La conjecture de Novikov pour les feuilletages hyperboliques

Tu, Jean Louis

doi:10.1023/a:1007756501903

Cited by 86 publications

(105 citation statements)

References 5 publications

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“…We will now prove the following extension of Theorem 2.4, which validates Definition 2.3 4 To construct such a section, start with any non-vanishing section, then average over G, using a cutoff function [16] as in the proof of the vanishing theorem for groupoid cohomology in [6]. …”

Section: The Differentiable Casementioning

confidence: 67%

The Volume of a Differentiable Stack

Weinstein

2009

Lett Math Phys

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Abstract. We extend the notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associated stack rather than of the groupoid itself. Since there is no natural measure in the smooth case like the counting measure in the discrete case, we need extra data to define the volume. This data has the form of an invariant section of a natural line bundle over the base of the groupoid. Invariant sections of a square root of this line bundle constitute an "intrinsic Hilbert space" of the stack. Mathematics Subject Classification (2000). Primary 58H05; Secondary 53D17.

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Section: The Differentiable Casementioning

confidence: 67%

The Volume of a Differentiable Stack

Weinstein

2009

Lett Math Phys

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“…If G has a γ-element as above, then, by work of Kasparov and Tu [17,21] (extended in [4, Theorem 1.11] to the weaker notion of a γ-element used here) the Baum-Connes assembly map is known to be split injective with image…”

Section: Examplesmentioning

confidence: 99%

Fibrations with noncommutative fibers

Echterhoff¹,

Nest²,

Oyono‐Oyono³

2009

J. Noncommut. Geom.

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Abstract. We study an analogue of fibrations of topological spaces with the homotopy lifting property in the setting of C * -algebra bundles. We then derive an analogue of the Leray-Serre spectral sequence to compute the K-theory of the fibration in terms of the cohomology of the base and the K-theory of the fibres. We present many examples which show that fibrations with noncommutative fibres appear in abundance in nature. IntroductionIn recent years the study of the topological properties of C*-algebra bundles plays a more and more prominent rôle in the field of Operator algebras. The main reason for this is two-fold: on one side there are many important examples of C*-algebras which do come with a canonical bundle structure. On the other side, the study of C*-algebra bundles over a locally compact Hausdorff base space X is the natural next step in classification theory, after the far reaching results which have been obtained in the classification of simple C*-algebras. To fix notation, by a C*-algebra bundle A(X) over X we shall simply mean a C 0 (X)-algebra in the sense of Kasparov (see [17]): it is a C*-algebra A together with a non-degenerate * -homomorphismwhere ZM(A) denotes the center of the multiplier algebra M(A) of A. For such C 0 (X)-algebra A, the fibre over x ∈ X is then A x = A/I x , where I x = {Φ(f ) · a; a ∈ A and f ∈ C 0 (X) such that f (x) = 0}, and the canonical quotient map q x : A → A x is called the evaluation map at x. We shall often write A(X) to indicate the given C 0 (X)-structure of A. We shall recall the basic constructions and properties of C 0 (X)-algebras in the preliminary section below. We refer to [8] for further notations concerning C 0 (X)-algebras.The main problem when studying bundles from the topological point of view is to provide good topological invariants which help to understand the local and global structure of the bundles. A good example is given by the class of separable continuous-trace C*-algebras, which are, up to Morita equivalence, just the section algebras of locally trivial bundles over X with fibres the compact This work was partially supported by the Deutsche Forschungsgemeinschaft (SFB 478).

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“…the appendix in [47] for the construction of such functions). We now check that the following formula defines a contraction of C * d (G; E):…”

Section: For Any G ∈ G(x Y) and Anymentioning

confidence: 99%

Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes

Crainic

2003

Comment. Math. Helv.

161

280

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Abstract. In the first section we discuss Morita invariance of differentiable/algebroid cohomology.In the second section we extend the Van Est isomorphism to groupoids. As a first application we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and conjectured in degree 2 by Weinstein-Xu [50]). As a second application we extend Van Est's argument for the integrability of Lie algebras. Applied to Poisson manifolds, this immediately implies the integrability criterion of Hector-Dazord [14].In the third section we describe the relevant characteristic classes of representations, living in algebroid cohomology, as well as their relation to the Van Est map. This extends EvensLu-Weinstein's characteristic class θ L [20] (hence, in particular, the modular class of Poisson manifolds), and also the classical characteristic classes of flat vector bundles [2,30].In the last section we describe applications to Poisson geometry. Mathematics Subject Classification (2000). 58H05, 57R20, 53D17.

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La conjecture de Novikov pour les feuilletages hyperboliques

Cited by 86 publications

References 5 publications

The Volume of a Differentiable Stack

The Volume of a Differentiable Stack

Fibrations with noncommutative fibers

Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes

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