Higher invariants in noncommutative geometry

Xie, Zhizhang; Yu, Guoliang

doi:10.1007/978-3-030-29597-4_12

Cited by 8 publications

(7 citation statements)

References 116 publications

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“…This theory was first introduced by Higson and Roe [161,163]. We refer to the splendid survey by Zhizhang Xie and Guoliang Yu [281] for the remarkable recent developments in this domain. We shall not attempt (for lack of space) to do justice to these considerable progress but briefly mention the papers [231] of P. Piazza and T. Schick and [277,279,281] of G. Yu and Z. Xie who develop sophisticated secondary invariants involving the group C * -algebra of Galois covers to analyse the connected components of the space of metrics of positive scalar curvature.…”

Section: Higher Index Theorymentioning

confidence: 99%

“…Moreover a whole subject has developed concerning noncommutative manifolds and for lack of space we shall not cover it here but refer to the papers [69-72, 99, 111, 112, 189]. We apologise for the brevity of our account of the topics covered in the present survey and the reader will find a more complete treatment and bibliography in the collection of surveys [35,90,104,114,117,124,134,195,199,272,276,281] to be published by Springer.…”

Section: New Paradigm: Spectral Geometrymentioning

confidence: 99%

“…We now turn to the recent developments [136,137,215,231,277,279,281] in the secondary higher index theory. When the primary index vanishes as in the case of positive scalar curvature, or in the relative situation of homotopic manifolds for the signature, the secondary theory comes into play, and one obtains a secondary invariant called the rho-invariant.…”

Section: Higher Index Theorymentioning

confidence: 99%

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Noncommutative Geometry, the spectral standpoint

Connes

2019

Preprint

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Section: Higher Index Theorymentioning

confidence: 99%

Section: New Paradigm: Spectral Geometrymentioning

confidence: 99%

Section: Higher Index Theorymentioning

confidence: 99%

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Noncommutative Geometry, the spectral standpoint

Connes

2019

Preprint

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“…K theory plays an key role in studying the topology especially in classify C * − algebra and index theory. This section is based on the textbook of [36] , [30] and a survey [37].…”

Section: Preliminariesmentioning

confidence: 99%

C*-Algebraic higher Signature on Non-Witt space

Liu

2021

Preprint

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Signature plays an important role in geometry and topology. In the space with singularity, Goresky and MacPherson extend the signatures to oriented pseudomanifolds with only even codimensional stratums by using generalized Poincare duality of intersection homology. After that Siegel extended the signature on Witt spaces. Higson and Xie study the C * -higher signature on Witt space. Non-Witt spaces need a refined intersection homology to hold Poincare duality. Followed by the combinatorial framework developed by Higson and Roe, this paper construct the C * -signature on non Witt space with noncommutative geometric methods. In conical singular case, we compare analytical signature of smooth stratified non Witt space by Albin, Leichtnam, Mazzeo and Piazza.

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“…In particular, if a delocalized eta invariant is transcendental, then it would lead to a counterexample to the Baum-Connes conjecture. We refer the reader to [32] for a more detailed survey of the delocalized eta invariant and its higher analogues.…”

Section: Introductionmentioning

confidence: 99%

Approximations of delocalized eta invariants by their finite analogues

Wang¹,

Xie²,

Yu³

2020

Preprint

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For a given self-adjoint first order elliptic differential operator on a closed smooth manifold, we prove a list of results on when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces. One of our main results is the following. Suppose M is a closed smooth spin manifold and M is a Γ-regular covering space of M . Let α be the conjugacy class of a non-identity element α ∈ Γ. Suppose {Γ i } is a sequence of finite-index normal subgroups of Γ that distinguishes α . Let π Γi be the quotient map from Γ to Γ/Γ i and π Γi (α) the conjugacy class of π Γi (α) in Γ/Γ i . If the scalar curvature on M is everywhere bounded below by a sufficiently large positive number, then the delocalized eta invariant for the Dirac operator of M at the conjugacy class α is equal to the limit of the delocalized eta invariants for the Dirac operators of M Γi at the conjugacy class π Γi (α) , where M Γi = M /Γ i is the finite-sheeted covering space of M determined by Γ i .1 There is also an extra technical assumption that the conjugacy class α used in the definition of the delocalized eta invariant is required to have polynomial growth.

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Higher invariants in noncommutative geometry

Cited by 8 publications

References 116 publications

Noncommutative Geometry, the spectral standpoint

Noncommutative Geometry, the spectral standpoint

C*-Algebraic higher Signature on Non-Witt space

Approximations of delocalized eta invariants by their finite analogues

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