Homology of formal deformations of proper etale Lie groupoids
Nikolai Neumaier, Markus J. Pflaum, Hessel Posthuma, Xiang Tang
备注：
A few minor corrections have been made in the preliminaries, and Section 6.3 has been improved. The article will appear in Journal fuer die Reine und Angewandte Mathematik
In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a proper etale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it in terms of alternating multi-vector fields on the associated inertia groupoid. We introduce a noncommutative Poisson homology whose computation enables us to determine the Hochschild homology of formal deformations of the convolution algebra. Then it is shown that the cyclic (co)homology of such formal deformations can be described by an appropriate sheaf cohomology theory. This enables us to determine the corresponding cyclic homology groups in terms of orbifold cohomology of the underlying orbifold. Using the thus obtained description of cyclic cohomology of the deformed convolution algebra, we give a complete classification of all traces on this formal deformation, and provide an explicit construction.
Generalizing Karoubi's multiplicative K-theory and multiplicative cohomology groups for smooth manifolds we define secondary theories and characteristic classes for smooth etale groupoids. As special cases we obtain versions of the groups of differential characters for smooth etale groupoids and for orbifolds.
Etale 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.
We introduce the dual Roe algebras for proper étale groupoid actions and deduce the expected Higson-Roe short exact sequence. When the action is cocompact, we show that the Roe $C^*$-ideal of locally compact operators is Morita equivalent to the reduced $C^*$-algebra of our groupoid, and we further identify the boundary map of the associated periodic six-term exact sequence with the Baum-Connes map, via a Paschke-Higson map for groupoids. For proper actions on continuous families of manifolds of bounded geometry, we associate with any $G$-equivariant Dirac-type family, a coarse index class whi…
We introduce a graded homology theory for graded étale groupoids. For $\mathbb Z$-graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we prove that the graded zeroth homology group with constant coefficients $\mathbb Z$ is isomorphic to the graded Grothendieck group of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra of the covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading…
We study the connection between the Baum-Connes conjecture for an ample groupoid $G$ with coefficient $A$ and the Künneth formula for the K-theory of tensor products by the crossed product $A\rtimes_r G$. To do so we develop the machinery of Going-Down functors for ample groupoids. As an application we prove that both the uniform Roe algebra of a coarse space which uniformly embeds into a Hilbert space and the maximal Roe algebra of a space admitting a fibred coarse embedding into a Hilbert space satisfy the Künneth formula. We also provide a stability result for the Künneth formula using cont…
We investigate the homology of ample Hausdorff groupoids. We establish that a number of notions of equivalence of groupoids appearing in the literature coincide for ample Hausdorff groupoids, and deduce that they all preserve groupoid homology. We compute the homology of a Deaconu{Renault groupoid associated to k pairwisecommuting local homeomorphisms of a zero-dimensional space, and show that Matui's HK conjecture holds for such a groupoid when k is one or two. We specialise to k-graph groupoids, and show that their homology can be computed in terms of the adjacency matrices, using a chain co…
We give an example of a locally compact effective Hausdorff, minimal ample groupoid such that its rational homology differs from the $K$-theory of its reduced groupoid $C^*$-algebra. Moreover, we prove that such example satisfies Matui's AH-conjecture.
The classical Serre-Swan's theorem defines a bijective correspondence between vector bundles and finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an etale Lie groupoid and the category of modules over its convolution algebra that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita bicategory of etale Lie groupoids and the given correspondence represents a natural equivalen…
We study the Hochschild homology of the convolution algebra of a proper Lie groupoid by introducing a convolution sheaf over the space of orbits. We develop a localization result for the associated Hochschild homology sheaf, and prove that the Hochschild homology sheaf at each stalk is quasi-isomorphic to the stalk at the origin of the Hochschild homology of the convolution algebra of its linearization, which is the transformation groupoid of a linear action of a compact isotropy group on a vector space. We then explain Brylinski's ansatz to compute the Hochschild homology of the transformatio…
Let $Γ$ be a discrete finitely generated group. Let $\hat{M}\to T$ be a $Γ$-equivariant fibration, with fibers diffeomorphic to a fixed even dimensional manifold with boundary $Z$. We assume that $Γ\to \hat{M}\to \hat{M}/Γ$ is a Galois covering of a compact manifold with boundary. Let $(D^+ (θ))_{θ\in T}$ be a $Γ$-equivariant family of Dirac-type operators. Under the assumption that the boundary family is $L^2$-invertible, we define an index class in the K-theory of the cross-product algebra, $K_0 (C^0 (T)\rtimes_r Γ)$. If, in addition, $Γ$ is of polynomial growth, we define higher indeces by …
In this paper we study the Lie groupoids which appear in foliation theory. A foliation groupoid is a Lie groupoid which integrates a foliation, or, equivalently, whose anchor map is injective. The first theorem shows that, for a Lie groupoid G, the following are equivalent: - G is a foliation groupoid, - G has discrete isotropy groups, - G is Morita equivalent to an etale groupoid. Moreover, we show that among the Lie groupoids integrating a given foliation, the holonomy and the monodromy groupoids are extreme examples. The second theorem shows that the cyclic homology of convolution algebras …
Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K-groups of the (reduced) groupoid C*-algebra, provided the groupoid has torsion-free stabilizers and satisfies a strong form of the Baum-Connes conjecture. The construction is based on the triangulated category approach to the Baum-Connes conjecture developed by Meyer and Nest. We also present a few applications to topological dynamics and discuss the HK conjecture of Matui.
We prove that Matui's AH conjecture holds for graph groupoids of infinite graphs. This is a conjecture which relates the topological full group of an ample groupoid with the homology of the groupoid. Our main result complements Matui's result in the finite case, which makes the AH conjecture true for all graph groupoids covered by the assumptions of said conjecture. Furthermore, we observe that for arbitrary graphs, the homology of a graph groupoid coincides with the $K$-theory of its groupoid $C^*$-algebra.
