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Reebless leaves are pi-one-injective and transverse loops are essential

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a $C^{2}$ transversely oriented codimension-one foliation of a closed oriented $3$ -manifold $M$ containing no Reeb component. Then: (i) for every leaf $L$ of $F$ the inclusion-induced homomorphism $π_{1} (L) \to π_{1} (M)$ is injective; and (ii) every closed transversal to $F$ represents a nontrivial class in $π_{1} (M)$ .

Facts & Assumptions

Given: A $C^{2}$ transversely oriented codimension-one foliation $F$ of a closed oriented three-manifold $M$ with no Reeb component (Reeb components of a codimension-one foliation).

[F1]

If some leaf has non-injective inclusion-induced homomorphism $π_{1} (L) \to π_{1} (M)$ , or some closed transversal is null-homotopic in $M$ , then $F$ contains a Reeb component (Novikov's Reeb component theorem).

[F2]

The inclusion-induced homomorphism on fundamental groups is defined by $π_{1}$ -functoriality (The homomorphism on fundamental groups induced by a pointed continuous map, Based loops and the fundamental group).

[F3]

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

If a leaf inclusion $π_{1} (L) \to π_{1} (M)$ were not injective, alternative (a) of [F1] would produce a Reeb component in $F$ , contradicting Reeblessness; hence every leaf inclusion is injective.

1.2F1F2given

If a closed transversal were null-homotopic in $M$ , alternative (b) of [F1] would produce a Reeb component in $F$ , again contradicting Reeblessness; hence every closed transversal represents a nontrivial class in $π_{1} (M)$ .

2.1F1F3step 1.1step 1.2∎

Both asserted conclusions therefore hold under the same $C^{2}$ , coorientation, closedness and ${A C}_{ω}$ hypotheses, the proof being the two contrapositives of Novikov's Reeb component theorem and consuming only the standing countable choice from [F3].

Depends on

Used by

Dependency tree · two levels

26 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

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