Suppose M is a compact connected odd-dimensional manifold with boundary, whose interior M comes with a complete hyperbolic metric of finite volume. We will show that the L 2 -topological torsion of M and the L 2 -analytic torsion of the Riemannian manifold M are equal. In particular, the L 2 -topological torsion of M is proportional to the hyperbolic volume of M , with a constant of proportionality which depends only on the dimension and which is known to be nonzero in odd dimensions [14]. In dimension 3 this proves the conjecture [18, Conjecture 2.3] or [16, conjecture 7.7] which gives a complete calculation of the L 2 -topological torsion of compact L 2 -acyclic 3-manifolds which admit a geometric JSJT-decomposition.In an appendix we give a counterexample to an extension of the Cheeger-Müller theorem to manifolds with boundary: if the metric is not a product near the boundary, in general analytic and topological torsion are not equal, even if the Euler characteristic of the boundary vanishes.Key words: L 2 -torsion, hyperbolic manifolds, 3-manifolds, manifolds with boundary AMS-classification number: 58G11 IntroductionIn this paper we study L 2 -analytic torsion of a compact connected manifold M with boundary such that the interior M comes with a complete hyperbolic metric of finite volume.Notation 0.1. Let m be the dimension of M. From hyperbolic geometry we know [1, Chapter D3] that M can be written aswhere M 0 is a compact manifold with boundary and E 0 is a finite disjoint union of hyperbolic ends [0, ∞) × F j . Here each F j is a closed flat manifold and the metric on the end is the warped product du 2 + e −2u dx 2 1
Abstract. We investigate how one can twist L 2 -invariants such as L 2 -Betti numbers and L 2 -torsion with finite-dimensional representations. As a special case we assign to the universal covering X of a finite connected CW -complex X together with an element φ ∈ H 1 (X; R) a φ-twisted L 2 -torsion function R >0 → R, provided that the fundamental group of X is residually finite and X is L 2 -acyclic.
For a normal covering over a closed oriented topological manifold we give a proof of the L 2 -signature theorem with twisted coefficients, using Lipschitz structures and the Lipschitz signature operator introduced by Teleman. We also prove that the L-theory isomorphism conjecture as well as the C * max -version of the Baum-Connes conjecture imply the L 2signature theorem for a normal covering over a Poincaré space, provided that the group of deck transformations is torsion-free.We discuss the various possible definitions of L 2 -signatures (using the signature operator, using the cap product of differential forms, using a cap product in cellular L 2 -cohomology, . . . ) in this situation, and prove that they all coincide.
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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