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24
Differentiable and algebroid cohomology, van Est . . . 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 49 (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
Twisted Ktheory of differentiable stacks
 ANN. SCI. ÉCOLE NORM. SUP
, 2004
"... In this paper, we develop twisted Ktheory for stacks, where the twisted class is given by an S 1gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framew ..."
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Cited by 42 (12 self)
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In this paper, we develop twisted Ktheory for stacks, where the twisted class is given by an S 1gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framework for studying various twisted Ktheories including the usual twisted Ktheory of topological spaces, twisted equivariant Ktheory, and the twisted Ktheory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted Kgroups can be expressed by socalled “twisted vector bundles”. Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of Ktheory (KKtheory) of C ∗algebras.
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 17 (6 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
Étale groupoids and their quantales
, 2004
"... We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantal ..."
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Cited by 16 (7 self)
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We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic étale groupoids and their quantales, which are given a rather simple characterization and are here called inverse quantal
Local index theory over etale groupoids
 J. Reine Angew. Math
"... Abstract. We give a superconnection proof of Connes ’ index theorem for proper cocompact actions of étale groupoids. This includes Connes ’ general foliation index theorem for foliations with Hausdorff holonomy groupoid. 1. ..."
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Cited by 13 (2 self)
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Abstract. We give a superconnection proof of Connes ’ index theorem for proper cocompact actions of étale groupoids. This includes Connes ’ general foliation index theorem for foliations with Hausdorff holonomy groupoid. 1.
Gerbes over orbifolds and twisted Ktheory
"... Abstract. In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H 3 (X) via ChernWeil theory. For an arbitrary gerbe L, a twisting L Korb(X) of the orbifold Ktheory of X is constru ..."
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Cited by 9 (2 self)
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Abstract. In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H 3 (X) via ChernWeil theory. For an arbitrary gerbe L, a twisting L Korb(X) of the orbifold Ktheory of X is constructed, and shown to generalize previous twisting by Rosenberg [28], Witten [35], AtiyahSegal [2] and Bowknegt et. al. [4] in the smooth case and by AdemRuan [1] for discrete torsion on an orbifold. Contents
Linearization of Regular Proper Groupoids
, 2001
"... Let G be a Lie groupoid over M such that the targetsource map from G to M × M is proper. We show that, if O is an orbit of finite type (i.e. which admits a proper function with finitely many critical points), then the restriction GU of G to some neighborhood U of O in M is isomorphic to a similar ..."
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Cited by 9 (0 self)
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Let G be a Lie groupoid over M such that the targetsource map from G to M × M is proper. We show that, if O is an orbit of finite type (i.e. which admits a proper function with finitely many critical points), then the restriction GU of G to some neighborhood U of O in M is isomorphic to a similar restriction of the action groupoid for the linear action of the transitive groupoid GO on the normal bundle NO. The proof uses a deformation argument based on a cohomology vanishing theorem, along with a slice theorem which is derived from a new result on submersions with a fibre of finite type.
CechDe Rham theory for leaf spaces of foliations
"... This paper is concerned with characteristic classes in the cohomology of leaf spaces of foliations. For a manifold M equipped with a foliation F it is wellknown that the coarse (naive) leaf space M/F, obtained from M by identifying each leaf to a point, ..."
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Cited by 9 (3 self)
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This paper is concerned with characteristic classes in the cohomology of leaf spaces of foliations. For a manifold M equipped with a foliation F it is wellknown that the coarse (naive) leaf space M/F, obtained from M by identifying each leaf to a point,