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Pseudodifferential calculus on manifolds with corners and groupoids
Abstract
We associate to any manifold with corners (even with non-embedded hyperfaces) a (non-Hausdorff) longitudinally smooth Lie groupoid, on which we define a pseudodifferential calculus. This calculus generalizes the b-calculus of Melrose, defined for manifolds with embedded corners. The groupoid of a manifold with corners is shown to be unique up to equivalence for manifolds with corners of same codimension. Using tools from the theory of -algebras of groupoids, we also obtain new proofs for the study of b-calculus.
... Premièrement, il existe un calcul pseudodifférentiel défini de manière général sur les groupoïdes de Lie. Celui-ci a été d'abord introduit par Connes [56] pour le cas particulier du groupoïde d'holonomie, puis généralisé indépendamment par Monthubert [140] et Nistor, Weinstein et Xu [152]. Le calcul pseudodifférentiel sur G V contient en particulier les opérateurs différentiels engendrés par V. Le fibré dual (AG V ) * prend le rôle joué usuellement par le fibré cotangent T * M : le symbole principal donne ainsi une suite exacte ...
... Comme pour le b-calcul, les opérateurs pseudodifférentiels sur G b sont des distributions conormales à l'inclusion de la diagonale ∆ M ⊂ G b (voir Figure 3.6). Si l'on impose une condition de décroissance rapide à l'infini de ces noyaux, Monthubert [140] a montré que l'on retrouve exactement le petit b-calcul de Melrose. Cet exemple est discuté plus en détail partie 4. 3.4. ...
... Pseudodifferential operators on Lie groupoids were introduced by Connes in the late 1970's for the specific case of the holonomy groupoid of a foliation [56], and subsequently generalized by Monthubert [140] and Nistor, Weinstein and Xu [152] to any Lie groupoid. This calculus gives a very broad geometric setting to understand a number of different constructions (Melrose's b-calculus, scattering operators on R n , Schulze's calculi on edges and cusps. . . ...
This thesis is set in the general context of extending the theory of elliptic operators, well-understood in the smooth setting, to so-called singular domains. The methods used rely on operator algebras and tools coming from non commutative geometry, together with suitable pseudodifferential calculi that are often built from a groupoid adapted to the particular geometry of the problem. The first part of the thesis deals with the general investigation of a particular class of such groupoids, called Fredholm, that provide a very good setting for the study of elliptic operators. One of the major results proved here is that this Fredholm property is local, in the sense that it only depends on the restrictions of the groupoid to sufficiently many open subsets. In the same spirit, we study with C. Carvalho and Y. Qiao groupoids whose local structure is given by gluing group actions, and consider in particular a groupoid suited to the study of layer potential operators. This part concludes with a well-posedness result for a boundary value problem on a domain with a rotational cusp. The second part deals with equivariant operators on a compact manifold, acted upon by a finite group. We answer the following question: given an irreducible representation of the group, under which condition is a differential operator Fredholm between the corresponding isotypical components of the Sobolev spaces? In a joint work with A. Baldare, M. Lesch and V. Nistor, we introduce a corresponding notion of ellipticity associated with some fixed irreducible representation, and show that it characterizes Fredholm operators.
... (4) The b-calculus on manifolds with corners [23]. We use the b-groupoid [25]. (5) The calculus on manifolds with fibred boundary or with iterated fibred corners [20,8]. ...
... to the algebra of uniformly supported and equivariant C ∞ family of pseudodifferential (resp. equivariant C ∞ family of differential) operators on the fibers of s [28,25,17]. ...
... That √ ∆ = P c + S ∈ Ψ 1 G with σ 0 G (P c ) = ξ 2 j follows from [32] and we get here the existence of a C ∞ family U t of right invariant F IO on G [27,18] such that ( ∂ ∂t + i √ ∆)U t ∈ C ∞ (G) ∩ C * r (G) for any t. (4) The groupoid G b ⇒ X of the b-calculus of a manifold with embedded corners X [25]. ...
Using the calculus of Fourier integral operators on Lie groupoids developped in [18], we study the fundamental solution of the evolution equation ( 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].
... Under such assumptions, f preserves the codimension of points and its fibers have no boundary. If E j → M are vector bundles, we denote by Ψ * G (E 0 , E 1 ) the space of compactly Gpseudodifferential operators [14,36,34,41]. The principal symbol map is denoted by: ...
... By Diagram (32) and the five lemma, π nc ! is an isomorphism too. Now Diagrams (34) and (33) give: ...
... As a consequence, if (M, G) is a Lie manifold, the absolute faces of W are not saturated for the pullback groupoid Φ G. We will replace the latter by [34,22] : ...
We prove that the computation of the Fredholm index for fully elliptic pseudodifferential operators on Lie manifolds can be reduced to the computation of the index of Dirac operators perturbed by smoothing operators. 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 classified into a variant of the famous geometric K-homology groups, for the specific situation here. We also define comparison maps between this geometric K-homology theory and relative K-theory.
... Lie groupoids have been proven to be an effective tool to obtain Fredholmness results and to model analysis on singular spaces in general (see for instance [1,2,15,16,31,43,44,47] and the references therein for a small sample of applications). One general advantage of this strategy is that, by associating a Lie groupoid to a given singular problem, not only are we able to use groupoid techniques, but we also get automatically a groupoid C * -algebra and well-behaved pseudodifferential calculi naturally affiliated to this C * -algebra [3,30,31,42,53,64]. In many situations, the family of limit operators can be obtained from suitable representations of the groupoid C * -algebra, so Fredholmness may be studied through representation theory. ...
... Pseudodifferential operators on Lie groupoids. We recall in this subsection the construction of pseudodifferential operators on Lie groupoids [30,31,42,43,45,53]. Let P = (P x ) x∈M be a smooth family of pseudodifferential operators acting on ...
... Let M be a manifold with smooth boundary and let V b denote the class of vector fields on M that are tangent to the boundary. The associated groupoid was defined in [40,42,53]. Let ...
This paper is a merge of arXiv:1807.05418 and arXiv:1808.01442. We introduce a new class of groupoids, called "boundary action groupoids", which are obtained by gluing reductions of action groupoids. We show that such groupoids model the analysis on many singular spaces, and we give several examples. Under some conditions on the action of the groupoid, we obtain Fredholm criteria for the pseudodifferential operators generated by boundary action groupoids. Moreover, we show that layer potential groupoids for conical domains constructed in an earlier paper (Carvalho-Qiao, Central European J. Math., 2013) are both Fredholm groupoids and boundary action groupoids, which enables us to deal with many analysis problems on singular spaces in a unified way. As an application, we obtain Fredholm criteria for operators on layer potential groupoids.
... Lie groupoids are effective tools to model analysis problems on singular spaces, for a small sample of applications see, for instance, [1,2,4,11,12,13,24,36,38,40,44] and references therein. One general advantage behind this strategy is that, by associating a Lie groupoid to a given singular problem, not only we are able to apply groupoid techniques, but also get automatically a groupoid C * -algebra and well-behaved pseudodifferential calculi naturally affiliated to this C * -algebra [5,23,24,37,46,58]. ...
... Pseudodifferential operators on Lie groupoids. We recall in this subsection the construction of pseudodifferential operators on Lie groupoids [23,24,36,37,39,46]. Let P = (P x ), x ∈ M be a smooth family of pseudodifferential operators acting on G ...
... Let M be a manifold with smooth boundary and let V b denote the class of vector fields on M that are tangent to the boundary. The associated groupoid was defined in [32,37,46]. ...
We show that layer potential groupoids for conical domains constructed in an earlier paper (Carvalho-Qiao, Central European J. Math., 2013) are Fredholm groupoids, which enables us to deal with many analysis problems on singular spaces in a unified treatment. As an application, we obtain Fredholm criteria for operators on layer potential groupoids.
... See [19,24,18,12,29] for a detailed presentation of pseudodifferential calculus on groupoids. ...
... Let X be a compact manifold with embedded corners, so by definition we are assuming there is a smooth compact manifold (of the same dimension)X with X ⊂X and ρ 1 , ..., ρ n defining functions of the faces. In [18], Monthubert constructed a Lie groupoid (called Puff groupoid) associated to any decoupage (X, (ρ i )), it has the following expression ...
... The b−groupoid was introduced by B. Monthubert in order to give a groupoid description for the Melrose's algebra of b-pseudodifferential operators. We summarize below the main properties we will be using about this groupoid: Theorem 2.5 (Monthubert [18]) Let X be a manifold with corners as above, we have that ...
For every connected manifold with corners we introduce a very computable 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), , 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. . 2) If the even Periodic conormal homology group vanishes, i.e. , 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 is torsion free and if the even Euler corner character of X vanishes, i.e. 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 theory groups of the algebra 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).
... For this section we refer the reader to [23] and [29, 3.1]. Let X be a manifold with boundary ∂X. ...
... Notice that by [23,Proposition 3.5] the Lie groupoid Γ(X, ∂X) is amenable and then C * (Γ(X, ∂X)) = C * r (Γ(X, ∂X)). Definition 1.9. ...
... Hence C * r (Γ(R + , {0})) C * r (R + R * + ) C 0 (R + ) R * + and, by the Connes-Thom isomorphism, K * (C 0 (R + ) R) K * −1 (C 0 (R + )) = 0. Remark 1.13. By [23,Proposition 3.5], Γ(R + , {0}) is amenable. Then we have the following short exact sequence ...
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.
... In fact, we will see that the natural Riemannian metric of a manifold with (asymptotically) cylindrical ends. This example was studied also by Debord and Lescure [42,45], Melrose and Piazza [99], Monthubert [105], Schulze [143], and many others. ...
... See [8] for details. Let us just say that it is obtained by realizing Ψ * V (M ) as the image of a groupoid pseudodifferential operator algebra [8,105,107,118] for any Lie groupoid integrating the Lie algebroid A defining the Lie manifold (M , A) [43,44,117]. The algebra Ψ * V (M ) has the property that its subset of differential operators coincides with Diff(V). ...
... The groups are G α ≃ R k , where k is the codimension of the corresponding face (so all are commutative Lie groups). In the case of a smooth boundary, the Z α 's are the connected components of the boundary, G α = R, and P α is the restriction of I(P ) to a translation invariant operator on Z α × R. See also [85,99,100,105,107,142] for just a sample of the many papers on this particular class of manifolds. ...
We discuss and develop some connections between analysis on singular spaces
and operator algebras, as presented in my sequence of four lectures at the
conference "Noncommutative geometry and applications," Frascati, Italy, June
16-21, 2014. Therefore this paper is mostly a survey paper, but the
presentation is new, and there are included some new results as well. In
particular, Sections 3 and 4 provide a complete short introduction to analysis
on noncompact manifolds that is geared towards a class of manifolds--called
"Lie manifolds"--that often appears in practice. Our interest in Lie manifolds
is due to the fact that they provide the link between analysis on singular
spaces and operator algebras. The groupoids integrating Lie manifolds play an
important background role in establishing this link because they provide
operator algebras whose structure is often well understood. The initial
motivation for the work surveyed here--work that spans over close to two
decades--was to develop the index theory of stratified singular spaces.
Meanwhile, several other applications have emerged as well, including
applications to Partial Differential Equations and Numerical Methods. These
will be mentioned only briefly, however, due to the lack of space. Instead, we
shall concentrate on the applications to Index theory.
... It only depends on the Lie structure itself and the assumption that it leads to algebroids which are integrable via compatible groupoids. We recall first the definition of the b-groupoid from [24] (see also [20]). For this consider the case V = V b where V b denotes the module of vector fields which are tangent to all hyperfaces of X. ...
... The assertion follows from the longitudinal smoothness of the groupoid G where r, s are surjective submersions in the sense of [27]. We have the actions Since p, q are restrictions of r ∂ , s we obtain that X has smooth fibers by the local triviality property [24] of the groupoid G. The same reasoning applies to X t . ...
... Instead we use the groupoids as given by Monthubert (cf. [24], see also [20]) and defined in the last section. ...
We consider general pseudodifferential boundary value problems on a Lie
manifold with boundary. This is accomplished by constructing a suitable
generalization of the Boutet de Monvel calculus for boundary value problems.
The data consists of a compact manifold with corners M which is endowed with
a Lie structure of vector fields , a so-called Lie manifold as
introduced by Bernd Ammann, Robert Lauter and Victor Nistor. The Lie manifold
M is split into two equal parts and each of which are Lie
manifolds which intersect in an embedded hypersurface . In
this setup our goal is to describe a transmission Boutet de Monvel calculus for
boundary value problems. Starting with the example of b-vector fields we show
that there are two groupoids integrating the Lie structure on M and Y
respectively which form a bimodule structure (a groupoid correspondence) and in
mild cases these groupoids are isomorphic inside the category of Lie groupoids
(Morita equivalent). With the help of the bimodule structure and canonically
defined manifolds with corners, which are blow-ups in particular cases, we
define a class of extended Boutet de Monvel operators. Then we describe the
restricted transmission Boutet de Monvel calculus by truncation of the extended
operators. We define the representation for restricted operators and show
closedness under composition with the help of an analog of the Ammann, Lauter,
Nistor representation theorem. Finally, we analyze the parametrix construction
and in the last section state the index problem for boundary value problems on
Lie manifolds.
... We then have the following result [29,37,46]. ...
... To [33,45] and will be denoted by G M . For G M the vector representation π is injective [37,31]. We shall also denote by C * (M ) = C * (G M ). ...
... We shall also denote by C * (M ) = C * (G M ). Recall that C * (G M ) = Ψ −1 (G), [31,37]. ...
We compute the K-theory of the comparison C *-algebra associated to a manifold with corners. These comparison algebras are an example of the abstract pseudodifferential algebras introduced by Connes and Moscovici [16]. Our calculation is obtained by showing that the comparison algebra associated to a manifold with corners is a homomorphic image of an explicit groupoid C *-algebra. We then prove an index theorem with values in the K-theory groups of the comparison algebra.
... Let X be a manifold with embedded corners, so by definition we are assuming there is a smooth manifold (of the same dimension)X with X ⊂X and ρ 1 , ..., ρ n defining functions of the faces. In [19], Monthubert constructed a Lie groupoid (called Puff groupoid) associated to any decoupage (X, (ρ i )), it has the following expression ...
... The b−groupoid was introduced by B. Monthubert in order to give a groupoid description for the Melrose's algebra of b-pseudodifferential operators, we will come to this in the last section. In any case, as a C ∞,0 −groupoid the b-groupoid is amenable, in particular its reduced and maximal C * -algebras coincide, see [19]. In this paper we will simply denote by K b (X) = C * (Γ b (X)) this algebra and by ...
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 . Using our previous works we define an index morphism 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 pseudodifferential operator in the associated 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 (where is the principal symbol class) vanishes in the even conormal homology group . The main result of this paper is the explicit computation of the even and odd conormal index morphisms for X a manifold with corners of codimension less or equal to three. The coefficients of the conormal corner cycles 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.
... Filtered groupoids and the osculating groupoid. For simplicity, we shall work here with Hausdorff Lie groupoids, although the construction works equally well for almost differentiable [NWX99] or longitudinally smooth [Mon99] groupoids-i.e., groupoids in which the fibres are manifolds while the base is a manifold with corners. ...
... This construction leads one naturally to imagine questions of analysis on manifolds with boundaries or corners in the presence of a Lie filtration. Compare, e.g., [Mon99], [ALN07], [DLR15] for the unfiltered case. We hope to return to this in a future work. ...
We give an intrinsic (coordinate-free) construction of the tangent groupoid of a filtered manifold.
... Many authors have worked in several different cases and with very different methods. One of the tools that have produced interesting results in the last few years is the use of Lie groupoids (Lie manifolds [1,2], singular foliations [3,4,12], manifolds with corners [18,19], stratified pseudomanifolds [14] and implicitly in cases where there are nice integrable Lie algebroids to mention some cases). These examples shows that having a "good" groupoid for a particular geometric situation is a very good first step to start doing index theory, in the sense that it allows one to construct appropriate algebra of differential and pseudodifferential operators and consider pseudodifferential calculus. ...
... By the work of Claire Debord [12], these almost regular foliations can be integrated into holonomy groupoids (what she calls quasi-graphoids following Bigonnet and Pradines). In her paper, for the particular case we are treating in this article, she studies further the case of M = R 2 and M 1 = { * } (as she recalls the codimension 1 case are essentially the groupoids studied before her by Monthubert [18]). ...
We consider Lie groupoids of the form G(M,M1):=M0×M0⊔H×M1×M1⇉M, where M0=M\M1 and the isotropy subgroup H is an exponential Lie group of dimension equal to the codimension of the manifold M1 in M. The existence of such Lie groupoids follows from integration of almost injective Lie algebroids by Claire Debord. They correspond to (the s-connected version of) the problem of the existence of a holonomy groupoid associated to the singular foliation whose leaves are the connnected components of M1 and the connected components of M0. We study the Lie groupoid structure of these groupoids, and verify that they are amenable and Fredholm in the sense recently introduced by Carvalho, Nistor and Qiao. We compute explicitly the K-groups of these groupoid’s C∗-algebras, we obtain K0(C∗(G(M,M1)))≅Z,K1(C∗(G(M,M1)))≅Z for M1 of odd codimension, and K0(C∗(G(M,M1)))≅Z⊕Z,K1(C∗(G(M,M1)))≅{0} for M1 of even codimension. When M and M1 are compact we obtain, as an application of our previous K-theory computations, that in the odd codimensional case every elliptic operator (in the groupoid pseudodifferential calculus) can be perturbed (up to stabilization by an identity operator) with a regularizing operator, such that the perturbed operator is Fredholm; and in the even case, given an elliptic operator there is a topological obstruction to satisfy the previous Fredholm perturbation property given by the Atiyah-Singer topological index of the restriction operator to M1.
... We are interested in Fredholm groupoids because of their applications to Fredholm conditions. Let G be a continuous family groupoid (see [12] and the references therein) and Ψ m (G) be the space of order m, classical pseudodifferential operators P = (P x ) x∈M on G [12] (see [2,3,16,20,27] for Lie groupoids, which are continuous family groupoids). Recall that, by definition, each P x ∈ Ψ m (G x ), x ∈ M and P x = π x (P ), for the regular representation π x , x ∈ M . ...
... Let M be a manifold with corners. Then the b-groupoid of [13,16,20] is a Lie groupoid such that as a set is given by ...
... Androulidakis and Skandalis [6], Debord and Skandalis [27,28,29], Debord, Lescure, and Rochon [26], Monthubert [66,67], van Erp and Yuncken [92,93], the first named author [71], the second named author (with collaborators) [22,56,81,82], and by others. The groupoids arising from crossed product C * -algebras have been used even for a longer time. ...
... Then Ξ ω is also a Lie groupoid (respectively, a longitudinally smooth groupoid), provided that the cocycle is smooth enough. This allows us to consider the pseudo-differential calculus (algebra) Ψ ∞ (Ξ ω ) [54,66,67,82]. Using the formalism above, one should be able to introduce and study a twisted pseudo-differential algebra Ψ ∞ (Ξ; ω) , such that Ψ 0 (Ξ; ω) ⊂ C * (Ξ, ω) . ...
... The so-called Puff groupoid [28] is then defined by: ...
... This is a Lie subgroupoid ofX ×X × R k . The b-groupoid Γ b (X) [28] is then defined as the sconnected component of the restriction of the Puff groupoid to (the saturared closed subspace) X. It is again a Lie amenable groupoid (in the extended sense of [16]) whose Lie algebroid identifies in a canonical way with the compressed tangent bundle b T X. ...
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 between the theory group of the algebra 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 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 theory groups of the algebra .
... With a few exceptions, including [7,57,59], twisted groupoids have not yet been used before in spectral theory. On the other hand, regular (untwisted) groupoids have recently been used in relation to spectral theory (mainly in connection with Fredholm and index properties) by Androulidakis and Skandalis [5], Debord and Skandalis [23,24,25], Debord, Lescure, and Rochon [22], Monthubert [63,64], van Erp and Yuncken [79,78], the second named author (with collaborators) [18,47,70,71], and by others. The groupoids arising from crossed product C * -algebras have been used even for a longer time. ...
... Then Ξ ω is also a Lie groupoid (respectively, a longitudinally smooth groupoid). This allows us to consider the pseudo-differential calculus (algebra) Ψ ∞ (Ξ ω ) [63,64,71,45]. Using the formalism above, one should be able to introduce and study a twisted pseudo-differential algebra ...
We study bounded operators defined in terms of the regular representations of the -algebra of an amenable, Hausdorff, second countable locally compact groupoid endowed with a continuous 2-cocycle. We concentrate on spectral quantities associated to natural quotients of this twisted algebra, such as the essential spectrum, the essential numerical range, and Fredholm properties. We obtain decompositions for the regular representations associated to units of the groupoid belonging to a free locally closed orbit, in terms of spectral quantities attached to points (or orbits) in the boundary of this main orbit. As examples, we discuss various classes of magnetic pseudo-differential operators on nilpotent groups. We also prove localization and non-propagation properties associated to suitable parts of the essential spectrum. These are applied to twisted groupoids having a totally intransitive groupoid restriction at the boundary.
... Recently, there were quite a few papers using C * -algebras to obtain Fredholm conditions, see, for instance, [17,19,33,35,39] among many others. Often these were groupoid algebras [13,20,42,53]. A powerful related technique is that of "limit operators" [31,37,38,52]. ...
Let Γ be a finite abelian group acting on a smooth, compact manifold M without boundary and let P∈ψm(M;E0,E1) be a Γ-invariant, classical, pseudodifferential operator acting between sections of two Γ-equivariant vector bundles. Let α be an irreducible representation of Γ. We obtain necessary and sufficient conditions for the restriction πα(P):Hs(M;E0)α→Hs−m(M;E1)α of P between the α-isotypical components of Sobolev spaces to be Fredholm.
... Here one considers any Lie groupoid G ⇒ M with M compact (the classical case corresponds to the pair groupoid G = M × M ). On G one constructs the pseudo-differential calculus [1,4,8,24,25,38,44]. The sub-algebras of operators of order 0 and −∞ can be completed to C * -algebras U(G) and C * (G), respectively. ...
We consider the index problem of certain boundary groupoids of the form \cG = M _0 \times M _0 \cup \mathbb{R}^q \times M _1 \times M _1. Since it has been shown that when q is odd and , K _0 (C^* (\cG)) \cong \bbZ , and moreover the K-theoretic index coincides with the Fredholm index, in this paper we attempt to derive a numerical formula. Our approach is similar to that of renormalized trace of Moroianu and Nistor \cite{Nistor;Hom2}. However, we find that when , the eta term vanishes, and hence the K-theoretic and Fredholm indexes of elliptic (respectively fully elliptic) pseudo-differential operators on these groupoids are given only by the Atiyah-Singer term. As for the q=1 case we find that the result depends on how the singularity set lies in M.
... D'abord, dans le cas d'une variété singulière, il faut trouver un groupoïde de Lie qui va pouvoir remplacer le groupoïde des paires. Plusieurs solutions ontété trouvées : B. Monthubert [Mon03] construit un groupoïde permettant de faire du calcul pseudodifférentiel dans le cas des variétés a coins, C. Debord, J-M. Lescure et V. Nistor [DLN09] donnent un théorème de l'indice pour les variétéà singularités coniques. ...
Dans cette thèse, on utilise le Blup de Debord-Skandalis afin de généraliser la construction d'Alain Connes pour le cas classique, et de P. Carrillo-Rouse, J-M. Lescure et B. Monthubert pour le cas à bord de théorèmes de l'indices via groupoïdes de déformations. Ainsi, on donne un théorème de l'indice classique et un théorème de l'indice de Fredholm pour un grand nombre de groupoïdes de Lie, réalisés comme des blow up de groupoïdes de Lie au sens de Debord-Skandalis. Il s'agit donc de donner une construction fonctorielle des théorèmes dans ce cadre, et pour se faire, nous définissons les quadruplé b-principaux comme la donnée de d'une immersion de groupoïdes de Lie, munis chacun d'un morphisme vers un groupoïde des paires d'un espace euclidien, vérifiant certaines propriétés. together with properties. Beaucoup d'exemples très classiques entrent dans ce contexte.
... respectively. The groupoid corresponding to b-calculus is introduced by [Mon99], and a general construction by [Nis00] includes the and e cases. Here we use the description using the blowup construction of groupoids. ...
In this paper, we investigate topological aspects of indices of twisted geometric operators on manifolds equipped with fibered boundaries. We define K-groups relative to the pushforward for boundary fibration, and show that indices of twisted geometric operators, defined by complete Φ or edge metrics, can be regarded as the index pairing over these K-groups. We also prove various properties of these indices using groupoid deformation techniques. Using these properties, we give an application to the localization problem of signature operators for singular fiber bundles.
... C * -algebras were used recently to obtain Fredholm conditions in [31,33,50,62], for example. Some of the algebras involved were groupoid algebras [4,3,25,34,59,70]. Fredholm conditions play an important role in the study of the essential spectrum of Quantum Hamiltonians [12,37,36,42,49]. ...
Let be a finite group acting on a smooth, compact manifold M, let be a -invariant, classical pseudodifferential operator acting between sections of two equivariant vector bundles , , and let be an irreducible representation of the group . Then P induces a map between the -isotypical components of the corresponding Sobolev spaces of sections. We prove that the map is Fredholm if, and only if, P is "-elliptic", an explicit condition that we define in terms of the principal symbol of P and the action of on the vector bundles . The result is not true for non-discrete groups. In the process, we also obtain several results on the structure of the algebra of invariant pseudodifferential operators on , especially in relation to induced representations. We include applications to Hodge theory and to index theory of singular quotient spaces.
... The module of smooth vector fields on M that are tangent to V was considered by Melrose for the study of b-operators for manifold with boundary [83]. This module is the module of sections of a Lie algebroid A b over M which integrates into the b-groupoid G b [90]. ...
Alain Connes introduced the use of Lie groupoids in noncommutative geometry in his pioneering work on the index theory of foliations. In the present paper, we recall the basic notion involved: groupoids, their C∗-algebras, their pseudodifferential calculus, etc. We review several recent and older advances on the involvement of Lie groupoids in noncommutative geometry. We then propose some open questions and possible developments of the subject.
... Recently, there were quite a few papers using C * -algebras to obtain Fredholm conditions, see, for instance, [17,19,33,35,39] among many others. Often these were groupoid algebras [13,20,42,53]. A powerful related technique is that of "limit operators" [31,37,38,52]. ...
We answer the question of when an invariant pseudodifferential operator is Fredholm on a fixed, given isotypical component. More precisely, let be a compact group acting on a smooth, compact, manifold M without boundary and let be a -invariant, classical, pseudodifferential operator acting between sections of two -equivariant vector bundles and . Let be an irreducible representation of the group . Then P induces by restriction a map between the -isotypical components of the corresponding Sobolev spaces of sections. We study in this paper conditions on the map to be Fredholm. It turns out that the discrete and non-discrete cases are quite different. Additionally, the discrete abelian case, which provides some of the most interesting applications, presents some special features and is much easier than the general case. We thus concentrate in this paper on the case when is finite abelian. We prove then that the restriction is Fredholm if, and only if, P is "-elliptic", a condition defined in terms of the principal symbol of P. If P is elliptic, then P is also -elliptic, but the converse is not true in general. However, if acts freely on a dense open subset of M, then P is -elliptic for the given fixed if, and only if, it is elliptic. The proofs are based on the study of the structure of the algebra of classical, -invariant pseudodifferential operators acting on sections of the vector bundle and of the structure of its restrictions to the isotypical components of . These structures are described in terms of the isotropy groups of the action of the group on .
... Remark 3.3. Some references on manifolds with corners are Cerf [5], Douady [6], Gillam and Molcho [13, §6.7], Kottke and Melrose [30], Margalef-Roig and Outerelo Dominguez [34], Melrose [35][36][37], Monthubert [41], and the second author [18,23]. Just as objects, without considering morphisms, most authors define manifolds with corners X as in Definition 3.2. ...
If X is a manifold then the set of smooth functions is a -ring, a rich algebraic structure with many operations. -schemes are schemes over -rings, a way of using Algebro-Geometric techniques in Differential Geometry. They include smooth manifolds, but also many singular and infinite-dimensional spaces. They have applications to Synthetic Differential Geometry, and to derived manifolds. In this book, a sequel to the second author's monograph on -algebraic geometry arXiv:1001.0023, we define and study new categories of -rings with corners and -schemes with corners, which generalize manifolds with corners in the same way that -rings and -schemes generalize manifolds. These will be used in future work as the foundations of theories of derived manifolds and derived orbifolds with corners. This book is based on the PhD thesis of the first author, supervised by the second author.
... The index map is formulated using the trace density, which gives a map from the Hochschild or negative cyclic homology of the deformation quantization to the de Rham coho-mology of the manifold. We refer to [14,221] for this line of development as well as [229,230] for index theory in the framework of Lie groupoids and to [212][213][214]226] for the analysis on singular spaces consisting of a smooth interior part and a tower of foliated structures at the boundary, including corner and cone points and edges of all possible dimensions. This leads us to the next topic which is the geometrization of the pseudodifferential calculus and of deformations. ...
We report on the following highlights from among the many discoveries made in Noncommutative Geometry since year 2000: 1) The interplay of the geometry with the modular theory for noncommutative tori, 2) Advances on the Baum-Connes conjecture, on coarse geometry and on higher index theory, 3) The geometrization of the pseudo-differential calculi using smooth groupoids, 4) The development of Hopf cyclic cohomology, 5) The increasing role of topological cyclic homology in number theory, and of the lambda operations in archimedean cohomology, 6) The understanding of the renormalization group as a motivic Galois group, 7) The development of quantum field theory on noncommutative spaces, 8) The discovery of a simple equation whose irreducible representations correspond to 4-dimensional spin geometries with quantized volume and give an explanation of the Lagrangian of the standard model coupled to gravity, 9) The discovery that very natural toposes such as the scaling site provide the missing algebro-geometric structure on the noncommutative adele class space underlying the spectral realization of zeros of L-functions.
... This groupoid G 0 is described more precisely in [5]. Let G 1 ⇒ X be the groupoid obtained by blowing up X 2 ⇒ X at (∂ 1 X) 2 ⇒ ∂ 1 X, which is called a puff groupoid in [18]. Then G is the fiber product of G 0 and G 1 . ...
Let X be a smooth compact manifold with corners which has two embedded boundary hypersurfaces , and a fiber bundle is given. By using the method of blowing up, we define a pseudodifferential culculus generalizing the -calculus of Mazzeo and Melrose and the (small) b-calculus of Melrose. We discuss the Fredholm condition of such operators and prove the relative index theorem. And as its application, the index theorem of "non-closed" - manifolds is proved.
... The module of smooth vector fields on M that are tangent to V was considered by Melrose for the study of b-operators for manifold with boundary [83]. This module is the module of sections of a Lie algebroid A b over M which integrates into the b-groupoid G b [90]. ...
Alain Connes introduced the use of Lie groupoids in noncommutative geometry in his pioneering work on the index theory of foliations. In the present paper, we recall the basic notion involved: groupoids, their C*-algebras, their pseudodifferential calculus... We review several recent and older advances on the involvement of Lie groupoids in noncommutative geometry. We then propose some open questions and possible developments of the subject.
... the product of the pair groupoid V × V and the transformation groupoid[0, ∞) k F (0, ∞) k F .If M has embedded faces, that is, if each hyperface H has a defining function r H , then we can identify G b with an open subset of Monthubert's realization of the b-groupoid[65, Proposition 4.5] ...
We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a “user’s guide” to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with poly-cylindrical ends. The reader interested in applications should be able to read right away the results related to those examples, beginning with Section 5. Our general, theoretical results are that an operator adapted to the geometry is Fredholm if, and only if, it is elliptic and all its limit operators (in a sense to be made precise) are invertible. Central to our theoretical results is the concept of a “Fredholm groupoid.” By definition, a Fredholm groupoid is one for which this characterization of the Fredholm condition is valid. We use the notions of exhaustive and strictly spectral families of representations to obtain a general characterization of Fredholm groupoids. In particular, we introduce the class of the so-called groupoids with Exel’s property as the groupoids for which the regular representations are exhaustive. We show that the class of “stratified submersion groupoids” has Exel’s property, where stratified submersion groupoids are defined by gluing fibered pull-backs of bundles of Lie groups. We prove that a stratified submersion groupoid is Fredholm whenever its isotropy groups are amenable. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. This fact is exploited to yield Fredholm conditions not only in the above-mentioned classes, but also on manifolds that are obtained by desingularization or by blow-up of singular sets.
... [31,38]). They appear in many geometric situations as manifolds with boundary or with corners [26,27,28,23,30,7,6], stratified manifolds [13,14], Lie manifolds [37] (and literature there). ...
We study various exact sequences associated with a closed saturated subset in the space of units of a Lie groupoid: the corresponding exact sequence of groupoid C *-algebras, the associated index maps and full index maps. Moreover we study Connes–Thom type isomorphisms of Lie groupoid C *-algebras.
... We recall briefly the construction of the space of pseudodifferential operators associated to a Lie groupoid G with units M [33,34,54,55,58]. The dimension of M is n ≥ 1. ...
We use an approach based on pseudodifferential operators on Lie groupoids to study the double layer potentials on plane polygons. Let be a simply connected polygon in . Denote by K the double layer potential operator on associated with the Laplace operator . We show that the operator K belongs to the groupoid -algebra that the first named author has constructed in an earlier paper (Carvalho and Qiao in Cent Eur J Math 11(1):27–54, 2013). By combining this result with general results in groupoid -algebras, we prove that the operators are Fredholm between appropriate weighted Sobolev spaces, where I is the identity operator. Furthermore, we establish that the operators are invertible between suitable weighted Sobolev spaces through techniques from Mellin transform. The invertibility of these operators implies a solvability result in weighted Sobolev spaces for the interior and exterior Dirichlet problems on .
... Pseudodifferential operators and groupoid C˚-algebras. We recall here the construction of the space of pseudodifferential operators associated to a Lie groupoid G ⇒ M [30,31,42,41,43,45]. ...
Let W be a three-dimensional wedge, and K be the double layer potential operator associated to W and the Laplacian. We show that 12±K are isomorphisms between suitable weighted Sobolev spaces, which implies a solvability result in weighted Sobolev spaces for the Dirichlet problem on W. Furthermore, we show that the double layer potential operator K is an element in C⁎(G)⊗M2(C), where G is the action (transformation) groupoid M⋊G, with G={(10ab):a∈R,b∈R+}, which is a Lie group, and M is a kind of compactification of G. This result can be used to prove the Fredholmness of 12+KΩ, where Ω is “a domain with edge singularities” and KΩ the double layer potential operator associated to the Laplacian and Ω.
... See [19,22,18,14,27] for a detailed presentation of pseudodifferential calculus on groupoids. ...
Using recently introduced Debord-Skandalis Blup's groupoids we study index theory for a compact foliated manifold with boundary inducing a foliation in its boundary. For this we consider first a blup groupoid whose Lie algebroid has sections consisting of vector fields tangent to the leaves in the interior and tangent to the leaves of the foliation in the boundary. In particular the holonomy b-groupoid allows us to consider the appropriate pseudodifferential calculus and the appropriate index problems. In this paper we further use the blup groupoids, and in particular its functoriality properties, to actually get index theorems. For the previous geometric situtation we have two index morphisms, one related to ellipticity and a second one related to fully ellipticity (generalized Fredholmness). For the first we are able to extend the longitudinal Connes-Skandalis index theorem and use it to get that every b-longitudinal elliptic operator can be perturbed (up to stable homotopy) with a regularizing operator in the calculus to get a fully elliptic operator. For the second index morphism, the one related to fully elliptic operators, we restrict ourselves to the case of families of manifolds with boundary (particular case of foliatons as above) and we prove a new K-theoretical index theorem, i.e. construct a topological index and prove the equality with the analytic-Fredholm index, and use it to get a cohomological index formula for every fully elliptic operator. Previous formulas where obtained for Dirac operators by Bismut and Cheeger, and, Melrose and Piazza. In particular, for a perturbed (or with a choice of a spectral section) family of generalized Dirac operators we can compare our formula with the one by Melrose-Piazza to get a new geometric expression for the eta form of the family.
... It is therefore necessary to study particular cases of V and explicitely construct a suitable groupoid and orbit foliation associated to the Lie structure. This program is still underway, it has been solved for a few particular cases [16], [30], [38] and we also refer to [18] were a general framework is developed which includes constructions of minimal integrating Lie groupoids for many cases of Lie manifolds. We give below some examples for the case of embeddings of Lie manifolds which include the appropriate groupoid correspondences. ...
We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study the problem for embeddings in the category of compact manifolds with corners. The construction of a calculus for elliptic problems is achieved using the theory of Fourier integral operators on Lie groupoids. We show that our calculus is closed under composition and furnishes a so-called noncommutative completion of the given embedding. A representation of the algebra is defined and the continuity of the operators in the algebra on suitable Sobolev spaces is established.
... Iterating (at least locally) this construction, we obtain the b-groupoid of Monthubert for manifolds with corners -cf. [38,39]. ...
We present natural and general ways of building Lie groupoids, by using the classical procedures of blowups and of deformations to the normal cone. Our constructions are seen to recover many known ones involved in index theory. The deformation and blowup groupoids obtained give rise to several extensions of -algebras and to full index problems. We compute the corresponding K-theory maps. Finally, the blowup of a manifold sitting in a transverse way in the space of objects of a Lie groupoid leads to a calculus, quite similar to the Boutet de Monvel calculus for manifolds with boundary.
... the product of the pair groupoid V × V and the transformation groupoid [0, ∞) kF ⋊ (0, ∞) kF .If M has embedded faces, that is, if each hyperface H has a defining function r H , then we can identify G b with an open subset of Monthubert's realization of the b-groupoid[62, Proposition 4.5] ...
We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a "user's guide" to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with poly-cylindrical ends. The reader interested in applications should be able read right away the results related to those examples, beginning with Section 5. Our general, theoretical results are that an operator adapted to the geometry is Fredholm if, and only if, it is elliptic and all its limit operators, in a sense to be made precise, are invertible. Central to our theoretical results is the concept of a Fredholm groupoid, which is the class of groupoids for which this characterization of the Fredholm condition is valid. We use the notions of exhaustive and strictly spectral families of representations to obtain a general characterization of Fredholm groupoids. In particular, we introduce the class of the so-called groupoids with Exel's property as the groupoids for which the regular representations are exhaustive. We show that the class of "stratified submersion groupoids" has Exel's property, where stratified submersion groupoids are defined by glueing fibered pull-backs of bundles of Lie groups. We prove that a stratified submersion groupoid is Fredholm whenever its isotropy groups are amenable. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. This fact is explored to yield Fredholm conditions not only in the above mentioned classes, but also on manifolds that are obtained by desingularization or by blow-up of singular sets.
... amenable and Hausdorff) for the given Lie structure, see e.g. [31]. We make in this work the common assumption that A |M 0 ∼ = T M 0 and G |M 0 ∼ = M 0 × M 0 are the tangent bundle and pair groupoid on the interior respectively. ...
In this work we prove a local index theorem of the Atiyah-Singer type for Dirac operators on so-called manifolds with a Lie structure at infinity (Lie manifolds for short). We introduce the renormalized supertrace on Lie manifolds with spin structure defined on a suitable class of rapidly decaying functions. The proof of the index theorem then relies on a rescaling technique similar in spirit to Getzler's rescaling trick. To a given Lie manifold we use an appropriate integrating Lie groupoid and introduce a rescaling of the heat kernel encoded in a vector bundle over the adiabatic deformation groupoid. The heat kernel is described using a functional calculus with values in the convolution algebra of sections of the rescaled bundle over the adiabatic groupoid. Then we calculate the right coefficient in the heat kernel expansion using the Lichnerowicz theorem on the fibers of the groupoid and the Lie manifold.
... We are interested in Fredholm groupoids because of their applications to Fredholm conditions. Let G be a continuous family groupoid [15] and Ψ m (G) be the space of order m, classical pseudodifferential operators P = (P x ) x∈M on G [15] (see [2,1,22,27,39] for Lie groupoids, which are continuous family groupoids). Recall that, by definition, each P x ∈ Ψ m (G x ), x ∈ M . ...
We characterize the groupoids for which an operator is Fredholm if, and only if, its principal symbol and all its boundary restrictions are invertible. A groupoid with this property is called {\em Fredholm}. Using results on the Effros-Hahn conjecture, we show that an almost amenable, Hausdorff, second countable groupoid is Fredholm. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. In particular, one can use these results to characterize the Fredholm operators on manifolds with cylindrical and poly-cylindrical ends, on manifolds that are asymptotically Euclidean or asymptotically hyperbolic, on products of such manifolds, and on many others. Moreover, we show that the desingularization of Lie groupoids preserves the class of Fredholm groupoids.
... There are, of course, other approaches to the analytic index in the context of groupoids, notably those using quasi-isomorphisms and the " deformation " approach using the tangent groupoid (e.g. [6, 13, 14, 16, 17, 21, 22]). However, these approaches apply (at the present time) only to the case where X = G, and in particular, do not cover the classical families index theorem. ...
Many index theorems (both classical and in noncommutative geometry) can be interpreted in terms of a Lie groupoid acting properly on a manifold and leaving an elliptic family of pseudodifferential operators invariant. Alain Connes in his book raised the question of an index theorem in this general context. In this paper, an analytic index for many such situations is constructed. The approach is inspired by the classical families theorem of Atiyah and Singer, and the proof generalizes, to the case of proper Lie groupoid actions, some of the results proved for proper locally compact group actions by N. C. Phillips.
... Also, for any manifold X, the manifold X × X inherits a canonical structure of Lie groupoid with unit space X and multiplication given by (x, y).(y, z) = (x, z). The reader can find in [27,21,2,19,17,5,16,7,6,23,24,4] further examples of groupoids as well as applications. ...
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 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.
... In fact, our definition of desingularization groupoid provides the necessary results for the construction of integral kernel operators on the resulting blown-up spaces, since the kernels of the resulting integral operators will be defined on the groupoid. It turns out that quite general operators can be obtained using invariant pseudodifferential operators on the groupoid [3,4,41,47]. For instance, by combining this desingularization construction with the construction of psedodifferential operators on groupoids, one can recover the pseudodifferential calculi of Grushin [20], Mazzeo [38], and Schulze [54,55]. ...
We introduce and study a "desingularization" of a Lie groupoid G along an
"A(G)-tame" submanifold L of the space of units M. An A(G)-tame
submanifold is one that has, by definition, a tubular
neighborhood on which A(G) becomes a thick pull-back Lie algebroid. The
construction of the desingularization [[G:L]] of G along L is based on a
canonical fibered pull-back groupoid structure result for G in a neighborhood
of the tame A(G)-submanifold . This local structure result is
obtained by integrating a certain groupoid morphism, using results of Moerdijk
and Mrcun (Amer. J. Math. 2002). Locally, the desingularization [[G:L]] is
defined using a construction of Debord and Skandalis (Advances in Math., 2014).
The space of units of the desingularization [[G:L]] is [M:L], the blow up
of M along L. The space of units and the desingularization groupoid
[[G:L]] are constructed using a gluing construction of Gualtieri and Li (IMRN
2014). We provide an explicit description of the structure of the
desingularized groupoid and we identify its Lie algebroid, which is important
in analysis applications. We also discuss a variant of our construction that is
useful for analysis on asymptotically hyperbolic manifolds. We conclude with an
example relating our constructions to the so called "edge pseudodifferential
calculus." The paper is written such that it also provides an introduction to
Lie groupoids designed for applications to analysis on singular spaces.
... As mentioned in the Introduction, operators invariant with respect to groups appear in analysis on singular spaces, see [1,2,3,9,29,36,51], for example. It would be quite important to extend the results of this paper to operators invariant with respect to bundles of solvable Lie groups [15,25,30,34,49,50]. ...
We continue our study of gauge equivariant K-theory. We thus study the analysis of complexes endowed with the action of a family of compact Lie groups and their index in gauge equivariant K-theory. We introduce various index functions, including an axiomatic one, and show that all index functions coincide. As an application, we prove a topological index theorem for a family D = (D b ) b∈B of gauge-invariant elliptic operators on a G-bundle X → B, where G → B is a locally trivial bundle of compact groups, with typical fiber G. More precisely, one of our main results states that a-ind(D) = t-ind(D) ∈ K G 0 (X), that is, the equality of the analytic index and of the topological index of the family D in the gauge-equivariant K-theory groups of X. The analytic index ind a (D) is defined using analytic properties of the family D and is essentially the difference of the kernel and cokernel K G -classes of D. The topological index is defined purely in terms of the principal symbol of D.
... Remark 2.4. Some references on manifolds with corners are Cerf [2], Douady [3], Gillam and Molcho [7, §4.7], Kottke and Melrose [18], Margalef-Roig and Outerelo Dominguez [20], Melrose [24][25][26], Monthubert [27], and the author [9], [10, §5]. Just as objects, without considering morphisms, most authors define manifolds with corners as in Definition 2.2. ...
In conventional Differential Geometry one studies manifolds, locally modelled
on , manifolds with boundary, locally modelled on
, and manifolds with corners, locally
modelled on . They form categories . Manifolds with corners X have
boundaries , also manifolds with corners, with .
We introduce a new notion of 'manifolds with generalized corners', or
'manifolds with g-corners', extending manifolds with corners, which form a
category with . Manifolds with g-corners are locally modelled on
for P a weakly toric monoid,
where for .
Most differential geometry of manifolds with corners extends nicely to
manifolds with g-corners, including well-behaved boundaries . In
some ways manifolds with g-corners have better properties than manifolds with
corners; in particular, transverse fibre products in exist under
much weaker conditions than in .
This paper was motivated by future applications in symplectic geometry, in
which some moduli spaces of J-holomorphic curves can be manifolds or
Kuranishi spaces with g-corners (see the author arXiv:1409.6908) rather than
ordinary corners.
Our manifolds with g-corners are related to the 'interior binomial varieties'
of Kottke and Melrose in arXiv:1107.3320.
We prove a local index theorem of Atiyah-Singer type for Dirac operators on manifolds with a Lie structure at infinity (Lie manifolds for short). With the help of a renormalized supertrace, defined on a suitable class of regularizing operators, the proof of the index theorem relies on a rescaling technique similar in spirit to Getzler's rescaling. With a given Lie manifold we associate an appropriate integrating Lie groupoid. We then describe the heat kernel of a geometric Dirac operator via a functional calculus with values in the convolution algebra of sections of a rescaled bundle over the adiabatic groupoid. Finally, we calculate the right coefficient in the heat kernel expansion by deforming the Dirac operator into a polynomial coefficient operator over the rescaled bundle and applying the Lichnerowicz theorem to the fibers of the groupoid and the Lie manifold.
In this paper, we investigate topological aspects of indices of twisted geometric operators on manifolds equipped with fibered boundaries. We define K-groups relative to the pushforward for boundary fibration, and show that indices of twisted geometric operators, defined by complete or edge metrics, can be regarded as the index pairing over these K-groups. We also prove various properties of these indices using groupoid deformation techniques. Using these properties, we give an application to the localization problem of signature operators for singular fiber bundles.
Manifolds with boundary and with corners form categories . A manifold with corners X has two notions of tangent bundle: the tangent bundle TX, and the b-tangent bundle . The usual definition of smooth structure uses TX, as is defined to be smooth if exists as a continuous section of for all . We define 'manifolds with analytic corners', or 'manifolds with a-corners', with a different smooth structure, in which roughly is smooth if exists as a continuous section of for all . These are different from manifolds with corners even when , for instance is smooth for all real when has a-corners. Manifolds with a-boundary and with a-corners form categories , with well behaved differential geometry. Partial differential equations on manifolds with boundary may have boundary conditions of two kinds: (i) 'at finite distance', e.g. Dirichlet or Neumann boundary conditions, or (ii) 'at infinity', prescribing the asymptotic behaviour of the solution. We argue that manifolds with corners should be used for (i), and with a-corners for (ii). We discuss many applications of manifolds with a-corners in boundary problems of type (ii), and to singular p.d.e. problems involving 'bubbling', 'neck-stretching' and 'gluing'.
We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid. As applications, we give a new proof of the Poincare-Birkhoff-Witt theorem for Lie algebroids and a concrete quantization of the Lie-Poisson structure on the dual A* of a Lie algebroid.
Mémoire d'habilitation à diriger des recherches, soutenue en novembre 2014.
We announce a reformulation of the conjecture in (8,9,10). The advantage of the new version is that it is simpler and applies more generally than the earlier statement. A key point is to use the universal example for proper actions introduced in (10). There, the universal example seemed somewhat peripheral to the main issue. Here, however, it will play a central role.
1 Transverse measure for flows 4 2 Transverse measure for foliations 6
We compute K-theory invariants of algebras of pseudodifferential operators on manifolds with corners and prove an equivariant index theorem
for operators invariant with respect to an action of . We briefly discuss the relation between our results and the -invariant.
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 construct algebras of pseudodifferential operators on a continuous family groupoid G that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on G as a dense subalgebra, and reflect the smooth structure of the groupoid G, when G is smooth. As an application, we get a better understanding on the structure of inverses of elliptic pseudodifferential operators on classes of non-compact manifolds. For the construction of these algebras closed under holomorphic functional calculus, we develop three methods: one using two-sided semi-ideals, one using commutators, and one based on Schwartz spaces on the groupoid.
We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid. As applications, we give a new proof of the Poincar\'e-Birkhoff-Witt theorem for Lie algebroids and a concrete quantization of the Lie-Poisson structure on the dual of a Lie algebroid.
We compute K-theory invariants of algebras of pseudodifferential operators on manifolds with corners and prove an equivariant index theorem for operators invariant with respect to an action of We discuss the relation between our results and the -invariant.
The Hochschild and cyclic homology groups are computed for the algebra of `cusp' pseudodifferential operators on any compact manifold with boundary. The index functional for this algebra is interpreted as a Hochschild 1-cocycle and evaluated in terms of extensions of the trace functionals on the two natural ideals, corresponding to the two filtrations by interior order and vanishing degree at the boundary, together with the exterior derivations of the algebra. This leads to an index formula which is a pseudodifferential extension of that of Atiyah, Patodi and Singer for Dirac operators; together with a symbolic term it involves the `eta' invariant on the suspended algebra over the boundary previously introduced by the first author.
1. Introduction . The main purpose of this paper is to present a generalization of Hirzebruch's signature theorem for the case of manifolds with boundary. Our result is in the framework of Riemannian geometry and can be viewed as analogous to the Gauss–Bonnet theorem for manifolds with boundary, although there is a very significant difference between the two cases which is, in a sense, the central topic of the paper. To explain this difference let us begin by recalling that the classical Gauss–Bonnet theorem for a surface X with boundary Y asserts that the Euler characteristic E(X) is given by a formula:
where K is the Gauss curvature of X and σ is the geodesic curvature of Y in X . In particular if, near the boundary, X is isometric to the product Y x R ⁺ , the boundary integral in (1.1) vanishes and the formula is the same as for closed surfaces. Similar remarks hold in higher dimensions. Now if X is a closed oriented Riemannian manifold of dimension 4, there is another formula relating cohomological invariants with curvature in addition to the Gauss–Bonnet formula. This expresses the signature of the quadratic form on H ² ( X , R) by an integral formula
where p 1 is the differential 4-form representing the first Pontrjagin class and is given in terms of the curvature matrix R by p 1 = (2π) ⁻² Tr R ² . It is natural to ask if (1.2) continues to hold for manifolds with boundary, provided the metric is a product near the boundary. Simple examples show that this is false, so that in general
Let G be a Lie groupoid. The pseudodifferential calculus over G defines the analytic index. The tangent groupoid induces a morphism in K-theory; we prove that it coincides with the analytic index. Thus the latter may be defined without considering the pseudodifferential calculus.
Thesis (Ph. D. in Mathematics)--University of California, Berkeley, Dec. 1979. Includes bibliographical references (leaves 213-218).
f problems to be attacked is that of compact manifolds with corners; the morphisms are the b--maps, discussed below. Such a space is a compact topological manifold with boundary, X, with a structure locally modeled on the spaces (1) R k = [0, #) n-k . This fixes the space # (X) of smooth functions on X ; locally these are just the functions on R k which are the restrictions of # functions on R . For simplicity we also insist that all the boundary hypersurfaces of X are embedded. If we let M 1 (X) be the set of boundary hypersurfaces then this means that each H M 1 (X) has a defining function (2) #H # (X) s.t. H = {#H = 0}, #H 0, d#H 0 at H. Typeset by A M S-T E X We denote by M (1) (X) the set of true boundary faces, the connected closed submanifolds locally given by intersection of the boundary hypersurfaces and set M(X) = M (1) (X)#{X}; all these submanifolds are embedded. Limiting the analysis to compact manifolds with corners implies that in m
The aim of this paper is to explain, mostly through examples, what groupoids are and how they describe symmetry. We will begin with elementary examples, with discrete symmetry, and end with examples in the differentiable setting which involve Lie groupoids and their corresponding infinitesimal objects, Lie algebroids.
Groupo. ıdes et calcul pseudo-diffe´ sur les varie´ varie´eá coins
- B Monthubert
B. Monthubert, Groupo. ıdes et calcul pseudo-diffe´ sur les varie´ varie´eá coins, Ph.D. Thesis, Universite´ 7, 1998.
The Analysis of Linear Partial Differential Operators III: Pseudodifferential Operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathema-tical Sciences
- L Ho¨
L. Ho¨, The Analysis of Linear Partial Differential Operators III: Pseudodifferential Operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathema-tical Sciences], Vol. 274, Springer, Berlin, 1994.