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Closed defining forms have vanishing Godbillon-Vey class

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a transversely oriented codimension-one foliation of a smooth manifold $M$ defined by a closed nowhere-vanishing $1$ -form $ω$ with $T F = \ker ω$ (so that $\ker ω$ is integrable by Closed constant-rank one-forms define integrable hyperplane fields). Then $G V (F) = 0$ in $H_{d R}^{3} (M; R)$ .

Facts & Assumptions

Given: A transversely oriented codimension-one foliation $F$ of a smooth manifold $M$ defined by a closed nowhere-vanishing one-form $ω$ with $T F = \ker ω$ , and the standing countable choice assumption.

[F1]

For a defining form $ω$ and a one-form $η$ with $d ω = η \land ω$ , the Godbillon-Vey class is the de Rham class $G V (F) = [η \land d η]$ . (The Godbillon-Vey class of a codimension-one foliation).

[F2]

For a transversely oriented codimension-one foliation with nowhere-vanishing defining form $ω$ there is a smooth one-form $η$ with $d ω = η \land ω$ . (Frobenius divisibility: d omega equals eta wedge omega).

Proof

technique · direct

1.1F2given

For the closed defining form $ω$ the choice $η = 0$ satisfies $d ω = 0 = 0 \land ω$ , so it is one of the forms whose existence the divisibility lemma [F2] guarantees.

2.1F1step 1.1∎

The Godbillon-Vey form of this choice is $η \land d η = 0 \land 0 = 0$ , so the class defined in [F1] is the class of the zero form, namely $G V (F) = 0$ in $H_{d R}^{3} (M; R)$ ; this applies in particular to fibre foliations of bundles over $S^{1}$ defined by pullbacks of volume forms on the circle, and no choice principle is used.

Depends on

Used by

A fibration over the circle has zero Godbillon-Vey class Example

Dependency tree · two levels

22 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

Steven Hurder and Remi Langevin, Dynamics and the Godbillon-Vey Class of C1 Foliations (complete author-hosted PDF) (standard reference, not scraped)