A Cheeger-Müller theorem for delocalized analytic torsion (Q873924)

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 \(L^2\) torsions are simpler than \(L^2\) torsions, by constructing the delocalized \(L^2\) combinatorial torsions and by establishing their relation in clear way.