Elliptic Operators on Manifolds with Singularities and K-Homology

Savin, Anton

doi:10.1007/s10977-005-1515-1

Cited by 24 publications

(29 citation statements)

References 31 publications

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“…Proof. This can be seen as a particular case of a result of Savin [46,Theorem 4]. Alternatively, since Theorem 9.11 identifies FE S (X) with the relative K-group K(q) associated to the homomorphism q : A 0 → A/K (see for instance [4] or [16] for a definition of K(q)), we can follow instead the approach in [16,Theorem 3.29].…”

Section: The Semiclassical S-calculusmentioning

confidence: 99%

Pseudodifferential operators on manifolds with fibred corners

Debord¹,

Lescure²,

Rochon³

2015

Annales De l'Institut Fourier

103

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One way to geometrically encode the singularities of a stratified pseudomanifold is to endow its interior with an iterated fibred cusp metric. For such a metric, we develop and study a pseudodifferential calculus generalizing the Φ-calculus of Mazzeo and Melrose. Our starting point is the observation, going back to Melrose, that a stratified pseudomanifold can be 'resolved' into a manifold with fibred corners. This allows us to define pseudodifferential operators as conormal distributions on a suitably blown-up double space. Various symbol maps are introduced, leading to the notion of full ellipticity. This is used to construct refined parametrices and to provide criteria for the mapping properties of operators such as Fredholmness or compactness. We also introduce a semiclassical version of the calculus and use it to establish a Poincaré duality between the K-homology of the stratified pseudomanifold and the K-group of fully elliptic operators. Contents 10 3. The definition of S-pseudodifferential operators 13 4. Groupoids 17 5. Action of S-pseudodifferential operators 21 6. Suspended S-operators 24 7. Symbol Maps 28 8. Composition 31 9. Mapping properties 36 10. The semiclassical S-calculus 46 11. Poincaré duality 53 References 62

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Section: The Semiclassical S-calculusmentioning

confidence: 99%

Pseudodifferential operators on manifolds with fibred corners

Debord¹,

Lescure²,

Rochon³

2015

Annales De l'Institut Fourier

103

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“…One of the main results in [40], see also [47,52,51,59] for similar results using different methods, is that the inverse of the map Σ X of Equation (0.2) can be realized, as in the smooth case, by a map that assigns to each element in K 0 (A X ) an elliptic operator. Thus elements of K 0 (A X ) can be viewed as the symbols of some natural elliptic pseudodifferential operators realizing the K-homology of X.…”

Section: Introductionmentioning

confidence: 99%

Groupoids and an index theorem for conical pseudo-manifolds

Debord¹,

Lescure²,

Nistor³

2009

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold M . A main new ingredient in our proof is a non-commutative algebra that plays in our setting the role of C 0 (T * M ). We prove a Thom isomorphism between non-commutative algebras which gives a new example of wrong way functoriality in K-theory. We then give a new proof of the Atiyah-Singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds.

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“…The homotopy classification of elliptic operators in Boutet de Monvel's algebra was obtained on a manifold with a boundary [3,4]. The homotopy classification was also obtained for many classes of manifolds with singularities [5]- [7]. There were considered the applications of the classification to calculating obstruction of Atiyah-Bott type [8] for the existence of elliptic problems on manifolds with singularities, to the description of Poincaré duality on manifolds with singularities and others [9]- [13].…”

Section: Introductionmentioning

confidence: 99%

Homotopy classification of elliptic problems associated with discrete group actions on manifolds with boundary

Savin¹,

Sternin²

2016

Ufimskii Matematicheskii Zhurnal

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Given an action of a discrete group on a smooth compact manifold with a boundary, we consider a class of operators generated by pseudodifferential operators on and shift operators associated with the group action. For elliptic operators in this class, we obtain a classification up to stable homotopies and show that the group of stable homotopy classes of such problems is isomorphic to the-group of the crossed product of the algebra of continuous functions on the cotangent bundle over the interior of the manifold and the group acting on this algebra by automorphisms.

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Elliptic Operators on Manifolds with Singularities and K-Homology

Cited by 24 publications

References 31 publications

Pseudodifferential operators on manifolds with fibred corners

Pseudodifferential operators on manifolds with fibred corners

Groupoids and an index theorem for conical pseudo-manifolds

Homotopy classification of elliptic problems associated with discrete group actions on manifolds with boundary

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