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
ρ
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
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 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
Serre
dR
(f)
. We prove $\rho_{an}(E;V) = \int_B \rho_{an}(F_b;V) \cdot \Pf_B + \sum_{q} (-1)^q \cdot \rho_{an}(B;H^q_{dR}(F;V)) + \rho^{Serre}_{dR}(f)$.