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Groupoids of manifolds with corners and index theory
... Groupoids and groupoid C * -algebras have appeared useful in the analysis over singular spaces and, in particular, spaces with conical singularities, see for instance [1,2,9,10,11,20,21,37,39,42]. ...
... Pseudodifferential operators and groupoid C * -algebras. We recall here the construction of the space of pseudodifferential operators associated to a Lie groupoid G with units M [20,21,38,37,40,47]. The dimension of M is n > 0. ...
... where M 2 0 denotes the pair groupoid of M 0 := int(M ) and ∂ j M denote the connected components of ∂M . Then G b can be given the structure of a Lie groupoid with units M and we have that it integrates b T M , that is, [30,38,37,47] for details. ...
To a domain with conical points \Omega, we associate a natural C*-algebra
that is motivated by the study of boundary value problems on \Omega, especially
using the method of layer potentials. In two dimensions, we allow \Omega to be
a domain with ramified cracks. We construct an explicit groupoid associated to
the boundary of \Omega and use the theory of pseudodifferential operators on
groupoids and its representations to obtain our layer potentials C*-algebra. We
study its structure, compute the associated K-groups, and prove Fredholm
conditions for the natural pseudodifferential operators affiliated to this
C*-algebra.
... induce isomorphisms in K-theory, the K-theory long exact sequence induced by (31) can be written as the top row of the diagram ...
... Therefore, the nonvanishing of ∂ϕ(a) is a necessary and sufficient condition for the existence of a lifting of a ∈ Ell(M, X) to Ell(M). The boundary map in K-homology plays a similar role in other problems (see Baum-Douglas [29], Roe [30], and Monthubert [31]). ...
We find the stable homotopy classification of elliptic operators on stratified manifolds. Namely, we establish an isomorphism of the set of elliptic operators modulo stable homotopy and the K-homology group of the singular manifold. As a corollary, we obtain an explicit formula for the obstruction of Atiyah--Bott type to making interior elliptic operators Fredholm.
... 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]. ...
... 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 ...
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 ...
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.
... induce isomorphisms in K-theory, the K-theory long exact sequence induced by (31) can be written as the top row of the diagram ...
... Therefore, the nonvanishing of ∂ϕ(a) is a necessary and sufficient condition for the existence of a lifting of a ∈ Ell(M, X) to Ell(M). The boundary map in K-homology plays a similar role in other problems (see Baum-Douglas [29], Roe [30], and Monthubert [31]). ...
The problem of homotopy classification of elliptic operators on an arbitrary stratified manifold is discussed. A classification of elliptic operators on smooth manifolds up to homotopy is an essential point in the solution of the index problem by Atiyah and Singer and the classification is in terms of the K homology of a stratified manifold. The ideas of Atiyah allow to avoid the difficulties caused by the fact that the In the situation under consideration,the operators module compact operators are determined by whole sets of symbols of strata rather than by only one symbol, and the ellipticity condition is of an infinite-dimensional character. As application an index formula and a topological obstruction for Fredholm problems is obtained and to calculate the K group algebras of pseudodifferential operators(PDOs). Stable homotopy is an equivalence on the set of elliptic PDOs acting on sections of bundles.
... However, recent counterexamples [36] show that other ways of attacking the problem need to be discov-ered. Applications of the K-theory of groupoids include: tilings and gap labeling (see for instance [40]), index theorems, and pseudodifferential calculi [50,57]. ...
In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an -gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure are derived. Our approach provides a uniform framework for studying various twisted K-theories including the usual twisted K-theory of topological spaces, twisted equivariant K-theory, and the twisted K-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted K-groups can be expressed by so-called "twisted vector bundles". Our approach is to work on presentations of stacks, namely \emph{groupoids}, and relies heavily on the machinery of K-theory (KK-theory) of -algebras.
... A number of papers are relevant to this construction in the context where X = G, e.g. [25,26,27,33,34,32,35,41,43]. A version of our construction under special assumptions -including the assumptions that G 0 is a proper, G-compact, G-space and that X is a fiber bundle over G 0 with compact smooth manifold as fiber -is proved in [46]. ...
The paper constructs the analytic index for an elliptic pseudodifferential family of L^{m}_{\rho,\de}-operators invariant under the proper action of a continuous family groupoid on a G-compact, G-space.
... 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.
... Граничное отображение в K-гомологиях играет аналогичную роль препятствия в других задачах (см. Баум-Дуглас [31] , Роэ [32], Монтубер [33]). ...
28 декабря 2006 г. Аннотация В работе дана гомотопическая классификация эллиптических операторов на стратифицированных многообразиях. Именно, получен изоморфизм множе-ства эллиптических операторов по модулю стабильных гомотопий и группы K-гомологий многообразия. В качестве приложения классификации дается явная формула для препятствия типа Атьи–Ботта к постановке фредгольмовых задач в ситуации стратифицированных многообразий. 1 Введение В классической работе Атьи [1] было замечено, что абстрактные эллиптические опе-раторы на компактном пространстве X (напомним, что так называют фредгольмовы операторы в C(X)-модулях перестановочные с операторами умножения на функции с точностью до компактных операторов) определяют элементы группы K-гомологий пространства X. Более того, Каспаров [2] и Браун–Дуглас–Филмур [3] показали, что можно получить не только элементы K-гомологических групп, но и реализацию K-гомологий как обобщенной теории гомологий, если профакторизовать абстрактные эллиптические операторы по отношению эквивалентности стабильной гомотопии. Оказывается, однако, что для гладких многообразий группу K-гомологий мож-но получить, если вместо абстрактных операторов ограничиться дифференциальны-ми (псевдодифференциальными) операторами, естественными в теории уравнений с частными производными. Более того, если при этом многообразие дополнительно имеет spin c -структуру, то вообще достаточно рассматривать только (скрученные) операторы Дирака. Этот пример служит мотивировкой естественной задачи срав-нения группы K-гомологий и группы, порожденной эллиптическими псевдодиффе-ренциальными операторами (в дальнейшем ПДО) для негладких пространств (ср. с проблемой Зингера в [4]), в частности, для стратифицированных многообразий. К настоящему времени решение этой задачи известно в некоторых частных слу-чаях. Так, для многообразий с изолированными особенностями классификация об-щих эллиптических ПДО в терминах K-гомологий была установлена в [5]. Этот ре-зультат был независимо доказан с использованием группоидов и KK-теории в [6, 7]. 1 Для стратифицированных многообразий с двумя стратами классификация реберно-вырождающихся эллиптических операторов получена в [8], [9]. Несмотря на то, что исчисление ПДО на общих стратифицированных многообра-зиях не является чем-то новым (оно было построено, например, в [10] или, в рамках предложенного в [11] общего подхода к построению ПДО, ассоциированных с задан-ной алгеброй Ли векторных полей, и с использованием аппарата теории группоидов, в [12, 13] и других работах), результаты о гомотопической классификации эллип-тических операторов на таких многообразиях до сих пор известны не были. Такие результаты устанавливаются в настоящей работе. Основная теорема настоящей работы утверждает, что для компактного страти-фицированного многообразия X с произвольным конечным числом стратов имеет место изоморфизм групп Ell(X) K 0 (X), (1) где через Ell(X) обозначена группа, порожденная эллиптическими псевдодифферен-циальными операторами на многообразии X по модулю стабильных гомотопий, а K 0 (X) обозначает группу четных K-гомологий многообразия X. Частные случаи этого изоморфизма были получены в цитированных выше работах [5–9]. Изоморфизм (1) позволяет перенести топологические методы из теории K-гомо-логий в эллиптическую теорию. В качестве примеров приложения топологических методов мы вычисляем препятствие типа Атьи–Ботта к постановке фредгольмовых задач на стратифицированных многообразиях, а также даем обобщение теоремы об инвариантности индекса относительно кобордизма (см. параграф 8). Кроме указанных приложений к эллиптическим операторам изоморфизм (1) име-ет интересную интерпретацию в рамках некоммутативной геометрии. Дело здесь в том, что с точки зрения некоммутативной геометрии алгебра ПДО на стратифи-цированном многообразии отвечает некоторому группоиду (см. [15, 16]). При этом группа Ell(X) связана с K-группой C * -алгебры группоида [13]. Известная гипотеза Баума–Конна [14] утверждает, что последняя K-группа C * -алгебры изоморфна то-пологической K-группе для классифицирующего пространства группоида (см. [17]). Явные вычисления для простейших стратифицированных многообразий показывают, что K-группа классифицирующего пространства изоморфна K 0 (X) правой части соотношения (1). Интересным представляется дальнейшее сравнение изоморфизма (1) с отображением Баума–Конна. Заключая введение, мы хотели бы выразить признательность профессору Т. Фа-ку (Лион), который указал на возможную связь изоморфизма (1) с изоморфизмом Баума–Конна для группоидов. Изложенные здесь результаты были доложены на международной конференции "Workshop on index theory", Münster (4–8 ноября 2005 г.). Работа частично поддержа-на грантами РФФИ 05-01-00928, 06-01-00098, грантом президента МК-1713.2005.1, а также грантом DFG 436 RUS 113/849/0-1 and noncommutative geometry of stratified manifolds".
... The problem was to define tools which would be independent of particular cases of applications. While in global analysis each pseudodifferential calculus is defined in the context of a certain type of manifolds, we wanted to define once for all the pseudodifferential calculus on a groupoid so that it is sufficient to define the groupoid adapted to the type of manifold of interest (see section 2.1, and [27] [23] [24] [22] [25]). The first studies have been made in parallel to and independently of those of V. Nistor, A. Weinstein and P. Xu. ([29]). ...
This survey of the work of the author with several collaborators presents the way groupoids appear and can be used in index theory. We define the general tools, and apply them to the case of manifolds with corners, ending with a topological index theorem.
... However, recent counterexamples [36] show that other ways of attacking the problem need to be discov-ered. Applications of the K-theory of groupoids include: tilings and gap labeling (see for instance [40]), index theorems, and pseudodifferential calculi [50,57]. ...
In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an S1-gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure are derived. Our approach provides a uniform framework for studying various twisted K-theories including the usual twisted K-theory of topological spaces, twisted equivariant K-theory, and the twisted K-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted K-groups can be expressed by so-called “twisted vector bundles”.Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of K-theory (KK-theory) of C∗-algebras.RésuméDans cet article, nous développons la K-théorie tordue pour les champs différentiables, où la torsion s'effectue par une S1-gerbe sur le champ en question. Nous en établissons les propriétés générales telles que les suites exactes de Mayer–Vietoris, la périodicité de Bott, et le produit . Notre approche fournit un cadre général permettant d'étudier diverses K-théories tordues, en particulier la K-théorie tordue usuelle des espaces topologiques, la K-théorie tordue équivariante, et la K-théorie tordue des orbifolds. Nous donnons également une définition équivalente utilisant des opérateurs de Fredholm, et nous discutons les conditions sous lesquelles les groupes de K-théorie tordue peuvent être réalisés à partir de “fibrés vectoriels tordus”.Notre approche consiste à travailler sur les réalisations concrètes des champs, à savoir les groupoïdes, et s'appuie de façon importante sur les techniques de K-théorie (KK-théorie) des C∗-algèbres.
... If F = E we get an algebra denoted by Ψ * c (G, E). Basic examples of the usefulness of these operators are the case of foliations [6, 28] and manifolds with corners [16]. This calculus is also used in [7] to define KK-theory classes and to compute some Kasparov products. ...
In a earlier work of Claire Debord and the author, a notion of noncommutative tangent space isdefined for a conical pseudomanifold and the Poincar\'e 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 Poincar\'e duality in the singular setting as a principal symbol map.
... A number of papers are relevant to this construction in the context where X = G, e.g. [25,26,27,33,34,32,35,41,43]. A version of our construction under special assumptions -including the assumptions that G 0 is a proper, G-compact, G-space and that X is a fiber bundle over G 0 with compact smooth manifold as fiber -is proved in [46]. ...
The paper constructs the analytic index for an elliptic pseudodifferential family of L^{m}_{\rho,\de}-operators invariant under the proper action of a continuous family groupoid on a G-compact, G-space.
The computation of a stable homotopic classification of elliptic operators is an important problem of elliptic theory. The
classical solution of this problem is given by Atiyah and Singer for the case of smooth compact manifolds. It is formulated
in terms of K-theory for a cotangent fibering of the given manifold. It cannot be extended for the case of nonsmooth manifolds because
their cotangent fiberings do not contain all necessary information. Another Atiyah definition might fit in such a case: it
is based on the concept of abstract elliptic operators and is given in term of K-homologies of the manifold itself (instead of its fiberings). Indeed, this theorem is recently extended for manifolds with
conic singularities, ribs, and general so-called stratified manifolds: it suffices just to replace the phrase “smooth manifold”
by the phrase “stratified manifold” (of the corresponding class). Thus, stratified manifolds is a strange phenomenon in a
way: the algebra of symbols of differential (pseudodifferential) operators is quite noncommutative on such manifolds (the
symbol components corresponding to strata of positive codimensions are operator-valued functions), but the solution of the
classification problem can be found in purely geometric terms. In general, it is impossible for other classes of nonsmooth
manifolds. In particular, the authors recently found that, for manifolds with angles, the classification is given by a K-group of a noncommutative C* -algebra and it cannot be reduced to a commutative algebra if normal fiberings of faces of the considered manifold are nontrivial.
Note that the proofs are based on noncommutative geometry (more exactly, the K-theory of C* -algebras) even in the case of stratified manifolds though the results are “classical.” In this paper, we provide a review
of the abovementioned classification results for elliptic operators on manifolds with singularities and corresponding methods
of noncommutative geometry (in particular, the localization principle in C* -algebras).
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