Abstract:
Reidemeister torsion (or R-torsion) was originally introduced by K. Reidemeister in 1935, who
used it to classify 3-dimensional lens spaces. R-torsion is a homeomorphism invariant which
may be defined using core concepts in algebraic topology and linear algebra.
Later, in 1971, D. Ray and I. Singer defined an analytic analogue of R-torsion, which involved
using the zeta function to define a regularized determinant of the Laplacian on the space of
differential forms. After proving that their analytic torsion (which has come to be known as
Ray-Singer torsion, or RS-torsion) satisfies many of the same properties of R-torsion, Ray and
Singer conjectured that RS-torsion and R-torsion are equal for closed Riemannian manifolds,
and provided computational evidence. This conjecture was proven independently in celebrated
papers by W. M¨uller and J. Cheeger.
In 1994, J. M. Bismut and W. Zhang gave an analytic proof of a generalization of the Cheeger-
M¨uller theorem. Their approach utilizes the Witten deformation of the Laplacian to factorize
the Ray-Singer torsion into large and small components, which then may be analyzed separately.
In 2003, M. Braverman gave another proof which uses Bismut and Zhang’s analysis of
the small component of the RS-torsion, but introduces a clever comparison analysis of the large
component of the RS-torsion.
In this thesis we present Braverman’s analytic approach. However, we also provide original
proofs for some of the results which are used.