[Paper Review] Cheeger-Mueller Theorem on manifolds with cusps

Boris Vertman|arXiv (Cornell University)|Nov 3, 2014

Geometric Analysis and Curvature Flows47 references4 citations

TL;DR

This paper establishes the Cheeger-Müller theorem for Witt manifolds with cusps by proving equality between the renormalized Ray-Singer analytic torsion and the intersection R-torsion, up to an error term depending on the Betti numbers of the cusp cross-section and the torsion of a model cone. The proof combines explicit computation of analytic torsion on a model cusp and a general gluing formula for non-compact manifolds with asymptotically conical ends.

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-Mueller theorem on cusps follows by a combination of the previous explicit computation and the gluing formula.

Motivation & Objective

• To extend the Cheeger-Müller theorem to non-compact Witt manifolds with cusp singularities.
• To resolve the challenge of spectral contributions in analytic torsion for singular spaces by introducing renormalization.
• To establish a gluing formula for analytic torsion on non-compact manifolds with asymptotically conical ends.
• To compute the renormalized Ray-Singer analytic torsion explicitly for a model cusp in general dimension.
• To prove equality between analytic and intersection R-torsion on Witt manifolds with cusps, up to a topologically determined error term.

Proposed method

• Explicitly computes the zeta-regularized determinant of scalar cuspidal operators using asymptotics of modified Bessel functions.
• Derives a gluing formula for renormalized Ray-Singer analytic torsion on a general class of non-compact manifolds, including those with cusp and scattering ends.
• Applies Spreafico’s double summation method to compute the analytic torsion of a model infinite cusp.
• Uses the gluing formula to relate the analytic torsion on the full manifold to that on a truncated cusp and the interior region.
• Relies on the intersection cohomology framework and Dar’s intersection R-torsion for stratified spaces.
• Combines explicit computation on the model cusp with the gluing formula to deduce the Cheeger-Müller equality.

Experimental results

Research questions

• RQ1 Does the Cheeger-Müller theorem hold for non-compact Witt manifolds with cusp singularities?
• RQ2 How does the renormalized Ray-Singer analytic torsion behave on a model cusp in arbitrary dimension?
• RQ3 Can a general gluing formula for analytic torsion be established on non-compact manifolds with asymptotically conical ends?
• RQ4 What is the precise error term relating analytic and intersection R-torsion on Witt manifolds with cusps?
• RQ5 How do the Betti numbers of the cusp cross-section and the torsion of a model cone affect the discrepancy between analytic and combinatorial torsions?

Key findings

• The renormalized Ray-Singer analytic torsion on a model cusp is computed explicitly in general dimension without requiring the Witt condition.
• A general gluing formula for analytic torsion is established for non-compact manifolds with asymptotically conical ends, including Witt manifolds with cusps.
• The Cheeger-Müller theorem holds on Witt manifolds with cusps, with the analytic and intersection R-torsions differing by a term involving the Betti numbers of the cusp cross-section and the torsion of a model cone.
• The error term in the Cheeger-Müller equality is explicitly given by a sum over cohomology degrees involving the absolute difference between p and n/2, weighted by Betti numbers and logarithmic factors.
• The result confirms a topological interpretation of analytic torsion on singular spaces, extending the classical theorem beyond compact manifolds.
• The final formula expresses the logarithmic ratio of analytic to intersection torsion as a sum over cohomology degrees, with coefficients involving |n/2 - p| and log(2|n/2 - p|).