Approximations of delocalized eta invariants by their finite analogues

Wang, Jinmin; Xie, Zhizhang; Yu, Guoliang

doi:10.48550/arxiv.2003.03401

Cited by 2 publications

(2 citation statements)

References 35 publications

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“…A positive answer to the above question can be used to compute certain secondary index theoretic invariants, such as delocalized eta invariants and higher rho invariants associated to positive scalar curvature metrics on the boundary of a spin manifold, cf. [20,23].…”

Section: Introductionmentioning

confidence: 99%

$l^1$-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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Section: Introductionmentioning

confidence: 99%

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

Wang¹,

Xie²,

Yu³

2022

Preprint

Self Cite

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show abstract

“…This result can be applied to the problem of computing the higher rho invariant, along with other related invariants. For instance it is a key tool in the recent work of Wang, Xie, and Yu [25], who computed the delocalized eta invariant on the universal cover of a closed manifold by showing that, under suitable conditions, it can be expressed as a limit of delocalized eta invariants associated to finite-sheeted covering spaces.…”

Section: Introductionmentioning

confidence: 99%

Functoriality for higher rho invariants of elliptic operators

Guo

Xie

2020

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Let N be a closed spin manifold with positive scalar curvature and DN the Dirac operator on N . Let M1 and M2 be two Galois covers of N such that M2 is a quotient of M1. Then the quotient map from of M1 to M2 naturally induces maps between the geometric C * -algebras associated to the two manifolds. We prove, by a finite-propagation argument, that the maximal higher rho invariants of the lifts of DN to M1 and M2 behave functorially with respect to the above quotient map. This can be applied to the computation of higher rho invariants, along with other related invariants.

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Approximations of delocalized eta invariants by their finite analogues

Cited by 2 publications

References 35 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

Functoriality for higher rho invariants of elliptic operators

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