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Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 761–819 | Cite as

Cheeger–Müller theorem on manifolds with cusps

  • Boris VertmanEmail author
Article
  • 75 Downloads

Abstract

We prove equality between the renormalized Ray–Singer analytic torsion and the intersection R-torsion on a Witt-manifold with cusps, up to an error term determined explicitly by the Betti numbers of the cross section of the cusp and the intersection R-torsion of a model cone. In the first step of the proof we compute explicitly the renormalized Ray–Singer analytic torsion of a model cusp in general dimension and without the Witt-condition. In the second step we establish a gluing formula for renormalized Ray–Singer analytic torsion on a general class of non-compact manifolds in any dimension that includes Witt-manifolds with cusps, but also scattering manifolds with asymptotically conical ends. In the final step, a Cheeger–Müller theorem on cusps follows by a combination of the previous explicit computation and the gluing formula.

Mathematics Subject Classification

58J52 34B24 

Notes

Acknowledgements

I thank Werner Müller for suggesting the topic and encouragement, Jonathan Pfaff, Matthias Lesch, Xianzhe Dai, Ulrich Bunke and Frederik Rochon for helpful discussions. I am grateful to Luiz Hartmann for careful reading of the manuscript and the explicit computations on the model cusp. I greatly appreciate the careful reading and valuable improvements suggested by the anonymous referee. I thank the Hausdorff Center for Mathematics in Bonn and University Münster for hospitality and financial support.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Carl von Ossietzky Universität Oldenburg, Institut für MathematikOldenburgGermany

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