A Cheeger-Müller theorem for delocalized analytic torsion
type: Item
instance of: scholarly article
title: A Cheeger-Müller theorem for delocalized analytic torsion
zbMATH Open document ID: 1120.58021
author: Guangxiang Su
published in: Annals of Global Analysis and Geometry
publication date: 2007-03-20T00:00:00Z
review text: Let \(F\) be a flat complex vector bundle on a closed Riemannian manifold \(M\). If \(F\) is unitary and \(H^*(M,F)=0\), Reidemeister (R-)torsion is a real number defined in a combinatorial way. R-torsion is the first topological invariant which is invariant under homeomorphisms but not a homotopy invariant. If \(H^*(M,F)\) is not reduced to zero, we can instead define a Reidemeister metric on the determinant of the cohomology of \(F\), it is still a topological invariant. In 1973, Ray and Singer asked whether there exists an analytic version of R-torsion, and they introduced the famous Ray-Singer analytic torsion which is defined as some combination of the spectrum of the associated Laplacian acting on forms with values in \(F\). Their conjecture on the equality of these two torsions was proved by \textit{J. Cheeger} [Ann. Math. (2) 109, 259--322 (1979; Zbl 0412.58026)] and \textit{W. Müller} [Adv. Math. 28, 233--305 (1978; Zbl 0395.57011)] independently. Finally, \textit{J.-M. Bismut} and \textit{W. Zhang} [An extension of a theorem by Cheeger and Müller, Astérisque 205. Paris (1992; Zbl 0781.58039)] established the general relation for these two torsions for any flat vector bundle. Recently, \textit{J. Lott} [J. Funct. Anal. 169, No. 1, 1--31 (1999; Zbl 0958.58027)] introduced the delocalized \(L^2\) analytic torsion \(T_{\langle g\rangle}\) for any conjugacy class \(\langle g\rangle\) of the fundamental group \(\Gamma=\pi_1(M)\). If \(g=e\), then \(T_{\langle e\rangle}\) is the \(L^2\) analytic torsion. Thus it is a natural and interesting problem to ask for its combinatorial version and to study their relation. In this paper, for \(\langle g\rangle\) a nontrivial finite conjugacy class, the author introduces the delocalized \(L^2\) combinatorial torsion, and the main result is Theorem 5.1, where the author establishes the relation of two delocalized \(L^2\) torsions by following the strategy of Bismut-Zhang. In a certain sense, the author shows the delocalized …
reviewed by: Xiaonan Ma
Mathematics Subject Classification ID: 58J52
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 5135832
zbMATH Keywords: delocalized \(L^2\)-analytic torsion
zbMATH Keywords: delocalized \(L^2\)-combinatorial torsion
zbMATH Keywords: Novikov-Shubin invariant
MaRDI profile type: Publication
full work available at URL: s10455-006-9043-7
OpenAlex ID: W1990268991
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DOI: 10.1007/S10455-006-9043-7
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description: scientific article; zbMATH DE number 5135832