  It is shown that for any piecewise-linear closed orientable manifold of odd
dimension there exists an invariantly defined metric on the determinant line of
cohomology with coefficients in an arbitrary flat bundle E over the manifold (E
is not required to be unimodular). The construction of this metric (called
Poincare - Reidemeister metric) is purely combinatorial; it combines the
standard Reidemeister type construction with Poincare duality. The main result
of the paper states that the Poincare-Reidemeister metric computes
combinatorially the Ray-Singer metric. It is shown also that the Ray-Singer
metrics on some relative determinant lines can be computed combinatorially
(including the even-dimensional case) in terms of metrics determined by
correspondences.
