Reduction of the calculus of pseudodifferential operators on a noncompact manifold to the calculus on a compact manifold of doubled dimension

For three classes of elliptic pseudodifferential operators on a compact manifold with boundary which have "geometric K-theory", namely the "transmission algebra" introduced by Boutet de Monvel [5], the "zero algebra" introduced by Mazzeo in [9, 10] and the "scattering algebra" from [16], we give explicit formulas for the Chern character of the index bundle in terms of the symbols (including normal operators at the boundary) of a Fredholm family of fiber operators. This involves appropriate descriptions, in each case, of the cohomology with compact supports in the interior of the total space of a vector bundle over a manifold with boundary in which the Chern character, mapping from the corresponding realization of K-theory, naturally takes values.

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On equivalence of two approaches in index theory

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is008001004jkt026 ◽

2008 ◽

Vol 2 (3) ◽

pp. 445-452

Author(s):

V. Manuilov

Keyword(s):

Commutative Algebra ◽

Compact Manifold ◽

Pseudodifferential Operators ◽

Index Theory ◽

Index Theorem ◽

Continuous Functions ◽

Zero Section ◽

Order Zero ◽

The One

AbstractThe algebra Ψ(M) of order zero pseudodifferential operators on a compact manifoldMdefines a well-knownC*-extension of the algebraC(S*M) of continuous functions on the cospherical bundleS*M⊂T*Mby the algebra К of compact operators. In his proof of the index theorem, Higson defined and used an asymptotic homomorphismTfromC0(T*M) to К, which plays the role of a deformation for the commutative algebraC0(T*M). Similar constructions exist also for operators and symbols with coefficients in aC*-algebra. Recently we have shown that the image of the above extension under the Connes–Higson construction isTand that this extension can be reconstructed out ofT. That is why the classical approach to the index theory coincides with the one based on asymptotic homomorphisms. But the image of the above extension is defined only outside the zero section ofT*(M), so it may seem that the information encoded in the extension is not the same as that in the asymptotic homomorphism. We show that this is not the case.

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Pseudodifferential operators on differential groupoids

Pacific Journal of Mathematics ◽

10.2140/pjm.1999.189.117 ◽

1999 ◽

Vol 189 (1) ◽

pp. 117-152 ◽

Cited By ~ 77

Author(s):

Victor Nistor ◽

Alan Weinstein ◽

Ping Xu

Keyword(s):

Pseudodifferential Operators

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Renormalised energies and renormalisable singular harmonic maps into a compact manifold on planar domains

Mathematische Annalen ◽

10.1007/s00208-021-02204-8 ◽

2021 ◽

Author(s):

Antonin Monteil ◽

Rémy Rodiac ◽

Jean Van Schaftingen

Keyword(s):

Compact Manifold ◽

Harmonic Maps ◽

Planar Domains

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Diffeomorphism cocycles over partially hyperbolic systems

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.131 ◽

2020 ◽

pp. 1-24

Author(s):

VICTORIA SADOVSKAYA

Keyword(s):

Hyperbolic System ◽

Compact Manifold ◽

Hyperbolic Systems ◽

Riemannian Metrics ◽

Hölder Continuous ◽

Group Of Diffeomorphisms ◽

Small Loss ◽

Partially Hyperbolic ◽

Loss Of Regularity

Abstract We consider Hölder continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold $\mathcal {M}$ . We obtain several results for this setting. If a cocycle is bounded in $C^{1+\gamma }$ , we show that it has a continuous invariant family of $\gamma $ -Hölder Riemannian metrics on $\mathcal {M}$ . We establish continuity of a measurable conjugacy between two cocycles assuming bunching or existence of holonomies for both and pre-compactness in $C^0$ for one of them. We give conditions for existence of a continuous conjugacy between two cocycles in terms of their cycle weights. We also study the relation between the conjugacy and holonomies of the cocycles. Our results give arbitrarily small loss of regularity of the conjugacy along the fiber compared to that of the holonomies and of the cocycle.

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