Twisted index theory for foliations

For any Lie groupoid with a twisting, we define an analytic index morphism using the Connes tangent groupoid. This morphism agrees with the one of the Lie groupoid when the twisting is trivial. We discuss a longitudinal index theorem, geometric cycles, push-forward maps and Baum–Connes assembly maps for foliations with a twisting on the space of leaves.     

Index Theory and Non-Commutative Geometry I. Higher Families Index Theory

We prove an index theorem for foliated manifolds. We do so by constructing a push forward map in cohomology for a k-oriented map from an arbitrary manifold to the space of leaves of an oriented foliation, and by constructing a Chern–Connes character from the k-theory of the compactly supported smooth functions on the holonomy groupoid of the foliation to the Haefliger cohomology of the foliation. Combining these with the Connes–Skandalis topological index map and the classical Chern character gives a commutative diagram from which the index theorem follows immediately.     

Bivariant K-theory via correspondences

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. Our construction uses no special features of equivariant K-theory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories.We formulate necessary and sufficient conditions for certain duality isomorphisms in the topological bivariant K-theory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, the topological and analytic bivariant K-theories agree if there is such a duality isomorphism.     

Longitudinal smoothness of the holonomy groupoid

Iakovos Androulidakis and Georges Skandalis have defined a holonomy groupoid for any singular foliation. This groupoid, whose topology is usually quite bad, is the starting point for the study of longitudinal pseudodifferential calculus on such foliation and its associated index theory. These studies can be highly simplified under the assumption of the holonomy groupoid being longitudinally smooth. In this note, we rephrase the period bounding lemma that asserts that a vector field on a compact manifold admits a strictly positive lower bound for its periodic orbits in order to prove that the holonomy groupoid is always longitudinally smooth.     

La conjecture de Baum–Connes pour les feuilletages moyennables

We show, using the construction of Higson and Kasparov, that the Baum–Connes Conjecture holds for foliations whose holonomy groupoid is Hausdorff and amenable. More generally, for every locally compact, -compact and Hausdorff groupoid G acting continuously and isometrically on a continuous field of affine Euclidean spaces, the Baum–Connes conjecture with coefficients is an isomorphism, and G amenable in K-theory. In addition, we show that C^*(G) satisfies the Universal Coefficient Theorem.     

Local index theory over foliation groupoids

We give a local proof of an index theorem for a Dirac-type operator that is invariant with respect to the action of a foliation groupoid G. If M denotes the space of units of G then the input is a G-equivariant fiber bundle P→M along with a G-invariant fiberwise Dirac-type operator D on P. The index theorem is a formula for the pairing of the index of D, as an element of a certain K-theory group, with a closed graded trace on a certain noncommutative de Rham algebra Ω^*B associated to G. The proof is by means of superconnections in the framework of noncommutative geometry.     

Fibred coarse embeddings,a-T-menability and the coarse analogue of the Novikov conjecture

The connection between the coarse geometry of metric spaces and analytic properties of topological groupoids is well known. One of the main results of Skandalis, Tu and Yu is that a space admits a coarse embedding into Hilbert space if and only if a certain associated topological groupoid is a-T-menable. This groupoid characterisation then reduces the proof that the coarse Baum–Connes conjecture holds for a coarsely embeddable space to known results for a-T-menable groupoids. The property of admitting a fibred coarse embedding into Hilbert space was introduced by Chen, Wang and Yu to provide a property that is sufficient for the maximal analogue to the coarse Baum–Connes conjecture and in this paper we connect this property to the traditional coarse Baum–Connes conjecture using a restriction of the coarse groupoid and homological algebra. Additionally we use this results to give a characterisation of the a-T-menability for residually finite discrete groups.     

Index Theorem for Equivariant Dirac Operators on Noncompact Manifolds

Let D be a (generalized) Dirac operator on a noncompact complete Riemannian manifold M acted on by a compact Lie group G. Let v: M g = Lie G be an equivariant map, such that the corresponding vector field on M does not vanish outside of a compact subset. These data define an element of K-theory of the transversal cotangent bundle to M. Hence, by embedding of M into a compact manifold, one can define a topological index of the pair (D,v) as an element of the completed ring of characters of G. We define an analytic index of (D,v) as an index space of certain deformation of D and we prove that the analytic and topological indexes coincide. As a main step of the proof, we show that index is an invariant of a certain class of cobordisms, similar to the one considered by Ginzburg, Guillemin and Karshon. In particular, this means that the topological index of Atiyah is also invariant under this class of noncompact cobordisms. As an application, we extend the Atiyah–Segal–Singer equivariant index theorem to our noncompact setting. In particular, we obtain a new proof of this theorem for compact manifolds.     

Functoriality of the Bimodule Associated to a Hilsum–Skandalis Map

We extend the functoriality of the Connes convolution algebra to the category of Hilsum–Skandalis maps between separated smooth étale groupoids. Thereby we establish that Morita equivalent separated smooth étale groupoids have Morita equivalent convolution algebras, analogously to the results of Muhly, Renault and Williams, and others, on the C^*algebras of Morita equivalent groupoids.     

Equivariant eta forms and equivariant differential K-theory

In this paper, for a compact Lie group action, we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes. In order to prove them, we extend the Melrose-Piazza spectral section and its main properties to the equivariant case and introduce the equivariant version of the Dai-Zhang higher spectral flow for arbitrarydimensional fibers. Using these results, we construct a new analytic model of the equivariant differential K-theory for compact manifolds when the group action has finite stabilizers only, which modifies the Bunke-Schick model of the differential K-theory. This model could also be regarded as an analytic model of the differential Ktheory for compact orbifolds. Especially, we answer a question proposed by Bunke and Schick(2009) about the well-definedness of the push-forward map.     

Rigid geometric structures,isometric actions,and algebraic quotients

By using a Borel density theorem for algebraic quotients, we prove a theorem concerning isometric actions of a Lie group G on a smooth or analytic manifold M with a rigid A-structure σ. It generalizes Gromov’s centralizer and representation theorems to the case where R(G) is split solvable and G/R(G) has no compact factors, strengthens a special case of Gromov’s open dense orbit theorem, and implies that for smooth M and simple G, if Gromov’s representation theorem does not hold, then the local Killing fields on [(M)\tilde]{\widetilde{M}} are highly non-extendable. As applications of the generalized centralizer and representation theorems, we prove (1) a structural property of Iso(M) for simply connected compact analytic M with unimodular σ, (2) three results illustrating the phenomenon that if G is split solvable and large then π ₁(M) is also large, and (3) two fixed point theorems for split solvable G and compact analytic M with non-unimodular σ.     

Bivariant Chern character and longitudinal index

In this paper we consider a family of Dirac-type operators on fibration P→B equivariant with respect to an action of an étale groupoid. Such a family defines an element in the bivariant K theory. We compute the action of the bivariant Chern character of this element on the image of Connes' map Φ in the cyclic cohomology. A particular case of this result is Connes' index theorem for étale groupoids [A. Connes, Noncommutative Geometry, Academic Press, 1994] in the case of fibrations.     

Equivariant uniformization theorem for subanalytic sets

In this paper we prove an equivariant version of the uniformization theorem for closed subanalytic sets: Let G be a Lie group and let M be a proper real analytic G-manifold. Let X be a closed subanalytic G-invariant subset of M. We show that there exist a proper real analytic G-manifold N of the same dimension as X and a proper real analytic G-equivariant map such that .      

Inverse systems of groupoids,with applications to groupoid C⁎-algebras

We define what it means for a proper continuous morphism between groupoids to be Haar system preserving, and show that such a morphism induces (via pullback) a *-morphism between the corresponding convolution algebras. We proceed to provide a plethora of examples of Haar system preserving morphisms and discuss connections to noncommutative CW-complexes and interval algebras. We prove that an inverse system of groupoids with Haar system preserving bonding maps has a limit, and that we get a corresponding direct system of groupoid $C^{?}$-algebras. An explicit construction of an inverse system of groupoids is used to approximate a σ-compact groupoid G by second countable groupoids; if G is equipped with a Haar system and 2-cocycle then so are the approximation groupoids, and the maps in the inverse system are Haar system preserving. As an application of this construction, we show how to easily extend the Maximal Equivalence Theorem of Jean Renault to σ-compact groupoids.     

Chern–Weil map for principal bundles over groupoids

The theory of principal G-bundles over a Lie groupoid is an important one unifying various types of principal G-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal G-bundles. In this paper, we study differential geometry of these objects, including connections and holonomy maps. We also introduce a Chern–Weil map for these principal bundles and prove that the characteristic classes obtained coincide with the universal characteristic classes. As an application, we recover the equivariant Chern–Weil map of Bott–Tu. We also obtain an explicit chain map between the Weil model and the simplicial model of equivariant cohomology which reduces to the Bott–Shulman map when the manifold is a point. P. Xu Research partially supported by NSF grant DMS-03-06665.     

The homotopy limit problem for Hermitian K-theory, equivariant motivic homotopy theory and motivic Real cobordism

The homotopy limit problem for Karoubi?s Hermitian K-theory (Karoubi, 1980) [26] was posed by Thomason (1983) [44]. There is a canonical map from algebraic Hermitian K-theory to the Z/2-homotopy fixed points of algebraic K-theory. The problem asks, roughly, how close this map is to being an isomorphism, specifically after completion at 2. In this paper, we solve this problem completely for fields of characteristic 0 (Theorems 16, 20). We show that the 2-completed map is an isomorphism for fields F of characteristic 0 which satisfy cd₂(F[i])<∞, but not in general.     

La conjecture de Novikov pour les feuilletages hyperboliques

Nous définissons la notion de bolicité pour les feuilletages, qui est une notion plus faible que l'hyperbolicité de Gromov, et nous démontrons la conjecture de Novikov pour les feuilletages boliques à base compacte dont le groupoï de d'holonomie est séparé en établissant l'injectivité de l'application de Baum–Connes. Ce résultat généralise celui de Kasparov et Skandalis obtenu dans le cas des groupes boliques.We define the notion of bolicity for foliations, which is a weaker notion than Gromov's hyperbolicity, and we prove the Novikov conjecture for foliations with compact base and whose holonomy groupoid is Hausdorff, by showing that the Baum–Connes map is injective. This result generalizes that of Kasparov and Skandalis in the case of bolic groups.     

A Berger type normal holonomy theorem for complex submanifolds

We prove a Berger type theorem for the normal holonomy F^{^}{\Phi^\perp} (i.e., the holonomy group of the normal connection) of a full complete complex submanifold M of the complex projective space \mathbbC Pⁿ{\mathbb{C} P^n}. Namely, if F^{^}{\Phi^\perp} does not act transitively, then M is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible Hermitian symmetric space of rank greater or equal to 3. Moreover, we show that for complete irreducible complex submanifolds of \mathbbCⁿ{\mathbb{C}^n} the normal holonomy is generic, i.e., it acts transitively on the unit sphere of the normal space. The methods in the proofs rely heavily on the singular data of appropriate holonomy tubes (after lifting the submanifold to the complex Euclidean space, in the \mathbbC Pⁿ{\mathbb{C} P^n} case) and basic facts of complex submanifolds.     

A van Kampen theorem for equivariant fundamental groupoids

The equivariant fundamental groupoid of a G-space X is a category which generalizes the fundamental groupoid of a space to the equivariant setting. In this paper, we prove a van Kampen theorem for these categories: the equivariant fundamental groupoid of X can be obtained as a pushout of the categories associated to two open G-subsets covering X. This is proved by interpreting the equivariant fundamental groupoid as a Grothendieck semidirect product construction, and combining general properties of this construction with the ordinary (non-equivariant) van Kampen theorem. We then illustrate applications of this theorem by showing that the equivariant fundamental groupoid of a G-CW complex only depends on the 2-skeleton and also by using the theorem to compute an example.