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Godbillon-Vey invariance under smooth foliated concordance

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $M$ be a closed smooth manifold and let $F_{0}, F_{1}$ be smoothly foliated-concordant transversely oriented codimension-one foliations of $M$ (Smooth foliated concordance of codimension-one foliations). Then $G V (F_{0}) = G V (F_{1})$ in $H_{d R}^{3} (M; R)$ .

Facts & Assumptions

Given: A closed smooth manifold $M$ , smoothly foliated-concordant transversely oriented codimension-one foliations $F_{0}, F_{1}$ of $M$ , a concordance $(G, ω)$ on $W = M \times [0, 1]$ , and the standing countable choice assumption.

[F1]

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

[F2]

A codimension-one foliation of a manifold with boundary transverse to the boundary restricts to a codimension-one foliation of the boundary, and if $d ω = η \land ω$ then $d (ω ∣_{\partial W}) = (η ∣_{\partial W}) \land (ω ∣_{\partial W})$ . (Restriction of a foliation transverse to the boundary).

[F3]

Smoothly homotopic maps induce the same map on de Rham cohomology. (Smoothly homotopic maps induce the same de rham map).

[F4]

The de Rham complex, wedge identities and natural pullback extend to smooth manifolds with boundary (The de Rham complex and pullback extend to manifolds with boundary).

Proof

technique · direct

1.1F1given

Let $(G, ω)$ be the concordance on $W = M \times [0, 1]$ and choose $η$ with $d ω = η \land ω$ by the divisibility lemma, so that $[η \land d η] = G V (G) \in H_{d R}^{3} (W; R)$ according to [F1].

2.1F2step 1.1

By [F2] the inclusions $i_{j} : M \to W$ , $i_{j} (x) = (x, j)$ , pull the defining data back to defining data of $F_{j}$ : $i_{j}^{*} ω$ is a defining form for $F_{j}$ and $d (i_{j}^{*} ω) = (i_{j}^{*} η) \land (i_{j}^{*} ω)$ , so $i_{j}^{*} (η \land d η) = (i_{j}^{*} η) \land d (i_{j}^{*} η)$ represents $G V (F_{j})$ , that is $i_{j}^{*} G V (G) = G V (F_{j})$ .

3.1F3F4step 2.1∎

For completeness the endpoint equality holds on the boundary cylinder as follows. Write the closed form $α = η \land d η$ on $M \times [0, 1]$ as $α_{s} + d s \land γ_{s}$ . By [F4], $d α = 0$ gives $\partial_{s} α_{s} = d_{M} γ_{s}$ . Integrating the smooth coefficients in s yields $i_{1}^{*} α - i_{0}^{*} α = d_{M} \int_{0}^{1} γ_{s} d s$ . Thus the endpoint forms represent the same de Rham class, and step 2.1 identifies them with the two Godbillon–Vey classes. This is the homotopy identity of [F3], here derived explicitly on the cylinder.

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Sources

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