L’invariant de Godbillon-Vey. (The Godbillon-Vey invariant).

*(French)*Zbl 0707.57015
Sémin. Bourbaki, Vol. 1988/89, 41e année, Exp. No. 706, Astérisque 177-178, 155-181 (1989).

Let F be a smooth foliation defined by \(\omega =0\) on a smooth manifold M. The Godbillon-Vey invariant GV(F) is defined as the de Rham cohomology class represented by a closed 3-form \(\alpha\wedge d\alpha\), where \(\alpha\) is a 1-form on M satisfying \(d\omega =\omega \wedge \alpha\). Since C. Godbillon and J. Vey introduced this invariant in 1971, it immediately was generalized to the theory of the characteristic classes of foliations and many contributions were made by various authors. An interesting and important question, which is the main theme of this paper, has been “What is the geometric meaning of the characteristic classes of foliations, especially of the GV-invariant?”. The present paper makes a good surveying article on the progress of the research in this direction from the beginning of the study up to the recent results. To make the problem clear, the author focuses most of his discussions on the original GV-invariant and tries to make the expositions self- contained.

The paper is divided into seven sections. Following is a rough indication of the contents. In §1, starting with the definitions, the author explains the works of W. Thurston and of D. Sullivan on the geometric interpretation of the GV-invariant. §2, §3 and §4 are devoted to the exposition of the ideas and the resultsof G. Duminy’s work which was a great step in this area of research. In §2, we see how Duminy introduced the Godbillon invariant (Godbillon measure) and the Vey invariant (Vey homomorphism) obtaining a vanishing theorem on GV(F) when F is a codimension one foliation without holonomy. §3 discusses the localization of the Godbillon invariant to saturated Borel sets, i.e. Borel sets of M which are unions of leaves of F. In §4, the notion of resilient leaves, the growth and the level of leaves are introduced. Then Duminy’s vanishing theorem to the effect that the GV-invariant of a codimension one foliation without resilient leaves is zero is stated. In §5, we can see how S. Hurder generalized the method of Duminy to the cases of higher codimensions obtaining similar results in these cases. In §6 we find a theorem of A. Connes, which reveals a tight relation between GV(F) and invariant measure of the flow of weights of ‘relation’ F. The author gives an adequate account of the theorem when F is a codimension one foliation obtained from a foliated bundle structure (a suspension foliation). In the last §7, concerning the topological invariance of the GV-invariant, which is an open problem in itself, the author gives his own results and discusses the problem of extending the GV-invariant to foliations of lower differentiability and refers to recent results due to S. Hurder and A. Katok.

For the entire collection see [Zbl 0691.00001].

The paper is divided into seven sections. Following is a rough indication of the contents. In §1, starting with the definitions, the author explains the works of W. Thurston and of D. Sullivan on the geometric interpretation of the GV-invariant. §2, §3 and §4 are devoted to the exposition of the ideas and the resultsof G. Duminy’s work which was a great step in this area of research. In §2, we see how Duminy introduced the Godbillon invariant (Godbillon measure) and the Vey invariant (Vey homomorphism) obtaining a vanishing theorem on GV(F) when F is a codimension one foliation without holonomy. §3 discusses the localization of the Godbillon invariant to saturated Borel sets, i.e. Borel sets of M which are unions of leaves of F. In §4, the notion of resilient leaves, the growth and the level of leaves are introduced. Then Duminy’s vanishing theorem to the effect that the GV-invariant of a codimension one foliation without resilient leaves is zero is stated. In §5, we can see how S. Hurder generalized the method of Duminy to the cases of higher codimensions obtaining similar results in these cases. In §6 we find a theorem of A. Connes, which reveals a tight relation between GV(F) and invariant measure of the flow of weights of ‘relation’ F. The author gives an adequate account of the theorem when F is a codimension one foliation obtained from a foliated bundle structure (a suspension foliation). In the last §7, concerning the topological invariance of the GV-invariant, which is an open problem in itself, the author gives his own results and discusses the problem of extending the GV-invariant to foliations of lower differentiability and refers to recent results due to S. Hurder and A. Katok.

For the entire collection see [Zbl 0691.00001].

Reviewer: T.Mizutani

##### MSC:

57R30 | Foliations in differential topology; geometric theory |

57R32 | Classifying spaces for foliations; Gelfand-Fuks cohomology |