Higher rho invariant and delocalized eta invariant at infinity

Chen, Xiaoman; Liu, Hongzhi; Wang, Hang; Yu, Guoliang

doi:10.48550/arxiv.2004.00486

Cited by 2 publications

(5 citation statements)

References 37 publications

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“…as long as ℓ(γ) is sufficiently large. It follows that G (2) ( D) 1,K is also finite. This finishes the proof.…”

Section: 4mentioning

confidence: 95%

“…We write G as a sum of two functions G = G (1) + G (2) , where the Fourier transforms of G (1) and G (2) are…”

Section: 4mentioning

confidence: 99%

“…Therefore G (1) ( D) 1,K is finite. On the other hand, G (2) is a Schwartz function with Gaussian decay. By Lemma 2.9, there exists ε > 0 such that…”

Section: 4mentioning

confidence: 99%

“…An alternative proof of Theorem 5.4. As pointed out in the introduction, Theorem 5.4 is an strengthening of the higher Atiyah-Patodi-Singer index formula in [2,Theorem 3.36]. Since [2, Theorem 3.36] was stated and proved using the language of b-calculus.…”

mentioning

confidence: 93%

“…Definition 2. 16), and g + dt 2 is the product metric on X × R induced by g and the standard Euclidean metric on R.…”

Section: Introductionmentioning

confidence: 99%

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$l^1$-higher index, $l^1$-higher rho invariant and cyclic cohomology

Wang¹,

Xie²,

Yu³

2022

Preprint

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In this paper, we study l 1 -higher index theory and its pairing with cyclic cohomology for both closed manifolds and compact manifolds with boundary. We first give a sufficient geometric condition for the vanishing of the l 1 -higher indices of Dirac-type operators on closed manifolds. This leads us to define an l 1 -version of higher rho invariants. We prove a product formula for these l 1 -higher rho invariants. A main novelty of our product formula is that it works in the general Banach algebra setting, in particular, the l 1 -setting.On compact spin manifolds with boundary, we also give a sufficient geometric condition for Dirac operators to have well-defined l 1 -higher indices. More precisely, we show that, on a compact spin manifold M with boundary equipped with a Riemannian metric which has product structure near the boundary, if the scalar curvature on the boundary is sufficiently large, then the l 1 -higher index of its Dirac operator D M is well-defined and lies in the Ktheory of the l 1 -algebra of the fundamental group. As an immediate corollary, we see that if the Bost conjecture holds for the fundamental group of M , then the C * -algebraic higher index of D M lies in the image of the Baum-Connes assembly map.By pairing the above K-theoretic l 1 -index results with cyclic cocycles, we prove an l 1 -version of the higher Atiyah-Patodi-Singer index theorem for manifolds with boundary. A key ingredient of its proof is the product formula for l 1 -higher rho invariants mentioned above.

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“…as long as ℓ(γ) is sufficiently large. It follows that G (2) ( D) 1,K is also finite. This finishes the proof.…”

Section: 4mentioning

confidence: 95%

“…We write G as a sum of two functions G = G (1) + G (2) , where the Fourier transforms of G (1) and G (2) are…”

Section: 4mentioning

confidence: 99%

“…Therefore G (1) ( D) 1,K is finite. On the other hand, G (2) is a Schwartz function with Gaussian decay. By Lemma 2.9, there exists ε > 0 such that…”

Section: 4mentioning

confidence: 99%

mentioning

confidence: 93%

“…Definition 2. 16), and g + dt 2 is the product metric on X × R induced by g and the standard Euclidean metric on R.…”

Section: Introductionmentioning

confidence: 99%

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$l^1$-higher index, $l^1$-higher rho invariant and cyclic cohomology

Wang¹,

Xie²,

Yu³

2022

Preprint

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An absolute version of the Gromov-Lawson relative index theorem

Hochs

Wang

2021

Preprint

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A Dirac operator on a complete manifold is Fredholm if it is invertible outside a compact set. Assuming a compact group to act on all relevant structure, we express the equivariant index of such a Dirac operator as an Atiyah-Segal-Singer type contribution from inside this compact set, and a contribution from outside this set. Consequences include equivariant versions of the relative index theorem of Gromov and Lawson and the Atiyah-Patodi-Singer index theorem.

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Higher rho invariant and delocalized eta invariant at infinity

Cited by 2 publications

References 37 publications

$l^1$-higher index, $l^1$-higher rho invariant and cyclic cohomology

$l^1$-higher index, $l^1$-higher rho invariant and cyclic cohomology

An absolute version of the Gromov-Lawson relative index theorem

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