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The Godbillon-Vey class of a codimension-one foliation

Definition

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a transversely oriented codimension-one foliation of a smooth manifold $M$ , let $ω$ be a nowhere-vanishing defining $1$ -form with $T F = \ker ω$ , and let $η$ be a smooth $1$ -form with $d ω = η \land ω$ , whose existence is guaranteed by Frobenius divisibility: d omega equals eta wedge omega. The Godbillon-Vey class of $F$ is $G V (F) : = [η \land d η] \in H_{d R}^{3} (M; R)$ . It is well defined: $η \land d η$ is closed (The Godbillon-Vey form eta wedge d eta is closed), and the class is unchanged by replacing $η$ by $η + f ω$ (Independence of the auxiliary form eta up to exact forms) and by rescaling $ω$ (Rescaling the defining form changes the Godbillon-Vey form by an exact form). Only the de Rham class over $R$ is named; no integral refinement is defined on this page.

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Used by

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28 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)