K-duality for pseudomanifolds with isolated singularities

Debord, Claire; Lescure, Jean-Marie

doi:10.1016/j.jfa.2004.03.017

Cited by 21 publications

(56 citation statements)

References 17 publications

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“…where V is a closed smooth manifold ( [21,8], see also [13] for a description of the Dirac element in terms of groupoids). This duality allows to recover that the usual quantification and principal symbol maps are mutually inverse isomorphisms in K -theory:…”

Section: Example 1 a Basic Example Ismentioning

confidence: 99%

“…It is thus a sequel of [13], but can be read independently. At first glance, one should have expected that the technics of [13] iterate easily to give the general result.…”

Section: Introductionmentioning

confidence: 99%

“…It is thus a sequel of [13], but can be read independently. At first glance, one should have expected that the technics of [13] iterate easily to give the general result. In fact, although the definition of the groupoid giving the noncommutative tangent space itself is natural and intuitive in the general case, its smoothness is quite intricate and brings issues that did not exist in the conical case.…”

Section: Introductionmentioning

confidence: 99%

“…We have given here a detailed treatment of this point, since we believe that this material will be usefull in further studies about the geometry of stratified spaces. Another difference with [13] is that we have given up the explicit construction of a dual Dirac element. Instead, we use an easily defined Dirac element and then prove the Poincaré duality by an induction, based on an operation called unfolding which consists in removing the minimal strata in a pseudomanifold and then "doubling" it to get a new pseudomanifold, less singular.…”

Section: Introductionmentioning

confidence: 99%

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K–duality for stratified pseudomanifolds

Debord

Lescure

2009

Geom. Topol.

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This paper continues the project started in [13] where Poincaré duality in K -theory was studied for singular manifolds with isolated conical singularities. Here, we extend the study and the results to general stratified pseudomanifolds. We review the axiomatic definition of a smooth stratification S of a topological space X and we define a groupoid T S X , called the S-tangent space. This groupoid is made of different pieces encoding the tangent spaces of strata, and these pieces are glued into the smooth noncommutative groupoid T S X using the familiar procedure introduced by A. Connes for the tangent groupoid of a manifold. The main result is that C * (T S X) is Poincaré dual to C(X), in other words, the S-tangent space plays the role in K -theory of a tangent space for X . 58B34, 46L80, 19K35, 58H05, 57N80; 19K33, 19K56, 58A35 IntroductionThis paper takes place in a longstanding project aiming to study index theory and related questions on stratified pseudomanifolds using tools and concepts from noncommutative geometry.The key observation at the beginning of this project is that in its K -theoritic form, the Atiyah-Singer index theorem [2] involves ingredients that should survive to the singularities allowed in a stratified pseudomanifold. This is possible, from our opinion, as soon as one accepts reasonable generalizations and new presentation of certain classical objects on smooth manifolds, making sense on stratified pseudomanifolds.The first instance of these classical objects that need to be adapted to singularities is the notion of tangent space. Since index maps in [2] are defined on the K -theory of the tangent spaces of smooth manifolds, one must have a similar space adapted to stratified pseudomanifolds. Moreover, such a space should satisfy natural requirements. It should coincide with the usual notion on the regular part of the pseudomanifold and incorporate in some way copies of usual tangent spaces of strata, while keeping enough smoothness to allow interesting computations. Moreover, it should be Poincaré dual In [13], we introduced a candidate to be the tangent space of a pseudomanifold with isolated conical singularities. It appeared to be a smooth groupoid, leading to a noncommutative C * -algebra, and we proved that it fulfills the expected K -duality.In [22], the second author interpreted the duality proved in [13] as a principal symbol map, thus recovering the classical picture of Poincaré duality in K -theory for smooth manifolds. This interpretation used a notion of noncommutative elliptic symbols, which appeared to be the cycles of the K -theory of the noncommutative tangent space.In [14], the noncommutative tangent space together with other deformation groupoids was used to construct analytical and topological index maps, and their equality was proved. As expected, these index maps are straight generalizations of those of [2] for manifolds.The present paper is devoted to the construction of the noncommutative tangent space for a general stratified pseudomanifold and the proof of ...

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Section: Example 1 a Basic Example Ismentioning

confidence: 99%

“…It is thus a sequel of [13], but can be read independently. At first glance, one should have expected that the technics of [13] iterate easily to give the general result.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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K–duality for stratified pseudomanifolds

Debord

Lescure

2009

Geom. Topol.

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“…We can improve this result (for manifolds with boundary for the moment) by using the works of C. Debord and J. M. Lescure ( [9]). Indeed, the groupoid G 1 is KKequivalent to their "tangent space" ( [9], remark 4). A totally elliptic pseudo-differential operator leads then to an element of the K-homology of the manifold with conical singularities associated.…”

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confidence: 99%

Contribution of noncommutative geometry to index theory on singular manifolds

Monthubert

2007

Banach Center Publications

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Abstract. 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.1. Introduction. Quantum physics, by putting the notion of operator at the very centre of its mathematical development, generated numerous index problems.In its simplest form, the index of an operator is a number. It is the difference between the dimension of the space of solutions of the equation associated to this operator, and the restrictions it imposes on the image space of this operator:

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Elliptic Operators on Manifolds with Singularities and K-Homology

Savin¹

2005

K-Theory

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Elliptic operators on smooth compact manifolds are classified by K-homology. We prove that a similar classification is valid also for manifolds with simplest singularities: isolated conical points and edges. The main ingredients of the proof of these results are: Atiyah-Singer difference construction in the noncommutative case and Poincare isomorphism in K-theory for (our) singular manifolds. As an application we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with edges. MSC2000: 58J05(Primary) 19K33 35S35 47L15(Secondary)

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K-duality for pseudomanifolds with isolated singularities

Cited by 21 publications

References 17 publications

K–duality for stratified pseudomanifolds

K–duality for stratified pseudomanifolds

Contribution of noncommutative geometry to index theory on singular manifolds

Elliptic Operators on Manifolds with Singularities and K-Homology

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