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Morphismes K-orientés d'espaces de feuilles et fonctorialité en théorie de Kasparov (d'après une conjecture d'A. Connes)
... In the context of noncommutative differential geometry, these operators appeared in [6] to provide examples of Fredholm modules, associated with foliated manifolds. Namely, it was proved there that any zeroth order transversally elliptic operator with the holonomy invariant transversal principal symbol gives rise to a finitedimensional Fredholm module over the foliation algebra C ∞ c (G F ) (see also [7,17]). Theorem 1 provides an extension of the above mentioned result to the case of transversally elliptic operators of positive order. ...
... Since R j (λ) ∈ Ψ 0 (M, E) Ψ −∞ (N * F , E), j = 1, 2, by (17), R j defines a continuous map from H t,−t (M, F , E) to H s,−s (M, F , E) for any s and t, that implies the same is true for T (λ). The desired norm estimates for operators P (λ) and T (λ) follow immediately from the symbol estimates (23) and (22). ...
... Since k ∈ OP −n/q (F ), by Proposition 4, the first term, R E (k)P , in the right-hand side of (33) defines a continuous mapping from H s,t (M, F , E) to H s+1,t+n/q (M, F , E) for any s and t. By (17), the operator R E (k)R 2 A −1 defines a continuous mapping from L 2 (M, E) to H N,n/q−N (M, F , E) for any N . So we get that the operator R E (k)A −1 defines a continuous map from L 2 (M, E) to H 1,p/q (M, F , E). ...
We construct spectral triples in a sense of noncommutative differential geometry, associated with a Riemannian foliation on a compact manifold, and describe its dimension spectrum.
... Let (ΠX , ΛX ) be any quasi-Poisson structure on Γ1[X] ∼ = Γ2[X]⇒X representing the class (19). By construction, we have P ois(φ1)(ΛX ⊕ ΠX ) = Λ1 ⊕ Π1 and P ois(φ2)(ΛX ⊕ ΠX ) = Λ2 ⊕ Π2 . ...
... We will refer to anchor and multiplication maps as the structure maps of X. A generalized morphism where X is also a left Γ1-torsor, i.e. the left Γ1-action on ϕ2 : X → M2 is principal is referred to as a Lie groupoid bitorsor (see [19]). Definition 4.13. ...
... The results described below are completely analogous to the Lie groupoid case and can be proved in the same way (see [19]). Let E1 ← Z1 → E2 and E2 ← Z2 → E3 be generalized VB-morphisms from V1⇒E1 to V2⇒E2 and from V2⇒E2 to V3⇒E3 respectively. ...
The purpose of this paper is to investigate shifted (+1) Poisson structures in context of differential geometry. The relevant notion is shifted (+1) Poisson structures on differentiable stacks. More precisely, we develop the notion of Morita equivalence of quasi-Poisson groupoids. Thus isomorphism classes of (+1) Poisson stack correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following programs of independent interests: (1) We introduce a -graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its homotopy equivalence class is invariant under Morita equivalence of Lie groupoids, thus can be considered as polyvector fields on the corresponding differentiable stack . It turns out that shifted (+1) Poisson structures on correspond exactly to elements of the Maurer-Cartan moduli set of the corresponding dgla. (2) We introduce the notion of tangent complex and cotangent complex of a differentiable stack in terms of any Lie groupoid representing . They correspond to homotopy class of 2-term homotopy -modules and , respectively. We prove that a (+1)-shifted Poisson structure on a differentiable stack , defines a morphism .
... There is an equivalent definition ( [22], [14], [27]) of the analytical index for a foliated manifold (M, F ) using the Connes tangent groupoid 1] of the holonomy groupoid G M . The definition works for any Lie groupoid G . ...
... The definition works for any Lie groupoid G . Let AG be the associated Lie algebroid which can be viewed as a Lie groupoid given by the vector bundle structure, then one has the following exact sequence ( [22]) ...
... As we mentioned, the analytic index morphism associated to a Lie groupoid arises naturally from a geometric construction, that of the Connes tangent groupoid, [22,14,27,29]. This groupoid encodes the deformation of the groupoid to the Lie algebroid. ...
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 construct the pushforward map in twisted K-theory associated to any smooth (generalized) map and a twisting on the holonomy groupoid M/F, next we use the longitudinal index theorem to prove the functoriality of this construction. We generalize in this way the wrong way functoriality results of Connes-Skandalis when the twisting is trivial and of Carey-Wang for manifolds.
... [1,6,19,25] We follow Antonini, Azzali, Skandalis [1] definition of KK-theory with real coefficients. By using their work on the α-inavraints [1], we construct an element in the equivariant KK theory of the groupoid U n ⋊ U δ n defined by Kasparov [27] and Le Gall [30] which when pulled back by the classifying map (seen as a generalised homomorphism in the sense of Hilsum-Skandalis [21]) of a trivialised unitary flat vector bundle f : M → U n ⋊ U δ n gives the relative α-invariant. ...
... Where the first map is of degree 0 and the second is of degree 1. Functoriality and Morita equivalence hold that is if f : G → G ′ is a generalised morphism, in the sense of [21], of groupoids then ...
... See[21] for the definition of a generalised morphism. ...
For a unitary representation of the fundamental group of a compact smooth manifold, Atiyah, Patodi, Singer defined the so called alpha-invariant of the representation using Chern-Simons invariants. In this article using traces on C*-algebras, we define intrinsically(i.e without using Chern character) an element in KK-theory with real coefficients theory whose pullback by the representation is the alpha-invariant.
... Here groupoids are assumed to be Lie groupoids although most of the discussions can be easily adapted to general locally compact groupoids. Let us recall the definition below [33,38,51]. ...
... We now return to the diagram (38), and show that V is well-defined. We first need two preliminary lemmas. ...
... Finally, we prove that all maps in (38) are isomorphisms. For i and V ′ , this follows from Theorem 5.28 and Corollary 5.39. ...
In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an -gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure are derived. Our approach provides a uniform framework for studying various twisted K-theories including the usual twisted K-theory of topological spaces, twisted equivariant K-theory, and the twisted K-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted K-groups can be expressed by so-called "twisted vector bundles". Our approach is to work on presentations of stacks, namely \emph{groupoids}, and relies heavily on the machinery of K-theory (KK-theory) of -algebras.
... The proof uses the normal groupoid of Hilsum and Skandalis [9] (also cf. [33,15]), re-interpreted in terms of the Lie algebroid. ...
... [33,15]), re-interpreted in terms of the Lie algebroid. We recall the definition; our construction of the smooth structure is different from the one in [9]. The essence is to regard the vector bundle G as a Lie groupoid under addition in each fiber, and glue it to G so as to obtain a new Lie groupoid containing both G and G. ...
... Furthermore, one has C * r (G N )/I 0 ≃ C * r (G) ≃ C 0 (G * ); the second isomorphism is established by the fiberwise Fourier transform (20) below (also cf. [9,2]). Hence Prim(C * (G N )/I 0 ) ≃ G * . ...
For any Lie groupoid G, the vector bundle dual to the associated Lie algebroid g is canonically a Poisson manifold. The (reduced) C*-algebra of G (as defined by A. Connes) is shown to be a strict quantization (in the sense of M. Rieffel) of . This is proved using a generalization of Weyl's quantization prescription on flat space. Many other known strict quantizations are a special case of this procedure; on a Riemannian manifold, one recovers Connes' tangent groupoid as well as a recent generalization of Weyl's prescription. When G is the gauge groupoid of a principal bundle one is led to the Weyl quantization of a particle moving in an external Yang-Mills field. In case that G is a Lie group (with Lie algebra g) one recovers Rieffel's quantization of the Lie-Poisson structure on . A transformation group C*-algebra defined by a smooth action of a Lie group on a manifold Q turns out to be the quantization of the semidirect product Poisson manifold defined by this action.
... Finally, let us explain the link between our constructions and the wrong way functoriality maps defined by Hilsum and Skandalis for foliations in [HS87] and recently extended in [Ze23] to exact sequences associated with adiabatic extension of pseudodifferential operators on smooth manifolds. According to [HS87], we need to use the maximal crossed product completions¸m ax Γ. ...
... Finally, let us explain the link between our constructions and the wrong way functoriality maps defined by Hilsum and Skandalis for foliations in [HS87] and recently extended in [Ze23] to exact sequences associated with adiabatic extension of pseudodifferential operators on smooth manifolds. According to [HS87], we need to use the maximal crossed product completions¸m ax Γ. If we assume that X is a smooth closed even dimensional oriented manifold and that Γ acts by smooth orientation preserving diffeomorphisms, then for any smooth closed manifold M with π 1 pM q " Γ, we may consider the usual suspension of this action, say the foliated bundle V " Ă MˆΓ X Ñ M where Ă M is the universal cover of M , with the leaves being given by the projections of the submanifolds Ă Mˆtxu, so are transverse to the fibers. ...
... The monodromy groupoid of the foliation of V is then Morita equivalent to our groupoid G " X¸Γ, while the monodromy groupoid of the trivial foliation of V with one leaf is Morita equivalent to Γ 0 . According to [HS87], which can easily be extended to monodromy as opposed to holonomy, we have a well defined Hilsum-Skandalis class in KKpCpXq¸m ax Γ, Cm ax pΓ 0 qq representing Connes' transverse fundamental class in K-theory and precisely well defined since we are using the maximal completion CpXq¸m ax Γ. Using the functoriality morphism Cm ax pΓ 0 q Ñ Cm ax pΓq we can pushforward this class to get a class, denoted rV {F s in KKpCpXq¸m ax Γ, Cm ax pΓqq. ...
Given a finitely generated discrete group {\Gamma}, we construct for any admissible crossed product completion and for any metrizable finite dimensional compact {\Gamma}-space X, a universal Higson-Roe six-term exact sequence for the transformation groupoid X\rtimes {\Gamma}. In particular, we generalize the maximal Higson- Roe sequence to such groupoids. In the case where the groupoid X\rtimes {\Gamma} satisfies the rectified Baum-Connes conjecture, this yields some rigidity consequences.
... Connes' tangent groupoid was an inspiration for many papers (cf. [61,91,94]...) where this idea was generalized to various geometric contexts. Its natural setting is the deformation to the normal cone (DNC) construction. ...
... One can associate useful distributions to much more general symbols. In [61] were used symbols of type (ρ, δ) and the associated pseudodifferential operators. ...
... Polyhomogeneous symbols, i.e. the ones considered above, are particular cases of symbols of type (1, 0). These symbols of type (ρ, δ) were used in [61] in order to construct holonomy almost invariant transversally elliptic operators on any foliation, i.e. holonomy invariant up to lower order. Restricting to a transversal this amounts to finding operators on a manifold almost invariant under the action of a (pseudo)group Γ. Thanks to the work of Connes ([31]), one may assume that the (tangent) bundle E has an invariant subbundle F which has an invariant euclidean metric as well as the quotient E/F . ...
... The second part details the restriction of symbols to the transverse part in the setting of filtered calculus which allows to define a "transversally Rockland" condition. Finally we give two constructions of the transverse cycle, one from the symbol and using KK-theoretic tools and the other one more analytic, similar to the one in [18] using a pseudodifferential operator instead of its symbol and we prove the equality between the two KK-classes, yielding a Poincare duality type result. ...
... We thus need to extend the algebra of symbols to the one of symbols "bounded" at infinity. More precisely, following [18], let ...
... In this section we give a more explicit construction of the cycle representing j Hol(H 0 ) ([σ]) ⊗ ind hol H with H-pseudodifferential operators. We first recall their definition and then follow the reasoning of [18] appendix A to show that a H-pseudodifferential operator with symbol σ (which, as in the previous section, is transversally Rockland) defines a KK-cycle whose class is j Hol (Hol(H 0 )), C). ...
We use filtrations of the tangent bundle of a manifold starting with an integrable subbundle to define transverse symbols to the corresponding foliation, define a condition of transversally Rockland and prove that transversally Rockland operators yield a K-homology class. We construct an equivariant KK-class for transversally Rockland transverse symbols and show a Poincare duality type result linking the class of an operator and its symbol.
... In the present article we cope the task of putting the ad hoc construction given in [15] into the more systematic setup offered by [14]. In this classical paper, Hilsum and Skandalis provide the construction of a KK-element associated with a K-oriented map of foliated manifolds and they proved the functoriality (namely what we are referring to here with interior product formulas) of their construction for the index classes associated with longitudinal operators. ...
... The focus of this paper is to construct a mapping of the exact sequence (1.2) associated with G to the one associated with H, isomorphic to the classical Higson-Roe exact sequence as follows the middle term of (1.2). Moreover, whereas the lower shriek maps of Hilsum and Skandalis where defined only for ameanable groupoids by means of KK-elements, our maps are defined for general general groupoids: this is only due to the fact that we can implement these mapping through asymptotic morphisms in E-theory whereas, at the time when [14] was written, E-theory did not exist yet. ...
... The paper is organized as follows: in Section 2 we introduce the elementary definitions and notations about Lie groupoids and their C*-algebras; in Section 3 the elementary KK-theoretic tools are recalled: the KK-theory of mapping cone C*-algebras, the definition of the bimodules implementing the Bott periodicity, the Thom isomorphisms in KK-theory defined in [14], the asymptotic morphism associated with a deformation groupoid; in Section 4 we recall the definitions of the primary invariants, the secondary invariants and the fundamental classes as elements of the first, the second and the third K-theory group in (1.2) respectively; Section 5 is dedicated to the construction of the elements (1.6) both in the case of an isometric and an almost isometric immersion of foliations; Section 6 is devoted to set the previous constructions in the equivariant setting and to provide a precise relation between equivariant objects with associated non-equivariant objects over quotients; in Section 7 we give the proof of the main results of the paper, namely the interior product formulas, and of the functoriality of -classes for general surjectiveétale maps of foliation groupoids; finally, in Section 8 we list the problems which remain unsolved and we propose further directions of research starting from this paper. ...
Let be a suitably oriented inclusion of foliations over a manifold M, then we extend the construction of the lower shriek maps given by Hilsum and Skandalis to adiabatic deformation groupoid C*-algebras: we construct an asymptotic morphism , where G and H are the monodromy groupoids associated with and respectively. Furthermore, we prove an interior Kasparov product formula for foliated -classes associated with longitudinal metrics of positive scalar curvature in the case of Riemannian foliated bundles.
... In this case, F := G (0) \ O is a saturated closed subset of G (0) and the restriction of functions induces a surjective morphism r F from C * (G) to C * (G| F ). Moreover, according to [10], the following sequence of C * -algebras is exact : ...
... We explain here a classical construction [5,10]. A smooth groupoid G is called a deformation groupoid if : ...
We associate to a pseudomanifold X with isolated singularities a differentiable groupoid G which plays the role of the tangent space of X. We construct a Dirac element D and a Dual Dirac element such that D and induce a Poincar{\'e} duality in K-theory between the -algebras C(X) and .
... 1.6 Bundles. (see, for example, [4,18,22,30,32,33]) Let B be a "base" space, and G anétale groupoid. A left G-bundle over B consists of a space P , a map π : P −→ B, and a left action of G on P (see 1.3.6) ...
... It follows from this description and the corresponding fact for spaces that ϕ ! maps the exact sequence (22) into an exact sequence: ...
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.
... Following [20] and [24], we associate with such family of G-transversely elliptic operators an index class, now living in the bivariant Kasparov group KK(C * G, C(B)). For a given elliptic operator Q on the base B and using results from [17], we compute the Kaspavov product of our index class with the K-homology class of Q by exploiting the convenient description of the index class through the morphism induced from the representation ring R(G) to the topological K-theory K(B). This latter intersection product computation is crucial to define a distributional index à la Atiyah for families of G-transversally elliptic operators, and to prove a Berline-Vergne formula for families. ...
... This was not rigorously needed in the previous sections. We compute now the Kasparov product of the index class of family of G-transversally elliptic operators with the K-homology class of an elliptic operator on the base B. We begin by recalling some definitions and results from [17]. ...
We define and study the index map for families of G-transversally elliptic operators and introduce the multiplicity for a given irreducible representation as a virtual bundle over the base of the fibration. We then prove the usual axiomatic properties for the index map extending the Atiyah-Singer results [1]. Finally, we compute the Kasparov intersection product of our index class against the K-homology class of an elliptic operator on the base. Our approach is based on the functorial properties of the intersection product, and relies on some constructions due to Connes-Skandalis and to Hilsum-Skandalis.
... In order to define its smooth structure, we fix an exponential map, which is a diffeomorphism θ from a neighbourhood V of the zero section given by (m, ξ, t) → (θ(m, tξ), t), for t = 0, and by (m, ξ, 0) → (m, ξ, 0), for t = 0, is a diffeomorphism. One can verify that the transition map on the overlap of these two open sets is smooth, see for instance [22]. ...
... In this appendix we are going to recall some basic construction from [15,22]. Let G ⇒ X be a topological groupoid. ...
In this paper we define K-theoretic secondary invariants attached to a Lie groupoid G. The K-theory of (where 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 adiabatic deformation groupoid K-groups with respect to transverse maps. This extends the construction of the lower shriek map given by Connes and Skandalis. Moreover we attach a secondary invariant to the two following operators: the signature operator on a pair of homotopically equivalent Lie groupoids; the Dirac operator on a Lie groupoid equipped with a metric that has positive scalar curvature s-fiber-wise. Furthermore we prove a Lie groupoid version of the Delocalized APS Index Theorem of Piazza and Schick. Finally we give a product formula for the secondary invariants and we state stability results about cobordism classes of Lie groupoid structures and bordism classes of Lie groupoid metric with positive scalar curvature along the s-fibers. This is the revised version accepted by Advances in Mathematics.
... Equivalence bibundles are a particular case of the notion of generalized morphisms. In the context of foliations and Lie groupoids this last notion seemed to appear first in the work of Hilsum and Skandalis, [3]. ...
Let p be a prime number and the p-solenoid. For we consider in this paper a naturally associated action groupoid whose algebra is a model for the noncommutative solenoid studied by Latremoli\`ere and Packer. Following the geometric ideas of Connes and Rieffel to describe the Morita equivalences of noncommutative torus using the Kronecker foliation on the torus, we give an explicit description of the geometric/topologic equivalence bibundle for groupoids and whenever are in the same orbit of the action by linear fractional transformations. As a corollary, for as above we get an explicit description of the imprimitivity bimodules for the associated noncommutative solenoids.
... . 8 The idea later resurfaced in various incarnations in the study of foliations and generalised (Morita) morphisms of groupoids, in particular in the works of Hilsum and Skandalis [HiS83,HiS87] (leading to the related notion of Hilsum-Skandalis maps), and Haefliger [H84]. ...
We present a novel generalisation of principal bundles -- principaloid bundles: These are fibre bundles where the typical fibre is the arrow manifold G of a Lie groupoid and the structure group is reduced to the latter's group of bisections. Each such bundle canonically comes with a bundle map to another fibre bundle F over the base B, with typical fibre M. Examples of principaloid bundles include ordinary principal -bundles, obtained for , bundles associated to them, obtained for action groupoids , and general fibre bundles if G is a pair groupoid. While is far from being a principal G-bundle, we prove that D is one. Connections on the principaloid bundle are thus required to be G-invariant Ehresmann connections. In the three examples mentioned above, this reproduces the usual types of connection for each of them. In a local description over a trivialising cover of B, the connection gives rise to Lie algebroid-valued objects living over bundle trivialisations of F. Their behaviour under bundle automorphisms, including gauge transformations, is studied in detail. Finally, we construct the Atiyah-Ehresmann groupoid which governs symmetries of P, this time mapping distinct D-fibres to one another in general. It is a fibre-bundle object in the category of Lie groupoids, with typical fibre and base . We show that those of its bisections which project to bisections of its base are in a one-to-one correspondence with automorphisms of .
... A ⋆-diagram M ← P → M ′ promotes a Hausdorff-Morita equivalence between the G-varieties M and M ′ , as firstly remarked in [11] and proved in [15]. We hope our geometric realizations shed light on the study of Lie groupoid Morita equivalence via Hilsum-Skandalis maps [13], [16]. This specifies the study of leaf spaces [17], and in the particular case of Riemannian foliations, the isospectrality question for the basic Laplacian is rather natural [18]. ...
This paper explores the existence and properties of basic eigenvalues and eigenfunctions associated with the Riemannian Laplacian on closed, connected Riemannian manifolds featuring an effective isometric action by a compact Lie group. Our primary focus is on investigating the potential existence of homeomorphic yet not diffeomorphic smooth manifolds that can accommodate invariant metrics sharing common basic spectra. We establish the occurrence of such scenarios for specific homotopy spheres and connected sums. Moreover, the developed theory demonstrates that the ring of invariant admissible scalar curvature functions fails to recover the smooth structure in many examples. We show the existence of homotopy spheres with identical rings of invariant scalar curvature functions, irrespective of the underlying smooth structure.
... Remark 5.9. Hilsum and Skandalis in [31] define equivariant K-orientability of a smooth but potentially non-isometric group Γ action on a smooth manifold X, using the metaplectic group in place of the spin group, and they construct a corresponding analytic morphism ...
This paper studies the K-homology of a crossed product of a discrete group acting smoothly on a manifold, with a better understanding of the noncommutative geometry of the crossed-product as the primary goal, and the Baum-Connes apparatus as the main tool. Examples suggest that the correct notion of the `Dirac class' of such a noncommutative space is the image under the equivalence determined by Baum-Connes of the fibre of the fibration of the Borel space associated to the action and a smooth model for the classifying space of the group. We give a systematic study of such fibre, or `Dirac classes,' with applications to the construction of interesting spectral triples and computation of their K-theory functionals, and we prove in particular that both the well-known deformation of the Dolbeault operator on the noncommutative torus, and the class of the boundary extension of a hyperbolic group, are both Dirac classes in this sense and therefore can be treated topologically in the same way.
... associated to any (K-oriented) smooth map f : X → Y between smooth manifolds. It is an element in the bivariant K-theory KK(C(X), C(Y )) of Kasparov [23] and is defined using the principal symbol of a suitably defined pseudodifferential operator of order 0. We refer to [8,11] (and [20] for the general case of maps between foliations) for full details. For the special case that Y is a point the shriek map is the fundamental class [X] of the manifold in the K-homology group KK(C(X), C). ...
We establish the factorization of Dirac operators on Riemannian submersions of compact spin manifolds in unbounded KK-theory. More precisely, we show that the Dirac operator on the total space of such a submersion is unitarily equivalent to the tensor sum of a family of Dirac operators with the Dirac operator on the base space, up to an explicit bounded curvature term. Thus, the latter is an obstruction to having a factorization in unbounded KK-theory. We show that our tensor sum represents the bounded KK-product of the corresponding KK-cycles and connect to the early work of Connes and Skandalis.
... These groups are the right receptacles of indices of longitudinally elliptic operators (see [Con82,Section 7] and Section 10) and there is a wrong way functoriality f ! : K * C (M 1 /F 1 ) → K * C (M 2 /F 2 ) under K-oriented smooth maps of leaf spaces f : M 1 /F 1 → M 2 /F 2 [HS87]. ...
Pursuing conjectures of John Roe, we use the stable Higson corona of foliated cones to construct a new K-theory model for the leaf space of a foliation. This new K-theory model is -- in contrast to Alain Connes' K-theory model -- a ring. We show that Connes' K-theory model is a module over this ring and develop an interpretation of the module multiplication in terms of indices of twisted longitudinally elliptic operators.
... Indeed, the orbits of an isometric Lie group action constitute an example of singular Riemannian foliations [45,1] and, more generally, orbits of any Lie group action are an example of the almost-regular foliations that are analyzed using holonomy groupoids in [2,15]. We expect that -at least in the context of these examples-our factorization results of Dirac operators in unbounded KK-theory and the appearance of curvature can be applied to index theory on singular foliations as studied in [9,13,26,17,18]. ...
We establish the factorization of the Dirac operator on an almost-regular fibration of spin manifolds in unbounded KK-theory. As a first intermediate result we establish that any vertically elliptic and symmetric first-order differential operator on a proper submersion defines an unbounded Kasparov module, and thus represents a class in KK-theory. Then, we generalize our previous results on factorizations of Dirac operators to proper Riemannian submersions of spin manifolds. This allows us to show that the Dirac operator on the total space of an almost-regular fibration can be written as the tensor sum of a vertically elliptic family of Dirac operators with the horizontal Dirac operator, up to an explicit `obstructing' curvature term. We conclude by showing that the tensor sum factorization represents the interior Kasparov product in bivariant K-theory.
... In particular, a hypoelliptic, L m ρ,δ calculus is required in the study of foliations. This calculus is developed in the paper of Hilsum and Skandalis ( [17]), where a K-homology ...
The paper constructs the analytic index for an elliptic pseudodifferential family of L^{m}_{\rho,\de}-operators invariant under the proper action of a continuous family groupoid on a G-compact, G-space.
... For example, the foliation groupoids (and in particular the holonomy groupoids of foliations) may be represented byétale Lie groupoids [7,17]. It turns out that two Lie groupoids are weakly equivalent if and only if they are isomorphic in the Morita category of Lie groupoids, the category in which morphisms are isomorphism classes of principal bundles [8,10,15,18,19]. We are therefore primarily interested in those algebraic invariants of Lie groupoids which are functorially defined on the Morita category, thus respecting the weak equivalence. ...
Any etale Lie groupoid G is completely determined by its associated convolution algebra C_c(G) equipped with the natural Hopf algebroid structure. We extend this result to the generalized morphisms between etale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated C_c(G)-C_c(H)-bimodule C_c(P) equipped with the natural coalgebra structure. Furthermore, we prove that the functor C_c gives an equivalence between the Morita category of etale Lie groupoids and the Morita category of locally grouplike Hopf algebroids.
... see also [15,24,43]. Out of the adiabatic groupoid we construct the F -Fredholm groupoid : ...
We review recent progress regarding the index theory of operators defined on non-compact manifolds that can be modeled by Lie groupoids. The structure of a particular type of almost regular foliation is recalled and the construction of the corresponding accompanying holonomy Lie groupoid. Using deformation groupoids, K-theoretical invariants can be defined and compared. We summarize how questions in index theory are addressed via the geometrization made possible by the use of deformation groupoids. The discussion is motivated by examples and applications to degenerate PDE’s, diffusion processes, evolution equations and geometry.
... Our strategy will be to show that any generalised morphism is equivalent to a P-anafunctor induced by a bibundle. The theory of bibundles offers another method of localising LieGpoid with a more geometric flavour; see [11,13,16] for details. We do not require the full theory here, but simply borrow the necessary concepts. ...
We prove that the 2-category of action Lie groupoids localised in the following three different ways yield equivalent bicategories: localising at equivariant weak equivalences à la Pronk, localising using surjective submersive equivariant weak equivalences and anafunctors à la Roberts, and localising at all weak equivalences. These constructions generalise the known case of representable orbifold groupoids. We also show that any weak equivalence between action Lie groupoids is isomorphic to the composition of two particularly nice forms of equivariant weak equivalences.
... Morita equivalence is an equivalence relation between Lie groupoids that is weaker than isomorphism. The approach that we take here, originally introduced in [25,27,45], is based on the use of principal groupoid bibundles. ...
Using tools from the theory of Lie groupoids, we study the category of logarithmic flat connections on principal G-bundles, where G is a complex reductive structure group. Flat connections on the affine line with a logarithmic singularity at the origin are equivalent to representations of a groupoid associated to the exponentiated action of C. We show that such representations admit a canonical Jordan–Chevalley decomposition and may be linearized by converting the C-action to a C∗-action. We then apply these results to give a functorial classification. Flat connections on a complex manifold with logarithmic singularities along a hypersurface are equivalent to representations of a twisted fundamental groupoid. Using a Morita equivalence, whose construction is inspired by Deligne’s notion of paths with tangential basepoints, we prove a van Kampen type theorem for this groupoid. This allows us to show that the category of representations of the twisted fundamental groupoid can be localized to the normal bundle of the hypersurface. As a result, we obtain a functorial Riemann–Hilbert correspondence for logarithmic connections in terms of generalized monodromy data.
... These manifolds have an alternate pseudodifferential calculus which allows the construction of parametrices for hypoelliptic but not elliptic differential operators. Classical examples gather contact [17,3], CR [20] and Heisenberg manifolds [4] but also foliated manifolds [23,9] and more recently, manifolds with parabolic geometries were also considered [13]. These new advances are all thanks to a new approach on the pseudodifferential calculus using groupoid methods [37,1] (there is another approach relying on harmonic analysis techniques in [19]). ...
We show an isomorphism between the kernel of the C*-algebra of the tangent groupoid of a filtered manifold and the crossed product of the order 0 pseudodifferential operators in the associated filtered calculus by a natural R-action. This isomorphism is constructed in the same way as in the classical pseudodifferential calculus by Debord and Skandalis. The proof however relies on a structure result for the C*-algebra of graded nilpotent Lie groups which did not appear in the commutative case. A consequence of this structure result is a decomposition of the principal symbol algebra, generalizing the decomposition of Epstein and Melrose in the case of contact manifolds.
... 1/ M OE0; 1/ is open and invariant. By [32], the kernel of p 0 is C .T H M j M .0;1/ /. ...
On filtered manifolds one can define a different notion of order for the differential operators. In this paper, we use generalized fixed point algebras to construct a pseudodifferential extension that reflects this behaviour. In the corresponding calculus, the principal symbol of an operator is a family of operators acting on certain nilpotent Lie groups. The role of ellipticity as a Fredholm condition is replaced by the Rockland condition on these groups. Our approach allows to understand this in terms of the representations of the corresponding algebra of principal symbols. Moreover, we compute the K-theory of this algebra.
... The composition of bibundles is defined using a construction known as the "balanced tensor product". In the literature, this is also known as the Hilsum-Skandalis tensor product, first appearing in the context of topological groupoids in a paper by Hilsum and Skandalis [HS87]. ...
We study three variants of bicategories of diffeological groupoids, and show that they are all equivalent: the bicategory with generalised morphisms, the bicategory with anafunctors, and the bicategory with bibundles. Moreover, we show that the corresponding sub-bicategories of locally subductive diffeological groupoids, as well as Lie groupoids, are full sub-bicategories. As applications, we obtain equivalences with the corresponding 2-categories of geometric stacks, and present some Morita invariants.
... It is well known that these form a bicategory and that taking groupoid C * -algebras is a homomorphism from this bicategory to the bicategory of C * -algebras and Morita-Rieffel equivalences; this goes back already to the seminal work of Muhly-Renault-Williams [26], except that they do not use the language of bicategories and allow the more general case of locally compact groupoids with Haar systems. Another variant of groupoid correspondences was studied by Hilsum-Skandalis [14] to construct wrong-way functoriality maps between the K-theory groups of groupoid C * -algebras. These, however, usually fail to induce C * -correspondences. ...
We define a bicategory with \'etale, locally compact groupoids as objects and suitable correspondences, that is, spaces with two commuting actions as arrows; the 2-arrows are injective, equivariant continuous maps. We prove that the usual recipe for composition makes this a bicategory, carefully treating also non-Hausdorff groupoids and correspondences. We extend the groupoid C*-algebra construction to a homomorphism from this bicategory to that of C*-algebra correspondences. We describe the C*-algebras of self-similar groups, higher-rank graphs, and discrete Conduch\'e fibrations in our setup.
... is open and invariant. By [HS87], the kernel of p 0 is C * (T H M | M×(0,∞) ). The fibre projections p t from (8) for t > 0 combine to an isomorphism ...
On filtered manifolds one can define a different notion of order for the differential operators. In this paper, we use generalized fixed point algebras to construct a pseudodifferential extension that reflects this behaviour. In the corresponding calculus, the principal symbol of an operator is a family of operators acting on certain nilpotent Lie groups. The role of ellipticity as a Fredholm condition is replaced by the Rockland condition on these groups. Our approach allows to understand this in terms of the representation of the corresponding algebra of principal symbols. Moreover, we compute the K-theory of this algebra.
... For interesting results concerning index theory, Lie groups and more generally groupoids, we refer the reader to [9,10,13,22,23,24,30,31,32,38,47,64] and the references therein. In particular, we point out the similar setting of gauge-invariant operators investigated in [48,49,50]. ...
Following [44], we introduce the notion of families of projective operators on fibrations equipped with an Azumaya bundle . We define and compute the index of such families using the cohomological index formula from [7]. More precisely, a family of projective operators A can be pulled back in a family of SU(N)-transversally elliptic operators on the PU(N)-principal bundle of trivialisations of . Through the distributional index of , we can define an index for the family A of projective operators and using the cohomological index formula from [7], we obtain an explicit cohomological index formula. Let be a central extension by an abelian finite group. As a preliminary result, we compute the index of families of -transversally elliptic operators on a G-principal bundle P.
We define and study, under suitable assumptions, the fundamental class, the index class and the rho class of a spin Dirac operator on the regular part of a spin stratified pseudomanifold. More singular structures, such as singular foliations, are also treated. We employ groupoid techniques in a crucial way; however, an effort has been made in order to make this article accessible to readers with only a minimal knowledge of groupoids. Finally, whenever appropriate, a comparison between classical microlocal methods and groupoids methods has been provided.
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.
We study multivariate generalisations of the classical Wiener--Hopf algebra, which is the C-algebra generated by the Wiener--Hopf operators, given by the convolutions restricted to convex cones. By the work of Muhly and Renault, this C-algebra is known to be isomorphic to the reduced C-algebra of a certain restricted action groupoid, given by the action of Euclidean space on a certain compactification. Using groupoid methods, we construct composition series for the Wiener--Hopf C-algebra by a detailed study of this compactification. We compute the spectrum, and express homomorphisms in K-theory induced by the symbol maps which arise by the subquotients of the composition series in analytical terms. Namely, these symbols maps turn out to be given by an analytical family index of a continuous family of Fredholm operators. In a subsequent paper, we also obtain a topological expression of these indices.
We give a complete and explicit description of the kinematical data of higher gauge theory on principal 2-bundles with the string 2-group model of Schommer-Pries as structure 2-group. We start with a self-contained review of the weak 2-category Bibun of Lie groupoids, bibundles and bibundle morphisms. We then construct categories internal to Bibun, which allow us to define principal 2-bundles with 2-groups internal to Bibun as structure 2-groups. Using these, we Lie-differentiate the 2-group model of the string group and we obtain the well-known string Lie 2-algebra. Generalizing the differentiation process, we find Maurer-Cartan forms leading us to higher non-abelian Deligne cohomology, encoding the kinematical data of higher gauge theory together with their (finite) gauge symmetries. We end by discussing an example of non-abelian self-dual strings in this setting.
Locality is implemented in an arbitrary category using Grothendieck topologies. We explore how different Grothendieck topologies on one category can be related, and, more general, how functors between categories can preserve them. As applications of locality, we review geometric objects such as sheaves, groupoids, functors, bibundles, and anafunctors internal to an arbitrary Grothendieck site. We give definitions such that all these objects are invariant under equivalences of Grothendieck topologies and certain functors between sites. As examples of sites, we look at categories of smooth manifolds, diffeological spaces, topological spaces, and sheaves, and we study properties of various functors between those.
This is a review of
Farsi, Carla; Scull, Laura; Watts, Jordan
Bicategories of action groupoids. (English) Zbl 07870820
The meeting displayed the cyclic theory as a fundamental mathematical tool with applications in diverse domains such as analysis, algebraic K-theory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory.
Let be vector fields satisfying H\"ormander's Lie bracket generating condition on a smooth manifold M. We generalise Connes's tangent groupoid, by constructing a completion of the space using the sub-Riemannian metric. We use our space to calculate all the tangent cones of the sub-Riemannian metric in the sense of the Gromov-Hausdorff distance. This generalises a result of Bella\"iche.
We prove that the 2-category of action Lie groupoids localised in the following three different ways yield equivalent bicategories: localising at equivariant weak equivalences \`a la Pronk, localising using surjective submersive equivariant weak equivalences and anafunctors \`a la Roberts, and localising at all weak equivalences. This generalises the known result for representable orbifold groupoids. As an application, we show that any weak equivalence between action Lie groupoids is isomorphic to the composition of two special equivariant weak equivalences, again extending a result known for representable orbifold groupoids.
Let \widetilde{X} be a smooth Riemannian manifold equipped with a proper, free, isometric, and cocompact action of a discrete group \Gamma . In this paper, we prove that the analytic surgery exact sequence of Higson–Roe for \widetilde{X} is isomorphic to the exact sequence associated to the adiabatic deformation of the Lie groupoid \widetilde{X}\times_\Gamma\widetilde{X} . We then generalize this result to the context of smoothly stratified manifolds. Finally, we show, by means of the aforementioned isomorphism, that the \varrho -classes associated to a metric with a positive scalar curvature defined by Piazza and Schick (2014) correspond to the \varrho -classes defined by Zenobi (2019).
This is a review of
Quijano, Juan Pablo; Resende, Pedro
Functoriality of groupoid quantales. II. (English) Zbl 07382619
Appl. Categ. Struct. 29, No. 4, 629-670 (2021).
In this paper, we use the bivariant K theory of Kasparov ([19]) as a basic tool to prove the K-theoretical 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 Atiyah-Singer index theorem for families ([2], Theorem 3.1). It implies the index theorem for measured
We calculate the cohomology of the Lie algebra of formal vector fields
at the origin in a euclidean space. The results are applied to the investigation of the
Lie algebra of tangent vector fields on a smooth manifold.
In [ 11 ] G. G. Kasparov defined the “operator K -functor” KK(A, B) associated with the graded C *-algebras A and B . If the algebras A and B are trivially graded and A is nuclear he proves six term exact sequence theorems. He asks whether this extends to the graded case.
Here we prove such “six-term exact sequence” results in the graded case. Our proof does not use nuclearity of the algebra A . This condition is replaced by a completely positive lifting condition (Theorem 1.1).
Using our result we may extend the results by M. Pimsner and D. Voiculescu on the K groups of crossed products by free groups to KK groups [ 15 ]. We give however a different way of computing these groups using the equivariant KK -theory developed by G. G. Kasparov in [ 12 ]. This method also allows us to compute the KK groups of crossed products by PSL 2 ( Z ).
We give a criterion that an ergodic action be amenable in terms of the operator algebra associated to it by the Murray-von Neumann construction.
In this paper the homological K-functor is defined on the category of involutory Banach algebras, and Bott periodicity is proved, along with a series of theorems corresponding to the Eilenberg-Steenrod axioms. As an application, a generalization of the Atiyah-Singer index theorem is obtained,and some problems connected with representation rings of discrete groups and higher signatures of smooth manifolds are discussed.Bibliography: 16 items.
Nous définissons une notion de nucléarité en K-théorie pour des C*-algèbres, moins restrictive que la nucléarité et analogue à la moyennabilité en K-théorie de J. Cuntz. Nous montrons que les groupes de Kasparov des algèbres nucléaires en K-théorie se comportent vis-à-vis des produits tensoriels et des suites exactes comme ceux des algèbres nucléaires. Un exemple
d'algèbre non nucléaire en K-théorie a quelques conséquences intéressantes.
We define a notion of K-theoretic nuclearity for C
*-algebras. Less restrictive than nuclearity, this notion is analogous to J. Cuntz's K-theoretic amenability. We prove that the Kasparov groups of K-theoretically nuclear C
*-algebras behave like those of nuclear algebras with respect to tensor products and exact sequences. An example of a non-K-theoretically
nuclear C
*-algebra has some interesting consequences.
Lecture 1: Transversally elliptic operators.- Lecture 2: The index of transversally elliptic operators.- Lecture 3: The excision and multiplicative properties.- Lecture 4: The naturality of the index and the localization theorem.- Lecture 5: The index homomorphism for G = S1.- Lecture 6: The operators for G = S1.- Lecture 7: Toral actions with finite isotropy groups.- Lecture 8: The index homomorphism for G = Tn.- Lecture 9: The cohomology formula.- Lecture 10: Applications.
Fermionic quantization, or Clifford algebra, is combined with pseudodifferential operators to simplify the proof of the Atiyah-Singer index theorem for the Dirac operator on a spin manifold.
Using a recently introduced index for supersymmetric theories, we present a simple derivation of the Atiyah-Singer index theorem for classical complexes and itsG-index generalization using elementary properties of quantum mechanical supersymmetric systems.
Représentations des produits croisés d'algèbre de groupô'des. Prépublication Université Pierre-etMarie-Curie
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On définit un élément de multiplicité
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- Un Fibre Complexe G-Équivariant
Soit L un fibre complexe G-équivariant. On définit un élément de multiplicité
[L]€KK(C*(G); C*(G)) (2.11) (cf. [10]).
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des formes quadratiques définies positives sur E. On retrouve les éléments aEeKK
- W Soit
Soit W le G-fibré des formes quadratiques définies positives sur E. On retrouve les
éléments aEeKK(C*(G); (^(G^) et J^ e KK (C* (G^; C*(G)) (2.10) de [10], § V.
On thé Index of Pseudodifferential Operators, Elliptische Differential Gleischungen Band II
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Collège de France, Annexe, Équipe d'A. Connes, 3, rue d
- M Hilsum
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M. HILSUM,
G. SKANDALIS,
Collège de France, Annexe,
Équipe d'A. Connes,
3, rue d'Ulm,
75005 Paris.
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On thé Invariants of Serre and Dixmier-Douady, Preprint I
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Math., vol. 44, 1980, p. 571-636).
encore un symbole transverse invariant et que, par le lemme A. 9.2 [P®1, Q"] ne dépend module les compacts que du symbole transverse de Q". Par un argument de densité on peut supposer que ^^
- Q Notons Que
Notons que Q // =^Q / a encore un symbole transverse invariant et que, par le
lemme A. 9.2 [P®1, Q"] ne dépend module les compacts que du symbole transverse de
Q". Par un argument de densité on peut supposer que ^^(fQ // —Pb) = ^ °ù
be^^lïrxRyxY, (lîTxtR")*; ^(E^E^)) est de la forme
&(Xi, X2, Xi, X'2, ^ ^i, ^^Eiê/^OX^l-^l)^^ ^2» ^ ^1. ^2)
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Theory for Lie Groups and Foliations Preprint I
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Thé Graph of a Foliation
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SKANDALIS Terminons cet appendice par une remarque
- M Hilsum
M. HILSUM ET G. SKANDALIS
Terminons cet appendice par une remarque :
existence sur X d'un feuilletage « parallèle » à F n'implique pas une action de G comme nous l'avons déjà vu dans la remarque 1.4 (c) en construisant une immersion au sens ensembliste de V\/Fi-> V^i ^i nîest P^ une immersion au sens de la définition 1.3. et [P®1
- Remarquons Que La Réciproque Est Fausse
Remarquons que la réciproque est fausse : l'existence sur X d'un feuilletage « parallèle »
à F n'implique pas une action de G comme nous l'avons déjà vu dans la remarque 1.4
(c) en construisant une immersion au sens ensembliste de V\/Fi-> V^i ^i nîest P^
une immersion au sens de la définition 1.3.
et
[P®1, Q]-[P8)1, Q / ]=PÔl)((pQ-cpQ / )-(Q(p-Q / (p)(P(2)l)eJf(^).
Familles of Pseudo Differential Operators, in Topics in Functional Analysis I
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