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 of foliation groupoids is invariant under Morita equivalence of groupoids, and we give explicit formulas. Combined with the previous results of Brylinski, Nistor and the authors, this theorem completes the computation of cyclic homology for various foliation groupoids, like the (full) holonomy/monodromy groupoid, Lie groupoids modeling orbifolds, and crossed products by actions of Lie groups with finite stabilizers. Some parts of the proof, such as the H-unitality of convolution algebras, apply to general Lie groupoids.
Since one of our motivation is a better understanding of various approaches to longitudinal index theorems for foliations, we have added a few brief comments at the end of the second section.
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.
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 Hochschild homology groups of the algebra of complete symbols on a foliated manifold $(M,F)$. The first step is to relate these groups to the Poisson homology of $(M,F)$ and of other related foliated manifolds. We then establish several general properties of the Poisson homology groups of foliated manifolds. As an example, we completely determine these Hochschild homology groups for the algebra of complete symbols on the irrational slope foliation of a torus (under some diophantine approximation assumptions). We also use our calculations to determine all residue traces on algebras…
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…
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.
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.
For a Lie groupoid G with a twisting (a PU(H)-principal bundle over G), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid. For a smooth foliated manifold with twistings on the holonomy groupoid we prove the twisted analog of Connes-Skandalis longitudinal index theorem. When the foliation is given by fibers of a fibration, our index coincides with the one recently introduced by Mathai-Melrose-Singer. We…
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.
We consider singular foliations whose holonomy groupoid may be nicely decomposed using Lie groupoids (of unequal dimension). We show that the Baum-Connes conjecture can be formulated in this setting. This conjecture is shown to hold under assumptions of amenability. We examine several examples that can be described in this way and make explicit computations of their K-theory.
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 c…
We have previously shown that the truncated Weil algebra of any Lie algebra is a Hopf-cyclic type complex with nontrivial coefficients. In this paper we apply this result to transfer the characteristic classes of transversely orientable foliations into the cyclic cohomology of the groupoid action algebra. Our result in codimension 1 matches with the only existing explicit computation done by Connes-Moscovici. In codimension 2 case, we carry out a constructive and explicit computation, by which we present the transverse fundamental class, the Godbillon-Vey class, and the other four residual cla…
We introduce the concept of Roe C*-algebra for a locally compact groupoid whose unit space is in general not compact, and that is equipped with an appropriate coarse structure and Haar system. Using Connes' tangent groupoid method, we introduce an analytic index for an elliptic differential operator on a Lie groupoid equipped with additional metric structure, which takes values in the K-theory of the Roe C*-algebra. We apply our theory to derive a Lichnerowicz type vanishing result for foliations on open manifolds.
In this paper we define K-theoretic secondary invariants attached to a Lie groupoid $G$. The K-theory of $C^*_r(G_{ad}^0)$ (where $G_{ad}^0$ is the adiabatic deformation $G$ restricted to the interval $[0,1)$) is the receptacle for K-theoretic secondary invariants. We give a Lie groupoid version of construction given by Piazza and Schick in the setting of the Coarse Geometry. Our construction directly generalises to more involved geometrical situation, such as foliations, well encoded by a Lie groupoid. Along the way we tackle the problem of producing a wrong-way functoriality between adiabati…
It is proven that Matui's AH~conjecture is true for Katsura--Exel--Pardo groupoids $\mathcal{G}_{A,B}$ associated to integral matrices $A$ and $B$. This conjecture relates the topological full group of an ample groupoid with the homology groups of the groupoid. We also give a criterion under which the topological full group $[[\mathcal{G}_{A,B}]]$ is finitely generated.
For any Lie groupoid we construct an analytic index morphism taking values in a modified $K-theory$ group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constructed in \cite{Ca2}. This allows in particular to prove a more primitive version of the Connes-Skandalis Longitudinal index Theorem for foliations, that is, an index theorem taking values in a group which pairs with Cyclic cocycles. As other application, for $D$ a $\gr-$PDO ellipt…
We study invariants associated to Smale spaces obtained from an expanding endomorphism on a (closed connected Riemannian) flat manifold. Specifically, the relevant invariants are the $K$-theory of the associated $C^*$-algebras and Putnam's homology theory for Smale spaces. The latter is isomorphic to the groupoid homology of the groupoids used to construct the $C^*$-algebras.
Given a gerbe $L$, on the holonomy groupoid $\mathcal G$ of the foliation $(M, \mathcal F)$, whose pull-back to $M$ is torsion, we construct a Connes $Φ$-map from the twisted Dupont-Sullivan bicomplex of $\mathcal G$ to the cyclic complex of the $L$-projective leafwise smoothing operators on $(M, \mathcal F)$. Our construction allows to couple the $K$-theory analytic indices of $L$-projective leafwise elliptic operators with the twisted cohomology of $B\mathcal G$ producing scalar higher invariants. Finally by adapting the Bismut-Quillen superconnection approach, we compute these higher twiste…
We construct the geometric Baum-Connes assembly map for twisted Lie groupoids, that means for Lie groupoids together with a given groupoid equivariant $PU(H)-$principle bundle. The construction is based on the use of geometric deformation groupoids, these objects allow in particular to give a geometric construction of the associated pushforward maps and to establish the functoriality. The main results in this paper are to define the geometric twisted K-homology groups and to construct the assembly map. Even in the untwisted case the fact that the geometric twisted K-homology groups and the geo…
Gysin maps and wrong way functoriality via geometric deformation groupoids
Institut de Mathématiques de Toulouse, Université Toulouse III（图卢兹数学研究所，图卢兹第三大学）；Universidad Nacional Autónoma de México, Instituto de Matemáticas, Unidad Cuernavaca（墨西哥国立自治大学，数学研究所，库埃纳瓦卡分部）
We pursue the study of local index theory for operators of Fourier-integral type associated to non-proper and non-isometric actions of Lie groupoids, initiated in a previous work. We introduce the notion of geometric cocycles for Lie groupoids, which allow to represent fairly general cyclic cohomology classes of the convolution algebra of Lie groupoids localized at isotropic submanifolds. Then we compute the image of geometric cocycles localized at units under the excision map of the fundamental pseudodifferential extension. As an illustrative example, we prove an equivariant longitudinal inde…
We view the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model. Then we define the cyclic-homology Chern-Weil homomorphism by extending the Chern-Galois character from the characters of finite-dimensional comodules to arbitrary cotraces. To reduce the cyclic-homology Chern-Weil homomorphism to a tautological natural transformation, we replace the unital coaction-invariant subalgebra by its certain natural H-unital nilpotent extension (row extension), and prove that their cyclic-homology groups are isomorphic. In t…
