  For a closed manifold equipped with a Riemannian metric, a triangulation, a
representation of its fundamental group on an Hilbert module of finite type
(over of finite von Neumann algebra), and a Hermitian structure on the flat
bundle associated to the representation, one defines a numerical invariant, the
relative torsion. The relative torsion is a positive real number and unlike the
analytic torsion or the Reidemeister torsion, which are defined only when the
pair manifold- representation is of determinant class, is always defined. When
the pair is of determinant class the relative torsionis equal to the quotient
of the analytic and the Reidemeister torsion.We calculate the relative torsion.
