We study the behaviour of analytic torsion under smooth fibrations. Namely, let F → E f − → B be a smooth fiber bundle 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 ρ an (E; V ) be the analytic torsion of E with coefficients in V , let ρ 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 ∈ 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 ρ 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 ρ LS dR (f ). We proveThis formula simplifies in special cases such as bundles with S n as fiber or base, in which case the correction terms ρ LS dR (f ) reduces to the torsion of the associated Gysin or Wang sequence, resp.
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