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Area functionals and Godbillon-Vey cocycles

Takashi Tsuboi (1992)

Annales de l'institut Fourier

We investigate the natural domain of definition of the Godbillon-Vey 2- dimensional cohomology class of the group of diffeomorphisms of the circle. We introduce the notion of area functionals on a space of functions on the circle, we give a sufficiently large space of functions with nontrivial area functional and we give a sufficiently large group of Lipschitz homeomorphisms of the circle where the Godbillon-Vey class is defined.


Francis Sergeraert (1977/1978)

Séminaire Bourbaki

Bilipschitz invariance of the first transverse characteristic map

Michel Hilsum (2012)

Banach Center Publications

Given a smooth S¹-foliated bundle, A. Connes has shown the existence of an additive morphism ϕ from the K-theory group of the foliation C*-algebra to the scalar field, which factorizes, via the assembly map, the Godbillon-Vey class, which is the first secondary characteristic class, of the classifying space. We prove the invariance of this map under a bilipschitz homeomorphism, extending a previous result for maps of class C¹ by H. Natsume.

Characteristic classes of subfoliations

Luis A. Cordero, X. Masa (1981)

Annales de l'institut Fourier

This paper is devoted to define a characteristic homomorphism for a subfoliation ( F 1 , F 2 ) and to study its relation with the usual characteristic homomorphism for each foliation (as defined by Bott). Moreover, two applications are given: 1) the Yamato’s 2-codimensional foliation is shown to be no homotopic to F 2 in a (1,2)-codimensional subfoliation; 2) an obstruction to the existence of d everywhere independent and transverse infinitesimal transformations of a foliation F 2 is obtained, when F 2 and these...

Characteristic homomorphism for ( F 1 , F 2 ) -foliated bundles over subfoliated manifolds

José Manuel Carballés (1984)

Annales de l'institut Fourier

In this paper a construction of characteristic classes for a subfoliation ( F 1 , F 2 ) is given by using Kamber-Tondeur’s techniques. For this purpose, the notion of ( F 1 , F 2 ) -foliated principal bundle, and the definition of its associated characteristic homomorphism, are introduced. The relation with the characteristic homomorphism of F i -foliated bundles, i = 1 , 2 , the results of Kamber-Tondeur on the cohomology of g - D G -algebras. Finally, Goldman’s results on the restriction of foliated bundles to the leaves of a foliation...

Classifying toposes and foliations

Ieke Moerdijk (1991)

Annales de l'institut Fourier

For any etale topological groupoid G (for example, the holonomy groupoid of a foliation), it is shown that its classifying topos is homotopy equivalent to its classifying space. As an application, we prove that the fundamental group of Haefliger for the (leaf space of) a foliation agrees with the one introduced by Van Est. We also give a new proof of Segal’s theorem on Haefliger’s classifying space B Γ q .

Groupoïde fondamental et d'holonomie de certains feuilletages réguliers

María C. Lasso de la Vega (1989)

Publicacions Matemàtiques

Let M be a manifold with a regular foliation F. We recall the construction of the fundamental groupoid and the homotopy groupoid associated to F. We describe some interesting particular cases and give some glueing techniques. We characterize the cases where these groupoids are Hausdorff spaces.We study in particular both groupoids associated to foliations with Reeb components.

Leibniz cohomology for differentiable manifolds

Jerry M. Lodder (1998)

Annales de l'institut Fourier

We propose a definition of Leibniz cohomology, H L * , for differentiable manifolds. Then H L * becomes a non-commutative version of Gelfand-Fuks cohomology. The calculations of H L * ( R n ; R ) reduce to those of formal vector fields, and can be identified with certain invariants of foliations.

L'espace des feuilletages d'un espace analytique compact

Daniel Barlet (1987)

Annales de l'institut Fourier

Nous construisons sur l’ensemble des feuilletages (avec singulariés) d’un espace analytique compact normal une structure analytique complexe. Dans le cas faiblement kählérien, nous montrons qu’à un point frontière de la compactification naturelle de l’espace des feuilletages est encore associé un feuilletage.

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