Lück W., Schick, T. and Thielmann, T.: Torsion and fibrations

• Autor: Wolfgang Lück, Thomas Schick and Thomas Thielmann
• Titel: Torsion and fibrations
• J. Reine Angew. Math. 498 (1998), 1--33
• Abstract: We study the behaviour of analytic torsion under smooth fibrations. Namely, let f:E to B be a smooth fiber bundle with fiber F of connected closed oriented smooth manifolds and let V be a flat vector bundle over E. Assume that E and B come with Riemannian metrics. Suppose that dim(E) is odd and V is unimodular and comes with an arbitrary Riemannian metric or that dim(E) is even and V comes with a unimodular (not necessarily flat) Riemannian metric. Let rho_an(E;V) be the analytic torsion of E with coefficients in V, let rho_an(F_b;V) be the analytic torsion of the fiber over b with coefficients in V restricted to F_b and let Pf_B be the Pfaffian dim(B)-form. Let H^q_dR(F;V) be the flat vector bundle over $B$ whose fiber over b in B is H^q_dR(F_b;V) with the Riemannian metric which comes from the Hodge-deRham decomposition and the Hilbert space structure on the space of harmonic forms induced by the Riemannian metrics. Let $rho_{an}(B;H^q_{dR}(F;V))$ be the analytic torsion of $B$ with coefficients in this bundle. The Leray-Serre spectral sequence for deRham cohomology determines a certain correction term $rho^{Serre}_{dR}(f)$. We prove rho_an(E;V) = int_B rho_an(F_b;V) cdot Pf_B + sum_{q} (-1)^q cdot rho_an}(B;H^q_dR(F;V)) + rho^{Serre}_{dR}(f). This formula simplifies in special cases such as bundles with S^n as fiber or base, in which case the correction terms $rho^{Serre}_{dR}(f)$ reduces to the torsion of the associated Gysin or Wang sequence, resp.