some closed transversal $γ : S^{1} \to M$ is null-homotopic in $M$ , then $F$ contains a Reeb component (Reeb components of a codimension-one foliation). Equivalently, a Reebless $C^{2}$ transversely oriented foliation of a closed oriented $3$ -manifold has all leaves $π_{1}$ -injective and all closed transversals essential (cor-reebless-leaves-are-pi-one-injective-under-novikov-hypotheses).

Facts & Assumptions

Given: A $C^{2}$ transversely oriented codimension-one foliation $F$ of a closed oriented three-manifold $M$ , and either alternative (a) or (b) of the statement.

[F1]

A compressible leaf, that is one whose inclusion-induced homomorphism $π_{1} (L) \to π_{1} (M)$ is not injective, yields a vanishing cycle (A compressible leaf yields a vanishing cycle, The homomorphism on fundamental groups induced by a pointed continuous map, Vanishing cycles of a codimension-one foliation).

[F2]

A closed transversal that is null-homotopic in $M$ yields a vanishing cycle (A null-homotopic closed transversal yields a vanishing cycle).

[F3]

A vanishing cycle produces a compact leaf $L_{1}$ which bounds a Reeb component: there is a compact saturated submanifold $R$ diffeomorphic to $D^{2} \times S^{1}$ with $\partial R = L_{1}$ , every interior leaf a plane, foliated-homeomorphic to the standard Reeb component (A nonzero limitwise-nullhomotopy class yields a compact boundary leaf, The compact leaf produced by a vanishing cycle bounds a Reeb component, Reeb components of a codimension-one foliation).

[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

1.1F1F2given

In branch (a) [F1] supplies a vanishing cycle from the non-injective leaf inclusion; in branch (b) [F2] supplies a vanishing cycle from the null-homotopic closed transversal. The two branches are independent and cover the two hypotheses of the theorem.

2.1F3step 1.1

Either vanishing cycle, together with its leafwise family and characteristic structure, satisfies the hypotheses of the compact-leaf-and-Reeb-component result [F3]: applying A nonzero limitwise-nullhomotopy class yields a compact boundary leaf produces a compact boundary leaf $L_{1}$ , and applying The compact leaf produced by a vanishing cycle bounds a Reeb component produces a compact saturated solid torus $R$ with $\partial R = L_{1}$ whose foliation is foliated-homeomorphic to the standard Reeb model, so $R$ is a Reeb component of $F$ in the sense of the definition.

3.1F3F4step 2.1∎

Therefore both alternatives (a) and (b) force the existence of a Reeb component; equivalently, a Reebless $C^{2}$ transversely oriented foliation of a closed oriented three-manifold has all leaf inclusions $π_{1}$ -injective and all closed transversals essential, the equivalence being the contrapositive of the two alternatives. The extra $Π$ -side limit-set clause of the Reeb-component construction is unnecessary for this existence conclusion, and the proof consumes only the two branch suppliers, one vanishing-cycle chain and the standing countable choice from [F4].

Depends on

Used by

Dependency tree · two levels

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