Induced cyclic cocycles and higher eta invariants
type: Item
instance of: scholarly article
title: Induced cyclic cocycles and higher eta invariants
zbMATH Open document ID: 0882.58050
author: Ronald G. Douglas
author: Jerome Kaminker
published in: Journal of Functional Analysis
publication date: 1998-02-03T00:00:00Z
review text: Let \(M\) be a closed, odd-dimensional Riemannian manifold with \(\text{spin}^c\) structure. Having a representation \(\alpha:\pi_1(M)\to G\) of the fundamental group of \(M\) into a compact connected Lie group \(G\), the product \(M\times G\) has two foliations, the obvious one \({\mathcal F}_G\) and, transversally, \({\mathcal F}_{\alpha}\) induced by the representation \(\alpha\). Furthermore, the \(\text{spin}^c\) Dirac operator \(\eth\) of \(M\) induces a leafwise elliptic Dirac operator \(\eth_{\alpha}\) over \(M\times G\). The paper gives a construction of a cyclic cocycle \(c_H\) on the foliation algebra \(C_c^{\infty}({\mathcal F}_{\alpha})\) for any closed normal subgroup \(H\) of \(G\), such that the pairing with the index of the induced Dirac operator \(\eth_{\alpha}\) gives a spectral invariant, and motivates the notion of these invariants as higher eta invariants.
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 1046928
zbMATH Keywords: leafwise elliptic Dirac operator on a foliated manifold
zbMATH Keywords: cyclic cocycles
zbMATH Keywords: K-theory
zbMATH Keywords: index theory
zbMATH Keywords: higher eta invariants
reviewed by: Katharina Habermann
MaRDI profile type: Publication
full work available at URL: jfan.1996.3049
OpenAlex ID: W2059782747
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DOI: 10.1006/JFAN.1996.3049
description: scientific article; zbMATH DE number 1046928