Analytic Torsion of a Bounded Generalized Cone

TL;DR

This paper computes the analytic torsion of a bounded generalized cone, extending methods from previous work to manifolds with conical singularities, and relates it to classical torsion invariants.

Contribution

It generalizes the computation of analytic torsion to bounded generalized cones, building on Spreafico's methods and Lesch's symmetry results.

Findings

Computed the analytic torsion for bounded generalized cones.

Extended existing methods to manifolds with conical singularities.

Provided insights into the relationship between analytic and combinatorial torsion.

Abstract

Torsion invariants for manifolds which are not simply connected were introduced by K. Reidemeister and generalized to higher dimensions by W. Franz. The Reidemeister torsion, was the first invariant of manifolds which was not a homotopy invariant. The analytic counterpart of the combinatorial Reidemeister torsion was introduced by D. B. Ray and I. M. Singer in form of a weighted product of zeta-regularized determinants of Laplace operators on differential forms. The celebrated Cheeger-Mueller Theorem, established independently by J. Cheeger and W. Mueller, proved equality between the analytic Ray-Singer torsion and the combinatorial Reidemeister torsion for any smooth closed manifold with an orthogonal representation of its fundamental group. Motivated by the vision of a Cheeger-Mueller type result on manifolds with conical singularities, we compute the analytic torsion of a bounded generalized cone by generalizing the computational methods of M. Spreafico and using the symmetry of the de Rham complex, as established by M. Lesch.