Torsion and fibrations

Abstract

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.

Analytic Torsion for Twisted De Rham Complexes

“…It would be interesting to establish the behavior of the analytic torsion (form) under a general smooth fibration, analogous to [7,21,36], for the twisted de Rham or other Z 2 -graded complexes.…”

“…When X is a 3-manifold, Proposition 5.1 relates τ (X, H) to τ (X), which can be calculated by the spectral sequence of fibration [21,22,36,38]. Proof.…”

L2-Invariants of Regular Coverings of Compact Manifolds and CW-Complexes

“…This conjecture was proven independently by Cheeger [50] and Müller [184]. Manifolds with boundary and manifolds with symmetries, sum (= glueing) formulas and fibration formulas are treated in [67], [149], [153], [165], [242], [243], [244]. Non-unitary coefficient systems are studied in [25], [26], [186].…”

Hilbert modules and modules over finite von Neumann algebras and applications to $L^2$ -invariants

“…This allows for instance to carry over the results in [36] 8. The "Zero in the spectrum" conjecture The "Zero in the spectrum" conjecture says that there is no contractible closed Riemannian manifold M with an isometric proper cocompact action of a discrete group Γ such that for all p zero is not in the spectrum of the Laplace operator in dimension p [18, page 238], [25].…”