arXiv subjects
Publications and source records attributed to Jean-Marie Lescure.
We use Toeplitz operators to define a star-product on Poisson manifolds whose Poisson structure is induced by a symplectic Lie algebroid. The Toeplitz operators we consider are defined on groupoids whose algebroid can be endowed with a Heisenberg group structure on the fibers. This generalizes an approach due to Guillemin and Melrose in the symplectic case.
Given a connected manifold with corners $X$ of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles, these conormal homology groups are denoted by $H^{cn}_*(X)$. Using our previous works we define an index morphism $$K^0(^bT^*X)\stackrel{Ind_{ev,cn}^X}{\longrightarrow}H_{ev}^{cn}(X)$$ for $X$ a manifold with corners of codimension less or equal to three and called here the even conormal index morphism. In the case that $X$ is compact and connected and $D$ is an elliptic $b-$pseudodifferential operator in the associated $b-$calculus of $X$ we know, by our previous works and other authors works, that, up to adding an identity operator, $D$ can be perturbed (with a regularizing operator in the calculus) to a Fredholm operator iff $Ind_{ev,cn}^X([σ_D])$ (where $[σ_D]\in K^0(^bT^*X)$ is the principal symbol class) vanishes in the even conormal homology group $H_{ev}^{cn}(X)$. The main result of this paper is the explicit computation of the even and odd conormal index morphisms $Ind_{ev/odd,cn}^X(σ)\in H_{ev/odd}^{cn}(X)$ for $X$ a manifold with corners of codimension less or equal to three. The coefficients of the conormal corner cycles $Ind_{ev/odd,cn}^X(σ)$ are given in terms of some suspended Atiyah-Singer indices of the maximal codimension faces of $X$ and in terms of some suspended Atiyah-Patodi-Singer indices of the non-maximal codimension faces of $X$. As a corollary we give a complete caracterization to the obstruction of the Fredholm perturbation property for closed manifolds with corners of codimension less or equal to three in terms of the above mentioned indices of the faces, this allows us as well to give such a characterization in terms of the respective topological indices.
We show that the computation of the Fredholm index of a fully elliptic pseudodifferential operator on an integrated Lie manifold can be reduced to the computation of the index of a Dirac operator, perturbed by a smoothing operator, canonically associated, via the so-called clutching map. To this end we adapt to our framework ideas coming from Baum-Douglas geometric $K$-homology and in particular we introduce a notion of geometric cycles, that can be categorized as a variant of the famous geometric $K$-homology groups, for the specific situation here. We also define a comparison map between this geometric $K$-homology theory and a relative $K$-theory group, directly associated to a fully elliptic pseudodifferential operator.
Using the calculus of Fourier integral operators on Lie groupoids developped in [18], we study the fundamental solution of the evolution equation ($\partial$ $\partial$t + iP)u = 0 where P is a self adjoint elliptic order one G-pseudodifferential operator on the Lie groupoid G. Along the way, we continue the study of distributions on Lie groupoids done in [17] by adding the reduced C *-algebra of G in the picture and we investigate the local nature of the regularizing operators of [32].
Given a connected manifold with corners of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles. Our main theorem is that, for any manifold with corners $X$ of any codimension, there is a natural and explicit morphism $$K_*(\mathcal{K}_b(X)) \stackrel{T}{\longrightarrow} H^{pcn}_*(X,\mathbb{Q})$$ between the $K-$theory group of the algebra $\mathcal{K}_b(X)$ of $b$-compact operators for $X$ and the periodic conormal homology group with rational coeficients, and that $T$ is a rational isomorphism. As shown by the first two authors in a previous paper this computation implies that the rational groups $H^{pcn}_{ev}(X,\mathbb{Q})$ provide an obstruction to the Fredholm perturbation property for compact connected manifold with corners. The difference with respect to the previous article of the first two authors in which they solve this problem for low codimensions is that we overcome in the present article the problem of computing the higher spectral sequence K-theory differentials associated to the canonical filtration by codimension by introducing an explicit topological space whose singular cohomology is canonically isomorphic to the conormal homology and whose K-theory is naturally isomorphic to the $K-$theory groups of the algebra $\mathcal{K}_b(X)$.
For every connected manifold with corners we use a homology theory called conormal homology, defined in terms of faces and incidences and whose cycles correspond geometrically to corner's cycles. Its Euler characteristic (over the rationals, dimension of the total even space minus the dimension of the total odd space), $χ_{cn}:=χ_0-χ_1$, is given by the alternated sum of the number of (open) faces of a given codimension. The main result of the present paper is that for a compact connected manifold with corners $X$ given as a finite product of manifolds with corners of codimension less or equal to three we have that 1) If $X$ satisfies the Fredholm Perturbation property (every elliptic pseudodifferential b-operator on $X$ can be perturbed by a b-regularizing operator so it becomes Fredholm) then the even Euler corner character of $X$ vanishes, i.e. $χ_0(X)=0$. 2) If the even Periodic conormal homology group vanishes, i.e. $H_0^{pcn}(X)=0$, then $X$ satisfies the stably homotopic Fredholm Perturbation property (i.e. every elliptic pseudodifferential b-operator on $X$ satisfies the same named property up to stable homotopy among elliptic operators). 3) If $H_0^{pcn}(X)$ is torsion free and if the even Euler corner character of $X$ vanishes, i.e. $χ_0(X)=0$ then $X$ satisfies the stably homotopic Fredholm Perturbation property. For example for every finite product of manifolds with corners of codimension at most two the conormal homology groups are torsion free. The main theorem behind the above result is the explicit computation in terms of conormal homology of the $K-$theory groups of the algebra $\mathcal{K}_b(X)$ of $b$-compact operators for $X$ as above. Our computation unifies the only general cases covered before, for codimension zero (smooth manifolds) and for codimension 1 (smooth manifolds with boundary).
As announced in [12], we develop a calculus of Fourier integral G-operators on any Lie groupoid G. For that purpose, we study convolability and invertibility of Lagrangian conic submanifolds of the symplectic groupoid T * G. We also identify those Lagrangian which correspond to equivariant families parametrized by the unit space G (0) of homogeneous canonical relations in (T * Gx \ 0) x (T * G x \ 0). This allows us to select a subclass of Lagrangian distributions on any Lie groupoid G that deserve the name of Fourier integral G-operators (G-FIO). By construction, the class of G-FIO contains the class of equivariant families of ordinary Fourier integral operators on the manifolds Gx, x $\in$ G (0). We then develop for G-FIO the first stages of the calculus in the spirit of Hormander's work. Finally, we work out an example proving the efficiency of the present approach for studying Fourier integral operators on singular manifolds.
We review the properties of transversality of distributions with respect to submersions. This allows us to construct a convolution product for a large class of distributions on Lie groupoids. We get a unital involutive algebra $\cE\_{r,s}'(G,Ω^{1/2})$ enlarging the convolution algebra $C^\infty\_c(G,Ω^{1/2})$ associated with any Lie groupoid $G$. We prove that $G$-operators are convolution operators by transversal distributions. We also investigate the microlocal aspects of the convolution product. We give conditions on wave front sets sufficient to compute the convolution product and we show that the wave front set of the convolution product of two distributions is essentially the product of their wave front sets in the symplectic groupoid $T^*G$ of Coste-Dazord-Weinstein. This also leads to a subalgebra $\cE\_{a}'(G,Ω^{1/2})$ of $\cE\_{r,s}'(G,Ω^{1/2})$ which contains for instance the algebra of pseudodifferential $G$-operators and a class of Fourier integral $G$-operators which will be the central theme of a forthcoming paper.
We give a cohomological formula for the index of a fully elliptic pseudodifferential operator on a manifold with boundary. As in the classic case of Atiyah-Singer, we use an embedding into an euclidean space to express the index as the integral of a cohomology class depending in this case on a noncommutative symbol, the integral being over a $C^\infty$-manifold called the singular normal bundle associated to the embedding. The formula is based on a K-theoretical Atiyah-Patodi-Singer theorem for manifolds with boundary that is drawn from Connes' tangent groupoid approach.
One way to geometrically encode the singularities of a stratified pseudomanifold is to endow its interior with an iterated fibred cusp metric. For such a metric, we develop and study a pseudodifferential calculus generalizing the Φ-calculus of Mazzeo and Melrose. Our starting point is the observation, going back to Melrose, that a stratified pseudomanifold can be `resolved' into a manifold with fibred corners. This allows us to define pseudodifferential operators as conormal distributions on a suitably blown-up double space. Various symbol maps are introduced, leading to the notion of full ellipticity. This is used to construct refined parametrices and to provide criteria for the mapping properties of operators such as Fredholmness or compactness. We also introduce a semiclassical version of the calculus and use it to establish a Poincaré duality between the K-homology of the stratified pseudomanifold and the K-group of fully elliptic operators.
This paper is devoted to the study of Poincaré duality in K-theory for general stratified pseudomanifolds. We review the axiomatic definition of a smooth stratification $\fS$ of a topological space $X$ and we define a groupoid $T^{\fS}X$, called the $\fS$-tangent space. This groupoid is made of different pieces encoding the tangent spaces of the strata, and these pieces are glued into the smooth noncommutative groupoid $T^{\fS}X$ using the familiar procedure introduced by A. Connes for the tangent groupoid of a manifold. The main result is that $C^{*}(T^{\fS}X)$ is Poincaré dual to $C(X)$, in other words, the $\fS$-tangent space plays the role in $K$-theory of a tangent space for $X$.
This paper collects the notes of a serie of lectures given by the two authors during the summer school "Geometric and topological methods for Quantum Field Theory" at Villa de Leyva, Colombia, summer 2007. These lecture notes are mainly devoted to a proof using groupoids and $KK$-theory of Atiyah-Singer index theorem on compact smooth manifolds. We will present an elementary introduction to groupoids, $C^*$-algebras, $KK$-theory and pseudodifferential calculus on groupoids. We will finish by showing that the point of view adopted here generalizes to the case of conical pseudo-manifolds.
In a earlier work of Claire Debord and the author, a notion of noncommutative tangent space isdefined for a conical pseudomanifold and the Poincaré duality in $K$-theory is proved between this space and the pseudomanifold. The present paper continues this work. We show that an appropriate and natural presentation of the notion of symbols on a manifold generalizes right away to conical pseudomanifolds and that it enables us to interpret the Poincaré duality in the singular setting as a principal symbol map.
We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold $M$. A main ingredient is a non-commutative algebra that plays in our setting the role of $C_0(T^*M)$. We prove a Thom isomorphism between non-commutative algebras which gives a new example of wrong way functoriality in $K$-theory. We then give a new proof of the Atiyah-Singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds.