Torsion and fibrations

Abstract

We study the behaviour of analytic torsion under smooth fibrations. Namely, let F \to E \to^{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 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 $V$ and let $\Pf_B$ be the Pfaffian $\dim (B)$-form. Let $H_{dR}^q(F;V)$ be the flat vector bundle over $B$ whose fiber over $b\in B$ is $H_{dR}^q(F_b;V)$ with the Riemannian metric which comes from the Hilbert space structure on the space of harmonic forms induced by the Riemannian metrics. Let ${\rho }_{an}(B;H_{dR}^q(F;V))$ be the analytic torsion of $B$ with coefficients in this bundle. The Leray-Serre spectral sequence for deRham cohomology determines a certain correction term ${\rho }_{dR}^{Serre}(f)$. We prove $ρ_{an}(E;V) = \int_B ρ_{an}(F_b;V) \cdot \Pf_B + \sum_{q} (-1)^q \cdot ρ_{an}(B;H^q_{dR}(F;V)) + ρ^{Serre}_{dR}(f)$.

34 pages, AMS-Latex2e

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