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74
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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From groups to groupoids: a brief survey
 Bull. London Math. Soc
, 1987
"... A groupoid should be thought of as a group with many objects, or with many identities. A precise definition is given below. A groupoid with one object is essentially just a group. So the notion of groupoid is an extension of that of groups. It gives an additional convenience, flexibility and range o ..."
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Cited by 94 (9 self)
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A groupoid should be thought of as a group with many objects, or with many identities. A precise definition is given below. A groupoid with one object is essentially just a group. So the notion of groupoid is an extension of that of groups. It gives an additional convenience, flexibility and range of applications, so that even for purely grouptheoretical work, it can be useful to take a path through the world of groupoids.
Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes
, 2000
"... In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. As a first application we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and ..."
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Cited by 86 (16 self)
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In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of 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 WeinsteinXu [47]). As a second application we extend van Est’s argument for the integrability of Lie algebras. Applied to Poisson manifolds, this immediately gives a slight improvement of HectorDazord’s integrability criterion [12]. 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 EvensLuWeinstein’s characteristic class θL [17] (hence, in particular, the modular class of Poisson manifolds), and also the classical characteristic classes of flat vector bundles
The longitudinal index theorem for foliations
 PROC. RES. INST. MATH. SCI., KYOTO UNIV
, 1984
"... In this paper, we use the bivariant K theory of Kasparov ([19]) as a basic tool to prove the Ktheoretical version of the index theorem for longitudinal elliptic differential operators for foliations which is stated as a problem in [10], Section 10. When the foliation is by the fibers of a fibration ..."
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Cited by 78 (3 self)
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In this paper, we use the bivariant K theory of Kasparov ([19]) as a basic tool to prove the Ktheoretical version of the index theorem for longitudinal elliptic differential operators for foliations which is stated as a problem in [10], Section 10. When the foliation is by the fibers of a fibration, this theorem reduces to the AtiyahSinger index theorem for families ([2], Theorem 3.1). It implies the index theorem for measured
A Homology Theory for Étale Groupoids
"... Étale groupoids arise naturally as models for leaf spaces of foliations, for orbifolds, and for orbit spaces of discrete group actions. In this paper we introduce a sheaf homology theory for etale groupoids. We prove its invariance under Morita equivalence, as well as Verdier duality between Haeflig ..."
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Cited by 38 (5 self)
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Étale groupoids arise naturally as models for leaf spaces of foliations, for orbifolds, and for orbit spaces of discrete group actions. In this paper we introduce a sheaf homology theory for etale groupoids. We prove its invariance under Morita equivalence, as well as Verdier duality between Haefliger cohomology and this homology. We also discuss the relation to the cyclic and Hochschild homologies of Connes' convolution algebra of the groupoid, and derive some spectral sequences which serve as a tool for the computation of these homologies.
On orbit equivalence of measure preserving actions
 In Rigidity in dynamics and geometry (Cambridge
, 2002
"... We give a brief survey of some classification results on orbit equivalence of probability measure preserving countable group actions. The notion of 2 Betti numbers for groups is gently introduced. An account of orbit equivalence invariance for 2 Betti numbers is presented together with a description ..."
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Cited by 27 (4 self)
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We give a brief survey of some classification results on orbit equivalence of probability measure preserving countable group actions. The notion of 2 Betti numbers for groups is gently introduced. An account of orbit equivalence invariance for 2 Betti numbers is presented together with a description of the theory of equivalence relation actions on simplicial complexes. We relate orbit equivalence to a measure theoretic analogue of commensurability and quasiisometry of groups: measure equivalence. Rather than a complete description of these subjects, a lot of examples are provided. 1 Equivalence Relations 1.1 Equivalence Relation defined by an action Let (X,µ) be a standard Borel space, where µ is a probability measure without atoms. Remember that it is Borel isomorphic to the unit interval of the reals, with Lebesgue measure. Let Γ be a countable group and α an action of Γ on (X,µ) by µpreserving Borel automorphisms. Consider the orbit equivalence relation on X: Rα = {(x, γ.x) : x ∈ X, γ ∈ Γ}. As a subset of X ×X, this equivalence relation is just the union of the graphs of the γ ∈ Γ. In this measured context, null sets are neglected. Thus the action is free if the only element of Γ with a fixedpoint set of positive measure is the identity element. Examples 1.1 The first examples to keep in mind are the following:
Foliation groupoids and their cyclic homology
 Advances of Mathematics
"... The purpose of this paper is to prove two theorems which concern the position of étale groupoids among general smooth (or ”Lie”) groupoids. Our motivation comes from the noncommutative geometry and algebraic topology concerning leaf spaces of foliations. Here, one is concerned with invariants of th ..."
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Cited by 25 (5 self)
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The purpose of this paper is to prove two theorems which concern the position of étale groupoids among general smooth (or ”Lie”) groupoids. Our motivation comes from the noncommutative geometry and algebraic topology concerning leaf spaces of foliations. Here, one is concerned with invariants of the holonomy groupoid of a foliation
Cyclic Cohomology of Étale Groupoids; The General Case
 Ktheory
, 1999
"... We give a general method for computing the cyclic cohomology of crossed products by 'etale groupoids, extending the FeiginTsyganNistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor and Tsygan for the convolution ..."
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Cited by 24 (1 self)
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We give a general method for computing the cyclic cohomology of crossed products by 'etale groupoids, extending the FeiginTsyganNistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor and Tsygan for the convolution algebra C 1 c (G) of an 'etale groupoid, removing the Hausdorffness condition and including the computation of hyperbolic components. Examples like group actions on manifolds and foliations are considered. Keywords: cyclic cohomology, groupoids, crossed products, duality, foliations. Contents 1 Introduction 3 2 Homology and Cohomology of Sheaves on ' Etale Groupoids 4 2.1 ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 \Gamma c in the nonHausdorff case : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3 Homology and Cohomology of ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : 8 3 Cyclic Homologies of Sheaves ...