The longitudinal index theorem for foliations (original) (raw)

The higher twisted index theorem for foliations

Journal of Functional Analysis, 2017

Given a gerbe L, on the holonomy groupoid G of the foliation (M, F), whose pull-back to M is torsion, we construct a Connes Φ-map from the twisted Dupont-Sullivan bicomplex of G to the cyclic complex of the L-projective leafwise smoothing operators on (M, F). Our construction allows to couple the K-theory analytic indices of L-projective leafwise elliptic operators with the twisted cohomology of BG producing scalar higher invariants. Finally by adapting the Bismut-Quillen superconnection approach, we compute these higher twisted indices as integrals over the ambiant manifold of the expected twisted characteristic classes.

The Index of Operators on Foliated Bundles

Journal of Functional Analysis, 1996

We compute the equivariant cohomology Chern character of the index of elliptic operators along the leaves of the foliation of a flat bundle. The proof is based on the study of certain algebras of pseudodifferential operators and uses techniques for analizing noncommutative algebras similar to those developed in Algebraic Topology, but in the framework of cyclic cohomology and noncommutative geometry.

The index of leafwise G-transversally elliptic operators on foliations

Journal of Geometry and Physics, 2021

We introduce and study the index morphism for leafwise G-transversally elliptic operators on smooth closed foliated manifolds. We prove the usual axioms of excision, multiplicativity and induction for closed subgroups. In the case of free actions, we relate our index class with the Connes-Skandalis index class of the corresponding leafwise elliptic operator on the quotient foliation. Finally we prove the compatibility of our index morphism with the Gysin morphism and reduce its computation to the case of tori actions. We also construct a topological candidate for an index theorem using the Kasparov Dirac element for euclidean G-representations.

Atiyah covering index theorem for Riemannian foliations

Transactions of the American Mathematical Society, 2018

We use the symbol calculus for foliations developed in [BH17a] to derive a cohomological formula for the Connes-Chern character of the Type II spectral triple given in [BH17b]. The same proof works for the Type I spectral triple of Connes-Moscovici. The cohomology classes of the two Connes-Chern characters induce the same map on the image of the maximal Baum-Connes map in K-theory, thereby proving an Atiyah L 2 covering index theorem.

Index theorems for holomorphic maps and foliations

Indiana University Mathematics Journal, 2008

We describe a general construction providing index theorems localizing the Chern classes of the normal bundle of a subvariety inside a complex manifold. As particular instances of our construction we recover both Lehmann-Suwa's generalization of the classical Camacho-Sad index theorem for holomorphic foliations and our index theorem for holomorphic maps with positive dimensional fixed point set. Furthermore, we also obtain generalizations of recent index theorems of Camacho-Movasati-Sad and Camacho-Lehmann for holomorphic foliations transversal to a subvariety.

Local index theory over foliation groupoids

Advances in Mathematics, 2006

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 * B associated to G. The proof is by means of superconnections in the framework of noncommutative geometry.

Transversality and Lefschetz numbers for foliation maps

Journal of Fixed Point Theory and Applications, 2008

Let F be a smooth foliation on a closed Riemannian manifold M , and let Λ be a transverse invariant measure of F . Suppose that Λ is absolutely continuous with respect to the Lebesgue measure on smooth transversals. Then a topological definition of the Λ-Lefschetz number of any leaf preserving diffeomorphism (M, F ) → (M, F ) is given. For this purpose, standard results about smooth approximation and transversality are extended to the case of foliation maps. It is asked whether this topological Λ-Lefschetz number is equal to the analytic Λ-Lefschetz number defined by Heitsch and Lazarov which would be a version of the Lefschetz trace formula. Heitsch and Lazarov have shown such a trace formula when the fixed point set is transverse to F .

A symbol calculus for foliations

Journal of Noncommutative Geometry, 2017

The classical Getzler rescaling theorem of [G83] is extended to the transverse geometry of foliations. More precisely, a Getzler rescaling calculus, [G83], as well as a Block-Fox calculus of asymptotic pseudodifferential operators (AΨDOs), [BlF90], is constructed for all transversely spin foliations. This calculus applies to operators of degree m globally times degree ℓ in the leaf directions, and is thus an appropriate tool for a better understanding of the index theory of transversely elliptic operators on foliations [CM95]. The main result is that the composition of AΨDOs is again an AΨDO, and includes a formula for the leading symbol. Our formula is more complicated due to its wide generality but its form is essentially the same, and it simplifies notably for Riemannian foliations. In short, we construct an asymptotic pseudodifferential calculus for the "leaf space" of any foliation. Applications will be derived in [BH16a, BH16b] where we give a Getzler-like proof of a local topological formula for the Connes-Chern character of the Connes-Moscovici spectral triple of [K97], as well as the (semi-finite) spectral triple given in [BH16a], yielding an extension of the seminal Atiyah-Singer L 2 covering index theorem, [A76], to coverings of "leaf spaces" of foliations. Contents 1. Introduction 1 2. Symbols adapted to foliations 3 3. Symbols of differential operators 6 4. Composition of polynomial symbols 10 5. Asymptotic pseudodifferential operators and their symbol calculus 15 6. The main theorem 24 Appendix A. proof of Lemma 5.10 29 Appendix B. Bifiltered calculus on complete foliations 32 References 34