A Topological Approach to Indices of Geometric Operators on Manifolds with Fibered Boundaries

Yamashita, Mayuko

doi:10.1007/s00220-019-03595-1

Cited by 2 publications

(1 citation statement)

References 27 publications

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“…Piazza and Zenobi ([97]) actually realized that the groupoid constructed in [44] integrating this algebroid can be obtained via a double blowup construction. See also [117] for topological aspects of indices in this context.…”

Section: Let Us Outline Specific Examplesmentioning

confidence: 99%

Lie groupoids, pseudodifferential calculus, and index theory

Debord¹,

Skandalis²

2019

Advances in Noncommutative Geometry

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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. Lie groupoids and their operators algebrasWe refer to [78,80] for the classical definitions and constructions related to groupoids and their Lie algebroids. The construction of the C * -algebra of a groupoid is due to Jean Renault [100], one can look at his course [101] which is mainly devoted to locally compact groupoids. Lie groupoids GeneralitiesA groupoid is a small category in which every morphism is an isomorphism. Thus a groupoid G is a pair (G (0) , G (1) ) of sets together with structural morphisms:Units and arrows. The set G (0) denotes the set of objects (or units) of the groupoid, whereas the set G (1) is the set of morphisms (or arrows). The unit map u : G (0) → G (1) is the injective map which assigns to any object of G its identity morphism.The source and range maps s, r : G (1) → G (0) are (surjective) maps equal to identity in restriction toThe inverse ι : G (1) → G (1) is an involutive map which exchanges the source and range:for α ∈ G, (α −1 ) −1 = α and s(α −1 ) = r(α), where α −1 denotes ι(α)

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Section: Let Us Outline Specific Examplesmentioning

confidence: 99%

Lie groupoids, pseudodifferential calculus, and index theory

Debord¹,

Skandalis²

2019

Advances in Noncommutative Geometry

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Renormalized Index Formulas for Elliptic Differential Operators on Boundary Groupoids

Qiao¹,

So²

2021

Preprint

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We consider the index problem of certain boundary groupoids of the formSince it has been shown that when q is odd and ≥ 3, K 0 (C * (G)) ∼ = Z, 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 [41]. However, we find that when q ≥ 3, 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 M 1 lies in M .

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A Topological Approach to Indices of Geometric Operators on Manifolds with Fibered Boundaries

Cited by 2 publications

References 27 publications

Lie groupoids, pseudodifferential calculus, and index theory

Lie groupoids, pseudodifferential calculus, and index theory

Renormalized Index Formulas for Elliptic Differential Operators on Boundary Groupoids

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