ISSN:
1420-8970
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract. Suppose $ \bar{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 $ \bar{M} $ and the $ L^2 $ -analytic torsion of the Riemannian manifold M are equal. In particular, the $ L^2 $ -topological torsion of $ \bar{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 [HS]. In dimension 3 this proves the conjecture [Lü2, Conjecture 2.3] or [LLü, 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s000390050095
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