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A compressible leaf yields a vanishing cycle

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a $C^{2}$ cooriented codimension- one foliation of a closed oriented $3$ -manifold $M$ and let $L$ be a leaf such that the inclusion-induced homomorphism $π_{1} (L) \to π_{1} (M)$ is not injective. Then $F$ admits a vanishing cycle.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ . A $C^{2}$ cooriented codimension-one foliation $F$ of a closed oriented $3$ -manifold $M$ and a leaf $L$ whose inclusion-induced homomorphism $π_{1} (L) \to π_{1} (M)$ is not injective.

[F1]

A vanishing cycle supported on a leaf $L_{1}$ is a jointly $C^{2}$ family of loops $σ_{t}$ lying in leaves $L_{t}$ , with $[σ_{1}]$ nonzero in $π_{1} (L_{1})$ , each $σ_{t}$ null-homotopic in $L_{t}$ for $t < 1$ , and transverse trace, and it determines a nonzero class in the appropriate $Π_{1}^{j}$ . (Vanishing cycles of a codimension-one foliation).

[F2]

Under ${A C}_{ω}$ , the embedding and neighborhood retraction constructed in Relative Whitney approximation for manifold-valued maps, Facts L1 and Proof 1.1, can be fixed for the smooth ambient target. Whitney approximation for Euclidean-valued maps approximates a continuous Euclidean map uniformly on a compact disk. Finite general position for a leafwise loop supplies regular C² representatives of intrinsic leaf-loop classes.

[F3]

The generic-position supplier assumes a C² defining form, but a C² atlas supplies only C¹ forms $d z$ . Its proof still applies: singularities are critical points of $u = z \circ h$ ; collar adjustment uses a smooth positive transverse flow and continuity; interior perturbations are $u \mapsto u + ρ a \cdot x$ . The gradient is C¹, so Sard applies in equal source and target dimension two. Compactness preserves earlier nondegenerate cores. No operation differentiates the defining form twice. The cited proof therefore supplies the required genericity from a C² atlas and C¹ defining form.

Proof

technique · direct

1.1F2givenconstruct

Noninjectivity gives an essential kernel class. F2 represents it by a regular C² leaf loop $γ$ , with a continuous ambient filling. Compress that filling into a smaller concentric disk and set it equal to $γ (θ)$ on an outer radial collar, extended slightly beyond the disk. Fix the target embedding and smooth neighborhood retraction of F2. Approximate the continuous embedded filling by a smooth Euclidean map; blend it with the original C² collar map using a cutoff supported in that collar and equal to one near the boundary. Sufficiently small uniform error keeps the blend in the retraction neighborhood. Retraction gives a C² ambient disk with boundary exactly $γ$ , establishing the differentiable filling from the continuous nullhomotopy.

2.1F3step 1.1

Use the leafwise boundary adjustment and finite gradient perturbations of F3, the proof of Relative generic position for characteristic disk maps. They keep the boundary loop fixed, make its collar characteristic-regular and give finitely many nondegenerate interior centers and saddles. This verifies the C²-atlas hypotheses without assuming a C² defining form.

3.1F1step 2.1

The essential-leafwise-boundary alternative of the finite characteristic-disk supplier (A characteristic disk with essential boundary data produces a vanishing cycle) then produces a vanishing cycle in the sense of [F1]; the spanning disk is used only as a characteristic map and is not claimed to be a leafwise cap.

4.1step 3.1∎

Hence a foliation satisfying the stated hypotheses admits a vanishing cycle, and only the standing countable choice and the two cited disk suppliers were used.

Depends on

Used by

Novikov's Reeb component theorem Theorem

Dependency tree · two levels

68 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)
Mark Brittenham, Foliations and the Topology of 3-Manifolds, class 11, author-hosted lecture notes (standard reference, not scraped)
Alberto Candel and Lawrence Conlon, Foliations II, Graduate Studies in Mathematics 60, American Mathematical Society (2003) (standard reference, not scraped)