The \(\eta\)-invariant for Lipschitz manifolds
type: Item
instance of: scholarly article
title: The \(\eta\)-invariant for Lipschitz manifolds
zbMATH Open document ID: 1036.58022
author: Michel Hilsum
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: The \(\eta\)-invariant has been defined for \(C^\infty\)-manifolds by \textit{M. F. Atiyah, V. K. Patodi} and \textit{I. M. Singer} in [Math. Proc. Camb. Philos. Soc. 77, 43--69 (1975; Zbl 0297.58008)] as follows: let \(W\) be an oriented and odd-dimensional \(C^\infty\)-manifold, \(g\) a Riemannian structure on \(W\), \((E,l,\nabla)\) a Hermitian bundle with a unitary connection and \(A\) the associated signature operator. Then, the function \(\eta(s,A)= \text{Tr} (A| A|^{-s-1})\) is analytic for \(\mathbb{R}(s)> \dim W\) and admits a meromorphic extension to \(\mathbb{C}\). By definition \(\eta(W,E)= \frac12 \eta(0,A)\). In the quoted paper one establishes the formula \[ \operatorname {Ind}D= \int_V \text{ch} (\widetilde{\nabla}) L(V,r)- \eta(W,E)- \frac12 \dim \ker A, \] where \(V\) is an oriented manifold with \(\partial V=W\), \(r\) is a Riemannian structure such that \(r= dt^2\oplus g\) in a tubular neighbourhood of \(W\), \((\widetilde{E}, \widetilde{l}, \widetilde{\nabla})\) is an extension of \((E,l,\nabla)\), \(L(V,r)\) is the Hirzebruch polynomial of Pontryagin forms of the Riemannian connection \(r\) and \(D\) is a selfadjoint extension of the closing \(D_0\) of \(A\) in \(\Omega(V)\), defined by the spectral projector of the positive part of the spectrum of \(A\) (\(\text{ch} (\widetilde{\nabla})\) is the Chern character). In the present paper the author defines the \(\eta\)-invariant for Lipschitz Riemannian manifolds by the above formula given sense to \(\operatorname {Ind}D\) and \(L(V,r)\). Moreover he extends from smooth to topological manifolds the Atiyah-Patodi-Singer index theorem for flat bundles.
reviewed by: Enrique Outerelo
Mathematics Subject Classification ID: 58J28
Mathematics Subject Classification ID: 58J20
zbMATH DE Number: 1782648
zbMATH Keywords: invariants
zbMATH Keywords: index theorem
zbMATH Keywords: Lipschitz Riemannian manifolds
Wikidata QID: Q115203247
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090340565
OpenAlex ID: W1485472476
DOI: 10.4310/JDG/1090340565
description: scientific article; zbMATH DE number 1782648