Foliation Groupoids and Their Cyclic Homology

By M. Crainic, Ieke Moerdijk. 2000

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The purpose of this paper is to prove two theorems which concern the position of etale 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 such as the cohomology of its classifying space the cyclic homology of its smooth convolution algebra or the Ktheory of the Cconvolution algebras Many results here depend on the fact that such a holonomy groupoid can be reduced to what is called a complete transversal of the foliation giving rise to an equivalent etale groupoid For etale groupoids sometimes called rdiscrete groupoids in the literature the cyclic homology sheaf theory and classifying spaces are each well understood as is the relation between these

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e50bd8e7631de4f97bd47a01a77c08a16e6e1361

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Orbifolds as groupoids: an introduction (2002, unknown): “roper. Moreover, the space S G is in fact a manifold. This is clear from the local charts (3.4) in the case Gis ´etale, and it follows for general orbifold groupoids by considering local charts as in [CM1]. The space S G has the structure of a G-space, with the action deﬁned by conjugation. Let Λ(G)”
The transverse index theorem for proper cocompact actions of Lie groupoids (2013, background): “ve for principal G-bundles is connected to the index theorems of Connes, c.f. [Co, Sec. III.7.γ], and Gorokhovsky–Lott, c.f. [GoLo2]. For this we assume G to be a foliation groupoid. This means, c.f. [CrMo] that the anchor of its Lie algebroid ρ: A→ TMis injective, and its image deﬁnes the foliation”
Riemannian metrics on Lie groupoids (2014, background): “Alternative characterizations of foliations groupoids are: (i) a Lie groupoid which is Morita equivalent to an étale groupoid, or (ii) a Lie groupoid whose Lie algebroid has injective anchor map (see [6]).”
Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes (2000, background): “f they have the same space of orbits (i.e. the same transversal geometry). This notion is important in the theory of foliations (where groupoids are often replaced by Morita equivalent ones; see e.g. [13, 27]), non-commutative geometry (recall that the C∗-algebras deﬁned by two Morita equivalent gro”